diff options
Diffstat (limited to 'theories7')
202 files changed, 55869 insertions, 0 deletions
diff --git a/theories7/Arith/Arith.v b/theories7/Arith/Arith.v new file mode 100755 index 00000000..181fadbc --- /dev/null +++ b/theories7/Arith/Arith.v @@ -0,0 +1,21 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Arith.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*) + +Require Export Le. +Require Export Lt. +Require Export Plus. +Require Export Gt. +Require Export Minus. +Require Export Mult. +Require Export Between. +Require Export Minus. +Require Export Peano_dec. +Require Export Compare_dec. +Require Export Factorial. diff --git a/theories7/Arith/Between.v b/theories7/Arith/Between.v new file mode 100755 index 00000000..b3fef325 --- /dev/null +++ b/theories7/Arith/Between.v @@ -0,0 +1,185 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Between.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*) + +Require Le. +Require Lt. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type k,l,p,q,r:nat. + +Section Between. +Variables P,Q : nat -> Prop. + +Inductive between [k:nat] : nat -> Prop + := bet_emp : (between k k) + | bet_S : (l:nat)(between k l)->(P l)->(between k (S l)). + +Hint constr_between : arith v62 := Constructors between. + +Lemma bet_eq : (k,l:nat)(l=k)->(between k l). +Proof. +NewInduction 1; Auto with arith. +Qed. + +Hints Resolve bet_eq : arith v62. + +Lemma between_le : (k,l:nat)(between k l)->(le k l). +Proof. +NewInduction 1; Auto with arith. +Qed. +Hints Immediate between_le : arith v62. + +Lemma between_Sk_l : (k,l:nat)(between k l)->(le (S k) l)->(between (S k) l). +Proof. +NewInduction 1. +Intros; Absurd (le (S k) k); Auto with arith. +NewDestruct H; Auto with arith. +Qed. +Hints Resolve between_Sk_l : arith v62. + +Lemma between_restr : + (k,l,m:nat)(le k l)->(le l m)->(between k m)->(between l m). +Proof. +NewInduction 1; Auto with arith. +Qed. + +Inductive exists [k:nat] : nat -> Prop + := exists_S : (l:nat)(exists k l)->(exists k (S l)) + | exists_le: (l:nat)(le k l)->(Q l)->(exists k (S l)). + +Hint constr_exists : arith v62 := Constructors exists. + +Lemma exists_le_S : (k,l:nat)(exists k l)->(le (S k) l). +Proof. +NewInduction 1; Auto with arith. +Qed. + +Lemma exists_lt : (k,l:nat)(exists k l)->(lt k l). +Proof exists_le_S. +Hints Immediate exists_le_S exists_lt : arith v62. + +Lemma exists_S_le : (k,l:nat)(exists k (S l))->(le k l). +Proof. +Intros; Apply le_S_n; Auto with arith. +Qed. +Hints Immediate exists_S_le : arith v62. + +Definition in_int := [p,q,r:nat](le p r)/\(lt r q). + +Lemma in_int_intro : (p,q,r:nat)(le p r)->(lt r q)->(in_int p q r). +Proof. +Red; Auto with arith. +Qed. +Hints Resolve in_int_intro : arith v62. + +Lemma in_int_lt : (p,q,r:nat)(in_int p q r)->(lt p q). +Proof. +NewInduction 1; Intros. +Apply le_lt_trans with r; Auto with arith. +Qed. + +Lemma in_int_p_Sq : + (p,q,r:nat)(in_int p (S q) r)->((in_int p q r) \/ <nat>r=q). +Proof. +NewInduction 1; Intros. +Elim (le_lt_or_eq r q); Auto with arith. +Qed. + +Lemma in_int_S : (p,q,r:nat)(in_int p q r)->(in_int p (S q) r). +Proof. +NewInduction 1;Auto with arith. +Qed. +Hints Resolve in_int_S : arith v62. + +Lemma in_int_Sp_q : (p,q,r:nat)(in_int (S p) q r)->(in_int p q r). +Proof. +NewInduction 1; Auto with arith. +Qed. +Hints Immediate in_int_Sp_q : arith v62. + +Lemma between_in_int : (k,l:nat)(between k l)->(r:nat)(in_int k l r)->(P r). +Proof. +NewInduction 1; Intros. +Absurd (lt k k); Auto with arith. +Apply in_int_lt with r; Auto with arith. +Elim (in_int_p_Sq k l r); Intros; Auto with arith. +Rewrite H2; Trivial with arith. +Qed. + +Lemma in_int_between : + (k,l:nat)(le k l)->((r:nat)(in_int k l r)->(P r))->(between k l). +Proof. +NewInduction 1; Auto with arith. +Qed. + +Lemma exists_in_int : + (k,l:nat)(exists k l)->(EX m:nat | (in_int k l m) & (Q m)). +Proof. +NewInduction 1. +Case IHexists; Intros p inp Qp; Exists p; Auto with arith. +Exists l; Auto with arith. +Qed. + +Lemma in_int_exists : (k,l,r:nat)(in_int k l r)->(Q r)->(exists k l). +Proof. +NewDestruct 1; Intros. +Elim H0; Auto with arith. +Qed. + +Lemma between_or_exists : + (k,l:nat)(le k l)->((n:nat)(in_int k l n)->((P n)\/(Q n))) + ->((between k l)\/(exists k l)). +Proof. +NewInduction 1; Intros; Auto with arith. +Elim IHle; Intro; Auto with arith. +Elim (H0 m); Auto with arith. +Qed. + +Lemma between_not_exists : (k,l:nat)(between k l)-> + ((n:nat)(in_int k l n) -> (P n) -> ~(Q n)) + -> ~(exists k l). +Proof. +NewInduction 1; Red; Intros. +Absurd (lt k k); Auto with arith. +Absurd (Q l); Auto with arith. +Elim (exists_in_int k (S l)); Auto with arith; Intros l' inl' Ql'. +Replace l with l'; Auto with arith. +Elim inl'; Intros. +Elim (le_lt_or_eq l' l); Auto with arith; Intros. +Absurd (exists k l); Auto with arith. +Apply in_int_exists with l'; Auto with arith. +Qed. + +Inductive P_nth [init:nat] : nat->nat->Prop + := nth_O : (P_nth init init O) + | nth_S : (k,l:nat)(n:nat)(P_nth init k n)->(between (S k) l) + ->(Q l)->(P_nth init l (S n)). + +Lemma nth_le : (init,l,n:nat)(P_nth init l n)->(le init l). +Proof. +NewInduction 1; Intros; Auto with arith. +Apply le_trans with (S k); Auto with arith. +Qed. + +Definition eventually := [n:nat](EX k:nat | (le k n) & (Q k)). + +Lemma event_O : (eventually O)->(Q O). +Proof. +NewInduction 1; Intros. +Replace O with x; Auto with arith. +Qed. + +End Between. + +Hints Resolve nth_O bet_S bet_emp bet_eq between_Sk_l exists_S exists_le + in_int_S in_int_intro : arith v62. +Hints Immediate in_int_Sp_q exists_le_S exists_S_le : arith v62. diff --git a/theories7/Arith/Bool_nat.v b/theories7/Arith/Bool_nat.v new file mode 100644 index 00000000..c36f8f15 --- /dev/null +++ b/theories7/Arith/Bool_nat.v @@ -0,0 +1,43 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(* $Id: Bool_nat.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ *) + +Require Export Compare_dec. +Require Export Peano_dec. +Require Sumbool. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,x,y:nat. + +(** The decidability of equality and order relations over + type [nat] give some boolean functions with the adequate specification. *) + +Definition notzerop := [n:nat] (sumbool_not ? ? (zerop n)). +Definition lt_ge_dec : (x,y:nat){(lt x y)}+{(ge x y)} := + [n,m:nat] (sumbool_not ? ? (le_lt_dec m n)). + +Definition nat_lt_ge_bool := + [x,y:nat](bool_of_sumbool (lt_ge_dec x y)). +Definition nat_ge_lt_bool := + [x,y:nat](bool_of_sumbool (sumbool_not ? ? (lt_ge_dec x y))). + +Definition nat_le_gt_bool := + [x,y:nat](bool_of_sumbool (le_gt_dec x y)). +Definition nat_gt_le_bool := + [x,y:nat](bool_of_sumbool (sumbool_not ? ? (le_gt_dec x y))). + +Definition nat_eq_bool := + [x,y:nat](bool_of_sumbool (eq_nat_dec x y)). +Definition nat_noteq_bool := + [x,y:nat](bool_of_sumbool (sumbool_not ? ? (eq_nat_dec x y))). + +Definition zerop_bool := [x:nat](bool_of_sumbool (zerop x)). +Definition notzerop_bool := [x:nat](bool_of_sumbool (notzerop x)). diff --git a/theories7/Arith/Compare.v b/theories7/Arith/Compare.v new file mode 100755 index 00000000..1bca3fbe --- /dev/null +++ b/theories7/Arith/Compare.v @@ -0,0 +1,60 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Compare.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*) + +(** Equality is decidable on [nat] *) +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +(* +Lemma not_eq_sym : (A:Set)(p,q:A)(~p=q) -> ~(q=p). +Proof sym_not_eq. +Hints Immediate not_eq_sym : arith. +*) +Notation not_eq_sym := sym_not_eq. + +Implicit Variables Type m,n,p,q:nat. + +Require Arith. +Require Peano_dec. +Require Compare_dec. + +Definition le_or_le_S := le_le_S_dec. + +Definition compare := gt_eq_gt_dec. + +Lemma le_dec : (n,m:nat) {le n m} + {le m n}. +Proof le_ge_dec. + +Definition lt_or_eq := [n,m:nat]{(gt m n)}+{n=m}. + +Lemma le_decide : (n,m:nat)(le n m)->(lt_or_eq n m). +Proof le_lt_eq_dec. + +Lemma le_le_S_eq : (p,q:nat)(le p q)->((le (S p) q)\/(p=q)). +Proof le_lt_or_eq. + +(* By special request of G. Kahn - Used in Group Theory *) +Lemma discrete_nat : (m, n: nat) (lt m n) -> + (S m) = n \/ (EX r: nat | n = (S (S (plus m r)))). +Proof. +Intros m n H. +LApply (lt_le_S m n); Auto with arith. +Intro H'; LApply (le_lt_or_eq (S m) n); Auto with arith. +NewInduction 1; Auto with arith. +Right; Exists (minus n (S (S m))); Simpl. +Rewrite (plus_sym m (minus n (S (S m)))). +Rewrite (plus_n_Sm (minus n (S (S m))) m). +Rewrite (plus_n_Sm (minus n (S (S m))) (S m)). +Rewrite (plus_sym (minus n (S (S m))) (S (S m))); Auto with arith. +Qed. + +Require Export Wf_nat. + +Require Export Min. diff --git a/theories7/Arith/Compare_dec.v b/theories7/Arith/Compare_dec.v new file mode 100755 index 00000000..504c0562 --- /dev/null +++ b/theories7/Arith/Compare_dec.v @@ -0,0 +1,109 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Compare_dec.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*) + +Require Le. +Require Lt. +Require Gt. +Require Decidable. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,x,y:nat. + +Definition zerop : (n:nat){n=O}+{lt O n}. +NewDestruct n; Auto with arith. +Defined. + +Definition lt_eq_lt_dec : (n,m:nat){(lt n m)}+{n=m}+{(lt m n)}. +Proof. +NewInduction n; Destruct m; Auto with arith. +Intros m0; Elim (IHn m0); Auto with arith. +NewInduction 1; Auto with arith. +Defined. + +Lemma gt_eq_gt_dec : (n,m:nat)({(gt m n)}+{n=m})+{(gt n m)}. +Proof lt_eq_lt_dec. + +Lemma le_lt_dec : (n,m:nat) {le n m} + {lt m n}. +Proof. +NewInduction n. +Auto with arith. +NewInduction m. +Auto with arith. +Elim (IHn m); Auto with arith. +Defined. + +Definition le_le_S_dec : (n,m:nat) {le n m} + {le (S m) n}. +Proof. +Exact le_lt_dec. +Defined. + +Definition le_ge_dec : (n,m:nat) {le n m} + {ge n m}. +Proof. +Intros; Elim (le_lt_dec n m); Auto with arith. +Defined. + +Definition le_gt_dec : (n,m:nat){(le n m)}+{(gt n m)}. +Proof. +Exact le_lt_dec. +Defined. + +Definition le_lt_eq_dec : (n,m:nat)(le n m)->({(lt n m)}+{n=m}). +Proof. +Intros; Elim (lt_eq_lt_dec n m); Auto with arith. +Intros; Absurd (lt m n); Auto with arith. +Defined. + +(** Proofs of decidability *) + +Theorem dec_le:(x,y:nat)(decidable (le x y)). +Intros x y; Unfold decidable ; Elim (le_gt_dec x y); [ + Auto with arith +| Intro; Right; Apply gt_not_le; Assumption]. +Qed. + +Theorem dec_lt:(x,y:nat)(decidable (lt x y)). +Intros x y; Unfold lt; Apply dec_le. +Qed. + +Theorem dec_gt:(x,y:nat)(decidable (gt x y)). +Intros x y; Unfold gt; Apply dec_lt. +Qed. + +Theorem dec_ge:(x,y:nat)(decidable (ge x y)). +Intros x y; Unfold ge; Apply dec_le. +Qed. + +Theorem not_eq : (x,y:nat) ~ x=y -> (lt x y) \/ (lt y x). +Intros x y H; Elim (lt_eq_lt_dec x y); [ + Intros H1; Elim H1; [ Auto with arith | Intros H2; Absurd x=y; Assumption] +| Auto with arith]. +Qed. + + +Theorem not_le : (x,y:nat) ~(le x y) -> (gt x y). +Intros x y H; Elim (le_gt_dec x y); + [ Intros H1; Absurd (le x y); Assumption | Trivial with arith ]. +Qed. + +Theorem not_gt : (x,y:nat) ~(gt x y) -> (le x y). +Intros x y H; Elim (le_gt_dec x y); + [ Trivial with arith | Intros H1; Absurd (gt x y); Assumption]. +Qed. + +Theorem not_ge : (x,y:nat) ~(ge x y) -> (lt x y). +Intros x y H; Exact (not_le y x H). +Qed. + +Theorem not_lt : (x,y:nat) ~(lt x y) -> (ge x y). +Intros x y H; Exact (not_gt y x H). +Qed. + diff --git a/theories7/Arith/Div.v b/theories7/Arith/Div.v new file mode 100755 index 00000000..59694628 --- /dev/null +++ b/theories7/Arith/Div.v @@ -0,0 +1,64 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Div.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*) + +(** Euclidean division *) + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Require Le. +Require Euclid_def. +Require Compare_dec. + +Implicit Variables Type n,a,b,q,r:nat. + +Fixpoint inf_dec [n:nat] : nat->bool := + [m:nat] Cases n m of + O _ => true + | (S n') O => false + | (S n') (S m') => (inf_dec n' m') + end. + +Theorem div1 : (b:nat)(gt b O)->(a:nat)(diveucl a b). +Realizer Fix div1 {div1/2: nat->nat->diveucl := + [b,a]Cases a of + O => (O,O) + | (S n) => + let (q,r) = (div1 b n) in + if (le_gt_dec b (S r)) then ((S q),O) + else (q,(S r)) + end}. +Program_all. +Rewrite e. +Replace b with (S r). +Simpl. +Elim plus_n_O; Auto with arith. +Apply le_antisym; Auto with arith. +Elim plus_n_Sm; Auto with arith. +Qed. + +Theorem div2 : (b:nat)(gt b O)->(a:nat)(diveucl a b). +Realizer Fix div1 {div1/2: nat->nat->diveucl := + [b,a]Cases a of + O => (O,O) + | (S n) => + let (q,r) = (div1 b n) in + if (inf_dec b (S r)) :: :: { {(le b (S r))}+{(gt b (S r))} } + then ((S q),O) + else (q,(S r)) + end}. +Program_all. +Rewrite e. +Replace b with (S r). +Simpl. +Elim plus_n_O; Auto with arith. +Apply le_antisym; Auto with arith. +Elim plus_n_Sm; Auto with arith. +Qed. diff --git a/theories7/Arith/Div2.v b/theories7/Arith/Div2.v new file mode 100644 index 00000000..8bd0160f --- /dev/null +++ b/theories7/Arith/Div2.v @@ -0,0 +1,174 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Div2.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +Require Lt. +Require Plus. +Require Compare_dec. +Require Even. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type n:nat. + +(** Here we define [n/2] and prove some of its properties *) + +Fixpoint div2 [n:nat] : nat := + Cases n of + O => O + | (S O) => O + | (S (S n')) => (S (div2 n')) + end. + +(** Since [div2] is recursively defined on [0], [1] and [(S (S n))], it is + useful to prove the corresponding induction principle *) + +Lemma ind_0_1_SS : (P:nat->Prop) + (P O) -> (P (S O)) -> ((n:nat)(P n)->(P (S (S n)))) -> (n:nat)(P n). +Proof. +Intros. +Cut (n:nat)(P n)/\(P (S n)). +Intros. Elim (H2 n). Auto with arith. + +NewInduction n0. Auto with arith. +Intros. Elim IHn0; Auto with arith. +Qed. + +(** [0 <n => n/2 < n] *) + +Lemma lt_div2 : (n:nat) (lt O n) -> (lt (div2 n) n). +Proof. +Intro n. Pattern n. Apply ind_0_1_SS. +Intro. Inversion H. +Auto with arith. +Intros. Simpl. +Case (zerop n0). +Intro. Rewrite e. Auto with arith. +Auto with arith. +Qed. + +Hints Resolve lt_div2 : arith. + +(** Properties related to the parity *) + +Lemma even_odd_div2 : (n:nat) + ((even n)<->(div2 n)=(div2 (S n))) /\ ((odd n)<->(S (div2 n))=(div2 (S n))). +Proof. +Intro n. Pattern n. Apply ind_0_1_SS. +(* n = 0 *) +Split. Split; Auto with arith. +Split. Intro H. Inversion H. +Intro H. Absurd (S (div2 O))=(div2 (S O)); Auto with arith. +(* n = 1 *) +Split. Split. Intro. Inversion H. Inversion H1. +Intro H. Absurd (div2 (S O))=(div2 (S (S O))). +Simpl. Discriminate. Assumption. +Split; Auto with arith. +(* n = (S (S n')) *) +Intros. Decompose [and] H. Unfold iff in H0 H1. +Decompose [and] H0. Decompose [and] H1. Clear H H0 H1. +Split; Split; Auto with arith. +Intro H. Inversion H. Inversion H1. +Change (S (div2 n0))=(S (div2 (S n0))). Auto with arith. +Intro H. Inversion H. Inversion H1. +Change (S (S (div2 n0)))=(S (div2 (S n0))). Auto with arith. +Qed. + +(** Specializations *) + +Lemma even_div2 : (n:nat) (even n) -> (div2 n)=(div2 (S n)). +Proof [n:nat](proj1 ? ? (proj1 ? ? (even_odd_div2 n))). + +Lemma div2_even : (n:nat) (div2 n)=(div2 (S n)) -> (even n). +Proof [n:nat](proj2 ? ? (proj1 ? ? (even_odd_div2 n))). + +Lemma odd_div2 : (n:nat) (odd n) -> (S (div2 n))=(div2 (S n)). +Proof [n:nat](proj1 ? ? (proj2 ? ? (even_odd_div2 n))). + +Lemma div2_odd : (n:nat) (S (div2 n))=(div2 (S n)) -> (odd n). +Proof [n:nat](proj2 ? ? (proj2 ? ? (even_odd_div2 n))). + +Hints Resolve even_div2 div2_even odd_div2 div2_odd : arith. + +(** Properties related to the double ([2n]) *) + +Definition double := [n:nat](plus n n). + +Hints Unfold double : arith. + +Lemma double_S : (n:nat) (double (S n))=(S (S (double n))). +Proof. +Intro. Unfold double. Simpl. Auto with arith. +Qed. + +Lemma double_plus : (m,n:nat) (double (plus m n))=(plus (double m) (double n)). +Proof. +Intros m n. Unfold double. +Do 2 Rewrite -> plus_assoc_r. Rewrite -> (plus_permute n). +Reflexivity. +Qed. + +Hints Resolve double_S : arith. + +Lemma even_odd_double : (n:nat) + ((even n)<->n=(double (div2 n))) /\ ((odd n)<->n=(S (double (div2 n)))). +Proof. +Intro n. Pattern n. Apply ind_0_1_SS. +(* n = 0 *) +Split; Split; Auto with arith. +Intro H. Inversion H. +(* n = 1 *) +Split; Split; Auto with arith. +Intro H. Inversion H. Inversion H1. +(* n = (S (S n')) *) +Intros. Decompose [and] H. Unfold iff in H0 H1. +Decompose [and] H0. Decompose [and] H1. Clear H H0 H1. +Split; Split. +Intro H. Inversion H. Inversion H1. +Simpl. Rewrite (double_S (div2 n0)). Auto with arith. +Simpl. Rewrite (double_S (div2 n0)). Intro H. Injection H. Auto with arith. +Intro H. Inversion H. Inversion H1. +Simpl. Rewrite (double_S (div2 n0)). Auto with arith. +Simpl. Rewrite (double_S (div2 n0)). Intro H. Injection H. Auto with arith. +Qed. + + +(** Specializations *) + +Lemma even_double : (n:nat) (even n) -> n=(double (div2 n)). +Proof [n:nat](proj1 ? ? (proj1 ? ? (even_odd_double n))). + +Lemma double_even : (n:nat) n=(double (div2 n)) -> (even n). +Proof [n:nat](proj2 ? ? (proj1 ? ? (even_odd_double n))). + +Lemma odd_double : (n:nat) (odd n) -> n=(S (double (div2 n))). +Proof [n:nat](proj1 ? ? (proj2 ? ? (even_odd_double n))). + +Lemma double_odd : (n:nat) n=(S (double (div2 n))) -> (odd n). +Proof [n:nat](proj2 ? ? (proj2 ? ? (even_odd_double n))). + +Hints Resolve even_double double_even odd_double double_odd : arith. + +(** Application: + - if [n] is even then there is a [p] such that [n = 2p] + - if [n] is odd then there is a [p] such that [n = 2p+1] + + (Immediate: it is [n/2]) *) + +Lemma even_2n : (n:nat) (even n) -> { p:nat | n=(double p) }. +Proof. +Intros n H. Exists (div2 n). Auto with arith. +Qed. + +Lemma odd_S2n : (n:nat) (odd n) -> { p:nat | n=(S (double p)) }. +Proof. +Intros n H. Exists (div2 n). Auto with arith. +Qed. + diff --git a/theories7/Arith/EqNat.v b/theories7/Arith/EqNat.v new file mode 100755 index 00000000..9f5ee7ee --- /dev/null +++ b/theories7/Arith/EqNat.v @@ -0,0 +1,78 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: EqNat.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +(** Equality on natural numbers *) + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,x,y:nat. + +Fixpoint eq_nat [n:nat] : nat -> Prop := + [m:nat]Cases n m of + O O => True + | O (S _) => False + | (S _) O => False + | (S n1) (S m1) => (eq_nat n1 m1) + end. + +Theorem eq_nat_refl : (n:nat)(eq_nat n n). +NewInduction n; Simpl; Auto. +Qed. +Hints Resolve eq_nat_refl : arith v62. + +Theorem eq_eq_nat : (n,m:nat)(n=m)->(eq_nat n m). +NewInduction 1; Trivial with arith. +Qed. +Hints Immediate eq_eq_nat : arith v62. + +Theorem eq_nat_eq : (n,m:nat)(eq_nat n m)->(n=m). +NewInduction n; NewInduction m; Simpl; Contradiction Orelse Auto with arith. +Qed. +Hints Immediate eq_nat_eq : arith v62. + +Theorem eq_nat_elim : (n:nat)(P:nat->Prop)(P n)->(m:nat)(eq_nat n m)->(P m). +Intros; Replace m with n; Auto with arith. +Qed. + +Theorem eq_nat_decide : (n,m:nat){(eq_nat n m)}+{~(eq_nat n m)}. +NewInduction n. +NewDestruct m. +Auto with arith. +Intros; Right; Red; Trivial with arith. +NewDestruct m. +Right; Red; Auto with arith. +Intros. +Simpl. +Apply IHn. +Defined. + +Fixpoint beq_nat [n:nat] : nat -> bool := + [m:nat]Cases n m of + O O => true + | O (S _) => false + | (S _) O => false + | (S n1) (S m1) => (beq_nat n1 m1) + end. + +Lemma beq_nat_refl : (x:nat)true=(beq_nat x x). +Proof. + Intro x; NewInduction x; Simpl; Auto. +Qed. + +Definition beq_nat_eq : (x,y:nat)true=(beq_nat x y)->x=y. +Proof. + Double Induction x y; Simpl. + Reflexivity. + Intros; Discriminate H0. + Intros; Discriminate H0. + Intros; Case (H0 ? H1); Reflexivity. +Defined. + diff --git a/theories7/Arith/Euclid.v b/theories7/Arith/Euclid.v new file mode 100644 index 00000000..adeaf713 --- /dev/null +++ b/theories7/Arith/Euclid.v @@ -0,0 +1,65 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Euclid.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +Require Mult. +Require Compare_dec. +Require Wf_nat. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type a,b,n,q,r:nat. + +Inductive diveucl [a,b:nat] : Set + := divex : (q,r:nat)(gt b r)->(a=(plus (mult q b) r))->(diveucl a b). + + +Lemma eucl_dev : (b:nat)(gt b O)->(a:nat)(diveucl a b). +Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0. +Elim (le_gt_dec b n). +Intro lebn. +Elim (H0 (minus n b)); Auto with arith. +Intros q r g e. +Apply divex with (S q) r; Simpl; Auto with arith. +Elim plus_assoc_l. +Elim e; Auto with arith. +Intros gtbn. +Apply divex with O n; Simpl; Auto with arith. +Qed. + +Lemma quotient : (b:nat)(gt b O)-> + (a:nat){q:nat|(EX r:nat | (a=(plus (mult q b) r))/\(gt b r))}. +Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0. +Elim (le_gt_dec b n). +Intro lebn. +Elim (H0 (minus n b)); Auto with arith. +Intros q Hq; Exists (S q). +Elim Hq; Intros r Hr. +Exists r; Simpl; Elim Hr; Intros. +Elim plus_assoc_l. +Elim H1; Auto with arith. +Intros gtbn. +Exists O; Exists n; Simpl; Auto with arith. +Qed. + +Lemma modulo : (b:nat)(gt b O)-> + (a:nat){r:nat|(EX q:nat | (a=(plus (mult q b) r))/\(gt b r))}. +Intros b H a; Pattern a; Apply gt_wf_rec; Intros n H0. +Elim (le_gt_dec b n). +Intro lebn. +Elim (H0 (minus n b)); Auto with arith. +Intros r Hr; Exists r. +Elim Hr; Intros q Hq. +Elim Hq; Intros; Exists (S q); Simpl. +Elim plus_assoc_l. +Elim H1; Auto with arith. +Intros gtbn. +Exists n; Exists O; Simpl; Auto with arith. +Qed. diff --git a/theories7/Arith/Even.v b/theories7/Arith/Even.v new file mode 100644 index 00000000..bcc413f5 --- /dev/null +++ b/theories7/Arith/Even.v @@ -0,0 +1,310 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Even.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +(** Here we define the predicates [even] and [odd] by mutual induction + and we prove the decidability and the exclusion of those predicates. + The main results about parity are proved in the module Div2. *) + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n:nat. + +Inductive even : nat->Prop := + even_O : (even O) + | even_S : (n:nat)(odd n)->(even (S n)) +with odd : nat->Prop := + odd_S : (n:nat)(even n)->(odd (S n)). + +Hint constr_even : arith := Constructors even. +Hint constr_odd : arith := Constructors odd. + +Lemma even_or_odd : (n:nat) (even n)\/(odd n). +Proof. +NewInduction n. +Auto with arith. +Elim IHn; Auto with arith. +Qed. + +Lemma even_odd_dec : (n:nat) { (even n) }+{ (odd n) }. +Proof. +NewInduction n. +Auto with arith. +Elim IHn; Auto with arith. +Qed. + +Lemma not_even_and_odd : (n:nat) (even n) -> (odd n) -> False. +Proof. +NewInduction n. +Intros. Inversion H0. +Intros. Inversion H. Inversion H0. Auto with arith. +Qed. + +Lemma even_plus_aux: + (n,m:nat) + (iff (odd (plus n m)) (odd n) /\ (even m) \/ (even n) /\ (odd m)) /\ + (iff (even (plus n m)) (even n) /\ (even m) \/ (odd n) /\ (odd m)). +Proof. +Intros n; Elim n; Simpl; Auto with arith. +Intros m; Split; Auto. +Split. +Intros H; Right; Split; Auto with arith. +Intros H'; Case H'; Auto with arith. +Intros H'0; Elim H'0; Intros H'1 H'2; Inversion H'1. +Intros H; Elim H; Auto. +Split; Auto with arith. +Intros H'; Elim H'; Auto with arith. +Intros H; Elim H; Auto. +Intros H'0; Elim H'0; Intros H'1 H'2; Inversion H'1. +Intros n0 H' m; Elim (H' m); Intros H'1 H'2; Elim H'1; Intros E1 E2; Elim H'2; + Intros E3 E4; Clear H'1 H'2. +Split; Split. +Intros H'0; Case E3. +Inversion H'0; Auto. +Intros H; Elim H; Intros H0 H1; Clear H; Auto with arith. +Intros H; Elim H; Intros H0 H1; Clear H; Auto with arith. +Intros H'0; Case H'0; Intros C0; Case C0; Intros C1 C2. +Apply odd_S. +Apply E4; Left; Split; Auto with arith. +Inversion C1; Auto. +Apply odd_S. +Apply E4; Right; Split; Auto with arith. +Inversion C1; Auto. +Intros H'0. +Case E1. +Inversion H'0; Auto. +Intros H; Elim H; Intros H0 H1; Clear H; Auto with arith. +Intros H; Elim H; Intros H0 H1; Clear H; Auto with arith. +Intros H'0; Case H'0; Intros C0; Case C0; Intros C1 C2. +Apply even_S. +Apply E2; Left; Split; Auto with arith. +Inversion C1; Auto. +Apply even_S. +Apply E2; Right; Split; Auto with arith. +Inversion C1; Auto. +Qed. + +Lemma even_even_plus : (n,m:nat) (even n) -> (even m) -> (even (plus n m)). +Proof. +Intros n m; Case (even_plus_aux n m). +Intros H H0; Case H0; Auto. +Qed. + +Lemma odd_even_plus : (n,m:nat) (odd n) -> (odd m) -> (even (plus n m)). +Proof. +Intros n m; Case (even_plus_aux n m). +Intros H H0; Case H0; Auto. +Qed. + +Lemma even_plus_even_inv_r : + (n,m:nat) (even (plus n m)) -> (even n) -> (even m). +Proof. +Intros n m H; Case (even_plus_aux n m). +Intros H' H'0; Elim H'0. +Intros H'1; Case H'1; Auto. +Intros H0; Elim H0; Auto. +Intros H0 H1 H2; Case (not_even_and_odd n); Auto. +Case H0; Auto. +Qed. + +Lemma even_plus_even_inv_l : + (n,m:nat) (even (plus n m)) -> (even m) -> (even n). +Proof. +Intros n m H; Case (even_plus_aux n m). +Intros H' H'0; Elim H'0. +Intros H'1; Case H'1; Auto. +Intros H0; Elim H0; Auto. +Intros H0 H1 H2; Case (not_even_and_odd m); Auto. +Case H0; Auto. +Qed. + +Lemma even_plus_odd_inv_r : (n,m:nat) (even (plus n m)) -> (odd n) -> (odd m). +Proof. +Intros n m H; Case (even_plus_aux n m). +Intros H' H'0; Elim H'0. +Intros H'1; Case H'1; Auto. +Intros H0 H1 H2; Case (not_even_and_odd n); Auto. +Case H0; Auto. +Intros H0; Case H0; Auto. +Qed. + +Lemma even_plus_odd_inv_l : (n,m:nat) (even (plus n m)) -> (odd m) -> (odd n). +Proof. +Intros n m H; Case (even_plus_aux n m). +Intros H' H'0; Elim H'0. +Intros H'1; Case H'1; Auto. +Intros H0 H1 H2; Case (not_even_and_odd m); Auto. +Case H0; Auto. +Intros H0; Case H0; Auto. +Qed. +Hints Resolve even_even_plus odd_even_plus :arith. + +Lemma odd_plus_l : (n,m:nat) (odd n) -> (even m) -> (odd (plus n m)). +Proof. +Intros n m; Case (even_plus_aux n m). +Intros H; Case H; Auto. +Qed. + +Lemma odd_plus_r : (n,m:nat) (even n) -> (odd m) -> (odd (plus n m)). +Proof. +Intros n m; Case (even_plus_aux n m). +Intros H; Case H; Auto. +Qed. + +Lemma odd_plus_even_inv_l : (n,m:nat) (odd (plus n m)) -> (odd m) -> (even n). +Proof. +Intros n m H; Case (even_plus_aux n m). +Intros H' H'0; Elim H'. +Intros H'1; Case H'1; Auto. +Intros H0 H1 H2; Case (not_even_and_odd m); Auto. +Case H0; Auto. +Intros H0; Case H0; Auto. +Qed. + +Lemma odd_plus_even_inv_r : (n,m:nat) (odd (plus n m)) -> (odd n) -> (even m). +Proof. +Intros n m H; Case (even_plus_aux n m). +Intros H' H'0; Elim H'. +Intros H'1; Case H'1; Auto. +Intros H0; Case H0; Auto. +Intros H0 H1 H2; Case (not_even_and_odd n); Auto. +Case H0; Auto. +Qed. + +Lemma odd_plus_odd_inv_l : (n,m:nat) (odd (plus n m)) -> (even m) -> (odd n). +Proof. +Intros n m H; Case (even_plus_aux n m). +Intros H' H'0; Elim H'. +Intros H'1; Case H'1; Auto. +Intros H0; Case H0; Auto. +Intros H0 H1 H2; Case (not_even_and_odd m); Auto. +Case H0; Auto. +Qed. + +Lemma odd_plus_odd_inv_r : (n,m:nat) (odd (plus n m)) -> (even n) -> (odd m). +Proof. +Intros n m H; Case (even_plus_aux n m). +Intros H' H'0; Elim H'. +Intros H'1; Case H'1; Auto. +Intros H0 H1 H2; Case (not_even_and_odd n); Auto. +Case H0; Auto. +Intros H0; Case H0; Auto. +Qed. +Hints Resolve odd_plus_l odd_plus_r :arith. + +Lemma even_mult_aux : + (n,m:nat) + (iff (odd (mult n m)) (odd n) /\ (odd m)) /\ + (iff (even (mult n m)) (even n) \/ (even m)). +Proof. +Intros n; Elim n; Simpl; Auto with arith. +Intros m; Split; Split; Auto with arith. +Intros H'; Inversion H'. +Intros H'; Elim H'; Auto. +Intros n0 H' m; Split; Split; Auto with arith. +Intros H'0. +Elim (even_plus_aux m (mult n0 m)); Intros H'3 H'4; Case H'3; Intros H'1 H'2; + Case H'1; Auto. +Intros H'5; Elim H'5; Intros H'6 H'7; Auto with arith. +Split; Auto with arith. +Case (H' m). +Intros H'8 H'9; Case H'9. +Intros H'10; Case H'10; Auto with arith. +Intros H'11 H'12; Case (not_even_and_odd m); Auto with arith. +Intros H'5; Elim H'5; Intros H'6 H'7; Case (not_even_and_odd (mult n0 m)); Auto. +Case (H' m). +Intros H'8 H'9; Case H'9; Auto. +Intros H'0; Elim H'0; Intros H'1 H'2; Clear H'0. +Elim (even_plus_aux m (mult n0 m)); Auto. +Intros H'0 H'3. +Elim H'0. +Intros H'4 H'5; Apply H'5; Auto. +Left; Split; Auto with arith. +Case (H' m). +Intros H'6 H'7; Elim H'7. +Intros H'8 H'9; Apply H'9. +Left. +Inversion H'1; Auto. +Intros H'0. +Elim (even_plus_aux m (mult n0 m)); Intros H'3 H'4; Case H'4. +Intros H'1 H'2. +Elim H'1; Auto. +Intros H; Case H; Auto. +Intros H'5; Elim H'5; Intros H'6 H'7; Auto with arith. +Left. +Case (H' m). +Intros H'8; Elim H'8. +Intros H'9; Elim H'9; Auto with arith. +Intros H'0; Elim H'0; Intros H'1. +Case (even_or_odd m); Intros H'2. +Apply even_even_plus; Auto. +Case (H' m). +Intros H H0; Case H0; Auto. +Apply odd_even_plus; Auto. +Inversion H'1; Case (H' m); Auto. +Intros H1; Case H1; Auto. +Apply even_even_plus; Auto. +Case (H' m). +Intros H H0; Case H0; Auto. +Qed. + +Lemma even_mult_l : (n,m:nat) (even n) -> (even (mult n m)). +Proof. +Intros n m; Case (even_mult_aux n m); Auto. +Intros H H0; Case H0; Auto. +Qed. + +Lemma even_mult_r: (n,m:nat) (even m) -> (even (mult n m)). +Proof. +Intros n m; Case (even_mult_aux n m); Auto. +Intros H H0; Case H0; Auto. +Qed. +Hints Resolve even_mult_l even_mult_r :arith. + +Lemma even_mult_inv_r: (n,m:nat) (even (mult n m)) -> (odd n) -> (even m). +Proof. +Intros n m H' H'0. +Case (even_mult_aux n m). +Intros H'1 H'2; Elim H'2. +Intros H'3; Elim H'3; Auto. +Intros H; Case (not_even_and_odd n); Auto. +Qed. + +Lemma even_mult_inv_l : (n,m:nat) (even (mult n m)) -> (odd m) -> (even n). +Proof. +Intros n m H' H'0. +Case (even_mult_aux n m). +Intros H'1 H'2; Elim H'2. +Intros H'3; Elim H'3; Auto. +Intros H; Case (not_even_and_odd m); Auto. +Qed. + +Lemma odd_mult : (n,m:nat) (odd n) -> (odd m) -> (odd (mult n m)). +Proof. +Intros n m; Case (even_mult_aux n m); Intros H; Case H; Auto. +Qed. +Hints Resolve even_mult_l even_mult_r odd_mult :arith. + +Lemma odd_mult_inv_l : (n,m:nat) (odd (mult n m)) -> (odd n). +Proof. +Intros n m H'. +Case (even_mult_aux n m). +Intros H'1 H'2; Elim H'1. +Intros H'3; Elim H'3; Auto. +Qed. + +Lemma odd_mult_inv_r : (n,m:nat) (odd (mult n m)) -> (odd m). +Proof. +Intros n m H'. +Case (even_mult_aux n m). +Intros H'1 H'2; Elim H'1. +Intros H'3; Elim H'3; Auto. +Qed. + diff --git a/theories7/Arith/Factorial.v b/theories7/Arith/Factorial.v new file mode 100644 index 00000000..a8a60c98 --- /dev/null +++ b/theories7/Arith/Factorial.v @@ -0,0 +1,51 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Factorial.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +Require Plus. +Require Mult. +Require Lt. +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +(** Factorial *) + +Fixpoint fact [n:nat]:nat:= + Cases n of + O => (S O) + |(S n) => (mult (S n) (fact n)) + end. + +Arguments Scope fact [ nat_scope ]. + +Lemma lt_O_fact : (n:nat)(lt O (fact n)). +Proof. +Induction n; Unfold lt; Simpl; Auto with arith. +Qed. + +Lemma fact_neq_0:(n:nat)~(fact n)=O. +Proof. +Intro. +Apply sym_not_eq. +Apply lt_O_neq. +Apply lt_O_fact. +Qed. + +Lemma fact_growing : (n,m:nat) (le n m) -> (le (fact n) (fact m)). +Proof. +NewInduction 1. +Apply le_n. +Assert (le (mult (S O) (fact n)) (mult (S m) (fact m))). +Apply le_mult_mult. +Apply lt_le_S; Apply lt_O_Sn. +Assumption. +Simpl (mult (S O) (fact n)) in H0. +Rewrite <- plus_n_O in H0. +Assumption. +Qed. diff --git a/theories7/Arith/Gt.v b/theories7/Arith/Gt.v new file mode 100755 index 00000000..16b6f203 --- /dev/null +++ b/theories7/Arith/Gt.v @@ -0,0 +1,149 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Gt.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +Require Le. +Require Lt. +Require Plus. +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,p:nat. + +(** Order and successor *) + +Theorem gt_Sn_O : (n:nat)(gt (S n) O). +Proof. + Auto with arith. +Qed. +Hints Resolve gt_Sn_O : arith v62. + +Theorem gt_Sn_n : (n:nat)(gt (S n) n). +Proof. + Auto with arith. +Qed. +Hints Resolve gt_Sn_n : arith v62. + +Theorem gt_n_S : (n,m:nat)(gt n m)->(gt (S n) (S m)). +Proof. + Auto with arith. +Qed. +Hints Resolve gt_n_S : arith v62. + +Lemma gt_S_n : (n,p:nat)(gt (S p) (S n))->(gt p n). +Proof. + Auto with arith. +Qed. +Hints Immediate gt_S_n : arith v62. + +Theorem gt_S : (n,m:nat)(gt (S n) m)->((gt n m)\/(m=n)). +Proof. + Intros n m H; Unfold gt; Apply le_lt_or_eq; Auto with arith. +Qed. + +Lemma gt_pred : (n,p:nat)(gt p (S n))->(gt (pred p) n). +Proof. + Auto with arith. +Qed. +Hints Immediate gt_pred : arith v62. + +(** Irreflexivity *) + +Lemma gt_antirefl : (n:nat)~(gt n n). +Proof lt_n_n. +Hints Resolve gt_antirefl : arith v62. + +(** Asymmetry *) + +Lemma gt_not_sym : (n,m:nat)(gt n m) -> ~(gt m n). +Proof [n,m:nat](lt_not_sym m n). + +Hints Resolve gt_not_sym : arith v62. + +(** Relating strict and large orders *) + +Lemma le_not_gt : (n,m:nat)(le n m) -> ~(gt n m). +Proof le_not_lt. +Hints Resolve le_not_gt : arith v62. + +Lemma gt_not_le : (n,m:nat)(gt n m) -> ~(le n m). +Proof. +Auto with arith. +Qed. + +Hints Resolve gt_not_le : arith v62. + +Theorem le_S_gt : (n,m:nat)(le (S n) m)->(gt m n). +Proof. + Auto with arith. +Qed. +Hints Immediate le_S_gt : arith v62. + +Lemma gt_S_le : (n,p:nat)(gt (S p) n)->(le n p). +Proof. + Intros n p; Exact (lt_n_Sm_le n p). +Qed. +Hints Immediate gt_S_le : arith v62. + +Lemma gt_le_S : (n,p:nat)(gt p n)->(le (S n) p). +Proof. + Auto with arith. +Qed. +Hints Resolve gt_le_S : arith v62. + +Lemma le_gt_S : (n,p:nat)(le n p)->(gt (S p) n). +Proof. + Auto with arith. +Qed. +Hints Resolve le_gt_S : arith v62. + +(** Transitivity *) + +Theorem le_gt_trans : (n,m,p:nat)(le m n)->(gt m p)->(gt n p). +Proof. + Red; Intros; Apply lt_le_trans with m; Auto with arith. +Qed. + +Theorem gt_le_trans : (n,m,p:nat)(gt n m)->(le p m)->(gt n p). +Proof. + Red; Intros; Apply le_lt_trans with m; Auto with arith. +Qed. + +Lemma gt_trans : (n,m,p:nat)(gt n m)->(gt m p)->(gt n p). +Proof. + Red; Intros n m p H1 H2. + Apply lt_trans with m; Auto with arith. +Qed. + +Theorem gt_trans_S : (n,m,p:nat)(gt (S n) m)->(gt m p)->(gt n p). +Proof. + Red; Intros; Apply lt_le_trans with m; Auto with arith. +Qed. + +Hints Resolve gt_trans_S le_gt_trans gt_le_trans : arith v62. + +(** Comparison to 0 *) + +Theorem gt_O_eq : (n:nat)((gt n O)\/(O=n)). +Proof. + Intro n ; Apply gt_S ; Auto with arith. +Qed. + +(** Simplification and compatibility *) + +Lemma simpl_gt_plus_l : (n,m,p:nat)(gt (plus p n) (plus p m))->(gt n m). +Proof. + Red; Intros n m p H; Apply simpl_lt_plus_l with p; Auto with arith. +Qed. + +Lemma gt_reg_l : (n,m,p:nat)(gt n m)->(gt (plus p n) (plus p m)). +Proof. + Auto with arith. +Qed. +Hints Resolve gt_reg_l : arith v62. diff --git a/theories7/Arith/Le.v b/theories7/Arith/Le.v new file mode 100755 index 00000000..cdb98645 --- /dev/null +++ b/theories7/Arith/Le.v @@ -0,0 +1,122 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Le.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +(** Order on natural numbers *) +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,p:nat. + +(** Reflexivity *) + +Theorem le_refl : (n:nat)(le n n). +Proof. +Exact le_n. +Qed. + +(** Transitivity *) + +Theorem le_trans : (n,m,p:nat)(le n m)->(le m p)->(le n p). +Proof. + NewInduction 2; Auto. +Qed. +Hints Resolve le_trans : arith v62. + +(** Order, successor and predecessor *) + +Theorem le_n_S : (n,m:nat)(le n m)->(le (S n) (S m)). +Proof. + NewInduction 1; Auto. +Qed. + +Theorem le_n_Sn : (n:nat)(le n (S n)). +Proof. + Auto. +Qed. + +Theorem le_O_n : (n:nat)(le O n). +Proof. + NewInduction n ; Auto. +Qed. + +Hints Resolve le_n_S le_n_Sn le_O_n le_n_S : arith v62. + +Theorem le_pred_n : (n:nat)(le (pred n) n). +Proof. +NewInduction n ; Auto with arith. +Qed. +Hints Resolve le_pred_n : arith v62. + +Theorem le_trans_S : (n,m:nat)(le (S n) m)->(le n m). +Proof. +Intros n m H ; Apply le_trans with (S n); Auto with arith. +Qed. +Hints Immediate le_trans_S : arith v62. + +Theorem le_S_n : (n,m:nat)(le (S n) (S m))->(le n m). +Proof. +Intros n m H ; Change (le (pred (S n)) (pred (S m))). +Elim H ; Simpl ; Auto with arith. +Qed. +Hints Immediate le_S_n : arith v62. + +Theorem le_pred : (n,m:nat)(le n m)->(le (pred n) (pred m)). +Proof. +NewInduction n as [|n IHn]. Simpl. Auto with arith. +NewDestruct m as [|m]. Simpl. Intro H. Inversion H. +Simpl. Auto with arith. +Qed. + +(** Comparison to 0 *) + +Theorem le_Sn_O : (n:nat)~(le (S n) O). +Proof. +Red ; Intros n H. +Change (IsSucc O) ; Elim H ; Simpl ; Auto with arith. +Qed. +Hints Resolve le_Sn_O : arith v62. + +Theorem le_n_O_eq : (n:nat)(le n O)->(O=n). +Proof. +NewInduction n; Auto with arith. +Intro; Contradiction le_Sn_O with n. +Qed. +Hints Immediate le_n_O_eq : arith v62. + +(** Negative properties *) + +Theorem le_Sn_n : (n:nat)~(le (S n) n). +Proof. +NewInduction n; Auto with arith. +Qed. +Hints Resolve le_Sn_n : arith v62. + +(** Antisymmetry *) + +Theorem le_antisym : (n,m:nat)(le n m)->(le m n)->(n=m). +Proof. +Intros n m h ; NewDestruct h as [|m0]; Auto with arith. +Intros H1. +Absurd (le (S m0) m0) ; Auto with arith. +Apply le_trans with n ; Auto with arith. +Qed. +Hints Immediate le_antisym : arith v62. + +(** A different elimination principle for the order on natural numbers *) + +Lemma le_elim_rel : (P:nat->nat->Prop) + ((p:nat)(P O p))-> + ((p,q:nat)(le p q)->(P p q)->(P (S p) (S q)))-> + (n,m:nat)(le n m)->(P n m). +Proof. +NewInduction n; Auto with arith. +Intros m Le. +Elim Le; Auto with arith. +Qed. diff --git a/theories7/Arith/Lt.v b/theories7/Arith/Lt.v new file mode 100755 index 00000000..9bb1d564 --- /dev/null +++ b/theories7/Arith/Lt.v @@ -0,0 +1,176 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Lt.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +Require Le. +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,p:nat. + +(** Irreflexivity *) + +Theorem lt_n_n : (n:nat)~(lt n n). +Proof le_Sn_n. +Hints Resolve lt_n_n : arith v62. + +(** Relationship between [le] and [lt] *) + +Theorem lt_le_S : (n,p:nat)(lt n p)->(le (S n) p). +Proof. +Auto with arith. +Qed. +Hints Immediate lt_le_S : arith v62. + +Theorem lt_n_Sm_le : (n,m:nat)(lt n (S m))->(le n m). +Proof. +Auto with arith. +Qed. +Hints Immediate lt_n_Sm_le : arith v62. + +Theorem le_lt_n_Sm : (n,m:nat)(le n m)->(lt n (S m)). +Proof. +Auto with arith. +Qed. +Hints Immediate le_lt_n_Sm : arith v62. + +Theorem le_not_lt : (n,m:nat)(le n m) -> ~(lt m n). +Proof. +NewInduction 1; Auto with arith. +Qed. + +Theorem lt_not_le : (n,m:nat)(lt n m) -> ~(le m n). +Proof. +Red; Intros n m Lt Le; Exact (le_not_lt m n Le Lt). +Qed. +Hints Immediate le_not_lt lt_not_le : arith v62. + +(** Asymmetry *) + +Theorem lt_not_sym : (n,m:nat)(lt n m) -> ~(lt m n). +Proof. +NewInduction 1; Auto with arith. +Qed. + +(** Order and successor *) + +Theorem lt_n_Sn : (n:nat)(lt n (S n)). +Proof. +Auto with arith. +Qed. +Hints Resolve lt_n_Sn : arith v62. + +Theorem lt_S : (n,m:nat)(lt n m)->(lt n (S m)). +Proof. +Auto with arith. +Qed. +Hints Resolve lt_S : arith v62. + +Theorem lt_n_S : (n,m:nat)(lt n m)->(lt (S n) (S m)). +Proof. +Auto with arith. +Qed. +Hints Resolve lt_n_S : arith v62. + +Theorem lt_S_n : (n,m:nat)(lt (S n) (S m))->(lt n m). +Proof. +Auto with arith. +Qed. +Hints Immediate lt_S_n : arith v62. + +Theorem lt_O_Sn : (n:nat)(lt O (S n)). +Proof. +Auto with arith. +Qed. +Hints Resolve lt_O_Sn : arith v62. + +Theorem lt_n_O : (n:nat)~(lt n O). +Proof le_Sn_O. +Hints Resolve lt_n_O : arith v62. + +(** Predecessor *) + +Lemma S_pred : (n,m:nat)(lt m n)->n=(S (pred n)). +Proof. +NewInduction 1; Auto with arith. +Qed. + +Lemma lt_pred : (n,p:nat)(lt (S n) p)->(lt n (pred p)). +Proof. +NewInduction 1; Simpl; Auto with arith. +Qed. +Hints Immediate lt_pred : arith v62. + +Lemma lt_pred_n_n : (n:nat)(lt O n)->(lt (pred n) n). +NewDestruct 1; Simpl; Auto with arith. +Qed. +Hints Resolve lt_pred_n_n : arith v62. + +(** Transitivity properties *) + +Theorem lt_trans : (n,m,p:nat)(lt n m)->(lt m p)->(lt n p). +Proof. +NewInduction 2; Auto with arith. +Qed. + +Theorem lt_le_trans : (n,m,p:nat)(lt n m)->(le m p)->(lt n p). +Proof. +NewInduction 2; Auto with arith. +Qed. + +Theorem le_lt_trans : (n,m,p:nat)(le n m)->(lt m p)->(lt n p). +Proof. +NewInduction 2; Auto with arith. +Qed. + +Hints Resolve lt_trans lt_le_trans le_lt_trans : arith v62. + +(** Large = strict or equal *) + +Theorem le_lt_or_eq : (n,m:nat)(le n m)->((lt n m) \/ n=m). +Proof. +NewInduction 1; Auto with arith. +Qed. + +Theorem lt_le_weak : (n,m:nat)(lt n m)->(le n m). +Proof. +Auto with arith. +Qed. +Hints Immediate lt_le_weak : arith v62. + +(** Dichotomy *) + +Theorem le_or_lt : (n,m:nat)((le n m)\/(lt m n)). +Proof. +Intros n m; Pattern n m; Apply nat_double_ind; Auto with arith. +NewInduction 1; Auto with arith. +Qed. + +Theorem nat_total_order: (m,n: nat) ~ m = n -> (lt m n) \/ (lt n m). +Proof. +Intros m n diff. +Elim (le_or_lt n m); [Intro H'0 | Auto with arith]. +Elim (le_lt_or_eq n m); Auto with arith. +Intro H'; Elim diff; Auto with arith. +Qed. + +(** Comparison to 0 *) + +Theorem neq_O_lt : (n:nat)(~O=n)->(lt O n). +Proof. +NewInduction n; Auto with arith. +Intros; Absurd O=O; Trivial with arith. +Qed. +Hints Immediate neq_O_lt : arith v62. + +Theorem lt_O_neq : (n:nat)(lt O n)->(~O=n). +Proof. +NewInduction 1; Auto with arith. +Qed. +Hints Immediate lt_O_neq : arith v62. diff --git a/theories7/Arith/Max.v b/theories7/Arith/Max.v new file mode 100755 index 00000000..aea389d1 --- /dev/null +++ b/theories7/Arith/Max.v @@ -0,0 +1,87 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Max.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +Require Arith. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n:nat. + +(** maximum of two natural numbers *) + +Fixpoint max [n:nat] : nat -> nat := +[m:nat]Cases n m of + O _ => m + | (S n') O => n + | (S n') (S m') => (S (max n' m')) + end. + +(** Simplifications of [max] *) + +Lemma max_SS : (n,m:nat)((S (max n m))=(max (S n) (S m))). +Proof. +Auto with arith. +Qed. + +Lemma max_sym : (n,m:nat)(max n m)=(max m n). +Proof. +NewInduction n;NewInduction m;Simpl;Auto with arith. +Qed. + +(** [max] and [le] *) + +Lemma max_l : (n,m:nat)(le m n)->(max n m)=n. +Proof. +NewInduction n;NewInduction m;Simpl;Auto with arith. +Qed. + +Lemma max_r : (n,m:nat)(le n m)->(max n m)=m. +Proof. +NewInduction n;NewInduction m;Simpl;Auto with arith. +Qed. + +Lemma le_max_l : (n,m:nat)(le n (max n m)). +Proof. +NewInduction n; Intros; Simpl; Auto with arith. +Elim m; Intros; Simpl; Auto with arith. +Qed. + +Lemma le_max_r : (n,m:nat)(le m (max n m)). +Proof. +NewInduction n; Simpl; Auto with arith. +NewInduction m; Simpl; Auto with arith. +Qed. +Hints Resolve max_r max_l le_max_l le_max_r: arith v62. + + +(** [max n m] is equal to [n] or [m] *) + +Lemma max_dec : (n,m:nat){(max n m)=n}+{(max n m)=m}. +Proof. +NewInduction n;NewInduction m;Simpl;Auto with arith. +Elim (IHn m);Intro H;Elim H;Auto. +Qed. + +Lemma max_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (max n m)). +Proof. +NewInduction n; Simpl; Auto with arith. +NewInduction m; Intros; Simpl; Auto with arith. +Pattern (max n m); Apply IHn ; Auto with arith. +Qed. + +Lemma max_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (max n m)). +Proof. +NewInduction n; Simpl; Auto with arith. +NewInduction m; Intros; Simpl; Auto with arith. +Pattern (max n m); Apply IHn ; Auto with arith. +Qed. + + diff --git a/theories7/Arith/Min.v b/theories7/Arith/Min.v new file mode 100755 index 00000000..fd5da61a --- /dev/null +++ b/theories7/Arith/Min.v @@ -0,0 +1,84 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Min.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +Require Arith. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n:nat. + +(** minimum of two natural numbers *) + +Fixpoint min [n:nat] : nat -> nat := +[m:nat]Cases n m of + O _ => O + | (S n') O => O + | (S n') (S m') => (S (min n' m')) + end. + +(** Simplifications of [min] *) + +Lemma min_SS : (n,m:nat)((S (min n m))=(min (S n) (S m))). +Proof. +Auto with arith. +Qed. + +Lemma min_sym : (n,m:nat)(min n m)=(min m n). +Proof. +NewInduction n;NewInduction m;Simpl;Auto with arith. +Qed. + +(** [min] and [le] *) + +Lemma min_l : (n,m:nat)(le n m)->(min n m)=n. +Proof. +NewInduction n;NewInduction m;Simpl;Auto with arith. +Qed. + +Lemma min_r : (n,m:nat)(le m n)->(min n m)=m. +Proof. +NewInduction n;NewInduction m;Simpl;Auto with arith. +Qed. + +Lemma le_min_l : (n,m:nat)(le (min n m) n). +Proof. +NewInduction n; Intros; Simpl; Auto with arith. +Elim m; Intros; Simpl; Auto with arith. +Qed. + +Lemma le_min_r : (n,m:nat)(le (min n m) m). +Proof. +NewInduction n; Simpl; Auto with arith. +NewInduction m; Simpl; Auto with arith. +Qed. +Hints Resolve min_l min_r le_min_l le_min_r : arith v62. + +(** [min n m] is equal to [n] or [m] *) + +Lemma min_dec : (n,m:nat){(min n m)=n}+{(min n m)=m}. +Proof. +NewInduction n;NewInduction m;Simpl;Auto with arith. +Elim (IHn m);Intro H;Elim H;Auto. +Qed. + +Lemma min_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (min n m)). +Proof. +NewInduction n; Simpl; Auto with arith. +NewInduction m; Intros; Simpl; Auto with arith. +Pattern (min n m); Apply IHn ; Auto with arith. +Qed. + +Lemma min_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (min n m)). +Proof. +NewInduction n; Simpl; Auto with arith. +NewInduction m; Intros; Simpl; Auto with arith. +Pattern (min n m); Apply IHn ; Auto with arith. +Qed. diff --git a/theories7/Arith/Minus.v b/theories7/Arith/Minus.v new file mode 100755 index 00000000..709d5f0b --- /dev/null +++ b/theories7/Arith/Minus.v @@ -0,0 +1,120 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Minus.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) + +(** Subtraction (difference between two natural numbers) *) + +Require Lt. +Require Le. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,p:nat. + +(** 0 is right neutral *) + +Lemma minus_n_O : (n:nat)(n=(minus n O)). +Proof. +NewInduction n; Simpl; Auto with arith. +Qed. +Hints Resolve minus_n_O : arith v62. + +(** Permutation with successor *) + +Lemma minus_Sn_m : (n,m:nat)(le m n)->((S (minus n m))=(minus (S n) m)). +Proof. +Intros n m Le; Pattern m n; Apply le_elim_rel; Simpl; Auto with arith. +Qed. +Hints Resolve minus_Sn_m : arith v62. + +Theorem pred_of_minus : (x:nat)(pred x)=(minus x (S O)). +Intro x; NewInduction x; Simpl; Auto with arith. +Qed. + +(** Diagonal *) + +Lemma minus_n_n : (n:nat)(O=(minus n n)). +Proof. +NewInduction n; Simpl; Auto with arith. +Qed. +Hints Resolve minus_n_n : arith v62. + +(** Simplification *) + +Lemma minus_plus_simpl : + (n,m,p:nat)((minus n m)=(minus (plus p n) (plus p m))). +Proof. + NewInduction p; Simpl; Auto with arith. +Qed. +Hints Resolve minus_plus_simpl : arith v62. + +(** Relation with plus *) + +Lemma plus_minus : (n,m,p:nat)(n=(plus m p))->(p=(minus n m)). +Proof. +Intros n m p; Pattern m n; Apply nat_double_ind; Simpl; Intros. +Replace (minus n0 O) with n0; Auto with arith. +Absurd O=(S (plus n0 p)); Auto with arith. +Auto with arith. +Qed. +Hints Immediate plus_minus : arith v62. + +Lemma minus_plus : (n,m:nat)(minus (plus n m) n)=m. +Symmetry; Auto with arith. +Qed. +Hints Resolve minus_plus : arith v62. + +Lemma le_plus_minus : (n,m:nat)(le n m)->(m=(plus n (minus m n))). +Proof. +Intros n m Le; Pattern n m; Apply le_elim_rel; Simpl; Auto with arith. +Qed. +Hints Resolve le_plus_minus : arith v62. + +Lemma le_plus_minus_r : (n,m:nat)(le n m)->(plus n (minus m n))=m. +Proof. +Symmetry; Auto with arith. +Qed. +Hints Resolve le_plus_minus_r : arith v62. + +(** Relation with order *) + +Theorem le_minus: (i,h:nat) (le (minus i h) i). +Proof. +Intros i h;Pattern i h; Apply nat_double_ind; [ + Auto +| Auto +| Intros m n H; Simpl; Apply le_trans with m:=m; Auto ]. +Qed. + +Lemma lt_minus : (n,m:nat)(le m n)->(lt O m)->(lt (minus n m) n). +Proof. +Intros n m Le; Pattern m n; Apply le_elim_rel; Simpl; Auto with arith. +Intros; Absurd (lt O O); Auto with arith. +Intros p q lepq Hp gtp. +Elim (le_lt_or_eq O p); Auto with arith. +Auto with arith. +NewInduction 1; Elim minus_n_O; Auto with arith. +Qed. +Hints Resolve lt_minus : arith v62. + +Lemma lt_O_minus_lt : (n,m:nat)(lt O (minus n m))->(lt m n). +Proof. +Intros n m; Pattern n m; Apply nat_double_ind; Simpl; Auto with arith. +Intros; Absurd (lt O O); Trivial with arith. +Qed. +Hints Immediate lt_O_minus_lt : arith v62. + +Theorem inj_minus_aux: (x,y:nat) ~(le y x) -> (minus x y) = O. +Intros y x; Pattern y x ; Apply nat_double_ind; [ + Simpl; Trivial with arith +| Intros n H; Absurd (le O (S n)); [ Assumption | Apply le_O_n] +| Simpl; Intros n m H1 H2; Apply H1; + Unfold not ; Intros H3; Apply H2; Apply le_n_S; Assumption]. +Qed. diff --git a/theories7/Arith/Mult.v b/theories7/Arith/Mult.v new file mode 100755 index 00000000..9bd4aaf9 --- /dev/null +++ b/theories7/Arith/Mult.v @@ -0,0 +1,224 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Mult.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +Require Export Plus. +Require Export Minus. +Require Export Lt. +Require Export Le. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,p:nat. + +(** Zero property *) + +Lemma mult_0_r : (n:nat) (mult n O)=O. +Proof. +Intro; Symmetry; Apply mult_n_O. +Qed. + +Lemma mult_0_l : (n:nat) (mult O n)=O. +Proof. +Reflexivity. +Qed. + +(** Distributivity *) + +Lemma mult_plus_distr : + (n,m,p:nat)((mult (plus n m) p)=(plus (mult n p) (mult m p))). +Proof. +Intros; Elim n; Simpl; Intros; Auto with arith. +Elim plus_assoc_l; Elim H; Auto with arith. +Qed. +Hints Resolve mult_plus_distr : arith v62. + +Lemma mult_plus_distr_r : (n,m,p:nat) (mult n (plus m p))=(plus (mult n m) (mult n p)). +Proof. + NewInduction n. Trivial. + Intros. Simpl. Rewrite (IHn m p). Apply sym_eq. Apply plus_permute_2_in_4. +Qed. + +Lemma mult_minus_distr : (n,m,p:nat)((mult (minus n m) p)=(minus (mult n p) (mult m p))). +Proof. +Intros; Pattern n m; Apply nat_double_ind; Simpl; Intros; Auto with arith. +Elim minus_plus_simpl; Auto with arith. +Qed. +Hints Resolve mult_minus_distr : arith v62. + +(** Associativity *) + +Lemma mult_assoc_r : (n,m,p:nat)((mult (mult n m) p) = (mult n (mult m p))). +Proof. +Intros; Elim n; Intros; Simpl; Auto with arith. +Rewrite mult_plus_distr. +Elim H; Auto with arith. +Qed. +Hints Resolve mult_assoc_r : arith v62. + +Lemma mult_assoc_l : (n,m,p:nat)(mult n (mult m p)) = (mult (mult n m) p). +Proof. +Auto with arith. +Qed. +Hints Resolve mult_assoc_l : arith v62. + +(** Commutativity *) + +Lemma mult_sym : (n,m:nat)(mult n m)=(mult m n). +Proof. +Intros; Elim n; Intros; Simpl; Auto with arith. +Elim mult_n_Sm. +Elim H; Apply plus_sym. +Qed. +Hints Resolve mult_sym : arith v62. + +(** 1 is neutral *) + +Lemma mult_1_n : (n:nat)(mult (S O) n)=n. +Proof. +Simpl; Auto with arith. +Qed. +Hints Resolve mult_1_n : arith v62. + +Lemma mult_n_1 : (n:nat)(mult n (S O))=n. +Proof. +Intro; Elim mult_sym; Auto with arith. +Qed. +Hints Resolve mult_n_1 : arith v62. + +(** Compatibility with orders *) + +Lemma mult_O_le : (n,m:nat)(m=O)\/(le n (mult m n)). +Proof. +NewInduction m; Simpl; Auto with arith. +Qed. +Hints Resolve mult_O_le : arith v62. + +Lemma mult_le_compat_l : (n,m,p:nat) (le n m) -> (le (mult p n) (mult p m)). +Proof. + NewInduction p as [|p IHp]. Intros. Simpl. Apply le_n. + Intros. Simpl. Apply le_plus_plus. Assumption. + Apply IHp. Assumption. +Qed. +Hints Resolve mult_le_compat_l : arith. +V7only [ +Notation mult_le := [m,n,p:nat](mult_le_compat_l p n m). +]. + + +Lemma le_mult_right : (m,n,p:nat)(le m n)->(le (mult m p) (mult n p)). +Intros m n p H. +Rewrite mult_sym. Rewrite (mult_sym n). +Auto with arith. +Qed. + +Lemma le_mult_mult : + (m,n,p,q:nat)(le m n)->(le p q)->(le (mult m p) (mult n q)). +Proof. +Intros m n p q Hmn Hpq; NewInduction Hmn. +NewInduction Hpq. +(* m*p<=m*p *) +Apply le_n. +(* m*p<=m*m0 -> m*p<=m*(S m0) *) +Rewrite <- mult_n_Sm; Apply le_trans with (mult m m0). +Assumption. +Apply le_plus_l. +(* m*p<=m0*q -> m*p<=(S m0)*q *) +Simpl; Apply le_trans with (mult m0 q). +Assumption. +Apply le_plus_r. +Qed. + +Lemma mult_lt : (m,n,p:nat) (lt n p) -> (lt (mult (S m) n) (mult (S m) p)). +Proof. + Intro m; NewInduction m. Intros. Simpl. Rewrite <- plus_n_O. Rewrite <- plus_n_O. Assumption. + Intros. Exact (lt_plus_plus ? ? ? ? H (IHm ? ? H)). +Qed. + +Hints Resolve mult_lt : arith. +V7only [ +Notation lt_mult_left := mult_lt. +(* Theorem lt_mult_left : + (x,y,z:nat) (lt x y) -> (lt (mult (S z) x) (mult (S z) y)). +*) +]. + +Lemma lt_mult_right : + (m,n,p:nat) (lt m n) -> (lt (0) p) -> (lt (mult m p) (mult n p)). +Intros m n p H H0. +NewInduction p. +Elim (lt_n_n ? H0). +Rewrite mult_sym. +Replace (mult n (S p)) with (mult (S p) n); Auto with arith. +Qed. + +Lemma mult_le_conv_1 : (m,n,p:nat) (le (mult (S m) n) (mult (S m) p)) -> (le n p). +Proof. + Intros m n p H. Elim (le_or_lt n p). Trivial. + Intro H0. Cut (lt (mult (S m) n) (mult (S m) n)). Intro. Elim (lt_n_n ? H1). + Apply le_lt_trans with m:=(mult (S m) p). Assumption. + Apply mult_lt. Assumption. +Qed. + +(** n|->2*n and n|->2n+1 have disjoint image *) + +V7only [ (* From Zdivides *) ]. +Theorem odd_even_lem: + (p, q : ?) ~ (plus (mult (2) p) (1)) = (mult (2) q). +Intros p; Elim p; Auto. +Intros q; Case q; Simpl. +Red; Intros; Discriminate. +Intros q'; Rewrite [x, y : ?] (plus_sym x (S y)); Simpl; Red; Intros; + Discriminate. +Intros p' H q; Case q. +Simpl; Red; Intros; Discriminate. +Intros q'; Red; Intros H0; Case (H q'). +Replace (mult (S (S O)) q') with (minus (mult (S (S O)) (S q')) (2)). +Rewrite <- H0; Simpl; Auto. +Repeat Rewrite [x, y : ?] (plus_sym x (S y)); Simpl; Auto. +Simpl; Repeat Rewrite [x, y : ?] (plus_sym x (S y)); Simpl; Auto. +Case q'; Simpl; Auto. +Qed. + + +(** Tail-recursive mult *) + +(** [tail_mult] is an alternative definition for [mult] which is + tail-recursive, whereas [mult] is not. This can be useful + when extracting programs. *) + +Fixpoint mult_acc [s,m,n:nat] : nat := + Cases n of + O => s + | (S p) => (mult_acc (tail_plus m s) m p) + end. + +Lemma mult_acc_aux : (n,s,m:nat)(plus s (mult n m))= (mult_acc s m n). +Proof. +NewInduction n as [|p IHp]; Simpl;Auto. +Intros s m; Rewrite <- plus_tail_plus; Rewrite <- IHp. +Rewrite <- plus_assoc_r; Apply (f_equal2 nat nat);Auto. +Rewrite plus_sym;Auto. +Qed. + +Definition tail_mult := [n,m:nat](mult_acc O m n). + +Lemma mult_tail_mult : (n,m:nat)(mult n m)=(tail_mult n m). +Proof. +Intros; Unfold tail_mult; Rewrite <- mult_acc_aux;Auto. +Qed. + +(** [TailSimpl] transforms any [tail_plus] and [tail_mult] into [plus] + and [mult] and simplify *) + +Tactic Definition TailSimpl := + Repeat Rewrite <- plus_tail_plus; + Repeat Rewrite <- mult_tail_mult; + Simpl. diff --git a/theories7/Arith/Peano_dec.v b/theories7/Arith/Peano_dec.v new file mode 100755 index 00000000..6646545a --- /dev/null +++ b/theories7/Arith/Peano_dec.v @@ -0,0 +1,36 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Peano_dec.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +Require Decidable. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,x,y:nat. + +Theorem O_or_S : (n:nat)({m:nat|(S m)=n})+{O=n}. +Proof. +NewInduction n. +Auto. +Left; Exists n; Auto. +Defined. + +Theorem eq_nat_dec : (n,m:nat){n=m}+{~(n=m)}. +Proof. +NewInduction n; NewInduction m; Auto. +Elim (IHn m); Auto. +Defined. + +Hints Resolve O_or_S eq_nat_dec : arith. + +Theorem dec_eq_nat:(x,y:nat)(decidable (x=y)). +Intros x y; Unfold decidable; Elim (eq_nat_dec x y); Auto with arith. +Defined. + diff --git a/theories7/Arith/Plus.v b/theories7/Arith/Plus.v new file mode 100755 index 00000000..23488b4c --- /dev/null +++ b/theories7/Arith/Plus.v @@ -0,0 +1,223 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Plus.v,v 1.5.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +(** Properties of addition *) + +Require Le. +Require Lt. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,p,q:nat. + +(** Zero is neutral *) + +Lemma plus_0_l : (n:nat) (O+n)=n. +Proof. +Reflexivity. +Qed. + +Lemma plus_0_r : (n:nat) (n+O)=n. +Proof. +Intro; Symmetry; Apply plus_n_O. +Qed. + +(** Commutativity *) + +Lemma plus_sym : (n,m:nat)(n+m)=(m+n). +Proof. +Intros n m ; Elim n ; Simpl ; Auto with arith. +Intros y H ; Elim (plus_n_Sm m y) ; Auto with arith. +Qed. +Hints Immediate plus_sym : arith v62. + +(** Associativity *) + +Lemma plus_Snm_nSm : (n,m:nat)((S n)+m)=(n+(S m)). +Intros. +Simpl. +Rewrite -> (plus_sym n m). +Rewrite -> (plus_sym n (S m)). +Trivial with arith. +Qed. + +Lemma plus_assoc_l : (n,m,p:nat)((n+(m+p))=((n+m)+p)). +Proof. +Intros n m p; Elim n; Simpl; Auto with arith. +Qed. +Hints Resolve plus_assoc_l : arith v62. + +Lemma plus_permute : (n,m,p:nat) ((n+(m+p))=(m+(n+p))). +Proof. +Intros; Rewrite (plus_assoc_l m n p); Rewrite (plus_sym m n); Auto with arith. +Qed. + +Lemma plus_assoc_r : (n,m,p:nat)(((n+m)+p)=(n+(m+p))). +Proof. +Auto with arith. +Qed. +Hints Resolve plus_assoc_r : arith v62. + +(** Simplification *) + +Lemma plus_reg_l : (m,p,n:nat)((n+m)=(n+p))->(m=p). +Proof. +Intros m p n; NewInduction n ; Simpl ; Auto with arith. +Qed. +V7only [ +(* Compatibility order of arguments *) +Notation "'simpl_plus_l' c" := [a,b:nat](plus_reg_l a b c) + (at level 10, c at next level). +Notation "'simpl_plus_l' c a" := [b:nat](plus_reg_l a b c) + (at level 10, a, c at next level). +Notation "'simpl_plus_l' c a b" := (plus_reg_l a b c) + (at level 10, a, b, c at next level). +Notation simpl_plus_l := plus_reg_l. +]. + +Lemma plus_le_reg_l : (n,m,p:nat)((p+n)<=(p+m))->(n<=m). +Proof. +NewInduction p; Simpl; Auto with arith. +Qed. +V7only [ +(* Compatibility order of arguments *) +Notation "'simpl_le_plus_l' c" := [a,b:nat](plus_le_reg_l a b c) + (at level 10, c at next level). +Notation "'simpl_le_plus_l' c a" := [b:nat](plus_le_reg_l a b c) + (at level 10, a, c at next level). +Notation "'simpl_le_plus_l' c a b" := (plus_le_reg_l a b c) + (at level 10, a, b, c at next level). +Notation simpl_le_plus_l := [p,n,m:nat](plus_le_reg_l n m p). +]. + +Lemma simpl_lt_plus_l : (n,m,p:nat) (p+n)<(p+m) -> n<m. +Proof. +NewInduction p; Simpl; Auto with arith. +Qed. + +(** Compatibility with order *) + +Lemma le_reg_l : (n,m,p:nat) n<=m -> (p+n)<=(p+m). +Proof. +NewInduction p; Simpl; Auto with arith. +Qed. +Hints Resolve le_reg_l : arith v62. + +Lemma le_reg_r : (a,b,c:nat) a<=b -> (a+c)<=(b+c). +Proof. +NewInduction 1 ; Simpl; Auto with arith. +Qed. +Hints Resolve le_reg_r : arith v62. + +Lemma le_plus_l : (n,m:nat) n<=(n+m). +Proof. +NewInduction n; Simpl; Auto with arith. +Qed. +Hints Resolve le_plus_l : arith v62. + +Lemma le_plus_r : (n,m:nat) m<=(n+m). +Proof. +Intros n m; Elim n; Simpl; Auto with arith. +Qed. +Hints Resolve le_plus_r : arith v62. + +Theorem le_plus_trans : (n,m,p:nat) n<=m -> n<=(m+p). +Proof. +Intros; Apply le_trans with m:=m; Auto with arith. +Qed. +Hints Resolve le_plus_trans : arith v62. + +Theorem lt_plus_trans : (n,m,p:nat) n<m -> n<(m+p). +Proof. +Intros; Apply lt_le_trans with m:=m; Auto with arith. +Qed. +Hints Immediate lt_plus_trans : arith v62. + +Lemma lt_reg_l : (n,m,p:nat) n<m -> (p+n)<(p+m). +Proof. +NewInduction p; Simpl; Auto with arith. +Qed. +Hints Resolve lt_reg_l : arith v62. + +Lemma lt_reg_r : (n,m,p:nat) n<m -> (n+p)<(m+p). +Proof. +Intros n m p H ; Rewrite (plus_sym n p) ; Rewrite (plus_sym m p). +Elim p; Auto with arith. +Qed. +Hints Resolve lt_reg_r : arith v62. + +Lemma le_plus_plus : (n,m,p,q:nat) n<=m -> p<=q -> (n+p)<=(m+q). +Proof. +Intros n m p q H H0. +Elim H; Simpl; Auto with arith. +Qed. + +Lemma le_lt_plus_plus : (n,m,p,q:nat) n<=m -> p<q -> (n+p)<(m+q). +Proof. + Unfold lt. Intros. Change ((S n)+p)<=(m+q). Rewrite plus_Snm_nSm. + Apply le_plus_plus; Assumption. +Qed. + +Lemma lt_le_plus_plus : (n,m,p,q:nat) n<m -> p<=q -> (n+p)<(m+q). +Proof. + Unfold lt. Intros. Change ((S n)+p)<=(m+q). Apply le_plus_plus; Assumption. +Qed. + +Lemma lt_plus_plus : (n,m,p,q:nat) n<m -> p<q -> (n+p)<(m+q). +Proof. + Intros. Apply lt_le_plus_plus. Assumption. + Apply lt_le_weak. Assumption. +Qed. + +(** Inversion lemmas *) + +Lemma plus_is_O : (m,n:nat) (m+n)=O -> m=O /\ n=O. +Proof. + Intro m; NewDestruct m; Auto. + Intros. Discriminate H. +Qed. + +Definition plus_is_one : + (m,n:nat) (m+n)=(S O) -> {m=O /\ n=(S O)}+{m=(S O) /\ n=O}. +Proof. + Intro m; NewDestruct m; Auto. + NewDestruct n; Auto. + Intros. + Simpl in H. Discriminate H. +Defined. + +(** Derived properties *) + +Lemma plus_permute_2_in_4 : (m,n,p,q:nat) ((m+n)+(p+q))=((m+p)+(n+q)). +Proof. + Intros m n p q. + Rewrite <- (plus_assoc_l m n (p+q)). Rewrite (plus_assoc_l n p q). + Rewrite (plus_sym n p). Rewrite <- (plus_assoc_l p n q). Apply plus_assoc_l. +Qed. + +(** Tail-recursive plus *) + +(** [tail_plus] is an alternative definition for [plus] which is + tail-recursive, whereas [plus] is not. This can be useful + when extracting programs. *) + +Fixpoint plus_acc [q,n:nat] : nat := + Cases n of + O => q + | (S p) => (plus_acc (S q) p) + end. + +Definition tail_plus := [n,m:nat](plus_acc m n). + +Lemma plus_tail_plus : (n,m:nat)(n+m)=(tail_plus n m). +Unfold tail_plus; NewInduction n as [|n IHn]; Simpl; Auto. +Intro m; Rewrite <- IHn; Simpl; Auto. +Qed. diff --git a/theories7/Arith/Wf_nat.v b/theories7/Arith/Wf_nat.v new file mode 100755 index 00000000..be1003ce --- /dev/null +++ b/theories7/Arith/Wf_nat.v @@ -0,0 +1,200 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Wf_nat.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +(** Well-founded relations and natural numbers *) + +Require Lt. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Implicit Variables Type m,n,p:nat. + +Chapter Well_founded_Nat. + +Variable A : Set. + +Variable f : A -> nat. +Definition ltof := [a,b:A](lt (f a) (f b)). +Definition gtof := [a,b:A](gt (f b) (f a)). + +Theorem well_founded_ltof : (well_founded A ltof). +Proof. +Red. +Cut (n:nat)(a:A)(lt (f a) n)->(Acc A ltof a). +Intros H a; Apply (H (S (f a))); Auto with arith. +NewInduction n. +Intros; Absurd (lt (f a) O); Auto with arith. +Intros a ltSma. +Apply Acc_intro. +Unfold ltof; Intros b ltfafb. +Apply IHn. +Apply lt_le_trans with (f a); Auto with arith. +Qed. + +Theorem well_founded_gtof : (well_founded A gtof). +Proof well_founded_ltof. + +(** It is possible to directly prove the induction principle going + back to primitive recursion on natural numbers ([induction_ltof1]) + or to use the previous lemmas to extract a program with a fixpoint + ([induction_ltof2]) + +the ML-like program for [induction_ltof1] is : [[ + let induction_ltof1 F a = indrec ((f a)+1) a + where rec indrec = + function 0 -> (function a -> error) + |(S m) -> (function a -> (F a (function y -> indrec y m)));; +]] + +the ML-like program for [induction_ltof2] is : [[ + let induction_ltof2 F a = indrec a + where rec indrec a = F a indrec;; +]] *) + +Theorem induction_ltof1 + : (P:A->Set)((x:A)((y:A)(ltof y x)->(P y))->(P x))->(a:A)(P a). +Proof. +Intros P F; Cut (n:nat)(a:A)(lt (f a) n)->(P a). +Intros H a; Apply (H (S (f a))); Auto with arith. +NewInduction n. +Intros; Absurd (lt (f a) O); Auto with arith. +Intros a ltSma. +Apply F. +Unfold ltof; Intros b ltfafb. +Apply IHn. +Apply lt_le_trans with (f a); Auto with arith. +Defined. + +Theorem induction_gtof1 + : (P:A->Set)((x:A)((y:A)(gtof y x)->(P y))->(P x))->(a:A)(P a). +Proof. +Exact induction_ltof1. +Defined. + +Theorem induction_ltof2 + : (P:A->Set)((x:A)((y:A)(ltof y x)->(P y))->(P x))->(a:A)(P a). +Proof. +Exact (well_founded_induction A ltof well_founded_ltof). +Defined. + +Theorem induction_gtof2 + : (P:A->Set)((x:A)((y:A)(gtof y x)->(P y))->(P x))->(a:A)(P a). +Proof. +Exact induction_ltof2. +Defined. + +(** If a relation [R] is compatible with [lt] i.e. if [x R y => f(x) < f(y)] + then [R] is well-founded. *) + +Variable R : A->A->Prop. + +Hypothesis H_compat : (x,y:A) (R x y) -> (lt (f x) (f y)). + +Theorem well_founded_lt_compat : (well_founded A R). +Proof. +Red. +Cut (n:nat)(a:A)(lt (f a) n)->(Acc A R a). +Intros H a; Apply (H (S (f a))); Auto with arith. +NewInduction n. +Intros; Absurd (lt (f a) O); Auto with arith. +Intros a ltSma. +Apply Acc_intro. +Intros b ltfafb. +Apply IHn. +Apply lt_le_trans with (f a); Auto with arith. +Qed. + +End Well_founded_Nat. + +Lemma lt_wf : (well_founded nat lt). +Proof (well_founded_ltof nat [m:nat]m). + +Lemma lt_wf_rec1 : (p:nat)(P:nat->Set) + ((n:nat)((m:nat)(lt m n)->(P m))->(P n)) -> (P p). +Proof. +Exact [p:nat][P:nat->Set][F:(n:nat)((m:nat)(lt m n)->(P m))->(P n)] + (induction_ltof1 nat [m:nat]m P F p). +Defined. + +Lemma lt_wf_rec : (p:nat)(P:nat->Set) + ((n:nat)((m:nat)(lt m n)->(P m))->(P n)) -> (P p). +Proof. +Exact [p:nat][P:nat->Set][F:(n:nat)((m:nat)(lt m n)->(P m))->(P n)] + (induction_ltof2 nat [m:nat]m P F p). +Defined. + +Lemma lt_wf_ind : (p:nat)(P:nat->Prop) + ((n:nat)((m:nat)(lt m n)->(P m))->(P n)) -> (P p). +Intro p; Intros; Elim (lt_wf p); Auto with arith. +Qed. + +Lemma gt_wf_rec : (p:nat)(P:nat->Set) + ((n:nat)((m:nat)(gt n m)->(P m))->(P n)) -> (P p). +Proof. +Exact lt_wf_rec. +Defined. + +Lemma gt_wf_ind : (p:nat)(P:nat->Prop) + ((n:nat)((m:nat)(gt n m)->(P m))->(P n)) -> (P p). +Proof lt_wf_ind. + +Lemma lt_wf_double_rec : + (P:nat->nat->Set) + ((n,m:nat)((p,q:nat)(lt p n)->(P p q))->((p:nat)(lt p m)->(P n p))->(P n m)) + -> (p,q:nat)(P p q). +Intros P Hrec p; Pattern p; Apply lt_wf_rec. +Intros n H q; Pattern q; Apply lt_wf_rec; Auto with arith. +Defined. + +Lemma lt_wf_double_ind : + (P:nat->nat->Prop) + ((n,m:nat)((p,q:nat)(lt p n)->(P p q))->((p:nat)(lt p m)->(P n p))->(P n m)) + -> (p,q:nat)(P p q). +Intros P Hrec p; Pattern p; Apply lt_wf_ind. +Intros n H q; Pattern q; Apply lt_wf_ind; Auto with arith. +Qed. + +Hints Resolve lt_wf : arith. +Hints Resolve well_founded_lt_compat : arith. + +Section LT_WF_REL. +Variable A :Set. +Variable R:A->A->Prop. + +(* Relational form of inversion *) +Variable F : A -> nat -> Prop. +Definition inv_lt_rel + [x,y]:=(EX n | (F x n) & (m:nat)(F y m)->(lt n m)). + +Hypothesis F_compat : (x,y:A) (R x y) -> (inv_lt_rel x y). +Remark acc_lt_rel : + (x:A)(EX n | (F x n))->(Acc A R x). +Intros x (n,fxn); Generalize x fxn; Clear x fxn. +Pattern n; Apply lt_wf_ind; Intros. +Constructor; Intros. +Case (F_compat y x); Trivial; Intros. +Apply (H x0); Auto. +Save. + +Theorem well_founded_inv_lt_rel_compat : (well_founded A R). +Constructor; Intros. +Case (F_compat y a); Trivial; Intros. +Apply acc_lt_rel; Trivial. +Exists x; Trivial. +Save. + + +End LT_WF_REL. + +Lemma well_founded_inv_rel_inv_lt_rel + : (A:Set)(F:A->nat->Prop)(well_founded A (inv_lt_rel A F)). +Intros; Apply (well_founded_inv_lt_rel_compat A (inv_lt_rel A F) F); Trivial. +Save. diff --git a/theories7/Bool/Bool.v b/theories7/Bool/Bool.v new file mode 100755 index 00000000..cd75cf30 --- /dev/null +++ b/theories7/Bool/Bool.v @@ -0,0 +1,544 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Bool.v,v 1.2.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +(** Booleans *) + +(** The type [bool] is defined in the prelude as + [Inductive bool : Set := true : bool | false : bool] *) + +(** Interpretation of booleans as Proposition *) +Definition Is_true := [b:bool](Cases b of + true => True + | false => False + end). +Hints Unfold Is_true : bool. + +Lemma Is_true_eq_left : (x:bool)x=true -> (Is_true x). +Proof. + Intros; Rewrite H; Auto with bool. +Qed. + +Lemma Is_true_eq_right : (x:bool)true=x -> (Is_true x). +Proof. + Intros; Rewrite <- H; Auto with bool. +Qed. + +Hints Immediate Is_true_eq_right Is_true_eq_left : bool. + +(*******************) +(** Discrimination *) +(*******************) + +Lemma diff_true_false : ~true=false. +Proof. +Unfold not; Intro contr; Change (Is_true false). +Elim contr; Simpl; Trivial with bool. +Qed. +Hints Resolve diff_true_false : bool v62. + +Lemma diff_false_true : ~false=true. +Proof. +Red; Intros H; Apply diff_true_false. +Symmetry. +Assumption. +Qed. +Hints Resolve diff_false_true : bool v62. + +Lemma eq_true_false_abs : (b:bool)(b=true)->(b=false)->False. +Intros b H; Rewrite H; Auto with bool. +Qed. +Hints Resolve eq_true_false_abs : bool. + +Lemma not_true_is_false : (b:bool)~b=true->b=false. +NewDestruct b. +Intros. +Red in H; Elim H. +Reflexivity. +Intros abs. +Reflexivity. +Qed. + +Lemma not_false_is_true : (b:bool)~b=false->b=true. +NewDestruct b. +Intros. +Reflexivity. +Intro H; Red in H; Elim H. +Reflexivity. +Qed. + +(**********************) +(** Order on booleans *) +(**********************) + +Definition leb := [b1,b2:bool] + Cases b1 of + | true => b2=true + | false => True + end. +Hints Unfold leb : bool v62. + +(*************) +(** Equality *) +(*************) + +Definition eqb : bool->bool->bool := + [b1,b2:bool] + Cases b1 b2 of + true true => true + | true false => false + | false true => false + | false false => true + end. + +Lemma eqb_refl : (x:bool)(Is_true (eqb x x)). +NewDestruct x; Simpl; Auto with bool. +Qed. + +Lemma eqb_eq : (x,y:bool)(Is_true (eqb x y))->x=y. +NewDestruct x; NewDestruct y; Simpl; Tauto. +Qed. + +Lemma Is_true_eq_true : (x:bool) (Is_true x) -> x=true. +NewDestruct x; Simpl; Tauto. +Qed. + +Lemma Is_true_eq_true2 : (x:bool) x=true -> (Is_true x). +NewDestruct x; Simpl; Auto with bool. +Qed. + +Lemma eqb_subst : + (P:bool->Prop)(b1,b2:bool)(eqb b1 b2)=true->(P b1)->(P b2). +Unfold eqb . +Intros P b1. +Intros b2. +Case b1. +Case b2. +Trivial with bool. +Intros H. +Inversion_clear H. +Case b2. +Intros H. +Inversion_clear H. +Trivial with bool. +Qed. + +Lemma eqb_reflx : (b:bool)(eqb b b)=true. +Intro b. +Case b. +Trivial with bool. +Trivial with bool. +Qed. + +Lemma eqb_prop : (a,b:bool)(eqb a b)=true -> a=b. +NewDestruct a; NewDestruct b; Simpl; Intro; + Discriminate H Orelse Reflexivity. +Qed. + + +(************************) +(** Logical combinators *) +(************************) + +Definition ifb : bool -> bool -> bool -> bool + := [b1,b2,b3:bool](Cases b1 of true => b2 | false => b3 end). + +Definition andb : bool -> bool -> bool + := [b1,b2:bool](ifb b1 b2 false). + +Definition orb : bool -> bool -> bool + := [b1,b2:bool](ifb b1 true b2). + +Definition implb : bool -> bool -> bool + := [b1,b2:bool](ifb b1 b2 true). + +Definition xorb : bool -> bool -> bool + := [b1,b2:bool] + Cases b1 b2 of + true true => false + | true false => true + | false true => true + | false false => false + end. + +Definition negb := [b:bool]Cases b of + true => false + | false => true + end. + +Infix "||" orb (at level 4, left associativity) : bool_scope. +Infix "&&" andb (at level 3, no associativity) : bool_scope + V8only (at level 40, left associativity). + +Open Scope bool_scope. + +Delimits Scope bool_scope with bool. + +Bind Scope bool_scope with bool. + +(**************************) +(** Lemmas about [negb] *) +(**************************) + +Lemma negb_intro : (b:bool)b=(negb (negb b)). +Proof. +NewDestruct b; Reflexivity. +Qed. + +Lemma negb_elim : (b:bool)(negb (negb b))=b. +Proof. +NewDestruct b; Reflexivity. +Qed. + +Lemma negb_orb : (b1,b2:bool) + (negb (orb b1 b2)) = (andb (negb b1) (negb b2)). +Proof. + NewDestruct b1; NewDestruct b2; Simpl; Reflexivity. +Qed. + +Lemma negb_andb : (b1,b2:bool) + (negb (andb b1 b2)) = (orb (negb b1) (negb b2)). +Proof. + NewDestruct b1; NewDestruct b2; Simpl; Reflexivity. +Qed. + +Lemma negb_sym : (b,b':bool)(b'=(negb b))->(b=(negb b')). +Proof. +NewDestruct b; NewDestruct b'; Intros; Simpl; Trivial with bool. +Qed. + +Lemma no_fixpoint_negb : (b:bool)~(negb b)=b. +Proof. +NewDestruct b; Simpl; Intro; Apply diff_true_false; Auto with bool. +Qed. + +Lemma eqb_negb1 : (b:bool)(eqb (negb b) b)=false. +NewDestruct b. +Trivial with bool. +Trivial with bool. +Qed. + +Lemma eqb_negb2 : (b:bool)(eqb b (negb b))=false. +NewDestruct b. +Trivial with bool. +Trivial with bool. +Qed. + + +Lemma if_negb : (A:Set) (b:bool) (x,y:A) (if (negb b) then x else y)=(if b then y else x). +Proof. + NewDestruct b;Trivial. +Qed. + + +(****************************) +(** A few lemmas about [or] *) +(****************************) + +Lemma orb_prop : + (a,b:bool)(orb a b)=true -> (a = true)\/(b = true). +NewDestruct a; NewDestruct b; Simpl; Try (Intro H;Discriminate H); Auto with bool. +Qed. + +Lemma orb_prop2 : + (a,b:bool)(Is_true (orb a b)) -> (Is_true a)\/(Is_true b). +NewDestruct a; NewDestruct b; Simpl; Try (Intro H;Discriminate H); Auto with bool. +Qed. + +Lemma orb_true_intro + : (b1,b2:bool)(b1=true)\/(b2=true)->(orb b1 b2)=true. +NewDestruct b1; Auto with bool. +NewDestruct 1; Intros. +Elim diff_true_false; Auto with bool. +Rewrite H; Trivial with bool. +Qed. +Hints Resolve orb_true_intro : bool v62. + +Lemma orb_b_true : (b:bool)(orb b true)=true. +Auto with bool. +Qed. +Hints Resolve orb_b_true : bool v62. + +Lemma orb_true_b : (b:bool)(orb true b)=true. +Trivial with bool. +Qed. + +Definition orb_true_elim : (b1,b2:bool)(orb b1 b2)=true -> {b1=true}+{b2=true}. +NewDestruct b1; Simpl; Auto with bool. +Defined. + +Lemma orb_false_intro + : (b1,b2:bool)(b1=false)->(b2=false)->(orb b1 b2)=false. +Intros b1 b2 H1 H2; Rewrite H1; Rewrite H2; Trivial with bool. +Qed. +Hints Resolve orb_false_intro : bool v62. + +Lemma orb_b_false : (b:bool)(orb b false)=b. +Proof. + NewDestruct b; Trivial with bool. +Qed. +Hints Resolve orb_b_false : bool v62. + +Lemma orb_false_b : (b:bool)(orb false b)=b. +Proof. + NewDestruct b; Trivial with bool. +Qed. +Hints Resolve orb_false_b : bool v62. + +Lemma orb_false_elim : + (b1,b2:bool)(orb b1 b2)=false -> (b1=false)/\(b2=false). +Proof. + NewDestruct b1. + Intros; Elim diff_true_false; Auto with bool. + NewDestruct b2. + Intros; Elim diff_true_false; Auto with bool. + Auto with bool. +Qed. + +Lemma orb_neg_b : + (b:bool)(orb b (negb b))=true. +Proof. + NewDestruct b; Reflexivity. +Qed. +Hints Resolve orb_neg_b : bool v62. + +Lemma orb_sym : (b1,b2:bool)(orb b1 b2)=(orb b2 b1). +NewDestruct b1; NewDestruct b2; Reflexivity. +Qed. + +Lemma orb_assoc : (b1,b2,b3:bool)(orb b1 (orb b2 b3))=(orb (orb b1 b2) b3). +Proof. + NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity. +Qed. + +Hints Resolve orb_sym orb_assoc orb_b_false orb_false_b : bool v62. + +(*****************************) +(** A few lemmas about [and] *) +(*****************************) + +Lemma andb_prop : + (a,b:bool)(andb a b) = true -> (a = true)/\(b = true). + +Proof. + NewDestruct a; NewDestruct b; Simpl; Try (Intro H;Discriminate H); + Auto with bool. +Qed. +Hints Resolve andb_prop : bool v62. + +Definition andb_true_eq : (a,b:bool) true = (andb a b) -> true = a /\ true = b. +Proof. + NewDestruct a; NewDestruct b; Auto. +Defined. + +Lemma andb_prop2 : + (a,b:bool)(Is_true (andb a b)) -> (Is_true a)/\(Is_true b). +Proof. + NewDestruct a; NewDestruct b; Simpl; Try (Intro H;Discriminate H); + Auto with bool. +Qed. +Hints Resolve andb_prop2 : bool v62. + +Lemma andb_true_intro : (b1,b2:bool)(b1=true)/\(b2=true)->(andb b1 b2)=true. +Proof. + NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool. +Qed. +Hints Resolve andb_true_intro : bool v62. + +Lemma andb_true_intro2 : + (b1,b2:bool)(Is_true b1)->(Is_true b2)->(Is_true (andb b1 b2)). +Proof. + NewDestruct b1; NewDestruct b2; Simpl; Tauto. +Qed. +Hints Resolve andb_true_intro2 : bool v62. + +Lemma andb_false_intro1 + : (b1,b2:bool)(b1=false)->(andb b1 b2)=false. +NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool. +Qed. + +Lemma andb_false_intro2 + : (b1,b2:bool)(b2=false)->(andb b1 b2)=false. +NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool. +Qed. + +Lemma andb_b_false : (b:bool)(andb b false)=false. +NewDestruct b; Auto with bool. +Qed. + +Lemma andb_false_b : (b:bool)(andb false b)=false. +Trivial with bool. +Qed. + +Lemma andb_b_true : (b:bool)(andb b true)=b. +NewDestruct b; Auto with bool. +Qed. + +Lemma andb_true_b : (b:bool)(andb true b)=b. +Trivial with bool. +Qed. + +Definition andb_false_elim : + (b1,b2:bool)(andb b1 b2)=false -> {b1=false}+{b2=false}. +NewDestruct b1; Simpl; Auto with bool. +Defined. +Hints Resolve andb_false_elim : bool v62. + +Lemma andb_neg_b : + (b:bool)(andb b (negb b))=false. +NewDestruct b; Reflexivity. +Qed. +Hints Resolve andb_neg_b : bool v62. + +Lemma andb_sym : (b1,b2:bool)(andb b1 b2)=(andb b2 b1). +NewDestruct b1; NewDestruct b2; Reflexivity. +Qed. + +Lemma andb_assoc : (b1,b2,b3:bool)(andb b1 (andb b2 b3))=(andb (andb b1 b2) b3). +NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity. +Qed. + +Hints Resolve andb_sym andb_assoc : bool v62. + +(*******************************) +(** Properties of [xorb] *) +(*******************************) + +Lemma xorb_false : (b:bool) (xorb b false)=b. +Proof. + NewDestruct b; Trivial. +Qed. + +Lemma false_xorb : (b:bool) (xorb false b)=b. +Proof. + NewDestruct b; Trivial. +Qed. + +Lemma xorb_true : (b:bool) (xorb b true)=(negb b). +Proof. + Trivial. +Qed. + +Lemma true_xorb : (b:bool) (xorb true b)=(negb b). +Proof. + NewDestruct b; Trivial. +Qed. + +Lemma xorb_nilpotent : (b:bool) (xorb b b)=false. +Proof. + NewDestruct b; Trivial. +Qed. + +Lemma xorb_comm : (b,b':bool) (xorb b b')=(xorb b' b). +Proof. + NewDestruct b; NewDestruct b'; Trivial. +Qed. + +Lemma xorb_assoc : (b,b',b'':bool) (xorb (xorb b b') b'')=(xorb b (xorb b' b'')). +Proof. + NewDestruct b; NewDestruct b'; NewDestruct b''; Trivial. +Qed. + +Lemma xorb_eq : (b,b':bool) (xorb b b')=false -> b=b'. +Proof. + NewDestruct b; NewDestruct b'; Trivial. + Unfold xorb. Intros. Rewrite H. Reflexivity. +Qed. + +Lemma xorb_move_l_r_1 : (b,b',b'':bool) (xorb b b')=b'' -> b'=(xorb b b''). +Proof. + Intros. Rewrite <- (false_xorb b'). Rewrite <- (xorb_nilpotent b). Rewrite xorb_assoc. + Rewrite H. Reflexivity. +Qed. + +Lemma xorb_move_l_r_2 : (b,b',b'':bool) (xorb b b')=b'' -> b=(xorb b'' b'). +Proof. + Intros. Rewrite xorb_comm in H. Rewrite (xorb_move_l_r_1 b' b b'' H). Apply xorb_comm. +Qed. + +Lemma xorb_move_r_l_1 : (b,b',b'':bool) b=(xorb b' b'') -> (xorb b' b)=b''. +Proof. + Intros. Rewrite H. Rewrite <- xorb_assoc. Rewrite xorb_nilpotent. Apply false_xorb. +Qed. + +Lemma xorb_move_r_l_2 : (b,b',b'':bool) b=(xorb b' b'') -> (xorb b b'')=b'. +Proof. + Intros. Rewrite H. Rewrite xorb_assoc. Rewrite xorb_nilpotent. Apply xorb_false. +Qed. + +(*******************************) +(** De Morgan's law *) +(*******************************) + +Lemma demorgan1 : (b1,b2,b3:bool) + (andb b1 (orb b2 b3)) = (orb (andb b1 b2) (andb b1 b3)). +NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity. +Qed. + +Lemma demorgan2 : (b1,b2,b3:bool) + (andb (orb b1 b2) b3) = (orb (andb b1 b3) (andb b2 b3)). +NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity. +Qed. + +Lemma demorgan3 : (b1,b2,b3:bool) + (orb b1 (andb b2 b3)) = (andb (orb b1 b2) (orb b1 b3)). +NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity. +Qed. + +Lemma demorgan4 : (b1,b2,b3:bool) + (orb (andb b1 b2) b3) = (andb (orb b1 b3) (orb b2 b3)). +NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity. +Qed. + +Lemma absoption_andb : (b1,b2:bool) + (andb b1 (orb b1 b2)) = b1. +Proof. + NewDestruct b1; NewDestruct b2; Simpl; Reflexivity. +Qed. + +Lemma absoption_orb : (b1,b2:bool) + (orb b1 (andb b1 b2)) = b1. +Proof. + NewDestruct b1; NewDestruct b2; Simpl; Reflexivity. +Qed. + + +(** Misc. equalities between booleans (to be used by Auto) *) + +Lemma bool_1 : (b1,b2:bool)(b1=true <-> b2=true) -> b1=b2. +Proof. + Intros b1 b2; Case b1; Case b2; Intuition. +Qed. + +Lemma bool_2 : (b1,b2:bool)b1=b2 -> b1=true -> b2=true. +Proof. + Intros b1 b2; Case b1; Case b2; Intuition. +Qed. + +Lemma bool_3 : (b:bool) ~(negb b)=true -> b=true. +Proof. + NewDestruct b; Intuition. +Qed. + +Lemma bool_4 : (b:bool) b=true -> ~(negb b)=true. +Proof. + NewDestruct b; Intuition. +Qed. + +Lemma bool_5 : (b:bool) (negb b)=true -> ~b=true. +Proof. + NewDestruct b; Intuition. +Qed. + +Lemma bool_6 : (b:bool) ~b=true -> (negb b)=true. +Proof. + NewDestruct b; Intuition. +Qed. + +Hints Resolve bool_1 bool_2 bool_3 bool_4 bool_5 bool_6. diff --git a/theories7/Bool/BoolEq.v b/theories7/Bool/BoolEq.v new file mode 100644 index 00000000..b670dbdd --- /dev/null +++ b/theories7/Bool/BoolEq.v @@ -0,0 +1,72 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: BoolEq.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) +(* Cuihtlauac Alvarado - octobre 2000 *) + +(** Properties of a boolean equality *) + + +Require Export Bool. + +Section Bool_eq_dec. + + Variable A : Set. + + Variable beq : A -> A -> bool. + + Variable beq_refl : (x:A)true=(beq x x). + + Variable beq_eq : (x,y:A)true=(beq x y)->x=y. + + Definition beq_eq_true : (x,y:A)x=y->true=(beq x y). + Proof. + Intros x y H. + Case H. + Apply beq_refl. + Defined. + + Definition beq_eq_not_false : (x,y:A)x=y->~false=(beq x y). + Proof. + Intros x y e. + Rewrite <- beq_eq_true; Trivial; Discriminate. + Defined. + + Definition beq_false_not_eq : (x,y:A)false=(beq x y)->~x=y. + Proof. + Exact [x,y:A; H:(false=(beq x y)); e:(x=y)](beq_eq_not_false x y e H). + Defined. + + Definition exists_beq_eq : (x,y:A){b:bool | b=(beq x y)}. + Proof. + Intros. + Exists (beq x y). + Constructor. + Defined. + + Definition not_eq_false_beq : (x,y:A)~x=y->false=(beq x y). + Proof. + Intros x y H. + Symmetry. + Apply not_true_is_false. + Intro. + Apply H. + Apply beq_eq. + Symmetry. + Assumption. + Defined. + + Definition eq_dec : (x,y:A){x=y}+{~x=y}. + Proof. + Intros x y; Case (exists_beq_eq x y). + Intros b; Case b; Intro H. + Left; Apply beq_eq; Assumption. + Right; Apply beq_false_not_eq; Assumption. + Defined. + +End Bool_eq_dec. diff --git a/theories7/Bool/Bvector.v b/theories7/Bool/Bvector.v new file mode 100644 index 00000000..e6545381 --- /dev/null +++ b/theories7/Bool/Bvector.v @@ -0,0 +1,266 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Bvector.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +(** Bit vectors. Contribution by Jean Duprat (ENS Lyon). *) + +Require Export Bool. +Require Export Sumbool. +Require Arith. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +(* +On s'inspire de PolyList pour fabriquer les vecteurs de bits. +La dimension du vecteur est un paramètre trop important pour +se contenter de la fonction "length". +La première idée est de faire un record avec la liste et la longueur. +Malheureusement, cette verification a posteriori amene a faire +de nombreux lemmes pour gerer les longueurs. +La seconde idée est de faire un type dépendant dans lequel la +longueur est un paramètre de construction. Cela complique un +peu les inductions structurelles, la solution qui a ma préférence +est alors d'utiliser un terme de preuve comme définition. + +(En effet une définition comme : +Fixpoint Vunaire [n:nat; v:(vector n)]: (vector n) := +Cases v of + | Vnil => Vnil + | (Vcons a p v') => (Vcons (f a) p (Vunaire p v')) +end. +provoque ce message d'erreur : +Coq < Error: Inference of annotation not yet implemented in this case). + + + Inductive list [A : Set] : Set := + nil : (list A) | cons : A->(list A)->(list A). + head = [A:Set; l:(list A)] Cases l of + | nil => Error + | (cons x _) => (Value x) + end + : (A:Set)(list A)->(option A). + tail = [A:Set; l:(list A)]Cases l of + | nil => (nil A) + | (cons _ m) => m + end + : (A:Set)(list A)->(list A). + length = [A:Set] Fix length {length [l:(list A)] : nat := + Cases l of + | nil => O + | (cons _ m) => (S (length m)) + end} + : (A:Set)(list A)->nat. + map = [A,B:Set; f:(A->B)] Fix map {map [l:(list A)] : (list B) := + Cases l of + | nil => (nil B) + | (cons a t) => (cons (f a) (map t)) + end} + : (A,B:Set)(A->B)->(list A)->(list B) +*) + +Section VECTORS. + +(* +Un vecteur est une liste de taille n d'éléments d'un ensemble A. +Si la taille est non nulle, on peut extraire la première composante et +le reste du vecteur, la dernière composante ou rajouter ou enlever +une composante (carry) ou repeter la dernière composante en fin de vecteur. +On peut aussi tronquer le vecteur de ses p dernières composantes ou +au contraire l'étendre (concaténer) d'un vecteur de longueur p. +Une fonction unaire sur A génère une fonction des vecteurs de taille n +dans les vecteurs de taille n en appliquant f terme à terme. +Une fonction binaire sur A génère une fonction des couple de vecteurs +de taille n dans les vecteurs de taille n en appliquant f terme à terme. +*) + +Variable A : Set. + +Inductive vector: nat -> Set := + | Vnil : (vector O) + | Vcons : (a:A) (n:nat) (vector n) -> (vector (S n)). + +Definition Vhead : (n:nat) (vector (S n)) -> A. +Proof. + Intros n v; Inversion v; Exact a. +Defined. + +Definition Vtail : (n:nat) (vector (S n)) -> (vector n). +Proof. + Intros n v; Inversion v; Exact H0. +Defined. + +Definition Vlast : (n:nat) (vector (S n)) -> A. +Proof. + NewInduction n as [|n f]; Intro v. + Inversion v. + Exact a. + + Inversion v. + Exact (f H0). +Defined. + +Definition Vconst : (a:A) (n:nat) (vector n). +Proof. + NewInduction n as [|n v]. + Exact Vnil. + + Exact (Vcons a n v). +Defined. + +Lemma Vshiftout : (n:nat) (vector (S n)) -> (vector n). +Proof. + NewInduction n as [|n f]; Intro v. + Exact Vnil. + + Inversion v. + Exact (Vcons a n (f H0)). +Defined. + +Lemma Vshiftin : (n:nat) A -> (vector n) -> (vector (S n)). +Proof. + NewInduction n as [|n f]; Intros a v. + Exact (Vcons a O v). + + Inversion v. + Exact (Vcons a (S n) (f a H0)). +Defined. + +Lemma Vshiftrepeat : (n:nat) (vector (S n)) -> (vector (S (S n))). +Proof. + NewInduction n as [|n f]; Intro v. + Inversion v. + Exact (Vcons a (1) v). + + Inversion v. + Exact (Vcons a (S (S n)) (f H0)). +Defined. + +(* +Lemma S_minus_S : (n,p:nat) (gt n (S p)) -> (S (minus n (S p)))=(minus n p). +Proof. + Intros. +Save. +*) + +Lemma Vtrunc : (n,p:nat) (gt n p) -> (vector n) -> (vector (minus n p)). +Proof. + NewInduction p as [|p f]; Intros H v. + Rewrite <- minus_n_O. + Exact v. + + Apply (Vshiftout (minus n (S p))). + +Rewrite minus_Sn_m. +Apply f. +Auto with *. +Exact v. +Auto with *. +Defined. + +Lemma Vextend : (n,p:nat) (vector n) -> (vector p) -> (vector (plus n p)). +Proof. + NewInduction n as [|n f]; Intros p v v0. + Simpl; Exact v0. + + Inversion v. + Simpl; Exact (Vcons a (plus n p) (f p H0 v0)). +Defined. + +Variable f : A -> A. + +Lemma Vunary : (n:nat)(vector n)->(vector n). +Proof. + NewInduction n as [|n g]; Intro v. + Exact Vnil. + + Inversion v. + Exact (Vcons (f a) n (g H0)). +Defined. + +Variable g : A -> A -> A. + +Lemma Vbinary : (n:nat)(vector n)->(vector n)->(vector n). +Proof. + NewInduction n as [|n h]; Intros v v0. + Exact Vnil. + + Inversion v; Inversion v0. + Exact (Vcons (g a a0) n (h H0 H2)). +Defined. + +End VECTORS. + +Section BOOLEAN_VECTORS. + +(* +Un vecteur de bits est un vecteur sur l'ensemble des booléens de longueur fixe. +ATTENTION : le stockage s'effectue poids FAIBLE en tête. +On en extrait le bit de poids faible (head) et la fin du vecteur (tail). +On calcule la négation d'un vecteur, le et, le ou et le xor bit à bit de 2 vecteurs. +On calcule les décalages d'une position vers la gauche (vers les poids forts, on +utilise donc Vshiftout, vers la droite (vers les poids faibles, on utilise Vshiftin) en +insérant un bit 'carry' (logique) ou en répétant le bit de poids fort (arithmétique). +ATTENTION : Tous les décalages prennent la taille moins un comme paramètre +(ils ne travaillent que sur des vecteurs au moins de longueur un). +*) + +Definition Bvector := (vector bool). + +Definition Bnil := (Vnil bool). + +Definition Bcons := (Vcons bool). + +Definition Bvect_true := (Vconst bool true). + +Definition Bvect_false := (Vconst bool false). + +Definition Blow := (Vhead bool). + +Definition Bhigh := (Vtail bool). + +Definition Bsign := (Vlast bool). + +Definition Bneg := (Vunary bool negb). + +Definition BVand := (Vbinary bool andb). + +Definition BVor := (Vbinary bool orb). + +Definition BVxor := (Vbinary bool xorb). + +Definition BshiftL := [n:nat; bv : (Bvector (S n)); carry:bool] + (Bcons carry n (Vshiftout bool n bv)). + +Definition BshiftRl := [n:nat; bv : (Bvector (S n)); carry:bool] + (Bhigh (S n) (Vshiftin bool (S n) carry bv)). + +Definition BshiftRa := [n:nat; bv : (Bvector (S n))] + (Bhigh (S n) (Vshiftrepeat bool n bv)). + +Fixpoint BshiftL_iter [n:nat; bv:(Bvector (S n)); p:nat]:(Bvector (S n)) := +Cases p of + | O => bv + | (S p') => (BshiftL n (BshiftL_iter n bv p') false) +end. + +Fixpoint BshiftRl_iter [n:nat; bv:(Bvector (S n)); p:nat]:(Bvector (S n)) := +Cases p of + | O => bv + | (S p') => (BshiftRl n (BshiftRl_iter n bv p') false) +end. + +Fixpoint BshiftRa_iter [n:nat; bv:(Bvector (S n)); p:nat]:(Bvector (S n)) := +Cases p of + | O => bv + | (S p') => (BshiftRa n (BshiftRa_iter n bv p')) +end. + +End BOOLEAN_VECTORS. + diff --git a/theories7/Bool/DecBool.v b/theories7/Bool/DecBool.v new file mode 100755 index 00000000..c22cd032 --- /dev/null +++ b/theories7/Bool/DecBool.v @@ -0,0 +1,27 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: DecBool.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +Set Implicit Arguments. + +Definition ifdec : (A,B:Prop)(C:Set)({A}+{B})->C->C->C + := [A,B,C,H,x,y]if H then [_]x else [_]y. + + +Theorem ifdec_left : (A,B:Prop)(C:Set)(H:{A}+{B})~B->(x,y:C)(ifdec H x y)=x. +Intros; Case H; Auto. +Intro; Absurd B; Trivial. +Qed. + +Theorem ifdec_right : (A,B:Prop)(C:Set)(H:{A}+{B})~A->(x,y:C)(ifdec H x y)=y. +Intros; Case H; Auto. +Intro; Absurd A; Trivial. +Qed. + +Unset Implicit Arguments. diff --git a/theories7/Bool/IfProp.v b/theories7/Bool/IfProp.v new file mode 100755 index 00000000..bcfa4be3 --- /dev/null +++ b/theories7/Bool/IfProp.v @@ -0,0 +1,49 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: IfProp.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +Require Bool. + +Inductive IfProp [A,B:Prop] : bool-> Prop + := Iftrue : A -> (IfProp A B true) + | Iffalse : B -> (IfProp A B false). + +Hints Resolve Iftrue Iffalse : bool v62. + +Lemma Iftrue_inv : (A,B:Prop)(b:bool) (IfProp A B b) -> b=true -> A. +NewDestruct 1; Intros; Auto with bool. +Case diff_true_false; Auto with bool. +Qed. + +Lemma Iffalse_inv : (A,B:Prop)(b:bool) (IfProp A B b) -> b=false -> B. +NewDestruct 1; Intros; Auto with bool. +Case diff_true_false; Trivial with bool. +Qed. + +Lemma IfProp_true : (A,B:Prop)(IfProp A B true) -> A. +Intros. +Inversion H. +Assumption. +Qed. + +Lemma IfProp_false : (A,B:Prop)(IfProp A B false) -> B. +Intros. +Inversion H. +Assumption. +Qed. + +Lemma IfProp_or : (A,B:Prop)(b:bool)(IfProp A B b) -> A\/B. +NewDestruct 1; Auto with bool. +Qed. + +Lemma IfProp_sum : (A,B:Prop)(b:bool)(IfProp A B b) -> {A}+{B}. +NewDestruct b; Intro H. +Left; Inversion H; Auto with bool. +Right; Inversion H; Auto with bool. +Qed. diff --git a/theories7/Bool/Sumbool.v b/theories7/Bool/Sumbool.v new file mode 100644 index 00000000..8d55cbb6 --- /dev/null +++ b/theories7/Bool/Sumbool.v @@ -0,0 +1,77 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Sumbool.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +(** Here are collected some results about the type sumbool (see INIT/Specif.v) + [sumbool A B], which is written [{A}+{B}], is the informative + disjunction "A or B", where A and B are logical propositions. + Its extraction is isomorphic to the type of booleans. *) + +(** A boolean is either [true] or [false], and this is decidable *) + +Definition sumbool_of_bool : (b:bool) {b=true}+{b=false}. +Proof. + NewDestruct b; Auto. +Defined. + +Hints Resolve sumbool_of_bool : bool. + +Definition bool_eq_rec : (b:bool)(P:bool->Set) + ((b=true)->(P true))->((b=false)->(P false))->(P b). +NewDestruct b; Auto. +Defined. + +Definition bool_eq_ind : (b:bool)(P:bool->Prop) + ((b=true)->(P true))->((b=false)->(P false))->(P b). +NewDestruct b; Auto. +Defined. + + +(*i pourquoi ce machin-la est dans BOOL et pas dans LOGIC ? Papageno i*) + +(** Logic connectives on type [sumbool] *) + +Section connectives. + +Variables A,B,C,D : Prop. + +Hypothesis H1 : {A}+{B}. +Hypothesis H2 : {C}+{D}. + +Definition sumbool_and : {A/\C}+{B\/D}. +Proof. +Case H1; Case H2; Auto. +Defined. + +Definition sumbool_or : {A\/C}+{B/\D}. +Proof. +Case H1; Case H2; Auto. +Defined. + +Definition sumbool_not : {B}+{A}. +Proof. +Case H1; Auto. +Defined. + +End connectives. + +Hints Resolve sumbool_and sumbool_or sumbool_not : core. + + +(** Any decidability function in type [sumbool] can be turned into a function + returning a boolean with the corresponding specification: *) + +Definition bool_of_sumbool : + (A,B:Prop) {A}+{B} -> { b:bool | if b then A else B }. +Proof. +Intros A B H. +Elim H; [ Intro; Exists true; Assumption + | Intro; Exists false; Assumption ]. +Defined. +Implicits bool_of_sumbool. diff --git a/theories7/Bool/Zerob.v b/theories7/Bool/Zerob.v new file mode 100755 index 00000000..24e48c28 --- /dev/null +++ b/theories7/Bool/Zerob.v @@ -0,0 +1,36 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zerob.v,v 1.1.2.1 2004/07/16 19:31:25 herbelin Exp $ i*) + +Require Arith. +Require Bool. + +V7only [Import nat_scope.]. +Open Local Scope nat_scope. + +Definition zerob : nat->bool + := [n:nat]Cases n of O => true | (S _) => false end. + +Lemma zerob_true_intro : (n:nat)(n=O)->(zerob n)=true. +NewDestruct n; [Trivial with bool | Inversion 1]. +Qed. +Hints Resolve zerob_true_intro : bool. + +Lemma zerob_true_elim : (n:nat)(zerob n)=true->(n=O). +NewDestruct n; [Trivial with bool | Inversion 1]. +Qed. + +Lemma zerob_false_intro : (n:nat)~(n=O)->(zerob n)=false. +NewDestruct n; [NewDestruct 1; Auto with bool | Trivial with bool]. +Qed. +Hints Resolve zerob_false_intro : bool. + +Lemma zerob_false_elim : (n:nat)(zerob n)=false -> ~(n=O). +NewDestruct n; [Intro H; Inversion H | Auto with bool]. +Qed. diff --git a/theories7/Init/Datatypes.v b/theories7/Init/Datatypes.v new file mode 100755 index 00000000..006ec08e --- /dev/null +++ b/theories7/Init/Datatypes.v @@ -0,0 +1,125 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Datatypes.v,v 1.3.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +Require Notations. +Require Logic. + +Set Implicit Arguments. +V7only [Unset Implicit Arguments.]. + +(** [unit] is a singleton datatype with sole inhabitant [tt] *) + +Inductive unit : Set := tt : unit. + +(** [bool] is the datatype of the booleans values [true] and [false] *) + +Inductive bool : Set := true : bool + | false : bool. + +Add Printing If bool. + +(** [nat] is the datatype of natural numbers built from [O] and successor [S]; + note that zero is the letter O, not the numeral 0 *) + +Inductive nat : Set := O : nat + | S : nat->nat. + +Delimits Scope nat_scope with nat. +Bind Scope nat_scope with nat. +Arguments Scope S [ nat_scope ]. + +(** [Empty_set] has no inhabitant *) + +Inductive Empty_set:Set :=. + +(** [identity A a] is the family of datatypes on [A] whose sole non-empty + member is the singleton datatype [identity A a a] whose + sole inhabitant is denoted [refl_identity A a] *) + +Inductive identity [A:Type; a:A] : A->Set := + refl_identity: (identity A a a). +Hints Resolve refl_identity : core v62. + +Implicits identity_ind [1]. +Implicits identity_rec [1]. +Implicits identity_rect [1]. +V7only [ +Implicits identity_ind []. +Implicits identity_rec []. +Implicits identity_rect []. +]. + +(** [option A] is the extension of A with a dummy element None *) + +Inductive option [A:Set] : Set := Some : A -> (option A) | None : (option A). + +Implicits None [1]. +V7only [Implicits None [].]. + +(** [sum A B], equivalently [A + B], is the disjoint sum of [A] and [B] *) +(* Syntax defined in Specif.v *) +Inductive sum [A,B:Set] : Set + := inl : A -> (sum A B) + | inr : B -> (sum A B). + +Notation "x + y" := (sum x y) : type_scope. + +(** [prod A B], written [A * B], is the product of [A] and [B]; + the pair [pair A B a b] of [a] and [b] is abbreviated [(a,b)] *) + +Inductive prod [A,B:Set] : Set := pair : A -> B -> (prod A B). +Add Printing Let prod. + +Notation "x * y" := (prod x y) : type_scope. +V7only [Notation "( x , y )" := (pair ? ? x y) : core_scope.]. +V8Notation "( x , y , .. , z )" := (pair ? ? .. (pair ? ? x y) .. z) : core_scope. + +Section projections. + Variables A,B:Set. + Definition fst := [p:(prod A B)]Cases p of (pair x y) => x end. + Definition snd := [p:(prod A B)]Cases p of (pair x y) => y end. +End projections. + +V7only [ +Notation Fst := (fst ? ?). +Notation Snd := (snd ? ?). +]. +Hints Resolve pair inl inr : core v62. + +Lemma surjective_pairing : (A,B:Set;p:A*B)p=(pair A B (Fst p) (Snd p)). +Proof. +NewDestruct p; Reflexivity. +Qed. + +Lemma injective_projections : + (A,B:Set;p1,p2:A*B)(Fst p1)=(Fst p2)->(Snd p1)=(Snd p2)->p1=p2. +Proof. +NewDestruct p1; NewDestruct p2; Simpl; Intros Hfst Hsnd. +Rewrite Hfst; Rewrite Hsnd; Reflexivity. +Qed. + +V7only[ +(** Parsing only of things in [Datatypes.v] *) +Notation "< A , B > ( x , y )" := (pair A B x y) (at level 1, only parsing, A annot). +Notation "< A , B > 'Fst' ( p )" := (fst A B p) (at level 1, only parsing, A annot). +Notation "< A , B > 'Snd' ( p )" := (snd A B p) (at level 1, only parsing, A annot). +]. + +(** Comparison *) + +Inductive relation : Set := + EGAL :relation | INFERIEUR : relation | SUPERIEUR : relation. + +Definition Op := [r:relation] + Cases r of + EGAL => EGAL + | INFERIEUR => SUPERIEUR + | SUPERIEUR => INFERIEUR + end. diff --git a/theories7/Init/Logic.v b/theories7/Init/Logic.v new file mode 100755 index 00000000..6ba9c7a1 --- /dev/null +++ b/theories7/Init/Logic.v @@ -0,0 +1,306 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Logic.v,v 1.6.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +Set Implicit Arguments. +V7only [Unset Implicit Arguments.]. + +Require Notations. + +(** [True] is the always true proposition *) +Inductive True : Prop := I : True. + +(** [False] is the always false proposition *) +Inductive False : Prop := . + +(** [not A], written [~A], is the negation of [A] *) +Definition not := [A:Prop]A->False. + +Notation "~ x" := (not x) : type_scope. + +Hints Unfold not : core. + +Inductive and [A,B:Prop] : Prop := conj : A -> B -> A /\ B + +where "A /\ B" := (and A B) : type_scope. + +V7only[ +Notation "< P , Q > { p , q }" := (conj P Q p q) (P annot, at level 1). +]. + +Section Conjunction. + + (** [and A B], written [A /\ B], is the conjunction of [A] and [B] + + [conj A B p q], written [<p,q>] is a proof of [A /\ B] as soon as + [p] is a proof of [A] and [q] a proof of [B] + + [proj1] and [proj2] are first and second projections of a conjunction *) + + Variables A,B : Prop. + + Theorem proj1 : (and A B) -> A. + Proof. + NewDestruct 1; Trivial. + Qed. + + Theorem proj2 : (and A B) -> B. + Proof. + NewDestruct 1; Trivial. + Qed. + +End Conjunction. + +(** [or A B], written [A \/ B], is the disjunction of [A] and [B] *) + +Inductive or [A,B:Prop] : Prop := + or_introl : A -> A \/ B + | or_intror : B -> A \/ B + +where "A \/ B" := (or A B) : type_scope. + +(** [iff A B], written [A <-> B], expresses the equivalence of [A] and [B] *) + +Definition iff := [A,B:Prop] (and (A->B) (B->A)). + +Notation "A <-> B" := (iff A B) : type_scope. + +Section Equivalence. + +Theorem iff_refl : (A:Prop) (iff A A). + Proof. + Split; Auto. + Qed. + +Theorem iff_trans : (a,b,c:Prop) (iff a b) -> (iff b c) -> (iff a c). + Proof. + Intros A B C (H1,H2) (H3,H4); Split; Auto. + Qed. + +Theorem iff_sym : (A,B:Prop) (iff A B) -> (iff B A). + Proof. + Intros A B (H1,H2); Split; Auto. + Qed. + +End Equivalence. + +(** [(IF P Q R)], or more suggestively [(either P and_then Q or_else R)], + denotes either [P] and [Q], or [~P] and [Q] *) +Definition IF_then_else := [P,Q,R:Prop] (or (and P Q) (and (not P) R)). +V7only [Notation IF:=IF_then_else.]. + +Notation "'IF' c1 'then' c2 'else' c3" := (IF c1 c2 c3) + (at level 1, c1, c2, c3 at level 8) : type_scope + V8only (at level 200). + +(** First-order quantifiers *) + + (** [ex A P], or simply [exists x, P x], expresses the existence of an + [x] of type [A] which satisfies the predicate [P] ([A] is of type + [Set]). This is existential quantification. *) + + (** [ex2 A P Q], or simply [exists2 x, P x & Q x], expresses the + existence of an [x] of type [A] which satisfies both the predicates + [P] and [Q] *) + + (** Universal quantification (especially first-order one) is normally + written [forall x:A, P x]. For duality with existential quantification, + the construction [all P] is provided too *) + +Inductive ex [A:Type;P:A->Prop] : Prop + := ex_intro : (x:A)(P x)->(ex A P). + +Inductive ex2 [A:Type;P,Q:A->Prop] : Prop + := ex_intro2 : (x:A)(P x)->(Q x)->(ex2 A P Q). + +Definition all := [A:Type][P:A->Prop](x:A)(P x). + +(* Rule order is important to give printing priority to fully typed exists *) + +V7only [ Notation Ex := (ex ?). ]. +Notation "'EX' x | p" := (ex ? [x]p) + (at level 10, p at level 8) : type_scope + V8only "'exists' x , p" (at level 200, x ident, p at level 99). +Notation "'EX' x : t | p" := (ex ? [x:t]p) + (at level 10, p at level 8) : type_scope + V8only "'exists' x : t , p" (at level 200, x ident, p at level 99, format + "'exists' '/ ' x : t , '/ ' p"). + +V7only [ Notation Ex2 := (ex2 ?). ]. +Notation "'EX' x | p & q" := (ex2 ? [x]p [x]q) + (at level 10, p, q at level 8) : type_scope + V8only "'exists2' x , p & q" (at level 200, x ident, p, q at level 99). +Notation "'EX' x : t | p & q" := (ex2 ? [x:t]p [x:t]q) + (at level 10, p, q at level 8) : type_scope + V8only "'exists2' x : t , p & q" + (at level 200, x ident, t at level 200, p, q at level 99, format + "'exists2' '/ ' x : t , '/ ' '[' p & '/' q ']'"). + +V7only [Notation All := (all ?). +Notation "'ALL' x | p" := (all ? [x]p) + (at level 10, p at level 8) : type_scope + V8only (at level 200, x ident, p at level 200). +Notation "'ALL' x : t | p" := (all ? [x:t]p) + (at level 10, p at level 8) : type_scope + V8only (at level 200, x ident, t, p at level 200). +]. + +(** Universal quantification *) + +Section universal_quantification. + + Variable A : Type. + Variable P : A->Prop. + + Theorem inst : (x:A)(all ? [x](P x))->(P x). + Proof. + Unfold all; Auto. + Qed. + + Theorem gen : (B:Prop)(f:(y:A)B->(P y))B->(all A P). + Proof. + Red; Auto. + Qed. + + End universal_quantification. + +(** Equality *) + +(** [eq A x y], or simply [x=y], expresses the (Leibniz') equality + of [x] and [y]. Both [x] and [y] must belong to the same type [A]. + The definition is inductive and states the reflexivity of the equality. + The others properties (symmetry, transitivity, replacement of + equals) are proved below *) + +Inductive eq [A:Type;x:A] : A->Prop + := refl_equal : x = x :> A + +where "x = y :> A" := (!eq A x y) : type_scope. + +Notation "x = y" := (eq ? x y) : type_scope. +Notation "x <> y :> T" := ~ (!eq T x y) : type_scope. +Notation "x <> y" := ~ x=y : type_scope. + +Implicits eq_ind [1]. +Implicits eq_rec [1]. +Implicits eq_rect [1]. +V7only [ +Implicits eq_ind []. +Implicits eq_rec []. +Implicits eq_rect []. +]. + +Hints Resolve I conj or_introl or_intror refl_equal : core v62. +Hints Resolve ex_intro ex_intro2 : core v62. + +Section Logic_lemmas. + + Theorem absurd : (A:Prop)(C:Prop) A -> (not A) -> C. + Proof. + Unfold not; Intros A C h1 h2. + NewDestruct (h2 h1). + Qed. + + Section equality. + Variable A,B : Type. + Variable f : A->B. + Variable x,y,z : A. + + Theorem sym_eq : (eq ? x y) -> (eq ? y x). + Proof. + NewDestruct 1; Trivial. + Defined. + Opaque sym_eq. + + Theorem trans_eq : (eq ? x y) -> (eq ? y z) -> (eq ? x z). + Proof. + NewDestruct 2; Trivial. + Defined. + Opaque trans_eq. + + Theorem f_equal : (eq ? x y) -> (eq ? (f x) (f y)). + Proof. + NewDestruct 1; Trivial. + Defined. + Opaque f_equal. + + Theorem sym_not_eq : (not (eq ? x y)) -> (not (eq ? y x)). + Proof. + Red; Intros h1 h2; Apply h1; NewDestruct h2; Trivial. + Qed. + + Definition sym_equal := sym_eq. + Definition sym_not_equal := sym_not_eq. + Definition trans_equal := trans_eq. + + End equality. + +(* Is now a primitive principle + Theorem eq_rect: (A:Type)(x:A)(P:A->Type)(P x)->(y:A)(eq ? x y)->(P y). + Proof. + Intros. + Cut (identity A x y). + NewDestruct 1; Auto. + NewDestruct H; Auto. + Qed. +*) + + Definition eq_ind_r : (A:Type)(x:A)(P:A->Prop)(P x)->(y:A)(eq ? y x)->(P y). + Intros A x P H y H0; Elim sym_eq with 1:= H0; Assumption. + Defined. + + Definition eq_rec_r : (A:Type)(x:A)(P:A->Set)(P x)->(y:A)(eq ? y x)->(P y). + Intros A x P H y H0; Elim sym_eq with 1:= H0; Assumption. + Defined. + + Definition eq_rect_r : (A:Type)(x:A)(P:A->Type)(P x)->(y:A)(eq ? y x)->(P y). + Intros A x P H y H0; Elim sym_eq with 1:= H0; Assumption. + Defined. +End Logic_lemmas. + +Theorem f_equal2 : (A1,A2,B:Type)(f:A1->A2->B)(x1,y1:A1)(x2,y2:A2) + (eq ? x1 y1) -> (eq ? x2 y2) -> (eq ? (f x1 x2) (f y1 y2)). +Proof. + NewDestruct 1; NewDestruct 1; Reflexivity. +Qed. + +Theorem f_equal3 : (A1,A2,A3,B:Type)(f:A1->A2->A3->B)(x1,y1:A1)(x2,y2:A2) + (x3,y3:A3)(eq ? x1 y1) -> (eq ? x2 y2) -> (eq ? x3 y3) + -> (eq ? (f x1 x2 x3) (f y1 y2 y3)). +Proof. + NewDestruct 1; NewDestruct 1; NewDestruct 1; Reflexivity. +Qed. + +Theorem f_equal4 : (A1,A2,A3,A4,B:Type)(f:A1->A2->A3->A4->B) + (x1,y1:A1)(x2,y2:A2)(x3,y3:A3)(x4,y4:A4) + (eq ? x1 y1) -> (eq ? x2 y2) -> (eq ? x3 y3) -> (eq ? x4 y4) + -> (eq ? (f x1 x2 x3 x4) (f y1 y2 y3 y4)). +Proof. + NewDestruct 1; NewDestruct 1; NewDestruct 1; NewDestruct 1; Reflexivity. +Qed. + +Theorem f_equal5 : (A1,A2,A3,A4,A5,B:Type)(f:A1->A2->A3->A4->A5->B) + (x1,y1:A1)(x2,y2:A2)(x3,y3:A3)(x4,y4:A4)(x5,y5:A5) + (eq ? x1 y1) -> (eq ? x2 y2) -> (eq ? x3 y3) -> (eq ? x4 y4) -> (eq ? x5 y5) + -> (eq ? (f x1 x2 x3 x4 x5) (f y1 y2 y3 y4 y5)). +Proof. + NewDestruct 1; NewDestruct 1; NewDestruct 1; NewDestruct 1; NewDestruct 1; + Reflexivity. +Qed. + +Hints Immediate sym_eq sym_not_eq : core v62. + +V7only[ +(** Parsing only of things in [Logic.v] *) +Notation "< A > 'All' ( P )" :=(all A P) (A annot, at level 1, only parsing). +Notation "< A > x = y" := (eq A x y) + (A annot, at level 1, x at level 0, only parsing). +Notation "< A > x <> y" := ~(eq A x y) + (A annot, at level 1, x at level 0, only parsing). +]. diff --git a/theories7/Init/Logic_Type.v b/theories7/Init/Logic_Type.v new file mode 100755 index 00000000..793b671c --- /dev/null +++ b/theories7/Init/Logic_Type.v @@ -0,0 +1,304 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Logic_Type.v,v 1.3.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +Set Implicit Arguments. +V7only [Unset Implicit Arguments.]. + +(** This module defines quantification on the world [Type] + ([Logic.v] was defining it on the world [Set]) *) + +Require Datatypes. +Require Export Logic. + +V7only [ +(* +(** [allT A P], or simply [(ALLT x | P(x))], stands for [(x:A)(P x)] + when [A] is of type [Type] *) + +Definition allT := [A:Type][P:A->Prop](x:A)(P x). +*) + +Notation allT := all (only parsing). +Notation inst := Logic.inst (only parsing). +Notation gen := Logic.gen (only parsing). + +(* Order is important to give printing priority to fully typed ALL and EX *) + +Notation AllT := (all ?). +Notation "'ALLT' x | p" := (all ? [x]p) (at level 10, p at level 8). +Notation "'ALLT' x : t | p" := (all ? [x:t]p) (at level 10, p at level 8). + +(* +Section universal_quantification. + +Variable A : Type. +Variable P : A->Prop. + +Theorem inst : (x:A)(allT ? [x](P x))->(P x). +Proof. +Unfold all; Auto. +Qed. + +Theorem gen : (B:Prop)(f:(y:A)B->(P y))B->(allT A P). +Proof. +Red; Auto. +Qed. + +End universal_quantification. +*) + +(* +(** * Existential Quantification *) + +(** [exT A P], or simply [(EXT x | P(x))], stands for the existential + quantification on the predicate [P] when [A] is of type [Type] *) + +(** [exT2 A P Q], or simply [(EXT x | P(x) & Q(x))], stands for the + existential quantification on both [P] and [Q] when [A] is of + type [Type] *) +Inductive exT [A:Type;P:A->Prop] : Prop + := exT_intro : (x:A)(P x)->(exT A P). +*) + +Notation exT := ex (only parsing). +Notation exT_intro := ex_intro (only parsing). +Notation exT_ind := ex_ind (only parsing). + +Notation ExT := (ex ?). +Notation "'EXT' x | p" := (ex ? [x]p) + (at level 10, p at level 8, only parsing). +Notation "'EXT' x : t | p" := (ex ? [x:t]p) + (at level 10, p at level 8, only parsing). + +(* +Inductive exT2 [A:Type;P,Q:A->Prop] : Prop + := exT_intro2 : (x:A)(P x)->(Q x)->(exT2 A P Q). +*) + +Notation exT2 := ex2 (only parsing). +Notation exT_intro2 := ex_intro2 (only parsing). +Notation exT2_ind := ex2_ind (only parsing). + +Notation ExT2 := (ex2 ?). +Notation "'EXT' x | p & q" := (ex2 ? [x]p [x]q) + (at level 10, p, q at level 8). +Notation "'EXT' x : t | p & q" := (ex2 ? [x:t]p [x:t]q) + (at level 10, p, q at level 8). + +(* +(** Leibniz equality : [A:Type][x,y:A] (P:A->Prop)(P x)->(P y) + + [eqT A x y], or simply [x==y], is Leibniz' equality when [A] is of + type [Type]. This equality satisfies reflexivity (by definition), + symmetry, transitivity and stability by congruence *) + + +Inductive eqT [A:Type;x:A] : A -> Prop + := refl_eqT : (eqT A x x). + +Hints Resolve refl_eqT (* exT_intro2 exT_intro *) : core v62. +*) + +Notation eqT := eq (only parsing). +Notation refl_eqT := refl_equal (only parsing). +Notation eqT_ind := eq_ind (only parsing). +Notation eqT_rect := eq_rect (only parsing). +Notation eqT_rec := eq_rec (only parsing). + +Notation "x == y" := (eq ? x y) (at level 5, no associativity, only parsing). + +(** Parsing only of things in [Logic_type.v] *) + +Notation "< A > x == y" := (eq A x y) + (A annot, at level 1, x at level 0, only parsing). + +(* +Section Equality_is_a_congruence. + + Variables A,B : Type. + Variable f : A->B. + + Variable x,y,z : A. + + Lemma sym_eqT : (eqT ? x y) -> (eqT ? y x). + Proof. + NewDestruct 1; Trivial. + Qed. + + Lemma trans_eqT : (eqT ? x y) -> (eqT ? y z) -> (eqT ? x z). + Proof. + NewDestruct 2; Trivial. + Qed. + + Lemma congr_eqT : (eqT ? x y)->(eqT ? (f x) (f y)). + Proof. + NewDestruct 1; Trivial. + Qed. + + Lemma sym_not_eqT : ~(eqT ? x y) -> ~(eqT ? y x). + Proof. + Red; Intros H H'; Apply H; NewDestruct H'; Trivial. + Qed. + +End Equality_is_a_congruence. +*) + +Notation sym_eqT := sym_eq (only parsing). +Notation trans_eqT := trans_eq (only parsing). +Notation congr_eqT := f_equal (only parsing). +Notation sym_not_eqT := sym_not_eq (only parsing). + +(* +Hints Immediate sym_eqT sym_not_eqT : core v62. +*) + +(** This states the replacement of equals by equals *) + +(* +Definition eqT_ind_r : (A:Type)(x:A)(P:A->Prop)(P x)->(y:A)(eqT ? y x)->(P y). +Intros A x P H y H0; Case sym_eqT with 1:=H0; Trivial. +Defined. + +Definition eqT_rec_r : (A:Type)(x:A)(P:A->Set)(P x)->(y:A)(eqT ? y x)->(P y). +Intros A x P H y H0; Case sym_eqT with 1:=H0; Trivial. +Defined. + +Definition eqT_rect_r : (A:Type)(x:A)(P:A->Type)(P x)->(y:A)(eqT ? y x)->(P y). +Intros A x P H y H0; Case sym_eqT with 1:=H0; Trivial. +Defined. +*) + +Notation eqT_ind_r := eq_ind_r (only parsing). +Notation eqT_rec_r := eq_rec_r (only parsing). +Notation eqT_rect_r := eq_rect_r (only parsing). + +(** Some datatypes at the [Type] level *) +(* +Inductive EmptyT: Type :=. +Inductive UnitT : Type := IT : UnitT. +*) + +Notation EmptyT := False (only parsing). +Notation UnitT := unit (only parsing). +Notation IT := tt. +]. +Definition notT := [A:Type] A->EmptyT. + +V7only [ +(** Have you an idea of what means [identityT A a b]? No matter! *) + +(* +Inductive identityT [A:Type; a:A] : A -> Type := + refl_identityT : (identityT A a a). +*) + +Notation identityT := identity (only parsing). +Notation refl_identityT := refl_identity (only parsing). + +Notation "< A > x === y" := (!identityT A x y) + (A annot, at level 1, x at level 0, only parsing) : type_scope. + +Notation "x === y" := (identityT ? x y) + (at level 5, no associativity, only parsing) : type_scope. + +(* +Hints Resolve refl_identityT : core v62. +*) +]. +Section identity_is_a_congruence. + + Variables A,B : Type. + Variable f : A->B. + + Variable x,y,z : A. + + Lemma sym_id : (identityT ? x y) -> (identityT ? y x). + Proof. + NewDestruct 1; Trivial. + Qed. + + Lemma trans_id : (identityT ? x y) -> (identityT ? y z) -> (identityT ? x z). + Proof. + NewDestruct 2; Trivial. + Qed. + + Lemma congr_id : (identityT ? x y)->(identityT ? (f x) (f y)). + Proof. + NewDestruct 1; Trivial. + Qed. + + Lemma sym_not_id : (notT (identityT ? x y)) -> (notT (identityT ? y x)). + Proof. + Red; Intros H H'; Apply H; NewDestruct H'; Trivial. + Qed. + +End identity_is_a_congruence. + +Definition identity_ind_r : + (A:Type) + (a:A) + (P:A->Prop) + (P a)->(y:A)(identityT ? y a)->(P y). + Intros A x P H y H0; Case sym_id with 1:= H0; Trivial. +Defined. + +Definition identity_rec_r : + (A:Type) + (a:A) + (P:A->Set) + (P a)->(y:A)(identityT ? y a)->(P y). + Intros A x P H y H0; Case sym_id with 1:= H0; Trivial. +Defined. + +Definition identity_rect_r : + (A:Type) + (a:A) + (P:A->Type) + (P a)->(y:A)(identityT ? y a)->(P y). + Intros A x P H y H0; Case sym_id with 1:= H0; Trivial. +Defined. + +V7only [ +Notation sym_idT := sym_id (only parsing). +Notation trans_idT := trans_id (only parsing). +Notation congr_idT := congr_id (only parsing). +Notation sym_not_idT := sym_not_id (only parsing). +Notation identityT_ind_r := identity_ind_r (only parsing). +Notation identityT_rec_r := identity_rec_r (only parsing). +Notation identityT_rect_r := identity_rect_r (only parsing). +]. +Inductive prodT [A,B:Type] : Type := pairT : A -> B -> (prodT A B). + +Section prodT_proj. + + Variables A, B : Type. + + Definition fstT := [H:(prodT A B)]Cases H of (pairT x _) => x end. + Definition sndT := [H:(prodT A B)]Cases H of (pairT _ y) => y end. + +End prodT_proj. + +Definition prodT_uncurry : (A,B,C:Type)((prodT A B)->C)->A->B->C := + [A,B,C:Type; f:((prodT A B)->C); x:A; y:B] + (f (pairT A B x y)). + +Definition prodT_curry : (A,B,C:Type)(A->B->C)->(prodT A B)->C := + [A,B,C:Type; f:(A->B->C); p:(prodT A B)] + Cases p of + | (pairT x y) => (f x y) + end. + +Hints Immediate sym_id sym_not_id : core v62. + +V7only [ +Implicits fstT [1 2]. +Implicits sndT [1 2]. +Implicits pairT [1 2]. +]. diff --git a/theories7/Init/Notations.v b/theories7/Init/Notations.v new file mode 100644 index 00000000..34bfcbfa --- /dev/null +++ b/theories7/Init/Notations.v @@ -0,0 +1,94 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Notations.v,v 1.5.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +(** These are the notations whose level and associativity is imposed by Coq *) + +(** Notations for logical connectives *) + +Uninterpreted Notation "x <-> y" (at level 8, right associativity) + V8only (at level 95, no associativity). +Uninterpreted Notation "x /\ y" (at level 6, right associativity) + V8only (at level 80, right associativity). +Uninterpreted Notation "x \/ y" (at level 7, right associativity) + V8only (at level 85, right associativity). +Uninterpreted Notation "~ x" (at level 5, right associativity) + V8only (at level 75, right associativity). + +(** Notations for equality and inequalities *) + +Uninterpreted Notation "x = y :> T" + (at level 5, y at next level, no associativity). +Uninterpreted Notation "x = y" + (at level 5, no associativity). +Uninterpreted Notation "x = y = z" + (at level 5, no associativity, y at next level). + +Uninterpreted Notation "x <> y :> T" + (at level 5, y at next level, no associativity). +Uninterpreted Notation "x <> y" + (at level 5, no associativity). + +Uninterpreted V8Notation "x <= y" (at level 70, no associativity). +Uninterpreted V8Notation "x < y" (at level 70, no associativity). +Uninterpreted V8Notation "x >= y" (at level 70, no associativity). +Uninterpreted V8Notation "x > y" (at level 70, no associativity). + +Uninterpreted V8Notation "x <= y <= z" (at level 70, y at next level). +Uninterpreted V8Notation "x <= y < z" (at level 70, y at next level). +Uninterpreted V8Notation "x < y < z" (at level 70, y at next level). +Uninterpreted V8Notation "x < y <= z" (at level 70, y at next level). + +(** Arithmetical notations (also used for type constructors) *) + +Uninterpreted Notation "x + y" (at level 4, left associativity). +Uninterpreted V8Notation "x - y" (at level 50, left associativity). +Uninterpreted Notation "x * y" (at level 3, right associativity) + V8only (at level 40, left associativity). +Uninterpreted V8Notation "x / y" (at level 40, left associativity). +Uninterpreted V8Notation "- x" (at level 35, right associativity). +Uninterpreted V8Notation "/ x" (at level 35, right associativity). +Uninterpreted V8Notation "x ^ y" (at level 30, right associativity). + +(** Notations for pairs *) + +V7only [Uninterpreted Notation "( x , y )" (at level 0) V8only.]. +Uninterpreted V8Notation "( x , y , .. , z )" (at level 0). + +(** Notation "{ x }" is reserved and has a special status as component + of other notations; it is at level 1 to factor with {x:A|P} etc *) + +Uninterpreted Notation "{ x }" (at level 1) + V8only (at level 0, x at level 99). + +(** Notations for sum-types *) + +Uninterpreted Notation "{ A } + { B }" (at level 4, left associativity) + V8only (at level 50, left associativity). + +Uninterpreted Notation "A + { B }" (at level 4, left associativity) + V8only (at level 50, left associativity). + +(** Notations for sigma-types or subsets *) + +Uninterpreted Notation "{ x : A | P }" (at level 1) + V8only (at level 0, x at level 99). +Uninterpreted Notation "{ x : A | P & Q }" (at level 1) + V8only (at level 0, x at level 99). + +Uninterpreted Notation "{ x : A & P }" (at level 1) + V8only (at level 0, x at level 99). +Uninterpreted Notation "{ x : A & P & Q }" (at level 1) + V8only (at level 0, x at level 99). + +Delimits Scope type_scope with type. +Delimits Scope core_scope with core. + +Open Scope core_scope. +Open Scope type_scope. diff --git a/theories7/Init/Peano.v b/theories7/Init/Peano.v new file mode 100755 index 00000000..72d19399 --- /dev/null +++ b/theories7/Init/Peano.v @@ -0,0 +1,218 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Peano.v,v 1.1.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +(** Natural numbers [nat] built from [O] and [S] are defined in Datatypes.v *) + +(** This module defines the following operations on natural numbers : + - predecessor [pred] + - addition [plus] + - multiplication [mult] + - less or equal order [le] + - less [lt] + - greater or equal [ge] + - greater [gt] + + This module states various lemmas and theorems about natural numbers, + including Peano's axioms of arithmetic (in Coq, these are in fact provable) + Case analysis on [nat] and induction on [nat * nat] are provided too *) + +Require Notations. +Require Datatypes. +Require Logic. + +Open Scope nat_scope. + +Definition eq_S := (f_equal nat nat S). + +Hint eq_S : v62 := Resolve (f_equal nat nat S). +Hint eq_nat_unary : core := Resolve (f_equal nat). + +(** The predecessor function *) + +Definition pred : nat->nat := [n:nat](Cases n of O => O | (S u) => u end). +Hint eq_pred : v62 := Resolve (f_equal nat nat pred). + +Theorem pred_Sn : (m:nat) m=(pred (S m)). +Proof. + Auto. +Qed. + +Theorem eq_add_S : (n,m:nat) (S n)=(S m) -> n=m. +Proof. + Intros n m H ; Change (pred (S n))=(pred (S m)); Auto. +Qed. + +Hints Immediate eq_add_S : core v62. + +(** A consequence of the previous axioms *) + +Theorem not_eq_S : (n,m:nat) ~(n=m) -> ~((S n)=(S m)). +Proof. + Red; Auto. +Qed. +Hints Resolve not_eq_S : core v62. + +Definition IsSucc : nat->Prop + := [n:nat]Cases n of O => False | (S p) => True end. + + +Theorem O_S : (n:nat)~(O=(S n)). +Proof. + Red;Intros n H. + Change (IsSucc O). + Rewrite <- (sym_eq nat O (S n));[Exact I | Assumption]. +Qed. +Hints Resolve O_S : core v62. + +Theorem n_Sn : (n:nat) ~(n=(S n)). +Proof. + NewInduction n ; Auto. +Qed. +Hints Resolve n_Sn : core v62. + +(** Addition *) + +Fixpoint plus [n:nat] : nat -> nat := + [m:nat]Cases n of + O => m + | (S p) => (S (plus p m)) end. +Hint eq_plus : v62 := Resolve (f_equal2 nat nat nat plus). +Hint eq_nat_binary : core := Resolve (f_equal2 nat nat). + +V8Infix "+" plus : nat_scope. + +Lemma plus_n_O : (n:nat) n=(plus n O). +Proof. + NewInduction n ; Simpl ; Auto. +Qed. +Hints Resolve plus_n_O : core v62. + +Lemma plus_O_n : (n:nat) (plus O n)=n. +Proof. + Auto. +Qed. + +Lemma plus_n_Sm : (n,m:nat) (S (plus n m))=(plus n (S m)). +Proof. + Intros n m; NewInduction n; Simpl; Auto. +Qed. +Hints Resolve plus_n_Sm : core v62. + +Lemma plus_Sn_m : (n,m:nat)(plus (S n) m)=(S (plus n m)). +Proof. + Auto. +Qed. + +(** Multiplication *) + +Fixpoint mult [n:nat] : nat -> nat := + [m:nat]Cases n of O => O + | (S p) => (plus m (mult p m)) end. +Hint eq_mult : core v62 := Resolve (f_equal2 nat nat nat mult). + +V8Infix "*" mult : nat_scope. + +Lemma mult_n_O : (n:nat) O=(mult n O). +Proof. + NewInduction n; Simpl; Auto. +Qed. +Hints Resolve mult_n_O : core v62. + +Lemma mult_n_Sm : (n,m:nat) (plus (mult n m) n)=(mult n (S m)). +Proof. + Intros; NewInduction n as [|p H]; Simpl; Auto. + NewDestruct H; Rewrite <- plus_n_Sm; Apply (f_equal nat nat S). + Pattern 1 3 m; Elim m; Simpl; Auto. +Qed. +Hints Resolve mult_n_Sm : core v62. + +(** Definition of subtraction on [nat] : [m-n] is [0] if [n>=m] *) + +Fixpoint minus [n:nat] : nat -> nat := + [m:nat]Cases n m of + O _ => O + | (S k) O => (S k) + | (S k) (S l) => (minus k l) + end. + +V8Infix "-" minus : nat_scope. + +(** Definition of the usual orders, the basic properties of [le] and [lt] + can be found in files Le and Lt *) + +(** An inductive definition to define the order *) + +Inductive le [n:nat] : nat -> Prop + := le_n : (le n n) + | le_S : (m:nat)(le n m)->(le n (S m)). + +V8Infix "<=" le : nat_scope. + +Hint constr_le : core v62 := Constructors le. +(*i equivalent to : "Hints Resolve le_n le_S : core v62." i*) + +Definition lt := [n,m:nat](le (S n) m). +Hints Unfold lt : core v62. + +V8Infix "<" lt : nat_scope. + +Definition ge := [n,m:nat](le m n). +Hints Unfold ge : core v62. + +V8Infix ">=" ge : nat_scope. + +Definition gt := [n,m:nat](lt m n). +Hints Unfold gt : core v62. + +V8Infix ">" gt : nat_scope. + +V8Notation "x <= y <= z" := (le x y)/\(le y z) : nat_scope. +V8Notation "x <= y < z" := (le x y)/\(lt y z) : nat_scope. +V8Notation "x < y < z" := (lt x y)/\(lt y z) : nat_scope. +V8Notation "x < y <= z" := (lt x y)/\(le y z) : nat_scope. + +(** Pattern-Matching on natural numbers *) + +Theorem nat_case : (n:nat)(P:nat->Prop)(P O)->((m:nat)(P (S m)))->(P n). +Proof. + NewInduction n ; Auto. +Qed. + +(** Principle of double induction *) + +Theorem nat_double_ind : (R:nat->nat->Prop) + ((n:nat)(R O n)) -> ((n:nat)(R (S n) O)) + -> ((n,m:nat)(R n m)->(R (S n) (S m))) + -> (n,m:nat)(R n m). +Proof. + NewInduction n; Auto. + NewDestruct m; Auto. +Qed. + +(** Notations *) +V7only[ +Syntax constr + level 0: + S [ (S $p) ] -> [$p:"nat_printer":9] + | O [ O ] -> ["(0)"]. +]. + +V7only [ +(* For parsing/printing based on scopes *) +Module nat_scope. +Infix 4 "+" plus : nat_scope. +Infix 3 "*" mult : nat_scope. +Infix 4 "-" minus : nat_scope. +Infix NONA 5 "<=" le : nat_scope. +Infix NONA 5 "<" lt : nat_scope. +Infix NONA 5 ">=" ge : nat_scope. +Infix NONA 5 ">" gt : nat_scope. +End nat_scope. +]. diff --git a/theories7/Init/Prelude.v b/theories7/Init/Prelude.v new file mode 100755 index 00000000..2752f462 --- /dev/null +++ b/theories7/Init/Prelude.v @@ -0,0 +1,16 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Prelude.v,v 1.1.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +Require Export Notations. +Require Export Logic. +Require Export Datatypes. +Require Export Specif. +Require Export Peano. +Require Export Wf. diff --git a/theories7/Init/Specif.v b/theories7/Init/Specif.v new file mode 100755 index 00000000..c39e5ed8 --- /dev/null +++ b/theories7/Init/Specif.v @@ -0,0 +1,204 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Specif.v,v 1.2.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +Set Implicit Arguments. +V7only [Unset Implicit Arguments.]. + +(** Basic specifications : Sets containing logical information *) + +Require Notations. +Require Datatypes. +Require Logic. + +(** Subsets *) + +(** [(sig A P)], or more suggestively [{x:A | (P x)}], denotes the subset + of elements of the Set [A] which satisfy the predicate [P]. + Similarly [(sig2 A P Q)], or [{x:A | (P x) & (Q x)}], denotes the subset + of elements of the Set [A] which satisfy both [P] and [Q]. *) + +Inductive sig [A:Set;P:A->Prop] : Set + := exist : (x:A)(P x) -> (sig A P). + +Inductive sig2 [A:Set;P,Q:A->Prop] : Set + := exist2 : (x:A)(P x) -> (Q x) -> (sig2 A P Q). + +(** [(sigS A P)], or more suggestively [{x:A & (P x)}], is a subtle variant + of subset where [P] is now of type [Set]. + Similarly for [(sigS2 A P Q)], also written [{x:A & (P x) & (Q x)}]. *) + +Inductive sigS [A:Set;P:A->Set] : Set + := existS : (x:A)(P x) -> (sigS A P). + +Inductive sigS2 [A:Set;P,Q:A->Set] : Set + := existS2 : (x:A)(P x) -> (Q x) -> (sigS2 A P Q). + +Arguments Scope sig [type_scope type_scope]. +Arguments Scope sig2 [type_scope type_scope type_scope]. +Arguments Scope sigS [type_scope type_scope]. +Arguments Scope sigS2 [type_scope type_scope type_scope]. + +Notation "{ x : A | P }" := (sig A [x:A]P) : type_scope. +Notation "{ x : A | P & Q }" := (sig2 A [x:A]P [x:A]Q) : type_scope. +Notation "{ x : A & P }" := (sigS A [x:A]P) : type_scope. +Notation "{ x : A & P & Q }" := (sigS2 A [x:A]P [x:A]Q) : type_scope. + +Add Printing Let sig. +Add Printing Let sig2. +Add Printing Let sigS. +Add Printing Let sigS2. + + +(** Projections of sig *) + +Section Subset_projections. + + Variable A:Set. + Variable P:A->Prop. + + Definition proj1_sig := + [e:(sig A P)]Cases e of (exist a b) => a end. + + Definition proj2_sig := + [e:(sig A P)] + <[e:(sig A P)](P (proj1_sig e))>Cases e of (exist a b) => b end. + +End Subset_projections. + + +(** Projections of sigS *) + +Section Projections. + + Variable A:Set. + Variable P:A->Set. + + (** An element [y] of a subset [{x:A & (P x)}] is the pair of an [a] of + type [A] and of a proof [h] that [a] satisfies [P]. + Then [(projS1 y)] is the witness [a] + and [(projS2 y)] is the proof of [(P a)] *) + + Definition projS1 : (sigS A P) -> A + := [x:(sigS A P)]Cases x of (existS a _) => a end. + Definition projS2 : (x:(sigS A P))(P (projS1 x)) + := [x:(sigS A P)]<[x:(sigS A P)](P (projS1 x))> + Cases x of (existS _ h) => h end. + +End Projections. + + +(** Extended_booleans *) + +Inductive sumbool [A,B:Prop] : Set + := left : A -> {A}+{B} + | right : B -> {A}+{B} + +where "{ A } + { B }" := (sumbool A B) : type_scope. + +Inductive sumor [A:Set;B:Prop] : Set + := inleft : A -> A+{B} + | inright : B -> A+{B} + +where "A + { B }" := (sumor A B) : type_scope. + +(** Choice *) + +Section Choice_lemmas. + + (** The following lemmas state various forms of the axiom of choice *) + + Variables S,S':Set. + Variable R:S->S'->Prop. + Variable R':S->S'->Set. + Variables R1,R2 :S->Prop. + + Lemma Choice : ((x:S)(sig ? [y:S'](R x y))) -> + (sig ? [f:S->S'](z:S)(R z (f z))). + Proof. + Intro H. + Exists [z:S]Cases (H z) of (exist y _) => y end. + Intro z; NewDestruct (H z); Trivial. + Qed. + + Lemma Choice2 : ((x:S)(sigS ? [y:S'](R' x y))) -> + (sigS ? [f:S->S'](z:S)(R' z (f z))). + Proof. + Intro H. + Exists [z:S]Cases (H z) of (existS y _) => y end. + Intro z; NewDestruct (H z); Trivial. + Qed. + + Lemma bool_choice : + ((x:S)(sumbool (R1 x) (R2 x))) -> + (sig ? [f:S->bool] (x:S)( ((f x)=true /\ (R1 x)) + \/ ((f x)=false /\ (R2 x)))). + Proof. + Intro H. + Exists [z:S]Cases (H z) of (left _) => true | (right _) => false end. + Intro z; NewDestruct (H z); Auto. + Qed. + +End Choice_lemmas. + + (** A result of type [(Exc A)] is either a normal value of type [A] or + an [error] : + [Inductive Exc [A:Set] : Set := value : A->(Exc A) | error : (Exc A)] + it is implemented using the option type. *) + +Definition Exc := option. +Definition value := Some. +Definition error := !None. + +Implicits error [1]. + +Definition except := False_rec. (* for compatibility with previous versions *) + +Implicits except [1]. + +V7only [ +Notation Except := (!except ?) (only parsing). +Notation Error := (!error ?) (only parsing). +V7only [Implicits error [].]. +V7only [Implicits except [].]. +]. +Theorem absurd_set : (A:Prop)(C:Set)A->(~A)->C. +Proof. + Intros A C h1 h2. + Apply False_rec. + Apply (h2 h1). +Qed. + +Hints Resolve left right inleft inright : core v62. + +(** Sigma Type at Type level [sigT] *) + +Inductive sigT [A:Type;P:A->Type] : Type + := existT : (x:A)(P x) -> (sigT A P). + +Section projections_sigT. + + Variable A:Type. + Variable P:A->Type. + + Definition projT1 : (sigT A P) -> A + := [H:(sigT A P)]Cases H of (existT x _) => x end. + + Definition projT2 : (x:(sigT A P))(P (projT1 x)) + := [H:(sigT A P)]<[H:(sigT A P)](P (projT1 H))> + Cases H of (existT x h) => h end. + +End projections_sigT. + +V7only [ +Notation ProjS1 := (projS1 ? ?). +Notation ProjS2 := (projS2 ? ?). +Notation Value := (value ?). +]. + diff --git a/theories7/Init/Wf.v b/theories7/Init/Wf.v new file mode 100755 index 00000000..b65057eb --- /dev/null +++ b/theories7/Init/Wf.v @@ -0,0 +1,158 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +Set Implicit Arguments. +V7only [Unset Implicit Arguments.]. + +(*i $Id: Wf.v,v 1.1.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +(** This module proves the validity of + - well-founded recursion (also called course of values) + - well-founded induction + + from a well-founded ordering on a given set *) + +Require Notations. +Require Logic. +Require Datatypes. + +(** Well-founded induction principle on Prop *) + +Chapter Well_founded. + + Variable A : Set. + Variable R : A -> A -> Prop. + + (** The accessibility predicate is defined to be non-informative *) + + Inductive Acc : A -> Prop + := Acc_intro : (x:A)((y:A)(R y x)->(Acc y))->(Acc x). + + Lemma Acc_inv : (x:A)(Acc x) -> (y:A)(R y x) -> (Acc y). + NewDestruct 1; Trivial. + Defined. + + (** the informative elimination : + [let Acc_rec F = let rec wf x = F x wf in wf] *) + + Section AccRecType. + Variable P : A -> Type. + Variable F : (x:A)((y:A)(R y x)->(Acc y))->((y:A)(R y x)->(P y))->(P x). + + Fixpoint Acc_rect [x:A;a:(Acc x)] : (P x) + := (F x (Acc_inv x a) ([y:A][h:(R y x)](Acc_rect y (Acc_inv x a y h)))). + + End AccRecType. + + Definition Acc_rec [P:A->Set] := (Acc_rect P). + + (** A simplified version of Acc_rec(t) *) + + Section AccIter. + Variable P : A -> Type. + Variable F : (x:A)((y:A)(R y x)-> (P y))->(P x). + + Fixpoint Acc_iter [x:A;a:(Acc x)] : (P x) + := (F x ([y:A][h:(R y x)](Acc_iter y (Acc_inv x a y h)))). + + End AccIter. + + (** A relation is well-founded if every element is accessible *) + + Definition well_founded := (a:A)(Acc a). + + (** well-founded induction on Set and Prop *) + + Hypothesis Rwf : well_founded. + + Theorem well_founded_induction_type : + (P:A->Type)((x:A)((y:A)(R y x)->(P y))->(P x))->(a:A)(P a). + Proof. + Intros; Apply (Acc_iter P); Auto. + Defined. + + Theorem well_founded_induction : + (P:A->Set)((x:A)((y:A)(R y x)->(P y))->(P x))->(a:A)(P a). + Proof. + Exact [P:A->Set](well_founded_induction_type P). + Defined. + + Theorem well_founded_ind : + (P:A->Prop)((x:A)((y:A)(R y x)->(P y))->(P x))->(a:A)(P a). + Proof. + Exact [P:A->Prop](well_founded_induction_type P). + Defined. + +(** Building fixpoints *) + +Section FixPoint. + +Variable P : A -> Set. +Variable F : (x:A)((y:A)(R y x)->(P y))->(P x). + +Fixpoint Fix_F [x:A;r:(Acc x)] : (P x) := + (F x [y:A][p:(R y x)](Fix_F y (Acc_inv x r y p))). + +Definition fix := [x:A](Fix_F x (Rwf x)). + +(** Proof that [well_founded_induction] satisfies the fixpoint equation. + It requires an extra property of the functional *) + +Hypothesis F_ext : + (x:A)(f,g:(y:A)(R y x)->(P y)) + ((y:A)(p:(R y x))((f y p)=(g y p)))->(F x f)=(F x g). + +Scheme Acc_inv_dep := Induction for Acc Sort Prop. + +Lemma Fix_F_eq + : (x:A)(r:(Acc x)) + (F x [y:A][p:(R y x)](Fix_F y (Acc_inv x r y p)))=(Fix_F x r). +NewDestruct r using Acc_inv_dep; Auto. +Qed. + +Lemma Fix_F_inv : (x:A)(r,s:(Acc x))(Fix_F x r)=(Fix_F x s). +Intro x; NewInduction (Rwf x); Intros. +Rewrite <- (Fix_F_eq x r); Rewrite <- (Fix_F_eq x s); Intros. +Apply F_ext; Auto. +Qed. + + +Lemma Fix_eq : (x:A)(fix x)=(F x [y:A][p:(R y x)](fix y)). +Intro x; Unfold fix. +Rewrite <- (Fix_F_eq x). +Apply F_ext; Intros. +Apply Fix_F_inv. +Qed. + +End FixPoint. + +End Well_founded. + +(** A recursor over pairs *) + +Chapter Well_founded_2. + + Variable A,B : Set. + Variable R : A * B -> A * B -> Prop. + + Variable P : A -> B -> Type. + Variable F : (x:A)(x':B)((y:A)(y':B)(R (y,y') (x,x'))-> (P y y'))->(P x x'). + + Fixpoint Acc_iter_2 [x:A;x':B;a:(Acc ? R (x,x'))] : (P x x') + := (F x x' ([y:A][y':B][h:(R (y,y') (x,x'))](Acc_iter_2 y y' (Acc_inv ? ? (x,x') a (y,y') h)))). + + Hypothesis Rwf : (well_founded ? R). + + Theorem well_founded_induction_type_2 : + ((x:A)(x':B)((y:A)(y':B)(R (y,y') (x,x'))->(P y y'))->(P x x'))->(a:A)(b:B)(P a b). + Proof. + Intros; Apply Acc_iter_2; Auto. + Defined. + +End Well_founded_2. + diff --git a/theories7/IntMap/Adalloc.v b/theories7/IntMap/Adalloc.v new file mode 100644 index 00000000..9e8dd1b3 --- /dev/null +++ b/theories7/IntMap/Adalloc.v @@ -0,0 +1,339 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Adalloc.v,v 1.1.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require ZArith. +Require Arith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Fset. + +Section AdAlloc. + + Variable A : Set. + + Definition nat_of_ad := [a:ad] Cases a of + ad_z => O + | (ad_x p) => (convert p) + end. + + Fixpoint nat_le [m:nat] : nat -> bool := + Cases m of + O => [_:nat] true + | (S m') => [n:nat] Cases n of + O => false + | (S n') => (nat_le m' n') + end + end. + + Lemma nat_le_correct : (m,n:nat) (le m n) -> (nat_le m n)=true. + Proof. + NewInduction m as [|m IHm]. Trivial. + NewDestruct n. Intro H. Elim (le_Sn_O ? H). + Intros. Simpl. Apply IHm. Apply le_S_n. Assumption. + Qed. + + Lemma nat_le_complete : (m,n:nat) (nat_le m n)=true -> (le m n). + Proof. + NewInduction m. Trivial with arith. + NewDestruct n. Intro H. Discriminate H. + Auto with arith. + Qed. + + Lemma nat_le_correct_conv : (m,n:nat) (lt m n) -> (nat_le n m)=false. + Proof. + Intros. Elim (sumbool_of_bool (nat_le n m)). Intro H0. + Elim (lt_n_n ? (lt_le_trans ? ? ? H (nat_le_complete ? ? H0))). + Trivial. + Qed. + + Lemma nat_le_complete_conv : (m,n:nat) (nat_le n m)=false -> (lt m n). + Proof. + Intros. Elim (le_or_lt n m). Intro. Conditional Trivial Rewrite nat_le_correct in H. Discriminate H. + Trivial. + Qed. + + Definition ad_of_nat := [n:nat] Cases n of + O => ad_z + | (S n') => (ad_x (anti_convert n')) + end. + + Lemma ad_of_nat_of_ad : (a:ad) (ad_of_nat (nat_of_ad a))=a. + Proof. + NewDestruct a as [|p]. Reflexivity. + Simpl. Elim (ZL4 p). Intros n H. Rewrite H. Simpl. Rewrite <- bij1 in H. + Rewrite convert_intro with 1:=H. Reflexivity. + Qed. + + Lemma nat_of_ad_of_nat : (n:nat) (nat_of_ad (ad_of_nat n))=n. + Proof. + NewInduction n. Trivial. + Intros. Simpl. Apply bij1. + Qed. + + Definition ad_le := [a,b:ad] (nat_le (nat_of_ad a) (nat_of_ad b)). + + Lemma ad_le_refl : (a:ad) (ad_le a a)=true. + Proof. + Intro. Unfold ad_le. Apply nat_le_correct. Apply le_n. + Qed. + + Lemma ad_le_antisym : (a,b:ad) (ad_le a b)=true -> (ad_le b a)=true -> a=b. + Proof. + Unfold ad_le. Intros. Rewrite <- (ad_of_nat_of_ad a). Rewrite <- (ad_of_nat_of_ad b). + Rewrite (le_antisym ? ? (nat_le_complete ? ? H) (nat_le_complete ? ? H0)). Reflexivity. + Qed. + + Lemma ad_le_trans : (a,b,c:ad) (ad_le a b)=true -> (ad_le b c)=true -> + (ad_le a c)=true. + Proof. + Unfold ad_le. Intros. Apply nat_le_correct. Apply le_trans with m:=(nat_of_ad b). + Apply nat_le_complete. Assumption. + Apply nat_le_complete. Assumption. + Qed. + + Lemma ad_le_lt_trans : (a,b,c:ad) (ad_le a b)=true -> (ad_le c b)=false -> + (ad_le c a)=false. + Proof. + Unfold ad_le. Intros. Apply nat_le_correct_conv. Apply le_lt_trans with m:=(nat_of_ad b). + Apply nat_le_complete. Assumption. + Apply nat_le_complete_conv. Assumption. + Qed. + + Lemma ad_lt_le_trans : (a,b,c:ad) (ad_le b a)=false -> (ad_le b c)=true -> + (ad_le c a)=false. + Proof. + Unfold ad_le. Intros. Apply nat_le_correct_conv. Apply lt_le_trans with m:=(nat_of_ad b). + Apply nat_le_complete_conv. Assumption. + Apply nat_le_complete. Assumption. + Qed. + + Lemma ad_lt_trans : (a,b,c:ad) (ad_le b a)=false -> (ad_le c b)=false -> + (ad_le c a)=false. + Proof. + Unfold ad_le. Intros. Apply nat_le_correct_conv. Apply lt_trans with m:=(nat_of_ad b). + Apply nat_le_complete_conv. Assumption. + Apply nat_le_complete_conv. Assumption. + Qed. + + Lemma ad_lt_le_weak : (a,b:ad) (ad_le b a)=false -> (ad_le a b)=true. + Proof. + Unfold ad_le. Intros. Apply nat_le_correct. Apply lt_le_weak. + Apply nat_le_complete_conv. Assumption. + Qed. + + Definition ad_min := [a,b:ad] if (ad_le a b) then a else b. + + Lemma ad_min_choice : (a,b:ad) {(ad_min a b)=a}+{(ad_min a b)=b}. + Proof. + Unfold ad_min. Intros. Elim (sumbool_of_bool (ad_le a b)). Intro H. Left . Rewrite H. + Reflexivity. + Intro H. Right . Rewrite H. Reflexivity. + Qed. + + Lemma ad_min_le_1 : (a,b:ad) (ad_le (ad_min a b) a)=true. + Proof. + Unfold ad_min. Intros. Elim (sumbool_of_bool (ad_le a b)). Intro H. Rewrite H. + Apply ad_le_refl. + Intro H. Rewrite H. Apply ad_lt_le_weak. Assumption. + Qed. + + Lemma ad_min_le_2 : (a,b:ad) (ad_le (ad_min a b) b)=true. + Proof. + Unfold ad_min. Intros. Elim (sumbool_of_bool (ad_le a b)). Intro H. Rewrite H. Assumption. + Intro H. Rewrite H. Apply ad_le_refl. + Qed. + + Lemma ad_min_le_3 : (a,b,c:ad) (ad_le a (ad_min b c))=true -> (ad_le a b)=true. + Proof. + Unfold ad_min. Intros. Elim (sumbool_of_bool (ad_le b c)). Intro H0. Rewrite H0 in H. + Assumption. + Intro H0. Rewrite H0 in H. Apply ad_lt_le_weak. Apply ad_le_lt_trans with b:=c; Assumption. + Qed. + + Lemma ad_min_le_4 : (a,b,c:ad) (ad_le a (ad_min b c))=true -> (ad_le a c)=true. + Proof. + Unfold ad_min. Intros. Elim (sumbool_of_bool (ad_le b c)). Intro H0. Rewrite H0 in H. + Apply ad_le_trans with b:=b; Assumption. + Intro H0. Rewrite H0 in H. Assumption. + Qed. + + Lemma ad_min_le_5 : (a,b,c:ad) (ad_le a b)=true -> (ad_le a c)=true -> + (ad_le a (ad_min b c))=true. + Proof. + Intros. Elim (ad_min_choice b c). Intro H1. Rewrite H1. Assumption. + Intro H1. Rewrite H1. Assumption. + Qed. + + Lemma ad_min_lt_3 : (a,b,c:ad) (ad_le (ad_min b c) a)=false -> (ad_le b a)=false. + Proof. + Unfold ad_min. Intros. Elim (sumbool_of_bool (ad_le b c)). Intro H0. Rewrite H0 in H. + Assumption. + Intro H0. Rewrite H0 in H. Apply ad_lt_trans with b:=c; Assumption. + Qed. + + Lemma ad_min_lt_4 : (a,b,c:ad) (ad_le (ad_min b c) a)=false -> (ad_le c a)=false. + Proof. + Unfold ad_min. Intros. Elim (sumbool_of_bool (ad_le b c)). Intro H0. Rewrite H0 in H. + Apply ad_lt_le_trans with b:=b; Assumption. + Intro H0. Rewrite H0 in H. Assumption. + Qed. + + (** Allocator: returns an address not in the domain of [m]. + This allocator is optimal in that it returns the lowest possible address, + in the usual ordering on integers. It is not the most efficient, however. *) + Fixpoint ad_alloc_opt [m:(Map A)] : ad := + Cases m of + M0 => ad_z + | (M1 a _) => if (ad_eq a ad_z) + then (ad_x xH) + else ad_z + | (M2 m1 m2) => (ad_min (ad_double (ad_alloc_opt m1)) + (ad_double_plus_un (ad_alloc_opt m2))) + end. + + Lemma ad_alloc_opt_allocates_1 : (m:(Map A)) (MapGet A m (ad_alloc_opt m))=(NONE A). + Proof. + NewInduction m as [|a|m0 H m1 H0]. Reflexivity. + Simpl. Elim (sumbool_of_bool (ad_eq a ad_z)). Intro H. Rewrite H. + Rewrite (ad_eq_complete ? ? H). Reflexivity. + Intro H. Rewrite H. Rewrite H. Reflexivity. + Intros. Change (ad_alloc_opt (M2 A m0 m1)) with + (ad_min (ad_double (ad_alloc_opt m0)) (ad_double_plus_un (ad_alloc_opt m1))). + Elim (ad_min_choice (ad_double (ad_alloc_opt m0)) (ad_double_plus_un (ad_alloc_opt m1))). + Intro H1. Rewrite H1. Rewrite MapGet_M2_bit_0_0. Rewrite ad_double_div_2. Assumption. + Apply ad_double_bit_0. + Intro H1. Rewrite H1. Rewrite MapGet_M2_bit_0_1. Rewrite ad_double_plus_un_div_2. Assumption. + Apply ad_double_plus_un_bit_0. + Qed. + + Lemma ad_alloc_opt_allocates : (m:(Map A)) (in_dom A (ad_alloc_opt m) m)=false. + Proof. + Unfold in_dom. Intro. Rewrite (ad_alloc_opt_allocates_1 m). Reflexivity. + Qed. + + (** Moreover, this is optimal: all addresses below [(ad_alloc_opt m)] + are in [dom m]: *) + + Lemma nat_of_ad_double : (a:ad) (nat_of_ad (ad_double a))=(mult (2) (nat_of_ad a)). + Proof. + NewDestruct a as [|p]. Trivial. + Exact (convert_xO p). + Qed. + + Lemma nat_of_ad_double_plus_un : (a:ad) + (nat_of_ad (ad_double_plus_un a))=(S (mult (2) (nat_of_ad a))). + Proof. + NewDestruct a as [|p]. Trivial. + Exact (convert_xI p). + Qed. + + Lemma ad_le_double_mono : (a,b:ad) (ad_le a b)=true -> + (ad_le (ad_double a) (ad_double b))=true. + Proof. + Unfold ad_le. Intros. Rewrite nat_of_ad_double. Rewrite nat_of_ad_double. Apply nat_le_correct. + Simpl. Apply le_plus_plus. Apply nat_le_complete. Assumption. + Apply le_plus_plus. Apply nat_le_complete. Assumption. + Apply le_n. + Qed. + + Lemma ad_le_double_plus_un_mono : (a,b:ad) (ad_le a b)=true -> + (ad_le (ad_double_plus_un a) (ad_double_plus_un b))=true. + Proof. + Unfold ad_le. Intros. Rewrite nat_of_ad_double_plus_un. Rewrite nat_of_ad_double_plus_un. + Apply nat_le_correct. Apply le_n_S. Simpl. Apply le_plus_plus. Apply nat_le_complete. + Assumption. + Apply le_plus_plus. Apply nat_le_complete. Assumption. + Apply le_n. + Qed. + + Lemma ad_le_double_mono_conv : (a,b:ad) (ad_le (ad_double a) (ad_double b))=true -> + (ad_le a b)=true. + Proof. + Unfold ad_le. Intros a b. Rewrite nat_of_ad_double. Rewrite nat_of_ad_double. Intro. + Apply nat_le_correct. Apply (mult_le_conv_1 (1)). Apply nat_le_complete. Assumption. + Qed. + + Lemma ad_le_double_plus_un_mono_conv : (a,b:ad) + (ad_le (ad_double_plus_un a) (ad_double_plus_un b))=true -> (ad_le a b)=true. + Proof. + Unfold ad_le. Intros a b. Rewrite nat_of_ad_double_plus_un. Rewrite nat_of_ad_double_plus_un. + Intro. Apply nat_le_correct. Apply (mult_le_conv_1 (1)). Apply le_S_n. Apply nat_le_complete. + Assumption. + Qed. + + Lemma ad_lt_double_mono : (a,b:ad) (ad_le a b)=false -> + (ad_le (ad_double a) (ad_double b))=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_le (ad_double a) (ad_double b))). Intro H0. + Rewrite (ad_le_double_mono_conv ? ? H0) in H. Discriminate H. + Trivial. + Qed. + + Lemma ad_lt_double_plus_un_mono : (a,b:ad) (ad_le a b)=false -> + (ad_le (ad_double_plus_un a) (ad_double_plus_un b))=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_le (ad_double_plus_un a) (ad_double_plus_un b))). Intro H0. + Rewrite (ad_le_double_plus_un_mono_conv ? ? H0) in H. Discriminate H. + Trivial. + Qed. + + Lemma ad_lt_double_mono_conv : (a,b:ad) (ad_le (ad_double a) (ad_double b))=false -> + (ad_le a b)=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_le a b)). Intro H0. Rewrite (ad_le_double_mono ? ? H0) in H. + Discriminate H. + Trivial. + Qed. + + Lemma ad_lt_double_plus_un_mono_conv : (a,b:ad) + (ad_le (ad_double_plus_un a) (ad_double_plus_un b))=false -> (ad_le a b)=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_le a b)). Intro H0. + Rewrite (ad_le_double_plus_un_mono ? ? H0) in H. Discriminate H. + Trivial. + Qed. + + Lemma ad_alloc_opt_optimal_1 : (m:(Map A)) (a:ad) (ad_le (ad_alloc_opt m) a)=false -> + {y:A | (MapGet A m a)=(SOME A y)}. + Proof. + NewInduction m as [|a y|m0 H m1 H0]. Simpl. Unfold ad_le. Simpl. Intros. Discriminate H. + Simpl. Intros b H. Elim (sumbool_of_bool (ad_eq a ad_z)). Intro H0. Rewrite H0 in H. + Unfold ad_le in H. Cut ad_z=b. Intro. Split with y. Rewrite <- H1. Rewrite H0. Reflexivity. + Rewrite <- (ad_of_nat_of_ad b). + Rewrite <- (le_n_O_eq ? (le_S_n ? ? (nat_le_complete_conv ? ? H))). Reflexivity. + Intro H0. Rewrite H0 in H. Discriminate H. + Intros. Simpl in H1. Elim (ad_double_or_double_plus_un a). Intro H2. Elim H2. Intros a0 H3. + Rewrite H3 in H1. Elim (H ? (ad_lt_double_mono_conv ? ? (ad_min_lt_3 ? ? ? H1))). Intros y H4. + Split with y. Rewrite H3. Rewrite MapGet_M2_bit_0_0. Rewrite ad_double_div_2. Assumption. + Apply ad_double_bit_0. + Intro H2. Elim H2. Intros a0 H3. Rewrite H3 in H1. + Elim (H0 ? (ad_lt_double_plus_un_mono_conv ? ? (ad_min_lt_4 ? ? ? H1))). Intros y H4. + Split with y. Rewrite H3. Rewrite MapGet_M2_bit_0_1. Rewrite ad_double_plus_un_div_2. + Assumption. + Apply ad_double_plus_un_bit_0. + Qed. + + Lemma ad_alloc_opt_optimal : (m:(Map A)) (a:ad) (ad_le (ad_alloc_opt m) a)=false -> + (in_dom A a m)=true. + Proof. + Intros. Unfold in_dom. Elim (ad_alloc_opt_optimal_1 m a H). Intros y H0. Rewrite H0. + Reflexivity. + Qed. + +End AdAlloc. + +V7only [ +(* Moved to NArith *) +Notation positive_to_nat_2 := positive_to_nat_2. +Notation positive_to_nat_4 := positive_to_nat_4. +]. diff --git a/theories7/IntMap/Addec.v b/theories7/IntMap/Addec.v new file mode 100644 index 00000000..50dc1480 --- /dev/null +++ b/theories7/IntMap/Addec.v @@ -0,0 +1,179 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Addec.v,v 1.1.2.1 2004/07/16 19:31:26 herbelin Exp $ i*) + +(** Equality on adresses *) + +Require Bool. +Require Sumbool. +Require ZArith. +Require Addr. + +Fixpoint ad_eq_1 [p1,p2:positive] : bool := + Cases p1 p2 of + xH xH => true + | (xO p'1) (xO p'2) => (ad_eq_1 p'1 p'2) + | (xI p'1) (xI p'2) => (ad_eq_1 p'1 p'2) + | _ _ => false + end. + +Definition ad_eq := [a,a':ad] + Cases a a' of + ad_z ad_z => true + | (ad_x p) (ad_x p') => (ad_eq_1 p p') + | _ _ => false + end. + +Lemma ad_eq_correct : (a:ad) (ad_eq a a)=true. +Proof. + NewDestruct a; Trivial. + NewInduction p; Trivial. +Qed. + +Lemma ad_eq_complete : (a,a':ad) (ad_eq a a')=true -> a=a'. +Proof. + NewDestruct a. NewDestruct a'; Trivial. NewDestruct p. + Discriminate 1. + Discriminate 1. + Discriminate 1. + NewDestruct a'. Intros. Discriminate H. + Unfold ad_eq. Intros. Cut p=p0. Intros. Rewrite H0. Reflexivity. + Generalize Dependent p0. + NewInduction p as [p IHp|p IHp|]. NewDestruct p0; Intro H. + Rewrite (IHp p0). Reflexivity. + Exact H. + Discriminate H. + Discriminate H. + NewDestruct p0; Intro H. Discriminate H. + Rewrite (IHp p0 H). Reflexivity. + Discriminate H. + NewDestruct p0; Intro H. Discriminate H. + Discriminate H. + Trivial. +Qed. + +Lemma ad_eq_comm : (a,a':ad) (ad_eq a a')=(ad_eq a' a). +Proof. + Intros. Cut (b,b':bool)(ad_eq a a')=b->(ad_eq a' a)=b'->b=b'. + Intros. Apply H. Reflexivity. + Reflexivity. + NewDestruct b. Intros. Cut a=a'. + Intro. Rewrite H1 in H0. Rewrite (ad_eq_correct a') in H0. Exact H0. + Apply ad_eq_complete. Exact H. + NewDestruct b'. Intros. Cut a'=a. + Intro. Rewrite H1 in H. Rewrite H1 in H0. Rewrite <- H. Exact H0. + Apply ad_eq_complete. Exact H0. + Trivial. +Qed. + +Lemma ad_xor_eq_true : (a,a':ad) (ad_xor a a')=ad_z -> (ad_eq a a')=true. +Proof. + Intros. Rewrite (ad_xor_eq a a' H). Apply ad_eq_correct. +Qed. + +Lemma ad_xor_eq_false : + (a,a':ad) (p:positive) (ad_xor a a')=(ad_x p) -> (ad_eq a a')=false. +Proof. + Intros. Elim (sumbool_of_bool (ad_eq a a')). Intro H0. + Rewrite (ad_eq_complete a a' H0) in H. Rewrite (ad_xor_nilpotent a') in H. Discriminate H. + Trivial. +Qed. + +Lemma ad_bit_0_1_not_double : (a:ad) (ad_bit_0 a)=true -> + (a0:ad) (ad_eq (ad_double a0) a)=false. +Proof. + Intros. Elim (sumbool_of_bool (ad_eq (ad_double a0) a)). Intro H0. + Rewrite <- (ad_eq_complete ? ? H0) in H. Rewrite (ad_double_bit_0 a0) in H. Discriminate H. + Trivial. +Qed. + +Lemma ad_not_div_2_not_double : (a,a0:ad) (ad_eq (ad_div_2 a) a0)=false -> + (ad_eq a (ad_double a0))=false. +Proof. + Intros. Elim (sumbool_of_bool (ad_eq (ad_double a0) a)). Intro H0. + Rewrite <- (ad_eq_complete ? ? H0) in H. Rewrite (ad_double_div_2 a0) in H. + Rewrite (ad_eq_correct a0) in H. Discriminate H. + Intro. Rewrite ad_eq_comm. Assumption. +Qed. + +Lemma ad_bit_0_0_not_double_plus_un : (a:ad) (ad_bit_0 a)=false -> + (a0:ad) (ad_eq (ad_double_plus_un a0) a)=false. +Proof. + Intros. Elim (sumbool_of_bool (ad_eq (ad_double_plus_un a0) a)). Intro H0. + Rewrite <- (ad_eq_complete ? ? H0) in H. Rewrite (ad_double_plus_un_bit_0 a0) in H. + Discriminate H. + Trivial. +Qed. + +Lemma ad_not_div_2_not_double_plus_un : (a,a0:ad) (ad_eq (ad_div_2 a) a0)=false -> + (ad_eq (ad_double_plus_un a0) a)=false. +Proof. + Intros. Elim (sumbool_of_bool (ad_eq a (ad_double_plus_un a0))). Intro H0. + Rewrite (ad_eq_complete ? ? H0) in H. Rewrite (ad_double_plus_un_div_2 a0) in H. + Rewrite (ad_eq_correct a0) in H. Discriminate H. + Intro H0. Rewrite ad_eq_comm. Assumption. +Qed. + +Lemma ad_bit_0_neq : + (a,a':ad) (ad_bit_0 a)=false -> (ad_bit_0 a')=true -> (ad_eq a a')=false. +Proof. + Intros. Elim (sumbool_of_bool (ad_eq a a')). Intro H1. Rewrite (ad_eq_complete ? ? H1) in H. + Rewrite H in H0. Discriminate H0. + Trivial. +Qed. + +Lemma ad_div_eq : + (a,a':ad) (ad_eq a a')=true -> (ad_eq (ad_div_2 a) (ad_div_2 a'))=true. +Proof. + Intros. Cut a=a'. Intros. Rewrite H0. Apply ad_eq_correct. + Apply ad_eq_complete. Exact H. +Qed. + +Lemma ad_div_neq : (a,a':ad) (ad_eq (ad_div_2 a) (ad_div_2 a'))=false -> + (ad_eq a a')=false. +Proof. + Intros. Elim (sumbool_of_bool (ad_eq a a')). Intro H0. + Rewrite (ad_eq_complete ? ? H0) in H. Rewrite (ad_eq_correct (ad_div_2 a')) in H. Discriminate H. + Trivial. +Qed. + +Lemma ad_div_bit_eq : (a,a':ad) (ad_bit_0 a)=(ad_bit_0 a') -> + (ad_div_2 a)=(ad_div_2 a') -> a=a'. +Proof. + Intros. Apply ad_faithful. Unfold eqf. NewDestruct n. + Rewrite ad_bit_0_correct. Rewrite ad_bit_0_correct. Assumption. + Rewrite <- ad_div_2_correct. Rewrite <- ad_div_2_correct. + Rewrite H0. Reflexivity. +Qed. + +Lemma ad_div_bit_neq : (a,a':ad) (ad_eq a a')=false -> (ad_bit_0 a)=(ad_bit_0 a') -> + (ad_eq (ad_div_2 a) (ad_div_2 a'))=false. +Proof. + Intros. Elim (sumbool_of_bool (ad_eq (ad_div_2 a) (ad_div_2 a'))). Intro H1. + Rewrite (ad_div_bit_eq ? ? H0 (ad_eq_complete ? ? H1)) in H. + Rewrite (ad_eq_correct a') in H. Discriminate H. + Trivial. +Qed. + +Lemma ad_neq : (a,a':ad) (ad_eq a a')=false -> + (ad_bit_0 a)=(negb (ad_bit_0 a')) \/ (ad_eq (ad_div_2 a) (ad_div_2 a'))=false. +Proof. + Intros. Cut (ad_bit_0 a)=(ad_bit_0 a')\/(ad_bit_0 a)=(negb (ad_bit_0 a')). + Intros. Elim H0. Intro. Right . Apply ad_div_bit_neq. Assumption. + Assumption. + Intro. Left . Assumption. + Case (ad_bit_0 a); Case (ad_bit_0 a'); Auto. +Qed. + +Lemma ad_double_or_double_plus_un : (a:ad) + {a0:ad | a=(ad_double a0)}+{a1:ad | a=(ad_double_plus_un a1)}. +Proof. + Intro. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H. Right . Split with (ad_div_2 a). + Rewrite (ad_div_2_double_plus_un a H). Reflexivity. + Intro H. Left . Split with (ad_div_2 a). Rewrite (ad_div_2_double a H). Reflexivity. +Qed. diff --git a/theories7/IntMap/Addr.v b/theories7/IntMap/Addr.v new file mode 100644 index 00000000..9f362772 --- /dev/null +++ b/theories7/IntMap/Addr.v @@ -0,0 +1,456 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Addr.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*) + +(** Representation of adresses by the [positive] type of binary numbers *) + +Require Bool. +Require ZArith. + +Inductive ad : Set := + ad_z : ad + | ad_x : positive -> ad. + +Lemma ad_sum : (a:ad) {p:positive | a=(ad_x p)}+{a=ad_z}. +Proof. + NewDestruct a; Auto. + Left; Exists p; Trivial. +Qed. + +Fixpoint p_xor [p:positive] : positive -> ad := + [p2] Cases p of + xH => Cases p2 of + xH => ad_z + | (xO p'2) => (ad_x (xI p'2)) + | (xI p'2) => (ad_x (xO p'2)) + end + | (xO p') => Cases p2 of + xH => (ad_x (xI p')) + | (xO p'2) => Cases (p_xor p' p'2) of + ad_z => ad_z + | (ad_x p'') => (ad_x (xO p'')) + end + | (xI p'2) => Cases (p_xor p' p'2) of + ad_z => (ad_x xH) + | (ad_x p'') => (ad_x (xI p'')) + end + end + | (xI p') => Cases p2 of + xH => (ad_x (xO p')) + | (xO p'2) => Cases (p_xor p' p'2) of + ad_z => (ad_x xH) + | (ad_x p'') => (ad_x (xI p'')) + end + | (xI p'2) => Cases (p_xor p' p'2) of + ad_z => ad_z + | (ad_x p'') => (ad_x (xO p'')) + end + end + end. + +Definition ad_xor := [a,a':ad] + Cases a of + ad_z => a' + | (ad_x p) => Cases a' of + ad_z => a + | (ad_x p') => (p_xor p p') + end + end. + +Lemma ad_xor_neutral_left : (a:ad) (ad_xor ad_z a)=a. +Proof. + Trivial. +Qed. + +Lemma ad_xor_neutral_right : (a:ad) (ad_xor a ad_z)=a. +Proof. + NewDestruct a; Trivial. +Qed. + +Lemma ad_xor_comm : (a,a':ad) (ad_xor a a')=(ad_xor a' a). +Proof. + NewDestruct a; NewDestruct a'; Simpl; Auto. + Generalize p0; Clear p0; NewInduction p as [p Hrecp|p Hrecp|]; Simpl; Auto. + NewDestruct p0; Simpl; Trivial; Intros. + Rewrite Hrecp; Trivial. + Rewrite Hrecp; Trivial. + NewDestruct p0; Simpl; Trivial; Intros. + Rewrite Hrecp; Trivial. + Rewrite Hrecp; Trivial. + NewDestruct p0; Simpl; Auto. +Qed. + +Lemma ad_xor_nilpotent : (a:ad) (ad_xor a a)=ad_z. +Proof. + NewDestruct a; Trivial. + Simpl. NewInduction p as [p IHp|p IHp|]; Trivial. + Simpl. Rewrite IHp; Reflexivity. + Simpl. Rewrite IHp; Reflexivity. +Qed. + +Fixpoint ad_bit_1 [p:positive] : nat -> bool := + Cases p of + xH => [n:nat] Cases n of + O => true + | (S _) => false + end + | (xO p) => [n:nat] Cases n of + O => false + | (S n') => (ad_bit_1 p n') + end + | (xI p) => [n:nat] Cases n of + O => true + | (S n') => (ad_bit_1 p n') + end + end. + +Definition ad_bit := [a:ad] + Cases a of + ad_z => [_:nat] false + | (ad_x p) => (ad_bit_1 p) + end. + +Definition eqf := [f,g:nat->bool] (n:nat) (f n)=(g n). + +Lemma ad_faithful_1 : (a:ad) (eqf (ad_bit ad_z) (ad_bit a)) -> ad_z=a. +Proof. + NewDestruct a. Trivial. + NewInduction p as [p IHp|p IHp|];Intro H. Absurd ad_z=(ad_x p). Discriminate. + Exact (IHp [n:nat](H (S n))). + Absurd ad_z=(ad_x p). Discriminate. + Exact (IHp [n:nat](H (S n))). + Absurd false=true. Discriminate. + Exact (H O). +Qed. + +Lemma ad_faithful_2 : (a:ad) (eqf (ad_bit (ad_x xH)) (ad_bit a)) -> (ad_x xH)=a. +Proof. + NewDestruct a. Intros. Absurd true=false. Discriminate. + Exact (H O). + NewDestruct p. Intro H. Absurd ad_z=(ad_x p). Discriminate. + Exact (ad_faithful_1 (ad_x p) [n:nat](H (S n))). + Intros. Absurd true=false. Discriminate. + Exact (H O). + Trivial. +Qed. + +Lemma ad_faithful_3 : + (a:ad) (p:positive) + ((p':positive) (eqf (ad_bit (ad_x p)) (ad_bit (ad_x p'))) -> p=p') -> + (eqf (ad_bit (ad_x (xO p))) (ad_bit a)) -> + (ad_x (xO p))=a. +Proof. + NewDestruct a. Intros. Cut (eqf (ad_bit ad_z) (ad_bit (ad_x (xO p)))). + Intro. Rewrite (ad_faithful_1 (ad_x (xO p)) H1). Reflexivity. + Unfold eqf. Intro. Unfold eqf in H0. Rewrite H0. Reflexivity. + Case p. Intros. Absurd false=true. Discriminate. + Exact (H0 O). + Intros. Rewrite (H p0 [n:nat](H0 (S n))). Reflexivity. + Intros. Absurd false=true. Discriminate. + Exact (H0 O). +Qed. + +Lemma ad_faithful_4 : + (a:ad) (p:positive) + ((p':positive) (eqf (ad_bit (ad_x p)) (ad_bit (ad_x p'))) -> p=p') -> + (eqf (ad_bit (ad_x (xI p))) (ad_bit a)) -> + (ad_x (xI p))=a. +Proof. + NewDestruct a. Intros. Cut (eqf (ad_bit ad_z) (ad_bit (ad_x (xI p)))). + Intro. Rewrite (ad_faithful_1 (ad_x (xI p)) H1). Reflexivity. + Unfold eqf. Intro. Unfold eqf in H0. Rewrite H0. Reflexivity. + Case p. Intros. Rewrite (H p0 [n:nat](H0 (S n))). Reflexivity. + Intros. Absurd true=false. Discriminate. + Exact (H0 O). + Intros. Absurd ad_z=(ad_x p0). Discriminate. + Cut (eqf (ad_bit (ad_x xH)) (ad_bit (ad_x (xI p0)))). + Intro. Exact (ad_faithful_1 (ad_x p0) [n:nat](H1 (S n))). + Unfold eqf. Unfold eqf in H0. Intro. Rewrite H0. Reflexivity. +Qed. + +Lemma ad_faithful : (a,a':ad) (eqf (ad_bit a) (ad_bit a')) -> a=a'. +Proof. + NewDestruct a. Exact ad_faithful_1. + NewInduction p. Intros a' H. Apply ad_faithful_4. Intros. Cut (ad_x p)=(ad_x p'). + Intro. Inversion H1. Reflexivity. + Exact (IHp (ad_x p') H0). + Assumption. + Intros. Apply ad_faithful_3. Intros. Cut (ad_x p)=(ad_x p'). Intro. Inversion H1. Reflexivity. + Exact (IHp (ad_x p') H0). + Assumption. + Exact ad_faithful_2. +Qed. + +Definition adf_xor := [f,g:nat->bool; n:nat] (xorb (f n) (g n)). + +Lemma ad_xor_sem_1 : (a':ad) (ad_bit (ad_xor ad_z a') O)=(ad_bit a' O). +Proof. + Trivial. +Qed. + +Lemma ad_xor_sem_2 : (a':ad) (ad_bit (ad_xor (ad_x xH) a') O)=(negb (ad_bit a' O)). +Proof. + Intro. Case a'. Trivial. + Simpl. Intro. + Case p; Trivial. +Qed. + +Lemma ad_xor_sem_3 : + (p:positive) (a':ad) (ad_bit (ad_xor (ad_x (xO p)) a') O)=(ad_bit a' O). +Proof. + Intros. Case a'. Trivial. + Simpl. Intro. + Case p0; Trivial. Intro. + Case (p_xor p p1); Trivial. + Intro. Case (p_xor p p1); Trivial. +Qed. + +Lemma ad_xor_sem_4 : (p:positive) (a':ad) + (ad_bit (ad_xor (ad_x (xI p)) a') O)=(negb (ad_bit a' O)). +Proof. + Intros. Case a'. Trivial. + Simpl. Intro. Case p0; Trivial. Intro. + Case (p_xor p p1); Trivial. + Intro. + Case (p_xor p p1); Trivial. +Qed. + +Lemma ad_xor_sem_5 : + (a,a':ad) (ad_bit (ad_xor a a') O)=(adf_xor (ad_bit a) (ad_bit a') O). +Proof. + NewDestruct a. Intro. Change (ad_bit a' O)=(xorb false (ad_bit a' O)). Rewrite false_xorb. Trivial. + Case p. Exact ad_xor_sem_4. + Intros. Change (ad_bit (ad_xor (ad_x (xO p0)) a') O)=(xorb false (ad_bit a' O)). + Rewrite false_xorb. Apply ad_xor_sem_3. Exact ad_xor_sem_2. +Qed. + +Lemma ad_xor_sem_6 : (n:nat) + ((a,a':ad) (ad_bit (ad_xor a a') n)=(adf_xor (ad_bit a) (ad_bit a') n)) -> + (a,a':ad) (ad_bit (ad_xor a a') (S n))=(adf_xor (ad_bit a) (ad_bit a') (S n)). +Proof. + Intros. Case a. Unfold adf_xor. Unfold 2 ad_bit. Rewrite false_xorb. Reflexivity. + Case a'. Unfold adf_xor. Unfold 3 ad_bit. Intro. Rewrite xorb_false. Reflexivity. + Intros. Case p0. Case p. Intros. + Change (ad_bit (ad_xor (ad_x (xI p2)) (ad_x (xI p1))) (S n)) + =(adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n). + Rewrite <- H. Simpl. + Case (p_xor p2 p1); Trivial. + Intros. + Change (ad_bit (ad_xor (ad_x (xI p2)) (ad_x (xO p1))) (S n)) + =(adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n). + Rewrite <- H. Simpl. + Case (p_xor p2 p1); Trivial. + Intro. Unfold adf_xor. Unfold 3 ad_bit. Unfold ad_bit_1. Rewrite xorb_false. Reflexivity. + Case p. Intros. + Change (ad_bit (ad_xor (ad_x (xO p2)) (ad_x (xI p1))) (S n)) + =(adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n). + Rewrite <- H. Simpl. + Case (p_xor p2 p1); Trivial. + Intros. + Change (ad_bit (ad_xor (ad_x (xO p2)) (ad_x (xO p1))) (S n)) + =(adf_xor (ad_bit (ad_x p2)) (ad_bit (ad_x p1)) n). + Rewrite <- H. Simpl. + Case (p_xor p2 p1); Trivial. + Intro. Unfold adf_xor. Unfold 3 ad_bit. Unfold ad_bit_1. Rewrite xorb_false. Reflexivity. + Unfold adf_xor. Unfold 2 ad_bit. Unfold ad_bit_1. Rewrite false_xorb. Simpl. Case p; Trivial. +Qed. + +Lemma ad_xor_semantics : + (a,a':ad) (eqf (ad_bit (ad_xor a a')) (adf_xor (ad_bit a) (ad_bit a'))). +Proof. + Unfold eqf. Intros. Generalize a a'. Elim n. Exact ad_xor_sem_5. + Exact ad_xor_sem_6. +Qed. + +Lemma eqf_sym : (f,f':nat->bool) (eqf f f') -> (eqf f' f). +Proof. + Unfold eqf. Intros. Rewrite H. Reflexivity. +Qed. + +Lemma eqf_refl : (f:nat->bool) (eqf f f). +Proof. + Unfold eqf. Trivial. +Qed. + +Lemma eqf_trans : (f,f',f'':nat->bool) (eqf f f') -> (eqf f' f'') -> (eqf f f''). +Proof. + Unfold eqf. Intros. Rewrite H. Exact (H0 n). +Qed. + +Lemma adf_xor_eq : (f,f':nat->bool) (eqf (adf_xor f f') [n:nat] false) -> (eqf f f'). +Proof. + Unfold eqf. Unfold adf_xor. Intros. Apply xorb_eq. Apply H. +Qed. + +Lemma ad_xor_eq : (a,a':ad) (ad_xor a a')=ad_z -> a=a'. +Proof. + Intros. Apply ad_faithful. Apply adf_xor_eq. Apply eqf_trans with f':=(ad_bit (ad_xor a a')). + Apply eqf_sym. Apply ad_xor_semantics. + Rewrite H. Unfold eqf. Trivial. +Qed. + +Lemma adf_xor_assoc : (f,f',f'':nat->bool) + (eqf (adf_xor (adf_xor f f') f'') (adf_xor f (adf_xor f' f''))). +Proof. + Unfold eqf. Unfold adf_xor. Intros. Apply xorb_assoc. +Qed. + +Lemma eqf_xor_1 : (f,f',f'',f''':nat->bool) (eqf f f') -> (eqf f'' f''') -> + (eqf (adf_xor f f'') (adf_xor f' f''')). +Proof. + Unfold eqf. Intros. Unfold adf_xor. Rewrite H. Rewrite H0. Reflexivity. +Qed. + +Lemma ad_xor_assoc : + (a,a',a'':ad) (ad_xor (ad_xor a a') a'')=(ad_xor a (ad_xor a' a'')). +Proof. + Intros. Apply ad_faithful. + Apply eqf_trans with f':=(adf_xor (adf_xor (ad_bit a) (ad_bit a')) (ad_bit a'')). + Apply eqf_trans with f':=(adf_xor (ad_bit (ad_xor a a')) (ad_bit a'')). + Apply ad_xor_semantics. + Apply eqf_xor_1. Apply ad_xor_semantics. + Apply eqf_refl. + Apply eqf_trans with f':=(adf_xor (ad_bit a) (adf_xor (ad_bit a') (ad_bit a''))). + Apply adf_xor_assoc. + Apply eqf_trans with f':=(adf_xor (ad_bit a) (ad_bit (ad_xor a' a''))). + Apply eqf_xor_1. Apply eqf_refl. + Apply eqf_sym. Apply ad_xor_semantics. + Apply eqf_sym. Apply ad_xor_semantics. +Qed. + +Definition ad_double := [a:ad] + Cases a of + ad_z => ad_z + | (ad_x p) => (ad_x (xO p)) + end. + +Definition ad_double_plus_un := [a:ad] + Cases a of + ad_z => (ad_x xH) + | (ad_x p) => (ad_x (xI p)) + end. + +Definition ad_div_2 := [a:ad] + Cases a of + ad_z => ad_z + | (ad_x xH) => ad_z + | (ad_x (xO p)) => (ad_x p) + | (ad_x (xI p)) => (ad_x p) + end. + +Lemma ad_double_div_2 : (a:ad) (ad_div_2 (ad_double a))=a. +Proof. + NewDestruct a; Trivial. +Qed. + +Lemma ad_double_plus_un_div_2 : (a:ad) (ad_div_2 (ad_double_plus_un a))=a. +Proof. + NewDestruct a; Trivial. +Qed. + +Lemma ad_double_inj : (a0,a1:ad) (ad_double a0)=(ad_double a1) -> a0=a1. +Proof. + Intros. Rewrite <- (ad_double_div_2 a0). Rewrite H. Apply ad_double_div_2. +Qed. + +Lemma ad_double_plus_un_inj : + (a0,a1:ad) (ad_double_plus_un a0)=(ad_double_plus_un a1) -> a0=a1. +Proof. + Intros. Rewrite <- (ad_double_plus_un_div_2 a0). Rewrite H. Apply ad_double_plus_un_div_2. +Qed. + +Definition ad_bit_0 := [a:ad] + Cases a of + ad_z => false + | (ad_x (xO _)) => false + | _ => true + end. + +Lemma ad_double_bit_0 : (a:ad) (ad_bit_0 (ad_double a))=false. +Proof. + NewDestruct a; Trivial. +Qed. + +Lemma ad_double_plus_un_bit_0 : (a:ad) (ad_bit_0 (ad_double_plus_un a))=true. +Proof. + NewDestruct a; Trivial. +Qed. + +Lemma ad_div_2_double : (a:ad) (ad_bit_0 a)=false -> (ad_double (ad_div_2 a))=a. +Proof. + NewDestruct a. Trivial. NewDestruct p. Intro H. Discriminate H. + Intros. Reflexivity. + Intro H. Discriminate H. +Qed. + +Lemma ad_div_2_double_plus_un : + (a:ad) (ad_bit_0 a)=true -> (ad_double_plus_un (ad_div_2 a))=a. +Proof. + NewDestruct a. Intro. Discriminate H. + NewDestruct p. Intros. Reflexivity. + Intro H. Discriminate H. + Intro. Reflexivity. +Qed. + +Lemma ad_bit_0_correct : (a:ad) (ad_bit a O)=(ad_bit_0 a). +Proof. + NewDestruct a; Trivial. + NewDestruct p; Trivial. +Qed. + +Lemma ad_div_2_correct : (a:ad) (n:nat) (ad_bit (ad_div_2 a) n)=(ad_bit a (S n)). +Proof. + NewDestruct a; Trivial. + NewDestruct p; Trivial. +Qed. + +Lemma ad_xor_bit_0 : + (a,a':ad) (ad_bit_0 (ad_xor a a'))=(xorb (ad_bit_0 a) (ad_bit_0 a')). +Proof. + Intros. Rewrite <- ad_bit_0_correct. Rewrite (ad_xor_semantics a a' O). + Unfold adf_xor. Rewrite ad_bit_0_correct. Rewrite ad_bit_0_correct. Reflexivity. +Qed. + +Lemma ad_xor_div_2 : + (a,a':ad) (ad_div_2 (ad_xor a a'))=(ad_xor (ad_div_2 a) (ad_div_2 a')). +Proof. + Intros. Apply ad_faithful. Unfold eqf. Intro. + Rewrite (ad_xor_semantics (ad_div_2 a) (ad_div_2 a') n). + Rewrite ad_div_2_correct. + Rewrite (ad_xor_semantics a a' (S n)). + Unfold adf_xor. Rewrite ad_div_2_correct. Rewrite ad_div_2_correct. + Reflexivity. +Qed. + +Lemma ad_neg_bit_0 : (a,a':ad) (ad_bit_0 (ad_xor a a'))=true -> + (ad_bit_0 a)=(negb (ad_bit_0 a')). +Proof. + Intros. Rewrite <- true_xorb. Rewrite <- H. Rewrite ad_xor_bit_0. + Rewrite xorb_assoc. Rewrite xorb_nilpotent. Rewrite xorb_false. Reflexivity. +Qed. + +Lemma ad_neg_bit_0_1 : + (a,a':ad) (ad_xor a a')=(ad_x xH) -> (ad_bit_0 a)=(negb (ad_bit_0 a')). +Proof. + Intros. Apply ad_neg_bit_0. Rewrite H. Reflexivity. +Qed. + +Lemma ad_neg_bit_0_2 : (a,a':ad) (p:positive) (ad_xor a a')=(ad_x (xI p)) -> + (ad_bit_0 a)=(negb (ad_bit_0 a')). +Proof. + Intros. Apply ad_neg_bit_0. Rewrite H. Reflexivity. +Qed. + +Lemma ad_same_bit_0 : (a,a':ad) (p:positive) (ad_xor a a')=(ad_x (xO p)) -> + (ad_bit_0 a)=(ad_bit_0 a'). +Proof. + Intros. Rewrite <- (xorb_false (ad_bit_0 a)). Cut (ad_bit_0 (ad_x (xO p)))=false. + Intro. Rewrite <- H0. Rewrite <- H. Rewrite ad_xor_bit_0. Rewrite <- xorb_assoc. + Rewrite xorb_nilpotent. Rewrite false_xorb. Reflexivity. + Reflexivity. +Qed. diff --git a/theories7/IntMap/Adist.v b/theories7/IntMap/Adist.v new file mode 100644 index 00000000..a7948c72 --- /dev/null +++ b/theories7/IntMap/Adist.v @@ -0,0 +1,321 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Adist.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*) + +Require Bool. +Require ZArith. +Require Arith. +Require Min. +Require Addr. + +Fixpoint ad_plength_1 [p:positive] : nat := + Cases p of + xH => O + | (xI _) => O + | (xO p') => (S (ad_plength_1 p')) + end. + +Inductive natinf : Set := + infty : natinf + | ni : nat -> natinf. + +Definition ad_plength := [a:ad] + Cases a of + ad_z => infty + | (ad_x p) => (ni (ad_plength_1 p)) + end. + +Lemma ad_plength_infty : (a:ad) (ad_plength a)=infty -> a=ad_z. +Proof. + Induction a; Trivial. + Unfold ad_plength; Intros; Discriminate H. +Qed. + +Lemma ad_plength_zeros : (a:ad) (n:nat) (ad_plength a)=(ni n) -> + (k:nat) (lt k n) -> (ad_bit a k)=false. +Proof. + Induction a; Trivial. + Induction p. Induction n. Intros. Inversion H1. + Induction k. Simpl in H1. Discriminate H1. + Intros. Simpl in H1. Discriminate H1. + Induction k. Trivial. + Generalize H0. Case n. Intros. Inversion H3. + Intros. Simpl. Unfold ad_bit in H. Apply (H n0). Simpl in H1. Inversion H1. Reflexivity. + Exact (lt_S_n n1 n0 H3). + Simpl. Intros n H. Inversion H. Intros. Inversion H0. +Qed. + +Lemma ad_plength_one : (a:ad) (n:nat) (ad_plength a)=(ni n) -> (ad_bit a n)=true. +Proof. + Induction a. Intros. Inversion H. + Induction p. Intros. Simpl in H0. Inversion H0. Reflexivity. + Intros. Simpl in H0. Inversion H0. Simpl. Unfold ad_bit in H. Apply H. Reflexivity. + Intros. Simpl in H. Inversion H. Reflexivity. +Qed. + +Lemma ad_plength_first_one : (a:ad) (n:nat) + ((k:nat) (lt k n) -> (ad_bit a k)=false) -> (ad_bit a n)=true -> + (ad_plength a)=(ni n). +Proof. + Induction a. Intros. Simpl in H0. Discriminate H0. + Induction p. Intros. Generalize H0. Case n. Intros. Reflexivity. + Intros. Absurd (ad_bit (ad_x (xI p0)) O)=false. Trivial with bool. + Auto with bool arith. + Intros. Generalize H0 H1. Case n. Intros. Simpl in H3. Discriminate H3. + Intros. Simpl. Unfold ad_plength in H. + Cut (ni (ad_plength_1 p0))=(ni n0). Intro. Inversion H4. Reflexivity. + Apply H. Intros. Change (ad_bit (ad_x (xO p0)) (S k))=false. Apply H2. Apply lt_n_S. Exact H4. + Exact H3. + Intro. Case n. Trivial. + Intros. Simpl in H0. Discriminate H0. +Qed. + +Definition ni_min := [d,d':natinf] + Cases d of + infty => d' + | (ni n) => Cases d' of + infty => d + | (ni n') => (ni (min n n')) + end + end. + +Lemma ni_min_idemp : (d:natinf) (ni_min d d)=d. +Proof. + Induction d; Trivial. + Unfold ni_min. + Induction n; Trivial. + Intros. + Simpl. + Inversion H. + Rewrite H1. + Rewrite H1. + Reflexivity. +Qed. + +Lemma ni_min_comm : (d,d':natinf) (ni_min d d')=(ni_min d' d). +Proof. + Induction d. Induction d'; Trivial. + Induction d'; Trivial. Elim n. Induction n0; Trivial. + Intros. Elim n1; Trivial. Intros. Unfold ni_min in H. Cut (min n0 n2)=(min n2 n0). + Intro. Unfold ni_min. Simpl. Rewrite H1. Reflexivity. + Cut (ni (min n0 n2))=(ni (min n2 n0)). Intros. + Inversion H1; Trivial. + Exact (H n2). +Qed. + +Lemma ni_min_assoc : (d,d',d'':natinf) (ni_min (ni_min d d') d'')=(ni_min d (ni_min d' d'')). +Proof. + Induction d; Trivial. Induction d'; Trivial. + Induction d''; Trivial. + Unfold ni_min. Intro. Cut (min (min n n0) n1)=(min n (min n0 n1)). + Intro. Rewrite H. Reflexivity. + Generalize n0 n1. Elim n; Trivial. + Induction n3; Trivial. Induction n5; Trivial. + Intros. Simpl. Auto. +Qed. + +Lemma ni_min_O_l : (d:natinf) (ni_min (ni O) d)=(ni O). +Proof. + Induction d; Trivial. +Qed. + +Lemma ni_min_O_r : (d:natinf) (ni_min d (ni O))=(ni O). +Proof. + Intros. Rewrite ni_min_comm. Apply ni_min_O_l. +Qed. + +Lemma ni_min_inf_l : (d:natinf) (ni_min infty d)=d. +Proof. + Trivial. +Qed. + +Lemma ni_min_inf_r : (d:natinf) (ni_min d infty)=d. +Proof. + Induction d; Trivial. +Qed. + +Definition ni_le := [d,d':natinf] (ni_min d d')=d. + +Lemma ni_le_refl : (d:natinf) (ni_le d d). +Proof. + Exact ni_min_idemp. +Qed. + +Lemma ni_le_antisym : (d,d':natinf) (ni_le d d') -> (ni_le d' d) -> d=d'. +Proof. + Unfold ni_le. Intros d d'. Rewrite ni_min_comm. Intro H. Rewrite H. Trivial. +Qed. + +Lemma ni_le_trans : (d,d',d'':natinf) (ni_le d d') -> (ni_le d' d'') -> (ni_le d d''). +Proof. + Unfold ni_le. Intros. Rewrite <- H. Rewrite ni_min_assoc. Rewrite H0. Reflexivity. +Qed. + +Lemma ni_le_min_1 : (d,d':natinf) (ni_le (ni_min d d') d). +Proof. + Unfold ni_le. Intros. Rewrite (ni_min_comm d d'). Rewrite ni_min_assoc. + Rewrite ni_min_idemp. Reflexivity. +Qed. + +Lemma ni_le_min_2 : (d,d':natinf) (ni_le (ni_min d d') d'). +Proof. + Unfold ni_le. Intros. Rewrite ni_min_assoc. Rewrite ni_min_idemp. Reflexivity. +Qed. + +Lemma ni_min_case : (d,d':natinf) (ni_min d d')=d \/ (ni_min d d')=d'. +Proof. + Induction d. Intro. Right . Exact (ni_min_inf_l d'). + Induction d'. Left . Exact (ni_min_inf_r (ni n)). + Unfold ni_min. Cut (n0:nat)(min n n0)=n\/(min n n0)=n0. + Intros. Case (H n0). Intro. Left . Rewrite H0. Reflexivity. + Intro. Right . Rewrite H0. Reflexivity. + Elim n. Intro. Left . Reflexivity. + Induction n1. Right . Reflexivity. + Intros. Case (H n2). Intro. Left . Simpl. Rewrite H1. Reflexivity. + Intro. Right . Simpl. Rewrite H1. Reflexivity. +Qed. + +Lemma ni_le_total : (d,d':natinf) (ni_le d d') \/ (ni_le d' d). +Proof. + Unfold ni_le. Intros. Rewrite (ni_min_comm d' d). Apply ni_min_case. +Qed. + +Lemma ni_le_min_induc : (d,d',dm:natinf) (ni_le dm d) -> (ni_le dm d') -> + ((d'':natinf) (ni_le d'' d) -> (ni_le d'' d') -> (ni_le d'' dm)) -> + (ni_min d d')=dm. +Proof. + Intros. Case (ni_min_case d d'). Intro. Rewrite H2. + Apply ni_le_antisym. Apply H1. Apply ni_le_refl. + Exact H2. + Exact H. + Intro. Rewrite H2. Apply ni_le_antisym. Apply H1. Unfold ni_le. Rewrite ni_min_comm. Exact H2. + Apply ni_le_refl. + Exact H0. +Qed. + +Lemma le_ni_le : (m,n:nat) (le m n) -> (ni_le (ni m) (ni n)). +Proof. + Cut (m,n:nat)(le m n)->(min m n)=m. + Intros. Unfold ni_le ni_min. Rewrite (H m n H0). Reflexivity. + Induction m. Trivial. + Induction n0. Intro. Inversion H0. + Intros. Simpl. Rewrite (H n1 (le_S_n n n1 H1)). Reflexivity. +Qed. + +Lemma ni_le_le : (m,n:nat) (ni_le (ni m) (ni n)) -> (le m n). +Proof. + Unfold ni_le. Unfold ni_min. Intros. Inversion H. Apply le_min_r. +Qed. + +Lemma ad_plength_lb : (a:ad) (n:nat) ((k:nat) (lt k n) -> (ad_bit a k)=false) -> + (ni_le (ni n) (ad_plength a)). +Proof. + Induction a. Intros. Exact (ni_min_inf_r (ni n)). + Intros. Unfold ad_plength. Apply le_ni_le. Case (le_or_lt n (ad_plength_1 p)). Trivial. + Intro. Absurd (ad_bit (ad_x p) (ad_plength_1 p))=false. + Rewrite (ad_plength_one (ad_x p) (ad_plength_1 p) + (refl_equal natinf (ad_plength (ad_x p)))). + Discriminate. + Apply H. Exact H0. +Qed. + +Lemma ad_plength_ub : (a:ad) (n:nat) (ad_bit a n)=true -> + (ni_le (ad_plength a) (ni n)). +Proof. + Induction a. Intros. Discriminate H. + Intros. Unfold ad_plength. Apply le_ni_le. Case (le_or_lt (ad_plength_1 p) n). Trivial. + Intro. Absurd (ad_bit (ad_x p) n)=true. + Rewrite (ad_plength_zeros (ad_x p) (ad_plength_1 p) + (refl_equal natinf (ad_plength (ad_x p))) n H0). + Discriminate. + Exact H. +Qed. + + +(** We define an ultrametric distance between addresses: + $d(a,a')=1/2^pd(a,a')$, + where $pd(a,a')$ is the number of identical bits at the beginning + of $a$ and $a'$ (infinity if $a=a'$). + Instead of working with $d$, we work with $pd$, namely + [ad_pdist]: *) + +Definition ad_pdist := [a,a':ad] (ad_plength (ad_xor a a')). + +(** d is a distance, so $d(a,a')=0$ iff $a=a'$; this means that + $pd(a,a')=infty$ iff $a=a'$: *) + +Lemma ad_pdist_eq_1 : (a:ad) (ad_pdist a a)=infty. +Proof. + Intros. Unfold ad_pdist. Rewrite ad_xor_nilpotent. Reflexivity. +Qed. + +Lemma ad_pdist_eq_2 : (a,a':ad) (ad_pdist a a')=infty -> a=a'. +Proof. + Intros. Apply ad_xor_eq. Apply ad_plength_infty. Exact H. +Qed. + +(** $d$ is a distance, so $d(a,a')=d(a',a)$: *) + +Lemma ad_pdist_comm : (a,a':ad) (ad_pdist a a')=(ad_pdist a' a). +Proof. + Unfold ad_pdist. Intros. Rewrite ad_xor_comm. Reflexivity. +Qed. + +(** $d$ is an ultrametric distance, that is, not only $d(a,a')\leq + d(a,a'')+d(a'',a')$, + but in fact $d(a,a')\leq max(d(a,a''),d(a'',a'))$. + This means that $min(pd(a,a''),pd(a'',a'))<=pd(a,a')$ (lemma [ad_pdist_ultra] below). + This follows from the fact that $a ~Ra~|a| = 1/2^{\texttt{ad\_plength}}(a))$ + is an ultrametric norm, i.e. that $|a-a'| \leq max (|a-a''|, |a''-a'|)$, + or equivalently that $|a+b|<=max(|a|,|b|)$, i.e. that + min $(\texttt{ad\_plength}(a), \texttt{ad\_plength}(b)) \leq + \texttt{ad\_plength} (a~\texttt{xor}~ b)$ + (lemma [ad_plength_ultra]). +*) + +Lemma ad_plength_ultra_1 : (a,a':ad) + (ni_le (ad_plength a) (ad_plength a')) -> + (ni_le (ad_plength a) (ad_plength (ad_xor a a'))). +Proof. + Induction a. Intros. Unfold ni_le in H. Unfold 1 3 ad_plength in H. + Rewrite (ni_min_inf_l (ad_plength a')) in H. + Rewrite (ad_plength_infty a' H). Simpl. Apply ni_le_refl. + Intros. Unfold 1 ad_plength. Apply ad_plength_lb. Intros. + Cut (a'':ad)(ad_xor (ad_x p) a')=a''->(ad_bit a'' k)=false. + Intros. Apply H1. Reflexivity. + Intro a''. Case a''. Intro. Reflexivity. + Intros. Rewrite <- H1. Rewrite (ad_xor_semantics (ad_x p) a' k). Unfold adf_xor. + Rewrite (ad_plength_zeros (ad_x p) (ad_plength_1 p) + (refl_equal natinf (ad_plength (ad_x p))) k H0). + Generalize H. Case a'. Trivial. + Intros. Cut (ad_bit (ad_x p1) k)=false. Intros. Rewrite H3. Reflexivity. + Apply ad_plength_zeros with n:=(ad_plength_1 p1). Reflexivity. + Apply (lt_le_trans k (ad_plength_1 p) (ad_plength_1 p1)). Exact H0. + Apply ni_le_le. Exact H2. +Qed. + +Lemma ad_plength_ultra : (a,a':ad) + (ni_le (ni_min (ad_plength a) (ad_plength a')) (ad_plength (ad_xor a a'))). +Proof. + Intros. Case (ni_le_total (ad_plength a) (ad_plength a')). Intro. + Cut (ni_min (ad_plength a) (ad_plength a'))=(ad_plength a). + Intro. Rewrite H0. Apply ad_plength_ultra_1. Exact H. + Exact H. + Intro. Cut (ni_min (ad_plength a) (ad_plength a'))=(ad_plength a'). + Intro. Rewrite H0. Rewrite ad_xor_comm. Apply ad_plength_ultra_1. Exact H. + Rewrite ni_min_comm. Exact H. +Qed. + +Lemma ad_pdist_ultra : (a,a',a'':ad) + (ni_le (ni_min (ad_pdist a a'') (ad_pdist a'' a')) (ad_pdist a a')). +Proof. + Intros. Unfold ad_pdist. Cut (ad_xor (ad_xor a a'') (ad_xor a'' a'))=(ad_xor a a'). + Intro. Rewrite <- H. Apply ad_plength_ultra. + Rewrite ad_xor_assoc. Rewrite <- (ad_xor_assoc a'' a'' a'). Rewrite ad_xor_nilpotent. + Rewrite ad_xor_neutral_left. Reflexivity. +Qed. diff --git a/theories7/IntMap/Allmaps.v b/theories7/IntMap/Allmaps.v new file mode 100644 index 00000000..e76e210f --- /dev/null +++ b/theories7/IntMap/Allmaps.v @@ -0,0 +1,26 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Allmaps.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*) + +Require Export Addr. +Require Export Adist. +Require Export Addec. +Require Export Map. + +Require Export Fset. +Require Export Mapaxioms. +Require Export Mapiter. + +Require Export Mapsubset. +Require Export Lsort. +Require Export Mapfold. +Require Export Mapcard. +Require Export Mapcanon. +Require Export Mapc. +Require Export Maplists. +Require Export Adalloc. diff --git a/theories7/IntMap/Fset.v b/theories7/IntMap/Fset.v new file mode 100644 index 00000000..545c1716 --- /dev/null +++ b/theories7/IntMap/Fset.v @@ -0,0 +1,338 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Fset.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*) + +(*s Sets operations on maps *) + +Require Bool. +Require Sumbool. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. + +Section Dom. + + Variable A, B : Set. + + Fixpoint MapDomRestrTo [m:(Map A)] : (Map B) -> (Map A) := + Cases m of + M0 => [_:(Map B)] (M0 A) + | (M1 a y) => [m':(Map B)] Cases (MapGet B m' a) of + NONE => (M0 A) + | _ => m + end + | (M2 m1 m2) => [m':(Map B)] Cases m' of + M0 => (M0 A) + | (M1 a' y') => Cases (MapGet A m a') of + NONE => (M0 A) + | (SOME y) => (M1 A a' y) + end + | (M2 m'1 m'2) => (makeM2 A (MapDomRestrTo m1 m'1) + (MapDomRestrTo m2 m'2)) + end + end. + + Lemma MapDomRestrTo_semantics : (m:(Map A)) (m':(Map B)) + (eqm A (MapGet A (MapDomRestrTo m m')) + [a0:ad] Cases (MapGet B m' a0) of + NONE => (NONE A) + | _ => (MapGet A m a0) + end). + Proof. + Unfold eqm. Induction m. Simpl. Intros. Case (MapGet B m' a); Trivial. + Intros. Simpl. Elim (sumbool_of_bool (ad_eq a a1)). Intro H. Rewrite H. + Rewrite <- (ad_eq_complete ? ? H). Case (MapGet B m' a). Reflexivity. + Intro. Apply M1_semantics_1. + Intro H. Rewrite H. Case (MapGet B m' a). + Case (MapGet B m' a1); Reflexivity. + Case (MapGet B m' a1); Intros; Exact (M1_semantics_2 A a a1 a0 H). + Induction m'. Trivial. + Unfold MapDomRestrTo. Intros. Elim (sumbool_of_bool (ad_eq a a1)). + Intro H1. + Rewrite (ad_eq_complete ? ? H1). Rewrite (M1_semantics_1 B a1 a0). + Case (MapGet A (M2 A m0 m1) a1). Reflexivity. + Intro. Apply M1_semantics_1. + Intro H1. Rewrite (M1_semantics_2 B a a1 a0 H1). Case (MapGet A (M2 A m0 m1) a). Reflexivity. + Intro. Exact (M1_semantics_2 A a a1 a2 H1). + Intros. Change (MapGet A (makeM2 A (MapDomRestrTo m0 m2) (MapDomRestrTo m1 m3)) a) + =(Cases (MapGet B (M2 B m2 m3) a) of + NONE => (NONE A) + | (SOME _) => (MapGet A (M2 A m0 m1) a) + end). + Rewrite (makeM2_M2 A (MapDomRestrTo m0 m2) (MapDomRestrTo m1 m3) a). + Rewrite MapGet_M2_bit_0_if. Rewrite (H0 m3 (ad_div_2 a)). Rewrite (H m2 (ad_div_2 a)). + Rewrite (MapGet_M2_bit_0_if B m2 m3 a). Rewrite (MapGet_M2_bit_0_if A m0 m1 a). + Case (ad_bit_0 a); Reflexivity. + Qed. + + Fixpoint MapDomRestrBy [m:(Map A)] : (Map B) -> (Map A) := + Cases m of + M0 => [_:(Map B)] (M0 A) + | (M1 a y) => [m':(Map B)] Cases (MapGet B m' a) of + NONE => m + | _ => (M0 A) + end + | (M2 m1 m2) => [m':(Map B)] Cases m' of + M0 => m + | (M1 a' y') => (MapRemove A m a') + | (M2 m'1 m'2) => (makeM2 A (MapDomRestrBy m1 m'1) + (MapDomRestrBy m2 m'2)) + end + end. + + Lemma MapDomRestrBy_semantics : (m:(Map A)) (m':(Map B)) + (eqm A (MapGet A (MapDomRestrBy m m')) + [a0:ad] Cases (MapGet B m' a0) of + NONE => (MapGet A m a0) + | _ => (NONE A) + end). + Proof. + Unfold eqm. Induction m. Simpl. Intros. Case (MapGet B m' a); Trivial. + Intros. Simpl. Elim (sumbool_of_bool (ad_eq a a1)). Intro H. Rewrite H. + Rewrite (ad_eq_complete ? ? H). Case (MapGet B m' a1). Apply M1_semantics_1. + Trivial. + Intro H. Rewrite H. Case (MapGet B m' a). Rewrite (M1_semantics_2 A a a1 a0 H). + Case (MapGet B m' a1); Trivial. + Case (MapGet B m' a1); Trivial. + Induction m'. Trivial. + Unfold MapDomRestrBy. Intros. Rewrite (MapRemove_semantics A (M2 A m0 m1) a a1). + Elim (sumbool_of_bool (ad_eq a a1)). Intro H1. Rewrite H1. Rewrite (ad_eq_complete ? ? H1). + Rewrite (M1_semantics_1 B a1 a0). Reflexivity. + Intro H1. Rewrite H1. Rewrite (M1_semantics_2 B a a1 a0 H1). Reflexivity. + Intros. Change (MapGet A (makeM2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3)) a) + =(Cases (MapGet B (M2 B m2 m3) a) of + NONE => (MapGet A (M2 A m0 m1) a) + | (SOME _) => (NONE A) + end). + Rewrite (makeM2_M2 A (MapDomRestrBy m0 m2) (MapDomRestrBy m1 m3) a). + Rewrite MapGet_M2_bit_0_if. Rewrite (H0 m3 (ad_div_2 a)). Rewrite (H m2 (ad_div_2 a)). + Rewrite (MapGet_M2_bit_0_if B m2 m3 a). Rewrite (MapGet_M2_bit_0_if A m0 m1 a). + Case (ad_bit_0 a); Reflexivity. + Qed. + + Definition in_dom := [a:ad; m:(Map A)] + Cases (MapGet A m a) of + NONE => false + | _ => true + end. + + Lemma in_dom_M0 : (a:ad) (in_dom a (M0 A))=false. + Proof. + Trivial. + Qed. + + Lemma in_dom_M1 : (a,a0:ad) (y:A) (in_dom a0 (M1 A a y))=(ad_eq a a0). + Proof. + Unfold in_dom. Intros. Simpl. Case (ad_eq a a0); Reflexivity. + Qed. + + Lemma in_dom_M1_1 : (a:ad) (y:A) (in_dom a (M1 A a y))=true. + Proof. + Intros. Rewrite in_dom_M1. Apply ad_eq_correct. + Qed. + + Lemma in_dom_M1_2 : (a,a0:ad) (y:A) (in_dom a0 (M1 A a y))=true -> a=a0. + Proof. + Intros. Apply (ad_eq_complete a a0). Rewrite (in_dom_M1 a a0 y) in H. Assumption. + Qed. + + Lemma in_dom_some : (m:(Map A)) (a:ad) (in_dom a m)=true -> + {y:A | (MapGet A m a)=(SOME A y)}. + Proof. + Unfold in_dom. Intros. Elim (option_sum ? (MapGet A m a)). Trivial. + Intro H0. Rewrite H0 in H. Discriminate H. + Qed. + + Lemma in_dom_none : (m:(Map A)) (a:ad) (in_dom a m)=false -> + (MapGet A m a)=(NONE A). + Proof. + Unfold in_dom. Intros. Elim (option_sum ? (MapGet A m a)). Intro H0. Elim H0. + Intros y H1. Rewrite H1 in H. Discriminate H. + Trivial. + Qed. + + Lemma in_dom_put : (m:(Map A)) (a0:ad) (y0:A) (a:ad) + (in_dom a (MapPut A m a0 y0))=(orb (ad_eq a a0) (in_dom a m)). + Proof. + Unfold in_dom. Intros. Rewrite (MapPut_semantics A m a0 y0 a). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. + Rewrite H. Rewrite orb_true_b. Reflexivity. + Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H. Rewrite orb_false_b. + Reflexivity. + Qed. + + Lemma in_dom_put_behind : (m:(Map A)) (a0:ad) (y0:A) (a:ad) + (in_dom a (MapPut_behind A m a0 y0))=(orb (ad_eq a a0) (in_dom a m)). + Proof. + Unfold in_dom. Intros. Rewrite (MapPut_behind_semantics A m a0 y0 a). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. + Rewrite H. Case (MapGet A m a); Reflexivity. + Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H. Case (MapGet A m a); Trivial. + Qed. + + Lemma in_dom_remove : (m:(Map A)) (a0:ad) (a:ad) + (in_dom a (MapRemove A m a0))=(andb (negb (ad_eq a a0)) (in_dom a m)). + Proof. + Unfold in_dom. Intros. Rewrite (MapRemove_semantics A m a0 a). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. + Rewrite H. Reflexivity. + Intro H. Rewrite H. Rewrite (ad_eq_comm a a0) in H. Rewrite H. + Case (MapGet A m a); Reflexivity. + Qed. + + Lemma in_dom_merge : (m,m':(Map A)) (a:ad) + (in_dom a (MapMerge A m m'))=(orb (in_dom a m) (in_dom a m')). + Proof. + Unfold in_dom. Intros. Rewrite (MapMerge_semantics A m m' a). + Elim (option_sum A (MapGet A m' a)). Intro H. Elim H. Intros y H0. Rewrite H0. + Case (MapGet A m a); Reflexivity. + Intro H. Rewrite H. Rewrite orb_b_false. Reflexivity. + Qed. + + Lemma in_dom_delta : (m,m':(Map A)) (a:ad) + (in_dom a (MapDelta A m m'))=(xorb (in_dom a m) (in_dom a m')). + Proof. + Unfold in_dom. Intros. Rewrite (MapDelta_semantics A m m' a). + Elim (option_sum A (MapGet A m' a)). Intro H. Elim H. Intros y H0. Rewrite H0. + Case (MapGet A m a); Reflexivity. + Intro H. Rewrite H. Case (MapGet A m a); Reflexivity. + Qed. + +End Dom. + +Section InDom. + + Variable A, B : Set. + + Lemma in_dom_restrto : (m:(Map A)) (m':(Map B)) (a:ad) + (in_dom A a (MapDomRestrTo A B m m'))=(andb (in_dom A a m) (in_dom B a m')). + Proof. + Unfold in_dom. Intros. Rewrite (MapDomRestrTo_semantics A B m m' a). + Elim (option_sum B (MapGet B m' a)). Intro H. Elim H. Intros y H0. Rewrite H0. + Rewrite andb_b_true. Reflexivity. + Intro H. Rewrite H. Rewrite andb_b_false. Reflexivity. + Qed. + + Lemma in_dom_restrby : (m:(Map A)) (m':(Map B)) (a:ad) + (in_dom A a (MapDomRestrBy A B m m'))=(andb (in_dom A a m) (negb (in_dom B a m'))). + Proof. + Unfold in_dom. Intros. Rewrite (MapDomRestrBy_semantics A B m m' a). + Elim (option_sum B (MapGet B m' a)). Intro H. Elim H. Intros y H0. Rewrite H0. + Unfold negb. Rewrite andb_b_false. Reflexivity. + Intro H. Rewrite H. Unfold negb. Rewrite andb_b_true. Reflexivity. + Qed. + +End InDom. + +Definition FSet := (Map unit). + +Section FSetDefs. + + Variable A : Set. + + Definition in_FSet : ad -> FSet -> bool := (in_dom unit). + + Fixpoint MapDom [m:(Map A)] : FSet := + Cases m of + M0 => (M0 unit) + | (M1 a _) => (M1 unit a tt) + | (M2 m m') => (M2 unit (MapDom m) (MapDom m')) + end. + + Lemma MapDom_semantics_1 : (m:(Map A)) (a:ad) + (y:A) (MapGet A m a)=(SOME A y) -> (in_FSet a (MapDom m))=true. + Proof. + Induction m. Intros. Discriminate H. + Unfold MapDom. Unfold in_FSet. Unfold in_dom. Unfold MapGet. Intros a y a0 y0. + Case (ad_eq a a0). Trivial. + Intro. Discriminate H. + Intros m0 H m1 H0 a y. Rewrite (MapGet_M2_bit_0_if A m0 m1 a). Simpl. Unfold in_FSet. + Unfold in_dom. Rewrite (MapGet_M2_bit_0_if unit (MapDom m0) (MapDom m1) a). + Case (ad_bit_0 a). Unfold in_FSet in_dom in H0. Intro. Apply H0 with y:=y. Assumption. + Unfold in_FSet in_dom in H. Intro. Apply H with y:=y. Assumption. + Qed. + + Lemma MapDom_semantics_2 : (m:(Map A)) (a:ad) + (in_FSet a (MapDom m))=true -> {y:A | (MapGet A m a)=(SOME A y)}. + Proof. + Induction m. Intros. Discriminate H. + Unfold MapDom. Unfold in_FSet. Unfold in_dom. Unfold MapGet. Intros a y a0. Case (ad_eq a a0). + Intro. Split with y. Reflexivity. + Intro. Discriminate H. + Intros m0 H m1 H0 a. Rewrite (MapGet_M2_bit_0_if A m0 m1 a). Simpl. Unfold in_FSet. + Unfold in_dom. Rewrite (MapGet_M2_bit_0_if unit (MapDom m0) (MapDom m1) a). + Case (ad_bit_0 a). Unfold in_FSet in_dom in H0. Intro. Apply H0. Assumption. + Unfold in_FSet in_dom in H. Intro. Apply H. Assumption. + Qed. + + Lemma MapDom_semantics_3 : (m:(Map A)) (a:ad) + (MapGet A m a)=(NONE A) -> (in_FSet a (MapDom m))=false. + Proof. + Intros. Elim (sumbool_of_bool (in_FSet a (MapDom m))). Intro H0. + Elim (MapDom_semantics_2 m a H0). Intros y H1. Rewrite H in H1. Discriminate H1. + Trivial. + Qed. + + Lemma MapDom_semantics_4 : (m:(Map A)) (a:ad) + (in_FSet a (MapDom m))=false -> (MapGet A m a)=(NONE A). + Proof. + Intros. Elim (option_sum A (MapGet A m a)). Intro H0. Elim H0. Intros y H1. + Rewrite (MapDom_semantics_1 m a y H1) in H. Discriminate H. + Trivial. + Qed. + + Lemma MapDom_Dom : (m:(Map A)) (a:ad) (in_dom A a m)=(in_FSet a (MapDom m)). + Proof. + Intros. Elim (sumbool_of_bool (in_FSet a (MapDom m))). Intro H. + Elim (MapDom_semantics_2 m a H). Intros y H0. Rewrite H. Unfold in_dom. Rewrite H0. + Reflexivity. + Intro H. Rewrite H. Unfold in_dom. Rewrite (MapDom_semantics_4 m a H). Reflexivity. + Qed. + + Definition FSetUnion : FSet -> FSet -> FSet := [s,s':FSet] (MapMerge unit s s'). + + Lemma in_FSet_union : (s,s':FSet) (a:ad) + (in_FSet a (FSetUnion s s'))=(orb (in_FSet a s) (in_FSet a s')). + Proof. + Exact (in_dom_merge unit). + Qed. + + Definition FSetInter : FSet -> FSet -> FSet := [s,s':FSet] (MapDomRestrTo unit unit s s'). + + Lemma in_FSet_inter : (s,s':FSet) (a:ad) + (in_FSet a (FSetInter s s'))=(andb (in_FSet a s) (in_FSet a s')). + Proof. + Exact (in_dom_restrto unit unit). + Qed. + + Definition FSetDiff : FSet -> FSet -> FSet := [s,s':FSet] (MapDomRestrBy unit unit s s'). + + Lemma in_FSet_diff : (s,s':FSet) (a:ad) + (in_FSet a (FSetDiff s s'))=(andb (in_FSet a s) (negb (in_FSet a s'))). + Proof. + Exact (in_dom_restrby unit unit). + Qed. + + Definition FSetDelta : FSet -> FSet -> FSet := [s,s':FSet] (MapDelta unit s s'). + + Lemma in_FSet_delta : (s,s':FSet) (a:ad) + (in_FSet a (FSetDelta s s'))=(xorb (in_FSet a s) (in_FSet a s')). + Proof. + Exact (in_dom_delta unit). + Qed. + +End FSetDefs. + +Lemma FSet_Dom : (s:FSet) (MapDom unit s)=s. +Proof. + Induction s. Trivial. + Simpl. Intros a t. Elim t. Reflexivity. + Intros. Simpl. Rewrite H. Rewrite H0. Reflexivity. +Qed. diff --git a/theories7/IntMap/Lsort.v b/theories7/IntMap/Lsort.v new file mode 100644 index 00000000..31b71c62 --- /dev/null +++ b/theories7/IntMap/Lsort.v @@ -0,0 +1,537 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Lsort.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require Arith. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require PolyList. +Require Mapiter. + +Section LSort. + + Variable A : Set. + + Fixpoint ad_less_1 [a,a':ad; p:positive] : bool := + Cases p of + (xO p') => (ad_less_1 (ad_div_2 a) (ad_div_2 a') p') + | _ => (andb (negb (ad_bit_0 a)) (ad_bit_0 a')) + end. + + Definition ad_less := [a,a':ad] Cases (ad_xor a a') of + ad_z => false + | (ad_x p) => (ad_less_1 a a' p) + end. + + Lemma ad_bit_0_less : (a,a':ad) (ad_bit_0 a)=false -> (ad_bit_0 a')=true -> + (ad_less a a')=true. + Proof. + Intros. Elim (ad_sum (ad_xor a a')). Intro H1. Elim H1. Intros p H2. Unfold ad_less. + Rewrite H2. Generalize H2. Elim p. Intros. Simpl. Rewrite H. Rewrite H0. Reflexivity. + Intros. Cut (ad_bit_0 (ad_xor a a'))=false. Intro. Rewrite (ad_xor_bit_0 a a') in H5. + Rewrite H in H5. Rewrite H0 in H5. Discriminate H5. + Rewrite H4. Reflexivity. + Intro. Simpl. Rewrite H. Rewrite H0. Reflexivity. + Intro H1. Cut (ad_bit_0 (ad_xor a a'))=false. Intro. Rewrite (ad_xor_bit_0 a a') in H2. + Rewrite H in H2. Rewrite H0 in H2. Discriminate H2. + Rewrite H1. Reflexivity. + Qed. + + Lemma ad_bit_0_gt : (a,a':ad) (ad_bit_0 a)=true -> (ad_bit_0 a')=false -> + (ad_less a a')=false. + Proof. + Intros. Elim (ad_sum (ad_xor a a')). Intro H1. Elim H1. Intros p H2. Unfold ad_less. + Rewrite H2. Generalize H2. Elim p. Intros. Simpl. Rewrite H. Rewrite H0. Reflexivity. + Intros. Cut (ad_bit_0 (ad_xor a a'))=false. Intro. Rewrite (ad_xor_bit_0 a a') in H5. + Rewrite H in H5. Rewrite H0 in H5. Discriminate H5. + Rewrite H4. Reflexivity. + Intro. Simpl. Rewrite H. Rewrite H0. Reflexivity. + Intro H1. Unfold ad_less. Rewrite H1. Reflexivity. + Qed. + + Lemma ad_less_not_refl : (a:ad) (ad_less a a)=false. + Proof. + Intro. Unfold ad_less. Rewrite (ad_xor_nilpotent a). Reflexivity. + Qed. + + Lemma ad_ind_double : + (a:ad)(P:ad->Prop) (P ad_z) -> + ((a:ad) (P a) -> (P (ad_double a))) -> + ((a:ad) (P a) -> (P (ad_double_plus_un a))) -> (P a). + Proof. + Intros; Elim a. Trivial. + Induction p. Intros. + Apply (H1 (ad_x p0)); Trivial. + Intros; Apply (H0 (ad_x p0)); Trivial. + Intros; Apply (H1 ad_z); Assumption. + Qed. + + Lemma ad_rec_double : + (a:ad)(P:ad->Set) (P ad_z) -> + ((a:ad) (P a) -> (P (ad_double a))) -> + ((a:ad) (P a) -> (P (ad_double_plus_un a))) -> (P a). + Proof. + Intros; Elim a. Trivial. + Induction p. Intros. + Apply (H1 (ad_x p0)); Trivial. + Intros; Apply (H0 (ad_x p0)); Trivial. + Intros; Apply (H1 ad_z); Assumption. + Qed. + + Lemma ad_less_def_1 : (a,a':ad) (ad_less (ad_double a) (ad_double a'))=(ad_less a a'). + Proof. + Induction a. Induction a'. Reflexivity. + Trivial. + Induction a'. Unfold ad_less. Simpl. (Elim p; Trivial). + Unfold ad_less. Simpl. Intro. Case (p_xor p p0). Reflexivity. + Trivial. + Qed. + + Lemma ad_less_def_2 : (a,a':ad) + (ad_less (ad_double_plus_un a) (ad_double_plus_un a'))=(ad_less a a'). + Proof. + Induction a. Induction a'. Reflexivity. + Trivial. + Induction a'. Unfold ad_less. Simpl. (Elim p; Trivial). + Unfold ad_less. Simpl. Intro. Case (p_xor p p0). Reflexivity. + Trivial. + Qed. + + Lemma ad_less_def_3 : (a,a':ad) (ad_less (ad_double a) (ad_double_plus_un a'))=true. + Proof. + Intros. Apply ad_bit_0_less. Apply ad_double_bit_0. + Apply ad_double_plus_un_bit_0. + Qed. + + Lemma ad_less_def_4 : (a,a':ad) (ad_less (ad_double_plus_un a) (ad_double a'))=false. + Proof. + Intros. Apply ad_bit_0_gt. Apply ad_double_plus_un_bit_0. + Apply ad_double_bit_0. + Qed. + + Lemma ad_less_z : (a:ad) (ad_less a ad_z)=false. + Proof. + Induction a. Reflexivity. + Unfold ad_less. Intro. Rewrite (ad_xor_neutral_right (ad_x p)). (Elim p; Trivial). + Qed. + + Lemma ad_z_less_1 : (a:ad) (ad_less ad_z a)=true -> {p:positive | a=(ad_x p)}. + Proof. + Induction a. Intro. Discriminate H. + Intros. Split with p. Reflexivity. + Qed. + + Lemma ad_z_less_2 : (a:ad) (ad_less ad_z a)=false -> a=ad_z. + Proof. + Induction a. Trivial. + Unfold ad_less. Simpl. Cut (p:positive)(ad_less_1 ad_z (ad_x p) p)=false->False. + Intros. Elim (H p H0). + Induction p. Intros. Discriminate H0. + Intros. Exact (H H0). + Intro. Discriminate H. + Qed. + + Lemma ad_less_trans : (a,a',a'':ad) + (ad_less a a')=true -> (ad_less a' a'')=true -> (ad_less a a'')=true. + Proof. + Intro a. Apply ad_ind_double with P:=[a:ad] + (a',a'':ad) + (ad_less a a')=true + ->(ad_less a' a'')=true->(ad_less a a'')=true. + Intros. Elim (sumbool_of_bool (ad_less ad_z a'')). Trivial. + Intro H1. Rewrite (ad_z_less_2 a'' H1) in H0. Rewrite (ad_less_z a') in H0. Discriminate H0. + Intros a0 H a'. Apply ad_ind_double with P:=[a':ad] + (a'':ad) + (ad_less (ad_double a0) a')=true + ->(ad_less a' a'')=true->(ad_less (ad_double a0) a'')=true. + Intros. Rewrite (ad_less_z (ad_double a0)) in H0. Discriminate H0. + Intros a1 H0 a'' H1. Rewrite (ad_less_def_1 a0 a1) in H1. + Apply ad_ind_double with P:=[a'':ad] + (ad_less (ad_double a1) a'')=true + ->(ad_less (ad_double a0) a'')=true. + Intro. Rewrite (ad_less_z (ad_double a1)) in H2. Discriminate H2. + Intros. Rewrite (ad_less_def_1 a1 a2) in H3. Rewrite (ad_less_def_1 a0 a2). + Exact (H a1 a2 H1 H3). + Intros. Apply ad_less_def_3. + Intros a1 H0 a'' H1. Apply ad_ind_double with P:=[a'':ad] + (ad_less (ad_double_plus_un a1) a'')=true + ->(ad_less (ad_double a0) a'')=true. + Intro. Rewrite (ad_less_z (ad_double_plus_un a1)) in H2. Discriminate H2. + Intros. Rewrite (ad_less_def_4 a1 a2) in H3. Discriminate H3. + Intros. Apply ad_less_def_3. + Intros a0 H a'. Apply ad_ind_double with P:=[a':ad] + (a'':ad) + (ad_less (ad_double_plus_un a0) a')=true + ->(ad_less a' a'')=true + ->(ad_less (ad_double_plus_un a0) a'')=true. + Intros. Rewrite (ad_less_z (ad_double_plus_un a0)) in H0. Discriminate H0. + Intros. Rewrite (ad_less_def_4 a0 a1) in H1. Discriminate H1. + Intros a1 H0 a'' H1. Apply ad_ind_double with P:=[a'':ad] + (ad_less (ad_double_plus_un a1) a'')=true + ->(ad_less (ad_double_plus_un a0) a'')=true. + Intro. Rewrite (ad_less_z (ad_double_plus_un a1)) in H2. Discriminate H2. + Intros. Rewrite (ad_less_def_4 a1 a2) in H3. Discriminate H3. + Rewrite (ad_less_def_2 a0 a1) in H1. Intros. Rewrite (ad_less_def_2 a1 a2) in H3. + Rewrite (ad_less_def_2 a0 a2). Exact (H a1 a2 H1 H3). + Qed. + + Fixpoint alist_sorted [l:(alist A)] : bool := + Cases l of + nil => true + | (cons (a, _) l') => Cases l' of + nil => true + | (cons (a', y') l'') => (andb (ad_less a a') + (alist_sorted l')) + end + end. + + Fixpoint alist_nth_ad [n:nat; l:(alist A)] : ad := + Cases l of + nil => ad_z (* dummy *) + | (cons (a, y) l') => Cases n of + O => a + | (S n') => (alist_nth_ad n' l') + end + end. + + Definition alist_sorted_1 := [l:(alist A)] + (n:nat) (le (S (S n)) (length l)) -> + (ad_less (alist_nth_ad n l) (alist_nth_ad (S n) l))=true. + + Lemma alist_sorted_imp_1 : (l:(alist A)) (alist_sorted l)=true -> (alist_sorted_1 l). + Proof. + Unfold alist_sorted_1. Induction l. Intros. Elim (le_Sn_O (S n) H0). + Intro r. Elim r. Intros a y. Induction l0. Intros. Simpl in H1. + Elim (le_Sn_O n (le_S_n (S n) O H1)). + Intro r0. Elim r0. Intros a0 y0. Induction n. Intros. Simpl. Simpl in H1. + Exact (proj1 ? ? (andb_prop ? ? H1)). + Intros. Change (ad_less (alist_nth_ad n0 (cons (a0,y0) l1)) + (alist_nth_ad (S n0) (cons (a0,y0) l1)))=true. + Apply H0. Exact (proj2 ? ? (andb_prop ? ? H1)). + Apply le_S_n. Exact H3. + Qed. + + Definition alist_sorted_2 := [l:(alist A)] + (m,n:nat) (lt m n) -> (le (S n) (length l)) -> + (ad_less (alist_nth_ad m l) (alist_nth_ad n l))=true. + + Lemma alist_sorted_1_imp_2 : (l:(alist A)) (alist_sorted_1 l) -> (alist_sorted_2 l). + Proof. + Unfold alist_sorted_1 alist_sorted_2 lt. Intros l H m n H0. Elim H0. Exact (H m). + Intros. Apply ad_less_trans with a':=(alist_nth_ad m0 l). Apply H2. Apply le_trans_S. + Assumption. + Apply H. Assumption. + Qed. + + Lemma alist_sorted_2_imp : (l:(alist A)) (alist_sorted_2 l) -> (alist_sorted l)=true. + Proof. + Unfold alist_sorted_2 lt. Induction l. Trivial. + Intro r. Elim r. Intros a y. Induction l0. Trivial. + Intro r0. Elim r0. Intros a0 y0. Intros. + Change (andb (ad_less a a0) (alist_sorted (cons (a0,y0) l1)))=true. + Apply andb_true_intro. Split. Apply (H1 (0) (1)). Apply le_n. + Simpl. Apply le_n_S. Apply le_n_S. Apply le_O_n. + Apply H0. Intros. Apply (H1 (S m) (S n)). Apply le_n_S. Assumption. + Exact (le_n_S ? ? H3). + Qed. + + Lemma app_length : (C:Set) (l,l':(list C)) (length (app l l'))=(plus (length l) (length l')). + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite (H l'). Reflexivity. + Qed. + + Lemma aapp_length : (l,l':(alist A)) (length (aapp A l l'))=(plus (length l) (length l')). + Proof. + Exact (app_length ad*A). + Qed. + + Lemma alist_nth_ad_aapp_1 : (l,l':(alist A)) (n:nat) + (le (S n) (length l)) -> (alist_nth_ad n (aapp A l l'))=(alist_nth_ad n l). + Proof. + Induction l. Intros. Elim (le_Sn_O n H). + Intro r. Elim r. Intros a y l' H l''. Induction n. Trivial. + Intros. Simpl. Apply H. Apply le_S_n. Exact H1. + Qed. + + Lemma alist_nth_ad_aapp_2 : (l,l':(alist A)) (n:nat) + (le (S n) (length l')) -> + (alist_nth_ad (plus (length l) n) (aapp A l l'))=(alist_nth_ad n l'). + Proof. + Induction l. Trivial. + Intro r. Elim r. Intros a y l' H l'' n H0. Simpl. Apply H. Exact H0. + Qed. + + Lemma interval_split : (p,q,n:nat) (le (S n) (plus p q)) -> + {n' : nat | (le (S n') q) /\ n=(plus p n')}+{(le (S n) p)}. + Proof. + Induction p. Simpl. Intros. Left . Split with n. (Split; [ Assumption | Reflexivity ]). + Intros p' H q. Induction n. Intros. Right . Apply le_n_S. Apply le_O_n. + Intros. Elim (H ? ? (le_S_n ? ? H1)). Intro H2. Left . Elim H2. Intros n' H3. + Elim H3. Intros H4 H5. Split with n'. (Split; [ Assumption | Rewrite H5; Reflexivity ]). + Intro H2. Right . Apply le_n_S. Assumption. + Qed. + + Lemma alist_conc_sorted : (l,l':(alist A)) (alist_sorted_2 l) -> (alist_sorted_2 l') -> + ((n,n':nat) (le (S n) (length l)) -> (le (S n') (length l')) -> + (ad_less (alist_nth_ad n l) (alist_nth_ad n' l'))=true) -> + (alist_sorted_2 (aapp A l l')). + Proof. + Unfold alist_sorted_2 lt. Intros. Rewrite (aapp_length l l') in H3. + Elim (interval_split (length l) (length l') m + (le_trans ? ? ? (le_n_S ? ? (lt_le_weak m n H2)) H3)). + Intro H4. Elim H4. Intros m' H5. Elim H5. Intros. Rewrite H7. + Rewrite (alist_nth_ad_aapp_2 l l' m' H6). Elim (interval_split (length l) (length l') n H3). + Intro H8. Elim H8. Intros n' H9. Elim H9. Intros. Rewrite H11. + Rewrite (alist_nth_ad_aapp_2 l l' n' H10). Apply H0. Rewrite H7 in H2. Rewrite H11 in H2. + Change (le (plus (S (length l)) m') (plus (length l) n')) in H2. + Rewrite (plus_Snm_nSm (length l) m') in H2. Exact (simpl_le_plus_l (length l) (S m') n' H2). + Exact H10. + Intro H8. Rewrite H7 in H2. Cut (le (S (length l)) (length l)). Intros. Elim (le_Sn_n ? H9). + Apply le_trans with m:=(S n). Apply le_n_S. Apply le_trans with m:=(S (plus (length l) m')). + Apply le_trans with m:=(plus (length l) m'). Apply le_plus_l. + Apply le_n_Sn. + Exact H2. + Exact H8. + Intro H4. Rewrite (alist_nth_ad_aapp_1 l l' m H4). + Elim (interval_split (length l) (length l') n H3). Intro H5. Elim H5. Intros n' H6. Elim H6. + Intros. Rewrite H8. Rewrite (alist_nth_ad_aapp_2 l l' n' H7). Exact (H1 m n' H4 H7). + Intro H5. Rewrite (alist_nth_ad_aapp_1 l l' n H5). Exact (H m n H2 H5). + Qed. + + Lemma alist_nth_ad_semantics : (l:(alist A)) (n:nat) (le (S n) (length l)) -> + {y:A | (alist_semantics A l (alist_nth_ad n l))=(SOME A y)}. + Proof. + Induction l. Intros. Elim (le_Sn_O ? H). + Intro r. Elim r. Intros a y l0 H. Induction n. Simpl. Intro. Split with y. + Rewrite (ad_eq_correct a). Reflexivity. + Intros. Elim (H ? (le_S_n ? ? H1)). Intros y0 H2. + Elim (sumbool_of_bool (ad_eq a (alist_nth_ad n0 l0))). Intro H3. Split with y. + Rewrite (ad_eq_complete ? ? H3). Simpl. Rewrite (ad_eq_correct (alist_nth_ad n0 l0)). + Reflexivity. + Intro H3. Split with y0. Simpl. Rewrite H3. Assumption. + Qed. + + Lemma alist_of_Map_nth_ad : (m:(Map A)) (pf:ad->ad) + (l:(alist A)) l=(MapFold1 A (alist A) (anil A) (aapp A) + [a0:ad][y:A](acons A (a0,y) (anil A)) pf m) -> + (n:nat) (le (S n) (length l)) -> {a':ad | (alist_nth_ad n l)=(pf a')}. + Proof. + Intros. Elim (alist_nth_ad_semantics l n H0). Intros y H1. + Apply (alist_of_Map_semantics_1_1 A m pf (alist_nth_ad n l) y). + Rewrite <- H. Assumption. + Qed. + + Definition ad_monotonic := [pf:ad->ad] (a,a':ad) + (ad_less a a')=true -> (ad_less (pf a) (pf a'))=true. + + Lemma ad_double_monotonic : (ad_monotonic ad_double). + Proof. + Unfold ad_monotonic. Intros. Rewrite ad_less_def_1. Assumption. + Qed. + + Lemma ad_double_plus_un_monotonic : (ad_monotonic ad_double_plus_un). + Proof. + Unfold ad_monotonic. Intros. Rewrite ad_less_def_2. Assumption. + Qed. + + Lemma ad_comp_monotonic : (pf,pf':ad->ad) (ad_monotonic pf) -> (ad_monotonic pf') -> + (ad_monotonic [a0:ad] (pf (pf' a0))). + Proof. + Unfold ad_monotonic. Intros. Apply H. Apply H0. Exact H1. + Qed. + + Lemma ad_comp_double_monotonic : (pf:ad->ad) (ad_monotonic pf) -> + (ad_monotonic [a0:ad] (pf (ad_double a0))). + Proof. + Intros. Apply ad_comp_monotonic. Assumption. + Exact ad_double_monotonic. + Qed. + + Lemma ad_comp_double_plus_un_monotonic : (pf:ad->ad) (ad_monotonic pf) -> + (ad_monotonic [a0:ad] (pf (ad_double_plus_un a0))). + Proof. + Intros. Apply ad_comp_monotonic. Assumption. + Exact ad_double_plus_un_monotonic. + Qed. + + Lemma alist_of_Map_sorts_1 : (m:(Map A)) (pf:ad->ad) (ad_monotonic pf) -> + (alist_sorted_2 (MapFold1 A (alist A) (anil A) (aapp A) + [a:ad][y:A](acons A (a,y) (anil A)) pf m)). + Proof. + Induction m. Simpl. Intros. Apply alist_sorted_1_imp_2. Apply alist_sorted_imp_1. Reflexivity. + Intros. Simpl. Apply alist_sorted_1_imp_2. Apply alist_sorted_imp_1. Reflexivity. + Intros. Simpl. Apply alist_conc_sorted. + Exact (H [a0:ad](pf (ad_double a0)) (ad_comp_double_monotonic pf H1)). + Exact (H0 [a0:ad](pf (ad_double_plus_un a0)) (ad_comp_double_plus_un_monotonic pf H1)). + Intros. Elim (alist_of_Map_nth_ad m0 [a0:ad](pf (ad_double a0)) + (MapFold1 A (alist A) (anil A) (aapp A) + [a0:ad][y:A](acons A (a0,y) (anil A)) + [a0:ad](pf (ad_double a0)) m0) (refl_equal ? ?) n H2). + Intros a H4. Rewrite H4. Elim (alist_of_Map_nth_ad m1 [a0:ad](pf (ad_double_plus_un a0)) + (MapFold1 A (alist A) (anil A) (aapp A) + [a0:ad][y:A](acons A (a0,y) (anil A)) + [a0:ad](pf (ad_double_plus_un a0)) m1) (refl_equal ? ?) n' H3). + Intros a' H5. Rewrite H5. Unfold ad_monotonic in H1. Apply H1. Apply ad_less_def_3. + Qed. + + Lemma alist_of_Map_sorts : (m:(Map A)) (alist_sorted (alist_of_Map A m))=true. + Proof. + Intro. Apply alist_sorted_2_imp. + Exact (alist_of_Map_sorts_1 m [a0:ad]a0 [a,a':ad][p:(ad_less a a')=true]p). + Qed. + + Lemma alist_of_Map_sorts1 : (m:(Map A)) (alist_sorted_1 (alist_of_Map A m)). + Proof. + Intro. Apply alist_sorted_imp_1. Apply alist_of_Map_sorts. + Qed. + + Lemma alist_of_Map_sorts2 : (m:(Map A)) (alist_sorted_2 (alist_of_Map A m)). + Proof. + Intro. Apply alist_sorted_1_imp_2. Apply alist_of_Map_sorts1. + Qed. + + Lemma ad_less_total : (a,a':ad) {(ad_less a a')=true}+{(ad_less a' a)=true}+{a=a'}. + Proof. + Intro a. Refine (ad_rec_double a [a:ad] (a':ad){(ad_less a a')=true}+{(ad_less a' a)=true}+{a=a'} + ? ? ?). + Intro. Elim (sumbool_of_bool (ad_less ad_z a')). Intro H. Left . Left . Assumption. + Intro H. Right . Rewrite (ad_z_less_2 a' H). Reflexivity. + Intros a0 H a'. Refine (ad_rec_double a' [a':ad] {(ad_less (ad_double a0) a')=true} + +{(ad_less a' (ad_double a0))=true}+{(ad_double a0)=a'} ? ? ?). + Elim (sumbool_of_bool (ad_less ad_z (ad_double a0))). Intro H0. Left . Right . Assumption. + Intro H0. Right . Exact (ad_z_less_2 ? H0). + Intros a1 H0. Rewrite ad_less_def_1. Rewrite ad_less_def_1. Elim (H a1). Intro H1. + Left . Assumption. + Intro H1. Right . Rewrite H1. Reflexivity. + Intros a1 H0. Left . Left . Apply ad_less_def_3. + Intros a0 H a'. Refine (ad_rec_double a' [a':ad] {(ad_less (ad_double_plus_un a0) a')=true} + +{(ad_less a' (ad_double_plus_un a0))=true} + +{(ad_double_plus_un a0)=a'} ? ? ?). + Left . Right . (Case a0; Reflexivity). + Intros a1 H0. Left . Right . Apply ad_less_def_3. + Intros a1 H0. Rewrite ad_less_def_2. Rewrite ad_less_def_2. Elim (H a1). Intro H1. + Left . Assumption. + Intro H1. Right . Rewrite H1. Reflexivity. + Qed. + + Lemma alist_too_low : (l:(alist A)) (a,a':ad) (y:A) + (ad_less a a')=true -> (alist_sorted_2 (cons (a',y) l)) -> + (alist_semantics A (cons (a',y) l) a)=(NONE A). + Proof. + Induction l. Intros. Simpl. Elim (sumbool_of_bool (ad_eq a' a)). Intro H1. + Rewrite (ad_eq_complete ? ? H1) in H. Rewrite (ad_less_not_refl a) in H. Discriminate H. + Intro H1. Rewrite H1. Reflexivity. + Intro r. Elim r. Intros a y l0 H a0 a1 y0 H0 H1. + Change (Case (ad_eq a1 a0) of + (SOME A y0) + (alist_semantics A (cons (a,y) l0) a0) + end)=(NONE A). + Elim (sumbool_of_bool (ad_eq a1 a0)). Intro H2. Rewrite (ad_eq_complete ? ? H2) in H0. + Rewrite (ad_less_not_refl a0) in H0. Discriminate H0. + Intro H2. Rewrite H2. Apply H. Apply ad_less_trans with a':=a1. Assumption. + Unfold alist_sorted_2 in H1. Apply (H1 (0) (1)). Apply lt_n_Sn. + Simpl. Apply le_n_S. Apply le_n_S. Apply le_O_n. + Apply alist_sorted_1_imp_2. Apply alist_sorted_imp_1. + Cut (alist_sorted (cons (a1,y0) (cons (a,y) l0)))=true. Intro H3. + Exact (proj2 ? ? (andb_prop ? ? H3)). + Apply alist_sorted_2_imp. Assumption. + Qed. + + Lemma alist_semantics_nth_ad : (l:(alist A)) (a:ad) (y:A) + (alist_semantics A l a)=(SOME A y) -> + {n:nat | (le (S n) (length l)) /\ (alist_nth_ad n l)=a}. + Proof. + Induction l. Intros. Discriminate H. + Intro r. Elim r. Intros a y l0 H a0 y0 H0. Simpl in H0. Elim (sumbool_of_bool (ad_eq a a0)). + Intro H1. Rewrite H1 in H0. Split with O. Split. Simpl. Apply le_n_S. Apply le_O_n. + Simpl. Exact (ad_eq_complete ? ? H1). + Intro H1. Rewrite H1 in H0. Elim (H a0 y0 H0). Intros n' H2. Split with (S n'). Split. + Simpl. Apply le_n_S. Exact (proj1 ? ? H2). + Exact (proj2 ? ? H2). + Qed. + + Lemma alist_semantics_tail : (l:(alist A)) (a:ad) (y:A) + (alist_sorted_2 (cons (a,y) l)) -> + (eqm A (alist_semantics A l) [a0:ad] if (ad_eq a a0) + then (NONE A) + else (alist_semantics A (cons (a,y) l) a0)). + Proof. + Unfold eqm. Intros. Elim (sumbool_of_bool (ad_eq a a0)). Intro H0. Rewrite H0. + Rewrite <- (ad_eq_complete ? ? H0). Unfold alist_sorted_2 in H. + Elim (option_sum A (alist_semantics A l a)). Intro H1. Elim H1. Intros y0 H2. + Elim (alist_semantics_nth_ad l a y0 H2). Intros n H3. Elim H3. Intros. + Cut (ad_less (alist_nth_ad (0) (cons (a,y) l)) (alist_nth_ad (S n) (cons (a,y) l)))=true. + Intro. Simpl in H6. Rewrite H5 in H6. Rewrite (ad_less_not_refl a) in H6. Discriminate H6. + Apply H. Apply lt_O_Sn. + Simpl. Apply le_n_S. Assumption. + Trivial. + Intro H0. Simpl. Rewrite H0. Reflexivity. + Qed. + + Lemma alist_semantics_same_tail : (l,l':(alist A)) (a:ad) (y:A) + (alist_sorted_2 (cons (a,y) l)) -> (alist_sorted_2 (cons (a,y) l')) -> + (eqm A (alist_semantics A (cons (a,y) l)) (alist_semantics A (cons (a,y) l'))) -> + (eqm A (alist_semantics A l) (alist_semantics A l')). + Proof. + Unfold eqm. Intros. Rewrite (alist_semantics_tail ? ? ? H a0). + Rewrite (alist_semantics_tail ? ? ? H0 a0). Case (ad_eq a a0). Reflexivity. + Exact (H1 a0). + Qed. + + Lemma alist_sorted_tail : (l:(alist A)) (a:ad) (y:A) + (alist_sorted_2 (cons (a,y) l)) -> (alist_sorted_2 l). + Proof. + Unfold alist_sorted_2. Intros. Apply (H (S m) (S n)). Apply lt_n_S. Assumption. + Simpl. Apply le_n_S. Assumption. + Qed. + + Lemma alist_canonical : (l,l':(alist A)) + (eqm A (alist_semantics A l) (alist_semantics A l')) -> + (alist_sorted_2 l) -> (alist_sorted_2 l') -> l=l'. + Proof. + Unfold eqm. Induction l. Induction l'. Trivial. + Intro r. Elim r. Intros a y l0 H H0 H1 H2. Simpl in H0. + Cut (NONE A)=(Case (ad_eq a a) of (SOME A y) + (alist_semantics A l0 a) + end). + Rewrite (ad_eq_correct a). Intro. Discriminate H3. + Exact (H0 a). + Intro r. Elim r. Intros a y l0 H. Induction l'. Intros. Simpl in H0. + Cut (Case (ad_eq a a) of (SOME A y) + (alist_semantics A l0 a) + end)=(NONE A). + Rewrite (ad_eq_correct a). Intro. Discriminate H3. + Exact (H0 a). + Intro r'. Elim r'. Intros a' y' l'0 H0 H1 H2 H3. Elim (ad_less_total a a'). Intro H4. + Elim H4. Intro H5. + Cut (alist_semantics A (cons (a,y) l0) a)=(alist_semantics A (cons (a',y') l'0) a). + Intro. Rewrite (alist_too_low l'0 a a' y' H5 H3) in H6. Simpl in H6. + Rewrite (ad_eq_correct a) in H6. Discriminate H6. + Exact (H1 a). + Intro H5. Cut (alist_semantics A (cons (a,y) l0) a')=(alist_semantics A (cons (a',y') l'0) a'). + Intro. Rewrite (alist_too_low l0 a' a y H5 H2) in H6. Simpl in H6. + Rewrite (ad_eq_correct a') in H6. Discriminate H6. + Exact (H1 a'). + Intro H4. Rewrite H4. + Cut (alist_semantics A (cons (a,y) l0) a)=(alist_semantics A (cons (a',y') l'0) a). + Intro. Simpl in H5. Rewrite H4 in H5. Rewrite (ad_eq_correct a') in H5. Inversion H5. + Rewrite H4 in H1. Rewrite H7 in H1. Cut l0=l'0. Intro. Rewrite H6. Reflexivity. + Apply H. Rewrite H4 in H2. Rewrite H7 in H2. + Exact (alist_semantics_same_tail l0 l'0 a' y' H2 H3 H1). + Exact (alist_sorted_tail ? ? ? H2). + Exact (alist_sorted_tail ? ? ? H3). + Exact (H1 a). + Qed. + +End LSort. diff --git a/theories7/IntMap/Map.v b/theories7/IntMap/Map.v new file mode 100644 index 00000000..00ba3f8a --- /dev/null +++ b/theories7/IntMap/Map.v @@ -0,0 +1,786 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Map.v,v 1.1.2.1 2004/07/16 19:31:27 herbelin Exp $ i*) + +(** Definition of finite sets as trees indexed by adresses *) + +Require Bool. +Require Sumbool. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. + + +Section MapDefs. + +(** We define maps from ad to A. *) + Variable A : Set. + + Inductive Map : Set := + M0 : Map + | M1 : ad -> A -> Map + | M2 : Map -> Map -> Map. + + Inductive option : Set := + NONE : option + | SOME : A -> option. + + Lemma option_sum : (o:option) {y:A | o=(SOME y)}+{o=NONE}. + Proof. + Induction o. Right . Reflexivity. + Left . Split with a. Reflexivity. + Qed. + + (** The semantics of maps is given by the function [MapGet]. + The semantics of a map [m] is a partial, finite function from + [ad] to [A]: *) + + Fixpoint MapGet [m:Map] : ad -> option := + Cases m of + M0 => [a:ad] NONE + | (M1 x y) => [a:ad] + if (ad_eq x a) + then (SOME y) + else NONE + | (M2 m1 m2) => [a:ad] + Cases a of + ad_z => (MapGet m1 ad_z) + | (ad_x xH) => (MapGet m2 ad_z) + | (ad_x (xO p)) => (MapGet m1 (ad_x p)) + | (ad_x (xI p)) => (MapGet m2 (ad_x p)) + end + end. + + Definition newMap := M0. + + Definition MapSingleton := M1. + + Definition eqm := [g,g':ad->option] (a:ad) (g a)=(g' a). + + Lemma newMap_semantics : (eqm (MapGet newMap) [a:ad] NONE). + Proof. + Simpl. Unfold eqm. Trivial. + Qed. + + Lemma MapSingleton_semantics : (a:ad) (y:A) + (eqm (MapGet (MapSingleton a y)) [a':ad] if (ad_eq a a') then (SOME y) else NONE). + Proof. + Simpl. Unfold eqm. Trivial. + Qed. + + Lemma M1_semantics_1 : (a:ad) (y:A) (MapGet (M1 a y) a)=(SOME y). + Proof. + Unfold MapGet. Intros. Rewrite (ad_eq_correct a). Reflexivity. + Qed. + + Lemma M1_semantics_2 : + (a,a':ad) (y:A) (ad_eq a a')=false -> (MapGet (M1 a y) a')=NONE. + Proof. + Intros. Simpl. Rewrite H. Reflexivity. + Qed. + + Lemma Map2_semantics_1 : + (m,m':Map) (eqm (MapGet m) [a:ad] (MapGet (M2 m m') (ad_double a))). + Proof. + Unfold eqm. Induction a; Trivial. + Qed. + + Lemma Map2_semantics_1_eq : (m,m':Map) (f:ad->option) (eqm (MapGet (M2 m m')) f) + -> (eqm (MapGet m) [a:ad] (f (ad_double a))). + Proof. + Unfold eqm. + Intros. + Rewrite <- (H (ad_double a)). + Exact (Map2_semantics_1 m m' a). + Qed. + + Lemma Map2_semantics_2 : + (m,m':Map) (eqm (MapGet m') [a:ad] (MapGet (M2 m m') (ad_double_plus_un a))). + Proof. + Unfold eqm. Induction a; Trivial. + Qed. + + Lemma Map2_semantics_2_eq : (m,m':Map) (f:ad->option) (eqm (MapGet (M2 m m')) f) + -> (eqm (MapGet m') [a:ad] (f (ad_double_plus_un a))). + Proof. + Unfold eqm. + Intros. + Rewrite <- (H (ad_double_plus_un a)). + Exact (Map2_semantics_2 m m' a). + Qed. + + Lemma MapGet_M2_bit_0_0 : (a:ad) (ad_bit_0 a)=false + -> (m,m':Map) (MapGet (M2 m m') a)=(MapGet m (ad_div_2 a)). + Proof. + Induction a; Trivial. Induction p. Intros. Discriminate H0. + Trivial. + Intros. Discriminate H. + Qed. + + Lemma MapGet_M2_bit_0_1 : (a:ad) (ad_bit_0 a)=true + -> (m,m':Map) (MapGet (M2 m m') a)=(MapGet m' (ad_div_2 a)). + Proof. + Induction a. Intros. Discriminate H. + Induction p. Trivial. + Intros. Discriminate H0. + Trivial. + Qed. + + Lemma MapGet_M2_bit_0_if : (m,m':Map) (a:ad) (MapGet (M2 m m') a)= + (if (ad_bit_0 a) then (MapGet m' (ad_div_2 a)) else (MapGet m (ad_div_2 a))). + Proof. + Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H. Rewrite H. + Apply MapGet_M2_bit_0_1; Assumption. + Intro H. Rewrite H. Apply MapGet_M2_bit_0_0; Assumption. + Qed. + + Lemma MapGet_M2_bit_0 : (m,m',m'':Map) + (a:ad) (if (ad_bit_0 a) then (MapGet (M2 m' m) a) else (MapGet (M2 m m'') a))= + (MapGet m (ad_div_2 a)). + Proof. + Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H. Rewrite H. + Apply MapGet_M2_bit_0_1; Assumption. + Intro H. Rewrite H. Apply MapGet_M2_bit_0_0; Assumption. + Qed. + + Lemma Map2_semantics_3 : (m,m':Map) (eqm (MapGet (M2 m m')) + [a:ad] Cases (ad_bit_0 a) of + false => (MapGet m (ad_div_2 a)) + | true => (MapGet m' (ad_div_2 a)) + end). + Proof. + Unfold eqm. + Induction a; Trivial. + Induction p; Trivial. + Qed. + + Lemma Map2_semantics_3_eq : (m,m':Map) (f,f':ad->option) + (eqm (MapGet m) f) -> (eqm (MapGet m') f') -> (eqm (MapGet (M2 m m')) + [a:ad] Cases (ad_bit_0 a) of + false => (f (ad_div_2 a)) + | true => (f' (ad_div_2 a)) + end). + Proof. + Unfold eqm. + Intros. + Rewrite <- (H (ad_div_2 a)). + Rewrite <- (H0 (ad_div_2 a)). + Exact (Map2_semantics_3 m m' a). + Qed. + + Fixpoint MapPut1 [a:ad; y:A; a':ad; y':A; p:positive] : Map := + Cases p of + (xO p') => let m = (MapPut1 (ad_div_2 a) y (ad_div_2 a') y' p') in + Cases (ad_bit_0 a) of + false => (M2 m M0) + | true => (M2 M0 m) + end + | _ => Cases (ad_bit_0 a) of + false => (M2 (M1 (ad_div_2 a) y) (M1 (ad_div_2 a') y')) + | true => (M2 (M1 (ad_div_2 a') y') (M1 (ad_div_2 a) y)) + end + end. + + Lemma MapGet_if_commute : (b:bool) (m,m':Map) (a:ad) + (MapGet (if b then m else m') a)=(if b then (MapGet m a) else (MapGet m' a)). + Proof. + Intros. Case b; Trivial. + Qed. + + (*i + Lemma MapGet_M2_bit_0_1' : (m,m',m'',m''':Map) + (a:ad) (MapGet (if (ad_bit_0 a) then (M2 m m') else (M2 m'' m''')) a)= + (MapGet (if (ad_bit_0 a) then m' else m'') (ad_div_2 a)). + Proof. + Intros. Rewrite (MapGet_if_commute (ad_bit_0 a)). Rewrite (MapGet_if_commute (ad_bit_0 a)). + Cut (ad_bit_0 a)=false\/(ad_bit_0 a)=true. Intros. Elim H. Intros. Rewrite H0. + Apply MapGet_M2_bit_0_0. Assumption. + Intros. Rewrite H0. Apply MapGet_M2_bit_0_1. Assumption. + Case (ad_bit_0 a); Auto. + Qed. + i*) + + Lemma MapGet_if_same : (m:Map) (b:bool) (a:ad) + (MapGet (if b then m else m) a)=(MapGet m a). + Proof. + Induction b;Trivial. + Qed. + + Lemma MapGet_M2_bit_0_2 : (m,m',m'':Map) + (a:ad) (MapGet (if (ad_bit_0 a) then (M2 m m') else (M2 m' m'')) a)= + (MapGet m' (ad_div_2 a)). + Proof. + Intros. Rewrite MapGet_if_commute. Apply MapGet_M2_bit_0. + Qed. + + Lemma MapPut1_semantics_1 : (p:positive) (a,a':ad) (y,y':A) + (ad_xor a a')=(ad_x p) + -> (MapGet (MapPut1 a y a' y' p) a)=(SOME y). + Proof. + Induction p. Intros. Unfold MapPut1. Rewrite MapGet_M2_bit_0_2. Apply M1_semantics_1. + Intros. Simpl. Rewrite MapGet_M2_bit_0_2. Apply H. Rewrite <- ad_xor_div_2. Rewrite H0. + Reflexivity. + Intros. Unfold MapPut1. Rewrite MapGet_M2_bit_0_2. Apply M1_semantics_1. + Qed. + + Lemma MapPut1_semantics_2 : (p:positive) (a,a':ad) (y,y':A) + (ad_xor a a')=(ad_x p) + -> (MapGet (MapPut1 a y a' y' p) a')=(SOME y'). + Proof. + Induction p. Intros. Unfold MapPut1. Rewrite (ad_neg_bit_0_2 a a' p0 H0). + Rewrite if_negb. Rewrite MapGet_M2_bit_0_2. Apply M1_semantics_1. + Intros. Simpl. Rewrite (ad_same_bit_0 a a' p0 H0). Rewrite MapGet_M2_bit_0_2. + Apply H. Rewrite <- ad_xor_div_2. Rewrite H0. Reflexivity. + Intros. Unfold MapPut1. Rewrite (ad_neg_bit_0_1 a a' H). Rewrite if_negb. + Rewrite MapGet_M2_bit_0_2. Apply M1_semantics_1. + Qed. + + Lemma MapGet_M2_both_NONE : (m,m':Map) (a:ad) + (MapGet m (ad_div_2 a))=NONE -> (MapGet m' (ad_div_2 a))=NONE -> + (MapGet (M2 m m') a)=NONE. + Proof. + Intros. Rewrite (Map2_semantics_3 m m' a). + Case (ad_bit_0 a); Assumption. + Qed. + + Lemma MapPut1_semantics_3 : (p:positive) (a,a',a0:ad) (y,y':A) + (ad_xor a a')=(ad_x p) -> (ad_eq a a0)=false -> (ad_eq a' a0)=false -> + (MapGet (MapPut1 a y a' y' p) a0)=NONE. + Proof. + Induction p. Intros. Unfold MapPut1. Elim (ad_neq a a0 H1). Intro. Rewrite H3. Rewrite if_negb. + Rewrite MapGet_M2_bit_0_2. Apply M1_semantics_2. Apply ad_div_bit_neq. Assumption. + Rewrite (ad_neg_bit_0_2 a a' p0 H0) in H3. Rewrite (negb_intro (ad_bit_0 a')). + Rewrite (negb_intro (ad_bit_0 a0)). Rewrite H3. Reflexivity. + Intro. Elim (ad_neq a' a0 H2). Intro. Rewrite (ad_neg_bit_0_2 a a' p0 H0). Rewrite H4. + Rewrite (negb_elim (ad_bit_0 a0)). Rewrite MapGet_M2_bit_0_2. + Apply M1_semantics_2; Assumption. + Intro; Case (ad_bit_0 a); Apply MapGet_M2_both_NONE; + Apply M1_semantics_2; Assumption. + Intros. Simpl. Elim (ad_neq a a0 H1). Intro. Rewrite H3. Rewrite if_negb. + Rewrite MapGet_M2_bit_0_2. Reflexivity. + Intro. Elim (ad_neq a' a0 H2). Intro. Rewrite (ad_same_bit_0 a a' p0 H0). Rewrite H4. + Rewrite if_negb. Rewrite MapGet_M2_bit_0_2. Reflexivity. + Intro. Cut (ad_xor (ad_div_2 a) (ad_div_2 a'))=(ad_x p0). Intro. + Case (ad_bit_0 a); Apply MapGet_M2_both_NONE; Trivial; + Apply H; Assumption. + Rewrite <- ad_xor_div_2. Rewrite H0. Reflexivity. + Intros. Simpl. Elim (ad_neq a a0 H0). Intro. Rewrite H2. Rewrite if_negb. + Rewrite MapGet_M2_bit_0_2. Apply M1_semantics_2. Apply ad_div_bit_neq. Assumption. + Rewrite (ad_neg_bit_0_1 a a' H) in H2. Rewrite (negb_intro (ad_bit_0 a')). + Rewrite (negb_intro (ad_bit_0 a0)). Rewrite H2. Reflexivity. + Intro. Elim (ad_neq a' a0 H1). Intro. Rewrite (ad_neg_bit_0_1 a a' H). Rewrite H3. + Rewrite (negb_elim (ad_bit_0 a0)). Rewrite MapGet_M2_bit_0_2. + Apply M1_semantics_2; Assumption. + Intro. Case (ad_bit_0 a); Apply MapGet_M2_both_NONE; Apply M1_semantics_2; Assumption. + Qed. + + Lemma MapPut1_semantics : (p:positive) (a,a':ad) (y,y':A) + (ad_xor a a')=(ad_x p) + -> (eqm (MapGet (MapPut1 a y a' y' p)) + [a0:ad] if (ad_eq a a0) then (SOME y) + else if (ad_eq a' a0) then (SOME y') else NONE). + Proof. + Unfold eqm. Intros. Elim (sumbool_of_bool (ad_eq a a0)). Intro H0. Rewrite H0. + Rewrite <- (ad_eq_complete ? ? H0). Exact (MapPut1_semantics_1 p a a' y y' H). + Intro H0. Rewrite H0. Elim (sumbool_of_bool (ad_eq a' a0)). Intro H1. + Rewrite <- (ad_eq_complete ? ? H1). Rewrite (ad_eq_correct a'). + Exact (MapPut1_semantics_2 p a a' y y' H). + Intro H1. Rewrite H1. Exact (MapPut1_semantics_3 p a a' a0 y y' H H0 H1). + Qed. + + Lemma MapPut1_semantics' : (p:positive) (a,a':ad) (y,y':A) + (ad_xor a a')=(ad_x p) + -> (eqm (MapGet (MapPut1 a y a' y' p)) + [a0:ad] if (ad_eq a' a0) then (SOME y') + else if (ad_eq a a0) then (SOME y) else NONE). + Proof. + Unfold eqm. Intros. Rewrite (MapPut1_semantics p a a' y y' H a0). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H0. Rewrite H0. + Rewrite <- (ad_eq_complete a a0 H0). Rewrite (ad_eq_comm a' a). + Rewrite (ad_xor_eq_false a a' p H). Reflexivity. + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Fixpoint MapPut [m:Map] : ad -> A -> Map := + Cases m of + M0 => M1 + | (M1 a y) => [a':ad; y':A] + Cases (ad_xor a a') of + ad_z => (M1 a' y') + | (ad_x p) => (MapPut1 a y a' y' p) + end + | (M2 m1 m2) => [a:ad; y:A] + Cases a of + ad_z => (M2 (MapPut m1 ad_z y) m2) + | (ad_x xH) => (M2 m1 (MapPut m2 ad_z y)) + | (ad_x (xO p)) => (M2 (MapPut m1 (ad_x p) y) m2) + | (ad_x (xI p)) => (M2 m1 (MapPut m2 (ad_x p) y)) + end + end. + + Lemma MapPut_semantics_1 : (a:ad) (y:A) (a0:ad) + (MapGet (MapPut M0 a y) a0)=(MapGet (M1 a y) a0). + Proof. + Trivial. + Qed. + + Lemma MapPut_semantics_2_1 : (a:ad) (y,y':A) (a0:ad) + (MapGet (MapPut (M1 a y) a y') a0)=(if (ad_eq a a0) then (SOME y') else NONE). + Proof. + Simpl. Intros. Rewrite (ad_xor_nilpotent a). Trivial. + Qed. + + Lemma MapPut_semantics_2_2 : (a,a':ad) (y,y':A) (a0:ad) (a'':ad) (ad_xor a a')=a'' -> + (MapGet (MapPut (M1 a y) a' y') a0)= + (if (ad_eq a' a0) then (SOME y') else + if (ad_eq a a0) then (SOME y) else NONE). + Proof. + Induction a''. Intro. Rewrite (ad_xor_eq ? ? H). Rewrite MapPut_semantics_2_1. + Case (ad_eq a' a0); Trivial. + Intros. Simpl. Rewrite H. Rewrite (MapPut1_semantics p a a' y y' H a0). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H0. Rewrite H0. Rewrite <- (ad_eq_complete ? ? H0). + Rewrite (ad_eq_comm a' a). Rewrite (ad_xor_eq_false ? ? ? H). Reflexivity. + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Lemma MapPut_semantics_2 : (a,a':ad) (y,y':A) (a0:ad) + (MapGet (MapPut (M1 a y) a' y') a0)= + (if (ad_eq a' a0) then (SOME y') else + if (ad_eq a a0) then (SOME y) else NONE). + Proof. + Intros. Apply MapPut_semantics_2_2 with a'':=(ad_xor a a'); Trivial. + Qed. + + Lemma MapPut_semantics_3_1 : (m,m':Map) (a:ad) (y:A) + (MapPut (M2 m m') a y)=(if (ad_bit_0 a) then (M2 m (MapPut m' (ad_div_2 a) y)) + else (M2 (MapPut m (ad_div_2 a) y) m')). + Proof. + Induction a. Trivial. + Induction p; Trivial. + Qed. + + Lemma MapPut_semantics : (m:Map) (a:ad) (y:A) + (eqm (MapGet (MapPut m a y)) [a':ad] if (ad_eq a a') then (SOME y) else (MapGet m a')). + Proof. + Unfold eqm. Induction m. Exact MapPut_semantics_1. + Intros. Unfold 2 MapGet. Apply MapPut_semantics_2; Assumption. + Intros. Rewrite MapPut_semantics_3_1. Rewrite (MapGet_M2_bit_0_if m0 m1 a0). + Elim (sumbool_of_bool (ad_bit_0 a)). Intro H1. Rewrite H1. Rewrite MapGet_M2_bit_0_if. + Elim (sumbool_of_bool (ad_bit_0 a0)). Intro H2. Rewrite H2. + Rewrite (H0 (ad_div_2 a) y (ad_div_2 a0)). Elim (sumbool_of_bool (ad_eq a a0)). + Intro H3. Rewrite H3. Rewrite (ad_div_eq ? ? H3). Reflexivity. + Intro H3. Rewrite H3. Rewrite <- H2 in H1. Rewrite (ad_div_bit_neq ? ? H3 H1). Reflexivity. + Intro H2. Rewrite H2. Rewrite (ad_eq_comm a a0). Rewrite (ad_bit_0_neq a0 a H2 H1). + Reflexivity. + Intro H1. Rewrite H1. Rewrite MapGet_M2_bit_0_if. Elim (sumbool_of_bool (ad_bit_0 a0)). + Intro H2. Rewrite H2. Rewrite (ad_bit_0_neq a a0 H1 H2). Reflexivity. + Intro H2. Rewrite H2. Rewrite (H (ad_div_2 a) y (ad_div_2 a0)). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H3. Rewrite H3. + Rewrite (ad_div_eq a a0 H3). Reflexivity. + Intro H3. Rewrite H3. Rewrite <- H2 in H1. Rewrite (ad_div_bit_neq a a0 H3 H1). Reflexivity. + Qed. + + Fixpoint MapPut_behind [m:Map] : ad -> A -> Map := + Cases m of + M0 => M1 + | (M1 a y) => [a':ad; y':A] + Cases (ad_xor a a') of + ad_z => m + | (ad_x p) => (MapPut1 a y a' y' p) + end + | (M2 m1 m2) => [a:ad; y:A] + Cases a of + ad_z => (M2 (MapPut_behind m1 ad_z y) m2) + | (ad_x xH) => (M2 m1 (MapPut_behind m2 ad_z y)) + | (ad_x (xO p)) => (M2 (MapPut_behind m1 (ad_x p) y) m2) + | (ad_x (xI p)) => (M2 m1 (MapPut_behind m2 (ad_x p) y)) + end + end. + + Lemma MapPut_behind_semantics_3_1 : (m,m':Map) (a:ad) (y:A) + (MapPut_behind (M2 m m') a y)= + (if (ad_bit_0 a) then (M2 m (MapPut_behind m' (ad_div_2 a) y)) + else (M2 (MapPut_behind m (ad_div_2 a) y) m')). + Proof. + Induction a. Trivial. + Induction p; Trivial. + Qed. + + Lemma MapPut_behind_as_before_1 : (a,a',a0:ad) (ad_eq a' a0)=false -> + (y,y':A) (MapGet (MapPut (M1 a y) a' y') a0) + =(MapGet (MapPut_behind (M1 a y) a' y') a0). + Proof. + Intros a a' a0. Simpl. Intros H y y'. Elim (ad_sum (ad_xor a a')). Intro H0. Elim H0. + Intros p H1. Rewrite H1. Reflexivity. + Intro H0. Rewrite H0. Rewrite (ad_xor_eq ? ? H0). Rewrite (M1_semantics_2 a' a0 y H). + Exact (M1_semantics_2 a' a0 y' H). + Qed. + + Lemma MapPut_behind_as_before : (m:Map) (a:ad) (y:A) + (a0:ad) (ad_eq a a0)=false -> + (MapGet (MapPut m a y) a0)=(MapGet (MapPut_behind m a y) a0). + Proof. + Induction m. Trivial. + Intros a y a' y' a0 H. Exact (MapPut_behind_as_before_1 a a' a0 H y y'). + Intros. Rewrite MapPut_semantics_3_1. Rewrite MapPut_behind_semantics_3_1. + Elim (sumbool_of_bool (ad_bit_0 a)). Intro H2. Rewrite H2. Rewrite MapGet_M2_bit_0_if. + Rewrite MapGet_M2_bit_0_if. Elim (sumbool_of_bool (ad_bit_0 a0)). Intro H3. + Rewrite H3. Apply H0. Rewrite <- H3 in H2. Exact (ad_div_bit_neq a a0 H1 H2). + Intro H3. Rewrite H3. Reflexivity. + Intro H2. Rewrite H2. Rewrite MapGet_M2_bit_0_if. Rewrite MapGet_M2_bit_0_if. + Elim (sumbool_of_bool (ad_bit_0 a0)). Intro H3. Rewrite H3. Reflexivity. + Intro H3. Rewrite H3. Apply H. Rewrite <- H3 in H2. Exact (ad_div_bit_neq a a0 H1 H2). + Qed. + + Lemma MapPut_behind_new : (m:Map) (a:ad) (y:A) + (MapGet (MapPut_behind m a y) a)=(Cases (MapGet m a) of + (SOME y') => (SOME y') + | _ => (SOME y) + end). + Proof. + Induction m. Simpl. Intros. Rewrite (ad_eq_correct a). Reflexivity. + Intros. Elim (ad_sum (ad_xor a a1)). Intro H. Elim H. Intros p H0. Simpl. + Rewrite H0. Rewrite (ad_xor_eq_false a a1 p). Exact (MapPut1_semantics_2 p a a1 a0 y H0). + Assumption. + Intro H. Simpl. Rewrite H. Rewrite <- (ad_xor_eq ? ? H). Rewrite (ad_eq_correct a). + Exact (M1_semantics_1 a a0). + Intros. Rewrite MapPut_behind_semantics_3_1. Rewrite (MapGet_M2_bit_0_if m0 m1 a). + Elim (sumbool_of_bool (ad_bit_0 a)). Intro H1. Rewrite H1. Rewrite (MapGet_M2_bit_0_1 a H1). + Exact (H0 (ad_div_2 a) y). + Intro H1. Rewrite H1. Rewrite (MapGet_M2_bit_0_0 a H1). Exact (H (ad_div_2 a) y). + Qed. + + Lemma MapPut_behind_semantics : (m:Map) (a:ad) (y:A) + (eqm (MapGet (MapPut_behind m a y)) + [a':ad] Cases (MapGet m a') of + (SOME y') => (SOME y') + | _ => if (ad_eq a a') then (SOME y) else NONE + end). + Proof. + Unfold eqm. Intros. Elim (sumbool_of_bool (ad_eq a a0)). Intro H. Rewrite H. + Rewrite (ad_eq_complete ? ? H). Apply MapPut_behind_new. + Intro H. Rewrite H. Rewrite <- (MapPut_behind_as_before m a y a0 H). + Rewrite (MapPut_semantics m a y a0). Rewrite H. Case (MapGet m a0); Trivial. + Qed. + + Definition makeM2 := [m,m':Map] Cases m m' of + M0 M0 => M0 + | M0 (M1 a y) => (M1 (ad_double_plus_un a) y) + | (M1 a y) M0 => (M1 (ad_double a) y) + | _ _ => (M2 m m') + end. + + Lemma makeM2_M2 : (m,m':Map) (eqm (MapGet (makeM2 m m')) (MapGet (M2 m m'))). + Proof. + Unfold eqm. Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H. + Rewrite (MapGet_M2_bit_0_1 a H m m'). Case m'. Case m. Reflexivity. + Intros a0 y. Simpl. Rewrite (ad_bit_0_1_not_double a H a0). Reflexivity. + Intros m1 m2. Unfold makeM2. Rewrite MapGet_M2_bit_0_1. Reflexivity. + Assumption. + Case m. Intros a0 y. Simpl. Elim (sumbool_of_bool (ad_eq a0 (ad_div_2 a))). + Intro H0. Rewrite H0. Rewrite (ad_eq_complete ? ? H0). Rewrite (ad_div_2_double_plus_un a H). + Rewrite (ad_eq_correct a). Reflexivity. + Intro H0. Rewrite H0. Rewrite (ad_eq_comm a0 (ad_div_2 a)) in H0. + Rewrite (ad_not_div_2_not_double_plus_un a a0 H0). Reflexivity. + Intros a0 y0 a1 y1. Unfold makeM2. Rewrite MapGet_M2_bit_0_1. Reflexivity. + Assumption. + Intros m1 m2 a0 y. Unfold makeM2. Rewrite MapGet_M2_bit_0_1. Reflexivity. + Assumption. + Intros m1 m2. Unfold makeM2. + Cut (MapGet (M2 m (M2 m1 m2)) a)=(MapGet (M2 m1 m2) (ad_div_2 a)). + Case m; Trivial. + Exact (MapGet_M2_bit_0_1 a H m (M2 m1 m2)). + Intro H. Rewrite (MapGet_M2_bit_0_0 a H m m'). Case m. Case m'. Reflexivity. + Intros a0 y. Simpl. Rewrite (ad_bit_0_0_not_double_plus_un a H a0). Reflexivity. + Intros m1 m2. Unfold makeM2. Rewrite MapGet_M2_bit_0_0. Reflexivity. + Assumption. + Case m'. Intros a0 y. Simpl. Elim (sumbool_of_bool (ad_eq a0 (ad_div_2 a))). Intro H0. + Rewrite H0. Rewrite (ad_eq_complete ? ? H0). Rewrite (ad_div_2_double a H). + Rewrite (ad_eq_correct a). Reflexivity. + Intro H0. Rewrite H0. Rewrite (ad_eq_comm (ad_double a0) a). + Rewrite (ad_eq_comm a0 (ad_div_2 a)) in H0. Rewrite (ad_not_div_2_not_double a a0 H0). + Reflexivity. + Intros a0 y0 a1 y1. Unfold makeM2. Rewrite MapGet_M2_bit_0_0. Reflexivity. + Assumption. + Intros m1 m2 a0 y. Unfold makeM2. Rewrite MapGet_M2_bit_0_0. Reflexivity. + Assumption. + Intros m1 m2. Unfold makeM2. Exact (MapGet_M2_bit_0_0 a H (M2 m1 m2) m'). + Qed. + + Fixpoint MapRemove [m:Map] : ad -> Map := + Cases m of + M0 => [_:ad] M0 + | (M1 a y) => [a':ad] + Cases (ad_eq a a') of + true => M0 + | false => m + end + | (M2 m1 m2) => [a:ad] + if (ad_bit_0 a) + then (makeM2 m1 (MapRemove m2 (ad_div_2 a))) + else (makeM2 (MapRemove m1 (ad_div_2 a)) m2) + end. + + Lemma MapRemove_semantics : (m:Map) (a:ad) + (eqm (MapGet (MapRemove m a)) [a':ad] if (ad_eq a a') then NONE else (MapGet m a')). + Proof. + Unfold eqm. Induction m. Simpl. Intros. Case (ad_eq a a0); Trivial. + Intros. Simpl. Elim (sumbool_of_bool (ad_eq a1 a2)). Intro H. Rewrite H. + Elim (sumbool_of_bool (ad_eq a a1)). Intro H0. Rewrite H0. Reflexivity. + Intro H0. Rewrite H0. Rewrite (ad_eq_complete ? ? H) in H0. Exact (M1_semantics_2 a a2 a0 H0). + Intro H. Elim (sumbool_of_bool (ad_eq a a1)). Intro H0. Rewrite H0. Rewrite H. + Rewrite <- (ad_eq_complete ? ? H0) in H. Rewrite H. Reflexivity. + Intro H0. Rewrite H0. Rewrite H. Reflexivity. + Intros. Change (MapGet (if (ad_bit_0 a) + then (makeM2 m0 (MapRemove m1 (ad_div_2 a))) + else (makeM2 (MapRemove m0 (ad_div_2 a)) m1)) + a0) + =(if (ad_eq a a0) then NONE else (MapGet (M2 m0 m1) a0)). + Elim (sumbool_of_bool (ad_bit_0 a)). Intro H1. Rewrite H1. + Rewrite (makeM2_M2 m0 (MapRemove m1 (ad_div_2 a)) a0). Elim (sumbool_of_bool (ad_bit_0 a0)). + Intro H2. Rewrite MapGet_M2_bit_0_1. Rewrite (H0 (ad_div_2 a) (ad_div_2 a0)). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H3. Rewrite H3. Rewrite (ad_div_eq ? ? H3). + Reflexivity. + Intro H3. Rewrite H3. Rewrite <- H2 in H1. Rewrite (ad_div_bit_neq ? ? H3 H1). + Rewrite (MapGet_M2_bit_0_1 a0 H2 m0 m1). Reflexivity. + Assumption. + Intro H2. Rewrite (MapGet_M2_bit_0_0 a0 H2 m0 (MapRemove m1 (ad_div_2 a))). + Rewrite (ad_eq_comm a a0). Rewrite (ad_bit_0_neq ? ? H2 H1). + Rewrite (MapGet_M2_bit_0_0 a0 H2 m0 m1). Reflexivity. + Intro H1. Rewrite H1. Rewrite (makeM2_M2 (MapRemove m0 (ad_div_2 a)) m1 a0). + Elim (sumbool_of_bool (ad_bit_0 a0)). Intro H2. Rewrite MapGet_M2_bit_0_1. + Rewrite (MapGet_M2_bit_0_1 a0 H2 m0 m1). Rewrite (ad_bit_0_neq a a0 H1 H2). Reflexivity. + Assumption. + Intro H2. Rewrite MapGet_M2_bit_0_0. Rewrite (H (ad_div_2 a) (ad_div_2 a0)). + Rewrite (MapGet_M2_bit_0_0 a0 H2 m0 m1). Elim (sumbool_of_bool (ad_eq a a0)). Intro H3. + Rewrite H3. Rewrite (ad_div_eq ? ? H3). Reflexivity. + Intro H3. Rewrite H3. Rewrite <- H2 in H1. Rewrite (ad_div_bit_neq ? ? H3 H1). Reflexivity. + Assumption. + Qed. + + Fixpoint MapCard [m:Map] : nat := + Cases m of + M0 => O + | (M1 _ _) => (S O) + | (M2 m m') => (plus (MapCard m) (MapCard m')) + end. + + Fixpoint MapMerge [m:Map] : Map -> Map := + Cases m of + M0 => [m':Map] m' + | (M1 a y) => [m':Map] (MapPut_behind m' a y) + | (M2 m1 m2) => [m':Map] Cases m' of + M0 => m + | (M1 a' y') => (MapPut m a' y') + | (M2 m'1 m'2) => (M2 (MapMerge m1 m'1) + (MapMerge m2 m'2)) + end + end. + + Lemma MapMerge_semantics : (m,m':Map) + (eqm (MapGet (MapMerge m m')) + [a0:ad] Cases (MapGet m' a0) of + (SOME y') => (SOME y') + | NONE => (MapGet m a0) + end). + Proof. + Unfold eqm. Induction m. Intros. Simpl. Case (MapGet m' a); Trivial. + Intros. Simpl. Rewrite (MapPut_behind_semantics m' a a0 a1). Reflexivity. + Induction m'. Trivial. + Intros. Unfold MapMerge. Rewrite (MapPut_semantics (M2 m0 m1) a a0 a1). + Elim (sumbool_of_bool (ad_eq a a1)). Intro H1. Rewrite H1. Rewrite (ad_eq_complete ? ? H1). + Rewrite (M1_semantics_1 a1 a0). Reflexivity. + Intro H1. Rewrite H1. Rewrite (M1_semantics_2 a a1 a0 H1). Reflexivity. + Intros. Cut (MapMerge (M2 m0 m1) (M2 m2 m3))=(M2 (MapMerge m0 m2) (MapMerge m1 m3)). + Intro. Rewrite H3. Rewrite MapGet_M2_bit_0_if. Rewrite (H0 m3 (ad_div_2 a)). + Rewrite (H m2 (ad_div_2 a)). Rewrite (MapGet_M2_bit_0_if m2 m3 a). + Rewrite (MapGet_M2_bit_0_if m0 m1 a). Case (ad_bit_0 a); Trivial. + Reflexivity. + Qed. + + (** [MapInter], [MapRngRestrTo], [MapRngRestrBy], [MapInverse] + not implemented: need a decidable equality on [A]. *) + + Fixpoint MapDelta [m:Map] : Map -> Map := + Cases m of + M0 => [m':Map] m' + | (M1 a y) => [m':Map] Cases (MapGet m' a) of + NONE => (MapPut m' a y) + | _ => (MapRemove m' a) + end + | (M2 m1 m2) => [m':Map] Cases m' of + M0 => m + | (M1 a' y') => Cases (MapGet m a') of + NONE => (MapPut m a' y') + | _ => (MapRemove m a') + end + | (M2 m'1 m'2) => (makeM2 (MapDelta m1 m'1) + (MapDelta m2 m'2)) + end + end. + + Lemma MapDelta_semantics_comm : (m,m':Map) + (eqm (MapGet (MapDelta m m')) (MapGet (MapDelta m' m))). + Proof. + Unfold eqm. Induction m. Induction m'; Reflexivity. + Induction m'. Reflexivity. + Unfold MapDelta. Intros. Elim (sumbool_of_bool (ad_eq a a1)). Intro H. + Rewrite <- (ad_eq_complete ? ? H). Rewrite (M1_semantics_1 a a2). + Rewrite (M1_semantics_1 a a0). Simpl. Rewrite (ad_eq_correct a). Reflexivity. + Intro H. Rewrite (M1_semantics_2 a a1 a0 H). Rewrite (ad_eq_comm a a1) in H. + Rewrite (M1_semantics_2 a1 a a2 H). Rewrite (MapPut_semantics (M1 a a0) a1 a2 a3). + Rewrite (MapPut_semantics (M1 a1 a2) a a0 a3). Elim (sumbool_of_bool (ad_eq a a3)). + Intro H0. Rewrite H0. Rewrite (ad_eq_complete ? ? H0) in H. Rewrite H. + Rewrite (ad_eq_complete ? ? H0). Rewrite (M1_semantics_1 a3 a0). Reflexivity. + Intro H0. Rewrite H0. Rewrite (M1_semantics_2 a a3 a0 H0). + Elim (sumbool_of_bool (ad_eq a1 a3)). Intro H1. Rewrite H1. + Rewrite (ad_eq_complete ? ? H1). Exact (M1_semantics_1 a3 a2). + Intro H1. Rewrite H1. Exact (M1_semantics_2 a1 a3 a2 H1). + Intros. Reflexivity. + Induction m'. Reflexivity. + Reflexivity. + Intros. Simpl. Rewrite (makeM2_M2 (MapDelta m0 m2) (MapDelta m1 m3) a). + Rewrite (makeM2_M2 (MapDelta m2 m0) (MapDelta m3 m1) a). + Rewrite (MapGet_M2_bit_0_if (MapDelta m0 m2) (MapDelta m1 m3) a). + Rewrite (MapGet_M2_bit_0_if (MapDelta m2 m0) (MapDelta m3 m1) a). + Rewrite (H0 m3 (ad_div_2 a)). Rewrite (H m2 (ad_div_2 a)). Reflexivity. + Qed. + + Lemma MapDelta_semantics_1_1 : (a:ad) (y:A) (m':Map) (a0:ad) + (MapGet (M1 a y) a0)=NONE -> (MapGet m' a0)=NONE -> + (MapGet (MapDelta (M1 a y) m') a0)=NONE. + Proof. + Intros. Unfold MapDelta. Elim (sumbool_of_bool (ad_eq a a0)). Intro H1. + Rewrite (ad_eq_complete ? ? H1) in H. Rewrite (M1_semantics_1 a0 y) in H. Discriminate H. + Intro H1. Case (MapGet m' a). Rewrite (MapPut_semantics m' a y a0). Rewrite H1. Assumption. + Rewrite (MapRemove_semantics m' a a0). Rewrite H1. Trivial. + Qed. + + Lemma MapDelta_semantics_1 : (m,m':Map) (a:ad) + (MapGet m a)=NONE -> (MapGet m' a)=NONE -> + (MapGet (MapDelta m m') a)=NONE. + Proof. + Induction m. Trivial. + Exact MapDelta_semantics_1_1. + Induction m'. Trivial. + Intros. Rewrite (MapDelta_semantics_comm (M2 m0 m1) (M1 a a0) a1). + Apply MapDelta_semantics_1_1; Trivial. + Intros. Simpl. Rewrite (makeM2_M2 (MapDelta m0 m2) (MapDelta m1 m3) a). + Rewrite MapGet_M2_bit_0_if. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H5. Rewrite H5. + Apply H0. Rewrite (MapGet_M2_bit_0_1 a H5 m0 m1) in H3. Exact H3. + Rewrite (MapGet_M2_bit_0_1 a H5 m2 m3) in H4. Exact H4. + Intro H5. Rewrite H5. Apply H. Rewrite (MapGet_M2_bit_0_0 a H5 m0 m1) in H3. Exact H3. + Rewrite (MapGet_M2_bit_0_0 a H5 m2 m3) in H4. Exact H4. + Qed. + + Lemma MapDelta_semantics_2_1 : (a:ad) (y:A) (m':Map) (a0:ad) (y0:A) + (MapGet (M1 a y) a0)=NONE -> (MapGet m' a0)=(SOME y0) -> + (MapGet (MapDelta (M1 a y) m') a0)=(SOME y0). + Proof. + Intros. Unfold MapDelta. Elim (sumbool_of_bool (ad_eq a a0)). Intro H1. + Rewrite (ad_eq_complete ? ? H1) in H. Rewrite (M1_semantics_1 a0 y) in H. Discriminate H. + Intro H1. Case (MapGet m' a). Rewrite (MapPut_semantics m' a y a0). Rewrite H1. Assumption. + Rewrite (MapRemove_semantics m' a a0). Rewrite H1. Trivial. + Qed. + + Lemma MapDelta_semantics_2_2 : (a:ad) (y:A) (m':Map) (a0:ad) (y0:A) + (MapGet (M1 a y) a0)=(SOME y0) -> (MapGet m' a0)=NONE -> + (MapGet (MapDelta (M1 a y) m') a0)=(SOME y0). + Proof. + Intros. Unfold MapDelta. Elim (sumbool_of_bool (ad_eq a a0)). Intro H1. + Rewrite (ad_eq_complete ? ? H1) in H. Rewrite (ad_eq_complete ? ? H1). + Rewrite H0. Rewrite (MapPut_semantics m' a0 y a0). Rewrite (ad_eq_correct a0). + Rewrite (M1_semantics_1 a0 y) in H. Simple Inversion H. Assumption. + Intro H1. Rewrite (M1_semantics_2 a a0 y H1) in H. Discriminate H. + Qed. + + Lemma MapDelta_semantics_2 : (m,m':Map) (a:ad) (y:A) + (MapGet m a)=NONE -> (MapGet m' a)=(SOME y) -> + (MapGet (MapDelta m m') a)=(SOME y). + Proof. + Induction m. Trivial. + Exact MapDelta_semantics_2_1. + Induction m'. Intros. Discriminate H2. + Intros. Rewrite (MapDelta_semantics_comm (M2 m0 m1) (M1 a a0) a1). + Apply MapDelta_semantics_2_2; Assumption. + Intros. Simpl. Rewrite (makeM2_M2 (MapDelta m0 m2) (MapDelta m1 m3) a). + Rewrite MapGet_M2_bit_0_if. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H5. Rewrite H5. + Apply H0. Rewrite <- (MapGet_M2_bit_0_1 a H5 m0 m1). Assumption. + Rewrite <- (MapGet_M2_bit_0_1 a H5 m2 m3). Assumption. + Intro H5. Rewrite H5. Apply H. Rewrite <- (MapGet_M2_bit_0_0 a H5 m0 m1). Assumption. + Rewrite <- (MapGet_M2_bit_0_0 a H5 m2 m3). Assumption. + Qed. + + Lemma MapDelta_semantics_3_1 : (a0:ad) (y0:A) (m':Map) (a:ad) (y,y':A) + (MapGet (M1 a0 y0) a)=(SOME y) -> (MapGet m' a)=(SOME y') -> + (MapGet (MapDelta (M1 a0 y0) m') a)=NONE. + Proof. + Intros. Unfold MapDelta. Elim (sumbool_of_bool (ad_eq a0 a)). Intro H1. + Rewrite (ad_eq_complete a0 a H1). Rewrite H0. Rewrite (MapRemove_semantics m' a a). + Rewrite (ad_eq_correct a). Reflexivity. + Intro H1. Rewrite (M1_semantics_2 a0 a y0 H1) in H. Discriminate H. + Qed. + + Lemma MapDelta_semantics_3 : (m,m':Map) (a:ad) (y,y':A) + (MapGet m a)=(SOME y) -> (MapGet m' a)=(SOME y') -> + (MapGet (MapDelta m m') a)=NONE. + Proof. + Induction m. Intros. Discriminate H. + Exact MapDelta_semantics_3_1. + Induction m'. Intros. Discriminate H2. + Intros. Rewrite (MapDelta_semantics_comm (M2 m0 m1) (M1 a a0) a1). + Exact (MapDelta_semantics_3_1 a a0 (M2 m0 m1) a1 y' y H2 H1). + Intros. Simpl. Rewrite (makeM2_M2 (MapDelta m0 m2) (MapDelta m1 m3) a). + Rewrite MapGet_M2_bit_0_if. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H5. Rewrite H5. + Apply (H0 m3 (ad_div_2 a) y y'). Rewrite <- (MapGet_M2_bit_0_1 a H5 m0 m1). Assumption. + Rewrite <- (MapGet_M2_bit_0_1 a H5 m2 m3). Assumption. + Intro H5. Rewrite H5. Apply (H m2 (ad_div_2 a) y y'). + Rewrite <- (MapGet_M2_bit_0_0 a H5 m0 m1). Assumption. + Rewrite <- (MapGet_M2_bit_0_0 a H5 m2 m3). Assumption. + Qed. + + Lemma MapDelta_semantics : (m,m':Map) + (eqm (MapGet (MapDelta m m')) + [a0:ad] Cases (MapGet m a0) (MapGet m' a0) of + NONE (SOME y') => (SOME y') + | (SOME y) NONE => (SOME y) + | _ _ => NONE + end). + Proof. + Unfold eqm. Intros. Elim (option_sum (MapGet m' a)). Intro H. Elim H. Intros a0 H0. + Rewrite H0. Elim (option_sum (MapGet m a)). Intro H1. Elim H1. Intros a1 H2. Rewrite H2. + Exact (MapDelta_semantics_3 m m' a a1 a0 H2 H0). + Intro H1. Rewrite H1. Exact (MapDelta_semantics_2 m m' a a0 H1 H0). + Intro H. Rewrite H. Elim (option_sum (MapGet m a)). Intro H0. Elim H0. Intros a0 H1. + Rewrite H1. Rewrite (MapDelta_semantics_comm m m' a). + Exact (MapDelta_semantics_2 m' m a a0 H H1). + Intro H0. Rewrite H0. Exact (MapDelta_semantics_1 m m' a H0 H). + Qed. + + Definition MapEmptyp := [m:Map] + Cases m of + M0 => true + | _ => false + end. + + Lemma MapEmptyp_correct : (MapEmptyp M0)=true. + Proof. + Reflexivity. + Qed. + + Lemma MapEmptyp_complete : (m:Map) (MapEmptyp m)=true -> m=M0. + Proof. + Induction m; Trivial. Intros. Discriminate H. + Intros. Discriminate H1. + Qed. + + (** [MapSplit] not implemented: not the preferred way of recursing over Maps + (use [MapSweep], [MapCollect], or [MapFold] in Mapiter.v. *) + +End MapDefs. diff --git a/theories7/IntMap/Mapaxioms.v b/theories7/IntMap/Mapaxioms.v new file mode 100644 index 00000000..085afd69 --- /dev/null +++ b/theories7/IntMap/Mapaxioms.v @@ -0,0 +1,670 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Mapaxioms.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Fset. + +Section MapAxioms. + + Variable A, B, C : Set. + + Lemma eqm_sym : (f,f':ad->(option A)) (eqm A f f') -> (eqm A f' f). + Proof. + Unfold eqm. Intros. Rewrite H. Reflexivity. + Qed. + + Lemma eqm_refl : (f:ad->(option A)) (eqm A f f). + Proof. + Unfold eqm. Trivial. + Qed. + + Lemma eqm_trans : (f,f',f'':ad->(option A)) (eqm A f f') -> (eqm A f' f'') -> (eqm A f f''). + Proof. + Unfold eqm. Intros. Rewrite H. Exact (H0 a). + Qed. + + Definition eqmap := [m,m':(Map A)] (eqm A (MapGet A m) (MapGet A m')). + + Lemma eqmap_sym : (m,m':(Map A)) (eqmap m m') -> (eqmap m' m). + Proof. + Intros. Unfold eqmap. Apply eqm_sym. Assumption. + Qed. + + Lemma eqmap_refl : (m:(Map A)) (eqmap m m). + Proof. + Intros. Unfold eqmap. Apply eqm_refl. + Qed. + + Lemma eqmap_trans : (m,m',m'':(Map A)) (eqmap m m') -> (eqmap m' m'') -> (eqmap m m''). + Proof. + Intros. Exact (eqm_trans (MapGet A m) (MapGet A m') (MapGet A m'') H H0). + Qed. + + Lemma MapPut_as_Merge : (m:(Map A)) (a:ad) (y:A) + (eqmap (MapPut A m a y) (MapMerge A m (M1 A a y))). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapPut_semantics A m a y a0). + Rewrite (MapMerge_semantics A m (M1 A a y) a0). Unfold 2 MapGet. + Elim (sumbool_of_bool (ad_eq a a0)); Intro H; Rewrite H; Reflexivity. + Qed. + + Lemma MapPut_ext : (m,m':(Map A)) (eqmap m m') -> + (a:ad) (y:A) (eqmap (MapPut A m a y) (MapPut A m' a y)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapPut_semantics A m' a y a0). + Rewrite (MapPut_semantics A m a y a0). + Case (ad_eq a a0); [ Reflexivity | Apply H ]. + Qed. + + Lemma MapPut_behind_as_Merge : (m:(Map A)) (a:ad) (y:A) + (eqmap (MapPut_behind A m a y) (MapMerge A (M1 A a y) m)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapPut_behind_semantics A m a y a0). + Rewrite (MapMerge_semantics A (M1 A a y) m a0). Reflexivity. + Qed. + + Lemma MapPut_behind_ext : (m,m':(Map A)) (eqmap m m') -> + (a:ad) (y:A) (eqmap (MapPut_behind A m a y) (MapPut_behind A m' a y)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapPut_behind_semantics A m' a y a0). + Rewrite (MapPut_behind_semantics A m a y a0). Rewrite (H a0). Reflexivity. + Qed. + + Lemma MapMerge_empty_m_1 : (m:(Map A)) (MapMerge A (M0 A) m)=m. + Proof. + Trivial. + Qed. + + Lemma MapMerge_empty_m : (m:(Map A)) (eqmap (MapMerge A (M0 A) m) m). + Proof. + Unfold eqmap eqm. Trivial. + Qed. + + Lemma MapMerge_m_empty_1 : (m:(Map A)) (MapMerge A m (M0 A))=m. + Proof. + Induction m;Trivial. + Qed. + + Lemma MapMerge_m_empty : (m:(Map A)) (eqmap (MapMerge A m (M0 A)) m). + Proof. + Unfold eqmap eqm. Intros. Rewrite MapMerge_m_empty_1. Reflexivity. + Qed. + + Lemma MapMerge_empty_l : (m,m':(Map A)) (eqmap (MapMerge A m m') (M0 A)) -> + (eqmap m (M0 A)). + Proof. + Unfold eqmap eqm. Intros. Cut (MapGet A (MapMerge A m m') a)=(MapGet A (M0 A) a). + Rewrite (MapMerge_semantics A m m' a). Case (MapGet A m' a). Trivial. + Intros. Discriminate H0. + Exact (H a). + Qed. + + Lemma MapMerge_empty_r : (m,m':(Map A)) (eqmap (MapMerge A m m') (M0 A)) -> + (eqmap m' (M0 A)). + Proof. + Unfold eqmap eqm. Intros. Cut (MapGet A (MapMerge A m m') a)=(MapGet A (M0 A) a). + Rewrite (MapMerge_semantics A m m' a). Case (MapGet A m' a). Trivial. + Intros. Discriminate H0. + Exact (H a). + Qed. + + Lemma MapMerge_assoc : (m,m',m'':(Map A)) (eqmap + (MapMerge A (MapMerge A m m') m'') + (MapMerge A m (MapMerge A m' m''))). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapMerge_semantics A (MapMerge A m m') m'' a). + Rewrite (MapMerge_semantics A m (MapMerge A m' m'') a). Rewrite (MapMerge_semantics A m m' a). + Rewrite (MapMerge_semantics A m' m'' a). + Case (MapGet A m'' a); Case (MapGet A m' a); Trivial. + Qed. + + Lemma MapMerge_idempotent : (m:(Map A)) (eqmap (MapMerge A m m) m). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapMerge_semantics A m m a). + Case (MapGet A m a); Trivial. + Qed. + + Lemma MapMerge_ext : (m1,m2,m'1,m'2:(Map A)) + (eqmap m1 m'1) -> (eqmap m2 m'2) -> + (eqmap (MapMerge A m1 m2) (MapMerge A m'1 m'2)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapMerge_semantics A m1 m2 a). + Rewrite (MapMerge_semantics A m'1 m'2 a). Rewrite (H a). Rewrite (H0 a). Reflexivity. + Qed. + + Lemma MapMerge_ext_l : (m1,m'1,m2:(Map A)) + (eqmap m1 m'1) -> (eqmap (MapMerge A m1 m2) (MapMerge A m'1 m2)). + Proof. + Intros. Apply MapMerge_ext. Assumption. + Apply eqmap_refl. + Qed. + + Lemma MapMerge_ext_r : (m1,m2,m'2:(Map A)) + (eqmap m2 m'2) -> (eqmap (MapMerge A m1 m2) (MapMerge A m1 m'2)). + Proof. + Intros. Apply MapMerge_ext. Apply eqmap_refl. + Assumption. + Qed. + + Lemma MapMerge_RestrTo_l : (m,m',m'':(Map A)) + (eqmap (MapMerge A (MapDomRestrTo A A m m') m'') + (MapDomRestrTo A A (MapMerge A m m'') (MapMerge A m' m''))). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapMerge_semantics A (MapDomRestrTo A A m m') m'' a). + Rewrite (MapDomRestrTo_semantics A A m m' a). + Rewrite (MapDomRestrTo_semantics A A (MapMerge A m m'') (MapMerge A m' m'') a). + Rewrite (MapMerge_semantics A m' m'' a). Rewrite (MapMerge_semantics A m m'' a). + Case (MapGet A m'' a); Case (MapGet A m' a); Reflexivity. + Qed. + + Lemma MapRemove_as_RestrBy : (m:(Map A)) (a:ad) (y:B) + (eqmap (MapRemove A m a) (MapDomRestrBy A B m (M1 B a y))). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapRemove_semantics A m a a0). + Rewrite (MapDomRestrBy_semantics A B m (M1 B a y) a0). Elim (sumbool_of_bool (ad_eq a a0)). + Intro H. Rewrite H. Rewrite (ad_eq_complete a a0 H). Rewrite (M1_semantics_1 B a0 y). + Reflexivity. + Intro H. Rewrite H. Rewrite (M1_semantics_2 B a a0 y H). Reflexivity. + Qed. + + Lemma MapRemove_ext : (m,m':(Map A)) (eqmap m m') -> + (a:ad) (eqmap (MapRemove A m a) (MapRemove A m' a)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapRemove_semantics A m' a a0). + Rewrite (MapRemove_semantics A m a a0). + Case (ad_eq a a0); [ Reflexivity | Apply H ]. + Qed. + + Lemma MapDomRestrTo_empty_m_1 : + (m:(Map B)) (MapDomRestrTo A B (M0 A) m)=(M0 A). + Proof. + Trivial. + Qed. + + Lemma MapDomRestrTo_empty_m : + (m:(Map B)) (eqmap (MapDomRestrTo A B (M0 A) m) (M0 A)). + Proof. + Unfold eqmap eqm. Trivial. + Qed. + + Lemma MapDomRestrTo_m_empty_1 : + (m:(Map A)) (MapDomRestrTo A B m (M0 B))=(M0 A). + Proof. + Induction m;Trivial. + Qed. + + Lemma MapDomRestrTo_m_empty : + (m:(Map A)) (eqmap (MapDomRestrTo A B m (M0 B)) (M0 A)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDomRestrTo_m_empty_1 m). Reflexivity. + Qed. + + Lemma MapDomRestrTo_assoc : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (eqmap (MapDomRestrTo A C (MapDomRestrTo A B m m') m'') + (MapDomRestrTo A B m (MapDomRestrTo B C m' m''))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrTo_semantics A C (MapDomRestrTo A B m m') m'' a). + Rewrite (MapDomRestrTo_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A B m (MapDomRestrTo B C m' m'') a). + Rewrite (MapDomRestrTo_semantics B C m' m'' a). + Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. + Qed. + + Lemma MapDomRestrTo_idempotent : (m:(Map A)) (eqmap (MapDomRestrTo A A m m) m). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDomRestrTo_semantics A A m m a). + Case (MapGet A m a); Trivial. + Qed. + + Lemma MapDomRestrTo_Dom : (m:(Map A)) (m':(Map B)) + (eqmap (MapDomRestrTo A B m m') (MapDomRestrTo A unit m (MapDom B m'))). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDomRestrTo_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A unit m (MapDom B m') a). + Elim (sumbool_of_bool (in_FSet a (MapDom B m'))). Intro H. + Elim (MapDom_semantics_2 B m' a H). Intros y H0. Rewrite H0. Unfold in_FSet in_dom in H. + Generalize H. Case (MapGet unit (MapDom B m') a); Trivial. Intro H1. Discriminate H1. + Intro H. Rewrite (MapDom_semantics_4 B m' a H). Unfold in_FSet in_dom in H. + Generalize H. Case (MapGet unit (MapDom B m') a). Trivial. + Intros H0 H1. Discriminate H1. + Qed. + + Lemma MapDomRestrBy_empty_m_1 : + (m:(Map B)) (MapDomRestrBy A B (M0 A) m)=(M0 A). + Proof. + Trivial. + Qed. + + Lemma MapDomRestrBy_empty_m : + (m:(Map B)) (eqmap (MapDomRestrBy A B (M0 A) m) (M0 A)). + Proof. + Unfold eqmap eqm. Trivial. + Qed. + + Lemma MapDomRestrBy_m_empty_1 : (m:(Map A)) (MapDomRestrBy A B m (M0 B))=m. + Proof. + Induction m;Trivial. + Qed. + + Lemma MapDomRestrBy_m_empty : (m:(Map A)) (eqmap (MapDomRestrBy A B m (M0 B)) m). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDomRestrBy_m_empty_1 m). Reflexivity. + Qed. + + Lemma MapDomRestrBy_Dom : (m:(Map A)) (m':(Map B)) + (eqmap (MapDomRestrBy A B m m') (MapDomRestrBy A unit m (MapDom B m'))). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDomRestrBy_semantics A B m m' a). + Rewrite (MapDomRestrBy_semantics A unit m (MapDom B m') a). + Elim (sumbool_of_bool (in_FSet a (MapDom B m'))). Intro H. + Elim (MapDom_semantics_2 B m' a H). Intros y H0. Rewrite H0. + Unfold in_FSet in_dom in H. Generalize H. Case (MapGet unit (MapDom B m') a); Trivial. + Intro H1. Discriminate H1. + Intro H. Rewrite (MapDom_semantics_4 B m' a H). Unfold in_FSet in_dom in H. + Generalize H. Case (MapGet unit (MapDom B m') a). Trivial. + Intros H0 H1. Discriminate H1. + Qed. + + Lemma MapDomRestrBy_m_m_1 : (m:(Map A)) (eqmap (MapDomRestrBy A A m m) (M0 A)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDomRestrBy_semantics A A m m a). + Case (MapGet A m a); Trivial. + Qed. + + Lemma MapDomRestrBy_By : (m:(Map A)) (m':(Map B)) (m'':(Map B)) + (eqmap (MapDomRestrBy A B (MapDomRestrBy A B m m') m'') + (MapDomRestrBy A B m (MapMerge B m' m''))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrBy_semantics A B (MapDomRestrBy A B m m') m'' a). + Rewrite (MapDomRestrBy_semantics A B m m' a). + Rewrite (MapDomRestrBy_semantics A B m (MapMerge B m' m'') a). + Rewrite (MapMerge_semantics B m' m'' a). + Case (MapGet B m'' a); Case (MapGet B m' a); Trivial. + Qed. + + Lemma MapDomRestrBy_By_comm : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (eqmap (MapDomRestrBy A C (MapDomRestrBy A B m m') m'') + (MapDomRestrBy A B (MapDomRestrBy A C m m'') m')). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrBy_semantics A C (MapDomRestrBy A B m m') m'' a). + Rewrite (MapDomRestrBy_semantics A B m m' a). + Rewrite (MapDomRestrBy_semantics A B (MapDomRestrBy A C m m'') m' a). + Rewrite (MapDomRestrBy_semantics A C m m'' a). + Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. + Qed. + + Lemma MapDomRestrBy_To : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (eqmap (MapDomRestrBy A C (MapDomRestrTo A B m m') m'') + (MapDomRestrTo A B m (MapDomRestrBy B C m' m''))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrBy_semantics A C (MapDomRestrTo A B m m') m'' a). + Rewrite (MapDomRestrTo_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A B m (MapDomRestrBy B C m' m'') a). + Rewrite (MapDomRestrBy_semantics B C m' m'' a). + Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. + Qed. + + Lemma MapDomRestrBy_To_comm : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (eqmap (MapDomRestrBy A C (MapDomRestrTo A B m m') m'') + (MapDomRestrTo A B (MapDomRestrBy A C m m'') m')). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrBy_semantics A C (MapDomRestrTo A B m m') m'' a). + Rewrite (MapDomRestrTo_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A B (MapDomRestrBy A C m m'') m' a). + Rewrite (MapDomRestrBy_semantics A C m m'' a). + Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. + Qed. + + Lemma MapDomRestrTo_By : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (eqmap (MapDomRestrTo A C (MapDomRestrBy A B m m') m'') + (MapDomRestrTo A C m (MapDomRestrBy C B m'' m'))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrTo_semantics A C (MapDomRestrBy A B m m') m'' a). + Rewrite (MapDomRestrBy_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A C m (MapDomRestrBy C B m'' m') a). + Rewrite (MapDomRestrBy_semantics C B m'' m' a). + Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. + Qed. + + Lemma MapDomRestrTo_By_comm : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (eqmap (MapDomRestrTo A C (MapDomRestrBy A B m m') m'') + (MapDomRestrBy A B (MapDomRestrTo A C m m'') m')). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrTo_semantics A C (MapDomRestrBy A B m m') m'' a). + Rewrite (MapDomRestrBy_semantics A B m m' a). + Rewrite (MapDomRestrBy_semantics A B (MapDomRestrTo A C m m'') m' a). + Rewrite (MapDomRestrTo_semantics A C m m'' a). + Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. + Qed. + + Lemma MapDomRestrTo_To_comm : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (eqmap (MapDomRestrTo A C (MapDomRestrTo A B m m') m'') + (MapDomRestrTo A B (MapDomRestrTo A C m m'') m')). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrTo_semantics A C (MapDomRestrTo A B m m') m'' a). + Rewrite (MapDomRestrTo_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A B (MapDomRestrTo A C m m'') m' a). + Rewrite (MapDomRestrTo_semantics A C m m'' a). + Case (MapGet C m'' a); Case (MapGet B m' a); Trivial. + Qed. + + Lemma MapMerge_DomRestrTo : (m,m':(Map A)) (m'':(Map B)) + (eqmap (MapDomRestrTo A B (MapMerge A m m') m'') + (MapMerge A (MapDomRestrTo A B m m'') (MapDomRestrTo A B m' m''))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrTo_semantics A B (MapMerge A m m') m'' a). + Rewrite (MapMerge_semantics A m m' a). + Rewrite (MapMerge_semantics A (MapDomRestrTo A B m m'') (MapDomRestrTo A B m' m'') a). + Rewrite (MapDomRestrTo_semantics A B m' m'' a). + Rewrite (MapDomRestrTo_semantics A B m m'' a). + Case (MapGet B m'' a); Case (MapGet A m' a); Trivial. + Qed. + + Lemma MapMerge_DomRestrBy : (m,m':(Map A)) (m'':(Map B)) + (eqmap (MapDomRestrBy A B (MapMerge A m m') m'') + (MapMerge A (MapDomRestrBy A B m m'') (MapDomRestrBy A B m' m''))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrBy_semantics A B (MapMerge A m m') m'' a). + Rewrite (MapMerge_semantics A m m' a). + Rewrite (MapMerge_semantics A (MapDomRestrBy A B m m'') (MapDomRestrBy A B m' m'') a). + Rewrite (MapDomRestrBy_semantics A B m' m'' a). + Rewrite (MapDomRestrBy_semantics A B m m'' a). + Case (MapGet B m'' a); Case (MapGet A m' a); Trivial. + Qed. + + Lemma MapDelta_empty_m_1 : (m:(Map A)) (MapDelta A (M0 A) m)=m. + Proof. + Trivial. + Qed. + + Lemma MapDelta_empty_m : (m:(Map A)) (eqmap (MapDelta A (M0 A) m) m). + Proof. + Unfold eqmap eqm. Trivial. + Qed. + + Lemma MapDelta_m_empty_1 : (m:(Map A)) (MapDelta A m (M0 A))=m. + Proof. + Induction m;Trivial. + Qed. + + Lemma MapDelta_m_empty : (m:(Map A)) (eqmap (MapDelta A m (M0 A)) m). + Proof. + Unfold eqmap eqm. Intros. Rewrite MapDelta_m_empty_1. Reflexivity. + Qed. + + Lemma MapDelta_nilpotent : (m:(Map A)) (eqmap (MapDelta A m m) (M0 A)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m m a). + Case (MapGet A m a); Trivial. + Qed. + + Lemma MapDelta_as_Merge : (m,m':(Map A)) (eqmap (MapDelta A m m') + (MapMerge A (MapDomRestrBy A A m m') (MapDomRestrBy A A m' m))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDelta_semantics A m m' a). + Rewrite (MapMerge_semantics A (MapDomRestrBy A A m m') (MapDomRestrBy A A m' m) a). + Rewrite (MapDomRestrBy_semantics A A m' m a). + Rewrite (MapDomRestrBy_semantics A A m m' a). + Case (MapGet A m a); Case (MapGet A m' a); Trivial. + Qed. + + Lemma MapDelta_as_DomRestrBy : (m,m':(Map A)) (eqmap (MapDelta A m m') + (MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m m'))). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m m' a). + Rewrite (MapDomRestrBy_semantics A A (MapMerge A m m') (MapDomRestrTo A A m m') a). + Rewrite (MapDomRestrTo_semantics A A m m' a). Rewrite (MapMerge_semantics A m m' a). + Case (MapGet A m a); Case (MapGet A m' a); Trivial. + Qed. + + Lemma MapDelta_as_DomRestrBy_2 : (m,m':(Map A)) (eqmap (MapDelta A m m') + (MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m' m))). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m m' a). + Rewrite (MapDomRestrBy_semantics A A (MapMerge A m m') (MapDomRestrTo A A m' m) a). + Rewrite (MapDomRestrTo_semantics A A m' m a). Rewrite (MapMerge_semantics A m m' a). + Case (MapGet A m a); Case (MapGet A m' a); Trivial. + Qed. + + Lemma MapDelta_sym : (m,m':(Map A)) (eqmap (MapDelta A m m') (MapDelta A m' m)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m m' a). + Rewrite (MapDelta_semantics A m' m a). + Case (MapGet A m a); Case (MapGet A m' a); Trivial. + Qed. + + Lemma MapDelta_ext : (m1,m2,m'1,m'2:(Map A)) + (eqmap m1 m'1) -> (eqmap m2 m'2) -> + (eqmap (MapDelta A m1 m2) (MapDelta A m'1 m'2)). + Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics A m1 m2 a). + Rewrite (MapDelta_semantics A m'1 m'2 a). Rewrite (H a). Rewrite (H0 a). Reflexivity. + Qed. + + Lemma MapDelta_ext_l : (m1,m'1,m2:(Map A)) + (eqmap m1 m'1) -> (eqmap (MapDelta A m1 m2) (MapDelta A m'1 m2)). + Proof. + Intros. Apply MapDelta_ext. Assumption. + Apply eqmap_refl. + Qed. + + Lemma MapDelta_ext_r : (m1,m2,m'2:(Map A)) + (eqmap m2 m'2) -> (eqmap (MapDelta A m1 m2) (MapDelta A m1 m'2)). + Proof. + Intros. Apply MapDelta_ext. Apply eqmap_refl. + Assumption. + Qed. + + Lemma MapDom_Split_1 : (m:(Map A)) (m':(Map B)) + (eqmap m (MapMerge A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m'))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapMerge_semantics A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m') a). + Rewrite (MapDomRestrBy_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A B m m' a). + Case (MapGet B m' a); Case (MapGet A m a); Trivial. + Qed. + + Lemma MapDom_Split_2 : (m:(Map A)) (m':(Map B)) + (eqmap m (MapMerge A (MapDomRestrBy A B m m') (MapDomRestrTo A B m m'))). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapMerge_semantics A (MapDomRestrBy A B m m') (MapDomRestrTo A B m m') a). + Rewrite (MapDomRestrBy_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A B m m' a). + Case (MapGet B m' a); Case (MapGet A m a); Trivial. + Qed. + + Lemma MapDom_Split_3 : (m:(Map A)) (m':(Map B)) + (eqmap (MapDomRestrTo A A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m')) + (M0 A)). + Proof. + Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrTo_semantics A A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m') a). + Rewrite (MapDomRestrBy_semantics A B m m' a). + Rewrite (MapDomRestrTo_semantics A B m m' a). + Case (MapGet B m' a); Case (MapGet A m a); Trivial. + Qed. + +End MapAxioms. + +Lemma MapDomRestrTo_ext : (A,B:Set) + (m1:(Map A)) (m2:(Map B)) (m'1:(Map A)) (m'2:(Map B)) + (eqmap A m1 m'1) -> (eqmap B m2 m'2) -> + (eqmap A (MapDomRestrTo A B m1 m2) (MapDomRestrTo A B m'1 m'2)). +Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDomRestrTo_semantics A B m1 m2 a). + Rewrite (MapDomRestrTo_semantics A B m'1 m'2 a). Rewrite (H a). Rewrite (H0 a). Reflexivity. +Qed. + +Lemma MapDomRestrTo_ext_l : (A,B:Set) (m1:(Map A)) (m2:(Map B)) (m'1:(Map A)) + (eqmap A m1 m'1) -> + (eqmap A (MapDomRestrTo A B m1 m2) (MapDomRestrTo A B m'1 m2)). +Proof. + Intros. Apply MapDomRestrTo_ext; [ Assumption | Apply eqmap_refl ]. +Qed. + +Lemma MapDomRestrTo_ext_r : (A,B:Set) (m1:(Map A)) (m2:(Map B)) (m'2:(Map B)) + (eqmap B m2 m'2) -> + (eqmap A (MapDomRestrTo A B m1 m2) (MapDomRestrTo A B m1 m'2)). +Proof. + Intros. Apply MapDomRestrTo_ext; [ Apply eqmap_refl | Assumption ]. +Qed. + +Lemma MapDomRestrBy_ext : (A,B:Set) + (m1:(Map A)) (m2:(Map B)) (m'1:(Map A)) (m'2:(Map B)) + (eqmap A m1 m'1) -> (eqmap B m2 m'2) -> + (eqmap A (MapDomRestrBy A B m1 m2) (MapDomRestrBy A B m'1 m'2)). +Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDomRestrBy_semantics A B m1 m2 a). + Rewrite (MapDomRestrBy_semantics A B m'1 m'2 a). Rewrite (H a). Rewrite (H0 a). Reflexivity. +Qed. + +Lemma MapDomRestrBy_ext_l : (A,B:Set) (m1:(Map A)) (m2:(Map B)) (m'1:(Map A)) + (eqmap A m1 m'1) -> + (eqmap A (MapDomRestrBy A B m1 m2) (MapDomRestrBy A B m'1 m2)). +Proof. + Intros. Apply MapDomRestrBy_ext; [ Assumption | Apply eqmap_refl ]. +Qed. + +Lemma MapDomRestrBy_ext_r : (A,B:Set) (m1:(Map A)) (m2:(Map B)) (m'2:(Map B)) + (eqmap B m2 m'2) -> + (eqmap A (MapDomRestrBy A B m1 m2) (MapDomRestrBy A B m1 m'2)). +Proof. + Intros. Apply MapDomRestrBy_ext; [ Apply eqmap_refl | Assumption ]. +Qed. + +Lemma MapDomRestrBy_m_m : (A:Set) (m:(Map A)) + (eqmap A (MapDomRestrBy A unit m (MapDom A m)) (M0 A)). +Proof. + Intros. Apply eqmap_trans with m':=(MapDomRestrBy A A m m). Apply eqmap_sym. + Apply MapDomRestrBy_Dom. + Apply MapDomRestrBy_m_m_1. +Qed. + +Lemma FSetDelta_assoc : (s,s',s'':FSet) + (eqmap unit (MapDelta ? (MapDelta ? s s') s'') (MapDelta ? s (MapDelta ? s' s''))). +Proof. + Unfold eqmap eqm. Intros. Rewrite (MapDelta_semantics unit (MapDelta unit s s') s'' a). + Rewrite (MapDelta_semantics unit s s' a). + Rewrite (MapDelta_semantics unit s (MapDelta unit s' s'') a). + Rewrite (MapDelta_semantics unit s' s'' a). + Case (MapGet ? s a); Case (MapGet ? s' a); Case (MapGet ? s'' a); Trivial. + Intros. Elim u. Elim u1. Reflexivity. +Qed. + +Lemma FSet_ext : (s,s':FSet) ((a:ad) (in_FSet a s)=(in_FSet a s')) -> (eqmap unit s s'). +Proof. + Unfold in_FSet eqmap eqm. Intros. Elim (sumbool_of_bool (in_dom ? a s)). Intro H0. + Elim (in_dom_some ? s a H0). Intros y H1. Rewrite (H a) in H0. Elim (in_dom_some ? s' a H0). + Intros y' H2. Rewrite H1. Rewrite H2. Elim y. Elim y'. Reflexivity. + Intro H0. Rewrite (in_dom_none ? s a H0). Rewrite (H a) in H0. Rewrite (in_dom_none ? s' a H0). + Reflexivity. +Qed. + +Lemma FSetUnion_comm : (s,s':FSet) (eqmap unit (FSetUnion s s') (FSetUnion s' s)). +Proof. + Intros. Apply FSet_ext. Intro. Rewrite in_FSet_union. Rewrite in_FSet_union. Apply orb_sym. +Qed. + +Lemma FSetUnion_assoc : (s,s',s'':FSet) (eqmap unit + (FSetUnion (FSetUnion s s') s'') (FSetUnion s (FSetUnion s' s''))). +Proof. + Exact (MapMerge_assoc unit). +Qed. + +Lemma FSetUnion_M0_s : (s:FSet) (eqmap unit (FSetUnion (M0 unit) s) s). +Proof. + Exact (MapMerge_empty_m unit). +Qed. + +Lemma FSetUnion_s_M0 : (s:FSet) (eqmap unit (FSetUnion s (M0 unit)) s). +Proof. + Exact (MapMerge_m_empty unit). +Qed. + +Lemma FSetUnion_idempotent : (s:FSet) (eqmap unit (FSetUnion s s) s). +Proof. + Exact (MapMerge_idempotent unit). +Qed. + +Lemma FSetInter_comm : (s,s':FSet) (eqmap unit (FSetInter s s') (FSetInter s' s)). +Proof. + Intros. Apply FSet_ext. Intro. Rewrite in_FSet_inter. Rewrite in_FSet_inter. Apply andb_sym. +Qed. + +Lemma FSetInter_assoc : (s,s',s'':FSet) (eqmap unit + (FSetInter (FSetInter s s') s'') (FSetInter s (FSetInter s' s''))). +Proof. + Exact (MapDomRestrTo_assoc unit unit unit). +Qed. + +Lemma FSetInter_M0_s : (s:FSet) (eqmap unit (FSetInter (M0 unit) s) (M0 unit)). +Proof. + Exact (MapDomRestrTo_empty_m unit unit). +Qed. + +Lemma FSetInter_s_M0 : (s:FSet) (eqmap unit (FSetInter s (M0 unit)) (M0 unit)). +Proof. + Exact (MapDomRestrTo_m_empty unit unit). +Qed. + +Lemma FSetInter_idempotent : (s:FSet) (eqmap unit (FSetInter s s) s). +Proof. + Exact (MapDomRestrTo_idempotent unit). +Qed. + +Lemma FSetUnion_Inter_l : (s,s',s'':FSet) (eqmap unit + (FSetUnion (FSetInter s s') s'') (FSetInter (FSetUnion s s'') (FSetUnion s' s''))). +Proof. + Intros. Apply FSet_ext. Intro. Rewrite in_FSet_union. Rewrite in_FSet_inter. + Rewrite in_FSet_inter. Rewrite in_FSet_union. Rewrite in_FSet_union. + Case (in_FSet a s); Case (in_FSet a s'); Case (in_FSet a s''); Reflexivity. +Qed. + +Lemma FSetUnion_Inter_r : (s,s',s'':FSet) (eqmap unit + (FSetUnion s (FSetInter s' s'')) (FSetInter (FSetUnion s s') (FSetUnion s s''))). +Proof. + Intros. Apply FSet_ext. Intro. Rewrite in_FSet_union. Rewrite in_FSet_inter. + Rewrite in_FSet_inter. Rewrite in_FSet_union. Rewrite in_FSet_union. + Case (in_FSet a s); Case (in_FSet a s'); Case (in_FSet a s''); Reflexivity. +Qed. + +Lemma FSetInter_Union_l : (s,s',s'':FSet) (eqmap unit + (FSetInter (FSetUnion s s') s'') (FSetUnion (FSetInter s s'') (FSetInter s' s''))). +Proof. + Intros. Apply FSet_ext. Intro. Rewrite in_FSet_inter. Rewrite in_FSet_union. + Rewrite in_FSet_union. Rewrite in_FSet_inter. Rewrite in_FSet_inter. + Case (in_FSet a s); Case (in_FSet a s'); Case (in_FSet a s''); Reflexivity. +Qed. + +Lemma FSetInter_Union_r : (s,s',s'':FSet) (eqmap unit + (FSetInter s (FSetUnion s' s'')) (FSetUnion (FSetInter s s') (FSetInter s s''))). +Proof. + Intros. Apply FSet_ext. Intro. Rewrite in_FSet_inter. Rewrite in_FSet_union. + Rewrite in_FSet_union. Rewrite in_FSet_inter. Rewrite in_FSet_inter. + Case (in_FSet a s); Case (in_FSet a s'); Case (in_FSet a s''); Reflexivity. +Qed. diff --git a/theories7/IntMap/Mapc.v b/theories7/IntMap/Mapc.v new file mode 100644 index 00000000..181050b1 --- /dev/null +++ b/theories7/IntMap/Mapc.v @@ -0,0 +1,457 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Mapc.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require Arith. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Mapaxioms. +Require Fset. +Require Mapiter. +Require Mapsubset. +Require PolyList. +Require Lsort. +Require Mapcard. +Require Mapcanon. + +Section MapC. + + Variable A, B, C : Set. + + Lemma MapPut_as_Merge_c : (m:(Map A)) (mapcanon A m) -> + (a:ad) (y:A) (MapPut A m a y)=(MapMerge A m (M1 A a y)). + Proof. + Intros. Apply mapcanon_unique. Exact (MapPut_canon A m H a y). + Apply MapMerge_canon. Assumption. + Apply M1_canon. + Apply MapPut_as_Merge. + Qed. + + Lemma MapPut_behind_as_Merge_c : (m:(Map A)) (mapcanon A m) -> + (a:ad) (y:A) (MapPut_behind A m a y)=(MapMerge A (M1 A a y) m). + Proof. + Intros. Apply mapcanon_unique. Exact (MapPut_behind_canon A m H a y). + Apply MapMerge_canon. Apply M1_canon. + Assumption. + Apply MapPut_behind_as_Merge. + Qed. + + Lemma MapMerge_empty_m_c : (m:(Map A)) (MapMerge A (M0 A) m)=m. + Proof. + Trivial. + Qed. + + Lemma MapMerge_assoc_c : (m,m',m'':(Map A)) + (mapcanon A m) -> (mapcanon A m') -> (mapcanon A m'') -> + (MapMerge A (MapMerge A m m') m'')=(MapMerge A m (MapMerge A m' m'')). + Proof. + Intros. Apply mapcanon_unique. + (Apply MapMerge_canon; Try Assumption). (Apply MapMerge_canon; Try Assumption). + (Apply MapMerge_canon; Try Assumption). (Apply MapMerge_canon; Try Assumption). + Apply MapMerge_assoc. + Qed. + + Lemma MapMerge_idempotent_c : (m:(Map A)) (mapcanon A m) -> (MapMerge A m m)=m. + Proof. + Intros. Apply mapcanon_unique. (Apply MapMerge_canon; Assumption). + Assumption. + Apply MapMerge_idempotent. + Qed. + + Lemma MapMerge_RestrTo_l_c : (m,m',m'':(Map A)) + (mapcanon A m) -> (mapcanon A m'') -> + (MapMerge A (MapDomRestrTo A A m m') m'')= + (MapDomRestrTo A A (MapMerge A m m'') (MapMerge A m' m'')). + Proof. + Intros. Apply mapcanon_unique. Apply MapMerge_canon. Apply MapDomRestrTo_canon; Assumption. + Assumption. + Apply MapDomRestrTo_canon; Apply MapMerge_canon; Assumption. + Apply MapMerge_RestrTo_l. + Qed. + + Lemma MapRemove_as_RestrBy_c : (m:(Map A)) (mapcanon A m) -> + (a:ad) (y:B) (MapRemove A m a)=(MapDomRestrBy A B m (M1 B a y)). + Proof. + Intros. Apply mapcanon_unique. (Apply MapRemove_canon; Assumption). + (Apply MapDomRestrBy_canon; Assumption). + Apply MapRemove_as_RestrBy. + Qed. + + Lemma MapDomRestrTo_assoc_c : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (mapcanon A m) -> + (MapDomRestrTo A C (MapDomRestrTo A B m m') m'')= + (MapDomRestrTo A B m (MapDomRestrTo B C m' m'')). + Proof. + Intros. Apply mapcanon_unique. (Apply MapDomRestrTo_canon; Try Assumption). + (Apply MapDomRestrTo_canon; Try Assumption). + (Apply MapDomRestrTo_canon; Try Assumption). + Apply MapDomRestrTo_assoc. + Qed. + + Lemma MapDomRestrTo_idempotent_c : (m:(Map A)) (mapcanon A m) -> + (MapDomRestrTo A A m m)=m. + Proof. + Intros. Apply mapcanon_unique. (Apply MapDomRestrTo_canon; Assumption). + Assumption. + Apply MapDomRestrTo_idempotent. + Qed. + + Lemma MapDomRestrTo_Dom_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) -> + (MapDomRestrTo A B m m')=(MapDomRestrTo A unit m (MapDom B m')). + Proof. + Intros. Apply mapcanon_unique. (Apply MapDomRestrTo_canon; Assumption). + (Apply MapDomRestrTo_canon; Assumption). + Apply MapDomRestrTo_Dom. + Qed. + + Lemma MapDomRestrBy_Dom_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) -> + (MapDomRestrBy A B m m')=(MapDomRestrBy A unit m (MapDom B m')). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon; Assumption. + Apply MapDomRestrBy_canon; Assumption. + Apply MapDomRestrBy_Dom. + Qed. + + Lemma MapDomRestrBy_By_c : (m:(Map A)) (m':(Map B)) (m'':(Map B)) + (mapcanon A m) -> + (MapDomRestrBy A B (MapDomRestrBy A B m m') m'')= + (MapDomRestrBy A B m (MapMerge B m' m'')). + Proof. + Intros. Apply mapcanon_unique. (Apply MapDomRestrBy_canon; Try Assumption). + (Apply MapDomRestrBy_canon; Try Assumption). + (Apply MapDomRestrBy_canon; Try Assumption). + Apply MapDomRestrBy_By. + Qed. + + Lemma MapDomRestrBy_By_comm_c : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (mapcanon A m) -> + (MapDomRestrBy A C (MapDomRestrBy A B m m') m'')= + (MapDomRestrBy A B (MapDomRestrBy A C m m'') m'). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon. + (Apply MapDomRestrBy_canon; Assumption). + Apply MapDomRestrBy_canon. (Apply MapDomRestrBy_canon; Assumption). + Apply MapDomRestrBy_By_comm. + Qed. + + Lemma MapDomRestrBy_To_c : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (mapcanon A m) -> + (MapDomRestrBy A C (MapDomRestrTo A B m m') m'')= + (MapDomRestrTo A B m (MapDomRestrBy B C m' m'')). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon. + (Apply MapDomRestrTo_canon; Assumption). + (Apply MapDomRestrTo_canon; Assumption). + Apply MapDomRestrBy_To. + Qed. + + Lemma MapDomRestrBy_To_comm_c : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (mapcanon A m) -> + (MapDomRestrBy A C (MapDomRestrTo A B m m') m'')= + (MapDomRestrTo A B (MapDomRestrBy A C m m'') m'). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon. + Apply MapDomRestrTo_canon; Assumption. + Apply MapDomRestrTo_canon. Apply MapDomRestrBy_canon; Assumption. + Apply MapDomRestrBy_To_comm. + Qed. + + Lemma MapDomRestrTo_By_c : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (mapcanon A m) -> + (MapDomRestrTo A C (MapDomRestrBy A B m m') m'')= + (MapDomRestrTo A C m (MapDomRestrBy C B m'' m')). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon. + Apply MapDomRestrBy_canon; Assumption. + Apply MapDomRestrTo_canon; Assumption. + Apply MapDomRestrTo_By. + Qed. + + Lemma MapDomRestrTo_By_comm_c : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (mapcanon A m) -> + (MapDomRestrTo A C (MapDomRestrBy A B m m') m'')= + (MapDomRestrBy A B (MapDomRestrTo A C m m'') m'). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon. + (Apply MapDomRestrBy_canon; Assumption). + Apply MapDomRestrBy_canon. (Apply MapDomRestrTo_canon; Assumption). + Apply MapDomRestrTo_By_comm. + Qed. + + Lemma MapDomRestrTo_To_comm_c : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (mapcanon A m) -> + (MapDomRestrTo A C (MapDomRestrTo A B m m') m'')= + (MapDomRestrTo A B (MapDomRestrTo A C m m'') m'). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon. + Apply MapDomRestrTo_canon; Assumption. + Apply MapDomRestrTo_canon. Apply MapDomRestrTo_canon; Assumption. + Apply MapDomRestrTo_To_comm. + Qed. + + Lemma MapMerge_DomRestrTo_c : (m,m':(Map A)) (m'':(Map B)) + (mapcanon A m) -> (mapcanon A m') -> + (MapDomRestrTo A B (MapMerge A m m') m'')= + (MapMerge A (MapDomRestrTo A B m m'') (MapDomRestrTo A B m' m'')). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon. + (Apply MapMerge_canon; Assumption). + Apply MapMerge_canon. (Apply MapDomRestrTo_canon; Assumption). + (Apply MapDomRestrTo_canon; Assumption). + Apply MapMerge_DomRestrTo. + Qed. + + Lemma MapMerge_DomRestrBy_c : (m,m':(Map A)) (m'':(Map B)) + (mapcanon A m) -> (mapcanon A m') -> + (MapDomRestrBy A B (MapMerge A m m') m'')= + (MapMerge A (MapDomRestrBy A B m m'') (MapDomRestrBy A B m' m'')). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon. Apply MapMerge_canon; Assumption. + Apply MapMerge_canon. Apply MapDomRestrBy_canon; Assumption. + Apply MapDomRestrBy_canon; Assumption. + Apply MapMerge_DomRestrBy. + Qed. + + Lemma MapDelta_nilpotent_c : (m:(Map A)) (mapcanon A m) -> + (MapDelta A m m)=(M0 A). + Proof. + Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption). + Apply M0_canon. + Apply MapDelta_nilpotent. + Qed. + + Lemma MapDelta_as_Merge_c : (m,m':(Map A)) + (mapcanon A m) -> (mapcanon A m') -> + (MapDelta A m m')= + (MapMerge A (MapDomRestrBy A A m m') (MapDomRestrBy A A m' m)). + Proof. + Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption). + (Apply MapMerge_canon; Apply MapDomRestrBy_canon; Assumption). + Apply MapDelta_as_Merge. + Qed. + + Lemma MapDelta_as_DomRestrBy_c : (m,m':(Map A)) + (mapcanon A m) -> (mapcanon A m') -> + (MapDelta A m m')= + (MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m m')). + Proof. + Intros. Apply mapcanon_unique. Apply MapDelta_canon; Assumption. + Apply MapDomRestrBy_canon. (Apply MapMerge_canon; Assumption). + Apply MapDelta_as_DomRestrBy. + Qed. + + Lemma MapDelta_as_DomRestrBy_2_c : (m,m':(Map A)) + (mapcanon A m) -> (mapcanon A m') -> + (MapDelta A m m')= + (MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m' m)). + Proof. + Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption). + Apply MapDomRestrBy_canon. Apply MapMerge_canon; Assumption. + Apply MapDelta_as_DomRestrBy_2. + Qed. + + Lemma MapDelta_sym_c : (m,m':(Map A)) + (mapcanon A m) -> (mapcanon A m') -> (MapDelta A m m')=(MapDelta A m' m). + Proof. + Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption). + (Apply MapDelta_canon; Assumption). Apply MapDelta_sym. + Qed. + + Lemma MapDom_Split_1_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) -> + m=(MapMerge A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m')). + Proof. + Intros. Apply mapcanon_unique. Assumption. + Apply MapMerge_canon. Apply MapDomRestrTo_canon; Assumption. + Apply MapDomRestrBy_canon; Assumption. + Apply MapDom_Split_1. + Qed. + + Lemma MapDom_Split_2_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) -> + m=(MapMerge A (MapDomRestrBy A B m m') (MapDomRestrTo A B m m')). + Proof. + Intros. Apply mapcanon_unique. Assumption. + Apply MapMerge_canon. (Apply MapDomRestrBy_canon; Assumption). + (Apply MapDomRestrTo_canon; Assumption). + Apply MapDom_Split_2. + Qed. + + Lemma MapDom_Split_3_c : (m:(Map A)) (m':(Map B)) (mapcanon A m) -> + (MapDomRestrTo A A (MapDomRestrTo A B m m') (MapDomRestrBy A B m m'))= + (M0 A). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrTo_canon. + Apply MapDomRestrTo_canon; Assumption. + Apply M0_canon. + Apply MapDom_Split_3. + Qed. + + Lemma Map_of_alist_of_Map_c : (m:(Map A)) (mapcanon A m) -> + (Map_of_alist A (alist_of_Map A m))=m. + Proof. + Intros. (Apply mapcanon_unique; Try Assumption). Apply Map_of_alist_canon. + Apply Map_of_alist_of_Map. + Qed. + + Lemma alist_of_Map_of_alist_c : (l:(alist A)) (alist_sorted_2 A l) -> + (alist_of_Map A (Map_of_alist A l))=l. + Proof. + Intros. Apply alist_canonical. Apply alist_of_Map_of_alist. + Apply alist_of_Map_sorts2. + Assumption. + Qed. + + Lemma MapSubset_antisym_c : (m:(Map A)) (m':(Map B)) + (mapcanon A m) -> (mapcanon B m') -> + (MapSubset A B m m') -> (MapSubset B A m' m) -> (MapDom A m)=(MapDom B m'). + Proof. + Intros. Apply (mapcanon_unique unit). (Apply MapDom_canon; Assumption). + (Apply MapDom_canon; Assumption). + (Apply MapSubset_antisym; Assumption). + Qed. + + Lemma FSubset_antisym_c : (s,s':FSet) (mapcanon unit s) -> (mapcanon unit s') -> + (MapSubset ? ? s s') -> (MapSubset ? ? s' s) -> s=s'. + Proof. + Intros. Apply (mapcanon_unique unit); Try Assumption. Apply FSubset_antisym; Assumption. + Qed. + + Lemma MapDisjoint_empty_c : (m:(Map A)) (mapcanon A m) -> + (MapDisjoint A A m m) -> m=(M0 A). + Proof. + Intros. Apply mapcanon_unique; Try Assumption; Try Apply M0_canon. + Apply MapDisjoint_empty; Assumption. + Qed. + + Lemma MapDelta_disjoint_c : (m,m':(Map A)) (mapcanon A m) -> (mapcanon A m') -> + (MapDisjoint A A m m') -> (MapDelta A m m')=(MapMerge A m m'). + Proof. + Intros. Apply mapcanon_unique. (Apply MapDelta_canon; Assumption). + (Apply MapMerge_canon; Assumption). Apply MapDelta_disjoint; Assumption. + Qed. + +End MapC. + +Lemma FSetDelta_assoc_c : (s,s',s'':FSet) + (mapcanon unit s) -> (mapcanon unit s') -> (mapcanon unit s'') -> + (MapDelta ? (MapDelta ? s s') s'')=(MapDelta ? s (MapDelta ? s' s'')). +Proof. + Intros. Apply (mapcanon_unique unit). Apply MapDelta_canon. (Apply MapDelta_canon; Assumption). + Assumption. + Apply MapDelta_canon. Assumption. + (Apply MapDelta_canon; Assumption). + Apply FSetDelta_assoc; Assumption. +Qed. + +Lemma FSet_ext_c : (s,s':FSet) (mapcanon unit s) -> (mapcanon unit s') -> + ((a:ad) (in_FSet a s)=(in_FSet a s')) -> s=s'. +Proof. + Intros. (Apply (mapcanon_unique unit); Try Assumption). Apply FSet_ext. Assumption. +Qed. + +Lemma FSetUnion_comm_c : (s,s':FSet) (mapcanon unit s) -> (mapcanon unit s') -> + (FSetUnion s s')=(FSetUnion s' s). +Proof. + Intros. + Apply (mapcanon_unique unit); Try (Unfold FSetUnion; Apply MapMerge_canon; Assumption). + Apply FSetUnion_comm. +Qed. + +Lemma FSetUnion_assoc_c : (s,s',s'':FSet) + (mapcanon unit s) -> (mapcanon unit s') -> (mapcanon unit s'') -> + (FSetUnion (FSetUnion s s') s'')=(FSetUnion s (FSetUnion s' s'')). +Proof. + Exact (MapMerge_assoc_c unit). +Qed. + +Lemma FSetUnion_M0_s_c : (s:FSet) (FSetUnion (M0 unit) s)=s. +Proof. + Exact (MapMerge_empty_m_c unit). +Qed. + +Lemma FSetUnion_s_M0_c : (s:FSet) (FSetUnion s (M0 unit))=s. +Proof. + Exact (MapMerge_m_empty_1 unit). +Qed. + +Lemma FSetUnion_idempotent : (s:FSet) (mapcanon unit s) -> (FSetUnion s s)=s. +Proof. + Exact (MapMerge_idempotent_c unit). +Qed. + +Lemma FSetInter_comm_c : (s,s':FSet) (mapcanon unit s) -> (mapcanon unit s') -> + (FSetInter s s')=(FSetInter s' s). +Proof. + Intros. + Apply (mapcanon_unique unit); Try (Unfold FSetInter; Apply MapDomRestrTo_canon; Assumption). + Apply FSetInter_comm. +Qed. + +Lemma FSetInter_assoc_c : (s,s',s'':FSet) + (mapcanon unit s) -> + (FSetInter (FSetInter s s') s'')=(FSetInter s (FSetInter s' s'')). +Proof. + Exact (MapDomRestrTo_assoc_c unit unit unit). +Qed. + +Lemma FSetInter_M0_s_c : (s:FSet) (FSetInter (M0 unit) s)=(M0 unit). +Proof. + Trivial. +Qed. + +Lemma FSetInter_s_M0_c : (s:FSet) (FSetInter s (M0 unit))=(M0 unit). +Proof. + Exact (MapDomRestrTo_m_empty_1 unit unit). +Qed. + +Lemma FSetInter_idempotent : (s:FSet) (mapcanon unit s) -> (FSetInter s s)=s. +Proof. + Exact (MapDomRestrTo_idempotent_c unit). +Qed. + +Lemma FSetUnion_Inter_l_c : (s,s',s'':FSet) (mapcanon unit s) -> (mapcanon unit s'') -> + (FSetUnion (FSetInter s s') s'')=(FSetInter (FSetUnion s s'') (FSetUnion s' s'')). +Proof. + Intros. Apply (mapcanon_unique unit). Unfold FSetUnion. (Apply MapMerge_canon; Try Assumption). + Unfold FSetInter. (Apply MapDomRestrTo_canon; Assumption). + Unfold FSetInter; Unfold FSetUnion; Apply MapDomRestrTo_canon; Apply MapMerge_canon; Assumption. + Apply FSetUnion_Inter_l. +Qed. + +Lemma FSetUnion_Inter_r : (s,s',s'':FSet) (mapcanon unit s) -> (mapcanon unit s') -> + (FSetUnion s (FSetInter s' s''))=(FSetInter (FSetUnion s s') (FSetUnion s s'')). +Proof. + Intros. Apply (mapcanon_unique unit). Unfold FSetUnion. (Apply MapMerge_canon; Try Assumption). + Unfold FSetInter. (Apply MapDomRestrTo_canon; Assumption). + Unfold FSetInter; Unfold FSetUnion; Apply MapDomRestrTo_canon; Apply MapMerge_canon; Assumption. + Apply FSetUnion_Inter_r. +Qed. + +Lemma FSetInter_Union_l_c : (s,s',s'':FSet) (mapcanon unit s) -> (mapcanon unit s') -> + (FSetInter (FSetUnion s s') s'')=(FSetUnion (FSetInter s s'') (FSetInter s' s'')). +Proof. + Intros. Apply (mapcanon_unique unit). Unfold FSetInter. + Apply MapDomRestrTo_canon; Try Assumption. Unfold FSetUnion. + Apply MapMerge_canon; Assumption. + Unfold FSetUnion; Unfold FSetInter; Apply MapMerge_canon; Apply MapDomRestrTo_canon; + Assumption. + Apply FSetInter_Union_l. +Qed. + +Lemma FSetInter_Union_r : (s,s',s'':FSet) (mapcanon unit s) -> (mapcanon unit s') -> + (FSetInter s (FSetUnion s' s''))=(FSetUnion (FSetInter s s') (FSetInter s s'')). +Proof. + Intros. Apply (mapcanon_unique unit). Unfold FSetInter. + Apply MapDomRestrTo_canon; Try Assumption. + Unfold FSetUnion. Apply MapMerge_canon; Unfold FSetInter; Apply MapDomRestrTo_canon; Assumption. + Apply FSetInter_Union_r. +Qed. diff --git a/theories7/IntMap/Mapcanon.v b/theories7/IntMap/Mapcanon.v new file mode 100644 index 00000000..7beb1fd4 --- /dev/null +++ b/theories7/IntMap/Mapcanon.v @@ -0,0 +1,376 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Mapcanon.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require Arith. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Mapaxioms. +Require Mapiter. +Require Fset. +Require PolyList. +Require Lsort. +Require Mapsubset. +Require Mapcard. + +Section MapCanon. + + Variable A : Set. + + Inductive mapcanon : (Map A) -> Prop := + M0_canon : (mapcanon (M0 A)) + | M1_canon : (a:ad) (y:A) (mapcanon (M1 A a y)) + | M2_canon : (m1,m2:(Map A)) (mapcanon m1) -> (mapcanon m2) -> + (le (2) (MapCard A (M2 A m1 m2))) -> (mapcanon (M2 A m1 m2)). + + Lemma mapcanon_M2 : + (m1,m2:(Map A)) (mapcanon (M2 A m1 m2)) -> (le (2) (MapCard A (M2 A m1 m2))). + Proof. + Intros. Inversion H. Assumption. + Qed. + + Lemma mapcanon_M2_1 : (m1,m2:(Map A)) (mapcanon (M2 A m1 m2)) -> (mapcanon m1). + Proof. + Intros. Inversion H. Assumption. + Qed. + + Lemma mapcanon_M2_2 : (m1,m2:(Map A)) (mapcanon (M2 A m1 m2)) -> (mapcanon m2). + Proof. + Intros. Inversion H. Assumption. + Qed. + + Lemma M2_eqmap_1 : (m0,m1,m2,m3:(Map A)) + (eqmap A (M2 A m0 m1) (M2 A m2 m3)) -> (eqmap A m0 m2). + Proof. + Unfold eqmap eqm. Intros. Rewrite <- (ad_double_div_2 a). + Rewrite <- (MapGet_M2_bit_0_0 A ? (ad_double_bit_0 a) m0 m1). + Rewrite <- (MapGet_M2_bit_0_0 A ? (ad_double_bit_0 a) m2 m3). + Exact (H (ad_double a)). + Qed. + + Lemma M2_eqmap_2 : (m0,m1,m2,m3:(Map A)) + (eqmap A (M2 A m0 m1) (M2 A m2 m3)) -> (eqmap A m1 m3). + Proof. + Unfold eqmap eqm. Intros. Rewrite <- (ad_double_plus_un_div_2 a). + Rewrite <- (MapGet_M2_bit_0_1 A ? (ad_double_plus_un_bit_0 a) m0 m1). + Rewrite <- (MapGet_M2_bit_0_1 A ? (ad_double_plus_un_bit_0 a) m2 m3). + Exact (H (ad_double_plus_un a)). + Qed. + + Lemma mapcanon_unique : (m,m':(Map A)) (mapcanon m) -> (mapcanon m') -> + (eqmap A m m') -> m=m'. + Proof. + Induction m. Induction m'. Trivial. + Intros a y H H0 H1. Cut (NONE A)=(MapGet A (M1 A a y) a). Simpl. Rewrite (ad_eq_correct a). + Intro. Discriminate H2. + Exact (H1 a). + Intros. Cut (le (2) (MapCard A (M0 A))). Intro. Elim (le_Sn_O ? H4). + Rewrite (MapCard_ext A ? ? H3). Exact (mapcanon_M2 ? ? H2). + Intros a y. Induction m'. Intros. Cut (MapGet A (M1 A a y) a)=(NONE A). Simpl. + Rewrite (ad_eq_correct a). Intro. Discriminate H2. + Exact (H1 a). + Intros a0 y0 H H0 H1. Cut (MapGet A (M1 A a y) a)=(MapGet A (M1 A a0 y0) a). Simpl. + Rewrite (ad_eq_correct a). Intro. Elim (sumbool_of_bool (ad_eq a0 a)). Intro H3. + Rewrite H3 in H2. Inversion H2. Rewrite (ad_eq_complete ? ? H3). Reflexivity. + Intro H3. Rewrite H3 in H2. Discriminate H2. + Exact (H1 a). + Intros. Cut (le (2) (MapCard A (M1 A a y))). Intro. Elim (le_Sn_O ? (le_S_n ? ? H4)). + Rewrite (MapCard_ext A ? ? H3). Exact (mapcanon_M2 ? ? H2). + Induction m'. Intros. Cut (le (2) (MapCard A (M0 A))). Intro. Elim (le_Sn_O ? H4). + Rewrite <- (MapCard_ext A ? ? H3). Exact (mapcanon_M2 ? ? H1). + Intros a y H1 H2 H3. Cut (le (2) (MapCard A (M1 A a y))). Intro. + Elim (le_Sn_O ? (le_S_n ? ? H4)). + Rewrite <- (MapCard_ext A ? ? H3). Exact (mapcanon_M2 ? ? H1). + Intros. Rewrite (H m2). Rewrite (H0 m3). Reflexivity. + Exact (mapcanon_M2_2 ? ? H3). + Exact (mapcanon_M2_2 ? ? H4). + Exact (M2_eqmap_2 ? ? ? ? H5). + Exact (mapcanon_M2_1 ? ? H3). + Exact (mapcanon_M2_1 ? ? H4). + Exact (M2_eqmap_1 ? ? ? ? H5). + Qed. + + Lemma MapPut1_canon : + (p:positive) (a,a':ad) (y,y':A) (mapcanon (MapPut1 A a y a' y' p)). + Proof. + Induction p. Simpl. Intros. Case (ad_bit_0 a). Apply M2_canon. Apply M1_canon. + Apply M1_canon. + Apply le_n. + Apply M2_canon. Apply M1_canon. + Apply M1_canon. + Apply le_n. + Simpl. Intros. Case (ad_bit_0 a). Apply M2_canon. Apply M0_canon. + Apply H. + Simpl. Rewrite MapCard_Put1_equals_2. Apply le_n. + Apply M2_canon. Apply H. + Apply M0_canon. + Simpl. Rewrite MapCard_Put1_equals_2. Apply le_n. + Simpl. Simpl. Intros. Case (ad_bit_0 a). Apply M2_canon. Apply M1_canon. + Apply M1_canon. + Simpl. Apply le_n. + Apply M2_canon. Apply M1_canon. + Apply M1_canon. + Simpl. Apply le_n. + Qed. + + Lemma MapPut_canon : + (m:(Map A)) (mapcanon m) -> (a:ad) (y:A) (mapcanon (MapPut A m a y)). + Proof. + Induction m. Intros. Simpl. Apply M1_canon. + Intros a0 y0 H a y. Simpl. Case (ad_xor a0 a). Apply M1_canon. + Intro. Apply MapPut1_canon. + Intros. Simpl. Elim a. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). + Exact (mapcanon_M2_2 m0 m1 H1). + Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). Exact (mapcanon_M2 ? ? H1). + Apply le_plus_plus. Exact (MapCard_Put_lb A m0 ad_z y). + Apply le_n. + Intro. Case p. Intro. Apply M2_canon. Exact (mapcanon_M2_1 m0 m1 H1). + Apply H0. Exact (mapcanon_M2_2 m0 m1 H1). + Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). + Exact (mapcanon_M2 m0 m1 H1). + Apply le_reg_l. Exact (MapCard_Put_lb A m1 (ad_x p0) y). + Intro. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). + Exact (mapcanon_M2_2 m0 m1 H1). + Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). + Exact (mapcanon_M2 m0 m1 H1). + Apply le_reg_r. Exact (MapCard_Put_lb A m0 (ad_x p0) y). + Apply M2_canon. Apply (mapcanon_M2_1 m0 m1 H1). + Apply H0. Apply (mapcanon_M2_2 m0 m1 H1). + Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). + Exact (mapcanon_M2 m0 m1 H1). + Apply le_reg_l. Exact (MapCard_Put_lb A m1 ad_z y). + Qed. + + Lemma MapPut_behind_canon : (m:(Map A)) (mapcanon m) -> + (a:ad) (y:A) (mapcanon (MapPut_behind A m a y)). + Proof. + Induction m. Intros. Simpl. Apply M1_canon. + Intros a0 y0 H a y. Simpl. Case (ad_xor a0 a). Apply M1_canon. + Intro. Apply MapPut1_canon. + Intros. Simpl. Elim a. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). + Exact (mapcanon_M2_2 m0 m1 H1). + Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). Exact (mapcanon_M2 ? ? H1). + Apply le_plus_plus. Rewrite MapCard_Put_behind_Put. Exact (MapCard_Put_lb A m0 ad_z y). + Apply le_n. + Intro. Case p. Intro. Apply M2_canon. Exact (mapcanon_M2_1 m0 m1 H1). + Apply H0. Exact (mapcanon_M2_2 m0 m1 H1). + Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). + Exact (mapcanon_M2 m0 m1 H1). + Apply le_reg_l. Rewrite MapCard_Put_behind_Put. Exact (MapCard_Put_lb A m1 (ad_x p0) y). + Intro. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). + Exact (mapcanon_M2_2 m0 m1 H1). + Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). + Exact (mapcanon_M2 m0 m1 H1). + Apply le_reg_r. Rewrite MapCard_Put_behind_Put. Exact (MapCard_Put_lb A m0 (ad_x p0) y). + Apply M2_canon. Apply (mapcanon_M2_1 m0 m1 H1). + Apply H0. Apply (mapcanon_M2_2 m0 m1 H1). + Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). + Exact (mapcanon_M2 m0 m1 H1). + Apply le_reg_l. Rewrite MapCard_Put_behind_Put. Exact (MapCard_Put_lb A m1 ad_z y). + Qed. + + Lemma makeM2_canon : + (m,m':(Map A)) (mapcanon m) -> (mapcanon m') -> (mapcanon (makeM2 A m m')). + Proof. + Intro. Case m. Intro. Case m'. Intros. Exact M0_canon. + Intros a y H H0. Exact (M1_canon (ad_double_plus_un a) y). + Intros. Simpl. (Apply M2_canon; Try Assumption). Exact (mapcanon_M2 m0 m1 H0). + Intros a y m'. Case m'. Intros. Exact (M1_canon (ad_double a) y). + Intros a0 y0 H H0. Simpl. (Apply M2_canon; Try Assumption). Apply le_n. + Intros. Simpl. (Apply M2_canon; Try Assumption). + Apply le_trans with m:=(MapCard A (M2 A m0 m1)). Exact (mapcanon_M2 ? ? H0). + Exact (le_plus_r (MapCard A (M1 A a y)) (MapCard A (M2 A m0 m1))). + Simpl. Intros. (Apply M2_canon; Try Assumption). + Apply le_trans with m:=(MapCard A (M2 A m0 m1)). Exact (mapcanon_M2 ? ? H). + Exact (le_plus_l (MapCard A (M2 A m0 m1)) (MapCard A m')). + Qed. + + Fixpoint MapCanonicalize [m:(Map A)] : (Map A) := + Cases m of + (M2 m0 m1) => (makeM2 A (MapCanonicalize m0) (MapCanonicalize m1)) + | _ => m + end. + + Lemma mapcanon_exists_1 : (m:(Map A)) (eqmap A m (MapCanonicalize m)). + Proof. + Induction m. Apply eqmap_refl. + Intros. Apply eqmap_refl. + Intros. Simpl. Unfold eqmap eqm. Intro. + Rewrite (makeM2_M2 A (MapCanonicalize m0) (MapCanonicalize m1) a). + Rewrite MapGet_M2_bit_0_if. Rewrite MapGet_M2_bit_0_if. + Rewrite <- (H (ad_div_2 a)). Rewrite <- (H0 (ad_div_2 a)). Reflexivity. + Qed. + + Lemma mapcanon_exists_2 : (m:(Map A)) (mapcanon (MapCanonicalize m)). + Proof. + Induction m. Apply M0_canon. + Intros. Simpl. Apply M1_canon. + Intros. Simpl. (Apply makeM2_canon; Assumption). + Qed. + + Lemma mapcanon_exists : + (m:(Map A)) {m':(Map A) | (eqmap A m m') /\ (mapcanon m')}. + Proof. + Intro. Split with (MapCanonicalize m). Split. Apply mapcanon_exists_1. + Apply mapcanon_exists_2. + Qed. + + Lemma MapRemove_canon : + (m:(Map A)) (mapcanon m) -> (a:ad) (mapcanon (MapRemove A m a)). + Proof. + Induction m. Intros. Exact M0_canon. + Intros a y H a0. Simpl. Case (ad_eq a a0). Exact M0_canon. + Assumption. + Intros. Simpl. Case (ad_bit_0 a). Apply makeM2_canon. Exact (mapcanon_M2_1 ? ? H1). + Apply H0. Exact (mapcanon_M2_2 ? ? H1). + Apply makeM2_canon. Apply H. Exact (mapcanon_M2_1 ? ? H1). + Exact (mapcanon_M2_2 ? ? H1). + Qed. + + Lemma MapMerge_canon : (m,m':(Map A)) (mapcanon m) -> (mapcanon m') -> + (mapcanon (MapMerge A m m')). + Proof. + Induction m. Intros. Exact H0. + Simpl. Intros a y m' H H0. Exact (MapPut_behind_canon m' H0 a y). + Induction m'. Intros. Exact H1. + Intros a y H1 H2. Unfold MapMerge. Exact (MapPut_canon ? H1 a y). + Intros. Simpl. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 ? ? H3). + Exact (mapcanon_M2_1 ? ? H4). + Apply H0. Exact (mapcanon_M2_2 ? ? H3). + Exact (mapcanon_M2_2 ? ? H4). + Change (le (2) (MapCard A (MapMerge A (M2 A m0 m1) (M2 A m2 m3)))). + Apply le_trans with m:=(MapCard A (M2 A m0 m1)). Exact (mapcanon_M2 ? ? H3). + Exact (MapMerge_Card_lb_l A (M2 A m0 m1) (M2 A m2 m3)). + Qed. + + Lemma MapDelta_canon : (m,m':(Map A)) (mapcanon m) -> (mapcanon m') -> + (mapcanon (MapDelta A m m')). + Proof. + Induction m. Intros. Exact H0. + Simpl. Intros a y m' H H0. Case (MapGet A m' a). Exact (MapPut_canon m' H0 a y). + Intro. Exact (MapRemove_canon m' H0 a). + Induction m'. Intros. Exact H1. + Unfold MapDelta. Intros a y H1 H2. Case (MapGet A (M2 A m0 m1) a). + Exact (MapPut_canon ? H1 a y). + Intro. Exact (MapRemove_canon ? H1 a). + Intros. Simpl. Apply makeM2_canon. Apply H. Exact (mapcanon_M2_1 ? ? H3). + Exact (mapcanon_M2_1 ? ? H4). + Apply H0. Exact (mapcanon_M2_2 ? ? H3). + Exact (mapcanon_M2_2 ? ? H4). + Qed. + + Variable B : Set. + + Lemma MapDomRestrTo_canon : (m:(Map A)) (mapcanon m) -> + (m':(Map B)) (mapcanon (MapDomRestrTo A B m m')). + Proof. + Induction m. Intros. Exact M0_canon. + Simpl. Intros a y H m'. Case (MapGet B m' a). Exact M0_canon. + Intro. Apply M1_canon. + Induction m'. Exact M0_canon. + Unfold MapDomRestrTo. Intros a y. Case (MapGet A (M2 A m0 m1) a). Exact M0_canon. + Intro. Apply M1_canon. + Intros. Simpl. Apply makeM2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). + Apply H0. Exact (mapcanon_M2_2 m0 m1 H1). + Qed. + + Lemma MapDomRestrBy_canon : (m:(Map A)) (mapcanon m) -> + (m':(Map B)) (mapcanon (MapDomRestrBy A B m m')). + Proof. + Induction m. Intros. Exact M0_canon. + Simpl. Intros a y H m'. Case (MapGet B m' a). Assumption. + Intro. Exact M0_canon. + Induction m'. Exact H1. + Intros a y. Simpl. Case (ad_bit_0 a). Apply makeM2_canon. Exact (mapcanon_M2_1 ? ? H1). + Apply MapRemove_canon. Exact (mapcanon_M2_2 ? ? H1). + Apply makeM2_canon. Apply MapRemove_canon. Exact (mapcanon_M2_1 ? ? H1). + Exact (mapcanon_M2_2 ? ? H1). + Intros. Simpl. Apply makeM2_canon. Apply H. Exact (mapcanon_M2_1 ? ? H1). + Apply H0. Exact (mapcanon_M2_2 ? ? H1). + Qed. + + Lemma Map_of_alist_canon : (l:(alist A)) (mapcanon (Map_of_alist A l)). + Proof. + Induction l. Exact M0_canon. + Intro r. Elim r. Intros a y l0 H. Simpl. Apply MapPut_canon. Assumption. + Qed. + + Lemma MapSubset_c_1 : (m:(Map A)) (m':(Map B)) (mapcanon m) -> + (MapSubset A B m m') -> (MapDomRestrBy A B m m')=(M0 A). + Proof. + Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon. Assumption. + Apply M0_canon. + Exact (MapSubset_imp_2 ? ? m m' H0). + Qed. + + Lemma MapSubset_c_2 : (m:(Map A)) (m':(Map B)) + (MapDomRestrBy A B m m')=(M0 A) -> (MapSubset A B m m'). + Proof. + Intros. Apply MapSubset_2_imp. Unfold MapSubset_2. Rewrite H. Apply eqmap_refl. + Qed. + +End MapCanon. + +Section FSetCanon. + + Variable A : Set. + + Lemma MapDom_canon : (m:(Map A)) (mapcanon A m) -> (mapcanon unit (MapDom A m)). + Proof. + Induction m. Intro. Exact (M0_canon unit). + Intros a y H. Exact (M1_canon unit a ?). + Intros. Simpl. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 A ? ? H1). + Apply H0. Exact (mapcanon_M2_2 A ? ? H1). + Change (le (2) (MapCard unit (MapDom A (M2 A m0 m1)))). Rewrite <- MapCard_Dom. + Exact (mapcanon_M2 A ? ? H1). + Qed. + +End FSetCanon. + +Section MapFoldCanon. + + Variable A, B : Set. + + Lemma MapFold_canon_1 : (m0:(Map B)) (mapcanon B m0) -> + (op : (Map B) -> (Map B) -> (Map B)) + ((m1:(Map B)) (mapcanon B m1) -> (m2:(Map B)) (mapcanon B m2) -> + (mapcanon B (op m1 m2))) -> + (f : ad->A->(Map B)) ((a:ad) (y:A) (mapcanon B (f a y))) -> + (m:(Map A)) (pf : ad->ad) (mapcanon B (MapFold1 A (Map B) m0 op f pf m)). + Proof. + Induction m. Intro. Exact H. + Intros a y pf. Simpl. Apply H1. + Intros. Simpl. Apply H0. Apply H2. + Apply H3. + Qed. + + Lemma MapFold_canon : (m0:(Map B)) (mapcanon B m0) -> + (op : (Map B) -> (Map B) -> (Map B)) + ((m1:(Map B)) (mapcanon B m1) -> (m2:(Map B)) (mapcanon B m2) -> + (mapcanon B (op m1 m2))) -> + (f : ad->A->(Map B)) ((a:ad) (y:A) (mapcanon B (f a y))) -> + (m:(Map A)) (mapcanon B (MapFold A (Map B) m0 op f m)). + Proof. + Intros. Exact (MapFold_canon_1 m0 H op H0 f H1 m [a:ad]a). + Qed. + + Lemma MapCollect_canon : + (f : ad->A->(Map B)) ((a:ad) (y:A) (mapcanon B (f a y))) -> + (m:(Map A)) (mapcanon B (MapCollect A B f m)). + Proof. + Intros. Rewrite MapCollect_as_Fold. Apply MapFold_canon. Apply M0_canon. + Intros. Exact (MapMerge_canon B m1 m2 H0 H1). + Assumption. + Qed. + +End MapFoldCanon. diff --git a/theories7/IntMap/Mapcard.v b/theories7/IntMap/Mapcard.v new file mode 100644 index 00000000..5c5e2a93 --- /dev/null +++ b/theories7/IntMap/Mapcard.v @@ -0,0 +1,670 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Mapcard.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require Arith. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Mapaxioms. +Require Mapiter. +Require Fset. +Require Mapsubset. +Require PolyList. +Require Lsort. +Require Peano_dec. + +Section MapCard. + + Variable A, B : Set. + + Lemma MapCard_M0 : (MapCard A (M0 A))=O. + Proof. + Trivial. + Qed. + + Lemma MapCard_M1 : (a:ad) (y:A) (MapCard A (M1 A a y))=(1). + Proof. + Trivial. + Qed. + + Lemma MapCard_is_O : (m:(Map A)) (MapCard A m)=O -> + (a:ad) (MapGet A m a)=(NONE A). + Proof. + Induction m. Trivial. + Intros a y H. Discriminate H. + Intros. Simpl in H1. Elim (plus_is_O ? ? H1). Intros. Rewrite (MapGet_M2_bit_0_if A m0 m1 a). + Case (ad_bit_0 a). Apply H0. Assumption. + Apply H. Assumption. + Qed. + + Lemma MapCard_is_not_O : (m:(Map A)) (a:ad) (y:A) (MapGet A m a)=(SOME A y) -> + {n:nat | (MapCard A m)=(S n)}. + Proof. + Induction m. Intros. Discriminate H. + Intros a y a0 y0 H. Simpl in H. Elim (sumbool_of_bool (ad_eq a a0)). Intro H0. Split with O. + Reflexivity. + Intro H0. Rewrite H0 in H. Discriminate H. + Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H2. + Rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1) in H1. Elim (H0 (ad_div_2 a) y H1). Intros n H3. + Simpl. Rewrite H3. Split with (plus (MapCard A m0) n). + Rewrite <- (plus_Snm_nSm (MapCard A m0) n). Reflexivity. + Intro H2. Rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1) in H1. Elim (H (ad_div_2 a) y H1). + Intros n H3. Simpl. Rewrite H3. Split with (plus n (MapCard A m1)). Reflexivity. + Qed. + + Lemma MapCard_is_one : (m:(Map A)) (MapCard A m)=(1) -> + {a:ad & {y:A | (MapGet A m a)=(SOME A y)}}. + Proof. + Induction m. Intro. Discriminate H. + Intros a y H. Split with a. Split with y. Apply M1_semantics_1. + Intros. Simpl in H1. Elim (plus_is_one (MapCard A m0) (MapCard A m1) H1). + Intro H2. Elim H2. Intros. Elim (H0 H4). Intros a H5. Split with (ad_double_plus_un a). + Rewrite (MapGet_M2_bit_0_1 A ? (ad_double_plus_un_bit_0 a) m0 m1). + Rewrite ad_double_plus_un_div_2. Exact H5. + Intro H2. Elim H2. Intros. Elim (H H3). Intros a H5. Split with (ad_double a). + Rewrite (MapGet_M2_bit_0_0 A ? (ad_double_bit_0 a) m0 m1). + Rewrite ad_double_div_2. Exact H5. + Qed. + + Lemma MapCard_is_one_unique : (m:(Map A)) (MapCard A m)=(1) -> (a,a':ad) (y,y':A) + (MapGet A m a)=(SOME A y) -> (MapGet A m a')=(SOME A y') -> + a=a' /\ y=y'. + Proof. + Induction m. Intro. Discriminate H. + Intros. Elim (sumbool_of_bool (ad_eq a a1)). Intro H2. Rewrite (ad_eq_complete ? ? H2) in H0. + Rewrite (M1_semantics_1 A a1 a0) in H0. Inversion H0. Elim (sumbool_of_bool (ad_eq a a')). + Intro H5. Rewrite (ad_eq_complete ? ? H5) in H1. Rewrite (M1_semantics_1 A a' a0) in H1. + Inversion H1. Rewrite <- (ad_eq_complete ? ? H2). Rewrite <- (ad_eq_complete ? ? H5). + Rewrite <- H4. Rewrite <- H6. (Split; Reflexivity). + Intro H5. Rewrite (M1_semantics_2 A a a' a0 H5) in H1. Discriminate H1. + Intro H2. Rewrite (M1_semantics_2 A a a1 a0 H2) in H0. Discriminate H0. + Intros. Simpl in H1. Elim (plus_is_one ? ? H1). Intro H4. Elim H4. Intros. + Rewrite (MapGet_M2_bit_0_if A m0 m1 a) in H2. Elim (sumbool_of_bool (ad_bit_0 a)). + Intro H7. Rewrite H7 in H2. Rewrite (MapGet_M2_bit_0_if A m0 m1 a') in H3. + Elim (sumbool_of_bool (ad_bit_0 a')). Intro H8. Rewrite H8 in H3. Elim (H0 H6 ? ? ? ? H2 H3). + Intros. Split. Rewrite <- (ad_div_2_double_plus_un a H7). + Rewrite <- (ad_div_2_double_plus_un a' H8). Rewrite H9. Reflexivity. + Assumption. + Intro H8. Rewrite H8 in H3. Rewrite (MapCard_is_O m0 H5 (ad_div_2 a')) in H3. + Discriminate H3. + Intro H7. Rewrite H7 in H2. Rewrite (MapCard_is_O m0 H5 (ad_div_2 a)) in H2. + Discriminate H2. + Intro H4. Elim H4. Intros. Rewrite (MapGet_M2_bit_0_if A m0 m1 a) in H2. + Elim (sumbool_of_bool (ad_bit_0 a)). Intro H7. Rewrite H7 in H2. + Rewrite (MapCard_is_O m1 H6 (ad_div_2 a)) in H2. Discriminate H2. + Intro H7. Rewrite H7 in H2. Rewrite (MapGet_M2_bit_0_if A m0 m1 a') in H3. + Elim (sumbool_of_bool (ad_bit_0 a')). Intro H8. Rewrite H8 in H3. + Rewrite (MapCard_is_O m1 H6 (ad_div_2 a')) in H3. Discriminate H3. + Intro H8. Rewrite H8 in H3. Elim (H H5 ? ? ? ? H2 H3). Intros. Split. + Rewrite <- (ad_div_2_double a H7). Rewrite <- (ad_div_2_double a' H8). + Rewrite H9. Reflexivity. + Assumption. + Qed. + + Lemma length_as_fold : (C:Set) (l:(list C)) + (length l)=(fold_right [_:C][n:nat](S n) O l). + Proof. + Induction l. Reflexivity. + Intros. Simpl. Rewrite H. Reflexivity. + Qed. + + Lemma length_as_fold_2 : (l:(alist A)) + (length l)=(fold_right [r:ad*A][n:nat]let (a,y)=r in (plus (1) n) O l). + Proof. + Induction l. Reflexivity. + Intros. Simpl. Rewrite H. (Elim a; Reflexivity). + Qed. + + Lemma MapCard_as_Fold_1 : (m:(Map A)) (pf:ad->ad) + (MapCard A m)=(MapFold1 A nat O plus [_:ad][_:A](1) pf m). + Proof. + Induction m. Trivial. + Trivial. + Intros. Simpl. Rewrite <- (H [a0:ad](pf (ad_double a0))). + Rewrite <- (H0 [a0:ad](pf (ad_double_plus_un a0))). Reflexivity. + Qed. + + Lemma MapCard_as_Fold : + (m:(Map A)) (MapCard A m)=(MapFold A nat O plus [_:ad][_:A](1) m). + Proof. + Intro. Exact (MapCard_as_Fold_1 m [a0:ad]a0). + Qed. + + Lemma MapCard_as_length : (m:(Map A)) (MapCard A m)=(length (alist_of_Map A m)). + Proof. + Intro. Rewrite MapCard_as_Fold. Rewrite length_as_fold_2. + Apply MapFold_as_fold with op:=plus neutral:=O f:=[_:ad][_:A](1). Exact plus_assoc_r. + Trivial. + Intro. Rewrite <- plus_n_O. Reflexivity. + Qed. + + Lemma MapCard_Put1_equals_2 : (p:positive) (a,a':ad) (y,y':A) + (MapCard A (MapPut1 A a y a' y' p))=(2). + Proof. + Induction p. Intros. Simpl. (Case (ad_bit_0 a); Reflexivity). + Intros. Simpl. Case (ad_bit_0 a). Exact (H (ad_div_2 a) (ad_div_2 a') y y'). + Simpl. Rewrite <- plus_n_O. Exact (H (ad_div_2 a) (ad_div_2 a') y y'). + Intros. Simpl. (Case (ad_bit_0 a); Reflexivity). + Qed. + + Lemma MapCard_Put_sum : (m,m':(Map A)) (a:ad) (y:A) (n,n':nat) + m'=(MapPut A m a y) -> n=(MapCard A m) -> n'=(MapCard A m') -> + {n'=n}+{n'=(S n)}. + Proof. + Induction m. Simpl. Intros. Rewrite H in H1. Simpl in H1. Right . + Rewrite H0. Rewrite H1. Reflexivity. + Intros a y m' a0 y0 n n' H H0 H1. Simpl in H. Elim (ad_sum (ad_xor a a0)). Intro H2. + Elim H2. Intros p H3. Rewrite H3 in H. Rewrite H in H1. + Rewrite (MapCard_Put1_equals_2 p a a0 y y0) in H1. Simpl in H0. Right . + Rewrite H0. Rewrite H1. Reflexivity. + Intro H2. Rewrite H2 in H. Rewrite H in H1. Simpl in H1. Simpl in H0. Left . + Rewrite H0. Rewrite H1. Reflexivity. + Intros. Simpl in H2. Rewrite (MapPut_semantics_3_1 A m0 m1 a y) in H1. + Elim (sumbool_of_bool (ad_bit_0 a)). Intro H4. Rewrite H4 in H1. + Elim (H0 (MapPut A m1 (ad_div_2 a) y) (ad_div_2 a) y (MapCard A m1) + (MapCard A (MapPut A m1 (ad_div_2 a) y)) (refl_equal ? ?) + (refl_equal ? ?) (refl_equal ? ?)). + Intro H5. Rewrite H1 in H3. Simpl in H3. Rewrite H5 in H3. Rewrite <- H2 in H3. Left . + Assumption. + Intro H5. Rewrite H1 in H3. Simpl in H3. Rewrite H5 in H3. + Rewrite <- (plus_Snm_nSm (MapCard A m0) (MapCard A m1)) in H3. + Simpl in H3. Rewrite <- H2 in H3. Right . Assumption. + Intro H4. Rewrite H4 in H1. + Elim (H (MapPut A m0 (ad_div_2 a) y) (ad_div_2 a) y (MapCard A m0) + (MapCard A (MapPut A m0 (ad_div_2 a) y)) (refl_equal ? ?) + (refl_equal ? ?) (refl_equal ? ?)). + Intro H5. Rewrite H1 in H3. Simpl in H3. Rewrite H5 in H3. Rewrite <- H2 in H3. + Left . Assumption. + Intro H5. Rewrite H1 in H3. Simpl in H3. Rewrite H5 in H3. Simpl in H3. Rewrite <- H2 in H3. + Right . Assumption. + Qed. + + Lemma MapCard_Put_lb : (m:(Map A)) (a:ad) (y:A) + (ge (MapCard A (MapPut A m a y)) (MapCard A m)). + Proof. + Unfold ge. Intros. + Elim (MapCard_Put_sum m (MapPut A m a y) a y (MapCard A m) + (MapCard A (MapPut A m a y)) (refl_equal ? ?) (refl_equal ? ?) + (refl_equal ? ?)). + Intro H. Rewrite H. Apply le_n. + Intro H. Rewrite H. Apply le_n_Sn. + Qed. + + Lemma MapCard_Put_ub : (m:(Map A)) (a:ad) (y:A) + (le (MapCard A (MapPut A m a y)) (S (MapCard A m))). + Proof. + Intros. + Elim (MapCard_Put_sum m (MapPut A m a y) a y (MapCard A m) + (MapCard A (MapPut A m a y)) (refl_equal ? ?) (refl_equal ? ?) + (refl_equal ? ?)). + Intro H. Rewrite H. Apply le_n_Sn. + Intro H. Rewrite H. Apply le_n. + Qed. + + Lemma MapCard_Put_1 : (m:(Map A)) (a:ad) (y:A) + (MapCard A (MapPut A m a y))=(MapCard A m) -> + {y:A | (MapGet A m a)=(SOME A y)}. + Proof. + Induction m. Intros. Discriminate H. + Intros a y a0 y0 H. Simpl in H. Elim (ad_sum (ad_xor a a0)). Intro H0. Elim H0. + Intros p H1. Rewrite H1 in H. Rewrite (MapCard_Put1_equals_2 p a a0 y y0) in H. + Discriminate H. + Intro H0. Rewrite H0 in H. Rewrite (ad_xor_eq ? ? H0). Split with y. Apply M1_semantics_1. + Intros. Rewrite (MapPut_semantics_3_1 A m0 m1 a y) in H1. Elim (sumbool_of_bool (ad_bit_0 a)). + Intro H2. Rewrite H2 in H1. Simpl in H1. Elim (H0 (ad_div_2 a) y (simpl_plus_l ? ? ? H1)). + Intros y0 H3. Split with y0. Rewrite <- H3. Exact (MapGet_M2_bit_0_1 A a H2 m0 m1). + Intro H2. Rewrite H2 in H1. Simpl in H1. + Rewrite (plus_sym (MapCard A (MapPut A m0 (ad_div_2 a) y)) (MapCard A m1)) in H1. + Rewrite (plus_sym (MapCard A m0) (MapCard A m1)) in H1. + Elim (H (ad_div_2 a) y (simpl_plus_l ? ? ? H1)). Intros y0 H3. Split with y0. + Rewrite <- H3. Exact (MapGet_M2_bit_0_0 A a H2 m0 m1). + Qed. + + Lemma MapCard_Put_2 : (m:(Map A)) (a:ad) (y:A) + (MapCard A (MapPut A m a y))=(S (MapCard A m)) -> (MapGet A m a)=(NONE A). + Proof. + Induction m. Trivial. + Intros. Simpl in H. Elim (sumbool_of_bool (ad_eq a a1)). Intro H0. + Rewrite (ad_eq_complete ? ? H0) in H. Rewrite (ad_xor_nilpotent a1) in H. Discriminate H. + Intro H0. Exact (M1_semantics_2 A a a1 a0 H0). + Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H2. + Rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1). Apply (H0 (ad_div_2 a) y). + Apply simpl_plus_l with n:=(MapCard A m0). + Rewrite <- (plus_Snm_nSm (MapCard A m0) (MapCard A m1)). Simpl in H1. Simpl. Rewrite <- H1. + Clear H1. + NewInduction a. Discriminate H2. + NewInduction p. Reflexivity. + Discriminate H2. + Reflexivity. + Intro H2. Rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1). Apply (H (ad_div_2 a) y). + Cut (plus (MapCard A (MapPut A m0 (ad_div_2 a) y)) (MapCard A m1)) + =(plus (S (MapCard A m0)) (MapCard A m1)). + Intro. Rewrite (plus_sym (MapCard A (MapPut A m0 (ad_div_2 a) y)) (MapCard A m1)) in H3. + Rewrite (plus_sym (S (MapCard A m0)) (MapCard A m1)) in H3. Exact (simpl_plus_l ? ? ? H3). + Simpl. Simpl in H1. Rewrite <- H1. NewInduction a. Trivial. + NewInduction p. Discriminate H2. + Reflexivity. + Discriminate H2. + Qed. + + Lemma MapCard_Put_1_conv : (m:(Map A)) (a:ad) (y,y':A) + (MapGet A m a)=(SOME A y) -> (MapCard A (MapPut A m a y'))=(MapCard A m). + Proof. + Intros. + Elim (MapCard_Put_sum m (MapPut A m a y') a y' (MapCard A m) + (MapCard A (MapPut A m a y')) (refl_equal ? ?) (refl_equal ? ?) + (refl_equal ? ?)). + Trivial. + Intro H0. Rewrite (MapCard_Put_2 m a y' H0) in H. Discriminate H. + Qed. + + Lemma MapCard_Put_2_conv : (m:(Map A)) (a:ad) (y:A) + (MapGet A m a)=(NONE A) -> (MapCard A (MapPut A m a y))=(S (MapCard A m)). + Proof. + Intros. + Elim (MapCard_Put_sum m (MapPut A m a y) a y (MapCard A m) + (MapCard A (MapPut A m a y)) (refl_equal ? ?) (refl_equal ? ?) + (refl_equal ? ?)). + Intro H0. Elim (MapCard_Put_1 m a y H0). Intros y' H1. Rewrite H1 in H. Discriminate H. + Trivial. + Qed. + + Lemma MapCard_ext : (m,m':(Map A)) + (eqm A (MapGet A m) (MapGet A m')) -> (MapCard A m)=(MapCard A m'). + Proof. + Unfold eqm. Intros. Rewrite (MapCard_as_length m). Rewrite (MapCard_as_length m'). + Rewrite (alist_canonical A (alist_of_Map A m) (alist_of_Map A m')). Reflexivity. + Unfold eqm. Intro. Rewrite (Map_of_alist_semantics A (alist_of_Map A m) a). + Rewrite (Map_of_alist_semantics A (alist_of_Map A m') a). Rewrite (Map_of_alist_of_Map A m' a). + Rewrite (Map_of_alist_of_Map A m a). Exact (H a). + Apply alist_of_Map_sorts2. + Apply alist_of_Map_sorts2. + Qed. + + Lemma MapCard_Dom : (m:(Map A)) (MapCard A m)=(MapCard unit (MapDom A m)). + Proof. + (Induction m; Trivial). Intros. Simpl. Rewrite H. Rewrite H0. Reflexivity. + Qed. + + Lemma MapCard_Dom_Put_behind : (m:(Map A)) (a:ad) (y:A) + (MapDom A (MapPut_behind A m a y))=(MapDom A (MapPut A m a y)). + Proof. + Induction m. Trivial. + Intros a y a0 y0. Simpl. Elim (ad_sum (ad_xor a a0)). Intro H. Elim H. + Intros p H0. Rewrite H0. Reflexivity. + Intro H. Rewrite H. Rewrite (ad_xor_eq ? ? H). Reflexivity. + Intros. Simpl. Elim (ad_sum a). Intro H1. Elim H1. Intros p H2. Rewrite H2. Case p. + Intro p0. Simpl. Rewrite H0. Reflexivity. + Intro p0. Simpl. Rewrite H. Reflexivity. + Simpl. Rewrite H0. Reflexivity. + Intro H1. Rewrite H1. Simpl. Rewrite H. Reflexivity. + Qed. + + Lemma MapCard_Put_behind_Put : (m:(Map A)) (a:ad) (y:A) + (MapCard A (MapPut_behind A m a y))=(MapCard A (MapPut A m a y)). + Proof. + Intros. Rewrite MapCard_Dom. Rewrite MapCard_Dom. Rewrite MapCard_Dom_Put_behind. + Reflexivity. + Qed. + + Lemma MapCard_Put_behind_sum : (m,m':(Map A)) (a:ad) (y:A) (n,n':nat) + m'=(MapPut_behind A m a y) -> n=(MapCard A m) -> n'=(MapCard A m') -> + {n'=n}+{n'=(S n)}. + Proof. + Intros. (Apply (MapCard_Put_sum m (MapPut A m a y) a y n n'); Trivial). + Rewrite <- MapCard_Put_behind_Put. Rewrite <- H. Assumption. + Qed. + + Lemma MapCard_makeM2 : (m,m':(Map A)) + (MapCard A (makeM2 A m m'))=(plus (MapCard A m) (MapCard A m')). + Proof. + Intros. Rewrite (MapCard_ext ? ? (makeM2_M2 A m m')). Reflexivity. + Qed. + + Lemma MapCard_Remove_sum : (m,m':(Map A)) (a:ad) (n,n':nat) + m'=(MapRemove A m a) -> n=(MapCard A m) -> n'=(MapCard A m') -> + {n=n'}+{n=(S n')}. + Proof. + Induction m. Simpl. Intros. Rewrite H in H1. Simpl in H1. Left . Rewrite H1. Assumption. + Simpl. Intros. Elim (sumbool_of_bool (ad_eq a a1)). Intro H2. Rewrite H2 in H. + Rewrite H in H1. Simpl in H1. Right . Rewrite H1. Assumption. + Intro H2. Rewrite H2 in H. Rewrite H in H1. Simpl in H1. Left . Rewrite H1. Assumption. + Intros. Simpl in H1. Simpl in H2. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H4. + Rewrite H4 in H1. Rewrite H1 in H3. + Rewrite (MapCard_makeM2 m0 (MapRemove A m1 (ad_div_2 a))) in H3. + Elim (H0 (MapRemove A m1 (ad_div_2 a)) (ad_div_2 a) (MapCard A m1) + (MapCard A (MapRemove A m1 (ad_div_2 a))) (refl_equal ? ?) + (refl_equal ? ?) (refl_equal ? ?)). + Intro H5. Rewrite H5 in H2. Left . Rewrite H3. Exact H2. + Intro H5. Rewrite H5 in H2. + Rewrite <- (plus_Snm_nSm (MapCard A m0) (MapCard A (MapRemove A m1 (ad_div_2 a)))) in H2. + Right . Rewrite H3. Exact H2. + Intro H4. Rewrite H4 in H1. Rewrite H1 in H3. + Rewrite (MapCard_makeM2 (MapRemove A m0 (ad_div_2 a)) m1) in H3. + Elim (H (MapRemove A m0 (ad_div_2 a)) (ad_div_2 a) (MapCard A m0) + (MapCard A (MapRemove A m0 (ad_div_2 a))) (refl_equal ? ?) + (refl_equal ? ?) (refl_equal ? ?)). + Intro H5. Rewrite H5 in H2. Left . Rewrite H3. Exact H2. + Intro H5. Rewrite H5 in H2. Right . Rewrite H3. Exact H2. + Qed. + + Lemma MapCard_Remove_ub : (m:(Map A)) (a:ad) + (le (MapCard A (MapRemove A m a)) (MapCard A m)). + Proof. + Intros. + Elim (MapCard_Remove_sum m (MapRemove A m a) a (MapCard A m) + (MapCard A (MapRemove A m a)) (refl_equal ? ?) (refl_equal ? ?) + (refl_equal ? ?)). + Intro H. Rewrite H. Apply le_n. + Intro H. Rewrite H. Apply le_n_Sn. + Qed. + + Lemma MapCard_Remove_lb : (m:(Map A)) (a:ad) + (ge (S (MapCard A (MapRemove A m a))) (MapCard A m)). + Proof. + Unfold ge. Intros. + Elim (MapCard_Remove_sum m (MapRemove A m a) a (MapCard A m) + (MapCard A (MapRemove A m a)) (refl_equal ? ?) (refl_equal ? ?) + (refl_equal ? ?)). + Intro H. Rewrite H. Apply le_n_Sn. + Intro H. Rewrite H. Apply le_n. + Qed. + + Lemma MapCard_Remove_1 : (m:(Map A)) (a:ad) + (MapCard A (MapRemove A m a))=(MapCard A m) -> (MapGet A m a)=(NONE A). + Proof. + Induction m. Trivial. + Simpl. Intros a y a0 H. Elim (sumbool_of_bool (ad_eq a a0)). Intro H0. + Rewrite H0 in H. Discriminate H. + Intro H0. Rewrite H0. Reflexivity. + Intros. Simpl in H1. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H2. Rewrite H2 in H1. + Rewrite (MapCard_makeM2 m0 (MapRemove A m1 (ad_div_2 a))) in H1. + Rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1). Apply H0. Exact (simpl_plus_l ? ? ? H1). + Intro H2. Rewrite H2 in H1. + Rewrite (MapCard_makeM2 (MapRemove A m0 (ad_div_2 a)) m1) in H1. + Rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1). Apply H. + Rewrite (plus_sym (MapCard A (MapRemove A m0 (ad_div_2 a))) (MapCard A m1)) in H1. + Rewrite (plus_sym (MapCard A m0) (MapCard A m1)) in H1. Exact (simpl_plus_l ? ? ? H1). + Qed. + + Lemma MapCard_Remove_2 : (m:(Map A)) (a:ad) + (S (MapCard A (MapRemove A m a)))=(MapCard A m) -> + {y:A | (MapGet A m a)=(SOME A y)}. + Proof. + Induction m. Intros. Discriminate H. + Intros a y a0 H. Simpl in H. Elim (sumbool_of_bool (ad_eq a a0)). Intro H0. + Rewrite (ad_eq_complete ? ? H0). Split with y. Exact (M1_semantics_1 A a0 y). + Intro H0. Rewrite H0 in H. Discriminate H. + Intros. Simpl in H1. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H2. Rewrite H2 in H1. + Rewrite (MapCard_makeM2 m0 (MapRemove A m1 (ad_div_2 a))) in H1. + Rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1). Apply H0. + Change (plus (S (MapCard A m0)) (MapCard A (MapRemove A m1 (ad_div_2 a)))) + =(plus (MapCard A m0) (MapCard A m1)) in H1. + Rewrite (plus_Snm_nSm (MapCard A m0) (MapCard A (MapRemove A m1 (ad_div_2 a)))) in H1. + Exact (simpl_plus_l ? ? ? H1). + Intro H2. Rewrite H2 in H1. Rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1). Apply H. + Rewrite (MapCard_makeM2 (MapRemove A m0 (ad_div_2 a)) m1) in H1. + Change (plus (S (MapCard A (MapRemove A m0 (ad_div_2 a)))) (MapCard A m1)) + =(plus (MapCard A m0) (MapCard A m1)) in H1. + Rewrite (plus_sym (S (MapCard A (MapRemove A m0 (ad_div_2 a)))) (MapCard A m1)) in H1. + Rewrite (plus_sym (MapCard A m0) (MapCard A m1)) in H1. Exact (simpl_plus_l ? ? ? H1). + Qed. + + Lemma MapCard_Remove_1_conv : (m:(Map A)) (a:ad) + (MapGet A m a)=(NONE A) -> (MapCard A (MapRemove A m a))=(MapCard A m). + Proof. + Intros. + Elim (MapCard_Remove_sum m (MapRemove A m a) a (MapCard A m) + (MapCard A (MapRemove A m a)) (refl_equal ? ?) (refl_equal ? ?) + (refl_equal ? ?)). + Intro H0. Rewrite H0. Reflexivity. + Intro H0. Elim (MapCard_Remove_2 m a (sym_eq ? ? ? H0)). Intros y H1. Rewrite H1 in H. + Discriminate H. + Qed. + + Lemma MapCard_Remove_2_conv : (m:(Map A)) (a:ad) (y:A) + (MapGet A m a)=(SOME A y) -> + (S (MapCard A (MapRemove A m a)))=(MapCard A m). + Proof. + Intros. + Elim (MapCard_Remove_sum m (MapRemove A m a) a (MapCard A m) + (MapCard A (MapRemove A m a)) (refl_equal ? ?) (refl_equal ? ?) + (refl_equal ? ?)). + Intro H0. Rewrite (MapCard_Remove_1 m a (sym_eq ? ? ? H0)) in H. Discriminate H. + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Lemma MapMerge_Restr_Card : (m,m':(Map A)) + (plus (MapCard A m) (MapCard A m'))= + (plus (MapCard A (MapMerge A m m')) (MapCard A (MapDomRestrTo A A m m'))). + Proof. + Induction m. Simpl. Intro. Apply plus_n_O. + Simpl. Intros a y m'. Elim (option_sum A (MapGet A m' a)). Intro H. Elim H. Intros y0 H0. + Rewrite H0. Rewrite MapCard_Put_behind_Put. Rewrite (MapCard_Put_1_conv m' a y0 y H0). + Simpl. Rewrite <- plus_Snm_nSm. Apply plus_n_O. + Intro H. Rewrite H. Rewrite MapCard_Put_behind_Put. Rewrite (MapCard_Put_2_conv m' a y H). + Apply plus_n_O. + Intros. + Change (plus (plus (MapCard A m0) (MapCard A m1)) (MapCard A m')) + =(plus (MapCard A (MapMerge A (M2 A m0 m1) m')) + (MapCard A (MapDomRestrTo A A (M2 A m0 m1) m'))). + Elim m'. Reflexivity. + Intros a y. Unfold MapMerge. Unfold MapDomRestrTo. + Elim (option_sum A (MapGet A (M2 A m0 m1) a)). Intro H1. Elim H1. Intros y0 H2. Rewrite H2. + Rewrite (MapCard_Put_1_conv (M2 A m0 m1) a y0 y H2). Reflexivity. + Intro H1. Rewrite H1. Rewrite (MapCard_Put_2_conv (M2 A m0 m1) a y H1). Simpl. + Rewrite <- (plus_Snm_nSm (plus (MapCard A m0) (MapCard A m1)) O). Reflexivity. + Intros. Simpl. + Rewrite (plus_permute_2_in_4 (MapCard A m0) (MapCard A m1) (MapCard A m2) (MapCard A m3)). + Rewrite (H m2). Rewrite (H0 m3). + Rewrite (MapCard_makeM2 (MapDomRestrTo A A m0 m2) (MapDomRestrTo A A m1 m3)). + Apply plus_permute_2_in_4. + Qed. + + Lemma MapMerge_disjoint_Card : (m,m':(Map A)) (MapDisjoint A A m m') -> + (MapCard A (MapMerge A m m'))=(plus (MapCard A m) (MapCard A m')). + Proof. + Intros. Rewrite (MapMerge_Restr_Card m m'). + Rewrite (MapCard_ext ? ? (MapDisjoint_imp_2 ? ? ? ? H)). Apply plus_n_O. + Qed. + + Lemma MapSplit_Card : (m:(Map A)) (m':(Map B)) + (MapCard A m)=(plus (MapCard A (MapDomRestrTo A B m m')) + (MapCard A (MapDomRestrBy A B m m'))). + Proof. + Intros. Rewrite (MapCard_ext ? ? (MapDom_Split_1 A B m m')). Apply MapMerge_disjoint_Card. + Apply MapDisjoint_2_imp. Unfold MapDisjoint_2. Apply MapDom_Split_3. + Qed. + + Lemma MapMerge_Card_ub : (m,m':(Map A)) + (le (MapCard A (MapMerge A m m')) (plus (MapCard A m) (MapCard A m'))). + Proof. + Intros. Rewrite MapMerge_Restr_Card. Apply le_plus_l. + Qed. + + Lemma MapDomRestrTo_Card_ub_l : (m:(Map A)) (m':(Map B)) + (le (MapCard A (MapDomRestrTo A B m m')) (MapCard A m)). + Proof. + Intros. Rewrite (MapSplit_Card m m'). Apply le_plus_l. + Qed. + + Lemma MapDomRestrBy_Card_ub_l : (m:(Map A)) (m':(Map B)) + (le (MapCard A (MapDomRestrBy A B m m')) (MapCard A m)). + Proof. + Intros. Rewrite (MapSplit_Card m m'). Apply le_plus_r. + Qed. + + Lemma MapMerge_Card_disjoint : (m,m':(Map A)) + (MapCard A (MapMerge A m m'))=(plus (MapCard A m) (MapCard A m')) -> + (MapDisjoint A A m m'). + Proof. + Induction m. Intros. Apply Map_M0_disjoint. + Simpl. Intros. Rewrite (MapCard_Put_behind_Put m' a a0) in H. Unfold MapDisjoint in_dom. + Simpl. Intros. Elim (sumbool_of_bool (ad_eq a a1)). Intro H2. + Rewrite (ad_eq_complete ? ? H2) in H. Rewrite (MapCard_Put_2 m' a1 a0 H) in H1. + Discriminate H1. + Intro H2. Rewrite H2 in H0. Discriminate H0. + Induction m'. Intros. Apply Map_disjoint_M0. + Intros a y H1. Rewrite <- (MapCard_ext ? ? (MapPut_as_Merge A (M2 A m0 m1) a y)) in H1. + Unfold 3 MapCard in H1. Rewrite <- (plus_Snm_nSm (MapCard A (M2 A m0 m1)) O) in H1. + Rewrite <- (plus_n_O (S (MapCard A (M2 A m0 m1)))) in H1. Unfold MapDisjoint in_dom. + Unfold 2 MapGet. Intros. Elim (sumbool_of_bool (ad_eq a a0)). Intro H4. + Rewrite <- (ad_eq_complete ? ? H4) in H2. Rewrite (MapCard_Put_2 ? ? ? H1) in H2. + Discriminate H2. + Intro H4. Rewrite H4 in H3. Discriminate H3. + Intros. Unfold MapDisjoint. Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H6. + Unfold MapDisjoint in H0. Apply H0 with m':=m3 a:=(ad_div_2 a). Apply le_antisym. + Apply MapMerge_Card_ub. + Apply simpl_le_plus_l with p:=(plus (MapCard A m0) (MapCard A m2)). + Rewrite (plus_permute_2_in_4 (MapCard A m0) (MapCard A m2) (MapCard A m1) (MapCard A m3)). + Change (MapCard A (M2 A (MapMerge A m0 m2) (MapMerge A m1 m3))) + =(plus (plus (MapCard A m0) (MapCard A m1)) (plus (MapCard A m2) (MapCard A m3))) in H3. + Rewrite <- H3. Simpl. Apply le_reg_r. Apply MapMerge_Card_ub. + Elim (in_dom_some ? ? ? H4). Intros y H7. Rewrite (MapGet_M2_bit_0_1 ? a H6 m0 m1) in H7. + Unfold in_dom. Rewrite H7. Reflexivity. + Elim (in_dom_some ? ? ? H5). Intros y H7. Rewrite (MapGet_M2_bit_0_1 ? a H6 m2 m3) in H7. + Unfold in_dom. Rewrite H7. Reflexivity. + Intro H6. Unfold MapDisjoint in H. Apply H with m':=m2 a:=(ad_div_2 a). Apply le_antisym. + Apply MapMerge_Card_ub. + Apply simpl_le_plus_l with p:=(plus (MapCard A m1) (MapCard A m3)). + Rewrite (plus_sym (plus (MapCard A m1) (MapCard A m3)) (plus (MapCard A m0) (MapCard A m2))). + Rewrite (plus_permute_2_in_4 (MapCard A m0) (MapCard A m2) (MapCard A m1) (MapCard A m3)). + Rewrite (plus_sym (plus (MapCard A m1) (MapCard A m3)) (MapCard A (MapMerge A m0 m2))). + Change (plus (MapCard A (MapMerge A m0 m2)) (MapCard A (MapMerge A m1 m3))) + =(plus (plus (MapCard A m0) (MapCard A m1)) (plus (MapCard A m2) (MapCard A m3))) in H3. + Rewrite <- H3. Apply le_reg_l. Apply MapMerge_Card_ub. + Elim (in_dom_some ? ? ? H4). Intros y H7. Rewrite (MapGet_M2_bit_0_0 ? a H6 m0 m1) in H7. + Unfold in_dom. Rewrite H7. Reflexivity. + Elim (in_dom_some ? ? ? H5). Intros y H7. Rewrite (MapGet_M2_bit_0_0 ? a H6 m2 m3) in H7. + Unfold in_dom. Rewrite H7. Reflexivity. + Qed. + + Lemma MapCard_is_Sn : (m:(Map A)) (n:nat) (MapCard ? m)=(S n) -> + {a:ad | (in_dom ? a m)=true}. + Proof. + Induction m. Intros. Discriminate H. + Intros a y n H. Split with a. Unfold in_dom. Rewrite (M1_semantics_1 ? a y). Reflexivity. + Intros. Simpl in H1. Elim (O_or_S (MapCard ? m0)). Intro H2. Elim H2. Intros m2 H3. + Elim (H ? (sym_eq ? ? ? H3)). Intros a H4. Split with (ad_double a). Unfold in_dom. + Rewrite (MapGet_M2_bit_0_0 A (ad_double a) (ad_double_bit_0 a) m0 m1). + Rewrite (ad_double_div_2 a). Elim (in_dom_some ? ? ? H4). Intros y H5. Rewrite H5. Reflexivity. + Intro H2. Rewrite <- H2 in H1. Simpl in H1. Elim (H0 ? H1). Intros a H3. + Split with (ad_double_plus_un a). Unfold in_dom. + Rewrite (MapGet_M2_bit_0_1 A (ad_double_plus_un a) (ad_double_plus_un_bit_0 a) m0 m1). + Rewrite (ad_double_plus_un_div_2 a). Elim (in_dom_some ? ? ? H3). Intros y H4. Rewrite H4. + Reflexivity. + Qed. + +End MapCard. + +Section MapCard2. + + Variable A, B : Set. + + Lemma MapSubset_card_eq_1 : (n:nat) (m:(Map A)) (m':(Map B)) + (MapSubset ? ? m m') -> (MapCard ? m)=n -> (MapCard ? m')=n -> + (MapSubset ? ? m' m). + Proof. + Induction n. Intros. Unfold MapSubset in_dom. Intro. Rewrite (MapCard_is_O ? m H0 a). + Rewrite (MapCard_is_O ? m' H1 a). Intro H2. Discriminate H2. + Intros. Elim (MapCard_is_Sn A m n0 H1). Intros a H3. Elim (in_dom_some ? ? ? H3). + Intros y H4. Elim (in_dom_some ? ? ? (H0 ? H3)). Intros y' H6. + Cut (eqmap ? (MapPut ? (MapRemove ? m a) a y) m). Intro. + Cut (eqmap ? (MapPut ? (MapRemove ? m' a) a y') m'). Intro. + Apply MapSubset_ext with m0:=(MapPut ? (MapRemove ? m' a) a y') + m2:=(MapPut ? (MapRemove ? m a) a y). + Assumption. + Assumption. + Apply MapSubset_Put_mono. Apply H. Apply MapSubset_Remove_mono. Assumption. + Rewrite <- (MapCard_Remove_2_conv ? m a y H4) in H1. Inversion_clear H1. Reflexivity. + Rewrite <- (MapCard_Remove_2_conv ? m' a y' H6) in H2. Inversion_clear H2. Reflexivity. + Unfold eqmap eqm. Intro. Rewrite (MapPut_semantics ? (MapRemove B m' a) a y' a0). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H7. Rewrite H7. Rewrite <- (ad_eq_complete ? ? H7). + Apply sym_eq. Assumption. + Intro H7. Rewrite H7. Rewrite (MapRemove_semantics ? m' a a0). Rewrite H7. Reflexivity. + Unfold eqmap eqm. Intro. Rewrite (MapPut_semantics ? (MapRemove A m a) a y a0). + Elim (sumbool_of_bool (ad_eq a a0)). Intro H7. Rewrite H7. Rewrite <- (ad_eq_complete ? ? H7). + Apply sym_eq. Assumption. + Intro H7. Rewrite H7. Rewrite (MapRemove_semantics A m a a0). Rewrite H7. Reflexivity. + Qed. + + Lemma MapDomRestrTo_Card_ub_r : (m:(Map A)) (m':(Map B)) + (le (MapCard A (MapDomRestrTo A B m m')) (MapCard B m')). + Proof. + Induction m. Intro. Simpl. Apply le_O_n. + Intros a y m'. Simpl. Elim (option_sum B (MapGet B m' a)). Intro H. Elim H. Intros y0 H0. + Rewrite H0. Elim (MapCard_is_not_O B m' a y0 H0). Intros n H1. Rewrite H1. Simpl. + Apply le_n_S. Apply le_O_n. + Intro H. Rewrite H. Simpl. Apply le_O_n. + Induction m'. Simpl. Apply le_O_n. + + Intros a y. Unfold MapDomRestrTo. Case (MapGet A (M2 A m0 m1) a). Simpl. Apply le_O_n. + Intro. Simpl. Apply le_n. + Intros. Simpl. Rewrite (MapCard_makeM2 A (MapDomRestrTo A B m0 m2) (MapDomRestrTo A B m1 m3)). + Apply le_plus_plus. Apply H. + Apply H0. + Qed. + +End MapCard2. + +Section MapCard3. + + Variable A, B : Set. + + Lemma MapMerge_Card_lb_l : (m,m':(Map A)) + (ge (MapCard A (MapMerge A m m')) (MapCard A m)). + Proof. + Unfold ge. Intros. Apply (simpl_le_plus_l (MapCard A m')). + Rewrite (plus_sym (MapCard A m') (MapCard A m)). + Rewrite (plus_sym (MapCard A m') (MapCard A (MapMerge A m m'))). + Rewrite (MapMerge_Restr_Card A m m'). Apply le_reg_l. Apply MapDomRestrTo_Card_ub_r. + Qed. + + Lemma MapMerge_Card_lb_r : (m,m':(Map A)) + (ge (MapCard A (MapMerge A m m')) (MapCard A m')). + Proof. + Unfold ge. Intros. Apply (simpl_le_plus_l (MapCard A m)). Rewrite (MapMerge_Restr_Card A m m'). + Rewrite (plus_sym (MapCard A (MapMerge A m m')) (MapCard A (MapDomRestrTo A A m m'))). + Apply le_reg_r. Apply MapDomRestrTo_Card_ub_l. + Qed. + + Lemma MapDomRestrBy_Card_lb : (m:(Map A)) (m':(Map B)) + (ge (plus (MapCard B m') (MapCard A (MapDomRestrBy A B m m'))) (MapCard A m)). + Proof. + Unfold ge. Intros. Rewrite (MapSplit_Card A B m m'). Apply le_reg_r. + Apply MapDomRestrTo_Card_ub_r. + Qed. + + Lemma MapSubset_Card_le : (m:(Map A)) (m':(Map B)) + (MapSubset A B m m') -> (le (MapCard A m) (MapCard B m')). + Proof. + Intros. Apply le_trans with m:=(plus (MapCard B m') (MapCard A (MapDomRestrBy A B m m'))). + Exact (MapDomRestrBy_Card_lb m m'). + Rewrite (MapCard_ext ? ? ? (MapSubset_imp_2 ? ? ? ? H)). Simpl. Rewrite <- plus_n_O. + Apply le_n. + Qed. + + Lemma MapSubset_card_eq : (m:(Map A)) (m':(Map B)) + (MapSubset ? ? m m') -> (le (MapCard ? m') (MapCard ? m)) -> + (eqmap ? (MapDom ? m) (MapDom ? m')). + Proof. + Intros. Apply MapSubset_antisym. Assumption. + Cut (MapCard B m')=(MapCard A m). Intro. Apply (MapSubset_card_eq_1 A B (MapCard A m)). + Assumption. + Reflexivity. + Assumption. + Apply le_antisym. Assumption. + Apply MapSubset_Card_le. Assumption. + Qed. + +End MapCard3. diff --git a/theories7/IntMap/Mapfold.v b/theories7/IntMap/Mapfold.v new file mode 100644 index 00000000..8061f253 --- /dev/null +++ b/theories7/IntMap/Mapfold.v @@ -0,0 +1,381 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Mapfold.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Fset. +Require Mapaxioms. +Require Mapiter. +Require Lsort. +Require Mapsubset. +Require PolyList. + +Section MapFoldResults. + + Variable A : Set. + + Variable M : Set. + Variable neutral : M. + Variable op : M -> M -> M. + + Variable nleft : (a:M) (op neutral a)=a. + Variable nright : (a:M) (op a neutral)=a. + Variable assoc : (a,b,c:M) (op (op a b) c)=(op a (op b c)). + + Lemma MapFold_ext : (f:ad->A->M) (m,m':(Map A)) (eqmap A m m') -> + (MapFold ? ? neutral op f m)=(MapFold ? ? neutral op f m'). + Proof. + Intros. Rewrite (MapFold_as_fold A M neutral op assoc nleft nright f m). + Rewrite (MapFold_as_fold A M neutral op assoc nleft nright f m'). + Cut (alist_of_Map A m)=(alist_of_Map A m'). Intro. Rewrite H0. Reflexivity. + Apply alist_canonical. Unfold eqmap in H. Apply eqm_trans with f':=(MapGet A m). + Apply eqm_sym. Apply alist_of_Map_semantics. + Apply eqm_trans with f':=(MapGet A m'). Assumption. + Apply alist_of_Map_semantics. + Apply alist_of_Map_sorts2. + Apply alist_of_Map_sorts2. + Qed. + + Lemma MapFold_ext_f_1 : (m:(Map A)) (f,g:ad->A->M) (pf:ad->ad) + ((a:ad) (y:A) (MapGet ? m a)=(SOME ? y) -> (f (pf a) y)=(g (pf a) y)) -> + (MapFold1 ? ? neutral op f pf m)=(MapFold1 ? ? neutral op g pf m). + Proof. + Induction m. Trivial. + Simpl. Intros. Apply H. Rewrite (ad_eq_correct a). Reflexivity. + Intros. Simpl. Rewrite (H f g [a0:ad](pf (ad_double a0))). + Rewrite (H0 f g [a0:ad](pf (ad_double_plus_un a0))). Reflexivity. + Intros. Apply H1. Rewrite MapGet_M2_bit_0_1. Rewrite ad_double_plus_un_div_2. Assumption. + Apply ad_double_plus_un_bit_0. + Intros. Apply H1. Rewrite MapGet_M2_bit_0_0. Rewrite ad_double_div_2. Assumption. + Apply ad_double_bit_0. + Qed. + + Lemma MapFold_ext_f : (f,g:ad->A->M) (m:(Map A)) + ((a:ad) (y:A) (MapGet ? m a)=(SOME ? y) -> (f a y)=(g a y)) -> + (MapFold ? ? neutral op f m)=(MapFold ? ? neutral op g m). + Proof. + Intros. Exact (MapFold_ext_f_1 m f g [a0:ad]a0 H). + Qed. + + Lemma MapFold1_as_Fold_1 : (m:(Map A)) (f,f':ad->A->M) (pf, pf':ad->ad) + ((a:ad) (y:A) (f (pf a) y)=(f' (pf' a) y)) -> + (MapFold1 ? ? neutral op f pf m)=(MapFold1 ? ? neutral op f' pf' m). + Proof. + Induction m. Trivial. + Intros. Simpl. Apply H. + Intros. Simpl. + Rewrite (H f f' [a0:ad](pf (ad_double a0)) [a0:ad](pf' (ad_double a0))). + Rewrite (H0 f f' [a0:ad](pf (ad_double_plus_un a0)) [a0:ad](pf' (ad_double_plus_un a0))). + Reflexivity. + Intros. Apply H1. + Intros. Apply H1. + Qed. + + Lemma MapFold1_as_Fold : (f:ad->A->M) (pf:ad->ad) (m:(Map A)) + (MapFold1 ? ? neutral op f pf m)=(MapFold ? ? neutral op [a:ad][y:A] (f (pf a) y) m). + Proof. + Intros. Unfold MapFold. Apply MapFold1_as_Fold_1. Trivial. + Qed. + + Lemma MapFold1_ext : (f:ad->A->M) (m,m':(Map A)) (eqmap A m m') -> (pf:ad->ad) + (MapFold1 ? ? neutral op f pf m)=(MapFold1 ? ? neutral op f pf m'). + Proof. + Intros. Rewrite MapFold1_as_Fold. Rewrite MapFold1_as_Fold. Apply MapFold_ext. Assumption. + Qed. + + Variable comm : (a,b:M) (op a b)=(op b a). + + Lemma MapFold_Put_disjoint_1 : (p:positive) + (f:ad->A->M) (pf:ad->ad) (a1,a2:ad) (y1,y2:A) + (ad_xor a1 a2)=(ad_x p) -> + (MapFold1 A M neutral op f pf (MapPut1 A a1 y1 a2 y2 p))= + (op (f (pf a1) y1) (f (pf a2) y2)). + Proof. + Induction p. Intros. Simpl. Elim (sumbool_of_bool (ad_bit_0 a1)). Intro H1. Rewrite H1. + Simpl. Rewrite ad_div_2_double_plus_un. Rewrite ad_div_2_double. Apply comm. + Change (ad_bit_0 a2)=(negb true). Rewrite <- H1. Rewrite (ad_neg_bit_0_2 ? ? ? H0). + Rewrite negb_elim. Reflexivity. + Assumption. + Intro H1. Rewrite H1. Simpl. Rewrite ad_div_2_double. Rewrite ad_div_2_double_plus_un. + Reflexivity. + Change (ad_bit_0 a2)=(negb false). Rewrite <- H1. Rewrite (ad_neg_bit_0_2 ? ? ? H0). + Rewrite negb_elim. Reflexivity. + Assumption. + Simpl. Intros. Elim (sumbool_of_bool (ad_bit_0 a1)). Intro H1. Rewrite H1. Simpl. + Rewrite nleft. + Rewrite (H f [a0:ad](pf (ad_double_plus_un a0)) (ad_div_2 a1) (ad_div_2 a2) y1 y2). + Rewrite ad_div_2_double_plus_un. Rewrite ad_div_2_double_plus_un. Reflexivity. + Rewrite <- (ad_same_bit_0 ? ? ? H0). Assumption. + Assumption. + Rewrite <- ad_xor_div_2. Rewrite H0. Reflexivity. + Intro H1. Rewrite H1. Simpl. Rewrite nright. + Rewrite (H f [a0:ad](pf (ad_double a0)) (ad_div_2 a1) (ad_div_2 a2) y1 y2). + Rewrite ad_div_2_double. Rewrite ad_div_2_double. Reflexivity. + Rewrite <- (ad_same_bit_0 ? ? ? H0). Assumption. + Assumption. + Rewrite <- ad_xor_div_2. Rewrite H0. Reflexivity. + Intros. Simpl. Elim (sumbool_of_bool (ad_bit_0 a1)). Intro H0. Rewrite H0. Simpl. + Rewrite ad_div_2_double. Rewrite ad_div_2_double_plus_un. Apply comm. + Assumption. + Change (ad_bit_0 a2)=(negb true). Rewrite <- H0. Rewrite (ad_neg_bit_0_1 ? ? H). + Rewrite negb_elim. Reflexivity. + Intro H0. Rewrite H0. Simpl. Rewrite ad_div_2_double. Rewrite ad_div_2_double_plus_un. + Reflexivity. + Change (ad_bit_0 a2)=(negb false). Rewrite <- H0. Rewrite (ad_neg_bit_0_1 ? ? H). + Rewrite negb_elim. Reflexivity. + Assumption. + Qed. + + Lemma MapFold_Put_disjoint_2 : + (f:ad->A->M) (m:(Map A)) (a:ad) (y:A) (pf:ad->ad) + (MapGet A m a)=(NONE A) -> + (MapFold1 A M neutral op f pf (MapPut A m a y))= + (op (f (pf a) y) (MapFold1 A M neutral op f pf m)). + Proof. + Induction m. Intros. Simpl. Rewrite (nright (f (pf a) y)). Reflexivity. + Intros a1 y1 a2 y2 pf H. Simpl. Elim (ad_sum (ad_xor a1 a2)). Intro H0. Elim H0. + Intros p H1. Rewrite H1. Rewrite comm. Exact (MapFold_Put_disjoint_1 p f pf a1 a2 y1 y2 H1). + Intro H0. Rewrite (ad_eq_complete ? ? (ad_xor_eq_true ? ? H0)) in H. + Rewrite (M1_semantics_1 A a2 y1) in H. Discriminate H. + Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H2. + Cut (MapPut A (M2 A m0 m1) a y)=(M2 A m0 (MapPut A m1 (ad_div_2 a) y)). Intro. + Rewrite H3. Simpl. Rewrite (H0 (ad_div_2 a) y [a0:ad](pf (ad_double_plus_un a0))). + Rewrite ad_div_2_double_plus_un. Rewrite <- assoc. + Rewrite (comm (MapFold1 A M neutral op f [a0:ad](pf (ad_double a0)) m0) (f (pf a) y)). + Rewrite assoc. Reflexivity. + Assumption. + Rewrite (MapGet_M2_bit_0_1 A a H2 m0 m1) in H1. Assumption. + Simpl. Elim (ad_sum a). Intro H3. Elim H3. Intro p. Elim p. Intros p0 H4 H5. Rewrite H5. + Reflexivity. + Intros p0 H4 H5. Rewrite H5 in H2. Discriminate H2. + Intro H4. Rewrite H4. Reflexivity. + Intro H3. Rewrite H3 in H2. Discriminate H2. + Intro H2. Cut (MapPut A (M2 A m0 m1) a y)=(M2 A (MapPut A m0 (ad_div_2 a) y) m1). + Intro. Rewrite H3. Simpl. Rewrite (H (ad_div_2 a) y [a0:ad](pf (ad_double a0))). + Rewrite ad_div_2_double. Rewrite <- assoc. Reflexivity. + Assumption. + Rewrite (MapGet_M2_bit_0_0 A a H2 m0 m1) in H1. Assumption. + Simpl. Elim (ad_sum a). Intro H3. Elim H3. Intro p. Elim p. Intros p0 H4 H5. Rewrite H5 in H2. + Discriminate H2. + Intros p0 H4 H5. Rewrite H5. Reflexivity. + Intro H4. Rewrite H4 in H2. Discriminate H2. + Intro H3. Rewrite H3. Reflexivity. + Qed. + + Lemma MapFold_Put_disjoint : + (f:ad->A->M) (m:(Map A)) (a:ad) (y:A) + (MapGet A m a)=(NONE A) -> + (MapFold A M neutral op f (MapPut A m a y))= + (op (f a y) (MapFold A M neutral op f m)). + Proof. + Intros. Exact (MapFold_Put_disjoint_2 f m a y [a0:ad]a0 H). + Qed. + + Lemma MapFold_Put_behind_disjoint_2 : + (f:ad->A->M) (m:(Map A)) (a:ad) (y:A) (pf:ad->ad) + (MapGet A m a)=(NONE A) -> + (MapFold1 A M neutral op f pf (MapPut_behind A m a y))= + (op (f (pf a) y) (MapFold1 A M neutral op f pf m)). + Proof. + Intros. Cut (eqmap A (MapPut_behind A m a y) (MapPut A m a y)). Intro. + Rewrite (MapFold1_ext f ? ? H0 pf). Apply MapFold_Put_disjoint_2. Assumption. + Apply eqmap_trans with m':=(MapMerge A (M1 A a y) m). Apply MapPut_behind_as_Merge. + Apply eqmap_trans with m':=(MapMerge A m (M1 A a y)). + Apply eqmap_trans with m':=(MapDelta A (M1 A a y) m). Apply eqmap_sym. Apply MapDelta_disjoint. + Unfold MapDisjoint. Unfold in_dom. Simpl. Intros. Elim (sumbool_of_bool (ad_eq a a0)). + Intro H2. Rewrite (ad_eq_complete ? ? H2) in H. Rewrite H in H1. Discriminate H1. + Intro H2. Rewrite H2 in H0. Discriminate H0. + Apply eqmap_trans with m':=(MapDelta A m (M1 A a y)). Apply MapDelta_sym. + Apply MapDelta_disjoint. Unfold MapDisjoint. Unfold in_dom. Simpl. Intros. + Elim (sumbool_of_bool (ad_eq a a0)). Intro H2. Rewrite (ad_eq_complete ? ? H2) in H. + Rewrite H in H0. Discriminate H0. + Intro H2. Rewrite H2 in H1. Discriminate H1. + Apply eqmap_sym. Apply MapPut_as_Merge. + Qed. + + Lemma MapFold_Put_behind_disjoint : + (f:ad->A->M) (m:(Map A)) (a:ad) (y:A) + (MapGet A m a)=(NONE A) -> + (MapFold A M neutral op f (MapPut_behind A m a y)) + =(op (f a y) (MapFold A M neutral op f m)). + Proof. + Intros. Exact (MapFold_Put_behind_disjoint_2 f m a y [a0:ad]a0 H). + Qed. + + Lemma MapFold_Merge_disjoint_1 : + (f:ad->A->M) (m1,m2:(Map A)) (pf:ad->ad) + (MapDisjoint A A m1 m2) -> + (MapFold1 A M neutral op f pf (MapMerge A m1 m2))= + (op (MapFold1 A M neutral op f pf m1) (MapFold1 A M neutral op f pf m2)). + Proof. + Induction m1. Simpl. Intros. Rewrite nleft. Reflexivity. + Intros. Unfold MapMerge. Apply (MapFold_Put_behind_disjoint_2 f m2 a a0 pf). + Apply in_dom_none. Exact (MapDisjoint_M1_l ? ? m2 a a0 H). + Induction m2. Intros. Simpl. Rewrite nright. Reflexivity. + Intros. Unfold MapMerge. Rewrite (MapFold_Put_disjoint_2 f (M2 A m m0) a a0 pf). Apply comm. + Apply in_dom_none. Exact (MapDisjoint_M1_r ? ? (M2 A m m0) a a0 H1). + Intros. Simpl. Rewrite (H m3 [a0:ad](pf (ad_double a0))). + Rewrite (H0 m4 [a0:ad](pf (ad_double_plus_un a0))). + Cut (a,b,c,d:M)(op (op a b) (op c d))=(op (op a c) (op b d)). Intro. Apply H4. + Intros. Rewrite assoc. Rewrite <- (assoc b c d). Rewrite (comm b c). Rewrite (assoc c b d). + Rewrite assoc. Reflexivity. + Exact (MapDisjoint_M2_r ? ? ? ? ? ? H3). + Exact (MapDisjoint_M2_l ? ? ? ? ? ? H3). + Qed. + + Lemma MapFold_Merge_disjoint : + (f:ad->A->M) (m1,m2:(Map A)) + (MapDisjoint A A m1 m2) -> + (MapFold A M neutral op f (MapMerge A m1 m2))= + (op (MapFold A M neutral op f m1) (MapFold A M neutral op f m2)). + Proof. + Intros. Exact (MapFold_Merge_disjoint_1 f m1 m2 [a0:ad]a0 H). + Qed. + +End MapFoldResults. + +Section MapFoldDistr. + + Variable A : Set. + + Variable M : Set. + Variable neutral : M. + Variable op : M -> M -> M. + + Variable M' : Set. + Variable neutral' : M'. + Variable op' : M' -> M' -> M'. + + Variable N : Set. + + Variable times : M -> N -> M'. + + Variable absorb : (c:N)(times neutral c)=neutral'. + Variable distr : (a,b:M) (c:N) (times (op a b) c) = (op' (times a c) (times b c)). + + Lemma MapFold_distr_r_1 : (f:ad->A->M) (m:(Map A)) (c:N) (pf:ad->ad) + (times (MapFold1 A M neutral op f pf m) c)= + (MapFold1 A M' neutral' op' [a:ad][y:A] (times (f a y) c) pf m). + Proof. + Induction m. Intros. Exact (absorb c). + Trivial. + Intros. Simpl. Rewrite distr. Rewrite H. Rewrite H0. Reflexivity. + Qed. + + Lemma MapFold_distr_r : (f:ad->A->M) (m:(Map A)) (c:N) + (times (MapFold A M neutral op f m) c)= + (MapFold A M' neutral' op' [a:ad][y:A] (times (f a y) c) m). + Proof. + Intros. Exact (MapFold_distr_r_1 f m c [a:ad]a). + Qed. + +End MapFoldDistr. + +Section MapFoldDistrL. + + Variable A : Set. + + Variable M : Set. + Variable neutral : M. + Variable op : M -> M -> M. + + Variable M' : Set. + Variable neutral' : M'. + Variable op' : M' -> M' -> M'. + + Variable N : Set. + + Variable times : N -> M -> M'. + + Variable absorb : (c:N)(times c neutral)=neutral'. + Variable distr : (a,b:M) (c:N) (times c (op a b)) = (op' (times c a) (times c b)). + + Lemma MapFold_distr_l : (f:ad->A->M) (m:(Map A)) (c:N) + (times c (MapFold A M neutral op f m))= + (MapFold A M' neutral' op' [a:ad][y:A] (times c (f a y)) m). + Proof. + Intros. Apply MapFold_distr_r with times:=[a:M][b:N](times b a); Assumption. + Qed. + +End MapFoldDistrL. + +Section MapFoldExists. + + Variable A : Set. + + Lemma MapFold_orb_1 : (f:ad->A->bool) (m:(Map A)) (pf:ad->ad) + (MapFold1 A bool false orb f pf m)= + (Cases (MapSweep1 A f pf m) of + (SOME _) => true + | _ => false + end). + Proof. + Induction m. Trivial. + Intros a y pf. Simpl. Unfold MapSweep2. (Case (f (pf a) y); Reflexivity). + Intros. Simpl. Rewrite (H [a0:ad](pf (ad_double a0))). + Rewrite (H0 [a0:ad](pf (ad_double_plus_un a0))). + Case (MapSweep1 A f [a0:ad](pf (ad_double a0)) m0); Reflexivity. + Qed. + + Lemma MapFold_orb : (f:ad->A->bool) (m:(Map A)) (MapFold A bool false orb f m)= + (Cases (MapSweep A f m) of + (SOME _) => true + | _ => false + end). + Proof. + Intros. Exact (MapFold_orb_1 f m [a:ad]a). + Qed. + +End MapFoldExists. + +Section DMergeDef. + + Variable A : Set. + + Definition DMerge := (MapFold (Map A) (Map A) (M0 A) (MapMerge A) [_:ad][m:(Map A)] m). + + Lemma in_dom_DMerge_1 : (m:(Map (Map A))) (a:ad) (in_dom A a (DMerge m))= + (Cases (MapSweep ? [_:ad][m0:(Map A)] (in_dom A a m0) m) of + (SOME _) => true + | _ => false + end). + Proof. + Unfold DMerge. Intros. + Rewrite (MapFold_distr_l (Map A) (Map A) (M0 A) (MapMerge A) bool false + orb ad (in_dom A) [c:ad](refl_equal ? ?) (in_dom_merge A)). + Apply MapFold_orb. + Qed. + + Lemma in_dom_DMerge_2 : (m:(Map (Map A))) (a:ad) (in_dom A a (DMerge m))=true -> + {b:ad & {m0:(Map A) | (MapGet ? m b)=(SOME ? m0) /\ + (in_dom A a m0)=true}}. + Proof. + Intros m a. Rewrite in_dom_DMerge_1. + Elim (option_sum ? (MapSweep (Map A) [_:ad][m0:(Map A)](in_dom A a m0) m)). + Intro H. Elim H. Intro r. Elim r. Intros b m0 H0. Intro. Split with b. Split with m0. + Split. Exact (MapSweep_semantics_2 ? ? ? ? ? H0). + Exact (MapSweep_semantics_1 ? ? ? ? ? H0). + Intro H. Rewrite H. Intro. Discriminate H0. + Qed. + + Lemma in_dom_DMerge_3 : (m:(Map (Map A))) (a,b:ad) (m0:(Map A)) + (MapGet ? m a)=(SOME ? m0) -> (in_dom A b m0)=true -> + (in_dom A b (DMerge m))=true. + Proof. + Intros m a b m0 H H0. Rewrite in_dom_DMerge_1. + Elim (MapSweep_semantics_4 ? [_:ad][m'0:(Map A)](in_dom A b m'0) ? ? ? H H0). + Intros a' H1. Elim H1. Intros m'0 H2. Rewrite H2. Reflexivity. + Qed. + +End DMergeDef. diff --git a/theories7/IntMap/Mapiter.v b/theories7/IntMap/Mapiter.v new file mode 100644 index 00000000..144572fd --- /dev/null +++ b/theories7/IntMap/Mapiter.v @@ -0,0 +1,527 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Mapiter.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Mapaxioms. +Require Fset. +Require PolyList. + +Section MapIter. + + Variable A : Set. + + Section MapSweepDef. + + Variable f:ad->A->bool. + + Definition MapSweep2 := [a0:ad; y:A] if (f a0 y) then (SOME ? (a0, y)) else (NONE ?). + + Fixpoint MapSweep1 [pf:ad->ad; m:(Map A)] : (option (ad * A)) := + Cases m of + M0 => (NONE ?) + | (M1 a y) => (MapSweep2 (pf a) y) + | (M2 m m') => Cases (MapSweep1 ([a:ad] (pf (ad_double a))) m) of + (SOME r) => (SOME ? r) + | NONE => (MapSweep1 ([a:ad] (pf (ad_double_plus_un a))) m') + end + end. + + Definition MapSweep := [m:(Map A)] (MapSweep1 ([a:ad] a) m). + + Lemma MapSweep_semantics_1_1 : (m:(Map A)) (pf:ad->ad) (a:ad) (y:A) + (MapSweep1 pf m)=(SOME ? (a, y)) -> (f a y)=true. + Proof. + Induction m. Intros. Discriminate H. + Simpl. Intros a y pf a0 y0. Elim (sumbool_of_bool (f (pf a) y)). Intro H. Unfold MapSweep2. + Rewrite H. Intro H0. Inversion H0. Rewrite <- H3. Assumption. + Intro H. Unfold MapSweep2. Rewrite H. Intro H0. Discriminate H0. + Simpl. Intros. Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)). + Intro H2. Elim H2. Intros r H3. Rewrite H3 in H1. Inversion H1. Rewrite H5 in H3. + Exact (H [a0:ad](pf (ad_double a0)) a y H3). + Intro H2. Rewrite H2 in H1. Exact (H0 [a0:ad](pf (ad_double_plus_un a0)) a y H1). + Qed. + + Lemma MapSweep_semantics_1 : (m:(Map A)) (a:ad) (y:A) + (MapSweep m)=(SOME ? (a, y)) -> (f a y)=true. + Proof. + Intros. Exact (MapSweep_semantics_1_1 m [a:ad]a a y H). + Qed. + + Lemma MapSweep_semantics_2_1 : (m:(Map A)) (pf:ad->ad) (a:ad) (y:A) + (MapSweep1 pf m)=(SOME ? (a, y)) -> {a':ad | a=(pf a')}. + Proof. + Induction m. Intros. Discriminate H. + Simpl. Unfold MapSweep2. Intros a y pf a0 y0. Case (f (pf a) y). Intros. Split with a. + Inversion H. Reflexivity. + Intro. Discriminate H. + Intros m0 H m1 H0 pf a y. Simpl. + Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)). Intro H1. Elim H1. + Intros r H2. Rewrite H2. Intro H3. Inversion H3. Rewrite H5 in H2. + Elim (H [a0:ad](pf (ad_double a0)) a y H2). Intros a0 H6. Split with (ad_double a0). + Assumption. + Intro H1. Rewrite H1. Intro H2. Elim (H0 [a0:ad](pf (ad_double_plus_un a0)) a y H2). + Intros a0 H3. Split with (ad_double_plus_un a0). Assumption. + Qed. + + Lemma MapSweep_semantics_2_2 : (m:(Map A)) + (pf,fp:ad->ad) ((a0:ad) (fp (pf a0))=a0) -> (a:ad) (y:A) + (MapSweep1 pf m)=(SOME ? (a, y)) -> (MapGet A m (fp a))=(SOME ? y). + Proof. + Induction m. Intros. Discriminate H0. + Simpl. Intros a y pf fp H a0 y0. Unfold MapSweep2. Elim (sumbool_of_bool (f (pf a) y)). + Intro H0. Rewrite H0. Intro H1. Inversion H1. Rewrite (H a). Rewrite (ad_eq_correct a). + Reflexivity. + Intro H0. Rewrite H0. Intro H1. Discriminate H1. + Intros. Rewrite (MapGet_M2_bit_0_if A m0 m1 (fp a)). Elim (sumbool_of_bool (ad_bit_0 (fp a))). + Intro H3. Rewrite H3. Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)). + Intro H4. Simpl in H2. Apply (H0 [a0:ad](pf (ad_double_plus_un a0)) [a0:ad](ad_div_2 (fp a0))). + Intro. Rewrite H1. Apply ad_double_plus_un_div_2. + Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)). Intro H5. Elim H5. + Intros r H6. Rewrite H6 in H2. Inversion H2. Rewrite H8 in H6. + Elim (MapSweep_semantics_2_1 m0 [a0:ad](pf (ad_double a0)) a y H6). Intros a0 H9. + Rewrite H9 in H3. Rewrite (H1 (ad_double a0)) in H3. Rewrite (ad_double_bit_0 a0) in H3. + Discriminate H3. + Intro H5. Rewrite H5 in H2. Assumption. + Intro H4. Simpl in H2. Rewrite H4 in H2. + Apply (H0 [a0:ad](pf (ad_double_plus_un a0)) [a0:ad](ad_div_2 (fp a0))). Intro. + Rewrite H1. Apply ad_double_plus_un_div_2. + Assumption. + Intro H3. Rewrite H3. Simpl in H2. + Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)). Intro H4. Elim H4. + Intros r H5. Rewrite H5 in H2. Inversion H2. Rewrite H7 in H5. + Apply (H [a0:ad](pf (ad_double a0)) [a0:ad](ad_div_2 (fp a0))). Intro. Rewrite H1. + Apply ad_double_div_2. + Assumption. + Intro H4. Rewrite H4 in H2. + Elim (MapSweep_semantics_2_1 m1 [a0:ad](pf (ad_double_plus_un a0)) a y H2). + Intros a0 H5. Rewrite H5 in H3. Rewrite (H1 (ad_double_plus_un a0)) in H3. + Rewrite (ad_double_plus_un_bit_0 a0) in H3. Discriminate H3. + Qed. + + Lemma MapSweep_semantics_2 : (m:(Map A)) (a:ad) (y:A) + (MapSweep m)=(SOME ? (a, y)) -> (MapGet A m a)=(SOME ? y). + Proof. + Intros. + Exact (MapSweep_semantics_2_2 m [a0:ad]a0 [a0:ad]a0 [a0:ad](refl_equal ad a0) a y H). + Qed. + + Lemma MapSweep_semantics_3_1 : (m:(Map A)) (pf:ad->ad) + (MapSweep1 pf m)=(NONE ?) -> + (a:ad) (y:A) (MapGet A m a)=(SOME ? y) -> (f (pf a) y)=false. + Proof. + Induction m. Intros. Discriminate H0. + Simpl. Unfold MapSweep2. Intros a y pf. Elim (sumbool_of_bool (f (pf a) y)). Intro H. + Rewrite H. Intro. Discriminate H0. + Intro H. Rewrite H. Intros H0 a0 y0. Elim (sumbool_of_bool (ad_eq a a0)). Intro H1. Rewrite H1. + Intro H2. Inversion H2. Rewrite <- H4. Rewrite <- (ad_eq_complete ? ? H1). Assumption. + Intro H1. Rewrite H1. Intro. Discriminate H2. + Intros. Simpl in H1. Elim (option_sum ad*A (MapSweep1 [a:ad](pf (ad_double a)) m0)). + Intro H3. Elim H3. Intros r H4. Rewrite H4 in H1. Discriminate H1. + Intro H3. Rewrite H3 in H1. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H4. + Rewrite (MapGet_M2_bit_0_1 A a H4 m0 m1) in H2. Rewrite <- (ad_div_2_double_plus_un a H4). + Exact (H0 [a:ad](pf (ad_double_plus_un a)) H1 (ad_div_2 a) y H2). + Intro H4. Rewrite (MapGet_M2_bit_0_0 A a H4 m0 m1) in H2. Rewrite <- (ad_div_2_double a H4). + Exact (H [a:ad](pf (ad_double a)) H3 (ad_div_2 a) y H2). + Qed. + + Lemma MapSweep_semantics_3 : (m:(Map A)) + (MapSweep m)=(NONE ?) -> (a:ad) (y:A) (MapGet A m a)=(SOME ? y) -> + (f a y)=false. + Proof. + Intros. + Exact (MapSweep_semantics_3_1 m [a0:ad]a0 H a y H0). + Qed. + + Lemma MapSweep_semantics_4_1 : (m:(Map A)) (pf:ad->ad) (a:ad) (y:A) + (MapGet A m a)=(SOME A y) -> (f (pf a) y)=true -> + {a':ad & {y':A | (MapSweep1 pf m)=(SOME ? (a', y'))}}. + Proof. + Induction m. Intros. Discriminate H. + Intros. Elim (sumbool_of_bool (ad_eq a a1)). Intro H1. Split with (pf a1). Split with y. + Rewrite (ad_eq_complete ? ? H1). Unfold MapSweep1 MapSweep2. + Rewrite (ad_eq_complete ? ? H1) in H. Rewrite (M1_semantics_1 ? a1 a0) in H. + Inversion H. Rewrite H0. Reflexivity. + + Intro H1. Rewrite (M1_semantics_2 ? a a1 a0 H1) in H. Discriminate H. + + Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H3. + Rewrite (MapGet_M2_bit_0_1 ? ? H3 m0 m1) in H1. + Rewrite <- (ad_div_2_double_plus_un a H3) in H2. + Elim (H0 [a0:ad](pf (ad_double_plus_un a0)) (ad_div_2 a) y H1 H2). Intros a'' H4. Elim H4. + Intros y'' H5. Simpl. Elim (option_sum ? (MapSweep1 [a:ad](pf (ad_double a)) m0)). + Intro H6. Elim H6. Intro r. Elim r. Intros a''' y''' H7. Rewrite H7. Split with a'''. + Split with y'''. Reflexivity. + Intro H6. Rewrite H6. Split with a''. Split with y''. Assumption. + Intro H3. Rewrite (MapGet_M2_bit_0_0 ? ? H3 m0 m1) in H1. + Rewrite <- (ad_div_2_double a H3) in H2. + Elim (H [a0:ad](pf (ad_double a0)) (ad_div_2 a) y H1 H2). Intros a'' H4. Elim H4. + Intros y'' H5. Split with a''. Split with y''. Simpl. Rewrite H5. Reflexivity. + Qed. + + Lemma MapSweep_semantics_4 : (m:(Map A)) (a:ad) (y:A) + (MapGet A m a)=(SOME A y) -> (f a y)=true -> + {a':ad & {y':A | (MapSweep m)=(SOME ? (a', y'))}}. + Proof. + Intros. Exact (MapSweep_semantics_4_1 m [a0:ad]a0 a y H H0). + Qed. + + End MapSweepDef. + + Variable B : Set. + + Fixpoint MapCollect1 [f:ad->A->(Map B); pf:ad->ad; m:(Map A)] : (Map B) := + Cases m of + M0 => (M0 B) + | (M1 a y) => (f (pf a) y) + | (M2 m1 m2) => (MapMerge B (MapCollect1 f [a0:ad] (pf (ad_double a0)) m1) + (MapCollect1 f [a0:ad] (pf (ad_double_plus_un a0)) m2)) + end. + + Definition MapCollect := [f:ad->A->(Map B); m:(Map A)] (MapCollect1 f [a:ad]a m). + + Section MapFoldDef. + + Variable M : Set. + Variable neutral : M. + Variable op : M -> M -> M. + + Fixpoint MapFold1 [f:ad->A->M; pf:ad->ad; m:(Map A)] : M := + Cases m of + M0 => neutral + | (M1 a y) => (f (pf a) y) + | (M2 m1 m2) => (op (MapFold1 f [a0:ad] (pf (ad_double a0)) m1) + (MapFold1 f [a0:ad] (pf (ad_double_plus_un a0)) m2)) + end. + + Definition MapFold := [f:ad->A->M; m:(Map A)] (MapFold1 f [a:ad]a m). + + Lemma MapFold_empty : (f:ad->A->M) (MapFold f (M0 A))=neutral. + Proof. + Trivial. + Qed. + + Lemma MapFold_M1 : (f:ad->A->M) (a:ad) (y:A) (MapFold f (M1 A a y)) = (f a y). + Proof. + Trivial. + Qed. + + Variable State : Set. + Variable f:State -> ad -> A -> State * M. + + Fixpoint MapFold1_state [state:State; pf:ad->ad; m:(Map A)] + : State * M := + Cases m of + M0 => (state, neutral) + | (M1 a y) => (f state (pf a) y) + | (M2 m1 m2) => + Cases (MapFold1_state state [a0:ad] (pf (ad_double a0)) m1) of + (state1, x1) => + Cases (MapFold1_state state1 [a0:ad] (pf (ad_double_plus_un a0)) m2) of + (state2, x2) => (state2, (op x1 x2)) + end + end + end. + + Definition MapFold_state := [state:State] (MapFold1_state state [a:ad]a). + + Lemma pair_sp : (B,C:Set) (x:B*C) x=(Fst x, Snd x). + Proof. + Induction x. Trivial. + Qed. + + Lemma MapFold_state_stateless_1 : (m:(Map A)) (g:ad->A->M) (pf:ad->ad) + ((state:State) (a:ad) (y:A) (Snd (f state a y))=(g a y)) -> + (state:State) + (Snd (MapFold1_state state pf m))=(MapFold1 g pf m). + Proof. + Induction m. Trivial. + Intros. Simpl. Apply H. + Intros. Simpl. Rewrite (pair_sp ? ? + (MapFold1_state state [a0:ad](pf (ad_double a0)) m0)). + Rewrite (H g [a0:ad](pf (ad_double a0)) H1 state). + Rewrite (pair_sp ? ? + (MapFold1_state + (Fst (MapFold1_state state [a0:ad](pf (ad_double a0)) m0)) + [a0:ad](pf (ad_double_plus_un a0)) m1)). + Simpl. + Rewrite (H0 g [a0:ad](pf (ad_double_plus_un a0)) H1 + (Fst (MapFold1_state state [a0:ad](pf (ad_double a0)) m0))). + Reflexivity. + Qed. + + Lemma MapFold_state_stateless : (g:ad->A->M) + ((state:State) (a:ad) (y:A) (Snd (f state a y))=(g a y)) -> + (state:State) (m:(Map A)) + (Snd (MapFold_state state m))=(MapFold g m). + Proof. + Intros. Exact (MapFold_state_stateless_1 m g [a0:ad]a0 H state). + Qed. + + End MapFoldDef. + + Lemma MapCollect_as_Fold : (f:ad->A->(Map B)) (m:(Map A)) + (MapCollect f m)=(MapFold (Map B) (M0 B) (MapMerge B) f m). + Proof. + Induction m;Trivial. + Qed. + + Definition alist := (list (ad*A)). + Definition anil := (nil (ad*A)). + Definition acons := (!cons (ad*A)). + Definition aapp := (!app (ad*A)). + + Definition alist_of_Map := (MapFold alist anil aapp [a:ad;y:A] (acons (pair ? ? a y) anil)). + + Fixpoint alist_semantics [l:alist] : ad -> (option A) := + Cases l of + nil => [_:ad] (NONE A) + | (cons (a, y) l') => [a0:ad] if (ad_eq a a0) then (SOME A y) else (alist_semantics l' a0) + end. + + Lemma alist_semantics_app : (l,l':alist) (a:ad) + (alist_semantics (aapp l l') a)= + (Cases (alist_semantics l a) of + NONE => (alist_semantics l' a) + | (SOME y) => (SOME A y) + end). + Proof. + Unfold aapp. Induction l. Trivial. + Intros. Elim a. Intros a1 y1. Simpl. Case (ad_eq a1 a0). Reflexivity. + Apply H. + Qed. + + Lemma alist_of_Map_semantics_1_1 : (m:(Map A)) (pf:ad->ad) (a:ad) (y:A) + (alist_semantics (MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil) pf m) a) + =(SOME A y) -> {a':ad | a=(pf a')}. + Proof. + Induction m. Simpl. Intros. Discriminate H. + Simpl. Intros a y pf a0 y0. Elim (sumbool_of_bool (ad_eq (pf a) a0)). Intro H. Rewrite H. + Intro H0. Split with a. Rewrite (ad_eq_complete ? ? H). Reflexivity. + Intro H. Rewrite H. Intro H0. Discriminate H0. + Intros. Change (alist_semantics + (aapp + (MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil) + [a0:ad](pf (ad_double a0)) m0) + (MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil) + [a0:ad](pf (ad_double_plus_un a0)) m1)) a)=(SOME A y) in H1. + Rewrite (alist_semantics_app + (MapFold1 alist anil aapp [a0:ad][y0:A](acons (a0,y0) anil) + [a0:ad](pf (ad_double a0)) m0) + (MapFold1 alist anil aapp [a0:ad][y0:A](acons (a0,y0) anil) + [a0:ad](pf (ad_double_plus_un a0)) m1) a) in H1. + Elim (option_sum A + (alist_semantics + (MapFold1 alist anil aapp [a0:ad][y0:A](acons (a0,y0) anil) + [a0:ad](pf (ad_double a0)) m0) a)). + Intro H2. Elim H2. Intros y0 H3. Elim (H [a0:ad](pf (ad_double a0)) a y0 H3). Intros a0 H4. + Split with (ad_double a0). Assumption. + Intro H2. Rewrite H2 in H1. Elim (H0 [a0:ad](pf (ad_double_plus_un a0)) a y H1). + Intros a0 H3. Split with (ad_double_plus_un a0). Assumption. + Qed. + + Definition ad_inj := [pf:ad->ad] (a0,a1:ad) (pf a0)=(pf a1) -> a0=a1. + + Lemma ad_comp_double_inj : + (pf:ad->ad) (ad_inj pf) -> (ad_inj [a0:ad] (pf (ad_double a0))). + Proof. + Unfold ad_inj. Intros. Apply ad_double_inj. Exact (H ? ? H0). + Qed. + + Lemma ad_comp_double_plus_un_inj : (pf:ad->ad) (ad_inj pf) -> + (ad_inj [a0:ad] (pf (ad_double_plus_un a0))). + Proof. + Unfold ad_inj. Intros. Apply ad_double_plus_un_inj. Exact (H ? ? H0). + Qed. + + Lemma alist_of_Map_semantics_1 : (m:(Map A)) (pf:ad->ad) (ad_inj pf) -> + (a:ad) (MapGet A m a)=(alist_semantics (MapFold1 alist anil aapp + [a0:ad;y:A] (acons (pair ? ? a0 y) anil) pf m) + (pf a)). + Proof. + Induction m. Trivial. + Simpl. Intros. Elim (sumbool_of_bool (ad_eq a a1)). Intro H0. Rewrite H0. + Rewrite (ad_eq_complete ? ? H0). Rewrite (ad_eq_correct (pf a1)). Reflexivity. + Intro H0. Rewrite H0. Elim (sumbool_of_bool (ad_eq (pf a) (pf a1))). Intro H1. + Rewrite (H a a1 (ad_eq_complete ? ? H1)) in H0. Rewrite (ad_eq_correct a1) in H0. + Discriminate H0. + Intro H1. Rewrite H1. Reflexivity. + Intros. Change (MapGet A (M2 A m0 m1) a) + =(alist_semantics + (aapp + (MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil) + [a0:ad](pf (ad_double a0)) m0) + (MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil) + [a0:ad](pf (ad_double_plus_un a0)) m1)) (pf a)). + Rewrite alist_semantics_app. Rewrite (MapGet_M2_bit_0_if A m0 m1 a). + Elim (ad_double_or_double_plus_un a). Intro H2. Elim H2. Intros a0 H3. Rewrite H3. + Rewrite (ad_double_bit_0 a0). + Rewrite <- (H [a1:ad](pf (ad_double a1)) (ad_comp_double_inj pf H1) a0). + Rewrite ad_double_div_2. Case (MapGet A m0 a0). + Elim (option_sum A + (alist_semantics + (MapFold1 alist anil aapp [a1:ad][y:A](acons (a1,y) anil) + [a1:ad](pf (ad_double_plus_un a1)) m1) (pf (ad_double a0)))). + Intro H4. Elim H4. Intros y H5. + Elim (alist_of_Map_semantics_1_1 m1 [a1:ad](pf (ad_double_plus_un a1)) + (pf (ad_double a0)) y H5). + Intros a1 H6. Cut (ad_bit_0 (ad_double a0))=(ad_bit_0 (ad_double_plus_un a1)). + Intro. Rewrite (ad_double_bit_0 a0) in H7. Rewrite (ad_double_plus_un_bit_0 a1) in H7. + Discriminate H7. + Rewrite (H1 (ad_double a0) (ad_double_plus_un a1) H6). Reflexivity. + Intro H4. Rewrite H4. Reflexivity. + Trivial. + Intro H2. Elim H2. Intros a0 H3. Rewrite H3. Rewrite (ad_double_plus_un_bit_0 a0). + Rewrite <- (H0 [a1:ad](pf (ad_double_plus_un a1)) (ad_comp_double_plus_un_inj pf H1) a0). + Rewrite ad_double_plus_un_div_2. + Elim (option_sum A + (alist_semantics + (MapFold1 alist anil aapp [a1:ad][y:A](acons (a1,y) anil) + [a1:ad](pf (ad_double a1)) m0) (pf (ad_double_plus_un a0)))). + Intro H4. Elim H4. Intros y H5. + Elim (alist_of_Map_semantics_1_1 m0 [a1:ad](pf (ad_double a1)) + (pf (ad_double_plus_un a0)) y H5). + Intros a1 H6. Cut (ad_bit_0 (ad_double_plus_un a0))=(ad_bit_0 (ad_double a1)). + Intro H7. Rewrite (ad_double_plus_un_bit_0 a0) in H7. Rewrite (ad_double_bit_0 a1) in H7. + Discriminate H7. + Rewrite (H1 (ad_double_plus_un a0) (ad_double a1) H6). Reflexivity. + Intro H4. Rewrite H4. Reflexivity. + Qed. + + Lemma alist_of_Map_semantics : (m:(Map A)) + (eqm A (MapGet A m) (alist_semantics (alist_of_Map m))). + Proof. + Unfold eqm. Intros. Exact (alist_of_Map_semantics_1 m [a0:ad]a0 [a0,a1:ad][p:a0=a1]p a). + Qed. + + Fixpoint Map_of_alist [l:alist] : (Map A) := + Cases l of + nil => (M0 A) + | (cons (a, y) l') => (MapPut A (Map_of_alist l') a y) + end. + + Lemma Map_of_alist_semantics : (l:alist) + (eqm A (alist_semantics l) (MapGet A (Map_of_alist l))). + Proof. + Unfold eqm. Induction l. Trivial. + Intros r l0 H a. Elim r. Intros a0 y0. Simpl. Elim (sumbool_of_bool (ad_eq a0 a)). + Intro H0. Rewrite H0. Rewrite (ad_eq_complete ? ? H0). + Rewrite (MapPut_semantics A (Map_of_alist l0) a y0 a). Rewrite (ad_eq_correct a). + Reflexivity. + Intro H0. Rewrite H0. Rewrite (MapPut_semantics A (Map_of_alist l0) a0 y0 a). + Rewrite H0. Apply H. + Qed. + + Lemma Map_of_alist_of_Map : (m:(Map A)) (eqmap A (Map_of_alist (alist_of_Map m)) m). + Proof. + Unfold eqmap. Intro. Apply eqm_trans with f':=(alist_semantics (alist_of_Map m)). + Apply eqm_sym. Apply Map_of_alist_semantics. + Apply eqm_sym. Apply alist_of_Map_semantics. + Qed. + + Lemma alist_of_Map_of_alist : (l:alist) + (eqm A (alist_semantics (alist_of_Map (Map_of_alist l))) (alist_semantics l)). + Proof. + Intro. Apply eqm_trans with f':=(MapGet A (Map_of_alist l)). + Apply eqm_sym. Apply alist_of_Map_semantics. + Apply eqm_sym. Apply Map_of_alist_semantics. + Qed. + + Lemma fold_right_aapp : (M:Set) (neutral:M) (op:M->M->M) + ((a,b,c:M) (op (op a b) c)=(op a (op b c))) -> + ((a:M) (op neutral a)=a) -> + (f:ad->A->M) (l,l':alist) + (fold_right [r:ad*A][m:M] let (a,y)=r in (op (f a y) m) neutral + (aapp l l'))= + (op (fold_right [r:ad*A][m:M] let (a,y)=r in (op (f a y) m) neutral l) + (fold_right [r:ad*A][m:M] let (a,y)=r in (op (f a y) m) neutral l')) +. + Proof. + Induction l. Simpl. Intro. Rewrite H0. Reflexivity. + Intros r l0 H1 l'. Elim r. Intros a y. Simpl. Rewrite H. Rewrite (H1 l'). Reflexivity. + Qed. + + Lemma MapFold_as_fold_1 : (M:Set) (neutral:M) (op:M->M->M) + ((a,b,c:M) (op (op a b) c)=(op a (op b c))) -> + ((a:M) (op neutral a)=a) -> + ((a:M) (op a neutral)=a) -> + (f:ad->A->M) (m:(Map A)) (pf:ad->ad) + (MapFold1 M neutral op f pf m)= + (fold_right [r:(ad*A)][m:M] let (a,y)=r in (op (f a y) m) neutral + (MapFold1 alist anil aapp [a:ad;y:A] (acons (pair ? ? +a y) anil) pf m)). + Proof. + Induction m. Trivial. + Intros. Simpl. Rewrite H1. Reflexivity. + Intros. Simpl. Rewrite (fold_right_aapp M neutral op H H0 f). + Rewrite (H2 [a0:ad](pf (ad_double a0))). Rewrite (H3 [a0:ad](pf (ad_double_plus_un a0))). + Reflexivity. + Qed. + + Lemma MapFold_as_fold : (M:Set) (neutral:M) (op:M->M->M) + ((a,b,c:M) (op (op a b) c)=(op a (op b c))) -> + ((a:M) (op neutral a)=a) -> + ((a:M) (op a neutral)=a) -> + (f:ad->A->M) (m:(Map A)) + (MapFold M neutral op f m)= + (fold_right [r:(ad*A)][m:M] let (a,y)=r in (op (f a y) m) neutral + (alist_of_Map m)). + Proof. + Intros. Exact (MapFold_as_fold_1 M neutral op H H0 H1 f m [a0:ad]a0). + Qed. + + Lemma alist_MapMerge_semantics : (m,m':(Map A)) + (eqm A (alist_semantics (aapp (alist_of_Map m') (alist_of_Map m))) + (alist_semantics (alist_of_Map (MapMerge A m m')))). + Proof. + Unfold eqm. Intros. Rewrite alist_semantics_app. Rewrite <- (alist_of_Map_semantics m a). + Rewrite <- (alist_of_Map_semantics m' a). + Rewrite <- (alist_of_Map_semantics (MapMerge A m m') a). + Rewrite (MapMerge_semantics A m m' a). Reflexivity. + Qed. + + Lemma alist_MapMerge_semantics_disjoint : (m,m':(Map A)) + (eqmap A (MapDomRestrTo A A m m') (M0 A)) -> + (eqm A (alist_semantics (aapp (alist_of_Map m) (alist_of_Map m'))) + (alist_semantics (alist_of_Map (MapMerge A m m')))). + Proof. + Unfold eqm. Intros. Rewrite alist_semantics_app. Rewrite <- (alist_of_Map_semantics m a). + Rewrite <- (alist_of_Map_semantics m' a). + Rewrite <- (alist_of_Map_semantics (MapMerge A m m') a). Rewrite (MapMerge_semantics A m m' a). + Elim (option_sum ? (MapGet A m a)). Intro H0. Elim H0. Intros y H1. Rewrite H1. + Elim (option_sum ? (MapGet A m' a)). Intro H2. Elim H2. Intros y' H3. + Cut (MapGet A (MapDomRestrTo A A m m') a)=(NONE A). + Rewrite (MapDomRestrTo_semantics A A m m' a). Rewrite H3. Rewrite H1. Intro. Discriminate H4. + Exact (H a). + Intro H2. Rewrite H2. Reflexivity. + Intro H0. Rewrite H0. Case (MapGet A m' a); Trivial. + Qed. + + Lemma alist_semantics_disjoint_comm : (l,l':alist) + (eqmap A (MapDomRestrTo A A (Map_of_alist l) (Map_of_alist l')) (M0 A)) -> + (eqm A (alist_semantics (aapp l l')) (alist_semantics (aapp l' l))). + Proof. + Unfold eqm. Intros. Rewrite (alist_semantics_app l l' a). Rewrite (alist_semantics_app l' l a). + Rewrite <- (alist_of_Map_of_alist l a). Rewrite <- (alist_of_Map_of_alist l' a). + Rewrite <- (alist_semantics_app (alist_of_Map (Map_of_alist l)) + (alist_of_Map (Map_of_alist l')) a). + Rewrite <- (alist_semantics_app (alist_of_Map (Map_of_alist l')) + (alist_of_Map (Map_of_alist l)) a). + Rewrite (alist_MapMerge_semantics (Map_of_alist l) (Map_of_alist l') a). + Rewrite (alist_MapMerge_semantics_disjoint (Map_of_alist l) (Map_of_alist l') H a). + Reflexivity. + Qed. + +End MapIter. + diff --git a/theories7/IntMap/Maplists.v b/theories7/IntMap/Maplists.v new file mode 100644 index 00000000..f01ee3d8 --- /dev/null +++ b/theories7/IntMap/Maplists.v @@ -0,0 +1,399 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Maplists.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Addr. +Require Addec. +Require Map. +Require Fset. +Require Mapaxioms. +Require Mapsubset. +Require Mapcard. +Require Mapcanon. +Require Mapc. +Require Bool. +Require Sumbool. +Require PolyList. +Require Arith. +Require Mapiter. +Require Mapfold. + +Section MapLists. + + Fixpoint ad_in_list [a:ad;l:(list ad)] : bool := + Cases l of + nil => false + | (cons a' l') => (orb (ad_eq a a') (ad_in_list a l')) + end. + + Fixpoint ad_list_stutters [l:(list ad)] : bool := + Cases l of + nil => false + | (cons a l') => (orb (ad_in_list a l') (ad_list_stutters l')) + end. + + Lemma ad_in_list_forms_circuit : (x:ad) (l:(list ad)) (ad_in_list x l)=true -> + {l1 : (list ad) & {l2 : (list ad) | l=(app l1 (cons x l2))}}. + Proof. + Induction l. Intro. Discriminate H. + Intros. Elim (sumbool_of_bool (ad_eq x a)). Intro H1. Simpl in H0. Split with (nil ad). + Split with l0. Rewrite (ad_eq_complete ? ? H1). Reflexivity. + Intro H2. Simpl in H0. Rewrite H2 in H0. Simpl in H0. Elim (H H0). Intros l'1 H3. + Split with (cons a l'1). Elim H3. Intros l2 H4. Split with l2. Rewrite H4. Reflexivity. + Qed. + + Lemma ad_list_stutters_has_circuit : (l:(list ad)) (ad_list_stutters l)=true -> + {x:ad & {l0 : (list ad) & {l1 : (list ad) & {l2 : (list ad) | + l=(app l0 (cons x (app l1 (cons x l2))))}}}}. + Proof. + Induction l. Intro. Discriminate H. + Intros. Simpl in H0. Elim (orb_true_elim ? ? H0). Intro H1. Split with a. + Split with (nil ad). Simpl. Elim (ad_in_list_forms_circuit a l0 H1). Intros l1 H2. + Split with l1. Elim H2. Intros l2 H3. Split with l2. Rewrite H3. Reflexivity. + Intro H1. Elim (H H1). Intros x H2. Split with x. Elim H2. Intros l1 H3. + Split with (cons a l1). Elim H3. Intros l2 H4. Split with l2. Elim H4. Intros l3 H5. + Split with l3. Rewrite H5. Reflexivity. + Qed. + + Fixpoint Elems [l:(list ad)] : FSet := + Cases l of + nil => (M0 unit) + | (cons a l') => (MapPut ? (Elems l') a tt) + end. + + Lemma Elems_canon : (l:(list ad)) (mapcanon ? (Elems l)). + Proof. + Induction l. Exact (M0_canon unit). + Intros. Simpl. Apply MapPut_canon. Assumption. + Qed. + + Lemma Elems_app : (l,l':(list ad)) (Elems (app l l'))=(FSetUnion (Elems l) (Elems l')). + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite (MapPut_as_Merge_c unit (Elems l0)). + Rewrite (MapPut_as_Merge_c unit (Elems (app l0 l'))). + Change (FSetUnion (Elems (app l0 l')) (M1 unit a tt)) + =(FSetUnion (FSetUnion (Elems l0) (M1 unit a tt)) (Elems l')). + Rewrite FSetUnion_comm_c. Rewrite (FSetUnion_comm_c (Elems l0) (M1 unit a tt)). + Rewrite FSetUnion_assoc_c. Rewrite (H l'). Reflexivity. + Apply M1_canon. + Apply Elems_canon. + Apply Elems_canon. + Apply Elems_canon. + Apply M1_canon. + Apply Elems_canon. + Apply M1_canon. + Apply Elems_canon. + Apply Elems_canon. + Qed. + + Lemma Elems_rev : (l:(list ad)) (Elems (rev l))=(Elems l). + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite Elems_app. Simpl. Rewrite (MapPut_as_Merge_c unit (Elems l0)). + Rewrite H. Reflexivity. + Apply Elems_canon. + Qed. + + Lemma ad_in_elems_in_list : (l:(list ad)) (a:ad) (in_FSet a (Elems l))=(ad_in_list a l). + Proof. + Induction l. Trivial. + Simpl. Unfold in_FSet. Intros. Rewrite (in_dom_put ? (Elems l0) a tt a0). + Rewrite (H a0). Reflexivity. + Qed. + + Lemma ad_list_not_stutters_card : (l:(list ad)) (ad_list_stutters l)=false -> + (length l)=(MapCard ? (Elems l)). + Proof. + Induction l. Trivial. + Simpl. Intros. Rewrite MapCard_Put_2_conv. Rewrite H. Reflexivity. + Elim (orb_false_elim ? ? H0). Trivial. + Elim (sumbool_of_bool (in_FSet a (Elems l0))). Rewrite ad_in_elems_in_list. + Intro H1. Rewrite H1 in H0. Discriminate H0. + Exact (in_dom_none unit (Elems l0) a). + Qed. + + Lemma ad_list_card : (l:(list ad)) (le (MapCard ? (Elems l)) (length l)). + Proof. + Induction l. Trivial. + Intros. Simpl. Apply le_trans with m:=(S (MapCard ? (Elems l0))). Apply MapCard_Put_ub. + Apply le_n_S. Assumption. + Qed. + + Lemma ad_list_stutters_card : (l:(list ad)) (ad_list_stutters l)=true -> + (lt (MapCard ? (Elems l)) (length l)). + Proof. + Induction l. Intro. Discriminate H. + Intros. Simpl. Simpl in H0. Elim (orb_true_elim ? ? H0). Intro H1. + Rewrite <- (ad_in_elems_in_list l0 a) in H1. Elim (in_dom_some ? ? ? H1). Intros y H2. + Rewrite (MapCard_Put_1_conv ? ? ? ? tt H2). Apply le_lt_trans with m:=(length l0). + Apply ad_list_card. + Apply lt_n_Sn. + Intro H1. Apply le_lt_trans with m:=(S (MapCard ? (Elems l0))). Apply MapCard_Put_ub. + Apply lt_n_S. Apply H. Assumption. + Qed. + + Lemma ad_list_not_stutters_card_conv : (l:(list ad)) (length l)=(MapCard ? (Elems l)) -> + (ad_list_stutters l)=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_list_stutters l)). Intro H0. + Cut (lt (MapCard ? (Elems l)) (length l)). Intro. Rewrite H in H1. Elim (lt_n_n ? H1). + Exact (ad_list_stutters_card ? H0). + Trivial. + Qed. + + Lemma ad_list_stutters_card_conv : (l:(list ad)) (lt (MapCard ? (Elems l)) (length l)) -> + (ad_list_stutters l)=true. + Proof. + Intros. Elim (sumbool_of_bool (ad_list_stutters l)). Trivial. + Intro H0. Rewrite (ad_list_not_stutters_card ? H0) in H. Elim (lt_n_n ? H). + Qed. + + Lemma ad_in_list_l : (l,l':(list ad)) (a:ad) (ad_in_list a l)=true -> + (ad_in_list a (app l l'))=true. + Proof. + Induction l. Intros. Discriminate H. + Intros. Simpl. Simpl in H0. Elim (orb_true_elim ? ? H0). Intro H1. Rewrite H1. Reflexivity. + Intro H1. Rewrite (H l' a0 H1). Apply orb_b_true. + Qed. + + Lemma ad_list_stutters_app_l : (l,l':(list ad)) (ad_list_stutters l)=true -> + (ad_list_stutters (app l l'))=true. + Proof. + Induction l. Intros. Discriminate H. + Intros. Simpl. Simpl in H0. Elim (orb_true_elim ? ? H0). Intro H1. + Rewrite (ad_in_list_l l0 l' a H1). Reflexivity. + Intro H1. Rewrite (H l' H1). Apply orb_b_true. + Qed. + + Lemma ad_in_list_r : (l,l':(list ad)) (a:ad) (ad_in_list a l')=true -> + (ad_in_list a (app l l'))=true. + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite (H l' a0 H0). Apply orb_b_true. + Qed. + + Lemma ad_list_stutters_app_r : (l,l':(list ad)) (ad_list_stutters l')=true -> + (ad_list_stutters (app l l'))=true. + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite (H l' H0). Apply orb_b_true. + Qed. + + Lemma ad_list_stutters_app_conv_l : (l,l':(list ad)) (ad_list_stutters (app l l'))=false -> + (ad_list_stutters l)=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_list_stutters l)). Intro H0. + Rewrite (ad_list_stutters_app_l l l' H0) in H. Discriminate H. + Trivial. + Qed. + + Lemma ad_list_stutters_app_conv_r : (l,l':(list ad)) (ad_list_stutters (app l l'))=false -> + (ad_list_stutters l')=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_list_stutters l')). Intro H0. + Rewrite (ad_list_stutters_app_r l l' H0) in H. Discriminate H. + Trivial. + Qed. + + Lemma ad_in_list_app_1 : (l,l':(list ad)) (x:ad) (ad_in_list x (app l (cons x l')))=true. + Proof. + Induction l. Simpl. Intros. Rewrite (ad_eq_correct x). Reflexivity. + Intros. Simpl. Rewrite (H l' x). Apply orb_b_true. + Qed. + + Lemma ad_in_list_app : (l,l':(list ad)) (x:ad) + (ad_in_list x (app l l'))=(orb (ad_in_list x l) (ad_in_list x l')). + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite <- orb_assoc. Rewrite (H l' x). Reflexivity. + Qed. + + Lemma ad_in_list_rev : (l:(list ad)) (x:ad) + (ad_in_list x (rev l))=(ad_in_list x l). + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite ad_in_list_app. Rewrite (H x). Simpl. Rewrite orb_b_false. + Apply orb_sym. + Qed. + + Lemma ad_list_has_circuit_stutters : (l0,l1,l2:(list ad)) (x:ad) + (ad_list_stutters (app l0 (cons x (app l1 (cons x l2)))))=true. + Proof. + Induction l0. Simpl. Intros. Rewrite (ad_in_list_app_1 l1 l2 x). Reflexivity. + Intros. Simpl. Rewrite (H l1 l2 x). Apply orb_b_true. + Qed. + + Lemma ad_list_stutters_prev_l : (l,l':(list ad)) (x:ad) (ad_in_list x l)=true -> + (ad_list_stutters (app l (cons x l')))=true. + Proof. + Intros. Elim (ad_in_list_forms_circuit ? ? H). Intros l0 H0. Elim H0. Intros l1 H1. + Rewrite H1. Rewrite app_ass. Simpl. Apply ad_list_has_circuit_stutters. + Qed. + + Lemma ad_list_stutters_prev_conv_l : (l,l':(list ad)) (x:ad) + (ad_list_stutters (app l (cons x l')))=false -> (ad_in_list x l)=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_in_list x l)). Intro H0. + Rewrite (ad_list_stutters_prev_l l l' x H0) in H. Discriminate H. + Trivial. + Qed. + + Lemma ad_list_stutters_prev_r : (l,l':(list ad)) (x:ad) (ad_in_list x l')=true -> + (ad_list_stutters (app l (cons x l')))=true. + Proof. + Intros. Elim (ad_in_list_forms_circuit ? ? H). Intros l0 H0. Elim H0. Intros l1 H1. + Rewrite H1. Apply ad_list_has_circuit_stutters. + Qed. + + Lemma ad_list_stutters_prev_conv_r : (l,l':(list ad)) (x:ad) + (ad_list_stutters (app l (cons x l')))=false -> (ad_in_list x l')=false. + Proof. + Intros. Elim (sumbool_of_bool (ad_in_list x l')). Intro H0. + Rewrite (ad_list_stutters_prev_r l l' x H0) in H. Discriminate H. + Trivial. + Qed. + + Lemma ad_list_Elems : (l,l':(list ad)) (MapCard ? (Elems l))=(MapCard ? (Elems l')) -> + (length l)=(length l') -> + (ad_list_stutters l)=(ad_list_stutters l'). + Proof. + Intros. Elim (sumbool_of_bool (ad_list_stutters l)). Intro H1. Rewrite H1. Apply sym_eq. + Apply ad_list_stutters_card_conv. Rewrite <- H. Rewrite <- H0. Apply ad_list_stutters_card. + Assumption. + Intro H1. Rewrite H1. Apply sym_eq. Apply ad_list_not_stutters_card_conv. Rewrite <- H. + Rewrite <- H0. Apply ad_list_not_stutters_card. Assumption. + Qed. + + Lemma ad_list_app_length : (l,l':(list ad)) (length (app l l'))=(plus (length l) (length l')). + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite (H l'). Reflexivity. + Qed. + + Lemma ad_list_stutters_permute : (l,l':(list ad)) + (ad_list_stutters (app l l'))=(ad_list_stutters (app l' l)). + Proof. + Intros. Apply ad_list_Elems. Rewrite Elems_app. Rewrite Elems_app. + Rewrite (FSetUnion_comm_c ? ? (Elems_canon l) (Elems_canon l')). Reflexivity. + Rewrite ad_list_app_length. Rewrite ad_list_app_length. Apply plus_sym. + Qed. + + Lemma ad_list_rev_length : (l:(list ad)) (length (rev l))=(length l). + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite ad_list_app_length. Simpl. Rewrite H. Rewrite <- plus_Snm_nSm. + Rewrite <- plus_n_O. Reflexivity. + Qed. + + Lemma ad_list_stutters_rev : (l:(list ad)) (ad_list_stutters (rev l))=(ad_list_stutters l). + Proof. + Intros. Apply ad_list_Elems. Rewrite Elems_rev. Reflexivity. + Apply ad_list_rev_length. + Qed. + + Lemma ad_list_app_rev : (l,l':(list ad)) (x:ad) + (app (rev l) (cons x l'))=(app (rev (cons x l)) l'). + Proof. + Induction l. Trivial. + Intros. Simpl. Rewrite (app_ass (rev l0) (cons a (nil ad)) (cons x l')). Simpl. + Rewrite (H (cons x l') a). Simpl. + Rewrite (app_ass (rev l0) (cons a (nil ad)) (cons x (nil ad))). Simpl. + Rewrite app_ass. Simpl. Rewrite app_ass. Reflexivity. + Qed. + + Section ListOfDomDef. + + Variable A : Set. + + Definition ad_list_of_dom := + (MapFold A (list ad) (nil ad) (!app ad) [a:ad][_:A] (cons a (nil ad))). + + Lemma ad_in_list_of_dom_in_dom : (m:(Map A)) (a:ad) + (ad_in_list a (ad_list_of_dom m))=(in_dom A a m). + Proof. + Unfold ad_list_of_dom. Intros. + Rewrite (MapFold_distr_l A (list ad) (nil ad) (!app ad) bool false orb + ad [a:ad][l:(list ad)](ad_in_list a l) [c:ad](refl_equal ? ?) + ad_in_list_app [a0:ad][_:A](cons a0 (nil ad)) m a). + Simpl. Rewrite (MapFold_orb A [a0:ad][_:A](orb (ad_eq a a0) false) m). + Elim (option_sum ? (MapSweep A [a0:ad][_:A](orb (ad_eq a a0) false) m)). Intro H. Elim H. + Intro r. Elim r. Intros a0 y H0. Rewrite H0. Unfold in_dom. + Elim (orb_prop ? ? (MapSweep_semantics_1 ? ? ? ? ? H0)). Intro H1. + Rewrite (ad_eq_complete ? ? H1). Rewrite (MapSweep_semantics_2 A ? ? ? ? H0). Reflexivity. + Intro H1. Discriminate H1. + Intro H. Rewrite H. Elim (sumbool_of_bool (in_dom A a m)). Intro H0. + Elim (in_dom_some A m a H0). Intros y H1. + Elim (orb_false_elim ? ? (MapSweep_semantics_3 ? ? ? H ? ? H1)). Intro H2. + Rewrite (ad_eq_correct a) in H2. Discriminate H2. + Exact (sym_eq ? ? ?). + Qed. + + Lemma Elems_of_list_of_dom : + (m:(Map A)) (eqmap unit (Elems (ad_list_of_dom m)) (MapDom A m)). + Proof. + Unfold eqmap eqm. Intros. Elim (sumbool_of_bool (in_FSet a (Elems (ad_list_of_dom m)))). + Intro H. Elim (in_dom_some ? ? ? H). Intro t. Elim t. Intro H0. + Rewrite (ad_in_elems_in_list (ad_list_of_dom m) a) in H. + Rewrite (ad_in_list_of_dom_in_dom m a) in H. Rewrite (MapDom_Dom A m a) in H. + Elim (in_dom_some ? ? ? H). Intro t'. Elim t'. Intro H1. Rewrite H1. Assumption. + Intro H. Rewrite (in_dom_none ? ? ? H). + Rewrite (ad_in_elems_in_list (ad_list_of_dom m) a) in H. + Rewrite (ad_in_list_of_dom_in_dom m a) in H. Rewrite (MapDom_Dom A m a) in H. + Rewrite (in_dom_none ? ? ? H). Reflexivity. + Qed. + + Lemma Elems_of_list_of_dom_c : (m:(Map A)) (mapcanon A m) -> + (Elems (ad_list_of_dom m))=(MapDom A m). + Proof. + Intros. Apply (mapcanon_unique unit). Apply Elems_canon. + Apply MapDom_canon. Assumption. + Apply Elems_of_list_of_dom. + Qed. + + Lemma ad_list_of_dom_card_1 : (m:(Map A)) (pf:ad->ad) + (length (MapFold1 A (list ad) (nil ad) (app 1!ad) [a:ad][_:A](cons a (nil ad)) pf m))= + (MapCard A m). + Proof. + Induction m; Try Trivial. Simpl. Intros. Rewrite ad_list_app_length. + Rewrite (H [a0:ad](pf (ad_double a0))). Rewrite (H0 [a0:ad](pf (ad_double_plus_un a0))). + Reflexivity. + Qed. + + Lemma ad_list_of_dom_card : (m:(Map A)) (length (ad_list_of_dom m))=(MapCard A m). + Proof. + Exact [m:(Map A)](ad_list_of_dom_card_1 m [a:ad]a). + Qed. + + Lemma ad_list_of_dom_not_stutters : + (m:(Map A)) (ad_list_stutters (ad_list_of_dom m))=false. + Proof. + Intro. Apply ad_list_not_stutters_card_conv. Rewrite ad_list_of_dom_card. Apply sym_eq. + Rewrite (MapCard_Dom A m). Apply MapCard_ext. Exact (Elems_of_list_of_dom m). + Qed. + + End ListOfDomDef. + + Lemma ad_list_of_dom_Dom_1 : (A:Set) + (m:(Map A)) (pf:ad->ad) + (MapFold1 A (list ad) (nil ad) (app 1!ad) + [a:ad][_:A](cons a (nil ad)) pf m)= + (MapFold1 unit (list ad) (nil ad) (app 1!ad) + [a:ad][_:unit](cons a (nil ad)) pf (MapDom A m)). + Proof. + Induction m; Try Trivial. Simpl. Intros. Rewrite (H [a0:ad](pf (ad_double a0))). + Rewrite (H0 [a0:ad](pf (ad_double_plus_un a0))). Reflexivity. + Qed. + + Lemma ad_list_of_dom_Dom : (A:Set) (m:(Map A)) + (ad_list_of_dom A m)=(ad_list_of_dom unit (MapDom A m)). + Proof. + Intros. Exact (ad_list_of_dom_Dom_1 A m [a0:ad]a0). + Qed. + +End MapLists. diff --git a/theories7/IntMap/Mapsubset.v b/theories7/IntMap/Mapsubset.v new file mode 100644 index 00000000..c0b1cccd --- /dev/null +++ b/theories7/IntMap/Mapsubset.v @@ -0,0 +1,554 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Mapsubset.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Bool. +Require Sumbool. +Require Arith. +Require ZArith. +Require Addr. +Require Adist. +Require Addec. +Require Map. +Require Fset. +Require Mapaxioms. +Require Mapiter. + +Section MapSubsetDef. + + Variable A, B : Set. + + Definition MapSubset := [m:(Map A)] [m':(Map B)] + (a:ad) (in_dom A a m)=true -> (in_dom B a m')=true. + + Definition MapSubset_1 := [m:(Map A)] [m':(Map B)] + Cases (MapSweep A [a:ad][_:A] (negb (in_dom B a m')) m) of + NONE => true + | _ => false + end. + + Definition MapSubset_2 := [m:(Map A)] [m':(Map B)] + (eqmap A (MapDomRestrBy A B m m') (M0 A)). + + Lemma MapSubset_imp_1 : (m:(Map A)) (m':(Map B)) + (MapSubset m m') -> (MapSubset_1 m m')=true. + Proof. + Unfold MapSubset MapSubset_1. Intros. + Elim (option_sum ? (MapSweep A [a:ad][_:A](negb (in_dom B a m')) m)). + Intro H0. Elim H0. Intro r. Elim r. Intros a y H1. Cut (negb (in_dom B a m'))=true. + Intro. Cut (in_dom A a m)=false. Intro. Unfold in_dom in H3. + Rewrite (MapSweep_semantics_2 ? ? m a y H1) in H3. Discriminate H3. + Elim (sumbool_of_bool (in_dom A a m)). Intro H3. Rewrite (H a H3) in H2. Discriminate H2. + Trivial. + Exact (MapSweep_semantics_1 ? ? m a y H1). + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Lemma MapSubset_1_imp : (m:(Map A)) (m':(Map B)) + (MapSubset_1 m m')=true -> (MapSubset m m'). + Proof. + Unfold MapSubset MapSubset_1. Unfold 2 in_dom. Intros. Elim (option_sum ? (MapGet A m a)). + Intro H1. Elim H1. Intros y H2. + Elim (option_sum ? (MapSweep A [a:ad][_:A](negb (in_dom B a m')) m)). Intro H3. + Elim H3. Intro r. Elim r. Intros a' y' H4. Rewrite H4 in H. Discriminate H. + Intro H3. Cut (negb (in_dom B a m'))=false. Intro. Rewrite (negb_intro (in_dom B a m')). + Rewrite H4. Reflexivity. + Exact (MapSweep_semantics_3 ? ? m H3 a y H2). + Intro H1. Rewrite H1 in H0. Discriminate H0. + Qed. + + Lemma map_dom_empty_1 : + (m:(Map A)) (eqmap A m (M0 A)) -> (a:ad) (in_dom ? a m)=false. + Proof. + Unfold eqmap eqm in_dom. Intros. Rewrite (H a). Reflexivity. + Qed. + + Lemma map_dom_empty_2 : + (m:(Map A)) ((a:ad) (in_dom ? a m)=false) -> (eqmap A m (M0 A)). + Proof. + Unfold eqmap eqm in_dom. Intros. + Cut (Cases (MapGet A m a) of NONE => false | (SOME _) => true end)=false. + Case (MapGet A m a). Trivial. + Intros. Discriminate H0. + Exact (H a). + Qed. + + Lemma MapSubset_imp_2 : + (m:(Map A)) (m':(Map B)) (MapSubset m m') -> (MapSubset_2 m m'). + Proof. + Unfold MapSubset MapSubset_2. Intros. Apply map_dom_empty_2. Intro. Rewrite in_dom_restrby. + Elim (sumbool_of_bool (in_dom A a m)). Intro H0. Rewrite H0. Rewrite (H a H0). Reflexivity. + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Lemma MapSubset_2_imp : + (m:(Map A)) (m':(Map B)) (MapSubset_2 m m') -> (MapSubset m m'). + Proof. + Unfold MapSubset MapSubset_2. Intros. Cut (in_dom ? a (MapDomRestrBy A B m m'))=false. + Rewrite in_dom_restrby. Intro. Elim (andb_false_elim ? ? H1). Rewrite H0. + Intro H2. Discriminate H2. + Intro H2. Rewrite (negb_intro (in_dom B a m')). Rewrite H2. Reflexivity. + Exact (map_dom_empty_1 ? H a). + Qed. + +End MapSubsetDef. + +Section MapSubsetOrder. + + Variable A, B, C : Set. + + Lemma MapSubset_refl : (m:(Map A)) (MapSubset A A m m). + Proof. + Unfold MapSubset. Trivial. + Qed. + + Lemma MapSubset_antisym : (m:(Map A)) (m':(Map B)) + (MapSubset A B m m') -> (MapSubset B A m' m) -> + (eqmap unit (MapDom A m) (MapDom B m')). + Proof. + Unfold MapSubset eqmap eqm. Intros. Elim (option_sum ? (MapGet ? (MapDom A m) a)). + Intro H1. Elim H1. Intro t. Elim t. Intro H2. Elim (option_sum ? (MapGet ? (MapDom B m') a)). + Intro H3. Elim H3. Intro t'. Elim t'. Intro H4. Rewrite H4. Exact H2. + Intro H3. Cut (in_dom B a m')=true. Intro. Rewrite (MapDom_Dom B m' a) in H4. + Unfold in_FSet in_dom in H4. Rewrite H3 in H4. Discriminate H4. + Apply H. Rewrite (MapDom_Dom A m a). Unfold in_FSet in_dom. Rewrite H2. Reflexivity. + Intro H1. Elim (option_sum ? (MapGet ? (MapDom B m') a)). Intro H2. Elim H2. Intros t H3. + Cut (in_dom A a m)=true. Intro. Rewrite (MapDom_Dom A m a) in H4. Unfold in_FSet in_dom in H4. + Rewrite H1 in H4. Discriminate H4. + Apply H0. Rewrite (MapDom_Dom B m' a). Unfold in_FSet in_dom. Rewrite H3. Reflexivity. + Intro H2. Rewrite H2. Exact H1. + Qed. + + Lemma MapSubset_trans : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (MapSubset A B m m') -> (MapSubset B C m' m'') -> (MapSubset A C m m''). + Proof. + Unfold MapSubset. Intros. Apply H0. Apply H. Assumption. + Qed. + +End MapSubsetOrder. + +Section FSubsetOrder. + + Lemma FSubset_refl : (s:FSet) (MapSubset ? ? s s). + Proof. + Exact (MapSubset_refl unit). + Qed. + + Lemma FSubset_antisym : (s,s':FSet) + (MapSubset ? ? s s') -> (MapSubset ? ? s' s) -> (eqmap unit s s'). + Proof. + Intros. Rewrite <- (FSet_Dom s). Rewrite <- (FSet_Dom s'). + Exact (MapSubset_antisym ? ? s s' H H0). + Qed. + + Lemma FSubset_trans : (s,s',s'':FSet) + (MapSubset ? ? s s') -> (MapSubset ? ? s' s'') -> (MapSubset ? ? s s''). + Proof. + Exact (MapSubset_trans unit unit unit). + Qed. + +End FSubsetOrder. + +Section MapSubsetExtra. + + Variable A, B : Set. + + Lemma MapSubset_Dom_1 : (m:(Map A)) (m':(Map B)) + (MapSubset A B m m') -> (MapSubset unit unit (MapDom A m) (MapDom B m')). + Proof. + Unfold MapSubset. Intros. Elim (MapDom_semantics_2 ? m a H0). Intros y H1. + Cut (in_dom A a m)=true->(in_dom B a m')=true. Intro. Unfold in_dom in H2. + Rewrite H1 in H2. Elim (option_sum ? (MapGet B m' a)). Intro H3. Elim H3. + Intros y' H4. Exact (MapDom_semantics_1 ? m' a y' H4). + Intro H3. Rewrite H3 in H2. Cut false=true. Intro. Discriminate H4. + Apply H2. Reflexivity. + Exact (H a). + Qed. + + Lemma MapSubset_Dom_2 : (m:(Map A)) (m':(Map B)) + (MapSubset unit unit (MapDom A m) (MapDom B m')) -> (MapSubset A B m m'). + Proof. + Unfold MapSubset. Intros. Unfold in_dom in H0. Elim (option_sum ? (MapGet A m a)). + Intro H1. Elim H1. Intros y H2. + Elim (MapDom_semantics_2 ? ? ? (H a (MapDom_semantics_1 ? ? ? ? H2))). Intros y' H3. + Unfold in_dom. Rewrite H3. Reflexivity. + Intro H1. Rewrite H1 in H0. Discriminate H0. + Qed. + + Lemma MapSubset_1_Dom : (m:(Map A)) (m':(Map B)) + (MapSubset_1 A B m m')=(MapSubset_1 unit unit (MapDom A m) (MapDom B m')). + Proof. + Intros. Elim (sumbool_of_bool (MapSubset_1 A B m m')). Intro H. Rewrite H. + Apply sym_eq. Apply MapSubset_imp_1. Apply MapSubset_Dom_1. Exact (MapSubset_1_imp ? ? ? ? H). + Intro H. Rewrite H. Elim (sumbool_of_bool (MapSubset_1 unit unit (MapDom A m) (MapDom B m'))). + Intro H0. + Rewrite (MapSubset_imp_1 ? ? ? ? (MapSubset_Dom_2 ? ? (MapSubset_1_imp ? ? ? ? H0))) in H. + Discriminate H. + Intro. Apply sym_eq. Assumption. + Qed. + + Lemma MapSubset_Put : (m:(Map A)) (a:ad) (y:A) (MapSubset A A m (MapPut A m a y)). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_put. Rewrite H. Apply orb_b_true. + Qed. + + Lemma MapSubset_Put_mono : (m:(Map A)) (m':(Map B)) (a:ad) (y:A) (y':B) + (MapSubset A B m m') -> (MapSubset A B (MapPut A m a y) (MapPut B m' a y')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_put. Rewrite (in_dom_put A m a y a0) in H0. + Elim (orb_true_elim ? ? H0). Intro H1. Rewrite H1. Reflexivity. + Intro H1. Rewrite (H ? H1). Apply orb_b_true. + Qed. + + Lemma MapSubset_Put_behind : + (m:(Map A)) (a:ad) (y:A) (MapSubset A A m (MapPut_behind A m a y)). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_put_behind. Rewrite H. Apply orb_b_true. + Qed. + + Lemma MapSubset_Put_behind_mono : (m:(Map A)) (m':(Map B)) (a:ad) (y:A) (y':B) + (MapSubset A B m m') -> + (MapSubset A B (MapPut_behind A m a y) (MapPut_behind B m' a y')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_put_behind. + Rewrite (in_dom_put_behind A m a y a0) in H0. + Elim (orb_true_elim ? ? H0). Intro H1. Rewrite H1. Reflexivity. + Intro H1. Rewrite (H ? H1). Apply orb_b_true. + Qed. + + Lemma MapSubset_Remove : (m:(Map A)) (a:ad) (MapSubset A A (MapRemove A m a) m). + Proof. + Unfold MapSubset. Intros. Unfold MapSubset. Intros. Rewrite (in_dom_remove ? m a a0) in H. + Elim (andb_prop ? ? H). Trivial. + Qed. + + Lemma MapSubset_Remove_mono : (m:(Map A)) (m':(Map B)) (a:ad) + (MapSubset A B m m') -> (MapSubset A B (MapRemove A m a) (MapRemove B m' a)). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_remove. Rewrite (in_dom_remove A m a a0) in H0. + Elim (andb_prop ? ? H0). Intros. Rewrite H1. Rewrite (H ? H2). Reflexivity. + Qed. + + Lemma MapSubset_Merge_l : (m,m':(Map A)) (MapSubset A A m (MapMerge A m m')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_merge. Rewrite H. Reflexivity. + Qed. + + Lemma MapSubset_Merge_r : (m,m':(Map A)) (MapSubset A A m' (MapMerge A m m')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_merge. Rewrite H. Apply orb_b_true. + Qed. + + Lemma MapSubset_Merge_mono : (m,m':(Map A)) (m'',m''':(Map B)) + (MapSubset A B m m'') -> (MapSubset A B m' m''') -> + (MapSubset A B (MapMerge A m m') (MapMerge B m'' m''')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_merge. Rewrite (in_dom_merge A m m' a) in H1. + Elim (orb_true_elim ? ? H1). Intro H2. Rewrite (H ? H2). Reflexivity. + Intro H2. Rewrite (H0 ? H2). Apply orb_b_true. + Qed. + + Lemma MapSubset_DomRestrTo_l : (m:(Map A)) (m':(Map B)) + (MapSubset A A (MapDomRestrTo A B m m') m). + Proof. + Unfold MapSubset. Intros. Rewrite (in_dom_restrto ? ? m m' a) in H. Elim (andb_prop ? ? H). + Trivial. + Qed. + + Lemma MapSubset_DomRestrTo_r: (m:(Map A)) (m':(Map B)) + (MapSubset A B (MapDomRestrTo A B m m') m'). + Proof. + Unfold MapSubset. Intros. Rewrite (in_dom_restrto ? ? m m' a) in H. Elim (andb_prop ? ? H). + Trivial. + Qed. + + Lemma MapSubset_ext : (m0,m1:(Map A)) (m2,m3:(Map B)) + (eqmap A m0 m1) -> (eqmap B m2 m3) -> + (MapSubset A B m0 m2) -> (MapSubset A B m1 m3). + Proof. + Intros. Apply MapSubset_2_imp. Unfold MapSubset_2. + Apply eqmap_trans with m':=(MapDomRestrBy A B m0 m2). Apply MapDomRestrBy_ext. Apply eqmap_sym. + Assumption. + Apply eqmap_sym. Assumption. + Exact (MapSubset_imp_2 ? ? ? ? H1). + Qed. + + Variable C, D : Set. + + Lemma MapSubset_DomRestrTo_mono : + (m:(Map A)) (m':(Map B)) (m'':(Map C)) (m''':(Map D)) + (MapSubset ? ? m m'') -> (MapSubset ? ? m' m''') -> + (MapSubset ? ? (MapDomRestrTo ? ? m m') (MapDomRestrTo ? ? m'' m''')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_restrto. Rewrite (in_dom_restrto A B m m' a) in H1. + Elim (andb_prop ? ? H1). Intros. Rewrite (H ? H2). Rewrite (H0 ? H3). Reflexivity. + Qed. + + Lemma MapSubset_DomRestrBy_l : (m:(Map A)) (m':(Map B)) + (MapSubset A A (MapDomRestrBy A B m m') m). + Proof. + Unfold MapSubset. Intros. Rewrite (in_dom_restrby ? ? m m' a) in H. Elim (andb_prop ? ? H). + Trivial. + Qed. + + Lemma MapSubset_DomRestrBy_mono : + (m:(Map A)) (m':(Map B)) (m'':(Map C)) (m''':(Map D)) + (MapSubset ? ? m m'') -> (MapSubset ? ? m''' m') -> + (MapSubset ? ? (MapDomRestrBy ? ? m m') (MapDomRestrBy ? ? m'' m''')). + Proof. + Unfold MapSubset. Intros. Rewrite in_dom_restrby. Rewrite (in_dom_restrby A B m m' a) in H1. + Elim (andb_prop ? ? H1). Intros. Rewrite (H ? H2). Elim (sumbool_of_bool (in_dom D a m''')). + Intro H4. Rewrite (H0 ? H4) in H3. Discriminate H3. + Intro H4. Rewrite H4. Reflexivity. + Qed. + +End MapSubsetExtra. + +Section MapDisjointDef. + + Variable A, B : Set. + + Definition MapDisjoint := [m:(Map A)] [m':(Map B)] + (a:ad) (in_dom A a m)=true -> (in_dom B a m')=true -> False. + + Definition MapDisjoint_1 := [m:(Map A)] [m':(Map B)] + Cases (MapSweep A [a:ad][_:A] (in_dom B a m') m) of + NONE => true + | _ => false + end. + + Definition MapDisjoint_2 := [m:(Map A)] [m':(Map B)] + (eqmap A (MapDomRestrTo A B m m') (M0 A)). + + Lemma MapDisjoint_imp_1 : (m:(Map A)) (m':(Map B)) + (MapDisjoint m m') -> (MapDisjoint_1 m m')=true. + Proof. + Unfold MapDisjoint MapDisjoint_1. Intros. + Elim (option_sum ? (MapSweep A [a:ad][_:A](in_dom B a m') m)). Intro H0. Elim H0. + Intro r. Elim r. Intros a y H1. Cut (in_dom A a m)=true->(in_dom B a m')=true->False. + Intro. Unfold 1 in_dom in H2. Rewrite (MapSweep_semantics_2 ? ? ? ? ? H1) in H2. + Rewrite (MapSweep_semantics_1 ? ? ? ? ? H1) in H2. Elim (H2 (refl_equal ? ?) (refl_equal ? ?)). + Exact (H a). + Intro H0. Rewrite H0. Reflexivity. + Qed. + + Lemma MapDisjoint_1_imp : (m:(Map A)) (m':(Map B)) + (MapDisjoint_1 m m')=true -> (MapDisjoint m m'). + Proof. + Unfold MapDisjoint MapDisjoint_1. Intros. + Elim (option_sum ? (MapSweep A [a:ad][_:A](in_dom B a m') m)). Intro H2. Elim H2. + Intro r. Elim r. Intros a' y' H3. Rewrite H3 in H. Discriminate H. + Intro H2. Unfold in_dom in H0. Elim (option_sum ? (MapGet A m a)). Intro H3. Elim H3. + Intros y H4. Rewrite (MapSweep_semantics_3 ? ? ? H2 a y H4) in H1. Discriminate H1. + Intro H3. Rewrite H3 in H0. Discriminate H0. + Qed. + + Lemma MapDisjoint_imp_2 : (m:(Map A)) (m':(Map B)) (MapDisjoint m m') -> + (MapDisjoint_2 m m'). + Proof. + Unfold MapDisjoint MapDisjoint_2. Unfold eqmap eqm. Intros. + Rewrite (MapDomRestrTo_semantics A B m m' a). + Cut (in_dom A a m)=true->(in_dom B a m')=true->False. Intro. + Elim (option_sum ? (MapGet A m a)). Intro H1. Elim H1. Intros y H2. Unfold 1 in_dom in H0. + Elim (option_sum ? (MapGet B m' a)). Intro H3. Elim H3. Intros y' H4. Unfold 1 in_dom in H0. + Rewrite H4 in H0. Rewrite H2 in H0. Elim (H0 (refl_equal ? ?) (refl_equal ? ?)). + Intro H3. Rewrite H3. Reflexivity. + Intro H1. Rewrite H1. Case (MapGet B m' a); Reflexivity. + Exact (H a). + Qed. + + Lemma MapDisjoint_2_imp : (m:(Map A)) (m':(Map B)) (MapDisjoint_2 m m') -> + (MapDisjoint m m'). + Proof. + Unfold MapDisjoint MapDisjoint_2. Unfold eqmap eqm. Intros. Elim (in_dom_some ? ? ? H0). + Intros y H2. Elim (in_dom_some ? ? ? H1). Intros y' H3. + Cut (MapGet A (MapDomRestrTo A B m m') a)=(NONE A). Intro. + Rewrite (MapDomRestrTo_semantics ? ? m m' a) in H4. Rewrite H3 in H4. Rewrite H2 in H4. + Discriminate H4. + Exact (H a). + Qed. + + Lemma Map_M0_disjoint : (m:(Map B)) (MapDisjoint (M0 A) m). + Proof. + Unfold MapDisjoint in_dom. Intros. Discriminate H. + Qed. + + Lemma Map_disjoint_M0 : (m:(Map A)) (MapDisjoint m (M0 B)). + Proof. + Unfold MapDisjoint in_dom. Intros. Discriminate H0. + Qed. + +End MapDisjointDef. + +Section MapDisjointExtra. + + Variable A, B : Set. + + Lemma MapDisjoint_ext : (m0,m1:(Map A)) (m2,m3:(Map B)) + (eqmap A m0 m1) -> (eqmap B m2 m3) -> + (MapDisjoint A B m0 m2) -> (MapDisjoint A B m1 m3). + Proof. + Intros. Apply MapDisjoint_2_imp. Unfold MapDisjoint_2. + Apply eqmap_trans with m':=(MapDomRestrTo A B m0 m2). Apply eqmap_sym. Apply MapDomRestrTo_ext. + Assumption. + Assumption. + Exact (MapDisjoint_imp_2 ? ? ? ? H1). + Qed. + + Lemma MapMerge_disjoint : (m,m':(Map A)) (MapDisjoint A A m m') -> + (a:ad) (in_dom A a (MapMerge A m m'))= + (orb (andb (in_dom A a m) (negb (in_dom A a m'))) + (andb (in_dom A a m') (negb (in_dom A a m)))). + Proof. + Unfold MapDisjoint. Intros. Rewrite in_dom_merge. Elim (sumbool_of_bool (in_dom A a m)). + Intro H0. Rewrite H0. Elim (sumbool_of_bool (in_dom A a m')). Intro H1. Elim (H a H0 H1). + Intro H1. Rewrite H1. Reflexivity. + Intro H0. Rewrite H0. Simpl. Rewrite andb_b_true. Reflexivity. + Qed. + + Lemma MapDisjoint_M2_l : (m0,m1:(Map A)) (m2,m3:(Map B)) + (MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3)) -> (MapDisjoint A B m0 m2). + Proof. + Unfold MapDisjoint in_dom. Intros. Elim (option_sum ? (MapGet A m0 a)). Intro H2. + Elim H2. Intros y H3. Elim (option_sum ? (MapGet B m2 a)). Intro H4. Elim H4. + Intros y' H5. Apply (H (ad_double a)). + Rewrite (MapGet_M2_bit_0_0 ? (ad_double a) (ad_double_bit_0 a) m0 m1). + Rewrite (ad_double_div_2 a). Rewrite H3. Reflexivity. + Rewrite (MapGet_M2_bit_0_0 ? (ad_double a) (ad_double_bit_0 a) m2 m3). + Rewrite (ad_double_div_2 a). Rewrite H5. Reflexivity. + Intro H4. Rewrite H4 in H1. Discriminate H1. + Intro H2. Rewrite H2 in H0. Discriminate H0. + Qed. + + Lemma MapDisjoint_M2_r : (m0,m1:(Map A)) (m2,m3:(Map B)) + (MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3)) -> (MapDisjoint A B m1 m3). + Proof. + Unfold MapDisjoint in_dom. Intros. Elim (option_sum ? (MapGet A m1 a)). Intro H2. + Elim H2. Intros y H3. Elim (option_sum ? (MapGet B m3 a)). Intro H4. Elim H4. + Intros y' H5. Apply (H (ad_double_plus_un a)). + Rewrite (MapGet_M2_bit_0_1 ? (ad_double_plus_un a) (ad_double_plus_un_bit_0 a) m0 m1). + Rewrite (ad_double_plus_un_div_2 a). Rewrite H3. Reflexivity. + Rewrite (MapGet_M2_bit_0_1 ? (ad_double_plus_un a) (ad_double_plus_un_bit_0 a) m2 m3). + Rewrite (ad_double_plus_un_div_2 a). Rewrite H5. Reflexivity. + Intro H4. Rewrite H4 in H1. Discriminate H1. + Intro H2. Rewrite H2 in H0. Discriminate H0. + Qed. + + Lemma MapDisjoint_M2 : (m0,m1:(Map A)) (m2,m3:(Map B)) + (MapDisjoint A B m0 m2) -> (MapDisjoint A B m1 m3) -> + (MapDisjoint A B (M2 A m0 m1) (M2 B m2 m3)). + Proof. + Unfold MapDisjoint in_dom. Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H3. + Rewrite (MapGet_M2_bit_0_1 A a H3 m0 m1) in H1. + Rewrite (MapGet_M2_bit_0_1 B a H3 m2 m3) in H2. Exact (H0 (ad_div_2 a) H1 H2). + Intro H3. Rewrite (MapGet_M2_bit_0_0 A a H3 m0 m1) in H1. + Rewrite (MapGet_M2_bit_0_0 B a H3 m2 m3) in H2. Exact (H (ad_div_2 a) H1 H2). + Qed. + + Lemma MapDisjoint_M1_l : (m:(Map A)) (a:ad) (y:B) + (MapDisjoint B A (M1 B a y) m) -> (in_dom A a m)=false. + Proof. + Unfold MapDisjoint. Intros. Elim (sumbool_of_bool (in_dom A a m)). Intro H0. + Elim (H a (in_dom_M1_1 B a y) H0). + Trivial. + Qed. + + Lemma MapDisjoint_M1_r : (m:(Map A)) (a:ad) (y:B) + (MapDisjoint A B m (M1 B a y)) -> (in_dom A a m)=false. + Proof. + Unfold MapDisjoint. Intros. Elim (sumbool_of_bool (in_dom A a m)). Intro H0. + Elim (H a H0 (in_dom_M1_1 B a y)). + Trivial. + Qed. + + Lemma MapDisjoint_M1_conv_l : (m:(Map A)) (a:ad) (y:B) + (in_dom A a m)=false -> (MapDisjoint B A (M1 B a y) m). + Proof. + Unfold MapDisjoint. Intros. Rewrite (in_dom_M1_2 B a a0 y H0) in H. Rewrite H1 in H. + Discriminate H. + Qed. + + Lemma MapDisjoint_M1_conv_r : (m:(Map A)) (a:ad) (y:B) + (in_dom A a m)=false -> (MapDisjoint A B m (M1 B a y)). + Proof. + Unfold MapDisjoint. Intros. Rewrite (in_dom_M1_2 B a a0 y H1) in H. Rewrite H0 in H. + Discriminate H. + Qed. + + Lemma MapDisjoint_sym : (m:(Map A)) (m':(Map B)) + (MapDisjoint A B m m') -> (MapDisjoint B A m' m). + Proof. + Unfold MapDisjoint. Intros. Exact (H ? H1 H0). + Qed. + + Lemma MapDisjoint_empty : (m:(Map A)) (MapDisjoint A A m m) -> (eqmap A m (M0 A)). + Proof. + Unfold eqmap eqm. Intros. Rewrite <- (MapDomRestrTo_idempotent A m a). + Exact (MapDisjoint_imp_2 A A m m H a). + Qed. + + Lemma MapDelta_disjoint : (m,m':(Map A)) (MapDisjoint A A m m') -> + (eqmap A (MapDelta A m m') (MapMerge A m m')). + Proof. + Intros. + Apply eqmap_trans with m':=(MapDomRestrBy A A (MapMerge A m m') (MapDomRestrTo A A m m')). + Apply MapDelta_as_DomRestrBy. + Apply eqmap_trans with m':=(MapDomRestrBy A A (MapMerge A m m') (M0 A)). + Apply MapDomRestrBy_ext. Apply eqmap_refl. + Exact (MapDisjoint_imp_2 A A m m' H). + Apply MapDomRestrBy_m_empty. + Qed. + + Variable C : Set. + + Lemma MapDomRestr_disjoint : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (MapDisjoint A B (MapDomRestrTo A C m m'') (MapDomRestrBy B C m' m'')). + Proof. + Unfold MapDisjoint. Intros m m' m'' a. Rewrite in_dom_restrto. Rewrite in_dom_restrby. + Intros. Elim (andb_prop ? ? H). Elim (andb_prop ? ? H0). Intros. Rewrite H4 in H2. + Discriminate H2. + Qed. + + Lemma MapDelta_RestrTo_disjoint : (m,m':(Map A)) + (MapDisjoint A A (MapDelta A m m') (MapDomRestrTo A A m m')). + Proof. + Unfold MapDisjoint. Intros m m' a. Rewrite in_dom_delta. Rewrite in_dom_restrto. + Intros. Elim (andb_prop ? ? H0). Intros. Rewrite H1 in H. Rewrite H2 in H. Discriminate H. + Qed. + + Lemma MapDelta_RestrTo_disjoint_2 : (m,m':(Map A)) + (MapDisjoint A A (MapDelta A m m') (MapDomRestrTo A A m' m)). + Proof. + Unfold MapDisjoint. Intros m m' a. Rewrite in_dom_delta. Rewrite in_dom_restrto. + Intros. Elim (andb_prop ? ? H0). Intros. Rewrite H1 in H. Rewrite H2 in H. Discriminate H. + Qed. + + Variable D : Set. + + Lemma MapSubset_Disjoint : (m:(Map A)) (m':(Map B)) (m'':(Map C)) (m''':(Map D)) + (MapSubset ? ? m m') -> (MapSubset ? ? m'' m''') -> (MapDisjoint ? ? m' m''') -> + (MapDisjoint ? ? m m''). + Proof. + Unfold MapSubset MapDisjoint. Intros. Exact (H1 ? (H ? H2) (H0 ? H3)). + Qed. + + Lemma MapSubset_Disjoint_l : (m:(Map A)) (m':(Map B)) (m'':(Map C)) + (MapSubset ? ? m m') -> (MapDisjoint ? ? m' m'') -> + (MapDisjoint ? ? m m''). + Proof. + Unfold MapSubset MapDisjoint. Intros. Exact (H0 ? (H ? H1) H2). + Qed. + + Lemma MapSubset_Disjoint_r : (m:(Map A)) (m'':(Map C)) (m''':(Map D)) + (MapSubset ? ? m'' m''') -> (MapDisjoint ? ? m m''') -> + (MapDisjoint ? ? m m''). + Proof. + Unfold MapSubset MapDisjoint. Intros. Exact (H0 ? H1 (H ? H2)). + Qed. + +End MapDisjointExtra. diff --git a/theories7/Lists/List.v b/theories7/Lists/List.v new file mode 100755 index 00000000..574b2688 --- /dev/null +++ b/theories7/Lists/List.v @@ -0,0 +1,261 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: List.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +(* This file is a copy of file MonoList.v *) + +(** THIS IS A OLD CONTRIB. IT IS NO LONGER MAINTAINED ***) + +Require Le. + +Parameter List_Dom:Set. +Definition A := List_Dom. + +Inductive list : Set := nil : list | cons : A -> list -> list. + +Fixpoint app [l:list] : list -> list + := [m:list]<list>Cases l of + nil => m + | (cons a l1) => (cons a (app l1 m)) + end. + + +Lemma app_nil_end : (l:list)(l=(app l nil)). +Proof. + Intro l ; Elim l ; Simpl ; Auto. + Induction 1; Auto. +Qed. +Hints Resolve app_nil_end : list v62. + +Lemma app_ass : (l,m,n : list)(app (app l m) n)=(app l (app m n)). +Proof. + Intros l m n ; Elim l ; Simpl ; Auto with list. + Induction 1; Auto with list. +Qed. +Hints Resolve app_ass : list v62. + +Lemma ass_app : (l,m,n : list)(app l (app m n))=(app (app l m) n). +Proof. + Auto with list. +Qed. +Hints Resolve ass_app : list v62. + +Definition tail := + [l:list] <list>Cases l of (cons _ m) => m | _ => nil end : list->list. + + +Lemma nil_cons : (a:A)(m:list)~nil=(cons a m). + Intros; Discriminate. +Qed. + +(****************************************) +(* Length of lists *) +(****************************************) + +Fixpoint length [l:list] : nat + := <nat>Cases l of (cons _ m) => (S (length m)) | _ => O end. + +(******************************) +(* Length order of lists *) +(******************************) + +Section length_order. +Definition lel := [l,m:list](le (length l) (length m)). + +Hints Unfold lel : list. + +Variables a,b:A. +Variables l,m,n:list. + +Lemma lel_refl : (lel l l). +Proof. + Unfold lel ; Auto with list. +Qed. + +Lemma lel_trans : (lel l m)->(lel m n)->(lel l n). +Proof. + Unfold lel ; Intros. + Apply le_trans with (length m) ; Auto with list. +Qed. + +Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)). +Proof. + Unfold lel ; Simpl ; Auto with list arith. +Qed. + +Lemma lel_cons : (lel l m)->(lel l (cons b m)). +Proof. + Unfold lel ; Simpl ; Auto with list arith. +Qed. + +Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m). +Proof. + Unfold lel ; Simpl ; Auto with list arith. +Qed. + +Lemma lel_nil : (l':list)(lel l' nil)->(nil=l'). +Proof. + Intro l' ; Elim l' ; Auto with list arith. + Intros a' y H H0. + (* <list>nil=(cons a' y) + ============================ + H0 : (lel (cons a' y) nil) + H : (lel y nil)->(<list>nil=y) + y : list + a' : A + l' : list *) + Absurd (le (S (length y)) O); Auto with list arith. +Qed. +End length_order. + +Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons : list v62. + +Fixpoint In [a:A;l:list] : Prop := + Cases l of + nil => False + | (cons b m) => (b=a)\/(In a m) + end. + +Lemma in_eq : (a:A)(l:list)(In a (cons a l)). +Proof. + Simpl ; Auto with list. +Qed. +Hints Resolve in_eq : list v62. + +Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)). +Proof. + Simpl ; Auto with list. +Qed. +Hints Resolve in_cons : list v62. + +Lemma in_app_or : (l,m:list)(a:A)(In a (app l m))->((In a l)\/(In a m)). +Proof. + Intros l m a. + Elim l ; Simpl ; Auto with list. + Intros a0 y H H0. + (* ((<A>a0=a)\/(In a y))\/(In a m) + ============================ + H0 : (<A>a0=a)\/(In a (app y m)) + H : (In a (app y m))->((In a y)\/(In a m)) + y : list + a0 : A + a : A + m : list + l : list *) + Elim H0 ; Auto with list. + Intro H1. + (* ((<A>a0=a)\/(In a y))\/(In a m) + ============================ + H1 : (In a (app y m)) *) + Elim (H H1) ; Auto with list. +Qed. +Hints Immediate in_app_or : list v62. + +Lemma in_or_app : (l,m:list)(a:A)((In a l)\/(In a m))->(In a (app l m)). +Proof. + Intros l m a. + Elim l ; Simpl ; Intro H. + (* 1 (In a m) + ============================ + H : False\/(In a m) + a : A + m : list + l : list *) + Elim H ; Auto with list ; Intro H0. + (* (In a m) + ============================ + H0 : False *) + Elim H0. (* subProof completed *) + Intros y H0 H1. + (* 2 (<A>H=a)\/(In a (app y m)) + ============================ + H1 : ((<A>H=a)\/(In a y))\/(In a m) + H0 : ((In a y)\/(In a m))->(In a (app y m)) + y : list *) + Elim H1 ; Auto 4 with list. + Intro H2. + (* (<A>H=a)\/(In a (app y m)) + ============================ + H2 : (<A>H=a)\/(In a y) *) + Elim H2 ; Auto with list. +Qed. +Hints Resolve in_or_app : list v62. + +Definition incl := [l,m:list](a:A)(In a l)->(In a m). + +Hints Unfold incl : list v62. + +Lemma incl_refl : (l:list)(incl l l). +Proof. + Auto with list. +Qed. +Hints Resolve incl_refl : list v62. + +Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)). +Proof. + Auto with list. +Qed. +Hints Immediate incl_tl : list v62. + +Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n). +Proof. + Auto with list. +Qed. + +Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (app n m)). +Proof. + Auto with list. +Qed. +Hints Immediate incl_appl : list v62. + +Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (app m n)). +Proof. + Auto with list. +Qed. +Hints Immediate incl_appr : list v62. + +Lemma incl_cons : (a:A)(l,m:list)(In a m)->(incl l m)->(incl (cons a l) m). +Proof. + Unfold incl ; Simpl ; Intros a l m H H0 a0 H1. + (* (In a0 m) + ============================ + H1 : (<A>a=a0)\/(In a0 l) + a0 : A + H0 : (a:A)(In a l)->(In a m) + H : (In a m) + m : list + l : list + a : A *) + Elim H1. + (* 1 (<A>a=a0)->(In a0 m) *) + Elim H1 ; Auto with list ; Intro H2. + (* (<A>a=a0)->(In a0 m) + ============================ + H2 : <A>a=a0 *) + Elim H2 ; Auto with list. (* solves subgoal *) + (* 2 (In a0 l)->(In a0 m) *) + Auto with list. +Qed. +Hints Resolve incl_cons : list v62. + +Lemma incl_app : (l,m,n:list)(incl l n)->(incl m n)->(incl (app l m) n). +Proof. + Unfold incl ; Simpl ; Intros l m n H H0 a H1. + (* (In a n) + ============================ + H1 : (In a (app l m)) + a : A + H0 : (a:A)(In a m)->(In a n) + H : (a:A)(In a l)->(In a n) + n : list + m : list + l : list *) + Elim (in_app_or l m a) ; Auto with list. +Qed. +Hints Resolve incl_app : list v62. diff --git a/theories7/Lists/ListSet.v b/theories7/Lists/ListSet.v new file mode 100644 index 00000000..9bf259da --- /dev/null +++ b/theories7/Lists/ListSet.v @@ -0,0 +1,389 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ListSet.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +(** A Library for finite sets, implemented as lists + A Library with similar interface will soon be available under + the name TreeSet in the theories/Trees directory *) + +(** PolyList is loaded, but not exported. + This allow to "hide" the definitions, functions and theorems of PolyList + and to see only the ones of ListSet *) + +Require PolyList. + +Set Implicit Arguments. +V7only [Implicits nil [1].]. + +Section first_definitions. + + Variable A : Set. + Hypothesis Aeq_dec : (x,y:A){x=y}+{~x=y}. + + Definition set := (list A). + + Definition empty_set := (!nil ?) : set. + + Fixpoint set_add [a:A; x:set] : set := + Cases x of + | nil => (cons a nil) + | (cons a1 x1) => Cases (Aeq_dec a a1) of + | (left _) => (cons a1 x1) + | (right _) => (cons a1 (set_add a x1)) + end + end. + + + Fixpoint set_mem [a:A; x:set] : bool := + Cases x of + | nil => false + | (cons a1 x1) => Cases (Aeq_dec a a1) of + | (left _) => true + | (right _) => (set_mem a x1) + end + end. + + (** If [a] belongs to [x], removes [a] from [x]. If not, does nothing *) + Fixpoint set_remove [a:A; x:set] : set := + Cases x of + | nil => empty_set + | (cons a1 x1) => Cases (Aeq_dec a a1) of + | (left _) => x1 + | (right _) => (cons a1 (set_remove a x1)) + end + end. + + Fixpoint set_inter [x:set] : set -> set := + Cases x of + | nil => [y]nil + | (cons a1 x1) => [y]if (set_mem a1 y) + then (cons a1 (set_inter x1 y)) + else (set_inter x1 y) + end. + + Fixpoint set_union [x,y:set] : set := + Cases y of + | nil => x + | (cons a1 y1) => (set_add a1 (set_union x y1)) + end. + + (** returns the set of all els of [x] that does not belong to [y] *) + Fixpoint set_diff [x:set] : set -> set := + [y]Cases x of + | nil => nil + | (cons a1 x1) => if (set_mem a1 y) + then (set_diff x1 y) + else (set_add a1 (set_diff x1 y)) + end. + + + Definition set_In : A -> set -> Prop := (In 1!A). + + Lemma set_In_dec : (a:A; x:set){(set_In a x)}+{~(set_In a x)}. + + Proof. + Unfold set_In. + (*** Realizer set_mem. Program_all. ***) + Induction x. + Auto. + Intros a0 x0 Ha0. Case (Aeq_dec a a0); Intro eq. + Rewrite eq; Simpl; Auto with datatypes. + Elim Ha0. + Auto with datatypes. + Right; Simpl; Unfold not; Intros [Hc1 | Hc2 ]; Auto with datatypes. + Qed. + + Lemma set_mem_ind : + (B:Set)(P:B->Prop)(y,z:B)(a:A)(x:set) + ((set_In a x) -> (P y)) + ->(P z) + ->(P (if (set_mem a x) then y else z)). + + Proof. + Induction x; Simpl; Intros. + Assumption. + Elim (Aeq_dec a a0); Auto with datatypes. + Qed. + + Lemma set_mem_ind2 : + (B:Set)(P:B->Prop)(y,z:B)(a:A)(x:set) + ((set_In a x) -> (P y)) + ->(~(set_In a x) -> (P z)) + ->(P (if (set_mem a x) then y else z)). + + Proof. + Induction x; Simpl; Intros. + Apply H0; Red; Trivial. + Case (Aeq_dec a a0); Auto with datatypes. + Intro; Apply H; Intros; Auto. + Apply H1; Red; Intro. + Case H3; Auto. + Qed. + + + Lemma set_mem_correct1 : + (a:A)(x:set)(set_mem a x)=true -> (set_In a x). + Proof. + Induction x; Simpl. + Discriminate. + Intros a0 l; Elim (Aeq_dec a a0); Auto with datatypes. + Qed. + + Lemma set_mem_correct2 : + (a:A)(x:set)(set_In a x) -> (set_mem a x)=true. + Proof. + Induction x; Simpl. + Intro Ha; Elim Ha. + Intros a0 l; Elim (Aeq_dec a a0); Auto with datatypes. + Intros H1 H2 [H3 | H4]. + Absurd a0=a; Auto with datatypes. + Auto with datatypes. + Qed. + + Lemma set_mem_complete1 : + (a:A)(x:set)(set_mem a x)=false -> ~(set_In a x). + Proof. + Induction x; Simpl. + Tauto. + Intros a0 l; Elim (Aeq_dec a a0). + Intros; Discriminate H0. + Unfold not; Intros; Elim H1; Auto with datatypes. + Qed. + + Lemma set_mem_complete2 : + (a:A)(x:set)~(set_In a x) -> (set_mem a x)=false. + Proof. + Induction x; Simpl. + Tauto. + Intros a0 l; Elim (Aeq_dec a a0). + Intros; Elim H0; Auto with datatypes. + Tauto. + Qed. + + Lemma set_add_intro1 : (a,b:A)(x:set) + (set_In a x) -> (set_In a (set_add b x)). + + Proof. + Unfold set_In; Induction x; Simpl. + Auto with datatypes. + Intros a0 l H [ Ha0a | Hal ]. + Elim (Aeq_dec b a0); Left; Assumption. + Elim (Aeq_dec b a0); Right; [ Assumption | Auto with datatypes ]. + Qed. + + Lemma set_add_intro2 : (a,b:A)(x:set) + a=b -> (set_In a (set_add b x)). + + Proof. + Unfold set_In; Induction x; Simpl. + Auto with datatypes. + Intros a0 l H Hab. + Elim (Aeq_dec b a0); + [ Rewrite Hab; Intro Hba0; Rewrite Hba0; Simpl; Auto with datatypes + | Auto with datatypes ]. + Qed. + + Hints Resolve set_add_intro1 set_add_intro2. + + Lemma set_add_intro : (a,b:A)(x:set) + a=b\/(set_In a x) -> (set_In a (set_add b x)). + + Proof. + Intros a b x [H1 | H2] ; Auto with datatypes. + Qed. + + Lemma set_add_elim : (a,b:A)(x:set) + (set_In a (set_add b x)) -> a=b\/(set_In a x). + + Proof. + Unfold set_In. + Induction x. + Simpl; Intros [H1|H2]; Auto with datatypes. + Simpl; Do 3 Intro. + Elim (Aeq_dec b a0). + Simpl; Tauto. + Simpl; Intros; Elim H0. + Trivial with datatypes. + Tauto. + Tauto. + Qed. + + Lemma set_add_elim2 : (a,b:A)(x:set) + (set_In a (set_add b x)) -> ~(a=b) -> (set_In a x). + Intros a b x H; Case (set_add_elim H); Intros; Trivial. + Case H1; Trivial. + Qed. + + Hints Resolve set_add_intro set_add_elim set_add_elim2. + + Lemma set_add_not_empty : (a:A)(x:set)~(set_add a x)=empty_set. + Proof. + Induction x; Simpl. + Discriminate. + Intros; Elim (Aeq_dec a a0); Intros; Discriminate. + Qed. + + + Lemma set_union_intro1 : (a:A)(x,y:set) + (set_In a x) -> (set_In a (set_union x y)). + Proof. + Induction y; Simpl; Auto with datatypes. + Qed. + + Lemma set_union_intro2 : (a:A)(x,y:set) + (set_In a y) -> (set_In a (set_union x y)). + Proof. + Induction y; Simpl. + Tauto. + Intros; Elim H0; Auto with datatypes. + Qed. + + Hints Resolve set_union_intro2 set_union_intro1. + + Lemma set_union_intro : (a:A)(x,y:set) + (set_In a x)\/(set_In a y) -> (set_In a (set_union x y)). + Proof. + Intros; Elim H; Auto with datatypes. + Qed. + + Lemma set_union_elim : (a:A)(x,y:set) + (set_In a (set_union x y)) -> (set_In a x)\/(set_In a y). + Proof. + Induction y; Simpl. + Auto with datatypes. + Intros. + Generalize (set_add_elim H0). + Intros [H1 | H1]. + Auto with datatypes. + Tauto. + Qed. + + Lemma set_union_emptyL : (a:A)(x:set)(set_In a (set_union empty_set x)) -> (set_In a x). + Intros a x H; Case (set_union_elim H); Auto Orelse Contradiction. + Qed. + + + Lemma set_union_emptyR : (a:A)(x:set)(set_In a (set_union x empty_set)) -> (set_In a x). + Intros a x H; Case (set_union_elim H); Auto Orelse Contradiction. + Qed. + + + Lemma set_inter_intro : (a:A)(x,y:set) + (set_In a x) -> (set_In a y) -> (set_In a (set_inter x y)). + Proof. + Induction x. + Auto with datatypes. + Simpl; Intros a0 l Hrec y [Ha0a | Hal] Hy. + Simpl; Rewrite Ha0a. + Generalize (!set_mem_correct1 a y). + Generalize (!set_mem_complete1 a y). + Elim (set_mem a y); Simpl; Intros. + Auto with datatypes. + Absurd (set_In a y); Auto with datatypes. + Elim (set_mem a0 y); [ Right; Auto with datatypes | Auto with datatypes]. + Qed. + + Lemma set_inter_elim1 : (a:A)(x,y:set) + (set_In a (set_inter x y)) -> (set_In a x). + Proof. + Induction x. + Auto with datatypes. + Simpl; Intros a0 l Hrec y. + Generalize (!set_mem_correct1 a0 y). + Elim (set_mem a0 y); Simpl; Intros. + Elim H0; EAuto with datatypes. + EAuto with datatypes. + Qed. + + Lemma set_inter_elim2 : (a:A)(x,y:set) + (set_In a (set_inter x y)) -> (set_In a y). + Proof. + Induction x. + Simpl; Tauto. + Simpl; Intros a0 l Hrec y. + Generalize (!set_mem_correct1 a0 y). + Elim (set_mem a0 y); Simpl; Intros. + Elim H0; [ Intro Hr; Rewrite <- Hr; EAuto with datatypes | EAuto with datatypes ] . + EAuto with datatypes. + Qed. + + Hints Resolve set_inter_elim1 set_inter_elim2. + + Lemma set_inter_elim : (a:A)(x,y:set) + (set_In a (set_inter x y)) -> (set_In a x)/\(set_In a y). + Proof. + EAuto with datatypes. + Qed. + + Lemma set_diff_intro : (a:A)(x,y:set) + (set_In a x) -> ~(set_In a y) -> (set_In a (set_diff x y)). + Proof. + Induction x. + Simpl; Tauto. + Simpl; Intros a0 l Hrec y [Ha0a | Hal] Hay. + Rewrite Ha0a; Generalize (set_mem_complete2 Hay). + Elim (set_mem a y); [ Intro Habs; Discriminate Habs | Auto with datatypes ]. + Elim (set_mem a0 y); Auto with datatypes. + Qed. + + Lemma set_diff_elim1 : (a:A)(x,y:set) + (set_In a (set_diff x y)) -> (set_In a x). + Proof. + Induction x. + Simpl; Tauto. + Simpl; Intros a0 l Hrec y; Elim (set_mem a0 y). + EAuto with datatypes. + Intro; Generalize (set_add_elim H). + Intros [H1 | H2]; EAuto with datatypes. + Qed. + + Lemma set_diff_elim2 : (a:A)(x,y:set) + (set_In a (set_diff x y)) -> ~(set_In a y). + Intros a x y; Elim x; Simpl. + Intros; Contradiction. + Intros a0 l Hrec. + Apply set_mem_ind2; Auto. + Intros H1 H2; Case (set_add_elim H2); Intros; Auto. + Rewrite H; Trivial. + Qed. + + Lemma set_diff_trivial : (a:A)(x:set)~(set_In a (set_diff x x)). + Red; Intros a x H. + Apply (set_diff_elim2 H). + Apply (set_diff_elim1 H). + Qed. + +Hints Resolve set_diff_intro set_diff_trivial. + + +End first_definitions. + +Section other_definitions. + + Variables A,B : Set. + + Definition set_prod : (set A) -> (set B) -> (set A*B) := (list_prod 1!A 2!B). + + (** [B^A], set of applications from [A] to [B] *) + Definition set_power : (set A) -> (set B) -> (set (set A*B)) := + (list_power 1!A 2!B). + + Definition set_map : (A->B) -> (set A) -> (set B) := (map 1!A 2!B). + + Definition set_fold_left : (B -> A -> B) -> (set A) -> B -> B := + (fold_left 1!B 2!A). + + Definition set_fold_right : (A -> B -> B) -> (set A) -> B -> B := + [f][x][b](fold_right f b x). + + +End other_definitions. + +V7only [Implicits nil [].]. +Unset Implicit Arguments. diff --git a/theories7/Lists/MonoList.v b/theories7/Lists/MonoList.v new file mode 100755 index 00000000..2ab78f7f --- /dev/null +++ b/theories7/Lists/MonoList.v @@ -0,0 +1,259 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: MonoList.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +(** THIS IS A OLD CONTRIB. IT IS NO LONGER MAINTAINED ***) + +Require Le. + +Parameter List_Dom:Set. +Definition A := List_Dom. + +Inductive list : Set := nil : list | cons : A -> list -> list. + +Fixpoint app [l:list] : list -> list + := [m:list]<list>Cases l of + nil => m + | (cons a l1) => (cons a (app l1 m)) + end. + + +Lemma app_nil_end : (l:list)(l=(app l nil)). +Proof. + Intro l ; Elim l ; Simpl ; Auto. + Induction 1; Auto. +Qed. +Hints Resolve app_nil_end : list v62. + +Lemma app_ass : (l,m,n : list)(app (app l m) n)=(app l (app m n)). +Proof. + Intros l m n ; Elim l ; Simpl ; Auto with list. + Induction 1; Auto with list. +Qed. +Hints Resolve app_ass : list v62. + +Lemma ass_app : (l,m,n : list)(app l (app m n))=(app (app l m) n). +Proof. + Auto with list. +Qed. +Hints Resolve ass_app : list v62. + +Definition tail := + [l:list] <list>Cases l of (cons _ m) => m | _ => nil end : list->list. + + +Lemma nil_cons : (a:A)(m:list)~nil=(cons a m). + Intros; Discriminate. +Qed. + +(****************************************) +(* Length of lists *) +(****************************************) + +Fixpoint length [l:list] : nat + := <nat>Cases l of (cons _ m) => (S (length m)) | _ => O end. + +(******************************) +(* Length order of lists *) +(******************************) + +Section length_order. +Definition lel := [l,m:list](le (length l) (length m)). + +Hints Unfold lel : list. + +Variables a,b:A. +Variables l,m,n:list. + +Lemma lel_refl : (lel l l). +Proof. + Unfold lel ; Auto with list. +Qed. + +Lemma lel_trans : (lel l m)->(lel m n)->(lel l n). +Proof. + Unfold lel ; Intros. + Apply le_trans with (length m) ; Auto with list. +Qed. + +Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)). +Proof. + Unfold lel ; Simpl ; Auto with list arith. +Qed. + +Lemma lel_cons : (lel l m)->(lel l (cons b m)). +Proof. + Unfold lel ; Simpl ; Auto with list arith. +Qed. + +Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m). +Proof. + Unfold lel ; Simpl ; Auto with list arith. +Qed. + +Lemma lel_nil : (l':list)(lel l' nil)->(nil=l'). +Proof. + Intro l' ; Elim l' ; Auto with list arith. + Intros a' y H H0. + (* <list>nil=(cons a' y) + ============================ + H0 : (lel (cons a' y) nil) + H : (lel y nil)->(<list>nil=y) + y : list + a' : A + l' : list *) + Absurd (le (S (length y)) O); Auto with list arith. +Qed. +End length_order. + +Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons : list v62. + +Fixpoint In [a:A;l:list] : Prop := + Cases l of + nil => False + | (cons b m) => (b=a)\/(In a m) + end. + +Lemma in_eq : (a:A)(l:list)(In a (cons a l)). +Proof. + Simpl ; Auto with list. +Qed. +Hints Resolve in_eq : list v62. + +Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)). +Proof. + Simpl ; Auto with list. +Qed. +Hints Resolve in_cons : list v62. + +Lemma in_app_or : (l,m:list)(a:A)(In a (app l m))->((In a l)\/(In a m)). +Proof. + Intros l m a. + Elim l ; Simpl ; Auto with list. + Intros a0 y H H0. + (* ((<A>a0=a)\/(In a y))\/(In a m) + ============================ + H0 : (<A>a0=a)\/(In a (app y m)) + H : (In a (app y m))->((In a y)\/(In a m)) + y : list + a0 : A + a : A + m : list + l : list *) + Elim H0 ; Auto with list. + Intro H1. + (* ((<A>a0=a)\/(In a y))\/(In a m) + ============================ + H1 : (In a (app y m)) *) + Elim (H H1) ; Auto with list. +Qed. +Hints Immediate in_app_or : list v62. + +Lemma in_or_app : (l,m:list)(a:A)((In a l)\/(In a m))->(In a (app l m)). +Proof. + Intros l m a. + Elim l ; Simpl ; Intro H. + (* 1 (In a m) + ============================ + H : False\/(In a m) + a : A + m : list + l : list *) + Elim H ; Auto with list ; Intro H0. + (* (In a m) + ============================ + H0 : False *) + Elim H0. (* subProof completed *) + Intros y H0 H1. + (* 2 (<A>H=a)\/(In a (app y m)) + ============================ + H1 : ((<A>H=a)\/(In a y))\/(In a m) + H0 : ((In a y)\/(In a m))->(In a (app y m)) + y : list *) + Elim H1 ; Auto 4 with list. + Intro H2. + (* (<A>H=a)\/(In a (app y m)) + ============================ + H2 : (<A>H=a)\/(In a y) *) + Elim H2 ; Auto with list. +Qed. +Hints Resolve in_or_app : list v62. + +Definition incl := [l,m:list](a:A)(In a l)->(In a m). + +Hints Unfold incl : list v62. + +Lemma incl_refl : (l:list)(incl l l). +Proof. + Auto with list. +Qed. +Hints Resolve incl_refl : list v62. + +Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)). +Proof. + Auto with list. +Qed. +Hints Immediate incl_tl : list v62. + +Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n). +Proof. + Auto with list. +Qed. + +Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (app n m)). +Proof. + Auto with list. +Qed. +Hints Immediate incl_appl : list v62. + +Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (app m n)). +Proof. + Auto with list. +Qed. +Hints Immediate incl_appr : list v62. + +Lemma incl_cons : (a:A)(l,m:list)(In a m)->(incl l m)->(incl (cons a l) m). +Proof. + Unfold incl ; Simpl ; Intros a l m H H0 a0 H1. + (* (In a0 m) + ============================ + H1 : (<A>a=a0)\/(In a0 l) + a0 : A + H0 : (a:A)(In a l)->(In a m) + H : (In a m) + m : list + l : list + a : A *) + Elim H1. + (* 1 (<A>a=a0)->(In a0 m) *) + Elim H1 ; Auto with list ; Intro H2. + (* (<A>a=a0)->(In a0 m) + ============================ + H2 : <A>a=a0 *) + Elim H2 ; Auto with list. (* solves subgoal *) + (* 2 (In a0 l)->(In a0 m) *) + Auto with list. +Qed. +Hints Resolve incl_cons : list v62. + +Lemma incl_app : (l,m,n:list)(incl l n)->(incl m n)->(incl (app l m) n). +Proof. + Unfold incl ; Simpl ; Intros l m n H H0 a H1. + (* (In a n) + ============================ + H1 : (In a (app l m)) + a : A + H0 : (a:A)(In a m)->(In a n) + H : (a:A)(In a l)->(In a n) + n : list + m : list + l : list *) + Elim (in_app_or l m a) ; Auto with list. +Qed. +Hints Resolve incl_app : list v62. diff --git a/theories7/Lists/PolyList.v b/theories7/Lists/PolyList.v new file mode 100644 index 00000000..e69ecd10 --- /dev/null +++ b/theories7/Lists/PolyList.v @@ -0,0 +1,646 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: PolyList.v,v 1.2.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) + +Require Le. + + +Section Lists. + +Variable A : Set. + +Set Implicit Arguments. + +Inductive list : Set := nil : list | cons : A -> list -> list. + +Infix "::" cons (at level 7, right associativity) : list_scope + V8only (at level 60, right associativity). + +Open Scope list_scope. + +(*************************) +(** Discrimination *) +(*************************) + +Lemma nil_cons : (a:A)(m:list)~(nil=(cons a m)). +Proof. + Intros; Discriminate. +Qed. + +(*************************) +(** Concatenation *) +(*************************) + +Fixpoint app [l:list] : list -> list + := [m:list]Cases l of + nil => m + | (cons a l1) => (cons a (app l1 m)) + end. + +Infix RIGHTA 7 "^" app : list_scope + V8only RIGHTA 60 "++". + +Lemma app_nil_end : (l:list)l=(l^nil). +Proof. + NewInduction l ; Simpl ; Auto. + Rewrite <- IHl; Auto. +Qed. +Hints Resolve app_nil_end. + +Tactic Definition now_show c := Change c. +V7only [Tactic Definition NowShow := now_show.]. + +Lemma app_ass : (l,m,n : list)((l^m)^ n)=(l^(m^n)). +Proof. + Intros. NewInduction l ; Simpl ; Auto. + NowShow '(cons a (app (app l m) n))=(cons a (app l (app m n))). + Rewrite <- IHl; Auto. +Qed. +Hints Resolve app_ass. + +Lemma ass_app : (l,m,n : list)(l^(m^n))=((l^m)^n). +Proof. + Auto. +Qed. +Hints Resolve ass_app. + +Lemma app_comm_cons : (x,y:list)(a:A) (cons a (x^y))=((cons a x)^y). +Proof. + Auto. +Qed. + +Lemma app_eq_nil: (x,y:list) (x^y)=nil -> x=nil /\ y=nil. +Proof. + NewDestruct x;NewDestruct y;Simpl;Auto. + Intros H;Discriminate H. + Intros;Discriminate H. +Qed. + +Lemma app_cons_not_nil: (x,y:list)(a:A)~nil=(x^(cons a y)). +Proof. +Unfold not . + NewDestruct x;Simpl;Intros. + Discriminate H. + Discriminate H. +Qed. + +Lemma app_eq_unit:(x,y:list)(a:A) + (x^y)=(cons a nil)-> (x=nil)/\ y=(cons a nil) \/ x=(cons a nil)/\ y=nil. + +Proof. + NewDestruct x;NewDestruct y;Simpl. + Intros a H;Discriminate H. + Left;Split;Auto. + Right;Split;Auto. + Generalize H . + Generalize (app_nil_end l) ;Intros E. + Rewrite <- E;Auto. + Intros. + Injection H. + Intro. + Cut nil=(l^(cons a0 l0));Auto. + Intro. + Generalize (app_cons_not_nil H1); Intro. + Elim H2. +Qed. + +Lemma app_inj_tail : (x,y:list)(a,b:A) + (x^(cons a nil))=(y^(cons b nil)) -> x=y /\ a=b. +Proof. + NewInduction x as [|x l IHl];NewDestruct y;Simpl;Auto. + Intros a b H. + Injection H. + Auto. + Intros a0 b H. + Injection H;Intros. + Generalize (app_cons_not_nil H0) ;NewDestruct 1. + Intros a b H. + Injection H;Intros. + Cut nil=(l^(cons a nil));Auto. + Intro. + Generalize (app_cons_not_nil H2) ;NewDestruct 1. + Intros a0 b H. + Injection H;Intros. + NewDestruct (IHl l0 a0 b H0). + Split;Auto. + Rewrite <- H1;Rewrite <- H2;Reflexivity. +Qed. + +(*************************) +(** Head and tail *) +(*************************) + +Definition head := + [l:list]Cases l of + | nil => Error + | (cons x _) => (Value x) + end. + +Definition tail : list -> list := + [l:list]Cases l of + | nil => nil + | (cons a m) => m + end. + +(****************************************) +(** Length of lists *) +(****************************************) + +Fixpoint length [l:list] : nat + := Cases l of nil => O | (cons _ m) => (S (length m)) end. + +(******************************) +(** Length order of lists *) +(******************************) + +Section length_order. +Definition lel := [l,m:list](le (length l) (length m)). + +Variables a,b:A. +Variables l,m,n:list. + +Lemma lel_refl : (lel l l). +Proof. + Unfold lel ; Auto with arith. +Qed. + +Lemma lel_trans : (lel l m)->(lel m n)->(lel l n). +Proof. + Unfold lel ; Intros. + NowShow '(le (length l) (length n)). + Apply le_trans with (length m) ; Auto with arith. +Qed. + +Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)). +Proof. + Unfold lel ; Simpl ; Auto with arith. +Qed. + +Lemma lel_cons : (lel l m)->(lel l (cons b m)). +Proof. + Unfold lel ; Simpl ; Auto with arith. +Qed. + +Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m). +Proof. + Unfold lel ; Simpl ; Auto with arith. +Qed. + +Lemma lel_nil : (l':list)(lel l' nil)->(nil=l'). +Proof. + Intro l' ; Elim l' ; Auto with arith. + Intros a' y H H0. + NowShow 'nil=(cons a' y). + Absurd (le (S (length y)) O); Auto with arith. +Qed. +End length_order. + +Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons. + +(*********************************) +(** The [In] predicate *) +(*********************************) + +Fixpoint In [a:A;l:list] : Prop := + Cases l of nil => False | (cons b m) => (b=a)\/(In a m) end. + +Lemma in_eq : (a:A)(l:list)(In a (cons a l)). +Proof. + Simpl ; Auto. +Qed. +Hints Resolve in_eq. + +Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)). +Proof. + Simpl ; Auto. +Qed. +Hints Resolve in_cons. + +Lemma in_nil : (a:A)~(In a nil). +Proof. + Unfold not; Intros a H; Inversion_clear H. +Qed. + + +Lemma in_inv : (a,b:A)(l:list) + (In b (cons a l)) -> a=b \/ (In b l). +Proof. + Intros a b l H ; Inversion_clear H ; Auto. +Qed. + +Lemma In_dec : ((x,y:A){x=y}+{~x=y}) -> (a:A)(l:list){(In a l)}+{~(In a l)}. + +Proof. + NewInduction l as [|a0 l IHl]. + Right; Apply in_nil. + NewDestruct (H a0 a); Simpl; Auto. + NewDestruct IHl; Simpl; Auto. + Right; Unfold not; Intros [Hc1 | Hc2]; Auto. +Qed. + +Lemma in_app_or : (l,m:list)(a:A)(In a (l^m))->((In a l)\/(In a m)). +Proof. + Intros l m a. + Elim l ; Simpl ; Auto. + Intros a0 y H H0. + NowShow '(a0=a\/(In a y))\/(In a m). + Elim H0 ; Auto. + Intro H1. + NowShow '(a0=a\/(In a y))\/(In a m). + Elim (H H1) ; Auto. +Qed. +Hints Immediate in_app_or. + +Lemma in_or_app : (l,m:list)(a:A)((In a l)\/(In a m))->(In a (l^m)). +Proof. + Intros l m a. + Elim l ; Simpl ; Intro H. + NowShow '(In a m). + Elim H ; Auto ; Intro H0. + NowShow '(In a m). + Elim H0. (* subProof completed *) + Intros y H0 H1. + NowShow 'H=a\/(In a (app y m)). + Elim H1 ; Auto 4. + Intro H2. + NowShow 'H=a\/(In a (app y m)). + Elim H2 ; Auto. +Qed. +Hints Resolve in_or_app. + +(***************************) +(** Set inclusion on list *) +(***************************) + +Definition incl := [l,m:list](a:A)(In a l)->(In a m). +Hints Unfold incl. + +Lemma incl_refl : (l:list)(incl l l). +Proof. + Auto. +Qed. +Hints Resolve incl_refl. + +Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)). +Proof. + Auto. +Qed. +Hints Immediate incl_tl. + +Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n). +Proof. + Auto. +Qed. + +Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (n^m)). +Proof. + Auto. +Qed. +Hints Immediate incl_appl. + +Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (m^n)). +Proof. + Auto. +Qed. +Hints Immediate incl_appr. + +Lemma incl_cons : (a:A)(l,m:list)(In a m)->(incl l m)->(incl (cons a l) m). +Proof. + Unfold incl ; Simpl ; Intros a l m H H0 a0 H1. + NowShow '(In a0 m). + Elim H1. + NowShow 'a=a0->(In a0 m). + Elim H1 ; Auto ; Intro H2. + NowShow 'a=a0->(In a0 m). + Elim H2 ; Auto. (* solves subgoal *) + NowShow '(In a0 l)->(In a0 m). + Auto. +Qed. +Hints Resolve incl_cons. + +Lemma incl_app : (l,m,n:list)(incl l n)->(incl m n)->(incl (l^m) n). +Proof. + Unfold incl ; Simpl ; Intros l m n H H0 a H1. + NowShow '(In a n). + Elim (in_app_or H1); Auto. +Qed. +Hints Resolve incl_app. + +(**************************) +(** Nth element of a list *) +(**************************) + +Fixpoint nth [n:nat; l:list] : A->A := + [default]Cases n l of + O (cons x l') => x + | O other => default + | (S m) nil => default + | (S m) (cons x t) => (nth m t default) + end. + +Fixpoint nth_ok [n:nat; l:list] : A->bool := + [default]Cases n l of + O (cons x l') => true + | O other => false + | (S m) nil => false + | (S m) (cons x t) => (nth_ok m t default) + end. + +Lemma nth_in_or_default : + (n:nat)(l:list)(d:A){(In (nth n l d) l)}+{(nth n l d)=d}. +(* Realizer nth_ok. Program_all. *) +Proof. + Intros n l d; Generalize n; NewInduction l; Intro n0. + Right; Case n0; Trivial. + Case n0; Simpl. + Auto. + Intro n1; Elim (IHl n1); Auto. +Qed. + +Lemma nth_S_cons : + (n:nat)(l:list)(d:A)(a:A)(In (nth n l d) l) + ->(In (nth (S n) (cons a l) d) (cons a l)). +Proof. + Simpl; Auto. +Qed. + +Fixpoint nth_error [l:list;n:nat] : (Exc A) := + Cases n l of + | O (cons x _) => (Value x) + | (S n) (cons _ l) => (nth_error l n) + | _ _ => Error + end. + +Definition nth_default : A -> list -> nat -> A := + [default,l,n]Cases (nth_error l n) of + | (Some x) => x + | None => default + end. + +Lemma nth_In : + (n:nat)(l:list)(d:A)(lt n (length l))->(In (nth n l d) l). + +Proof. +Unfold lt; NewInduction n as [|n hn]; Simpl. +NewDestruct l ; Simpl ; [ Inversion 2 | Auto]. +NewDestruct l as [|a l hl] ; Simpl. +Inversion 2. +Intros d ie ; Right ; Apply hn ; Auto with arith. +Qed. + +(********************************) +(** Decidable equality on lists *) +(********************************) + + +Lemma list_eq_dec : ((x,y:A){x=y}+{~x=y})->(x,y:list){x=y}+{~x=y}. +Proof. + NewInduction x as [|a l IHl]; NewDestruct y as [|a0 l0]; Auto. + NewDestruct (H a a0) as [e|e]. + NewDestruct (IHl l0) as [e'|e']. + Left; Rewrite e; Rewrite e'; Trivial. + Right; Red; Intro. + Apply e'; Injection H0; Trivial. + Right; Red; Intro. + Apply e; Injection H0; Trivial. +Qed. + +(*************************) +(** Reverse *) +(*************************) + +Fixpoint rev [l:list] : list := + Cases l of + nil => nil + | (cons x l') => (rev l')^(cons x nil) + end. + +Lemma distr_rev : + (x,y:list) (rev (x^y))=((rev y)^(rev x)). +Proof. + NewInduction x as [|a l IHl]. + NewDestruct y. + Simpl. + Auto. + + Simpl. + Apply app_nil_end;Auto. + + Intro y. + Simpl. + Rewrite (IHl y). + Apply (app_ass (rev y) (rev l) (cons a nil)). +Qed. + +Remark rev_unit : (l:list)(a:A) (rev l^(cons a nil))= (cons a (rev l)). +Proof. + Intros. + Apply (distr_rev l (cons a nil));Simpl;Auto. +Qed. + +Lemma idempot_rev : (l:list)(rev (rev l))=l. +Proof. + NewInduction l as [|a l IHl]. + Simpl;Auto. + + Simpl. + Rewrite (rev_unit (rev l) a). + Rewrite -> IHl;Auto. +Qed. + +(*********************************************) +(** Reverse Induction Principle on Lists *) +(*********************************************) + +Section Reverse_Induction. + +Unset Implicit Arguments. + +Remark rev_list_ind: (P:list->Prop) + (P nil) + ->((a:A)(l:list)(P (rev l))->(P (rev (cons a l)))) + ->(l:list) (P (rev l)). +Proof. + NewInduction l; Auto. +Qed. +Set Implicit Arguments. + +Lemma rev_ind : + (P:list->Prop) + (P nil)-> + ((x:A)(l:list)(P l)->(P l^(cons x nil))) + ->(l:list)(P l). +Proof. + Intros. + Generalize (idempot_rev l) . + Intros E;Rewrite <- E. + Apply (rev_list_ind P). + Auto. + + Simpl. + Intros. + Apply (H0 a (rev l0)). + Auto. +Qed. + +End Reverse_Induction. + +End Lists. + +Implicits nil [1]. + +Hints Resolve nil_cons app_nil_end ass_app app_ass : datatypes v62. +Hints Resolve app_comm_cons app_cons_not_nil : datatypes v62. +Hints Immediate app_eq_nil : datatypes v62. +Hints Resolve app_eq_unit app_inj_tail : datatypes v62. +Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons + : datatypes v62. +Hints Resolve in_eq in_cons in_inv in_nil in_app_or in_or_app : datatypes v62. +Hints Resolve incl_refl incl_tl incl_tran incl_appl incl_appr incl_cons incl_app + : datatypes v62. + +Section Functions_on_lists. + +(****************************************************************) +(** Some generic functions on lists and basic functions of them *) +(****************************************************************) + +Section Map. +Variables A,B:Set. +Variable f:A->B. +Fixpoint map [l:(list A)] : (list B) := + Cases l of + nil => nil + | (cons a t) => (cons (f a) (map t)) + end. +End Map. + +Lemma in_map : (A,B:Set)(f:A->B)(l:(list A))(x:A) + (In x l) -> (In (f x) (map f l)). +Proof. + NewInduction l as [|a l IHl]; Simpl; + [ Auto + | NewDestruct 1; + [ Left; Apply f_equal with f:=f; Assumption + | Auto] + ]. +Qed. + +Fixpoint flat_map [A,B:Set; f:A->(list B); l:(list A)] : (list B) := + Cases l of + nil => nil + | (cons x t) => (app (f x) (flat_map f t)) + end. + +Fixpoint list_prod [A:Set; B:Set; l:(list A)] : (list B)->(list A*B) := + [l']Cases l of + nil => nil + | (cons x t) => (app (map [y:B](x,y) l') + (list_prod t l')) + end. + +Lemma in_prod_aux : + (A:Set)(B:Set)(x:A)(y:B)(l:(list B)) + (In y l) -> (In (x,y) (map [y0:B](x,y0) l)). +Proof. + NewInduction l; + [ Simpl; Auto + | Simpl; NewDestruct 1 as [H1|]; + [ Left; Rewrite H1; Trivial + | Right; Auto] + ]. +Qed. + +Lemma in_prod : (A:Set)(B:Set)(l:(list A))(l':(list B)) + (x:A)(y:B)(In x l)->(In y l')->(In (x,y) (list_prod l l')). +Proof. + NewInduction l; + [ Simpl; Tauto + | Simpl; Intros; Apply in_or_app; NewDestruct H; + [ Left; Rewrite H; Apply in_prod_aux; Assumption + | Right; Auto] + ]. +Qed. + +(** [(list_power x y)] is [y^x], or the set of sequences of elts of [y] + indexed by elts of [x], sorted in lexicographic order. *) + +Fixpoint list_power [A,B:Set; l:(list A)] : (list B)->(list (list A*B)) := + [l']Cases l of + nil => (cons nil nil) + | (cons x t) => (flat_map [f:(list A*B)](map [y:B](cons (x,y) f) l') + (list_power t l')) + end. + +(************************************) +(** Left-to-right iterator on lists *) +(************************************) + +Section Fold_Left_Recursor. +Variables A,B:Set. +Variable f:A->B->A. +Fixpoint fold_left[l:(list B)] : A -> A := +[a0]Cases l of + nil => a0 + | (cons b t) => (fold_left t (f a0 b)) + end. +End Fold_Left_Recursor. + +(************************************) +(** Right-to-left iterator on lists *) +(************************************) + +Section Fold_Right_Recursor. +Variables A,B:Set. +Variable f:B->A->A. +Variable a0:A. +Fixpoint fold_right [l:(list B)] : A := + Cases l of + nil => a0 + | (cons b t) => (f b (fold_right t)) + end. +End Fold_Right_Recursor. + +Theorem fold_symmetric : + (A:Set)(f:A->A->A) + ((x,y,z:A)(f x (f y z))=(f (f x y) z)) + ->((x,y:A)(f x y)=(f y x)) + ->(a0:A)(l:(list A))(fold_left f l a0)=(fold_right f a0 l). +Proof. +NewDestruct l as [|a l]. +Reflexivity. +Simpl. +Rewrite <- H0. +Generalize a0 a. +NewInduction l as [|a3 l IHl]; Simpl. +Trivial. +Intros. +Rewrite H. +Rewrite (H0 a2). +Rewrite <- (H a1). +Rewrite (H0 a1). +Rewrite IHl. +Reflexivity. +Qed. + +End Functions_on_lists. + +V7only [Implicits nil [].]. + +(** Exporting list notations *) + +V8Infix "::" cons (at level 60, right associativity) : list_scope. + +Infix RIGHTA 7 "^" app : list_scope V8only RIGHTA 60 "++". + +Open Scope list_scope. + +Delimits Scope list_scope with list. + +Bind Scope list_scope with list. diff --git a/theories7/Lists/PolyListSyntax.v b/theories7/Lists/PolyListSyntax.v new file mode 100644 index 00000000..15c57166 --- /dev/null +++ b/theories7/Lists/PolyListSyntax.v @@ -0,0 +1,10 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: PolyListSyntax.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + diff --git a/theories7/Lists/Streams.v b/theories7/Lists/Streams.v new file mode 100755 index 00000000..ccfc4895 --- /dev/null +++ b/theories7/Lists/Streams.v @@ -0,0 +1,170 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Streams.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +Set Implicit Arguments. + +(** Streams *) + +Section Streams. + +Variable A : Set. + +CoInductive Set Stream := Cons : A->Stream->Stream. + + +Definition hd := + [x:Stream] Cases x of (Cons a _) => a end. + +Definition tl := + [x:Stream] Cases x of (Cons _ s) => s end. + + +Fixpoint Str_nth_tl [n:nat] : Stream->Stream := + [s:Stream] Cases n of + O => s + |(S m) => (Str_nth_tl m (tl s)) + end. + +Definition Str_nth : nat->Stream->A := [n:nat][s:Stream](hd (Str_nth_tl n s)). + + +Lemma unfold_Stream :(x:Stream)x=(Cases x of (Cons a s) => (Cons a s) end). +Proof. + Intro x. + Case x. + Trivial. +Qed. + +Lemma tl_nth_tl : (n:nat)(s:Stream)(tl (Str_nth_tl n s))=(Str_nth_tl n (tl s)). +Proof. + Induction n; Simpl; Auto. +Qed. +Hints Resolve tl_nth_tl : datatypes v62. + +Lemma Str_nth_tl_plus +: (n,m:nat)(s:Stream)(Str_nth_tl n (Str_nth_tl m s))=(Str_nth_tl (plus n m) s). +Induction n; Simpl; Intros; Auto with datatypes. +Rewrite <- H. +Rewrite tl_nth_tl; Trivial with datatypes. +Qed. + +Lemma Str_nth_plus + : (n,m:nat)(s:Stream)(Str_nth n (Str_nth_tl m s))=(Str_nth (plus n m) s). +Intros; Unfold Str_nth; Rewrite Str_nth_tl_plus; Trivial with datatypes. +Qed. + +(** Extensional Equality between two streams *) + +CoInductive EqSt : Stream->Stream->Prop := + eqst : (s1,s2:Stream) + ((hd s1)=(hd s2))-> + (EqSt (tl s1) (tl s2)) + ->(EqSt s1 s2). + +(** A coinduction principle *) + +Tactic Definition CoInduction proof := + Cofix proof; Intros; Constructor; + [Clear proof | Try (Apply proof;Clear proof)]. + + +(** Extensional equality is an equivalence relation *) + +Theorem EqSt_reflex : (s:Stream)(EqSt s s). +CoInduction EqSt_reflex. +Reflexivity. +Qed. + +Theorem sym_EqSt : + (s1:Stream)(s2:Stream)(EqSt s1 s2)->(EqSt s2 s1). +(CoInduction Eq_sym). +Case H;Intros;Symmetry;Assumption. +Case H;Intros;Assumption. +Qed. + + +Theorem trans_EqSt : + (s1,s2,s3:Stream)(EqSt s1 s2)->(EqSt s2 s3)->(EqSt s1 s3). +(CoInduction Eq_trans). +Transitivity (hd s2). +Case H; Intros; Assumption. +Case H0; Intros; Assumption. +Apply (Eq_trans (tl s1) (tl s2) (tl s3)). +Case H; Trivial with datatypes. +Case H0; Trivial with datatypes. +Qed. + +(** The definition given is equivalent to require the elements at each + position to be equal *) + +Theorem eqst_ntheq : + (n:nat)(s1,s2:Stream)(EqSt s1 s2)->(Str_nth n s1)=(Str_nth n s2). +Unfold Str_nth; Induction n. +Intros s1 s2 H; Case H; Trivial with datatypes. +Intros m hypind. +Simpl. +Intros s1 s2 H. +Apply hypind. +Case H; Trivial with datatypes. +Qed. + +Theorem ntheq_eqst : + (s1,s2:Stream)((n:nat)(Str_nth n s1)=(Str_nth n s2))->(EqSt s1 s2). +(CoInduction Equiv2). +Apply (H O). +Intros n; Apply (H (S n)). +Qed. + +Section Stream_Properties. + +Variable P : Stream->Prop. + +(*i +Inductive Exists : Stream -> Prop := + | Here : forall x:Stream, P x -> Exists x + | Further : forall x:Stream, ~ P x -> Exists (tl x) -> Exists x. +i*) + +Inductive Exists : Stream -> Prop := + Here : (x:Stream)(P x) ->(Exists x) | + Further : (x:Stream)(Exists (tl x))->(Exists x). + +CoInductive ForAll : Stream -> Prop := + forall : (x:Stream)(P x)->(ForAll (tl x))->(ForAll x). + + +Section Co_Induction_ForAll. +Variable Inv : Stream -> Prop. +Hypothesis InvThenP : (x:Stream)(Inv x)->(P x). +Hypothesis InvIsStable: (x:Stream)(Inv x)->(Inv (tl x)). + +Theorem ForAll_coind : (x:Stream)(Inv x)->(ForAll x). +(CoInduction ForAll_coind);Auto. +Qed. +End Co_Induction_ForAll. + +End Stream_Properties. + +End Streams. + +Section Map. +Variables A,B : Set. +Variable f : A->B. +CoFixpoint map : (Stream A)->(Stream B) := + [s:(Stream A)](Cons (f (hd s)) (map (tl s))). +End Map. + +Section Constant_Stream. +Variable A : Set. +Variable a : A. +CoFixpoint const : (Stream A) := (Cons a const). +End Constant_Stream. + +Unset Implicit Arguments. diff --git a/theories7/Lists/TheoryList.v b/theories7/Lists/TheoryList.v new file mode 100755 index 00000000..f7adda70 --- /dev/null +++ b/theories7/Lists/TheoryList.v @@ -0,0 +1,386 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: TheoryList.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** Some programs and results about lists following CAML Manual *) + +Require Export PolyList. +Set Implicit Arguments. +Chapter Lists. + +Variable A : Set. + +(**********************) +(** The null function *) +(**********************) + +Definition Isnil : (list A) -> Prop := [l:(list A)](nil A)=l. + +Lemma Isnil_nil : (Isnil (nil A)). +Red; Auto. +Qed. +Hints Resolve Isnil_nil. + +Lemma not_Isnil_cons : (a:A)(l:(list A))~(Isnil (cons a l)). +Unfold Isnil. +Intros; Discriminate. +Qed. + +Hints Resolve Isnil_nil not_Isnil_cons. + +Lemma Isnil_dec : (l:(list A)){(Isnil l)}+{~(Isnil l)}. +Intro l; Case l;Auto. +(* +Realizer (fun l => match l with + | nil => true + | _ => false + end). +*) +Qed. + +(************************) +(** The Uncons function *) +(************************) + +Lemma Uncons : (l:(list A)){a : A & { m: (list A) | (cons a m)=l}}+{Isnil l}. +Intro l; Case l. +Auto. +Intros a m; Intros; Left; Exists a; Exists m; Reflexivity. +(* +Realizer (fun l => match l with + | nil => error + | (cons a m) => value (a,m) + end). +*) +Qed. + +(********************************) +(** The head function *) +(********************************) + +Lemma Hd : (l:(list A)){a : A | (EX m:(list A) |(cons a m)=l)}+{Isnil l}. +Intro l; Case l. +Auto. +Intros a m; Intros; Left; Exists a; Exists m; Reflexivity. +(* +Realizer (fun l => match l with + | nil => error + | (cons a m) => value a + end). +*) +Qed. + +Lemma Tl : (l:(list A)){m:(list A)| (EX a:A |(cons a m)=l) + \/ ((Isnil l) /\ (Isnil m)) }. +Intro l; Case l. +Exists (nil A); Auto. +Intros a m; Intros; Exists m; Left; Exists a; Reflexivity. +(* +Realizer (fun l => match l with + | nil => nil + | (cons a m) => m + end). +*) +Qed. + +(****************************************) +(** Length of lists *) +(****************************************) + +(* length is defined in List *) +Fixpoint Length_l [l:(list A)] : nat -> nat + := [n:nat] Cases l of + nil => n + | (cons _ m) => (Length_l m (S n)) + end. + +(* A tail recursive version *) +Lemma Length_l_pf : (l:(list A))(n:nat){m:nat|(plus n (length l))=m}. +NewInduction l as [|a m lrec]. +Intro n; Exists n; Simpl; Auto. +Intro n; Elim (lrec (S n)); Simpl; Intros. +Exists x; Transitivity (S (plus n (length m))); Auto. +(* +Realizer Length_l. +*) +Qed. + +Lemma Length : (l:(list A)){m:nat|(length l)=m}. +Intro l. Apply (Length_l_pf l O). +(* +Realizer (fun l -> Length_l_pf l O). +*) +Qed. + +(*******************************) +(** Members of lists *) +(*******************************) +Inductive In_spec [a:A] : (list A) -> Prop := + | in_hd : (l:(list A))(In_spec a (cons a l)) + | in_tl : (l:(list A))(b:A)(In a l)->(In_spec a (cons b l)). +Hints Resolve in_hd in_tl. +Hints Unfold In. +Hints Resolve in_cons. + +Theorem In_In_spec : (a:A)(l:(list A))(In a l) <-> (In_spec a l). +Split. +Elim l; [ Intros; Contradiction + | Intros; Elim H0; + [ Intros; Rewrite H1; Auto + | Auto ]]. +Intros; Elim H; Auto. +Qed. + +Inductive AllS [P:A->Prop] : (list A) -> Prop + := allS_nil : (AllS P (nil A)) + | allS_cons : (a:A)(l:(list A))(P a)->(AllS P l)->(AllS P (cons a l)). +Hints Resolve allS_nil allS_cons. + +Hypothesis eqA_dec : (a,b:A){a=b}+{~a=b}. + +Fixpoint mem [a:A; l:(list A)] : bool := + Cases l of + nil => false + | (cons b m) => if (eqA_dec a b) then [H]true else [H](mem a m) + end. + +Hints Unfold In. +Lemma Mem : (a:A)(l:(list A)){(In a l)}+{(AllS [b:A]~b=a l)}. +Intros a l. +NewInduction l. +Auto. +Elim (eqA_dec a a0). +Auto. +Simpl. Elim IHl; Auto. +(* +Realizer mem. +*) +Qed. + +(*********************************) +(** Index of elements *) +(*********************************) + +Require Le. +Require Lt. + +Inductive nth_spec : (list A)->nat->A->Prop := + nth_spec_O : (a:A)(l:(list A))(nth_spec (cons a l) (S O) a) +| nth_spec_S : (n:nat)(a,b:A)(l:(list A)) + (nth_spec l n a)->(nth_spec (cons b l) (S n) a). +Hints Resolve nth_spec_O nth_spec_S. + +Inductive fst_nth_spec : (list A)->nat->A->Prop := + fst_nth_O : (a:A)(l:(list A))(fst_nth_spec (cons a l) (S O) a) +| fst_nth_S : (n:nat)(a,b:A)(l:(list A))(~a=b)-> + (fst_nth_spec l n a)->(fst_nth_spec (cons b l) (S n) a). +Hints Resolve fst_nth_O fst_nth_S. + +Lemma fst_nth_nth : (l:(list A))(n:nat)(a:A)(fst_nth_spec l n a)->(nth_spec l n a). +NewInduction 1; Auto. +Qed. +Hints Immediate fst_nth_nth. + +Lemma nth_lt_O : (l:(list A))(n:nat)(a:A)(nth_spec l n a)->(lt O n). +NewInduction 1; Auto. +Qed. + +Lemma nth_le_length : (l:(list A))(n:nat)(a:A)(nth_spec l n a)->(le n (length l)). +NewInduction 1; Simpl; Auto with arith. +Qed. + +Fixpoint Nth_func [l:(list A)] : nat -> (Exc A) + := [n:nat] Cases l n of + (cons a _) (S O) => (value A a) + | (cons _ l') (S (S p)) => (Nth_func l' (S p)) + | _ _ => Error + end. + +Lemma Nth : (l:(list A))(n:nat) + {a:A|(nth_spec l n a)}+{(n=O)\/(lt (length l) n)}. +NewInduction l as [|a l IHl]. +Intro n; Case n; Simpl; Auto with arith. +Intro n; NewDestruct n as [|[|n1]]; Simpl; Auto. +Left; Exists a; Auto. +NewDestruct (IHl (S n1)) as [[b]|o]. +Left; Exists b; Auto. +Right; NewDestruct o. +Absurd (S n1)=O; Auto. +Auto with arith. +(* +Realizer Nth_func. +*) +Qed. + +Lemma Item : (l:(list A))(n:nat){a:A|(nth_spec l (S n) a)}+{(le (length l) n)}. +Intros l n; Case (Nth l (S n)); Intro. +Case s; Intro a; Left; Exists a; Auto. +Right; Case o; Intro. +Absurd (S n)=O; Auto. +Auto with arith. +Qed. + +Require Minus. +Require DecBool. + +Fixpoint index_p [a:A;l:(list A)] : nat -> (Exc nat) := + Cases l of nil => [p]Error + | (cons b m) => [p](ifdec (eqA_dec a b) (Value p) (index_p a m (S p))) + end. + +Lemma Index_p : (a:A)(l:(list A))(p:nat) + {n:nat|(fst_nth_spec l (minus (S n) p) a)}+{(AllS [b:A]~a=b l)}. +NewInduction l as [|b m irec]. +Auto. +Intro p. +NewDestruct (eqA_dec a b) as [e|e]. +Left; Exists p. +NewDestruct e; Elim minus_Sn_m; Trivial; Elim minus_n_n; Auto with arith. +NewDestruct (irec (S p)) as [[n H]|]. +Left; Exists n; Auto with arith. +Elim minus_Sn_m; Auto with arith. +Apply lt_le_weak; Apply lt_O_minus_lt; Apply nth_lt_O with m a; Auto with arith. +Auto. +Qed. + +Lemma Index : (a:A)(l:(list A)) + {n:nat|(fst_nth_spec l n a)}+{(AllS [b:A]~a=b l)}. + +Intros a l; Case (Index_p a l (S O)); Auto. +Intros (n,P); Left; Exists n; Auto. +Rewrite (minus_n_O n); Trivial. +(* +Realizer (fun a l -> Index_p a l (S O)). +*) +Qed. + +Section Find_sec. +Variable R,P : A -> Prop. + +Inductive InR : (list A) -> Prop + := inR_hd : (a:A)(l:(list A))(R a)->(InR (cons a l)) + | inR_tl : (a:A)(l:(list A))(InR l)->(InR (cons a l)). +Hints Resolve inR_hd inR_tl. + +Definition InR_inv := + [l:(list A)]Cases l of + nil => False + | (cons b m) => (R b)\/(InR m) + end. + +Lemma InR_INV : (l:(list A))(InR l)->(InR_inv l). +NewInduction 1; Simpl; Auto. +Qed. + +Lemma InR_cons_inv : (a:A)(l:(list A))(InR (cons a l))->((R a)\/(InR l)). +Intros a l H; Exact (InR_INV H). +Qed. + +Lemma InR_or_app : (l,m:(list A))((InR l)\/(InR m))->(InR (app l m)). +Intros l m [|]. +NewInduction 1; Simpl; Auto. +Intro. NewInduction l; Simpl; Auto. +Qed. + +Lemma InR_app_or : (l,m:(list A))(InR (app l m))->((InR l)\/(InR m)). +Intros l m; Elim l; Simpl; Auto. +Intros b l' Hrec IAc; Elim (InR_cons_inv IAc);Auto. +Intros; Elim Hrec; Auto. +Qed. + +Hypothesis RS_dec : (a:A){(R a)}+{(P a)}. + +Fixpoint find [l:(list A)] : (Exc A) := + Cases l of nil => Error + | (cons a m) => (ifdec (RS_dec a) (Value a) (find m)) + end. + +Lemma Find : (l:(list A)){a:A | (In a l) & (R a)}+{(AllS P l)}. +NewInduction l as [|a m [[b H1 H2]|H]]; Auto. +Left; Exists b; Auto. +NewDestruct (RS_dec a). +Left; Exists a; Auto. +Auto. +(* +Realizer find. +*) +Qed. + +Variable B : Set. +Variable T : A -> B -> Prop. + +Variable TS_dec : (a:A){c:B| (T a c)}+{(P a)}. + +Fixpoint try_find [l:(list A)] : (Exc B) := + Cases l of + nil => Error + | (cons a l1) => + Cases (TS_dec a) of + (inleft (exist c _)) => (Value c) + | (inright _) => (try_find l1) + end + end. + +Lemma Try_find : (l:(list A)){c:B|(EX a:A |(In a l) & (T a c))}+{(AllS P l)}. +NewInduction l as [|a m [[b H1]|H]]. +Auto. +Left; Exists b; NewDestruct H1 as [a' H2 H3]; Exists a'; Auto. +NewDestruct (TS_dec a) as [[c H1]|]. +Left; Exists c. +Exists a; Auto. +Auto. +(* +Realizer try_find. +*) +Qed. + +End Find_sec. + +Section Assoc_sec. + +Variable B : Set. +Fixpoint assoc [a:A;l:(list A*B)] : (Exc B) := + Cases l of nil => Error + | (cons (a',b) m) => (ifdec (eqA_dec a a') (Value b) (assoc a m)) + end. + +Inductive AllS_assoc [P:A -> Prop]: (list A*B) -> Prop := + allS_assoc_nil : (AllS_assoc P (nil A*B)) + | allS_assoc_cons : (a:A)(b:B)(l:(list A*B)) + (P a)->(AllS_assoc P l)->(AllS_assoc P (cons (a,b) l)). + +Hints Resolve allS_assoc_nil allS_assoc_cons. + +(* The specification seems too weak: it is enough to return b if the + list has at least an element (a,b); probably the intention is to have + the specification + + (a:A)(l:(list A*B)){b:B|(In_spec (a,b) l)}+{(AllS_assoc [a':A]~(a=a') l)}. +*) + +Lemma Assoc : (a:A)(l:(list A*B))(B+{(AllS_assoc [a':A]~(a=a') l)}). +NewInduction l as [|[a' b] m assrec]. Auto. +NewDestruct (eqA_dec a a'). +Left; Exact b. +NewDestruct assrec as [b'|]. +Left; Exact b'. +Right; Auto. +(* +Realizer assoc. +*) +Qed. + +End Assoc_sec. + +End Lists. + +Hints Resolve Isnil_nil not_Isnil_cons in_hd in_tl in_cons allS_nil allS_cons + : datatypes. +Hints Immediate fst_nth_nth : datatypes. + diff --git a/theories7/Logic/Berardi.v b/theories7/Logic/Berardi.v new file mode 100644 index 00000000..db9007ec --- /dev/null +++ b/theories7/Logic/Berardi.v @@ -0,0 +1,170 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Berardi.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** This file formalizes Berardi's paradox which says that in + the calculus of constructions, excluded middle (EM) and axiom of + choice (AC) implie proof irrelevenace (PI). + Here, the axiom of choice is not necessary because of the use + of inductive types. +<< +@article{Barbanera-Berardi:JFP96, + author = {F. Barbanera and S. Berardi}, + title = {Proof-irrelevance out of Excluded-middle and Choice + in the Calculus of Constructions}, + journal = {Journal of Functional Programming}, + year = {1996}, + volume = {6}, + number = {3}, + pages = {519-525} +} +>> *) + +Set Implicit Arguments. + +Section Berardis_paradox. + +(** Excluded middle *) +Hypothesis EM : (P:Prop) P \/ ~P. + +(** Conditional on any proposition. *) +Definition IFProp := [P,B:Prop][e1,e2:P] + Cases (EM B) of + (or_introl _) => e1 + | (or_intror _) => e2 + end. + +(** Axiom of choice applied to disjunction. + Provable in Coq because of dependent elimination. *) +Lemma AC_IF : (P,B:Prop)(e1,e2:P)(Q:P->Prop) + ( B -> (Q e1))-> + (~B -> (Q e2))-> + (Q (IFProp B e1 e2)). +Proof. +Intros P B e1 e2 Q p1 p2. +Unfold IFProp. +Case (EM B); Assumption. +Qed. + + +(** We assume a type with two elements. They play the role of booleans. + The main theorem under the current assumptions is that [T=F] *) +Variable Bool: Prop. +Variable T: Bool. +Variable F: Bool. + +(** The powerset operator *) +Definition pow [P:Prop] :=P->Bool. + + +(** A piece of theory about retracts *) +Section Retracts. + +Variable A,B: Prop. + +Record retract : Prop := { + i: A->B; + j: B->A; + inv: (a:A)(j (i a))==a + }. + +Record retract_cond : Prop := { + i2: A->B; + j2: B->A; + inv2: retract -> (a:A)(j2 (i2 a))==a + }. + + +(** The dependent elimination above implies the axiom of choice: *) +Lemma AC: (r:retract_cond) retract -> (a:A)((j2 r) ((i2 r) a))==a. +Proof. +Intros r. +Case r; Simpl. +Trivial. +Qed. + +End Retracts. + +(** This lemma is basically a commutation of implication and existential + quantification: (EX x | A -> P(x)) <=> (A -> EX x | P(x)) + which is provable in classical logic ( => is already provable in + intuitionnistic logic). *) + +Lemma L1 : (A,B:Prop)(retract_cond (pow A) (pow B)). +Proof. +Intros A B. +Elim (EM (retract (pow A) (pow B))). +Intros (f0, g0, e). +Exists f0 g0. +Trivial. + +Intros hf. +Exists ([x:(pow A); y:B]F) ([x:(pow B); y:A]F). +Intros; Elim hf; Auto. +Qed. + + +(** The paradoxical set *) +Definition U := (P:Prop)(pow P). + +(** Bijection between [U] and [(pow U)] *) +Definition f : U -> (pow U) := + [u](u U). + +Definition g : (pow U) -> U := + [h,X] + let lX = (j2 (L1 X U)) in + let rU = (i2 (L1 U U)) in + (lX (rU h)). + +(** We deduce that the powerset of [U] is a retract of [U]. + This lemma is stated in Berardi's article, but is not used + afterwards. *) +Lemma retract_pow_U_U : (retract (pow U) U). +Proof. +Exists g f. +Intro a. +Unfold f g; Simpl. +Apply AC. +Exists ([x:(pow U)]x) ([x:(pow U)]x). +Trivial. +Qed. + +(** Encoding of Russel's paradox *) + +(** The boolean negation. *) +Definition Not_b := [b:Bool](IFProp b==T F T). + +(** the set of elements not belonging to itself *) +Definition R : U := (g ([u:U](Not_b (u U u)))). + + +Lemma not_has_fixpoint : (R R)==(Not_b (R R)). +Proof. +Unfold 1 R. +Unfold g. +Rewrite AC with r:=(L1 U U) a:=[u:U](Not_b (u U u)). +Trivial. +Exists ([x:(pow U)]x) ([x:(pow U)]x); Trivial. +Qed. + + +Theorem classical_proof_irrelevence : T==F. +Proof. +Generalize not_has_fixpoint. +Unfold Not_b. +Apply AC_IF. +Intros is_true is_false. +Elim is_true; Elim is_false; Trivial. + +Intros not_true is_true. +Elim not_true; Trivial. +Qed. + +End Berardis_paradox. diff --git a/theories7/Logic/ChoiceFacts.v b/theories7/Logic/ChoiceFacts.v new file mode 100644 index 00000000..5b7e002a --- /dev/null +++ b/theories7/Logic/ChoiceFacts.v @@ -0,0 +1,134 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ChoiceFacts.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(* We show that the functional formulation of the axiom of Choice + (usual formulation in type theory) is equivalent to its relational + formulation (only formulation of set theory) + the axiom of + (parametric) definite description (aka axiom of unique choice) *) + +(* This shows that the axiom of choice can be assumed (under its + relational formulation) without known inconsistency with classical logic, + though definite description conflicts with classical logic *) + +Definition RelationalChoice := + (A:Type;B:Type;R: A->B->Prop) + ((x:A)(EX y:B|(R x y))) + -> (EXT R':A->B->Prop | + ((x:A)(EX y:B|(R x y)/\(R' x y)/\ ((y':B) (R' x y') -> y=y')))). + +Definition FunctionalChoice := + (A:Type;B:Type;R: A->B->Prop) + ((x:A)(EX y:B|(R x y))) -> (EX f:A->B | (x:A)(R x (f x))). + +Definition ParamDefiniteDescription := + (A:Type;B:Type;R: A->B->Prop) + ((x:A)(EX y:B|(R x y)/\ ((y':B)(R x y') -> y=y'))) + -> (EX f:A->B | (x:A)(R x (f x))). + +Lemma description_rel_choice_imp_funct_choice : + ParamDefiniteDescription->RelationalChoice->FunctionalChoice. +Intros Descr RelCh. +Red; Intros A B R H. +NewDestruct (RelCh A B R H) as [R' H0]. +NewDestruct (Descr A B R') as [f H1]. +Intro x. +Elim (H0 x); Intros y [H2 [H3 H4]]; Exists y; Split; [Exact H3 | Exact H4]. +Exists f; Intro x. +Elim (H0 x); Intros y [H2 [H3 H4]]. +Rewrite <- (H4 (f x) (H1 x)). +Exact H2. +Qed. + +Lemma funct_choice_imp_rel_choice : + FunctionalChoice->RelationalChoice. +Intros FunCh. +Red; Intros A B R H. +NewDestruct (FunCh A B R H) as [f H0]. +Exists [x,y]y=(f x). +Intro x; Exists (f x); +Split; [Apply H0| Split;[Reflexivity| Intros y H1; Symmetry; Exact H1]]. +Qed. + +Lemma funct_choice_imp_description : + FunctionalChoice->ParamDefiniteDescription. +Intros FunCh. +Red; Intros A B R H. +NewDestruct (FunCh A B R) as [f H0]. +(* 1 *) +Intro x. +Elim (H x); Intros y [H0 H1]. +Exists y; Exact H0. +(* 2 *) +Exists f; Exact H0. +Qed. + +Theorem FunChoice_Equiv_RelChoice_and_ParamDefinDescr : + FunctionalChoice <-> RelationalChoice /\ ParamDefiniteDescription. +Split. +Intro H; Split; [ + Exact (funct_choice_imp_rel_choice H) + | Exact (funct_choice_imp_description H)]. +Intros [H H0]; Exact (description_rel_choice_imp_funct_choice H0 H). +Qed. + +(* We show that the guarded relational formulation of the axiom of Choice + comes from the non guarded formulation in presence either of the + independance of premises or proof-irrelevance *) + +Definition GuardedRelationalChoice := + (A:Type;B:Type;P:A->Prop;R: A->B->Prop) + ((x:A)(P x)->(EX y:B|(R x y))) + -> (EXT R':A->B->Prop | + ((x:A)(P x)->(EX y:B|(R x y)/\(R' x y)/\ ((y':B) (R' x y') -> y=y')))). + +Definition ProofIrrelevance := (A:Prop)(a1,a2:A) a1==a2. + +Lemma rel_choice_and_proof_irrel_imp_guarded_rel_choice : + RelationalChoice -> ProofIrrelevance -> GuardedRelationalChoice. +Proof. +Intros rel_choice proof_irrel. +Red; Intros A B P R H. +NewDestruct (rel_choice ? ? [x:(sigT ? P);y:B](R (projT1 ? ? x) y)) as [R' H0]. +Intros [x HPx]. +NewDestruct (H x HPx) as [y HRxy]. +Exists y; Exact HRxy. +Pose R'':=[x:A;y:B](EXT H:(P x) | (R' (existT ? P x H) y)). +Exists R''; Intros x HPx. +NewDestruct (H0 (existT ? P x HPx)) as [y [HRxy [HR'xy Huniq]]]. +Exists y. Split. + Exact HRxy. + Split. + Red; Exists HPx; Exact HR'xy. + Intros y' HR''xy'. + Apply Huniq. + Unfold R'' in HR''xy'. + NewDestruct HR''xy' as [H'Px HR'xy']. + Rewrite proof_irrel with a1:=HPx a2:=H'Px. + Exact HR'xy'. +Qed. + +Definition IndependenceOfPremises := + (A:Type)(P:A->Prop)(Q:Prop)(Q->(EXT x|(P x)))->(EXT x|Q->(P x)). + +Lemma rel_choice_indep_of_premises_imp_guarded_rel_choice : + RelationalChoice -> IndependenceOfPremises -> GuardedRelationalChoice. +Proof. +Intros RelCh IndPrem. +Red; Intros A B P R H. +NewDestruct (RelCh A B [x,y](P x)->(R x y)) as [R' H0]. + Intro x. Apply IndPrem. + Apply H. + Exists R'. + Intros x HPx. + NewDestruct (H0 x) as [y [H1 H2]]. + Exists y. Split. + Apply (H1 HPx). + Exact H2. +Qed. diff --git a/theories7/Logic/Classical.v b/theories7/Logic/Classical.v new file mode 100755 index 00000000..8d7fe1d1 --- /dev/null +++ b/theories7/Logic/Classical.v @@ -0,0 +1,14 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Classical.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** Classical Logic *) + +Require Export Classical_Prop. +Require Export Classical_Pred_Type. diff --git a/theories7/Logic/ClassicalChoice.v b/theories7/Logic/ClassicalChoice.v new file mode 100644 index 00000000..5419e958 --- /dev/null +++ b/theories7/Logic/ClassicalChoice.v @@ -0,0 +1,31 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ClassicalChoice.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** This file provides classical logic and functional choice *) + +(** This file extends ClassicalDescription.v with the axiom of choice. + As ClassicalDescription.v, it implies the double-negation of + excluded-middle in Set and implies a strongly classical + world. Especially it conflicts with impredicativity of Set, knowing + that true<>false in Set. +*) + +Require Export ClassicalDescription. +Require Export RelationalChoice. +Require ChoiceFacts. + +Theorem choice : + (A:Type;B:Type;R: A->B->Prop) + ((x:A)(EX y:B|(R x y))) -> (EX f:A->B | (x:A)(R x (f x))). +Proof. +Apply description_rel_choice_imp_funct_choice. +Exact description. +Exact relational_choice. +Qed. diff --git a/theories7/Logic/ClassicalDescription.v b/theories7/Logic/ClassicalDescription.v new file mode 100644 index 00000000..85700c22 --- /dev/null +++ b/theories7/Logic/ClassicalDescription.v @@ -0,0 +1,76 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ClassicalDescription.v,v 1.2.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** This file provides classical logic and definite description *) + +(** Classical logic and definite description, as shown in [1], + implies the double-negation of excluded-middle in Set, hence it + implies a strongly classical world. Especially it conflicts with + impredicativity of Set, knowing that true<>false in Set. + + [1] Laurent Chicli, Loïc Pottier, Carlos Simpson, Mathematical + Quotients and Quotient Types in Coq, Proceedings of TYPES 2002, + Lecture Notes in Computer Science 2646, Springer Verlag. +*) + +Require Export Classical. + +Axiom dependent_description : + (A:Type;B:A->Type;R: (x:A)(B x)->Prop) + ((x:A)(EX y:(B x)|(R x y)/\ ((y':(B x))(R x y') -> y=y'))) + -> (EX f:(x:A)(B x) | (x:A)(R x (f x))). + +(** Principle of definite descriptions (aka axiom of unique choice) *) + +Theorem description : + (A:Type;B:Type;R: A->B->Prop) + ((x:A)(EX y:B|(R x y)/\ ((y':B)(R x y') -> y=y'))) + -> (EX f:A->B | (x:A)(R x (f x))). +Proof. +Intros A B. +Apply (dependent_description A [_]B). +Qed. + +(** The followig proof comes from [1] *) + +Theorem classic_set : (((P:Prop){P}+{~P}) -> False) -> False. +Proof. +Intro HnotEM. +Pose R:=[A,b]A/\true=b \/ ~A/\false=b. +Assert H:(EX f:Prop->bool|(A:Prop)(R A (f A))). +Apply description. +Intro A. +NewDestruct (classic A) as [Ha|Hnota]. + Exists true; Split. + Left; Split; [Assumption|Reflexivity]. + Intros y [[_ Hy]|[Hna _]]. + Assumption. + Contradiction. + Exists false; Split. + Right; Split; [Assumption|Reflexivity]. + Intros y [[Ha _]|[_ Hy]]. + Contradiction. + Assumption. +NewDestruct H as [f Hf]. +Apply HnotEM. +Intro P. +Assert HfP := (Hf P). +(* Elimination from Hf to Set is not allowed but from f to Set yes ! *) +NewDestruct (f P). + Left. + NewDestruct HfP as [[Ha _]|[_ Hfalse]]. + Assumption. + Discriminate. + Right. + NewDestruct HfP as [[_ Hfalse]|[Hna _]]. + Discriminate. + Assumption. +Qed. + diff --git a/theories7/Logic/ClassicalFacts.v b/theories7/Logic/ClassicalFacts.v new file mode 100644 index 00000000..1d37652e --- /dev/null +++ b/theories7/Logic/ClassicalFacts.v @@ -0,0 +1,214 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ClassicalFacts.v,v 1.2.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** Some facts and definitions about classical logic *) + +(** [prop_degeneracy] (also referred as propositional completeness) *) +(* asserts (up to consistency) that there are only two distinct formulas *) +Definition prop_degeneracy := (A:Prop) A==True \/ A==False. + +(** [prop_extensionality] asserts equivalent formulas are equal *) +Definition prop_extensionality := (A,B:Prop) (A<->B) -> A==B. + +(** [excluded_middle] asserts we can reason by case on the truth *) +(* or falsity of any formula *) +Definition excluded_middle := (A:Prop) A \/ ~A. + +(** [proof_irrelevance] asserts equality of all proofs of a given formula *) +Definition proof_irrelevance := (A:Prop)(a1,a2:A) a1==a2. + +(** We show [prop_degeneracy <-> (prop_extensionality /\ excluded_middle)] *) + +Lemma prop_degen_ext : prop_degeneracy -> prop_extensionality. +Proof. +Intros H A B (Hab,Hba). +NewDestruct (H A); NewDestruct (H B). + Rewrite H1; Exact H0. + Absurd B. + Rewrite H1; Exact [H]H. + Apply Hab; Rewrite H0; Exact I. + Absurd A. + Rewrite H0; Exact [H]H. + Apply Hba; Rewrite H1; Exact I. + Rewrite H1; Exact H0. +Qed. + +Lemma prop_degen_em : prop_degeneracy -> excluded_middle. +Proof. +Intros H A. +NewDestruct (H A). + Left; Rewrite H0; Exact I. + Right; Rewrite H0; Exact [x]x. +Qed. + +Lemma prop_ext_em_degen : + prop_extensionality -> excluded_middle -> prop_degeneracy. +Proof. +Intros Ext EM A. +NewDestruct (EM A). + Left; Apply (Ext A True); Split; [Exact [_]I | Exact [_]H]. + Right; Apply (Ext A False); Split; [Exact H | Apply False_ind]. +Qed. + +(** We successively show that: + + [prop_extensionality] + implies equality of [A] and [A->A] for inhabited [A], which + implies the existence of a (trivial) retract from [A->A] to [A] + (just take the identity), which + implies the existence of a fixpoint operator in [A] + (e.g. take the Y combinator of lambda-calculus) +*) + +Definition inhabited [A:Prop] := A. + +Lemma prop_ext_A_eq_A_imp_A : + prop_extensionality->(A:Prop)(inhabited A)->(A->A)==A. +Proof. +Intros Ext A a. +Apply (Ext A->A A); Split; [ Exact [_]a | Exact [_;_]a ]. +Qed. + +Record retract [A,B:Prop] : Prop := { + f1: A->B; + f2: B->A; + f1_o_f2: (x:B)(f1 (f2 x))==x +}. + +Lemma prop_ext_retract_A_A_imp_A : + prop_extensionality->(A:Prop)(inhabited A)->(retract A A->A). +Proof. +Intros Ext A a. +Rewrite -> (prop_ext_A_eq_A_imp_A Ext A a). +Exists [x:A]x [x:A]x. +Reflexivity. +Qed. + +Record has_fixpoint [A:Prop] : Prop := { + F : (A->A)->A; + fix : (f:A->A)(F f)==(f (F f)) +}. + +Lemma ext_prop_fixpoint : + prop_extensionality->(A:Prop)(inhabited A)->(has_fixpoint A). +Proof. +Intros Ext A a. +Case (prop_ext_retract_A_A_imp_A Ext A a); Intros g1 g2 g1_o_g2. +Exists [f]([x:A](f (g1 x x)) (g2 [x](f (g1 x x)))). +Intro f. +Pattern 1 (g1 (g2 [x:A](f (g1 x x)))). +Rewrite (g1_o_g2 [x:A](f (g1 x x))). +Reflexivity. +Qed. + +(** Assume we have booleans with the property that there is at most 2 + booleans (which is equivalent to dependent case analysis). Consider + the fixpoint of the negation function: it is either true or false by + dependent case analysis, but also the opposite by fixpoint. Hence + proof-irrelevance. + + We then map bool proof-irrelevance to all propositions. +*) + +Section Proof_irrelevance_gen. + +Variable bool : Prop. +Variable true : bool. +Variable false : bool. +Hypothesis bool_elim : (C:Prop)C->C->bool->C. +Hypothesis bool_elim_redl : (C:Prop)(c1,c2:C)c1==(bool_elim C c1 c2 true). +Hypothesis bool_elim_redr : (C:Prop)(c1,c2:C)c2==(bool_elim C c1 c2 false). +Local bool_dep_induction := (P:bool->Prop)(P true)->(P false)->(b:bool)(P b). + +Lemma aux : prop_extensionality -> bool_dep_induction -> true==false. +Proof. +Intros Ext Ind. +Case (ext_prop_fixpoint Ext bool true); Intros G Gfix. +Pose neg := [b:bool](bool_elim bool false true b). +Generalize (refl_eqT ? (G neg)). +Pattern 1 (G neg). +Apply Ind with b:=(G neg); Intro Heq. +Rewrite (bool_elim_redl bool false true). +Change true==(neg true); Rewrite -> Heq; Apply Gfix. +Rewrite (bool_elim_redr bool false true). +Change (neg false)==false; Rewrite -> Heq; Symmetry; Apply Gfix. +Qed. + +Lemma ext_prop_dep_proof_irrel_gen : + prop_extensionality -> bool_dep_induction -> proof_irrelevance. +Proof. +Intros Ext Ind A a1 a2. +Pose f := [b:bool](bool_elim A a1 a2 b). +Rewrite (bool_elim_redl A a1 a2). +Change (f true)==a2. +Rewrite (bool_elim_redr A a1 a2). +Change (f true)==(f false). +Rewrite (aux Ext Ind). +Reflexivity. +Qed. + +End Proof_irrelevance_gen. + +(** In the pure Calculus of Constructions, we can define the boolean + proposition bool = (C:Prop)C->C->C but we cannot prove that it has at + most 2 elements. +*) + +Section Proof_irrelevance_CC. + +Definition BoolP := (C:Prop)C->C->C. +Definition TrueP := [C][c1,c2]c1 : BoolP. +Definition FalseP := [C][c1,c2]c2 : BoolP. +Definition BoolP_elim := [C][c1,c2][b:BoolP](b C c1 c2). +Definition BoolP_elim_redl : (C:Prop)(c1,c2:C)c1==(BoolP_elim C c1 c2 TrueP) + := [C;c1,c2](refl_eqT C c1). +Definition BoolP_elim_redr : (C:Prop)(c1,c2:C)c2==(BoolP_elim C c1 c2 FalseP) + := [C;c1,c2](refl_eqT C c2). + +Definition BoolP_dep_induction := + (P:BoolP->Prop)(P TrueP)->(P FalseP)->(b:BoolP)(P b). + +Lemma ext_prop_dep_proof_irrel_cc : + prop_extensionality -> BoolP_dep_induction -> proof_irrelevance. +Proof (ext_prop_dep_proof_irrel_gen BoolP TrueP FalseP BoolP_elim + BoolP_elim_redl BoolP_elim_redr). + +End Proof_irrelevance_CC. + +(** In the Calculus of Inductive Constructions, inductively defined booleans + enjoy dependent case analysis, hence directly proof-irrelevance from + propositional extensionality. +*) + +Section Proof_irrelevance_CIC. + +Inductive boolP : Prop := trueP : boolP | falseP : boolP. +Definition boolP_elim_redl : (C:Prop)(c1,c2:C)c1==(boolP_ind C c1 c2 trueP) + := [C;c1,c2](refl_eqT C c1). +Definition boolP_elim_redr : (C:Prop)(c1,c2:C)c2==(boolP_ind C c1 c2 falseP) + := [C;c1,c2](refl_eqT C c2). +Scheme boolP_indd := Induction for boolP Sort Prop. + +Lemma ext_prop_dep_proof_irrel_cic : prop_extensionality -> proof_irrelevance. +Proof [pe](ext_prop_dep_proof_irrel_gen boolP trueP falseP boolP_ind + boolP_elim_redl boolP_elim_redr pe boolP_indd). + +End Proof_irrelevance_CIC. + +(** Can we state proof irrelevance from propositional degeneracy + (i.e. propositional extensionality + excluded middle) without + dependent case analysis ? + + Conjecture: it seems possible to build a model of CC interpreting + all non-empty types by the set of all lambda-terms. Such a model would + satisfy propositional degeneracy without satisfying proof-irrelevance + (nor dependent case analysis). This would imply that the previous + results cannot be refined. +*) diff --git a/theories7/Logic/Classical_Pred_Set.v b/theories7/Logic/Classical_Pred_Set.v new file mode 100755 index 00000000..b1c26e6d --- /dev/null +++ b/theories7/Logic/Classical_Pred_Set.v @@ -0,0 +1,64 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Classical_Pred_Set.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** Classical Predicate Logic on Set*) + +Require Classical_Prop. + +Section Generic. +Variable U: Set. + +(** de Morgan laws for quantifiers *) + +Lemma not_all_ex_not : (P:U->Prop)(~(n:U)(P n)) -> (EX n:U | ~(P n)). +Proof. +Unfold not; Intros P notall. +Apply NNPP; Unfold not. +Intro abs. +Cut ((n:U)(P n)); Auto. +Intro n; Apply NNPP. +Unfold not; Intros. +Apply abs; Exists n; Trivial. +Qed. + +Lemma not_all_not_ex : (P:U->Prop)(~(n:U)~(P n)) -> (EX n:U |(P n)). +Proof. +Intros P H. +Elim (not_all_ex_not [n:U]~(P n) H); Intros n Pn; Exists n. +Apply NNPP; Trivial. +Qed. + +Lemma not_ex_all_not : (P:U->Prop) (~(EX n:U |(P n))) -> (n:U)~(P n). +Proof. +Unfold not; Intros P notex n abs. +Apply notex. +Exists n; Trivial. +Qed. + +Lemma not_ex_not_all : (P:U->Prop)(~(EX n:U | ~(P n))) -> (n:U)(P n). +Proof. +Intros P H n. +Apply NNPP. +Red; Intro K; Apply H; Exists n; Trivial. +Qed. + +Lemma ex_not_not_all : (P:U->Prop) (EX n:U | ~(P n)) -> ~(n:U)(P n). +Proof. +Unfold not; Intros P exnot allP. +Elim exnot; Auto. +Qed. + +Lemma all_not_not_ex : (P:U->Prop) ((n:U)~(P n)) -> ~(EX n:U |(P n)). +Proof. +Unfold not; Intros P allnot exP; Elim exP; Intros n p. +Apply allnot with n; Auto. +Qed. + +End Generic. diff --git a/theories7/Logic/Classical_Pred_Type.v b/theories7/Logic/Classical_Pred_Type.v new file mode 100755 index 00000000..69175ec7 --- /dev/null +++ b/theories7/Logic/Classical_Pred_Type.v @@ -0,0 +1,64 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Classical_Pred_Type.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** Classical Predicate Logic on Type *) + +Require Classical_Prop. + +Section Generic. +Variable U: Type. + +(** de Morgan laws for quantifiers *) + +Lemma not_all_ex_not : (P:U->Prop)(~(n:U)(P n)) -> (EXT n:U | ~(P n)). +Proof. +Unfold not; Intros P notall. +Apply NNPP; Unfold not. +Intro abs. +Cut ((n:U)(P n)); Auto. +Intro n; Apply NNPP. +Unfold not; Intros. +Apply abs; Exists n; Trivial. +Qed. + +Lemma not_all_not_ex : (P:U->Prop)(~(n:U)~(P n)) -> (EXT n:U | (P n)). +Proof. +Intros P H. +Elim (not_all_ex_not [n:U]~(P n) H); Intros n Pn; Exists n. +Apply NNPP; Trivial. +Qed. + +Lemma not_ex_all_not : (P:U->Prop)(~(EXT n:U | (P n))) -> (n:U)~(P n). +Proof. +Unfold not; Intros P notex n abs. +Apply notex. +Exists n; Trivial. +Qed. + +Lemma not_ex_not_all : (P:U->Prop)(~(EXT n:U | ~(P n))) -> (n:U)(P n). +Proof. +Intros P H n. +Apply NNPP. +Red; Intro K; Apply H; Exists n; Trivial. +Qed. + +Lemma ex_not_not_all : (P:U->Prop) (EXT n:U | ~(P n)) -> ~(n:U)(P n). +Proof. +Unfold not; Intros P exnot allP. +Elim exnot; Auto. +Qed. + +Lemma all_not_not_ex : (P:U->Prop) ((n:U)~(P n)) -> ~(EXT n:U | (P n)). +Proof. +Unfold not; Intros P allnot exP; Elim exP; Intros n p. +Apply allnot with n; Auto. +Qed. + +End Generic. diff --git a/theories7/Logic/Classical_Prop.v b/theories7/Logic/Classical_Prop.v new file mode 100755 index 00000000..1dc7ec57 --- /dev/null +++ b/theories7/Logic/Classical_Prop.v @@ -0,0 +1,85 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Classical_Prop.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** Classical Propositional Logic *) + +Require ProofIrrelevance. + +Hints Unfold not : core. + +Axiom classic: (P:Prop)(P \/ ~(P)). + +Lemma NNPP : (p:Prop)~(~(p))->p. +Proof. +Unfold not; Intros; Elim (classic p); Auto. +Intro NP; Elim (H NP). +Qed. + +Lemma not_imply_elim : (P,Q:Prop)~(P->Q)->P. +Proof. +Intros; Apply NNPP; Red. +Intro; Apply H; Intro; Absurd P; Trivial. +Qed. + +Lemma not_imply_elim2 : (P,Q:Prop)~(P->Q) -> ~Q. +Proof. +Intros; Elim (classic Q); Auto. +Qed. + +Lemma imply_to_or : (P,Q:Prop)(P->Q) -> ~P \/ Q. +Proof. +Intros; Elim (classic P); Auto. +Qed. + +Lemma imply_to_and : (P,Q:Prop)~(P->Q) -> P /\ ~Q. +Proof. +Intros; Split. +Apply not_imply_elim with Q; Trivial. +Apply not_imply_elim2 with P; Trivial. +Qed. + +Lemma or_to_imply : (P,Q:Prop)(~P \/ Q) -> P->Q. +Proof. +Induction 1; Auto. +Intros H1 H2; Elim (H1 H2). +Qed. + +Lemma not_and_or : (P,Q:Prop)~(P/\Q)-> ~P \/ ~Q. +Proof. +Intros; Elim (classic P); Auto. +Qed. + +Lemma or_not_and : (P,Q:Prop)(~P \/ ~Q) -> ~(P/\Q). +Proof. +Induction 1; Red; Induction 2; Auto. +Qed. + +Lemma not_or_and : (P,Q:Prop)~(P\/Q)-> ~P /\ ~Q. +Proof. +Intros; Elim (classic P); Auto. +Qed. + +Lemma and_not_or : (P,Q:Prop)(~P /\ ~Q) -> ~(P\/Q). +Proof. +Induction 1; Red; Induction 3; Trivial. +Qed. + +Lemma imply_and_or: (P,Q:Prop)(P->Q) -> P \/ Q -> Q. +Proof. +Induction 2; Trivial. +Qed. + +Lemma imply_and_or2: (P,Q,R:Prop)(P->Q) -> P \/ R -> Q \/ R. +Proof. +Induction 2; Auto. +Qed. + +Lemma proof_irrelevance: (P:Prop)(p1,p2:P)p1==p2. +Proof (proof_irrelevance_cci classic). diff --git a/theories7/Logic/Classical_Type.v b/theories7/Logic/Classical_Type.v new file mode 100755 index 00000000..e34170cd --- /dev/null +++ b/theories7/Logic/Classical_Type.v @@ -0,0 +1,14 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Classical_Type.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** Classical Logic for Type *) + +Require Export Classical_Prop. +Require Export Classical_Pred_Type. diff --git a/theories7/Logic/Decidable.v b/theories7/Logic/Decidable.v new file mode 100644 index 00000000..537b5e88 --- /dev/null +++ b/theories7/Logic/Decidable.v @@ -0,0 +1,58 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Decidable.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** Properties of decidable propositions *) + +Definition decidable := [P:Prop] P \/ ~P. + +Theorem dec_not_not : (P:Prop)(decidable P) -> (~P -> False) -> P. +Unfold decidable; Tauto. +Qed. + +Theorem dec_True: (decidable True). +Unfold decidable; Auto. +Qed. + +Theorem dec_False: (decidable False). +Unfold decidable not; Auto. +Qed. + +Theorem dec_or: (A,B:Prop)(decidable A) -> (decidable B) -> (decidable (A\/B)). +Unfold decidable; Tauto. +Qed. + +Theorem dec_and: (A,B:Prop)(decidable A) -> (decidable B) ->(decidable (A/\B)). +Unfold decidable; Tauto. +Qed. + +Theorem dec_not: (A:Prop)(decidable A) -> (decidable ~A). +Unfold decidable; Tauto. +Qed. + +Theorem dec_imp: (A,B:Prop)(decidable A) -> (decidable B) ->(decidable (A->B)). +Unfold decidable; Tauto. +Qed. + +Theorem not_not : (P:Prop)(decidable P) -> (~(~P)) -> P. +Unfold decidable; Tauto. Qed. + +Theorem not_or : (A,B:Prop) ~(A\/B) -> ~A /\ ~B. +Tauto. Qed. + +Theorem not_and : (A,B:Prop) (decidable A) -> ~(A/\B) -> ~A \/ ~B. +Unfold decidable; Tauto. Qed. + +Theorem not_imp : (A,B:Prop) (decidable A) -> ~(A -> B) -> A /\ ~B. +Unfold decidable;Tauto. +Qed. + +Theorem imp_simp : (A,B:Prop) (decidable A) -> (A -> B) -> ~A \/ B. +Unfold decidable; Tauto. +Qed. + diff --git a/theories7/Logic/Diaconescu.v b/theories7/Logic/Diaconescu.v new file mode 100644 index 00000000..9f5f91a0 --- /dev/null +++ b/theories7/Logic/Diaconescu.v @@ -0,0 +1,133 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Diaconescu.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(* R. Diaconescu [Diaconescu] showed that the Axiom of Choice in Set Theory + entails Excluded-Middle; S. Lacas and B. Werner [LacasWerner] + adapted the proof to show that the axiom of choice in equivalence + classes entails Excluded-Middle in Type Theory. + + This is an adaptatation of the proof by Hugo Herbelin to show that + the relational form of the Axiom of Choice + Extensionality for + predicates entails Excluded-Middle + + [Diaconescu] R. Diaconescu, Axiom of Choice and Complementation, in + Proceedings of AMS, vol 51, pp 176-178, 1975. + + [LacasWerner] S. Lacas, B Werner, Which Choices imply the excluded middle?, + preprint, 1999. + +*) + +Section PredExt_GuardRelChoice_imp_EM. + +(* The axiom of extensionality for predicates *) + +Definition PredicateExtensionality := + (P,Q:bool->Prop)((b:bool)(P b)<->(Q b))->P==Q. + +(* From predicate extensionality we get propositional extensionality + hence proof-irrelevance *) + +Require ClassicalFacts. + +Variable pred_extensionality : PredicateExtensionality. + +Lemma prop_ext : (A,B:Prop) (A<->B) -> A==B. +Proof. + Intros A B H. + Change ([_]A true)==([_]B true). + Rewrite pred_extensionality with P:=[_:bool]A Q:=[_:bool]B. + Reflexivity. + Intros _; Exact H. +Qed. + +Lemma proof_irrel : (A:Prop)(a1,a2:A) a1==a2. +Proof. + Apply (ext_prop_dep_proof_irrel_cic prop_ext). +Qed. + +(* From proof-irrelevance and relational choice, we get guarded + relational choice *) + +Require ChoiceFacts. + +Variable rel_choice : RelationalChoice. + +Lemma guarded_rel_choice : + (A:Type)(B:Type)(P:A->Prop)(R:A->B->Prop) + ((x:A)(P x)->(EX y:B|(R x y)))-> + (EXT R':A->B->Prop | + ((x:A)(P x)->(EX y:B|(R x y)/\(R' x y)/\ ((y':B)(R' x y') -> y=y')))). +Proof. + Exact + (rel_choice_and_proof_irrel_imp_guarded_rel_choice rel_choice proof_irrel). +Qed. + +(* The form of choice we need: there is a functional relation which chooses + an element in any non empty subset of bool *) + +Require Bool. + +Lemma AC : + (EXT R:(bool->Prop)->bool->Prop | + (P:bool->Prop)(EX b : bool | (P b))-> + (EX b : bool | (P b) /\ (R P b) /\ ((b':bool)(R P b')->b=b'))). +Proof. + Apply guarded_rel_choice with + P:= [Q:bool->Prop](EX y | (Q y)) R:=[Q:bool->Prop;y:bool](Q y). + Exact [_;H]H. +Qed. + +(* The proof of the excluded middle *) +(* Remark: P could have been in Set or Type *) + +Theorem pred_ext_and_rel_choice_imp_EM : (P:Prop)P\/~P. +Proof. +Intro P. + +(* first we exhibit the choice functional relation R *) +NewDestruct AC as [R H]. + +Pose class_of_true := [b]b=true\/P. +Pose class_of_false := [b]b=false\/P. + +(* the actual "decision": is (R class_of_true) = true or false? *) +NewDestruct (H class_of_true) as [b0 [H0 [H0' H0'']]]. +Exists true; Left; Reflexivity. +NewDestruct H0. + +(* the actual "decision": is (R class_of_false) = true or false? *) +NewDestruct (H class_of_false) as [b1 [H1 [H1' H1'']]]. +Exists false; Left; Reflexivity. +NewDestruct H1. + +(* case where P is false: (R class_of_true)=true /\ (R class_of_false)=false *) +Right. +Intro HP. +Assert Hequiv:(b:bool)(class_of_true b)<->(class_of_false b). +Intro b; Split. +Unfold class_of_false; Right; Assumption. +Unfold class_of_true; Right; Assumption. +Assert Heq:class_of_true==class_of_false. +Apply pred_extensionality with 1:=Hequiv. +Apply diff_true_false. +Rewrite <- H0. +Rewrite <- H1. +Rewrite <- H0''. Reflexivity. +Rewrite Heq. +Assumption. + +(* cases where P is true *) +Left; Assumption. +Left; Assumption. + +Qed. + +End PredExt_GuardRelChoice_imp_EM. diff --git a/theories7/Logic/Eqdep.v b/theories7/Logic/Eqdep.v new file mode 100755 index 00000000..fc2dfe52 --- /dev/null +++ b/theories7/Logic/Eqdep.v @@ -0,0 +1,183 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Eqdep.v,v 1.2.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** This file defines dependent equality and shows its equivalence with + equality on dependent pairs (inhabiting sigma-types). It axiomatizes + the invariance by substitution of reflexive equality proofs and + shows the equivalence between the 4 following statements + + - Invariance by Substitution of Reflexive Equality Proofs. + - Injectivity of Dependent Equality + - Uniqueness of Identity Proofs + - Uniqueness of Reflexive Identity Proofs + - Streicher's Axiom K + + These statements are independent of the calculus of constructions [2]. + + References: + + [1] T. Streicher, Semantical Investigations into Intensional Type Theory, + Habilitationsschrift, LMU München, 1993. + [2] M. Hofmann, T. Streicher, The groupoid interpretation of type theory, + Proceedings of the meeting Twenty-five years of constructive + type theory, Venice, Oxford University Press, 1998 +*) + +Section Dependent_Equality. + +Variable U : Type. +Variable P : U->Type. + +(** Dependent equality *) + +Inductive eq_dep [p:U;x:(P p)] : (q:U)(P q)->Prop := + eq_dep_intro : (eq_dep p x p x). +Hint constr_eq_dep : core v62 := Constructors eq_dep. + +Lemma eq_dep_sym : (p,q:U)(x:(P p))(y:(P q))(eq_dep p x q y)->(eq_dep q y p x). +Proof. +NewDestruct 1; Auto. +Qed. +Hints Immediate eq_dep_sym : core v62. + +Lemma eq_dep_trans : (p,q,r:U)(x:(P p))(y:(P q))(z:(P r)) + (eq_dep p x q y)->(eq_dep q y r z)->(eq_dep p x r z). +Proof. +NewDestruct 1; Auto. +Qed. + +Inductive eq_dep1 [p:U;x:(P p);q:U;y:(P q)] : Prop := + eq_dep1_intro : (h:q=p) + (x=(eq_rect U q P y p h))->(eq_dep1 p x q y). + +Scheme eq_indd := Induction for eq Sort Prop. + +Lemma eq_dep1_dep : + (p:U)(x:(P p))(q:U)(y:(P q))(eq_dep1 p x q y)->(eq_dep p x q y). +Proof. +NewDestruct 1 as [eq_qp H]. +NewDestruct eq_qp using eq_indd. +Rewrite H. +Apply eq_dep_intro. +Qed. + +Lemma eq_dep_dep1 : + (p,q:U)(x:(P p))(y:(P q))(eq_dep p x q y)->(eq_dep1 p x q y). +Proof. +NewDestruct 1. +Apply eq_dep1_intro with (refl_equal U p). +Simpl; Trivial. +Qed. + +(** Invariance by Substitution of Reflexive Equality Proofs *) + +Axiom eq_rect_eq : (p:U)(Q:U->Type)(x:(Q p))(h:p=p) + x=(eq_rect U p Q x p h). + +(** Injectivity of Dependent Equality is a consequence of *) +(** Invariance by Substitution of Reflexive Equality Proof *) + +Lemma eq_dep1_eq : (p:U)(x,y:(P p))(eq_dep1 p x p y)->x=y. +Proof. +Destruct 1; Intro. +Rewrite <- eq_rect_eq; Auto. +Qed. + +Lemma eq_dep_eq : (p:U)(x,y:(P p))(eq_dep p x p y)->x=y. +Proof. +Intros; Apply eq_dep1_eq; Apply eq_dep_dep1; Trivial. +Qed. + +End Dependent_Equality. + +(** Uniqueness of Identity Proofs (UIP) is a consequence of *) +(** Injectivity of Dependent Equality *) + +Lemma UIP : (U:Type)(x,y:U)(p1,p2:x=y)p1=p2. +Proof. +Intros; Apply eq_dep_eq with P:=[y]x=y. +Elim p2 using eq_indd. +Elim p1 using eq_indd. +Apply eq_dep_intro. +Qed. + +(** Uniqueness of Reflexive Identity Proofs is a direct instance of UIP *) + +Lemma UIP_refl : (U:Type)(x:U)(p:x=x)p=(refl_equal U x). +Proof. +Intros; Apply UIP. +Qed. + +(** Streicher axiom K is a direct consequence of Uniqueness of + Reflexive Identity Proofs *) + +Lemma Streicher_K : (U:Type)(x:U)(P:x=x->Prop) + (P (refl_equal ? x))->(p:x=x)(P p). +Proof. +Intros; Rewrite UIP_refl; Assumption. +Qed. + +(** We finally recover eq_rec_eq (alternatively eq_rect_eq) from K *) + +Lemma eq_rec_eq : (U:Type)(P:U->Set)(p:U)(x:(P p))(h:p=p) + x=(eq_rec U p P x p h). +Proof. +Intros. +Apply Streicher_K with p:=h. +Reflexivity. +Qed. + +(** Dependent equality is equivalent to equality on dependent pairs *) + +Lemma equiv_eqex_eqdep : (U:Set)(P:U->Set)(p,q:U)(x:(P p))(y:(P q)) + (existS U P p x)=(existS U P q y) <-> (eq_dep U P p x q y). +Proof. +Split. +(* -> *) +Intro H. +Change p with (projS1 U P (existS U P p x)). +Change 2 x with (projS2 U P (existS U P p x)). +Rewrite H. +Apply eq_dep_intro. +(* <- *) +NewDestruct 1; Reflexivity. +Qed. + +(** UIP implies the injectivity of equality on dependent pairs *) + +Lemma inj_pair2: (U:Set)(P:U->Set)(p:U)(x,y:(P p)) + (existS U P p x)=(existS U P p y)-> x=y. +Proof. +Intros. +Apply (eq_dep_eq U P). +Generalize (equiv_eqex_eqdep U P p p x y) . +Induction 1. +Intros. +Auto. +Qed. + +(** UIP implies the injectivity of equality on dependent pairs *) + +Lemma inj_pairT2: (U:Type)(P:U->Type)(p:U)(x,y:(P p)) + (existT U P p x)=(existT U P p y)-> x=y. +Proof. +Intros. +Apply (eq_dep_eq U P). +Change 1 p with (projT1 U P (existT U P p x)). +Change 2 x with (projT2 U P (existT U P p x)). +Rewrite H. +Apply eq_dep_intro. +Qed. + +(** The main results to be exported *) + +Hints Resolve eq_dep_intro eq_dep_eq : core v62. +Hints Immediate eq_dep_sym : core v62. +Hints Resolve inj_pair2 inj_pairT2 : core. diff --git a/theories7/Logic/Eqdep_dec.v b/theories7/Logic/Eqdep_dec.v new file mode 100644 index 00000000..959395e3 --- /dev/null +++ b/theories7/Logic/Eqdep_dec.v @@ -0,0 +1,149 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Eqdep_dec.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** We prove that there is only one proof of [x=x], i.e [(refl_equal ? x)]. + This holds if the equality upon the set of [x] is decidable. + A corollary of this theorem is the equality of the right projections + of two equal dependent pairs. + + Author: Thomas Kleymann |<tms@dcs.ed.ac.uk>| in Lego + adapted to Coq by B. Barras + + Credit: Proofs up to [K_dec] follows an outline by Michael Hedberg +*) + + +(** We need some dependent elimination schemes *) + +Set Implicit Arguments. + + (** Bijection between [eq] and [eqT] *) + Definition eq2eqT: (A:Set)(x,y:A)x=y->x==y := + [A,x,_,eqxy]<[y:A]x==y>Cases eqxy of refl_equal => (refl_eqT ? x) end. + + Definition eqT2eq: (A:Set)(x,y:A)x==y->x=y := + [A,x,_,eqTxy]<[y:A]x=y>Cases eqTxy of refl_eqT => (refl_equal ? x) end. + + Lemma eq_eqT_bij: (A:Set)(x,y:A)(p:x=y)p==(eqT2eq (eq2eqT p)). +Intros. +Case p; Reflexivity. +Qed. + + Lemma eqT_eq_bij: (A:Set)(x,y:A)(p:x==y)p==(eq2eqT (eqT2eq p)). +Intros. +Case p; Reflexivity. +Qed. + + +Section DecidableEqDep. + + Variable A: Type. + + Local comp [x,y,y':A]: x==y->x==y'->y==y' := + [eq1,eq2](eqT_ind ? ? [a]a==y' eq2 ? eq1). + + Remark trans_sym_eqT: (x,y:A)(u:x==y)(comp u u)==(refl_eqT ? y). +Intros. +Case u; Trivial. +Qed. + + + + Variable eq_dec: (x,y:A) x==y \/ ~x==y. + + Variable x: A. + + + Local nu [y:A]: x==y->x==y := + [u]Cases (eq_dec x y) of + (or_introl eqxy) => eqxy + | (or_intror neqxy) => (False_ind ? (neqxy u)) + end. + + Local nu_constant : (y:A)(u,v:x==y) (nu u)==(nu v). +Intros. +Unfold nu. +Case (eq_dec x y); Intros. +Reflexivity. + +Case n; Trivial. +Qed. + + + Local nu_inv [y:A]: x==y->x==y := [v](comp (nu (refl_eqT ? x)) v). + + + Remark nu_left_inv : (y:A)(u:x==y) (nu_inv (nu u))==u. +Intros. +Case u; Unfold nu_inv. +Apply trans_sym_eqT. +Qed. + + + Theorem eq_proofs_unicity: (y:A)(p1,p2:x==y) p1==p2. +Intros. +Elim nu_left_inv with u:=p1. +Elim nu_left_inv with u:=p2. +Elim nu_constant with y p1 p2. +Reflexivity. +Qed. + + Theorem K_dec: (P:x==x->Prop)(P (refl_eqT ? x)) -> (p:x==x)(P p). +Intros. +Elim eq_proofs_unicity with x (refl_eqT ? x) p. +Trivial. +Qed. + + + (** The corollary *) + + Local proj: (P:A->Prop)(ExT P)->(P x)->(P x) := + [P,exP,def]Cases exP of + (exT_intro x' prf) => + Cases (eq_dec x' x) of + (or_introl eqprf) => (eqT_ind ? x' P prf x eqprf) + | _ => def + end + end. + + + Theorem inj_right_pair: (P:A->Prop)(y,y':(P x)) + (exT_intro ? P x y)==(exT_intro ? P x y') -> y==y'. +Intros. +Cut (proj (exT_intro A P x y) y)==(proj (exT_intro A P x y') y). +Simpl. +Case (eq_dec x x). +Intro e. +Elim e using K_dec; Trivial. + +Intros. +Case n; Trivial. + +Case H. +Reflexivity. +Qed. + +End DecidableEqDep. + + (** We deduce the [K] axiom for (decidable) Set *) + Theorem K_dec_set: (A:Set)((x,y:A){x=y}+{~x=y}) + ->(x:A)(P: x=x->Prop)(P (refl_equal ? x)) + ->(p:x=x)(P p). +Intros. +Rewrite eq_eqT_bij. +Elim (eq2eqT p) using K_dec. +Intros. +Case (H x0 y); Intros. +Elim e; Left ; Reflexivity. + +Right ; Red; Intro neq; Apply n; Elim neq; Reflexivity. + +Trivial. +Qed. diff --git a/theories7/Logic/Hurkens.v b/theories7/Logic/Hurkens.v new file mode 100644 index 00000000..066e51aa --- /dev/null +++ b/theories7/Logic/Hurkens.v @@ -0,0 +1,79 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(* Hurkens.v *) +(************************************************************************) + +(** This is Hurkens paradox [Hurkens] in system U-, adapted by Herman + Geuvers [Geuvers] to show the inconsistency in the pure calculus of + constructions of a retract from Prop into a small type. + + References: + + - [Hurkens] A. J. Hurkens, "A simplification of Girard's paradox", + Proceedings of the 2nd international conference Typed Lambda-Calculi + and Applications (TLCA'95), 1995. + + - [Geuvers] "Inconsistency of Classical Logic in Type Theory", 2001 + (see www.cs.kun.nl/~herman/note.ps.gz). +*) + +Section Paradox. + +Variable bool : Prop. +Variable p2b : Prop -> bool. +Variable b2p : bool -> Prop. +Hypothesis p2p1 : (A:Prop)(b2p (p2b A))->A. +Hypothesis p2p2 : (A:Prop)A->(b2p (p2b A)). +Variable B:Prop. + +Definition V := (A:Prop)((A->bool)->(A->bool))->(A->bool). +Definition U := V->bool. +Definition sb : V -> V := [z][A;r;a](r (z A r) a). +Definition le : (U->bool)->(U->bool) := [i][x](x [A;r;a](i [v](sb v A r a))). +Definition induct : (U->bool)->Prop := [i](x:U)(b2p (le i x))->(b2p (i x)). +Definition WF : U := [z](p2b (induct (z U le))). +Definition I : U->Prop := + [x]((i:U->bool)(b2p (le i x))->(b2p (i [v](sb v U le x))))->B. + +Lemma Omega : (i:U->bool)(induct i)->(b2p (i WF)). +Proof. +Intros i y. +Apply y. +Unfold le WF induct. +Apply p2p2. +Intros x H0. +Apply y. +Exact H0. +Qed. + +Lemma lemma1 : (induct [u](p2b (I u))). +Proof. +Unfold induct. +Intros x p. +Apply (p2p2 (I x)). +Intro q. +Apply (p2p1 (I [v:V](sb v U le x)) (q [u](p2b (I u)) p)). +Intro i. +Apply q with i:=[y:?](i [v:V](sb v U le y)). +Qed. + +Lemma lemma2 : ((i:U->bool)(induct i)->(b2p (i WF)))->B. +Proof. +Intro x. +Apply (p2p1 (I WF) (x [u](p2b (I u)) lemma1)). +Intros i H0. +Apply (x [y](i [v](sb v U le y))). +Apply (p2p1 ? H0). +Qed. + +Theorem paradox : B. +Proof. +Exact (lemma2 Omega). +Qed. + +End Paradox. diff --git a/theories7/Logic/JMeq.v b/theories7/Logic/JMeq.v new file mode 100644 index 00000000..38dfa5e6 --- /dev/null +++ b/theories7/Logic/JMeq.v @@ -0,0 +1,64 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: JMeq.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(** John Major's Equality as proposed by C. Mc Bride *) + +Set Implicit Arguments. + +Inductive JMeq [A:Set;x:A] : (B:Set)B->Prop := + JMeq_refl : (JMeq x x). +Reset JMeq_ind. + +Hints Resolve JMeq_refl. + +Lemma sym_JMeq : (A,B:Set)(x:A)(y:B)(JMeq x y)->(JMeq y x). +NewDestruct 1; Trivial. +Qed. + +Hints Immediate sym_JMeq. + +Lemma trans_JMeq : (A,B,C:Set)(x:A)(y:B)(z:C) + (JMeq x y)->(JMeq y z)->(JMeq x z). +NewDestruct 1; Trivial. +Qed. + +Axiom JMeq_eq : (A:Set)(x,y:A)(JMeq x y)->(x=y). + +Lemma JMeq_ind : (A:Set)(x,y:A)(P:A->Prop)(P x)->(JMeq x y)->(P y). +Intros A x y P H H'; Case JMeq_eq with 1:=H'; Trivial. +Qed. + +Lemma JMeq_rec : (A:Set)(x,y:A)(P:A->Set)(P x)->(JMeq x y)->(P y). +Intros A x y P H H'; Case JMeq_eq with 1:=H'; Trivial. +Qed. + +Lemma JMeq_ind_r : (A:Set)(x,y:A)(P:A->Prop)(P y)->(JMeq x y)->(P x). +Intros A x y P H H'; Case JMeq_eq with 1:=(sym_JMeq H'); Trivial. +Qed. + +Lemma JMeq_rec_r : (A:Set)(x,y:A)(P:A->Set)(P y)->(JMeq x y)->(P x). +Intros A x y P H H'; Case JMeq_eq with 1:=(sym_JMeq H'); Trivial. +Qed. + +(** [JMeq] is equivalent to [(eq_dep Set [X]X)] *) + +Require Eqdep. + +Lemma JMeq_eq_dep : (A,B:Set)(x:A)(y:B)(JMeq x y)->(eq_dep Set [X]X A x B y). +Proof. +NewDestruct 1. +Apply eq_dep_intro. +Qed. + +Lemma eq_dep_JMeq : (A,B:Set)(x:A)(y:B)(eq_dep Set [X]X A x B y)->(JMeq x y). +Proof. +NewDestruct 1. +Apply JMeq_refl. +Qed. diff --git a/theories7/Logic/ProofIrrelevance.v b/theories7/Logic/ProofIrrelevance.v new file mode 100644 index 00000000..3f031ff7 --- /dev/null +++ b/theories7/Logic/ProofIrrelevance.v @@ -0,0 +1,113 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(** This is a proof in the pure Calculus of Construction that + classical logic in Prop + dependent elimination of disjunction entails + proof-irrelevance. + + Since, dependent elimination is derivable in the Calculus of + Inductive Constructions (CCI), we get proof-irrelevance from classical + logic in the CCI. + + Reference: + + - [Coquand] T. Coquand, "Metamathematical Investigations of a + Calculus of Constructions", Proceedings of Logic in Computer Science + (LICS'90), 1990. + + Proof skeleton: classical logic + dependent elimination of + disjunction + discrimination of proofs implies the existence of a + retract from Prop into bool, hence inconsistency by encoding any + paradox of system U- (e.g. Hurkens' paradox). +*) + +Require Hurkens. + +Section Proof_irrelevance_CC. + +Variable or : Prop -> Prop -> Prop. +Variable or_introl : (A,B:Prop)A->(or A B). +Variable or_intror : (A,B:Prop)B->(or A B). +Hypothesis or_elim : (A,B:Prop)(C:Prop)(A->C)->(B->C)->(or A B)->C. +Hypothesis or_elim_redl : + (A,B:Prop)(C:Prop)(f:A->C)(g:B->C)(a:A) + (f a)==(or_elim A B C f g (or_introl A B a)). +Hypothesis or_elim_redr : + (A,B:Prop)(C:Prop)(f:A->C)(g:B->C)(b:B) + (g b)==(or_elim A B C f g (or_intror A B b)). +Hypothesis or_dep_elim : + (A,B:Prop)(P:(or A B)->Prop) + ((a:A)(P (or_introl A B a))) -> + ((b:B)(P (or_intror A B b))) -> (b:(or A B))(P b). + +Hypothesis em : (A:Prop)(or A ~A). +Variable B : Prop. +Variable b1,b2 : B. + +(** [p2b] and [b2p] form a retract if [~b1==b2] *) + +Definition p2b [A] := (or_elim A ~A B [_]b1 [_]b2 (em A)). +Definition b2p [b] := b1==b. + +Lemma p2p1 : (A:Prop) A -> (b2p (p2b A)). +Proof. + Unfold p2b; Intro A; Apply or_dep_elim with b:=(em A); Unfold b2p; Intros. + Apply (or_elim_redl A ~A B [_]b1 [_]b2). + NewDestruct (b H). +Qed. +Lemma p2p2 : ~b1==b2->(A:Prop) (b2p (p2b A)) -> A. +Proof. + Intro not_eq_b1_b2. + Unfold p2b; Intro A; Apply or_dep_elim with b:=(em A); Unfold b2p; Intros. + Assumption. + NewDestruct not_eq_b1_b2. + Rewrite <- (or_elim_redr A ~A B [_]b1 [_]b2) in H. + Assumption. +Qed. + +(** Using excluded-middle a second time, we get proof-irrelevance *) + +Theorem proof_irrelevance_cc : b1==b2. +Proof. + Refine (or_elim ? ? ? ? ? (em b1==b2));Intro H. + Trivial. + Apply (paradox B p2b b2p (p2p2 H) p2p1). +Qed. + +End Proof_irrelevance_CC. + + +(** The Calculus of Inductive Constructions (CCI) enjoys dependent + elimination, hence classical logic in CCI entails proof-irrelevance. +*) + +Section Proof_irrelevance_CCI. + +Hypothesis em : (A:Prop) A \/ ~A. + +Definition or_elim_redl : + (A,B:Prop)(C:Prop)(f:A->C)(g:B->C)(a:A) + (f a)==(or_ind A B C f g (or_introl A B a)) + := [A,B,C;f;g;a](refl_eqT C (f a)). +Definition or_elim_redr : + (A,B:Prop)(C:Prop)(f:A->C)(g:B->C)(b:B) + (g b)==(or_ind A B C f g (or_intror A B b)) + := [A,B,C;f;g;b](refl_eqT C (g b)). +Scheme or_indd := Induction for or Sort Prop. + +Theorem proof_irrelevance_cci : (B:Prop)(b1,b2:B)b1==b2. +Proof + (proof_irrelevance_cc or or_introl or_intror or_ind + or_elim_redl or_elim_redr or_indd em). + +End Proof_irrelevance_CCI. + +(** Remark: in CCI, [bool] can be taken in [Set] as well in the + paradox and since [~true=false] for [true] and [false] in + [bool], we get the inconsistency of [em : (A:Prop){A}+{~A}] in CCI +*) diff --git a/theories7/Logic/RelationalChoice.v b/theories7/Logic/RelationalChoice.v new file mode 100644 index 00000000..e61f3582 --- /dev/null +++ b/theories7/Logic/RelationalChoice.v @@ -0,0 +1,17 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: RelationalChoice.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) + +(* This file axiomatizes the relational form of the axiom of choice *) + +Axiom relational_choice : + (A:Type;B:Type;R: A->B->Prop) + ((x:A)(EX y:B|(R x y))) + -> (EXT R':A->B->Prop | + ((x:A)(EX y:B|(R x y)/\(R' x y)/\ ((y':B) (R' x y') -> y=y')))). diff --git a/theories7/NArith/BinNat.v b/theories7/NArith/BinNat.v new file mode 100644 index 00000000..5e04e22e --- /dev/null +++ b/theories7/NArith/BinNat.v @@ -0,0 +1,205 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: BinNat.v,v 1.1.2.1 2004/07/16 19:31:30 herbelin Exp $ i*) + +Require BinPos. + +(**********************************************************************) +(** Binary natural numbers *) + +Inductive entier: Set := Nul : entier | Pos : positive -> entier. + +(** Declare binding key for scope positive_scope *) + +Delimits Scope N_scope with N. + +(** Automatically open scope N_scope for the constructors of N *) + +Bind Scope N_scope with entier. +Arguments Scope Pos [ N_scope ]. + +Open Local Scope N_scope. + +(** Operation x -> 2*x+1 *) + +Definition Un_suivi_de := [x] + Cases x of Nul => (Pos xH) | (Pos p) => (Pos (xI p)) end. + +(** Operation x -> 2*x *) + +Definition Zero_suivi_de := + [n] Cases n of Nul => Nul | (Pos p) => (Pos (xO p)) end. + +(** Successor *) + +Definition Nsucc := + [n] Cases n of Nul => (Pos xH) | (Pos p) => (Pos (add_un p)) end. + +(** Addition *) + +Definition Nplus := [n,m] + Cases n m of + | Nul _ => m + | _ Nul => n + | (Pos p) (Pos q) => (Pos (add p q)) + end. + +V8Infix "+" Nplus : N_scope. + +(** Multiplication *) + +Definition Nmult := [n,m] + Cases n m of + | Nul _ => Nul + | _ Nul => Nul + | (Pos p) (Pos q) => (Pos (times p q)) + end. + +V8Infix "*" Nmult : N_scope. + +(** Order *) + +Definition Ncompare := [n,m] + Cases n m of + | Nul Nul => EGAL + | Nul (Pos m') => INFERIEUR + | (Pos n') Nul => SUPERIEUR + | (Pos n') (Pos m') => (compare n' m' EGAL) + end. + +V8Infix "?=" Ncompare (at level 70, no associativity) : N_scope. + +(** Peano induction on binary natural numbers *) + +Theorem Nind : (P:(entier ->Prop)) + (P Nul) ->((n:entier)(P n) ->(P (Nsucc n))) ->(n:entier)(P n). +Proof. +NewDestruct n. + Assumption. + Apply Pind with P := [p](P (Pos p)). +Exact (H0 Nul H). +Intro p'; Exact (H0 (Pos p')). +Qed. + +(** Properties of addition *) + +Theorem Nplus_0_l : (n:entier)(Nplus Nul n)=n. +Proof. +Reflexivity. +Qed. + +Theorem Nplus_0_r : (n:entier)(Nplus n Nul)=n. +Proof. +NewDestruct n; Reflexivity. +Qed. + +Theorem Nplus_comm : (n,m:entier)(Nplus n m)=(Nplus m n). +Proof. +Intros. +NewDestruct n; NewDestruct m; Simpl; Try Reflexivity. +Rewrite add_sym; Reflexivity. +Qed. + +Theorem Nplus_assoc : + (n,m,p:entier)(Nplus n (Nplus m p))=(Nplus (Nplus n m) p). +Proof. +Intros. +NewDestruct n; Try Reflexivity. +NewDestruct m; Try Reflexivity. +NewDestruct p; Try Reflexivity. +Simpl; Rewrite add_assoc; Reflexivity. +Qed. + +Theorem Nplus_succ : (n,m:entier)(Nplus (Nsucc n) m)=(Nsucc (Nplus n m)). +Proof. +NewDestruct n; NewDestruct m. + Simpl; Reflexivity. + Unfold Nsucc Nplus; Rewrite <- ZL12bis; Reflexivity. + Simpl; Reflexivity. + Simpl; Rewrite ZL14bis; Reflexivity. +Qed. + +Theorem Nsucc_inj : (n,m:entier)(Nsucc n)=(Nsucc m)->n=m. +Proof. +NewDestruct n; NewDestruct m; Simpl; Intro H; + Reflexivity Orelse Injection H; Clear H; Intro H. + Symmetry in H; Contradiction add_un_not_un with p. + Contradiction add_un_not_un with p. + Rewrite add_un_inj with 1:=H; Reflexivity. +Qed. + +Theorem Nplus_reg_l : (n,m,p:entier)(Nplus n m)=(Nplus n p)->m=p. +Proof. +Intro n; Pattern n; Apply Nind; Clear n; Simpl. + Trivial. + Intros n IHn m p H0; Do 2 Rewrite Nplus_succ in H0. + Apply IHn; Apply Nsucc_inj; Assumption. +Qed. + +(** Properties of multiplication *) + +Theorem Nmult_1_l : (n:entier)(Nmult (Pos xH) n)=n. +Proof. +NewDestruct n; Reflexivity. +Qed. + +Theorem Nmult_1_r : (n:entier)(Nmult n (Pos xH))=n. +Proof. +NewDestruct n; Simpl; Try Reflexivity. +Rewrite times_x_1; Reflexivity. +Qed. + +Theorem Nmult_comm : (n,m:entier)(Nmult n m)=(Nmult m n). +Proof. +Intros. +NewDestruct n; NewDestruct m; Simpl; Try Reflexivity. +Rewrite times_sym; Reflexivity. +Qed. + +Theorem Nmult_assoc : + (n,m,p:entier)(Nmult n (Nmult m p))=(Nmult (Nmult n m) p). +Proof. +Intros. +NewDestruct n; Try Reflexivity. +NewDestruct m; Try Reflexivity. +NewDestruct p; Try Reflexivity. +Simpl; Rewrite times_assoc; Reflexivity. +Qed. + +Theorem Nmult_plus_distr_r : + (n,m,p:entier)(Nmult (Nplus n m) p)=(Nplus (Nmult n p) (Nmult m p)). +Proof. +Intros. +NewDestruct n; Try Reflexivity. +NewDestruct m; NewDestruct p; Try Reflexivity. +Simpl; Rewrite times_add_distr_l; Reflexivity. +Qed. + +Theorem Nmult_reg_r : (n,m,p:entier) ~p=Nul->(Nmult n p)=(Nmult m p) -> n=m. +Proof. +NewDestruct p; Intros Hp H. +Contradiction Hp; Reflexivity. +NewDestruct n; NewDestruct m; Reflexivity Orelse Try Discriminate H. +Injection H; Clear H; Intro H; Rewrite simpl_times_r with 1:=H; Reflexivity. +Qed. + +Theorem Nmult_0_l : (n:entier) (Nmult Nul n) = Nul. +Proof. +Reflexivity. +Qed. + +(** Properties of comparison *) + +Theorem Ncompare_Eq_eq : (n,m:entier) (Ncompare n m) = EGAL -> n = m. +Proof. +NewDestruct n as [|n]; NewDestruct m as [|m]; Simpl; Intro H; + Reflexivity Orelse Try Discriminate H. + Rewrite (compare_convert_EGAL n m H); Reflexivity. +Qed. + diff --git a/theories7/NArith/BinPos.v b/theories7/NArith/BinPos.v new file mode 100644 index 00000000..ae61587d --- /dev/null +++ b/theories7/NArith/BinPos.v @@ -0,0 +1,894 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: BinPos.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +(**********************************************************************) +(** Binary positive numbers *) + +(** Original development by Pierre Crégut, CNET, Lannion, France *) + +Inductive positive : Set := + xI : positive -> positive +| xO : positive -> positive +| xH : positive. + +(** Declare binding key for scope positive_scope *) + +Delimits Scope positive_scope with positive. + +(** Automatically open scope positive_scope for type positive, xO and xI *) + +Bind Scope positive_scope with positive. +Arguments Scope xO [ positive_scope ]. +Arguments Scope xI [ positive_scope ]. + +(** Successor *) + +Fixpoint add_un [x:positive]:positive := + Cases x of + (xI x') => (xO (add_un x')) + | (xO x') => (xI x') + | xH => (xO xH) + end. + +(** Addition *) + +Fixpoint add [x:positive]:positive -> positive := [y:positive] + Cases x y of + | (xI x') (xI y') => (xO (add_carry x' y')) + | (xI x') (xO y') => (xI (add x' y')) + | (xI x') xH => (xO (add_un x')) + | (xO x') (xI y') => (xI (add x' y')) + | (xO x') (xO y') => (xO (add x' y')) + | (xO x') xH => (xI x') + | xH (xI y') => (xO (add_un y')) + | xH (xO y') => (xI y') + | xH xH => (xO xH) + end +with add_carry [x:positive]:positive -> positive := [y:positive] + Cases x y of + | (xI x') (xI y') => (xI (add_carry x' y')) + | (xI x') (xO y') => (xO (add_carry x' y')) + | (xI x') xH => (xI (add_un x')) + | (xO x') (xI y') => (xO (add_carry x' y')) + | (xO x') (xO y') => (xI (add x' y')) + | (xO x') xH => (xO (add_un x')) + | xH (xI y') => (xI (add_un y')) + | xH (xO y') => (xO (add_un y')) + | xH xH => (xI xH) + end. + +V7only [Notation "x + y" := (add x y) : positive_scope.]. +V8Infix "+" add : positive_scope. + +Open Local Scope positive_scope. + +(** From binary positive numbers to Peano natural numbers *) + +Fixpoint positive_to_nat [x:positive]:nat -> nat := + [pow2:nat] + Cases x of + (xI x') => (plus pow2 (positive_to_nat x' (plus pow2 pow2))) + | (xO x') => (positive_to_nat x' (plus pow2 pow2)) + | xH => pow2 + end. + +Definition convert := [x:positive] (positive_to_nat x (S O)). + +(** From Peano natural numbers to binary positive numbers *) + +Fixpoint anti_convert [n:nat]: positive := + Cases n of + O => xH + | (S x') => (add_un (anti_convert x')) + end. + +(** Operation x -> 2*x-1 *) + +Fixpoint double_moins_un [x:positive]:positive := + Cases x of + (xI x') => (xI (xO x')) + | (xO x') => (xI (double_moins_un x')) + | xH => xH + end. + +(** Predecessor *) + +Definition sub_un := [x:positive] + Cases x of + (xI x') => (xO x') + | (xO x') => (double_moins_un x') + | xH => xH + end. + +(** An auxiliary type for subtraction *) + +Inductive positive_mask: Set := + IsNul : positive_mask + | IsPos : positive -> positive_mask + | IsNeg : positive_mask. + +(** Operation x -> 2*x+1 *) + +Definition Un_suivi_de_mask := [x:positive_mask] + Cases x of IsNul => (IsPos xH) | IsNeg => IsNeg | (IsPos p) => (IsPos (xI p)) end. + +(** Operation x -> 2*x *) + +Definition Zero_suivi_de_mask := [x:positive_mask] + Cases x of IsNul => IsNul | IsNeg => IsNeg | (IsPos p) => (IsPos (xO p)) end. + +(** Operation x -> 2*x-2 *) + +Definition double_moins_deux := + [x:positive] Cases x of + (xI x') => (IsPos (xO (xO x'))) + | (xO x') => (IsPos (xO (double_moins_un x'))) + | xH => IsNul + end. + +(** Subtraction of binary positive numbers into a positive numbers mask *) + +Fixpoint sub_pos[x,y:positive]:positive_mask := + Cases x y of + | (xI x') (xI y') => (Zero_suivi_de_mask (sub_pos x' y')) + | (xI x') (xO y') => (Un_suivi_de_mask (sub_pos x' y')) + | (xI x') xH => (IsPos (xO x')) + | (xO x') (xI y') => (Un_suivi_de_mask (sub_neg x' y')) + | (xO x') (xO y') => (Zero_suivi_de_mask (sub_pos x' y')) + | (xO x') xH => (IsPos (double_moins_un x')) + | xH xH => IsNul + | xH _ => IsNeg + end +with sub_neg [x,y:positive]:positive_mask := + Cases x y of + (xI x') (xI y') => (Un_suivi_de_mask (sub_neg x' y')) + | (xI x') (xO y') => (Zero_suivi_de_mask (sub_pos x' y')) + | (xI x') xH => (IsPos (double_moins_un x')) + | (xO x') (xI y') => (Zero_suivi_de_mask (sub_neg x' y')) + | (xO x') (xO y') => (Un_suivi_de_mask (sub_neg x' y')) + | (xO x') xH => (double_moins_deux x') + | xH _ => IsNeg + end. + +(** Subtraction of binary positive numbers x and y, returns 1 if x<=y *) + +Definition true_sub := [x,y:positive] + Cases (sub_pos x y) of (IsPos z) => z | _ => xH end. + +V8Infix "-" true_sub : positive_scope. + +(** Multiplication on binary positive numbers *) + +Fixpoint times [x:positive] : positive -> positive:= + [y:positive] + Cases x of + (xI x') => (add y (xO (times x' y))) + | (xO x') => (xO (times x' y)) + | xH => y + end. + +V8Infix "*" times : positive_scope. + +(** Division by 2 rounded below but for 1 *) + +Definition Zdiv2_pos := + [z:positive]Cases z of xH => xH + | (xO p) => p + | (xI p) => p + end. + +V8Infix "/" Zdiv2_pos : positive_scope. + +(** Comparison on binary positive numbers *) + +Fixpoint compare [x,y:positive]: relation -> relation := + [r:relation] + Cases x y of + | (xI x') (xI y') => (compare x' y' r) + | (xI x') (xO y') => (compare x' y' SUPERIEUR) + | (xI x') xH => SUPERIEUR + | (xO x') (xI y') => (compare x' y' INFERIEUR) + | (xO x') (xO y') => (compare x' y' r) + | (xO x') xH => SUPERIEUR + | xH (xI y') => INFERIEUR + | xH (xO y') => INFERIEUR + | xH xH => r + end. + +V8Infix "?=" compare (at level 70, no associativity) : positive_scope. + +(**********************************************************************) +(** Miscellaneous properties of binary positive numbers *) + +Lemma ZL11: (x:positive) (x=xH) \/ ~(x=xH). +Proof. +Intros x;Case x;Intros; (Left;Reflexivity) Orelse (Right;Discriminate). +Qed. + +(**********************************************************************) +(** Properties of successor on binary positive numbers *) + +(** Specification of [xI] in term of [Psucc] and [xO] *) + +Lemma xI_add_un_xO : (x:positive)(xI x) = (add_un (xO x)). +Proof. +Reflexivity. +Qed. + +Lemma add_un_discr : (x:positive)x<>(add_un x). +Proof. +Intro x; NewDestruct x; Discriminate. +Qed. + +(** Successor and double *) + +Lemma is_double_moins_un : (x:positive) (add_un (double_moins_un x)) = (xO x). +Proof. +Intro x; NewInduction x as [x IHx|x|]; Simpl; Try Rewrite IHx; Reflexivity. +Qed. + +Lemma double_moins_un_add_un_xI : + (x:positive)(double_moins_un (add_un x))=(xI x). +Proof. +Intro x;NewInduction x as [x IHx|x|]; Simpl; Try Rewrite IHx; Reflexivity. +Qed. + +Lemma ZL1: (y:positive)(xO (add_un y)) = (add_un (add_un (xO y))). +Proof. +Intro y; Induction y; Simpl; Auto. +Qed. + +Lemma double_moins_un_xO_discr : (x:positive)(double_moins_un x)<>(xO x). +Proof. +Intro x; NewDestruct x; Discriminate. +Qed. + +(** Successor and predecessor *) + +Lemma add_un_not_un : (x:positive) (add_un x) <> xH. +Proof. +Intro x; NewDestruct x as [x|x|]; Discriminate. +Qed. + +Lemma sub_add_one : (x:positive) (sub_un (add_un x)) = x. +Proof. +(Intro x; NewDestruct x as [p|p|]; [Idtac | Idtac | Simpl;Auto]); +(NewInduction p as [p IHp||]; [Idtac | Reflexivity | Reflexivity ]); +Simpl; Simpl in IHp; Try Rewrite <- IHp; Reflexivity. +Qed. + +Lemma add_sub_one : (x:positive) (x=xH) \/ (add_un (sub_un x)) = x. +Proof. +Intro x; Induction x; [ + Simpl; Auto +| Simpl; Intros;Right;Apply is_double_moins_un +| Auto ]. +Qed. + +(** Injectivity of successor *) + +Lemma add_un_inj : (x,y:positive) (add_un x)=(add_un y) -> x=y. +Proof. +Intro x;NewInduction x; Intro y; NewDestruct y as [y|y|]; Simpl; + Intro H; Discriminate H Orelse Try (Injection H; Clear H; Intro H). +Rewrite (IHx y H); Reflexivity. +Absurd (add_un x)=xH; [ Apply add_un_not_un | Assumption ]. +Apply f_equal with 1:=H; Assumption. +Absurd (add_un y)=xH; [ Apply add_un_not_un | Symmetry; Assumption ]. +Reflexivity. +Qed. + +(**********************************************************************) +(** Properties of addition on binary positive numbers *) + +(** Specification of [Psucc] in term of [Pplus] *) + +Lemma ZL12: (q:positive) (add_un q) = (add q xH). +Proof. +Intro q; NewDestruct q; Reflexivity. +Qed. + +Lemma ZL12bis: (q:positive) (add_un q) = (add xH q). +Proof. +Intro q; NewDestruct q; Reflexivity. +Qed. + +(** Specification of [Pplus_carry] *) + +Theorem ZL13: (x,y:positive)(add_carry x y) = (add_un (add x y)). +Proof. +(Intro x; NewInduction x as [p IHp|p IHp|];Intro y; NewDestruct y;Simpl;Auto); + Rewrite IHp; Auto. +Qed. + +(** Commutativity *) + +Theorem add_sym : (x,y:positive) (add x y) = (add y x). +Proof. +Intro x; NewInduction x as [p IHp|p IHp|];Intro y; NewDestruct y;Simpl;Auto; + Try Do 2 Rewrite ZL13; Rewrite IHp;Auto. +Qed. + +(** Permutation of [Pplus] and [Psucc] *) + +Theorem ZL14: (x,y:positive)(add x (add_un y)) = (add_un (add x y)). +Proof. +Intro x; NewInduction x as [p IHp|p IHp|];Intro y; NewDestruct y;Simpl;Auto; [ + Rewrite ZL13; Rewrite IHp; Auto +| Rewrite ZL13; Auto +| NewDestruct p;Simpl;Auto +| Rewrite IHp;Auto +| NewDestruct p;Simpl;Auto ]. +Qed. + +Theorem ZL14bis: (x,y:positive)(add (add_un x) y) = (add_un (add x y)). +Proof. +Intros x y; Rewrite add_sym; Rewrite add_sym with x:=x; Apply ZL14. +Qed. + +Theorem ZL15: (q,z:positive) ~z=xH -> (add_carry q (sub_un z)) = (add q z). +Proof. +Intros q z H; Elim (add_sub_one z); [ + Intro;Absurd z=xH;Auto +| Intros E;Pattern 2 z ;Rewrite <- E; Rewrite ZL14; Rewrite ZL13; Trivial ]. +Qed. + +(** No neutral for addition on strictly positive numbers *) + +Lemma add_no_neutral : (x,y:positive) ~(add y x)=x. +Proof. +Intro x;NewInduction x; Intro y; NewDestruct y as [y|y|]; Simpl; Intro H; + Discriminate H Orelse Injection H; Clear H; Intro H; Apply (IHx y H). +Qed. + +Lemma add_carry_not_add_un : (x,y:positive) ~(add_carry y x)=(add_un x). +Proof. +Intros x y H; Absurd (add y x)=x; + [ Apply add_no_neutral + | Apply add_un_inj; Rewrite <- ZL13; Assumption ]. +Qed. + +(** Simplification *) + +Lemma add_carry_add : + (x,y,z,t:positive) (add_carry x z)=(add_carry y t) -> (add x z)=(add y t). +Proof. +Intros x y z t H; Apply add_un_inj; Do 2 Rewrite <- ZL13; Assumption. +Qed. + +Lemma simpl_add_r : (x,y,z:positive) (add x z)=(add y z) -> x=y. +Proof. +Intros x y z; Generalize x y; Clear x y. +NewInduction z as [z|z|]. + NewDestruct x as [x|x|]; Intro y; NewDestruct y as [y|y|]; Simpl; Intro H; + Discriminate H Orelse Try (Injection H; Clear H; Intro H). + Rewrite IHz with 1:=(add_carry_add ? ? ? ? H); Reflexivity. + Absurd (add_carry x z)=(add_un z); + [ Apply add_carry_not_add_un | Assumption ]. + Rewrite IHz with 1:=H; Reflexivity. + Symmetry in H; Absurd (add_carry y z)=(add_un z); + [ Apply add_carry_not_add_un | Assumption ]. + Reflexivity. + NewDestruct x as [x|x|]; Intro y; NewDestruct y as [y|y|]; Simpl; Intro H; + Discriminate H Orelse Try (Injection H; Clear H; Intro H). + Rewrite IHz with 1:=H; Reflexivity. + Absurd (add x z)=z; [ Apply add_no_neutral | Assumption ]. + Rewrite IHz with 1:=H; Reflexivity. + Symmetry in H; Absurd y+z=z; [ Apply add_no_neutral | Assumption ]. + Reflexivity. + Intros H x y; Apply add_un_inj; Do 2 Rewrite ZL12; Assumption. +Qed. + +Lemma simpl_add_l : (x,y,z:positive) (add x y)=(add x z) -> y=z. +Proof. +Intros x y z H;Apply simpl_add_r with z:=x; + Rewrite add_sym with x:=z; Rewrite add_sym with x:=y; Assumption. +Qed. + +Lemma simpl_add_carry_r : + (x,y,z:positive) (add_carry x z)=(add_carry y z) -> x=y. +Proof. +Intros x y z H; Apply simpl_add_r with z:=z; Apply add_carry_add; Assumption. +Qed. + +Lemma simpl_add_carry_l : + (x,y,z:positive) (add_carry x y)=(add_carry x z) -> y=z. +Proof. +Intros x y z H;Apply simpl_add_r with z:=x; +Rewrite add_sym with x:=z; Rewrite add_sym with x:=y; Apply add_carry_add; +Assumption. +Qed. + +(** Addition on positive is associative *) + +Theorem add_assoc: (x,y,z:positive)(add x (add y z)) = (add (add x y) z). +Proof. +Intros x y; Generalize x; Clear x. +NewInduction y as [y|y|]; Intro x. + NewDestruct x as [x|x|]; + Intro z; NewDestruct z as [z|z|]; Simpl; Repeat Rewrite ZL13; + Repeat Rewrite ZL14; Repeat Rewrite ZL14bis; Reflexivity Orelse + Repeat Apply f_equal with A:=positive; Apply IHy. + NewDestruct x as [x|x|]; + Intro z; NewDestruct z as [z|z|]; Simpl; Repeat Rewrite ZL13; + Repeat Rewrite ZL14; Repeat Rewrite ZL14bis; Reflexivity Orelse + Repeat Apply f_equal with A:=positive; Apply IHy. + Intro z; Rewrite add_sym with x:=xH; Do 2 Rewrite <- ZL12; Rewrite ZL14bis; Rewrite ZL14; Reflexivity. +Qed. + +(** Commutation of addition with the double of a positive number *) + +Lemma add_xI_double_moins_un : + (p,q:positive)(xO (add p q)) = (add (xI p) (double_moins_un q)). +Proof. +Intros; Change (xI p) with (add (xO p) xH). +Rewrite <- add_assoc; Rewrite <- ZL12bis; Rewrite is_double_moins_un. +Reflexivity. +Qed. + +Lemma add_xO_double_moins_un : + (p,q:positive) (double_moins_un (add p q)) = (add (xO p) (double_moins_un q)). +Proof. +NewInduction p as [p IHp|p IHp|]; NewDestruct q as [q|q|]; + Simpl; Try Rewrite ZL13; Try Rewrite double_moins_un_add_un_xI; + Try Rewrite IHp; Try Rewrite add_xI_double_moins_un; Try Reflexivity. + Rewrite <- is_double_moins_un; Rewrite ZL12bis; Reflexivity. +Qed. + +(** Misc *) + +Lemma add_x_x : (x:positive) (add x x) = (xO x). +Proof. +Intro x;NewInduction x; Simpl; Try Rewrite ZL13; Try Rewrite IHx; Reflexivity. +Qed. + +(**********************************************************************) +(** Peano induction on binary positive positive numbers *) + +Fixpoint plus_iter [x:positive] : positive -> positive := + [y]Cases x of + | xH => (add_un y) + | (xO x) => (plus_iter x (plus_iter x y)) + | (xI x) => (plus_iter x (plus_iter x (add_un y))) + end. + +Lemma plus_iter_add : (x,y:positive)(plus_iter x y)=(add x y). +Proof. +Intro x;NewInduction x as [p IHp|p IHp|]; Intro y; NewDestruct y; Simpl; + Reflexivity Orelse Do 2 Rewrite IHp; Rewrite add_assoc; Rewrite add_x_x; + Try Reflexivity. +Rewrite ZL13; Rewrite <- ZL14; Reflexivity. +Rewrite ZL12; Reflexivity. +Qed. + +Lemma plus_iter_xO : (x:positive)(plus_iter x x)=(xO x). +Proof. +Intro; Rewrite <- add_x_x; Apply plus_iter_add. +Qed. + +Lemma plus_iter_xI : (x:positive)(add_un (plus_iter x x))=(xI x). +Proof. +Intro; Rewrite xI_add_un_xO; Rewrite <- add_x_x; + Apply (f_equal positive); Apply plus_iter_add. +Qed. + +Lemma iterate_add : (P:(positive->Type)) + ((n:positive)(P n) ->(P (add_un n)))->(p,n:positive)(P n) -> + (P (plus_iter p n)). +Proof. +Intros P H; NewInduction p; Simpl; Intros. +Apply IHp; Apply IHp; Apply H; Assumption. +Apply IHp; Apply IHp; Assumption. +Apply H; Assumption. +Defined. + +(** Peano induction *) + +Theorem Pind : (P:(positive->Prop)) + (P xH) ->((n:positive)(P n) ->(P (add_un n))) ->(n:positive)(P n). +Proof. +Intros P H1 Hsucc n; NewInduction n. +Rewrite <- plus_iter_xI; Apply Hsucc; Apply iterate_add; Assumption. +Rewrite <- plus_iter_xO; Apply iterate_add; Assumption. +Assumption. +Qed. + +(** Peano recursion *) + +Definition Prec : (A:Set)A->(positive->A->A)->positive->A := + [A;a;f]Fix Prec { Prec [p:positive] : A := + Cases p of + | xH => a + | (xO p) => (iterate_add [_]A f p p (Prec p)) + | (xI p) => (f (plus_iter p p) (iterate_add [_]A f p p (Prec p))) + end}. + +(** Peano case analysis *) + +Theorem Pcase : (P:(positive->Prop)) + (P xH) ->((n:positive)(P (add_un n))) ->(n:positive)(P n). +Proof. +Intros; Apply Pind; Auto. +Qed. + +Check + let fact = (Prec positive xH [p;r](times (add_un p) r)) in + let seven = (xI (xI xH)) in + let five_thousand_forty= (xO(xO(xO(xO(xI(xI(xO(xI(xI(xI(xO(xO xH)))))))))))) + in ((refl_equal ? ?) :: (fact seven) = five_thousand_forty). + +(**********************************************************************) +(** Properties of multiplication on binary positive numbers *) + +(** One is right neutral for multiplication *) + +Lemma times_x_1 : (x:positive) (times x xH) = x. +Proof. +Intro x;NewInduction x; Simpl. + Rewrite IHx; Reflexivity. + Rewrite IHx; Reflexivity. + Reflexivity. +Qed. + +(** Right reduction properties for multiplication *) + +Lemma times_x_double : (x,y:positive) (times x (xO y)) = (xO (times x y)). +Proof. +Intros x y; NewInduction x; Simpl. + Rewrite IHx; Reflexivity. + Rewrite IHx; Reflexivity. + Reflexivity. +Qed. + +Lemma times_x_double_plus_one : + (x,y:positive) (times x (xI y)) = (add x (xO (times x y))). +Proof. +Intros x y; NewInduction x; Simpl. + Rewrite IHx; Do 2 Rewrite add_assoc; Rewrite add_sym with x:=y; Reflexivity. + Rewrite IHx; Reflexivity. + Reflexivity. +Qed. + +(** Commutativity of multiplication *) + +Theorem times_sym : (x,y:positive) (times x y) = (times y x). +Proof. +Intros x y; NewInduction y; Simpl. + Rewrite <- IHy; Apply times_x_double_plus_one. + Rewrite <- IHy; Apply times_x_double. + Apply times_x_1. +Qed. + +(** Distributivity of multiplication over addition *) + +Theorem times_add_distr: + (x,y,z:positive) (times x (add y z)) = (add (times x y) (times x z)). +Proof. +Intros x y z; NewInduction x; Simpl. + Rewrite IHx; Rewrite <- add_assoc with y := (xO (times x y)); + Rewrite -> add_assoc with x := (xO (times x y)); + Rewrite -> add_sym with x := (xO (times x y)); + Rewrite <- add_assoc with y := (xO (times x y)); + Rewrite -> add_assoc with y := z; Reflexivity. + Rewrite IHx; Reflexivity. + Reflexivity. +Qed. + +Theorem times_add_distr_l: + (x,y,z:positive) (times (add x y) z) = (add (times x z) (times y z)). +Proof. +Intros x y z; Do 3 Rewrite times_sym with y:=z; Apply times_add_distr. +Qed. + +(** Associativity of multiplication *) + +Theorem times_assoc : + ((x,y,z:positive) (times x (times y z))= (times (times x y) z)). +Proof. +Intro x;NewInduction x as [x|x|]; Simpl; Intros y z. + Rewrite IHx; Rewrite times_add_distr_l; Reflexivity. + Rewrite IHx; Reflexivity. + Reflexivity. +Qed. + +(** Parity properties of multiplication *) + +Lemma times_discr_xO_xI : + (x,y,z:positive)(times (xI x) z)<>(times (xO y) z). +Proof. +Intros x y z; NewInduction z as [|z IHz|]; Try Discriminate. +Intro H; Apply IHz; Clear IHz. +Do 2 Rewrite times_x_double in H. +Injection H; Clear H; Intro H; Exact H. +Qed. + +Lemma times_discr_xO : (x,y:positive)(times (xO x) y)<>y. +Proof. +Intros x y; NewInduction y; Try Discriminate. +Rewrite times_x_double; Injection; Assumption. +Qed. + +(** Simplification properties of multiplication *) + +Theorem simpl_times_r : (x,y,z:positive) (times x z)=(times y z) -> x=y. +Proof. +Intro x;NewInduction x as [p IHp|p IHp|]; Intro y; NewDestruct y as [q|q|]; Intros z H; + Reflexivity Orelse Apply (f_equal positive) Orelse Apply False_ind. + Simpl in H; Apply IHp with (xO z); Simpl; Do 2 Rewrite times_x_double; + Apply simpl_add_l with 1 := H. + Apply times_discr_xO_xI with 1 := H. + Simpl in H; Rewrite add_sym in H; Apply add_no_neutral with 1 := H. + Symmetry in H; Apply times_discr_xO_xI with 1 := H. + Apply IHp with (xO z); Simpl; Do 2 Rewrite times_x_double; Assumption. + Apply times_discr_xO with 1:=H. + Simpl in H; Symmetry in H; Rewrite add_sym in H; + Apply add_no_neutral with 1 := H. + Symmetry in H; Apply times_discr_xO with 1:=H. +Qed. + +Theorem simpl_times_l : (x,y,z:positive) (times z x)=(times z y) -> x=y. +Proof. +Intros x y z H; Apply simpl_times_r with z:=z. +Rewrite times_sym with x:=x; Rewrite times_sym with x:=y; Assumption. +Qed. + +(** Inversion of multiplication *) + +Lemma times_one_inversion_l : (x,y:positive) (times x y)=xH -> x=xH. +Proof. +Intros x y; NewDestruct x; Simpl. + NewDestruct y; Intro; Discriminate. + Intro; Discriminate. + Reflexivity. +Qed. + +(**********************************************************************) +(** Properties of comparison on binary positive numbers *) + +Theorem compare_convert1 : + (x,y:positive) + ~(compare x y SUPERIEUR) = EGAL /\ ~(compare x y INFERIEUR) = EGAL. +Proof. +Intro x; NewInduction x as [p IHp|p IHp|]; Intro y; NewDestruct y as [q|q|]; + Split;Simpl;Auto; + Discriminate Orelse (Elim (IHp q); Auto). +Qed. + +Theorem compare_convert_EGAL : (x,y:positive) (compare x y EGAL) = EGAL -> x=y. +Proof. +Intro x; NewInduction x as [p IHp|p IHp|]; + Intro y; NewDestruct y as [q|q|];Simpl;Auto; Intro H; [ + Rewrite (IHp q); Trivial +| Absurd (compare p q SUPERIEUR)=EGAL ; + [ Elim (compare_convert1 p q);Auto | Assumption ] +| Discriminate H +| Absurd (compare p q INFERIEUR) = EGAL; + [ Elim (compare_convert1 p q);Auto | Assumption ] +| Rewrite (IHp q);Auto +| Discriminate H +| Discriminate H +| Discriminate H ]. +Qed. + +Lemma ZLSI: + (x,y:positive) (compare x y SUPERIEUR) = INFERIEUR -> + (compare x y EGAL) = INFERIEUR. +Proof. +Intro x; Induction x;Intro y; Induction y;Simpl;Auto; + Discriminate Orelse Intros H;Discriminate H. +Qed. + +Lemma ZLIS: + (x,y:positive) (compare x y INFERIEUR) = SUPERIEUR -> + (compare x y EGAL) = SUPERIEUR. +Proof. +Intro x; Induction x;Intro y; Induction y;Simpl;Auto; + Discriminate Orelse Intros H;Discriminate H. +Qed. + +Lemma ZLII: + (x,y:positive) (compare x y INFERIEUR) = INFERIEUR -> + (compare x y EGAL) = INFERIEUR \/ x = y. +Proof. +(Intro x; NewInduction x as [p IHp|p IHp|]; + Intro y; NewDestruct y as [q|q|];Simpl;Auto;Try Discriminate); + Intro H2; Elim (IHp q H2);Auto; Intros E;Rewrite E; + Auto. +Qed. + +Lemma ZLSS: + (x,y:positive) (compare x y SUPERIEUR) = SUPERIEUR -> + (compare x y EGAL) = SUPERIEUR \/ x = y. +Proof. +(Intro x; NewInduction x as [p IHp|p IHp|]; + Intro y; NewDestruct y as [q|q|];Simpl;Auto;Try Discriminate); + Intro H2; Elim (IHp q H2);Auto; Intros E;Rewrite E; + Auto. +Qed. + +Lemma Dcompare : (r:relation) r=EGAL \/ r = INFERIEUR \/ r = SUPERIEUR. +Proof. +Induction r; Auto. +Qed. + +Tactic Definition ElimPcompare c1 c2:= + Elim (Dcompare (compare c1 c2 EGAL)); [ Idtac | + Let x = FreshId "H" In Intro x; Case x; Clear x ]. + +Theorem convert_compare_EGAL: (x:positive)(compare x x EGAL)=EGAL. +Intro x; Induction x; Auto. +Qed. + +Lemma Pcompare_antisym : + (x,y:positive)(r:relation) (Op (compare x y r)) = (compare y x (Op r)). +Proof. +Intro x; NewInduction x as [p IHp|p IHp|]; Intro y; NewDestruct y; +Intro r; Reflexivity Orelse (Symmetry; Assumption) Orelse Discriminate H +Orelse Simpl; Apply IHp Orelse Try Rewrite IHp; Try Reflexivity. +Qed. + +Lemma ZC1: + (x,y:positive)(compare x y EGAL)=SUPERIEUR -> (compare y x EGAL)=INFERIEUR. +Proof. +Intros; Change EGAL with (Op EGAL). +Rewrite <- Pcompare_antisym; Rewrite H; Reflexivity. +Qed. + +Lemma ZC2: + (x,y:positive)(compare x y EGAL)=INFERIEUR -> (compare y x EGAL)=SUPERIEUR. +Proof. +Intros; Change EGAL with (Op EGAL). +Rewrite <- Pcompare_antisym; Rewrite H; Reflexivity. +Qed. + +Lemma ZC3: (x,y:positive)(compare x y EGAL)=EGAL -> (compare y x EGAL)=EGAL. +Proof. +Intros; Change EGAL with (Op EGAL). +Rewrite <- Pcompare_antisym; Rewrite H; Reflexivity. +Qed. + +Lemma ZC4: (x,y:positive) (compare x y EGAL) = (Op (compare y x EGAL)). +Proof. +Intros; Change 1 EGAL with (Op EGAL). +Symmetry; Apply Pcompare_antisym. +Qed. + +(**********************************************************************) +(** Properties of subtraction on binary positive numbers *) + +Lemma ZS: (p:positive_mask) (Zero_suivi_de_mask p) = IsNul -> p = IsNul. +Proof. +NewDestruct p; Simpl; [ Trivial | Discriminate 1 | Discriminate 1 ]. +Qed. + +Lemma US: (p:positive_mask) ~(Un_suivi_de_mask p)=IsNul. +Proof. +Induction p; Intros; Discriminate. +Qed. + +Lemma USH: (p:positive_mask) (Un_suivi_de_mask p) = (IsPos xH) -> p = IsNul. +Proof. +NewDestruct p; Simpl; [ Trivial | Discriminate 1 | Discriminate 1 ]. +Qed. + +Lemma ZSH: (p:positive_mask) ~(Zero_suivi_de_mask p)= (IsPos xH). +Proof. +Induction p; Intros; Discriminate. +Qed. + +Theorem sub_pos_x_x : (x:positive) (sub_pos x x) = IsNul. +Proof. +Intro x; NewInduction x as [p IHp|p IHp|]; [ + Simpl; Rewrite IHp;Simpl; Trivial +| Simpl; Rewrite IHp;Auto +| Auto ]. +Qed. + +Lemma ZL10: (x,y:positive) + (sub_pos x y) = (IsPos xH) -> (sub_neg x y) = IsNul. +Proof. +Intro x; NewInduction x as [p|p|]; Intro y; NewDestruct y as [q|q|]; Simpl; + Intro H; Try Discriminate H; [ + Absurd (Zero_suivi_de_mask (sub_pos p q))=(IsPos xH); + [ Apply ZSH | Assumption ] +| Assert Heq : (sub_pos p q)=IsNul; + [ Apply USH;Assumption | Rewrite Heq; Reflexivity ] +| Assert Heq : (sub_neg p q)=IsNul; + [ Apply USH;Assumption | Rewrite Heq; Reflexivity ] +| Absurd (Zero_suivi_de_mask (sub_pos p q))=(IsPos xH); + [ Apply ZSH | Assumption ] +| NewDestruct p; Simpl; [ Discriminate H | Discriminate H | Reflexivity ] ]. +Qed. + +(** Properties of subtraction valid only for x>y *) + +Lemma sub_pos_SUPERIEUR: + (x,y:positive)(compare x y EGAL)=SUPERIEUR -> + (EX h:positive | (sub_pos x y) = (IsPos h) /\ (add y h) = x /\ + (h = xH \/ (sub_neg x y) = (IsPos (sub_un h)))). +Proof. +Intro x;NewInduction x as [p|p|];Intro y; NewDestruct y as [q|q|]; Simpl; Intro H; + Try Discriminate H. + NewDestruct (IHp q H) as [z [H4 [H6 H7]]]; Exists (xO z); Split. + Rewrite H4; Reflexivity. + Split. + Simpl; Rewrite H6; Reflexivity. + Right; Clear H6; NewDestruct (ZL11 z) as [H8|H8]; [ + Rewrite H8; Rewrite H8 in H4; + Rewrite ZL10; [ Reflexivity | Assumption ] + | Clear H4; NewDestruct H7 as [H9|H9]; [ + Absurd z=xH; Assumption + | Rewrite H9; Clear H9; NewDestruct z; + [ Reflexivity | Reflexivity | Absurd xH=xH; Trivial ]]]. + Case ZLSS with 1:=H; [ + Intros H3;Elim (IHp q H3); Intros z H4; Exists (xI z); + Elim H4;Intros H5 H6;Elim H6;Intros H7 H8; Split; [ + Simpl;Rewrite H5;Auto + | Split; [ + Simpl; Rewrite H7; Trivial + | Right; + Change (Zero_suivi_de_mask (sub_pos p q))=(IsPos (sub_un (xI z))); + Rewrite H5; Auto ]] + | Intros H3; Exists xH; Rewrite H3; Split; [ + Simpl; Rewrite sub_pos_x_x; Auto + | Split; Auto ]]. + Exists (xO p); Auto. + NewDestruct (IHp q) as [z [H4 [H6 H7]]]. + Apply ZLIS; Assumption. + NewDestruct (ZL11 z) as [vZ|]; [ + Exists xH; Split; [ + Rewrite ZL10; [ Reflexivity | Rewrite vZ in H4;Assumption ] + | Split; [ + Simpl; Rewrite ZL12; Rewrite <- vZ; Rewrite H6; Trivial + | Auto ]] + | Exists (xI (sub_un z)); NewDestruct H7 as [|H8];[ + Absurd z=xH;Assumption + | Split; [ + Rewrite H8; Trivial + | Split; [ Simpl; Rewrite ZL15; [ + Rewrite H6;Trivial + | Assumption ] + | Right; Rewrite H8; Reflexivity]]]]. + NewDestruct (IHp q H) as [z [H4 [H6 H7]]]. + Exists (xO z); Split; [ + Rewrite H4;Auto + | Split; [ + Simpl;Rewrite H6;Reflexivity + | Right; + Change (Un_suivi_de_mask (sub_neg p q))=(IsPos (double_moins_un z)); + NewDestruct (ZL11 z) as [H8|H8]; [ + Rewrite H8; Simpl; + Assert H9:(sub_neg p q)=IsNul;[ + Apply ZL10;Rewrite <- H8;Assumption + | Rewrite H9;Reflexivity ] + | NewDestruct H7 as [H9|H9]; [ + Absurd z=xH;Auto + | Rewrite H9; NewDestruct z; Simpl; + [ Reflexivity + | Reflexivity + | Absurd xH=xH; [Assumption | Reflexivity]]]]]]. + Exists (double_moins_un p); Split; [ + Reflexivity + | Clear IHp; Split; [ + NewDestruct p; Simpl; [ + Reflexivity + | Rewrite is_double_moins_un; Reflexivity + | Reflexivity ] + | NewDestruct p; [Right|Right|Left]; Reflexivity ]]. +Qed. + +Theorem sub_add: +(x,y:positive) (compare x y EGAL) = SUPERIEUR -> (add y (true_sub x y)) = x. +Proof. +Intros x y H;Elim sub_pos_SUPERIEUR with 1:=H; +Intros z H1;Elim H1;Intros H2 H3; Elim H3;Intros H4 H5; +Unfold true_sub ;Rewrite H2; Exact H4. +Qed. + diff --git a/theories7/NArith/NArith.v b/theories7/NArith/NArith.v new file mode 100644 index 00000000..d924ae2e --- /dev/null +++ b/theories7/NArith/NArith.v @@ -0,0 +1,14 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(* $Id: NArith.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ *) + +(** Library for binary natural numbers *) + +Require Export BinPos. +Require Export BinNat. diff --git a/theories7/NArith/Pnat.v b/theories7/NArith/Pnat.v new file mode 100644 index 00000000..d62661ed --- /dev/null +++ b/theories7/NArith/Pnat.v @@ -0,0 +1,472 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Pnat.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require BinPos. + +(**********************************************************************) +(** Properties of the injection from binary positive numbers to Peano + natural numbers *) + +(** Original development by Pierre Crégut, CNET, Lannion, France *) + +Require Le. +Require Lt. +Require Gt. +Require Plus. +Require Mult. +Require Minus. + +(** [nat_of_P] is a morphism for addition *) + +Lemma convert_add_un : + (x:positive)(m:nat) + (positive_to_nat (add_un x) m) = (plus m (positive_to_nat x m)). +Proof. +Intro x; NewInduction x as [p IHp|p IHp|]; Simpl; Auto; Intro m; Rewrite IHp; +Rewrite plus_assoc_l; Trivial. +Qed. + +Lemma cvt_add_un : + (p:positive) (convert (add_un p)) = (S (convert p)). +Proof. + Intro; Change (S (convert p)) with (plus (S O) (convert p)); + Unfold convert; Apply convert_add_un. +Qed. + +Theorem convert_add_carry : + (x,y:positive)(m:nat) + (positive_to_nat (add_carry x y) m) = + (plus m (positive_to_nat (add x y) m)). +Proof. +Intro x; NewInduction x as [p IHp|p IHp|]; + Intro y; NewDestruct y; Simpl; Auto with arith; Intro m; [ + Rewrite IHp; Rewrite plus_assoc_l; Trivial with arith +| Rewrite IHp; Rewrite plus_assoc_l; Trivial with arith +| Rewrite convert_add_un; Rewrite plus_assoc_l; Trivial with arith +| Rewrite convert_add_un; Apply plus_assoc_r ]. +Qed. + +Theorem cvt_carry : + (x,y:positive)(convert (add_carry x y)) = (S (convert (add x y))). +Proof. +Intros;Unfold convert; Rewrite convert_add_carry; Simpl; Trivial with arith. +Qed. + +Theorem add_verif : + (x,y:positive)(m:nat) + (positive_to_nat (add x y) m) = + (plus (positive_to_nat x m) (positive_to_nat y m)). +Proof. +Intro x; NewInduction x as [p IHp|p IHp|]; + Intro y; NewDestruct y;Simpl;Auto with arith; [ + Intros m;Rewrite convert_add_carry; Rewrite IHp; + Rewrite plus_assoc_r; Rewrite plus_assoc_r; + Rewrite (plus_permute m (positive_to_nat p (plus m m))); Trivial with arith +| Intros m; Rewrite IHp; Apply plus_assoc_l +| Intros m; Rewrite convert_add_un; + Rewrite (plus_sym (plus m (positive_to_nat p (plus m m)))); + Apply plus_assoc_r +| Intros m; Rewrite IHp; Apply plus_permute +| Intros m; Rewrite convert_add_un; Apply plus_assoc_r ]. +Qed. + +Theorem convert_add: + (x,y:positive) (convert (add x y)) = (plus (convert x) (convert y)). +Proof. +Intros x y; Exact (add_verif x y (S O)). +Qed. + +(** [Pmult_nat] is a morphism for addition *) + +Lemma ZL2: + (y:positive)(m:nat) + (positive_to_nat y (plus m m)) = + (plus (positive_to_nat y m) (positive_to_nat y m)). +Proof. +Intro y; NewInduction y as [p H|p H|]; Intro m; [ + Simpl; Rewrite H; Rewrite plus_assoc_r; + Rewrite (plus_permute m (positive_to_nat p (plus m m))); + Rewrite plus_assoc_r; Auto with arith +| Simpl; Rewrite H; Auto with arith +| Simpl; Trivial with arith ]. +Qed. + +Lemma ZL6: + (p:positive) (positive_to_nat p (S (S O))) = (plus (convert p) (convert p)). +Proof. +Intro p;Change (2) with (plus (S O) (S O)); Rewrite ZL2; Trivial. +Qed. + +(** [nat_of_P] is a morphism for multiplication *) + +Theorem times_convert : + (x,y:positive) (convert (times x y)) = (mult (convert x) (convert y)). +Proof. +Intros x y; NewInduction x as [ x' H | x' H | ]; [ + Change (times (xI x') y) with (add y (xO (times x' y))); Rewrite convert_add; + Unfold 2 3 convert; Simpl; Do 2 Rewrite ZL6; Rewrite H; + Rewrite -> mult_plus_distr; Reflexivity +| Unfold 1 2 convert; Simpl; Do 2 Rewrite ZL6; + Rewrite H; Rewrite mult_plus_distr; Reflexivity +| Simpl; Rewrite <- plus_n_O; Reflexivity ]. +Qed. +V7only [ + Comments "Compatibility with the old version of times and times_convert". + Syntactic Definition times1 := + [x:positive;_:positive->positive;y:positive](times x y). + Syntactic Definition times1_convert := + [x,y:positive;_:positive->positive](times_convert x y). +]. + +(** [nat_of_P] maps to the strictly positive subset of [nat] *) + +Lemma ZL4: (y:positive) (EX h:nat |(convert y)=(S h)). +Proof. +Intro y; NewInduction y as [p H|p H|]; [ + NewDestruct H as [x H1]; Exists (plus (S x) (S x)); + Unfold convert ;Simpl; Change (2) with (plus (1) (1)); Rewrite ZL2; Unfold convert in H1; + Rewrite H1; Auto with arith +| NewDestruct H as [x H2]; Exists (plus x (S x)); Unfold convert; + Simpl; Change (2) with (plus (1) (1)); Rewrite ZL2;Unfold convert in H2; Rewrite H2; Auto with arith +| Exists O ;Auto with arith ]. +Qed. + +(** Extra lemmas on [lt] on Peano natural numbers *) + +Lemma ZL7: + (m,n:nat) (lt m n) -> (lt (plus m m) (plus n n)). +Proof. +Intros m n H; Apply lt_trans with m:=(plus m n); [ + Apply lt_reg_l with 1:=H +| Rewrite (plus_sym m n); Apply lt_reg_l with 1:=H ]. +Qed. + +Lemma ZL8: + (m,n:nat) (lt m n) -> (lt (S (plus m m)) (plus n n)). +Proof. +Intros m n H; Apply le_lt_trans with m:=(plus m n); [ + Change (lt (plus m m) (plus m n)) ; Apply lt_reg_l with 1:=H +| Rewrite (plus_sym m n); Apply lt_reg_l with 1:=H ]. +Qed. + +(** [nat_of_P] is a morphism from [positive] to [nat] for [lt] (expressed + from [compare] on [positive]) + + Part 1: [lt] on [positive] is finer than [lt] on [nat] +*) + +Lemma compare_convert_INFERIEUR : + (x,y:positive) (compare x y EGAL) = INFERIEUR -> + (lt (convert x) (convert y)). +Proof. +Intro x; NewInduction x as [p H|p H|];Intro y; NewDestruct y as [q|q|]; + Intro H2; [ + Unfold convert ;Simpl; Apply lt_n_S; + Do 2 Rewrite ZL6; Apply ZL7; Apply H; Simpl in H2; Assumption +| Unfold convert ;Simpl; Do 2 Rewrite ZL6; + Apply ZL8; Apply H;Simpl in H2; Apply ZLSI;Assumption +| Simpl; Discriminate H2 +| Simpl; Unfold convert ;Simpl;Do 2 Rewrite ZL6; + Elim (ZLII p q H2); [ + Intros H3;Apply lt_S;Apply ZL7; Apply H;Apply H3 + | Intros E;Rewrite E;Apply lt_n_Sn] +| Simpl; Unfold convert ;Simpl;Do 2 Rewrite ZL6; + Apply ZL7;Apply H;Assumption +| Simpl; Discriminate H2 +| Unfold convert ;Simpl; Apply lt_n_S; Rewrite ZL6; + Elim (ZL4 q);Intros h H3; Rewrite H3;Simpl; Apply lt_O_Sn +| Unfold convert ;Simpl; Rewrite ZL6; Elim (ZL4 q);Intros h H3; + Rewrite H3; Simpl; Rewrite <- plus_n_Sm; Apply lt_n_S; Apply lt_O_Sn +| Simpl; Discriminate H2 ]. +Qed. + +(** [nat_of_P] is a morphism from [positive] to [nat] for [gt] (expressed + from [compare] on [positive]) + + Part 1: [gt] on [positive] is finer than [gt] on [nat] +*) + +Lemma compare_convert_SUPERIEUR : + (x,y:positive) (compare x y EGAL)=SUPERIEUR -> (gt (convert x) (convert y)). +Proof. +Unfold gt; Intro x; NewInduction x as [p H|p H|]; + Intro y; NewDestruct y as [q|q|]; Intro H2; [ + Simpl; Unfold convert ;Simpl;Do 2 Rewrite ZL6; + Apply lt_n_S; Apply ZL7; Apply H;Assumption +| Simpl; Unfold convert ;Simpl; Do 2 Rewrite ZL6; + Elim (ZLSS p q H2); [ + Intros H3;Apply lt_S;Apply ZL7;Apply H;Assumption + | Intros E;Rewrite E;Apply lt_n_Sn] +| Unfold convert ;Simpl; Rewrite ZL6;Elim (ZL4 p); + Intros h H3;Rewrite H3;Simpl; Apply lt_n_S; Apply lt_O_Sn +| Simpl;Unfold convert ;Simpl;Do 2 Rewrite ZL6; + Apply ZL8; Apply H; Apply ZLIS; Assumption +| Simpl; Unfold convert ;Simpl;Do 2 Rewrite ZL6; + Apply ZL7;Apply H;Assumption +| Unfold convert ;Simpl; Rewrite ZL6; Elim (ZL4 p); + Intros h H3;Rewrite H3;Simpl; Rewrite <- plus_n_Sm;Apply lt_n_S; + Apply lt_O_Sn +| Simpl; Discriminate H2 +| Simpl; Discriminate H2 +| Simpl; Discriminate H2 ]. +Qed. + +(** [nat_of_P] is a morphism from [positive] to [nat] for [lt] (expressed + from [compare] on [positive]) + + Part 2: [lt] on [nat] is finer than [lt] on [positive] +*) + +Lemma convert_compare_INFERIEUR : + (x,y:positive)(lt (convert x) (convert y)) -> (compare x y EGAL) = INFERIEUR. +Proof. +Intros x y; Unfold gt; Elim (Dcompare (compare x y EGAL)); [ + Intros E; Rewrite (compare_convert_EGAL x y E); + Intros H;Absurd (lt (convert y) (convert y)); [ Apply lt_n_n | Assumption ] +| Intros H;Elim H; [ + Auto + | Intros H1 H2; Absurd (lt (convert x) (convert y)); [ + Apply lt_not_sym; Change (gt (convert x) (convert y)); + Apply compare_convert_SUPERIEUR; Assumption + | Assumption ]]]. +Qed. + +(** [nat_of_P] is a morphism from [positive] to [nat] for [gt] (expressed + from [compare] on [positive]) + + Part 2: [gt] on [nat] is finer than [gt] on [positive] +*) + +Lemma convert_compare_SUPERIEUR : + (x,y:positive)(gt (convert x) (convert y)) -> (compare x y EGAL) = SUPERIEUR. +Proof. +Intros x y; Unfold gt; Elim (Dcompare (compare x y EGAL)); [ + Intros E; Rewrite (compare_convert_EGAL x y E); + Intros H;Absurd (lt (convert y) (convert y)); [ Apply lt_n_n | Assumption ] +| Intros H;Elim H; [ + Intros H1 H2; Absurd (lt (convert y) (convert x)); [ + Apply lt_not_sym; Apply compare_convert_INFERIEUR; Assumption + | Assumption ] + | Auto]]. +Qed. + +(** [nat_of_P] is strictly positive *) + +Lemma compare_positive_to_nat_O : + (p:positive)(m:nat)(le m (positive_to_nat p m)). +NewInduction p; Simpl; Auto with arith. +Intro m; Apply le_trans with (plus m m); Auto with arith. +Qed. + +Lemma compare_convert_O : (p:positive)(lt O (convert p)). +Intro; Unfold convert; Apply lt_le_trans with (S O); Auto with arith. +Apply compare_positive_to_nat_O. +Qed. + +(** Pmult_nat permutes with multiplication *) + +Lemma positive_to_nat_mult : (p:positive) (n,m:nat) + (positive_to_nat p (mult m n))=(mult m (positive_to_nat p n)). +Proof. + Induction p. Intros. Simpl. Rewrite mult_plus_distr_r. Rewrite <- (mult_plus_distr_r m n n). + Rewrite (H (plus n n) m). Reflexivity. + Intros. Simpl. Rewrite <- (mult_plus_distr_r m n n). Apply H. + Trivial. +Qed. + +Lemma positive_to_nat_2 : (p:positive) + (positive_to_nat p (2))=(mult (2) (positive_to_nat p (1))). +Proof. + Intros. Rewrite <- positive_to_nat_mult. Reflexivity. +Qed. + +Lemma positive_to_nat_4 : (p:positive) + (positive_to_nat p (4))=(mult (2) (positive_to_nat p (2))). +Proof. + Intros. Rewrite <- positive_to_nat_mult. Reflexivity. +Qed. + +(** Mapping of xH, xO and xI through [nat_of_P] *) + +Lemma convert_xH : (convert xH)=(1). +Proof. + Reflexivity. +Qed. + +Lemma convert_xO : (p:positive) (convert (xO p))=(mult (2) (convert p)). +Proof. + Induction p. Unfold convert. Simpl. Intros. Rewrite positive_to_nat_2. + Rewrite positive_to_nat_4. Rewrite H. Simpl. Rewrite <- plus_Snm_nSm. Reflexivity. + Unfold convert. Simpl. Intros. Rewrite positive_to_nat_2. Rewrite positive_to_nat_4. + Rewrite H. Reflexivity. + Reflexivity. +Qed. + +Lemma convert_xI : (p:positive) (convert (xI p))=(S (mult (2) (convert p))). +Proof. + Induction p. Unfold convert. Simpl. Intro p0. Intro. Rewrite positive_to_nat_2. + Rewrite positive_to_nat_4; Injection H; Intro H1; Rewrite H1; Rewrite <- plus_Snm_nSm; Reflexivity. + Unfold convert. Simpl. Intros. Rewrite positive_to_nat_2. Rewrite positive_to_nat_4. + Injection H; Intro H1; Rewrite H1; Reflexivity. + Reflexivity. +Qed. + +(**********************************************************************) +(** Properties of the shifted injection from Peano natural numbers to + binary positive numbers *) + +(** Composition of [P_of_succ_nat] and [nat_of_P] is successor on [nat] *) + +Theorem bij1 : (m:nat) (convert (anti_convert m)) = (S m). +Proof. +Intro m; NewInduction m as [|n H]; [ + Reflexivity +| Simpl; Rewrite cvt_add_un; Rewrite H; Auto ]. +Qed. + +(** Miscellaneous lemmas on [P_of_succ_nat] *) + +Lemma ZL3: (x:nat) (add_un (anti_convert (plus x x))) = (xO (anti_convert x)). +Proof. +Intro x; NewInduction x as [|n H]; [ + Simpl; Auto with arith +| Simpl; Rewrite plus_sym; Simpl; Rewrite H; Rewrite ZL1;Auto with arith]. +Qed. + +Lemma ZL5: (x:nat) (anti_convert (plus (S x) (S x))) = (xI (anti_convert x)). +Proof. +Intro x; NewInduction x as [|n H];Simpl; [ + Auto with arith +| Rewrite <- plus_n_Sm; Simpl; Simpl in H; Rewrite H; Auto with arith]. +Qed. + +(** Composition of [nat_of_P] and [P_of_succ_nat] is successor on [positive] *) + +Theorem bij2 : (x:positive) (anti_convert (convert x)) = (add_un x). +Proof. +Intro x; NewInduction x as [p H|p H|]; [ + Simpl; Rewrite <- H; Change (2) with (plus (1) (1)); + Rewrite ZL2; Elim (ZL4 p); + Unfold convert; Intros n H1;Rewrite H1; Rewrite ZL3; Auto with arith +| Unfold convert ;Simpl; Change (2) with (plus (1) (1)); + Rewrite ZL2; + Rewrite <- (sub_add_one + (anti_convert + (plus (positive_to_nat p (S O)) (positive_to_nat p (S O))))); + Rewrite <- (sub_add_one (xI p)); + Simpl;Rewrite <- H;Elim (ZL4 p); Unfold convert ;Intros n H1;Rewrite H1; + Rewrite ZL5; Simpl; Trivial with arith +| Unfold convert; Simpl; Auto with arith ]. +Qed. + +(** Composition of [nat_of_P], [P_of_succ_nat] and [Ppred] is identity + on [positive] *) + +Theorem bij3: (x:positive)(sub_un (anti_convert (convert x))) = x. +Proof. +Intros x; Rewrite bij2; Rewrite sub_add_one; Trivial with arith. +Qed. + +(**********************************************************************) +(** Extra properties of the injection from binary positive numbers to Peano + natural numbers *) + +(** [nat_of_P] is a morphism for subtraction on positive numbers *) + +Theorem true_sub_convert: + (x,y:positive) (compare x y EGAL) = SUPERIEUR -> + (convert (true_sub x y)) = (minus (convert x) (convert y)). +Proof. +Intros x y H; Apply plus_reg_l with (convert y); +Rewrite le_plus_minus_r; [ + Rewrite <- convert_add; Rewrite sub_add; Auto with arith +| Apply lt_le_weak; Exact (compare_convert_SUPERIEUR x y H)]. +Qed. + +(** [nat_of_P] is injective *) + +Lemma convert_intro : (x,y:positive)(convert x)=(convert y) -> x=y. +Proof. +Intros x y H;Rewrite <- (bij3 x);Rewrite <- (bij3 y); Rewrite H; Trivial with arith. +Qed. + +Lemma ZL16: (p,q:positive)(lt (minus (convert p) (convert q)) (convert p)). +Proof. +Intros p q; Elim (ZL4 p);Elim (ZL4 q); Intros h H1 i H2; +Rewrite H1;Rewrite H2; Simpl;Unfold lt; Apply le_n_S; Apply le_minus. +Qed. + +Lemma ZL17: (p,q:positive)(lt (convert p) (convert (add p q))). +Proof. +Intros p q; Rewrite convert_add;Unfold lt;Elim (ZL4 q); Intros k H;Rewrite H; +Rewrite plus_sym;Simpl; Apply le_n_S; Apply le_plus_r. +Qed. + +(** Comparison and subtraction *) + +Lemma compare_true_sub_right : + (p,q,z:positive) + (compare q p EGAL)=INFERIEUR-> + (compare z p EGAL)=SUPERIEUR-> + (compare z q EGAL)=SUPERIEUR-> + (compare (true_sub z p) (true_sub z q) EGAL)=INFERIEUR. +Proof. +Intros; Apply convert_compare_INFERIEUR; Rewrite true_sub_convert; [ + Rewrite true_sub_convert; [ + Apply simpl_lt_plus_l with p:=(convert q); Rewrite le_plus_minus_r; [ + Rewrite plus_sym; Apply simpl_lt_plus_l with p:=(convert p); + Rewrite plus_assoc_l; Rewrite le_plus_minus_r; [ + Rewrite (plus_sym (convert p)); Apply lt_reg_l; + Apply compare_convert_INFERIEUR; Assumption + | Apply lt_le_weak; Apply compare_convert_INFERIEUR; + Apply ZC1; Assumption ] + | Apply lt_le_weak;Apply compare_convert_INFERIEUR; + Apply ZC1; Assumption ] + | Assumption ] + | Assumption ]. +Qed. + +Lemma compare_true_sub_left : + (p,q,z:positive) + (compare q p EGAL)=INFERIEUR-> + (compare p z EGAL)=SUPERIEUR-> + (compare q z EGAL)=SUPERIEUR-> + (compare (true_sub q z) (true_sub p z) EGAL)=INFERIEUR. +Proof. +Intros p q z; Intros; + Apply convert_compare_INFERIEUR; Rewrite true_sub_convert; [ + Rewrite true_sub_convert; [ + Unfold gt; Apply simpl_lt_plus_l with p:=(convert z); + Rewrite le_plus_minus_r; [ + Rewrite le_plus_minus_r; [ + Apply compare_convert_INFERIEUR;Assumption + | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Apply ZC1;Assumption] + | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Apply ZC1; Assumption] + | Assumption] +| Assumption]. +Qed. + +(** Distributivity of multiplication over subtraction *) + +Theorem times_true_sub_distr: + (x,y,z:positive) (compare y z EGAL) = SUPERIEUR -> + (times x (true_sub y z)) = (true_sub (times x y) (times x z)). +Proof. +Intros x y z H; Apply convert_intro; +Rewrite times_convert; Rewrite true_sub_convert; [ + Rewrite true_sub_convert; [ + Do 2 Rewrite times_convert; + Do 3 Rewrite (mult_sym (convert x));Apply mult_minus_distr + | Apply convert_compare_SUPERIEUR; Do 2 Rewrite times_convert; + Unfold gt; Elim (ZL4 x);Intros h H1;Rewrite H1; Apply lt_mult_left; + Exact (compare_convert_SUPERIEUR y z H) ] +| Assumption ]. +Qed. + diff --git a/theories7/Reals/Alembert.v b/theories7/Reals/Alembert.v new file mode 100644 index 00000000..702daffc --- /dev/null +++ b/theories7/Reals/Alembert.v @@ -0,0 +1,549 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Alembert.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require SeqProp. +Require PartSum. +Require Max. + +Open Local Scope R_scope. + +(***************************************************) +(* Various versions of the criterion of D'Alembert *) +(***************************************************) + +Lemma Alembert_C1 : (An:nat->R) ((n:nat)``0<(An n)``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) R0) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intros An H H0. +Cut (sigTT R [l:R](is_lub (EUn [N:nat](sum_f_R0 An N)) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intro; Apply X. +Apply complet. +Unfold Un_cv in H0; Unfold bound; Cut ``0</2``; [Intro | Apply Rlt_Rinv; Sup0]. +Elim (H0 ``/2`` H1); Intros. +Exists ``(sum_f_R0 An x)+2*(An (S x))``. +Unfold is_upper_bound; Intros; Unfold EUn in H3; Elim H3; Intros. +Rewrite H4; Assert H5 := (lt_eq_lt_dec x1 x). +Elim H5; Intros. +Elim a; Intro. +Replace (sum_f_R0 An x) with (Rplus (sum_f_R0 An x1) (sum_f_R0 [i:nat](An (plus (S x1) i)) (minus x (S x1)))). +Pattern 1 (sum_f_R0 An x1); Rewrite <- Rplus_Or; Rewrite Rplus_assoc; Apply Rle_compatibility. +Left; Apply gt0_plus_gt0_is_gt0. +Apply tech1; Intros; Apply H. +Apply Rmult_lt_pos; [Sup0 | Apply H]. +Symmetry; Apply tech2; Assumption. +Rewrite b; Pattern 1 (sum_f_R0 An x); Rewrite <- Rplus_Or; Apply Rle_compatibility. +Left; Apply Rmult_lt_pos; [Sup0 | Apply H]. +Replace (sum_f_R0 An x1) with (Rplus (sum_f_R0 An x) (sum_f_R0 [i:nat](An (plus (S x) i)) (minus x1 (S x)))). +Apply Rle_compatibility. +Cut (Rle (sum_f_R0 [i:nat](An (plus (S x) i)) (minus x1 (S x))) (Rmult (An (S x)) (sum_f_R0 [i:nat](pow ``/2`` i) (minus x1 (S x))))). +Intro; Apply Rle_trans with (Rmult (An (S x)) (sum_f_R0 [i:nat](pow ``/2`` i) (minus x1 (S x)))). +Assumption. +Rewrite <- (Rmult_sym (An (S x))); Apply Rle_monotony. +Left; Apply H. +Rewrite tech3. +Replace ``1-/2`` with ``/2``. +Unfold Rdiv; Rewrite Rinv_Rinv. +Pattern 3 ``2``; Rewrite <- Rmult_1r; Rewrite <- (Rmult_sym ``2``); Apply Rle_monotony. +Left; Sup0. +Left; Apply Rlt_anti_compatibility with ``(pow (/2) (S (minus x1 (S x))))``. +Replace ``(pow (/2) (S (minus x1 (S x))))+(1-(pow (/2) (S (minus x1 (S x)))))`` with R1; [Idtac | Ring]. +Rewrite <- (Rplus_sym ``1``); Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility. +Apply pow_lt; Apply Rlt_Rinv; Sup0. +DiscrR. +Apply r_Rmult_mult with ``2``. +Rewrite Rminus_distr; Rewrite <- Rinv_r_sym. +Ring. +DiscrR. +DiscrR. +Pattern 3 R1; Replace R1 with ``/1``; [Apply tech7; DiscrR | Apply Rinv_R1]. +Replace (An (S x)) with (An (plus (S x) O)). +Apply (tech6 [i:nat](An (plus (S x) i)) ``/2``). +Left; Apply Rlt_Rinv; Sup0. +Intro; Cut (n:nat)(ge n x)->``(An (S n))</2*(An n)``. +Intro; Replace (plus (S x) (S i)) with (S (plus (S x) i)). +Apply H6; Unfold ge; Apply tech8. +Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. +Intros; Unfold R_dist in H2; Apply Rlt_monotony_contra with ``/(An n)``. +Apply Rlt_Rinv; Apply H. +Do 2 Rewrite (Rmult_sym ``/(An n)``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Replace ``(An (S n))*/(An n)`` with ``(Rabsolu ((Rabsolu ((An (S n))/(An n)))-0))``. +Apply H2; Assumption. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Rewrite Rabsolu_right. +Unfold Rdiv; Reflexivity. +Left; Unfold Rdiv; Change ``0<(An (S n))*/(An n)``; Apply Rmult_lt_pos; [Apply H | Apply Rlt_Rinv; Apply H]. +Red; Intro; Assert H8 := (H n); Rewrite H7 in H8; Elim (Rlt_antirefl ? H8). +Replace (plus (S x) O) with (S x); [Reflexivity | Ring]. +Symmetry; Apply tech2; Assumption. +Exists (sum_f_R0 An O); Unfold EUn; Exists O; Reflexivity. +Intro; Elim X; Intros. +Apply Specif.existT with x; Apply tech10; [Unfold Un_growing; Intro; Rewrite tech5; Pattern 1 (sum_f_R0 An n); Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply H | Apply p]. +Qed. + +Lemma Alembert_C2 : (An:nat->R) ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) R0) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intros. +Pose Vn := [i:nat]``(2*(Rabsolu (An i))+(An i))/2``. +Pose Wn := [i:nat]``(2*(Rabsolu (An i))-(An i))/2``. +Cut (n:nat)``0<(Vn n)``. +Intro; Cut (n:nat)``0<(Wn n)``. +Intro; Cut (Un_cv [n:nat](Rabsolu ``(Vn (S n))/(Vn n)``) ``0``). +Intro; Cut (Un_cv [n:nat](Rabsolu ``(Wn (S n))/(Wn n)``) ``0``). +Intro; Assert H5 := (Alembert_C1 Vn H1 H3). +Assert H6 := (Alembert_C1 Wn H2 H4). +Elim H5; Intros. +Elim H6; Intros. +Apply Specif.existT with ``x-x0``; Unfold Un_cv; Unfold Un_cv in p; Unfold Un_cv in p0; Intros; Cut ``0<eps/2``. +Intro; Elim (p ``eps/2`` H8); Clear p; Intros. +Elim (p0 ``eps/2`` H8); Clear p0; Intros. +Pose N := (max x1 x2). +Exists N; Intros; Replace (sum_f_R0 An n) with (Rminus (sum_f_R0 Vn n) (sum_f_R0 Wn n)). +Unfold R_dist; Replace (Rminus (Rminus (sum_f_R0 Vn n) (sum_f_R0 Wn n)) (Rminus x x0)) with (Rplus (Rminus (sum_f_R0 Vn n) x) (Ropp (Rminus (sum_f_R0 Wn n) x0))); [Idtac | Ring]; Apply Rle_lt_trans with (Rplus (Rabsolu (Rminus (sum_f_R0 Vn n) x)) (Rabsolu (Ropp (Rminus (sum_f_R0 Wn n) x0)))). +Apply Rabsolu_triang. +Rewrite Rabsolu_Ropp; Apply Rlt_le_trans with ``eps/2+eps/2``. +Apply Rplus_lt. +Unfold R_dist in H9; Apply H9; Unfold ge; Apply le_trans with N; [Unfold N; Apply le_max_l | Assumption]. +Unfold R_dist in H10; Apply H10; Unfold ge; Apply le_trans with N; [Unfold N; Apply le_max_r | Assumption]. +Right; Symmetry; Apply double_var. +Symmetry; Apply tech11; Intro; Unfold Vn Wn; Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/2``); Apply r_Rmult_mult with ``2``. +Rewrite Rminus_distr; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Ring. +DiscrR. +DiscrR. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Cut (n:nat)``/2*(Rabsolu (An n))<=(Wn n)<=(3*/2)*(Rabsolu (An n))``. +Intro; Cut (n:nat)``/(Wn n)<=2*/(Rabsolu (An n))``. +Intro; Cut (n:nat)``(Wn (S n))/(Wn n)<=3*(Rabsolu (An (S n))/(An n))``. +Intro; Unfold Un_cv; Intros; Unfold Un_cv in H0; Cut ``0<eps/3``. +Intro; Elim (H0 ``eps/3`` H8); Intros. +Exists x; Intros. +Assert H11 := (H9 n H10). +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Unfold R_dist in H11; Unfold Rminus in H11; Rewrite Ropp_O in H11; Rewrite Rplus_Or in H11; Rewrite Rabsolu_Rabsolu in H11; Rewrite Rabsolu_right. +Apply Rle_lt_trans with ``3*(Rabsolu ((An (S n))/(An n)))``. +Apply H6. +Apply Rlt_monotony_contra with ``/3``. +Apply Rlt_Rinv; Sup0. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H11; Exact H11. +Left; Change ``0<(Wn (S n))/(Wn n)``; Unfold Rdiv; Apply Rmult_lt_pos. +Apply H2. +Apply Rlt_Rinv; Apply H2. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Intro; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite <- Rmult_assoc; Replace ``3`` with ``2*(3*/2)``; [Idtac | Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR]; Apply Rle_trans with ``(Wn (S n))*2*/(Rabsolu (An n))``. +Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply H2. +Apply H5. +Rewrite Rabsolu_Rinv. +Replace ``(Wn (S n))*2*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*(Wn (S n))``; [Idtac | Ring]; Replace ``2*(3*/2)*(Rabsolu (An (S n)))*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*((3*/2)*(Rabsolu (An (S n))))``; [Idtac | Ring]; Apply Rle_monotony. +Left; Apply Rmult_lt_pos. +Sup0. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply H. +Elim (H4 (S n)); Intros; Assumption. +Apply H. +Intro; Apply Rle_monotony_contra with (Wn n). +Apply H2. +Rewrite <- Rinv_r_sym. +Apply Rle_monotony_contra with (Rabsolu (An n)). +Apply Rabsolu_pos_lt; Apply H. +Rewrite Rmult_1r; Replace ``(Rabsolu (An n))*((Wn n)*(2*/(Rabsolu (An n))))`` with ``2*(Wn n)*((Rabsolu (An n))*/(Rabsolu (An n)))``; [Idtac | Ring]; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Apply Rle_monotony_contra with ``/2``. +Apply Rlt_Rinv; Sup0. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Elim (H4 n); Intros; Assumption. +DiscrR. +Apply Rabsolu_no_R0; Apply H. +Red; Intro; Assert H6 := (H2 n); Rewrite H5 in H6; Elim (Rlt_antirefl ? H6). +Intro; Split. +Unfold Wn; Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Apply Rle_monotony. +Left; Apply Rlt_Rinv; Sup0. +Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Rewrite double; Unfold Rminus; Rewrite Rplus_assoc; Apply Rle_compatibility. +Apply Rle_anti_compatibility with (An n). +Rewrite Rplus_Or; Rewrite (Rplus_sym (An n)); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rle_Rabsolu. +Unfold Wn; Unfold Rdiv; Repeat Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply Rlt_Rinv; Sup0. +Unfold Rminus; Rewrite double; Replace ``3*(Rabsolu (An n))`` with ``(Rabsolu (An n))+(Rabsolu (An n))+(Rabsolu (An n))``; [Idtac | Ring]; Repeat Rewrite Rplus_assoc; Repeat Apply Rle_compatibility. +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Cut (n:nat)``/2*(Rabsolu (An n))<=(Vn n)<=(3*/2)*(Rabsolu (An n))``. +Intro; Cut (n:nat)``/(Vn n)<=2*/(Rabsolu (An n))``. +Intro; Cut (n:nat)``(Vn (S n))/(Vn n)<=3*(Rabsolu (An (S n))/(An n))``. +Intro; Unfold Un_cv; Intros; Unfold Un_cv in H1; Cut ``0<eps/3``. +Intro; Elim (H0 ``eps/3`` H7); Intros. +Exists x; Intros. +Assert H10 := (H8 n H9). +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Unfold R_dist in H10; Unfold Rminus in H10; Rewrite Ropp_O in H10; Rewrite Rplus_Or in H10; Rewrite Rabsolu_Rabsolu in H10; Rewrite Rabsolu_right. +Apply Rle_lt_trans with ``3*(Rabsolu ((An (S n))/(An n)))``. +Apply H5. +Apply Rlt_monotony_contra with ``/3``. +Apply Rlt_Rinv; Sup0. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H10; Exact H10. +Left; Change ``0<(Vn (S n))/(Vn n)``; Unfold Rdiv; Apply Rmult_lt_pos. +Apply H1. +Apply Rlt_Rinv; Apply H1. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Intro; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite <- Rmult_assoc; Replace ``3`` with ``2*(3*/2)``; [Idtac | Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR]; Apply Rle_trans with ``(Vn (S n))*2*/(Rabsolu (An n))``. +Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply H1. +Apply H4. +Rewrite Rabsolu_Rinv. +Replace ``(Vn (S n))*2*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*(Vn (S n))``; [Idtac | Ring]; Replace ``2*(3*/2)*(Rabsolu (An (S n)))*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*((3*/2)*(Rabsolu (An (S n))))``; [Idtac | Ring]; Apply Rle_monotony. +Left; Apply Rmult_lt_pos. +Sup0. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply H. +Elim (H3 (S n)); Intros; Assumption. +Apply H. +Intro; Apply Rle_monotony_contra with (Vn n). +Apply H1. +Rewrite <- Rinv_r_sym. +Apply Rle_monotony_contra with (Rabsolu (An n)). +Apply Rabsolu_pos_lt; Apply H. +Rewrite Rmult_1r; Replace ``(Rabsolu (An n))*((Vn n)*(2*/(Rabsolu (An n))))`` with ``2*(Vn n)*((Rabsolu (An n))*/(Rabsolu (An n)))``; [Idtac | Ring]; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Apply Rle_monotony_contra with ``/2``. +Apply Rlt_Rinv; Sup0. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Elim (H3 n); Intros; Assumption. +DiscrR. +Apply Rabsolu_no_R0; Apply H. +Red; Intro; Assert H5 := (H1 n); Rewrite H4 in H5; Elim (Rlt_antirefl ? H5). +Intro; Split. +Unfold Vn; Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Apply Rle_monotony. +Left; Apply Rlt_Rinv; Sup0. +Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Rewrite double; Rewrite Rplus_assoc; Apply Rle_compatibility. +Apply Rle_anti_compatibility with ``-(An n)``; Rewrite Rplus_Or; Rewrite <- (Rplus_sym (An n)); Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Unfold Vn; Unfold Rdiv; Repeat Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply Rlt_Rinv; Sup0. +Unfold Rminus; Rewrite double; Replace ``3*(Rabsolu (An n))`` with ``(Rabsolu (An n))+(Rabsolu (An n))+(Rabsolu (An n))``; [Idtac | Ring]; Repeat Rewrite Rplus_assoc; Repeat Apply Rle_compatibility; Apply Rle_Rabsolu. +Intro; Unfold Wn; Unfold Rdiv; Rewrite <- (Rmult_Or ``/2``); Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony. +Apply Rlt_Rinv; Sup0. +Apply Rlt_anti_compatibility with (An n); Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym (An n)); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rle_lt_trans with (Rabsolu (An n)). +Apply Rle_Rabsolu. +Rewrite double; Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rabsolu_pos_lt; Apply H. +Intro; Unfold Vn; Unfold Rdiv; Rewrite <- (Rmult_Or ``/2``); Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony. +Apply Rlt_Rinv; Sup0. +Apply Rlt_anti_compatibility with ``-(An n)``; Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym ``-(An n)``); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Apply Rle_lt_trans with (Rabsolu (An n)). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Rewrite double; Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rabsolu_pos_lt; Apply H. +Qed. + +Lemma AlembertC3_step1 : (An:nat->R;x:R) ``x<>0`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) ``0``) -> (SigT R [l:R](Pser An x l)). +Intros; Pose Bn := [i:nat]``(An i)*(pow x i)``. +Cut (n:nat)``(Bn n)<>0``. +Intro; Cut (Un_cv [n:nat](Rabsolu ``(Bn (S n))/(Bn n)``) ``0``). +Intro; Assert H4 := (Alembert_C2 Bn H2 H3). +Elim H4; Intros. +Apply Specif.existT with x0; Unfold Bn in p; Apply tech12; Assumption. +Unfold Un_cv; Intros; Unfold Un_cv in H1; Cut ``0<eps/(Rabsolu x)``. +Intro; Elim (H1 ``eps/(Rabsolu x)`` H4); Intros. +Exists x0; Intros; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Unfold Bn; Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``. +Rewrite Rabsolu_mult; Apply Rlt_monotony_contra with ``/(Rabsolu x)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Rewrite <- (Rmult_sym (Rabsolu x)); Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H5; Replace ``(Rabsolu ((An (S n))/(An n)))`` with ``(R_dist (Rabsolu ((An (S n))*/(An n))) 0)``. +Apply H5; Assumption. +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Unfold Rdiv; Reflexivity. +Apply Rabsolu_no_R0; Assumption. +Replace (S n) with (plus n (1)); [Idtac | Ring]; Rewrite pow_add; Unfold Rdiv; Rewrite Rinv_Rmult. +Replace ``(An (plus n (S O)))*((pow x n)*(pow x (S O)))*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*(pow x (S O))*/(An n)*((pow x n)*/(pow x n))``; [Idtac | Ring]; Rewrite <- Rinv_r_sym. +Simpl; Ring. +Apply pow_nonzero; Assumption. +Apply H0. +Apply pow_nonzero; Assumption. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption]. +Intro; Unfold Bn; Apply prod_neq_R0; [Apply H0 | Apply pow_nonzero; Assumption]. +Qed. + +Lemma AlembertC3_step2 : (An:nat->R;x:R) ``x==0`` -> (SigT R [l:R](Pser An x l)). +Intros; Apply Specif.existT with (An O). +Unfold Pser; Unfold infinit_sum; Intros; Exists O; Intros; Replace (sum_f_R0 [n0:nat]``(An n0)*(pow x n0)`` n) with (An O). +Unfold R_dist; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Induction n. +Simpl; Ring. +Rewrite tech5; Rewrite Hrecn; [Rewrite H; Simpl; Ring | Unfold ge; Apply le_O_n]. +Qed. + +(* An useful criterion of convergence for power series *) +Theorem Alembert_C3 : (An:nat->R;x:R) ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) ``0``) -> (SigT R [l:R](Pser An x l)). +Intros; Case (total_order_T x R0); Intro. +Elim s; Intro. +Cut ``x<>0``. +Intro; Apply AlembertC3_step1; Assumption. +Red; Intro; Rewrite H1 in a; Elim (Rlt_antirefl ? a). +Apply AlembertC3_step2; Assumption. +Cut ``x<>0``. +Intro; Apply AlembertC3_step1; Assumption. +Red; Intro; Rewrite H1 in r; Elim (Rlt_antirefl ? r). +Qed. + +Lemma Alembert_C4 : (An:nat->R;k:R) ``0<=k<1`` -> ((n:nat)``0<(An n)``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intros An k Hyp H H0. +Cut (sigTT R [l:R](is_lub (EUn [N:nat](sum_f_R0 An N)) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intro; Apply X. +Apply complet. +Assert H1 := (tech13 ? ? Hyp H0). +Elim H1; Intros. +Elim H2; Intros. +Elim H4; Intros. +Unfold bound; Exists ``(sum_f_R0 An x0)+/(1-x)*(An (S x0))``. +Unfold is_upper_bound; Intros; Unfold EUn in H6. +Elim H6; Intros. +Rewrite H7. +Assert H8 := (lt_eq_lt_dec x2 x0). +Elim H8; Intros. +Elim a; Intro. +Replace (sum_f_R0 An x0) with (Rplus (sum_f_R0 An x2) (sum_f_R0 [i:nat](An (plus (S x2) i)) (minus x0 (S x2)))). +Pattern 1 (sum_f_R0 An x2); Rewrite <- Rplus_Or. +Rewrite Rplus_assoc; Apply Rle_compatibility. +Left; Apply gt0_plus_gt0_is_gt0. +Apply tech1. +Intros; Apply H. +Apply Rmult_lt_pos. +Apply Rlt_Rinv; Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Replace ``x+(1-x)`` with R1; [Elim H3; Intros; Assumption | Ring]. +Apply H. +Symmetry; Apply tech2; Assumption. +Rewrite b; Pattern 1 (sum_f_R0 An x0); Rewrite <- Rplus_Or; Apply Rle_compatibility. +Left; Apply Rmult_lt_pos. +Apply Rlt_Rinv; Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Replace ``x+(1-x)`` with R1; [Elim H3; Intros; Assumption | Ring]. +Apply H. +Replace (sum_f_R0 An x2) with (Rplus (sum_f_R0 An x0) (sum_f_R0 [i:nat](An (plus (S x0) i)) (minus x2 (S x0)))). +Apply Rle_compatibility. +Cut (Rle (sum_f_R0 [i:nat](An (plus (S x0) i)) (minus x2 (S x0))) (Rmult (An (S x0)) (sum_f_R0 [i:nat](pow x i) (minus x2 (S x0))))). +Intro; Apply Rle_trans with (Rmult (An (S x0)) (sum_f_R0 [i:nat](pow x i) (minus x2 (S x0)))). +Assumption. +Rewrite <- (Rmult_sym (An (S x0))); Apply Rle_monotony. +Left; Apply H. +Rewrite tech3. +Unfold Rdiv; Apply Rle_monotony_contra with ``1-x``. +Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or. +Replace ``x+(1-x)`` with R1; [Elim H3; Intros; Assumption | Ring]. +Do 2 Rewrite (Rmult_sym ``1-x``). +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Apply Rle_anti_compatibility with ``(pow x (S (minus x2 (S x0))))``. +Replace ``(pow x (S (minus x2 (S x0))))+(1-(pow x (S (minus x2 (S x0)))))`` with R1; [Idtac | Ring]. +Rewrite <- (Rplus_sym R1); Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility. +Left; Apply pow_lt. +Apply Rle_lt_trans with k. +Elim Hyp; Intros; Assumption. +Elim H3; Intros; Assumption. +Apply Rminus_eq_contra. +Red; Intro. +Elim H3; Intros. +Rewrite H10 in H12; Elim (Rlt_antirefl ? H12). +Red; Intro. +Elim H3; Intros. +Rewrite H10 in H12; Elim (Rlt_antirefl ? H12). +Replace (An (S x0)) with (An (plus (S x0) O)). +Apply (tech6 [i:nat](An (plus (S x0) i)) x). +Left; Apply Rle_lt_trans with k. +Elim Hyp; Intros; Assumption. +Elim H3; Intros; Assumption. +Intro. +Cut (n:nat)(ge n x0)->``(An (S n))<x*(An n)``. +Intro. +Replace (plus (S x0) (S i)) with (S (plus (S x0) i)). +Apply H9. +Unfold ge. +Apply tech8. + Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. +Intros. +Apply Rlt_monotony_contra with ``/(An n)``. +Apply Rlt_Rinv; Apply H. +Do 2 Rewrite (Rmult_sym ``/(An n)``). +Rewrite Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r. +Replace ``(An (S n))*/(An n)`` with ``(Rabsolu ((An (S n))/(An n)))``. +Apply H5; Assumption. +Rewrite Rabsolu_right. +Unfold Rdiv; Reflexivity. +Left; Unfold Rdiv; Change ``0<(An (S n))*/(An n)``; Apply Rmult_lt_pos. +Apply H. +Apply Rlt_Rinv; Apply H. +Red; Intro. +Assert H11 := (H n). +Rewrite H10 in H11; Elim (Rlt_antirefl ? H11). +Replace (plus (S x0) O) with (S x0); [Reflexivity | Ring]. +Symmetry; Apply tech2; Assumption. +Exists (sum_f_R0 An O); Unfold EUn; Exists O; Reflexivity. +Intro; Elim X; Intros. +Apply Specif.existT with x; Apply tech10; [Unfold Un_growing; Intro; Rewrite tech5; Pattern 1 (sum_f_R0 An n); Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply H | Apply p]. +Qed. + +Lemma Alembert_C5 : (An:nat->R;k:R) ``0<=k<1`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intros. +Cut (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intro Hyp0; Apply Hyp0. +Apply cv_cauchy_2. +Apply cauchy_abs. +Apply cv_cauchy_1. +Cut (SigT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat](Rabsolu (An i)) N) l)) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat](Rabsolu (An i)) N) l)). +Intro Hyp; Apply Hyp. +Apply (Alembert_C4 [i:nat](Rabsolu (An i)) k). +Assumption. +Intro; Apply Rabsolu_pos_lt; Apply H0. +Unfold Un_cv. +Unfold Un_cv in H1. +Unfold Rdiv. +Intros. +Elim (H1 eps H2); Intros. +Exists x; Intros. +Rewrite <- Rabsolu_Rinv. +Rewrite <- Rabsolu_mult. +Rewrite Rabsolu_Rabsolu. +Unfold Rdiv in H3; Apply H3; Assumption. +Apply H0. +Intro. +Elim X; Intros. +Apply existTT with x. +Assumption. +Intro. +Elim X; Intros. +Apply Specif.existT with x. +Assumption. +Qed. + +(* Convergence of power series in D(O,1/k) *) +(* k=0 is described in Alembert_C3 *) +Lemma Alembert_C6 : (An:nat->R;x,k:R) ``0<k`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> ``(Rabsolu x)</k`` -> (SigT R [l:R](Pser An x l)). +Intros. +Cut (SigT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat]``(An i)*(pow x i)`` N) l)). +Intro. +Elim X; Intros. +Apply Specif.existT with x0. +Apply tech12; Assumption. +Case (total_order_T x R0); Intro. +Elim s; Intro. +EApply Alembert_C5 with ``k*(Rabsolu x)``. +Split. +Unfold Rdiv; Apply Rmult_le_pos. +Left; Assumption. +Left; Apply Rabsolu_pos_lt. +Red; Intro; Rewrite H3 in a; Elim (Rlt_antirefl ? a). +Apply Rlt_monotony_contra with ``/k``. +Apply Rlt_Rinv; Assumption. +Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Rewrite Rmult_1r; Assumption. +Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H). +Intro; Apply prod_neq_R0. +Apply H0. +Apply pow_nonzero. +Red; Intro; Rewrite H3 in a; Elim (Rlt_antirefl ? a). +Unfold Un_cv; Unfold Un_cv in H1. +Intros. +Cut ``0<eps/(Rabsolu x)``. +Intro. +Elim (H1 ``eps/(Rabsolu x)`` H4); Intros. +Exists x0. +Intros. +Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``. +Unfold R_dist. +Rewrite Rabsolu_mult. +Replace ``(Rabsolu ((An (S n))/(An n)))*(Rabsolu x)-k*(Rabsolu x)`` with ``(Rabsolu x)*((Rabsolu ((An (S n))/(An n)))-k)``; [Idtac | Ring]. +Rewrite Rabsolu_mult. +Rewrite Rabsolu_Rabsolu. +Apply Rlt_monotony_contra with ``/(Rabsolu x)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt. +Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a). +Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Rewrite <- (Rmult_sym eps). +Unfold R_dist in H5. +Unfold Rdiv; Unfold Rdiv in H5; Apply H5; Assumption. +Apply Rabsolu_no_R0. +Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a). +Unfold Rdiv; Replace (S n) with (plus n (1)); [Idtac | Ring]. +Rewrite pow_add. +Simpl. +Rewrite Rmult_1r. +Rewrite Rinv_Rmult. +Replace ``(An (plus n (S O)))*((pow x n)*x)*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*/(An n)*x*((pow x n)*/(pow x n))``; [Idtac | Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Reflexivity. +Apply pow_nonzero. +Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a). +Apply H0. +Apply pow_nonzero. +Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a). +Unfold Rdiv; Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt. +Red; Intro H7; Rewrite H7 in a; Elim (Rlt_antirefl ? a). +Apply Specif.existT with (An O). +Unfold Un_cv. +Intros. +Exists O. +Intros. +Unfold R_dist. +Replace (sum_f_R0 [i:nat]``(An i)*(pow x i)`` n) with (An O). +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Induction n. +Simpl; Ring. +Rewrite tech5. +Rewrite <- Hrecn. +Rewrite b; Simpl; Ring. +Unfold ge; Apply le_O_n. +EApply Alembert_C5 with ``k*(Rabsolu x)``. +Split. +Unfold Rdiv; Apply Rmult_le_pos. +Left; Assumption. +Left; Apply Rabsolu_pos_lt. +Red; Intro; Rewrite H3 in r; Elim (Rlt_antirefl ? r). +Apply Rlt_monotony_contra with ``/k``. +Apply Rlt_Rinv; Assumption. +Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Rewrite Rmult_1r; Assumption. +Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H). +Intro; Apply prod_neq_R0. +Apply H0. +Apply pow_nonzero. +Red; Intro; Rewrite H3 in r; Elim (Rlt_antirefl ? r). +Unfold Un_cv; Unfold Un_cv in H1. +Intros. +Cut ``0<eps/(Rabsolu x)``. +Intro. +Elim (H1 ``eps/(Rabsolu x)`` H4); Intros. +Exists x0. +Intros. +Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``. +Unfold R_dist. +Rewrite Rabsolu_mult. +Replace ``(Rabsolu ((An (S n))/(An n)))*(Rabsolu x)-k*(Rabsolu x)`` with ``(Rabsolu x)*((Rabsolu ((An (S n))/(An n)))-k)``; [Idtac | Ring]. +Rewrite Rabsolu_mult. +Rewrite Rabsolu_Rabsolu. +Apply Rlt_monotony_contra with ``/(Rabsolu x)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt. +Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r). +Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Rewrite <- (Rmult_sym eps). +Unfold R_dist in H5. +Unfold Rdiv; Unfold Rdiv in H5; Apply H5; Assumption. +Apply Rabsolu_no_R0. +Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r). +Unfold Rdiv; Replace (S n) with (plus n (1)); [Idtac | Ring]. +Rewrite pow_add. +Simpl. +Rewrite Rmult_1r. +Rewrite Rinv_Rmult. +Replace ``(An (plus n (S O)))*((pow x n)*x)*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*/(An n)*x*((pow x n)*/(pow x n))``; [Idtac | Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Reflexivity. +Apply pow_nonzero. +Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r). +Apply H0. +Apply pow_nonzero. +Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r). +Unfold Rdiv; Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt. +Red; Intro H7; Rewrite H7 in r; Elim (Rlt_antirefl ? r). +Qed. diff --git a/theories7/Reals/AltSeries.v b/theories7/Reals/AltSeries.v new file mode 100644 index 00000000..af4b558a --- /dev/null +++ b/theories7/Reals/AltSeries.v @@ -0,0 +1,362 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: AltSeries.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require SeqProp. +Require PartSum. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(**********) +Definition tg_alt [Un:nat->R] : nat->R := [i:nat]``(pow (-1) i)*(Un i)``. +Definition positivity_seq [Un:nat->R] : Prop := (n:nat)``0<=(Un n)``. + +Lemma CV_ALT_step0 : (Un:nat->R) (Un_decreasing Un) -> (Un_growing [N:nat](sum_f_R0 (tg_alt Un) (S (mult (2) N)))). +Intros; Unfold Un_growing; Intro. +Cut (mult (S (S O)) (S n)) = (S (S (mult (2) n))). +Intro; Rewrite H0. +Do 4 Rewrite tech5; Repeat Rewrite Rplus_assoc; Apply Rle_compatibility. +Pattern 1 (tg_alt Un (S (mult (S (S O)) n))); Rewrite <- Rplus_Or. +Apply Rle_compatibility. +Unfold tg_alt; Rewrite <- H0; Rewrite pow_1_odd; Rewrite pow_1_even; Rewrite Rmult_1l. +Apply Rle_anti_compatibility with ``(Un (S (mult (S (S O)) (S n))))``. +Rewrite Rplus_Or; Replace ``(Un (S (mult (S (S O)) (S n))))+((Un (mult (S (S O)) (S n)))+ -1*(Un (S (mult (S (S O)) (S n)))))`` with ``(Un (mult (S (S O)) (S n)))``; [Idtac | Ring]. +Apply H. +Cut (n:nat) (S n)=(plus n (1)); [Intro | Intro; Ring]. +Rewrite (H0 n); Rewrite (H0 (S (mult (2) n))); Rewrite (H0 (mult (2) n)); Ring. +Qed. + +Lemma CV_ALT_step1 : (Un:nat->R) (Un_decreasing Un) -> (Un_decreasing [N:nat](sum_f_R0 (tg_alt Un) (mult (2) N))). +Intros; Unfold Un_decreasing; Intro. +Cut (mult (S (S O)) (S n)) = (S (S (mult (2) n))). +Intro; Rewrite H0; Do 2 Rewrite tech5; Repeat Rewrite Rplus_assoc. +Pattern 2 (sum_f_R0 (tg_alt Un) (mult (S (S O)) n)); Rewrite <- Rplus_Or. +Apply Rle_compatibility. +Unfold tg_alt; Rewrite <- H0; Rewrite pow_1_odd; Rewrite pow_1_even; Rewrite Rmult_1l. +Apply Rle_anti_compatibility with ``(Un (S (mult (S (S O)) n)))``. +Rewrite Rplus_Or; Replace ``(Un (S (mult (S (S O)) n)))+( -1*(Un (S (mult (S (S O)) n)))+(Un (mult (S (S O)) (S n))))`` with ``(Un (mult (S (S O)) (S n)))``; [Idtac | Ring]. +Rewrite H0; Apply H. +Cut (n:nat) (S n)=(plus n (1)); [Intro | Intro; Ring]. +Rewrite (H0 n); Rewrite (H0 (S (mult (2) n))); Rewrite (H0 (mult (2) n)); Ring. +Qed. + +(**********) +Lemma CV_ALT_step2 : (Un:nat->R;N:nat) (Un_decreasing Un) -> (positivity_seq Un) -> (Rle (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (2) N))) R0). +Intros; Induction N. +Simpl; Unfold tg_alt; Simpl; Rewrite Rmult_1r. +Replace ``-1* -1*(Un (S (S O)))`` with (Un (S (S O))); [Idtac | Ring]. +Apply Rle_anti_compatibility with ``(Un (S O))``; Rewrite Rplus_Or. +Replace ``(Un (S O))+ (-1*(Un (S O))+(Un (S (S O))))`` with (Un (S (S O))); [Apply H | Ring]. +Cut (S (mult (2) (S N))) = (S (S (S (mult (2) N)))). +Intro; Rewrite H1; Do 2 Rewrite tech5. +Apply Rle_trans with (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (S (S O)) N))). +Pattern 2 (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (S (S O)) N))); Rewrite <- Rplus_Or. +Rewrite Rplus_assoc; Apply Rle_compatibility. +Unfold tg_alt; Rewrite <- H1. +Rewrite pow_1_odd. +Cut (S (S (mult (2) (S N)))) = (mult (2) (S (S N))). +Intro; Rewrite H2; Rewrite pow_1_even; Rewrite Rmult_1l; Rewrite <- H2. +Apply Rle_anti_compatibility with ``(Un (S (mult (S (S O)) (S N))))``. +Rewrite Rplus_Or; Replace ``(Un (S (mult (S (S O)) (S N))))+( -1*(Un (S (mult (S (S O)) (S N))))+(Un (S (S (mult (S (S O)) (S N))))))`` with ``(Un (S (S (mult (S (S O)) (S N)))))``; [Idtac | Ring]. +Apply H. +Apply INR_eq; Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply HrecN. +Apply INR_eq; Repeat Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Qed. + +(* A more general inequality *) +Lemma CV_ALT_step3 : (Un:nat->R;N:nat) (Un_decreasing Un) -> (positivity_seq Un) -> (Rle (sum_f_R0 [i:nat](tg_alt Un (S i)) N) R0). +Intros; Induction N. +Simpl; Unfold tg_alt; Simpl; Rewrite Rmult_1r. +Apply Rle_anti_compatibility with (Un (S O)). +Rewrite Rplus_Or; Replace ``(Un (S O))+ -1*(Un (S O))`` with R0; [Apply H0 | Ring]. +Assert H1 := (even_odd_cor N). +Elim H1; Intros. +Elim H2; Intro. +Rewrite H3; Apply CV_ALT_step2; Assumption. +Rewrite H3; Rewrite tech5. +Apply Rle_trans with (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (S (S O)) x))). +Pattern 2 (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (S (S O)) x))); Rewrite <- Rplus_Or. +Apply Rle_compatibility. +Unfold tg_alt; Simpl. +Replace (plus x (plus x O)) with (mult (2) x); [Idtac | Ring]. +Rewrite pow_1_even. +Replace `` -1*( -1*( -1*1))*(Un (S (S (S (mult (S (S O)) x)))))`` with ``-(Un (S (S (S (mult (S (S O)) x)))))``; [Idtac | Ring]. +Apply Rle_anti_compatibility with (Un (S (S (S (mult (S (S O)) x))))). +Rewrite Rplus_Or; Rewrite Rplus_Ropp_r. +Apply H0. +Apply CV_ALT_step2; Assumption. +Qed. + +(**********) +Lemma CV_ALT_step4 : (Un:nat->R) (Un_decreasing Un) -> (positivity_seq Un) -> (has_ub [N:nat](sum_f_R0 (tg_alt Un) (S (mult (2) N)))). +Intros; Unfold has_ub; Unfold bound. +Exists ``(Un O)``. +Unfold is_upper_bound; Intros; Elim H1; Intros. +Rewrite H2; Rewrite decomp_sum. +Replace (tg_alt Un O) with ``(Un O)``. +Pattern 2 ``(Un O)``; Rewrite <- Rplus_Or. +Apply Rle_compatibility. +Apply CV_ALT_step3; Assumption. +Unfold tg_alt; Simpl; Ring. +Apply lt_O_Sn. +Qed. + +(* This lemma gives an interesting result about alternated series *) +Lemma CV_ALT : (Un:nat->R) (Un_decreasing Un) -> (positivity_seq Un) -> (Un_cv Un R0) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 (tg_alt Un) N) l)). +Intros. +Assert H2 := (CV_ALT_step0 ? H). +Assert H3 := (CV_ALT_step4 ? H H0). +Assert X := (growing_cv ? H2 H3). +Elim X; Intros. +Apply existTT with x. +Unfold Un_cv; Unfold R_dist; Unfold Un_cv in H1; Unfold R_dist in H1; Unfold Un_cv in p; Unfold R_dist in p. +Intros; Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. +Elim (H1 ``eps/2`` H5); Intros N2 H6. +Elim (p ``eps/2`` H5); Intros N1 H7. +Pose N := (max (S (mult (2) N1)) N2). +Exists N; Intros. +Assert H9 := (even_odd_cor n). +Elim H9; Intros P H10. +Cut (le N1 P). +Intro; Elim H10; Intro. +Replace ``(sum_f_R0 (tg_alt Un) n)-x`` with ``((sum_f_R0 (tg_alt Un) (S n))-x)+(-(tg_alt Un (S n)))``. +Apply Rle_lt_trans with ``(Rabsolu ((sum_f_R0 (tg_alt Un) (S n))-x))+(Rabsolu (-(tg_alt Un (S n))))``. +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Rewrite H12; Apply H7; Assumption. +Rewrite Rabsolu_Ropp; Unfold tg_alt; Rewrite Rabsolu_mult; Rewrite pow_1_abs; Rewrite Rmult_1l; Unfold Rminus in H6; Rewrite Ropp_O in H6; Rewrite <- (Rplus_Or (Un (S n))); Apply H6. +Unfold ge; Apply le_trans with n. +Apply le_trans with N; [Unfold N; Apply le_max_r | Assumption]. +Apply le_n_Sn. +Rewrite tech5; Ring. +Rewrite H12; Apply Rlt_trans with ``eps/2``. +Apply H7; Assumption. +Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. +Sup0. +Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1r | DiscrR]. +Rewrite RIneq.double. +Pattern 1 eps; Rewrite <- (Rplus_Or eps); Apply Rlt_compatibility; Assumption. +Elim H10; Intro; Apply le_double. +Rewrite <- H11; Apply le_trans with N. +Unfold N; Apply le_trans with (S (mult (2) N1)); [Apply le_n_Sn | Apply le_max_l]. +Assumption. +Apply lt_n_Sm_le. +Rewrite <- H11. +Apply lt_le_trans with N. +Unfold N; Apply lt_le_trans with (S (mult (2) N1)). +Apply lt_n_Sn. +Apply le_max_l. +Assumption. +Qed. + +(************************************************) +(* Convergence of alternated series *) +(* *) +(* Applications: PI, cos, sin *) +(************************************************) +Theorem alternated_series : (Un:nat->R) (Un_decreasing Un) -> (Un_cv Un R0) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 (tg_alt Un) N) l)). +Intros; Apply CV_ALT. +Assumption. +Unfold positivity_seq; Apply decreasing_ineq; Assumption. +Assumption. +Qed. + +Theorem alternated_series_ineq : (Un:nat->R;l:R;N:nat) (Un_decreasing Un) -> (Un_cv Un R0) -> (Un_cv [N:nat](sum_f_R0 (tg_alt Un) N) l) -> ``(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) N)))<=l<=(sum_f_R0 (tg_alt Un) (mult (S (S O)) N))``. +Intros. +Cut (Un_cv [N:nat](sum_f_R0 (tg_alt Un) (mult (2) N)) l). +Cut (Un_cv [N:nat](sum_f_R0 (tg_alt Un) (S (mult (2) N))) l). +Intros; Split. +Apply (growing_ineq [N:nat](sum_f_R0 (tg_alt Un) (S (mult (2) N)))). +Apply CV_ALT_step0; Assumption. +Assumption. +Apply (decreasing_ineq [N:nat](sum_f_R0 (tg_alt Un) (mult (2) N))). +Apply CV_ALT_step1; Assumption. +Assumption. +Unfold Un_cv; Unfold R_dist; Unfold Un_cv in H1; Unfold R_dist in H1; Intros. +Elim (H1 eps H2); Intros. +Exists x; Intros. +Apply H3. +Unfold ge; Apply le_trans with (mult (2) n). +Apply le_trans with n. +Assumption. +Assert H5 := (mult_O_le n (2)). +Elim H5; Intro. +Cut ~(O)=(2); [Intro; Elim H7; Symmetry; Assumption | Discriminate]. +Assumption. +Apply le_n_Sn. +Unfold Un_cv; Unfold R_dist; Unfold Un_cv in H1; Unfold R_dist in H1; Intros. +Elim (H1 eps H2); Intros. +Exists x; Intros. +Apply H3. +Unfold ge; Apply le_trans with n. +Assumption. +Assert H5 := (mult_O_le n (2)). +Elim H5; Intro. +Cut ~(O)=(2); [Intro; Elim H7; Symmetry; Assumption | Discriminate]. +Assumption. +Qed. + +(************************************) +(* Application : construction of PI *) +(************************************) + +Definition PI_tg := [n:nat]``/(INR (plus (mult (S (S O)) n) (S O)))``. + +Lemma PI_tg_pos : (n:nat)``0<=(PI_tg n)``. +Intro; Unfold PI_tg; Left; Apply Rlt_Rinv; Apply lt_INR_0; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. +Qed. + +Lemma PI_tg_decreasing : (Un_decreasing PI_tg). +Unfold PI_tg Un_decreasing; Intro. +Apply Rle_monotony_contra with ``(INR (plus (mult (S (S O)) n) (S O)))``. +Apply lt_INR_0. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. +Rewrite <- Rinv_r_sym. +Apply Rle_monotony_contra with ``(INR (plus (mult (S (S O)) (S n)) (S O)))``. +Apply lt_INR_0. +Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Apply lt_O_Sn | Ring]. +Rewrite (Rmult_sym ``(INR (plus (mult (S (S O)) (S n)) (S O)))``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Do 2 Rewrite Rmult_1r; Apply le_INR. +Replace (plus (mult (2) (S n)) (1)) with (S (S (plus (mult (2) n) (1)))). +Apply le_trans with (S (plus (mult (2) n) (1))); Apply le_n_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Discriminate | Ring]. +Qed. + +Lemma PI_tg_cv : (Un_cv PI_tg R0). +Unfold Un_cv; Unfold R_dist; Intros. +Cut ``0<2*eps``; [Intro | Apply Rmult_lt_pos; [Sup0 | Assumption]]. +Assert H1 := (archimed ``/(2*eps)``). +Cut (Zle `0` ``(up (/(2*eps)))``). +Intro; Assert H3 := (IZN ``(up (/(2*eps)))`` H2). +Elim H3; Intros N H4. +Cut (lt O N). +Intro; Exists N; Intros. +Cut (lt O n). +Intro; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_right. +Unfold PI_tg; Apply Rlt_trans with ``/(INR (mult (S (S O)) n))``. +Apply Rlt_monotony_contra with ``(INR (mult (S (S O)) n))``. +Apply lt_INR_0. +Replace (mult (2) n) with (plus n n); [Idtac | Ring]. +Apply lt_le_trans with n. +Assumption. +Apply le_plus_l. +Rewrite <- Rinv_r_sym. +Apply Rlt_monotony_contra with ``(INR (plus (mult (S (S O)) n) (S O)))``. +Apply lt_INR_0. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. +Rewrite (Rmult_sym ``(INR (plus (mult (S (S O)) n) (S O)))``). +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Do 2 Rewrite Rmult_1r; Apply lt_INR. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_n_Sn | Ring]. +Apply not_O_INR; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Discriminate | Ring]. +Replace n with (S (pred n)). +Apply not_O_INR; Discriminate. +Symmetry; Apply S_pred with O. +Assumption. +Apply Rle_lt_trans with ``/(INR (mult (S (S O)) N))``. +Apply Rle_monotony_contra with ``(INR (mult (S (S O)) N))``. +Rewrite mult_INR; Apply Rmult_lt_pos; [Simpl; Sup0 | Apply lt_INR_0; Assumption]. +Rewrite <- Rinv_r_sym. +Apply Rle_monotony_contra with ``(INR (mult (S (S O)) n))``. +Rewrite mult_INR; Apply Rmult_lt_pos; [Simpl; Sup0 | Apply lt_INR_0; Assumption]. +Rewrite (Rmult_sym (INR (mult (S (S O)) n))); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Do 2 Rewrite Rmult_1r; Apply le_INR. +Apply mult_le; Assumption. +Replace n with (S (pred n)). +Apply not_O_INR; Discriminate. +Symmetry; Apply S_pred with O. +Assumption. +Replace N with (S (pred N)). +Apply not_O_INR; Discriminate. +Symmetry; Apply S_pred with O. +Assumption. +Rewrite mult_INR. +Rewrite Rinv_Rmult. +Replace (INR (S (S O))) with ``2``; [Idtac | Reflexivity]. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Idtac | DiscrR]. +Rewrite Rmult_1l; Apply Rlt_monotony_contra with (INR N). +Apply lt_INR_0; Assumption. +Rewrite <- Rinv_r_sym. +Apply Rlt_monotony_contra with ``/(2*eps)``. +Apply Rlt_Rinv; Assumption. +Rewrite Rmult_1r; Replace ``/(2*eps)*((INR N)*(2*eps))`` with ``(INR N)*((2*eps)*/(2*eps))``; [Idtac | Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Replace (INR N) with (IZR (INZ N)). +Rewrite <- H4. +Elim H1; Intros; Assumption. +Symmetry; Apply INR_IZR_INZ. +Apply prod_neq_R0; [DiscrR | Red; Intro; Rewrite H8 in H; Elim (Rlt_antirefl ? H)]. +Apply not_O_INR. +Red; Intro; Rewrite H8 in H5; Elim (lt_n_n ? H5). +Replace (INR (S (S O))) with ``2``; [DiscrR | Reflexivity]. +Apply not_O_INR. +Red; Intro; Rewrite H8 in H5; Elim (lt_n_n ? H5). +Apply Rle_sym1; Apply PI_tg_pos. +Apply lt_le_trans with N; Assumption. +Elim H1; Intros H5 _. +Assert H6 := (lt_eq_lt_dec O N). +Elim H6; Intro. +Elim a; Intro. +Assumption. +Rewrite <- b in H4. +Rewrite H4 in H5. +Simpl in H5. +Cut ``0</(2*eps)``; [Intro | Apply Rlt_Rinv; Assumption]. +Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H7 H5)). +Elim (lt_n_O ? b). +Apply le_IZR. +Simpl. +Left; Apply Rlt_trans with ``/(2*eps)``. +Apply Rlt_Rinv; Assumption. +Elim H1; Intros; Assumption. +Qed. + +Lemma exist_PI : (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 (tg_alt PI_tg) N) l)). +Apply alternated_series. +Apply PI_tg_decreasing. +Apply PI_tg_cv. +Qed. + +(* Now, PI is defined *) +Definition PI : R := (Rmult ``4`` (Cases exist_PI of (existTT a b) => a end)). + +(* We can get an approximation of PI with the following inequality *) +Lemma PI_ineq : (N:nat) ``(sum_f_R0 (tg_alt PI_tg) (S (mult (S (S O)) N)))<=PI/4<=(sum_f_R0 (tg_alt PI_tg) (mult (S (S O)) N))``. +Intro; Apply alternated_series_ineq. +Apply PI_tg_decreasing. +Apply PI_tg_cv. +Unfold PI; Case exist_PI; Intro. +Replace ``(4*x)/4`` with x. +Trivial. +Unfold Rdiv; Rewrite (Rmult_sym ``4``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r; Reflexivity | DiscrR]. +Qed. + +Lemma PI_RGT_0 : ``0<PI``. +Assert H := (PI_ineq O). +Apply Rlt_monotony_contra with ``/4``. +Apply Rlt_Rinv; Sup0. +Rewrite Rmult_Or; Rewrite Rmult_sym. +Elim H; Clear H; Intros H _. +Unfold Rdiv in H; Apply Rlt_le_trans with ``(sum_f_R0 (tg_alt PI_tg) (S (mult (S (S O)) O)))``. +Simpl; Unfold tg_alt; Simpl; Rewrite Rmult_1l; Rewrite Rmult_1r; Apply Rlt_anti_compatibility with ``(PI_tg (S O))``. +Rewrite Rplus_Or; Replace ``(PI_tg (S O))+((PI_tg O)+ -1*(PI_tg (S O)))`` with ``(PI_tg O)``; [Unfold PI_tg | Ring]. +Simpl; Apply Rinv_lt. +Rewrite Rmult_1l; Replace ``2+1`` with ``3``; [Sup0 | Ring]. +Rewrite Rplus_sym; Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Sup0. +Assumption. +Qed. diff --git a/theories7/Reals/ArithProp.v b/theories7/Reals/ArithProp.v new file mode 100644 index 00000000..468675ca --- /dev/null +++ b/theories7/Reals/ArithProp.v @@ -0,0 +1,134 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ArithProp.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rbasic_fun. +Require Even. +Require Div2. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope Z_scope. +Open Local Scope R_scope. + +Lemma minus_neq_O : (n,i:nat) (lt i n) -> ~(minus n i)=O. +Intros; Red; Intro. +Cut (n,m:nat) (le m n) -> (minus n m)=O -> n=m. +Intro; Assert H2 := (H1 ? ? (lt_le_weak ? ? H) H0); Rewrite H2 in H; Elim (lt_n_n ? H). +Pose R := [n,m:nat](le m n)->(minus n m)=(0)->n=m. +Cut ((n,m:nat)(R n m)) -> ((n0,m:nat)(le m n0)->(minus n0 m)=(0)->n0=m). +Intro; Apply H1. +Apply nat_double_ind. +Unfold R; Intros; Inversion H2; Reflexivity. +Unfold R; Intros; Simpl in H3; Assumption. +Unfold R; Intros; Simpl in H4; Assert H5 := (le_S_n ? ? H3); Assert H6 := (H2 H5 H4); Rewrite H6; Reflexivity. +Unfold R; Intros; Apply H1; Assumption. +Qed. + +Lemma le_minusni_n : (n,i:nat) (le i n)->(le (minus n i) n). +Pose R := [m,n:nat] (le n m) -> (le (minus m n) m). +Cut ((m,n:nat)(R m n)) -> ((n,i:nat)(le i n)->(le (minus n i) n)). +Intro; Apply H. +Apply nat_double_ind. +Unfold R; Intros; Simpl; Apply le_n. +Unfold R; Intros; Simpl; Apply le_n. +Unfold R; Intros; Simpl; Apply le_trans with n. +Apply H0; Apply le_S_n; Assumption. +Apply le_n_Sn. +Unfold R; Intros; Apply H; Assumption. +Qed. + +Lemma lt_minus_O_lt : (m,n:nat) (lt m n) -> (lt O (minus n m)). +Intros n m; Pattern n m; Apply nat_double_ind; [ + Intros; Rewrite <- minus_n_O; Assumption +| Intros; Elim (lt_n_O ? H) +| Intros; Simpl; Apply H; Apply lt_S_n; Assumption]. +Qed. + +Lemma even_odd_cor : (n:nat) (EX p : nat | n=(mult (2) p)\/n=(S (mult (2) p))). +Intro. +Assert H := (even_or_odd n). +Exists (div2 n). +Assert H0 := (even_odd_double n). +Elim H0; Intros. +Elim H1; Intros H3 _. +Elim H2; Intros H4 _. +Replace (mult (2) (div2 n)) with (Div2.double (div2 n)). +Elim H; Intro. +Left. +Apply H3; Assumption. +Right. +Apply H4; Assumption. +Unfold Div2.double; Ring. +Qed. + +(* 2m <= 2n => m<=n *) +Lemma le_double : (m,n:nat) (le (mult (2) m) (mult (2) n)) -> (le m n). +Intros; Apply INR_le. +Assert H1 := (le_INR ? ? H). +Do 2 Rewrite mult_INR in H1. +Apply Rle_monotony_contra with ``(INR (S (S O)))``. +Replace (INR (S (S O))) with ``2``; [Sup0 | Reflexivity]. +Assumption. +Qed. + +(* Here, we have the euclidian division *) +(* This lemma is used in the proof of sin_eq_0 : (sin x)=0<->x=kPI *) +Lemma euclidian_division : (x,y:R) ``y<>0`` -> (EXT k:Z | (EXT r : R | ``x==(IZR k)*y+r``/\``0<=r<(Rabsolu y)``)). +Intros. +Pose k0 := Cases (case_Rabsolu y) of + (leftT _) => (Zminus `1` (up ``x/-y``)) + | (rightT _) => (Zminus (up ``x/y``) `1`) end. +Exists k0. +Exists ``x-(IZR k0)*y``. +Split. +Ring. +Unfold k0; Case (case_Rabsolu y); Intro. +Assert H0 := (archimed ``x/-y``); Rewrite <- Z_R_minus; Simpl; Unfold Rminus. +Replace ``-((1+ -(IZR (up (x/( -y)))))*y)`` with ``((IZR (up (x/-y)))-1)*y``; [Idtac | Ring]. +Split. +Apply Rle_monotony_contra with ``/-y``. +Apply Rlt_Rinv; Apply Rgt_RO_Ropp; Exact r. +Rewrite Rmult_Or; Rewrite (Rmult_sym ``/-y``); Rewrite Rmult_Rplus_distrl; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. +Rewrite Rmult_assoc; Repeat Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]. +Apply Rle_anti_compatibility with ``(IZR (up (x/( -y))))-x/( -y)``. +Rewrite Rplus_Or; Unfold Rdiv; Pattern 4 ``/-y``; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. +Replace ``(IZR (up (x*/ -y)))-x* -/y+( -(x*/y)+ -((IZR (up (x*/ -y)))-1))`` with R1; [Idtac | Ring]. +Elim H0; Intros _ H1; Unfold Rdiv in H1; Exact H1. +Rewrite (Rabsolu_left ? r); Apply Rlt_monotony_contra with ``/-y``. +Apply Rlt_Rinv; Apply Rgt_RO_Ropp; Exact r. +Rewrite <- Rinv_l_sym. +Rewrite (Rmult_sym ``/-y``); Rewrite Rmult_Rplus_distrl; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. +Rewrite Rmult_assoc; Repeat Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rlt_anti_compatibility with ``((IZR (up (x/( -y))))-1)``. +Replace ``(IZR (up (x/( -y))))-1+1`` with ``(IZR (up (x/( -y))))``; [Idtac | Ring]. +Replace ``(IZR (up (x/( -y))))-1+( -(x*/y)+ -((IZR (up (x/( -y))))-1))`` with ``-(x*/y)``; [Idtac | Ring]. +Rewrite <- Ropp_mul3; Rewrite (Ropp_Rinv ? H); Elim H0; Unfold Rdiv; Intros H1 _; Exact H1. +Apply Ropp_neq; Assumption. +Assert H0 := (archimed ``x/y``); Rewrite <- Z_R_minus; Simpl; Cut ``0<y``. +Intro; Unfold Rminus; Replace ``-(((IZR (up (x/y)))+ -1)*y)`` with ``(1-(IZR (up (x/y))))*y``; [Idtac | Ring]. +Split. +Apply Rle_monotony_contra with ``/y``. +Apply Rlt_Rinv; Assumption. +Rewrite Rmult_Or; Rewrite (Rmult_sym ``/y``); Rewrite Rmult_Rplus_distrl; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rle_anti_compatibility with ``(IZR (up (x/y)))-x/y``; Rewrite Rplus_Or; Unfold Rdiv; Replace ``(IZR (up (x*/y)))-x*/y+(x*/y+(1-(IZR (up (x*/y)))))`` with R1; [Idtac | Ring]; Elim H0; Intros _ H2; Unfold Rdiv in H2; Exact H2. +Rewrite (Rabsolu_right ? r); Apply Rlt_monotony_contra with ``/y``. +Apply Rlt_Rinv; Assumption. +Rewrite <- (Rinv_l_sym ? H); Rewrite (Rmult_sym ``/y``); Rewrite Rmult_Rplus_distrl; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rlt_anti_compatibility with ``((IZR (up (x/y)))-1)``; Replace ``(IZR (up (x/y)))-1+1`` with ``(IZR (up (x/y)))``; [Idtac | Ring]; Replace ``(IZR (up (x/y)))-1+(x*/y+(1-(IZR (up (x/y)))))`` with ``x*/y``; [Idtac | Ring]; Elim H0; Unfold Rdiv; Intros H2 _; Exact H2. +Case (total_order_T R0 y); Intro. +Elim s; Intro. +Assumption. +Elim H; Symmetry; Exact b. +Assert H1 := (Rle_sym2 ? ? r); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 r0)). +Qed. + +Lemma tech8 : (n,i:nat) (le n (plus (S n) i)). +Intros; Induction i. +Replace (plus (S n) O) with (S n); [Apply le_n_Sn | Ring]. +Replace (plus (S n) (S i)) with (S (plus (S n) i)). +Apply le_S; Assumption. +Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. +Qed. diff --git a/theories7/Reals/Binomial.v b/theories7/Reals/Binomial.v new file mode 100644 index 00000000..1dfd2ec0 --- /dev/null +++ b/theories7/Reals/Binomial.v @@ -0,0 +1,181 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Binomial.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require PartSum. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Definition C [n,p:nat] : R := ``(INR (fact n))/((INR (fact p))*(INR (fact (minus n p))))``. + +Lemma pascal_step1 : (n,i:nat) (le i n) -> (C n i) == (C n (minus n i)). +Intros; Unfold C; Replace (minus n (minus n i)) with i. +Rewrite Rmult_sym. +Reflexivity. +Apply plus_minus; Rewrite plus_sym; Apply le_plus_minus; Assumption. +Qed. + +Lemma pascal_step2 : (n,i:nat) (le i n) -> (C (S n) i) == ``(INR (S n))/(INR (minus (S n) i))*(C n i)``. +Intros; Unfold C; Replace (minus (S n) i) with (S (minus n i)). +Cut (n:nat) (fact (S n))=(mult (S n) (fact n)). +Intro; Repeat Rewrite H0. +Unfold Rdiv; Repeat Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. +Ring. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply not_O_INR; Discriminate. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply prod_neq_R0. +Apply not_O_INR; Discriminate. +Apply INR_fact_neq_0. +Intro; Reflexivity. +Apply minus_Sn_m; Assumption. +Qed. + +Lemma pascal_step3 : (n,i:nat) (lt i n) -> (C n (S i)) == ``(INR (minus n i))/(INR (S i))*(C n i)``. +Intros; Unfold C. +Cut (n:nat) (fact (S n))=(mult (S n) (fact n)). +Intro. +Cut (minus n i) = (S (minus n (S i))). +Intro. +Pattern 2 (minus n i); Rewrite H1. +Repeat Rewrite H0; Unfold Rdiv; Repeat Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. +Rewrite <- H1; Rewrite (Rmult_sym ``/(INR (minus n i))``); Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym (INR (minus n i))); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Ring. +Apply not_O_INR; Apply minus_neq_O; Assumption. +Apply not_O_INR; Discriminate. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. +Apply not_O_INR; Discriminate. +Apply INR_fact_neq_0. +Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. +Apply INR_fact_neq_0. +Rewrite minus_Sn_m. +Simpl; Reflexivity. +Apply lt_le_S; Assumption. +Intro; Reflexivity. +Qed. + +(**********) +Lemma pascal : (n,i:nat) (lt i n) -> ``(C n i)+(C n (S i))==(C (S n) (S i))``. +Intros. +Rewrite pascal_step3; [Idtac | Assumption]. +Replace ``(C n i)+(INR (minus n i))/(INR (S i))*(C n i)`` with ``(C n i)*(1+(INR (minus n i))/(INR (S i)))``; [Idtac | Ring]. +Replace ``1+(INR (minus n i))/(INR (S i))`` with ``(INR (S n))/(INR (S i))``. +Rewrite pascal_step1. +Rewrite Rmult_sym; Replace (S i) with (minus (S n) (minus n i)). +Rewrite <- pascal_step2. +Apply pascal_step1. +Apply le_trans with n. +Apply le_minusni_n. +Apply lt_le_weak; Assumption. +Apply le_n_Sn. +Apply le_minusni_n. +Apply lt_le_weak; Assumption. +Rewrite <- minus_Sn_m. +Cut (minus n (minus n i))=i. +Intro; Rewrite H0; Reflexivity. +Symmetry; Apply plus_minus. +Rewrite plus_sym; Rewrite le_plus_minus_r. +Reflexivity. +Apply lt_le_weak; Assumption. +Apply le_minusni_n; Apply lt_le_weak; Assumption. +Apply lt_le_weak; Assumption. +Unfold Rdiv. +Repeat Rewrite S_INR. +Rewrite minus_INR. +Cut ``((INR i)+1)<>0``. +Intro. +Apply r_Rmult_mult with ``(INR i)+1``; [Idtac | Assumption]. +Rewrite Rmult_Rplus_distr. +Rewrite Rmult_1r. +Do 2 Rewrite (Rmult_sym ``(INR i)+1``). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym; [Idtac | Assumption]. +Ring. +Rewrite <- S_INR. +Apply not_O_INR; Discriminate. +Apply lt_le_weak; Assumption. +Qed. + +(*********************) +(*********************) +Lemma binomial : (x,y:R;n:nat) ``(pow (x+y) n)``==(sum_f_R0 [i:nat]``(C n i)*(pow x i)*(pow y (minus n i))`` n). +Intros; Induction n. +Unfold C; Simpl; Unfold Rdiv; Repeat Rewrite Rmult_1r; Rewrite Rinv_R1; Ring. +Pattern 1 (S n); Replace (S n) with (plus n (1)); [Idtac | Ring]. +Rewrite pow_add; Rewrite Hrecn. +Replace ``(pow (x+y) (S O))`` with ``x+y``; [Idtac | Simpl; Ring]. +Rewrite tech5. +Cut (p:nat)(C p p)==R1. +Cut (p:nat)(C p O)==R1. +Intros; Rewrite H0; Rewrite <- minus_n_n; Rewrite Rmult_1l. +Replace (pow y O) with R1; [Rewrite Rmult_1r | Simpl; Reflexivity]. +Induction n. +Simpl; Do 2 Rewrite H; Ring. +(* N >= 1 *) +Pose N := (S n). +Rewrite Rmult_Rplus_distr. +Replace (Rmult (sum_f_R0 ([i:nat]``(C N i)*(pow x i)*(pow y (minus N i))``) N) x) with (sum_f_R0 [i:nat]``(C N i)*(pow x (S i))*(pow y (minus N i))`` N). +Replace (Rmult (sum_f_R0 ([i:nat]``(C N i)*(pow x i)*(pow y (minus N i))``) N) y) with (sum_f_R0 [i:nat]``(C N i)*(pow x i)*(pow y (minus (S N) i))`` N). +Rewrite (decomp_sum [i:nat]``(C (S N) i)*(pow x i)*(pow y (minus (S N) i))`` N). +Rewrite H; Replace (pow x O) with R1; [Idtac | Reflexivity]. +Do 2 Rewrite Rmult_1l. +Replace (minus (S N) O) with (S N); [Idtac | Reflexivity]. +Pose An := [i:nat]``(C N i)*(pow x (S i))*(pow y (minus N i))``. +Pose Bn := [i:nat]``(C N (S i))*(pow x (S i))*(pow y (minus N i))``. +Replace (pred N) with n. +Replace (sum_f_R0 ([i:nat]``(C (S N) (S i))*(pow x (S i))*(pow y (minus (S N) (S i)))``) n) with (sum_f_R0 [i:nat]``(An i)+(Bn i)`` n). +Rewrite plus_sum. +Replace (pow x (S N)) with (An (S n)). +Rewrite (Rplus_sym (sum_f_R0 An n)). +Repeat Rewrite Rplus_assoc. +Rewrite <- tech5. +Fold N. +Pose Cn := [i:nat]``(C N i)*(pow x i)*(pow y (minus (S N) i))``. +Cut (i:nat) (lt i N)-> (Cn (S i))==(Bn i). +Intro; Replace (sum_f_R0 Bn n) with (sum_f_R0 [i:nat](Cn (S i)) n). +Replace (pow y (S N)) with (Cn O). +Rewrite <- Rplus_assoc; Rewrite (decomp_sum Cn N). +Replace (pred N) with n. +Ring. +Unfold N; Simpl; Reflexivity. +Unfold N; Apply lt_O_Sn. +Unfold Cn; Rewrite H; Simpl; Ring. +Apply sum_eq. +Intros; Apply H1. +Unfold N; Apply le_lt_trans with n; [Assumption | Apply lt_n_Sn]. +Intros; Unfold Bn Cn. +Replace (minus (S N) (S i)) with (minus N i); Reflexivity. +Unfold An; Fold N; Rewrite <- minus_n_n; Rewrite H0; Simpl; Ring. +Apply sum_eq. +Intros; Unfold An Bn; Replace (minus (S N) (S i)) with (minus N i); [Idtac | Reflexivity]. +Rewrite <- pascal; [Ring | Apply le_lt_trans with n; [Assumption | Unfold N; Apply lt_n_Sn]]. +Unfold N; Reflexivity. +Unfold N; Apply lt_O_Sn. +Rewrite <- (Rmult_sym y); Rewrite scal_sum; Apply sum_eq. +Intros; Replace (minus (S N) i) with (S (minus N i)). +Replace (S (minus N i)) with (plus (minus N i) (1)); [Idtac | Ring]. +Rewrite pow_add; Replace (pow y (S O)) with y; [Idtac | Simpl; Ring]; Ring. +Apply minus_Sn_m; Assumption. +Rewrite <- (Rmult_sym x); Rewrite scal_sum; Apply sum_eq. +Intros; Replace (S i) with (plus i (1)); [Idtac | Ring]; Rewrite pow_add; Replace (pow x (S O)) with x; [Idtac | Simpl; Ring]; Ring. +Intro; Unfold C. +Replace (INR (fact O)) with R1; [Idtac | Reflexivity]. +Replace (minus p O) with p; [Idtac | Apply minus_n_O]. +Rewrite Rmult_1l; Unfold Rdiv; Rewrite <- Rinv_r_sym; [Reflexivity | Apply INR_fact_neq_0]. +Intro; Unfold C. +Replace (minus p p) with O; [Idtac | Apply minus_n_n]. +Replace (INR (fact O)) with R1; [Idtac | Reflexivity]. +Rewrite Rmult_1r; Unfold Rdiv; Rewrite <- Rinv_r_sym; [Reflexivity | Apply INR_fact_neq_0]. +Qed. diff --git a/theories7/Reals/Cauchy_prod.v b/theories7/Reals/Cauchy_prod.v new file mode 100644 index 00000000..9442eff0 --- /dev/null +++ b/theories7/Reals/Cauchy_prod.v @@ -0,0 +1,347 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Cauchy_prod.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require PartSum. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(**********) +Lemma sum_N_predN : (An:nat->R;N:nat) (lt O N) -> (sum_f_R0 An N)==``(sum_f_R0 An (pred N)) + (An N)``. +Intros. +Replace N with (S (pred N)). +Rewrite tech5. +Reflexivity. +Symmetry; Apply S_pred with O; Assumption. +Qed. + +(**********) +Lemma sum_plus : (An,Bn:nat->R;N:nat) (sum_f_R0 [l:nat]``(An l)+(Bn l)`` N)==``(sum_f_R0 An N)+(sum_f_R0 Bn N)``. +Intros. +Induction N. +Reflexivity. +Do 3 Rewrite tech5. +Rewrite HrecN; Ring. +Qed. + +(* The main result *) +Theorem cauchy_finite : (An,Bn:nat->R;N:nat) (lt O N) -> (Rmult (sum_f_R0 An N) (sum_f_R0 Bn N)) == (Rplus (sum_f_R0 [k:nat](sum_f_R0 [p:nat]``(An p)*(Bn (minus k p))`` k) N) (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N))). +Intros; Induction N. +Elim (lt_n_n ? H). +Cut N=O\/(lt O N). +Intro; Elim H0; Intro. +Rewrite H1; Simpl; Ring. +Replace (pred (S N)) with (S (pred N)). +Do 5 Rewrite tech5. +Rewrite Rmult_Rplus_distrl; Rewrite Rmult_Rplus_distr; Rewrite (HrecN H1). +Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Replace (pred (minus (S N) (S (pred N)))) with (O). +Rewrite Rmult_Rplus_distr; Replace (sum_f_R0 [l:nat]``(An (S (plus l (S (pred N)))))*(Bn (minus (S N) l))`` O) with ``(An (S N))*(Bn (S N))``. +Repeat Rewrite <- Rplus_assoc; Do 2 Rewrite <- (Rplus_sym ``(An (S N))*(Bn (S N))``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Rewrite <- minus_n_n; Cut N=(1)\/(le (2) N). +Intro; Elim H2; Intro. +Rewrite H3; Simpl; Ring. +Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))) (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N))). +Replace (sum_f_R0 [p:nat]``(An p)*(Bn (minus (S N) p))`` N) with (Rplus (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N)) ``(An O)*(Bn (S N))``). +Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N))); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus (S N) l))`` (pred (minus (S N) k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) (Rmult (Bn (S N)) (sum_f_R0 [l:nat](An (S l)) (pred N)))). +Rewrite (decomp_sum An N H1); Rewrite Rmult_Rplus_distrl; Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym ``(An O)*(Bn (S N))``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym (Rmult (sum_f_R0 [i:nat](An (S i)) (pred N)) (Bn (S N)))); Rewrite <- (Rplus_sym (Rmult (Bn (S N)) (sum_f_R0 [i:nat](An (S i)) (pred N)))); Rewrite (Rmult_sym (Bn (S N))); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))) (Rmult (An (S N)) (sum_f_R0 [l:nat](Bn (S l)) (pred N)))). +Rewrite (decomp_sum Bn N H1); Rewrite Rmult_Rplus_distr. +Pose Z := (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))); Pose Z2 := (sum_f_R0 [i:nat](Bn (S i)) (pred N)); Ring. +Rewrite (sum_N_predN [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)). +Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred (pred N))) with (sum_f_R0 [k:nat](Rplus (sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) ``(An (S N))*(Bn (S k))``) (pred (pred N))). +Rewrite (sum_plus [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) [k:nat]``(An (S N))*(Bn (S k))`` (pred (pred N))). +Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Replace (pred (minus N (pred N))) with O. +Simpl; Rewrite <- minus_n_O. +Replace (S (pred N)) with N. +Replace (sum_f_R0 [k:nat]``(An (S N))*(Bn (S k))`` (pred (pred N))) with (sum_f_R0 [k:nat]``(Bn (S k))*(An (S N))`` (pred (pred N))). +Rewrite <- (scal_sum [l:nat](Bn (S l)) (pred (pred N)) (An (S N))); Rewrite (sum_N_predN [l:nat](Bn (S l)) (pred N)). +Replace (S (pred N)) with N. +Ring. +Apply S_pred with O; Assumption. +Apply lt_pred; Apply lt_le_trans with (2); [Apply lt_n_Sn | Assumption]. +Apply sum_eq; Intros; Apply Rmult_sym. +Apply S_pred with O; Assumption. +Replace (minus N (pred N)) with (1). +Reflexivity. +Pattern 1 N; Replace N with (S (pred N)). +Rewrite <- minus_Sn_m. +Rewrite <- minus_n_n; Reflexivity. +Apply le_n. +Symmetry; Apply S_pred with O; Assumption. +Apply sum_eq; Intros; Rewrite (sum_N_predN [l:nat]``(An (S (S (plus l i))))*(Bn (minus N l))`` (pred (minus N i))). +Replace (S (S (plus (pred (minus N i)) i))) with (S N). +Replace (minus N (pred (minus N i))) with (S i). +Ring. +Rewrite pred_of_minus; Apply INR_eq; Repeat Rewrite minus_INR. +Rewrite S_INR; Ring. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Apply INR_le; Rewrite minus_INR. +Apply Rle_anti_compatibility with ``(INR i)-1``. +Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring]. +Replace ``(INR i)-1+((INR N)-(INR i))`` with ``(INR N)-(INR (S O))``; [Idtac | Ring]. +Rewrite <- minus_INR. +Apply le_INR; Apply le_trans with (pred (pred N)). +Assumption. +Rewrite <- pred_of_minus; Apply le_pred_n. +Apply le_trans with (2). +Apply le_n_Sn. +Assumption. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Rewrite <- pred_of_minus. +Apply le_trans with (pred N). +Apply le_S_n. +Replace (S (pred N)) with N. +Replace (S (pred (minus N i))) with (minus N i). +Apply simpl_le_plus_l with i; Rewrite le_plus_minus_r. +Apply le_plus_r. +Apply le_trans with (pred (pred N)); [Assumption | Apply le_trans with (pred N); Apply le_pred_n]. +Apply S_pred with O. +Apply simpl_lt_plus_l with i; Rewrite le_plus_minus_r. +Replace (plus i O) with i; [Idtac | Ring]. +Apply le_lt_trans with (pred (pred N)); [Assumption | Apply lt_trans with (pred N); Apply lt_pred_n_n]. +Apply lt_S_n. +Replace (S (pred N)) with N. +Apply lt_le_trans with (2). +Apply lt_n_Sn. +Assumption. +Apply S_pred with O; Assumption. +Assumption. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Apply S_pred with O; Assumption. +Apply le_pred_n. +Apply INR_eq; Rewrite pred_of_minus; Do 3 Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite minus_INR. +Ring. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Apply INR_le. +Rewrite minus_INR. +Apply Rle_anti_compatibility with ``(INR i)-1``. +Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring]. +Replace ``(INR i)-1+((INR N)-(INR i))`` with ``(INR N)-(INR (S O))``; [Idtac | Ring]. +Rewrite <- minus_INR. +Apply le_INR. +Apply le_trans with (pred (pred N)). +Assumption. +Rewrite <- pred_of_minus. +Apply le_pred_n. +Apply le_trans with (2). +Apply le_n_Sn. +Assumption. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Apply lt_le_trans with (1). +Apply lt_O_Sn. +Apply INR_le. +Rewrite pred_of_minus. +Repeat Rewrite minus_INR. +Apply Rle_anti_compatibility with ``(INR i)-1``. +Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring]. +Replace ``(INR i)-1+((INR N)-(INR i)-(INR (S O)))`` with ``(INR N)-(INR (S O)) -(INR (S O))``. +Repeat Rewrite <- minus_INR. +Apply le_INR. +Apply le_trans with (pred (pred N)). +Assumption. +Do 2 Rewrite <- pred_of_minus. +Apply le_n. +Apply simpl_le_plus_l with (1). +Rewrite le_plus_minus_r. +Simpl; Assumption. +Apply le_trans with (2); [Apply le_n_Sn | Assumption]. +Apply le_trans with (2); [Apply le_n_Sn | Assumption]. +Ring. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Apply simpl_le_plus_l with i. +Rewrite le_plus_minus_r. +Replace (plus i (1)) with (S i). +Replace N with (S (pred N)). +Apply le_n_S. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_pred_n. +Symmetry; Apply S_pred with O; Assumption. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Reflexivity. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Apply lt_le_trans with (1). +Apply lt_O_Sn. +Apply le_S_n. +Replace (S (pred N)) with N. +Assumption. +Apply S_pred with O; Assumption. +Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus (S N) l))`` (pred (minus (S N) k))) (pred N)) with (sum_f_R0 [k:nat](Rplus (sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) ``(An (S k))*(Bn (S N))``) (pred N)). +Rewrite (sum_plus [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) [k:nat]``(An (S k))*(Bn (S N))``). +Apply Rplus_plus_r. +Rewrite scal_sum; Reflexivity. +Apply sum_eq; Intros; Rewrite Rplus_sym; Rewrite (decomp_sum [l:nat]``(An (S (plus l i)))*(Bn (minus (S N) l))`` (pred (minus (S N) i))). +Replace (plus O i) with i; [Idtac | Ring]. +Rewrite <- minus_n_O; Apply Rplus_plus_r. +Replace (pred (pred (minus (S N) i))) with (pred (minus N i)). +Apply sum_eq; Intros. +Replace (minus (S N) (S i0)) with (minus N i0); [Idtac | Reflexivity]. +Replace (plus (S i0) i) with (S (plus i0 i)). +Reflexivity. +Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring. +Cut (minus N i)=(pred (minus (S N) i)). +Intro; Rewrite H5; Reflexivity. +Rewrite pred_of_minus. +Apply INR_eq; Repeat Rewrite minus_INR. +Rewrite S_INR; Ring. +Apply le_trans with N. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply simpl_le_plus_l with i. +Rewrite le_plus_minus_r. +Replace (plus i (1)) with (S i). +Apply le_n_S. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_trans with N. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Replace (pred (minus (S N) i)) with (minus (S N) (S i)). +Replace (minus (S N) (S i)) with (minus N i); [Idtac | Reflexivity]. +Apply simpl_lt_plus_l with i. +Rewrite le_plus_minus_r. +Replace (plus i O) with i; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n. +Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Rewrite pred_of_minus. +Apply INR_eq; Repeat Rewrite minus_INR. +Repeat Rewrite S_INR; Ring. +Apply le_trans with N. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply simpl_le_plus_l with i. +Rewrite le_plus_minus_r. +Replace (plus i (1)) with (S i). +Apply le_n_S. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_trans with N. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply le_n_S. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Rewrite Rplus_sym. +Rewrite (decomp_sum [p:nat]``(An p)*(Bn (minus (S N) p))`` N). +Rewrite <- minus_n_O. +Apply Rplus_plus_r. +Apply sum_eq; Intros. +Reflexivity. +Assumption. +Rewrite Rplus_sym. +Rewrite (decomp_sum [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N)). +Rewrite <- minus_n_O. +Replace (sum_f_R0 [l:nat]``(An (S (plus l O)))*(Bn (minus N l))`` (pred N)) with (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N)). +Apply Rplus_plus_r. +Apply sum_eq; Intros. +Replace (pred (minus N (S i))) with (pred (pred (minus N i))). +Apply sum_eq; Intros. +Replace (plus i0 (S i)) with (S (plus i0 i)). +Reflexivity. +Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring. +Cut (pred (minus N i))=(minus N (S i)). +Intro; Rewrite H5; Reflexivity. +Rewrite pred_of_minus. +Apply INR_eq. +Repeat Rewrite minus_INR. +Repeat Rewrite S_INR; Ring. +Apply le_trans with (S (pred (pred N))). +Apply le_n_S; Assumption. +Replace (S (pred (pred N))) with (pred N). +Apply le_pred_n. +Apply S_pred with O. +Apply lt_S_n. +Replace (S (pred N)) with N. +Apply lt_le_trans with (2). +Apply lt_n_Sn. +Assumption. +Apply S_pred with O; Assumption. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Apply simpl_le_plus_l with i. +Rewrite le_plus_minus_r. +Replace (plus i (1)) with (S i). +Replace N with (S (pred N)). +Apply le_n_S. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_pred_n. +Symmetry; Apply S_pred with O; Assumption. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_trans with (pred (pred N)). +Assumption. +Apply le_trans with (pred N); Apply le_pred_n. +Apply sum_eq; Intros. +Replace (plus i O) with i; [Reflexivity | Trivial]. +Apply lt_S_n. +Replace (S (pred N)) with N. +Apply lt_le_trans with (2); [Apply lt_n_Sn | Assumption]. +Apply S_pred with O; Assumption. +Inversion H1. +Left; Reflexivity. +Right; Apply le_n_S; Assumption. +Simpl. +Replace (S (pred N)) with N. +Reflexivity. +Apply S_pred with O; Assumption. +Simpl. +Cut (minus N (pred N))=(1). +Intro; Rewrite H2; Reflexivity. +Rewrite pred_of_minus. +Apply INR_eq; Repeat Rewrite minus_INR. +Ring. +Apply lt_le_S; Assumption. +Rewrite <- pred_of_minus; Apply le_pred_n. +Simpl; Symmetry; Apply S_pred with O; Assumption. +Inversion H. +Left; Reflexivity. +Right; Apply lt_le_trans with (1); [Apply lt_n_Sn | Exact H1]. +Qed. diff --git a/theories7/Reals/Cos_plus.v b/theories7/Reals/Cos_plus.v new file mode 100644 index 00000000..481e51bf --- /dev/null +++ b/theories7/Reals/Cos_plus.v @@ -0,0 +1,1017 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Cos_plus.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo_def. +Require Cos_rel. +Require Max. +V7only [Import nat_scope.]. Open Local Scope nat_scope. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Definition Majxy [x,y:R] : nat->R := [n:nat](Rdiv (pow (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (4) (S n))) (INR (fact n))). + +Lemma Majxy_cv_R0 : (x,y:R) (Un_cv (Majxy x y) R0). +Intros. +Pose C := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). +Pose C0 := (pow C (4)). +Cut ``0<C``. +Intro. +Cut ``0<C0``. +Intro. +Assert H1 := (cv_speed_pow_fact C0). +Unfold Un_cv in H1; Unfold R_dist in H1. +Unfold Un_cv; Unfold R_dist; Intros. +Cut ``0<eps/C0``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Assumption]]. +Elim (H1 ``eps/C0`` H3); Intros N0 H4. +Exists N0; Intros. +Replace (Majxy x y n) with ``(pow C0 (S n))/(INR (fact n))``. +Simpl. +Apply Rlt_monotony_contra with ``(Rabsolu (/C0))``. +Apply Rabsolu_pos_lt. +Apply Rinv_neq_R0. +Red; Intro; Rewrite H6 in H0; Elim (Rlt_antirefl ? H0). +Rewrite <- Rabsolu_mult. +Unfold Rminus; Rewrite Rmult_Rplus_distr. +Rewrite Ropp_O; Rewrite Rmult_Or. +Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Rewrite (Rabsolu_right ``/C0``). +Rewrite <- (Rmult_sym eps). +Replace ``(pow C0 n)*/(INR (fact n))+0`` with ``(pow C0 n)*/(INR (fact n))-0``; [Idtac | Ring]. +Unfold Rdiv in H4; Apply H4; Assumption. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Assumption. +Red; Intro; Rewrite H6 in H0; Elim (Rlt_antirefl ? H0). +Unfold Majxy. +Unfold C0. +Rewrite pow_mult. +Unfold C; Reflexivity. +Unfold C0; Apply pow_lt; Assumption. +Apply Rlt_le_trans with R1. +Apply Rlt_R0_R1. +Unfold C. +Apply RmaxLess1. +Qed. + +Lemma reste1_maj : (x,y:R;N:nat) (lt O N) -> ``(Rabsolu (Reste1 x y N))<=(Majxy x y (pred N))``. +Intros. +Pose C := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). +Unfold Reste1. +Apply Rle_trans with (sum_f_R0 + [k:nat] + (Rabsolu (sum_f_R0 + [l:nat] + ``(pow ( -1) (S (plus l k)))/ + (INR (fact (mult (S (S O)) (S (plus l k)))))* + (pow x (mult (S (S O)) (S (plus l k))))* + (pow ( -1) (minus N l))/ + (INR (fact (mult (S (S O)) (minus N l))))* + (pow y (mult (S (S O)) (minus N l)))`` (pred (minus N k)))) + (pred N)). +Apply (sum_Rabsolu [k:nat] + (sum_f_R0 + [l:nat] + ``(pow ( -1) (S (plus l k)))/ + (INR (fact (mult (S (S O)) (S (plus l k)))))* + (pow x (mult (S (S O)) (S (plus l k))))* + (pow ( -1) (minus N l))/ + (INR (fact (mult (S (S O)) (minus N l))))* + (pow y (mult (S (S O)) (minus N l)))`` (pred (minus N k))) (pred N)). +Apply Rle_trans with (sum_f_R0 + [k:nat] + (sum_f_R0 + [l:nat] + (Rabsolu (``(pow ( -1) (S (plus l k)))/ + (INR (fact (mult (S (S O)) (S (plus l k)))))* + (pow x (mult (S (S O)) (S (plus l k))))* + (pow ( -1) (minus N l))/ + (INR (fact (mult (S (S O)) (minus N l))))* + (pow y (mult (S (S O)) (minus N l)))``)) (pred (minus N k))) + (pred N)). +Apply sum_Rle. +Intros. +Apply (sum_Rabsolu [l:nat] + ``(pow ( -1) (S (plus l n)))/ + (INR (fact (mult (S (S O)) (S (plus l n)))))* + (pow x (mult (S (S O)) (S (plus l n))))* + (pow ( -1) (minus N l))/ + (INR (fact (mult (S (S O)) (minus N l))))* + (pow y (mult (S (S O)) (minus N l)))`` (pred (minus N n))). +Apply Rle_trans with (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(INR (mult (fact (mult (S (S O)) (S (plus l k)))) (fact (mult (S (S O)) (minus N l)))))*(pow C (mult (S (S O)) (S (plus N k))))`` (pred (minus N k))) (pred N)). +Apply sum_Rle; Intros. +Apply sum_Rle; Intros. +Unfold Rdiv; Repeat Rewrite Rabsolu_mult. +Do 2 Rewrite pow_1_abs. +Do 2 Rewrite Rmult_1l. +Rewrite (Rabsolu_right ``/(INR (fact (mult (S (S O)) (S (plus n0 n)))))``). +Rewrite (Rabsolu_right ``/(INR (fact (mult (S (S O)) (minus N n0))))``). +Rewrite mult_INR. +Rewrite Rinv_Rmult. +Repeat Rewrite Rmult_assoc. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Rewrite <- Rmult_assoc. +Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (minus N n0))))``). +Rewrite Rmult_assoc. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Do 2 Rewrite <- Pow_Rabsolu. +Apply Rle_trans with ``(pow (Rabsolu x) (mult (S (S O)) (S (plus n0 n))))*(pow C (mult (S (S O)) (minus N n0)))``. +Apply Rle_monotony. +Apply pow_le; Apply Rabsolu_pos. +Apply pow_incr. +Split. +Apply Rabsolu_pos. +Unfold C. +Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)); Apply RmaxLess2. +Apply Rle_trans with ``(pow C (mult (S (S O)) (S (plus n0 n))))*(pow C (mult (S (S O)) (minus N n0)))``. +Do 2 Rewrite <- (Rmult_sym ``(pow C (mult (S (S O)) (minus N n0)))``). +Apply Rle_monotony. +Apply pow_le. +Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Unfold C; Apply RmaxLess1. +Apply pow_incr. +Split. +Apply Rabsolu_pos. +Unfold C; Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)). +Apply RmaxLess1. +Apply RmaxLess2. +Right. +Replace (mult (2) (S (plus N n))) with (plus (mult (2) (minus N n0)) (mult (2) (S (plus n0 n)))). +Rewrite pow_add. +Apply Rmult_sym. +Apply INR_eq; Rewrite plus_INR; Do 3 Rewrite mult_INR. +Rewrite minus_INR. +Repeat Rewrite S_INR; Do 2 Rewrite plus_INR; Ring. +Apply le_trans with (pred (minus N n)). +Exact H1. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_trans with (sum_f_R0 + [k:nat] + (sum_f_R0 + [l:nat] + ``/(INR + (mult (fact (mult (S (S O)) (S (plus l k)))) + (fact (mult (S (S O)) (minus N l)))))* + (pow C (mult (S (S (S (S O)))) N))`` (pred (minus N k))) + (pred N)). +Apply sum_Rle; Intros. +Apply sum_Rle; Intros. +Apply Rle_monotony. +Left; Apply Rlt_Rinv. +Rewrite mult_INR; Apply Rmult_lt_pos; Apply INR_fact_lt_0. +Apply Rle_pow. +Unfold C; Apply RmaxLess1. +Replace (mult (4) N) with (mult (2) (mult (2) N)); [Idtac | Ring]. +Apply mult_le. +Replace (mult (2) N) with (S (plus N (pred N))). +Apply le_n_S. +Apply le_reg_l; Assumption. +Rewrite pred_of_minus. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Rewrite minus_INR. +Repeat Rewrite S_INR; Ring. +Apply lt_le_S; Assumption. +Apply Rle_trans with (sum_f_R0 + [k:nat] + (sum_f_R0 + [l:nat] + ``(pow C (mult (S (S (S (S O)))) N))*(Rsqr (/(INR (fact (S (plus N k))))))`` (pred (minus N k))) + (pred N)). +Apply sum_Rle; Intros. +Apply sum_Rle; Intros. +Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) N))``). +Apply Rle_monotony. +Apply pow_le. +Left; Apply Rlt_le_trans with R1. +Apply Rlt_R0_R1. +Unfold C; Apply RmaxLess1. +Replace ``/(INR + (mult (fact (mult (S (S O)) (S (plus n0 n)))) + (fact (mult (S (S O)) (minus N n0)))))`` with ``(Binomial.C (mult (S (S O)) (S (plus N n))) (mult (S (S O)) (S (plus n0 n))))/(INR (fact (mult (S (S O)) (S (plus N n)))))``. +Apply Rle_trans with ``(Binomial.C (mult (S (S O)) (S (plus N n))) (S (plus N n)))/(INR (fact (mult (S (S O)) (S (plus N n)))))``. +Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (S (plus N n)))))``). +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply C_maj. +Apply mult_le. +Apply le_n_S. +Apply le_reg_r. +Apply le_trans with (pred (minus N n)). +Assumption. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Right. +Unfold Rdiv; Rewrite Rmult_sym. +Unfold Binomial.C. +Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Replace (minus (mult (2) (S (plus N n))) (S (plus N n))) with (S (plus N n)). +Rewrite Rinv_Rmult. +Unfold Rsqr; Reflexivity. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_eq; Rewrite S_INR; Rewrite minus_INR. +Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_n_2n. +Apply INR_fact_neq_0. +Unfold Rdiv; Rewrite Rmult_sym. +Unfold Binomial.C. +Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Replace (minus (mult (2) (S (plus N n))) (mult (2) (S (plus n0 n)))) with (mult (2) (minus N n0)). +Rewrite mult_INR. +Reflexivity. +Apply INR_eq; Rewrite minus_INR. +Do 3 Rewrite mult_INR; Repeat Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite minus_INR. +Ring. +Apply le_trans with (pred (minus N n)). +Assumption. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply mult_le. +Apply le_n_S. +Apply le_reg_r. +Apply le_trans with (pred (minus N n)). +Assumption. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply INR_fact_neq_0. +Apply Rle_trans with (sum_f_R0 [k:nat]``(INR N)/(INR (fact (S N)))*(pow C (mult (S (S (S (S O)))) N))`` (pred N)). +Apply sum_Rle; Intros. +Rewrite <- (scal_sum [_:nat]``(pow C (mult (S (S (S (S O)))) N))`` (pred (minus N n)) ``(Rsqr (/(INR (fact (S (plus N n))))))``). +Rewrite sum_cte. +Rewrite <- Rmult_assoc. +Do 2 Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) N))``). +Rewrite Rmult_assoc. +Apply Rle_monotony. +Apply pow_le. +Left; Apply Rlt_le_trans with R1. +Apply Rlt_R0_R1. +Unfold C; Apply RmaxLess1. +Apply Rle_trans with ``(Rsqr (/(INR (fact (S (plus N n))))))*(INR N)``. +Apply Rle_monotony. +Apply pos_Rsqr. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_INR. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Rewrite Rmult_sym; Unfold Rdiv; Apply Rle_monotony. +Apply pos_INR. +Apply Rle_trans with ``/(INR (fact (S (plus N n))))``. +Pattern 2 ``/(INR (fact (S (plus N n))))``; Rewrite <- Rmult_1r. +Unfold Rsqr. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_monotony_contra with ``(INR (fact (S (plus N n))))``. +Apply INR_fact_lt_0. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r. +Replace R1 with (INR (S O)). +Apply le_INR. +Apply lt_le_S. +Apply INR_lt; Apply INR_fact_lt_0. +Reflexivity. +Apply INR_fact_neq_0. +Apply Rle_monotony_contra with ``(INR (fact (S (plus N n))))``. +Apply INR_fact_lt_0. +Rewrite <- Rinv_r_sym. +Apply Rle_monotony_contra with ``(INR (fact (S N)))``. +Apply INR_fact_lt_0. +Rewrite Rmult_1r. +Rewrite (Rmult_sym (INR (fact (S N)))). +Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Apply le_INR. +Apply fact_growing. +Apply le_n_S. +Apply le_plus_l. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Rewrite sum_cte. +Apply Rle_trans with ``(pow C (mult (S (S (S (S O)))) N))/(INR (fact (pred N)))``. +Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) N))``). +Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. +Apply pow_le. +Left; Apply Rlt_le_trans with R1. +Apply Rlt_R0_R1. +Unfold C; Apply RmaxLess1. +Cut (S (pred N)) = N. +Intro; Rewrite H0. +Pattern 2 N; Rewrite <- H0. +Do 2 Rewrite fact_simpl. +Rewrite H0. +Repeat Rewrite mult_INR. +Repeat Rewrite Rinv_Rmult. +Rewrite (Rmult_sym ``/(INR (S N))``). +Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l. +Pattern 2 ``/(INR (fact (pred N)))``; Rewrite <- Rmult_1r. +Rewrite Rmult_assoc. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_monotony_contra with (INR (S N)). +Apply lt_INR_0; Apply lt_O_Sn. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Rewrite Rmult_1l. +Apply le_INR; Apply le_n_Sn. +Apply not_O_INR; Discriminate. +Apply not_O_INR. +Red; Intro; Rewrite H1 in H; Elim (lt_n_n ? H). +Apply not_O_INR. +Red; Intro; Rewrite H1 in H; Elim (lt_n_n ? H). +Apply INR_fact_neq_0. +Apply not_O_INR; Discriminate. +Apply prod_neq_R0. +Apply not_O_INR. +Red; Intro; Rewrite H1 in H; Elim (lt_n_n ? H). +Apply INR_fact_neq_0. +Symmetry; Apply S_pred with O; Assumption. +Right. +Unfold Majxy. +Unfold C. +Replace (S (pred N)) with N. +Reflexivity. +Apply S_pred with O; Assumption. +Qed. + +Lemma reste2_maj : (x,y:R;N:nat) (lt O N) -> ``(Rabsolu (Reste2 x y N))<=(Majxy x y N)``. +Intros. +Pose C := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). +Unfold Reste2. +Apply Rle_trans with (sum_f_R0 + [k:nat] + (Rabsolu (sum_f_R0 + [l:nat] + ``(pow ( -1) (S (plus l k)))/ + (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))* + (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))* + (pow ( -1) (minus N l))/ + (INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))* + (pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N k)))) + (pred N)). +Apply (sum_Rabsolu [k:nat] + (sum_f_R0 + [l:nat] + ``(pow ( -1) (S (plus l k)))/ + (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))* + (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))* + (pow ( -1) (minus N l))/ + (INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))* + (pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N k))) (pred N)). +Apply Rle_trans with (sum_f_R0 + [k:nat] + (sum_f_R0 + [l:nat] + (Rabsolu (``(pow ( -1) (S (plus l k)))/ + (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))* + (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))* + (pow ( -1) (minus N l))/ + (INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))* + (pow y (plus (mult (S (S O)) (minus N l)) (S O)))``)) (pred (minus N k))) + (pred N)). +Apply sum_Rle. +Intros. +Apply (sum_Rabsolu [l:nat] + ``(pow ( -1) (S (plus l n)))/ + (INR (fact (plus (mult (S (S O)) (S (plus l n))) (S O))))* + (pow x (plus (mult (S (S O)) (S (plus l n))) (S O)))* + (pow ( -1) (minus N l))/ + (INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))* + (pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N n))). +Apply Rle_trans with (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(INR (mult (fact (plus (mult (S (S O)) (S (plus l k))) (S O))) (fact (plus (mult (S (S O)) (minus N l)) (S O)))))*(pow C (mult (S (S O)) (S (S (plus N k)))))`` (pred (minus N k))) (pred N)). +Apply sum_Rle; Intros. +Apply sum_Rle; Intros. +Unfold Rdiv; Repeat Rewrite Rabsolu_mult. +Do 2 Rewrite pow_1_abs. +Do 2 Rewrite Rmult_1l. +Rewrite (Rabsolu_right ``/(INR (fact (plus (mult (S (S O)) (S (plus n0 n))) (S O))))``). +Rewrite (Rabsolu_right ``/(INR (fact (plus (mult (S (S O)) (minus N n0)) (S O))))``). +Rewrite mult_INR. +Rewrite Rinv_Rmult. +Repeat Rewrite Rmult_assoc. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Rewrite <- Rmult_assoc. +Rewrite <- (Rmult_sym ``/(INR (fact (plus (mult (S (S O)) (minus N n0)) (S O))))``). +Rewrite Rmult_assoc. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Do 2 Rewrite <- Pow_Rabsolu. +Apply Rle_trans with ``(pow (Rabsolu x) (plus (mult (S (S O)) (S (plus n0 n))) (S O)))*(pow C (plus (mult (S (S O)) (minus N n0)) (S O)))``. +Apply Rle_monotony. +Apply pow_le; Apply Rabsolu_pos. +Apply pow_incr. +Split. +Apply Rabsolu_pos. +Unfold C. +Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)); Apply RmaxLess2. +Apply Rle_trans with ``(pow C (plus (mult (S (S O)) (S (plus n0 n))) (S O)))*(pow C (plus (mult (S (S O)) (minus N n0)) (S O)))``. +Do 2 Rewrite <- (Rmult_sym ``(pow C (plus (mult (S (S O)) (minus N n0)) (S O)))``). +Apply Rle_monotony. +Apply pow_le. +Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Unfold C; Apply RmaxLess1. +Apply pow_incr. +Split. +Apply Rabsolu_pos. +Unfold C; Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)). +Apply RmaxLess1. +Apply RmaxLess2. +Right. +Replace (mult (2) (S (S (plus N n)))) with (plus (plus (mult (2) (minus N n0)) (S O)) (plus (mult (2) (S (plus n0 n))) (S O))). +Repeat Rewrite pow_add. +Ring. +Apply INR_eq; Repeat Rewrite plus_INR; Do 3 Rewrite mult_INR. +Rewrite minus_INR. +Repeat Rewrite S_INR; Do 2 Rewrite plus_INR; Ring. +Apply le_trans with (pred (minus N n)). +Exact H1. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply Rle_sym1; Left; Apply Rlt_Rinv. +Apply INR_fact_lt_0. +Apply Rle_sym1; Left; Apply Rlt_Rinv. +Apply INR_fact_lt_0. +Apply Rle_trans with (sum_f_R0 + [k:nat] + (sum_f_R0 + [l:nat] + ``/(INR + (mult (fact (plus (mult (S (S O)) (S (plus l k))) (S O))) + (fact (plus (mult (S (S O)) (minus N l)) (S O)))))* + (pow C (mult (S (S (S (S O)))) (S N)))`` (pred (minus N k))) + (pred N)). +Apply sum_Rle; Intros. +Apply sum_Rle; Intros. +Apply Rle_monotony. +Left; Apply Rlt_Rinv. +Rewrite mult_INR; Apply Rmult_lt_pos; Apply INR_fact_lt_0. +Apply Rle_pow. +Unfold C; Apply RmaxLess1. +Replace (mult (4) (S N)) with (mult (2) (mult (2) (S N))); [Idtac | Ring]. +Apply mult_le. +Replace (mult (2) (S N)) with (S (S (plus N N))). +Repeat Apply le_n_S. +Apply le_reg_l. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply INR_eq; Do 2Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR. +Repeat Rewrite S_INR; Ring. +Apply Rle_trans with (sum_f_R0 + [k:nat] + (sum_f_R0 + [l:nat] + ``(pow C (mult (S (S (S (S O)))) (S N)))*(Rsqr (/(INR (fact (S (S (plus N k)))))))`` (pred (minus N k))) + (pred N)). +Apply sum_Rle; Intros. +Apply sum_Rle; Intros. +Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) (S N)))``). +Apply Rle_monotony. +Apply pow_le. +Left; Apply Rlt_le_trans with R1. +Apply Rlt_R0_R1. +Unfold C; Apply RmaxLess1. +Replace ``/(INR + (mult (fact (plus (mult (S (S O)) (S (plus n0 n))) (S O))) + (fact (plus (mult (S (S O)) (minus N n0)) (S O)))))`` with ``(Binomial.C (mult (S (S O)) (S (S (plus N n)))) (plus (mult (S (S O)) (S (plus n0 n))) (S O)))/(INR (fact (mult (S (S O)) (S (S (plus N n))))))``. +Apply Rle_trans with ``(Binomial.C (mult (S (S O)) (S (S (plus N n)))) (S (S (plus N n))))/(INR (fact (mult (S (S O)) (S (S (plus N n))))))``. +Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (S (S (plus N n))))))``). +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply C_maj. +Apply le_trans with (mult (2) (S (S (plus n0 n)))). +Replace (mult (2) (S (S (plus n0 n)))) with (S (plus (mult (2) (S (plus n0 n))) (1))). +Apply le_n_Sn. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite plus_INR; Ring. +Apply mult_le. +Repeat Apply le_n_S. +Apply le_reg_r. +Apply le_trans with (pred (minus N n)). +Assumption. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Right. +Unfold Rdiv; Rewrite Rmult_sym. +Unfold Binomial.C. +Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Replace (minus (mult (2) (S (S (plus N n)))) (S (S (plus N n)))) with (S (S (plus N n))). +Rewrite Rinv_Rmult. +Unfold Rsqr; Reflexivity. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_eq; Do 2 Rewrite S_INR; Rewrite minus_INR. +Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_n_2n. +Apply INR_fact_neq_0. +Unfold Rdiv; Rewrite Rmult_sym. +Unfold Binomial.C. +Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Replace (minus (mult (2) (S (S (plus N n)))) (plus (mult (2) (S (plus n0 n))) (S O))) with (plus (mult (2) (minus N n0)) (S O)). +Rewrite mult_INR. +Reflexivity. +Apply INR_eq; Rewrite minus_INR. +Do 2 Rewrite plus_INR; Do 3 Rewrite mult_INR; Repeat Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite minus_INR. +Ring. +Apply le_trans with (pred (minus N n)). +Assumption. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_trans with (mult (2) (S (S (plus n0 n)))). +Replace (mult (2) (S (S (plus n0 n)))) with (S (plus (mult (2) (S (plus n0 n))) (1))). +Apply le_n_Sn. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite plus_INR; Ring. +Apply mult_le. +Repeat Apply le_n_S. +Apply le_reg_r. +Apply le_trans with (pred (minus N n)). +Assumption. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply INR_fact_neq_0. +Apply Rle_trans with (sum_f_R0 [k:nat]``(INR N)/(INR (fact (S (S N))))*(pow C (mult (S (S (S (S O)))) (S N)))`` (pred N)). +Apply sum_Rle; Intros. +Rewrite <- (scal_sum [_:nat]``(pow C (mult (S (S (S (S O)))) (S N)))`` (pred (minus N n)) ``(Rsqr (/(INR (fact (S (S (plus N n)))))))``). +Rewrite sum_cte. +Rewrite <- Rmult_assoc. +Do 2 Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) (S N)))``). +Rewrite Rmult_assoc. +Apply Rle_monotony. +Apply pow_le. +Left; Apply Rlt_le_trans with R1. +Apply Rlt_R0_R1. +Unfold C; Apply RmaxLess1. +Apply Rle_trans with ``(Rsqr (/(INR (fact (S (S (plus N n)))))))*(INR N)``. +Apply Rle_monotony. +Apply pos_Rsqr. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_INR. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n O) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Assumption. +Apply lt_pred_n_n; Assumption. +Apply le_trans with (pred N). +Assumption. +Apply le_pred_n. +Rewrite Rmult_sym; Unfold Rdiv; Apply Rle_monotony. +Apply pos_INR. +Apply Rle_trans with ``/(INR (fact (S (S (plus N n)))))``. +Pattern 2 ``/(INR (fact (S (S (plus N n)))))``; Rewrite <- Rmult_1r. +Unfold Rsqr. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_monotony_contra with ``(INR (fact (S (S (plus N n)))))``. +Apply INR_fact_lt_0. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r. +Replace R1 with (INR (S O)). +Apply le_INR. +Apply lt_le_S. +Apply INR_lt; Apply INR_fact_lt_0. +Reflexivity. +Apply INR_fact_neq_0. +Apply Rle_monotony_contra with ``(INR (fact (S (S (plus N n)))))``. +Apply INR_fact_lt_0. +Rewrite <- Rinv_r_sym. +Apply Rle_monotony_contra with ``(INR (fact (S (S N))))``. +Apply INR_fact_lt_0. +Rewrite Rmult_1r. +Rewrite (Rmult_sym (INR (fact (S (S N))))). +Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Apply le_INR. +Apply fact_growing. +Repeat Apply le_n_S. +Apply le_plus_l. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Rewrite sum_cte. +Apply Rle_trans with ``(pow C (mult (S (S (S (S O)))) (S N)))/(INR (fact N))``. +Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) (S N)))``). +Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. +Apply pow_le. +Left; Apply Rlt_le_trans with R1. +Apply Rlt_R0_R1. +Unfold C; Apply RmaxLess1. +Cut (S (pred N)) = N. +Intro; Rewrite H0. +Do 2 Rewrite fact_simpl. +Repeat Rewrite mult_INR. +Repeat Rewrite Rinv_Rmult. +Apply Rle_trans with ``(INR (S (S N)))*(/(INR (S (S N)))*(/(INR (S N))*/(INR (fact N))))* + (INR N)``. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym (INR N)). +Rewrite (Rmult_sym (INR (S (S N)))). +Apply Rle_monotony. +Repeat Apply Rmult_le_pos. +Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply lt_O_Sn. +Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply lt_O_Sn. +Left; Apply Rlt_Rinv. +Apply INR_fact_lt_0. +Apply pos_INR. +Apply le_INR. +Apply le_trans with (S N); Apply le_n_Sn. +Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l. +Apply Rle_trans with ``/(INR (S N))*/(INR (fact N))*(INR (S N))``. +Repeat Rewrite Rmult_assoc. +Repeat Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply lt_O_Sn. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply le_INR; Apply le_n_Sn. +Rewrite (Rmult_sym ``/(INR (S N))``). +Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Right; Reflexivity. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Apply INR_fact_neq_0. +Apply not_O_INR; Discriminate. +Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. +Symmetry; Apply S_pred with O; Assumption. +Right. +Unfold Majxy. +Unfold C. +Reflexivity. +Qed. + +Lemma reste1_cv_R0 : (x,y:R) (Un_cv (Reste1 x y) R0). +Intros. +Assert H := (Majxy_cv_R0 x y). +Unfold Un_cv in H; Unfold R_dist in H. +Unfold Un_cv; Unfold R_dist; Intros. +Elim (H eps H0); Intros N0 H1. +Exists (S N0); Intros. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. +Apply Rle_lt_trans with (Rabsolu (Majxy x y (pred n))). +Rewrite (Rabsolu_right (Majxy x y (pred n))). +Apply reste1_maj. +Apply lt_le_trans with (S N0). +Apply lt_O_Sn. +Assumption. +Apply Rle_sym1. +Unfold Majxy. +Unfold Rdiv; Apply Rmult_le_pos. +Apply pow_le. +Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Apply RmaxLess1. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Replace (Majxy x y (pred n)) with ``(Majxy x y (pred n))-0``; [Idtac | Ring]. +Apply H1. +Unfold ge; Apply le_S_n. +Replace (S (pred n)) with n. +Assumption. +Apply S_pred with O. +Apply lt_le_trans with (S N0); [Apply lt_O_Sn | Assumption]. +Qed. + +Lemma reste2_cv_R0 : (x,y:R) (Un_cv (Reste2 x y) R0). +Intros. +Assert H := (Majxy_cv_R0 x y). +Unfold Un_cv in H; Unfold R_dist in H. +Unfold Un_cv; Unfold R_dist; Intros. +Elim (H eps H0); Intros N0 H1. +Exists (S N0); Intros. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. +Apply Rle_lt_trans with (Rabsolu (Majxy x y n)). +Rewrite (Rabsolu_right (Majxy x y n)). +Apply reste2_maj. +Apply lt_le_trans with (S N0). +Apply lt_O_Sn. +Assumption. +Apply Rle_sym1. +Unfold Majxy. +Unfold Rdiv; Apply Rmult_le_pos. +Apply pow_le. +Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Apply RmaxLess1. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Replace (Majxy x y n) with ``(Majxy x y n)-0``; [Idtac | Ring]. +Apply H1. +Unfold ge; Apply le_trans with (S N0). +Apply le_n_Sn. +Exact H2. +Qed. + +Lemma reste_cv_R0 : (x,y:R) (Un_cv (Reste x y) R0). +Intros. +Unfold Reste. +Pose An := [n:nat](Reste2 x y n). +Pose Bn := [n:nat](Reste1 x y (S n)). +Cut (Un_cv [n:nat]``(An n)-(Bn n)`` ``0-0``) -> (Un_cv [N:nat]``(Reste2 x y N)-(Reste1 x y (S N))`` ``0``). +Intro. +Apply H. +Apply CV_minus. +Unfold An. +Replace [n:nat](Reste2 x y n) with (Reste2 x y). +Apply reste2_cv_R0. +Reflexivity. +Unfold Bn. +Assert H0 := (reste1_cv_R0 x y). +Unfold Un_cv in H0; Unfold R_dist in H0. +Unfold Un_cv; Unfold R_dist; Intros. +Elim (H0 eps H1); Intros N0 H2. +Exists N0; Intros. +Apply H2. +Unfold ge; Apply le_trans with (S N0). +Apply le_n_Sn. +Apply le_n_S; Assumption. +Unfold An Bn. +Intro. +Replace R0 with ``0-0``; [Idtac | Ring]. +Exact H. +Qed. + +Theorem cos_plus : (x,y:R) ``(cos (x+y))==(cos x)*(cos y)-(sin x)*(sin y)``. +Intros. +Cut (Un_cv (C1 x y) ``(cos x)*(cos y)-(sin x)*(sin y)``). +Cut (Un_cv (C1 x y) ``(cos (x+y))``). +Intros. +Apply UL_sequence with (C1 x y); Assumption. +Apply C1_cvg. +Unfold Un_cv; Unfold R_dist. +Intros. +Assert H0 := (A1_cvg x). +Assert H1 := (A1_cvg y). +Assert H2 := (B1_cvg x). +Assert H3 := (B1_cvg y). +Assert H4 := (CV_mult ? ? ? ? H0 H1). +Assert H5 := (CV_mult ? ? ? ? H2 H3). +Assert H6 := (reste_cv_R0 x y). +Unfold Un_cv in H4; Unfold Un_cv in H5; Unfold Un_cv in H6. +Unfold R_dist in H4; Unfold R_dist in H5; Unfold R_dist in H6. +Cut ``0<eps/3``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. +Elim (H4 ``eps/3`` H7); Intros N1 H8. +Elim (H5 ``eps/3`` H7); Intros N2 H9. +Elim (H6 ``eps/3`` H7); Intros N3 H10. +Pose N := (S (S (max (max N1 N2) N3))). +Exists N. +Intros. +Cut n = (S (pred n)). +Intro; Rewrite H12. +Rewrite <- cos_plus_form. +Rewrite <- H12. +Apply Rle_lt_trans with ``(Rabsolu ((A1 x n)*(A1 y n)-(cos x)*(cos y)))+(Rabsolu ((sin x)*(sin y)-(B1 x (pred n))*(B1 y (pred n))+(Reste x y (pred n))))``. +Replace ``(A1 x n)*(A1 y n)-(B1 x (pred n))*(B1 y (pred n))+ + (Reste x y (pred n))-((cos x)*(cos y)-(sin x)*(sin y))`` with ``((A1 x n)*(A1 y n)-(cos x)*(cos y))+((sin x)*(sin y)-(B1 x (pred n))*(B1 y (pred n))+(Reste x y (pred n)))``; [Apply Rabsolu_triang | Ring]. +Replace ``eps`` with ``eps/3+(eps/3+eps/3)``. +Apply Rplus_lt. +Apply H8. +Unfold ge; Apply le_trans with N. +Unfold N. +Apply le_trans with (max N1 N2). +Apply le_max_l. +Apply le_trans with (max (max N1 N2) N3). +Apply le_max_l. +Apply le_trans with (S (max (max N1 N2) N3)); Apply le_n_Sn. +Assumption. +Apply Rle_lt_trans with ``(Rabsolu ((sin x)*(sin y)-(B1 x (pred n))*(B1 y (pred n))))+(Rabsolu (Reste x y (pred n)))``. +Apply Rabsolu_triang. +Apply Rplus_lt. +Rewrite <- Rabsolu_Ropp. +Rewrite Ropp_distr2. +Apply H9. +Unfold ge; Apply le_trans with (max N1 N2). +Apply le_max_r. +Apply le_S_n. +Rewrite <- H12. +Apply le_trans with N. +Unfold N. +Apply le_n_S. +Apply le_trans with (max (max N1 N2) N3). +Apply le_max_l. +Apply le_n_Sn. +Assumption. +Replace (Reste x y (pred n)) with ``(Reste x y (pred n))-0``. +Apply H10. +Unfold ge. +Apply le_S_n. +Rewrite <- H12. +Apply le_trans with N. +Unfold N. +Apply le_n_S. +Apply le_trans with (max (max N1 N2) N3). +Apply le_max_r. +Apply le_n_Sn. +Assumption. +Ring. +Pattern 4 eps; Replace eps with ``3*eps/3``. +Ring. +Unfold Rdiv. +Rewrite <- Rmult_assoc. +Apply Rinv_r_simpl_m. +DiscrR. +Apply lt_le_trans with (pred N). +Unfold N; Simpl; Apply lt_O_Sn. +Apply le_S_n. +Rewrite <- H12. +Replace (S (pred N)) with N. +Assumption. +Unfold N; Simpl; Reflexivity. +Cut (lt O N). +Intro. +Cut (lt O n). +Intro. +Apply S_pred with O; Assumption. +Apply lt_le_trans with N; Assumption. +Unfold N; Apply lt_O_Sn. +Qed. diff --git a/theories7/Reals/Cos_rel.v b/theories7/Reals/Cos_rel.v new file mode 100644 index 00000000..e29825ab --- /dev/null +++ b/theories7/Reals/Cos_rel.v @@ -0,0 +1,360 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Cos_rel.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo_def. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Definition A1 [x:R] : nat->R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))*(pow x (mult (S (S O)) k))`` N). + +Definition B1 [x:R] : nat->R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow x (plus (mult (S (S O)) k) (S O)))`` N). + +Definition C1 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))*(pow (x+y) (mult (S (S O)) k))`` N). + +Definition Reste1 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(pow (-1) (S (plus l k)))/(INR (fact (mult (S (S O)) (S (plus l k)))))*(pow x (mult (S (S O)) (S (plus l k))))*(pow (-1) (minus N l))/(INR (fact (mult (S (S O)) (minus N l))))*(pow y (mult (S (S O)) (minus N l)))`` (pred (minus N k))) (pred N)). + +Definition Reste2 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(pow (-1) (S (plus l k)))/(INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))*(pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))*(pow (-1) (minus N l))/(INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))*(pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N k))) (pred N)). + +Definition Reste [x,y:R] : nat -> R := [N:nat]``(Reste2 x y N)-(Reste1 x y (S N))``. + +(* Here is the main result that will be used to prove that (cos (x+y))=(cos x)(cos y)-(sin x)(sin y) *) +Theorem cos_plus_form : (x,y:R;n:nat) (lt O n) -> ``(A1 x (S n))*(A1 y (S n))-(B1 x n)*(B1 y n)+(Reste x y n)``==(C1 x y (S n)). +Intros. +Unfold A1 B1. +Rewrite (cauchy_finite [k:nat] + ``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))* + (pow x (mult (S (S O)) k))`` [k:nat] + ``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))* + (pow y (mult (S (S O)) k))`` (S n)). +Rewrite (cauchy_finite [k:nat] + ``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))* + (pow x (plus (mult (S (S O)) k) (S O)))`` [k:nat] + ``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))* + (pow y (plus (mult (S (S O)) k) (S O)))`` n H). +Unfold Reste. +Replace (sum_f_R0 + [k:nat] + (sum_f_R0 + [l:nat] + ``(pow ( -1) (S (plus l k)))/ + (INR (fact (mult (S (S O)) (S (plus l k)))))* + (pow x (mult (S (S O)) (S (plus l k))))* + ((pow ( -1) (minus (S n) l))/ + (INR (fact (mult (S (S O)) (minus (S n) l))))* + (pow y (mult (S (S O)) (minus (S n) l))))`` + (pred (minus (S n) k))) (pred (S n))) with (Reste1 x y (S n)). +Replace (sum_f_R0 + [k:nat] + (sum_f_R0 + [l:nat] + ``(pow ( -1) (S (plus l k)))/ + (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))* + (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))* + ((pow ( -1) (minus n l))/ + (INR (fact (plus (mult (S (S O)) (minus n l)) (S O))))* + (pow y (plus (mult (S (S O)) (minus n l)) (S O))))`` + (pred (minus n k))) (pred n)) with (Reste2 x y n). +Ring. +Replace (sum_f_R0 + [k:nat] + (sum_f_R0 + [p:nat] + ``(pow ( -1) p)/(INR (fact (mult (S (S O)) p)))* + (pow x (mult (S (S O)) p))*((pow ( -1) (minus k p))/ + (INR (fact (mult (S (S O)) (minus k p))))* + (pow y (mult (S (S O)) (minus k p))))`` k) (S n)) with (sum_f_R0 [k:nat](Rmult ``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))`` (sum_f_R0 [l:nat]``(C (mult (S (S O)) k) (mult (S (S O)) l))*(pow x (mult (S (S O)) l))*(pow y (mult (S (S O)) (minus k l)))`` k)) (S n)). +Pose sin_nnn := [n:nat]Cases n of O => R0 | (S p) => (Rmult ``(pow (-1) (S p))/(INR (fact (mult (S (S O)) (S p))))`` (sum_f_R0 [l:nat]``(C (mult (S (S O)) (S p)) (S (mult (S (S O)) l)))*(pow x (S (mult (S (S O)) l)))*(pow y (S (mult (S (S O)) (minus p l))))`` p)) end. +Replace (Ropp (sum_f_R0 + [k:nat] + (sum_f_R0 + [p:nat] + ``(pow ( -1) p)/ + (INR (fact (plus (mult (S (S O)) p) (S O))))* + (pow x (plus (mult (S (S O)) p) (S O)))* + ((pow ( -1) (minus k p))/ + (INR (fact (plus (mult (S (S O)) (minus k p)) (S O))))* + (pow y (plus (mult (S (S O)) (minus k p)) (S O))))`` k) + n)) with (sum_f_R0 sin_nnn (S n)). +Rewrite <- sum_plus. +Unfold C1. +Apply sum_eq; Intros. +Induction i. +Simpl. +Rewrite Rplus_Ol. +Replace (C O O) with R1. +Unfold Rdiv; Rewrite Rinv_R1. +Ring. +Unfold C. +Rewrite <- minus_n_n. +Simpl. +Unfold Rdiv; Rewrite Rmult_1r; Rewrite Rinv_R1; Ring. +Unfold sin_nnn. +Rewrite <- Rmult_Rplus_distr. +Apply Rmult_mult_r. +Rewrite binomial. +Pose Wn := [i0:nat]``(C (mult (S (S O)) (S i)) i0)*(pow x i0)* + (pow y (minus (mult (S (S O)) (S i)) i0))``. +Replace (sum_f_R0 + [l:nat] + ``(C (mult (S (S O)) (S i)) (mult (S (S O)) l))* + (pow x (mult (S (S O)) l))* + (pow y (mult (S (S O)) (minus (S i) l)))`` (S i)) with (sum_f_R0 [l:nat](Wn (mult (2) l)) (S i)). +Replace (sum_f_R0 + [l:nat] + ``(C (mult (S (S O)) (S i)) (S (mult (S (S O)) l)))* + (pow x (S (mult (S (S O)) l)))* + (pow y (S (mult (S (S O)) (minus i l))))`` i) with (sum_f_R0 [l:nat](Wn (S (mult (2) l))) i). +Rewrite Rplus_sym. +Apply sum_decomposition. +Apply sum_eq; Intros. +Unfold Wn. +Apply Rmult_mult_r. +Replace (minus (mult (2) (S i)) (S (mult (2) i0))) with (S (mult (2) (minus i i0))). +Reflexivity. +Apply INR_eq. +Rewrite S_INR; Rewrite mult_INR. +Repeat Rewrite minus_INR. +Rewrite mult_INR; Repeat Rewrite S_INR. +Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Replace (mult (2) (S i)) with (S (S (mult (2) i))). +Apply le_n_S. +Apply le_trans with (mult (2) i). +Apply mult_le; Assumption. +Apply le_n_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Assumption. +Apply sum_eq; Intros. +Unfold Wn. +Apply Rmult_mult_r. +Replace (minus (mult (2) (S i)) (mult (2) i0)) with (mult (2) (minus (S i) i0)). +Reflexivity. +Apply INR_eq. +Rewrite mult_INR. +Repeat Rewrite minus_INR. +Rewrite mult_INR; Repeat Rewrite S_INR. +Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply mult_le; Assumption. +Assumption. +Rewrite <- (Ropp_Ropp (sum_f_R0 sin_nnn (S n))). +Apply eq_Ropp. +Replace ``-(sum_f_R0 sin_nnn (S n))`` with ``-1*(sum_f_R0 sin_nnn (S n))``; [Idtac | Ring]. +Rewrite scal_sum. +Rewrite decomp_sum. +Replace (sin_nnn O) with R0. +Rewrite Rmult_Ol; Rewrite Rplus_Ol. +Replace (pred (S n)) with n; [Idtac | Reflexivity]. +Apply sum_eq; Intros. +Rewrite Rmult_sym. +Unfold sin_nnn. +Rewrite scal_sum. +Rewrite scal_sum. +Apply sum_eq; Intros. +Unfold Rdiv. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym ``/(INR (fact (mult (S (S O)) (S i))))``). +Repeat Rewrite <- Rmult_assoc. +Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (S i))))``). +Repeat Rewrite <- Rmult_assoc. +Replace ``/(INR (fact (mult (S (S O)) (S i))))* + (C (mult (S (S O)) (S i)) (S (mult (S (S O)) i0)))`` with ``/(INR (fact (plus (mult (S (S O)) i0) (S O))))*/(INR (fact (plus (mult (S (S O)) (minus i i0)) (S O))))``. +Replace (S (mult (2) i0)) with (plus (mult (2) i0) (1)); [Idtac | Ring]. +Replace (S (mult (2) (minus i i0))) with (plus (mult (2) (minus i i0)) (1)); [Idtac | Ring]. +Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i0)*(pow (-1) (minus i i0))``. +Ring. +Simpl. +Pattern 2 i; Replace i with (plus i0 (minus i i0)). +Rewrite pow_add. +Ring. +Symmetry; Apply le_plus_minus; Assumption. +Unfold C. +Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Rewrite Rinv_Rmult. +Replace (S (mult (S (S O)) i0)) with (plus (mult (2) i0) (1)); [Apply Rmult_mult_r | Ring]. +Replace (minus (mult (2) (S i)) (plus (mult (2) i0) (1))) with (plus (mult (2) (minus i i0)) (1)). +Reflexivity. +Apply INR_eq. +Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite minus_INR. +Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) i0) (1)) with (S (mult (2) i0)). +Replace (mult (2) (S i)) with (S (S (mult (2) i))). +Apply le_n_S. +Apply le_trans with (mult (2) i). +Apply mult_le; Assumption. +Apply le_n_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Assumption. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Reflexivity. +Apply lt_O_Sn. +Apply sum_eq; Intros. +Rewrite scal_sum. +Apply sum_eq; Intros. +Unfold Rdiv. +Repeat Rewrite <- Rmult_assoc. +Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) i)))``). +Repeat Rewrite <- Rmult_assoc. +Replace ``/(INR (fact (mult (S (S O)) i)))* + (C (mult (S (S O)) i) (mult (S (S O)) i0))`` with ``/(INR (fact (mult (S (S O)) i0)))*/(INR (fact (mult (S (S O)) (minus i i0))))``. +Replace ``(pow (-1) i)`` with ``(pow (-1) i0)*(pow (-1) (minus i i0))``. +Ring. +Pattern 2 i; Replace i with (plus i0 (minus i i0)). +Rewrite pow_add. +Ring. +Symmetry; Apply le_plus_minus; Assumption. +Unfold C. +Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Rewrite Rinv_Rmult. +Replace (minus (mult (2) i) (mult (2) i0)) with (mult (2) (minus i i0)). +Reflexivity. +Apply INR_eq. +Rewrite mult_INR; Repeat Rewrite minus_INR. +Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply mult_le; Assumption. +Assumption. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Unfold Reste2; Apply sum_eq; Intros. +Apply sum_eq; Intros. +Unfold Rdiv; Ring. +Unfold Reste1; Apply sum_eq; Intros. +Apply sum_eq; Intros. +Unfold Rdiv; Ring. +Apply lt_O_Sn. +Qed. + +Lemma pow_sqr : (x:R;i:nat) (pow x (mult (2) i))==(pow ``x*x`` i). +Intros. +Assert H := (pow_Rsqr x i). +Unfold Rsqr in H; Exact H. +Qed. + +Lemma A1_cvg : (x:R) (Un_cv (A1 x) (cos x)). +Intro. +Assert H := (exist_cos ``x*x``). +Elim H; Intros. +Assert p_i := p. +Unfold cos_in in p. +Unfold cos_n infinit_sum in p. +Unfold R_dist in p. +Cut ``(cos x)==x0``. +Intro. +Rewrite H0. +Unfold Un_cv; Unfold R_dist; Intros. +Elim (p eps H1); Intros. +Exists x1; Intros. +Unfold A1. +Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))*(pow x (mult (S (S O)) k))``) n) with (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (mult (S (S O)) i)))*(pow (x*x) i)``) n). +Apply H2; Assumption. +Apply sum_eq. +Intros. +Replace ``(pow (x*x) i)`` with ``(pow x (mult (S (S O)) i))``. +Reflexivity. +Apply pow_sqr. +Unfold cos. +Case (exist_cos (Rsqr x)). +Unfold Rsqr; Intros. +Unfold cos_in in p_i. +Unfold cos_in in c. +Apply unicity_sum with [i:nat]``(cos_n i)*(pow (x*x) i)``; Assumption. +Qed. + +Lemma C1_cvg : (x,y:R) (Un_cv (C1 x y) (cos (Rplus x y))). +Intros. +Assert H := (exist_cos ``(x+y)*(x+y)``). +Elim H; Intros. +Assert p_i := p. +Unfold cos_in in p. +Unfold cos_n infinit_sum in p. +Unfold R_dist in p. +Cut ``(cos (x+y))==x0``. +Intro. +Rewrite H0. +Unfold Un_cv; Unfold R_dist; Intros. +Elim (p eps H1); Intros. +Exists x1; Intros. +Unfold C1. +Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))*(pow (x+y) (mult (S (S O)) k))``) n) with (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (mult (S (S O)) i)))*(pow ((x+y)*(x+y)) i)``) n). +Apply H2; Assumption. +Apply sum_eq. +Intros. +Replace ``(pow ((x+y)*(x+y)) i)`` with ``(pow (x+y) (mult (S (S O)) i))``. +Reflexivity. +Apply pow_sqr. +Unfold cos. +Case (exist_cos (Rsqr ``x+y``)). +Unfold Rsqr; Intros. +Unfold cos_in in p_i. +Unfold cos_in in c. +Apply unicity_sum with [i:nat]``(cos_n i)*(pow ((x+y)*(x+y)) i)``; Assumption. +Qed. + +Lemma B1_cvg : (x:R) (Un_cv (B1 x) (sin x)). +Intro. +Case (Req_EM x R0); Intro. +Rewrite H. +Rewrite sin_0. +Unfold B1. +Unfold Un_cv; Unfold R_dist; Intros; Exists O; Intros. +Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow 0 (plus (mult (S (S O)) k) (S O)))``) n) with R0. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Induction n. +Simpl; Ring. +Rewrite tech5; Rewrite <- Hrecn. +Simpl; Ring. +Unfold ge; Apply le_O_n. +Assert H0 := (exist_sin ``x*x``). +Elim H0; Intros. +Assert p_i := p. +Unfold sin_in in p. +Unfold sin_n infinit_sum in p. +Unfold R_dist in p. +Cut ``(sin x)==x*x0``. +Intro. +Rewrite H1. +Unfold Un_cv; Unfold R_dist; Intros. +Cut ``0<eps/(Rabsolu x)``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption]]. +Elim (p ``eps/(Rabsolu x)`` H3); Intros. +Exists x1; Intros. +Unfold B1. +Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow x (plus (mult (S (S O)) k) (S O)))``) n) with (Rmult x (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n)). +Replace (Rminus (Rmult x (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n)) (Rmult x x0)) with (Rmult x (Rminus (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n) x0)); [Idtac | Ring]. +Rewrite Rabsolu_mult. +Apply Rlt_monotony_contra with ``/(Rabsolu x)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H4; Apply H4; Assumption. +Apply Rabsolu_no_R0; Assumption. +Rewrite scal_sum. +Apply sum_eq. +Intros. +Rewrite pow_add. +Rewrite pow_sqr. +Simpl. +Ring. +Unfold sin. +Case (exist_sin (Rsqr x)). +Unfold Rsqr; Intros. +Unfold sin_in in p_i. +Unfold sin_in in s. +Assert H1 := (unicity_sum [i:nat]``(sin_n i)*(pow (x*x) i)`` x0 x1 p_i s). +Rewrite H1; Reflexivity. +Qed. diff --git a/theories7/Reals/DiscrR.v b/theories7/Reals/DiscrR.v new file mode 100644 index 00000000..31c90727 --- /dev/null +++ b/theories7/Reals/DiscrR.v @@ -0,0 +1,58 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: DiscrR.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require RIneq. +Require Omega. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Lemma Rlt_R0_R2 : ``0<2``. +Replace ``2`` with (INR (2)); [Apply lt_INR_0; Apply lt_O_Sn | Reflexivity]. +Qed. + +Lemma Rplus_lt_pos : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x+y``. +Intros. +Apply Rlt_trans with x. +Assumption. +Pattern 1 x; Rewrite <- Rplus_Or. +Apply Rlt_compatibility. +Assumption. +Qed. + +Lemma IZR_eq : (z1,z2:Z) z1=z2 -> (IZR z1)==(IZR z2). +Intros; Rewrite H; Reflexivity. +Qed. + +Lemma IZR_neq : (z1,z2:Z) `z1<>z2` -> ``(IZR z1)<>(IZR z2)``. +Intros; Red; Intro; Elim H; Apply eq_IZR; Assumption. +Qed. + +Tactic Definition DiscrR := + Try Match Context With + | [ |- ~(?1==?2) ] -> Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_neq; Try Discriminate | Reflexivity] | Reflexivity] | Reflexivity]. + +Recursive Tactic Definition Sup0 := + Match Context With + | [ |- ``0<1`` ] -> Apply Rlt_R0_R1 + | [ |- ``0<?1`` ] -> Repeat (Apply Rmult_lt_pos Orelse Apply Rplus_lt_pos; Try Apply Rlt_R0_R1 Orelse Apply Rlt_R0_R2) + | [ |- ``?1>0`` ] -> Change ``0<?1``; Sup0. + +Tactic Definition SupOmega := Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_lt; Omega | Reflexivity] | Reflexivity] | Reflexivity]. + +Recursive Tactic Definition Sup := + Match Context With + | [ |- (Rgt ?1 ?2) ] -> Change ``?2<?1``; Sup + | [ |- ``0<?1`` ] -> Sup0 + | [ |- (Rlt (Ropp ?1) R0) ] -> Rewrite <- Ropp_O; Sup + | [ |- (Rlt (Ropp ?1) (Ropp ?2)) ] -> Apply Rlt_Ropp; Sup + | [ |- (Rlt (Ropp ?1) ?2) ] -> Apply Rlt_trans with ``0``; Sup + | [ |- (Rlt ?1 ?2) ] -> SupOmega + | _ -> Idtac. + +Tactic Definition RCompute := Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_eq; Try Reflexivity | Reflexivity] | Reflexivity] | Reflexivity]. diff --git a/theories7/Reals/Exp_prop.v b/theories7/Reals/Exp_prop.v new file mode 100644 index 00000000..6ed9c00b --- /dev/null +++ b/theories7/Reals/Exp_prop.v @@ -0,0 +1,890 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Exp_prop.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis1. +Require PSeries_reg. +Require Div2. +Require Even. +Require Max. +V7only [Import R_scope.]. +Open Local Scope nat_scope. +V7only [Import nat_scope.]. +Open Local Scope R_scope. + +Definition E1 [x:R] : nat->R := [N:nat](sum_f_R0 [k:nat]``/(INR (fact k))*(pow x k)`` N). + +Lemma E1_cvg : (x:R) (Un_cv (E1 x) (exp x)). +Intro; Unfold exp; Unfold projT1. +Case (exist_exp x); Intro. +Unfold exp_in Un_cv; Unfold infinit_sum E1; Trivial. +Qed. + +Definition Reste_E [x,y:R] : nat->R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))`` (pred (minus N k))) (pred N)). + +Lemma exp_form : (x,y:R;n:nat) (lt O n) -> ``(E1 x n)*(E1 y n)-(Reste_E x y n)==(E1 (x+y) n)``. +Intros; Unfold E1. +Rewrite cauchy_finite. +Unfold Reste_E; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Apply sum_eq; Intros. +Rewrite binomial. +Rewrite scal_sum; Apply sum_eq; Intros. +Unfold C; Unfold Rdiv; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym (INR (fact i))); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite Rinv_Rmult. +Ring. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply H. +Qed. + +Definition maj_Reste_E [x,y:R] : nat->R := [N:nat]``4*(pow (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S O)) N))/(Rsqr (INR (fact (div2 (pred N)))))``. + +Lemma Rle_Rinv : (x,y:R) ``0<x`` -> ``0<y`` -> ``x<=y`` -> ``/y<=/x``. +Intros; Apply Rle_monotony_contra with x. +Apply H. +Rewrite <- Rinv_r_sym. +Apply Rle_monotony_contra with y. +Apply H0. +Rewrite Rmult_1r; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Apply H1. +Red; Intro; Rewrite H2 in H0; Elim (Rlt_antirefl ? H0). +Red; Intro; Rewrite H2 in H; Elim (Rlt_antirefl ? H). +Qed. + +(**********) +Lemma div2_double : (N:nat) (div2 (mult (2) N))=N. +Intro; Induction N. +Reflexivity. +Replace (mult (2) (S N)) with (S (S (mult (2) N))). +Simpl; Simpl in HrecN; Rewrite HrecN; Reflexivity. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Qed. + +Lemma div2_S_double : (N:nat) (div2 (S (mult (2) N)))=N. +Intro; Induction N. +Reflexivity. +Replace (mult (2) (S N)) with (S (S (mult (2) N))). +Simpl; Simpl in HrecN; Rewrite HrecN; Reflexivity. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Qed. + +Lemma div2_not_R0 : (N:nat) (lt (1) N) -> (lt O (div2 N)). +Intros; Induction N. +Elim (lt_n_O ? H). +Cut (lt (1) N)\/N=(1). +Intro; Elim H0; Intro. +Assert H2 := (even_odd_dec N). +Elim H2; Intro. +Rewrite <- (even_div2 ? a); Apply HrecN; Assumption. +Rewrite <- (odd_div2 ? b); Apply lt_O_Sn. +Rewrite H1; Simpl; Apply lt_O_Sn. +Inversion H. +Right; Reflexivity. +Left; Apply lt_le_trans with (2); [Apply lt_n_Sn | Apply H1]. +Qed. + +Lemma Reste_E_maj : (x,y:R;N:nat) (lt O N) -> ``(Rabsolu (Reste_E x y N))<=(maj_Reste_E x y N)``. +Intros; Pose M := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). +Apply Rle_trans with (Rmult (pow M (mult (2) N)) (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(Rsqr (INR (fact (div2 (S N)))))`` (pred (minus N k))) (pred N))). +Unfold Reste_E. +Apply Rle_trans with (sum_f_R0 [k:nat](Rabsolu (sum_f_R0 [l:nat]``/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))`` (pred (minus N k)))) (pred N)). +Apply (sum_Rabsolu [k:nat](sum_f_R0 [l:nat]``/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))`` (pred (minus N k))) (pred N)). +Apply Rle_trans with (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(Rabsolu (/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))))`` (pred (minus N k))) (pred N)). +Apply sum_Rle; Intros. +Apply (sum_Rabsolu [l:nat]``/(INR (fact (S (plus l n))))*(pow x (S (plus l n)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))``). +Apply Rle_trans with (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(pow M (mult (S (S O)) N))*/(INR (fact (S l)))*/(INR (fact (minus N l)))`` (pred (minus N k))) (pred N)). +Apply sum_Rle; Intros. +Apply sum_Rle; Intros. +Repeat Rewrite Rabsolu_mult. +Do 2 Rewrite <- Pow_Rabsolu. +Rewrite (Rabsolu_right ``/(INR (fact (S (plus n0 n))))``). +Rewrite (Rabsolu_right ``/(INR (fact (minus N n0)))``). +Replace ``/(INR (fact (S (plus n0 n))))*(pow (Rabsolu x) (S (plus n0 n)))* + (/(INR (fact (minus N n0)))*(pow (Rabsolu y) (minus N n0)))`` with ``/(INR (fact (minus N n0)))*/(INR (fact (S (plus n0 n))))*(pow (Rabsolu x) (S (plus n0 n)))*(pow (Rabsolu y) (minus N n0))``; [Idtac | Ring]. +Rewrite <- (Rmult_sym ``/(INR (fact (minus N n0)))``). +Repeat Rewrite Rmult_assoc. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_trans with ``/(INR (fact (S n0)))*(pow (Rabsolu x) (S (plus n0 n)))*(pow (Rabsolu y) (minus N n0))``. +Rewrite (Rmult_sym ``/(INR (fact (S (plus n0 n))))``); Rewrite (Rmult_sym ``/(INR (fact (S n0)))``); Repeat Rewrite Rmult_assoc; Apply Rle_monotony. +Apply pow_le; Apply Rabsolu_pos. +Rewrite (Rmult_sym ``/(INR (fact (S n0)))``); Apply Rle_monotony. +Apply pow_le; Apply Rabsolu_pos. +Apply Rle_Rinv. +Apply INR_fact_lt_0. +Apply INR_fact_lt_0. +Apply le_INR; Apply fact_growing; Apply le_n_S. +Apply le_plus_l. +Rewrite (Rmult_sym ``(pow M (mult (S (S O)) N))``); Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_trans with ``(pow M (S (plus n0 n)))*(pow (Rabsolu y) (minus N n0))``. +Do 2 Rewrite <- (Rmult_sym ``(pow (Rabsolu y) (minus N n0))``). +Apply Rle_monotony. +Apply pow_le; Apply Rabsolu_pos. +Apply pow_incr; Split. +Apply Rabsolu_pos. +Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)). +Apply RmaxLess1. +Unfold M; Apply RmaxLess2. +Apply Rle_trans with ``(pow M (S (plus n0 n)))*(pow M (minus N n0))``. +Apply Rle_monotony. +Apply pow_le; Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Unfold M; Apply RmaxLess1. +Apply pow_incr; Split. +Apply Rabsolu_pos. +Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)). +Apply RmaxLess2. +Unfold M; Apply RmaxLess2. +Rewrite <- pow_add; Replace (plus (S (plus n0 n)) (minus N n0)) with (plus N (S n)). +Apply Rle_pow. +Unfold M; Apply RmaxLess1. +Replace (mult (2) N) with (plus N N); [Idtac | Ring]. +Apply le_reg_l. +Replace N with (S (pred N)). +Apply le_n_S; Apply H0. +Symmetry; Apply S_pred with O; Apply H. +Apply INR_eq; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite plus_INR; Rewrite minus_INR. +Ring. +Apply le_trans with (pred (minus N n)). +Apply H1. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Apply H0. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n (0)) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Apply H0. +Apply lt_pred_n_n. +Apply H. +Apply le_trans with (pred N). +Apply H0. +Apply le_pred_n. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Rewrite scal_sum. +Apply sum_Rle; Intros. +Rewrite <- Rmult_sym. +Rewrite scal_sum. +Apply sum_Rle; Intros. +Rewrite (Rmult_sym ``/(Rsqr (INR (fact (div2 (S N)))))``). +Rewrite Rmult_assoc; Apply Rle_monotony. +Apply pow_le. +Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Unfold M; Apply RmaxLess1. +Assert H2 := (even_odd_cor N). +Elim H2; Intros N0 H3. +Elim H3; Intro. +Apply Rle_trans with ``/(INR (fact n0))*/(INR (fact (minus N n0)))``. +Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (minus N n0)))``). +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_Rinv. +Apply INR_fact_lt_0. +Apply INR_fact_lt_0. +Apply le_INR. +Apply fact_growing. +Apply le_n_Sn. +Replace ``/(INR (fact n0))*/(INR (fact (minus N n0)))`` with ``(C N n0)/(INR (fact N))``. +Pattern 1 N; Rewrite H4. +Apply Rle_trans with ``(C N N0)/(INR (fact N))``. +Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(INR (fact N))``). +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Rewrite H4. +Apply C_maj. +Rewrite <- H4; Apply le_trans with (pred (minus N n)). +Apply H1. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Apply H0. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n (0)) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Apply H0. +Apply lt_pred_n_n. +Apply H. +Apply le_trans with (pred N). +Apply H0. +Apply le_pred_n. +Replace ``(C N N0)/(INR (fact N))`` with ``/(Rsqr (INR (fact N0)))``. +Rewrite H4; Rewrite div2_S_double; Right; Reflexivity. +Unfold Rsqr C Rdiv. +Repeat Rewrite Rinv_Rmult. +Rewrite (Rmult_sym (INR (fact N))). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Replace (minus N N0) with N0. +Ring. +Replace N with (plus N0 N0). +Symmetry; Apply minus_plus. +Rewrite H4. +Apply INR_eq; Rewrite plus_INR; Rewrite mult_INR; Do 2 Rewrite S_INR; Ring. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Unfold C Rdiv. +Rewrite (Rmult_sym (INR (fact N))). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rinv_Rmult. +Rewrite Rmult_1r; Ring. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Replace ``/(INR (fact (S n0)))*/(INR (fact (minus N n0)))`` with ``(C (S N) (S n0))/(INR (fact (S N)))``. +Apply Rle_trans with ``(C (S N) (S N0))/(INR (fact (S N)))``. +Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (S N)))``). +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Cut (S N) = (mult (2) (S N0)). +Intro; Rewrite H5; Apply C_maj. +Rewrite <- H5; Apply le_n_S. +Apply le_trans with (pred (minus N n)). +Apply H1. +Apply le_S_n. +Replace (S (pred (minus N n))) with (minus N n). +Apply le_trans with N. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Apply H0. +Apply le_pred_n. +Apply le_n_Sn. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n (0)) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Apply H0. +Apply lt_pred_n_n. +Apply H. +Apply le_trans with (pred N). +Apply H0. +Apply le_pred_n. +Apply INR_eq; Rewrite H4. +Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Cut (S N) = (mult (2) (S N0)). +Intro. +Replace ``(C (S N) (S N0))/(INR (fact (S N)))`` with ``/(Rsqr (INR (fact (S N0))))``. +Rewrite H5; Rewrite div2_double. +Right; Reflexivity. +Unfold Rsqr C Rdiv. +Repeat Rewrite Rinv_Rmult. +Replace (minus (S N) (S N0)) with (S N0). +Rewrite (Rmult_sym (INR (fact (S N)))). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Reflexivity. +Apply INR_fact_neq_0. +Replace (S N) with (plus (S N0) (S N0)). +Symmetry; Apply minus_plus. +Rewrite H5; Ring. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_eq; Rewrite H4; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Unfold C Rdiv. +Rewrite (Rmult_sym (INR (fact (S N)))). +Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Rewrite Rinv_Rmult. +Reflexivity. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Unfold maj_Reste_E. +Unfold Rdiv; Rewrite (Rmult_sym ``4``). +Rewrite Rmult_assoc. +Apply Rle_monotony. +Apply pow_le. +Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Apply RmaxLess1. +Apply Rle_trans with (sum_f_R0 [k:nat]``(INR (minus N k))*/(Rsqr (INR (fact (div2 (S N)))))`` (pred N)). +Apply sum_Rle; Intros. +Rewrite sum_cte. +Replace (S (pred (minus N n))) with (minus N n). +Right; Apply Rmult_sym. +Apply S_pred with O. +Apply simpl_lt_plus_l with n. +Rewrite <- le_plus_minus. +Replace (plus n (0)) with n; [Idtac | Ring]. +Apply le_lt_trans with (pred N). +Apply H0. +Apply lt_pred_n_n. +Apply H. +Apply le_trans with (pred N). +Apply H0. +Apply le_pred_n. +Apply Rle_trans with (sum_f_R0 [k:nat]``(INR N)*/(Rsqr (INR (fact (div2 (S N)))))`` (pred N)). +Apply sum_Rle; Intros. +Do 2 Rewrite <- (Rmult_sym ``/(Rsqr (INR (fact (div2 (S N)))))``). +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt. +Apply INR_fact_neq_0. +Apply le_INR. +Apply simpl_le_plus_l with n. +Rewrite <- le_plus_minus. +Apply le_plus_r. +Apply le_trans with (pred N). +Apply H0. +Apply le_pred_n. +Rewrite sum_cte; Replace (S (pred N)) with N. +Cut (div2 (S N)) = (S (div2 (pred N))). +Intro; Rewrite H0. +Rewrite fact_simpl; Rewrite mult_sym; Rewrite mult_INR; Rewrite Rsqr_times. +Rewrite Rinv_Rmult. +Rewrite (Rmult_sym (INR N)); Repeat Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt; Apply INR_fact_neq_0. +Rewrite <- H0. +Cut ``(INR N)<=(INR (mult (S (S O)) (div2 (S N))))``. +Intro; Apply Rle_monotony_contra with ``(Rsqr (INR (div2 (S N))))``. +Apply Rsqr_pos_lt. +Apply not_O_INR; Red; Intro. +Cut (lt (1) (S N)). +Intro; Assert H4 := (div2_not_R0 ? H3). +Rewrite H2 in H4; Elim (lt_n_O ? H4). +Apply lt_n_S; Apply H. +Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l. +Replace ``(INR N)*(INR N)`` with (Rsqr (INR N)); [Idtac | Reflexivity]. +Rewrite Rmult_assoc. +Rewrite Rmult_sym. +Replace ``4`` with (Rsqr ``2``); [Idtac | SqRing]. +Rewrite <- Rsqr_times. +Apply Rsqr_incr_1. +Replace ``2`` with (INR (2)). +Rewrite <- mult_INR; Apply H1. +Reflexivity. +Left; Apply lt_INR_0; Apply H. +Left; Apply Rmult_lt_pos. +Sup0. +Apply lt_INR_0; Apply div2_not_R0. +Apply lt_n_S; Apply H. +Cut (lt (1) (S N)). +Intro; Unfold Rsqr; Apply prod_neq_R0; Apply not_O_INR; Intro; Assert H4 := (div2_not_R0 ? H2); Rewrite H3 in H4; Elim (lt_n_O ? H4). +Apply lt_n_S; Apply H. +Assert H1 := (even_odd_cor N). +Elim H1; Intros N0 H2. +Elim H2; Intro. +Pattern 2 N; Rewrite H3. +Rewrite div2_S_double. +Right; Rewrite H3; Reflexivity. +Pattern 2 N; Rewrite H3. +Replace (S (S (mult (2) N0))) with (mult (2) (S N0)). +Rewrite div2_double. +Rewrite H3. +Rewrite S_INR; Do 2 Rewrite mult_INR. +Rewrite (S_INR N0). +Rewrite Rmult_Rplus_distr. +Apply Rle_compatibility. +Rewrite Rmult_1r. +Simpl. +Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply Rlt_R0_R1. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Unfold Rsqr; Apply prod_neq_R0; Apply INR_fact_neq_0. +Unfold Rsqr; Apply prod_neq_R0; Apply not_O_INR; Discriminate. +Assert H0 := (even_odd_cor N). +Elim H0; Intros N0 H1. +Elim H1; Intro. +Cut (lt O N0). +Intro; Rewrite H2. +Rewrite div2_S_double. +Replace (mult (2) N0) with (S (S (mult (2) (pred N0)))). +Replace (pred (S (S (mult (2) (pred N0))))) with (S (mult (2) (pred N0))). +Rewrite div2_S_double. +Apply S_pred with O; Apply H3. +Reflexivity. +Replace N0 with (S (pred N0)). +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Symmetry; Apply S_pred with O; Apply H3. +Rewrite H2 in H. +Apply neq_O_lt. +Red; Intro. +Rewrite <- H3 in H. +Simpl in H. +Elim (lt_n_O ? H). +Rewrite H2. +Replace (pred (S (mult (2) N0))) with (mult (2) N0); [Idtac | Reflexivity]. +Replace (S (S (mult (2) N0))) with (mult (2) (S N0)). +Do 2 Rewrite div2_double. +Reflexivity. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply S_pred with O; Apply H. +Qed. + +Lemma maj_Reste_cv_R0 : (x,y:R) (Un_cv (maj_Reste_E x y) ``0``). +Intros; Assert H := (Majxy_cv_R0 x y). +Unfold Un_cv in H; Unfold Un_cv; Intros. +Cut ``0<eps/4``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. +Elim (H ? H1); Intros N0 H2. +Exists (max (mult (2) (S N0)) (2)); Intros. +Unfold R_dist in H2; Unfold R_dist; Rewrite minus_R0; Unfold Majxy in H2; Unfold maj_Reste_E. +Rewrite Rabsolu_right. +Apply Rle_lt_trans with ``4*(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S (S (S O)))) (S (div2 (pred n)))))/(INR (fact (div2 (pred n))))``. +Apply Rle_monotony. +Left; Sup0. +Unfold Rdiv Rsqr; Rewrite Rinv_Rmult. +Rewrite (Rmult_sym ``(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S O)) n))``); Rewrite (Rmult_sym ``(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S (S (S O)))) (S (div2 (pred n)))))``); Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Rle_trans with ``(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S O)) n))``. +Rewrite Rmult_sym; Pattern 2 (pow (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (2) n)); Rewrite <- Rmult_1r; Apply Rle_monotony. +Apply pow_le; Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Apply RmaxLess1. +Apply Rle_monotony_contra with ``(INR (fact (div2 (pred n))))``. +Apply INR_fact_lt_0. +Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. +Replace R1 with (INR (1)); [Apply le_INR | Reflexivity]. +Apply lt_le_S. +Apply INR_lt. +Apply INR_fact_lt_0. +Apply INR_fact_neq_0. +Apply Rle_pow. +Apply RmaxLess1. +Assert H4 := (even_odd_cor n). +Elim H4; Intros N1 H5. +Elim H5; Intro. +Cut (lt O N1). +Intro. +Rewrite H6. +Replace (pred (mult (2) N1)) with (S (mult (2) (pred N1))). +Rewrite div2_S_double. +Replace (S (pred N1)) with N1. +Apply INR_le. +Right. +Do 3 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply S_pred with O; Apply H7. +Replace (mult (2) N1) with (S (S (mult (2) (pred N1)))). +Reflexivity. +Pattern 2 N1; Replace N1 with (S (pred N1)). +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Symmetry ; Apply S_pred with O; Apply H7. +Apply INR_lt. +Apply Rlt_monotony_contra with (INR (2)). +Simpl; Sup0. +Rewrite Rmult_Or; Rewrite <- mult_INR. +Apply lt_INR_0. +Rewrite <- H6. +Apply lt_le_trans with (2). +Apply lt_O_Sn. +Apply le_trans with (max (mult (2) (S N0)) (2)). +Apply le_max_r. +Apply H3. +Rewrite H6. +Replace (pred (S (mult (2) N1))) with (mult (2) N1). +Rewrite div2_double. +Replace (mult (4) (S N1)) with (mult (2) (mult (2) (S N1))). +Apply mult_le. +Replace (mult (2) (S N1)) with (S (S (mult (2) N1))). +Apply le_n_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Ring. +Reflexivity. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply Rlt_monotony_contra with ``/4``. +Apply Rlt_Rinv; Sup0. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite Rmult_sym. +Replace ``(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S (S (S O)))) (S (div2 (pred n)))))/(INR (fact (div2 (pred n))))`` with ``(Rabsolu ((pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S (S (S O)))) (S (div2 (pred n)))))/(INR (fact (div2 (pred n))))-0))``. +Apply H2; Unfold ge. +Cut (le (mult (2) (S N0)) n). +Intro; Apply le_S_n. +Apply INR_le; Apply Rle_monotony_contra with (INR (2)). +Simpl; Sup0. +Do 2 Rewrite <- mult_INR; Apply le_INR. +Apply le_trans with n. +Apply H4. +Assert H5 := (even_odd_cor n). +Elim H5; Intros N1 H6. +Elim H6; Intro. +Cut (lt O N1). +Intro. +Rewrite H7. +Apply mult_le. +Replace (pred (mult (2) N1)) with (S (mult (2) (pred N1))). +Rewrite div2_S_double. +Replace (S (pred N1)) with N1. +Apply le_n. +Apply S_pred with O; Apply H8. +Replace (mult (2) N1) with (S (S (mult (2) (pred N1)))). +Reflexivity. +Pattern 2 N1; Replace N1 with (S (pred N1)). +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Symmetry; Apply S_pred with O; Apply H8. +Apply INR_lt. +Apply Rlt_monotony_contra with (INR (2)). +Simpl; Sup0. +Rewrite Rmult_Or; Rewrite <- mult_INR. +Apply lt_INR_0. +Rewrite <- H7. +Apply lt_le_trans with (2). +Apply lt_O_Sn. +Apply le_trans with (max (mult (2) (S N0)) (2)). +Apply le_max_r. +Apply H3. +Rewrite H7. +Replace (pred (S (mult (2) N1))) with (mult (2) N1). +Rewrite div2_double. +Replace (mult (2) (S N1)) with (S (S (mult (2) N1))). +Apply le_n_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Reflexivity. +Apply le_trans with (max (mult (2) (S N0)) (2)). +Apply le_max_l. +Apply H3. +Rewrite minus_R0; Apply Rabsolu_right. +Apply Rle_sym1. +Unfold Rdiv; Repeat Apply Rmult_le_pos. +Apply pow_le. +Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Apply RmaxLess1. +Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. +DiscrR. +Apply Rle_sym1. +Unfold Rdiv; Apply Rmult_le_pos. +Left; Sup0. +Apply Rmult_le_pos. +Apply pow_le. +Apply Rle_trans with R1. +Left; Apply Rlt_R0_R1. +Apply RmaxLess1. +Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt; Apply INR_fact_neq_0. +Qed. + +(**********) +Lemma Reste_E_cv : (x,y:R) (Un_cv (Reste_E x y) R0). +Intros; Assert H := (maj_Reste_cv_R0 x y). +Unfold Un_cv in H; Unfold Un_cv; Intros; Elim (H ? H0); Intros. +Exists (max x0 (1)); Intros. +Unfold R_dist; Rewrite minus_R0. +Apply Rle_lt_trans with (maj_Reste_E x y n). +Apply Reste_E_maj. +Apply lt_le_trans with (1). +Apply lt_O_Sn. +Apply le_trans with (max x0 (1)). +Apply le_max_r. +Apply H2. +Replace (maj_Reste_E x y n) with (R_dist (maj_Reste_E x y n) R0). +Apply H1. +Unfold ge; Apply le_trans with (max x0 (1)). +Apply le_max_l. +Apply H2. +Unfold R_dist; Rewrite minus_R0; Apply Rabsolu_right. +Apply Rle_sym1; Apply Rle_trans with (Rabsolu (Reste_E x y n)). +Apply Rabsolu_pos. +Apply Reste_E_maj. +Apply lt_le_trans with (1). +Apply lt_O_Sn. +Apply le_trans with (max x0 (1)). +Apply le_max_r. +Apply H2. +Qed. + +(**********) +Lemma exp_plus : (x,y:R) ``(exp (x+y))==(exp x)*(exp y)``. +Intros; Assert H0 := (E1_cvg x). +Assert H := (E1_cvg y). +Assert H1 := (E1_cvg ``x+y``). +EApply UL_sequence. +Apply H1. +Assert H2 := (CV_mult ? ? ? ? H0 H). +Assert H3 := (CV_minus ? ? ? ? H2 (Reste_E_cv x y)). +Unfold Un_cv; Unfold Un_cv in H3; Intros. +Elim (H3 ? H4); Intros. +Exists (S x0); Intros. +Rewrite <- (exp_form x y n). +Rewrite minus_R0 in H5. +Apply H5. +Unfold ge; Apply le_trans with (S x0). +Apply le_n_Sn. +Apply H6. +Apply lt_le_trans with (S x0). +Apply lt_O_Sn. +Apply H6. +Qed. + +(**********) +Lemma exp_pos_pos : (x:R) ``0<x`` -> ``0<(exp x)``. +Intros; Pose An := [N:nat]``/(INR (fact N))*(pow x N)``. +Cut (Un_cv [n:nat](sum_f_R0 An n) (exp x)). +Intro; Apply Rlt_le_trans with (sum_f_R0 An O). +Unfold An; Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Apply Rlt_R0_R1. +Apply sum_incr. +Assumption. +Intro; Unfold An; Left; Apply Rmult_lt_pos. +Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply (pow_lt ? n H). +Unfold exp; Unfold projT1; Case (exist_exp x); Intro. +Unfold exp_in; Unfold infinit_sum Un_cv; Trivial. +Qed. + +(**********) +Lemma exp_pos : (x:R) ``0<(exp x)``. +Intro; Case (total_order_T R0 x); Intro. +Elim s; Intro. +Apply (exp_pos_pos ? a). +Rewrite <- b; Rewrite exp_0; Apply Rlt_R0_R1. +Replace (exp x) with ``1/(exp (-x))``. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rlt_R0_R1. +Apply Rlt_Rinv; Apply exp_pos_pos. +Apply (Rgt_RO_Ropp ? r). +Cut ``(exp (-x))<>0``. +Intro; Unfold Rdiv; Apply r_Rmult_mult with ``(exp (-x))``. +Rewrite Rmult_1l; Rewrite <- Rinv_r_sym. +Rewrite <- exp_plus. +Rewrite Rplus_Ropp_l; Rewrite exp_0; Reflexivity. +Apply H. +Apply H. +Assert H := (exp_plus x ``-x``). +Rewrite Rplus_Ropp_r in H; Rewrite exp_0 in H. +Red; Intro; Rewrite H0 in H. +Rewrite Rmult_Or in H. +Elim R1_neq_R0; Assumption. +Qed. + +(* ((exp h)-1)/h -> 0 quand h->0 *) +Lemma derivable_pt_lim_exp_0 : (derivable_pt_lim exp ``0`` ``1``). +Unfold derivable_pt_lim; Intros. +Pose fn := [N:nat][x:R]``(pow x N)/(INR (fact (S N)))``. +Cut (CVN_R fn). +Intro; Cut (x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l)). +Intro cv; Cut ((n:nat)(continuity (fn n))). +Intro; Cut (continuity (SFL fn cv)). +Intro; Unfold continuity in H1. +Assert H2 := (H1 R0). +Unfold continuity_pt in H2; Unfold continue_in in H2; Unfold limit1_in in H2; Unfold limit_in in H2; Simpl in H2; Unfold R_dist in H2. +Elim (H2 ? H); Intros alp H3. +Elim H3; Intros. +Exists (mkposreal ? H4); Intros. +Rewrite Rplus_Ol; Rewrite exp_0. +Replace ``((exp h)-1)/h`` with (SFL fn cv h). +Replace R1 with (SFL fn cv R0). +Apply H5. +Split. +Unfold D_x no_cond; Split. +Trivial. +Apply (not_sym ? ? H6). +Rewrite minus_R0; Apply H7. +Unfold SFL. +Case (cv ``0``); Intros. +EApply UL_sequence. +Apply u. +Unfold Un_cv SP. +Intros; Exists (1); Intros. +Unfold R_dist; Rewrite decomp_sum. +Rewrite (Rplus_sym (fn O R0)). +Replace (fn O R0) with R1. +Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or. +Replace (sum_f_R0 [i:nat](fn (S i) ``0``) (pred n)) with R0. +Rewrite Rabsolu_R0; Apply H8. +Symmetry; Apply sum_eq_R0; Intros. +Unfold fn. +Simpl. +Unfold Rdiv; Do 2 Rewrite Rmult_Ol; Reflexivity. +Unfold fn; Simpl. +Unfold Rdiv; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. +Apply lt_le_trans with (1); [Apply lt_n_Sn | Apply H9]. +Unfold SFL exp. +Unfold projT1. +Case (cv h); Case (exist_exp h); Intros. +EApply UL_sequence. +Apply u. +Unfold Un_cv; Intros. +Unfold exp_in in e. +Unfold infinit_sum in e. +Cut ``0<eps0*(Rabsolu h)``. +Intro; Elim (e ? H9); Intros N0 H10. +Exists N0; Intros. +Unfold R_dist. +Apply Rlt_monotony_contra with ``(Rabsolu h)``. +Apply Rabsolu_pos_lt; Assumption. +Rewrite <- Rabsolu_mult. +Rewrite Rminus_distr. +Replace ``h*(x-1)/h`` with ``(x-1)``. +Unfold R_dist in H10. +Replace ``h*(SP fn n h)-(x-1)`` with (Rminus (sum_f_R0 [i:nat]``/(INR (fact i))*(pow h i)`` (S n)) x). +Rewrite (Rmult_sym (Rabsolu h)). +Apply H10. +Unfold ge. +Apply le_trans with (S N0). +Apply le_n_Sn. +Apply le_n_S; Apply H11. +Rewrite decomp_sum. +Replace ``/(INR (fact O))*(pow h O)`` with R1. +Unfold Rminus. +Rewrite Ropp_distr1. +Rewrite Ropp_Ropp. +Rewrite <- (Rplus_sym ``-x``). +Rewrite <- (Rplus_sym ``-x+1``). +Rewrite Rplus_assoc; Repeat Apply Rplus_plus_r. +Replace (pred (S n)) with n; [Idtac | Reflexivity]. +Unfold SP. +Rewrite scal_sum. +Apply sum_eq; Intros. +Unfold fn. +Replace (pow h (S i)) with ``h*(pow h i)``. +Unfold Rdiv; Ring. +Simpl; Ring. +Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. +Apply lt_O_Sn. +Unfold Rdiv. +Rewrite <- Rmult_assoc. +Symmetry; Apply Rinv_r_simpl_m. +Assumption. +Apply Rmult_lt_pos. +Apply H8. +Apply Rabsolu_pos_lt; Assumption. +Apply SFL_continuity; Assumption. +Intro; Unfold fn. +Replace [x:R]``(pow x n)/(INR (fact (S n)))`` with (div_fct (pow_fct n) (fct_cte (INR (fact (S n))))); [Idtac | Reflexivity]. +Apply continuity_div. +Apply derivable_continuous; Apply (derivable_pow n). +Apply derivable_continuous; Apply derivable_const. +Intro; Unfold fct_cte; Apply INR_fact_neq_0. +Apply (CVN_R_CVS ? X). +Assert H0 := Alembert_exp. +Unfold CVN_R. +Intro; Unfold CVN_r. +Apply Specif.existT with [N:nat]``(pow r N)/(INR (fact (S N)))``. +Cut (SigT ? [l:R](Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu ``(pow r k)/(INR (fact (S k)))``) n) l)). +Intro. +Elim X; Intros. +Exists x; Intros. +Split. +Apply p. +Unfold Boule; Intros. +Rewrite minus_R0 in H1. +Unfold fn. +Unfold Rdiv; Rewrite Rabsolu_mult. +Cut ``0<(INR (fact (S n)))``. +Intro. +Rewrite (Rabsolu_right ``/(INR (fact (S n)))``). +Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (S n)))``). +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply H2. +Rewrite <- Pow_Rabsolu. +Apply pow_maj_Rabs. +Rewrite Rabsolu_Rabsolu; Left; Apply H1. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply H2. +Apply INR_fact_lt_0. +Cut (r::R)<>``0``. +Intro; Apply Alembert_C2. +Intro; Apply Rabsolu_no_R0. +Unfold Rdiv; Apply prod_neq_R0. +Apply pow_nonzero; Assumption. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Unfold Un_cv in H0. +Unfold Un_cv; Intros. +Cut ``0<eps0/r``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply (cond_pos r)]]. +Elim (H0 ? H3); Intros N0 H4. +Exists N0; Intros. +Cut (ge (S n) N0). +Intro hyp_sn. +Assert H6 := (H4 ? hyp_sn). +Unfold R_dist in H6; Rewrite minus_R0 in H6. +Rewrite Rabsolu_Rabsolu in H6. +Unfold R_dist; Rewrite minus_R0. +Rewrite Rabsolu_Rabsolu. +Replace ``(Rabsolu ((pow r (S n))/(INR (fact (S (S n))))))/ + (Rabsolu ((pow r n)/(INR (fact (S n)))))`` with ``r*/(INR (fact (S (S n))))*//(INR (fact (S n)))``. +Rewrite Rmult_assoc; Rewrite Rabsolu_mult. +Rewrite (Rabsolu_right r). +Apply Rlt_monotony_contra with ``/r``. +Apply Rlt_Rinv; Apply (cond_pos r). +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps0). +Apply H6. +Assumption. +Apply Rle_sym1; Left; Apply (cond_pos r). +Unfold Rdiv. +Repeat Rewrite Rabsolu_mult. +Repeat Rewrite Rabsolu_Rinv. +Rewrite Rinv_Rmult. +Repeat Rewrite Rabsolu_right. +Rewrite Rinv_Rinv. +Rewrite (Rmult_sym r). +Rewrite (Rmult_sym (pow r (S n))). +Repeat Rewrite Rmult_assoc. +Apply Rmult_mult_r. +Rewrite (Rmult_sym r). +Rewrite <- Rmult_assoc; Rewrite <- (Rmult_sym (INR (fact (S n)))). +Apply Rmult_mult_r. +Simpl. +Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Ring. +Apply pow_nonzero; Assumption. +Apply INR_fact_neq_0. +Apply Rle_sym1; Left; Apply INR_fact_lt_0. +Apply Rle_sym1; Left; Apply pow_lt; Apply (cond_pos r). +Apply Rle_sym1; Left; Apply INR_fact_lt_0. +Apply Rle_sym1; Left; Apply pow_lt; Apply (cond_pos r). +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply Rinv_neq_R0; Apply Rabsolu_no_R0; Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Unfold ge; Apply le_trans with n. +Apply H5. +Apply le_n_Sn. +Assert H1 := (cond_pos r); Red; Intro; Rewrite H2 in H1; Elim (Rlt_antirefl ? H1). +Qed. + +(**********) +Lemma derivable_pt_lim_exp : (x:R) (derivable_pt_lim exp x (exp x)). +Intro; Assert H0 := derivable_pt_lim_exp_0. +Unfold derivable_pt_lim in H0; Unfold derivable_pt_lim; Intros. +Cut ``0<eps/(exp x)``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Apply H | Apply Rlt_Rinv; Apply exp_pos]]. +Elim (H0 ? H1); Intros del H2. +Exists del; Intros. +Assert H5 := (H2 ? H3 H4). +Rewrite Rplus_Ol in H5; Rewrite exp_0 in H5. +Replace ``((exp (x+h))-(exp x))/h-(exp x)`` with ``(exp x)*(((exp h)-1)/h-1)``. +Rewrite Rabsolu_mult; Rewrite (Rabsolu_right (exp x)). +Apply Rlt_monotony_contra with ``/(exp x)``. +Apply Rlt_Rinv; Apply exp_pos. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps). +Apply H5. +Assert H6 := (exp_pos x); Red; Intro; Rewrite H7 in H6; Elim (Rlt_antirefl ? H6). +Apply Rle_sym1; Left; Apply exp_pos. +Rewrite Rminus_distr. +Rewrite Rmult_1r; Unfold Rdiv; Rewrite <- Rmult_assoc; Rewrite Rminus_distr. +Rewrite Rmult_1r; Rewrite exp_plus; Reflexivity. +Qed. diff --git a/theories7/Reals/Integration.v b/theories7/Reals/Integration.v new file mode 100644 index 00000000..410429ed --- /dev/null +++ b/theories7/Reals/Integration.v @@ -0,0 +1,13 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Integration.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Export NewtonInt. +Require Export RiemannInt_SF. +Require Export RiemannInt.
\ No newline at end of file diff --git a/theories7/Reals/MVT.v b/theories7/Reals/MVT.v new file mode 100644 index 00000000..eae414b1 --- /dev/null +++ b/theories7/Reals/MVT.v @@ -0,0 +1,517 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: MVT.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +Require Rtopology. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(* The Mean Value Theorem *) +Theorem MVT : (f,g:R->R;a,b:R;pr1:(c:R)``a<c<b``->(derivable_pt f c);pr2:(c:R)``a<c<b``->(derivable_pt g c)) ``a<b`` -> ((c:R)``a<=c<=b``->(continuity_pt f c)) -> ((c:R)``a<=c<=b``->(continuity_pt g c)) -> (EXT c : R | (EXT P : ``a<c<b`` | ``((g b)-(g a))*(derive_pt f c (pr1 c P))==((f b)-(f a))*(derive_pt g c (pr2 c P))``)). +Intros; Assert H2 := (Rlt_le ? ? H). +Pose h := [y:R]``((g b)-(g a))*(f y)-((f b)-(f a))*(g y)``. +Cut (c:R)``a<c<b``->(derivable_pt h c). +Intro; Cut ((c:R)``a<=c<=b``->(continuity_pt h c)). +Intro; Assert H4 := (continuity_ab_maj h a b H2 H3). +Assert H5 := (continuity_ab_min h a b H2 H3). +Elim H4; Intros Mx H6. +Elim H5; Intros mx H7. +Cut (h a)==(h b). +Intro; Pose M := (h Mx); Pose m := (h mx). +Cut (c:R;P:``a<c<b``) (derive_pt h c (X c P))==``((g b)-(g a))*(derive_pt f c (pr1 c P))-((f b)-(f a))*(derive_pt g c (pr2 c P))``. +Intro; Case (Req_EM (h a) M); Intro. +Case (Req_EM (h a) m); Intro. +Cut ((c:R)``a<=c<=b``->(h c)==M). +Intro; Cut ``a<(a+b)/2<b``. +(*** h constant ***) +Intro; Exists ``(a+b)/2``. +Exists H13. +Apply Rminus_eq; Rewrite <- H9; Apply deriv_constant2 with a b. +Elim H13; Intros; Assumption. +Elim H13; Intros; Assumption. +Intros; Rewrite (H12 ``(a+b)/2``). +Apply H12; Split; Left; Assumption. +Elim H13; Intros; Split; Left; Assumption. +Split. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Apply H. +DiscrR. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite Rplus_sym; Rewrite double; Apply Rlt_compatibility; Apply H. +DiscrR. +Intros; Elim H6; Intros H13 _. +Elim H7; Intros H14 _. +Apply Rle_antisym. +Apply H13; Apply H12. +Rewrite H10 in H11; Rewrite H11; Apply H14; Apply H12. +Cut ``a<mx<b``. +(*** h admet un minimum global sur [a,b] ***) +Intro; Exists mx. +Exists H12. +Apply Rminus_eq; Rewrite <- H9; Apply deriv_minimum with a b. +Elim H12; Intros; Assumption. +Elim H12; Intros; Assumption. +Intros; Elim H7; Intros. +Apply H15; Split; Left; Assumption. +Elim H7; Intros _ H12; Elim H12; Intros; Split. +Inversion H13. +Apply H15. +Rewrite H15 in H11; Elim H11; Reflexivity. +Inversion H14. +Apply H15. +Rewrite H8 in H11; Rewrite <- H15 in H11; Elim H11; Reflexivity. +Cut ``a<Mx<b``. +(*** h admet un maximum global sur [a,b] ***) +Intro; Exists Mx. +Exists H11. +Apply Rminus_eq; Rewrite <- H9; Apply deriv_maximum with a b. +Elim H11; Intros; Assumption. +Elim H11; Intros; Assumption. +Intros; Elim H6; Intros; Apply H14. +Split; Left; Assumption. +Elim H6; Intros _ H11; Elim H11; Intros; Split. +Inversion H12. +Apply H14. +Rewrite H14 in H10; Elim H10; Reflexivity. +Inversion H13. +Apply H14. +Rewrite H8 in H10; Rewrite <- H14 in H10; Elim H10; Reflexivity. +Intros; Unfold h; Replace (derive_pt [y:R]``((g b)-(g a))*(f y)-((f b)-(f a))*(g y)`` c (X c P)) with (derive_pt (minus_fct (mult_fct (fct_cte ``(g b)-(g a)``) f) (mult_fct (fct_cte ``(f b)-(f a)``) g)) c (derivable_pt_minus ? ? ? (derivable_pt_mult ? ? ? (derivable_pt_const ``(g b)-(g a)`` c) (pr1 c P)) (derivable_pt_mult ? ? ? (derivable_pt_const ``(f b)-(f a)`` c) (pr2 c P)))); [Idtac | Apply pr_nu]. +Rewrite derive_pt_minus; Do 2 Rewrite derive_pt_mult; Do 2 Rewrite derive_pt_const; Do 2 Rewrite Rmult_Ol; Do 2 Rewrite Rplus_Ol; Reflexivity. +Unfold h; Ring. +Intros; Unfold h; Change (continuity_pt (minus_fct (mult_fct (fct_cte ``(g b)-(g a)``) f) (mult_fct (fct_cte ``(f b)-(f a)``) g)) c). +Apply continuity_pt_minus; Apply continuity_pt_mult. +Apply derivable_continuous_pt; Apply derivable_const. +Apply H0; Apply H3. +Apply derivable_continuous_pt; Apply derivable_const. +Apply H1; Apply H3. +Intros; Change (derivable_pt (minus_fct (mult_fct (fct_cte ``(g b)-(g a)``) f) (mult_fct (fct_cte ``(f b)-(f a)``) g)) c). +Apply derivable_pt_minus; Apply derivable_pt_mult. +Apply derivable_pt_const. +Apply (pr1 ? H3). +Apply derivable_pt_const. +Apply (pr2 ? H3). +Qed. + +(* Corollaries ... *) +Lemma MVT_cor1 : (f:(R->R); a,b:R; pr:(derivable f)) ``a < b``->(EXT c:R | ``(f b)-(f a) == (derive_pt f c (pr c))*(b-a)``/\``a < c < b``). +Intros f a b pr H; Cut (c:R)``a<c<b``->(derivable_pt f c); [Intro | Intros; Apply pr]. +Cut (c:R)``a<c<b``->(derivable_pt id c); [Intro | Intros; Apply derivable_pt_id]. +Cut ((c:R)``a<=c<=b``->(continuity_pt f c)); [Intro | Intros; Apply derivable_continuous_pt; Apply pr]. +Cut ((c:R)``a<=c<=b``->(continuity_pt id c)); [Intro | Intros; Apply derivable_continuous_pt; Apply derivable_id]. +Assert H2 := (MVT f id a b X X0 H H0 H1). +Elim H2; Intros c H3; Elim H3; Intros. +Exists c; Split. +Cut (derive_pt id c (X0 c x)) == (derive_pt id c (derivable_pt_id c)); [Intro | Apply pr_nu]. +Rewrite H5 in H4; Rewrite (derive_pt_id c) in H4; Rewrite Rmult_1r in H4; Rewrite <- H4; Replace (derive_pt f c (X c x)) with (derive_pt f c (pr c)); [Idtac | Apply pr_nu]; Apply Rmult_sym. +Apply x. +Qed. + +Theorem MVT_cor2 : (f,f':R->R;a,b:R) ``a<b`` -> ((c:R)``a<=c<=b``->(derivable_pt_lim f c (f' c))) -> (EXT c:R | ``(f b)-(f a)==(f' c)*(b-a)``/\``a<c<b``). +Intros f f' a b H H0; Cut ((c:R)``a<=c<=b``->(derivable_pt f c)). +Intro; Cut ((c:R)``a<c<b``->(derivable_pt f c)). +Intro; Cut ((c:R)``a<=c<=b``->(continuity_pt f c)). +Intro; Cut ((c:R)``a<=c<=b``->(derivable_pt id c)). +Intro; Cut ((c:R)``a<c<b``->(derivable_pt id c)). +Intro; Cut ((c:R)``a<=c<=b``->(continuity_pt id c)). +Intro; Elim (MVT f id a b X0 X2 H H1 H2); Intros; Elim H3; Clear H3; Intros; Exists x; Split. +Cut (derive_pt id x (X2 x x0))==R1. +Cut (derive_pt f x (X0 x x0))==(f' x). +Intros; Rewrite H4 in H3; Rewrite H5 in H3; Unfold id in H3; Rewrite Rmult_1r in H3; Rewrite Rmult_sym; Symmetry; Assumption. +Apply derive_pt_eq_0; Apply H0; Elim x0; Intros; Split; Left; Assumption. +Apply derive_pt_eq_0; Apply derivable_pt_lim_id. +Assumption. +Intros; Apply derivable_continuous_pt; Apply X1; Assumption. +Intros; Apply derivable_pt_id. +Intros; Apply derivable_pt_id. +Intros; Apply derivable_continuous_pt; Apply X; Assumption. +Intros; Elim H1; Intros; Apply X; Split; Left; Assumption. +Intros; Unfold derivable_pt; Apply Specif.existT with (f' c); Apply H0; Apply H1. +Qed. + +Lemma MVT_cor3 : (f,f':(R->R); a,b:R) ``a < b`` -> ((x:R)``a <= x`` -> ``x <= b``->(derivable_pt_lim f x (f' x))) -> (EXT c:R | ``a<=c``/\``c<=b``/\``(f b)==(f a) + (f' c)*(b-a)``). +Intros f f' a b H H0; Assert H1 : (EXT c:R | ``(f b) -(f a) == (f' c)*(b-a)``/\``a<c<b``); [Apply MVT_cor2; [Apply H | Intros; Elim H1; Intros; Apply (H0 ? H2 H3)] | Elim H1; Intros; Exists x; Elim H2; Intros; Elim H4; Intros; Split; [Left; Assumption | Split; [Left; Assumption | Rewrite <- H3; Ring]]]. +Qed. + +Lemma Rolle : (f:R->R;a,b:R;pr:(x:R)``a<x<b``->(derivable_pt f x)) ((x:R)``a<=x<=b``->(continuity_pt f x)) -> ``a<b`` -> (f a)==(f b) -> (EXT c:R | (EXT P: ``a<c<b`` | ``(derive_pt f c (pr c P))==0``)). +Intros; Assert H2 : (x:R)``a<x<b``->(derivable_pt id x). +Intros; Apply derivable_pt_id. +Assert H3 := (MVT f id a b pr H2 H0 H); Assert H4 : (x:R)``a<=x<=b``->(continuity_pt id x). +Intros; Apply derivable_continuous; Apply derivable_id. +Elim (H3 H4); Intros; Elim H5; Intros; Exists x; Exists x0; Rewrite H1 in H6; Unfold id in H6; Unfold Rminus in H6; Rewrite Rplus_Ropp_r in H6; Rewrite Rmult_Ol in H6; Apply r_Rmult_mult with ``b-a``; [Rewrite Rmult_Or; Apply H6 | Apply Rminus_eq_contra; Red; Intro; Rewrite H7 in H0; Elim (Rlt_antirefl ? H0)]. +Qed. + +(**********) +Lemma nonneg_derivative_1 : (f:R->R;pr:(derivable f)) ((x:R) ``0<=(derive_pt f x (pr x))``) -> (increasing f). +Intros. +Unfold increasing. +Intros. +Case (total_order_T x y); Intro. +Elim s; Intro. +Apply Rle_anti_compatibility with ``-(f x)``. +Rewrite Rplus_Ropp_l; Rewrite Rplus_sym. +Assert H1 := (MVT_cor1 f ? ? pr a). +Elim H1; Intros. +Elim H2; Intros. +Unfold Rminus in H3. +Rewrite H3. +Apply Rmult_le_pos. +Apply H. +Apply Rle_anti_compatibility with x. +Rewrite Rplus_Or; Replace ``x+(y+ -x)`` with y; [Assumption | Ring]. +Rewrite b; Right; Reflexivity. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 r)). +Qed. + +(**********) +Lemma nonpos_derivative_0 : (f:R->R;pr:(derivable f)) (decreasing f) -> ((x:R) ``(derive_pt f x (pr x))<=0``). +Intros f pr H x; Assert H0 :=H; Unfold decreasing in H0; Generalize (derivable_derive f x (pr x)); Intro; Elim H1; Intros l H2. +Rewrite H2; Case (total_order l R0); Intro. +Left; Assumption. +Elim H3; Intro. +Right; Assumption. +Generalize (derive_pt_eq_1 f x l (pr x) H2); Intros; Cut ``0< (l/2)``. +Intro; Elim (H5 ``(l/2)`` H6); Intros delta H7; Cut ``delta/2<>0``/\``0<delta/2``/\``(Rabsolu delta/2)<delta``. +Intro; Decompose [and] H8; Intros; Generalize (H7 ``delta/2`` H9 H12); Cut ``((f (x+delta/2))-(f x))/(delta/2)<=0``. +Intro; Cut ``0< -(((f (x+delta/2))-(f x))/(delta/2)-l)``. +Intro; Unfold Rabsolu; Case (case_Rabsolu ``((f (x+delta/2))-(f x))/(delta/2)-l``). +Intros; Generalize (Rlt_compatibility_r ``-l`` ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` ``(l/2)`` H14); Unfold Rminus. +Replace ``(l/2)+ -l`` with ``-(l/2)``. +Replace `` -(((f (x+delta/2))+ -(f x))/(delta/2)+ -l)+ -l`` with ``-(((f (x+delta/2))+ -(f x))/(delta/2))``. +Intro. +Generalize (Rlt_Ropp ``-(((f (x+delta/2))+ -(f x))/(delta/2))`` ``-(l/2)`` H15). +Repeat Rewrite Ropp_Ropp. +Intro. +Generalize (Rlt_trans ``0`` ``l/2`` ``((f (x+delta/2))-(f x))/(delta/2)`` H6 H16); Intro. +Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)`` ``0`` H17 H10)). +Ring. +Pattern 3 l; Rewrite double_var. +Ring. +Intros. +Generalize (Rge_Ropp ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``0`` r). +Rewrite Ropp_O. +Intro. +Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` ``0`` H13 H15)). +Replace ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` with ``(((f (x))-(f (x+delta/2)))/(delta/2)) +l``. +Unfold Rminus. +Apply ge0_plus_gt0_is_gt0. +Unfold Rdiv; Apply Rmult_le_pos. +Cut ``x<=(x+(delta*/2))``. +Intro; Generalize (H0 x ``x+(delta*/2)`` H13); Intro; Generalize (Rle_compatibility ``-(f (x+delta/2))`` ``(f (x+delta/2))`` ``(f x)`` H14); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. +Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_Rinv; Assumption. +Assumption. +Rewrite Ropp_distr2. +Unfold Rminus. +Rewrite (Rplus_sym l). +Unfold Rdiv. +Rewrite <- Ropp_mul1. +Rewrite Ropp_distr1. +Rewrite Ropp_Ropp. +Rewrite (Rplus_sym (f x)). +Reflexivity. +Replace ``((f (x+delta/2))-(f x))/(delta/2)`` with ``-(((f x)-(f (x+delta/2)))/(delta/2))``. +Rewrite <- Ropp_O. +Apply Rge_Ropp. +Apply Rle_sym1. +Unfold Rdiv; Apply Rmult_le_pos. +Cut ``x<=(x+(delta*/2))``. +Intro; Generalize (H0 x ``x+(delta*/2)`` H10); Intro. +Generalize (Rle_compatibility ``-(f (x+delta/2))`` ``(f (x+delta/2))`` ``(f x)`` H13); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. +Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_Rinv; Assumption. +Unfold Rdiv; Rewrite <- Ropp_mul1. +Rewrite Ropp_distr2. +Reflexivity. +Split. +Unfold Rdiv; Apply prod_neq_R0. +Generalize (cond_pos delta); Intro; Red; Intro H9; Rewrite H9 in H8; Elim (Rlt_antirefl ``0`` H8). +Apply Rinv_neq_R0; DiscrR. +Split. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. +Rewrite Rabsolu_right. +Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. +Sup0. +Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite double; Pattern 1 (pos delta); Rewrite <- Rplus_Or. +Apply Rlt_compatibility; Apply (cond_pos delta). +DiscrR. +Apply Rle_sym1; Unfold Rdiv; Left; Apply Rmult_lt_pos. +Apply (cond_pos delta). +Apply Rlt_Rinv; Sup0. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply H4 | Apply Rlt_Rinv; Sup0]. +Qed. + +(**********) +Lemma increasing_decreasing_opp : (f:R->R) (increasing f) -> (decreasing (opp_fct f)). +Unfold increasing decreasing opp_fct; Intros; Generalize (H x y H0); Intro; Apply Rge_Ropp; Apply Rle_sym1; Assumption. +Qed. + +(**********) +Lemma nonpos_derivative_1 : (f:R->R;pr:(derivable f)) ((x:R) ``(derive_pt f x (pr x))<=0``) -> (decreasing f). +Intros. +Cut (h:R)``-(-(f h))==(f h)``. +Intro. +Generalize (increasing_decreasing_opp (opp_fct f)). +Unfold decreasing. +Unfold opp_fct. +Intros. +Rewrite <- (H0 x); Rewrite <- (H0 y). +Apply H1. +Cut (x:R)``0<=(derive_pt (opp_fct f) x ((derivable_opp f pr) x))``. +Intros. +Replace [x:R]``-(f x)`` with (opp_fct f); [Idtac | Reflexivity]. +Apply (nonneg_derivative_1 (opp_fct f) (derivable_opp f pr) H3). +Intro. +Assert H3 := (derive_pt_opp f x0 (pr x0)). +Cut ``(derive_pt (opp_fct f) x0 (derivable_pt_opp f x0 (pr x0)))==(derive_pt (opp_fct f) x0 (derivable_opp f pr x0))``. +Intro. +Rewrite <- H4. +Rewrite H3. +Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Apply (H x0). +Apply pr_nu. +Assumption. +Intro; Ring. +Qed. + +(**********) +Lemma positive_derivative : (f:R->R;pr:(derivable f)) ((x:R) ``0<(derive_pt f x (pr x))``)->(strict_increasing f). +Intros. +Unfold strict_increasing. +Intros. +Apply Rlt_anti_compatibility with ``-(f x)``. +Rewrite Rplus_Ropp_l; Rewrite Rplus_sym. +Assert H1 := (MVT_cor1 f ? ? pr H0). +Elim H1; Intros. +Elim H2; Intros. +Unfold Rminus in H3. +Rewrite H3. +Apply Rmult_lt_pos. +Apply H. +Apply Rlt_anti_compatibility with x. +Rewrite Rplus_Or; Replace ``x+(y+ -x)`` with y; [Assumption | Ring]. +Qed. + +(**********) +Lemma strictincreasing_strictdecreasing_opp : (f:R->R) (strict_increasing f) -> +(strict_decreasing (opp_fct f)). +Unfold strict_increasing strict_decreasing opp_fct; Intros; Generalize (H x y H0); Intro; Apply Rlt_Ropp; Assumption. +Qed. + +(**********) +Lemma negative_derivative : (f:R->R;pr:(derivable f)) ((x:R) ``(derive_pt f x (pr x))<0``)->(strict_decreasing f). +Intros. +Cut (h:R)``- (-(f h))==(f h)``. +Intros. +Generalize (strictincreasing_strictdecreasing_opp (opp_fct f)). +Unfold strict_decreasing opp_fct. +Intros. +Rewrite <- (H0 x). +Rewrite <- (H0 y). +Apply H1; [Idtac | Assumption]. +Cut (x:R)``0<(derive_pt (opp_fct f) x (derivable_opp f pr x))``. +Intros; EApply positive_derivative; Apply H3. +Intro. +Assert H3 := (derive_pt_opp f x0 (pr x0)). +Cut ``(derive_pt (opp_fct f) x0 (derivable_pt_opp f x0 (pr x0)))==(derive_pt (opp_fct f) x0 (derivable_opp f pr x0))``. +Intro. +Rewrite <- H4; Rewrite H3. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Apply (H x0). +Apply pr_nu. +Intro; Ring. +Qed. + +(**********) +Lemma null_derivative_0 : (f:R->R;pr:(derivable f)) (constant f)->((x:R) ``(derive_pt f x (pr x))==0``). +Intros. +Unfold constant in H. +Apply derive_pt_eq_0. +Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Simpl; Intros. +Rewrite (H x ``x+h``); Unfold Rminus; Unfold Rdiv; Rewrite Rplus_Ropp_r; Rewrite Rmult_Ol; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Qed. + +(**********) +Lemma increasing_decreasing : (f:R->R) (increasing f) -> (decreasing f) -> (constant f). +Unfold increasing decreasing constant; Intros; Case (total_order x y); Intro. +Generalize (Rlt_le x y H1); Intro; Apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)). +Elim H1; Intro. +Rewrite H2; Reflexivity. +Generalize (Rlt_le y x H2); Intro; Symmetry; Apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)). +Qed. + +(**********) +Lemma null_derivative_1 : (f:R->R;pr:(derivable f)) ((x:R) ``(derive_pt f x (pr x))==0``)->(constant f). +Intros. +Cut (x:R)``(derive_pt f x (pr x)) <= 0``. +Cut (x:R)``0 <= (derive_pt f x (pr x))``. +Intros. +Assert H2 := (nonneg_derivative_1 f pr H0). +Assert H3 := (nonpos_derivative_1 f pr H1). +Apply increasing_decreasing; Assumption. +Intro; Right; Symmetry; Apply (H x). +Intro; Right; Apply (H x). +Qed. + +(**********) +Lemma derive_increasing_interv_ax : (a,b:R;f:R->R;pr:(derivable f)) ``a<b``-> (((t:R) ``a<t<b`` -> ``0<(derive_pt f t (pr t))``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<(f y)``)) /\ (((t:R) ``a<t<b`` -> ``0<=(derive_pt f t (pr t))``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<=(f y)``)). +Intros. +Split; Intros. +Apply Rlt_anti_compatibility with ``-(f x)``. +Rewrite Rplus_Ropp_l; Rewrite Rplus_sym. +Assert H4 := (MVT_cor1 f ? ? pr H3). +Elim H4; Intros. +Elim H5; Intros. +Unfold Rminus in H6. +Rewrite H6. +Apply Rmult_lt_pos. +Apply H0. +Elim H7; Intros. +Split. +Elim H1; Intros. +Apply Rle_lt_trans with x; Assumption. +Elim H2; Intros. +Apply Rlt_le_trans with y; Assumption. +Apply Rlt_anti_compatibility with x. +Rewrite Rplus_Or; Replace ``x+(y+ -x)`` with y; [Assumption | Ring]. +Apply Rle_anti_compatibility with ``-(f x)``. +Rewrite Rplus_Ropp_l; Rewrite Rplus_sym. +Assert H4 := (MVT_cor1 f ? ? pr H3). +Elim H4; Intros. +Elim H5; Intros. +Unfold Rminus in H6. +Rewrite H6. +Apply Rmult_le_pos. +Apply H0. +Elim H7; Intros. +Split. +Elim H1; Intros. +Apply Rle_lt_trans with x; Assumption. +Elim H2; Intros. +Apply Rlt_le_trans with y; Assumption. +Apply Rle_anti_compatibility with x. +Rewrite Rplus_Or; Replace ``x+(y+ -x)`` with y; [Left; Assumption | Ring]. +Qed. + +(**********) +Lemma derive_increasing_interv : (a,b:R;f:R->R;pr:(derivable f)) ``a<b``-> ((t:R) ``a<t<b`` -> ``0<(derive_pt f t (pr t))``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<(f y)``). +Intros. +Generalize (derive_increasing_interv_ax a b f pr H); Intro. +Elim H4; Intros H5 _; Apply (H5 H0 x y H1 H2 H3). +Qed. + +(**********) +Lemma derive_increasing_interv_var : (a,b:R;f:R->R;pr:(derivable f)) ``a<b``-> ((t:R) ``a<t<b`` -> ``0<=(derive_pt f t (pr t))``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<=(f y)``). +Intros a b f pr H H0 x y H1 H2 H3; Generalize (derive_increasing_interv_ax a b f pr H); Intro; Elim H4; Intros _ H5; Apply (H5 H0 x y H1 H2 H3). +Qed. + +(**********) +(**********) +Theorem IAF : (f:R->R;a,b,k:R;pr:(derivable f)) ``a<=b`` -> ((c:R) ``a<=c<=b`` -> ``(derive_pt f c (pr c))<=k``) -> ``(f b)-(f a)<=k*(b-a)``. +Intros. +Case (total_order_T a b); Intro. +Elim s; Intro. +Assert H1 := (MVT_cor1 f ? ? pr a0). +Elim H1; Intros. +Elim H2; Intros. +Rewrite H3. +Do 2 Rewrite <- (Rmult_sym ``(b-a)``). +Apply Rle_monotony. +Apply Rle_anti_compatibility with ``a``; Rewrite Rplus_Or. +Replace ``a+(b-a)`` with b; [Assumption | Ring]. +Apply H0. +Elim H4; Intros. +Split; Left; Assumption. +Rewrite b0. +Unfold Rminus; Do 2 Rewrite Rplus_Ropp_r. +Rewrite Rmult_Or; Right; Reflexivity. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Qed. + +Lemma IAF_var : (f,g:R->R;a,b:R;pr1:(derivable f);pr2:(derivable g)) ``a<=b`` -> ((c:R) ``a<=c<=b`` -> ``(derive_pt g c (pr2 c))<=(derive_pt f c (pr1 c))``) -> ``(g b)-(g a)<=(f b)-(f a)``. +Intros. +Cut (derivable (minus_fct g f)). +Intro. +Cut (c:R)``a<=c<=b``->``(derive_pt (minus_fct g f) c (X c))<=0``. +Intro. +Assert H2 := (IAF (minus_fct g f) a b R0 X H H1). +Rewrite Rmult_Ol in H2; Unfold minus_fct in H2. +Apply Rle_anti_compatibility with ``-(f b)+(f a)``. +Replace ``-(f b)+(f a)+((f b)-(f a))`` with R0; [Idtac | Ring]. +Replace ``-(f b)+(f a)+((g b)-(g a))`` with ``(g b)-(f b)-((g a)-(f a))``; [Apply H2 | Ring]. +Intros. +Cut (derive_pt (minus_fct g f) c (X c))==(derive_pt (minus_fct g f) c (derivable_pt_minus ? ? ? (pr2 c) (pr1 c))). +Intro. +Rewrite H2. +Rewrite derive_pt_minus. +Apply Rle_anti_compatibility with (derive_pt f c (pr1 c)). +Rewrite Rplus_Or. +Replace ``(derive_pt f c (pr1 c))+((derive_pt g c (pr2 c))-(derive_pt f c (pr1 c)))`` with ``(derive_pt g c (pr2 c))``; [Idtac | Ring]. +Apply H0; Assumption. +Apply pr_nu. +Apply derivable_minus; Assumption. +Qed. + +(* If f has a null derivative in ]a,b[ and is continue in [a,b], *) +(* then f is constant on [a,b] *) +Lemma null_derivative_loc : (f:R->R;a,b:R;pr:(x:R)``a<x<b``->(derivable_pt f x)) ((x:R)``a<=x<=b``->(continuity_pt f x)) -> ((x:R;P:``a<x<b``)(derive_pt f x (pr x P))==R0) -> (constant_D_eq f [x:R]``a<=x<=b`` (f a)). +Intros; Unfold constant_D_eq; Intros; Case (total_order_T a b); Intro. +Elim s; Intro. +Assert H2 : (y:R)``a<y<x``->(derivable_pt id y). +Intros; Apply derivable_pt_id. +Assert H3 : (y:R)``a<=y<=x``->(continuity_pt id y). +Intros; Apply derivable_continuous; Apply derivable_id. +Assert H4 : (y:R)``a<y<x``->(derivable_pt f y). +Intros; Apply pr; Elim H4; Intros; Split. +Assumption. +Elim H1; Intros; Apply Rlt_le_trans with x; Assumption. +Assert H5 : (y:R)``a<=y<=x``->(continuity_pt f y). +Intros; Apply H; Elim H5; Intros; Split. +Assumption. +Elim H1; Intros; Apply Rle_trans with x; Assumption. +Elim H1; Clear H1; Intros; Elim H1; Clear H1; Intro. +Assert H7 := (MVT f id a x H4 H2 H1 H5 H3). +Elim H7; Intros; Elim H8; Intros; Assert H10 : ``a<x0<b``. +Elim x1; Intros; Split. +Assumption. +Apply Rlt_le_trans with x; Assumption. +Assert H11 : ``(derive_pt f x0 (H4 x0 x1))==0``. +Replace (derive_pt f x0 (H4 x0 x1)) with (derive_pt f x0 (pr x0 H10)); [Apply H0 | Apply pr_nu]. +Assert H12 : ``(derive_pt id x0 (H2 x0 x1))==1``. +Apply derive_pt_eq_0; Apply derivable_pt_lim_id. +Rewrite H11 in H9; Rewrite H12 in H9; Rewrite Rmult_Or in H9; Rewrite Rmult_1r in H9; Apply Rminus_eq; Symmetry; Assumption. +Rewrite H1; Reflexivity. +Assert H2 : x==a. +Rewrite <- b0 in H1; Elim H1; Intros; Apply Rle_antisym; Assumption. +Rewrite H2; Reflexivity. +Elim H1; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (Rle_trans ? ? ? H2 H3) r)). +Qed. + +(* Unicity of the antiderivative *) +Lemma antiderivative_Ucte : (f,g1,g2:R->R;a,b:R) (antiderivative f g1 a b) -> (antiderivative f g2 a b) -> (EXT c:R | (x:R)``a<=x<=b``->``(g1 x)==(g2 x)+c``). +Unfold antiderivative; Intros; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros H0 _; Exists ``(g1 a)-(g2 a)``; Intros; Assert H3 : (x:R)``a<=x<=b``->(derivable_pt g1 x). +Intros; Unfold derivable_pt; Apply Specif.existT with (f x0); Elim (H x0 H3); Intros; EApply derive_pt_eq_1; Symmetry; Apply H4. +Assert H4 : (x:R)``a<=x<=b``->(derivable_pt g2 x). +Intros; Unfold derivable_pt; Apply Specif.existT with (f x0); Elim (H0 x0 H4); Intros; EApply derive_pt_eq_1; Symmetry; Apply H5. +Assert H5 : (x:R)``a<x<b``->(derivable_pt (minus_fct g1 g2) x). +Intros; Elim H5; Intros; Apply derivable_pt_minus; [Apply H3; Split; Left; Assumption | Apply H4; Split; Left; Assumption]. +Assert H6 : (x:R)``a<=x<=b``->(continuity_pt (minus_fct g1 g2) x). +Intros; Apply derivable_continuous_pt; Apply derivable_pt_minus; [Apply H3 | Apply H4]; Assumption. +Assert H7 : (x:R;P:``a<x<b``)(derive_pt (minus_fct g1 g2) x (H5 x P))==``0``. +Intros; Elim P; Intros; Apply derive_pt_eq_0; Replace R0 with ``(f x0)-(f x0)``; [Idtac | Ring]. +Assert H9 : ``a<=x0<=b``. +Split; Left; Assumption. +Apply derivable_pt_lim_minus; [Elim (H ? H9) | Elim (H0 ? H9)]; Intros; EApply derive_pt_eq_1; Symmetry; Apply H10. +Assert H8 := (null_derivative_loc (minus_fct g1 g2) a b H5 H6 H7); Unfold constant_D_eq in H8; Assert H9 := (H8 ? H2); Unfold minus_fct in H9; Rewrite <- H9; Ring. +Qed. diff --git a/theories7/Reals/NewtonInt.v b/theories7/Reals/NewtonInt.v new file mode 100644 index 00000000..56e5f15e --- /dev/null +++ b/theories7/Reals/NewtonInt.v @@ -0,0 +1,600 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: NewtonInt.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(*******************************************) +(* Newton's Integral *) +(*******************************************) + +Definition Newton_integrable [f:R->R;a,b:R] : Type := (sigTT ? [g:R->R](antiderivative f g a b)\/(antiderivative f g b a)). + +Definition NewtonInt [f:R->R;a,b:R;pr:(Newton_integrable f a b)] : R := let g = Cases pr of (existTT a b) => a end in ``(g b)-(g a)``. + +(* If f is differentiable, then f' is Newton integrable (Tautology ?) *) +Lemma FTCN_step1 : (f:Differential;a,b:R) (Newton_integrable [x:R](derive_pt f x (cond_diff f x)) a b). +Intros f a b; Unfold Newton_integrable; Apply existTT with (d1 f); Unfold antiderivative; Intros; Case (total_order_Rle a b); Intro; [Left; Split; [Intros; Exists (cond_diff f x); Reflexivity | Assumption] | Right; Split; [Intros; Exists (cond_diff f x); Reflexivity | Auto with real]]. +Defined. + +(* By definition, we have the Fondamental Theorem of Calculus *) +Lemma FTC_Newton : (f:Differential;a,b:R) (NewtonInt [x:R](derive_pt f x (cond_diff f x)) a b (FTCN_step1 f a b))==``(f b)-(f a)``. +Intros; Unfold NewtonInt; Reflexivity. +Qed. + +(* $\int_a^a f$ exists forall a:R and f:R->R *) +Lemma NewtonInt_P1 : (f:R->R;a:R) (Newton_integrable f a a). +Intros f a; Unfold Newton_integrable; Apply existTT with (mult_fct (fct_cte (f a)) id); Left; Unfold antiderivative; Split. +Intros; Assert H1 : (derivable_pt (mult_fct (fct_cte (f a)) id) x). +Apply derivable_pt_mult. +Apply derivable_pt_const. +Apply derivable_pt_id. +Exists H1; Assert H2 : x==a. +Elim H; Intros; Apply Rle_antisym; Assumption. +Symmetry; Apply derive_pt_eq_0; Replace (f x) with ``0*(id x)+(fct_cte (f a) x)*1``; [Apply (derivable_pt_lim_mult (fct_cte (f a)) id x); [Apply derivable_pt_lim_const | Apply derivable_pt_lim_id] | Unfold id fct_cte; Rewrite H2; Ring]. +Right; Reflexivity. +Defined. + +(* $\int_a^a f = 0$ *) +Lemma NewtonInt_P2 : (f:R->R;a:R) ``(NewtonInt f a a (NewtonInt_P1 f a))==0``. +Intros; Unfold NewtonInt; Simpl; Unfold mult_fct fct_cte id; Ring. +Qed. + +(* If $\int_a^b f$ exists, then $\int_b^a f$ exists too *) +Lemma NewtonInt_P3 : (f:R->R;a,b:R;X:(Newton_integrable f a b)) (Newton_integrable f b a). +Unfold Newton_integrable; Intros; Elim X; Intros g H; Apply existTT with g; Tauto. +Defined. + +(* $\int_a^b f = -\int_b^a f$ *) +Lemma NewtonInt_P4 : (f:R->R;a,b:R;pr:(Newton_integrable f a b)) ``(NewtonInt f a b pr)==-(NewtonInt f b a (NewtonInt_P3 f a b pr))``. +Intros; Unfold Newton_integrable in pr; Elim pr; Intros; Elim p; Intro. +Unfold NewtonInt; Case (NewtonInt_P3 f a b (existTT R->R [g:(R->R)](antiderivative f g a b)\/(antiderivative f g b a) x p)). +Intros; Elim o; Intro. +Unfold antiderivative in H0; Elim H0; Intros; Elim H2; Intro. +Unfold antiderivative in H; Elim H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H5 H3)). +Rewrite H3; Ring. +Assert H1 := (antiderivative_Ucte f x x0 a b H H0); Elim H1; Intros; Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. +Assert H3 : ``a<=a<=b``. +Split; [Right; Reflexivity | Assumption]. +Assert H4 : ``a<=b<=b``. +Split; [Assumption | Right; Reflexivity]. +Assert H5 := (H2 ? H3); Assert H6 := (H2 ? H4); Rewrite H5; Rewrite H6; Ring. +Unfold NewtonInt; Case (NewtonInt_P3 f a b (existTT R->R [g:(R->R)](antiderivative f g a b)\/(antiderivative f g b a) x p)); Intros; Elim o; Intro. +Assert H1 := (antiderivative_Ucte f x x0 b a H H0); Elim H1; Intros; Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. +Assert H3 : ``b<=a<=a``. +Split; [Assumption | Right; Reflexivity]. +Assert H4 : ``b<=b<=a``. +Split; [Right; Reflexivity | Assumption]. +Assert H5 := (H2 ? H3); Assert H6 := (H2 ? H4); Rewrite H5; Rewrite H6; Ring. +Unfold antiderivative in H0; Elim H0; Intros; Elim H2; Intro. +Unfold antiderivative in H; Elim H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H5 H3)). +Rewrite H3; Ring. +Qed. + +(* The set of Newton integrable functions is a vectorial space *) +Lemma NewtonInt_P5 : (f,g:R->R;l,a,b:R) (Newton_integrable f a b) -> (Newton_integrable g a b) -> (Newton_integrable [x:R]``l*(f x)+(g x)`` a b). +Unfold Newton_integrable; Intros; Elim X; Intros; Elim X0; Intros; Exists [y:R]``l*(x y)+(x0 y)``. +Elim p; Intro. +Elim p0; Intro. +Left; Unfold antiderivative; Unfold antiderivative in H H0; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros H0 _. +Split. +Intros; Elim (H ? H2); Elim (H0 ? H2); Intros. +Assert H5 : (derivable_pt [y:R]``l*(x y)+(x0 y)`` x1). +Reg. +Exists H5; Symmetry; Reg; Rewrite <- H3; Rewrite <- H4; Reflexivity. +Assumption. +Unfold antiderivative in H H0; Elim H; Elim H0; Intros; Elim H4; Intro. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H5 H2)). +Left; Rewrite <- H5; Unfold antiderivative; Split. +Intros; Elim H6; Intros; Assert H9 : ``x1==a``. +Apply Rle_antisym; Assumption. +Assert H10 : ``a<=x1<=b``. +Split; Right; [Symmetry; Assumption | Rewrite <- H5; Assumption]. +Assert H11 : ``b<=x1<=a``. +Split; Right; [Rewrite <- H5; Symmetry; Assumption | Assumption]. +Assert H12 : (derivable_pt x x1). +Unfold derivable_pt; Exists (f x1); Elim (H3 ? H10); Intros; EApply derive_pt_eq_1; Symmetry; Apply H12. +Assert H13 : (derivable_pt x0 x1). +Unfold derivable_pt; Exists (g x1); Elim (H1 ? H11); Intros; EApply derive_pt_eq_1; Symmetry; Apply H13. +Assert H14 : (derivable_pt [y:R]``l*(x y)+(x0 y)`` x1). +Reg. +Exists H14; Symmetry; Reg. +Assert H15 : ``(derive_pt x0 x1 H13)==(g x1)``. +Elim (H1 ? H11); Intros; Rewrite H15; Apply pr_nu. +Assert H16 : ``(derive_pt x x1 H12)==(f x1)``. +Elim (H3 ? H10); Intros; Rewrite H16; Apply pr_nu. +Rewrite H15; Rewrite H16; Ring. +Right; Reflexivity. +Elim p0; Intro. +Unfold antiderivative in H H0; Elim H; Elim H0; Intros; Elim H4; Intro. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H5 H2)). +Left; Rewrite H5; Unfold antiderivative; Split. +Intros; Elim H6; Intros; Assert H9 : ``x1==a``. +Apply Rle_antisym; Assumption. +Assert H10 : ``a<=x1<=b``. +Split; Right; [Symmetry; Assumption | Rewrite H5; Assumption]. +Assert H11 : ``b<=x1<=a``. +Split; Right; [Rewrite H5; Symmetry; Assumption | Assumption]. +Assert H12 : (derivable_pt x x1). +Unfold derivable_pt; Exists (f x1); Elim (H3 ? H11); Intros; EApply derive_pt_eq_1; Symmetry; Apply H12. +Assert H13 : (derivable_pt x0 x1). +Unfold derivable_pt; Exists (g x1); Elim (H1 ? H10); Intros; EApply derive_pt_eq_1; Symmetry; Apply H13. +Assert H14 : (derivable_pt [y:R]``l*(x y)+(x0 y)`` x1). +Reg. +Exists H14; Symmetry; Reg. +Assert H15 : ``(derive_pt x0 x1 H13)==(g x1)``. +Elim (H1 ? H10); Intros; Rewrite H15; Apply pr_nu. +Assert H16 : ``(derive_pt x x1 H12)==(f x1)``. +Elim (H3 ? H11); Intros; Rewrite H16; Apply pr_nu. +Rewrite H15; Rewrite H16; Ring. +Right; Reflexivity. +Right; Unfold antiderivative; Unfold antiderivative in H H0; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros H0 _; Split. +Intros; Elim (H ? H2); Elim (H0 ? H2); Intros. +Assert H5 : (derivable_pt [y:R]``l*(x y)+(x0 y)`` x1). +Reg. +Exists H5; Symmetry; Reg; Rewrite <- H3; Rewrite <- H4; Reflexivity. +Assumption. +Defined. + +(**********) +Lemma antiderivative_P1 : (f,g,F,G:R->R;l,a,b:R) (antiderivative f F a b) -> (antiderivative g G a b) -> (antiderivative [x:R]``l*(f x)+(g x)`` [x:R]``l*(F x)+(G x)`` a b). +Unfold antiderivative; Intros; Elim H; Elim H0; Clear H H0; Intros; Split. +Intros; Elim (H ? H3); Elim (H1 ? H3); Intros. +Assert H6 : (derivable_pt [x:R]``l*(F x)+(G x)`` x). +Reg. +Exists H6; Symmetry; Reg; Rewrite <- H4; Rewrite <- H5; Ring. +Assumption. +Qed. + +(* $\int_a^b \lambda f + g = \lambda \int_a^b f + \int_a^b f *) +Lemma NewtonInt_P6 : (f,g:R->R;l,a,b:R;pr1:(Newton_integrable f a b);pr2:(Newton_integrable g a b)) (NewtonInt [x:R]``l*(f x)+(g x)`` a b (NewtonInt_P5 f g l a b pr1 pr2))==``l*(NewtonInt f a b pr1)+(NewtonInt g a b pr2)``. +Intros f g l a b pr1 pr2; Unfold NewtonInt; Case (NewtonInt_P5 f g l a b pr1 pr2); Intros; Case pr1; Intros; Case pr2; Intros; Case (total_order_T a b); Intro. +Elim s; Intro. +Elim o; Intro. +Elim o0; Intro. +Elim o1; Intro. +Assert H2 := (antiderivative_P1 f g x0 x1 l a b H0 H1); Assert H3 := (antiderivative_Ucte ? ? ? ? ? H H2); Elim H3; Intros; Assert H5 : ``a<=a<=b``. +Split; [Right; Reflexivity | Left; Assumption]. +Assert H6 : ``a<=b<=b``. +Split; [Left; Assumption | Right; Reflexivity]. +Assert H7 := (H4 ? H5); Assert H8 := (H4 ? H6); Rewrite H7; Rewrite H8; Ring. +Unfold antiderivative in H1; Elim H1; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H3 a0)). +Unfold antiderivative in H0; Elim H0; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a0)). +Unfold antiderivative in H; Elim H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 a0)). +Rewrite b0; Ring. +Elim o; Intro. +Unfold antiderivative in H; Elim H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 r)). +Elim o0; Intro. +Unfold antiderivative in H0; Elim H0; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 r)). +Elim o1; Intro. +Unfold antiderivative in H1; Elim H1; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H3 r)). +Assert H2 := (antiderivative_P1 f g x0 x1 l b a H0 H1); Assert H3 := (antiderivative_Ucte ? ? ? ? ? H H2); Elim H3; Intros; Assert H5 : ``b<=a<=a``. +Split; [Left; Assumption | Right; Reflexivity]. +Assert H6 : ``b<=b<=a``. +Split; [Right; Reflexivity | Left; Assumption]. +Assert H7 := (H4 ? H5); Assert H8 := (H4 ? H6); Rewrite H7; Rewrite H8; Ring. +Qed. + +Lemma antiderivative_P2 : (f,F0,F1:R->R;a,b,c:R) (antiderivative f F0 a b) -> (antiderivative f F1 b c) -> (antiderivative f [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end) a c). +Unfold antiderivative; Intros; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros; Split. +2:Apply Rle_trans with b; Assumption. +Intros; Elim H3; Clear H3; Intros; Case (total_order_T x b); Intro. +Elim s; Intro. +Assert H5 : ``a<=x<=b``. +Split; [Assumption | Left; Assumption]. +Assert H6 := (H ? H5); Elim H6; Clear H6; Intros; Assert H7 : (derivable_pt_lim [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end) x (f x)). +Unfold derivable_pt_lim; Assert H7 : ``(derive_pt F0 x x0)==(f x)``. +Symmetry; Assumption. +Assert H8 := (derive_pt_eq_1 F0 x (f x) x0 H7); Unfold derivable_pt_lim in H8; Intros; Elim (H8 ? H9); Intros; Pose D := (Rmin x1 ``b-x``). +Assert H11 : ``0<D``. +Unfold D; Unfold Rmin; Case (total_order_Rle x1 ``b-x``); Intro. +Apply (cond_pos x1). +Apply Rlt_Rminus; Assumption. +Exists (mkposreal ? H11); Intros; Case (total_order_Rle x b); Intro. +Case (total_order_Rle ``x+h`` b); Intro. +Apply H10. +Assumption. +Apply Rlt_le_trans with D; [Assumption | Unfold D; Apply Rmin_l]. +Elim n; Left; Apply Rlt_le_trans with ``x+D``. +Apply Rlt_compatibility; Apply Rle_lt_trans with (Rabsolu h). +Apply Rle_Rabsolu. +Apply H13. +Apply Rle_anti_compatibility with ``-x``; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite Rplus_sym; Unfold D; Apply Rmin_r. +Elim n; Left; Assumption. +Assert H8 : (derivable_pt [x:R]Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end x). +Unfold derivable_pt; Apply Specif.existT with (f x); Apply H7. +Exists H8; Symmetry; Apply derive_pt_eq_0; Apply H7. +Assert H5 : ``a<=x<=b``. +Split; [Assumption | Right; Assumption]. +Assert H6 : ``b<=x<=c``. +Split; [Right; Symmetry; Assumption | Assumption]. +Elim (H ? H5); Elim (H0 ? H6); Intros; Assert H9 : (derive_pt F0 x x1)==(f x). +Symmetry; Assumption. +Assert H10 : (derive_pt F1 x x0)==(f x). +Symmetry; Assumption. +Assert H11 := (derive_pt_eq_1 F0 x (f x) x1 H9); Assert H12 := (derive_pt_eq_1 F1 x (f x) x0 H10); Assert H13 : (derivable_pt_lim [x:R]Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end x (f x)). +Unfold derivable_pt_lim; Unfold derivable_pt_lim in H11 H12; Intros; Elim (H11 ? H13); Elim (H12 ? H13); Intros; Pose D := (Rmin x2 x3); Assert H16 : ``0<D``. +Unfold D; Unfold Rmin; Case (total_order_Rle x2 x3); Intro. +Apply (cond_pos x2). +Apply (cond_pos x3). +Exists (mkposreal ? H16); Intros; Case (total_order_Rle x b); Intro. +Case (total_order_Rle ``x+h`` b); Intro. +Apply H15. +Assumption. +Apply Rlt_le_trans with D; [Assumption | Unfold D; Apply Rmin_r]. +Replace ``(F1 (x+h))+((F0 b)-(F1 b))-(F0 x)`` with ``(F1 (x+h))-(F1 x)``. +Apply H14. +Assumption. +Apply Rlt_le_trans with D; [Assumption | Unfold D; Apply Rmin_l]. +Rewrite b0; Ring. +Elim n; Right; Assumption. +Assert H14 : (derivable_pt [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end) x). +Unfold derivable_pt; Apply Specif.existT with (f x); Apply H13. +Exists H14; Symmetry; Apply derive_pt_eq_0; Apply H13. +Assert H5 : ``b<=x<=c``. +Split; [Left; Assumption | Assumption]. +Assert H6 := (H0 ? H5); Elim H6; Clear H6; Intros; Assert H7 : (derivable_pt_lim [x:R]Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end x (f x)). +Unfold derivable_pt_lim; Assert H7 : ``(derive_pt F1 x x0)==(f x)``. +Symmetry; Assumption. +Assert H8 := (derive_pt_eq_1 F1 x (f x) x0 H7); Unfold derivable_pt_lim in H8; Intros; Elim (H8 ? H9); Intros; Pose D := (Rmin x1 ``x-b``); Assert H11 : ``0<D``. +Unfold D; Unfold Rmin; Case (total_order_Rle x1 ``x-b``); Intro. +Apply (cond_pos x1). +Apply Rlt_Rminus; Assumption. +Exists (mkposreal ? H11); Intros; Case (total_order_Rle x b); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 r)). +Case (total_order_Rle ``x+h`` b); Intro. +Cut ``b<x+h``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 H14)). +Apply Rlt_anti_compatibility with ``-h-b``; Replace ``-h-b+b`` with ``-h``; [Idtac | Ring]; Replace ``-h-b+(x+h)`` with ``x-b``; [Idtac | Ring]; Apply Rle_lt_trans with (Rabsolu h). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Apply Rlt_le_trans with D. +Apply H13. +Unfold D; Apply Rmin_r. +Replace ``((F1 (x+h))+((F0 b)-(F1 b)))-((F1 x)+((F0 b)-(F1 b)))`` with ``(F1 (x+h))-(F1 x)``; [Idtac | Ring]; Apply H10. +Assumption. +Apply Rlt_le_trans with D. +Assumption. +Unfold D; Apply Rmin_l. +Assert H8 : (derivable_pt [x:R]Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end x). +Unfold derivable_pt; Apply Specif.existT with (f x); Apply H7. +Exists H8; Symmetry; Apply derive_pt_eq_0; Apply H7. +Qed. + +Lemma antiderivative_P3 : (f,F0,F1:R->R;a,b,c:R) (antiderivative f F0 a b) -> (antiderivative f F1 c b) -> (antiderivative f F1 c a)\/(antiderivative f F0 a c). +Intros; Unfold antiderivative in H H0; Elim H; Clear H; Elim H0; Clear H0; Intros; Case (total_order_T a c); Intro. +Elim s; Intro. +Right; Unfold antiderivative; Split. +Intros; Apply H1; Elim H3; Intros; Split; [Assumption | Apply Rle_trans with c; Assumption]. +Left; Assumption. +Right; Unfold antiderivative; Split. +Intros; Apply H1; Elim H3; Intros; Split; [Assumption | Apply Rle_trans with c; Assumption]. +Right; Assumption. +Left; Unfold antiderivative; Split. +Intros; Apply H; Elim H3; Intros; Split; [Assumption | Apply Rle_trans with a; Assumption]. +Left; Assumption. +Qed. + +Lemma antiderivative_P4 : (f,F0,F1:R->R;a,b,c:R) (antiderivative f F0 a b) -> (antiderivative f F1 a c) -> (antiderivative f F1 b c)\/(antiderivative f F0 c b). +Intros; Unfold antiderivative in H H0; Elim H; Clear H; Elim H0; Clear H0; Intros; Case (total_order_T c b); Intro. +Elim s; Intro. +Right; Unfold antiderivative; Split. +Intros; Apply H1; Elim H3; Intros; Split; [Apply Rle_trans with c; Assumption | Assumption]. +Left; Assumption. +Right; Unfold antiderivative; Split. +Intros; Apply H1; Elim H3; Intros; Split; [Apply Rle_trans with c; Assumption | Assumption]. +Right; Assumption. +Left; Unfold antiderivative; Split. +Intros; Apply H; Elim H3; Intros; Split; [Apply Rle_trans with b; Assumption | Assumption]. +Left; Assumption. +Qed. + +Lemma NewtonInt_P7 : (f:R->R;a,b,c:R) ``a<b`` -> ``b<c`` -> (Newton_integrable f a b) -> (Newton_integrable f b c) -> (Newton_integrable f a c). +Unfold Newton_integrable; Intros f a b c Hab Hbc X X0; Elim X; Clear X; Intros F0 H0; Elim X0; Clear X0; Intros F1 H1; Pose g := [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end); Apply existTT with g; Left; Unfold g; Apply antiderivative_P2. +Elim H0; Intro. +Assumption. +Unfold antiderivative in H; Elim H; Clear H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 Hab)). +Elim H1; Intro. +Assumption. +Unfold antiderivative in H; Elim H; Clear H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 Hbc)). +Qed. + +Lemma NewtonInt_P8 : (f:(R->R); a,b,c:R) (Newton_integrable f a b) -> (Newton_integrable f b c) -> (Newton_integrable f a c). +Intros. +Elim X; Intros F0 H0. +Elim X0; Intros F1 H1. +Case (total_order_T a b); Intro. +Elim s; Intro. +Case (total_order_T b c); Intro. +Elim s0; Intro. +(* a<b & b<c *) +Unfold Newton_integrable; Apply existTT with [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end). +Elim H0; Intro. +Elim H1; Intro. +Left; Apply antiderivative_P2; Assumption. +Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a1)). +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H a0)). +(* a<b & b=c *) +Rewrite b0 in X; Apply X. +(* a<b & b>c *) +Case (total_order_T a c); Intro. +Elim s0; Intro. +Unfold Newton_integrable; Apply existTT with F0. +Left. +Elim H1; Intro. +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Elim H0; Intro. +Assert H3 := (antiderivative_P3 f F0 F1 a b c H2 H). +Elim H3; Intro. +Unfold antiderivative in H4; Elim H4; Clear H4; Intros _ H4. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 a1)). +Assumption. +Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a0)). +Rewrite b0; Apply NewtonInt_P1. +Unfold Newton_integrable; Apply existTT with F1. +Right. +Elim H1; Intro. +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Elim H0; Intro. +Assert H3 := (antiderivative_P3 f F0 F1 a b c H2 H). +Elim H3; Intro. +Assumption. +Unfold antiderivative in H4; Elim H4; Clear H4; Intros _ H4. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 r0)). +Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a0)). +(* a=b *) +Rewrite b0; Apply X0. +Case (total_order_T b c); Intro. +Elim s; Intro. +(* a>b & b<c *) +Case (total_order_T a c); Intro. +Elim s0; Intro. +Unfold Newton_integrable; Apply existTT with F1. +Left. +Elim H1; Intro. +(*****************) +Elim H0; Intro. +Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 r)). +Assert H3 := (antiderivative_P4 f F0 F1 b a c H2 H). +Elim H3; Intro. +Assumption. +Unfold antiderivative in H4; Elim H4; Clear H4; Intros _ H4. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 a1)). +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H a0)). +Rewrite b0; Apply NewtonInt_P1. +Unfold Newton_integrable; Apply existTT with F0. +Right. +Elim H0; Intro. +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Elim H1; Intro. +Assert H3 := (antiderivative_P4 f F0 F1 b a c H H2). +Elim H3; Intro. +Unfold antiderivative in H4; Elim H4; Clear H4; Intros _ H4. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 r0)). +Assumption. +Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a0)). +(* a>b & b=c *) +Rewrite b0 in X; Apply X. +(* a>b & b>c *) +Assert X1 := (NewtonInt_P3 f a b X). +Assert X2 := (NewtonInt_P3 f b c X0). +Apply NewtonInt_P3. +Apply NewtonInt_P7 with b; Assumption. +Defined. + +(* Chasles' relation *) +Lemma NewtonInt_P9 : (f:R->R;a,b,c:R;pr1:(Newton_integrable f a b);pr2:(Newton_integrable f b c)) ``(NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2))==(NewtonInt f a b pr1)+(NewtonInt f b c pr2)``. +Intros; Unfold NewtonInt. +Case (NewtonInt_P8 f a b c pr1 pr2); Intros. +Case pr1; Intros. +Case pr2; Intros. +Case (total_order_T a b); Intro. +Elim s; Intro. +Case (total_order_T b c); Intro. +Elim s0; Intro. +(* a<b & b<c *) +Elim o0; Intro. +Elim o1; Intro. +Elim o; Intro. +Assert H2 := (antiderivative_P2 f x0 x1 a b c H H0). +Assert H3 := (antiderivative_Ucte f x [x:R] + Cases (total_order_Rle x b) of + (leftT _) => (x0 x) + | (rightT _) => ``(x1 x)+((x0 b)-(x1 b))`` + end a c H1 H2). +Elim H3; Intros. +Assert H5 : ``a<=a<=c``. +Split; [Right; Reflexivity | Left; Apply Rlt_trans with b; Assumption]. +Assert H6 : ``a<=c<=c``. +Split; [Left; Apply Rlt_trans with b; Assumption | Right; Reflexivity]. +Rewrite (H4 ? H5); Rewrite (H4 ? H6). +Case (total_order_Rle a b); Intro. +Case (total_order_Rle c b); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 a1)). +Ring. +Elim n; Left; Assumption. +Unfold antiderivative in H1; Elim H1; Clear H1; Intros _ H1. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 (Rlt_trans ? ? ? a0 a1))). +Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 a1)). +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H a0)). +(* a<b & b=c *) +Rewrite <- b0. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or. +Rewrite <- b0 in o. +Elim o0; Intro. +Elim o; Intro. +Assert H1 := (antiderivative_Ucte f x x0 a b H0 H). +Elim H1; Intros. +Rewrite (H2 b). +Rewrite (H2 a). +Ring. +Split; [Right; Reflexivity | Left; Assumption]. +Split; [Left; Assumption | Right; Reflexivity]. +Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 a0)). +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H a0)). +(* a<b & b>c *) +Elim o1; Intro. +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Elim o0; Intro. +Elim o; Intro. +Assert H2 := (antiderivative_P2 f x x1 a c b H1 H). +Assert H3 := (antiderivative_Ucte ? ? ? a b H0 H2). +Elim H3; Intros. +Rewrite (H4 a). +Rewrite (H4 b). +Case (total_order_Rle b c); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 r)). +Case (total_order_Rle a c); Intro. +Ring. +Elim n0; Unfold antiderivative in H1; Elim H1; Intros; Assumption. +Split; [Left; Assumption | Right; Reflexivity]. +Split; [Right; Reflexivity | Left; Assumption]. +Assert H2 := (antiderivative_P2 ? ? ? ? ? ? H1 H0). +Assert H3 := (antiderivative_Ucte ? ? ? c b H H2). +Elim H3; Intros. +Rewrite (H4 c). +Rewrite (H4 b). +Case (total_order_Rle b a); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 a0)). +Case (total_order_Rle c a); Intro. +Ring. +Elim n0; Unfold antiderivative in H1; Elim H1; Intros; Assumption. +Split; [Left; Assumption | Right; Reflexivity]. +Split; [Right; Reflexivity | Left; Assumption]. +Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 a0)). +(* a=b *) +Rewrite b0 in o; Rewrite b0. +Elim o; Intro. +Elim o1; Intro. +Assert H1 := (antiderivative_Ucte ? ? ? b c H H0). +Elim H1; Intros. +Assert H3 : ``b<=c``. +Unfold antiderivative in H; Elim H; Intros; Assumption. +Rewrite (H2 b). +Rewrite (H2 c). +Ring. +Split; [Assumption | Right; Reflexivity]. +Split; [Right; Reflexivity | Assumption]. +Assert H1 : ``b==c``. +Unfold antiderivative in H H0; Elim H; Elim H0; Intros; Apply Rle_antisym; Assumption. +Rewrite H1; Ring. +Elim o1; Intro. +Assert H1 : ``b==c``. +Unfold antiderivative in H H0; Elim H; Elim H0; Intros; Apply Rle_antisym; Assumption. +Rewrite H1; Ring. +Assert H1 := (antiderivative_Ucte ? ? ? c b H H0). +Elim H1; Intros. +Assert H3 : ``c<=b``. +Unfold antiderivative in H; Elim H; Intros; Assumption. +Rewrite (H2 c). +Rewrite (H2 b). +Ring. +Split; [Assumption | Right; Reflexivity]. +Split; [Right; Reflexivity | Assumption]. +(* a>b & b<c *) +Case (total_order_T b c); Intro. +Elim s; Intro. +Elim o0; Intro. +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Elim o1; Intro. +Elim o; Intro. +Assert H2 := (antiderivative_P2 ? ? ? ? ? ? H H1). +Assert H3 := (antiderivative_Ucte ? ? ? b c H0 H2). +Elim H3; Intros. +Rewrite (H4 b). +Rewrite (H4 c). +Case (total_order_Rle b a); Intro. +Case (total_order_Rle c a); Intro. +Assert H5 : ``a==c``. +Unfold antiderivative in H1; Elim H1; Intros; Apply Rle_antisym; Assumption. +Rewrite H5; Ring. +Ring. +Elim n; Left; Assumption. +Split; [Left; Assumption | Right; Reflexivity]. +Split; [Right; Reflexivity | Left; Assumption]. +Assert H2 := (antiderivative_P2 ? ? ? ? ? ? H0 H1). +Assert H3 := (antiderivative_Ucte ? ? ? b a H H2). +Elim H3; Intros. +Rewrite (H4 a). +Rewrite (H4 b). +Case (total_order_Rle b c); Intro. +Case (total_order_Rle a c); Intro. +Assert H5 : ``a==c``. +Unfold antiderivative in H1; Elim H1; Intros; Apply Rle_antisym; Assumption. +Rewrite H5; Ring. +Ring. +Elim n; Left; Assumption. +Split; [Right; Reflexivity | Left; Assumption]. +Split; [Left; Assumption | Right; Reflexivity]. +Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 a0)). +(* a>b & b=c *) +Rewrite <- b0. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or. +Rewrite <- b0 in o. +Elim o0; Intro. +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Elim o; Intro. +Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 r)). +Assert H1 := (antiderivative_Ucte f x x0 b a H0 H). +Elim H1; Intros. +Rewrite (H2 b). +Rewrite (H2 a). +Ring. +Split; [Left; Assumption | Right; Reflexivity]. +Split; [Right; Reflexivity | Left; Assumption]. +(* a>b & b>c *) +Elim o0; Intro. +Unfold antiderivative in H; Elim H; Clear H; Intros _ H. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Elim o1; Intro. +Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 r0)). +Elim o; Intro. +Unfold antiderivative in H1; Elim H1; Clear H1; Intros _ H1. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 (Rlt_trans ? ? ? r0 r))). +Assert H2 := (antiderivative_P2 ? ? ? ? ? ? H0 H). +Assert H3 := (antiderivative_Ucte ? ? ? c a H1 H2). +Elim H3; Intros. +Assert H5 : ``c<=a``. +Unfold antiderivative in H1; Elim H1; Intros; Assumption. +Rewrite (H4 c). +Rewrite (H4 a). +Case (total_order_Rle a b); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r1 r)). +Case (total_order_Rle c b); Intro. +Ring. +Elim n0; Left; Assumption. +Split; [Assumption | Right; Reflexivity]. +Split; [Right; Reflexivity | Assumption]. +Qed. + diff --git a/theories7/Reals/PSeries_reg.v b/theories7/Reals/PSeries_reg.v new file mode 100644 index 00000000..68645379 --- /dev/null +++ b/theories7/Reals/PSeries_reg.v @@ -0,0 +1,194 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: PSeries_reg.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Ranalysis1. +Require Max. +Require Even. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Definition Boule [x:R;r:posreal] : R -> Prop := [y:R]``(Rabsolu (y-x))<r``. + +(* Uniform convergence *) +Definition CVU [fn:nat->R->R;f:R->R;x:R;r:posreal] : Prop := (eps:R)``0<eps``->(EX N:nat | (n:nat;y:R) (le N n)->(Boule x r y)->``(Rabsolu ((f y)-(fn n y)))<eps``). + +(* Normal convergence *) +Definition CVN_r [fn:nat->R->R;r:posreal] : Type := (SigT ? [An:nat->R](sigTT R [l:R]((Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu (An k)) n) l)/\((n:nat)(y:R)(Boule R0 r y)->(Rle (Rabsolu (fn n y)) (An n)))))). + +Definition CVN_R [fn:nat->R->R] : Type := (r:posreal) (CVN_r fn r). + +Definition SFL [fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l))] : R-> R := [y:R](Cases (cv y) of (existTT a b) => a end). + +(* In a complete space, normal convergence implies uniform convergence *) +Lemma CVN_CVU : (fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l));r:posreal) (CVN_r fn r) -> (CVU [n:nat](SP fn n) (SFL fn cv) ``0`` r). +Intros; Unfold CVU; Intros. +Unfold CVN_r in X. +Elim X; Intros An X0. +Elim X0; Intros s H0. +Elim H0; Intros. +Cut (Un_cv [n:nat](Rminus (sum_f_R0 [k:nat]``(Rabsolu (An k))`` n) s) R0). +Intro; Unfold Un_cv in H3. +Elim (H3 eps H); Intros N0 H4. +Exists N0; Intros. +Apply Rle_lt_trans with (Rabsolu (Rminus (sum_f_R0 [k:nat]``(Rabsolu (An k))`` n) s)). +Rewrite <- (Rabsolu_Ropp (Rminus (sum_f_R0 [k:nat]``(Rabsolu (An k))`` n) s)); Rewrite Ropp_distr3; Rewrite (Rabsolu_right (Rminus s (sum_f_R0 [k:nat]``(Rabsolu (An k))`` n))). +EApply sum_maj1. +Unfold SFL; Case (cv y); Intro. +Trivial. +Apply H1. +Intro; Elim H0; Intros. +Rewrite (Rabsolu_right (An n0)). +Apply H8; Apply H6. +Apply Rle_sym1; Apply Rle_trans with (Rabsolu (fn n0 y)). +Apply Rabsolu_pos. +Apply H8; Apply H6. +Apply Rle_sym1; Apply Rle_anti_compatibility with (sum_f_R0 [k:nat](Rabsolu (An k)) n). +Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym s); Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Apply sum_incr. +Apply H1. +Intro; Apply Rabsolu_pos. +Unfold R_dist in H4; Unfold Rminus in H4; Rewrite Ropp_O in H4. +Assert H7 := (H4 n H5). +Rewrite Rplus_Or in H7; Apply H7. +Unfold Un_cv in H1; Unfold Un_cv; Intros. +Elim (H1? H3); Intros. +Exists x; Intros. +Unfold R_dist; Unfold R_dist in H4. +Rewrite minus_R0; Apply H4; Assumption. +Qed. + +(* Each limit of a sequence of functions which converges uniformly is continue *) +Lemma CVU_continuity : (fn:nat->R->R;f:R->R;x:R;r:posreal) (CVU fn f x r) -> ((n:nat)(y:R) (Boule x r y)->(continuity_pt (fn n) y)) -> ((y:R) (Boule x r y) -> (continuity_pt f y)). +Intros; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Unfold CVU in H. +Cut ``0<eps/3``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. +Elim (H ? H3); Intros N0 H4. +Assert H5 := (H0 N0 y H1). +Cut (EXT del : posreal | (h:R) ``(Rabsolu h)<del`` -> (Boule x r ``y+h``) ). +Intro. +Elim H6; Intros del1 H7. +Unfold continuity_pt in H5; Unfold continue_in in H5; Unfold limit1_in in H5; Unfold limit_in in H5; Simpl in H5; Unfold R_dist in H5. +Elim (H5 ? H3); Intros del2 H8. +Pose del := (Rmin del1 del2). +Exists del; Intros. +Split. +Unfold del; Unfold Rmin; Case (total_order_Rle del1 del2); Intro. +Apply (cond_pos del1). +Elim H8; Intros; Assumption. +Intros; Apply Rle_lt_trans with ``(Rabsolu ((f x0)-(fn N0 x0)))+(Rabsolu ((fn N0 x0)-(f y)))``. +Replace ``(f x0)-(f y)`` with ``((f x0)-(fn N0 x0))+((fn N0 x0)-(f y))``; [Apply Rabsolu_triang | Ring]. +Apply Rle_lt_trans with ``(Rabsolu ((f x0)-(fn N0 x0)))+(Rabsolu ((fn N0 x0)-(fn N0 y)))+(Rabsolu ((fn N0 y)-(f y)))``. +Rewrite Rplus_assoc; Apply Rle_compatibility. +Replace ``(fn N0 x0)-(f y)`` with ``((fn N0 x0)-(fn N0 y))+((fn N0 y)-(f y))``; [Apply Rabsolu_triang | Ring]. +Replace ``eps`` with ``eps/3+eps/3+eps/3``. +Repeat Apply Rplus_lt. +Apply H4. +Apply le_n. +Replace x0 with ``y+(x0-y)``; [Idtac | Ring]; Apply H7. +Elim H9; Intros. +Apply Rlt_le_trans with del. +Assumption. +Unfold del; Apply Rmin_l. +Elim H8; Intros. +Apply H11. +Split. +Elim H9; Intros; Assumption. +Elim H9; Intros; Apply Rlt_le_trans with del. +Assumption. +Unfold del; Apply Rmin_r. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr3; Apply H4. +Apply le_n. +Assumption. +Apply r_Rmult_mult with ``3``. +Do 2 Rewrite Rmult_Rplus_distr; Unfold Rdiv; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. +Ring. +DiscrR. +DiscrR. +Cut ``0<r-(Rabsolu (x-y))``. +Intro; Exists (mkposreal ? H6). +Simpl; Intros. +Unfold Boule; Replace ``y+h-x`` with ``h+(y-x)``; [Idtac | Ring]; Apply Rle_lt_trans with ``(Rabsolu h)+(Rabsolu (y-x))``. +Apply Rabsolu_triang. +Apply Rlt_anti_compatibility with ``-(Rabsolu (x-y))``. +Rewrite <- (Rabsolu_Ropp ``y-x``); Rewrite Ropp_distr3. +Replace ``-(Rabsolu (x-y))+r`` with ``r-(Rabsolu (x-y))``. +Replace ``-(Rabsolu (x-y))+((Rabsolu h)+(Rabsolu (x-y)))`` with (Rabsolu h). +Apply H7. +Ring. +Ring. +Unfold Boule in H1; Rewrite <- (Rabsolu_Ropp ``x-y``); Rewrite Ropp_distr3; Apply Rlt_anti_compatibility with ``(Rabsolu (y-x))``. +Rewrite Rplus_Or; Replace ``(Rabsolu (y-x))+(r-(Rabsolu (y-x)))`` with ``(pos r)``; [Apply H1 | Ring]. +Qed. + +(**********) +Lemma continuity_pt_finite_SF : (fn:nat->R->R;N:nat;x:R) ((n:nat)(le n N)->(continuity_pt (fn n) x)) -> (continuity_pt [y:R](sum_f_R0 [k:nat]``(fn k y)`` N) x). +Intros; Induction N. +Simpl; Apply (H O); Apply le_n. +Simpl; Replace [y:R](Rplus (sum_f_R0 [k:nat](fn k y) N) (fn (S N) y)) with (plus_fct [y:R](sum_f_R0 [k:nat](fn k y) N) [y:R](fn (S N) y)); [Idtac | Reflexivity]. +Apply continuity_pt_plus. +Apply HrecN. +Intros; Apply H. +Apply le_trans with N; [Assumption | Apply le_n_Sn]. +Apply (H (S N)); Apply le_n. +Qed. + +(* Continuity and normal convergence *) +Lemma SFL_continuity_pt : (fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l));r:posreal) (CVN_r fn r) -> ((n:nat)(y:R) (Boule ``0`` r y) -> (continuity_pt (fn n) y)) -> ((y:R) (Boule ``0`` r y) -> (continuity_pt (SFL fn cv) y)). +Intros; EApply CVU_continuity. +Apply CVN_CVU. +Apply X. +Intros; Unfold SP; Apply continuity_pt_finite_SF. +Intros; Apply H. +Apply H1. +Apply H0. +Qed. + +Lemma SFL_continuity : (fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l))) (CVN_R fn) -> ((n:nat)(continuity (fn n))) -> (continuity (SFL fn cv)). +Intros; Unfold continuity; Intro. +Cut ``0<(Rabsolu x)+1``; [Intro | Apply ge0_plus_gt0_is_gt0; [Apply Rabsolu_pos | Apply Rlt_R0_R1]]. +Cut (Boule ``0`` (mkposreal ? H0) x). +Intro; EApply SFL_continuity_pt with (mkposreal ? H0). +Apply X. +Intros; Apply (H n y). +Apply H1. +Unfold Boule; Simpl; Rewrite minus_R0; Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. +Qed. + +(* As R is complete, normal convergence implies that (fn) is simply-uniformly convergent *) +Lemma CVN_R_CVS : (fn:nat->R->R) (CVN_R fn) -> ((x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l))). +Intros; Apply R_complete. +Unfold SP; Pose An := [N:nat](fn N x). +Change (Cauchy_crit_series An). +Apply cauchy_abs. +Unfold Cauchy_crit_series; Apply CV_Cauchy. +Unfold CVN_R in X; Cut ``0<(Rabsolu x)+1``. +Intro; Assert H0 := (X (mkposreal ? H)). +Unfold CVN_r in H0; Elim H0; Intros Bn H1. +Elim H1; Intros l H2. +Elim H2; Intros. +Apply Rseries_CV_comp with Bn. +Intro; Split. +Apply Rabsolu_pos. +Unfold An; Apply H4; Unfold Boule; Simpl; Rewrite minus_R0. +Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. +Apply existTT with l. +Cut (n:nat)``0<=(Bn n)``. +Intro; Unfold Un_cv in H3; Unfold Un_cv; Intros. +Elim (H3 ? H6); Intros. +Exists x0; Intros. +Replace (sum_f_R0 Bn n) with (sum_f_R0 [k:nat](Rabsolu (Bn k)) n). +Apply H7; Assumption. +Apply sum_eq; Intros; Apply Rabsolu_right; Apply Rle_sym1; Apply H5. +Intro; Apply Rle_trans with (Rabsolu (An n)). +Apply Rabsolu_pos. +Unfold An; Apply H4; Unfold Boule; Simpl; Rewrite minus_R0; Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. +Apply ge0_plus_gt0_is_gt0; [Apply Rabsolu_pos | Apply Rlt_R0_R1]. +Qed. diff --git a/theories7/Reals/PartSum.v b/theories7/Reals/PartSum.v new file mode 100644 index 00000000..ee5fa498 --- /dev/null +++ b/theories7/Reals/PartSum.v @@ -0,0 +1,476 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: PartSum.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require Rcomplete. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Lemma tech1 : (An:nat->R;N:nat) ((n:nat)``(le n N)``->``0<(An n)``) -> ``0 < (sum_f_R0 An N)``. +Intros; Induction N. +Simpl; Apply H; Apply le_n. +Simpl; Apply gt0_plus_gt0_is_gt0. +Apply HrecN; Intros; Apply H; Apply le_S; Assumption. +Apply H; Apply le_n. +Qed. + +(* Chasles' relation *) +Lemma tech2 : (An:nat->R;m,n:nat) (lt m n) -> (sum_f_R0 An n) == (Rplus (sum_f_R0 An m) (sum_f_R0 [i:nat]``(An (plus (S m) i))`` (minus n (S m)))). +Intros; Induction n. +Elim (lt_n_O ? H). +Cut (lt m n)\/m=n. +Intro; Elim H0; Intro. +Replace (sum_f_R0 An (S n)) with ``(sum_f_R0 An n)+(An (S n))``; [Idtac | Reflexivity]. +Replace (minus (S n) (S m)) with (S (minus n (S m))). +Replace (sum_f_R0 [i:nat](An (plus (S m) i)) (S (minus n (S m)))) with (Rplus (sum_f_R0 [i:nat](An (plus (S m) i)) (minus n (S m))) (An (plus (S m) (S (minus n (S m)))))); [Idtac | Reflexivity]. +Replace (plus (S m) (S (minus n (S m)))) with (S n). +Rewrite (Hrecn H1). +Ring. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite minus_INR. +Rewrite S_INR; Ring. +Apply lt_le_S; Assumption. +Apply INR_eq; Rewrite S_INR; Repeat Rewrite minus_INR. +Repeat Rewrite S_INR; Ring. +Apply le_n_S; Apply lt_le_weak; Assumption. +Apply lt_le_S; Assumption. +Rewrite H1; Rewrite <- minus_n_n; Simpl. +Replace (plus n O) with n; [Reflexivity | Ring]. +Inversion H. +Right; Reflexivity. +Left; Apply lt_le_trans with (S m); [Apply lt_n_Sn | Assumption]. +Qed. + +(* Sum of geometric sequences *) +Lemma tech3 : (k:R;N:nat) ``k<>1`` -> (sum_f_R0 [i:nat](pow k i) N)==``(1-(pow k (S N)))/(1-k)``. +Intros; Cut ``1-k<>0``. +Intro; Induction N. +Simpl; Rewrite Rmult_1r; Unfold Rdiv; Rewrite <- Rinv_r_sym. +Reflexivity. +Apply H0. +Replace (sum_f_R0 ([i:nat](pow k i)) (S N)) with (Rplus (sum_f_R0 [i:nat](pow k i) N) (pow k (S N))); [Idtac | Reflexivity]; Rewrite HrecN; Replace ``(1-(pow k (S N)))/(1-k)+(pow k (S N))`` with ``((1-(pow k (S N)))+(1-k)*(pow k (S N)))/(1-k)``. +Apply r_Rmult_mult with ``1-k``. +Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(1-k)``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [ Do 2 Rewrite Rmult_1l; Simpl; Ring | Apply H0]. +Apply H0. +Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Rewrite (Rmult_sym ``1-k``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Reflexivity. +Apply H0. +Apply Rminus_eq_contra; Red; Intro; Elim H; Symmetry; Assumption. +Qed. + +Lemma tech4 : (An:nat->R;k:R;N:nat) ``0<=k`` -> ((i:nat)``(An (S i))<k*(An i)``) -> ``(An N)<=(An O)*(pow k N)``. +Intros; Induction N. +Simpl; Right; Ring. +Apply Rle_trans with ``k*(An N)``. +Left; Apply (H0 N). +Replace (S N) with (plus N (1)); [Idtac | Ring]. +Rewrite pow_add; Simpl; Rewrite Rmult_1r; Replace ``(An O)*((pow k N)*k)`` with ``k*((An O)*(pow k N))``; [Idtac | Ring]; Apply Rle_monotony. +Assumption. +Apply HrecN. +Qed. + +Lemma tech5 : (An:nat->R;N:nat) (sum_f_R0 An (S N))==``(sum_f_R0 An N)+(An (S N))``. +Intros; Reflexivity. +Qed. + +Lemma tech6 : (An:nat->R;k:R;N:nat) ``0<=k`` -> ((i:nat)``(An (S i))<k*(An i)``) -> (Rle (sum_f_R0 An N) (Rmult (An O) (sum_f_R0 [i:nat](pow k i) N))). +Intros; Induction N. +Simpl; Right; Ring. +Apply Rle_trans with (Rplus (Rmult (An O) (sum_f_R0 [i:nat](pow k i) N)) (An (S N))). +Rewrite tech5; Do 2 Rewrite <- (Rplus_sym (An (S N))); Apply Rle_compatibility. +Apply HrecN. +Rewrite tech5 ; Rewrite Rmult_Rplus_distr; Apply Rle_compatibility. +Apply tech4; Assumption. +Qed. + +Lemma tech7 : (r1,r2:R) ``r1<>0`` -> ``r2<>0`` -> ``r1<>r2`` -> ``/r1<>/r2``. +Intros; Red; Intro. +Assert H3 := (Rmult_mult_r r1 ? ? H2). +Rewrite <- Rinv_r_sym in H3; [Idtac | Assumption]. +Assert H4 := (Rmult_mult_r r2 ? ? H3). +Rewrite Rmult_1r in H4; Rewrite <- Rmult_assoc in H4. +Rewrite Rinv_r_simpl_m in H4; [Idtac | Assumption]. +Elim H1; Symmetry; Assumption. +Qed. + +Lemma tech11 : (An,Bn,Cn:nat->R;N:nat) ((i:nat) (An i)==``(Bn i)-(Cn i)``) -> (sum_f_R0 An N)==``(sum_f_R0 Bn N)-(sum_f_R0 Cn N)``. +Intros; Induction N. +Simpl; Apply H. +Do 3 Rewrite tech5; Rewrite HrecN; Rewrite (H (S N)); Ring. +Qed. + +Lemma tech12 : (An:nat->R;x:R;l:R) (Un_cv [N:nat](sum_f_R0 [i:nat]``(An i)*(pow x i)`` N) l) -> (Pser An x l). +Intros; Unfold Pser; Unfold infinit_sum; Unfold Un_cv in H; Assumption. +Qed. + +Lemma scal_sum : (An:nat->R;N:nat;x:R) (Rmult x (sum_f_R0 An N))==(sum_f_R0 [i:nat]``(An i)*x`` N). +Intros; Induction N. +Simpl; Ring. +Do 2 Rewrite tech5. +Rewrite Rmult_Rplus_distr; Rewrite <- HrecN; Ring. +Qed. + +Lemma decomp_sum : (An:nat->R;N:nat) (lt O N) -> (sum_f_R0 An N)==(Rplus (An O) (sum_f_R0 [i:nat](An (S i)) (pred N))). +Intros; Induction N. +Elim (lt_n_n ? H). +Cut (lt O N)\/N=O. +Intro; Elim H0; Intro. +Cut (S (pred N))=(pred (S N)). +Intro; Rewrite <- H2. +Do 2 Rewrite tech5. +Replace (S (S (pred N))) with (S N). +Rewrite (HrecN H1); Ring. +Rewrite H2; Simpl; Reflexivity. +Assert H2 := (O_or_S N). +Elim H2; Intros. +Elim a; Intros. +Rewrite <- p. +Simpl; Reflexivity. +Rewrite <- b in H1; Elim (lt_n_n ? H1). +Rewrite H1; Simpl; Reflexivity. +Inversion H. +Right; Reflexivity. +Left; Apply lt_le_trans with (1); [Apply lt_O_Sn | Assumption]. +Qed. + +Lemma plus_sum : (An,Bn:nat->R;N:nat) (sum_f_R0 [i:nat]``(An i)+(Bn i)`` N)==``(sum_f_R0 An N)+(sum_f_R0 Bn N)``. +Intros; Induction N. +Simpl; Ring. +Do 3 Rewrite tech5; Rewrite HrecN; Ring. +Qed. + +Lemma sum_eq : (An,Bn:nat->R;N:nat) ((i:nat)(le i N)->(An i)==(Bn i)) -> (sum_f_R0 An N)==(sum_f_R0 Bn N). +Intros; Induction N. +Simpl; Apply H; Apply le_n. +Do 2 Rewrite tech5; Rewrite HrecN. +Rewrite (H (S N)); [Reflexivity | Apply le_n]. +Intros; Apply H; Apply le_trans with N; [Assumption | Apply le_n_Sn]. +Qed. + +(* Unicity of the limit defined by convergent series *) +Lemma unicity_sum : (An:nat->R;l1,l2:R) (infinit_sum An l1) -> (infinit_sum An l2) -> l1 == l2. +Unfold infinit_sum; Intros. +Case (Req_EM l1 l2); Intro. +Assumption. +Cut ``0<(Rabsolu ((l1-l2)/2))``; [Intro | Apply Rabsolu_pos_lt]. +Elim (H ``(Rabsolu ((l1-l2)/2))`` H2); Intros. +Elim (H0 ``(Rabsolu ((l1-l2)/2))`` H2); Intros. +Pose N := (max x0 x); Cut (ge N x0). +Cut (ge N x). +Intros; Assert H7 := (H3 N H5); Assert H8 := (H4 N H6). +Cut ``(Rabsolu (l1-l2)) <= (R_dist (sum_f_R0 An N) l1) + (R_dist (sum_f_R0 An N) l2)``. +Intro; Assert H10 := (Rplus_lt ? ? ? ? H7 H8); Assert H11 := (Rle_lt_trans ? ? ? H9 H10); Unfold Rdiv in H11; Rewrite Rabsolu_mult in H11. +Cut ``(Rabsolu (/2))==/2``. +Intro; Rewrite H12 in H11; Assert H13 := double_var; Unfold Rdiv in H13; Rewrite <- H13 in H11. +Elim (Rlt_antirefl ? H11). +Apply Rabsolu_right; Left; Change ``0</2``; Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H20; Generalize (lt_INR_0 (2) (neq_O_lt (2) H20)); Unfold INR; Intro; Assumption | Discriminate]. +Unfold R_dist; Rewrite <- (Rabsolu_Ropp ``(sum_f_R0 An N)-l1``); Rewrite Ropp_distr3. +Replace ``l1-l2`` with ``((l1-(sum_f_R0 An N)))+((sum_f_R0 An N)-l2)``; [Idtac | Ring]. +Apply Rabsolu_triang. +Unfold ge; Unfold N; Apply le_max_r. +Unfold ge; Unfold N; Apply le_max_l. +Unfold Rdiv; Apply prod_neq_R0. +Apply Rminus_eq_contra; Assumption. +Apply Rinv_neq_R0; DiscrR. +Qed. + +Lemma minus_sum : (An,Bn:nat->R;N:nat) (sum_f_R0 [i:nat]``(An i)-(Bn i)`` N)==``(sum_f_R0 An N)-(sum_f_R0 Bn N)``. +Intros; Induction N. +Simpl; Ring. +Do 3 Rewrite tech5; Rewrite HrecN; Ring. +Qed. + +Lemma sum_decomposition : (An:nat->R;N:nat) (Rplus (sum_f_R0 [l:nat](An (mult (2) l)) (S N)) (sum_f_R0 [l:nat](An (S (mult (2) l))) N))==(sum_f_R0 An (mult (2) (S N))). +Intros. +Induction N. +Simpl; Ring. +Rewrite tech5. +Rewrite (tech5 [l:nat](An (S (mult (2) l))) N). +Replace (mult (2) (S (S N))) with (S (S (mult (2) (S N)))). +Rewrite (tech5 An (S (mult (2) (S N)))). +Rewrite (tech5 An (mult (2) (S N))). +Rewrite <- HrecN. +Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR;Repeat Rewrite S_INR. +Ring. +Qed. + +Lemma sum_Rle : (An,Bn:nat->R;N:nat) ((n:nat)(le n N)->``(An n)<=(Bn n)``) -> ``(sum_f_R0 An N)<=(sum_f_R0 Bn N)``. +Intros. +Induction N. +Simpl; Apply H. +Apply le_n. +Do 2 Rewrite tech5. +Apply Rle_trans with ``(sum_f_R0 An N)+(Bn (S N))``. +Apply Rle_compatibility. +Apply H. +Apply le_n. +Do 2 Rewrite <- (Rplus_sym ``(Bn (S N))``). +Apply Rle_compatibility. +Apply HrecN. +Intros; Apply H. +Apply le_trans with N; [Assumption | Apply le_n_Sn]. +Qed. + +Lemma sum_Rabsolu : (An:nat->R;N:nat) (Rle (Rabsolu (sum_f_R0 An N)) (sum_f_R0 [l:nat](Rabsolu (An l)) N)). +Intros. +Induction N. +Simpl. +Right; Reflexivity. +Do 2 Rewrite tech5. +Apply Rle_trans with ``(Rabsolu (sum_f_R0 An N))+(Rabsolu (An (S N)))``. +Apply Rabsolu_triang. +Do 2 Rewrite <- (Rplus_sym (Rabsolu (An (S N)))). +Apply Rle_compatibility. +Apply HrecN. +Qed. + +Lemma sum_cte : (x:R;N:nat) (sum_f_R0 [_:nat]x N) == ``x*(INR (S N))``. +Intros. +Induction N. +Simpl; Ring. +Rewrite tech5. +Rewrite HrecN; Repeat Rewrite S_INR; Ring. +Qed. + +(**********) +Lemma sum_growing : (An,Bn:nat->R;N:nat) ((n:nat)``(An n)<=(Bn n)``)->``(sum_f_R0 An N)<=(sum_f_R0 Bn N)``. +Intros. +Induction N. +Simpl; Apply H. +Do 2 Rewrite tech5. +Apply Rle_trans with ``(sum_f_R0 An N)+(Bn (S N))``. +Apply Rle_compatibility; Apply H. +Do 2 Rewrite <- (Rplus_sym (Bn (S N))). +Apply Rle_compatibility; Apply HrecN. +Qed. + +(**********) +Lemma Rabsolu_triang_gen : (An:nat->R;N:nat) (Rle (Rabsolu (sum_f_R0 An N)) (sum_f_R0 [i:nat](Rabsolu (An i)) N)). +Intros. +Induction N. +Simpl. +Right; Reflexivity. +Do 2 Rewrite tech5. +Apply Rle_trans with ``(Rabsolu ((sum_f_R0 An N)))+(Rabsolu (An (S N)))``. +Apply Rabsolu_triang. +Do 2 Rewrite <- (Rplus_sym (Rabsolu (An (S N)))). +Apply Rle_compatibility; Apply HrecN. +Qed. + +(**********) +Lemma cond_pos_sum : (An:nat->R;N:nat) ((n:nat)``0<=(An n)``) -> ``0<=(sum_f_R0 An N)``. +Intros. +Induction N. +Simpl; Apply H. +Rewrite tech5. +Apply ge0_plus_ge0_is_ge0. +Apply HrecN. +Apply H. +Qed. + +(* Cauchy's criterion for series *) +Definition Cauchy_crit_series [An:nat->R] : Prop := (Cauchy_crit [N:nat](sum_f_R0 An N)). + +(* If (|An|) satisfies the Cauchy's criterion for series, then (An) too *) +Lemma cauchy_abs : (An:nat->R) (Cauchy_crit_series [i:nat](Rabsolu (An i))) -> (Cauchy_crit_series An). +Unfold Cauchy_crit_series; Unfold Cauchy_crit. +Intros. +Elim (H eps H0); Intros. +Exists x. +Intros. +Cut (Rle (R_dist (sum_f_R0 An n) (sum_f_R0 An m)) (R_dist (sum_f_R0 [i:nat](Rabsolu (An i)) n) (sum_f_R0 [i:nat](Rabsolu (An i)) m))). +Intro. +Apply Rle_lt_trans with (R_dist (sum_f_R0 [i:nat](Rabsolu (An i)) n) (sum_f_R0 [i:nat](Rabsolu (An i)) m)). +Assumption. +Apply H1; Assumption. +Assert H4 := (lt_eq_lt_dec n m). +Elim H4; Intro. +Elim a; Intro. +Rewrite (tech2 An n m); [Idtac | Assumption]. +Rewrite (tech2 [i:nat](Rabsolu (An i)) n m); [Idtac | Assumption]. +Unfold R_dist. +Unfold Rminus. +Do 2 Rewrite Ropp_distr1. +Do 2 Rewrite <- Rplus_assoc. +Do 2 Rewrite Rplus_Ropp_r. +Do 2 Rewrite Rplus_Ol. +Do 2 Rewrite Rabsolu_Ropp. +Rewrite (Rabsolu_right (sum_f_R0 [i:nat](Rabsolu (An (plus (S n) i))) (minus m (S n)))). +Pose Bn:=[i:nat](An (plus (S n) i)). +Replace [i:nat](Rabsolu (An (plus (S n) i))) with [i:nat](Rabsolu (Bn i)). +Apply Rabsolu_triang_gen. +Unfold Bn; Reflexivity. +Apply Rle_sym1. +Apply cond_pos_sum. +Intro; Apply Rabsolu_pos. +Rewrite b. +Unfold R_dist. +Unfold Rminus; Do 2 Rewrite Rplus_Ropp_r. +Rewrite Rabsolu_R0; Right; Reflexivity. +Rewrite (tech2 An m n); [Idtac | Assumption]. +Rewrite (tech2 [i:nat](Rabsolu (An i)) m n); [Idtac | Assumption]. +Unfold R_dist. +Unfold Rminus. +Do 2 Rewrite Rplus_assoc. +Rewrite (Rplus_sym (sum_f_R0 An m)). +Rewrite (Rplus_sym (sum_f_R0 [i:nat](Rabsolu (An i)) m)). +Do 2 Rewrite Rplus_assoc. +Do 2 Rewrite Rplus_Ropp_l. +Do 2 Rewrite Rplus_Or. +Rewrite (Rabsolu_right (sum_f_R0 [i:nat](Rabsolu (An (plus (S m) i))) (minus n (S m)))). +Pose Bn:=[i:nat](An (plus (S m) i)). +Replace [i:nat](Rabsolu (An (plus (S m) i))) with [i:nat](Rabsolu (Bn i)). +Apply Rabsolu_triang_gen. +Unfold Bn; Reflexivity. +Apply Rle_sym1. +Apply cond_pos_sum. +Intro; Apply Rabsolu_pos. +Qed. + +(**********) +Lemma cv_cauchy_1 : (An:nat->R) (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)) -> (Cauchy_crit_series An). +Intros. +Elim X; Intros. +Unfold Un_cv in p. +Unfold Cauchy_crit_series; Unfold Cauchy_crit. +Intros. +Cut ``0<eps/2``. +Intro. +Elim (p ``eps/2`` H0); Intros. +Exists x0. +Intros. +Apply Rle_lt_trans with ``(R_dist (sum_f_R0 An n) x)+(R_dist (sum_f_R0 An m) x)``. +Unfold R_dist. +Replace ``(sum_f_R0 An n)-(sum_f_R0 An m)`` with ``((sum_f_R0 An n)-x)+ -((sum_f_R0 An m)-x)``; [Idtac | Ring]. +Rewrite <- (Rabsolu_Ropp ``(sum_f_R0 An m)-x``). +Apply Rabsolu_triang. +Apply Rlt_le_trans with ``eps/2+eps/2``. +Apply Rplus_lt. +Apply H1; Assumption. +Apply H1; Assumption. +Right; Symmetry; Apply double_var. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Qed. + +Lemma cv_cauchy_2 : (An:nat->R) (Cauchy_crit_series An) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intros. +Apply R_complete. +Unfold Cauchy_crit_series in H. +Exact H. +Qed. + +(**********) +Lemma sum_eq_R0 : (An:nat->R;N:nat) ((n:nat)(le n N)->``(An n)==0``) -> (sum_f_R0 An N)==R0. +Intros; Induction N. +Simpl; Apply H; Apply le_n. +Rewrite tech5; Rewrite HrecN; [Rewrite Rplus_Ol; Apply H; Apply le_n | Intros; Apply H; Apply le_trans with N; [Assumption | Apply le_n_Sn]]. +Qed. + +Definition SP [fn:nat->R->R;N:nat] : R->R := [x:R](sum_f_R0 [k:nat]``(fn k x)`` N). + +(**********) +Lemma sum_incr : (An:nat->R;N:nat;l:R) (Un_cv [n:nat](sum_f_R0 An n) l) -> ((n:nat)``0<=(An n)``) -> ``(sum_f_R0 An N)<=l``. +Intros; Case (total_order_T (sum_f_R0 An N) l); Intro. +Elim s; Intro. +Left; Apply a. +Right; Apply b. +Cut (Un_growing [n:nat](sum_f_R0 An n)). +Intro; Pose l1 := (sum_f_R0 An N). +Fold l1 in r. +Unfold Un_cv in H; Cut ``0<l1-l``. +Intro; Elim (H ? H2); Intros. +Pose N0 := (max x N); Cut (ge N0 x). +Intro; Assert H5 := (H3 N0 H4). +Cut ``l1<=(sum_f_R0 An N0)``. +Intro; Unfold R_dist in H5; Rewrite Rabsolu_right in H5. +Cut ``(sum_f_R0 An N0)<l1``. +Intro; Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H7 H6)). +Apply Rlt_anti_compatibility with ``-l``. +Do 2 Rewrite (Rplus_sym ``-l``). +Apply H5. +Apply Rle_sym1; Apply Rle_anti_compatibility with l. +Rewrite Rplus_Or; Replace ``l+((sum_f_R0 An N0)-l)`` with (sum_f_R0 An N0); [Idtac | Ring]; Apply Rle_trans with l1. +Left; Apply r. +Apply H6. +Unfold l1; Apply Rle_sym2; Apply (growing_prop [k:nat](sum_f_R0 An k)). +Apply H1. +Unfold ge N0; Apply le_max_r. +Unfold ge N0; Apply le_max_l. +Apply Rlt_anti_compatibility with l; Rewrite Rplus_Or; Replace ``l+(l1-l)`` with l1; [Apply r | Ring]. +Unfold Un_growing; Intro; Simpl; Pattern 1 (sum_f_R0 An n); Rewrite <- Rplus_Or; Apply Rle_compatibility; Apply H0. +Qed. + +(**********) +Lemma sum_cv_maj : (An:nat->R;fn:nat->R->R;x,l1,l2:R) (Un_cv [n:nat](SP fn n x) l1) -> (Un_cv [n:nat](sum_f_R0 An n) l2) -> ((n:nat)``(Rabsolu (fn n x))<=(An n)``) -> ``(Rabsolu l1)<=l2``. +Intros; Case (total_order_T (Rabsolu l1) l2); Intro. +Elim s; Intro. +Left; Apply a. +Right; Apply b. +Cut (n0:nat)``(Rabsolu (SP fn n0 x))<=(sum_f_R0 An n0)``. +Intro; Cut ``0<((Rabsolu l1)-l2)/2``. +Intro; Unfold Un_cv in H H0. +Elim (H ? H3); Intros Na H4. +Elim (H0 ? H3); Intros Nb H5. +Pose N := (max Na Nb). +Unfold R_dist in H4 H5. +Cut ``(Rabsolu ((sum_f_R0 An N)-l2))<((Rabsolu l1)-l2)/2``. +Intro; Cut ``(Rabsolu ((Rabsolu l1)-(Rabsolu (SP fn N x))))<((Rabsolu l1)-l2)/2``. +Intro; Cut ``(sum_f_R0 An N)<((Rabsolu l1)+l2)/2``. +Intro; Cut ``((Rabsolu l1)+l2)/2<(Rabsolu (SP fn N x))``. +Intro; Cut ``(sum_f_R0 An N)<(Rabsolu (SP fn N x))``. +Intro; Assert H11 := (H2 N). +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H11 H10)). +Apply Rlt_trans with ``((Rabsolu l1)+l2)/2``; Assumption. +Case (case_Rabsolu ``(Rabsolu l1)-(Rabsolu (SP fn N x))``); Intro. +Apply Rlt_trans with (Rabsolu l1). +Apply Rlt_monotony_contra with ``2``. +Sup0. +Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite double; Apply Rlt_compatibility; Apply r. +DiscrR. +Apply (Rminus_lt ? ? r0). +Rewrite (Rabsolu_right ? r0) in H7. +Apply Rlt_anti_compatibility with ``((Rabsolu l1)-l2)/2-(Rabsolu (SP fn N x))``. +Replace ``((Rabsolu l1)-l2)/2-(Rabsolu (SP fn N x))+((Rabsolu l1)+l2)/2`` with ``(Rabsolu l1)-(Rabsolu (SP fn N x))``. +Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H7. +Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Rewrite <- (Rmult_sym ``/2``); Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``/2``); Pattern 1 (Rabsolu l1); Rewrite double_var; Unfold Rdiv; Ring. +Case (case_Rabsolu ``(sum_f_R0 An N)-l2``); Intro. +Apply Rlt_trans with l2. +Apply (Rminus_lt ? ? r0). +Apply Rlt_monotony_contra with ``2``. +Sup0. +Rewrite (double l2); Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite (Rplus_sym (Rabsolu l1)); Apply Rlt_compatibility; Apply r. +DiscrR. +Rewrite (Rabsolu_right ? r0) in H6; Apply Rlt_anti_compatibility with ``-l2``. +Replace ``-l2+((Rabsolu l1)+l2)/2`` with ``((Rabsolu l1)-l2)/2``. +Rewrite Rplus_sym; Apply H6. +Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite Rminus_distr; Rewrite Rmult_Rplus_distrl; Pattern 2 l2; Rewrite double_var; Repeat Rewrite (Rmult_sym ``/2``); Rewrite Ropp_distr1; Unfold Rdiv; Ring. +Apply Rle_lt_trans with ``(Rabsolu ((SP fn N x)-l1))``. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr3; Apply Rabsolu_triang_inv2. +Apply H4; Unfold ge N; Apply le_max_l. +Apply H5; Unfold ge N; Apply le_max_r. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rlt_anti_compatibility with l2. +Rewrite Rplus_Or; Replace ``l2+((Rabsolu l1)-l2)`` with (Rabsolu l1); [Apply r | Ring]. +Apply Rlt_Rinv; Sup0. +Intros; Induction n0. +Unfold SP; Simpl; Apply H1. +Unfold SP; Simpl. +Apply Rle_trans with (Rplus (Rabsolu (sum_f_R0 [k:nat](fn k x) n0)) (Rabsolu (fn (S n0) x))). +Apply Rabsolu_triang. +Apply Rle_trans with ``(sum_f_R0 An n0)+(Rabsolu (fn (S n0) x))``. +Do 2 Rewrite <- (Rplus_sym (Rabsolu (fn (S n0) x))). +Apply Rle_compatibility; Apply Hrecn0. +Apply Rle_compatibility; Apply H1. +Qed. diff --git a/theories7/Reals/RIneq.v b/theories7/Reals/RIneq.v new file mode 100644 index 00000000..00d41c70 --- /dev/null +++ b/theories7/Reals/RIneq.v @@ -0,0 +1,1631 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: RIneq.v,v 1.2.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +(***************************************************************************) +(** Basic lemmas for the classical reals numbers *) +(***************************************************************************) + +Require Export Raxioms. +Require Export ZArithRing. +Require Omega. +Require Export Field. + +Open Local Scope Z_scope. +Open Local Scope R_scope. + +Implicit Variable Type r:R. + +(***************************************************************************) +(** Instantiating Ring tactic on reals *) +(***************************************************************************) + +Lemma RTheory : (Ring_Theory Rplus Rmult R1 R0 Ropp [x,y:R]false). + Split. + Exact Rplus_sym. + Symmetry; Apply Rplus_assoc. + Exact Rmult_sym. + Symmetry; Apply Rmult_assoc. + Intro; Apply Rplus_Ol. + Intro; Apply Rmult_1l. + Exact Rplus_Ropp_r. + Intros. + Rewrite Rmult_sym. + Rewrite (Rmult_sym n p). + Rewrite (Rmult_sym m p). + Apply Rmult_Rplus_distr. + Intros; Contradiction. +Defined. + +Add Field R Rplus Rmult R1 R0 Ropp [x,y:R]false Rinv RTheory Rinv_l + with minus:=Rminus div:=Rdiv. + +(**************************************************************************) +(** Relation between orders and equality *) +(**************************************************************************) + +(**********) +Lemma Rlt_antirefl:(r:R)~``r<r``. + Generalize Rlt_antisym. Intuition EAuto. +Qed. +Hints Resolve Rlt_antirefl : real. + +Lemma Rle_refl : (x:R) ``x<=x``. +Intro; Right; Reflexivity. +Qed. + +Lemma Rlt_not_eq:(r1,r2:R)``r1<r2``->``r1<>r2``. + Red; Intros r1 r2 H H0; Apply (Rlt_antirefl r1). + Pattern 2 r1; Rewrite H0; Trivial. +Qed. + +Lemma Rgt_not_eq:(r1,r2:R)``r1>r2``->``r1<>r2``. +Intros; Apply sym_not_eqT; Apply Rlt_not_eq; Auto with real. +Qed. + +(**********) +Lemma imp_not_Req:(r1,r2:R)(``r1<r2``\/ ``r1>r2``) -> ``r1<>r2``. +Generalize Rlt_not_eq Rgt_not_eq. Intuition EAuto. +Qed. +Hints Resolve imp_not_Req : real. + +(** Reasoning by case on equalities and order *) + +(**********) +Lemma Req_EM:(r1,r2:R)(r1==r2)\/``r1<>r2``. +Intros ; Generalize (total_order_T r1 r2) imp_not_Req ; Intuition EAuto 3. +Qed. +Hints Resolve Req_EM : real. + +(**********) +Lemma total_order:(r1,r2:R)``r1<r2``\/(r1==r2)\/``r1>r2``. +Intros;Generalize (total_order_T r1 r2);Tauto. +Qed. + +(**********) +Lemma not_Req:(r1,r2:R)``r1<>r2``->(``r1<r2``\/``r1>r2``). +Intros; Generalize (total_order_T r1 r2) ; Tauto. +Qed. + + +(*********************************************************************************) +(** Order Lemma : relating [<], [>], [<=] and [>=] *) +(*********************************************************************************) + +(**********) +Lemma Rlt_le:(r1,r2:R)``r1<r2``-> ``r1<=r2``. +Intros ; Red ; Tauto. +Qed. +Hints Resolve Rlt_le : real. + +(**********) +Lemma Rle_ge : (r1,r2:R)``r1<=r2`` -> ``r2>=r1``. +NewDestruct 1; Red; Auto with real. +Qed. + +Hints Immediate Rle_ge : real. + +(**********) +Lemma Rge_le : (r1,r2:R)``r1>=r2`` -> ``r2<=r1``. +NewDestruct 1; Red; Auto with real. +Qed. + +Hints Resolve Rge_le : real. + +(**********) +Lemma not_Rle:(r1,r2:R)~``r1<=r2`` -> ``r2<r1``. +Intros r1 r2 ; Generalize (total_order r1 r2) ; Unfold Rle; Tauto. +Qed. + +Hints Immediate not_Rle : real. + +Lemma not_Rge:(r1,r2:R)~``r1>=r2`` -> ``r1<r2``. +Intros; Apply not_Rle; Auto with real. +Qed. + +(**********) +Lemma Rlt_le_not:(r1,r2:R)``r2<r1`` -> ~``r1<=r2``. +Generalize Rlt_antisym imp_not_Req ; Unfold Rle. +Intuition EAuto 3. +Qed. + +Lemma Rle_not:(r1,r2:R)``r1>r2`` -> ~``r1<=r2``. +Proof Rlt_le_not. + +Hints Immediate Rlt_le_not : real. + +Lemma Rle_not_lt: (r1, r2:R) ``r2 <= r1`` -> ~``r1<r2``. +Intros r1 r2. Generalize (Rlt_antisym r1 r2) (imp_not_Req r1 r2). +Unfold Rle; Intuition. +Qed. + +(**********) +Lemma Rlt_ge_not:(r1,r2:R)``r1<r2`` -> ~``r1>=r2``. +Generalize Rlt_le_not. Unfold Rle Rge. Intuition EAuto 3. +Qed. + +Hints Immediate Rlt_ge_not : real. + +(**********) +Lemma eq_Rle:(r1,r2:R)r1==r2->``r1<=r2``. +Unfold Rle; Tauto. +Qed. +Hints Immediate eq_Rle : real. + +Lemma eq_Rge:(r1,r2:R)r1==r2->``r1>=r2``. +Unfold Rge; Tauto. +Qed. +Hints Immediate eq_Rge : real. + +Lemma eq_Rle_sym:(r1,r2:R)r2==r1->``r1<=r2``. +Unfold Rle; Auto. +Qed. +Hints Immediate eq_Rle_sym : real. + +Lemma eq_Rge_sym:(r1,r2:R)r2==r1->``r1>=r2``. +Unfold Rge; Auto. +Qed. +Hints Immediate eq_Rge_sym : real. + +Lemma Rle_antisym : (r1,r2:R)``r1<=r2`` -> ``r2<=r1``-> r1==r2. +Intros r1 r2; Generalize (Rlt_antisym r1 r2) ; Unfold Rle ; Intuition. +Qed. +Hints Resolve Rle_antisym : real. + +(**********) +Lemma Rle_le_eq:(r1,r2:R)(``r1<=r2``/\``r2<=r1``)<->(r1==r2). +Intuition. +Qed. + +Lemma Rlt_rew : (x,x',y,y':R)``x==x'``->``x'<y'`` -> `` y' == y`` -> ``x < y``. +Intros x x' y y'; Intros; Replace x with x'; Replace y with y'; Assumption. +Qed. + +(**********) +Lemma Rle_trans:(r1,r2,r3:R) ``r1<=r2``->``r2<=r3``->``r1<=r3``. +Generalize trans_eqT Rlt_trans Rlt_rew. +Unfold Rle. +Intuition EAuto 2. +Qed. + +(**********) +Lemma Rle_lt_trans:(r1,r2,r3:R)``r1<=r2``->``r2<r3``->``r1<r3``. +Generalize Rlt_trans Rlt_rew. +Unfold Rle. +Intuition EAuto 2. +Qed. + +(**********) +Lemma Rlt_le_trans:(r1,r2,r3:R)``r1<r2``->``r2<=r3``->``r1<r3``. +Generalize Rlt_trans Rlt_rew; Unfold Rle; Intuition EAuto 2. +Qed. + + +(** Decidability of the order *) +Lemma total_order_Rlt:(r1,r2:R)(sumboolT ``r1<r2`` ~(``r1<r2``)). +Intros;Generalize (total_order_T r1 r2) (imp_not_Req r1 r2) ; Intuition. +Qed. + +(**********) +Lemma total_order_Rle:(r1,r2:R)(sumboolT ``r1<=r2`` ~(``r1<=r2``)). +Intros r1 r2. +Generalize (total_order_T r1 r2) (imp_not_Req r1 r2). +Intuition EAuto 4 with real. +Qed. + +(**********) +Lemma total_order_Rgt:(r1,r2:R)(sumboolT ``r1>r2`` ~(``r1>r2``)). +Intros;Unfold Rgt;Intros;Apply total_order_Rlt. +Qed. + +(**********) +Lemma total_order_Rge:(r1,r2:R)(sumboolT (``r1>=r2``) ~(``r1>=r2``)). +Intros;Generalize (total_order_Rle r2 r1);Intuition. +Qed. + +Lemma total_order_Rlt_Rle:(r1,r2:R)(sumboolT ``r1<r2`` ``r2<=r1``). +Intros;Generalize (total_order_T r1 r2); Intuition. +Qed. + +Lemma Rle_or_lt: (n, m:R)(Rle n m) \/ (Rlt m n). +Intros n m; Elim (total_order_Rlt_Rle m n);Auto with real. +Qed. + +Lemma total_order_Rle_Rlt_eq :(r1,r2:R)``r1<=r2``-> + (sumboolT ``r1<r2`` ``r1==r2``). +Intros r1 r2 H;Generalize (total_order_T r1 r2); Intuition. +Qed. + +(**********) +Lemma inser_trans_R:(n,m,p,q:R)``n<=m<p``-> (sumboolT ``n<=m<q`` ``q<=m<p``). +Intros n m p q; Intros; Generalize (total_order_Rlt_Rle m q); Intuition. +Qed. + +(****************************************************************) +(** Field Lemmas *) +(* This part contains lemma involving the Fields operations *) +(****************************************************************) +(*********************************************************) +(** Addition *) +(*********************************************************) + +Lemma Rplus_ne:(r:R)``r+0==r``/\``0+r==r``. +Intro;Split;Ring. +Qed. +Hints Resolve Rplus_ne : real v62. + +Lemma Rplus_Or:(r:R)``r+0==r``. +Intro; Ring. +Qed. +Hints Resolve Rplus_Or : real. + +(**********) +Lemma Rplus_Ropp_l:(r:R)``(-r)+r==0``. + Intro; Ring. +Qed. +Hints Resolve Rplus_Ropp_l : real. + + +(**********) +Lemma Rplus_Ropp:(x,y:R)``x+y==0``->``y== -x``. + Intros x y H; Replace y with ``(-x+x)+y``; + [ Rewrite -> Rplus_assoc; Rewrite -> H; Ring + | Ring ]. +Qed. + +(*i New i*) +Hint eqT_R_congr : real := Resolve (congr_eqT R). + +Lemma Rplus_plus_r:(r,r1,r2:R)(r1==r2)->``r+r1==r+r2``. + Auto with real. +Qed. + +(*i Old i*)Hints Resolve Rplus_plus_r : v62. + +(**********) +Lemma r_Rplus_plus:(r,r1,r2:R)``r+r1==r+r2``->r1==r2. + Intros; Transitivity ``(-r+r)+r1``. + Ring. + Transitivity ``(-r+r)+r2``. + Repeat Rewrite -> Rplus_assoc; Rewrite <- H; Reflexivity. + Ring. +Qed. +Hints Resolve r_Rplus_plus : real. + +(**********) +Lemma Rplus_ne_i:(r,b:R)``r+b==r`` -> ``b==0``. + Intros r b; Pattern 2 r; Replace r with ``r+0``; + EAuto with real. +Qed. + +(***********************************************************) +(** Multiplication *) +(***********************************************************) + +(**********) +Lemma Rinv_r:(r:R)``r<>0``->``r* (/r)==1``. + Intros; Rewrite -> Rmult_sym; Auto with real. +Qed. +Hints Resolve Rinv_r : real. + +Lemma Rinv_l_sym:(r:R)``r<>0``->``1==(/r) * r``. + Symmetry; Auto with real. +Qed. + +Lemma Rinv_r_sym:(r:R)``r<>0``->``1==r* (/r)``. + Symmetry; Auto with real. +Qed. +Hints Resolve Rinv_l_sym Rinv_r_sym : real. + + +(**********) +Lemma Rmult_Or :(r:R) ``r*0==0``. +Intro; Ring. +Qed. +Hints Resolve Rmult_Or : real v62. + +(**********) +Lemma Rmult_Ol:(r:R) ``0*r==0``. +Intro; Ring. +Qed. +Hints Resolve Rmult_Ol : real v62. + +(**********) +Lemma Rmult_ne:(r:R)``r*1==r``/\``1*r==r``. +Intro;Split;Ring. +Qed. +Hints Resolve Rmult_ne : real v62. + +(**********) +Lemma Rmult_1r:(r:R)(``r*1==r``). +Intro; Ring. +Qed. +Hints Resolve Rmult_1r : real. + +(**********) +Lemma Rmult_mult_r:(r,r1,r2:R)r1==r2->``r*r1==r*r2``. + Auto with real. +Qed. + +(*i OLD i*)Hints Resolve Rmult_mult_r : v62. + +(**********) +Lemma r_Rmult_mult:(r,r1,r2:R)(``r*r1==r*r2``)->``r<>0``->(r1==r2). + Intros; Transitivity ``(/r * r)*r1``. + Rewrite Rinv_l; Auto with real. + Transitivity ``(/r * r)*r2``. + Repeat Rewrite Rmult_assoc; Rewrite H; Trivial. + Rewrite Rinv_l; Auto with real. +Qed. + +(**********) +Lemma without_div_Od:(r1,r2:R)``r1*r2==0`` -> ``r1==0`` \/ ``r2==0``. + Intros; Case (Req_EM r1 ``0``); [Intro Hz | Intro Hnotz]. + Auto. + Right; Apply r_Rmult_mult with r1; Trivial. + Rewrite H; Auto with real. +Qed. + +(**********) +Lemma without_div_Oi:(r1,r2:R) ``r1==0``\/``r2==0`` -> ``r1*r2==0``. + Intros r1 r2 [H | H]; Rewrite H; Auto with real. +Qed. + +Hints Resolve without_div_Oi : real. + +(**********) +Lemma without_div_Oi1:(r1,r2:R) ``r1==0`` -> ``r1*r2==0``. + Auto with real. +Qed. + +(**********) +Lemma without_div_Oi2:(r1,r2:R) ``r2==0`` -> ``r1*r2==0``. + Auto with real. +Qed. + + +(**********) +Lemma without_div_O_contr:(r1,r2:R)``r1*r2<>0`` -> ``r1<>0`` /\ ``r2<>0``. +Intros r1 r2 H; Split; Red; Intro; Apply H; Auto with real. +Qed. + +(**********) +Lemma mult_non_zero :(r1,r2:R)``r1<>0`` /\ ``r2<>0`` -> ``r1*r2<>0``. +Red; Intros r1 r2 (H1,H2) H. +Case (without_div_Od r1 r2); Auto with real. +Qed. +Hints Resolve mult_non_zero : real. + +(**********) +Lemma Rmult_Rplus_distrl: + (r1,r2,r3:R) ``(r1+r2)*r3 == (r1*r3)+(r2*r3)``. +Intros; Ring. +Qed. + +(** Square function *) + +(***********) +Definition Rsqr:R->R:=[r:R]``r*r``. +V7only[Notation "x ²" := (Rsqr x) (at level 2,left associativity).]. + +(***********) +Lemma Rsqr_O:(Rsqr ``0``)==``0``. + Unfold Rsqr; Auto with real. +Qed. + +(***********) +Lemma Rsqr_r_R0:(r:R)(Rsqr r)==``0``->``r==0``. +Unfold Rsqr;Intros;Elim (without_div_Od r r H);Trivial. +Qed. + +(*********************************************************) +(** Opposite *) +(*********************************************************) + +(**********) +Lemma eq_Ropp:(r1,r2:R)(r1==r2)->``-r1 == -r2``. + Auto with real. +Qed. +Hints Resolve eq_Ropp : real. + +(**********) +Lemma Ropp_O:``-0==0``. + Ring. +Qed. +Hints Resolve Ropp_O : real v62. + +(**********) +Lemma eq_RoppO:(r:R)``r==0``-> ``-r==0``. + Intros; Rewrite -> H; Auto with real. +Qed. +Hints Resolve eq_RoppO : real. + +(**********) +Lemma Ropp_Ropp:(r:R)``-(-r)==r``. + Intro; Ring. +Qed. +Hints Resolve Ropp_Ropp : real. + +(*********) +Lemma Ropp_neq:(r:R)``r<>0``->``-r<>0``. +Red;Intros r H H0. +Apply H. +Transitivity ``-(-r)``; Auto with real. +Qed. +Hints Resolve Ropp_neq : real. + +(**********) +Lemma Ropp_distr1:(r1,r2:R)``-(r1+r2)==(-r1 + -r2)``. + Intros; Ring. +Qed. +Hints Resolve Ropp_distr1 : real. + +(** Opposite and multiplication *) + +Lemma Ropp_mul1:(r1,r2:R)``(-r1)*r2 == -(r1*r2)``. + Intros; Ring. +Qed. +Hints Resolve Ropp_mul1 : real. + +(**********) +Lemma Ropp_mul2:(r1,r2:R)``(-r1)*(-r2)==r1*r2``. + Intros; Ring. +Qed. +Hints Resolve Ropp_mul2 : real. + +Lemma Ropp_mul3 : (r1,r2:R) ``r1*(-r2) == -(r1*r2)``. +Intros; Rewrite <- Ropp_mul1; Ring. +Qed. + +(** Substraction *) + +Lemma minus_R0:(r:R)``r-0==r``. +Intro;Ring. +Qed. +Hints Resolve minus_R0 : real. + +Lemma Rminus_Ropp:(r:R)``0-r==-r``. +Intro;Ring. +Qed. +Hints Resolve Rminus_Ropp : real. + +(**********) +Lemma Ropp_distr2:(r1,r2:R)``-(r1-r2)==r2-r1``. + Intros; Ring. +Qed. +Hints Resolve Ropp_distr2 : real. + +Lemma Ropp_distr3:(r1,r2:R)``-(r2-r1)==r1-r2``. +Intros; Ring. +Qed. +Hints Resolve Ropp_distr3 : real. + +(**********) +Lemma eq_Rminus:(r1,r2:R)(r1==r2)->``r1-r2==0``. + Intros; Rewrite H; Ring. +Qed. +Hints Resolve eq_Rminus : real. + +(**********) +Lemma Rminus_eq:(r1,r2:R)``r1-r2==0`` -> r1==r2. + Intros r1 r2; Unfold Rminus; Rewrite -> Rplus_sym; Intro. + Rewrite <- (Ropp_Ropp r2); Apply (Rplus_Ropp (Ropp r2) r1 H). +Qed. +Hints Immediate Rminus_eq : real. + +Lemma Rminus_eq_right:(r1,r2:R)``r2-r1==0`` -> r1==r2. +Intros;Generalize (Rminus_eq r2 r1 H);Clear H;Intro H;Rewrite H;Ring. +Qed. +Hints Immediate Rminus_eq_right : real. + +Lemma Rplus_Rminus: (p,q:R)``p+(q-p)``==q. +Intros; Ring. +Qed. +Hints Resolve Rplus_Rminus:real. + +(**********) +Lemma Rminus_eq_contra:(r1,r2:R)``r1<>r2``->``r1-r2<>0``. +Red; Intros r1 r2 H H0. +Apply H; Auto with real. +Qed. +Hints Resolve Rminus_eq_contra : real. + +Lemma Rminus_not_eq:(r1,r2:R)``r1-r2<>0``->``r1<>r2``. +Red; Intros; Elim H; Apply eq_Rminus; Auto. +Qed. +Hints Resolve Rminus_not_eq : real. + +Lemma Rminus_not_eq_right:(r1,r2:R)``r2-r1<>0`` -> ``r1<>r2``. +Red; Intros;Elim H;Rewrite H0; Ring. +Qed. +Hints Resolve Rminus_not_eq_right : real. + +V7only [Notation not_sym := (sym_not_eq R).]. + +(**********) +Lemma Rminus_distr: (x,y,z:R) ``x*(y-z)==(x*y) - (x*z)``. +Intros; Ring. +Qed. + +(** Inverse *) +Lemma Rinv_R1:``/1==1``. +Field;Auto with real. +Qed. +Hints Resolve Rinv_R1 : real. + +(*********) +Lemma Rinv_neq_R0:(r:R)``r<>0``->``(/r)<>0``. +Red; Intros; Apply R1_neq_R0. +Replace ``1`` with ``(/r) * r``; Auto with real. +Qed. +Hints Resolve Rinv_neq_R0 : real. + +(*********) +Lemma Rinv_Rinv:(r:R)``r<>0``->``/(/r)==r``. +Intros;Field;Auto with real. +Qed. +Hints Resolve Rinv_Rinv : real. + +(*********) +Lemma Rinv_Rmult:(r1,r2:R)``r1<>0``->``r2<>0``->``/(r1*r2)==(/r1)*(/r2)``. +Intros;Field;Auto with real. +Qed. + +(*********) +Lemma Ropp_Rinv:(r:R)``r<>0``->``-(/r)==/(-r)``. +Intros;Field;Auto with real. +Qed. + +Lemma Rinv_r_simpl_r : (r1,r2:R)``r1<>0``->``r1*(/r1)*r2==r2``. +Intros; Transitivity ``1*r2``; Auto with real. +Rewrite Rinv_r; Auto with real. +Qed. + +Lemma Rinv_r_simpl_l : (r1,r2:R)``r1<>0``->``r2*r1*(/r1)==r2``. +Intros; Transitivity ``r2*1``; Auto with real. +Transitivity ``r2*(r1*/r1)``; Auto with real. +Qed. + +Lemma Rinv_r_simpl_m : (r1,r2:R)``r1<>0``->``r1*r2*(/r1)==r2``. +Intros; Transitivity ``r2*1``; Auto with real. +Transitivity ``r2*(r1*/r1)``; Auto with real. +Ring. +Qed. +Hints Resolve Rinv_r_simpl_l Rinv_r_simpl_r Rinv_r_simpl_m : real. + +(*********) +Lemma Rinv_Rmult_simpl:(a,b,c:R)``a<>0``->``(a*(/b))*(c*(/a))==c*(/b)``. +Intros a b c; Intros. +Transitivity ``(a*/a)*(c*(/b))``; Auto with real. +Ring. +Qed. + +(** Order and addition *) + +Lemma Rlt_compatibility_r:(r,r1,r2:R)``r1<r2``->``r1+r<r2+r``. +Intros. +Rewrite (Rplus_sym r1 r); Rewrite (Rplus_sym r2 r); Auto with real. +Qed. + +Hints Resolve Rlt_compatibility_r : real. + +(**********) +Lemma Rlt_anti_compatibility: (r,r1,r2:R)``r+r1 < r+r2`` -> ``r1<r2``. +Intros; Cut ``(-r+r)+r1 < (-r+r)+r2``. +Rewrite -> Rplus_Ropp_l. +Elim (Rplus_ne r1); Elim (Rplus_ne r2); Intros; Rewrite <- H3; + Rewrite <- H1; Auto with zarith real. +Rewrite -> Rplus_assoc; Rewrite -> Rplus_assoc; + Apply (Rlt_compatibility ``-r`` ``r+r1`` ``r+r2`` H). +Qed. + +(**********) +Lemma Rle_compatibility:(r,r1,r2:R)``r1<=r2`` -> ``r+r1 <= r+r2 ``. +Unfold Rle; Intros; Elim H; Intro. +Left; Apply (Rlt_compatibility r r1 r2 H0). +Right; Rewrite <- H0; Auto with zarith real. +Qed. + +(**********) +Lemma Rle_compatibility_r:(r,r1,r2:R)``r1<=r2`` -> ``r1+r<=r2+r``. +Unfold Rle; Intros; Elim H; Intro. +Left; Apply (Rlt_compatibility_r r r1 r2 H0). +Right; Rewrite <- H0; Auto with real. +Qed. + +Hints Resolve Rle_compatibility Rle_compatibility_r : real. + +(**********) +Lemma Rle_anti_compatibility: (r,r1,r2:R)``r+r1<=r+r2`` -> ``r1<=r2``. +Unfold Rle; Intros; Elim H; Intro. +Left; Apply (Rlt_anti_compatibility r r1 r2 H0). +Right; Apply (r_Rplus_plus r r1 r2 H0). +Qed. + +(**********) +Lemma sum_inequa_Rle_lt:(a,x,b,c,y,d:R)``a<=x`` -> ``x<b`` -> + ``c<y`` -> ``y<=d`` -> ``a+c < x+y < b+d``. +Intros;Split. +Apply Rlt_le_trans with ``a+y``; Auto with real. +Apply Rlt_le_trans with ``b+y``; Auto with real. +Qed. + +(*********) +Lemma Rplus_lt:(r1,r2,r3,r4:R)``r1<r2`` -> ``r3<r4`` -> ``r1+r3 < r2+r4``. +Intros; Apply Rlt_trans with ``r2+r3``; Auto with real. +Qed. + +Lemma Rplus_le:(r1,r2,r3,r4:R)``r1<=r2`` -> ``r3<=r4`` -> ``r1+r3 <= r2+r4``. +Intros; Apply Rle_trans with ``r2+r3``; Auto with real. +Qed. + +(*********) +Lemma Rplus_lt_le_lt:(r1,r2,r3,r4:R)``r1<r2`` -> ``r3<=r4`` -> + ``r1+r3 < r2+r4``. +Intros; Apply Rlt_le_trans with ``r2+r3``; Auto with real. +Qed. + +(*********) +Lemma Rplus_le_lt_lt:(r1,r2,r3,r4:R)``r1<=r2`` -> ``r3<r4`` -> + ``r1+r3 < r2+r4``. +Intros; Apply Rle_lt_trans with ``r2+r3``; Auto with real. +Qed. + +Hints Immediate Rplus_lt Rplus_le Rplus_lt_le_lt Rplus_le_lt_lt : real. + +(** Order and Opposite *) + +(**********) +Lemma Rgt_Ropp:(r1,r2:R) ``r1 > r2`` -> ``-r1 < -r2``. +Unfold Rgt; Intros. +Apply (Rlt_anti_compatibility ``r2+r1``). +Replace ``r2+r1+(-r1)`` with r2. +Replace ``r2+r1+(-r2)`` with r1. +Trivial. +Ring. +Ring. +Qed. +Hints Resolve Rgt_Ropp. + +(**********) +Lemma Rlt_Ropp:(r1,r2:R) ``r1 < r2`` -> ``-r1 > -r2``. +Unfold Rgt; Auto with real. +Qed. +Hints Resolve Rlt_Ropp : real. + +Lemma Ropp_Rlt: (x,y:R) ``-y < -x`` ->``x<y``. +Intros x y H'. +Rewrite <- (Ropp_Ropp x); Rewrite <- (Ropp_Ropp y); Auto with real. +Qed. +Hints Immediate Ropp_Rlt : real. + +Lemma Rlt_Ropp1:(r1,r2:R) ``r2 < r1`` -> ``-r1 < -r2``. +Auto with real. +Qed. +Hints Resolve Rlt_Ropp1 : real. + +(**********) +Lemma Rle_Ropp:(r1,r2:R) ``r1 <= r2`` -> ``-r1 >= -r2``. +Unfold Rge; Intros r1 r2 [H|H]; Auto with real. +Qed. +Hints Resolve Rle_Ropp : real. + +Lemma Ropp_Rle: (x,y:R) ``-y <= -x`` ->``x <= y``. +Intros x y H. +Elim H;Auto with real. +Intro H1;Rewrite <-(Ropp_Ropp x);Rewrite <-(Ropp_Ropp y);Rewrite H1; + Auto with real. +Qed. +Hints Immediate Ropp_Rle : real. + +Lemma Rle_Ropp1:(r1,r2:R) ``r2 <= r1`` -> ``-r1 <= -r2``. +Intros r1 r2 H;Elim H;Auto with real. +Qed. +Hints Resolve Rle_Ropp1 : real. + +(**********) +Lemma Rge_Ropp:(r1,r2:R) ``r1 >= r2`` -> ``-r1 <= -r2``. +Unfold Rge; Intros r1 r2 [H|H]; Auto with real. +Qed. +Hints Resolve Rge_Ropp : real. + +(**********) +Lemma Rlt_RO_Ropp:(r:R) ``0 < r`` -> ``0 > -r``. +Intros; Replace ``0`` with ``-0``; Auto with real. +Qed. +Hints Resolve Rlt_RO_Ropp : real. + +(**********) +Lemma Rgt_RO_Ropp:(r:R) ``0 > r`` -> ``0 < -r``. +Intros; Replace ``0`` with ``-0``; Auto with real. +Qed. +Hints Resolve Rgt_RO_Ropp : real. + +(**********) +Lemma Rgt_RoppO:(r:R)``r>0``->``(-r)<0``. +Intros; Rewrite <- Ropp_O; Auto with real. +Qed. + +(**********) +Lemma Rlt_RoppO:(r:R)``r<0``->``-r>0``. +Intros; Rewrite <- Ropp_O; Auto with real. +Qed. +Hints Resolve Rgt_RoppO Rlt_RoppO: real. + +(**********) +Lemma Rle_RO_Ropp:(r:R) ``0 <= r`` -> ``0 >= -r``. +Intros; Replace ``0`` with ``-0``; Auto with real. +Qed. +Hints Resolve Rle_RO_Ropp : real. + +(**********) +Lemma Rge_RO_Ropp:(r:R) ``0 >= r`` -> ``0 <= -r``. +Intros; Replace ``0`` with ``-0``; Auto with real. +Qed. +Hints Resolve Rge_RO_Ropp : real. + +(** Order and multiplication *) + +Lemma Rlt_monotony_r:(r,r1,r2:R)``0<r`` -> ``r1 < r2`` -> ``r1*r < r2*r``. +Intros; Rewrite (Rmult_sym r1 r); Rewrite (Rmult_sym r2 r); Auto with real. +Qed. +Hints Resolve Rlt_monotony_r. + +Lemma Rlt_monotony_contra: (z, x, y:R) ``0<z`` ->``z*x<z*y`` ->``x<y``. +Intros z x y H H0. +Case (total_order x y); Intros Eq0; Auto; Elim Eq0; Clear Eq0; Intros Eq0. + Rewrite Eq0 in H0;ElimType False;Apply (Rlt_antirefl ``z*y``);Auto. +Generalize (Rlt_monotony z y x H Eq0);Intro;ElimType False; + Generalize (Rlt_trans ``z*x`` ``z*y`` ``z*x`` H0 H1);Intro; + Apply (Rlt_antirefl ``z*x``);Auto. +Qed. + +V7only [ +Notation Rlt_monotony_rev := Rlt_monotony_contra. +Notation "'Rlt_monotony_contra' a b c" := (Rlt_monotony_contra c a b) + (at level 10, a,b,c at level 9, only parsing). +]. + +Lemma Rlt_anti_monotony:(r,r1,r2:R)``r < 0`` -> ``r1 < r2`` -> ``r*r1 > r*r2``. +Intros; Replace r with ``-(-r)``; Auto with real. +Rewrite (Ropp_mul1 ``-r``); Rewrite (Ropp_mul1 ``-r``). +Apply Rlt_Ropp; Auto with real. +Qed. + +(**********) +Lemma Rle_monotony: + (r,r1,r2:R)``0 <= r`` -> ``r1 <= r2`` -> ``r*r1 <= r*r2``. +Intros r r1 r2 H H0; NewDestruct H; NewDestruct H0; Unfold Rle; Auto with real. +Right; Rewrite <- H; Do 2 Rewrite Rmult_Ol; Reflexivity. +Qed. +Hints Resolve Rle_monotony : real. + +Lemma Rle_monotony_r: + (r,r1,r2:R)``0 <= r`` -> ``r1 <= r2`` -> ``r1*r <= r2*r``. +Intros r r1 r2 H; +Rewrite (Rmult_sym r1 r); Rewrite (Rmult_sym r2 r); Auto with real. +Qed. +Hints Resolve Rle_monotony_r : real. + +Lemma Rmult_le_reg_l: + (z, x, y:R) ``0<z`` ->``z*x<=z*y`` ->``x<=y``. +Intros z x y H H0;Case H0; Auto with real. +Intros H1; Apply Rlt_le. +Apply Rlt_monotony_contra with z := z;Auto. +Intros H1;Replace x with (Rmult (Rinv z) (Rmult z x)); Auto with real. +Replace y with (Rmult (Rinv z) (Rmult z y)). + Rewrite H1;Auto with real. +Rewrite <- Rmult_assoc; Rewrite Rinv_l; Auto with real. +Rewrite <- Rmult_assoc; Rewrite Rinv_l; Auto with real. +Qed. + +V7only [ +Notation "'Rle_monotony_contra' a b c" := (Rmult_le_reg_l c a b) + (at level 10, a,b,c at level 9, only parsing). +Notation Rle_monotony_contra := Rmult_le_reg_l. +]. + + +Lemma Rle_anti_monotony1 + :(r,r1,r2:R)``r <= 0`` -> ``r1 <= r2`` -> ``r*r2 <= r*r1``. +Intros; Replace r with ``-(-r)``; Auto with real. +Do 2 Rewrite (Ropp_mul1 ``-r``). +Apply Rle_Ropp1; Auto with real. +Qed. +Hints Resolve Rle_anti_monotony1 : real. + +Lemma Rle_anti_monotony + :(r,r1,r2:R)``r <= 0`` -> ``r1 <= r2`` -> ``r*r1 >= r*r2``. +Intros; Apply Rle_ge; Auto with real. +Qed. +Hints Resolve Rle_anti_monotony : real. + +Lemma Rle_Rmult_comp: + (x, y, z, t:R) ``0 <= x`` -> ``0 <= z`` -> ``x <= y`` -> ``z <= t`` -> + ``x*z <= y*t``. +Intros x y z t H' H'0 H'1 H'2. +Apply Rle_trans with r2 := ``x*t``; Auto with real. +Repeat Rewrite [x:?](Rmult_sym x t). +Apply Rle_monotony; Auto. +Apply Rle_trans with z; Auto. +Qed. +Hints Resolve Rle_Rmult_comp :real. + +Lemma Rmult_lt:(r1,r2,r3,r4:R)``r3>0`` -> ``r2>0`` -> + `` r1 < r2`` -> ``r3 < r4`` -> ``r1*r3 < r2*r4``. +Intros; Apply Rlt_trans with ``r2*r3``; Auto with real. +Qed. + +(*********) +Lemma Rmult_lt_0 + :(r1,r2,r3,r4:R)``r3>=0``->``r2>0``->``r1<r2``->``r3<r4``->``r1*r3<r2*r4``. +Intros; Apply Rle_lt_trans with ``r2*r3``; Auto with real. +Qed. + +(** Order and Substractions *) +Lemma Rlt_minus:(r1,r2:R)``r1 < r2`` -> ``r1-r2 < 0``. +Intros; Apply (Rlt_anti_compatibility ``r2``). +Replace ``r2+(r1-r2)`` with r1. +Replace ``r2+0`` with r2; Auto with real. +Ring. +Qed. +Hints Resolve Rlt_minus : real. + +(**********) +Lemma Rle_minus:(r1,r2:R)``r1 <= r2`` -> ``r1-r2 <= 0``. +NewDestruct 1; Unfold Rle; Auto with real. +Qed. + +(**********) +Lemma Rminus_lt:(r1,r2:R)``r1-r2 < 0`` -> ``r1 < r2``. +Intros; Replace r1 with ``r1-r2+r2``. +Pattern 3 r2; Replace r2 with ``0+r2``; Auto with real. +Ring. +Qed. + +(**********) +Lemma Rminus_le:(r1,r2:R)``r1-r2 <= 0`` -> ``r1 <= r2``. +Intros; Replace r1 with ``r1-r2+r2``. +Pattern 3 r2; Replace r2 with ``0+r2``; Auto with real. +Ring. +Qed. + +(**********) +Lemma tech_Rplus:(r,s:R)``0<=r`` -> ``0<s`` -> ``r+s<>0``. +Intros; Apply sym_not_eqT; Apply Rlt_not_eq. +Rewrite Rplus_sym; Replace ``0`` with ``0+0``; Auto with real. +Qed. +Hints Immediate tech_Rplus : real. + +(** Order and the square function *) +Lemma pos_Rsqr:(r:R)``0<=(Rsqr r)``. +Intro; Case (total_order_Rlt_Rle r ``0``); Unfold Rsqr; Intro. +Replace ``r*r`` with ``(-r)*(-r)``; Auto with real. +Replace ``0`` with ``-r*0``; Auto with real. +Replace ``0`` with ``0*r``; Auto with real. +Qed. + +(***********) +Lemma pos_Rsqr1:(r:R)``r<>0``->``0<(Rsqr r)``. +Intros; Case (not_Req r ``0``); Trivial; Unfold Rsqr; Intro. +Replace ``r*r`` with ``(-r)*(-r)``; Auto with real. +Replace ``0`` with ``-r*0``; Auto with real. +Replace ``0`` with ``0*r``; Auto with real. +Qed. +Hints Resolve pos_Rsqr pos_Rsqr1 : real. + +(** Zero is less than one *) +Lemma Rlt_R0_R1:``0<1``. +Replace ``1`` with ``(Rsqr 1)``; Auto with real. +Unfold Rsqr; Auto with real. +Qed. +Hints Resolve Rlt_R0_R1 : real. + +Lemma Rle_R0_R1:``0<=1``. +Left. +Exact Rlt_R0_R1. +Qed. + +(** Order and inverse *) +Lemma Rlt_Rinv:(r:R)``0<r``->``0</r``. +Intros; Apply not_Rle; Red; Intros. +Absurd ``1<=0``; Auto with real. +Replace ``1`` with ``r*(/r)``; Auto with real. +Replace ``0`` with ``r*0``; Auto with real. +Qed. +Hints Resolve Rlt_Rinv : real. + +(*********) +Lemma Rlt_Rinv2:(r:R)``r < 0``->``/r < 0``. +Intros; Apply not_Rle; Red; Intros. +Absurd ``1<=0``; Auto with real. +Replace ``1`` with ``r*(/r)``; Auto with real. +Replace ``0`` with ``r*0``; Auto with real. +Qed. +Hints Resolve Rlt_Rinv2 : real. + +(*********) +Lemma Rinv_lt:(r1,r2:R)``0 < r1*r2`` -> ``r1 < r2`` -> ``/r2 < /r1``. +Intros; Apply Rlt_monotony_rev with ``r1*r2``; Auto with real. +Case (without_div_O_contr r1 r2 ); Intros; Auto with real. +Replace ``r1*r2*/r2`` with r1. +Replace ``r1*r2*/r1`` with r2; Trivial. +Symmetry; Auto with real. +Symmetry; Auto with real. +Qed. + +Lemma Rlt_Rinv_R1: (x, y:R) ``1 <= x`` -> ``x<y`` ->``/y< /x``. +Intros x y H' H'0. +Cut (Rlt R0 x); [Intros Lt0 | Apply Rlt_le_trans with r2 := R1]; + Auto with real. +Apply Rlt_monotony_contra with z := x; Auto with real. +Rewrite (Rmult_sym x (Rinv x)); Rewrite Rinv_l; Auto with real. +Apply Rlt_monotony_contra with z := y; Auto with real. +Apply Rlt_trans with r2:=x;Auto. +Cut ``y*(x*/y)==x``. +Intro H1;Rewrite H1;Rewrite (Rmult_1r y);Auto. +Rewrite (Rmult_sym x); Rewrite <- Rmult_assoc; Rewrite (Rmult_sym y (Rinv y)); + Rewrite Rinv_l; Auto with real. +Apply imp_not_Req; Right. +Red; Apply Rlt_trans with r2 := x; Auto with real. +Qed. +Hints Resolve Rlt_Rinv_R1 :real. + +(*********************************************************) +(** Greater *) +(*********************************************************) + +(**********) +Lemma Rge_ge_eq:(r1,r2:R)``r1 >= r2`` -> ``r2 >= r1`` -> r1==r2. +Intros; Apply Rle_antisym; Auto with real. +Qed. + +(**********) +Lemma Rlt_not_ge:(r1,r2:R)~(``r1<r2``)->``r1>=r2``. +Intros; Unfold Rge; Elim (total_order r1 r2); Intro. +Absurd ``r1<r2``; Trivial. +Case H0; Auto. +Qed. + +(**********) +Lemma Rnot_lt_le:(r1,r2:R)~(``r1<r2``)->``r2<=r1``. +Intros; Apply Rge_le; Apply Rlt_not_ge; Assumption. +Qed. + +(**********) +Lemma Rgt_not_le:(r1,r2:R)~(``r1>r2``)->``r1<=r2``. +Intros r1 r2 H; Apply Rge_le. +Exact (Rlt_not_ge r2 r1 H). +Qed. + +(**********) +Lemma Rgt_ge:(r1,r2:R)``r1>r2`` -> ``r1 >= r2``. +Red; Auto with real. +Qed. + +V7only [ +(**********) +Lemma Rlt_sym:(r1,r2:R)``r1<r2`` <-> ``r2>r1``. +Split; Unfold Rgt; Auto with real. +Qed. + +(**********) +Lemma Rle_sym1:(r1,r2:R)``r1<=r2``->``r2>=r1``. +Proof Rle_ge. + +Notation "'Rle_sym2' a b" := (Rge_le b a) + (at level 10, a,b at next level). +Notation "'Rle_sym2' a" := [b:R](Rge_le b a) + (at level 10, a at next level). +Notation Rle_sym2 := Rge_le. +(* +(**********) +Lemma Rle_sym2:(r1,r2:R)``r2>=r1`` -> ``r1<=r2``. +Proof [r1,r2](Rge_le r2 r1). +*) + +(**********) +Lemma Rle_sym:(r1,r2:R)``r1<=r2``<->``r2>=r1``. +Split; Auto with real. +Qed. +]. + +(**********) +Lemma Rge_gt_trans:(r1,r2,r3:R)``r1>=r2``->``r2>r3``->``r1>r3``. +Unfold Rgt; Intros; Apply Rlt_le_trans with r2; Auto with real. +Qed. + +(**********) +Lemma Rgt_ge_trans:(r1,r2,r3:R)``r1>r2`` -> ``r2>=r3`` -> ``r1>r3``. +Unfold Rgt; Intros; Apply Rle_lt_trans with r2; Auto with real. +Qed. + +(**********) +Lemma Rgt_trans:(r1,r2,r3:R)``r1>r2`` -> ``r2>r3`` -> ``r1>r3``. +Unfold Rgt; Intros; Apply Rlt_trans with r2; Auto with real. +Qed. + +(**********) +Lemma Rge_trans:(r1,r2,r3:R)``r1>=r2`` -> ``r2>=r3`` -> ``r1>=r3``. +Intros; Apply Rle_ge. +Apply Rle_trans with r2; Auto with real. +Qed. + +(**********) +Lemma Rlt_r_plus_R1:(r:R)``0<=r`` -> ``0<r+1``. +Intros. +Apply Rlt_le_trans with ``1``; Auto with real. +Pattern 1 ``1``; Replace ``1`` with ``0+1``; Auto with real. +Qed. +Hints Resolve Rlt_r_plus_R1: real. + +(**********) +Lemma Rlt_r_r_plus_R1:(r:R)``r<r+1``. +Intros. +Pattern 1 r; Replace r with ``r+0``; Auto with real. +Qed. +Hints Resolve Rlt_r_r_plus_R1: real. + +(**********) +Lemma tech_Rgt_minus:(r1,r2:R)``0<r2``->``r1>r1-r2``. +Red; Unfold Rminus; Intros. +Pattern 2 r1; Replace r1 with ``r1+0``; Auto with real. +Qed. + +(***********) +Lemma Rgt_plus_plus_r:(r,r1,r2:R)``r1>r2``->``r+r1 > r+r2``. +Unfold Rgt; Auto with real. +Qed. +Hints Resolve Rgt_plus_plus_r : real. + +(***********) +Lemma Rgt_r_plus_plus:(r,r1,r2:R)``r+r1 > r+r2`` -> ``r1 > r2``. +Unfold Rgt; Intros; Apply (Rlt_anti_compatibility r r2 r1 H). +Qed. + +(***********) +Lemma Rge_plus_plus_r:(r,r1,r2:R)``r1>=r2`` -> ``r+r1 >= r+r2``. +Intros; Apply Rle_ge; Auto with real. +Qed. +Hints Resolve Rge_plus_plus_r : real. + +(***********) +Lemma Rge_r_plus_plus:(r,r1,r2:R)``r+r1 >= r+r2`` -> ``r1>=r2``. +Intros; Apply Rle_ge; Apply Rle_anti_compatibility with r; Auto with real. +Qed. + +(***********) +Lemma Rmult_ge_compat_r: + (z,x,y:R) ``z>=0`` -> ``x>=y`` -> ``x*z >= y*z``. +Intros z x y; Intros; Apply Rle_ge; Apply Rle_monotony_r; Apply Rge_le; Assumption. +Qed. + +V7only [ +Notation "'Rge_monotony' a b c" := (Rmult_ge_compat_r c a b) + (at level 10, a,b,c at level 9, only parsing). +Notation Rge_monotony := Rmult_ge_compat_r. +]. + +(***********) +Lemma Rgt_minus:(r1,r2:R)``r1>r2`` -> ``r1-r2 > 0``. +Intros; Replace ``0`` with ``r2-r2``; Auto with real. +Unfold Rgt Rminus; Auto with real. +Qed. + +(*********) +Lemma minus_Rgt:(r1,r2:R)``r1-r2 > 0`` -> ``r1>r2``. +Intros; Replace r2 with ``r2+0``; Auto with real. +Intros; Replace r1 with ``r2+(r1-r2)``; Auto with real. +Qed. + +(**********) +Lemma Rge_minus:(r1,r2:R)``r1>=r2`` -> ``r1-r2 >= 0``. +Unfold Rge; Intros; Elim H; Intro. +Left; Apply (Rgt_minus r1 r2 H0). +Right; Apply (eq_Rminus r1 r2 H0). +Qed. + +(*********) +Lemma minus_Rge:(r1,r2:R)``r1-r2 >= 0`` -> ``r1>=r2``. +Intros; Replace r2 with ``r2+0``; Auto with real. +Intros; Replace r1 with ``r2+(r1-r2)``; Auto with real. +Qed. + + +(*********) +Lemma Rmult_gt:(r1,r2:R)``r1>0`` -> ``r2>0`` -> ``r1*r2>0``. +Unfold Rgt;Intros. +Replace ``0`` with ``0*r2``; Auto with real. +Qed. + +(*********) +Lemma Rmult_lt_pos:(x,y:R)``0<x`` -> ``0<y`` -> ``0<x*y``. +Proof Rmult_gt. + +(***********) +Lemma Rplus_eq_R0_l:(a,b:R)``0<=a`` -> ``0<=b`` -> ``a+b==0`` -> ``a==0``. +Intros a b [H|H] H0 H1; Auto with real. +Absurd ``0<a+b``. +Rewrite H1; Auto with real. +Replace ``0`` with ``0+0``; Auto with real. +Qed. + + +Lemma Rplus_eq_R0 + :(a,b:R)``0<=a`` -> ``0<=b`` -> ``a+b==0`` -> ``a==0``/\``b==0``. +Intros a b; Split. +Apply Rplus_eq_R0_l with b; Auto with real. +Apply Rplus_eq_R0_l with a; Auto with real. +Rewrite Rplus_sym; Auto with real. +Qed. + + +(***********) +Lemma Rplus_Rsr_eq_R0_l:(a,b:R)``(Rsqr a)+(Rsqr b)==0``->``a==0``. +Intros a b; Intros; Apply Rsqr_r_R0; Apply Rplus_eq_R0_l with (Rsqr b); Auto with real. +Qed. + +Lemma Rplus_Rsr_eq_R0:(a,b:R)``(Rsqr a)+(Rsqr b)==0``->``a==0``/\``b==0``. +Intros a b; Split. +Apply Rplus_Rsr_eq_R0_l with b; Auto with real. +Apply Rplus_Rsr_eq_R0_l with a; Auto with real. +Rewrite Rplus_sym; Auto with real. +Qed. + + +(**********************************************************) +(** Injection from [N] to [R] *) +(**********************************************************) + +(**********) +Lemma S_INR:(n:nat)(INR (S n))==``(INR n)+1``. +Intro; Case n; Auto with real. +Qed. + +(**********) +Lemma S_O_plus_INR:(n:nat) + (INR (plus (S O) n))==``(INR (S O))+(INR n)``. +Intro; Simpl; Case n; Intros; Auto with real. +Qed. + +(**********) +Lemma plus_INR:(n,m:nat)(INR (plus n m))==``(INR n)+(INR m)``. +Intros n m; Induction n. +Simpl; Auto with real. +Replace (plus (S n) m) with (S (plus n m)); Auto with arith. +Repeat Rewrite S_INR. +Rewrite Hrecn; Ring. +Qed. + +(**********) +Lemma minus_INR:(n,m:nat)(le m n)->(INR (minus n m))==``(INR n)-(INR m)``. +Intros n m le; Pattern m n; Apply le_elim_rel; Auto with real. +Intros; Rewrite <- minus_n_O; Auto with real. +Intros; Repeat Rewrite S_INR; Simpl. +Rewrite H0; Ring. +Qed. + +(*********) +Lemma mult_INR:(n,m:nat)(INR (mult n m))==(Rmult (INR n) (INR m)). +Intros n m; Induction n. +Simpl; Auto with real. +Intros; Repeat Rewrite S_INR; Simpl. +Rewrite plus_INR; Rewrite Hrecn; Ring. +Qed. + +Hints Resolve plus_INR minus_INR mult_INR : real. + +(*********) +Lemma lt_INR_0:(n:nat)(lt O n)->``0 < (INR n)``. +Induction 1; Intros; Auto with real. +Rewrite S_INR; Auto with real. +Qed. +Hints Resolve lt_INR_0: real. + +Lemma lt_INR:(n,m:nat)(lt n m)->``(INR n) < (INR m)``. +Induction 1; Intros; Auto with real. +Rewrite S_INR; Auto with real. +Rewrite S_INR; Apply Rlt_trans with (INR m0); Auto with real. +Qed. +Hints Resolve lt_INR: real. + +Lemma INR_lt_1:(n:nat)(lt (S O) n)->``1 < (INR n)``. +Intros;Replace ``1`` with (INR (S O));Auto with real. +Qed. +Hints Resolve INR_lt_1: real. + +(**********) +Lemma INR_pos : (p:positive)``0<(INR (convert p))``. +Intro; Apply lt_INR_0. +Simpl; Auto with real. +Apply compare_convert_O. +Qed. +Hints Resolve INR_pos : real. + +(**********) +Lemma pos_INR:(n:nat)``0 <= (INR n)``. +Intro n; Case n. +Simpl; Auto with real. +Auto with arith real. +Qed. +Hints Resolve pos_INR: real. + +Lemma INR_lt:(n,m:nat)``(INR n) < (INR m)``->(lt n m). +Double Induction n m;Intros. +Simpl;ElimType False;Apply (Rlt_antirefl R0);Auto. +Auto with arith. +Generalize (pos_INR (S n0));Intro;Cut (INR O)==R0; + [Intro H2;Rewrite H2 in H0;Idtac|Simpl;Trivial]. +Generalize (Rle_lt_trans ``0`` (INR (S n0)) ``0`` H1 H0);Intro; + ElimType False;Apply (Rlt_antirefl R0);Auto. +Do 2 Rewrite S_INR in H1;Cut ``(INR n1) < (INR n0)``. +Intro H2;Generalize (H0 n0 H2);Intro;Auto with arith. +Apply (Rlt_anti_compatibility ``1`` (INR n1) (INR n0)). +Rewrite Rplus_sym;Rewrite (Rplus_sym ``1`` (INR n0));Trivial. +Qed. +Hints Resolve INR_lt: real. + +(*********) +Lemma le_INR:(n,m:nat)(le n m)->``(INR n)<=(INR m)``. +Induction 1; Intros; Auto with real. +Rewrite S_INR. +Apply Rle_trans with (INR m0); Auto with real. +Qed. +Hints Resolve le_INR: real. + +(**********) +Lemma not_INR_O:(n:nat)``(INR n)<>0``->~n=O. +Red; Intros n H H1. +Apply H. +Rewrite H1; Trivial. +Qed. +Hints Immediate not_INR_O : real. + +(**********) +Lemma not_O_INR:(n:nat)~n=O->``(INR n)<>0``. +Intro n; Case n. +Intro; Absurd (0)=(0); Trivial. +Intros; Rewrite S_INR. +Apply Rgt_not_eq; Red; Auto with real. +Qed. +Hints Resolve not_O_INR : real. + +Lemma not_nm_INR:(n,m:nat)~n=m->``(INR n)<>(INR m)``. +Intros n m H; Case (le_or_lt n m); Intros H1. +Case (le_lt_or_eq ? ? H1); Intros H2. +Apply imp_not_Req; Auto with real. +ElimType False;Auto. +Apply sym_not_eqT; Apply imp_not_Req; Auto with real. +Qed. +Hints Resolve not_nm_INR : real. + +Lemma INR_eq: (n,m:nat)(INR n)==(INR m)->n=m. +Intros;Case (le_or_lt n m); Intros H1. +Case (le_lt_or_eq ? ? H1); Intros H2;Auto. +Cut ~n=m. +Intro H3;Generalize (not_nm_INR n m H3);Intro H4; + ElimType False;Auto. +Omega. +Symmetry;Cut ~m=n. +Intro H3;Generalize (not_nm_INR m n H3);Intro H4; + ElimType False;Auto. +Omega. +Qed. +Hints Resolve INR_eq : real. + +Lemma INR_le: (n, m : nat) (Rle (INR n) (INR m)) -> (le n m). +Intros;Elim H;Intro. +Generalize (INR_lt n m H0);Intro;Auto with arith. +Generalize (INR_eq n m H0);Intro;Rewrite H1;Auto. +Qed. +Hints Resolve INR_le : real. + +Lemma not_1_INR:(n:nat)~n=(S O)->``(INR n)<>1``. +Replace ``1`` with (INR (S O)); Auto with real. +Qed. +Hints Resolve not_1_INR : real. + +(**********************************************************) +(** Injection from [Z] to [R] *) +(**********************************************************) + +V7only [ +(**********) +Definition Z_of_nat := inject_nat. +Notation INZ:=Z_of_nat. +]. + +(**********) +Lemma IZN:(z:Z)(`0<=z`)->(Ex [m:nat] z=(INZ m)). +Intros z; Unfold INZ; Apply inject_nat_complete; Assumption. +Qed. + +(**********) +Lemma INR_IZR_INZ:(n:nat)(INR n)==(IZR (INZ n)). +Induction n; Auto with real. +Intros; Simpl; Rewrite bij1; Auto with real. +Qed. + +Lemma plus_IZR_NEG_POS : + (p,q:positive)(IZR `(POS p)+(NEG q)`)==``(IZR (POS p))+(IZR (NEG q))``. +Intros. +Case (lt_eq_lt_dec (convert p) (convert q)). +Intros [H | H]; Simpl. +Rewrite convert_compare_INFERIEUR; Simpl; Trivial. +Rewrite (true_sub_convert q p). +Rewrite minus_INR; Auto with arith; Ring. +Apply ZC2; Apply convert_compare_INFERIEUR; Trivial. +Rewrite (convert_intro p q); Trivial. +Rewrite convert_compare_EGAL; Simpl; Auto with real. +Intro H; Simpl. +Rewrite convert_compare_SUPERIEUR; Simpl; Auto with arith. +Rewrite (true_sub_convert p q). +Rewrite minus_INR; Auto with arith; Ring. +Apply ZC2; Apply convert_compare_INFERIEUR; Trivial. +Qed. + +(**********) +Lemma plus_IZR:(z,t:Z)(IZR `z+t`)==``(IZR z)+(IZR t)``. +Intro z; NewDestruct z; Intro t; NewDestruct t; Intros; Auto with real. +Simpl; Intros; Rewrite convert_add; Auto with real. +Apply plus_IZR_NEG_POS. +Rewrite Zplus_sym; Rewrite Rplus_sym; Apply plus_IZR_NEG_POS. +Simpl; Intros; Rewrite convert_add; Rewrite plus_INR; Auto with real. +Qed. + +(**********) +Lemma mult_IZR:(z,t:Z)(IZR `z*t`)==``(IZR z)*(IZR t)``. +Intros z t; Case z; Case t; Simpl; Auto with real. +Intros t1 z1; Rewrite times_convert; Auto with real. +Intros t1 z1; Rewrite times_convert; Auto with real. +Rewrite Rmult_sym. +Rewrite Ropp_mul1; Auto with real. +Apply eq_Ropp; Rewrite mult_sym; Auto with real. +Intros t1 z1; Rewrite times_convert; Auto with real. +Rewrite Ropp_mul1; Auto with real. +Intros t1 z1; Rewrite times_convert; Auto with real. +Rewrite Ropp_mul2; Auto with real. +Qed. + +(**********) +Lemma Ropp_Ropp_IZR:(z:Z)(IZR (`-z`))==``-(IZR z)``. +Intro z; Case z; Simpl; Auto with real. +Qed. + +(**********) +Lemma Z_R_minus:(z1,z2:Z)``(IZR z1)-(IZR z2)``==(IZR `z1-z2`). +Intros z1 z2; Unfold Rminus; Unfold Zminus. +Rewrite <-(Ropp_Ropp_IZR z2); Symmetry; Apply plus_IZR. +Qed. + +(**********) +Lemma lt_O_IZR:(z:Z)``0 < (IZR z)``->`0<z`. +Intro z; Case z; Simpl; Intros. +Absurd ``0<0``; Auto with real. +Unfold Zlt; Simpl; Trivial. +Case Rlt_le_not with 1:=H. +Replace ``0`` with ``-0``; Auto with real. +Qed. + +(**********) +Lemma lt_IZR:(z1,z2:Z)``(IZR z1)<(IZR z2)``->`z1<z2`. +Intros z1 z2 H; Apply Zlt_O_minus_lt. +Apply lt_O_IZR. +Rewrite <- Z_R_minus. +Exact (Rgt_minus (IZR z2) (IZR z1) H). +Qed. + +(**********) +Lemma eq_IZR_R0:(z:Z)``(IZR z)==0``->`z=0`. +Intro z; NewDestruct z; Simpl; Intros; Auto with zarith. +Case (Rlt_not_eq ``0`` (INR (convert p))); Auto with real. +Case (Rlt_not_eq ``-(INR (convert p))`` ``0`` ); Auto with real. +Apply Rgt_RoppO. Unfold Rgt; Apply INR_pos. +Qed. + +(**********) +Lemma eq_IZR:(z1,z2:Z)(IZR z1)==(IZR z2)->z1=z2. +Intros z1 z2 H;Generalize (eq_Rminus (IZR z1) (IZR z2) H); + Rewrite (Z_R_minus z1 z2);Intro;Generalize (eq_IZR_R0 `z1-z2` H0); + Intro;Omega. +Qed. + +(**********) +Lemma not_O_IZR:(z:Z)`z<>0`->``(IZR z)<>0``. +Intros z H; Red; Intros H0; Case H. +Apply eq_IZR; Auto. +Qed. + +(*********) +Lemma le_O_IZR:(z:Z)``0<= (IZR z)``->`0<=z`. +Unfold Rle; Intros z [H|H]. +Red;Intro;Apply (Zlt_le_weak `0` z (lt_O_IZR z H)); Assumption. +Rewrite (eq_IZR_R0 z); Auto with zarith real. +Qed. + +(**********) +Lemma le_IZR:(z1,z2:Z)``(IZR z1)<=(IZR z2)``->`z1<=z2`. +Unfold Rle; Intros z1 z2 [H|H]. +Apply (Zlt_le_weak z1 z2); Auto with real. +Apply lt_IZR; Trivial. +Rewrite (eq_IZR z1 z2); Auto with zarith real. +Qed. + +(**********) +Lemma le_IZR_R1:(z:Z)``(IZR z)<=1``-> `z<=1`. +Pattern 1 ``1``; Replace ``1`` with (IZR `1`); Intros; Auto. +Apply le_IZR; Trivial. +Qed. + +(**********) +Lemma IZR_ge: (m,n:Z) `m>= n` -> ``(IZR m)>=(IZR n)``. +Intros m n H; Apply Rlt_not_ge;Red;Intro. +Generalize (lt_IZR m n H0); Intro; Omega. +Qed. + +Lemma IZR_le: (m,n:Z) `m<= n` -> ``(IZR m)<=(IZR n)``. +Intros m n H;Apply Rgt_not_le;Red;Intro. +Unfold Rgt in H0;Generalize (lt_IZR n m H0); Intro; Omega. +Qed. + +Lemma IZR_lt: (m,n:Z) `m< n` -> ``(IZR m)<(IZR n)``. +Intros m n H;Cut `m<=n`. +Intro H0;Elim (IZR_le m n H0);Intro;Auto. +Generalize (eq_IZR m n H1);Intro;ElimType False;Omega. +Omega. +Qed. + +Lemma one_IZR_lt1 : (z:Z)``-1<(IZR z)<1``->`z=0`. +Intros z (H1,H2). +Apply Zle_antisym. +Apply Zlt_n_Sm_le; Apply lt_IZR; Trivial. +Replace `0` with (Zs `-1`); Trivial. +Apply Zlt_le_S; Apply lt_IZR; Trivial. +Qed. + +Lemma one_IZR_r_R1 + : (r:R)(z,x:Z)``r<(IZR z)<=r+1``->``r<(IZR x)<=r+1``->z=x. +Intros r z x (H1,H2) (H3,H4). +Cut `z-x=0`; Auto with zarith. +Apply one_IZR_lt1. +Rewrite <- Z_R_minus; Split. +Replace ``-1`` with ``r-(r+1)``. +Unfold Rminus; Apply Rplus_lt_le_lt; Auto with real. +Ring. +Replace ``1`` with ``(r+1)-r``. +Unfold Rminus; Apply Rplus_le_lt_lt; Auto with real. +Ring. +Qed. + + +(**********) +Lemma single_z_r_R1: + (r:R)(z,x:Z)``r<(IZR z)``->``(IZR z)<=r+1``->``r<(IZR x)``-> + ``(IZR x)<=r+1``->z=x. +Intros; Apply one_IZR_r_R1 with r; Auto. +Qed. + +(**********) +Lemma tech_single_z_r_R1 + :(r:R)(z:Z)``r<(IZR z)``->``(IZR z)<=r+1`` + -> (Ex [s:Z] (~s=z/\``r<(IZR s)``/\``(IZR s)<=r+1``))->False. +Intros r z H1 H2 (s, (H3,(H4,H5))). +Apply H3; Apply single_z_r_R1 with r; Trivial. +Qed. + +(*****************************************************************) +(** Definitions of new types *) +(*****************************************************************) + +Record nonnegreal : Type := mknonnegreal { +nonneg :> R; +cond_nonneg : ``0<=nonneg`` }. + +Record posreal : Type := mkposreal { +pos :> R; +cond_pos : ``0<pos`` }. + +Record nonposreal : Type := mknonposreal { +nonpos :> R; +cond_nonpos : ``nonpos<=0`` }. + +Record negreal : Type := mknegreal { +neg :> R; +cond_neg : ``neg<0`` }. + +Record nonzeroreal : Type := mknonzeroreal { +nonzero :> R; +cond_nonzero : ~``nonzero==0`` }. + +(**********) +Lemma prod_neq_R0 : (x,y:R) ~``x==0``->~``y==0``->~``x*y==0``. +Intros x y; Intros; Red; Intro; Generalize (without_div_Od x y H1); Intro; Elim H2; Intro; [Rewrite H3 in H; Elim H | Rewrite H3 in H0; Elim H0]; Reflexivity. +Qed. + +(*********) +Lemma Rmult_le_pos : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x*y``. +Intros x y H H0; Rewrite <- (Rmult_Ol x); Rewrite <- (Rmult_sym x); Apply (Rle_monotony x R0 y H H0). +Qed. + +Lemma double : (x:R) ``2*x==x+x``. +Intro; Ring. +Qed. + +Lemma double_var : (x:R) ``x == x/2 + x/2``. +Intro; Rewrite <- double; Unfold Rdiv; Rewrite <- Rmult_assoc; Symmetry; Apply Rinv_r_simpl_m. +Replace ``2`` with (INR (2)); [Apply not_O_INR; Discriminate | Unfold INR; Ring]. +Qed. + +(**********************************************************) +(** Other rules about < and <= *) +(**********************************************************) + +Lemma gt0_plus_gt0_is_gt0 : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x+y``. +Intros x y; Intros; Apply Rlt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption]. +Qed. + +Lemma ge0_plus_gt0_is_gt0 : (x,y:R) ``0<=x`` -> ``0<y`` -> ``0<x+y``. +Intros x y; Intros; Apply Rle_lt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption]. +Qed. + +Lemma gt0_plus_ge0_is_gt0 : (x,y:R) ``0<x`` -> ``0<=y`` -> ``0<x+y``. +Intros x y; Intros; Rewrite <- Rplus_sym; Apply ge0_plus_gt0_is_gt0; Assumption. +Qed. + +Lemma ge0_plus_ge0_is_ge0 : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x+y``. +Intros x y; Intros; Apply Rle_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption]. +Qed. + +Lemma plus_le_is_le : (x,y,z:R) ``0<=y`` -> ``x+y<=z`` -> ``x<=z``. +Intros x y z; Intros; Apply Rle_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption]. +Qed. + +Lemma plus_lt_is_lt : (x,y,z:R) ``0<=y`` -> ``x+y<z`` -> ``x<z``. +Intros x y z; Intros; Apply Rle_lt_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption]. +Qed. + +Lemma Rmult_lt2 : (r1,r2,r3,r4:R) ``0<=r1`` -> ``0<=r3`` -> ``r1<r2`` -> ``r3<r4`` -> ``r1*r3<r2*r4``. +Intros; Apply Rle_lt_trans with ``r2*r3``; [Apply Rle_monotony_r; [Assumption | Left; Assumption] | Apply Rlt_monotony; [Apply Rle_lt_trans with r1; Assumption | Assumption]]. +Qed. + +Lemma le_epsilon : (x,y:R) ((eps : R) ``0<eps``->``x<=y+eps``) -> ``x<=y``. +Intros x y; Intros; Elim (total_order x y); Intro. +Left; Assumption. +Elim H0; Intro. +Right; Assumption. +Clear H0; Generalize (Rgt_minus x y H1); Intro H2; Change ``0<x-y`` in H2. +Cut ``0<2``. +Intro. +Generalize (Rmult_lt_pos ``x-y`` ``/2`` H2 (Rlt_Rinv ``2`` H0)); Intro H3; Generalize (H ``(x-y)*/2`` H3); Replace ``y+(x-y)*/2`` with ``(y+x)*/2``. +Intro H4; Generalize (Rle_monotony ``2`` x ``(y+x)*/2`` (Rlt_le ``0`` ``2`` H0) H4); Rewrite <- (Rmult_sym ``((y+x)*/2)``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Replace ``2*x`` with ``x+x``. +Rewrite (Rplus_sym y); Intro H5; Apply Rle_anti_compatibility with x; Assumption. +Ring. +Replace ``2`` with (INR (S (S O))); [Apply not_O_INR; Discriminate | Ring]. +Pattern 2 y; Replace y with ``y/2+y/2``. +Unfold Rminus Rdiv. +Repeat Rewrite Rmult_Rplus_distrl. +Ring. +Cut (z:R) ``2*z == z + z``. +Intro. +Rewrite <- (H4 ``y/2``). +Unfold Rdiv. +Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. +Replace ``2`` with (INR (2)). +Apply not_O_INR. +Discriminate. +Unfold INR; Reflexivity. +Intro; Ring. +Cut ~(O=(2)); [Intro H0; Generalize (lt_INR_0 (2) (neq_O_lt (2) H0)); Unfold INR; Intro; Assumption | Discriminate]. +Qed. + +(**********) +Lemma complet_weak : (E:R->Prop) (bound E) -> (ExT [x:R] (E x)) -> (ExT [m:R] (is_lub E m)). +Intros; Elim (complet E H H0); Intros; Split with x; Assumption. +Qed. diff --git a/theories7/Reals/RList.v b/theories7/Reals/RList.v new file mode 100644 index 00000000..b89296fb --- /dev/null +++ b/theories7/Reals/RList.v @@ -0,0 +1,427 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: RList.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Inductive Rlist : Type := +| nil : Rlist +| cons : R -> Rlist -> Rlist. + +Fixpoint In [x:R;l:Rlist] : Prop := +Cases l of +| nil => False +| (cons a l') => ``x==a``\/(In x l') end. + +Fixpoint Rlength [l:Rlist] : nat := +Cases l of +| nil => O +| (cons a l') => (S (Rlength l')) end. + +Fixpoint MaxRlist [l:Rlist] : R := + Cases l of + | nil => R0 + | (cons a l1) => + Cases l1 of + | nil => a + | (cons a' l2) => (Rmax a (MaxRlist l1)) + end +end. + +Fixpoint MinRlist [l:Rlist] : R := +Cases l of + | nil => R1 + | (cons a l1) => + Cases l1 of + | nil => a + | (cons a' l2) => (Rmin a (MinRlist l1)) + end +end. + +Lemma MaxRlist_P1 : (l:Rlist;x:R) (In x l)->``x<=(MaxRlist l)``. +Intros; Induction l. +Simpl in H; Elim H. +Induction l. +Simpl in H; Elim H; Intro. +Simpl; Right; Assumption. +Elim H0. +Replace (MaxRlist (cons r (cons r0 l))) with (Rmax r (MaxRlist (cons r0 l))). +Simpl in H; Decompose [or] H. +Rewrite H0; Apply RmaxLess1. +Unfold Rmax; Case (total_order_Rle r (MaxRlist (cons r0 l))); Intro. +Apply Hrecl; Simpl; Tauto. +Apply Rle_trans with (MaxRlist (cons r0 l)); [Apply Hrecl; Simpl; Tauto | Left; Auto with real]. +Unfold Rmax; Case (total_order_Rle r (MaxRlist (cons r0 l))); Intro. +Apply Hrecl; Simpl; Tauto. +Apply Rle_trans with (MaxRlist (cons r0 l)); [Apply Hrecl; Simpl; Tauto | Left; Auto with real]. +Reflexivity. +Qed. + +Fixpoint AbsList [l:Rlist] : R->Rlist := +[x:R] Cases l of +| nil => nil +| (cons a l') => (cons ``(Rabsolu (a-x))/2`` (AbsList l' x)) +end. + +Lemma MinRlist_P1 : (l:Rlist;x:R) (In x l)->``(MinRlist l)<=x``. +Intros; Induction l. +Simpl in H; Elim H. +Induction l. +Simpl in H; Elim H; Intro. +Simpl; Right; Symmetry; Assumption. +Elim H0. +Replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). +Simpl in H; Decompose [or] H. +Rewrite H0; Apply Rmin_l. +Unfold Rmin; Case (total_order_Rle r (MinRlist (cons r0 l))); Intro. +Apply Rle_trans with (MinRlist (cons r0 l)). +Assumption. +Apply Hrecl; Simpl; Tauto. +Apply Hrecl; Simpl; Tauto. +Apply Rle_trans with (MinRlist (cons r0 l)). +Apply Rmin_r. +Apply Hrecl; Simpl; Tauto. +Reflexivity. +Qed. + +Lemma AbsList_P1 : (l:Rlist;x,y:R) (In y l) -> (In ``(Rabsolu (y-x))/2`` (AbsList l x)). +Intros; Induction l. +Elim H. +Simpl; Simpl in H; Elim H; Intro. +Left; Rewrite H0; Reflexivity. +Right; Apply Hrecl; Assumption. +Qed. + +Lemma MinRlist_P2 : (l:Rlist) ((y:R)(In y l)->``0<y``)->``0<(MinRlist l)``. +Intros; Induction l. +Apply Rlt_R0_R1. +Induction l. +Simpl; Apply H; Simpl; Tauto. +Replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). +Unfold Rmin; Case (total_order_Rle r (MinRlist (cons r0 l))); Intro. +Apply H; Simpl; Tauto. +Apply Hrecl; Intros; Apply H; Simpl; Simpl in H0; Tauto. +Reflexivity. +Qed. + +Lemma AbsList_P2 : (l:Rlist;x,y:R) (In y (AbsList l x)) -> (EXT z : R | (In z l)/\``y==(Rabsolu (z-x))/2``). +Intros; Induction l. +Elim H. +Elim H; Intro. +Exists r; Split. +Simpl; Tauto. +Assumption. +Assert H1 := (Hrecl H0); Elim H1; Intros; Elim H2; Clear H2; Intros; Exists x0; Simpl; Simpl in H2; Tauto. +Qed. + +Lemma MaxRlist_P2 : (l:Rlist) (EXT y:R | (In y l)) -> (In (MaxRlist l) l). +Intros; Induction l. +Simpl in H; Elim H; Trivial. +Induction l. +Simpl; Left; Reflexivity. +Change (In (Rmax r (MaxRlist (cons r0 l))) (cons r (cons r0 l))); Unfold Rmax; Case (total_order_Rle r (MaxRlist (cons r0 l))); Intro. +Right; Apply Hrecl; Exists r0; Left; Reflexivity. +Left; Reflexivity. +Qed. + +Fixpoint pos_Rl [l:Rlist] : nat->R := +[i:nat] Cases l of +| nil => R0 +| (cons a l') => + Cases i of + | O => a + | (S i') => (pos_Rl l' i') + end +end. + +Lemma pos_Rl_P1 : (l:Rlist;a:R) (lt O (Rlength l)) -> (pos_Rl (cons a l) (Rlength l))==(pos_Rl l (pred (Rlength l))). +Intros; Induction l; [Elim (lt_n_O ? H) | Simpl; Case (Rlength l); [Reflexivity | Intro; Reflexivity]]. +Qed. + +Lemma pos_Rl_P2 : (l:Rlist;x:R) (In x l)<->(EX i:nat | (lt i (Rlength l))/\x==(pos_Rl l i)). +Intros; Induction l. +Split; Intro; [Elim H | Elim H; Intros; Elim H0; Intros; Elim (lt_n_O ? H1)]. +Split; Intro. +Elim H; Intro. +Exists O; Split; [Simpl; Apply lt_O_Sn | Simpl; Apply H0]. +Elim Hrecl; Intros; Assert H3 := (H1 H0); Elim H3; Intros; Elim H4; Intros; Exists (S x0); Split; [Simpl; Apply lt_n_S; Assumption | Simpl; Assumption]. +Elim H; Intros; Elim H0; Intros; Elim (zerop x0); Intro. +Rewrite a in H2; Simpl in H2; Left; Assumption. +Right; Elim Hrecl; Intros; Apply H4; Assert H5 : (S (pred x0))=x0. +Symmetry; Apply S_pred with O; Assumption. +Exists (pred x0); Split; [Simpl in H1; Apply lt_S_n; Rewrite H5; Assumption | Rewrite <- H5 in H2; Simpl in H2; Assumption]. +Qed. + +Lemma Rlist_P1 : (l:Rlist;P:R->R->Prop) ((x:R)(In x l)->(EXT y:R | (P x y))) -> (EXT l':Rlist | (Rlength l)=(Rlength l')/\(i:nat) (lt i (Rlength l))->(P (pos_Rl l i) (pos_Rl l' i))). +Intros; Induction l. +Exists nil; Intros; Split; [Reflexivity | Intros; Simpl in H0; Elim (lt_n_O ? H0)]. +Assert H0 : (In r (cons r l)). +Simpl; Left; Reflexivity. +Assert H1 := (H ? H0); Assert H2 : (x:R)(In x l)->(EXT y:R | (P x y)). +Intros; Apply H; Simpl; Right; Assumption. +Assert H3 := (Hrecl H2); Elim H1; Intros; Elim H3; Intros; Exists (cons x x0); Intros; Elim H5; Clear H5; Intros; Split. +Simpl; Rewrite H5; Reflexivity. +Intros; Elim (zerop i); Intro. +Rewrite a; Simpl; Assumption. +Assert H8 : i=(S (pred i)). +Apply S_pred with O; Assumption. +Rewrite H8; Simpl; Apply H6; Simpl in H7; Apply lt_S_n; Rewrite <- H8; Assumption. +Qed. + +Definition ordered_Rlist [l:Rlist] : Prop := (i:nat) (lt i (pred (Rlength l))) -> (Rle (pos_Rl l i) (pos_Rl l (S i))). + +Fixpoint insert [l:Rlist] : R->Rlist := +[x:R] Cases l of +| nil => (cons x nil) +| (cons a l') => + Cases (total_order_Rle a x) of + | (leftT _) => (cons a (insert l' x)) + | (rightT _) => (cons x l) + end +end. + +Fixpoint cons_Rlist [l:Rlist] : Rlist->Rlist := +[k:Rlist] Cases l of +| nil => k +| (cons a l') => (cons a (cons_Rlist l' k)) end. + +Fixpoint cons_ORlist [k:Rlist] : Rlist->Rlist := +[l:Rlist] Cases k of +| nil => l +| (cons a k') => (cons_ORlist k' (insert l a)) +end. + +Fixpoint app_Rlist [l:Rlist] : (R->R)->Rlist := +[f:R->R] Cases l of +| nil => nil +| (cons a l') => (cons (f a) (app_Rlist l' f)) +end. + +Fixpoint mid_Rlist [l:Rlist] : R->Rlist := +[x:R] Cases l of +| nil => nil +| (cons a l') => (cons ``(x+a)/2`` (mid_Rlist l' a)) +end. + +Definition Rtail [l:Rlist] : Rlist := +Cases l of +| nil => nil +| (cons a l') => l' +end. + +Definition FF [l:Rlist;f:R->R] : Rlist := +Cases l of +| nil => nil +| (cons a l') => (app_Rlist (mid_Rlist l' a) f) +end. + +Lemma RList_P0 : (l:Rlist;a:R) ``(pos_Rl (insert l a) O) == a`` \/ ``(pos_Rl (insert l a) O) == (pos_Rl l O)``. +Intros; Induction l; [Left; Reflexivity | Simpl; Case (total_order_Rle r a); Intro; [Right; Reflexivity | Left; Reflexivity]]. +Qed. + +Lemma RList_P1 : (l:Rlist;a:R) (ordered_Rlist l) -> (ordered_Rlist (insert l a)). +Intros; Induction l. +Simpl; Unfold ordered_Rlist; Intros; Simpl in H0; Elim (lt_n_O ? H0). +Simpl; Case (total_order_Rle r a); Intro. +Assert H1 : (ordered_Rlist l). +Unfold ordered_Rlist; Unfold ordered_Rlist in H; Intros; Assert H1 : (lt (S i) (pred (Rlength (cons r l)))); [Simpl; Replace (Rlength l) with (S (pred (Rlength l))); [Apply lt_n_S; Assumption | Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H1 in H0; Simpl in H0; Elim (lt_n_O ? H0)] | Apply (H ? H1)]. +Assert H2 := (Hrecl H1); Unfold ordered_Rlist; Intros; Induction i. +Simpl; Assert H3 := (RList_P0 l a); Elim H3; Intro. +Rewrite H4; Assumption. +Induction l; [Simpl; Assumption | Rewrite H4; Apply (H O); Simpl; Apply lt_O_Sn]. +Simpl; Apply H2; Simpl in H0; Apply lt_S_n; Replace (S (pred (Rlength (insert l a)))) with (Rlength (insert l a)); [Assumption | Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H3 in H0; Elim (lt_n_O ? H0)]. +Unfold ordered_Rlist; Intros; Induction i; [Simpl; Auto with real | Change ``(pos_Rl (cons r l) i)<=(pos_Rl (cons r l) (S i))``; Apply H; Simpl in H0; Simpl; Apply (lt_S_n ? ? H0)]. +Qed. + +Lemma RList_P2 : (l1,l2:Rlist) (ordered_Rlist l2) ->(ordered_Rlist (cons_ORlist l1 l2)). +Induction l1; [Intros; Simpl; Apply H | Intros; Simpl; Apply H; Apply RList_P1; Assumption]. +Qed. + +Lemma RList_P3 : (l:Rlist;x:R) (In x l) <-> (EX i:nat | x==(pos_Rl l i)/\(lt i (Rlength l))). +Intros; Split; Intro; Induction l. +Elim H. +Elim H; Intro; [Exists O; Split; [Apply H0 | Simpl; Apply lt_O_Sn] | Elim (Hrecl H0); Intros; Elim H1; Clear H1; Intros; Exists (S x0); Split; [Apply H1 | Simpl; Apply lt_n_S; Assumption]]. +Elim H; Intros; Elim H0; Intros; Elim (lt_n_O ? H2). +Simpl; Elim H; Intros; Elim H0; Clear H0; Intros; Induction x0; [Left; Apply H0 | Right; Apply Hrecl; Exists x0; Split; [Apply H0 | Simpl in H1; Apply lt_S_n; Assumption]]. +Qed. + +Lemma RList_P4 : (l1:Rlist;a:R) (ordered_Rlist (cons a l1)) -> (ordered_Rlist l1). +Intros; Unfold ordered_Rlist; Intros; Apply (H (S i)); Simpl; Replace (Rlength l1) with (S (pred (Rlength l1))); [Apply lt_n_S; Assumption | Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H1 in H0; Elim (lt_n_O ? H0)]. +Qed. + +Lemma RList_P5 : (l:Rlist;x:R) (ordered_Rlist l) -> (In x l) -> ``(pos_Rl l O)<=x``. +Intros; Induction l; [Elim H0 | Simpl; Elim H0; Intro; [Rewrite H1; Right; Reflexivity | Apply Rle_trans with (pos_Rl l O); [Apply (H O); Simpl; Induction l; [Elim H1 | Simpl; Apply lt_O_Sn] | Apply Hrecl; [EApply RList_P4; Apply H | Assumption]]]]. +Qed. + +Lemma RList_P6 : (l:Rlist) (ordered_Rlist l)<->((i,j:nat)(le i j)->(lt j (Rlength l))->``(pos_Rl l i)<=(pos_Rl l j)``). +Induction l; Split; Intro. +Intros; Right; Reflexivity. +Unfold ordered_Rlist; Intros; Simpl in H0; Elim (lt_n_O ? H0). +Intros; Induction i; [Induction j; [Right; Reflexivity | Simpl; Apply Rle_trans with (pos_Rl r0 O); [Apply (H0 O); Simpl; Simpl in H2; Apply neq_O_lt; Red; Intro; Rewrite <- H3 in H2; Assert H4 := (lt_S_n ? ? H2); Elim (lt_n_O ? H4) | Elim H; Intros; Apply H3; [Apply RList_P4 with r; Assumption | Apply le_O_n | Simpl in H2; Apply lt_S_n; Assumption]]] | Induction j; [Elim (le_Sn_O ? H1) | Simpl; Elim H; Intros; Apply H3; [Apply RList_P4 with r; Assumption | Apply le_S_n; Assumption | Simpl in H2; Apply lt_S_n; Assumption]]]. +Unfold ordered_Rlist; Intros; Apply H0; [Apply le_n_Sn | Simpl; Simpl in H1; Apply lt_n_S; Assumption]. +Qed. + +Lemma RList_P7 : (l:Rlist;x:R) (ordered_Rlist l) -> (In x l) -> ``x<=(pos_Rl l (pred (Rlength l)))``. +Intros; Assert H1 := (RList_P6 l); Elim H1; Intros H2 _; Assert H3 := (H2 H); Clear H1 H2; Assert H1 := (RList_P3 l x); Elim H1; Clear H1; Intros; Assert H4 := (H1 H0); Elim H4; Clear H4; Intros; Elim H4; Clear H4; Intros; Rewrite H4; Assert H6 : (Rlength l)=(S (pred (Rlength l))). +Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H6 in H5; Elim (lt_n_O ? H5). +Apply H3; [Rewrite H6 in H5; Apply lt_n_Sm_le; Assumption | Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H7 in H5; Elim (lt_n_O ? H5)]. +Qed. + +Lemma RList_P8 : (l:Rlist;a,x:R) (In x (insert l a)) <-> x==a\/(In x l). +Induction l. +Intros; Split; Intro; Simpl in H; Apply H. +Intros; Split; Intro; [Simpl in H0; Generalize H0; Case (total_order_Rle r a); Intros; [Simpl in H1; Elim H1; Intro; [Right; Left; Assumption |Elim (H a x); Intros; Elim (H3 H2); Intro; [Left; Assumption | Right; Right; Assumption]] | Simpl in H1; Decompose [or] H1; [Left; Assumption | Right; Left; Assumption | Right; Right; Assumption]] | Simpl; Case (total_order_Rle r a); Intro; [Simpl in H0; Decompose [or] H0; [Right; Elim (H a x); Intros; Apply H3; Left | Left | Right; Elim (H a x); Intros; Apply H3; Right] | Simpl in H0; Decompose [or] H0; [Left | Right; Left | Right; Right]]; Assumption]. +Qed. + +Lemma RList_P9 : (l1,l2:Rlist;x:R) (In x (cons_ORlist l1 l2)) <-> (In x l1)\/(In x l2). +Induction l1. +Intros; Split; Intro; [Simpl in H; Right; Assumption | Simpl; Elim H; Intro; [Elim H0 | Assumption]]. +Intros; Split. +Simpl; Intros; Elim (H (insert l2 r) x); Intros; Assert H3 := (H1 H0); Elim H3; Intro; [Left; Right; Assumption | Elim (RList_P8 l2 r x); Intros H5 _; Assert H6 := (H5 H4); Elim H6; Intro; [Left; Left; Assumption | Right; Assumption]]. +Intro; Simpl; Elim (H (insert l2 r) x); Intros _ H1; Apply H1; Elim H0; Intro; [Elim H2; Intro; [Right; Elim (RList_P8 l2 r x); Intros _ H4; Apply H4; Left; Assumption | Left; Assumption] | Right; Elim (RList_P8 l2 r x); Intros _ H3; Apply H3; Right; Assumption]. +Qed. + +Lemma RList_P10 : (l:Rlist;a:R) (Rlength (insert l a))==(S (Rlength l)). +Intros; Induction l; [Reflexivity | Simpl; Case (total_order_Rle r a); Intro; [Simpl; Rewrite Hrecl; Reflexivity | Reflexivity]]. +Qed. + +Lemma RList_P11 : (l1,l2:Rlist) (Rlength (cons_ORlist l1 l2))=(plus (Rlength l1) (Rlength l2)). +Induction l1; [Intro; Reflexivity | Intros; Simpl; Rewrite (H (insert l2 r)); Rewrite RList_P10; Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring]. +Qed. + +Lemma RList_P12 : (l:Rlist;i:nat;f:R->R) (lt i (Rlength l)) -> (pos_Rl (app_Rlist l f) i)==(f (pos_Rl l i)). +Induction l; [Intros; Elim (lt_n_O ? H) | Intros; Induction i; [Reflexivity | Simpl; Apply H; Apply lt_S_n; Apply H0]]. +Qed. + +Lemma RList_P13 : (l:Rlist;i:nat;a:R) (lt i (pred (Rlength l))) -> ``(pos_Rl (mid_Rlist l a) (S i)) == ((pos_Rl l i)+(pos_Rl l (S i)))/2``. +Induction l. +Intros; Simpl in H; Elim (lt_n_O ? H). +Induction r0. +Intros; Simpl in H0; Elim (lt_n_O ? H0). +Intros; Simpl in H1; Induction i. +Reflexivity. +Change ``(pos_Rl (mid_Rlist (cons r1 r2) r) (S i)) == ((pos_Rl (cons r1 r2) i)+(pos_Rl (cons r1 r2) (S i)))/2``; Apply H0; Simpl; Apply lt_S_n; Assumption. +Qed. + +Lemma RList_P14 : (l:Rlist;a:R) (Rlength (mid_Rlist l a))=(Rlength l). +Induction l; Intros; [Reflexivity | Simpl; Rewrite (H r); Reflexivity]. +Qed. + +Lemma RList_P15 : (l1,l2:Rlist) (ordered_Rlist l1) -> (ordered_Rlist l2) -> (pos_Rl l1 O)==(pos_Rl l2 O) -> (pos_Rl (cons_ORlist l1 l2) O)==(pos_Rl l1 O). +Intros; Apply Rle_antisym. +Induction l1; [Simpl; Simpl in H1; Right; Symmetry; Assumption | Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (0))); Intros; Assert H4 : (In (pos_Rl (cons r l1) (0)) (cons r l1))\/(In (pos_Rl (cons r l1) (0)) l2); [Left; Left; Reflexivity | Assert H5 := (H3 H4); Apply RList_P5; [Apply RList_P2; Assumption | Assumption]]]. +Induction l1; [Simpl; Simpl in H1; Right; Assumption | Assert H2 : (In (pos_Rl (cons_ORlist (cons r l1) l2) (0)) (cons_ORlist (cons r l1) l2)); [Elim (RList_P3 (cons_ORlist (cons r l1) l2) (pos_Rl (cons_ORlist (cons r l1) l2) (0))); Intros; Apply H3; Exists O; Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_O_Sn] | Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) (0))); Intros; Assert H5 := (H3 H2); Elim H5; Intro; [Apply RList_P5; Assumption | Rewrite H1; Apply RList_P5; Assumption]]]. +Qed. + +Lemma RList_P16 : (l1,l2:Rlist) (ordered_Rlist l1) -> (ordered_Rlist l2) -> (pos_Rl l1 (pred (Rlength l1)))==(pos_Rl l2 (pred (Rlength l2))) -> (pos_Rl (cons_ORlist l1 l2) (pred (Rlength (cons_ORlist l1 l2))))==(pos_Rl l1 (pred (Rlength l1))). +Intros; Apply Rle_antisym. +Induction l1. +Simpl; Simpl in H1; Right; Symmetry; Assumption. +Assert H2 : (In (pos_Rl (cons_ORlist (cons r l1) l2) (pred (Rlength (cons_ORlist (cons r l1) l2)))) (cons_ORlist (cons r l1) l2)); [Elim (RList_P3 (cons_ORlist (cons r l1) l2) (pos_Rl (cons_ORlist (cons r l1) l2) (pred (Rlength (cons_ORlist (cons r l1) l2))))); Intros; Apply H3; Exists (pred (Rlength (cons_ORlist (cons r l1) l2))); Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_n_Sn] | Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) (pred (Rlength (cons_ORlist (cons r l1) l2))))); Intros; Assert H5 := (H3 H2); Elim H5; Intro; [Apply RList_P7; Assumption | Rewrite H1; Apply RList_P7; Assumption]]. +Induction l1. +Simpl; Simpl in H1; Right; Assumption. +Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); Intros; Assert H4 : (In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) (cons r l1))\/(In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) l2); [Left; Change (In (pos_Rl (cons r l1) (Rlength l1)) (cons r l1)); Elim (RList_P3 (cons r l1) (pos_Rl (cons r l1) (Rlength l1))); Intros; Apply H5; Exists (Rlength l1); Split; [Reflexivity | Simpl; Apply lt_n_Sn] | Assert H5 := (H3 H4); Apply RList_P7; [Apply RList_P2; Assumption | Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); Intros; Apply H7; Left; Elim (RList_P3 (cons r l1) (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); Intros; Apply H9; Exists (pred (Rlength (cons r l1))); Split; [Reflexivity | Simpl; Apply lt_n_Sn]]]. +Qed. + +Lemma RList_P17 : (l1:Rlist;x:R;i:nat) (ordered_Rlist l1) -> (In x l1) -> ``(pos_Rl l1 i)<x`` -> (lt i (pred (Rlength l1))) -> ``(pos_Rl l1 (S i))<=x``. +Induction l1. +Intros; Elim H0. +Intros; Induction i. +Simpl; Elim H1; Intro; [Simpl in H2; Rewrite H4 in H2; Elim (Rlt_antirefl ? H2) | Apply RList_P5; [Apply RList_P4 with r; Assumption | Assumption]]. +Simpl; Simpl in H2; Elim H1; Intro. +Rewrite H4 in H2; Assert H5 : ``r<=(pos_Rl r0 i)``; [Apply Rle_trans with (pos_Rl r0 O); [Apply (H0 O); Simpl; Simpl in H3; Apply neq_O_lt; Red; Intro; Rewrite <- H5 in H3; Elim (lt_n_O ? H3) | Elim (RList_P6 r0); Intros; Apply H5; [Apply RList_P4 with r; Assumption | Apply le_O_n | Simpl in H3; Apply lt_S_n; Apply lt_trans with (Rlength r0); [Apply H3 | Apply lt_n_Sn]]] | Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H5 H2))]. +Apply H; Try Assumption; [Apply RList_P4 with r; Assumption | Simpl in H3; Apply lt_S_n; Replace (S (pred (Rlength r0))) with (Rlength r0); [Apply H3 | Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H5 in H3; Elim (lt_n_O ? H3)]]. +Qed. + +Lemma RList_P18 : (l:Rlist;f:R->R) (Rlength (app_Rlist l f))=(Rlength l). +Induction l; Intros; [Reflexivity | Simpl; Rewrite H; Reflexivity]. +Qed. + +Lemma RList_P19 : (l:Rlist) ~l==nil -> (EXT r:R | (EXT r0:Rlist | l==(cons r r0))). +Intros; Induction l; [Elim H; Reflexivity | Exists r; Exists l; Reflexivity]. +Qed. + +Lemma RList_P20 : (l:Rlist) (le (2) (Rlength l)) -> (EXT r:R | (EXT r1:R | (EXT l':Rlist | l==(cons r (cons r1 l'))))). +Intros; Induction l; [Simpl in H; Elim (le_Sn_O ? H) | Induction l; [Simpl in H; Elim (le_Sn_O ? (le_S_n ? ? H)) | Exists r; Exists r0; Exists l; Reflexivity]]. +Qed. + +Lemma RList_P21 : (l,l':Rlist) l==l' -> (Rtail l)==(Rtail l'). +Intros; Rewrite H; Reflexivity. +Qed. + +Lemma RList_P22 : (l1,l2:Rlist) ~l1==nil -> (pos_Rl (cons_Rlist l1 l2) O)==(pos_Rl l1 O). +Induction l1; [Intros; Elim H; Reflexivity | Intros; Reflexivity]. +Qed. + +Lemma RList_P23 : (l1,l2:Rlist) (Rlength (cons_Rlist l1 l2))==(plus (Rlength l1) (Rlength l2)). +Induction l1; [Intro; Reflexivity | Intros; Simpl; Rewrite H; Reflexivity]. +Qed. + +Lemma RList_P24 : (l1,l2:Rlist) ~l2==nil -> (pos_Rl (cons_Rlist l1 l2) (pred (Rlength (cons_Rlist l1 l2)))) == (pos_Rl l2 (pred (Rlength l2))). +Induction l1. +Intros; Reflexivity. +Intros; Rewrite <- (H l2 H0); Induction l2. +Elim H0; Reflexivity. +Do 2 Rewrite RList_P23; Replace (plus (Rlength (cons r r0)) (Rlength (cons r1 l2))) with (S (S (plus (Rlength r0) (Rlength l2)))); [Replace (plus (Rlength r0) (Rlength (cons r1 l2))) with (S (plus (Rlength r0) (Rlength l2))); [Reflexivity | Simpl; Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring] | Simpl; Apply INR_eq; Do 3 Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring]. +Qed. + +Lemma RList_P25 : (l1,l2:Rlist) (ordered_Rlist l1) -> (ordered_Rlist l2) -> ``(pos_Rl l1 (pred (Rlength l1)))<=(pos_Rl l2 O)`` -> (ordered_Rlist (cons_Rlist l1 l2)). +Induction l1. +Intros; Simpl; Assumption. +Induction r0. +Intros; Simpl; Simpl in H2; Unfold ordered_Rlist; Intros; Simpl in H3. +Induction i. +Simpl; Assumption. +Change ``(pos_Rl l2 i)<=(pos_Rl l2 (S i))``; Apply (H1 i); Apply lt_S_n; Replace (S (pred (Rlength l2))) with (Rlength l2); [Assumption | Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H4 in H3; Elim (lt_n_O ? H3)]. +Intros; Clear H; Assert H : (ordered_Rlist (cons_Rlist (cons r1 r2) l2)). +Apply H0; Try Assumption. +Apply RList_P4 with r; Assumption. +Unfold ordered_Rlist; Intros; Simpl in H4; Induction i. +Simpl; Apply (H1 O); Simpl; Apply lt_O_Sn. +Change ``(pos_Rl (cons_Rlist (cons r1 r2) l2) i)<=(pos_Rl (cons_Rlist (cons r1 r2) l2) (S i))``; Apply (H i); Simpl; Apply lt_S_n; Assumption. +Qed. + +Lemma RList_P26 : (l1,l2:Rlist;i:nat) (lt i (Rlength l1)) -> (pos_Rl (cons_Rlist l1 l2) i)==(pos_Rl l1 i). +Induction l1. +Intros; Elim (lt_n_O ? H). +Intros; Induction i. +Apply RList_P22; Discriminate. +Apply (H l2 i); Simpl in H0; Apply lt_S_n; Assumption. +Qed. + +Lemma RList_P27 : (l1,l2,l3:Rlist) (cons_Rlist l1 (cons_Rlist l2 l3))==(cons_Rlist (cons_Rlist l1 l2) l3). +Induction l1; Intros; [Reflexivity | Simpl; Rewrite (H l2 l3); Reflexivity]. +Qed. + +Lemma RList_P28 : (l:Rlist) (cons_Rlist l nil)==l. +Induction l; [Reflexivity | Intros; Simpl; Rewrite H; Reflexivity]. +Qed. + +Lemma RList_P29 : (l2,l1:Rlist;i:nat) (le (Rlength l1) i) -> (lt i (Rlength (cons_Rlist l1 l2))) -> (pos_Rl (cons_Rlist l1 l2) i)==(pos_Rl l2 (minus i (Rlength l1))). +Induction l2. +Intros; Rewrite RList_P28 in H0; Elim (lt_n_n ? (le_lt_trans ? ? ? H H0)). +Intros; Replace (cons_Rlist l1 (cons r r0)) with (cons_Rlist (cons_Rlist l1 (cons r nil)) r0). +Inversion H0. +Rewrite <- minus_n_n; Simpl; Rewrite RList_P26. +Clear l2 r0 H i H0 H1 H2; Induction l1. +Reflexivity. +Simpl; Assumption. +Rewrite RList_P23; Rewrite plus_sym; Simpl; Apply lt_n_Sn. +Replace (minus (S m) (Rlength l1)) with (S (minus (S m) (S (Rlength l1)))). +Rewrite H3; Simpl; Replace (S (Rlength l1)) with (Rlength (cons_Rlist l1 (cons r nil))). +Apply (H (cons_Rlist l1 (cons r nil)) i). +Rewrite RList_P23; Rewrite plus_sym; Simpl; Rewrite <- H3; Apply le_n_S; Assumption. +Repeat Rewrite RList_P23; Simpl; Rewrite RList_P23 in H1; Rewrite plus_sym in H1; Simpl in H1; Rewrite (plus_sym (Rlength l1)); Simpl; Rewrite plus_sym; Apply H1. +Rewrite RList_P23; Rewrite plus_sym; Reflexivity. +Change (S (minus m (Rlength l1)))=(minus (S m) (Rlength l1)); Apply minus_Sn_m; Assumption. +Replace (cons r r0) with (cons_Rlist (cons r nil) r0); [Symmetry; Apply RList_P27 | Reflexivity]. +Qed. diff --git a/theories7/Reals/R_Ifp.v b/theories7/Reals/R_Ifp.v new file mode 100644 index 00000000..621cca64 --- /dev/null +++ b/theories7/Reals/R_Ifp.v @@ -0,0 +1,552 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: R_Ifp.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +(**********************************************************) +(** Complements for the reals.Integer and fractional part *) +(* *) +(**********************************************************) + +Require Rbase. +Require Omega. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(*********************************************************) +(** Fractional part *) +(*********************************************************) + +(**********) +Definition Int_part:R->Z:=[r:R](`(up r)-1`). + +(**********) +Definition frac_part:R->R:=[r:R](Rminus r (IZR (Int_part r))). + +(**********) +Lemma tech_up:(r:R)(z:Z)(Rlt r (IZR z))->(Rle (IZR z) (Rplus r R1))-> + z=(up r). +Intros;Generalize (archimed r);Intro;Elim H1;Intros;Clear H1; + Unfold Rgt in H2;Unfold Rminus in H3; +Generalize (Rle_compatibility r (Rplus (IZR (up r)) + (Ropp r)) R1 H3);Intro;Clear H3; + Rewrite (Rplus_sym (IZR (up r)) (Ropp r)) in H1; + Rewrite <-(Rplus_assoc r (Ropp r) (IZR (up r))) in H1; + Rewrite (Rplus_Ropp_r r) in H1;Elim (Rplus_ne (IZR (up r)));Intros a b; + Rewrite b in H1;Clear a b;Apply (single_z_r_R1 r z (up r));Auto with zarith real. +Qed. + +(**********) +Lemma up_tech:(r:R)(z:Z)(Rle (IZR z) r)->(Rlt r (IZR `z+1`))-> + `z+1`=(up r). +Intros;Generalize (Rle_compatibility R1 (IZR z) r H);Intro;Clear H; + Rewrite (Rplus_sym R1 (IZR z)) in H1;Rewrite (Rplus_sym R1 r) in H1; + Cut (R1==(IZR `1`));Auto with zarith real. +Intro;Generalize H1;Pattern 1 R1;Rewrite H;Intro;Clear H H1; + Rewrite <-(plus_IZR z `1`) in H2;Apply (tech_up r `z+1`);Auto with zarith real. +Qed. + +(**********) +Lemma fp_R0:(frac_part R0)==R0. +Unfold frac_part; Unfold Int_part; Elim (archimed R0); + Intros; Unfold Rminus; + Elim (Rplus_ne (Ropp (IZR `(up R0)-1`))); Intros a b; + Rewrite b;Clear a b;Rewrite <- Z_R_minus;Cut (up R0)=`1`. +Intro;Rewrite H1; + Rewrite (eq_Rminus (IZR `1`) (IZR `1`) (refl_eqT R (IZR `1`))); + Apply Ropp_O. +Elim (archimed R0);Intros;Clear H2;Unfold Rgt in H1; + Rewrite (minus_R0 (IZR (up R0))) in H0; + Generalize (lt_O_IZR (up R0) H1);Intro;Clear H1; + Generalize (le_IZR_R1 (up R0) H0);Intro;Clear H H0;Omega. +Qed. + +(**********) +Lemma for_base_fp:(r:R)(Rgt (Rminus (IZR (up r)) r) R0)/\ + (Rle (Rminus (IZR (up r)) r) R1). +Intro; Split; + Cut (Rgt (IZR (up r)) r)/\(Rle (Rminus (IZR (up r)) r) R1). +Intro; Elim H; Intros. +Apply (Rgt_minus (IZR (up r)) r H0). +Apply archimed. +Intro; Elim H; Intros. +Exact H1. +Apply archimed. +Qed. + +(**********) +Lemma base_fp:(r:R)(Rge (frac_part r) R0)/\(Rlt (frac_part r) R1). +Intro; Unfold frac_part; Unfold Int_part; Split. + (*sup a O*) +Cut (Rge (Rminus r (IZR (up r))) (Ropp R1)). +Rewrite <- Z_R_minus;Simpl;Intro; Unfold Rminus; + Rewrite Ropp_distr1;Rewrite <-Rplus_assoc; + Fold (Rminus r (IZR (up r))); + Fold (Rminus (Rminus r (IZR (up r))) (Ropp R1)); + Apply Rge_minus;Auto with zarith real. +Rewrite <- Ropp_distr2;Apply Rle_Ropp;Elim (for_base_fp r); Auto with zarith real. + (*inf a 1*) +Cut (Rlt (Rminus r (IZR (up r))) R0). +Rewrite <- Z_R_minus; Simpl;Intro; Unfold Rminus; + Rewrite Ropp_distr1;Rewrite <-Rplus_assoc; + Fold (Rminus r (IZR (up r)));Rewrite Ropp_Ropp; + Elim (Rplus_ne R1);Intros a b;Pattern 2 R1;Rewrite <-a;Clear a b; + Rewrite (Rplus_sym (Rminus r (IZR (up r))) R1); + Apply Rlt_compatibility;Auto with zarith real. +Elim (for_base_fp r);Intros;Rewrite <-Ropp_O; + Rewrite<-Ropp_distr2;Apply Rgt_Ropp;Auto with zarith real. +Qed. + +(*********************************************************) +(** Properties *) +(*********************************************************) + +(**********) +Lemma base_Int_part:(r:R)(Rle (IZR (Int_part r)) r)/\ + (Rgt (Rminus (IZR (Int_part r)) r) (Ropp R1)). +Intro;Unfold Int_part;Elim (archimed r);Intros. +Split;Rewrite <- (Z_R_minus (up r) `1`);Simpl. +Generalize (Rle_minus (Rminus (IZR (up r)) r) R1 H0);Intro; + Unfold Rminus in H1; + Rewrite (Rplus_assoc (IZR (up r)) (Ropp r) (Ropp R1)) in + H1;Rewrite (Rplus_sym (Ropp r) (Ropp R1)) in H1; + Rewrite <-(Rplus_assoc (IZR (up r)) (Ropp R1) (Ropp r)) in + H1;Fold (Rminus (IZR (up r)) R1) in H1; + Fold (Rminus (Rminus (IZR (up r)) R1) r) in H1; + Apply Rminus_le;Auto with zarith real. +Generalize (Rgt_plus_plus_r (Ropp R1) (IZR (up r)) r H);Intro; + Rewrite (Rplus_sym (Ropp R1) (IZR (up r))) in H1; + Generalize (Rgt_plus_plus_r (Ropp r) + (Rplus (IZR (up r)) (Ropp R1)) (Rplus (Ropp R1) r) H1); + Intro;Clear H H0 H1; + Rewrite (Rplus_sym (Ropp r) (Rplus (IZR (up r)) (Ropp R1))) + in H2;Fold (Rminus (IZR (up r)) R1) in H2; + Fold (Rminus (Rminus (IZR (up r)) R1) r) in H2; + Rewrite (Rplus_sym (Ropp r) (Rplus (Ropp R1) r)) in H2; + Rewrite (Rplus_assoc (Ropp R1) r (Ropp r)) in H2; + Rewrite (Rplus_Ropp_r r) in H2;Elim (Rplus_ne (Ropp R1));Intros a b; + Rewrite a in H2;Clear a b;Auto with zarith real. +Qed. + +(**********) +Lemma Int_part_INR:(n : nat) (Int_part (INR n)) = (inject_nat n). +Intros n; Unfold Int_part. +Cut (up (INR n)) = (Zplus (inject_nat n) (inject_nat (1))). +Intros H'; Rewrite H'; Simpl; Ring. +Apply sym_equal; Apply tech_up; Auto. +Replace (Zplus (inject_nat n) (inject_nat (1))) with (INZ (S n)). +Repeat Rewrite <- INR_IZR_INZ. +Apply lt_INR; Auto. +Rewrite Zplus_sym; Rewrite <- inj_plus; Simpl; Auto. +Rewrite plus_IZR; Simpl; Auto with real. +Repeat Rewrite <- INR_IZR_INZ; Auto with real. +Qed. + +(**********) +Lemma fp_nat:(r:R)(frac_part r)==R0->(Ex [c:Z](r==(IZR c))). +Unfold frac_part;Intros;Split with (Int_part r);Apply Rminus_eq; Auto with zarith real. +Qed. + +(**********) +Lemma R0_fp_O:(r:R)~R0==(frac_part r)->~R0==r. +Red;Intros;Rewrite <- H0 in H;Generalize fp_R0;Intro;Auto with zarith real. +Qed. + +(**********) +Lemma Rminus_Int_part1:(r1,r2:R)(Rge (frac_part r1) (frac_part r2))-> + (Int_part (Rminus r1 r2))=(Zminus (Int_part r1) (Int_part r2)). +Intros;Elim (base_fp r1);Elim (base_fp r2);Intros; + Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0; + Generalize (Rle_Ropp R0 (frac_part r2) H4);Intro;Clear H4; + Rewrite (Ropp_O) in H0; + Generalize (Rle_sym2 (Ropp (frac_part r2)) R0 H0);Intro;Clear H0; + Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2; + Generalize (Rlt_Ropp (frac_part r2) R1 H1);Intro;Clear H1; + Unfold Rgt in H2; + Generalize (sum_inequa_Rle_lt R0 (frac_part r1) R1 (Ropp R1) + (Ropp (frac_part r2)) R0 H0 H3 H2 H4);Intro;Elim H1;Intros; + Clear H1;Elim (Rplus_ne R1);Intros a b;Rewrite a in H6;Clear a b H5; + Generalize (Rge_minus (frac_part r1) (frac_part r2) H);Intro;Clear H; + Fold (Rminus (frac_part r1) (frac_part r2)) in H6; + Generalize (Rle_sym2 R0 (Rminus (frac_part r1) (frac_part r2)) H1); + Intro;Clear H1 H3 H4 H0 H2;Unfold frac_part in H6 H; + Unfold Rminus in H6 H; + Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H; + Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus (Ropp r2) (IZR (Int_part r2)))) in H; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) + (IZR (Int_part r2))) in H; + Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H; + Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) + (IZR (Int_part r2))) in H; + Rewrite <-(Rplus_assoc r1 (Ropp r2) + (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H; + Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H; + Fold (Rminus r1 r2) in H;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) + in H;Generalize (Rle_compatibility + (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R0 + (Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) H);Intro; + Clear H;Rewrite (Rplus_sym (Rminus r1 r2) + (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H0; + Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H0; + Unfold Rminus in H0;Fold (Rminus r1 r2) in H0; + Rewrite (Rplus_assoc (IZR (Int_part r1)) (Ropp (IZR (Int_part r2))) + (Rplus (IZR (Int_part r2)) (Ropp (IZR (Int_part r1))))) in H0; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r2))) (IZR (Int_part r2)) + (Ropp (IZR (Int_part r1)))) in H0;Rewrite (Rplus_Ropp_l (IZR (Int_part r2))) in + H0;Elim (Rplus_ne (Ropp (IZR (Int_part r1))));Intros a b;Rewrite b in H0; + Clear a b; + Elim (Rplus_ne (Rplus (IZR (Int_part r1)) (Ropp (IZR (Int_part r2))))); + Intros a b;Rewrite a in H0;Clear a b;Rewrite (Rplus_Ropp_r (IZR (Int_part r1))) + in H0;Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H0; + Clear a b;Fold (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) in H0; + Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H6; + Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H6; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus (Ropp r2) (IZR (Int_part r2)))) in H6; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) + (IZR (Int_part r2))) in H6; + Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H6; + Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) + (IZR (Int_part r2))) in H6; + Rewrite <-(Rplus_assoc r1 (Ropp r2) + (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H6; + Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H6; + Fold (Rminus r1 r2) in H6;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) + in H6;Generalize (Rlt_compatibility + (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) R1 H6); + Intro;Clear H6; + Rewrite (Rplus_sym (Rminus r1 r2) + (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H; + Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H; + Rewrite <-(Ropp_distr2 (IZR (Int_part r1)) (IZR (Int_part r2))) in H; + Rewrite (Rplus_Ropp_r (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H; + Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H;Clear a b; + Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; + Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; + Cut R1==(IZR `1`);Auto with zarith real. +Intro;Rewrite H1 in H;Clear H1; + Rewrite <-(plus_IZR `(Int_part r1)-(Int_part r2)` `1`) in H; + Generalize (up_tech (Rminus r1 r2) `(Int_part r1)-(Int_part r2)` + H0 H);Intros;Clear H H0;Unfold 1 Int_part;Omega. +Qed. + +(**********) +Lemma Rminus_Int_part2:(r1,r2:R)(Rlt (frac_part r1) (frac_part r2))-> + (Int_part (Rminus r1 r2))=(Zminus (Zminus (Int_part r1) (Int_part r2)) `1`). +Intros;Elim (base_fp r1);Elim (base_fp r2);Intros; + Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0; + Generalize (Rle_Ropp R0 (frac_part r2) H4);Intro;Clear H4; + Rewrite (Ropp_O) in H0; + Generalize (Rle_sym2 (Ropp (frac_part r2)) R0 H0);Intro;Clear H0; + Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2; + Generalize (Rlt_Ropp (frac_part r2) R1 H1);Intro;Clear H1; + Unfold Rgt in H2; + Generalize (sum_inequa_Rle_lt R0 (frac_part r1) R1 (Ropp R1) + (Ropp (frac_part r2)) R0 H0 H3 H2 H4);Intro;Elim H1;Intros; + Clear H1;Elim (Rplus_ne (Ropp R1));Intros a b;Rewrite b in H5; + Clear a b H6;Generalize (Rlt_minus (frac_part r1) (frac_part r2) H); + Intro;Clear H;Fold (Rminus (frac_part r1) (frac_part r2)) in H5; + Clear H3 H4 H0 H2;Unfold frac_part in H5 H1; + Unfold Rminus in H5 H1; + Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H5; + Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H5; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus (Ropp r2) (IZR (Int_part r2)))) in H5; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) + (IZR (Int_part r2))) in H5; + Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H5; + Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) + (IZR (Int_part r2))) in H5; + Rewrite <-(Rplus_assoc r1 (Ropp r2) + (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H5; + Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H5; + Fold (Rminus r1 r2) in H5;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) + in H5;Generalize (Rlt_compatibility + (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) (Ropp R1) + (Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) H5); + Intro;Clear H5;Rewrite (Rplus_sym (Rminus r1 r2) + (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H; + Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H; + Unfold Rminus in H;Fold (Rminus r1 r2) in H; + Rewrite (Rplus_assoc (IZR (Int_part r1)) (Ropp (IZR (Int_part r2))) + (Rplus (IZR (Int_part r2)) (Ropp (IZR (Int_part r1))))) in H; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r2))) (IZR (Int_part r2)) + (Ropp (IZR (Int_part r1)))) in H;Rewrite (Rplus_Ropp_l (IZR (Int_part r2))) in + H;Elim (Rplus_ne (Ropp (IZR (Int_part r1))));Intros a b;Rewrite b in H; + Clear a b;Rewrite (Rplus_Ropp_r (IZR (Int_part r1))) in H; + Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H; + Clear a b;Fold (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) in H; + Fold (Rminus (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R1) in H; + Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H1; + Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H1; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus (Ropp r2) (IZR (Int_part r2)))) in H1; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) + (IZR (Int_part r2))) in H1; + Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H1; + Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) + (IZR (Int_part r2))) in H1; + Rewrite <-(Rplus_assoc r1 (Ropp r2) + (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H1; + Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H1; + Fold (Rminus r1 r2) in H1;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) + in H1;Generalize (Rlt_compatibility + (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) R0 H1); + Intro;Clear H1; + Rewrite (Rplus_sym (Rminus r1 r2) + (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H0; + Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H0; + Rewrite <-(Ropp_distr2 (IZR (Int_part r1)) (IZR (Int_part r2))) in H0; + Rewrite (Rplus_Ropp_r (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0; + Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H0;Clear a b; + Rewrite <-(Rplus_Ropp_l R1) in H0; + Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Ropp R1) R1) in H0; + Fold (Rminus (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R1) in H0; + Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; + Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; + Cut R1==(IZR `1`);Auto with zarith real. +Intro;Rewrite H1 in H;Rewrite H1 in H0;Clear H1; + Rewrite (Z_R_minus `(Int_part r1)-(Int_part r2)` `1`) in H; + Rewrite (Z_R_minus `(Int_part r1)-(Int_part r2)` `1`) in H0; + Rewrite <-(plus_IZR `(Int_part r1)-(Int_part r2)-1` `1`) in H0; + Generalize (Rlt_le (IZR `(Int_part r1)-(Int_part r2)-1`) (Rminus r1 r2) H); + Intro;Clear H; + Generalize (up_tech (Rminus r1 r2) `(Int_part r1)-(Int_part r2)-1` + H1 H0);Intros;Clear H0 H1;Unfold 1 Int_part;Omega. +Qed. + +(**********) +Lemma Rminus_fp1:(r1,r2:R)(Rge (frac_part r1) (frac_part r2))-> + (frac_part (Rminus r1 r2))==(Rminus (frac_part r1) (frac_part r2)). +Intros;Unfold frac_part; + Generalize (Rminus_Int_part1 r1 r2 H);Intro;Rewrite -> H0; + Rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));Unfold Rminus; + Rewrite -> (Ropp_distr1 (IZR (Int_part r1)) (Ropp (IZR (Int_part r2)))); + Rewrite -> (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))); + Rewrite -> (Ropp_Ropp (IZR (Int_part r2))); + Rewrite -> (Rplus_assoc r1 (Ropp r2) + (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))); + Rewrite -> (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus (Ropp r2) (IZR (Int_part r2)))); + Rewrite <- (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) + (IZR (Int_part r2))); + Rewrite <- (Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) + (IZR (Int_part r2))); + Rewrite -> (Rplus_sym (Ropp r2) (Ropp (IZR (Int_part r1))));Auto with zarith real. +Qed. + +(**********) +Lemma Rminus_fp2:(r1,r2:R)(Rlt (frac_part r1) (frac_part r2))-> + (frac_part (Rminus r1 r2))== + (Rplus (Rminus (frac_part r1) (frac_part r2)) R1). +Intros;Unfold frac_part;Generalize (Rminus_Int_part2 r1 r2 H);Intro; + Rewrite -> H0; + Rewrite <- (Z_R_minus (Zminus (Int_part r1) (Int_part r2)) `1`); + Rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));Unfold Rminus; + Rewrite -> (Ropp_distr1 (Rplus (IZR (Int_part r1)) (Ropp (IZR (Int_part r2)))) + (Ropp (IZR `1`))); + Rewrite -> (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))); + Rewrite -> (Ropp_Ropp (IZR `1`)); + Rewrite -> (Ropp_Ropp (IZR (Int_part r2))); + Rewrite -> (Ropp_distr1 (IZR (Int_part r1))); + Rewrite -> (Ropp_Ropp (IZR (Int_part r2)));Simpl; + Rewrite <- (Rplus_assoc (Rplus r1 (Ropp r2)) + (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) R1); + Rewrite -> (Rplus_assoc r1 (Ropp r2) + (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))); + Rewrite -> (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus (Ropp r2) (IZR (Int_part r2)))); + Rewrite <- (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) + (IZR (Int_part r2))); + Rewrite <- (Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) + (IZR (Int_part r2))); + Rewrite -> (Rplus_sym (Ropp r2) (Ropp (IZR (Int_part r1))));Auto with zarith real. +Qed. + +(**********) +Lemma plus_Int_part1:(r1,r2:R)(Rge (Rplus (frac_part r1) (frac_part r2)) R1)-> + (Int_part (Rplus r1 r2))=(Zplus (Zplus (Int_part r1) (Int_part r2)) `1`). +Intros; + Generalize (Rle_sym2 R1 (Rplus (frac_part r1) (frac_part r2)) H); + Intro;Clear H;Elim (base_fp r1);Elim (base_fp r2);Intros;Clear H H2; + Generalize (Rlt_compatibility (frac_part r2) (frac_part r1) R1 H3); + Intro;Clear H3; + Generalize (Rlt_compatibility R1 (frac_part r2) R1 H1);Intro;Clear H1; + Rewrite (Rplus_sym R1 (frac_part r2)) in H2; + Generalize (Rlt_trans (Rplus (frac_part r2) (frac_part r1)) + (Rplus (frac_part r2) R1) (Rplus R1 R1) H H2);Intro;Clear H H2; + Rewrite (Rplus_sym (frac_part r2) (frac_part r1)) in H1; + Unfold frac_part in H0 H1;Unfold Rminus in H0 H1; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus r2 (Ropp (IZR (Int_part r2))))) in H1; + Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H1; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) + r2) in H1; + Rewrite (Rplus_sym + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H1; + Rewrite <-(Rplus_assoc r1 r2 + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H1; + Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H1; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus r2 (Ropp (IZR (Int_part r2))))) in H0; + Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H0; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) + r2) in H0; + Rewrite (Rplus_sym + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H0; + Rewrite <-(Rplus_assoc r1 r2 + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H0; + Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H0; + Generalize (Rle_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) + R1 (Rplus (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) H0);Intro; + Clear H0; + Generalize (Rlt_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Rplus (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) (Rplus R1 R1) H1); + Intro;Clear H1; + Rewrite (Rplus_sym (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H; + Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H; + Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H; + Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H;Clear a b; + Rewrite (Rplus_sym (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H0; + Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H0; + Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0; + Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H0;Clear a b; + Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) R1 R1) in + H0;Cut R1==(IZR `1`);Auto with zarith real. +Intro;Rewrite H1 in H0;Rewrite H1 in H;Clear H1; + Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H; + Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H0; + Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H; + Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H0; + Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)+1` `1`) in H0; + Generalize (up_tech (Rplus r1 r2) `(Int_part r1)+(Int_part r2)+1` H H0);Intro; + Clear H H0;Unfold 1 Int_part;Omega. +Qed. + +(**********) +Lemma plus_Int_part2:(r1,r2:R)(Rlt (Rplus (frac_part r1) (frac_part r2)) R1)-> + (Int_part (Rplus r1 r2))=(Zplus (Int_part r1) (Int_part r2)). +Intros;Elim (base_fp r1);Elim (base_fp r2);Intros;Clear H1 H3; + Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0; + Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2; + Generalize (Rle_compatibility (frac_part r1) R0 (frac_part r2) H1); + Intro;Clear H1;Elim (Rplus_ne (frac_part r1));Intros a b; + Rewrite a in H2;Clear a b;Generalize (Rle_trans R0 (frac_part r1) + (Rplus (frac_part r1) (frac_part r2)) H0 H2);Intro;Clear H0 H2; + Unfold frac_part in H H1;Unfold Rminus in H H1; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus r2 (Ropp (IZR (Int_part r2))))) in H1; + Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H1; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) + r2) in H1; + Rewrite (Rplus_sym + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H1; + Rewrite <-(Rplus_assoc r1 r2 + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H1; + Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H1; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus r2 (Ropp (IZR (Int_part r2))))) in H; + Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H; + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) + r2) in H; + Rewrite (Rplus_sym + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H; + Rewrite <-(Rplus_assoc r1 r2 + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H; + Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H; + Generalize (Rle_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) + R0 (Rplus (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) H1);Intro; + Clear H1; + Generalize (Rlt_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Rplus (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) R1 H); + Intro;Clear H; + Rewrite (Rplus_sym (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H1; + Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H1; + Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H1; + Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H1;Clear a b; + Rewrite (Rplus_sym (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H0; + Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H0; + Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0; + Elim (Rplus_ne (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))));Intros a b; + Rewrite a in H0;Clear a b;Elim (Rplus_ne (Rplus r1 r2));Intros a b; + Rewrite b in H0;Clear a b;Cut R1==(IZR `1`);Auto with zarith real. +Intro;Rewrite H in H1;Clear H; + Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H0; + Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H1; + Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H1; + Generalize (up_tech (Rplus r1 r2) `(Int_part r1)+(Int_part r2)` H0 H1);Intro; + Clear H0 H1;Unfold 1 Int_part;Omega. +Qed. + +(**********) +Lemma plus_frac_part1:(r1,r2:R) + (Rge (Rplus (frac_part r1) (frac_part r2)) R1)-> + (frac_part (Rplus r1 r2))== + (Rminus (Rplus (frac_part r1) (frac_part r2)) R1). +Intros;Unfold frac_part; + Generalize (plus_Int_part1 r1 r2 H);Intro;Rewrite H0; + Rewrite (plus_IZR `(Int_part r1)+(Int_part r2)` `1`); + Rewrite (plus_IZR (Int_part r1) (Int_part r2));Simpl;Unfold 3 4 Rminus; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus r2 (Ropp (IZR (Int_part r2))))); + Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))); + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) + r2); + Rewrite (Rplus_sym + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2); + Rewrite <-(Rplus_assoc r1 r2 + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))); + Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))); + Unfold Rminus; + Rewrite (Rplus_assoc (Rplus r1 r2) + (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) + (Ropp R1)); + Rewrite <-(Ropp_distr1 (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) R1); + Trivial with zarith real. +Qed. + +(**********) +Lemma plus_frac_part2:(r1,r2:R) + (Rlt (Rplus (frac_part r1) (frac_part r2)) R1)-> +(frac_part (Rplus r1 r2))==(Rplus (frac_part r1) (frac_part r2)). +Intros;Unfold frac_part; + Generalize (plus_Int_part2 r1 r2 H);Intro;Rewrite H0; + Rewrite (plus_IZR (Int_part r1) (Int_part r2));Unfold 2 3 Rminus; + Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) + (Rplus r2 (Ropp (IZR (Int_part r2))))); + Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))); + Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) + r2); + Rewrite (Rplus_sym + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2); + Rewrite <-(Rplus_assoc r1 r2 + (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))); + Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2)));Unfold Rminus; + Trivial with zarith real. +Qed. diff --git a/theories7/Reals/R_sqr.v b/theories7/Reals/R_sqr.v new file mode 100644 index 00000000..fc01a164 --- /dev/null +++ b/theories7/Reals/R_sqr.v @@ -0,0 +1,232 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: R_sqr.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rbasic_fun. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(****************************************************) +(* Rsqr : some results *) +(****************************************************) + +Tactic Definition SqRing := Unfold Rsqr; Ring. + +Lemma Rsqr_neg : (x:R) ``(Rsqr x)==(Rsqr (-x))``. +Intros; SqRing. +Qed. + +Lemma Rsqr_times : (x,y:R) ``(Rsqr (x*y))==(Rsqr x)*(Rsqr y)``. +Intros; SqRing. +Qed. + +Lemma Rsqr_plus : (x,y:R) ``(Rsqr (x+y))==(Rsqr x)+(Rsqr y)+2*x*y``. +Intros; SqRing. +Qed. + +Lemma Rsqr_minus : (x,y:R) ``(Rsqr (x-y))==(Rsqr x)+(Rsqr y)-2*x*y``. +Intros; SqRing. +Qed. + +Lemma Rsqr_neg_minus : (x,y:R) ``(Rsqr (x-y))==(Rsqr (y-x))``. +Intros; SqRing. +Qed. + +Lemma Rsqr_1 : ``(Rsqr 1)==1``. +SqRing. +Qed. + +Lemma Rsqr_gt_0_0 : (x:R) ``0<(Rsqr x)`` -> ~``x==0``. +Intros; Red; Intro; Rewrite H0 in H; Rewrite Rsqr_O in H; Elim (Rlt_antirefl ``0`` H). +Qed. + +Lemma Rsqr_pos_lt : (x:R) ~(x==R0)->``0<(Rsqr x)``. +Intros; Case (total_order R0 x); Intro; [Unfold Rsqr; Apply Rmult_lt_pos; Assumption | Elim H0; Intro; [Elim H; Symmetry; Exact H1 | Rewrite Rsqr_neg; Generalize (Rlt_Ropp x ``0`` H1); Rewrite Ropp_O; Intro; Unfold Rsqr; Apply Rmult_lt_pos; Assumption]]. +Qed. + +Lemma Rsqr_div : (x,y:R) ~``y==0`` -> ``(Rsqr (x/y))==(Rsqr x)/(Rsqr y)``. +Intros; Unfold Rsqr. +Unfold Rdiv. +Rewrite Rinv_Rmult. +Repeat Rewrite Rmult_assoc. +Apply Rmult_mult_r. +Pattern 2 x; Rewrite Rmult_sym. +Repeat Rewrite Rmult_assoc. +Apply Rmult_mult_r. +Reflexivity. +Assumption. +Assumption. +Qed. + +Lemma Rsqr_eq_0 : (x:R) ``(Rsqr x)==0`` -> ``x==0``. +Unfold Rsqr; Intros; Generalize (without_div_Od x x H); Intro; Elim H0; Intro ; Assumption. +Qed. + +Lemma Rsqr_minus_plus : (a,b:R) ``(a-b)*(a+b)==(Rsqr a)-(Rsqr b)``. +Intros; SqRing. +Qed. + +Lemma Rsqr_plus_minus : (a,b:R) ``(a+b)*(a-b)==(Rsqr a)-(Rsqr b)``. +Intros; SqRing. +Qed. + +Lemma Rsqr_incr_0 : (x,y:R) ``(Rsqr x)<=(Rsqr y)`` -> ``0<=x`` -> ``0<=y`` -> ``x<=y``. +Intros; Case (total_order_Rle x y); Intro; [Assumption | Cut ``y<x``; [Intro; Unfold Rsqr in H; Generalize (Rmult_lt2 y x y x H1 H1 H2 H2); Intro; Generalize (Rle_lt_trans ``x*x`` ``y*y`` ``x*x`` H H3); Intro; Elim (Rlt_antirefl ``x*x`` H4) | Auto with real]]. +Qed. + +Lemma Rsqr_incr_0_var : (x,y:R) ``(Rsqr x)<=(Rsqr y)`` -> ``0<=y`` -> ``x<=y``. +Intros; Case (total_order_Rle x y); Intro; [Assumption | Cut ``y<x``; [Intro; Unfold Rsqr in H; Generalize (Rmult_lt2 y x y x H0 H0 H1 H1); Intro; Generalize (Rle_lt_trans ``x*x`` ``y*y`` ``x*x`` H H2); Intro; Elim (Rlt_antirefl ``x*x`` H3) | Auto with real]]. +Qed. + +Lemma Rsqr_incr_1 : (x,y:R) ``x<=y``->``0<=x``->``0<= y``->``(Rsqr x)<=(Rsqr y)``. +Intros; Unfold Rsqr; Apply Rle_Rmult_comp; Assumption. +Qed. + +Lemma Rsqr_incrst_0 : (x,y:R) ``(Rsqr x)<(Rsqr y)``->``0<=x``->``0<=y``-> ``x<y``. +Intros; Case (total_order x y); Intro; [Assumption | Elim H2; Intro; [Rewrite H3 in H; Elim (Rlt_antirefl (Rsqr y) H) | Generalize (Rmult_lt2 y x y x H1 H1 H3 H3); Intro; Unfold Rsqr in H; Generalize (Rlt_trans ``x*x`` ``y*y`` ``x*x`` H H4); Intro; Elim (Rlt_antirefl ``x*x`` H5)]]. +Qed. + +Lemma Rsqr_incrst_1 : (x,y:R) ``x<y``->``0<=x``->``0<=y``->``(Rsqr x)<(Rsqr y)``. +Intros; Unfold Rsqr; Apply Rmult_lt2; Assumption. +Qed. + +Lemma Rsqr_neg_pos_le_0 : (x,y:R) ``(Rsqr x)<=(Rsqr y)``->``0<=y``->``-y<=x``. +Intros; Case (case_Rabsolu x); Intro. +Generalize (Rlt_Ropp x ``0`` r); Rewrite Ropp_O; Intro; Generalize (Rlt_le ``0`` ``-x`` H1); Intro; Rewrite (Rsqr_neg x) in H; Generalize (Rsqr_incr_0 (Ropp x) y H H2 H0); Intro; Rewrite <- (Ropp_Ropp x); Apply Rge_Ropp; Apply Rle_sym1; Assumption. +Apply Rle_trans with ``0``; [Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Assumption | Apply Rle_sym2; Assumption]. +Qed. + +Lemma Rsqr_neg_pos_le_1 : (x,y:R) ``(-y)<=x`` -> ``x<=y`` -> ``0<=y`` -> ``(Rsqr x)<=(Rsqr y)``. +Intros; Case (case_Rabsolu x); Intro. +Generalize (Rlt_Ropp x ``0`` r); Rewrite Ropp_O; Intro; Generalize (Rlt_le ``0`` ``-x`` H2); Intro; Generalize (Rle_Ropp ``-y`` x H); Rewrite Ropp_Ropp; Intro; Generalize (Rle_sym2 ``-x`` y H4); Intro; Rewrite (Rsqr_neg x); Apply Rsqr_incr_1; Assumption. +Generalize (Rle_sym2 ``0`` x r); Intro; Apply Rsqr_incr_1; Assumption. +Qed. + +Lemma neg_pos_Rsqr_le : (x,y:R) ``(-y)<=x``->``x<=y``->``(Rsqr x)<=(Rsqr y)``. +Intros; Case (case_Rabsolu x); Intro. +Generalize (Rlt_Ropp x ``0`` r); Rewrite Ropp_O; Intro; Generalize (Rle_Ropp ``-y`` x H); Rewrite Ropp_Ropp; Intro; Generalize (Rle_sym2 ``-x`` y H2); Intro; Generalize (Rlt_le ``0`` ``-x`` H1); Intro; Generalize (Rle_trans ``0`` ``-x`` y H4 H3); Intro; Rewrite (Rsqr_neg x); Apply Rsqr_incr_1; Assumption. +Generalize (Rle_sym2 ``0`` x r); Intro; Generalize (Rle_trans ``0`` x y H1 H0); Intro; Apply Rsqr_incr_1; Assumption. +Qed. + +Lemma Rsqr_abs : (x:R) ``(Rsqr x)==(Rsqr (Rabsolu x))``. +Intro; Unfold Rabsolu; Case (case_Rabsolu x); Intro; [Apply Rsqr_neg | Reflexivity]. +Qed. + +Lemma Rsqr_le_abs_0 : (x,y:R) ``(Rsqr x)<=(Rsqr y)`` -> ``(Rabsolu x)<=(Rabsolu y)``. +Intros; Apply Rsqr_incr_0; Repeat Rewrite <- Rsqr_abs; [Assumption | Apply Rabsolu_pos | Apply Rabsolu_pos]. +Qed. + +Lemma Rsqr_le_abs_1 : (x,y:R) ``(Rabsolu x)<=(Rabsolu y)`` -> ``(Rsqr x)<=(Rsqr y)``. +Intros; Rewrite (Rsqr_abs x); Rewrite (Rsqr_abs y); Apply (Rsqr_incr_1 (Rabsolu x) (Rabsolu y) H (Rabsolu_pos x) (Rabsolu_pos y)). +Qed. + +Lemma Rsqr_lt_abs_0 : (x,y:R) ``(Rsqr x)<(Rsqr y)`` -> ``(Rabsolu x)<(Rabsolu y)``. +Intros; Apply Rsqr_incrst_0; Repeat Rewrite <- Rsqr_abs; [Assumption | Apply Rabsolu_pos | Apply Rabsolu_pos]. +Qed. + +Lemma Rsqr_lt_abs_1 : (x,y:R) ``(Rabsolu x)<(Rabsolu y)`` -> ``(Rsqr x)<(Rsqr y)``. +Intros; Rewrite (Rsqr_abs x); Rewrite (Rsqr_abs y); Apply (Rsqr_incrst_1 (Rabsolu x) (Rabsolu y) H (Rabsolu_pos x) (Rabsolu_pos y)). +Qed. + +Lemma Rsqr_inj : (x,y:R) ``0<=x`` -> ``0<=y`` -> (Rsqr x)==(Rsqr y) -> x==y. +Intros; Generalize (Rle_le_eq (Rsqr x) (Rsqr y)); Intro; Elim H2; Intros _ H3; Generalize (H3 H1); Intro; Elim H4; Intros; Apply Rle_antisym; Apply Rsqr_incr_0; Assumption. +Qed. + +Lemma Rsqr_eq_abs_0 : (x,y:R) (Rsqr x)==(Rsqr y) -> (Rabsolu x)==(Rabsolu y). +Intros; Unfold Rabsolu; Case (case_Rabsolu x); Case (case_Rabsolu y); Intros. +Rewrite -> (Rsqr_neg x) in H; Rewrite -> (Rsqr_neg y) in H; Generalize (Rlt_Ropp y ``0`` r); Generalize (Rlt_Ropp x ``0`` r0); Rewrite Ropp_O; Intros; Generalize (Rlt_le ``0`` ``-x`` H0); Generalize (Rlt_le ``0`` ``-y`` H1); Intros; Apply Rsqr_inj; Assumption. +Rewrite -> (Rsqr_neg x) in H; Generalize (Rle_sym2 ``0`` y r); Intro; Generalize (Rlt_Ropp x ``0`` r0); Rewrite Ropp_O; Intro; Generalize (Rlt_le ``0`` ``-x`` H1); Intro; Apply Rsqr_inj; Assumption. +Rewrite -> (Rsqr_neg y) in H; Generalize (Rle_sym2 ``0`` x r0); Intro; Generalize (Rlt_Ropp y ``0`` r); Rewrite Ropp_O; Intro; Generalize (Rlt_le ``0`` ``-y`` H1); Intro; Apply Rsqr_inj; Assumption. +Generalize (Rle_sym2 ``0`` x r0); Generalize (Rle_sym2 ``0`` y r); Intros; Apply Rsqr_inj; Assumption. +Qed. + +Lemma Rsqr_eq_asb_1 : (x,y:R) (Rabsolu x)==(Rabsolu y) -> (Rsqr x)==(Rsqr y). +Intros; Cut ``(Rsqr (Rabsolu x))==(Rsqr (Rabsolu y))``. +Intro; Repeat Rewrite <- Rsqr_abs in H0; Assumption. +Rewrite H; Reflexivity. +Qed. + +Lemma triangle_rectangle : (x,y,z:R) ``0<=z``->``(Rsqr x)+(Rsqr y)<=(Rsqr z)``->``-z<=x<=z`` /\``-z<=y<=z``. +Intros; Generalize (plus_le_is_le (Rsqr x) (Rsqr y) (Rsqr z) (pos_Rsqr y) H0); Rewrite Rplus_sym in H0; Generalize (plus_le_is_le (Rsqr y) (Rsqr x) (Rsqr z) (pos_Rsqr x) H0); Intros; Split; [Split; [Apply Rsqr_neg_pos_le_0; Assumption | Apply Rsqr_incr_0_var; Assumption] | Split; [Apply Rsqr_neg_pos_le_0; Assumption | Apply Rsqr_incr_0_var; Assumption]]. +Qed. + +Lemma triangle_rectangle_lt : (x,y,z:R) ``(Rsqr x)+(Rsqr y)<(Rsqr z)`` -> ``(Rabsolu x)<(Rabsolu z)``/\``(Rabsolu y)<(Rabsolu z)``. +Intros; Split; [Generalize (plus_lt_is_lt (Rsqr x) (Rsqr y) (Rsqr z) (pos_Rsqr y) H); Intro; Apply Rsqr_lt_abs_0; Assumption | Rewrite Rplus_sym in H; Generalize (plus_lt_is_lt (Rsqr y) (Rsqr x) (Rsqr z) (pos_Rsqr x) H); Intro; Apply Rsqr_lt_abs_0; Assumption]. +Qed. + +Lemma triangle_rectangle_le : (x,y,z:R) ``(Rsqr x)+(Rsqr y)<=(Rsqr z)`` -> ``(Rabsolu x)<=(Rabsolu z)``/\``(Rabsolu y)<=(Rabsolu z)``. +Intros; Split; [Generalize (plus_le_is_le (Rsqr x) (Rsqr y) (Rsqr z) (pos_Rsqr y) H); Intro; Apply Rsqr_le_abs_0; Assumption | Rewrite Rplus_sym in H; Generalize (plus_le_is_le (Rsqr y) (Rsqr x) (Rsqr z) (pos_Rsqr x) H); Intro; Apply Rsqr_le_abs_0; Assumption]. +Qed. + +Lemma Rsqr_inv : (x:R) ~``x==0`` -> ``(Rsqr (/x))==/(Rsqr x)``. +Intros; Unfold Rsqr. +Rewrite Rinv_Rmult; Try Reflexivity Orelse Assumption. +Qed. + +Lemma canonical_Rsqr : (a:nonzeroreal;b,c,x:R) ``a*(Rsqr x)+b*x+c == a* (Rsqr (x+b/(2*a))) + (4*a*c - (Rsqr b))/(4*a)``. +Intros. +Rewrite Rsqr_plus. +Repeat Rewrite Rmult_Rplus_distr. +Repeat Rewrite Rplus_assoc. +Apply Rplus_plus_r. +Unfold Rdiv Rminus. +Replace ``2*1+2*1`` with ``4``; [Idtac | Ring]. +Rewrite (Rmult_Rplus_distrl ``4*a*c`` ``-(Rsqr b)`` ``/(4*a)``). +Rewrite Rsqr_times. +Repeat Rewrite Rinv_Rmult. +Repeat Rewrite (Rmult_sym a). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite (Rmult_sym ``2``). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite (Rmult_sym ``/2``). +Rewrite (Rmult_sym ``2``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite (Rmult_sym a). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite (Rmult_sym ``2``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Repeat Rewrite Rplus_assoc. +Rewrite (Rplus_sym ``(Rsqr b)*((Rsqr (/a*/2))*a)``). +Repeat Rewrite Rplus_assoc. +Rewrite (Rmult_sym x). +Apply Rplus_plus_r. +Rewrite (Rmult_sym ``/a``). +Unfold Rsqr; Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Ring. +Apply (cond_nonzero a). +DiscrR. +Apply (cond_nonzero a). +DiscrR. +DiscrR. +Apply (cond_nonzero a). +DiscrR. +DiscrR. +DiscrR. +Apply (cond_nonzero a). +DiscrR. +Apply (cond_nonzero a). +Qed. + +Lemma Rsqr_eq : (x,y:R) (Rsqr x)==(Rsqr y) -> x==y \/ x==``-y``. +Intros; Unfold Rsqr in H; Generalize (Rplus_plus_r ``-(y*y)`` ``x*x`` ``y*y`` H); Rewrite Rplus_Ropp_l; Replace ``-(y*y)+x*x`` with ``(x-y)*(x+y)``. +Intro; Generalize (without_div_Od ``x-y`` ``x+y`` H0); Intro; Elim H1; Intros. +Left; Apply Rminus_eq; Assumption. +Right; Apply Rminus_eq; Unfold Rminus; Rewrite Ropp_Ropp; Assumption. +Ring. +Qed. diff --git a/theories7/Reals/R_sqrt.v b/theories7/Reals/R_sqrt.v new file mode 100644 index 00000000..8c87659b --- /dev/null +++ b/theories7/Reals/R_sqrt.v @@ -0,0 +1,251 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: R_sqrt.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rsqrt_def. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(* Here is a continuous extension of Rsqrt on R *) +Definition sqrt : R->R := [x:R](Cases (case_Rabsolu x) of + (leftT _) => R0 + | (rightT a) => (Rsqrt (mknonnegreal x (Rle_sym2 ? ? a))) end). + +Lemma sqrt_positivity : (x:R) ``0<=x`` -> ``0<=(sqrt x)``. +Intros. +Unfold sqrt. +Case (case_Rabsolu x); Intro. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? r H)). +Apply Rsqrt_positivity. +Qed. + +Lemma sqrt_sqrt : (x:R) ``0<=x`` -> ``(sqrt x)*(sqrt x)==x``. +Intros. +Unfold sqrt. +Case (case_Rabsolu x); Intro. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? r H)). +Rewrite Rsqrt_Rsqrt; Reflexivity. +Qed. + +Lemma sqrt_0 : ``(sqrt 0)==0``. +Apply Rsqr_eq_0; Unfold Rsqr; Apply sqrt_sqrt; Right; Reflexivity. +Qed. + +Lemma sqrt_1 : ``(sqrt 1)==1``. +Apply (Rsqr_inj (sqrt R1) R1); [Apply sqrt_positivity; Left | Left | Unfold Rsqr; Rewrite -> sqrt_sqrt; [Ring | Left]]; Apply Rlt_R0_R1. +Qed. + +Lemma sqrt_eq_0 : (x:R) ``0<=x``->``(sqrt x)==0``->``x==0``. +Intros; Cut ``(Rsqr (sqrt x))==0``. +Intro; Unfold Rsqr in H1; Rewrite -> sqrt_sqrt in H1; Assumption. +Rewrite H0; Apply Rsqr_O. +Qed. + +Lemma sqrt_lem_0 : (x,y:R) ``0<=x``->``0<=y``->(sqrt x)==y->``y*y==x``. +Intros; Rewrite <- H1; Apply (sqrt_sqrt x H). +Qed. + +Lemma sqtr_lem_1 : (x,y:R) ``0<=x``->``0<=y``->``y*y==x``->(sqrt x)==y. +Intros; Apply Rsqr_inj; [Apply (sqrt_positivity x H) | Assumption | Unfold Rsqr; Rewrite -> H1; Apply (sqrt_sqrt x H)]. +Qed. + +Lemma sqrt_def : (x:R) ``0<=x``->``(sqrt x)*(sqrt x)==x``. +Intros; Apply (sqrt_sqrt x H). +Qed. + +Lemma sqrt_square : (x:R) ``0<=x``->``(sqrt (x*x))==x``. +Intros; Apply (Rsqr_inj (sqrt (Rsqr x)) x (sqrt_positivity (Rsqr x) (pos_Rsqr x)) H); Unfold Rsqr; Apply (sqrt_sqrt (Rsqr x) (pos_Rsqr x)). +Qed. + +Lemma sqrt_Rsqr : (x:R) ``0<=x``->``(sqrt (Rsqr x))==x``. +Intros; Unfold Rsqr; Apply sqrt_square; Assumption. +Qed. + +Lemma sqrt_Rsqr_abs : (x:R) (sqrt (Rsqr x))==(Rabsolu x). +Intro x; Rewrite -> Rsqr_abs; Apply sqrt_Rsqr; Apply Rabsolu_pos. +Qed. + +Lemma Rsqr_sqrt : (x:R) ``0<=x``->(Rsqr (sqrt x))==x. +Intros x H1; Unfold Rsqr; Apply (sqrt_sqrt x H1). +Qed. + +Lemma sqrt_times : (x,y:R) ``0<=x``->``0<=y``->``(sqrt (x*y))==(sqrt x)*(sqrt y)``. +Intros x y H1 H2; Apply (Rsqr_inj (sqrt (Rmult x y)) (Rmult (sqrt x) (sqrt y)) (sqrt_positivity (Rmult x y) (Rmult_le_pos x y H1 H2)) (Rmult_le_pos (sqrt x) (sqrt y) (sqrt_positivity x H1) (sqrt_positivity y H2))); Rewrite Rsqr_times; Repeat Rewrite Rsqr_sqrt; [Ring | Assumption |Assumption | Apply (Rmult_le_pos x y H1 H2)]. +Qed. + +Lemma sqrt_lt_R0 : (x:R) ``0<x`` -> ``0<(sqrt x)``. +Intros x H1; Apply Rsqr_incrst_0; [Rewrite Rsqr_O; Rewrite Rsqr_sqrt ; [Assumption | Left; Assumption] | Right; Reflexivity | Apply (sqrt_positivity x (Rlt_le R0 x H1))]. +Qed. + +Lemma sqrt_div : (x,y:R) ``0<=x``->``0<y``->``(sqrt (x/y))==(sqrt x)/(sqrt y)``. +Intros x y H1 H2; Apply Rsqr_inj; [ Apply sqrt_positivity; Apply (Rmult_le_pos x (Rinv y)); [ Assumption | Generalize (Rlt_Rinv y H2); Clear H2; Intro H2; Left; Assumption] | Apply (Rmult_le_pos (sqrt x) (Rinv (sqrt y))) ; [ Apply (sqrt_positivity x H1) | Generalize (sqrt_lt_R0 y H2); Clear H2; Intro H2; Generalize (Rlt_Rinv (sqrt y) H2); Clear H2; Intro H2; Left; Assumption] | Rewrite Rsqr_div; Repeat Rewrite Rsqr_sqrt; [ Reflexivity | Left; Assumption | Assumption | Generalize (Rlt_Rinv y H2); Intro H3; Generalize (Rlt_le R0 (Rinv y) H3); Intro H4; Apply (Rmult_le_pos x (Rinv y) H1 H4) |Red; Intro H3; Generalize (Rlt_le R0 y H2); Intro H4; Generalize (sqrt_eq_0 y H4 H3); Intro H5; Rewrite H5 in H2; Elim (Rlt_antirefl R0 H2)]]. +Qed. + +Lemma sqrt_lt_0 : (x,y:R) ``0<=x``->``0<=y``->``(sqrt x)<(sqrt y)``->``x<y``. +Intros x y H1 H2 H3; Generalize (Rsqr_incrst_1 (sqrt x) (sqrt y) H3 (sqrt_positivity x H1) (sqrt_positivity y H2)); Intro H4; Rewrite (Rsqr_sqrt x H1) in H4; Rewrite (Rsqr_sqrt y H2) in H4; Assumption. +Qed. + +Lemma sqrt_lt_1 : (x,y:R) ``0<=x``->``0<=y``->``x<y``->``(sqrt x)<(sqrt y)``. +Intros x y H1 H2 H3; Apply Rsqr_incrst_0; [Rewrite (Rsqr_sqrt x H1); Rewrite (Rsqr_sqrt y H2); Assumption | Apply (sqrt_positivity x H1) | Apply (sqrt_positivity y H2)]. +Qed. + +Lemma sqrt_le_0 : (x,y:R) ``0<=x``->``0<=y``->``(sqrt x)<=(sqrt y)``->``x<=y``. +Intros x y H1 H2 H3; Generalize (Rsqr_incr_1 (sqrt x) (sqrt y) H3 (sqrt_positivity x H1) (sqrt_positivity y H2)); Intro H4; Rewrite (Rsqr_sqrt x H1) in H4; Rewrite (Rsqr_sqrt y H2) in H4; Assumption. +Qed. + +Lemma sqrt_le_1 : (x,y:R) ``0<=x``->``0<=y``->``x<=y``->``(sqrt x)<=(sqrt y)``. +Intros x y H1 H2 H3; Apply Rsqr_incr_0; [ Rewrite (Rsqr_sqrt x H1); Rewrite (Rsqr_sqrt y H2); Assumption | Apply (sqrt_positivity x H1) | Apply (sqrt_positivity y H2)]. +Qed. + +Lemma sqrt_inj : (x,y:R) ``0<=x``->``0<=y``->(sqrt x)==(sqrt y)->x==y. +Intros; Cut ``(Rsqr (sqrt x))==(Rsqr (sqrt y))``. +Intro; Rewrite (Rsqr_sqrt x H) in H2; Rewrite (Rsqr_sqrt y H0) in H2; Assumption. +Rewrite H1; Reflexivity. +Qed. + +Lemma sqrt_less : (x:R) ``0<=x``->``1<x``->``(sqrt x)<x``. +Intros x H1 H2; Generalize (sqrt_lt_1 R1 x (Rlt_le R0 R1 (Rlt_R0_R1)) H1 H2); Intro H3; Rewrite sqrt_1 in H3; Generalize (Rmult_ne (sqrt x)); Intro H4; Elim H4; Intros H5 H6; Rewrite <- H5; Pattern 2 x; Rewrite <- (sqrt_def x H1); Apply (Rlt_monotony (sqrt x) R1 (sqrt x) (sqrt_lt_R0 x (Rlt_trans R0 R1 x Rlt_R0_R1 H2)) H3). +Qed. + +Lemma sqrt_more : (x:R) ``0<x``->``x<1``->``x<(sqrt x)``. +Intros x H1 H2; Generalize (sqrt_lt_1 x R1 (Rlt_le R0 x H1) (Rlt_le R0 R1 (Rlt_R0_R1)) H2); Intro H3; Rewrite sqrt_1 in H3; Generalize (Rmult_ne (sqrt x)); Intro H4; Elim H4; Intros H5 H6; Rewrite <- H5; Pattern 1 x; Rewrite <- (sqrt_def x (Rlt_le R0 x H1)); Apply (Rlt_monotony (sqrt x) (sqrt x) R1 (sqrt_lt_R0 x H1) H3). +Qed. + +Lemma sqrt_cauchy : (a,b,c,d:R) ``a*c+b*d<=(sqrt ((Rsqr a)+(Rsqr b)))*(sqrt ((Rsqr c)+(Rsqr d)))``. +Intros a b c d; Apply Rsqr_incr_0_var; [Rewrite Rsqr_times; Repeat Rewrite Rsqr_sqrt; Unfold Rsqr; [Replace ``(a*c+b*d)*(a*c+b*d)`` with ``(a*a*c*c+b*b*d*d)+(2*a*b*c*d)``; [Replace ``(a*a+b*b)*(c*c+d*d)`` with ``(a*a*c*c+b*b*d*d)+(a*a*d*d+b*b*c*c)``; [Apply Rle_compatibility; Replace ``a*a*d*d+b*b*c*c`` with ``(2*a*b*c*d)+(a*a*d*d+b*b*c*c-2*a*b*c*d)``; [Pattern 1 ``2*a*b*c*d``; Rewrite <- Rplus_Or; Apply Rle_compatibility; Replace ``a*a*d*d+b*b*c*c-2*a*b*c*d`` with (Rsqr (Rminus (Rmult a d) (Rmult b c))); [Apply pos_Rsqr | Unfold Rsqr; Ring] | Ring] | Ring] | Ring] | Apply (ge0_plus_ge0_is_ge0 (Rsqr c) (Rsqr d) (pos_Rsqr c) (pos_Rsqr d)) | Apply (ge0_plus_ge0_is_ge0 (Rsqr a) (Rsqr b) (pos_Rsqr a) (pos_Rsqr b))] | Apply Rmult_le_pos; Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr]. +Qed. + +(************************************************************) +(* Resolution of [a*X^2+b*X+c=0] *) +(************************************************************) + +Definition Delta [a:nonzeroreal;b,c:R] : R := ``(Rsqr b)-4*a*c``. + +Definition Delta_is_pos [a:nonzeroreal;b,c:R] : Prop := ``0<=(Delta a b c)``. + +Definition sol_x1 [a:nonzeroreal;b,c:R] : R := ``(-b+(sqrt (Delta a b c)))/(2*a)``. + +Definition sol_x2 [a:nonzeroreal;b,c:R] : R := ``(-b-(sqrt (Delta a b c)))/(2*a)``. + +Lemma Rsqr_sol_eq_0_1 : (a:nonzeroreal;b,c,x:R) (Delta_is_pos a b c) -> (x==(sol_x1 a b c))\/(x==(sol_x2 a b c)) -> ``a*(Rsqr x)+b*x+c==0``. +Intros; Elim H0; Intro. +Unfold sol_x1 in H1; Unfold Delta in H1; Rewrite H1; Unfold Rdiv; Repeat Rewrite Rsqr_times; Rewrite Rsqr_plus; Rewrite <- Rsqr_neg; Rewrite Rsqr_sqrt. +Rewrite Rsqr_inv. +Unfold Rsqr; Repeat Rewrite Rinv_Rmult. +Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym a). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite Rmult_Rplus_distrl. +Repeat Rewrite Rmult_assoc. +Pattern 2 ``2``; Rewrite (Rmult_sym ``2``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite (Rmult_Rplus_distrl ``-b`` ``(sqrt (b*b-(2*(2*(a*c)))))`` ``(/2*/a)``). +Rewrite Rmult_Rplus_distr; Repeat Rewrite Rplus_assoc. +Replace ``( -b*((sqrt (b*b-(2*(2*(a*c)))))*(/2*/a))+(b*( -b*(/2*/a))+(b*((sqrt (b*b-(2*(2*(a*c)))))*(/2*/a))+c)))`` with ``(b*( -b*(/2*/a)))+c``. +Unfold Rminus; Repeat Rewrite <- Rplus_assoc. +Replace ``b*b+b*b`` with ``2*(b*b)``. +Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite Ropp_mul1; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym a); Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite <- Ropp_mul2. +Ring. +Apply (cond_nonzero a). +DiscrR. +DiscrR. +DiscrR. +Ring. +Ring. +DiscrR. +Apply (cond_nonzero a). +DiscrR. +Apply (cond_nonzero a). +Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. +Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. +Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. +Assumption. +Unfold sol_x2 in H1; Unfold Delta in H1; Rewrite H1; Unfold Rdiv; Repeat Rewrite Rsqr_times; Rewrite Rsqr_minus; Rewrite <- Rsqr_neg; Rewrite Rsqr_sqrt. +Rewrite Rsqr_inv. +Unfold Rsqr; Repeat Rewrite Rinv_Rmult; Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym a); Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Unfold Rminus; Rewrite Rmult_Rplus_distrl. +Rewrite Ropp_mul1; Repeat Rewrite Rmult_assoc; Pattern 2 ``2``; Rewrite (Rmult_sym ``2``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite (Rmult_Rplus_distrl ``-b`` ``-(sqrt (b*b+ -(2*(2*(a*c))))) `` ``(/2*/a)``). +Rewrite Rmult_Rplus_distr; Repeat Rewrite Rplus_assoc. +Rewrite Ropp_mul1; Rewrite Ropp_Ropp. +Replace ``(b*((sqrt (b*b+ -(2*(2*(a*c)))))*(/2*/a))+(b*( -b*(/2*/a))+(b*( -(sqrt (b*b+ -(2*(2*(a*c)))))*(/2*/a))+c)))`` with ``(b*( -b*(/2*/a)))+c``. +Repeat Rewrite <- Rplus_assoc; Replace ``b*b+b*b`` with ``2*(b*b)``. +Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Ropp_mul1; Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym a); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite <- Ropp_mul2; Ring. +Apply (cond_nonzero a). +DiscrR. +DiscrR. +DiscrR. +Ring. +Ring. +DiscrR. +Apply (cond_nonzero a). +DiscrR. +DiscrR. +Apply (cond_nonzero a). +Apply prod_neq_R0; DiscrR Orelse Apply (cond_nonzero a). +Apply prod_neq_R0; DiscrR Orelse Apply (cond_nonzero a). +Apply prod_neq_R0; DiscrR Orelse Apply (cond_nonzero a). +Assumption. +Qed. + +Lemma Rsqr_sol_eq_0_0 : (a:nonzeroreal;b,c,x:R) (Delta_is_pos a b c) -> ``a*(Rsqr x)+b*x+c==0`` -> (x==(sol_x1 a b c))\/(x==(sol_x2 a b c)). +Intros; Rewrite (canonical_Rsqr a b c x) in H0; Rewrite Rplus_sym in H0; Generalize (Rplus_Ropp ``(4*a*c-(Rsqr b))/(4*a)`` ``a*(Rsqr (x+b/(2*a)))`` H0); Cut ``(Rsqr b)-4*a*c==(Delta a b c)``. +Intro; Replace ``-((4*a*c-(Rsqr b))/(4*a))`` with ``((Rsqr b)-4*a*c)/(4*a)``. +Rewrite H1; Intro; Generalize (Rmult_mult_r ``/a`` ``a*(Rsqr (x+b/(2*a)))`` ``(Delta a b c)/(4*a)`` H2); Replace ``/a*(a*(Rsqr (x+b/(2*a))))`` with ``(Rsqr (x+b/(2*a)))``. +Replace ``/a*(Delta a b c)/(4*a)`` with ``(Rsqr ((sqrt (Delta a b c))/(2*a)))``. +Intro; Generalize (Rsqr_eq ``(x+b/(2*a))`` ``((sqrt (Delta a b c))/(2*a))`` H3); Intro; Elim H4; Intro. +Left; Unfold sol_x1; Generalize (Rplus_plus_r ``-(b/(2*a))`` ``x+b/(2*a)`` ``(sqrt (Delta a b c))/(2*a)`` H5); Replace `` -(b/(2*a))+(x+b/(2*a))`` with x. +Intro; Rewrite H6; Unfold Rdiv; Ring. +Ring. +Right; Unfold sol_x2; Generalize (Rplus_plus_r ``-(b/(2*a))`` ``x+b/(2*a)`` ``-((sqrt (Delta a b c))/(2*a))`` H5); Replace `` -(b/(2*a))+(x+b/(2*a))`` with x. +Intro; Rewrite H6; Unfold Rdiv; Ring. +Ring. +Rewrite Rsqr_div. +Rewrite Rsqr_sqrt. +Unfold Rdiv. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym ``/a``). +Rewrite Rmult_assoc. +Rewrite <- Rinv_Rmult. +Replace ``(2*(2*a))*a`` with ``(Rsqr (2*a))``. +Reflexivity. +SqRing. +Rewrite <- Rmult_assoc; Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. +Apply (cond_nonzero a). +Assumption. +Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Symmetry; Apply Rmult_1l. +Apply (cond_nonzero a). +Unfold Rdiv; Rewrite <- Ropp_mul1. +Rewrite Ropp_distr2. +Reflexivity. +Reflexivity. +Qed. diff --git a/theories7/Reals/Ranalysis.v b/theories7/Reals/Ranalysis.v new file mode 100644 index 00000000..d5d84f50 --- /dev/null +++ b/theories7/Reals/Ranalysis.v @@ -0,0 +1,477 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Ranalysis.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rtrigo. +Require SeqSeries. +Require Export Ranalysis1. +Require Export Ranalysis2. +Require Export Ranalysis3. +Require Export Rtopology. +Require Export MVT. +Require Export PSeries_reg. +Require Export Exp_prop. +Require Export Rtrigo_reg. +Require Export Rsqrt_def. +Require Export R_sqrt. +Require Export Rtrigo_calc. +Require Export Rgeom. +Require Export RList. +Require Export Sqrt_reg. +Require Export Ranalysis4. +Require Export Rpower. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Axiom AppVar : R. + +(**********) +Recursive Tactic Definition IntroHypG trm := +Match trm With +|[(plus_fct ?1 ?2)] -> + (Match Context With + |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 + |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 + | _ -> Idtac) +|[(minus_fct ?1 ?2)] -> + (Match Context With + |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 + |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 + | _ -> Idtac) +|[(mult_fct ?1 ?2)] -> + (Match Context With + |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 + |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 + | _ -> Idtac) +|[(div_fct ?1 ?2)] -> Let aux = ?2 In + (Match Context With + |[_:(x0:R)``(aux x0)<>0``|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 + |[_:(x0:R)``(aux x0)<>0``|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 + |[|-(derivable ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1; IntroHypG ?2 | Try Assumption] + |[|-(continuity ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1; IntroHypG ?2 | Try Assumption] + | _ -> Idtac) +|[(comp ?1 ?2)] -> + (Match Context With + |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 + |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 + | _ -> Idtac) +|[(opp_fct ?1)] -> + (Match Context With + |[|-(derivable ?)] -> IntroHypG ?1 + |[|-(continuity ?)] -> IntroHypG ?1 + | _ -> Idtac) +|[(inv_fct ?1)] -> Let aux = ?1 In + (Match Context With + |[_:(x0:R)``(aux x0)<>0``|-(derivable ?)] -> IntroHypG ?1 + |[_:(x0:R)``(aux x0)<>0``|-(continuity ?)] -> IntroHypG ?1 + |[|-(derivable ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1 | Try Assumption] + |[|-(continuity ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1| Try Assumption] + | _ -> Idtac) +|[cos] -> Idtac +|[sin] -> Idtac +|[cosh] -> Idtac +|[sinh] -> Idtac +|[exp] -> Idtac +|[Rsqr] -> Idtac +|[sqrt] -> Idtac +|[id] -> Idtac +|[(fct_cte ?)] -> Idtac +|[(pow_fct ?)] -> Idtac +|[Rabsolu] -> Idtac +|[?1] -> Let p = ?1 In + (Match Context With + |[_:(derivable p)|- ?] -> Idtac + |[|-(derivable p)] -> Idtac + |[|-(derivable ?)] -> Cut True -> (derivable p); [Intro HYPPD; Cut (derivable p); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] + | [_:(continuity p)|- ?] -> Idtac + |[|-(continuity p)] -> Idtac + |[|-(continuity ?)] -> Cut True -> (continuity p); [Intro HYPPD; Cut (continuity p); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] + | _ -> Idtac). + +(**********) +Recursive Tactic Definition IntroHypL trm pt := +Match trm With +|[(plus_fct ?1 ?2)] -> + (Match Context With + |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + | _ -> Idtac) +|[(minus_fct ?1 ?2)] -> + (Match Context With + |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + | _ -> Idtac) +|[(mult_fct ?1 ?2)] -> + (Match Context With + |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + | _ -> Idtac) +|[(div_fct ?1 ?2)] -> Let aux = ?2 In + (Match Context With + |[_:``(aux pt)<>0``|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[_:``(aux pt)<>0``|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[_:``(aux pt)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt + |[id:(x0:R)``(aux x0)<>0``|-(derivable_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt + |[id:(x0:R)``(aux x0)<>0``|-(continuity_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt + |[id:(x0:R)``(aux x0)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt + |[|-(derivable_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] + |[|-(continuity_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] + | _ -> Idtac) +|[(comp ?1 ?2)] -> + (Match Context With + |[|-(derivable_pt ? ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt + |[|-(continuity_pt ? ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt + | _ -> Idtac) +|[(opp_fct ?1)] -> + (Match Context With + |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt + |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt + | _ -> Idtac) +|[(inv_fct ?1)] -> Let aux = ?1 In + (Match Context With + |[_:``(aux pt)<>0``|-(derivable_pt ? ?)] -> IntroHypL ?1 pt + |[_:``(aux pt)<>0``|-(continuity_pt ? ?)] -> IntroHypL ?1 pt + |[_:``(aux pt)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt + |[id:(x0:R)``(aux x0)<>0``|-(derivable_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt + |[id:(x0:R)``(aux x0)<>0``|-(continuity_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt + |[id:(x0:R)``(aux x0)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt + |[|-(derivable_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt | Try Assumption] + |[|-(continuity_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt| Try Assumption] + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt | Try Assumption] + | _ -> Idtac) +|[cos] -> Idtac +|[sin] -> Idtac +|[cosh] -> Idtac +|[sinh] -> Idtac +|[exp] -> Idtac +|[Rsqr] -> Idtac +|[id] -> Idtac +|[(fct_cte ?)] -> Idtac +|[(pow_fct ?)] -> Idtac +|[sqrt] -> + (Match Context With + |[|-(derivable_pt ? ?)] -> Cut ``0<pt``; [Intro | Try Assumption] + |[|-(continuity_pt ? ?)] -> Cut ``0<=pt``; [Intro | Try Assumption] + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``0<pt``; [Intro | Try Assumption] + | _ -> Idtac) +|[Rabsolu] -> + (Match Context With + |[|-(derivable_pt ? ?)] -> Cut ``pt<>0``; [Intro | Try Assumption] + | _ -> Idtac) +|[?1] -> Let p = ?1 In + (Match Context With + |[_:(derivable_pt p pt)|- ?] -> Idtac + |[|-(derivable_pt p pt)] -> Idtac + |[|-(derivable_pt ? ?)] -> Cut True -> (derivable_pt p pt); [Intro HYPPD; Cut (derivable_pt p pt); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] + |[_:(continuity_pt p pt)|- ?] -> Idtac + |[|-(continuity_pt p pt)] -> Idtac + |[|-(continuity_pt ? ?)] -> Cut True -> (continuity_pt p pt); [Intro HYPPD; Cut (continuity_pt p pt); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] + |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut True -> (derivable_pt p pt); [Intro HYPPD; Cut (derivable_pt p pt); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] + | _ -> Idtac). + +(**********) +Recursive Tactic Definition IsDiff_pt := +Match Context With + (* fonctions de base *) + [|-(derivable_pt Rsqr ?)] -> Apply derivable_pt_Rsqr +|[|-(derivable_pt id ?1)] -> Apply (derivable_pt_id ?1) +|[|-(derivable_pt (fct_cte ?) ?)] -> Apply derivable_pt_const +|[|-(derivable_pt sin ?)] -> Apply derivable_pt_sin +|[|-(derivable_pt cos ?)] -> Apply derivable_pt_cos +|[|-(derivable_pt sinh ?)] -> Apply derivable_pt_sinh +|[|-(derivable_pt cosh ?)] -> Apply derivable_pt_cosh +|[|-(derivable_pt exp ?)] -> Apply derivable_pt_exp +|[|-(derivable_pt (pow_fct ?) ?)] -> Unfold pow_fct; Apply derivable_pt_pow +|[|-(derivable_pt sqrt ?1)] -> Apply (derivable_pt_sqrt ?1); Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct +|[|-(derivable_pt Rabsolu ?1)] -> Apply (derivable_pt_Rabsolu ?1); Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct + (* regles de differentiabilite *) + (* PLUS *) +|[|-(derivable_pt (plus_fct ?1 ?2) ?3)] -> Apply (derivable_pt_plus ?1 ?2 ?3); IsDiff_pt + (* MOINS *) +|[|-(derivable_pt (minus_fct ?1 ?2) ?3)] -> Apply (derivable_pt_minus ?1 ?2 ?3); IsDiff_pt + (* OPPOSE *) +|[|-(derivable_pt (opp_fct ?1) ?2)] -> Apply (derivable_pt_opp ?1 ?2); IsDiff_pt + (* MULTIPLICATION PAR UN SCALAIRE *) +|[|-(derivable_pt (mult_real_fct ?1 ?2) ?3)] -> Apply (derivable_pt_scal ?2 ?1 ?3); IsDiff_pt + (* MULTIPLICATION *) +|[|-(derivable_pt (mult_fct ?1 ?2) ?3)] -> Apply (derivable_pt_mult ?1 ?2 ?3); IsDiff_pt + (* DIVISION *) + |[|-(derivable_pt (div_fct ?1 ?2) ?3)] -> Apply (derivable_pt_div ?1 ?2 ?3); [IsDiff_pt | IsDiff_pt | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp pow_fct id fct_cte] + (* INVERSION *) + |[|-(derivable_pt (inv_fct ?1) ?2)] -> Apply (derivable_pt_inv ?1 ?2); [Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp pow_fct id fct_cte | IsDiff_pt] + (* COMPOSITION *) +|[|-(derivable_pt (comp ?1 ?2) ?3)] -> Apply (derivable_pt_comp ?2 ?1 ?3); IsDiff_pt +|[_:(derivable_pt ?1 ?2)|-(derivable_pt ?1 ?2)] -> Assumption +|[_:(derivable ?1) |- (derivable_pt ?1 ?2)] -> Cut (derivable ?1); [Intro HypDDPT; Apply HypDDPT | Assumption] +|[|-True->(derivable_pt ? ?)] -> Intro HypTruE; Clear HypTruE; IsDiff_pt +| _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. + +(**********) +Recursive Tactic Definition IsDiff_glob := +Match Context With + (* fonctions de base *) + [|-(derivable Rsqr)] -> Apply derivable_Rsqr + |[|-(derivable id)] -> Apply derivable_id + |[|-(derivable (fct_cte ?))] -> Apply derivable_const + |[|-(derivable sin)] -> Apply derivable_sin + |[|-(derivable cos)] -> Apply derivable_cos + |[|-(derivable cosh)] -> Apply derivable_cosh + |[|-(derivable sinh)] -> Apply derivable_sinh + |[|-(derivable exp)] -> Apply derivable_exp + |[|-(derivable (pow_fct ?))] -> Unfold pow_fct; Apply derivable_pow + (* regles de differentiabilite *) + (* PLUS *) + |[|-(derivable (plus_fct ?1 ?2))] -> Apply (derivable_plus ?1 ?2); IsDiff_glob + (* MOINS *) + |[|-(derivable (minus_fct ?1 ?2))] -> Apply (derivable_minus ?1 ?2); IsDiff_glob + (* OPPOSE *) + |[|-(derivable (opp_fct ?1))] -> Apply (derivable_opp ?1); IsDiff_glob + (* MULTIPLICATION PAR UN SCALAIRE *) + |[|-(derivable (mult_real_fct ?1 ?2))] -> Apply (derivable_scal ?2 ?1); IsDiff_glob + (* MULTIPLICATION *) + |[|-(derivable (mult_fct ?1 ?2))] -> Apply (derivable_mult ?1 ?2); IsDiff_glob + (* DIVISION *) + |[|-(derivable (div_fct ?1 ?2))] -> Apply (derivable_div ?1 ?2); [IsDiff_glob | IsDiff_glob | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct] + (* INVERSION *) + |[|-(derivable (inv_fct ?1))] -> Apply (derivable_inv ?1); [Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct | IsDiff_glob] + (* COMPOSITION *) + |[|-(derivable (comp sqrt ?))] -> Unfold derivable; Intro; Try IsDiff_pt + |[|-(derivable (comp Rabsolu ?))] -> Unfold derivable; Intro; Try IsDiff_pt + |[|-(derivable (comp ?1 ?2))] -> Apply (derivable_comp ?2 ?1); IsDiff_glob + |[_:(derivable ?1)|-(derivable ?1)] -> Assumption + |[|-True->(derivable ?)] -> Intro HypTruE; Clear HypTruE; IsDiff_glob + | _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. + +(**********) +Recursive Tactic Definition IsCont_pt := +Match Context With + (* fonctions de base *) + [|-(continuity_pt Rsqr ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_Rsqr +|[|-(continuity_pt id ?1)] -> Apply derivable_continuous_pt; Apply (derivable_pt_id ?1) +|[|-(continuity_pt (fct_cte ?) ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_const +|[|-(continuity_pt sin ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_sin +|[|-(continuity_pt cos ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_cos +|[|-(continuity_pt sinh ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_sinh +|[|-(continuity_pt cosh ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_cosh +|[|-(continuity_pt exp ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_exp +|[|-(continuity_pt (pow_fct ?) ?)] -> Unfold pow_fct; Apply derivable_continuous_pt; Apply derivable_pt_pow +|[|-(continuity_pt sqrt ?1)] -> Apply continuity_pt_sqrt; Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct +|[|-(continuity_pt Rabsolu ?1)] -> Apply (continuity_Rabsolu ?1) + (* regles de differentiabilite *) + (* PLUS *) +|[|-(continuity_pt (plus_fct ?1 ?2) ?3)] -> Apply (continuity_pt_plus ?1 ?2 ?3); IsCont_pt + (* MOINS *) +|[|-(continuity_pt (minus_fct ?1 ?2) ?3)] -> Apply (continuity_pt_minus ?1 ?2 ?3); IsCont_pt + (* OPPOSE *) +|[|-(continuity_pt (opp_fct ?1) ?2)] -> Apply (continuity_pt_opp ?1 ?2); IsCont_pt + (* MULTIPLICATION PAR UN SCALAIRE *) +|[|-(continuity_pt (mult_real_fct ?1 ?2) ?3)] -> Apply (continuity_pt_scal ?2 ?1 ?3); IsCont_pt + (* MULTIPLICATION *) +|[|-(continuity_pt (mult_fct ?1 ?2) ?3)] -> Apply (continuity_pt_mult ?1 ?2 ?3); IsCont_pt + (* DIVISION *) + |[|-(continuity_pt (div_fct ?1 ?2) ?3)] -> Apply (continuity_pt_div ?1 ?2 ?3); [IsCont_pt | IsCont_pt | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp id fct_cte pow_fct] + (* INVERSION *) + |[|-(continuity_pt (inv_fct ?1) ?2)] -> Apply (continuity_pt_inv ?1 ?2); [IsCont_pt | Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp id fct_cte pow_fct] + (* COMPOSITION *) +|[|-(continuity_pt (comp ?1 ?2) ?3)] -> Apply (continuity_pt_comp ?2 ?1 ?3); IsCont_pt +|[_:(continuity_pt ?1 ?2)|-(continuity_pt ?1 ?2)] -> Assumption +|[_:(continuity ?1) |- (continuity_pt ?1 ?2)] -> Cut (continuity ?1); [Intro HypDDPT; Apply HypDDPT | Assumption] +|[_:(derivable_pt ?1 ?2)|-(continuity_pt ?1 ?2)] -> Apply derivable_continuous_pt; Assumption +|[_:(derivable ?1)|-(continuity_pt ?1 ?2)] -> Cut (continuity ?1); [Intro HypDDPT; Apply HypDDPT | Apply derivable_continuous; Assumption] +|[|-True->(continuity_pt ? ?)] -> Intro HypTruE; Clear HypTruE; IsCont_pt +| _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. + +(**********) +Recursive Tactic Definition IsCont_glob := +Match Context With + (* fonctions de base *) + [|-(continuity Rsqr)] -> Apply derivable_continuous; Apply derivable_Rsqr + |[|-(continuity id)] -> Apply derivable_continuous; Apply derivable_id + |[|-(continuity (fct_cte ?))] -> Apply derivable_continuous; Apply derivable_const + |[|-(continuity sin)] -> Apply derivable_continuous; Apply derivable_sin + |[|-(continuity cos)] -> Apply derivable_continuous; Apply derivable_cos + |[|-(continuity exp)] -> Apply derivable_continuous; Apply derivable_exp + |[|-(continuity (pow_fct ?))] -> Unfold pow_fct; Apply derivable_continuous; Apply derivable_pow + |[|-(continuity sinh)] -> Apply derivable_continuous; Apply derivable_sinh + |[|-(continuity cosh)] -> Apply derivable_continuous; Apply derivable_cosh + |[|-(continuity Rabsolu)] -> Apply continuity_Rabsolu + (* regles de continuite *) + (* PLUS *) +|[|-(continuity (plus_fct ?1 ?2))] -> Apply (continuity_plus ?1 ?2); Try IsCont_glob Orelse Assumption + (* MOINS *) +|[|-(continuity (minus_fct ?1 ?2))] -> Apply (continuity_minus ?1 ?2); Try IsCont_glob Orelse Assumption + (* OPPOSE *) +|[|-(continuity (opp_fct ?1))] -> Apply (continuity_opp ?1); Try IsCont_glob Orelse Assumption + (* INVERSE *) +|[|-(continuity (inv_fct ?1))] -> Apply (continuity_inv ?1); Try IsCont_glob Orelse Assumption + (* MULTIPLICATION PAR UN SCALAIRE *) +|[|-(continuity (mult_real_fct ?1 ?2))] -> Apply (continuity_scal ?2 ?1); Try IsCont_glob Orelse Assumption + (* MULTIPLICATION *) +|[|-(continuity (mult_fct ?1 ?2))] -> Apply (continuity_mult ?1 ?2); Try IsCont_glob Orelse Assumption + (* DIVISION *) + |[|-(continuity (div_fct ?1 ?2))] -> Apply (continuity_div ?1 ?2); [Try IsCont_glob Orelse Assumption | Try IsCont_glob Orelse Assumption | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte pow_fct] + (* COMPOSITION *) + |[|-(continuity (comp sqrt ?))] -> Unfold continuity_pt; Intro; Try IsCont_pt + |[|-(continuity (comp ?1 ?2))] -> Apply (continuity_comp ?2 ?1); Try IsCont_glob Orelse Assumption + |[_:(continuity ?1)|-(continuity ?1)] -> Assumption + |[|-True->(continuity ?)] -> Intro HypTruE; Clear HypTruE; IsCont_glob + |[_:(derivable ?1)|-(continuity ?1)] -> Apply derivable_continuous; Assumption + | _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. + +(**********) +Recursive Tactic Definition RewTerm trm := +Match trm With +| [(Rplus ?1 ?2)] -> Let p1= (RewTerm ?1) And p2 = (RewTerm ?2) In + (Match p1 With + [(fct_cte ?3)] -> + (Match p2 With + | [(fct_cte ?4)] -> '(fct_cte (Rplus ?3 ?4)) + | _ -> '(plus_fct p1 p2)) + | _ -> '(plus_fct p1 p2)) +| [(Rminus ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In + (Match p1 With + [(fct_cte ?3)] -> + (Match p2 With + | [(fct_cte ?4)] -> '(fct_cte (Rminus ?3 ?4)) + | _ -> '(minus_fct p1 p2)) + | _ -> '(minus_fct p1 p2)) +| [(Rdiv ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In + (Match p1 With + [(fct_cte ?3)] -> + (Match p2 With + | [(fct_cte ?4)] -> '(fct_cte (Rdiv ?3 ?4)) + | _ -> '(div_fct p1 p2)) + | _ -> + (Match p2 With + | [(fct_cte ?4)] -> '(mult_fct p1 (fct_cte (Rinv ?4))) + | _ -> '(div_fct p1 p2))) +| [(Rmult ?1 (Rinv ?2))] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In + (Match p1 With + [(fct_cte ?3)] -> + (Match p2 With + | [(fct_cte ?4)] -> '(fct_cte (Rdiv ?3 ?4)) + | _ -> '(div_fct p1 p2)) + | _ -> + (Match p2 With + | [(fct_cte ?4)] -> '(mult_fct p1 (fct_cte (Rinv ?4))) + | _ -> '(div_fct p1 p2))) +| [(Rmult ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In + (Match p1 With + [(fct_cte ?3)] -> + (Match p2 With + | [(fct_cte ?4)] -> '(fct_cte (Rmult ?3 ?4)) + | _ -> '(mult_fct p1 p2)) + | _ -> '(mult_fct p1 p2)) +| [(Ropp ?1)] -> Let p = (RewTerm ?1) In + (Match p With + [(fct_cte ?2)] -> '(fct_cte (Ropp ?2)) + | _ -> '(opp_fct p)) +| [(Rinv ?1)] -> Let p = (RewTerm ?1) In + (Match p With + [(fct_cte ?2)] -> '(fct_cte (Rinv ?2)) + | _ -> '(inv_fct p)) +| [(?1 AppVar)] -> '?1 +| [(?1 ?2)] -> Let p = (RewTerm ?2) In + (Match p With + | [(fct_cte ?3)] -> '(fct_cte (?1 ?3)) + | _ -> '(comp ?1 p)) +| [AppVar] -> 'id +| [(pow AppVar ?1)] -> '(pow_fct ?1) +| [(pow ?1 ?2)] -> Let p = (RewTerm ?1) In + (Match p With + | [(fct_cte ?3)] -> '(fct_cte (pow_fct ?2 ?3)) + | _ -> '(comp (pow_fct ?2) p)) +| [?1]-> '(fct_cte ?1). + +(**********) +Recursive Tactic Definition ConsProof trm pt := +Match trm With +| [(plus_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_plus ?1 ?2 pt p1 p2) +| [(minus_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_minus ?1 ?2 pt p1 p2) +| [(mult_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_mult ?1 ?2 pt p1 p2) +| [(div_fct ?1 ?2)] -> + (Match Context With + |[id:~((?2 pt)==R0) |- ?] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_div ?1 ?2 pt p1 p2 id) + | _ -> 'False) +| [(inv_fct ?1)] -> + (Match Context With + |[id:~((?1 pt)==R0) |- ?] -> Let p1 = (ConsProof ?1 pt) In '(derivable_pt_inv ?1 pt p1 id) + | _ -> 'False) +| [(comp ?1 ?2)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In Let p1 = (ConsProof ?1 pt_f1) And p2 = (ConsProof ?2 pt) In '(derivable_pt_comp ?2 ?1 pt p2 p1) +| [(opp_fct ?1)] -> Let p1 = (ConsProof ?1 pt) In '(derivable_pt_opp ?1 pt p1) +| [sin] -> '(derivable_pt_sin pt) +| [cos] -> '(derivable_pt_cos pt) +| [sinh] -> '(derivable_pt_sinh pt) +| [cosh] -> '(derivable_pt_cosh pt) +| [exp] -> '(derivable_pt_exp pt) +| [id] -> '(derivable_pt_id pt) +| [Rsqr] -> '(derivable_pt_Rsqr pt) +| [sqrt] -> + (Match Context With + |[id:(Rlt R0 pt) |- ?] -> '(derivable_pt_sqrt pt id) + | _ -> 'False) +| [(fct_cte ?1)] -> '(derivable_pt_const ?1 pt) +| [?1] -> Let aux = ?1 In + (Match Context With + [ id : (derivable_pt aux pt) |- ?] -> 'id + |[ id : (derivable aux) |- ?] -> '(id pt) + | _ -> 'False). + +(**********) +Recursive Tactic Definition SimplifyDerive trm pt := +Match trm With +| [(plus_fct ?1 ?2)] -> Try Rewrite derive_pt_plus; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt +| [(minus_fct ?1 ?2)] -> Try Rewrite derive_pt_minus; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt +| [(mult_fct ?1 ?2)] -> Try Rewrite derive_pt_mult; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt +| [(div_fct ?1 ?2)] -> Try Rewrite derive_pt_div; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt +| [(comp ?1 ?2)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In Try Rewrite derive_pt_comp; SimplifyDerive ?1 pt_f1; SimplifyDerive ?2 pt +| [(opp_fct ?1)] -> Try Rewrite derive_pt_opp; SimplifyDerive ?1 pt +| [(inv_fct ?1)] -> Try Rewrite derive_pt_inv; SimplifyDerive ?1 pt +| [(fct_cte ?1)] -> Try Rewrite derive_pt_const +| [id] -> Try Rewrite derive_pt_id +| [sin] -> Try Rewrite derive_pt_sin +| [cos] -> Try Rewrite derive_pt_cos +| [sinh] -> Try Rewrite derive_pt_sinh +| [cosh] -> Try Rewrite derive_pt_cosh +| [exp] -> Try Rewrite derive_pt_exp +| [Rsqr] -> Try Rewrite derive_pt_Rsqr +| [sqrt] -> Try Rewrite derive_pt_sqrt +| [?1] -> Let aux = ?1 In + (Match Context With + [ id : (eqT ? (derive_pt aux pt ?2) ?); H : (derivable aux) |- ? ] -> Try Replace (derive_pt aux pt (H pt)) with (derive_pt aux pt ?2); [Rewrite id | Apply pr_nu] + |[ id : (eqT ? (derive_pt aux pt ?2) ?); H : (derivable_pt aux pt) |- ? ] -> Try Replace (derive_pt aux pt H) with (derive_pt aux pt ?2); [Rewrite id | Apply pr_nu] + | _ -> Idtac ) +| _ -> Idtac. + +(**********) +Tactic Definition Reg := +Match Context With +| [|-(derivable_pt ?1 ?2)] -> +Let trm = Eval Cbv Beta in (?1 AppVar) In +Let aux = (RewTerm trm) In IntroHypL aux ?2; Try (Change (derivable_pt aux ?2); IsDiff_pt) Orelse IsDiff_pt +| [|-(derivable ?1)] -> +Let trm = Eval Cbv Beta in (?1 AppVar) In +Let aux = (RewTerm trm) In IntroHypG aux; Try (Change (derivable aux); IsDiff_glob) Orelse IsDiff_glob +| [|-(continuity ?1)] -> +Let trm = Eval Cbv Beta in (?1 AppVar) In +Let aux = (RewTerm trm) In IntroHypG aux; Try (Change (continuity aux); IsCont_glob) Orelse IsCont_glob +| [|-(continuity_pt ?1 ?2)] -> +Let trm = Eval Cbv Beta in (?1 AppVar) In +Let aux = (RewTerm trm) In IntroHypL aux ?2; Try (Change (continuity_pt aux ?2); IsCont_pt) Orelse IsCont_pt +| [|-(eqT ? (derive_pt ?1 ?2 ?3) ?4)] -> +Let trm = Eval Cbv Beta in (?1 AppVar) In +Let aux = (RewTerm trm) In +IntroHypL aux ?2; Let aux2 = (ConsProof aux ?2) In Try (Replace (derive_pt ?1 ?2 ?3) with (derive_pt aux ?2 aux2); [SimplifyDerive aux ?2; Try Unfold plus_fct minus_fct mult_fct div_fct id fct_cte inv_fct opp_fct; Try Ring | Try Apply pr_nu]) Orelse IsDiff_pt. diff --git a/theories7/Reals/Ranalysis1.v b/theories7/Reals/Ranalysis1.v new file mode 100644 index 00000000..8cb4c358 --- /dev/null +++ b/theories7/Reals/Ranalysis1.v @@ -0,0 +1,1046 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Ranalysis1.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Export Rlimit. +Require Export Rderiv. +V7only [Import R_scope.]. Open Local Scope R_scope. +Implicit Variable Type f:R->R. + +(****************************************************) +(** Basic operations on functions *) +(****************************************************) +Definition plus_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)+(f2 x)``. +Definition opp_fct [f:R->R] : R->R := [x:R] ``-(f x)``. +Definition mult_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)*(f2 x)``. +Definition mult_real_fct [a:R;f:R->R] : R->R := [x:R] ``a*(f x)``. +Definition minus_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)-(f2 x)``. +Definition div_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)/(f2 x)``. +Definition div_real_fct [a:R;f:R->R] : R->R := [x:R] ``a/(f x)``. +Definition comp [f1,f2:R->R] : R->R := [x:R] ``(f1 (f2 x))``. +Definition inv_fct [f:R->R] : R->R := [x:R]``/(f x)``. + +V8Infix "+" plus_fct : Rfun_scope. +V8Notation "- x" := (opp_fct x) : Rfun_scope. +V8Infix "*" mult_fct : Rfun_scope. +V8Infix "-" minus_fct : Rfun_scope. +V8Infix "/" div_fct : Rfun_scope. +Notation Local "f1 'o' f2" := (comp f1 f2) (at level 2, right associativity) + : Rfun_scope + V8only (at level 20, right associativity). +V8Notation "/ x" := (inv_fct x) : Rfun_scope. + +Delimits Scope Rfun_scope with F. + +Definition fct_cte [a:R] : R->R := [x:R]a. +Definition id := [x:R]x. + +(****************************************************) +(** Variations of functions *) +(****************************************************) +Definition increasing [f:R->R] : Prop := (x,y:R) ``x<=y``->``(f x)<=(f y)``. +Definition decreasing [f:R->R] : Prop := (x,y:R) ``x<=y``->``(f y)<=(f x)``. +Definition strict_increasing [f:R->R] : Prop := (x,y:R) ``x<y``->``(f x)<(f y)``. +Definition strict_decreasing [f:R->R] : Prop := (x,y:R) ``x<y``->``(f y)<(f x)``. +Definition constant [f:R->R] : Prop := (x,y:R) ``(f x)==(f y)``. + +(**********) +Definition no_cond : R->Prop := [x:R] True. + +(**********) +Definition constant_D_eq [f:R->R;D:R->Prop;c:R] : Prop := (x:R) (D x) -> (f x)==c. + +(***************************************************) +(** Definition of continuity as a limit *) +(***************************************************) + +(**********) +Definition continuity_pt [f:R->R; x0:R] : Prop := (continue_in f no_cond x0). +Definition continuity [f:R->R] : Prop := (x:R) (continuity_pt f x). + +Arguments Scope continuity_pt [Rfun_scope R_scope]. +Arguments Scope continuity [Rfun_scope]. + +(**********) +Lemma continuity_pt_plus : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (plus_fct f1 f2) x0). +Unfold continuity_pt plus_fct; Unfold continue_in; Intros; Apply limit_plus; Assumption. +Qed. + +Lemma continuity_pt_opp : (f:R->R; x0:R) (continuity_pt f x0) -> (continuity_pt (opp_fct f) x0). +Unfold continuity_pt opp_fct; Unfold continue_in; Intros; Apply limit_Ropp; Assumption. +Qed. + +Lemma continuity_pt_minus : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (minus_fct f1 f2) x0). +Unfold continuity_pt minus_fct; Unfold continue_in; Intros; Apply limit_minus; Assumption. +Qed. + +Lemma continuity_pt_mult : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (mult_fct f1 f2) x0). +Unfold continuity_pt mult_fct; Unfold continue_in; Intros; Apply limit_mul; Assumption. +Qed. + +Lemma continuity_pt_const : (f:R->R; x0:R) (constant f) -> (continuity_pt f x0). +Unfold constant continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Intros; Exists ``1``; Split; [Apply Rlt_R0_R1 | Intros; Generalize (H x x0); Intro; Rewrite H2; Simpl; Rewrite R_dist_eq; Assumption]. +Qed. + +Lemma continuity_pt_scal : (f:R->R;a:R; x0:R) (continuity_pt f x0) -> (continuity_pt (mult_real_fct a f) x0). +Unfold continuity_pt mult_real_fct; Unfold continue_in; Intros; Apply (limit_mul ([x:R] a) f (D_x no_cond x0) a (f x0) x0). +Unfold limit1_in; Unfold limit_in; Intros; Exists ``1``; Split. +Apply Rlt_R0_R1. +Intros; Rewrite R_dist_eq; Assumption. +Assumption. +Qed. + +Lemma continuity_pt_inv : (f:R->R; x0:R) (continuity_pt f x0) -> ~``(f x0)==0`` -> (continuity_pt (inv_fct f) x0). +Intros. +Replace (inv_fct f) with [x:R]``/(f x)``. +Unfold continuity_pt; Unfold continue_in; Intros; Apply limit_inv; Assumption. +Unfold inv_fct; Reflexivity. +Qed. + +Lemma div_eq_inv : (f1,f2:R->R) (div_fct f1 f2)==(mult_fct f1 (inv_fct f2)). +Intros; Reflexivity. +Qed. + +Lemma continuity_pt_div : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> ~``(f2 x0)==0`` -> (continuity_pt (div_fct f1 f2) x0). +Intros; Rewrite -> (div_eq_inv f1 f2); Apply continuity_pt_mult; [Assumption | Apply continuity_pt_inv; Assumption]. +Qed. + +Lemma continuity_pt_comp : (f1,f2:R->R;x:R) (continuity_pt f1 x) -> (continuity_pt f2 (f1 x)) -> (continuity_pt (comp f2 f1) x). +Unfold continuity_pt; Unfold continue_in; Intros; Unfold comp. +Cut (limit1_in [x0:R](f2 (f1 x0)) (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) +(f2 (f1 x)) x) -> (limit1_in [x0:R](f2 (f1 x0)) (D_x no_cond x) (f2 (f1 x)) x). +Intro; Apply H1. +EApply limit_comp. +Apply H. +Apply H0. +Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Assert H3 := (H1 eps H2). +Elim H3; Intros. +Exists x0. +Split. +Elim H4; Intros; Assumption. +Intros; Case (Req_EM (f1 x) (f1 x1)); Intro. +Rewrite H6; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Elim H4; Intros; Apply H8. +Split. +Unfold Dgf D_x no_cond. +Split. +Split. +Trivial. +Elim H5; Unfold D_x no_cond; Intros. +Elim H9; Intros; Assumption. +Split. +Trivial. +Assumption. +Elim H5; Intros; Assumption. +Qed. + +(**********) +Lemma continuity_plus : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (plus_fct f1 f2)). +Unfold continuity; Intros; Apply (continuity_pt_plus f1 f2 x (H x) (H0 x)). +Qed. + +Lemma continuity_opp : (f:R->R) (continuity f)->(continuity (opp_fct f)). +Unfold continuity; Intros; Apply (continuity_pt_opp f x (H x)). +Qed. + +Lemma continuity_minus : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (minus_fct f1 f2)). +Unfold continuity; Intros; Apply (continuity_pt_minus f1 f2 x (H x) (H0 x)). +Qed. + +Lemma continuity_mult : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (mult_fct f1 f2)). +Unfold continuity; Intros; Apply (continuity_pt_mult f1 f2 x (H x) (H0 x)). +Qed. + +Lemma continuity_const : (f:R->R) (constant f) -> (continuity f). +Unfold continuity; Intros; Apply (continuity_pt_const f x H). +Qed. + +Lemma continuity_scal : (f:R->R;a:R) (continuity f) -> (continuity (mult_real_fct a f)). +Unfold continuity; Intros; Apply (continuity_pt_scal f a x (H x)). +Qed. + +Lemma continuity_inv : (f:R->R) (continuity f)->((x:R) ~``(f x)==0``)->(continuity (inv_fct f)). +Unfold continuity; Intros; Apply (continuity_pt_inv f x (H x) (H0 x)). +Qed. + +Lemma continuity_div : (f1,f2:R->R) (continuity f1)->(continuity f2)->((x:R) ~``(f2 x)==0``)->(continuity (div_fct f1 f2)). +Unfold continuity; Intros; Apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)). +Qed. + +Lemma continuity_comp : (f1,f2:R->R) (continuity f1) -> (continuity f2) -> (continuity (comp f2 f1)). +Unfold continuity; Intros. +Apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))). +Qed. + + +(*****************************************************) +(** Derivative's definition using Landau's kernel *) +(*****************************************************) + +Definition derivable_pt_lim [f:R->R;x,l:R] : Prop := ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``))). + +Definition derivable_pt_abs [f:R->R;x:R] : R -> Prop := [l:R](derivable_pt_lim f x l). + +Definition derivable_pt [f:R->R;x:R] := (SigT R (derivable_pt_abs f x)). +Definition derivable [f:R->R] := (x:R)(derivable_pt f x). + +Definition derive_pt [f:R->R;x:R;pr:(derivable_pt f x)] := (projT1 ? ? pr). +Definition derive [f:R->R;pr:(derivable f)] := [x:R](derive_pt f x (pr x)). + +Arguments Scope derivable_pt_lim [Rfun_scope R_scope]. +Arguments Scope derivable_pt_abs [Rfun_scope R_scope R_scope]. +Arguments Scope derivable_pt [Rfun_scope R_scope]. +Arguments Scope derivable [Rfun_scope]. +Arguments Scope derive_pt [Rfun_scope R_scope _]. +Arguments Scope derive [Rfun_scope _]. + +Definition antiderivative [f,g:R->R;a,b:R] : Prop := ((x:R)``a<=x<=b``->(EXT pr : (derivable_pt g x) | (f x)==(derive_pt g x pr)))/\``a<=b``. +(************************************) +(** Class of differential functions *) +(************************************) +Record Differential : Type := mkDifferential { +d1 :> R->R; +cond_diff : (derivable d1) }. + +Record Differential_D2 : Type := mkDifferential_D2 { +d2 :> R->R; +cond_D1 : (derivable d2); +cond_D2 : (derivable (derive d2 cond_D1)) }. + +(**********) +Lemma unicite_step1 : (f:R->R;x,l1,l2:R) (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l1 R0) -> (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l2 R0) -> l1 == l2. +Intros; Apply (single_limit [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l1 l2 R0); Try Assumption. +Unfold adhDa; Intros; Exists ``alp/2``. +Split. +Unfold Rdiv; Apply prod_neq_R0. +Red; Intro; Rewrite H2 in H1; Elim (Rlt_antirefl ? H1). +Apply Rinv_neq_R0; DiscrR. +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rabsolu_mult. +Replace ``(Rabsolu (/2))`` with ``/2``. +Replace (Rabsolu alp) with alp. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1r; Rewrite double; Pattern 1 alp; Replace alp with ``alp+0``; [Idtac | Ring]; Apply Rlt_compatibility; Assumption. +Symmetry; Apply Rabsolu_right; Left; Assumption. +Symmetry; Apply Rabsolu_right; Left; Change ``0</2``; Apply Rlt_Rinv; Sup0. +Qed. + +Lemma unicite_step2 : (f:R->R;x,l:R) (derivable_pt_lim f x l) -> (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l R0). +Unfold derivable_pt_lim; Intros; Unfold limit1_in; Unfold limit_in; Intros. +Assert H1 := (H eps H0). +Elim H1 ; Intros. +Exists (pos x0). +Split. +Apply (cond_pos x0). +Simpl; Unfold R_dist; Intros. +Elim H3; Intros. +Apply H2; [Assumption |Unfold Rminus in H5; Rewrite Ropp_O in H5; Rewrite Rplus_Or in H5; Assumption]. +Qed. + +Lemma unicite_step3 : (f:R->R;x,l:R) (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l R0) -> (derivable_pt_lim f x l). +Unfold limit1_in derivable_pt_lim; Unfold limit_in; Unfold dist; Simpl; Intros. +Elim (H eps H0). +Intros; Elim H1; Intros. +Exists (mkposreal x0 H2). +Simpl; Intros; Unfold R_dist in H3; Apply (H3 h). +Split; [Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Assumption]. +Qed. + +Lemma unicite_limite : (f:R->R;x,l1,l2:R) (derivable_pt_lim f x l1) -> (derivable_pt_lim f x l2) -> l1==l2. +Intros. +Assert H1 := (unicite_step2 ? ? ? H). +Assert H2 := (unicite_step2 ? ? ? H0). +Assert H3 := (unicite_step1 ? ? ? ? H1 H2). +Assumption. +Qed. + +Lemma derive_pt_eq : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derive_pt f x pr)==l <-> (derivable_pt_lim f x l). +Intros; Split. +Intro; Assert H1 := (projT2 ? ? pr); Unfold derive_pt in H; Rewrite H in H1; Assumption. +Intro; Assert H1 := (projT2 ? ? pr); Unfold derivable_pt_abs in H1. +Assert H2 := (unicite_limite ? ? ? ? H H1). +Unfold derive_pt; Unfold derivable_pt_abs. +Symmetry; Assumption. +Qed. + +(**********) +Lemma derive_pt_eq_0 : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derivable_pt_lim f x l) -> (derive_pt f x pr)==l. +Intros; Elim (derive_pt_eq f x l pr); Intros. +Apply (H1 H). +Qed. + +(**********) +Lemma derive_pt_eq_1 : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derive_pt f x pr)==l -> (derivable_pt_lim f x l). +Intros; Elim (derive_pt_eq f x l pr); Intros. +Apply (H0 H). +Qed. + + +(********************************************************************) +(** Equivalence of this definition with the one using limit concept *) +(********************************************************************) +Lemma derive_pt_D_in : (f,df:R->R;x:R;pr:(derivable_pt f x)) (D_in f df no_cond x) <-> (derive_pt f x pr)==(df x). +Intros; Split. +Unfold D_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Apply derive_pt_eq_0. +Unfold derivable_pt_lim. +Intros; Elim (H eps H0); Intros alpha H1; Elim H1; Intros; Exists (mkposreal alpha H2); Intros; Generalize (H3 ``x+h``); Intro; Cut ``x+h-x==h``; [Intro; Cut ``(D_x no_cond x (x+h))``/\``(Rabsolu (x+h-x)) < alpha``; [Intro; Generalize (H6 H8); Rewrite H7; Intro; Assumption | Split; [Unfold D_x; Split; [Unfold no_cond; Trivial | Apply Rminus_not_eq_right; Rewrite H7; Assumption] | Rewrite H7; Assumption]] | Ring]. +Intro. +Assert H0 := (derive_pt_eq_1 f x (df x) pr H). +Unfold D_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Elim (H0 eps H1); Intros alpha H2; Exists (pos alpha); Split. +Apply (cond_pos alpha). +Intros; Elim H3; Intros; Unfold D_x in H4; Elim H4; Intros; Cut ``x0-x<>0``. +Intro; Generalize (H2 ``x0-x`` H8 H5); Replace ``x+(x0-x)`` with x0. +Intro; Assumption. +Ring. +Auto with real. +Qed. + +Lemma derivable_pt_lim_D_in : (f,df:R->R;x:R) (D_in f df no_cond x) <-> (derivable_pt_lim f x (df x)). +Intros; Split. +Unfold D_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Unfold derivable_pt_lim. +Intros; Elim (H eps H0); Intros alpha H1; Elim H1; Intros; Exists (mkposreal alpha H2); Intros; Generalize (H3 ``x+h``); Intro; Cut ``x+h-x==h``; [Intro; Cut ``(D_x no_cond x (x+h))``/\``(Rabsolu (x+h-x)) < alpha``; [Intro; Generalize (H6 H8); Rewrite H7; Intro; Assumption | Split; [Unfold D_x; Split; [Unfold no_cond; Trivial | Apply Rminus_not_eq_right; Rewrite H7; Assumption] | Rewrite H7; Assumption]] | Ring]. +Intro. +Unfold derivable_pt_lim in H. +Unfold D_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Elim (H eps H0); Intros alpha H2; Exists (pos alpha); Split. +Apply (cond_pos alpha). +Intros. +Elim H1; Intros; Unfold D_x in H3; Elim H3; Intros; Cut ``x0-x<>0``. +Intro; Generalize (H2 ``x0-x`` H7 H4); Replace ``x+(x0-x)`` with x0. +Intro; Assumption. +Ring. +Auto with real. +Qed. + + +(***********************************) +(** derivability -> continuity *) +(***********************************) +(**********) +Lemma derivable_derive : (f:R->R;x:R;pr:(derivable_pt f x)) (EXT l : R | (derive_pt f x pr)==l). +Intros; Exists (projT1 ? ? pr). +Unfold derive_pt; Reflexivity. +Qed. + +Theorem derivable_continuous_pt : (f:R->R;x:R) (derivable_pt f x) -> (continuity_pt f x). +Intros. +Generalize (derivable_derive f x X); Intro. +Elim H; Intros l H1. +Cut l==((fct_cte l) x). +Intro. +Rewrite H0 in H1. +Generalize (derive_pt_D_in f (fct_cte l) x); Intro. +Elim (H2 X); Intros. +Generalize (H4 H1); Intro. +Unfold continuity_pt. +Apply (cont_deriv f (fct_cte l) no_cond x H5). +Unfold fct_cte; Reflexivity. +Qed. + +Theorem derivable_continuous : (f:R->R) (derivable f) -> (continuity f). +Unfold derivable continuity; Intros. +Apply (derivable_continuous_pt f x (X x)). +Qed. + +(****************************************************************) +(** Main rules *) +(****************************************************************) + +Lemma derivable_pt_lim_plus : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (plus_fct f1 f2) x ``l1+l2``). +Intros. +Apply unicite_step3. +Assert H1 := (unicite_step2 ? ? ? H). +Assert H2 := (unicite_step2 ? ? ? H0). +Unfold plus_fct. +Cut (h:R)``((f1 (x+h))+(f2 (x+h))-((f1 x)+(f2 x)))/h``==``((f1 (x+h))-(f1 x))/h+((f2 (x+h))-(f2 x))/h``. +Intro. +Generalize(limit_plus [h':R]``((f1 (x+h'))-(f1 x))/h'`` [h':R]``((f2 (x+h'))-(f2 x))/h'`` [h:R]``h <> 0`` l1 l2 ``0`` H1 H2). +Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Elim (H4 eps H5); Intros. +Exists x0. +Elim H6; Intros. +Split. +Assumption. +Intros; Rewrite H3; Apply H8; Assumption. +Intro; Unfold Rdiv; Ring. +Qed. + +Lemma derivable_pt_lim_opp : (f:R->R;x,l:R) (derivable_pt_lim f x l) -> (derivable_pt_lim (opp_fct f) x (Ropp l)). +Intros. +Apply unicite_step3. +Assert H1 := (unicite_step2 ? ? ? H). +Unfold opp_fct. +Cut (h:R) ``( -(f (x+h))- -(f x))/h``==(Ropp ``((f (x+h))-(f x))/h``). +Intro. +Generalize (limit_Ropp [h:R]``((f (x+h))-(f x))/h``[h:R]``h <> 0`` l ``0`` H1). +Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Elim (H2 eps H3); Intros. +Exists x0. +Elim H4; Intros. +Split. +Assumption. +Intros; Rewrite H0; Apply H6; Assumption. +Intro; Unfold Rdiv; Ring. +Qed. + +Lemma derivable_pt_lim_minus : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (minus_fct f1 f2) x ``l1-l2``). +Intros. +Apply unicite_step3. +Assert H1 := (unicite_step2 ? ? ? H). +Assert H2 := (unicite_step2 ? ? ? H0). +Unfold minus_fct. +Cut (h:R)``((f1 (x+h))-(f1 x))/h-((f2 (x+h))-(f2 x))/h``==``((f1 (x+h))-(f2 (x+h))-((f1 x)-(f2 x)))/h``. +Intro. +Generalize (limit_minus [h':R]``((f1 (x+h'))-(f1 x))/h'`` [h':R]``((f2 (x+h'))-(f2 x))/h'`` [h:R]``h <> 0`` l1 l2 ``0`` H1 H2). +Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Elim (H4 eps H5); Intros. +Exists x0. +Elim H6; Intros. +Split. +Assumption. +Intros; Rewrite <- H3; Apply H8; Assumption. +Intro; Unfold Rdiv; Ring. +Qed. + +Lemma derivable_pt_lim_mult : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (mult_fct f1 f2) x ``l1*(f2 x)+(f1 x)*l2``). +Intros. +Assert H1 := (derivable_pt_lim_D_in f1 [y:R]l1 x). +Elim H1; Intros. +Assert H4 := (H3 H). +Assert H5 := (derivable_pt_lim_D_in f2 [y:R]l2 x). +Elim H5; Intros. +Assert H8 := (H7 H0). +Clear H1 H2 H3 H5 H6 H7. +Assert H1 := (derivable_pt_lim_D_in (mult_fct f1 f2) [y:R]``l1*(f2 x)+(f1 x)*l2`` x). +Elim H1; Intros. +Clear H1 H3. +Apply H2. +Unfold mult_fct. +Apply (Dmult no_cond [y:R]l1 [y:R]l2 f1 f2 x); Assumption. +Qed. + +Lemma derivable_pt_lim_const : (a,x:R) (derivable_pt_lim (fct_cte a) x ``0``). +Intros; Unfold fct_cte derivable_pt_lim. +Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Unfold Rminus; Rewrite Rplus_Ropp_r; Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Qed. + +Lemma derivable_pt_lim_scal : (f:R->R;a,x,l:R) (derivable_pt_lim f x l) -> (derivable_pt_lim (mult_real_fct a f) x ``a*l``). +Intros. +Assert H0 := (derivable_pt_lim_const a x). +Replace (mult_real_fct a f) with (mult_fct (fct_cte a) f). +Replace ``a*l`` with ``0*(f x)+a*l``; [Idtac | Ring]. +Apply (derivable_pt_lim_mult (fct_cte a) f x ``0`` l); Assumption. +Unfold mult_real_fct mult_fct fct_cte; Reflexivity. +Qed. + +Lemma derivable_pt_lim_id : (x:R) (derivable_pt_lim id x ``1``). +Intro; Unfold derivable_pt_lim. +Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Unfold id; Replace ``(x+h-x)/h-1`` with ``0``. +Rewrite Rabsolu_R0; Apply Rle_lt_trans with ``(Rabsolu h)``. +Apply Rabsolu_pos. +Assumption. +Unfold Rminus; Rewrite Rplus_assoc; Rewrite (Rplus_sym x); Rewrite Rplus_assoc. +Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Unfold Rdiv; Rewrite <- Rinv_r_sym. +Symmetry; Apply Rplus_Ropp_r. +Assumption. +Qed. + +Lemma derivable_pt_lim_Rsqr : (x:R) (derivable_pt_lim Rsqr x ``2*x``). +Intro; Unfold derivable_pt_lim. +Unfold Rsqr; Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Replace ``((x+h)*(x+h)-x*x)/h-2*x`` with ``h``. +Assumption. +Replace ``(x+h)*(x+h)-x*x`` with ``2*x*h+h*h``; [Idtac | Ring]. +Unfold Rdiv; Rewrite Rmult_Rplus_distrl. +Repeat Rewrite Rmult_assoc. +Repeat Rewrite <- Rinv_r_sym; [Idtac | Assumption]. +Ring. +Qed. + +Lemma derivable_pt_lim_comp : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 (f1 x) l2) -> (derivable_pt_lim (comp f2 f1) x ``l2*l1``). +Intros; Assert H1 := (derivable_pt_lim_D_in f1 [y:R]l1 x). +Elim H1; Intros. +Assert H4 := (H3 H). +Assert H5 := (derivable_pt_lim_D_in f2 [y:R]l2 (f1 x)). +Elim H5; Intros. +Assert H8 := (H7 H0). +Clear H1 H2 H3 H5 H6 H7. +Assert H1 := (derivable_pt_lim_D_in (comp f2 f1) [y:R]``l2*l1`` x). +Elim H1; Intros. +Clear H1 H3; Apply H2. +Unfold comp; Cut (D_in [x0:R](f2 (f1 x0)) [y:R]``l2*l1`` (Dgf no_cond no_cond f1) x) -> (D_in [x0:R](f2 (f1 x0)) [y:R]``l2*l1`` no_cond x). +Intro; Apply H1. +Rewrite Rmult_sym; Apply (Dcomp no_cond no_cond [y:R]l1 [y:R]l2 f1 f2 x); Assumption. +Unfold Dgf D_in no_cond; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Elim (H1 eps H3); Intros. +Exists x0; Intros; Split. +Elim H5; Intros; Assumption. +Intros; Elim H5; Intros; Apply H9; Split. +Unfold D_x; Split. +Split; Trivial. +Elim H6; Intros; Unfold D_x in H10; Elim H10; Intros; Assumption. +Elim H6; Intros; Assumption. +Qed. + +Lemma derivable_pt_plus : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (plus_fct f1 f2) x). +Unfold derivable_pt; Intros. +Elim X; Intros. +Elim X0; Intros. +Apply Specif.existT with ``x0+x1``. +Apply derivable_pt_lim_plus; Assumption. +Qed. + +Lemma derivable_pt_opp : (f:R->R;x:R) (derivable_pt f x) -> (derivable_pt (opp_fct f) x). +Unfold derivable_pt; Intros. +Elim X; Intros. +Apply Specif.existT with ``-x0``. +Apply derivable_pt_lim_opp; Assumption. +Qed. + +Lemma derivable_pt_minus : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (minus_fct f1 f2) x). +Unfold derivable_pt; Intros. +Elim X; Intros. +Elim X0; Intros. +Apply Specif.existT with ``x0-x1``. +Apply derivable_pt_lim_minus; Assumption. +Qed. + +Lemma derivable_pt_mult : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (mult_fct f1 f2) x). +Unfold derivable_pt; Intros. +Elim X; Intros. +Elim X0; Intros. +Apply Specif.existT with ``x0*(f2 x)+(f1 x)*x1``. +Apply derivable_pt_lim_mult; Assumption. +Qed. + +Lemma derivable_pt_const : (a,x:R) (derivable_pt (fct_cte a) x). +Intros; Unfold derivable_pt. +Apply Specif.existT with ``0``. +Apply derivable_pt_lim_const. +Qed. + +Lemma derivable_pt_scal : (f:R->R;a,x:R) (derivable_pt f x) -> (derivable_pt (mult_real_fct a f) x). +Unfold derivable_pt; Intros. +Elim X; Intros. +Apply Specif.existT with ``a*x0``. +Apply derivable_pt_lim_scal; Assumption. +Qed. + +Lemma derivable_pt_id : (x:R) (derivable_pt id x). +Unfold derivable_pt; Intro. +Exists ``1``. +Apply derivable_pt_lim_id. +Qed. + +Lemma derivable_pt_Rsqr : (x:R) (derivable_pt Rsqr x). +Unfold derivable_pt; Intro; Apply Specif.existT with ``2*x``. +Apply derivable_pt_lim_Rsqr. +Qed. + +Lemma derivable_pt_comp : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 (f1 x)) -> (derivable_pt (comp f2 f1) x). +Unfold derivable_pt; Intros. +Elim X; Intros. +Elim X0 ;Intros. +Apply Specif.existT with ``x1*x0``. +Apply derivable_pt_lim_comp; Assumption. +Qed. + +Lemma derivable_plus : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (plus_fct f1 f2)). +Unfold derivable; Intros. +Apply (derivable_pt_plus ? ? x (X ?) (X0 ?)). +Qed. + +Lemma derivable_opp : (f:R->R) (derivable f) -> (derivable (opp_fct f)). +Unfold derivable; Intros. +Apply (derivable_pt_opp ? x (X ?)). +Qed. + +Lemma derivable_minus : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (minus_fct f1 f2)). +Unfold derivable; Intros. +Apply (derivable_pt_minus ? ? x (X ?) (X0 ?)). +Qed. + +Lemma derivable_mult : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (mult_fct f1 f2)). +Unfold derivable; Intros. +Apply (derivable_pt_mult ? ? x (X ?) (X0 ?)). +Qed. + +Lemma derivable_const : (a:R) (derivable (fct_cte a)). +Unfold derivable; Intros. +Apply derivable_pt_const. +Qed. + +Lemma derivable_scal : (f:R->R;a:R) (derivable f) -> (derivable (mult_real_fct a f)). +Unfold derivable; Intros. +Apply (derivable_pt_scal ? a x (X ?)). +Qed. + +Lemma derivable_id : (derivable id). +Unfold derivable; Intro; Apply derivable_pt_id. +Qed. + +Lemma derivable_Rsqr : (derivable Rsqr). +Unfold derivable; Intro; Apply derivable_pt_Rsqr. +Qed. + +Lemma derivable_comp : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (comp f2 f1)). +Unfold derivable; Intros. +Apply (derivable_pt_comp ? ? x (X ?) (X0 ?)). +Qed. + +Lemma derive_pt_plus : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (plus_fct f1 f2) x (derivable_pt_plus ? ? ? pr1 pr2)) == (derive_pt f1 x pr1) + (derive_pt f2 x pr2)``. +Intros. +Assert H := (derivable_derive f1 x pr1). +Assert H0 := (derivable_derive f2 x pr2). +Assert H1 := (derivable_derive (plus_fct f1 f2) x (derivable_pt_plus ? ? ? pr1 pr2)). +Elim H; Clear H; Intros l1 H. +Elim H0; Clear H0; Intros l2 H0. +Elim H1; Clear H1; Intros l H1. +Rewrite H; Rewrite H0; Apply derive_pt_eq_0. +Assert H3 := (projT2 ? ? pr1). +Unfold derive_pt in H; Rewrite H in H3. +Assert H4 := (projT2 ? ? pr2). +Unfold derive_pt in H0; Rewrite H0 in H4. +Apply derivable_pt_lim_plus; Assumption. +Qed. + +Lemma derive_pt_opp : (f:R->R;x:R;pr1:(derivable_pt f x)) ``(derive_pt (opp_fct f) x (derivable_pt_opp ? ? pr1)) == -(derive_pt f x pr1)``. +Intros. +Assert H := (derivable_derive f x pr1). +Assert H0 := (derivable_derive (opp_fct f) x (derivable_pt_opp ? ? pr1)). +Elim H; Clear H; Intros l1 H. +Elim H0; Clear H0; Intros l2 H0. +Rewrite H; Apply derive_pt_eq_0. +Assert H3 := (projT2 ? ? pr1). +Unfold derive_pt in H; Rewrite H in H3. +Apply derivable_pt_lim_opp; Assumption. +Qed. + +Lemma derive_pt_minus : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (minus_fct f1 f2) x (derivable_pt_minus ? ? ? pr1 pr2)) == (derive_pt f1 x pr1) - (derive_pt f2 x pr2)``. +Intros. +Assert H := (derivable_derive f1 x pr1). +Assert H0 := (derivable_derive f2 x pr2). +Assert H1 := (derivable_derive (minus_fct f1 f2) x (derivable_pt_minus ? ? ? pr1 pr2)). +Elim H; Clear H; Intros l1 H. +Elim H0; Clear H0; Intros l2 H0. +Elim H1; Clear H1; Intros l H1. +Rewrite H; Rewrite H0; Apply derive_pt_eq_0. +Assert H3 := (projT2 ? ? pr1). +Unfold derive_pt in H; Rewrite H in H3. +Assert H4 := (projT2 ? ? pr2). +Unfold derive_pt in H0; Rewrite H0 in H4. +Apply derivable_pt_lim_minus; Assumption. +Qed. + +Lemma derive_pt_mult : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (mult_fct f1 f2) x (derivable_pt_mult ? ? ? pr1 pr2)) == (derive_pt f1 x pr1)*(f2 x) + (f1 x)*(derive_pt f2 x pr2)``. +Intros. +Assert H := (derivable_derive f1 x pr1). +Assert H0 := (derivable_derive f2 x pr2). +Assert H1 := (derivable_derive (mult_fct f1 f2) x (derivable_pt_mult ? ? ? pr1 pr2)). +Elim H; Clear H; Intros l1 H. +Elim H0; Clear H0; Intros l2 H0. +Elim H1; Clear H1; Intros l H1. +Rewrite H; Rewrite H0; Apply derive_pt_eq_0. +Assert H3 := (projT2 ? ? pr1). +Unfold derive_pt in H; Rewrite H in H3. +Assert H4 := (projT2 ? ? pr2). +Unfold derive_pt in H0; Rewrite H0 in H4. +Apply derivable_pt_lim_mult; Assumption. +Qed. + +Lemma derive_pt_const : (a,x:R) (derive_pt (fct_cte a) x (derivable_pt_const a x)) == R0. +Intros. +Apply derive_pt_eq_0. +Apply derivable_pt_lim_const. +Qed. + +Lemma derive_pt_scal : (f:R->R;a,x:R;pr:(derivable_pt f x)) ``(derive_pt (mult_real_fct a f) x (derivable_pt_scal ? ? ? pr)) == a * (derive_pt f x pr)``. +Intros. +Assert H := (derivable_derive f x pr). +Assert H0 := (derivable_derive (mult_real_fct a f) x (derivable_pt_scal ? ? ? pr)). +Elim H; Clear H; Intros l1 H. +Elim H0; Clear H0; Intros l2 H0. +Rewrite H; Apply derive_pt_eq_0. +Assert H3 := (projT2 ? ? pr). +Unfold derive_pt in H; Rewrite H in H3. +Apply derivable_pt_lim_scal; Assumption. +Qed. + +Lemma derive_pt_id : (x:R) (derive_pt id x (derivable_pt_id ?))==R1. +Intros. +Apply derive_pt_eq_0. +Apply derivable_pt_lim_id. +Qed. + +Lemma derive_pt_Rsqr : (x:R) (derive_pt Rsqr x (derivable_pt_Rsqr ?)) == ``2*x``. +Intros. +Apply derive_pt_eq_0. +Apply derivable_pt_lim_Rsqr. +Qed. + +Lemma derive_pt_comp : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 (f1 x))) ``(derive_pt (comp f2 f1) x (derivable_pt_comp ? ? ? pr1 pr2)) == (derive_pt f2 (f1 x) pr2) * (derive_pt f1 x pr1)``. +Intros. +Assert H := (derivable_derive f1 x pr1). +Assert H0 := (derivable_derive f2 (f1 x) pr2). +Assert H1 := (derivable_derive (comp f2 f1) x (derivable_pt_comp ? ? ? pr1 pr2)). +Elim H; Clear H; Intros l1 H. +Elim H0; Clear H0; Intros l2 H0. +Elim H1; Clear H1; Intros l H1. +Rewrite H; Rewrite H0; Apply derive_pt_eq_0. +Assert H3 := (projT2 ? ? pr1). +Unfold derive_pt in H; Rewrite H in H3. +Assert H4 := (projT2 ? ? pr2). +Unfold derive_pt in H0; Rewrite H0 in H4. +Apply derivable_pt_lim_comp; Assumption. +Qed. + +(* Pow *) +Definition pow_fct [n:nat] : R->R := [y:R](pow y n). + +Lemma derivable_pt_lim_pow_pos : (x:R;n:nat) (lt O n) -> (derivable_pt_lim [y:R](pow y n) x ``(INR n)*(pow x (pred n))``). +Intros. +Induction n. +Elim (lt_n_n ? H). +Cut n=O\/(lt O n). +Intro; Elim H0; Intro. +Rewrite H1; Simpl. +Replace [y:R]``y*1`` with (mult_fct id (fct_cte R1)). +Replace ``1*1`` with ``1*(fct_cte R1 x)+(id x)*0``. +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_const. +Unfold fct_cte id; Ring. +Reflexivity. +Replace [y:R](pow y (S n)) with [y:R]``y*(pow y n)``. +Replace (pred (S n)) with n; [Idtac | Reflexivity]. +Replace [y:R]``y*(pow y n)`` with (mult_fct id [y:R](pow y n)). +Pose f := [y:R](pow y n). +Replace ``(INR (S n))*(pow x n)`` with (Rplus (Rmult R1 (f x)) (Rmult (id x) (Rmult (INR n) (pow x (pred n))))). +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_id. +Unfold f; Apply Hrecn; Assumption. +Unfold f. +Pattern 1 5 n; Replace n with (S (pred n)). +Unfold id; Rewrite S_INR; Simpl. +Ring. +Symmetry; Apply S_pred with O; Assumption. +Unfold mult_fct id; Reflexivity. +Reflexivity. +Inversion H. +Left; Reflexivity. +Right. +Apply lt_le_trans with (1). +Apply lt_O_Sn. +Assumption. +Qed. + +Lemma derivable_pt_lim_pow : (x:R; n:nat) (derivable_pt_lim [y:R](pow y n) x ``(INR n)*(pow x (pred n))``). +Intros. +Induction n. +Simpl. +Rewrite Rmult_Ol. +Replace [_:R]``1`` with (fct_cte R1); [Apply derivable_pt_lim_const | Reflexivity]. +Apply derivable_pt_lim_pow_pos. +Apply lt_O_Sn. +Qed. + +Lemma derivable_pt_pow : (n:nat;x:R) (derivable_pt [y:R](pow y n) x). +Intros; Unfold derivable_pt. +Apply Specif.existT with ``(INR n)*(pow x (pred n))``. +Apply derivable_pt_lim_pow. +Qed. + +Lemma derivable_pow : (n:nat) (derivable [y:R](pow y n)). +Intro; Unfold derivable; Intro; Apply derivable_pt_pow. +Qed. + +Lemma derive_pt_pow : (n:nat;x:R) (derive_pt [y:R](pow y n) x (derivable_pt_pow n x))==``(INR n)*(pow x (pred n))``. +Intros; Apply derive_pt_eq_0. +Apply derivable_pt_lim_pow. +Qed. + +Lemma pr_nu : (f:R->R;x:R;pr1,pr2:(derivable_pt f x)) (derive_pt f x pr1)==(derive_pt f x pr2). +Intros. +Unfold derivable_pt in pr1. +Unfold derivable_pt in pr2. +Elim pr1; Intros. +Elim pr2; Intros. +Unfold derivable_pt_abs in p. +Unfold derivable_pt_abs in p0. +Simpl. +Apply (unicite_limite f x x0 x1 p p0). +Qed. + + +(************************************************************) +(** Local extremum's condition *) +(************************************************************) + +Theorem deriv_maximum : (f:R->R;a,b,c:R;pr:(derivable_pt f c)) ``a<c``->``c<b``->((x:R) ``a<x``->``x<b``->``(f x)<=(f c)``)->``(derive_pt f c pr)==0``. +Intros; Case (total_order R0 (derive_pt f c pr)); Intro. +Assert H3 := (derivable_derive f c pr). +Elim H3; Intros l H4; Rewrite H4 in H2. +Assert H5 := (derive_pt_eq_1 f c l pr H4). +Cut ``0<l/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. +Elim (H5 ``l/2`` H6); Intros delta H7. +Cut ``0<(b-c)/2``. +Intro; Cut ``(Rmin delta/2 ((b-c)/2))<>0``. +Intro; Cut ``(Rabsolu (Rmin delta/2 ((b-c)/2)))<delta``. +Intro. +Assert H11 := (H7 ``(Rmin delta/2 ((b-c)/2))`` H9 H10). +Cut ``0<(Rmin (delta/2) ((b-c)/2))``. +Intro; Cut ``a<c+(Rmin (delta/2) ((b-c)/2))``. +Intro; Cut ``c+(Rmin (delta/2) ((b-c)/2))<b``. +Intro; Assert H15 := (H1 ``c+(Rmin (delta/2) ((b-c)/2))`` H13 H14). +Cut ``((f (c+(Rmin (delta/2) ((b-c)/2))))-(f c))/(Rmin (delta/2) ((b-c)/2))<=0``. +Intro; Cut ``-l<0``. +Intro; Unfold Rminus in H11. +Cut ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l<0``. +Intro; Cut ``(Rabsolu (((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l)) < l/2``. +Unfold Rabsolu; Case (case_Rabsolu ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l``); Intro. +Replace `` -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l)`` with ``l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))``. +Intro; Generalize (Rlt_compatibility ``-l`` ``l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))`` ``l/2`` H19); Repeat Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Replace ``-l+l/2`` with ``-(l/2)``. +Intro; Generalize (Rlt_Ropp ``-(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))`` ``-(l/2)`` H20); Repeat Rewrite Ropp_Ropp; Intro; Generalize (Rlt_trans ``0`` ``l/2`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))`` H6 H21); Intro; Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))`` ``0`` H22 H16)). +Pattern 2 l; Rewrite double_var. +Ring. +Ring. +Intro. +Assert H20 := (Rle_sym2 ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` r). +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H20 H18)). +Assumption. +Rewrite <- Ropp_O; Replace ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` with ``-(l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))-(f c))/(Rmin (delta/2) ((b+ -c)/2))))``. +Apply Rgt_Ropp; Change ``0<l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))-(f c))/(Rmin (delta/2) ((b+ -c)/2)))``; Apply gt0_plus_ge0_is_gt0; [Assumption | Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Assumption]. +Ring. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Replace ``((f (c+(Rmin (delta/2) ((b-c)/2))))-(f c))/(Rmin (delta/2) ((b-c)/2))`` with ``- (((f c)-(f (c+(Rmin (delta/2) ((b-c)/2)))))/(Rmin (delta/2) ((b-c)/2)))``. +Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Unfold Rdiv; Apply Rmult_le_pos; [Generalize (Rle_compatibility_r ``-(f (c+(Rmin (delta*/2) ((b-c)*/2))))`` ``(f (c+(Rmin (delta*/2) ((b-c)*/2))))`` (f c) H15); Rewrite Rplus_Ropp_r; Intro; Assumption | Left; Apply Rlt_Rinv; Assumption]. +Unfold Rdiv. +Rewrite <- Ropp_mul1. +Repeat Rewrite <- (Rmult_sym ``/(Rmin (delta*/2) ((b-c)*/2))``). +Apply r_Rmult_mult with ``(Rmin (delta*/2) ((b-c)*/2))``. +Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_r_sym. +Repeat Rewrite Rmult_1l. +Ring. +Red; Intro. +Unfold Rdiv in H12; Rewrite H16 in H12; Elim (Rlt_antirefl ``0`` H12). +Red; Intro. +Unfold Rdiv in H12; Rewrite H16 in H12; Elim (Rlt_antirefl ``0`` H12). +Assert H14 := (Rmin_r ``(delta/2)`` ``((b-c)/2)``). +Assert H15 := (Rle_compatibility ``c`` ``(Rmin (delta/2) ((b-c)/2))`` ``(b-c)/2`` H14). +Apply Rle_lt_trans with ``c+(b-c)/2``. +Assumption. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Replace ``2*(c+(b-c)/2)`` with ``c+b``. +Replace ``2*b`` with ``b+b``. +Apply Rlt_compatibility_r; Assumption. +Ring. +Unfold Rdiv; Rewrite Rmult_Rplus_distr. +Repeat Rewrite (Rmult_sym ``2``). +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Ring. +DiscrR. +Apply Rlt_trans with c. +Assumption. +Pattern 1 c; Rewrite <- (Rplus_Or c); Apply Rlt_compatibility; Assumption. +Cut ``0<delta/2``. +Intro; Apply (Rmin_stable_in_posreal (mkposreal ``delta/2`` H12) (mkposreal ``(b-c)/2`` H8)). +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. +Unfold Rabsolu; Case (case_Rabsolu (Rmin ``delta/2`` ``(b-c)/2``)). +Intro. +Cut ``0<delta/2``. +Intro. +Generalize (Rmin_stable_in_posreal (mkposreal ``delta/2`` H10) (mkposreal ``(b-c)/2`` H8)); Simpl; Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(Rmin (delta/2) ((b-c)/2))`` ``0`` H11 r)). +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. +Intro; Apply Rle_lt_trans with ``delta/2``. +Apply Rmin_l. +Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. +Sup0. +Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l. +Replace ``2*delta`` with ``delta+delta``. +Pattern 2 delta; Rewrite <- (Rplus_Or delta); Apply Rlt_compatibility. +Rewrite Rplus_Or; Apply (cond_pos delta). +Symmetry; Apply double. +DiscrR. +Cut ``0<delta/2``. +Intro; Generalize (Rmin_stable_in_posreal (mkposreal ``delta/2`` H9) (mkposreal ``(b-c)/2`` H8)); Simpl; Intro; Red; Intro; Rewrite H11 in H10; Elim (Rlt_antirefl ``0`` H10). +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. +Unfold Rdiv; Apply Rmult_lt_pos. +Generalize (Rlt_compatibility_r ``-c`` c b H0); Rewrite Rplus_Ropp_r; Intro; Assumption. +Apply Rlt_Rinv; Sup0. +Elim H2; Intro. +Symmetry; Assumption. +Generalize (derivable_derive f c pr); Intro; Elim H4; Intros l H5. +Rewrite H5 in H3; Generalize (derive_pt_eq_1 f c l pr H5); Intro; Cut ``0< -(l/2)``. +Intro; Elim (H6 ``-(l/2)`` H7); Intros delta H9. +Cut ``0<(c-a)/2``. +Intro; Cut ``(Rmax (-(delta/2)) ((a-c)/2))<0``. +Intro; Cut ``(Rmax (-(delta/2)) ((a-c)/2))<>0``. +Intro; Cut ``(Rabsolu (Rmax (-(delta/2)) ((a-c)/2)))<delta``. +Intro; Generalize (H9 ``(Rmax (-(delta/2)) ((a-c)/2))`` H11 H12); Intro; Cut ``a<c+(Rmax (-(delta/2)) ((a-c)/2))``. +Cut ``c+(Rmax (-(delta/2)) ((a-c)/2))<b``. +Intros; Generalize (H1 ``c+(Rmax (-(delta/2)) ((a-c)/2))`` H15 H14); Intro; Cut ``0<=((f (c+(Rmax (-(delta/2)) ((a-c)/2))))-(f c))/(Rmax (-(delta/2)) ((a-c)/2))``. +Intro; Cut ``0< -l``. +Intro; Unfold Rminus in H13; Cut ``0<((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l``. +Intro; Cut ``(Rabsolu (((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l)) < -(l/2)``. +Unfold Rabsolu; Case (case_Rabsolu ``((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l``). +Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``((f (c+(Rmax ( -(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax ( -(delta/2)) ((a+ -c)/2))+ -l`` ``0`` H19 r)). +Intros; Generalize (Rlt_compatibility_r ``l`` ``(((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2)))+ -l`` ``-(l/2)`` H20); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Replace ``-(l/2)+l`` with ``l/2``. +Cut ``l/2<0``. +Intros; Generalize (Rlt_trans ``((f (c+(Rmax ( -(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax ( -(delta/2)) ((a+ -c)/2))`` ``l/2`` ``0`` H22 H21); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``((f (c+(Rmax ( -(delta/2)) ((a-c)/2))))-(f c))/(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H17 H23)). +Rewrite <- (Ropp_Ropp ``l/2``); Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Pattern 3 l; Rewrite double_var. +Ring. +Assumption. +Apply ge0_plus_gt0_is_gt0; Assumption. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Unfold Rdiv; Replace ``((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c))*/(Rmax ( -(delta*/2)) ((a-c)*/2))`` with ``(-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c)))*/(-(Rmax ( -(delta*/2)) ((a-c)*/2)))``. +Apply Rmult_le_pos. +Generalize (Rle_compatibility ``-(f (c+(Rmax (-(delta*/2)) ((a-c)*/2))))`` ``(f (c+(Rmax (-(delta*/2)) ((a-c)*/2))))`` (f c) H16); Rewrite Rplus_Ropp_l; Replace ``-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c))`` with ``-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2)))))+(f c)``. +Intro; Assumption. +Ring. +Left; Apply Rlt_Rinv; Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Unfold Rdiv. +Rewrite <- Ropp_Rinv. +Rewrite Ropp_mul2. +Reflexivity. +Unfold Rdiv in H11; Assumption. +Generalize (Rlt_compatibility c ``(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H10); Rewrite Rplus_Or; Intro; Apply Rlt_trans with ``c``; Assumption. +Generalize (RmaxLess2 ``(-(delta/2))`` ``((a-c)/2)``); Intro; Generalize (Rle_compatibility c ``(a-c)/2`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` H14); Intro; Apply Rlt_le_trans with ``c+(a-c)/2``. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Replace ``2*(c+(a-c)/2)`` with ``a+c``. +Rewrite double. +Apply Rlt_compatibility; Assumption. +Ring. +Rewrite <- Rplus_assoc. +Rewrite <- double_var. +Ring. +Assumption. +Unfold Rabsolu; Case (case_Rabsolu (Rmax ``-(delta/2)`` ``(a-c)/2``)). +Intro; Generalize (RmaxLess1 ``-(delta/2)`` ``(a-c)/2``); Intro; Generalize (Rle_Ropp ``-(delta/2)`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` H12); Rewrite Ropp_Ropp; Intro; Generalize (Rle_sym2 ``-(Rmax ( -(delta/2)) ((a-c)/2))`` ``delta/2`` H13); Intro; Apply Rle_lt_trans with ``delta/2``. +Assumption. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite double. +Pattern 2 delta; Rewrite <- (Rplus_Or delta); Apply Rlt_compatibility; Rewrite Rplus_Or; Apply (cond_pos delta). +DiscrR. +Cut ``-(delta/2) < 0``. +Cut ``(a-c)/2<0``. +Intros; Generalize (Rmax_stable_in_negreal (mknegreal ``-(delta/2)`` H13) (mknegreal ``(a-c)/2`` H12)); Simpl; Intro; Generalize (Rle_sym2 ``0`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` r); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H15 H14)). +Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp ``(a-c)/2``); Apply Rlt_Ropp; Replace ``-((a-c)/2)`` with ``(c-a)/2``. +Assumption. +Unfold Rdiv. +Rewrite <- Ropp_mul1. +Rewrite (Ropp_distr2 a c). +Reflexivity. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Assert Hyp : ``0<2``; [Sup0 | Apply (Rlt_Rinv ``2`` Hyp)]]. +Red; Intro; Rewrite H11 in H10; Elim (Rlt_antirefl ``0`` H10). +Cut ``(a-c)/2<0``. +Intro; Cut ``-(delta/2)<0``. +Intro; Apply (Rmax_stable_in_negreal (mknegreal ``-(delta/2)`` H11) (mknegreal ``(a-c)/2`` H10)). +Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Assert Hyp : ``0<2``; [Sup0 | Apply (Rlt_Rinv ``2`` Hyp)]]. +Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp ``(a-c)/2``); Apply Rlt_Ropp; Replace ``-((a-c)/2)`` with ``(c-a)/2``. +Assumption. +Unfold Rdiv. +Rewrite <- Ropp_mul1. +Rewrite (Ropp_distr2 a c). +Reflexivity. +Unfold Rdiv; Apply Rmult_lt_pos; [Generalize (Rlt_compatibility_r ``-a`` a c H); Rewrite Rplus_Ropp_r; Intro; Assumption | Assert Hyp : ``0<2``; [Sup0 | Apply (Rlt_Rinv ``2`` Hyp)]]. +Replace ``-(l/2)`` with ``(-l)/2``. +Unfold Rdiv; Apply Rmult_lt_pos. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Assert Hyp : ``0<2``; [Sup0 | Apply (Rlt_Rinv ``2`` Hyp)]. +Unfold Rdiv; Apply Ropp_mul1. +Qed. + +Theorem deriv_minimum : (f:R->R;a,b,c:R;pr:(derivable_pt f c)) ``a<c``->``c<b``->((x:R) ``a<x``->``x<b``->``(f c)<=(f x)``)->``(derive_pt f c pr)==0``. +Intros. +Rewrite <- (Ropp_Ropp (derive_pt f c pr)). +Apply eq_RoppO. +Rewrite <- (derive_pt_opp f c pr). +Cut (x:R)(``a<x``->``x<b``->``((opp_fct f) x)<=((opp_fct f) c)``). +Intro. +Apply (deriv_maximum (opp_fct f) a b c (derivable_pt_opp ? ? pr) H H0 H2). +Intros; Unfold opp_fct; Apply Rge_Ropp; Apply Rle_sym1. +Apply (H1 x H2 H3). +Qed. + +Theorem deriv_constant2 : (f:R->R;a,b,c:R;pr:(derivable_pt f c)) ``a<c``->``c<b``->((x:R) ``a<x``->``x<b``->``(f x)==(f c)``)->``(derive_pt f c pr)==0``. +Intros. +EApply deriv_maximum with a b; Try Assumption. +Intros; Right; Apply (H1 x H2 H3). +Qed. + +(**********) +Lemma nonneg_derivative_0 : (f:R->R;pr:(derivable f)) (increasing f) -> ((x:R) ``0<=(derive_pt f x (pr x))``). +Intros; Unfold increasing in H. +Assert H0 := (derivable_derive f x (pr x)). +Elim H0; Intros l H1. +Rewrite H1; Case (total_order R0 l); Intro. +Left; Assumption. +Elim H2; Intro. +Right; Assumption. +Assert H4 := (derive_pt_eq_1 f x l (pr x) H1). +Cut ``0< -(l/2)``. +Intro; Elim (H4 ``-(l/2)`` H5); Intros delta H6. +Cut ``delta/2<>0``/\``0<delta/2``/\``(Rabsolu delta/2)<delta``. +Intro; Decompose [and] H7; Intros; Generalize (H6 ``delta/2`` H8 H11); Cut ``0<=((f (x+delta/2))-(f x))/(delta/2)``. +Intro; Cut ``0<=((f (x+delta/2))-(f x))/(delta/2)-l``. +Intro; Unfold Rabsolu; Case (case_Rabsolu ``((f (x+delta/2))-(f x))/(delta/2)-l``). +Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``0`` H12 r)). +Intros; Generalize (Rlt_compatibility_r l ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``-(l/2)`` H13); Unfold Rminus; Replace ``-(l/2)+l`` with ``l/2``. +Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Intro; Generalize (Rle_lt_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)`` ``l/2`` H9 H14); Intro; Cut ``l/2<0``. +Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``l/2`` ``0`` H15 H16)). +Rewrite <- Ropp_O in H5; Generalize (Rlt_Ropp ``-0`` ``-(l/2)`` H5); Repeat Rewrite Ropp_Ropp; Intro; Assumption. +Pattern 3 l ; Rewrite double_var. +Ring. +Unfold Rminus; Apply ge0_plus_ge0_is_ge0. +Unfold Rdiv; Apply Rmult_le_pos. +Cut ``x<=(x+(delta*/2))``. +Intro; Generalize (H x ``x+(delta*/2)`` H12); Intro; Generalize (Rle_compatibility ``-(f x)`` ``(f x)`` ``(f (x+delta*/2))`` H13); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. +Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_Rinv; Assumption. +Left; Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Unfold Rdiv; Apply Rmult_le_pos. +Cut ``x<=(x+(delta*/2))``. +Intro; Generalize (H x ``x+(delta*/2)`` H9); Intro; Generalize (Rle_compatibility ``-(f x)`` ``(f x)`` ``(f (x+delta*/2))`` H12); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. +Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_Rinv; Assumption. +Split. +Unfold Rdiv; Apply prod_neq_R0. +Generalize (cond_pos delta); Intro; Red; Intro H9; Rewrite H9 in H7; Elim (Rlt_antirefl ``0`` H7). +Apply Rinv_neq_R0; DiscrR. +Split. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. +Replace ``(Rabsolu delta/2)`` with ``delta/2``. +Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. +Sup0. +Rewrite (Rmult_sym ``2``). +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. +Rewrite Rmult_1r. +Rewrite double. +Pattern 1 (pos delta); Rewrite <- Rplus_Or. +Apply Rlt_compatibility; Apply (cond_pos delta). +Symmetry; Apply Rabsolu_right. +Left; Change ``0<delta/2``; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. +Unfold Rdiv; Rewrite <- Ropp_mul1; Apply Rmult_lt_pos. +Apply Rlt_anti_compatibility with l. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Assumption. +Apply Rlt_Rinv; Sup0. +Qed. diff --git a/theories7/Reals/Ranalysis2.v b/theories7/Reals/Ranalysis2.v new file mode 100644 index 00000000..35fa58d5 --- /dev/null +++ b/theories7/Reals/Ranalysis2.v @@ -0,0 +1,302 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Ranalysis2.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(**********) +Lemma formule : (x,h,l1,l2:R;f1,f2:R->R) ``h<>0`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ``((f1 (x+h))/(f2 (x+h))-(f1 x)/(f2 x))/h-(l1*(f2 x)-l2*(f1 x))/(Rsqr (f2 x))`` == ``/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1) + l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))) - (f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2) + (l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))``. +Intros; Unfold Rdiv Rminus Rsqr. +Repeat Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_Rplus_distr; Repeat Rewrite Rinv_Rmult; Try Assumption. +Replace ``l1*(f2 x)*(/(f2 x)*/(f2 x))`` with ``l1*/(f2 x)*((f2 x)*/(f2 x))``; [Idtac | Ring]. +Replace ``l1*(/(f2 x)*/(f2 (x+h)))*(f2 x)`` with ``l1*/(f2 (x+h))*((f2 x)*/(f2 x))``; [Idtac | Ring]. +Replace ``l1*(/(f2 x)*/(f2 (x+h)))* -(f2 (x+h))`` with ``-(l1*/(f2 x)*((f2 (x+h))*/(f2 (x+h))))``; [Idtac | Ring]. +Replace ``(f1 x)*(/(f2 x)*/(f2 (x+h)))*((f2 (x+h))*/h)`` with ``(f1 x)*/(f2 x)*/h*((f2 (x+h))*/(f2 (x+h)))``; [Idtac | Ring]. +Replace ``(f1 x)*(/(f2 x)*/(f2 (x+h)))*( -(f2 x)*/h)`` with ``-((f1 x)*/(f2 (x+h))*/h*((f2 x)*/(f2 x)))``; [Idtac | Ring]. +Replace ``(l2*(f1 x)*(/(f2 x)*/(f2 x)*/(f2 (x+h)))*(f2 (x+h)))`` with ``l2*(f1 x)*/(f2 x)*/(f2 x)*((f2 (x+h))*/(f2 (x+h)))``; [Idtac | Ring]. +Replace ``l2*(f1 x)*(/(f2 x)*/(f2 x)*/(f2 (x+h)))* -(f2 x)`` with ``-(l2*(f1 x)*/(f2 x)*/(f2 (x+h))*((f2 x)*/(f2 x)))``; [Idtac | Ring]. +Repeat Rewrite <- Rinv_r_sym; Try Assumption Orelse Ring. +Apply prod_neq_R0; Assumption. +Qed. + +Lemma Rmin_pos : (x,y:R) ``0<x`` -> ``0<y`` -> ``0 < (Rmin x y)``. +Intros; Unfold Rmin. +Case (total_order_Rle x y); Intro; Assumption. +Qed. + +Lemma maj_term1 : (x,h,eps,l1,alp_f2:R;eps_f2,alp_f1d:posreal;f1,f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((h:R)``h <> 0``->``(Rabsolu h) < alp_f1d``->``(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < (Rabsolu ((eps*(f2 x))/8))``) -> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f1d`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) < eps/4``. +Intros. +Assert H7 := (H3 h H6). +Assert H8 := (H2 h H4 H5). +Apply Rle_lt_trans with ``2/(Rabsolu (f2 x))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1))``. +Rewrite Rabsolu_mult. +Apply Rle_monotony_r. +Apply Rabsolu_pos. +Rewrite Rabsolu_Rinv; [Left; Exact H7 | Assumption]. +Apply Rlt_le_trans with ``2/(Rabsolu (f2 x))*(Rabsolu ((eps*(f2 x))/8))``. +Apply Rlt_monotony. +Unfold Rdiv; Apply Rmult_lt_pos; [Sup0 | Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption]. +Exact H8. +Right; Unfold Rdiv. +Repeat Rewrite Rabsolu_mult. +Rewrite Rabsolu_Rinv; DiscrR. +Replace ``(Rabsolu 8)`` with ``8``. +Replace ``8`` with ``2*4``; [Idtac | Ring]. +Rewrite Rinv_Rmult; [Idtac | DiscrR | DiscrR]. +Replace ``2*/(Rabsolu (f2 x))*((Rabsolu eps)*(Rabsolu (f2 x))*(/2*/4))`` with ``(Rabsolu eps)*/4*(2*/2)*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))``; [Idtac | Ring]. +Replace (Rabsolu eps) with eps. +Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption). +Ring. +Symmetry; Apply Rabsolu_right; Left; Assumption. +Symmetry; Apply Rabsolu_right; Left; Sup. +Qed. + +Lemma maj_term2 : (x,h,eps,l1,alp_f2,alp_f2t2:R;eps_f2:posreal;f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((a:R)``(Rabsolu a) < alp_f2t2``->``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``)-> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f2t2`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``l1<>0`` -> ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) < eps/4``. +Intros. +Assert H8 := (H3 h H6). +Assert H9 := (H2 h H5). +Apply Rle_lt_trans with ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. +Rewrite Rabsolu_mult; Apply Rle_monotony. +Apply Rabsolu_pos. +Rewrite <- (Rabsolu_Ropp ``(f2 x)-(f2 (x+h))``); Rewrite Ropp_distr2. +Left; Apply H9. +Apply Rlt_le_trans with ``(Rabsolu (2*l1/((f2 x)*(f2 x))))*(Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. +Apply Rlt_monotony_r. +Apply Rabsolu_pos_lt. +Unfold Rdiv; Unfold Rsqr; Repeat Apply prod_neq_R0; Try Assumption Orelse DiscrR. +Red; Intro H10; Rewrite H10 in H; Elim (Rlt_antirefl ? H). +Apply Rinv_neq_R0; Apply prod_neq_R0; Try Assumption Orelse DiscrR. +Unfold Rdiv. +Repeat Rewrite Rinv_Rmult; Try Assumption. +Repeat Rewrite Rabsolu_mult. +Replace ``(Rabsolu 2)`` with ``2``. +Rewrite (Rmult_sym ``2``). +Replace ``(Rabsolu l1)*((Rabsolu (/(f2 x)))*(Rabsolu (/(f2 x))))*2`` with ``(Rabsolu l1)*((Rabsolu (/(f2 x)))*((Rabsolu (/(f2 x)))*2))``; [Idtac | Ring]. +Repeat Apply Rlt_monotony. +Apply Rabsolu_pos_lt; Assumption. +Apply Rabsolu_pos_lt; Apply Rinv_neq_R0; Assumption. +Repeat Rewrite Rabsolu_Rinv; Try Assumption. +Rewrite <- (Rmult_sym ``2``). +Unfold Rdiv in H8; Exact H8. +Symmetry; Apply Rabsolu_right; Left; Sup0. +Right. +Unfold Rsqr Rdiv. +Do 1 Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Do 1 Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Repeat Rewrite Rabsolu_mult. +Repeat Rewrite Rabsolu_Rinv; Try Assumption Orelse DiscrR. +Replace (Rabsolu eps) with eps. +Replace ``(Rabsolu (8))`` with ``8``. +Replace ``(Rabsolu 2)`` with ``2``. +Replace ``8`` with ``4*2``; [Idtac | Ring]. +Rewrite Rinv_Rmult; DiscrR. +Replace ``2*((Rabsolu l1)*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))))*(eps*((Rabsolu (f2 x))*(Rabsolu (f2 x)))*(/4*/2*/(Rabsolu l1)))`` with ``eps*/4*((Rabsolu l1)*/(Rabsolu l1))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*(2*/2)``; [Idtac | Ring]. +Repeat Rewrite <- Rinv_r_sym; Try (Apply Rabsolu_no_R0; Assumption) Orelse DiscrR. +Ring. +Symmetry; Apply Rabsolu_right; Left; Sup0. +Symmetry; Apply Rabsolu_right; Left; Sup. +Symmetry; Apply Rabsolu_right; Left; Assumption. +Qed. + +Lemma maj_term3 : (x,h,eps,l2,alp_f2:R;eps_f2,alp_f2d:posreal;f1,f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((h:R)``h <> 0``->``(Rabsolu h) < alp_f2d``->``(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < (Rabsolu (((Rsqr (f2 x))*eps)/(8*(f1 x))))``) -> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f2d`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``(f1 x)<>0`` -> ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) < eps/4``. +Intros. +Assert H8 := (H2 h H4 H5). +Assert H9 := (H3 h H6). +Apply Rle_lt_trans with ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((Rsqr (f2 x))*eps)/(8*(f1 x))))``. +Rewrite Rabsolu_mult. +Apply Rle_monotony. +Apply Rabsolu_pos. +Left; Apply H8. +Apply Rlt_le_trans with ``(Rabsolu (2*(f1 x)/((f2 x)*(f2 x))))*(Rabsolu (((Rsqr (f2 x))*eps)/(8*(f1 x))))``. +Apply Rlt_monotony_r. +Apply Rabsolu_pos_lt. +Unfold Rdiv; Unfold Rsqr; Repeat Apply prod_neq_R0; Try Assumption. +Red; Intro H10; Rewrite H10 in H; Elim (Rlt_antirefl ? H). +Apply Rinv_neq_R0; Apply prod_neq_R0; DiscrR Orelse Assumption. +Unfold Rdiv. +Repeat Rewrite Rinv_Rmult; Try Assumption. +Repeat Rewrite Rabsolu_mult. +Replace ``(Rabsolu 2)`` with ``2``. +Rewrite (Rmult_sym ``2``). +Replace ``(Rabsolu (f1 x))*((Rabsolu (/(f2 x)))*(Rabsolu (/(f2 x))))*2`` with ``(Rabsolu (f1 x))*((Rabsolu (/(f2 x)))*((Rabsolu (/(f2 x)))*2))``; [Idtac | Ring]. +Repeat Apply Rlt_monotony. +Apply Rabsolu_pos_lt; Assumption. +Apply Rabsolu_pos_lt; Apply Rinv_neq_R0; Assumption. +Repeat Rewrite Rabsolu_Rinv; Assumption Orelse Idtac. +Rewrite <- (Rmult_sym ``2``). +Unfold Rdiv in H9; Exact H9. +Symmetry; Apply Rabsolu_right; Left; Sup0. +Right. +Unfold Rsqr Rdiv. +Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Repeat Rewrite Rabsolu_mult. +Repeat Rewrite Rabsolu_Rinv; Try Assumption Orelse DiscrR. +Replace (Rabsolu eps) with eps. +Replace ``(Rabsolu (8))`` with ``8``. +Replace ``(Rabsolu 2)`` with ``2``. +Replace ``8`` with ``4*2``; [Idtac | Ring]. +Rewrite Rinv_Rmult; DiscrR. +Replace ``2*((Rabsolu (f1 x))*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))))*((Rabsolu (f2 x))*(Rabsolu (f2 x))*eps*(/4*/2*/(Rabsolu (f1 x))))`` with ``eps*/4*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f1 x))*/(Rabsolu (f1 x)))*(2*/2)``; [Idtac | Ring]. +Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption). +Ring. +Symmetry; Apply Rabsolu_right; Left; Sup0. +Symmetry; Apply Rabsolu_right; Left; Sup. +Symmetry; Apply Rabsolu_right; Left; Assumption. +Qed. + +Lemma maj_term4 : (x,h,eps,l2,alp_f2,alp_f2c:R;eps_f2:posreal;f1,f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((a:R)``(Rabsolu a) < alp_f2c`` -> ``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``) -> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f2c`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``(f1 x)<>0`` -> ``l2<>0`` -> ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x)))) < eps/4``. +Intros. +Assert H9 := (H2 h H5). +Assert H10 := (H3 h H6). +Apply Rle_lt_trans with ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. +Rewrite Rabsolu_mult. +Apply Rle_monotony. +Apply Rabsolu_pos. +Left; Apply H9. +Apply Rlt_le_trans with ``(Rabsolu (2*l2*(f1 x)/((Rsqr (f2 x))*(f2 x))))*(Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. +Apply Rlt_monotony_r. +Apply Rabsolu_pos_lt. +Unfold Rdiv; Unfold Rsqr; Repeat Apply prod_neq_R0; Assumption Orelse Idtac. +Red; Intro H11; Rewrite H11 in H; Elim (Rlt_antirefl ? H). +Apply Rinv_neq_R0; Apply prod_neq_R0. +Apply prod_neq_R0. +DiscrR. +Assumption. +Assumption. +Unfold Rdiv. +Repeat Rewrite Rinv_Rmult; Try Assumption Orelse (Unfold Rsqr; Apply prod_neq_R0; Assumption). +Repeat Rewrite Rabsolu_mult. +Replace ``(Rabsolu 2)`` with ``2``. +Replace ``2*(Rabsolu l2)*((Rabsolu (f1 x))*((Rabsolu (/(Rsqr (f2 x))))*(Rabsolu (/(f2 x)))))`` with ``(Rabsolu l2)*((Rabsolu (f1 x))*((Rabsolu (/(Rsqr (f2 x))))*((Rabsolu (/(f2 x)))*2)))``; [Idtac | Ring]. +Replace ``(Rabsolu l2)*(Rabsolu (f1 x))*((Rabsolu (/(Rsqr (f2 x))))*(Rabsolu (/(f2 (x+h)))))`` with ``(Rabsolu l2)*((Rabsolu (f1 x))*(((Rabsolu (/(Rsqr (f2 x))))*(Rabsolu (/(f2 (x+h)))))))``; [Idtac | Ring]. +Repeat Apply Rlt_monotony. +Apply Rabsolu_pos_lt; Assumption. +Apply Rabsolu_pos_lt; Assumption. +Apply Rabsolu_pos_lt; Apply Rinv_neq_R0; Unfold Rsqr; Apply prod_neq_R0; Assumption. +Repeat Rewrite Rabsolu_Rinv; [Idtac | Assumption | Assumption]. +Rewrite <- (Rmult_sym ``2``). +Unfold Rdiv in H10; Exact H10. +Symmetry; Apply Rabsolu_right; Left; Sup0. +Right; Unfold Rsqr Rdiv. +Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Repeat Rewrite Rabsolu_mult. +Repeat Rewrite Rabsolu_Rinv; Try Assumption Orelse DiscrR. +Replace (Rabsolu eps) with eps. +Replace ``(Rabsolu (8))`` with ``8``. +Replace ``(Rabsolu 2)`` with ``2``. +Replace ``8`` with ``4*2``; [Idtac | Ring]. +Rewrite Rinv_Rmult; DiscrR. +Replace ``2*(Rabsolu l2)*((Rabsolu (f1 x))*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))*/(Rabsolu (f2 x))))*((Rabsolu (f2 x))*(Rabsolu (f2 x))*(Rabsolu (f2 x))*eps*(/4*/2*/(Rabsolu (f1 x))*/(Rabsolu l2)))`` with ``eps*/4*((Rabsolu l2)*/(Rabsolu l2))*((Rabsolu (f1 x))*/(Rabsolu (f1 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*(2*/2)``; [Idtac | Ring]. +Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption). +Ring. +Symmetry; Apply Rabsolu_right; Left; Sup0. +Symmetry; Apply Rabsolu_right; Left; Sup. +Symmetry; Apply Rabsolu_right; Left; Assumption. +Apply prod_neq_R0; Assumption Orelse DiscrR. +Apply prod_neq_R0; Assumption. +Qed. + +Lemma D_x_no_cond : (x,a:R) ``a<>0`` -> (D_x no_cond x ``x+a``). +Intros. +Unfold D_x no_cond. +Split. +Trivial. +Apply Rminus_not_eq. +Unfold Rminus. +Rewrite Ropp_distr1. +Rewrite <- Rplus_assoc. +Rewrite Rplus_Ropp_r. +Rewrite Rplus_Ol. +Apply Ropp_neq; Assumption. +Qed. + +Lemma Rabsolu_4 : (a,b,c,d:R) ``(Rabsolu (a+b+c+d)) <= (Rabsolu a) + (Rabsolu b) + (Rabsolu c) + (Rabsolu d)``. +Intros. +Apply Rle_trans with ``(Rabsolu (a+b)) + (Rabsolu (c+d))``. +Replace ``a+b+c+d`` with ``(a+b)+(c+d)``; [Apply Rabsolu_triang | Ring]. +Apply Rle_trans with ``(Rabsolu a) + (Rabsolu b) + (Rabsolu (c+d))``. +Apply Rle_compatibility_r. +Apply Rabsolu_triang. +Repeat Rewrite Rplus_assoc; Repeat Apply Rle_compatibility. +Apply Rabsolu_triang. +Qed. + +Lemma Rlt_4 : (a,b,c,d,e,f,g,h:R) ``a < b`` -> ``c < d`` -> ``e < f `` -> ``g < h`` -> ``a+c+e+g < b+d+f+h``. +Intros; Apply Rlt_trans with ``b+c+e+g``. +Repeat Apply Rlt_compatibility_r; Assumption. +Repeat Rewrite Rplus_assoc; Apply Rlt_compatibility. +Apply Rlt_trans with ``d+e+g``. +Rewrite Rplus_assoc; Apply Rlt_compatibility_r; Assumption. +Rewrite Rplus_assoc; Apply Rlt_compatibility; Apply Rlt_trans with ``f+g``. +Apply Rlt_compatibility_r; Assumption. +Apply Rlt_compatibility; Assumption. +Qed. + +Lemma Rmin_2 : (a,b,c:R) ``a < b`` -> ``a < c`` -> ``a < (Rmin b c)``. +Intros; Unfold Rmin; Case (total_order_Rle b c); Intro; Assumption. +Qed. + +Lemma quadruple : (x:R) ``4*x == x + x + x + x``. +Intro; Ring. +Qed. + +Lemma quadruple_var : (x:R) `` x == x/4 + x/4 + x/4 + x/4``. +Intro; Rewrite <- quadruple. +Unfold Rdiv; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m; DiscrR. +Reflexivity. +Qed. + +(**********) +Lemma continuous_neq_0 : (f:R->R; x0:R) (continuity_pt f x0) -> ~``(f x0)==0`` -> (EXT eps : posreal | (h:R) ``(Rabsolu h) < eps`` -> ~``(f (x0+h))==0``). +Intros; Unfold continuity_pt in H; Unfold continue_in in H; Unfold limit1_in in H; Unfold limit_in in H; Elim (H ``(Rabsolu ((f x0)/2))``). +Intros; Elim H1; Intros. +Exists (mkposreal x H2). +Intros; Assert H5 := (H3 ``x0+h``). +Cut ``(dist R_met (x0+h) x0) < x`` -> ``(dist R_met (f (x0+h)) (f x0)) < (Rabsolu ((f x0)/2))``. +Unfold dist; Simpl; Unfold R_dist; Replace ``x0+h-x0`` with h. +Intros; Assert H7 := (H6 H4). +Red; Intro. +Rewrite H8 in H7; Unfold Rminus in H7; Rewrite Rplus_Ol in H7; Rewrite Rabsolu_Ropp in H7; Unfold Rdiv in H7; Rewrite Rabsolu_mult in H7; Pattern 1 ``(Rabsolu (f x0)) `` in H7; Rewrite <- Rmult_1r in H7. +Cut ``0<(Rabsolu (f x0))``. +Intro; Assert H10 := (Rlt_monotony_contra ? ? ? H9 H7). +Cut ``(Rabsolu (/2))==/2``. +Assert Hyp:``0<2``. +Sup0. +Intro; Rewrite H11 in H10; Assert H12 := (Rlt_monotony ``2`` ? ? Hyp H10); Rewrite Rmult_1r in H12; Rewrite <- Rinv_r_sym in H12; [Idtac | DiscrR]. +Cut (Rlt (IZR `1`) (IZR `2`)). +Unfold IZR; Unfold INR convert; Simpl; Intro; Elim (Rlt_antirefl ``1`` (Rlt_trans ? ? ? H13 H12)). +Apply IZR_lt; Omega. +Unfold Rabsolu; Case (case_Rabsolu ``/2``); Intro. +Assert Hyp:``0<2``. +Sup0. +Assert H11 := (Rlt_monotony ``2`` ? ? Hyp r); Rewrite Rmult_Or in H11; Rewrite <- Rinv_r_sym in H11; [Idtac | DiscrR]. +Elim (Rlt_antirefl ``0`` (Rlt_trans ? ? ? Rlt_R0_R1 H11)). +Reflexivity. +Apply (Rabsolu_pos_lt ? H0). +Ring. +Assert H6 := (Req_EM ``x0`` ``x0+h``); Elim H6; Intro. +Intro; Rewrite <- H7; Unfold dist R_met; Unfold R_dist; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rabsolu_pos_lt. +Unfold Rdiv; Apply prod_neq_R0; [Assumption | Apply Rinv_neq_R0; DiscrR]. +Intro; Apply H5. +Split. +Unfold D_x no_cond. +Split; Trivial Orelse Assumption. +Assumption. +Change ``0 < (Rabsolu ((f x0)/2))``. +Apply Rabsolu_pos_lt; Unfold Rdiv; Apply prod_neq_R0. +Assumption. +Apply Rinv_neq_R0; DiscrR. +Qed. diff --git a/theories7/Reals/Ranalysis3.v b/theories7/Reals/Ranalysis3.v new file mode 100644 index 00000000..6ce63bbc --- /dev/null +++ b/theories7/Reals/Ranalysis3.v @@ -0,0 +1,617 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Ranalysis3.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +Require Ranalysis2. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(* Division *) +Theorem derivable_pt_lim_div : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> ~``(f2 x)==0``-> (derivable_pt_lim (div_fct f1 f2) x ``(l1*(f2 x)-l2*(f1 x))/(Rsqr (f2 x))``). +Intros. +Cut (derivable_pt f2 x); [Intro | Unfold derivable_pt; Apply Specif.existT with l2; Exact H0]. +Assert H2 := ((continuous_neq_0 ? ? (derivable_continuous_pt ? ? X)) H1). +Elim H2; Clear H2; Intros eps_f2 H2. +Unfold div_fct. +Assert H3 := (derivable_continuous_pt ? ? X). +Unfold continuity_pt in H3; Unfold continue_in in H3; Unfold limit1_in in H3; Unfold limit_in in H3; Unfold dist in H3. +Simpl in H3; Unfold R_dist in H3. +Elim (H3 ``(Rabsolu (f2 x))/2``); [Idtac | Unfold Rdiv; Change ``0 < (Rabsolu (f2 x))*/2``; Apply Rmult_lt_pos; [Apply Rabsolu_pos_lt; Assumption | Apply Rlt_Rinv; Sup0]]. +Clear H3; Intros alp_f2 H3. +Cut (x0:R) ``(Rabsolu (x0-x)) < alp_f2`` ->``(Rabsolu ((f2 x0)-(f2 x))) < (Rabsolu (f2 x))/2``. +Intro H4. +Cut (a:R) ``(Rabsolu (a-x)) < alp_f2``->``(Rabsolu (f2 x))/2 < (Rabsolu (f2 a))``. +Intro H5. +Cut (a:R) ``(Rabsolu (a)) < (Rmin eps_f2 alp_f2)`` -> ``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``. +Intro Maj. +Unfold derivable_pt_lim; Intros. +Elim (H ``(Rabsolu ((eps*(f2 x))/8))``); [Idtac | Unfold Rdiv; Change ``0 < (Rabsolu (eps*(f2 x)*/8))``; Apply Rabsolu_pos_lt; Repeat Apply prod_neq_R0; [Red; Intro H7; Rewrite H7 in H6; Elim (Rlt_antirefl ? H6) | Assumption | Apply Rinv_neq_R0; DiscrR]]. +Intros alp_f1d H7. +Case (Req_EM (f1 x) R0); Intro. +Case (Req_EM l1 R0); Intro. +(***********************************) +(* Cas n° 1 *) +(* (f1 x)=0 l1 =0 *) +(***********************************) +Cut ``0 < (Rmin eps_f2 (Rmin alp_f2 alp_f1d))``; [Intro | Repeat Apply Rmin_pos; [Apply (cond_pos eps_f2) | Elim H3; Intros; Assumption | Apply (cond_pos alp_f1d)]]. +Exists (mkposreal (Rmin eps_f2 (Rmin alp_f2 alp_f1d)) H10). +Simpl; Intros. +Assert H13 := (Rlt_le_trans ? ? ? H12 (Rmin_r ? ?)). +Assert H14 := (Rlt_le_trans ? ? ? H12 (Rmin_l ? ?)). +Assert H15 := (Rlt_le_trans ? ? ? H13 (Rmin_r ? ?)). +Assert H16 := (Rlt_le_trans ? ? ? H13 (Rmin_l ? ?)). +Assert H17 := (H7 ? H11 H15). +Rewrite formule; [Idtac | Assumption | Assumption | Apply H2; Apply H14]. +Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. +Unfold Rminus. +Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). +Apply Rabsolu_4. +Repeat Rewrite Rabsolu_mult. +Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. +Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. +Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. +Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. +Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. +Intros. +Apply Rlt_4; Assumption. +Rewrite H8. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite H8. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite H9. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite <- Rabsolu_mult. +Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption Orelse Apply H2. +Apply H14. +Apply Rmin_2; Assumption. +Right; Symmetry; Apply quadruple_var. +(***********************************) +(* Cas n° 2 *) +(* (f1 x)=0 l1<>0 *) +(***********************************) +Assert H10 := (derivable_continuous_pt ? ? X). +Unfold continuity_pt in H10. +Unfold continue_in in H10. +Unfold limit1_in in H10. +Unfold limit_in in H10. +Unfold dist in H10. +Simpl in H10. +Unfold R_dist in H10. +Elim (H10 ``(Rabsolu (eps*(Rsqr (f2 x)))/(8*l1))``). +Clear H10; Intros alp_f2t2 H10. +Cut (a:R) ``(Rabsolu a) < alp_f2t2`` -> ``(Rabsolu ((f2 (x+a)) - (f2 x))) < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. +Intro H11. +Cut ``0 < (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2))``. +Intro. +Exists (mkposreal (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)) H12). +Simpl. +Intros. +Assert H15 := (Rlt_le_trans ? ? ? H14 (Rmin_r ? ?)). +Assert H16 := (Rlt_le_trans ? ? ? H14 (Rmin_l ? ?)). +Assert H17 := (Rlt_le_trans ? ? ? H15 (Rmin_l ? ?)). +Assert H18 := (Rlt_le_trans ? ? ? H15 (Rmin_r ? ?)). +Assert H19 := (Rlt_le_trans ? ? ? H16 (Rmin_l ? ?)). +Assert H20 := (Rlt_le_trans ? ? ? H16 (Rmin_r ? ?)). +Clear H14 H15 H16. +Rewrite formule; Try Assumption. +Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. +Unfold Rminus. +Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). +Apply Rabsolu_4. +Repeat Rewrite Rabsolu_mult. +Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. +Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. +Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. +Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. +Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. +Intros. +Apply Rlt_4; Assumption. +Rewrite H8. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite H8. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite <- Rabsolu_mult. +Apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Right; Symmetry; Apply quadruple_var. +Apply H2; Assumption. +Repeat Apply Rmin_pos. +Apply (cond_pos eps_f2). +Apply (cond_pos alp_f1d). +Elim H3; Intros; Assumption. +Elim H10; Intros; Assumption. +Intros. +Elim H10; Intros. +Case (Req_EM a R0); Intro. +Rewrite H14; Rewrite Rplus_Or. +Unfold Rminus; Rewrite Rplus_Ropp_r. +Rewrite Rabsolu_R0. +Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr; Repeat Rewrite Rmult_assoc. +Repeat Apply prod_neq_R0; Try Assumption. +Red; Intro; Rewrite H15 in H6; Elim (Rlt_antirefl ? H6). +Apply Rinv_neq_R0; Repeat Apply prod_neq_R0; DiscrR Orelse Assumption. +Apply H13. +Split. +Apply D_x_no_cond; Assumption. +Replace ``x+a-x`` with a; [Assumption | Ring]. +Change ``0<(Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. +Apply Rabsolu_pos_lt; Unfold Rdiv Rsqr; Repeat Rewrite Rmult_assoc; Repeat Apply prod_neq_R0. +Red; Intro; Rewrite H11 in H6; Elim (Rlt_antirefl ? H6). +Assumption. +Assumption. +Apply Rinv_neq_R0; Repeat Apply prod_neq_R0; [DiscrR | DiscrR | DiscrR | Assumption]. +(***********************************) +(* Cas n° 3 *) +(* (f1 x)<>0 l1=0 l2=0 *) +(***********************************) +Case (Req_EM l1 R0); Intro. +Case (Req_EM l2 R0); Intro. +Elim (H0 ``(Rabsolu ((Rsqr (f2 x))*eps)/(8*(f1 x)))``); [Idtac | Apply Rabsolu_pos_lt; Unfold Rdiv Rsqr; Repeat Rewrite Rmult_assoc; Repeat Apply prod_neq_R0; [Assumption | Assumption | Red; Intro; Rewrite H11 in H6; Elim (Rlt_antirefl ? H6) | Apply Rinv_neq_R0; Repeat Apply prod_neq_R0; DiscrR Orelse Assumption]]. +Intros alp_f2d H12. +Cut ``0 < (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))``. +Intro. +Exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) H11). +Simpl. +Intros. +Assert H15 := (Rlt_le_trans ? ? ? H14 (Rmin_l ? ?)). +Assert H16 := (Rlt_le_trans ? ? ? H14 (Rmin_r ? ?)). +Assert H17 := (Rlt_le_trans ? ? ? H15 (Rmin_l ? ?)). +Assert H18 := (Rlt_le_trans ? ? ? H15 (Rmin_r ? ?)). +Assert H19 := (Rlt_le_trans ? ? ? H16 (Rmin_l ? ?)). +Assert H20 := (Rlt_le_trans ? ? ? H16 (Rmin_r ? ?)). +Clear H15 H16. +Rewrite formule; Try Assumption. +Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. +Unfold Rminus. +Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). +Apply Rabsolu_4. +Repeat Rewrite Rabsolu_mult. +Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. +Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. +Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. +Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. +Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. +Intros. +Apply Rlt_4; Assumption. +Rewrite H10. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite <- Rabsolu_mult. +Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite H9. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite <- Rabsolu_mult. +Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Assumption Orelse Idtac. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Right; Symmetry; Apply quadruple_var. +Apply H2; Assumption. +Repeat Apply Rmin_pos. +Apply (cond_pos eps_f2). +Elim H3; Intros; Assumption. +Apply (cond_pos alp_f1d). +Apply (cond_pos alp_f2d). +(***********************************) +(* Cas n° 4 *) +(* (f1 x)<>0 l1=0 l2<>0 *) +(***********************************) +Elim (H0 ``(Rabsolu ((Rsqr (f2 x))*eps)/(8*(f1 x)))``); [Idtac | Apply Rabsolu_pos_lt; Unfold Rsqr Rdiv; Repeat Rewrite Rinv_Rmult; Repeat Apply prod_neq_R0; Try Assumption Orelse DiscrR]. +Intros alp_f2d H11. +Assert H12 := (derivable_continuous_pt ? ? X). +Unfold continuity_pt in H12. +Unfold continue_in in H12. +Unfold limit1_in in H12. +Unfold limit_in in H12. +Unfold dist in H12. +Simpl in H12. +Unfold R_dist in H12. +Elim (H12 ``(Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``). +Intros alp_f2c H13. +Cut ``0 < (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c)))``. +Intro. +Exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))) H14). +Simpl; Intros. +Assert H17 := (Rlt_le_trans ? ? ? H16 (Rmin_l ? ?)). +Assert H18 := (Rlt_le_trans ? ? ? H16 (Rmin_r ? ?)). +Assert H19 := (Rlt_le_trans ? ? ? H18 (Rmin_r ? ?)). +Assert H20 := (Rlt_le_trans ? ? ? H19 (Rmin_l ? ?)). +Assert H21 := (Rlt_le_trans ? ? ? H19 (Rmin_r ? ?)). +Assert H22 := (Rlt_le_trans ? ? ? H18 (Rmin_l ? ?)). +Assert H23 := (Rlt_le_trans ? ? ? H17 (Rmin_l ? ?)). +Assert H24 := (Rlt_le_trans ? ? ? H17 (Rmin_r ? ?)). +Clear H16 H17 H18 H19. +Cut (a:R) ``(Rabsolu a) < alp_f2c`` -> ``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. +Intro. +Rewrite formule; Try Assumption. +Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. +Unfold Rminus. +Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). +Apply Rabsolu_4. +Repeat Rewrite Rabsolu_mult. +Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. +Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. +Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. +Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. +Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. +Intros. +Apply Rlt_4; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite H9. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite <- Rabsolu_mult. +Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Right; Symmetry; Apply quadruple_var. +Apply H2; Assumption. +Intros. +Case (Req_EM a R0); Intro. +Rewrite H17; Rewrite Rplus_Or. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0. +Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr. +Repeat Rewrite Rinv_Rmult; Try Assumption. +Repeat Apply prod_neq_R0; Try Assumption. +Red; Intro H18; Rewrite H18 in H6; Elim (Rlt_antirefl ? H6). +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; Assumption. +Apply Rinv_neq_R0; Assumption. +DiscrR. +DiscrR. +DiscrR. +DiscrR. +DiscrR. +Apply prod_neq_R0; [DiscrR | Assumption]. +Elim H13; Intros. +Apply H19. +Split. +Apply D_x_no_cond; Assumption. +Replace ``x+a-x`` with a; [Assumption | Ring]. +Repeat Apply Rmin_pos. +Apply (cond_pos eps_f2). +Elim H3; Intros; Assumption. +Apply (cond_pos alp_f1d). +Apply (cond_pos alp_f2d). +Elim H13; Intros; Assumption. +Change ``0 < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. +Apply Rabsolu_pos_lt. +Unfold Rsqr Rdiv. +Repeat Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. +Repeat Apply prod_neq_R0; Try Assumption. +Red; Intro H13; Rewrite H13 in H6; Elim (Rlt_antirefl ? H6). +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; Assumption. +Apply Rinv_neq_R0; Assumption. +Apply prod_neq_R0; [DiscrR | Assumption]. +Red; Intro H11; Rewrite H11 in H6; Elim (Rlt_antirefl ? H6). +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; DiscrR. +Apply Rinv_neq_R0; Assumption. +(***********************************) +(* Cas n° 5 *) +(* (f1 x)<>0 l1<>0 l2=0 *) +(***********************************) +Case (Req_EM l2 R0); Intro. +Assert H11 := (derivable_continuous_pt ? ? X). +Unfold continuity_pt in H11. +Unfold continue_in in H11. +Unfold limit1_in in H11. +Unfold limit_in in H11. +Unfold dist in H11. +Simpl in H11. +Unfold R_dist in H11. +Elim (H11 ``(Rabsolu (eps*(Rsqr (f2 x)))/(8*l1))``). +Clear H11; Intros alp_f2t2 H11. +Elim (H0 ``(Rabsolu ((Rsqr (f2 x))*eps)/(8*(f1 x)))``). +Intros alp_f2d H12. +Cut ``0 < (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2)))``. +Intro. +Exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))) H13). +Simpl. +Intros. +Cut (a:R) ``(Rabsolu a)<alp_f2t2`` -> ``(Rabsolu ((f2 (x+a))-(f2 x)))<(Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. +Intro. +Assert H17 := (Rlt_le_trans ? ? ? H15 (Rmin_l ? ?)). +Assert H18 := (Rlt_le_trans ? ? ? H15 (Rmin_r ? ?)). +Assert H19 := (Rlt_le_trans ? ? ? H17 (Rmin_r ? ?)). +Assert H20 := (Rlt_le_trans ? ? ? H17 (Rmin_l ? ?)). +Assert H21 := (Rlt_le_trans ? ? ? H18 (Rmin_r ? ?)). +Assert H22 := (Rlt_le_trans ? ? ? H18 (Rmin_l ? ?)). +Assert H23 := (Rlt_le_trans ? ? ? H21 (Rmin_l ? ?)). +Assert H24 := (Rlt_le_trans ? ? ? H21 (Rmin_r ? ?)). +Clear H15 H17 H18 H21. +Rewrite formule; Try Assumption. +Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. +Unfold Rminus. +Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). +Apply Rabsolu_4. +Repeat Rewrite Rabsolu_mult. +Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. +Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. +Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. +Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. +Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. +Intros. +Apply Rlt_4; Assumption. +Rewrite H10. +Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. +Rewrite Rabsolu_R0; Rewrite Rmult_Ol. +Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. +Rewrite <- Rabsolu_mult. +Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Right; Symmetry; Apply quadruple_var. +Apply H2; Assumption. +Intros. +Case (Req_EM a R0); Intro. +Rewrite H17; Rewrite Rplus_Or; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0. +Apply Rabsolu_pos_lt. +Unfold Rdiv; Rewrite Rinv_Rmult; Try DiscrR Orelse Assumption. +Unfold Rsqr. +Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H18; Rewrite H18 in H6; Elim (Rlt_antirefl ? H6)). +Elim H11; Intros. +Apply H19. +Split. +Apply D_x_no_cond; Assumption. +Replace ``x+a-x`` with a; [Assumption | Ring]. +Repeat Apply Rmin_pos. +Apply (cond_pos eps_f2). +Elim H3; Intros; Assumption. +Apply (cond_pos alp_f1d). +Apply (cond_pos alp_f2d). +Elim H11; Intros; Assumption. +Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr; Rewrite Rinv_Rmult; Try DiscrR Orelse Assumption. +Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H12; Rewrite H12 in H6; Elim (Rlt_antirefl ? H6)). +Change ``0 < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. +Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr; Rewrite Rinv_Rmult; Try DiscrR Orelse Assumption. +Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H12; Rewrite H12 in H6; Elim (Rlt_antirefl ? H6)). +(***********************************) +(* Cas n° 6 *) +(* (f1 x)<>0 l1<>0 l2<>0 *) +(***********************************) +Elim (H0 ``(Rabsolu ((Rsqr (f2 x))*eps)/(8*(f1 x)))``). +Intros alp_f2d H11. +Assert H12 := (derivable_continuous_pt ? ? X). +Unfold continuity_pt in H12. +Unfold continue_in in H12. +Unfold limit1_in in H12. +Unfold limit_in in H12. +Unfold dist in H12. +Simpl in H12. +Unfold R_dist in H12. +Elim (H12 ``(Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``). +Intros alp_f2c H13. +Elim (H12 ``(Rabsolu (eps*(Rsqr (f2 x)))/(8*l1))``). +Intros alp_f2t2 H14. +Cut ``0 < (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) (Rmin alp_f2c alp_f2t2))``. +Intro. +Exists (mkposreal (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) (Rmin alp_f2c alp_f2t2)) H15). +Simpl. +Intros. +Assert H18 := (Rlt_le_trans ? ? ? H17 (Rmin_l ? ?)). +Assert H19 := (Rlt_le_trans ? ? ? H17 (Rmin_r ? ?)). +Assert H20 := (Rlt_le_trans ? ? ? H18 (Rmin_l ? ?)). +Assert H21 := (Rlt_le_trans ? ? ? H18 (Rmin_r ? ?)). +Assert H22 := (Rlt_le_trans ? ? ? H19 (Rmin_l ? ?)). +Assert H23 := (Rlt_le_trans ? ? ? H19 (Rmin_r ? ?)). +Assert H24 := (Rlt_le_trans ? ? ? H20 (Rmin_l ? ?)). +Assert H25 := (Rlt_le_trans ? ? ? H20 (Rmin_r ? ?)). +Assert H26 := (Rlt_le_trans ? ? ? H21 (Rmin_l ? ?)). +Assert H27 := (Rlt_le_trans ? ? ? H21 (Rmin_r ? ?)). +Clear H17 H18 H19 H20 H21. +Cut (a:R) ``(Rabsolu a) < alp_f2t2`` -> ``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. +Cut (a:R) ``(Rabsolu a) < alp_f2c`` -> ``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. +Intros. +Rewrite formule; Try Assumption. +Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. +Unfold Rminus. +Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). +Apply Rabsolu_4. +Repeat Rewrite Rabsolu_mult. +Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. +Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. +Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. +Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. +Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. +Intros. +Apply Rlt_4; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Rewrite <- Rabsolu_mult. +Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption. +Apply H2; Assumption. +Apply Rmin_2; Assumption. +Right; Symmetry; Apply quadruple_var. +Apply H2; Assumption. +Intros. +Case (Req_EM a R0); Intro. +Rewrite H18; Rewrite Rplus_Or; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr; Rewrite Rinv_Rmult. +Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H28; Rewrite H28 in H6; Elim (Rlt_antirefl ? H6)). +Apply prod_neq_R0; [DiscrR | Assumption]. +Apply prod_neq_R0; [DiscrR | Assumption]. +Assumption. +Elim H13; Intros. +Apply H20. +Split. +Apply D_x_no_cond; Assumption. +Replace ``x+a-x`` with a; [Assumption | Ring]. +Intros. +Case (Req_EM a R0); Intro. +Rewrite H18; Rewrite Rplus_Or; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr; Rewrite Rinv_Rmult. +Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H28; Rewrite H28 in H6; Elim (Rlt_antirefl ? H6)). +DiscrR. +Assumption. +Elim H14; Intros. +Apply H20. +Split. +Unfold D_x no_cond; Split. +Trivial. +Apply Rminus_not_eq_right. +Replace ``x+a-x`` with a; [Assumption | Ring]. +Replace ``x+a-x`` with a; [Assumption | Ring]. +Repeat Apply Rmin_pos. +Apply (cond_pos eps_f2). +Elim H3; Intros; Assumption. +Apply (cond_pos alp_f1d). +Apply (cond_pos alp_f2d). +Elim H13; Intros; Assumption. +Elim H14; Intros; Assumption. +Change ``0 < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``; Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr; Rewrite Rinv_Rmult; Try DiscrR Orelse Assumption. +Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H14; Rewrite H14 in H6; Elim (Rlt_antirefl ? H6)). +Change ``0 < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``; Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr; Rewrite Rinv_Rmult. +Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H13; Rewrite H13 in H6; Elim (Rlt_antirefl ? H6)). +Apply prod_neq_R0; [DiscrR | Assumption]. +Apply prod_neq_R0; [DiscrR | Assumption]. +Assumption. +Apply Rabsolu_pos_lt. +Unfold Rdiv Rsqr; Rewrite Rinv_Rmult; [Idtac | DiscrR | Assumption]. +Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H11; Rewrite H11 in H6; Elim (Rlt_antirefl ? H6)). +Intros. +Unfold Rdiv. +Apply Rlt_monotony_contra with ``(Rabsolu (f2 (x+a)))``. +Apply Rabsolu_pos_lt; Apply H2. +Apply Rlt_le_trans with (Rmin eps_f2 alp_f2). +Assumption. +Apply Rmin_l. +Rewrite <- Rinv_r_sym. +Apply Rlt_monotony_contra with (Rabsolu (f2 x)). +Apply Rabsolu_pos_lt; Assumption. +Rewrite Rmult_1r. +Rewrite (Rmult_sym (Rabsolu (f2 x))). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Apply Rlt_monotony_contra with ``/2``. +Apply Rlt_Rinv; Sup0. +Repeat Rewrite (Rmult_sym ``/2``). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r. +Unfold Rdiv in H5; Apply H5. +Replace ``x+a-x`` with a. +Assert H7 := (Rlt_le_trans ? ? ? H6 (Rmin_r ? ?)); Assumption. +Ring. +DiscrR. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Apply H2. +Assert H7 := (Rlt_le_trans ? ? ? H6 (Rmin_l ? ?)); Assumption. +Intros. +Assert H6 := (H4 a H5). +Rewrite <- (Rabsolu_Ropp ``(f2 a)-(f2 x)``) in H6. +Rewrite Ropp_distr2 in H6. +Assert H7 := (Rle_lt_trans ? ? ? (Rabsolu_triang_inv ? ?) H6). +Apply Rlt_anti_compatibility with ``-(Rabsolu (f2 a)) + (Rabsolu (f2 x))/2``. +Rewrite Rplus_assoc. +Rewrite <- double_var. +Do 2 Rewrite (Rplus_sym ``-(Rabsolu (f2 a))``). +Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. +Unfold Rminus in H7; Assumption. +Intros. +Case (Req_EM x x0); Intro. +Rewrite <- H5; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Unfold Rdiv; Apply Rmult_lt_pos; [Apply Rabsolu_pos_lt; Assumption | Apply Rlt_Rinv; Sup0]. +Elim H3; Intros. +Apply H7. +Split. +Unfold D_x no_cond; Split. +Trivial. +Assumption. +Assumption. +Qed. + +Lemma derivable_pt_div : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> ``(f2 x)<>0`` -> (derivable_pt (div_fct f1 f2) x). +Unfold derivable_pt. +Intros. +Elim X; Intros. +Elim X0; Intros. +Apply Specif.existT with ``(x0*(f2 x)-x1*(f1 x))/(Rsqr (f2 x))``. +Apply derivable_pt_lim_div; Assumption. +Qed. + +Lemma derivable_div : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> ((x:R)``(f2 x)<>0``) -> (derivable (div_fct f1 f2)). +Unfold derivable; Intros. +Apply (derivable_pt_div ? ? ? (X x) (X0 x) (H x)). +Qed. + +Lemma derive_pt_div : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x);na:``(f2 x)<>0``) ``(derive_pt (div_fct f1 f2) x (derivable_pt_div ? ? ? pr1 pr2 na)) == ((derive_pt f1 x pr1)*(f2 x)-(derive_pt f2 x pr2)*(f1 x))/(Rsqr (f2 x))``. +Intros. +Assert H := (derivable_derive f1 x pr1). +Assert H0 := (derivable_derive f2 x pr2). +Assert H1 := (derivable_derive (div_fct f1 f2) x (derivable_pt_div ? ? ? pr1 pr2 na)). +Elim H; Clear H; Intros l1 H. +Elim H0; Clear H0; Intros l2 H0. +Elim H1; Clear H1; Intros l H1. +Rewrite H; Rewrite H0; Apply derive_pt_eq_0. +Assert H3 := (projT2 ? ? pr1). +Unfold derive_pt in H; Rewrite H in H3. +Assert H4 := (projT2 ? ? pr2). +Unfold derive_pt in H0; Rewrite H0 in H4. +Apply derivable_pt_lim_div; Assumption. +Qed. diff --git a/theories7/Reals/Ranalysis4.v b/theories7/Reals/Ranalysis4.v new file mode 100644 index 00000000..061854dc --- /dev/null +++ b/theories7/Reals/Ranalysis4.v @@ -0,0 +1,313 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Ranalysis4.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis1. +Require Ranalysis3. +Require Exp_prop. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(**********) +Lemma derivable_pt_inv : (f:R->R;x:R) ``(f x)<>0`` -> (derivable_pt f x) -> (derivable_pt (inv_fct f) x). +Intros; Cut (derivable_pt (div_fct (fct_cte R1) f) x) -> (derivable_pt (inv_fct f) x). +Intro; Apply X0. +Apply derivable_pt_div. +Apply derivable_pt_const. +Assumption. +Assumption. +Unfold div_fct inv_fct fct_cte; Intro; Elim X0; Intros; Unfold derivable_pt; Apply Specif.existT with x0; Unfold derivable_pt_abs; Unfold derivable_pt_lim; Unfold derivable_pt_abs in p; Unfold derivable_pt_lim in p; Intros; Elim (p eps H0); Intros; Exists x1; Intros; Unfold Rdiv in H1; Unfold Rdiv; Rewrite <- (Rmult_1l ``/(f x)``); Rewrite <- (Rmult_1l ``/(f (x+h))``). +Apply H1; Assumption. +Qed. + +(**********) +Lemma pr_nu_var : (f,g:R->R;x:R;pr1:(derivable_pt f x);pr2:(derivable_pt g x)) f==g -> (derive_pt f x pr1) == (derive_pt g x pr2). +Unfold derivable_pt derive_pt; Intros. +Elim pr1; Intros. +Elim pr2; Intros. +Simpl. +Rewrite H in p. +Apply unicite_limite with g x; Assumption. +Qed. + +(**********) +Lemma pr_nu_var2 : (f,g:R->R;x:R;pr1:(derivable_pt f x);pr2:(derivable_pt g x)) ((h:R)(f h)==(g h)) -> (derive_pt f x pr1) == (derive_pt g x pr2). +Unfold derivable_pt derive_pt; Intros. +Elim pr1; Intros. +Elim pr2; Intros. +Simpl. +Assert H0 := (unicite_step2 ? ? ? p). +Assert H1 := (unicite_step2 ? ? ? p0). +Cut (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h <> 0`` x1 ``0``). +Intro; Assert H3 := (unicite_step1 ? ? ? ? H0 H2). +Assumption. +Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Unfold limit1_in in H1; Unfold limit_in in H1; Unfold dist in H1; Simpl in H1; Unfold R_dist in H1. +Intros; Elim (H1 eps H2); Intros. +Elim H3; Intros. +Exists x2. +Split. +Assumption. +Intros; Do 2 Rewrite H; Apply H5; Assumption. +Qed. + +(**********) +Lemma derivable_inv : (f:R->R) ((x:R)``(f x)<>0``)->(derivable f)->(derivable (inv_fct f)). +Intros. +Unfold derivable; Intro. +Apply derivable_pt_inv. +Apply (H x). +Apply (X x). +Qed. + +Lemma derive_pt_inv : (f:R->R;x:R;pr:(derivable_pt f x);na:``(f x)<>0``) (derive_pt (inv_fct f) x (derivable_pt_inv f x na pr)) == ``-(derive_pt f x pr)/(Rsqr (f x))``. +Intros; Replace (derive_pt (inv_fct f) x (derivable_pt_inv f x na pr)) with (derive_pt (div_fct (fct_cte R1) f) x (derivable_pt_div (fct_cte R1) f x (derivable_pt_const R1 x) pr na)). +Rewrite derive_pt_div; Rewrite derive_pt_const; Unfold fct_cte; Rewrite Rmult_Ol; Rewrite Rmult_1r; Unfold Rminus; Rewrite Rplus_Ol; Reflexivity. +Apply pr_nu_var2. +Intro; Unfold div_fct fct_cte inv_fct. +Unfold Rdiv; Ring. +Qed. + +(* Rabsolu *) +Lemma Rabsolu_derive_1 : (x:R) ``0<x`` -> (derivable_pt_lim Rabsolu x ``1``). +Intros. +Unfold derivable_pt_lim; Intros. +Exists (mkposreal x H); Intros. +Rewrite (Rabsolu_right x). +Rewrite (Rabsolu_right ``x+h``). +Rewrite Rplus_sym. +Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r. +Rewrite Rplus_Or; Unfold Rdiv; Rewrite <- Rinv_r_sym. +Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H0. +Apply H1. +Apply Rle_sym1. +Case (case_Rabsolu h); Intro. +Rewrite (Rabsolu_left h r) in H2. +Left; Rewrite Rplus_sym; Apply Rlt_anti_compatibility with ``-h``; Rewrite Rplus_Or; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Apply H2. +Apply ge0_plus_ge0_is_ge0. +Left; Apply H. +Apply Rle_sym2; Apply r. +Left; Apply H. +Qed. + +Lemma Rabsolu_derive_2 : (x:R) ``x<0`` -> (derivable_pt_lim Rabsolu x ``-1``). +Intros. +Unfold derivable_pt_lim; Intros. +Cut ``0< -x``. +Intro; Exists (mkposreal ``-x`` H1); Intros. +Rewrite (Rabsolu_left x). +Rewrite (Rabsolu_left ``x+h``). +Rewrite Rplus_sym. +Rewrite Ropp_distr1. +Unfold Rminus; Rewrite Ropp_Ropp; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l. +Rewrite Rplus_Or; Unfold Rdiv. +Rewrite Ropp_mul1. +Rewrite <- Rinv_r_sym. +Rewrite Ropp_Ropp; Rewrite Rplus_Ropp_l; Rewrite Rabsolu_R0; Apply H0. +Apply H2. +Case (case_Rabsolu h); Intro. +Apply Ropp_Rlt. +Rewrite Ropp_O; Rewrite Ropp_distr1; Apply gt0_plus_gt0_is_gt0. +Apply H1. +Apply Rgt_RO_Ropp; Apply r. +Rewrite (Rabsolu_right h r) in H3. +Apply Rlt_anti_compatibility with ``-x``; Rewrite Rplus_Or; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Apply H3. +Apply H. +Apply Rgt_RO_Ropp; Apply H. +Qed. + +(* Rabsolu is derivable for all x <> 0 *) +Lemma derivable_pt_Rabsolu : (x:R) ``x<>0`` -> (derivable_pt Rabsolu x). +Intros. +Case (total_order_T x R0); Intro. +Elim s; Intro. +Unfold derivable_pt; Apply Specif.existT with ``-1``. +Apply (Rabsolu_derive_2 x a). +Elim H; Exact b. +Unfold derivable_pt; Apply Specif.existT with ``1``. +Apply (Rabsolu_derive_1 x r). +Qed. + +(* Rabsolu is continuous for all x *) +Lemma continuity_Rabsolu : (continuity Rabsolu). +Unfold continuity; Intro. +Case (Req_EM x R0); Intro. +Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Exists eps; Split. +Apply H0. +Intros; Rewrite H; Rewrite Rabsolu_R0; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Elim H1; Intros; Rewrite H in H3; Unfold Rminus in H3; Rewrite Ropp_O in H3; Rewrite Rplus_Or in H3; Apply H3. +Apply derivable_continuous_pt; Apply (derivable_pt_Rabsolu x H). +Qed. + +(* Finite sums : Sum a_k x^k *) +Lemma continuity_finite_sum : (An:nat->R;N:nat) (continuity [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N)). +Intros; Unfold continuity; Intro. +Induction N. +Simpl. +Apply continuity_pt_const. +Unfold constant; Intros; Reflexivity. +Replace [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` (S N)) with (plus_fct [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) [y:R]``(An (S N))*(pow y (S N))``). +Apply continuity_pt_plus. +Apply HrecN. +Replace [y:R]``(An (S N))*(pow y (S N))`` with (mult_real_fct (An (S N)) [y:R](pow y (S N))). +Apply continuity_pt_scal. +Apply derivable_continuous_pt. +Apply derivable_pt_pow. +Reflexivity. +Reflexivity. +Qed. + +Lemma derivable_pt_lim_fs : (An:nat->R;x:R;N:nat) (lt O N) -> (derivable_pt_lim [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N))). +Intros; Induction N. +Elim (lt_n_n ? H). +Cut N=O\/(lt O N). +Intro; Elim H0; Intro. +Rewrite H1. +Simpl. +Replace [y:R]``(An O)*1+(An (S O))*(y*1)`` with (plus_fct (fct_cte ``(An O)*1``) (mult_real_fct ``(An (S O))`` (mult_fct id (fct_cte R1)))). +Replace ``1*(An (S O))*1`` with ``0+(An (S O))*(1*(fct_cte R1 x)+(id x)*0)``. +Apply derivable_pt_lim_plus. +Apply derivable_pt_lim_const. +Apply derivable_pt_lim_scal. +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_const. +Unfold fct_cte id; Ring. +Reflexivity. +Replace [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` (S N)) with (plus_fct [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) [y:R]``(An (S N))*(pow y (S N))``). +Replace (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred (S N))) with (Rplus (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N)) ``(An (S N))*((INR (S (pred (S N))))*(pow x (pred (S N))))``). +Apply derivable_pt_lim_plus. +Apply HrecN. +Assumption. +Replace [y:R]``(An (S N))*(pow y (S N))`` with (mult_real_fct (An (S N)) [y:R](pow y (S N))). +Apply derivable_pt_lim_scal. +Replace (pred (S N)) with N; [Idtac | Reflexivity]. +Pattern 3 N; Replace N with (pred (S N)). +Apply derivable_pt_lim_pow. +Reflexivity. +Reflexivity. +Cut (pred (S N)) = (S (pred N)). +Intro; Rewrite H2. +Rewrite tech5. +Apply Rplus_plus_r. +Rewrite <- H2. +Replace (pred (S N)) with N; [Idtac | Reflexivity]. +Ring. +Simpl. +Apply S_pred with O; Assumption. +Unfold plus_fct. +Simpl; Reflexivity. +Inversion H. +Left; Reflexivity. +Right; Apply lt_le_trans with (1); [Apply lt_O_Sn | Assumption]. +Qed. + +Lemma derivable_pt_lim_finite_sum : (An:(nat->R); x:R; N:nat) (derivable_pt_lim [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x (Cases N of O => R0 | _ => (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N)) end)). +Intros. +Induction N. +Simpl. +Rewrite Rmult_1r. +Replace [_:R]``(An O)`` with (fct_cte (An O)); [Apply derivable_pt_lim_const | Reflexivity]. +Apply derivable_pt_lim_fs; Apply lt_O_Sn. +Qed. + +Lemma derivable_pt_finite_sum : (An:nat->R;N:nat;x:R) (derivable_pt [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x). +Intros. +Unfold derivable_pt. +Assert H := (derivable_pt_lim_finite_sum An x N). +Induction N. +Apply Specif.existT with R0; Apply H. +Apply Specif.existT with (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred (S N))); Apply H. +Qed. + +Lemma derivable_finite_sum : (An:nat->R;N:nat) (derivable [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N)). +Intros; Unfold derivable; Intro; Apply derivable_pt_finite_sum. +Qed. + +(* Regularity of hyperbolic functions *) +Lemma derivable_pt_lim_cosh : (x:R) (derivable_pt_lim cosh x ``(sinh x)``). +Intro. +Unfold cosh sinh; Unfold Rdiv. +Replace [x0:R]``((exp x0)+(exp ( -x0)))*/2`` with (mult_fct (plus_fct exp (comp exp (opp_fct id))) (fct_cte ``/2``)); [Idtac | Reflexivity]. +Replace ``((exp x)-(exp ( -x)))*/2`` with ``((exp x)+((exp (-x))*-1))*((fct_cte (Rinv 2)) x)+((plus_fct exp (comp exp (opp_fct id))) x)*0``. +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_plus. +Apply derivable_pt_lim_exp. +Apply derivable_pt_lim_comp. +Apply derivable_pt_lim_opp. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_exp. +Apply derivable_pt_lim_const. +Unfold plus_fct mult_real_fct comp opp_fct id fct_cte; Ring. +Qed. + +Lemma derivable_pt_lim_sinh : (x:R) (derivable_pt_lim sinh x ``(cosh x)``). +Intro. +Unfold cosh sinh; Unfold Rdiv. +Replace [x0:R]``((exp x0)-(exp ( -x0)))*/2`` with (mult_fct (minus_fct exp (comp exp (opp_fct id))) (fct_cte ``/2``)); [Idtac | Reflexivity]. +Replace ``((exp x)+(exp ( -x)))*/2`` with ``((exp x)-((exp (-x))*-1))*((fct_cte (Rinv 2)) x)+((minus_fct exp (comp exp (opp_fct id))) x)*0``. +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_minus. +Apply derivable_pt_lim_exp. +Apply derivable_pt_lim_comp. +Apply derivable_pt_lim_opp. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_exp. +Apply derivable_pt_lim_const. +Unfold plus_fct mult_real_fct comp opp_fct id fct_cte; Ring. +Qed. + +Lemma derivable_pt_exp : (x:R) (derivable_pt exp x). +Intro. +Unfold derivable_pt. +Apply Specif.existT with (exp x). +Apply derivable_pt_lim_exp. +Qed. + +Lemma derivable_pt_cosh : (x:R) (derivable_pt cosh x). +Intro. +Unfold derivable_pt. +Apply Specif.existT with (sinh x). +Apply derivable_pt_lim_cosh. +Qed. + +Lemma derivable_pt_sinh : (x:R) (derivable_pt sinh x). +Intro. +Unfold derivable_pt. +Apply Specif.existT with (cosh x). +Apply derivable_pt_lim_sinh. +Qed. + +Lemma derivable_exp : (derivable exp). +Unfold derivable; Apply derivable_pt_exp. +Qed. + +Lemma derivable_cosh : (derivable cosh). +Unfold derivable; Apply derivable_pt_cosh. +Qed. + +Lemma derivable_sinh : (derivable sinh). +Unfold derivable; Apply derivable_pt_sinh. +Qed. + +Lemma derive_pt_exp : (x:R) (derive_pt exp x (derivable_pt_exp x))==(exp x). +Intro; Apply derive_pt_eq_0. +Apply derivable_pt_lim_exp. +Qed. + +Lemma derive_pt_cosh : (x:R) (derive_pt cosh x (derivable_pt_cosh x))==(sinh x). +Intro; Apply derive_pt_eq_0. +Apply derivable_pt_lim_cosh. +Qed. + +Lemma derive_pt_sinh : (x:R) (derive_pt sinh x (derivable_pt_sinh x))==(cosh x). +Intro; Apply derive_pt_eq_0. +Apply derivable_pt_lim_sinh. +Qed. diff --git a/theories7/Reals/Raxioms.v b/theories7/Reals/Raxioms.v new file mode 100644 index 00000000..caf8524c --- /dev/null +++ b/theories7/Reals/Raxioms.v @@ -0,0 +1,172 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Raxioms.v,v 1.2.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +(*********************************************************) +(** Axiomatisation of the classical reals *) +(*********************************************************) + +Require Export ZArith_base. +V7only [ +Require Export Rsyntax. +Import R_scope. +]. +Open Local Scope R_scope. + +V7only [ +(*********************************************************) +(* Compatibility *) +(*********************************************************) +Notation sumboolT := Specif.sumbool. +Notation leftT := Specif.left. +Notation rightT := Specif.right. +Notation sumorT := Specif.sumor. +Notation inleftT := Specif.inleft. +Notation inrightT := Specif.inright. +Notation sigTT := Specif.sigT. +Notation existTT := Specif.existT. +Notation SigT := Specif.sigT. +]. + +(*********************************************************) +(* Field axioms *) +(*********************************************************) + +(*********************************************************) +(** Addition *) +(*********************************************************) + +(**********) +Axiom Rplus_sym:(r1,r2:R)``r1+r2==r2+r1``. +Hints Resolve Rplus_sym : real. + +(**********) +Axiom Rplus_assoc:(r1,r2,r3:R)``(r1+r2)+r3==r1+(r2+r3)``. +Hints Resolve Rplus_assoc : real. + +(**********) +Axiom Rplus_Ropp_r:(r:R)``r+(-r)==0``. +Hints Resolve Rplus_Ropp_r : real v62. + +(**********) +Axiom Rplus_Ol:(r:R)``0+r==r``. +Hints Resolve Rplus_Ol : real. + +(***********************************************************) +(** Multiplication *) +(***********************************************************) + +(**********) +Axiom Rmult_sym:(r1,r2:R)``r1*r2==r2*r1``. +Hints Resolve Rmult_sym : real v62. + +(**********) +Axiom Rmult_assoc:(r1,r2,r3:R)``(r1*r2)*r3==r1*(r2*r3)``. +Hints Resolve Rmult_assoc : real v62. + +(**********) +Axiom Rinv_l:(r:R)``r<>0``->``(/r)*r==1``. +Hints Resolve Rinv_l : real. + +(**********) +Axiom Rmult_1l:(r:R)``1*r==r``. +Hints Resolve Rmult_1l : real. + +(**********) +Axiom R1_neq_R0:``1<>0``. +Hints Resolve R1_neq_R0 : real. + +(*********************************************************) +(** Distributivity *) +(*********************************************************) + +(**********) +Axiom Rmult_Rplus_distr:(r1,r2,r3:R)``r1*(r2+r3)==(r1*r2)+(r1*r3)``. +Hints Resolve Rmult_Rplus_distr : real v62. + +(*********************************************************) +(** Order axioms *) +(*********************************************************) +(*********************************************************) +(** Total Order *) +(*********************************************************) + +(**********) +Axiom total_order_T:(r1,r2:R)(sumorT (sumboolT ``r1<r2`` r1==r2) ``r1>r2``). + +(*********************************************************) +(** Lower *) +(*********************************************************) + +(**********) +Axiom Rlt_antisym:(r1,r2:R)``r1<r2`` -> ~ ``r2<r1``. + +(**********) +Axiom Rlt_trans:(r1,r2,r3:R) + ``r1<r2``->``r2<r3``->``r1<r3``. + +(**********) +Axiom Rlt_compatibility:(r,r1,r2:R)``r1<r2``->``r+r1<r+r2``. + +(**********) +Axiom Rlt_monotony:(r,r1,r2:R)``0<r``->``r1<r2``->``r*r1<r*r2``. + +Hints Resolve Rlt_antisym Rlt_compatibility Rlt_monotony : real. + +(**********************************************************) +(** Injection from N to R *) +(**********************************************************) + +(**********) +Fixpoint INR [n:nat]:R:=(Cases n of + O => ``0`` + |(S O) => ``1`` + |(S n) => ``(INR n)+1`` + end). +Arguments Scope INR [nat_scope]. + + +(**********************************************************) +(** Injection from [Z] to [R] *) +(**********************************************************) + +(**********) +Definition IZR:Z->R:=[z:Z](Cases z of + ZERO => ``0`` + |(POS n) => (INR (convert n)) + |(NEG n) => ``-(INR (convert n))`` + end). +Arguments Scope IZR [Z_scope]. + +(**********************************************************) +(** [R] Archimedian *) +(**********************************************************) + +(**********) +Axiom archimed:(r:R)``(IZR (up r)) > r``/\``(IZR (up r))-r <= 1``. + +(**********************************************************) +(** [R] Complete *) +(**********************************************************) + +(**********) +Definition is_upper_bound:=[E:R->Prop][m:R](x:R)(E x)->``x <= m``. + +(**********) +Definition bound:=[E:R->Prop](ExT [m:R](is_upper_bound E m)). + +(**********) +Definition is_lub:=[E:R->Prop][m:R] + (is_upper_bound E m)/\(b:R)(is_upper_bound E b)->``m <= b``. + +(**********) +Axiom complet:(E:R->Prop)(bound E)-> + (ExT [x:R] (E x))-> + (sigTT R [m:R](is_lub E m)). + diff --git a/theories7/Reals/Rbase.v b/theories7/Reals/Rbase.v new file mode 100644 index 00000000..54226206 --- /dev/null +++ b/theories7/Reals/Rbase.v @@ -0,0 +1,14 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rbase.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +Require Export Rdefinitions. +Require Export Raxioms. +Require Export RIneq. +Require Export DiscrR. diff --git a/theories7/Reals/Rbasic_fun.v b/theories7/Reals/Rbasic_fun.v new file mode 100644 index 00000000..3d143e34 --- /dev/null +++ b/theories7/Reals/Rbasic_fun.v @@ -0,0 +1,476 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rbasic_fun.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +(*********************************************************) +(** Complements for the real numbers *) +(* *) +(*********************************************************) + +Require Rbase. +Require R_Ifp. +Require Fourier. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Implicit Variable Type r:R. + +(*******************************) +(** Rmin *) +(*******************************) + +(*********) +Definition Rmin :R->R->R:=[x,y:R] + Cases (total_order_Rle x y) of + (leftT _) => x + | (rightT _) => y + end. + +(*********) +Lemma Rmin_Rgt_l:(r1,r2,r:R)(Rgt (Rmin r1 r2) r) -> + ((Rgt r1 r)/\(Rgt r2 r)). +Intros r1 r2 r;Unfold Rmin;Case (total_order_Rle r1 r2);Intros. +Split. +Assumption. +Unfold Rgt;Unfold Rgt in H;Exact (Rlt_le_trans r r1 r2 H r0). +Split. +Generalize (not_Rle r1 r2 n);Intro;Exact (Rgt_trans r1 r2 r H0 H). +Assumption. +Qed. + +(*********) +Lemma Rmin_Rgt_r:(r1,r2,r:R)(((Rgt r1 r)/\(Rgt r2 r)) -> + (Rgt (Rmin r1 r2) r)). +Intros;Unfold Rmin;Case (total_order_Rle r1 r2);Elim H;Clear H;Intros; + Assumption. +Qed. + +(*********) +Lemma Rmin_Rgt:(r1,r2,r:R)(Rgt (Rmin r1 r2) r)<-> + ((Rgt r1 r)/\(Rgt r2 r)). +Intros; Split. +Exact (Rmin_Rgt_l r1 r2 r). +Exact (Rmin_Rgt_r r1 r2 r). +Qed. + +(*********) +Lemma Rmin_l : (x,y:R) ``(Rmin x y)<=x``. +Intros; Unfold Rmin; Case (total_order_Rle x y); Intro H1; [Right; Reflexivity | Auto with real]. +Qed. + +(*********) +Lemma Rmin_r : (x,y:R) ``(Rmin x y)<=y``. +Intros; Unfold Rmin; Case (total_order_Rle x y); Intro H1; [Assumption | Auto with real]. +Qed. + +(*********) +Lemma Rmin_sym : (a,b:R) (Rmin a b)==(Rmin b a). +Intros; Unfold Rmin; Case (total_order_Rle a b); Case (total_order_Rle b a); Intros; Try Reflexivity Orelse (Apply Rle_antisym; Assumption Orelse Auto with real). +Qed. + +(*********) +Lemma Rmin_stable_in_posreal : (x,y:posreal) ``0<(Rmin x y)``. +Intros; Apply Rmin_Rgt_r; Split; [Apply (cond_pos x) | Apply (cond_pos y)]. +Qed. + +(*******************************) +(** Rmax *) +(*******************************) + +(*********) +Definition Rmax :R->R->R:=[x,y:R] + Cases (total_order_Rle x y) of + (leftT _) => y + | (rightT _) => x + end. + +(*********) +Lemma Rmax_Rle:(r1,r2,r:R)(Rle r (Rmax r1 r2))<-> + ((Rle r r1)\/(Rle r r2)). +Intros;Split. +Unfold Rmax;Case (total_order_Rle r1 r2);Intros;Auto. +Intro;Unfold Rmax;Case (total_order_Rle r1 r2);Elim H;Clear H;Intros;Auto. +Apply (Rle_trans r r1 r2);Auto. +Generalize (not_Rle r1 r2 n);Clear n;Intro;Unfold Rgt in H0; + Apply (Rlt_le r r1 (Rle_lt_trans r r2 r1 H H0)). +Qed. + +Lemma RmaxLess1: (r1, r2 : R) (Rle r1 (Rmax r1 r2)). +Intros r1 r2; Unfold Rmax; Case (total_order_Rle r1 r2); Auto with real. +Qed. + +Lemma RmaxLess2: (r1, r2 : R) (Rle r2 (Rmax r1 r2)). +Intros r1 r2; Unfold Rmax; Case (total_order_Rle r1 r2); Auto with real. +Qed. + +Lemma RmaxSym: (p, q : R) (Rmax p q) == (Rmax q p). +Intros p q; Unfold Rmax; + Case (total_order_Rle p q); Case (total_order_Rle q p); Auto; Intros H1 H2; + Apply Rle_antisym; Auto with real. +Qed. + +Lemma RmaxRmult: + (p, q, r : R) + (Rle R0 r) -> (Rmax (Rmult r p) (Rmult r q)) == (Rmult r (Rmax p q)). +Intros p q r H; Unfold Rmax. +Case (total_order_Rle p q); Case (total_order_Rle (Rmult r p) (Rmult r q)); + Auto; Intros H1 H2; Auto. +Case H; Intros E1. +Case H1; Auto with real. +Rewrite <- E1; Repeat Rewrite Rmult_Ol; Auto. +Case H; Intros E1. +Case H2; Auto with real. +Apply Rle_monotony_contra with z := r; Auto. +Rewrite <- E1; Repeat Rewrite Rmult_Ol; Auto. +Qed. + +Lemma Rmax_stable_in_negreal : (x,y:negreal) ``(Rmax x y)<0``. +Intros; Unfold Rmax; Case (total_order_Rle x y); Intro; [Apply (cond_neg y) | Apply (cond_neg x)]. +Qed. + +(*******************************) +(** Rabsolu *) +(*******************************) + +(*********) +Lemma case_Rabsolu:(r:R)(sumboolT (Rlt r R0) (Rge r R0)). +Intro;Generalize (total_order_Rle R0 r);Intro X;Elim X;Intro;Clear X. +Right;Apply (Rle_sym1 R0 r a). +Left;Fold (Rgt R0 r);Apply (not_Rle R0 r b). +Qed. + +(*********) +Definition Rabsolu:R->R:= + [r:R](Cases (case_Rabsolu r) of + (leftT _) => (Ropp r) + |(rightT _) => r + end). + +(*********) +Lemma Rabsolu_R0:(Rabsolu R0)==R0. +Unfold Rabsolu;Case (case_Rabsolu R0);Auto;Intro. +Generalize (Rlt_antirefl R0);Intro;ElimType False;Auto. +Qed. + +Lemma Rabsolu_R1: (Rabsolu R1)==R1. +Unfold Rabsolu; Case (case_Rabsolu R1); Auto with real. +Intros H; Absurd ``1 < 0``;Auto with real. +Qed. + +(*********) +Lemma Rabsolu_no_R0:(r:R)~r==R0->~(Rabsolu r)==R0. +Intros;Unfold Rabsolu;Case (case_Rabsolu r);Intro;Auto. +Apply Ropp_neq;Auto. +Qed. + +(*********) +Lemma Rabsolu_left: (r:R)(Rlt r R0)->((Rabsolu r) == (Ropp r)). +Intros;Unfold Rabsolu;Case (case_Rabsolu r);Trivial;Intro;Absurd (Rge r R0). +Exact (Rlt_ge_not r R0 H). +Assumption. +Qed. + +(*********) +Lemma Rabsolu_right: (r:R)(Rge r R0)->((Rabsolu r) == r). +Intros;Unfold Rabsolu;Case (case_Rabsolu r);Intro. +Absurd (Rge r R0). +Exact (Rlt_ge_not r R0 r0). +Assumption. +Trivial. +Qed. + +Lemma Rabsolu_left1: (a : R) (Rle a R0) -> (Rabsolu a) == (Ropp a). +Intros a H; Case H; Intros H1. +Apply Rabsolu_left; Auto. +Rewrite H1; Simpl; Rewrite Rabsolu_right; Auto with real. +Qed. + +(*********) +Lemma Rabsolu_pos:(x:R)(Rle R0 (Rabsolu x)). +Intros;Unfold Rabsolu;Case (case_Rabsolu x);Intro. +Generalize (Rlt_Ropp x R0 r);Intro;Unfold Rgt in H; + Rewrite Ropp_O in H;Unfold Rle;Left;Assumption. +Apply Rle_sym2;Assumption. +Qed. + +Lemma Rle_Rabsolu: + (x:R) (Rle x (Rabsolu x)). +Intro; Unfold Rabsolu;Case (case_Rabsolu x);Intros;Fourier. +Qed. + +(*********) +Lemma Rabsolu_pos_eq:(x:R)(Rle R0 x)->(Rabsolu x)==x. +Intros;Unfold Rabsolu;Case (case_Rabsolu x);Intro; + [Generalize (Rle_not R0 x r);Intro;ElimType False;Auto|Trivial]. +Qed. + +(*********) +Lemma Rabsolu_Rabsolu:(x:R)(Rabsolu (Rabsolu x))==(Rabsolu x). +Intro;Apply (Rabsolu_pos_eq (Rabsolu x) (Rabsolu_pos x)). +Qed. + +(*********) +Lemma Rabsolu_pos_lt:(x:R)(~x==R0)->(Rlt R0 (Rabsolu x)). +Intros;Generalize (Rabsolu_pos x);Intro;Unfold Rle in H0; + Elim H0;Intro;Auto. +ElimType False;Clear H0;Elim H;Clear H;Generalize H1; + Unfold Rabsolu;Case (case_Rabsolu x);Intros;Auto. +Clear r H1; Generalize (Rplus_plus_r x R0 (Ropp x) H0); + Rewrite (let (H1,H2)=(Rplus_ne x) in H1);Rewrite (Rplus_Ropp_r x);Trivial. +Qed. + +(*********) +Lemma Rabsolu_minus_sym:(x,y:R) + (Rabsolu (Rminus x y))==(Rabsolu (Rminus y x)). +Intros;Unfold Rabsolu;Case (case_Rabsolu (Rminus x y)); + Case (case_Rabsolu (Rminus y x));Intros. + Generalize (Rminus_lt y x r);Generalize (Rminus_lt x y r0);Intros; + Generalize (Rlt_antisym x y H);Intro;ElimType False;Auto. +Rewrite (Ropp_distr2 x y);Trivial. +Rewrite (Ropp_distr2 y x);Trivial. +Unfold Rge in r r0;Elim r;Elim r0;Intros;Clear r r0. +Generalize (Rgt_RoppO (Rminus x y) H);Rewrite (Ropp_distr2 x y); + Intro;Unfold Rgt in H0;Generalize (Rlt_antisym R0 (Rminus y x) H0); + Intro;ElimType False;Auto. +Rewrite (Rminus_eq x y H);Trivial. +Rewrite (Rminus_eq y x H0);Trivial. +Rewrite (Rminus_eq y x H0);Trivial. +Qed. + +(*********) +Lemma Rabsolu_mult:(x,y:R) + (Rabsolu (Rmult x y))==(Rmult (Rabsolu x) (Rabsolu y)). +Intros;Unfold Rabsolu;Case (case_Rabsolu (Rmult x y)); + Case (case_Rabsolu x);Case (case_Rabsolu y);Intros;Auto. +Generalize (Rlt_anti_monotony y x R0 r r0);Intro; + Rewrite (Rmult_Or y) in H;Generalize (Rlt_antisym (Rmult x y) R0 r1); + Intro;Unfold Rgt in H;ElimType False;Rewrite (Rmult_sym y x) in H; + Auto. +Rewrite (Ropp_mul1 x y);Trivial. +Rewrite (Rmult_sym x (Ropp y));Rewrite (Ropp_mul1 y x); + Rewrite (Rmult_sym x y);Trivial. +Unfold Rge in r r0;Elim r;Elim r0;Clear r r0;Intros;Unfold Rgt in H H0. +Generalize (Rlt_monotony x R0 y H H0);Intro;Rewrite (Rmult_Or x) in H1; + Generalize (Rlt_antisym (Rmult x y) R0 r1);Intro;ElimType False;Auto. +Rewrite H in r1;Rewrite (Rmult_Ol y) in r1;Generalize (Rlt_antirefl R0); + Intro;ElimType False;Auto. +Rewrite H0 in r1;Rewrite (Rmult_Or x) in r1;Generalize (Rlt_antirefl R0); + Intro;ElimType False;Auto. +Rewrite H0 in r1;Rewrite (Rmult_Or x) in r1;Generalize (Rlt_antirefl R0); + Intro;ElimType False;Auto. +Rewrite (Ropp_mul2 x y);Trivial. +Unfold Rge in r r1;Elim r;Elim r1;Clear r r1;Intros;Unfold Rgt in H0 H. +Generalize (Rlt_monotony y x R0 H0 r0);Intro;Rewrite (Rmult_Or y) in H1; + Rewrite (Rmult_sym y x) in H1; + Generalize (Rlt_antisym (Rmult x y) R0 H1);Intro;ElimType False;Auto. +Generalize (imp_not_Req x R0 (or_introl (Rlt x R0) (Rgt x R0) r0)); + Generalize (imp_not_Req y R0 (or_intror (Rlt y R0) (Rgt y R0) H0));Intros; + Generalize (without_div_Od x y H);Intro;Elim H3;Intro;ElimType False; + Auto. +Rewrite H0 in H;Rewrite (Rmult_Or x) in H;Unfold Rgt in H; + Generalize (Rlt_antirefl R0);Intro;ElimType False;Auto. +Rewrite H0;Rewrite (Rmult_Or x);Rewrite (Rmult_Or (Ropp x));Trivial. +Unfold Rge in r0 r1;Elim r0;Elim r1;Clear r0 r1;Intros;Unfold Rgt in H0 H. +Generalize (Rlt_monotony x y R0 H0 r);Intro;Rewrite (Rmult_Or x) in H1; + Generalize (Rlt_antisym (Rmult x y) R0 H1);Intro;ElimType False;Auto. +Generalize (imp_not_Req y R0 (or_introl (Rlt y R0) (Rgt y R0) r)); + Generalize (imp_not_Req R0 x (or_introl (Rlt R0 x) (Rgt R0 x) H0));Intros; + Generalize (without_div_Od x y H);Intro;Elim H3;Intro;ElimType False; + Auto. +Rewrite H0 in H;Rewrite (Rmult_Ol y) in H;Unfold Rgt in H; + Generalize (Rlt_antirefl R0);Intro;ElimType False;Auto. +Rewrite H0;Rewrite (Rmult_Ol y);Rewrite (Rmult_Ol (Ropp y));Trivial. +Qed. + +(*********) +Lemma Rabsolu_Rinv:(r:R)(~r==R0)->(Rabsolu (Rinv r))== + (Rinv (Rabsolu r)). +Intro;Unfold Rabsolu;Case (case_Rabsolu r); + Case (case_Rabsolu (Rinv r));Auto;Intros. +Apply Ropp_Rinv;Auto. +Generalize (Rlt_Rinv2 r r1);Intro;Unfold Rge in r0;Elim r0;Intros. +Unfold Rgt in H1;Generalize (Rlt_antisym R0 (Rinv r) H1);Intro; + ElimType False;Auto. +Generalize + (imp_not_Req (Rinv r) R0 + (or_introl (Rlt (Rinv r) R0) (Rgt (Rinv r) R0) H0));Intro; + ElimType False;Auto. +Unfold Rge in r1;Elim r1;Clear r1;Intro. +Unfold Rgt in H0;Generalize (Rlt_antisym R0 (Rinv r) + (Rlt_Rinv r H0));Intro;ElimType False;Auto. +ElimType False;Auto. +Qed. + +Lemma Rabsolu_Ropp: + (x:R) (Rabsolu (Ropp x))==(Rabsolu x). +Intro;Cut (Ropp x)==(Rmult (Ropp R1) x). +Intros; Rewrite H. +Rewrite Rabsolu_mult. +Cut (Rabsolu (Ropp R1))==R1. +Intros; Rewrite H0. +Ring. +Unfold Rabsolu; Case (case_Rabsolu (Ropp R1)). +Intro; Ring. +Intro H0;Generalize (Rle_sym2 R0 (Ropp R1) H0);Intros. +Generalize (Rle_Ropp R0 (Ropp R1) H1). +Rewrite Ropp_Ropp; Rewrite Ropp_O. +Intro;Generalize (Rle_not R1 R0 Rlt_R0_R1);Intro; + Generalize (Rle_sym2 R1 R0 H2);Intro; + ElimType False;Auto. +Ring. +Qed. + +(*********) +Lemma Rabsolu_triang:(a,b:R)(Rle (Rabsolu (Rplus a b)) + (Rplus (Rabsolu a) (Rabsolu b))). +Intros a b;Unfold Rabsolu;Case (case_Rabsolu (Rplus a b)); + Case (case_Rabsolu a);Case (case_Rabsolu b);Intros. +Apply (eq_Rle (Ropp (Rplus a b)) (Rplus (Ropp a) (Ropp b))); + Rewrite (Ropp_distr1 a b);Reflexivity. +(**) +Rewrite (Ropp_distr1 a b); + Apply (Rle_compatibility (Ropp a) (Ropp b) b); + Unfold Rle;Unfold Rge in r;Elim r;Intro. +Left;Unfold Rgt in H;Generalize (Rlt_compatibility (Ropp b) R0 b H); + Intro;Elim (Rplus_ne (Ropp b));Intros v w;Rewrite v in H0;Clear v w; + Rewrite (Rplus_Ropp_l b) in H0;Apply (Rlt_trans (Ropp b) R0 b H0 H). +Right;Rewrite H;Apply Ropp_O. +(**) +Rewrite (Ropp_distr1 a b); + Rewrite (Rplus_sym (Ropp a) (Ropp b)); + Rewrite (Rplus_sym a (Ropp b)); + Apply (Rle_compatibility (Ropp b) (Ropp a) a); + Unfold Rle;Unfold Rge in r0;Elim r0;Intro. +Left;Unfold Rgt in H;Generalize (Rlt_compatibility (Ropp a) R0 a H); + Intro;Elim (Rplus_ne (Ropp a));Intros v w;Rewrite v in H0;Clear v w; + Rewrite (Rplus_Ropp_l a) in H0;Apply (Rlt_trans (Ropp a) R0 a H0 H). +Right;Rewrite H;Apply Ropp_O. +(**) +ElimType False;Generalize (Rge_plus_plus_r a b R0 r);Intro; + Elim (Rplus_ne a);Intros v w;Rewrite v in H;Clear v w; + Generalize (Rge_trans (Rplus a b) a R0 H r0);Intro;Clear H; + Unfold Rge in H0;Elim H0;Intro;Clear H0. +Unfold Rgt in H;Generalize (Rlt_antisym (Rplus a b) R0 r1);Intro;Auto. +Absurd (Rplus a b)==R0;Auto. +Apply (imp_not_Req (Rplus a b) R0);Left;Assumption. +(**) +ElimType False;Generalize (Rlt_compatibility a b R0 r);Intro; + Elim (Rplus_ne a);Intros v w;Rewrite v in H;Clear v w; + Generalize (Rlt_trans (Rplus a b) a R0 H r0);Intro;Clear H; + Unfold Rge in r1;Elim r1;Clear r1;Intro. +Unfold Rgt in H; + Generalize (Rlt_trans (Rplus a b) R0 (Rplus a b) H0 H);Intro; + Apply (Rlt_antirefl (Rplus a b));Assumption. +Rewrite H in H0;Apply (Rlt_antirefl R0);Assumption. +(**) +Rewrite (Rplus_sym a b);Rewrite (Rplus_sym (Ropp a) b); + Apply (Rle_compatibility b a (Ropp a)); + Apply (Rminus_le a (Ropp a));Unfold Rminus;Rewrite (Ropp_Ropp a); + Generalize (Rlt_compatibility a a R0 r0);Clear r r1;Intro; + Elim (Rplus_ne a);Intros v w;Rewrite v in H;Clear v w; + Generalize (Rlt_trans (Rplus a a) a R0 H r0);Intro; + Apply (Rlt_le (Rplus a a) R0 H0). +(**) +Apply (Rle_compatibility a b (Ropp b)); + Apply (Rminus_le b (Ropp b));Unfold Rminus;Rewrite (Ropp_Ropp b); + Generalize (Rlt_compatibility b b R0 r);Clear r0 r1;Intro; + Elim (Rplus_ne b);Intros v w;Rewrite v in H;Clear v w; + Generalize (Rlt_trans (Rplus b b) b R0 H r);Intro; + Apply (Rlt_le (Rplus b b) R0 H0). +(**) +Unfold Rle;Right;Reflexivity. +Qed. + +(*********) +Lemma Rabsolu_triang_inv:(a,b:R)(Rle (Rminus (Rabsolu a) (Rabsolu b)) + (Rabsolu (Rminus a b))). +Intros; + Apply (Rle_anti_compatibility (Rabsolu b) + (Rminus (Rabsolu a) (Rabsolu b)) (Rabsolu (Rminus a b))); + Unfold Rminus; + Rewrite <- (Rplus_assoc (Rabsolu b) (Rabsolu a) (Ropp (Rabsolu b))); + Rewrite (Rplus_sym (Rabsolu b) (Rabsolu a)); + Rewrite (Rplus_assoc (Rabsolu a) (Rabsolu b) (Ropp (Rabsolu b))); + Rewrite (Rplus_Ropp_r (Rabsolu b)); + Rewrite (proj1 ? ? (Rplus_ne (Rabsolu a))); + Replace (Rabsolu a) with (Rabsolu (Rplus a R0)). + Rewrite <- (Rplus_Ropp_r b); + Rewrite <- (Rplus_assoc a b (Ropp b)); + Rewrite (Rplus_sym a b); + Rewrite (Rplus_assoc b a (Ropp b)). + Exact (Rabsolu_triang b (Rplus a (Ropp b))). + Rewrite (proj1 ? ? (Rplus_ne a));Trivial. +Qed. + +(* ||a|-|b||<=|a-b| *) +Lemma Rabsolu_triang_inv2 : (a,b:R) ``(Rabsolu ((Rabsolu a)-(Rabsolu b)))<=(Rabsolu (a-b))``. +Cut (a,b:R) ``(Rabsolu b)<=(Rabsolu a)``->``(Rabsolu ((Rabsolu a)-(Rabsolu b))) <= (Rabsolu (a-b))``. +Intros; NewDestruct (total_order (Rabsolu a) (Rabsolu b)) as [Hlt|[Heq|Hgt]]. +Rewrite <- (Rabsolu_Ropp ``(Rabsolu a)-(Rabsolu b)``); Rewrite <- (Rabsolu_Ropp ``a-b``); Do 2 Rewrite Ropp_distr2. +Apply H; Left; Assumption. +Rewrite Heq; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rabsolu_pos. +Apply H; Left; Assumption. +Intros; Replace ``(Rabsolu ((Rabsolu a)-(Rabsolu b)))`` with ``(Rabsolu a)-(Rabsolu b)``. +Apply Rabsolu_triang_inv. +Rewrite (Rabsolu_right ``(Rabsolu a)-(Rabsolu b)``); [Reflexivity | Apply Rle_sym1; Apply Rle_anti_compatibility with (Rabsolu b); Rewrite Rplus_Or; Replace ``(Rabsolu b)+((Rabsolu a)-(Rabsolu b))`` with (Rabsolu a); [Assumption | Ring]]. +Qed. + +(*********) +Lemma Rabsolu_def1:(x,a:R)(Rlt x a)->(Rlt (Ropp a) x)->(Rlt (Rabsolu x) a). +Unfold Rabsolu;Intros;Case (case_Rabsolu x);Intro. +Generalize (Rlt_Ropp (Ropp a) x H0);Unfold Rgt;Rewrite Ropp_Ropp;Intro; + Assumption. +Assumption. +Qed. + +(*********) +Lemma Rabsolu_def2:(x,a:R)(Rlt (Rabsolu x) a)->(Rlt x a)/\(Rlt (Ropp a) x). +Unfold Rabsolu;Intro x;Case (case_Rabsolu x);Intros. +Generalize (Rlt_RoppO x r);Unfold Rgt;Intro; + Generalize (Rlt_trans R0 (Ropp x) a H0 H);Intro;Split. +Apply (Rlt_trans x R0 a r H1). +Generalize (Rlt_Ropp (Ropp x) a H);Rewrite (Ropp_Ropp x);Unfold Rgt;Trivial. +Fold (Rgt a x) in H;Generalize (Rgt_ge_trans a x R0 H r);Intro; + Generalize (Rgt_RoppO a H0);Intro;Fold (Rgt R0 (Ropp a)); + Generalize (Rge_gt_trans x R0 (Ropp a) r H1);Unfold Rgt;Intro;Split; + Assumption. +Qed. + +Lemma RmaxAbs: + (p, q, r : R) + (Rle p q) -> (Rle q r) -> (Rle (Rabsolu q) (Rmax (Rabsolu p) (Rabsolu r))). +Intros p q r H' H'0; Case (Rle_or_lt R0 p); Intros H'1. +Repeat Rewrite Rabsolu_right; Auto with real. +Apply Rle_trans with r; Auto with real. +Apply RmaxLess2; Auto. +Apply Rge_trans with p; Auto with real; Apply Rge_trans with q; Auto with real. +Apply Rge_trans with p; Auto with real. +Rewrite (Rabsolu_left p); Auto. +Case (Rle_or_lt R0 q); Intros H'2. +Repeat Rewrite Rabsolu_right; Auto with real. +Apply Rle_trans with r; Auto. +Apply RmaxLess2; Auto. +Apply Rge_trans with q; Auto with real. +Rewrite (Rabsolu_left q); Auto. +Case (Rle_or_lt R0 r); Intros H'3. +Repeat Rewrite Rabsolu_right; Auto with real. +Apply Rle_trans with (Ropp p); Auto with real. +Apply RmaxLess1; Auto. +Rewrite (Rabsolu_left r); Auto. +Apply Rle_trans with (Ropp p); Auto with real. +Apply RmaxLess1; Auto. +Qed. + +Lemma Rabsolu_Zabs: (z : Z) (Rabsolu (IZR z)) == (IZR (Zabs z)). +Intros z; Case z; Simpl; Auto with real. +Apply Rabsolu_right; Auto with real. +Intros p0; Apply Rabsolu_right; Auto with real zarith. +Intros p0; Rewrite Rabsolu_Ropp. +Apply Rabsolu_right; Auto with real zarith. +Qed. + diff --git a/theories7/Reals/Rcomplete.v b/theories7/Reals/Rcomplete.v new file mode 100644 index 00000000..5985a382 --- /dev/null +++ b/theories7/Reals/Rcomplete.v @@ -0,0 +1,175 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rcomplete.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require SeqProp. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(****************************************************) +(* R is complete : *) +(* Each sequence which satisfies *) +(* the Cauchy's criterion converges *) +(* *) +(* Proof with adjacent sequences (Vn and Wn) *) +(****************************************************) + +Theorem R_complete : (Un:nat->R) (Cauchy_crit Un) -> (sigTT R [l:R](Un_cv Un l)). +Intros. +Pose Vn := (sequence_minorant Un (cauchy_min Un H)). +Pose Wn := (sequence_majorant Un (cauchy_maj Un H)). +Assert H0 := (maj_cv Un H). +Fold Wn in H0. +Assert H1 := (min_cv Un H). +Fold Vn in H1. +Elim H0; Intros. +Elim H1; Intros. +Cut x==x0. +Intros. +Apply existTT with x. +Rewrite <- H2 in p0. +Unfold Un_cv. +Intros. +Unfold Un_cv in p; Unfold Un_cv in p0. +Cut ``0<eps/3``. +Intro. +Elim (p ``eps/3`` H4); Intros. +Elim (p0 ``eps/3`` H4); Intros. +Exists (max x1 x2). +Intros. +Unfold R_dist. +Apply Rle_lt_trans with ``(Rabsolu ((Un n)-(Vn n)))+(Rabsolu ((Vn n)-x))``. +Replace ``(Un n)-x`` with ``((Un n)-(Vn n))+((Vn n)-x)``; [Apply Rabsolu_triang | Ring]. +Apply Rle_lt_trans with ``(Rabsolu ((Wn n)-(Vn n)))+(Rabsolu ((Vn n)-x))``. +Do 2 Rewrite <- (Rplus_sym ``(Rabsolu ((Vn n)-x))``). +Apply Rle_compatibility. +Repeat Rewrite Rabsolu_right. +Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-(Vn n)``); Apply Rle_compatibility. +Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). +Fold Vn Wn in H8. +Elim (H8 n); Intros. +Assumption. +Apply Rle_sym1. +Unfold Rminus; Apply Rle_anti_compatibility with (Vn n). +Rewrite Rplus_Or. +Replace ``(Vn n)+((Wn n)+ -(Vn n))`` with (Wn n); [Idtac | Ring]. +Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). +Fold Vn Wn in H8. +Elim (H8 n); Intros. +Apply Rle_trans with (Un n); Assumption. +Apply Rle_sym1. +Unfold Rminus; Apply Rle_anti_compatibility with (Vn n). +Rewrite Rplus_Or. +Replace ``(Vn n)+((Un n)+ -(Vn n))`` with (Un n); [Idtac | Ring]. +Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). +Fold Vn Wn in H8. +Elim (H8 n); Intros. +Assumption. +Apply Rle_lt_trans with ``(Rabsolu ((Wn n)-x))+(Rabsolu (x-(Vn n)))+(Rabsolu ((Vn n)-x))``. +Do 2 Rewrite <- (Rplus_sym ``(Rabsolu ((Vn n)-x))``). +Apply Rle_compatibility. +Replace ``(Wn n)-(Vn n)`` with ``((Wn n)-x)+(x-(Vn n))``; [Apply Rabsolu_triang | Ring]. +Apply Rlt_le_trans with ``eps/3+eps/3+eps/3``. +Repeat Apply Rplus_lt. +Unfold R_dist in H5. +Apply H5. +Unfold ge; Apply le_trans with (max x1 x2). +Apply le_max_l. +Assumption. +Rewrite <- Rabsolu_Ropp. +Replace ``-(x-(Vn n))`` with ``(Vn n)-x``; [Idtac | Ring]. +Unfold R_dist in H6. +Apply H6. +Unfold ge; Apply le_trans with (max x1 x2). +Apply le_max_r. +Assumption. +Unfold R_dist in H6. +Apply H6. +Unfold ge; Apply le_trans with (max x1 x2). +Apply le_max_r. +Assumption. +Right. +Pattern 4 eps; Replace ``eps`` with ``3*eps/3``. +Ring. +Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Apply cond_eq. +Intros. +Cut ``0<eps/5``. +Intro. +Unfold Un_cv in p; Unfold Un_cv in p0. +Unfold R_dist in p; Unfold R_dist in p0. +Elim (p ``eps/5`` H3); Intros N1 H4. +Elim (p0 ``eps/5`` H3); Intros N2 H5. +Unfold Cauchy_crit in H. +Unfold R_dist in H. +Elim (H ``eps/5`` H3); Intros N3 H6. +Pose N := (max (max N1 N2) N3). +Apply Rle_lt_trans with ``(Rabsolu (x-(Wn N)))+(Rabsolu ((Wn N)-x0))``. +Replace ``x-x0`` with ``(x-(Wn N))+((Wn N)-x0)``; [Apply Rabsolu_triang | Ring]. +Apply Rle_lt_trans with ``(Rabsolu (x-(Wn N)))+(Rabsolu ((Wn N)-(Vn N)))+(Rabsolu (((Vn N)-x0)))``. +Rewrite Rplus_assoc. +Apply Rle_compatibility. +Replace ``(Wn N)-x0`` with ``((Wn N)-(Vn N))+((Vn N)-x0)``; [Apply Rabsolu_triang | Ring]. +Replace ``eps`` with ``eps/5+3*eps/5+eps/5``. +Repeat Apply Rplus_lt. +Rewrite <- Rabsolu_Ropp. +Replace ``-(x-(Wn N))`` with ``(Wn N)-x``; [Apply H4 | Ring]. +Unfold ge N. +Apply le_trans with (max N1 N2); Apply le_max_l. +Unfold Wn Vn. +Unfold sequence_majorant sequence_minorant. +Assert H7 := (approx_maj [k:nat](Un (plus N k)) (maj_ss Un N (cauchy_maj Un H))). +Assert H8 := (approx_min [k:nat](Un (plus N k)) (min_ss Un N (cauchy_min Un H))). +Cut (Wn N)==(majorant ([k:nat](Un (plus N k))) (maj_ss Un N (cauchy_maj Un H))). +Cut (Vn N)==(minorant ([k:nat](Un (plus N k))) (min_ss Un N (cauchy_min Un H))). +Intros. +Rewrite <- H9; Rewrite <- H10. +Rewrite <- H9 in H8. +Rewrite <- H10 in H7. +Elim (H7 ``eps/5`` H3); Intros k2 H11. +Elim (H8 ``eps/5`` H3); Intros k1 H12. +Apply Rle_lt_trans with ``(Rabsolu ((Wn N)-(Un (plus N k2))))+(Rabsolu ((Un (plus N k2))-(Vn N)))``. +Replace ``(Wn N)-(Vn N)`` with ``((Wn N)-(Un (plus N k2)))+((Un (plus N k2))-(Vn N))``; [Apply Rabsolu_triang | Ring]. +Apply Rle_lt_trans with ``(Rabsolu ((Wn N)-(Un (plus N k2))))+(Rabsolu ((Un (plus N k2))-(Un (plus N k1))))+(Rabsolu ((Un (plus N k1))-(Vn N)))``. +Rewrite Rplus_assoc. +Apply Rle_compatibility. +Replace ``(Un (plus N k2))-(Vn N)`` with ``((Un (plus N k2))-(Un (plus N k1)))+((Un (plus N k1))-(Vn N))``; [Apply Rabsolu_triang | Ring]. +Replace ``3*eps/5`` with ``eps/5+eps/5+eps/5``; [Repeat Apply Rplus_lt | Ring]. +Assumption. +Apply H6. +Unfold ge. +Apply le_trans with N. +Unfold N; Apply le_max_r. +Apply le_plus_l. +Unfold ge. +Apply le_trans with N. +Unfold N; Apply le_max_r. +Apply le_plus_l. +Rewrite <- Rabsolu_Ropp. +Replace ``-((Un (plus N k1))-(Vn N))`` with ``(Vn N)-(Un (plus N k1))``; [Assumption | Ring]. +Reflexivity. +Reflexivity. +Apply H5. +Unfold ge; Apply le_trans with (max N1 N2). +Apply le_max_r. +Unfold N; Apply le_max_l. +Pattern 4 eps; Replace ``eps`` with ``5*eps/5``. +Ring. +Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. +DiscrR. +Unfold Rdiv; Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv. +Sup0; Try Apply lt_O_Sn. +Qed. diff --git a/theories7/Reals/Rdefinitions.v b/theories7/Reals/Rdefinitions.v new file mode 100644 index 00000000..79be0176 --- /dev/null +++ b/theories7/Reals/Rdefinitions.v @@ -0,0 +1,69 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Rdefinitions.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + + +(*********************************************************) +(** Definitions for the axiomatization *) +(* *) +(*********************************************************) + +Require Export ZArith_base. + +Parameter R:Set. + +(* Declare Scope positive_scope with Key R *) +Delimits Scope R_scope with R. + +(* Automatically open scope R_scope for arguments of type R *) +Bind Scope R_scope with R. + +Parameter R0:R. +Parameter R1:R. +Parameter Rplus:R->R->R. +Parameter Rmult:R->R->R. +Parameter Ropp:R->R. +Parameter Rinv:R->R. +Parameter Rlt:R->R->Prop. +Parameter up:R->Z. + +V8Infix "+" Rplus : R_scope. +V8Infix "*" Rmult : R_scope. +V8Notation "- x" := (Ropp x) : R_scope. +V8Notation "/ x" := (Rinv x) : R_scope. + +V8Infix "<" Rlt : R_scope. + +(*i*******************************************************i*) + +(**********) +Definition Rgt:R->R->Prop:=[r1,r2:R](Rlt r2 r1). + +(**********) +Definition Rle:R->R->Prop:=[r1,r2:R]((Rlt r1 r2)\/(r1==r2)). + +(**********) +Definition Rge:R->R->Prop:=[r1,r2:R]((Rgt r1 r2)\/(r1==r2)). + +(**********) +Definition Rminus:R->R->R:=[r1,r2:R](Rplus r1 (Ropp r2)). + +(**********) +Definition Rdiv:R->R->R:=[r1,r2:R](Rmult r1 (Rinv r2)). + +V8Infix "-" Rminus : R_scope. +V8Infix "/" Rdiv : R_scope. + +V8Infix "<=" Rle : R_scope. +V8Infix ">=" Rge : R_scope. +V8Infix ">" Rgt : R_scope. + +V8Notation "x <= y <= z" := (Rle x y)/\(Rle y z) : R_scope. +V8Notation "x <= y < z" := (Rle x y)/\(Rlt y z) : R_scope. +V8Notation "x < y < z" := (Rlt x y)/\(Rlt y z) : R_scope. +V8Notation "x < y <= z" := (Rlt x y)/\(Rle y z) : R_scope. diff --git a/theories7/Reals/Rderiv.v b/theories7/Reals/Rderiv.v new file mode 100644 index 00000000..b55aa6ea --- /dev/null +++ b/theories7/Reals/Rderiv.v @@ -0,0 +1,453 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rderiv.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +(*********************************************************) +(** Definition of the derivative,continuity *) +(* *) +(*********************************************************) + +Require Rbase. +Require Rfunctions. +Require Rlimit. +Require Fourier. +Require Classical_Prop. +Require Classical_Pred_Type. +Require Omega. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(*********) +Definition D_x:(R->Prop)->R->R->Prop:=[D:R->Prop][y:R][x:R] + (D x)/\(~y==x). + +(*********) +Definition continue_in:(R->R)->(R->Prop)->R->Prop:= + [f:R->R; D:R->Prop; x0:R](limit1_in f (D_x D x0) (f x0) x0). + +(*********) +Definition D_in:(R->R)->(R->R)->(R->Prop)->R->Prop:= + [f:R->R; d:R->R; D:R->Prop; x0:R](limit1_in + [x:R] (Rdiv (Rminus (f x) (f x0)) (Rminus x x0)) + (D_x D x0) (d x0) x0). + +(*********) +Lemma cont_deriv:(f,d:R->R;D:R->Prop;x0:R) + (D_in f d D x0)->(continue_in f D x0). +Unfold continue_in;Unfold D_in;Unfold limit1_in;Unfold limit_in; + Unfold Rdiv;Simpl;Intros;Elim (H eps H0); Clear H;Intros; + Elim H;Clear H;Intros; Elim (Req_EM (d x0) R0);Intro. +Split with (Rmin R1 x);Split. +Elim (Rmin_Rgt R1 x R0);Intros a b; + Apply (b (conj (Rgt R1 R0) (Rgt x R0) Rlt_R0_R1 H)). +Intros;Elim H3;Clear H3;Intros; +Generalize (let (H1,H2)=(Rmin_Rgt R1 x (R_dist x1 x0)) in H1); + Unfold Rgt;Intro;Elim (H5 H4);Clear H5;Intros; + Generalize (H1 x1 (conj (D_x D x0 x1) (Rlt (R_dist x1 x0) x) H3 H6)); + Clear H1;Intro;Unfold D_x in H3;Elim H3;Intros. +Rewrite H2 in H1;Unfold R_dist; Unfold R_dist in H1; + Cut (Rlt (Rabsolu (Rminus (f x1) (f x0))) + (Rmult eps (Rabsolu (Rminus x1 x0)))). +Intro;Unfold R_dist in H5; + Generalize (Rlt_monotony eps ``(Rabsolu (x1-x0))`` ``1`` H0 H5); +Rewrite Rmult_1r;Intro;Apply Rlt_trans with r2:=``eps*(Rabsolu (x1-x0))``; + Assumption. +Rewrite (minus_R0 ``((f x1)-(f x0))*/(x1-x0)``) in H1; + Rewrite Rabsolu_mult in H1; Cut ``x1-x0 <> 0``. +Intro;Rewrite (Rabsolu_Rinv (Rminus x1 x0) H9) in H1; + Generalize (Rlt_monotony ``(Rabsolu (x1-x0))`` + ``(Rabsolu ((f x1)-(f x0)))*/(Rabsolu (x1-x0))`` eps + (Rabsolu_pos_lt ``x1-x0`` H9) H1);Intro; Rewrite Rmult_sym in H10; + Rewrite Rmult_assoc in H10;Rewrite Rinv_l in H10. +Rewrite Rmult_1r in H10;Rewrite Rmult_sym;Assumption. +Apply Rabsolu_no_R0;Auto. +Apply Rminus_eq_contra;Auto. +(**) + Split with (Rmin (Rmin (Rinv (Rplus R1 R1)) x) + (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0)))))); + Split. +Cut (Rgt (Rmin (Rinv (Rplus R1 R1)) x) R0). +Cut (Rgt (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) R0). +Intros;Elim (Rmin_Rgt (Rmin (Rinv (Rplus R1 R1)) x) + (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) R0); + Intros a b; + Apply (b (conj (Rgt (Rmin (Rinv (Rplus R1 R1)) x) R0) + (Rgt (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) R0) + H4 H3)). +Apply Rmult_gt;Auto. +Unfold Rgt;Apply Rlt_Rinv;Apply Rabsolu_pos_lt;Apply mult_non_zero; + Split. +DiscrR. +Assumption. +Elim (Rmin_Rgt (Rinv (Rplus R1 R1)) x R0);Intros a b; + Cut (Rlt R0 (Rplus R1 R1)). +Intro;Generalize (Rlt_Rinv (Rplus R1 R1) H3);Intro; + Fold (Rgt (Rinv (Rplus R1 R1)) R0) in H4; + Apply (b (conj (Rgt (Rinv (Rplus R1 R1)) R0) (Rgt x R0) H4 H)). +Fourier. +Intros;Elim H3;Clear H3;Intros; + Generalize (let (H1,H2)=(Rmin_Rgt (Rmin (Rinv (Rplus R1 R1)) x) + (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) + (R_dist x1 x0)) in H1);Unfold Rgt;Intro;Elim (H5 H4);Clear H5; + Intros; + Generalize (let (H1,H2)=(Rmin_Rgt (Rinv (Rplus R1 R1)) x + (R_dist x1 x0)) in H1);Unfold Rgt;Intro;Elim (H7 H5);Clear H7; + Intros;Clear H4 H5; + Generalize (H1 x1 (conj (D_x D x0 x1) (Rlt (R_dist x1 x0) x) H3 H8)); + Clear H1;Intro;Unfold D_x in H3;Elim H3;Intros; + Generalize (sym_not_eqT R x0 x1 H5);Clear H5;Intro H5; + Generalize (Rminus_eq_contra x1 x0 H5); + Intro;Generalize H1;Pattern 1 (d x0); + Rewrite <-(let (H1,H2)=(Rmult_ne (d x0)) in H2); + Rewrite <-(Rinv_l (Rminus x1 x0) H9); Unfold R_dist;Unfold 1 Rminus; + Rewrite (Rmult_sym (Rminus (f x1) (f x0)) (Rinv (Rminus x1 x0))); + Rewrite (Rmult_sym (Rmult (Rinv (Rminus x1 x0)) (Rminus x1 x0)) (d x0)); + Rewrite <-(Ropp_mul1 (d x0) (Rmult (Rinv (Rminus x1 x0)) (Rminus x1 x0))); + Rewrite (Rmult_sym (Ropp (d x0)) + (Rmult (Rinv (Rminus x1 x0)) (Rminus x1 x0))); + Rewrite (Rmult_assoc (Rinv (Rminus x1 x0)) (Rminus x1 x0) (Ropp (d x0))); + Rewrite <-(Rmult_Rplus_distr (Rinv (Rminus x1 x0)) (Rminus (f x1) (f x0)) + (Rmult (Rminus x1 x0) (Ropp (d x0)))); + Rewrite (Rabsolu_mult (Rinv (Rminus x1 x0)) + (Rplus (Rminus (f x1) (f x0)) + (Rmult (Rminus x1 x0) (Ropp (d x0))))); + Clear H1;Intro;Generalize (Rlt_monotony (Rabsolu (Rminus x1 x0)) + (Rmult (Rabsolu (Rinv (Rminus x1 x0))) + (Rabsolu + (Rplus (Rminus (f x1) (f x0)) + (Rmult (Rminus x1 x0) (Ropp (d x0)))))) eps + (Rabsolu_pos_lt (Rminus x1 x0) H9) H1); + Rewrite <-(Rmult_assoc (Rabsolu (Rminus x1 x0)) + (Rabsolu (Rinv (Rminus x1 x0))) + (Rabsolu + (Rplus (Rminus (f x1) (f x0)) + (Rmult (Rminus x1 x0) (Ropp (d x0)))))); + Rewrite (Rabsolu_Rinv (Rminus x1 x0) H9); + Rewrite (Rinv_r (Rabsolu (Rminus x1 x0)) + (Rabsolu_no_R0 (Rminus x1 x0) H9)); + Rewrite (let (H1,H2)=(Rmult_ne (Rabsolu + (Rplus (Rminus (f x1) (f x0)) + (Rmult (Rminus x1 x0) (Ropp (d x0)))))) in H2); + Generalize (Rabsolu_triang_inv (Rminus (f x1) (f x0)) + (Rmult (Rminus x1 x0) (d x0)));Intro; + Rewrite (Rmult_sym (Rminus x1 x0) (Ropp (d x0))); + Rewrite (Ropp_mul1 (d x0) (Rminus x1 x0)); + Fold (Rminus (Rminus (f x1) (f x0)) (Rmult (d x0) (Rminus x1 x0))); + Rewrite (Rmult_sym (Rminus x1 x0) (d x0)) in H10; + Clear H1;Intro;Generalize (Rle_lt_trans + (Rminus (Rabsolu (Rminus (f x1) (f x0))) + (Rabsolu (Rmult (d x0) (Rminus x1 x0)))) + (Rabsolu + (Rminus (Rminus (f x1) (f x0)) (Rmult (d x0) (Rminus x1 x0)))) + (Rmult (Rabsolu (Rminus x1 x0)) eps) H10 H1); + Clear H1;Intro; + Generalize (Rlt_compatibility (Rabsolu (Rmult (d x0) (Rminus x1 x0))) + (Rminus (Rabsolu (Rminus (f x1) (f x0))) + (Rabsolu (Rmult (d x0) (Rminus x1 x0)))) + (Rmult (Rabsolu (Rminus x1 x0)) eps) H1); + Unfold 2 Rminus;Rewrite (Rplus_sym (Rabsolu (Rminus (f x1) (f x0))) + (Ropp (Rabsolu (Rmult (d x0) (Rminus x1 x0))))); + Rewrite <-(Rplus_assoc (Rabsolu (Rmult (d x0) (Rminus x1 x0))) + (Ropp (Rabsolu (Rmult (d x0) (Rminus x1 x0)))) + (Rabsolu (Rminus (f x1) (f x0)))); + Rewrite (Rplus_Ropp_r (Rabsolu (Rmult (d x0) (Rminus x1 x0)))); + Rewrite (let (H1,H2)=(Rplus_ne (Rabsolu (Rminus (f x1) (f x0)))) in H2); + Clear H1;Intro;Cut (Rlt (Rplus (Rabsolu (Rmult (d x0) (Rminus x1 x0))) + (Rmult (Rabsolu (Rminus x1 x0)) eps)) eps). +Intro;Apply (Rlt_trans (Rabsolu (Rminus (f x1) (f x0))) + (Rplus (Rabsolu (Rmult (d x0) (Rminus x1 x0))) + (Rmult (Rabsolu (Rminus x1 x0)) eps)) eps H1 H11). +Clear H1 H5 H3 H10;Generalize (Rabsolu_pos_lt (d x0) H2); + Intro;Unfold Rgt in H0;Generalize (Rlt_monotony eps (R_dist x1 x0) + (Rinv (Rplus R1 R1)) H0 H7);Clear H7;Intro; + Generalize (Rlt_monotony (Rabsolu (d x0)) (R_dist x1 x0) + (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) H1 H6); + Clear H6;Intro;Rewrite (Rmult_sym eps (R_dist x1 x0)) in H3; + Unfold R_dist in H3 H5; + Rewrite <-(Rabsolu_mult (d x0) (Rminus x1 x0)) in H5; + Rewrite (Rabsolu_mult (Rplus R1 R1) (d x0)) in H5; + Cut ~(Rabsolu (Rplus R1 R1))==R0. +Intro;Fold (Rgt (Rabsolu (d x0)) R0) in H1; + Rewrite (Rinv_Rmult (Rabsolu (Rplus R1 R1)) (Rabsolu (d x0)) + H6 (imp_not_Req (Rabsolu (d x0)) R0 + (or_intror (Rlt (Rabsolu (d x0)) R0) (Rgt (Rabsolu (d x0)) R0) H1))) + in H5; + Rewrite (Rmult_sym (Rabsolu (d x0)) (Rmult eps + (Rmult (Rinv (Rabsolu (Rplus R1 R1))) + (Rinv (Rabsolu (d x0)))))) in H5; + Rewrite <-(Rmult_assoc eps (Rinv (Rabsolu (Rplus R1 R1))) + (Rinv (Rabsolu (d x0)))) in H5; + Rewrite (Rmult_assoc (Rmult eps (Rinv (Rabsolu (Rplus R1 R1)))) + (Rinv (Rabsolu (d x0))) (Rabsolu (d x0))) in H5; + Rewrite (Rinv_l (Rabsolu (d x0)) (imp_not_Req (Rabsolu (d x0)) R0 + (or_intror (Rlt (Rabsolu (d x0)) R0) (Rgt (Rabsolu (d x0)) R0) H1))) + in H5; + Rewrite (let (H1,H2)=(Rmult_ne (Rmult eps (Rinv (Rabsolu (Rplus R1 R1))))) + in H1) in H5;Cut (Rabsolu (Rplus R1 R1))==(Rplus R1 R1). +Intro;Rewrite H7 in H5; + Generalize (Rplus_lt (Rabsolu (Rmult (d x0) (Rminus x1 x0))) + (Rmult eps (Rinv (Rplus R1 R1))) + (Rmult (Rabsolu (Rminus x1 x0)) eps) + (Rmult eps (Rinv (Rplus R1 R1))) H5 H3);Intro; + Rewrite eps2 in H10;Assumption. +Unfold Rabsolu;Case (case_Rabsolu (Rplus R1 R1));Auto. + Intro;Cut (Rlt R0 (Rplus R1 R1)). +Intro;Generalize (Rlt_antisym R0 (Rplus R1 R1) H7);Intro;ElimType False; + Auto. +Fourier. +Apply Rabsolu_no_R0. +DiscrR. +Qed. + + +(*********) +Lemma Dconst:(D:R->Prop)(y:R)(x0:R)(D_in [x:R]y [x:R]R0 D x0). +Unfold D_in;Intros;Unfold limit1_in;Unfold limit_in;Unfold Rdiv;Intros;Simpl; + Split with eps;Split;Auto. +Intros;Rewrite (eq_Rminus y y (refl_eqT R y)); + Rewrite Rmult_Ol;Unfold R_dist; + Rewrite (eq_Rminus R0 R0 (refl_eqT R R0));Unfold Rabsolu; + Case (case_Rabsolu R0);Intro. +Absurd (Rlt R0 R0);Auto. +Red;Intro;Apply (Rlt_antirefl R0 H1). +Unfold Rgt in H0;Assumption. +Qed. + +(*********) +Lemma Dx:(D:R->Prop)(x0:R)(D_in [x:R]x [x:R]R1 D x0). +Unfold D_in;Unfold Rdiv;Intros;Unfold limit1_in;Unfold limit_in;Intros;Simpl; + Split with eps;Split;Auto. +Intros;Elim H0;Clear H0;Intros;Unfold D_x in H0; + Elim H0;Intros; + Rewrite (Rinv_r (Rminus x x0) (Rminus_eq_contra x x0 + (sym_not_eqT R x0 x H3))); + Unfold R_dist; + Rewrite (eq_Rminus R1 R1 (refl_eqT R R1));Unfold Rabsolu; + Case (case_Rabsolu R0);Intro. +Absurd (Rlt R0 R0);Auto. +Red;Intro;Apply (Rlt_antirefl R0 r). +Unfold Rgt in H;Assumption. +Qed. + +(*********) +Lemma Dadd:(D:R->Prop)(df,dg:R->R)(f,g:R->R)(x0:R) + (D_in f df D x0)->(D_in g dg D x0)-> + (D_in [x:R](Rplus (f x) (g x)) [x:R](Rplus (df x) (dg x)) D x0). +Unfold D_in;Intros;Generalize (limit_plus + [x:R](Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) + [x:R](Rmult (Rminus (g x) (g x0)) (Rinv (Rminus x x0))) + (D_x D x0) (df x0) (dg x0) x0 H H0);Clear H H0; + Unfold limit1_in;Unfold limit_in;Simpl;Intros; + Elim (H eps H0);Clear H;Intros;Elim H;Clear H;Intros; + Split with x;Split;Auto;Intros;Generalize (H1 x1 H2);Clear H1;Intro; + Rewrite (Rmult_sym (Rminus (f x1) (f x0)) (Rinv (Rminus x1 x0))) in H1; + Rewrite (Rmult_sym (Rminus (g x1) (g x0)) (Rinv (Rminus x1 x0))) in H1; + Rewrite <-(Rmult_Rplus_distr (Rinv (Rminus x1 x0)) + (Rminus (f x1) (f x0)) + (Rminus (g x1) (g x0))) in H1; + Rewrite (Rmult_sym (Rinv (Rminus x1 x0)) + (Rplus (Rminus (f x1) (f x0)) (Rminus (g x1) (g x0)))) in H1; + Cut (Rplus (Rminus (f x1) (f x0)) (Rminus (g x1) (g x0)))== + (Rminus (Rplus (f x1) (g x1)) (Rplus (f x0) (g x0))). +Intro;Rewrite H3 in H1;Assumption. +Ring. +Qed. + +(*********) +Lemma Dmult:(D:R->Prop)(df,dg:R->R)(f,g:R->R)(x0:R) + (D_in f df D x0)->(D_in g dg D x0)-> + (D_in [x:R](Rmult (f x) (g x)) + [x:R](Rplus (Rmult (df x) (g x)) (Rmult (f x) (dg x))) D x0). +Intros;Unfold D_in;Generalize H H0;Intros;Unfold D_in in H H0; + Generalize (cont_deriv f df D x0 H1);Unfold continue_in;Intro; + Generalize (limit_mul + [x:R](Rmult (Rminus (g x) (g x0)) (Rinv (Rminus x x0))) + [x:R](f x) (D_x D x0) (dg x0) (f x0) x0 H0 H3);Intro; + Cut (limit1_in [x:R](g x0) (D_x D x0) (g x0) x0). +Intro;Generalize (limit_mul + [x:R](Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) + [_:R](g x0) (D_x D x0) (df x0) (g x0) x0 H H5);Clear H H0 H1 H2 H3 H5; + Intro;Generalize (limit_plus + [x:R](Rmult (Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) (g x0)) + [x:R](Rmult (Rmult (Rminus (g x) (g x0)) (Rinv (Rminus x x0))) + (f x)) (D_x D x0) (Rmult (df x0) (g x0)) + (Rmult (dg x0) (f x0)) x0 H H4); + Clear H4 H;Intro;Unfold limit1_in in H;Unfold limit_in in H; + Simpl in H;Unfold limit1_in;Unfold limit_in;Simpl;Intros; + Elim (H eps H0);Clear H;Intros;Elim H;Clear H;Intros; + Split with x;Split;Auto;Intros;Generalize (H1 x1 H2);Clear H1;Intro; + Rewrite (Rmult_sym (Rminus (f x1) (f x0)) (Rinv (Rminus x1 x0))) in H1; + Rewrite (Rmult_sym (Rminus (g x1) (g x0)) (Rinv (Rminus x1 x0))) in H1; + Rewrite (Rmult_assoc (Rinv (Rminus x1 x0)) (Rminus (f x1) (f x0)) + (g x0)) in H1; + Rewrite (Rmult_assoc (Rinv (Rminus x1 x0)) (Rminus (g x1) (g x0)) + (f x1)) in H1; + Rewrite <-(Rmult_Rplus_distr (Rinv (Rminus x1 x0)) + (Rmult (Rminus (f x1) (f x0)) (g x0)) + (Rmult (Rminus (g x1) (g x0)) (f x1))) in H1; + Rewrite (Rmult_sym (Rinv (Rminus x1 x0)) + (Rplus (Rmult (Rminus (f x1) (f x0)) (g x0)) + (Rmult (Rminus (g x1) (g x0)) (f x1)))) in H1; + Rewrite (Rmult_sym (dg x0) (f x0)) in H1; + Cut (Rplus (Rmult (Rminus (f x1) (f x0)) (g x0)) + (Rmult (Rminus (g x1) (g x0)) (f x1)))== + (Rminus (Rmult (f x1) (g x1)) (Rmult (f x0) (g x0))). +Intro;Rewrite H3 in H1;Assumption. +Ring. +Unfold limit1_in;Unfold limit_in;Simpl;Intros; + Split with eps;Split;Auto;Intros;Elim (R_dist_refl (g x0) (g x0)); + Intros a b;Rewrite (b (refl_eqT R (g x0)));Unfold Rgt in H;Assumption. +Qed. + +(*********) +Lemma Dmult_const:(D:R->Prop)(f,df:R->R)(x0:R)(a:R)(D_in f df D x0)-> + (D_in [x:R](Rmult a (f x)) ([x:R](Rmult a (df x))) D x0). +Intros;Generalize (Dmult D [_:R]R0 df [_:R]a f x0 (Dconst D a x0) H); + Unfold D_in;Intros; + Rewrite (Rmult_Ol (f x0)) in H0; + Rewrite (let (H1,H2)=(Rplus_ne (Rmult a (df x0))) in H2) in H0; + Assumption. +Qed. + +(*********) +Lemma Dopp:(D:R->Prop)(f,df:R->R)(x0:R)(D_in f df D x0)-> + (D_in [x:R](Ropp (f x)) ([x:R](Ropp (df x))) D x0). +Intros;Generalize (Dmult_const D f df x0 (Ropp R1) H); Unfold D_in; + Unfold limit1_in;Unfold limit_in;Intros; + Generalize (H0 eps H1);Clear H0;Intro;Elim H0;Clear H0;Intros; + Elim H0;Clear H0;Simpl;Intros;Split with x;Split;Auto. +Intros;Generalize (H2 x1 H3);Clear H2;Intro;Rewrite Ropp_mul1 in H2; + Rewrite Ropp_mul1 in H2;Rewrite Ropp_mul1 in H2; + Rewrite (let (H1,H2)=(Rmult_ne (f x1)) in H2) in H2; + Rewrite (let (H1,H2)=(Rmult_ne (f x0)) in H2) in H2; + Rewrite (let (H1,H2)=(Rmult_ne (df x0)) in H2) in H2;Assumption. +Qed. + +(*********) +Lemma Dminus:(D:R->Prop)(df,dg:R->R)(f,g:R->R)(x0:R) + (D_in f df D x0)->(D_in g dg D x0)-> + (D_in [x:R](Rminus (f x) (g x)) [x:R](Rminus (df x) (dg x)) D x0). +Unfold Rminus;Intros;Generalize (Dopp D g dg x0 H0);Intro; + Apply (Dadd D df [x:R](Ropp (dg x)) f [x:R](Ropp (g x)) x0);Assumption. +Qed. + +(*********) +Lemma Dx_pow_n:(n:nat)(D:R->Prop)(x0:R) + (D_in [x:R](pow x n) + [x:R](Rmult (INR n) (pow x (minus n (1)))) D x0). +Induction n;Intros. +Simpl; Rewrite Rmult_Ol; Apply Dconst. +Intros;Cut n0=(minus (S n0) (1)); + [ Intro a; Rewrite <- a;Clear a | Simpl; Apply minus_n_O ]. +Generalize (Dmult D [_:R]R1 + [x:R](Rmult (INR n0) (pow x (minus n0 (1)))) [x:R]x [x:R](pow x n0) + x0 (Dx D x0) (H D x0));Unfold D_in;Unfold limit1_in;Unfold limit_in; + Simpl;Intros; + Elim (H0 eps H1);Clear H0;Intros;Elim H0;Clear H0;Intros; + Split with x;Split;Auto. +Intros;Generalize (H2 x1 H3);Clear H2 H3;Intro; + Rewrite (let (H1,H2)=(Rmult_ne (pow x0 n0)) in H2) in H2; + Rewrite (tech_pow_Rmult x1 n0) in H2; + Rewrite (tech_pow_Rmult x0 n0) in H2; + Rewrite (Rmult_sym (INR n0) (pow x0 (minus n0 (1)))) in H2; + Rewrite <-(Rmult_assoc x0 (pow x0 (minus n0 (1))) (INR n0)) in H2; + Rewrite (tech_pow_Rmult x0 (minus n0 (1))) in H2; + Elim (classic (n0=O));Intro cond. +Rewrite cond in H2;Rewrite cond;Simpl in H2;Simpl; + Cut (Rplus R1 (Rmult (Rmult x0 R1) R0))==(Rmult R1 R1); + [Intro A; Rewrite A in H2; Assumption|Ring]. +Cut ~(n0=O)->(S (minus n0 (1)))=n0;[Intro|Omega]; + Rewrite (H3 cond) in H2; Rewrite (Rmult_sym (pow x0 n0) (INR n0)) in H2; + Rewrite (tech_pow_Rplus x0 n0 n0) in H2; Assumption. +Qed. + +(*********) +Lemma Dcomp:(Df,Dg:R->Prop)(df,dg:R->R)(f,g:R->R)(x0:R) + (D_in f df Df x0)->(D_in g dg Dg (f x0))-> + (D_in [x:R](g (f x)) [x:R](Rmult (df x) (dg (f x))) + (Dgf Df Dg f) x0). +Intros Df Dg df dg f g x0 H H0;Generalize H H0;Unfold D_in;Unfold Rdiv;Intros; +Generalize (limit_comp f [x:R](Rmult (Rminus (g x) (g (f x0))) + (Rinv (Rminus x (f x0)))) (D_x Df x0) + (D_x Dg (f x0)) + (f x0) (dg (f x0)) x0);Intro; + Generalize (cont_deriv f df Df x0 H);Intro;Unfold continue_in in H4; + Generalize (H3 H4 H2);Clear H3;Intro; + Generalize (limit_mul [x:R](Rmult (Rminus (g (f x)) (g (f x0))) + (Rinv (Rminus (f x) (f x0)))) + [x:R](Rmult (Rminus (f x) (f x0)) + (Rinv (Rminus x x0))) + (Dgf (D_x Df x0) (D_x Dg (f x0)) f) + (dg (f x0)) (df x0) x0 H3);Intro; + Cut (limit1_in + [x:R](Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) + (Dgf (D_x Df x0) (D_x Dg (f x0)) f) (df x0) x0). +Intro;Generalize (H5 H6);Clear H5;Intro; + Generalize (limit_mul + [x:R](Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) + [x:R](dg (f x0)) + (D_x Df x0) (df x0) (dg (f x0)) x0 H1 + (limit_free [x:R](dg (f x0)) (D_x Df x0) x0 x0)); + Intro; + Unfold limit1_in;Unfold limit_in;Simpl;Unfold limit1_in in H5 H7; + Unfold limit_in in H5 H7;Simpl in H5 H7;Intros;Elim (H5 eps H8); + Elim (H7 eps H8);Clear H5 H7;Intros;Elim H5;Elim H7;Clear H5 H7; + Intros;Split with (Rmin x x1);Split. +Elim (Rmin_Rgt x x1 R0);Intros a b; + Apply (b (conj (Rgt x R0) (Rgt x1 R0) H9 H5));Clear a b. +Intros;Elim H11;Clear H11;Intros;Elim (Rmin_Rgt x x1 (R_dist x2 x0)); + Intros a b;Clear b;Unfold Rgt in a;Elim (a H12);Clear H5 a;Intros; + Unfold D_x Dgf in H11 H7 H10;Clear H12; + Elim (classic (f x2)==(f x0));Intro. +Elim H11;Clear H11;Intros;Elim H11;Clear H11;Intros; + Generalize (H10 x2 (conj (Df x2)/\~x0==x2 (Rlt (R_dist x2 x0) x) + (conj (Df x2) ~x0==x2 H11 H14) H5));Intro; + Rewrite (eq_Rminus (f x2) (f x0) H12) in H16; + Rewrite (Rmult_Ol (Rinv (Rminus x2 x0))) in H16; + Rewrite (Rmult_Ol (dg (f x0))) in H16; + Rewrite H12; + Rewrite (eq_Rminus (g (f x0)) (g (f x0)) (refl_eqT R (g (f x0)))); + Rewrite (Rmult_Ol (Rinv (Rminus x2 x0)));Assumption. +Clear H10 H5;Elim H11;Clear H11;Intros;Elim H5;Clear H5;Intros; +Cut (((Df x2)/\~x0==x2)/\(Dg (f x2))/\~(f x0)==(f x2)) + /\(Rlt (R_dist x2 x0) x1);Auto;Intro; + Generalize (H7 x2 H14);Intro; + Generalize (Rminus_eq_contra (f x2) (f x0) H12);Intro; + Rewrite (Rmult_assoc (Rminus (g (f x2)) (g (f x0))) + (Rinv (Rminus (f x2) (f x0))) + (Rmult (Rminus (f x2) (f x0)) (Rinv (Rminus x2 x0)))) in H15; + Rewrite <-(Rmult_assoc (Rinv (Rminus (f x2) (f x0))) + (Rminus (f x2) (f x0)) (Rinv (Rminus x2 x0))) in H15; + Rewrite (Rinv_l (Rminus (f x2) (f x0)) H16) in H15; + Rewrite (let (H1,H2)=(Rmult_ne (Rinv (Rminus x2 x0))) in H2) in H15; + Rewrite (Rmult_sym (df x0) (dg (f x0)));Assumption. +Clear H5 H3 H4 H2;Unfold limit1_in;Unfold limit_in;Simpl; + Unfold limit1_in in H1;Unfold limit_in in H1;Simpl in H1;Intros; + Elim (H1 eps H2);Clear H1;Intros;Elim H1;Clear H1;Intros; + Split with x;Split;Auto;Intros;Unfold D_x Dgf in H4 H3; + Elim H4;Clear H4;Intros;Elim H4;Clear H4;Intros; + Exact (H3 x1 (conj (Df x1)/\~x0==x1 (Rlt (R_dist x1 x0) x) H4 H5)). +Qed. + +(*********) +Lemma D_pow_n:(n:nat)(D:R->Prop)(x0:R)(expr,dexpr:R->R) + (D_in expr dexpr D x0)-> (D_in [x:R](pow (expr x) n) + [x:R](Rmult (Rmult (INR n) (pow (expr x) (minus n (1)))) (dexpr x)) + (Dgf D D expr) x0). +Intros n D x0 expr dexpr H; + Generalize (Dcomp D D dexpr [x:R](Rmult (INR n) (pow x (minus n (1)))) + expr [x:R](pow x n) x0 H (Dx_pow_n n D (expr x0))); + Intro; Unfold D_in; Unfold limit1_in; Unfold limit_in;Simpl;Intros; + Unfold D_in in H0; Unfold limit1_in in H0; Unfold limit_in in H0;Simpl in H0; + Elim (H0 eps H1);Clear H0;Intros;Elim H0;Clear H0;Intros;Split with x;Split; + Intros; Auto. +Cut ``((dexpr x0)*((INR n)*(pow (expr x0) (minus n (S O)))))== + ((INR n)*(pow (expr x0) (minus n (S O)))*(dexpr x0))``; + [Intro Rew;Rewrite <- Rew;Exact (H2 x1 H3)|Ring]. +Qed. + diff --git a/theories7/Reals/Reals.v b/theories7/Reals/Reals.v new file mode 100644 index 00000000..d0f879ab --- /dev/null +++ b/theories7/Reals/Reals.v @@ -0,0 +1,32 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Reals.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +(* The library REALS is divided in 6 parts : + - Rbase: basic lemmas on R + equalities and inequalities + Ring and Field are instantiated on R + - Rfunctions: some useful functions (Rabsolu, Rmin, Rmax, fact...) + - SeqSeries: theory of sequences and series + - Rtrigo: theory of trigonometric functions + - Ranalysis: some topology and general results of real analysis (mean value theorem, intermediate value theorem,...) + - Integration: Newton and Riemann' integrals + + Tactics are: + - DiscrR: for goals like ``?1<>0`` + - Sup: for goals like ``?1<?2`` + - RCompute: for equalities with constants like ``10*10==100`` + - Reg: for goals like (continuity_pt ?1 ?2) or (derivable_pt ?1 ?2) *) + +Require Export Rbase. +Require Export Rfunctions. +Require Export SeqSeries. +Require Export Rtrigo. +Require Export Ranalysis. +Require Export Integration. diff --git a/theories7/Reals/Rfunctions.v b/theories7/Reals/Rfunctions.v new file mode 100644 index 00000000..fe6ccd96 --- /dev/null +++ b/theories7/Reals/Rfunctions.v @@ -0,0 +1,832 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rfunctions.v,v 1.2.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +(*i Some properties about pow and sum have been made with John Harrison i*) +(*i Some Lemmas (about pow and powerRZ) have been done by Laurent Thery i*) + +(********************************************************) +(** Definition of the sum functions *) +(* *) +(********************************************************) + +Require Rbase. +Require Export R_Ifp. +Require Export Rbasic_fun. +Require Export R_sqr. +Require Export SplitAbsolu. +Require Export SplitRmult. +Require Export ArithProp. +Require Omega. +Require Zpower. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +(*******************************) +(** Lemmas about factorial *) +(*******************************) +(*********) +Lemma INR_fact_neq_0:(n:nat)~(INR (fact n))==R0. +Proof. +Intro;Red;Intro;Apply (not_O_INR (fact n) (fact_neq_0 n));Assumption. +Qed. + +(*********) +Lemma fact_simpl : (n:nat) (fact (S n))=(mult (S n) (fact n)). +Proof. +Intro; Reflexivity. +Qed. + +(*********) +Lemma simpl_fact:(n:nat)(Rmult (Rinv (INR (fact (S n)))) + (Rinv (Rinv (INR (fact n)))))== + (Rinv (INR (S n))). +Proof. +Intro;Rewrite (Rinv_Rinv (INR (fact n)) (INR_fact_neq_0 n)); + Unfold 1 fact;Cbv Beta Iota;Fold fact; + Rewrite (mult_INR (S n) (fact n)); + Rewrite (Rinv_Rmult (INR (S n)) (INR (fact n))). +Rewrite (Rmult_assoc (Rinv (INR (S n))) (Rinv (INR (fact n))) + (INR (fact n)));Rewrite (Rinv_l (INR (fact n)) (INR_fact_neq_0 n)); + Apply (let (H1,H2)=(Rmult_ne (Rinv (INR (S n)))) in H1). +Apply not_O_INR;Auto. +Apply INR_fact_neq_0. +Qed. + +(*******************************) +(* Power *) +(*******************************) +(*********) +Fixpoint pow [r:R;n:nat]:R:= + Cases n of + O => R1 + |(S n) => (Rmult r (pow r n)) + end. + +V8Infix "^" pow : R_scope. + +Lemma pow_O: (x : R) (pow x O) == R1. +Proof. +Reflexivity. +Qed. + +Lemma pow_1: (x : R) (pow x (1)) == x. +Proof. +Simpl; Auto with real. +Qed. + +Lemma pow_add: + (x : R) (n, m : nat) (pow x (plus n m)) == (Rmult (pow x n) (pow x m)). +Proof. +Intros x n; Elim n; Simpl; Auto with real. +Intros n0 H' m; Rewrite H'; Auto with real. +Qed. + +Lemma pow_nonzero: + (x:R) (n:nat) ~(x==R0) -> ~((pow x n)==R0). +Proof. +Intro; Induction n; Simpl. +Intro; Red;Intro;Apply R1_neq_R0;Assumption. +Intros;Red; Intro;Elim (without_div_Od x (pow x n0) H1). +Intro; Auto. +Apply H;Assumption. +Qed. + +Hints Resolve pow_O pow_1 pow_add pow_nonzero:real. + +Lemma pow_RN_plus: + (x : R) + (n, m : nat) + ~ x == R0 -> (pow x n) == (Rmult (pow x (plus n m)) (Rinv (pow x m))). +Proof. +Intros x n; Elim n; Simpl; Auto with real. +Intros n0 H' m H'0. +Rewrite Rmult_assoc; Rewrite <- H'; Auto. +Qed. + +Lemma pow_lt: (x : R) (n : nat) (Rlt R0 x) -> (Rlt R0 (pow x n)). +Proof. +Intros x n; Elim n; Simpl; Auto with real. +Intros n0 H' H'0; Replace R0 with (Rmult x R0); Auto with real. +Qed. +Hints Resolve pow_lt :real. + +Lemma Rlt_pow_R1: + (x : R) (n : nat) (Rlt R1 x) -> (lt O n) -> (Rlt R1 (pow x n)). +Proof. +Intros x n; Elim n; Simpl; Auto with real. +Intros H' H'0; ElimType False; Omega. +Intros n0; Case n0. +Simpl; Rewrite Rmult_1r; Auto. +Intros n1 H' H'0 H'1. +Replace R1 with (Rmult R1 R1); Auto with real. +Apply Rlt_trans with r2 := (Rmult x R1); Auto with real. +Apply Rlt_monotony; Auto with real. +Apply Rlt_trans with r2 := R1; Auto with real. +Apply H'; Auto with arith. +Qed. +Hints Resolve Rlt_pow_R1 :real. + +Lemma Rlt_pow: + (x : R) (n, m : nat) (Rlt R1 x) -> (lt n m) -> (Rlt (pow x n) (pow x m)). +Proof. +Intros x n m H' H'0; Replace m with (plus (minus m n) n). +Rewrite pow_add. +Pattern 1 (pow x n); Replace (pow x n) with (Rmult R1 (pow x n)); + Auto with real. +Apply Rminus_lt. +Repeat Rewrite [y : R] (Rmult_sym y (pow x n)); Rewrite <- Rminus_distr. +Replace R0 with (Rmult (pow x n) R0); Auto with real. +Apply Rlt_monotony; Auto with real. +Apply pow_lt; Auto with real. +Apply Rlt_trans with r2 := R1; Auto with real. +Apply Rlt_minus; Auto with real. +Apply Rlt_pow_R1; Auto with arith. +Apply simpl_lt_plus_l with p := n; Auto with arith. +Rewrite le_plus_minus_r; Auto with arith; Rewrite <- plus_n_O; Auto. +Rewrite plus_sym; Auto with arith. +Qed. +Hints Resolve Rlt_pow :real. + +(*********) +Lemma tech_pow_Rmult:(x:R)(n:nat)(Rmult x (pow x n))==(pow x (S n)). +Proof. +Induction n; Simpl; Trivial. +Qed. + +(*********) +Lemma tech_pow_Rplus:(x:R)(a,n:nat) + (Rplus (pow x a) (Rmult (INR n) (pow x a)))== + (Rmult (INR (S n)) (pow x a)). +Proof. +Intros; Pattern 1 (pow x a); + Rewrite <-(let (H1,H2)=(Rmult_ne (pow x a)) in H1); + Rewrite (Rmult_sym (INR n) (pow x a)); + Rewrite <- (Rmult_Rplus_distr (pow x a) R1 (INR n)); + Rewrite (Rplus_sym R1 (INR n)); Rewrite <-(S_INR n); + Apply Rmult_sym. +Qed. + +Lemma poly: (n:nat)(x:R)(Rlt R0 x)-> + (Rle (Rplus R1 (Rmult (INR n) x)) (pow (Rplus R1 x) n)). +Proof. +Intros;Elim n. +Simpl;Cut (Rplus R1 (Rmult R0 x))==R1. +Intro;Rewrite H0;Unfold Rle;Right; Reflexivity. +Ring. +Intros;Unfold pow; Fold pow; + Apply (Rle_trans (Rplus R1 (Rmult (INR (S n0)) x)) + (Rmult (Rplus R1 x) (Rplus R1 (Rmult (INR n0) x))) + (Rmult (Rplus R1 x) (pow (Rplus R1 x) n0))). +Cut (Rmult (Rplus R1 x) (Rplus R1 (Rmult (INR n0) x)))== + (Rplus (Rplus R1 (Rmult (INR (S n0)) x)) + (Rmult (INR n0) (Rmult x x))). +Intro;Rewrite H1;Pattern 1 (Rplus R1 (Rmult (INR (S n0)) x)); + Rewrite <-(let (H1,H2)= + (Rplus_ne (Rplus R1 (Rmult (INR (S n0)) x))) in H1); + Apply Rle_compatibility;Elim n0;Intros. +Simpl;Rewrite Rmult_Ol;Unfold Rle;Right;Auto. +Unfold Rle;Left;Generalize Rmult_gt;Unfold Rgt;Intro; + Fold (Rsqr x);Apply (H3 (INR (S n1)) (Rsqr x) + (lt_INR_0 (S n1) (lt_O_Sn n1)));Fold (Rgt x R0) in H; + Apply (pos_Rsqr1 x (imp_not_Req x R0 + (or_intror (Rlt x R0) (Rgt x R0) H))). +Rewrite (S_INR n0);Ring. +Unfold Rle in H0;Elim H0;Intro. +Unfold Rle;Left;Apply Rlt_monotony. +Rewrite Rplus_sym; + Apply (Rlt_r_plus_R1 x (Rlt_le R0 x H)). +Assumption. +Rewrite H1;Unfold Rle;Right;Trivial. +Qed. + +Lemma Power_monotonic: + (x:R) (m,n:nat) (Rgt (Rabsolu x) R1) + -> (le m n) + -> (Rle (Rabsolu (pow x m)) (Rabsolu (pow x n))). +Proof. +Intros x m n H;Induction n;Intros;Inversion H0. +Unfold Rle; Right; Reflexivity. +Unfold Rle; Right; Reflexivity. +Apply (Rle_trans (Rabsolu (pow x m)) + (Rabsolu (pow x n)) + (Rabsolu (pow x (S n)))). +Apply Hrecn; Assumption. +Simpl;Rewrite Rabsolu_mult. +Pattern 1 (Rabsolu (pow x n)). +Rewrite <-Rmult_1r. +Rewrite (Rmult_sym (Rabsolu x) (Rabsolu (pow x n))). +Apply Rle_monotony. +Apply Rabsolu_pos. +Unfold Rgt in H. +Apply Rlt_le; Assumption. +Qed. + +Lemma Pow_Rabsolu: (x:R) (n:nat) + (pow (Rabsolu x) n)==(Rabsolu (pow x n)). +Proof. +Intro;Induction n;Simpl. +Apply sym_eqT;Apply Rabsolu_pos_eq;Apply Rlt_le;Apply Rlt_R0_R1. +Intros; Rewrite H;Apply sym_eqT;Apply Rabsolu_mult. +Qed. + + +Lemma Pow_x_infinity: + (x:R) (Rgt (Rabsolu x) R1) + -> (b:R) (Ex [N:nat] ((n:nat) (ge n N) + -> (Rge (Rabsolu (pow x n)) b ))). +Proof. +Intros;Elim (archimed (Rmult b (Rinv (Rminus (Rabsolu x) R1))));Intros; + Clear H1; + Cut (Ex[N:nat] (Rge (INR N) (Rmult b (Rinv (Rminus (Rabsolu x) R1))))). +Intro; Elim H1;Clear H1;Intros;Exists x0;Intros; + Apply (Rge_trans (Rabsolu (pow x n)) (Rabsolu (pow x x0)) b). +Apply Rle_sym1;Apply Power_monotonic;Assumption. +Rewrite <- Pow_Rabsolu;Cut (Rabsolu x)==(Rplus R1 (Rminus (Rabsolu x) R1)). +Intro; Rewrite H3; + Apply (Rge_trans (pow (Rplus R1 (Rminus (Rabsolu x) R1)) x0) + (Rplus R1 (Rmult (INR x0) + (Rminus (Rabsolu x) R1))) + b). +Apply Rle_sym1;Apply poly;Fold (Rgt (Rminus (Rabsolu x) R1) R0); + Apply Rgt_minus;Assumption. +Apply (Rge_trans + (Rplus R1 (Rmult (INR x0) (Rminus (Rabsolu x) R1))) + (Rmult (INR x0) (Rminus (Rabsolu x) R1)) + b). +Apply Rle_sym1; Apply Rlt_le;Rewrite (Rplus_sym R1 + (Rmult (INR x0) (Rminus (Rabsolu x) R1))); + Pattern 1 (Rmult (INR x0) (Rminus (Rabsolu x) R1)); + Rewrite <- (let (H1,H2) = (Rplus_ne + (Rmult (INR x0) (Rminus (Rabsolu x) R1))) in + H1); + Apply Rlt_compatibility; + Apply Rlt_R0_R1. +Cut b==(Rmult (Rmult b (Rinv (Rminus (Rabsolu x) R1))) + (Rminus (Rabsolu x) R1)). +Intros; Rewrite H4;Apply Rge_monotony. +Apply Rge_minus;Unfold Rge; Left; Assumption. +Assumption. +Rewrite Rmult_assoc;Rewrite Rinv_l. +Ring. +Apply imp_not_Req; Right;Apply Rgt_minus;Assumption. +Ring. +Cut `0<= (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))`\/ + `(up (Rmult b (Rinv (Rminus (Rabsolu x) R1)))) <= 0`. +Intros;Elim H1;Intro. +Elim (IZN (up (Rmult b (Rinv (Rminus (Rabsolu x) R1)))) H2);Intros;Exists x0; + Apply (Rge_trans + (INR x0) + (IZR (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))) + (Rmult b (Rinv (Rminus (Rabsolu x) R1)))). +Rewrite INR_IZR_INZ;Apply IZR_ge;Omega. +Unfold Rge; Left; Assumption. +Exists O;Apply (Rge_trans (INR (0)) + (IZR (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))) + (Rmult b (Rinv (Rminus (Rabsolu x) R1)))). +Rewrite INR_IZR_INZ;Apply IZR_ge;Simpl;Omega. +Unfold Rge; Left; Assumption. +Omega. +Qed. + +Lemma pow_ne_zero: + (n:nat) ~(n=(0))-> (pow R0 n) == R0. +Proof. +Induction n. +Simpl;Auto. +Intros;Elim H;Reflexivity. +Intros; Simpl;Apply Rmult_Ol. +Qed. + +Lemma Rinv_pow: + (x:R) (n:nat) ~(x==R0) -> (Rinv (pow x n))==(pow (Rinv x) n). +Proof. +Intros; Elim n; Simpl. +Apply Rinv_R1. +Intro m;Intro;Rewrite Rinv_Rmult. +Rewrite H0; Reflexivity;Assumption. +Assumption. +Apply pow_nonzero;Assumption. +Qed. + +Lemma pow_lt_1_zero: + (x:R) (Rlt (Rabsolu x) R1) + -> (y:R) (Rlt R0 y) + -> (Ex[N:nat] (n:nat) (ge n N) + -> (Rlt (Rabsolu (pow x n)) y)). +Proof. +Intros;Elim (Req_EM x R0);Intro. +Exists (1);Rewrite H1;Intros n GE;Rewrite pow_ne_zero. +Rewrite Rabsolu_R0;Assumption. +Inversion GE;Auto. +Cut (Rgt (Rabsolu (Rinv x)) R1). +Intros;Elim (Pow_x_infinity (Rinv x) H2 (Rplus (Rinv y) R1));Intros N. +Exists N;Intros;Rewrite <- (Rinv_Rinv y). +Rewrite <- (Rinv_Rinv (Rabsolu (pow x n))). +Apply Rinv_lt. +Apply Rmult_lt_pos. +Apply Rlt_Rinv. +Assumption. +Apply Rlt_Rinv. +Apply Rabsolu_pos_lt. +Apply pow_nonzero. +Assumption. +Rewrite <- Rabsolu_Rinv. +Rewrite Rinv_pow. +Apply (Rlt_le_trans (Rinv y) + (Rplus (Rinv y) R1) + (Rabsolu (pow (Rinv x) n))). +Pattern 1 (Rinv y). +Rewrite <- (let (H1,H2) = + (Rplus_ne (Rinv y)) in H1). +Apply Rlt_compatibility. +Apply Rlt_R0_R1. +Apply Rle_sym2. +Apply H3. +Assumption. +Assumption. +Apply pow_nonzero. +Assumption. +Apply Rabsolu_no_R0. +Apply pow_nonzero. +Assumption. +Apply imp_not_Req. +Right; Unfold Rgt; Assumption. +Rewrite <- (Rinv_Rinv R1). +Rewrite Rabsolu_Rinv. +Unfold Rgt; Apply Rinv_lt. +Apply Rmult_lt_pos. +Apply Rabsolu_pos_lt. +Assumption. +Rewrite Rinv_R1; Apply Rlt_R0_R1. +Rewrite Rinv_R1; Assumption. +Assumption. +Red;Intro; Apply R1_neq_R0;Assumption. +Qed. + +Lemma pow_R1: + (r : R) (n : nat) (pow r n) == R1 -> (Rabsolu r) == R1 \/ n = O. +Proof. +Intros r n H'. +Case (Req_EM (Rabsolu r) R1); Auto; Intros H'1. +Case (not_Req ? ? H'1); Intros H'2. +Generalize H'; Case n; Auto. +Intros n0 H'0. +Cut ~ r == R0; [Intros Eq1 | Idtac]. +Cut ~ (Rabsolu r) == R0; [Intros Eq2 | Apply Rabsolu_no_R0]; Auto. +Absurd (Rlt (pow (Rabsolu (Rinv r)) O) (pow (Rabsolu (Rinv r)) (S n0))); Auto. +Replace (pow (Rabsolu (Rinv r)) (S n0)) with R1. +Simpl; Apply Rlt_antirefl; Auto. +Rewrite Rabsolu_Rinv; Auto. +Rewrite <- Rinv_pow; Auto. +Rewrite Pow_Rabsolu; Auto. +Rewrite H'0; Rewrite Rabsolu_right; Auto with real. +Apply Rle_ge; Auto with real. +Apply Rlt_pow; Auto with arith. +Rewrite Rabsolu_Rinv; Auto. +Apply Rlt_monotony_contra with z := (Rabsolu r). +Case (Rabsolu_pos r); Auto. +Intros H'3; Case Eq2; Auto. +Rewrite Rmult_1r; Rewrite Rinv_r; Auto with real. +Red;Intro;Absurd ``(pow r (S n0)) == 1``;Auto. +Simpl; Rewrite H; Rewrite Rmult_Ol; Auto with real. +Generalize H'; Case n; Auto. +Intros n0 H'0. +Cut ~ r == R0; [Intros Eq1 | Auto with real]. +Cut ~ (Rabsolu r) == R0; [Intros Eq2 | Apply Rabsolu_no_R0]; Auto. +Absurd (Rlt (pow (Rabsolu r) O) (pow (Rabsolu r) (S n0))); + Auto with real arith. +Repeat Rewrite Pow_Rabsolu; Rewrite H'0; Simpl; Auto with real. +Red;Intro;Absurd ``(pow r (S n0)) == 1``;Auto. +Simpl; Rewrite H; Rewrite Rmult_Ol; Auto with real. +Qed. + +Lemma pow_Rsqr : (x:R;n:nat) (pow x (mult (2) n))==(pow (Rsqr x) n). +Proof. +Intros; Induction n. +Reflexivity. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Replace (pow x (S (S (mult (2) n)))) with ``x*x*(pow x (mult (S (S O)) n))``. +Rewrite Hrecn; Reflexivity. +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Qed. + +Lemma pow_le : (a:R;n:nat) ``0<=a`` -> ``0<=(pow a n)``. +Proof. +Intros; Induction n. +Simpl; Left; Apply Rlt_R0_R1. +Simpl; Apply Rmult_le_pos; Assumption. +Qed. + +(**********) +Lemma pow_1_even : (n:nat) ``(pow (-1) (mult (S (S O)) n))==1``. +Proof. +Intro; Induction n. +Reflexivity. +Replace (mult (2) (S n)) with (plus (2) (mult (2) n)). +Rewrite pow_add; Rewrite Hrecn; Simpl; Ring. +Replace (S n) with (plus n (1)); [Ring | Ring]. +Qed. + +(**********) +Lemma pow_1_odd : (n:nat) ``(pow (-1) (S (mult (S (S O)) n)))==-1``. +Proof. +Intro; Replace (S (mult (2) n)) with (plus (mult (2) n) (1)); [Idtac | Ring]. +Rewrite pow_add; Rewrite pow_1_even; Simpl; Ring. +Qed. + +(**********) +Lemma pow_1_abs : (n:nat) ``(Rabsolu (pow (-1) n))==1``. +Proof. +Intro; Induction n. +Simpl; Apply Rabsolu_R1. +Replace (S n) with (plus n (1)); [Rewrite pow_add | Ring]. +Rewrite Rabsolu_mult. +Rewrite Hrecn; Rewrite Rmult_1l; Simpl; Rewrite Rmult_1r; Rewrite Rabsolu_Ropp; Apply Rabsolu_R1. +Qed. + +Lemma pow_mult : (x:R;n1,n2:nat) (pow x (mult n1 n2))==(pow (pow x n1) n2). +Proof. +Intros; Induction n2. +Simpl; Replace (mult n1 O) with O; [Reflexivity | Ring]. +Replace (mult n1 (S n2)) with (plus (mult n1 n2) n1). +Replace (S n2) with (plus n2 (1)); [Idtac | Ring]. +Do 2 Rewrite pow_add. +Rewrite Hrecn2. +Simpl. +Ring. +Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite mult_INR; Rewrite S_INR; Ring. +Qed. + +Lemma pow_incr : (x,y:R;n:nat) ``0<=x<=y`` -> ``(pow x n)<=(pow y n)``. +Proof. +Intros. +Induction n. +Right; Reflexivity. +Simpl. +Elim H; Intros. +Apply Rle_trans with ``y*(pow x n)``. +Do 2 Rewrite <- (Rmult_sym (pow x n)). +Apply Rle_monotony. +Apply pow_le; Assumption. +Assumption. +Apply Rle_monotony. +Apply Rle_trans with x; Assumption. +Apply Hrecn. +Qed. + +Lemma pow_R1_Rle : (x:R;k:nat) ``1<=x`` -> ``1<=(pow x k)``. +Proof. +Intros. +Induction k. +Right; Reflexivity. +Simpl. +Apply Rle_trans with ``x*1``. +Rewrite Rmult_1r; Assumption. +Apply Rle_monotony. +Left; Apply Rlt_le_trans with R1; [Apply Rlt_R0_R1 | Assumption]. +Exact Hreck. +Qed. + +Lemma Rle_pow : (x:R;m,n:nat) ``1<=x`` -> (le m n) -> ``(pow x m)<=(pow x n)``. +Proof. +Intros. +Replace n with (plus (minus n m) m). +Rewrite pow_add. +Rewrite Rmult_sym. +Pattern 1 (pow x m); Rewrite <- Rmult_1r. +Apply Rle_monotony. +Apply pow_le; Left; Apply Rlt_le_trans with R1; [Apply Rlt_R0_R1 | Assumption]. +Apply pow_R1_Rle; Assumption. +Rewrite plus_sym. +Symmetry; Apply le_plus_minus; Assumption. +Qed. + +Lemma pow1 : (n:nat) (pow R1 n)==R1. +Proof. +Intro; Induction n. +Reflexivity. +Simpl; Rewrite Hrecn; Rewrite Rmult_1r; Reflexivity. +Qed. + +Lemma pow_Rabs : (x:R;n:nat) ``(pow x n)<=(pow (Rabsolu x) n)``. +Proof. +Intros; Induction n. +Right; Reflexivity. +Simpl; Case (case_Rabsolu x); Intro. +Apply Rle_trans with (Rabsolu ``x*(pow x n)``). +Apply Rle_Rabsolu. +Rewrite Rabsolu_mult. +Apply Rle_monotony. +Apply Rabsolu_pos. +Right; Symmetry; Apply Pow_Rabsolu. +Pattern 1 (Rabsolu x); Rewrite (Rabsolu_right x r); Apply Rle_monotony. +Apply Rle_sym2; Exact r. +Apply Hrecn. +Qed. + +Lemma pow_maj_Rabs : (x,y:R;n:nat) ``(Rabsolu y)<=x`` -> ``(pow y n)<=(pow x n)``. +Proof. +Intros; Cut ``0<=x``. +Intro; Apply Rle_trans with (pow (Rabsolu y) n). +Apply pow_Rabs. +Induction n. +Right; Reflexivity. +Simpl; Apply Rle_trans with ``x*(pow (Rabsolu y) n)``. +Do 2 Rewrite <- (Rmult_sym (pow (Rabsolu y) n)). +Apply Rle_monotony. +Apply pow_le; Apply Rabsolu_pos. +Assumption. +Apply Rle_monotony. +Apply H0. +Apply Hrecn. +Apply Rle_trans with (Rabsolu y); [Apply Rabsolu_pos | Exact H]. +Qed. + +(*******************************) +(** PowerRZ *) +(*******************************) +(*i Due to L.Thery i*) + +Tactic Definition CaseEqk name := +Generalize (refl_equal ? name); Pattern -1 name; Case name. + +Definition powerRZ := + [x : R] [n : Z] Cases n of + ZERO => R1 + | (POS p) => (pow x (convert p)) + | (NEG p) => (Rinv (pow x (convert p))) + end. + +Infix Local "^Z" powerRZ (at level 2, left associativity) : R_scope. + +Lemma Zpower_NR0: + (x : Z) (n : nat) (Zle ZERO x) -> (Zle ZERO (Zpower_nat x n)). +Proof. +NewInduction n; Unfold Zpower_nat; Simpl; Auto with zarith. +Qed. + +Lemma powerRZ_O: (x : R) (powerRZ x ZERO) == R1. +Proof. +Reflexivity. +Qed. + +Lemma powerRZ_1: (x : R) (powerRZ x (Zs ZERO)) == x. +Proof. +Simpl; Auto with real. +Qed. + +Lemma powerRZ_NOR: (x : R) (z : Z) ~ x == R0 -> ~ (powerRZ x z) == R0. +Proof. +NewDestruct z; Simpl; Auto with real. +Qed. + +Lemma powerRZ_add: + (x : R) + (n, m : Z) + ~ x == R0 -> (powerRZ x (Zplus n m)) == (Rmult (powerRZ x n) (powerRZ x m)). +Proof. +Intro x; NewDestruct n as [|n1|n1]; NewDestruct m as [|m1|m1]; Simpl; + Auto with real. +(* POS/POS *) +Rewrite convert_add; Auto with real. +(* POS/NEG *) +(CaseEqk '(compare n1 m1 EGAL)); Simpl; Auto with real. +Intros H' H'0; Rewrite compare_convert_EGAL with 1 := H'; Auto with real. +Intros H' H'0; Rewrite (true_sub_convert m1 n1); Auto with real. +Rewrite (pow_RN_plus x (minus (convert m1) (convert n1)) (convert n1)); + Auto with real. +Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. +Rewrite Rinv_Rmult; Auto with real. +Rewrite Rinv_Rinv; Auto with real. +Apply lt_le_weak. +Apply compare_convert_INFERIEUR; Auto. +Apply ZC2; Auto. +Intros H' H'0; Rewrite (true_sub_convert n1 m1); Auto with real. +Rewrite (pow_RN_plus x (minus (convert n1) (convert m1)) (convert m1)); + Auto with real. +Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. +Apply lt_le_weak. +Change (gt (convert n1) (convert m1)). +Apply compare_convert_SUPERIEUR; Auto. +(* NEG/POS *) +(CaseEqk '(compare n1 m1 EGAL)); Simpl; Auto with real. +Intros H' H'0; Rewrite compare_convert_EGAL with 1 := H'; Auto with real. +Intros H' H'0; Rewrite (true_sub_convert m1 n1); Auto with real. +Rewrite (pow_RN_plus x (minus (convert m1) (convert n1)) (convert n1)); + Auto with real. +Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. +Apply lt_le_weak. +Apply compare_convert_INFERIEUR; Auto. +Apply ZC2; Auto. +Intros H' H'0; Rewrite (true_sub_convert n1 m1); Auto with real. +Rewrite (pow_RN_plus x (minus (convert n1) (convert m1)) (convert m1)); + Auto with real. +Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. +Rewrite Rinv_Rmult; Auto with real. +Apply lt_le_weak. +Change (gt (convert n1) (convert m1)). +Apply compare_convert_SUPERIEUR; Auto. +(* NEG/NEG *) +Rewrite convert_add; Auto with real. +Intros H'; Rewrite pow_add; Auto with real. +Apply Rinv_Rmult; Auto. +Apply pow_nonzero; Auto. +Apply pow_nonzero; Auto. +Qed. +Hints Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add :real. + +Lemma Zpower_nat_powerRZ: + (n, m : nat) + (IZR (Zpower_nat (inject_nat n) m)) == (powerRZ (INR n) (inject_nat m)). +Proof. +Intros n m; Elim m; Simpl; Auto with real. +Intros m1 H'; Rewrite bij1; Simpl. +Replace (Zpower_nat (inject_nat n) (S m1)) + with (Zmult (inject_nat n) (Zpower_nat (inject_nat n) m1)). +Rewrite mult_IZR; Auto with real. +Repeat Rewrite <- INR_IZR_INZ; Simpl. +Rewrite H'; Simpl. +Case m1; Simpl; Auto with real. +Intros m2; Rewrite bij1; Auto. +Unfold Zpower_nat; Auto. +Qed. + +Lemma powerRZ_lt: (x : R) (z : Z) (Rlt R0 x) -> (Rlt R0 (powerRZ x z)). +Proof. +Intros x z; Case z; Simpl; Auto with real. +Qed. +Hints Resolve powerRZ_lt :real. + +Lemma powerRZ_le: (x : R) (z : Z) (Rlt R0 x) -> (Rle R0 (powerRZ x z)). +Proof. +Intros x z H'; Apply Rlt_le; Auto with real. +Qed. +Hints Resolve powerRZ_le :real. + +Lemma Zpower_nat_powerRZ_absolu: + (n, m : Z) + (Zle ZERO m) -> (IZR (Zpower_nat n (absolu m))) == (powerRZ (IZR n) m). +Proof. +Intros n m; Case m; Simpl; Auto with zarith. +Intros p H'; Elim (convert p); Simpl; Auto with zarith. +Intros n0 H'0; Rewrite <- H'0; Simpl; Auto with zarith. +Rewrite <- mult_IZR; Auto. +Intros p H'; Absurd `0 <= (NEG p)`;Auto with zarith. +Qed. + +Lemma powerRZ_R1: (n : Z) (powerRZ R1 n) == R1. +Proof. +Intros n; Case n; Simpl; Auto. +Intros p; Elim (convert p); Simpl; Auto; Intros n0 H'; Rewrite H'; Ring. +Intros p; Elim (convert p); Simpl. +Exact Rinv_R1. +Intros n1 H'; Rewrite Rinv_Rmult; Try Rewrite Rinv_R1; Try Rewrite H'; + Auto with real. +Qed. + +(*******************************) +(** Sum of n first naturals *) +(*******************************) +(*********) +Fixpoint sum_nat_f_O [f:nat->nat;n:nat]:nat:= + Cases n of + O => (f O) + |(S n') => (plus (sum_nat_f_O f n') (f (S n'))) + end. + +(*********) +Definition sum_nat_f [s,n:nat;f:nat->nat]:nat:= + (sum_nat_f_O [x:nat](f (plus x s)) (minus n s)). + +(*********) +Definition sum_nat_O [n:nat]:nat:= + (sum_nat_f_O [x:nat]x n). + +(*********) +Definition sum_nat [s,n:nat]:nat:= + (sum_nat_f s n [x:nat]x). + +(*******************************) +(** Sum *) +(*******************************) +(*********) +Fixpoint sum_f_R0 [f:nat->R;N:nat]:R:= + Cases N of + O => (f O) + |(S i) => (Rplus (sum_f_R0 f i) (f (S i))) + end. + +(*********) +Definition sum_f [s,n:nat;f:nat->R]:R:= + (sum_f_R0 [x:nat](f (plus x s)) (minus n s)). + +Lemma GP_finite: + (x:R) (n:nat) (Rmult (sum_f_R0 [n:nat] (pow x n) n) + (Rminus x R1)) == + (Rminus (pow x (plus n (1))) R1). +Proof. +Intros; Induction n; Simpl. +Ring. +Rewrite Rmult_Rplus_distrl;Rewrite Hrecn;Cut (plus n (1))=(S n). +Intro H;Rewrite H;Simpl;Ring. +Omega. +Qed. + +Lemma sum_f_R0_triangle: + (x:nat->R)(n:nat) (Rle (Rabsolu (sum_f_R0 x n)) + (sum_f_R0 [i:nat] (Rabsolu (x i)) n)). +Proof. +Intro; Induction n; Simpl. +Unfold Rle; Right; Reflexivity. +Intro m; Intro;Apply (Rle_trans + (Rabsolu (Rplus (sum_f_R0 x m) (x (S m)))) + (Rplus (Rabsolu (sum_f_R0 x m)) + (Rabsolu (x (S m)))) + (Rplus (sum_f_R0 [i:nat](Rabsolu (x i)) m) + (Rabsolu (x (S m))))). +Apply Rabsolu_triang. +Rewrite Rplus_sym;Rewrite (Rplus_sym + (sum_f_R0 [i:nat](Rabsolu (x i)) m) (Rabsolu (x (S m)))); + Apply Rle_compatibility;Assumption. +Qed. + +(*******************************) +(* Distance in R *) +(*******************************) + +(*********) +Definition R_dist:R->R->R:=[x,y:R](Rabsolu (Rminus x y)). + +(*********) +Lemma R_dist_pos:(x,y:R)(Rge (R_dist x y) R0). +Proof. +Intros;Unfold R_dist;Unfold Rabsolu;Case (case_Rabsolu (Rminus x y));Intro l. +Unfold Rge;Left;Apply (Rlt_RoppO (Rminus x y) l). +Trivial. +Qed. + +(*********) +Lemma R_dist_sym:(x,y:R)(R_dist x y)==(R_dist y x). +Proof. +Unfold R_dist;Intros;SplitAbsolu;Ring. +Generalize (Rlt_RoppO (Rminus y x) r); Intro; + Rewrite (Ropp_distr2 y x) in H; + Generalize (Rlt_antisym (Rminus x y) R0 r0); Intro;Unfold Rgt in H; + ElimType False; Auto. +Generalize (minus_Rge y x r); Intro; + Generalize (minus_Rge x y r0); Intro; + Generalize (Rge_ge_eq x y H0 H); Intro;Rewrite H1;Ring. +Qed. + +(*********) +Lemma R_dist_refl:(x,y:R)((R_dist x y)==R0<->x==y). +Proof. +Unfold R_dist;Intros;SplitAbsolu;Split;Intros. +Rewrite (Ropp_distr2 x y) in H;Apply sym_eqT; + Apply (Rminus_eq y x H). +Rewrite (Ropp_distr2 x y);Generalize (sym_eqT R x y H);Intro; + Apply (eq_Rminus y x H0). +Apply (Rminus_eq x y H). +Apply (eq_Rminus x y H). +Qed. + +Lemma R_dist_eq:(x:R)(R_dist x x)==R0. +Proof. +Unfold R_dist;Intros;SplitAbsolu;Intros;Ring. +Qed. + +(***********) +Lemma R_dist_tri:(x,y,z:R)(Rle (R_dist x y) + (Rplus (R_dist x z) (R_dist z y))). +Proof. +Intros;Unfold R_dist; Replace ``x-y`` with ``(x-z)+(z-y)``; + [Apply (Rabsolu_triang ``x-z`` ``z-y``)|Ring]. +Qed. + +(*********) +Lemma R_dist_plus: (a,b,c,d:R)(Rle (R_dist (Rplus a c) (Rplus b d)) + (Rplus (R_dist a b) (R_dist c d))). +Proof. +Intros;Unfold R_dist; + Replace (Rminus (Rplus a c) (Rplus b d)) + with (Rplus (Rminus a b) (Rminus c d)). +Exact (Rabsolu_triang (Rminus a b) (Rminus c d)). +Ring. +Qed. + +(*******************************) +(** Infinit Sum *) +(*******************************) +(*********) +Definition infinit_sum:(nat->R)->R->Prop:=[s:nat->R;l:R] + (eps:R)(Rgt eps R0)-> + (Ex[N:nat](n:nat)(ge n N)->(Rlt (R_dist (sum_f_R0 s n) l) eps)). diff --git a/theories7/Reals/Rgeom.v b/theories7/Reals/Rgeom.v new file mode 100644 index 00000000..12c52e37 --- /dev/null +++ b/theories7/Reals/Rgeom.v @@ -0,0 +1,84 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rgeom.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require R_sqrt. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Definition dist_euc [x0,y0,x1,y1:R] : R := ``(sqrt ((Rsqr (x0-x1))+(Rsqr (y0-y1))))``. + +Lemma distance_refl : (x0,y0:R) ``(dist_euc x0 y0 x0 y0)==0``. +Intros x0 y0; Unfold dist_euc; Apply Rsqr_inj; [Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0; [Apply pos_Rsqr | Apply pos_Rsqr] | Right; Reflexivity | Rewrite Rsqr_O; Rewrite Rsqr_sqrt; [Unfold Rsqr; Ring | Apply ge0_plus_ge0_is_ge0; [Apply pos_Rsqr | Apply pos_Rsqr]]]. +Qed. + +Lemma distance_symm : (x0,y0,x1,y1:R) ``(dist_euc x0 y0 x1 y1) == (dist_euc x1 y1 x0 y0)``. +Intros x0 y0 x1 y1; Unfold dist_euc; Apply Rsqr_inj; [ Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0 | Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0 | Repeat Rewrite Rsqr_sqrt; [Unfold Rsqr; Ring | Apply ge0_plus_ge0_is_ge0 |Apply ge0_plus_ge0_is_ge0]]; Apply pos_Rsqr. +Qed. + +Lemma law_cosines : (x0,y0,x1,y1,x2,y2,ac:R) let a = (dist_euc x1 y1 x0 y0) in let b=(dist_euc x2 y2 x0 y0) in let c=(dist_euc x2 y2 x1 y1) in ( ``a*c*(cos ac) == ((x0-x1)*(x2-x1) + (y0-y1)*(y2-y1))`` -> ``(Rsqr b)==(Rsqr c)+(Rsqr a)-2*(a*c*(cos ac))`` ). +Unfold dist_euc; Intros; Repeat Rewrite -> Rsqr_sqrt; [ Rewrite H; Unfold Rsqr; Ring | Apply ge0_plus_ge0_is_ge0 | Apply ge0_plus_ge0_is_ge0 | Apply ge0_plus_ge0_is_ge0]; Apply pos_Rsqr. +Qed. + +Lemma triangle : (x0,y0,x1,y1,x2,y2:R) ``(dist_euc x0 y0 x1 y1)<=(dist_euc x0 y0 x2 y2)+(dist_euc x2 y2 x1 y1)``. +Intros; Unfold dist_euc; Apply Rsqr_incr_0; [Rewrite Rsqr_plus; Repeat Rewrite Rsqr_sqrt; [Replace ``(Rsqr (x0-x1))`` with ``(Rsqr (x0-x2))+(Rsqr (x2-x1))+2*(x0-x2)*(x2-x1)``; [Replace ``(Rsqr (y0-y1))`` with ``(Rsqr (y0-y2))+(Rsqr (y2-y1))+2*(y0-y2)*(y2-y1)``; [Apply Rle_anti_compatibility with ``-(Rsqr (x0-x2))-(Rsqr (x2-x1))-(Rsqr (y0-y2))-(Rsqr (y2-y1))``; Replace `` -(Rsqr (x0-x2))-(Rsqr (x2-x1))-(Rsqr (y0-y2))-(Rsqr (y2-y1))+((Rsqr (x0-x2))+(Rsqr (x2-x1))+2*(x0-x2)*(x2-x1)+((Rsqr (y0-y2))+(Rsqr (y2-y1))+2*(y0-y2)*(y2-y1)))`` with ``2*((x0-x2)*(x2-x1)+(y0-y2)*(y2-y1))``; [Replace ``-(Rsqr (x0-x2))-(Rsqr (x2-x1))-(Rsqr (y0-y2))-(Rsqr (y2-y1))+((Rsqr (x0-x2))+(Rsqr (y0-y2))+((Rsqr (x2-x1))+(Rsqr (y2-y1)))+2*(sqrt ((Rsqr (x0-x2))+(Rsqr (y0-y2))))*(sqrt ((Rsqr (x2-x1))+(Rsqr (y2-y1)))))`` with ``2*((sqrt ((Rsqr (x0-x2))+(Rsqr (y0-y2))))*(sqrt ((Rsqr (x2-x1))+(Rsqr (y2-y1)))))``; [Apply Rle_monotony; [Left; Cut ~(O=(2)); [Intros; Generalize (lt_INR_0 (2) (neq_O_lt (2) H)); Intro H0; Assumption | Discriminate] | Apply sqrt_cauchy] | Ring] | Ring] | SqRing] | SqRing] | Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr | Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr | Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr] | Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr | Apply ge0_plus_ge0_is_ge0; Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr]. +Qed. + +(******************************************************************) +(** Translation *) +(******************************************************************) + +Definition xt[x,tx:R] : R := ``x+tx``. +Definition yt[y,ty:R] : R := ``y+ty``. + +Lemma translation_0 : (x,y:R) ``(xt x 0)==x``/\``(yt y 0)==y``. +Intros x y; Split; [Unfold xt | Unfold yt]; Ring. +Qed. + +Lemma isometric_translation : (x1,x2,y1,y2,tx,ty:R) ``(Rsqr (x1-x2))+(Rsqr (y1-y2))==(Rsqr ((xt x1 tx)-(xt x2 tx)))+(Rsqr ((yt y1 ty)-(yt y2 ty)))``. +Intros; Unfold Rsqr xt yt; Ring. +Qed. + +(******************************************************************) +(** Rotation *) +(******************************************************************) + +Definition xr [x,y,theta:R] : R := ``x*(cos theta)+y*(sin theta)``. +Definition yr [x,y,theta:R] : R := ``-x*(sin theta)+y*(cos theta)``. + +Lemma rotation_0 : (x,y:R) ``(xr x y 0)==x`` /\ ``(yr x y 0)==y``. +Intros x y; Unfold xr yr; Split; Rewrite cos_0; Rewrite sin_0; Ring. +Qed. + +Lemma rotation_PI2 : (x,y:R) ``(xr x y PI/2)==y`` /\ ``(yr x y PI/2)==-x``. +Intros x y; Unfold xr yr; Split; Rewrite cos_PI2; Rewrite sin_PI2; Ring. +Qed. + +Lemma isometric_rotation_0 : (x1,y1,x2,y2,theta:R) ``(Rsqr (x1-x2))+(Rsqr (y1-y2)) == (Rsqr ((xr x1 y1 theta))-(xr x2 y2 theta)) + (Rsqr ((yr x1 y1 theta))-(yr x2 y2 theta))``. +Intros; Unfold xr yr; Replace ``x1*(cos theta)+y1*(sin theta)-(x2*(cos theta)+y2*(sin theta))`` with ``(cos theta)*(x1-x2)+(sin theta)*(y1-y2)``; [Replace ``-x1*(sin theta)+y1*(cos theta)-( -x2*(sin theta)+y2*(cos theta))`` with ``(cos theta)*(y1-y2)+(sin theta)*(x2-x1)``; [Repeat Rewrite Rsqr_plus; Repeat Rewrite Rsqr_times; Repeat Rewrite cos2; Ring; Replace ``x2-x1`` with ``-(x1-x2)``; [Rewrite <- Rsqr_neg; Ring | Ring] |Ring] | Ring]. +Qed. + +Lemma isometric_rotation : (x1,y1,x2,y2,theta:R) ``(dist_euc x1 y1 x2 y2) == (dist_euc (xr x1 y1 theta) (yr x1 y1 theta) (xr x2 y2 theta) (yr x2 y2 theta))``. +Unfold dist_euc; Intros; Apply Rsqr_inj; [Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0 | Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0 | Repeat Rewrite Rsqr_sqrt; [ Apply isometric_rotation_0 | Apply ge0_plus_ge0_is_ge0 | Apply ge0_plus_ge0_is_ge0]]; Apply pos_Rsqr. +Qed. + +(******************************************************************) +(** Similarity *) +(******************************************************************) + +Lemma isometric_rot_trans : (x1,y1,x2,y2,tx,ty,theta:R) ``(Rsqr (x1-x2))+(Rsqr (y1-y2)) == (Rsqr ((xr (xt x1 tx) (yt y1 ty) theta)-(xr (xt x2 tx) (yt y2 ty) theta))) + (Rsqr ((yr (xt x1 tx) (yt y1 ty) theta)-(yr (xt x2 tx) (yt y2 ty) theta)))``. +Intros; Rewrite <- isometric_rotation_0; Apply isometric_translation. +Qed. + +Lemma isometric_trans_rot : (x1,y1,x2,y2,tx,ty,theta:R) ``(Rsqr (x1-x2))+(Rsqr (y1-y2)) == (Rsqr ((xt (xr x1 y1 theta) tx)-(xt (xr x2 y2 theta) tx))) + (Rsqr ((yt (yr x1 y1 theta) ty)-(yt (yr x2 y2 theta) ty)))``. +Intros; Rewrite <- isometric_translation; Apply isometric_rotation_0. +Qed. diff --git a/theories7/Reals/RiemannInt.v b/theories7/Reals/RiemannInt.v new file mode 100644 index 00000000..c9301b56 --- /dev/null +++ b/theories7/Reals/RiemannInt.v @@ -0,0 +1,1699 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: RiemannInt.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +Require Rfunctions. +Require SeqSeries. +Require Ranalysis. +Require Rbase. +Require RiemannInt_SF. +Require Classical_Prop. +Require Classical_Pred_Type. +Require Max. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Implicit Arguments On. + +(********************************************) +(* Riemann's Integral *) +(********************************************) + +Definition Riemann_integrable [f:R->R;a,b:R] : Type := (eps:posreal) (SigT ? [phi:(StepFun a b)](SigT ? [psi:(StepFun a b)]((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(phi t)))<=(psi t)``)/\``(Rabsolu (RiemannInt_SF psi))<eps``)). + +Definition phi_sequence [un:nat->posreal;f:R->R;a,b:R;pr:(Riemann_integrable f a b)] := [n:nat](projT1 ? ? (pr (un n))). + +Lemma phi_sequence_prop : (un:nat->posreal;f:R->R;a,b:R;pr:(Riemann_integrable f a b);N:nat) (SigT ? [psi:(StepFun a b)]((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-[(phi_sequence un pr N t)]))<=(psi t)``)/\``(Rabsolu (RiemannInt_SF psi))<(un N)``). +Intros; Apply (projT2 ? ? (pr (un N))). +Qed. + +Lemma RiemannInt_P1 : (f:R->R;a,b:R) (Riemann_integrable f a b) -> (Riemann_integrable f b a). +Unfold Riemann_integrable; Intros; Elim (X eps); Clear X; Intros; Elim p; Clear p; Intros; Apply Specif.existT with (mkStepFun (StepFun_P6 (pre x))); Apply Specif.existT with (mkStepFun (StepFun_P6 (pre x0))); Elim p; Clear p; Intros; Split. +Intros; Apply (H t); Elim H1; Clear H1; Intros; Split; [Apply Rle_trans with (Rmin b a); Try Assumption; Right; Unfold Rmin | Apply Rle_trans with (Rmax b a); Try Assumption; Right; Unfold Rmax]; (Case (total_order_Rle a b); Case (total_order_Rle b a); Intros; Try Reflexivity Orelse Apply Rle_antisym; [Assumption | Assumption | Auto with real | Auto with real]). +Generalize H0; Unfold RiemannInt_SF; Case (total_order_Rle a b); Case (total_order_Rle b a); Intros; (Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre x0)))) (subdivision (mkStepFun (StepFun_P6 (pre x0))))) with (Int_SF (subdivision_val x0) (subdivision x0)); [Idtac | Apply StepFun_P17 with (fe x0) a b; [Apply StepFun_P1 | Apply StepFun_P2; Apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre x0))))]]). +Apply H1. +Rewrite Rabsolu_Ropp; Apply H1. +Rewrite Rabsolu_Ropp in H1; Apply H1. +Apply H1. +Qed. + +Lemma RiemannInt_P2 : (f:R->R;a,b:R;un:nat->posreal;vn,wn:nat->(StepFun a b)) (Un_cv un R0) -> ``a<=b`` -> ((n:nat)((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(vn n t)))<=(wn n t)``)/\``(Rabsolu (RiemannInt_SF (wn n)))<(un n)``) -> (sigTT ? [l:R](Un_cv [N:nat](RiemannInt_SF (vn N)) l)). +Intros; Apply R_complete; Unfold Un_cv in H; Unfold Cauchy_crit; Intros; Assert H3 : ``0<eps/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H ? H3); Intros N0 H4; Exists N0; Intros; Unfold R_dist; Unfold R_dist in H4; Elim (H1 n); Elim (H1 m); Intros; Replace ``(RiemannInt_SF (vn n))-(RiemannInt_SF (vn m))`` with ``(RiemannInt_SF (vn n))+(-1)*(RiemannInt_SF (vn m))``; [Idtac | Ring]; Rewrite <- StepFun_P30; Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (vn n) (vn m)))))). +Apply StepFun_P34; Assumption. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 R1 (wn n) (wn m)))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Apply Rle_trans with ``(Rabsolu ((vn n x)-(f x)))+(Rabsolu ((f x)-(vn m x)))``. +Replace ``(vn n x)+-1*(vn m x)`` with ``((vn n x)-(f x))+((f x)-(vn m x))``; [Apply Rabsolu_triang | Ring]. +Assert H12 : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Assert H13 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Rewrite <- H12 in H11; Pattern 2 b in H11; Rewrite <- H13 in H11; Rewrite Rmult_1l; Apply Rplus_le. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H9. +Elim H11; Intros; Split; Left; Assumption. +Apply H7. +Elim H11; Intros; Split; Left; Assumption. +Rewrite StepFun_P30; Rewrite Rmult_1l; Apply Rlt_trans with ``(un n)+(un m)``. +Apply Rle_lt_trans with ``(Rabsolu (RiemannInt_SF (wn n)))+(Rabsolu (RiemannInt_SF (wn m)))``. +Apply Rplus_le; Apply Rle_Rabsolu. +Apply Rplus_lt; Assumption. +Apply Rle_lt_trans with ``(Rabsolu (un n))+(Rabsolu (un m))``. +Apply Rplus_le; Apply Rle_Rabsolu. +Replace (pos (un n)) with ``(un n)-0``; [Idtac | Ring]; Replace (pos (un m)) with ``(un m)-0``; [Idtac | Ring]; Rewrite (double_var eps); Apply Rplus_lt; Apply H4; Assumption. +Qed. + +Lemma RiemannInt_P3 : (f:R->R;a,b:R;un:nat->posreal;vn,wn:nat->(StepFun a b)) (Un_cv un R0) -> ((n:nat)((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(vn n t)))<=(wn n t)``)/\``(Rabsolu (RiemannInt_SF (wn n)))<(un n)``)->(sigTT R ([l:R](Un_cv ([N:nat](RiemannInt_SF (vn N))) l))). +Intros; Case (total_order_Rle a b); Intro. +Apply RiemannInt_P2 with f un wn; Assumption. +Assert H1 : ``b<=a``; Auto with real. +Pose vn' := [n:nat](mkStepFun (StepFun_P6 (pre (vn n)))); Pose wn' := [n:nat](mkStepFun (StepFun_P6 (pre (wn n)))); Assert H2 : (n:nat)((t:R)``(Rmin b a)<=t<=(Rmax b a)``->``(Rabsolu ((f t)-(vn' n t)))<=(wn' n t)``)/\``(Rabsolu (RiemannInt_SF (wn' n)))<(un n)``. +Intro; Elim (H0 n0); Intros; Split. +Intros; Apply (H2 t); Elim H4; Clear H4; Intros; Split; [Apply Rle_trans with (Rmin b a); Try Assumption; Right; Unfold Rmin | Apply Rle_trans with (Rmax b a); Try Assumption; Right; Unfold Rmax]; (Case (total_order_Rle a b); Case (total_order_Rle b a); Intros; Try Reflexivity Orelse Apply Rle_antisym; [Assumption | Assumption | Auto with real | Auto with real]). +Generalize H3; Unfold RiemannInt_SF; Case (total_order_Rle a b); Case (total_order_Rle b a); Unfold wn'; Intros; (Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (wn n0))))) (subdivision (mkStepFun (StepFun_P6 (pre (wn n0)))))) with (Int_SF (subdivision_val (wn n0)) (subdivision (wn n0))); [Idtac | Apply StepFun_P17 with (fe (wn n0)) a b; [Apply StepFun_P1 | Apply StepFun_P2; Apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (wn n0)))))]]). +Apply H4. +Rewrite Rabsolu_Ropp; Apply H4. +Rewrite Rabsolu_Ropp in H4; Apply H4. +Apply H4. +Assert H3 := (RiemannInt_P2 H H1 H2); Elim H3; Intros; Apply existTT with ``-x``; Unfold Un_cv; Unfold Un_cv in p; Intros; Elim (p ? H4); Intros; Exists x0; Intros; Generalize (H5 ? H6); Unfold R_dist RiemannInt_SF; Case (total_order_Rle b a); Case (total_order_Rle a b); Intros. +Elim n; Assumption. +Unfold vn' in H7; Replace (Int_SF (subdivision_val (vn n0)) (subdivision (vn n0))) with (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (vn n0))))) (subdivision (mkStepFun (StepFun_P6 (pre (vn n0)))))); [Unfold Rminus; Rewrite Ropp_Ropp; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Apply H7 | Symmetry; Apply StepFun_P17 with (fe (vn n0)) a b; [Apply StepFun_P1 | Apply StepFun_P2; Apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (vn n0)))))]]. +Elim n1; Assumption. +Elim n2; Assumption. +Qed. + +Lemma RiemannInt_exists : (f:R->R;a,b:R;pr:(Riemann_integrable f a b);un:nat->posreal) (Un_cv un R0) -> (sigTT ? [l:R](Un_cv [N:nat](RiemannInt_SF (phi_sequence un pr N)) l)). +Intros f; Intros; Apply RiemannInt_P3 with f un [n:nat](projT1 ? ? (phi_sequence_prop un pr n)); [Apply H | Intro; Apply (projT2 ? ? (phi_sequence_prop un pr n))]. +Qed. + +Lemma RiemannInt_P4 : (f:R->R;a,b,l:R;pr1,pr2:(Riemann_integrable f a b);un,vn:nat->posreal) (Un_cv un R0) -> (Un_cv vn R0) -> (Un_cv [N:nat](RiemannInt_SF (phi_sequence un pr1 N)) l) -> (Un_cv [N:nat](RiemannInt_SF (phi_sequence vn pr2 N)) l). +Unfold Un_cv; Unfold R_dist; Intros f; Intros; Assert H3 : ``0<eps/3``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H ? H3); Clear H; Intros N0 H; Elim (H0 ? H3); Clear H0; Intros N1 H0; Elim (H1 ? H3); Clear H1; Intros N2 H1; Pose N := (max (max N0 N1) N2); Exists N; Intros; Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence vn pr2 n)])-(RiemannInt_SF [(phi_sequence un pr1 n)])))+(Rabsolu ((RiemannInt_SF [(phi_sequence un pr1 n)])-l))``. +Replace ``(RiemannInt_SF [(phi_sequence vn pr2 n)])-l`` with ``((RiemannInt_SF [(phi_sequence vn pr2 n)])-(RiemannInt_SF [(phi_sequence un pr1 n)]))+((RiemannInt_SF [(phi_sequence un pr1 n)])-l)``; [Apply Rabsolu_triang | Ring]. +Replace ``eps`` with ``2*eps/3+eps/3``. +Apply Rplus_lt. +Elim (phi_sequence_prop vn pr2 n); Intros psi_vn H5; Elim (phi_sequence_prop un pr1 n); Intros psi_un H6; Replace ``(RiemannInt_SF [(phi_sequence vn pr2 n)])-(RiemannInt_SF [(phi_sequence un pr1 n)])`` with ``(RiemannInt_SF [(phi_sequence vn pr2 n)])+(-1)*(RiemannInt_SF [(phi_sequence un pr1 n)])``; [Idtac | Ring]; Rewrite <- StepFun_P30. +Case (total_order_Rle a b); Intro. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (phi_sequence vn pr2 n) (phi_sequence un pr1 n)))))). +Apply StepFun_P34; Assumption. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 R1 psi_un psi_vn))). +Apply StepFun_P37; Try Assumption; Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu ([(phi_sequence vn pr2 n x)]-(f x)))+(Rabsolu ((f x)-[(phi_sequence un pr1 n x)]))``. +Replace ``[(phi_sequence vn pr2 n x)]+-1*[(phi_sequence un pr1 n x)]`` with ``([(phi_sequence vn pr2 n x)]-(f x))+((f x)-[(phi_sequence un pr1 n x)])``; [Apply Rabsolu_triang | Ring]. +Assert H10 : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Assert H11 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Rewrite (Rplus_sym (psi_un x)); Apply Rplus_le. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Elim H5; Intros; Apply H8. +Rewrite H10; Rewrite H11; Elim H7; Intros; Split; Left; Assumption. +Elim H6; Intros; Apply H8. +Rewrite H10; Rewrite H11; Elim H7; Intros; Split; Left; Assumption. +Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Apply Rplus_lt. +Apply Rlt_trans with (pos (un n)). +Elim H6; Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi_un)). +Apply Rle_Rabsolu. +Assumption. +Replace (pos (un n)) with (Rabsolu ``(un n)-0``); [Apply H; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_trans with (max N0 N1); Apply le_max_l | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (un n))]. +Apply Rlt_trans with (pos (vn n)). +Elim H5; Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi_vn)). +Apply Rle_Rabsolu; Assumption. +Assumption. +Replace (pos (vn n)) with (Rabsolu ``(vn n)-0``); [Apply H0; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_trans with (max N0 N1); [Apply le_max_r | Apply le_max_l] | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (vn n))]. +Rewrite StepFun_P39; Rewrite Rabsolu_Ropp; Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 ``-1`` (phi_sequence vn pr2 n) (phi_sequence un pr1 n))))))))). +Apply StepFun_P34; Try Auto with real. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 R1 psi_vn psi_un)))))). +Apply StepFun_P37. +Auto with real. +Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu ([(phi_sequence vn pr2 n x)]-(f x)))+(Rabsolu ((f x)-[(phi_sequence un pr1 n x)]))``. +Replace ``[(phi_sequence vn pr2 n x)]+-1*[(phi_sequence un pr1 n x)]`` with ``([(phi_sequence vn pr2 n x)]-(f x))+((f x)-[(phi_sequence un pr1 n x)])``; [Apply Rabsolu_triang | Ring]. +Assert H10 : (Rmin a b)==b. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Elim n0; Assumption | Reflexivity]. +Assert H11 : (Rmax a b)==a. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Elim n0; Assumption | Reflexivity]. +Apply Rplus_le. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Elim H5; Intros; Apply H8. +Rewrite H10; Rewrite H11; Elim H7; Intros; Split; Left; Assumption. +Elim H6; Intros; Apply H8. +Rewrite H10; Rewrite H11; Elim H7; Intros; Split; Left; Assumption. +Rewrite <- (Ropp_Ropp (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 R1 psi_vn psi_un))))))); Rewrite <- StepFun_P39; Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Rewrite Ropp_distr1; Apply Rplus_lt. +Apply Rlt_trans with (pos (vn n)). +Elim H5; Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi_vn)). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Assumption. +Replace (pos (vn n)) with (Rabsolu ``(vn n)-0``); [Apply H0; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_trans with (max N0 N1); [Apply le_max_r | Apply le_max_l] | Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (vn n))]. +Apply Rlt_trans with (pos (un n)). +Elim H6; Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi_un)). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu; Assumption. +Assumption. +Replace (pos (un n)) with (Rabsolu ``(un n)-0``); [Apply H; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_trans with (max N0 N1); Apply le_max_l | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (un n))]. +Apply H1; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_r. +Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Qed. + +Lemma RinvN_pos : (n:nat) ``0</((INR n)+1)``. +Intro; Apply Rlt_Rinv; Apply ge0_plus_gt0_is_gt0; [Apply pos_INR | Apply Rlt_R0_R1]. +Qed. + +Definition RinvN : nat->posreal := [N:nat](mkposreal ? (RinvN_pos N)). + +Lemma RinvN_cv : (Un_cv RinvN R0). +Unfold Un_cv; Intros; Assert H0 := (archimed ``/eps``); Elim H0; Clear H0; Intros; Assert H2 : `0<=(up (Rinv eps))`. +Apply le_IZR; Left; Apply Rlt_trans with ``/eps``; [Apply Rlt_Rinv; Assumption | Assumption]. +Elim (IZN ? H2); Intros; Exists x; Intros; Unfold R_dist; Simpl; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Assert H5 : ``0<(INR n)+1``. +Apply ge0_plus_gt0_is_gt0; [Apply pos_INR | Apply Rlt_R0_R1]. +Rewrite Rabsolu_right; [Idtac | Left; Change ``0</((INR n)+1)``; Apply Rlt_Rinv; Assumption]; Apply Rle_lt_trans with ``/((INR x)+1)``. +Apply Rle_Rinv. +Apply ge0_plus_gt0_is_gt0; [Apply pos_INR | Apply Rlt_R0_R1]. +Assumption. +Do 2 Rewrite <- (Rplus_sym R1); Apply Rle_compatibility; Apply le_INR; Apply H4. +Rewrite <- (Rinv_Rinv eps). +Apply Rinv_lt. +Apply Rmult_lt_pos. +Apply Rlt_Rinv; Assumption. +Apply ge0_plus_gt0_is_gt0; [Apply pos_INR | Apply Rlt_R0_R1]. +Apply Rlt_trans with (INR x); [Rewrite INR_IZR_INZ; Rewrite <- H3; Apply H0 | Pattern 1 (INR x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1]. +Red; Intro; Rewrite H6 in H; Elim (Rlt_antirefl ? H). +Qed. + +(**********) +Definition RiemannInt [f:R->R;a,b:R;pr:(Riemann_integrable f a b)] : R := Cases +(RiemannInt_exists pr 5!RinvN RinvN_cv) of (existTT a' b') => a' end. + +Lemma RiemannInt_P5 : (f:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable f a b)) (RiemannInt pr1)==(RiemannInt pr2). +Intros; Unfold RiemannInt; Case (RiemannInt_exists pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists pr2 5!RinvN RinvN_cv); Intros; EApply UL_sequence; [Apply u0 | Apply RiemannInt_P4 with pr2 RinvN; Apply RinvN_cv Orelse Assumption]. +Qed. + +(**************************************) +(* C°([a,b]) is included in L1([a,b]) *) +(**************************************) + +Lemma maxN : (a,b:R;del:posreal) ``a<b`` -> (sigTT ? [n:nat]``a+(INR n)*del<b``/\``b<=a+(INR (S n))*del``). +Intros; Pose I := [n:nat]``a+(INR n)*del < b``; Assert H0 : (EX n:nat | (I n)). +Exists O; Unfold I; Rewrite Rmult_Ol; Rewrite Rplus_Or; Assumption. +Cut (Nbound I). +Intro; Assert H2 := (Nzorn H0 H1); Elim H2; Intros; Exists x; Elim p; Intros; Split. +Apply H3. +Case (total_order_T ``a+(INR (S x))*del`` b); Intro. +Elim s; Intro. +Assert H5 := (H4 (S x) a0); Elim (le_Sn_n ? H5). +Right; Symmetry; Assumption. +Left; Apply r. +Assert H1 : ``0<=(b-a)/del``. +Unfold Rdiv; Apply Rmult_le_pos; [Apply Rle_sym2; Apply Rge_minus; Apply Rle_sym1; Left; Apply H | Left; Apply Rlt_Rinv; Apply (cond_pos del)]. +Elim (archimed ``(b-a)/del``); Intros; Assert H4 : `0<=(up (Rdiv (Rminus b a) del))`. +Apply le_IZR; Simpl; Left; Apply Rle_lt_trans with ``(b-a)/del``; Assumption. +Assert H5 := (IZN ? H4); Elim H5; Clear H5; Intros N H5; Unfold Nbound; Exists N; Intros; Unfold I in H6; Apply INR_le; Rewrite H5 in H2; Rewrite <- INR_IZR_INZ in H2; Left; Apply Rle_lt_trans with ``(b-a)/del``; Try Assumption; Apply Rle_monotony_contra with (pos del); [Apply (cond_pos del) | Unfold Rdiv; Rewrite <- (Rmult_sym ``/del``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite Rmult_sym; Apply Rle_anti_compatibility with a; Replace ``a+(b-a)`` with b; [Left; Assumption | Ring] | Assert H7 := (cond_pos del); Red; Intro; Rewrite H8 in H7; Elim (Rlt_antirefl ? H7)]]. +Qed. + +Fixpoint SubEquiN [N:nat] : R->R->posreal->Rlist := +[x:R][y:R][del:posreal] Cases N of +| O => (cons y nil) +| (S p) => (cons x (SubEquiN p ``x+del`` y del)) +end. + +Definition max_N [a,b:R;del:posreal;h:``a<b``] : nat := Cases (maxN del h) of (existTT N H0) => N end. + +Definition SubEqui [a,b:R;del:posreal;h:``a<b``] :Rlist := (SubEquiN (S (max_N del h)) a b del). + +Lemma Heine_cor1 : (f:R->R;a,b:R) ``a<b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> (eps:posreal) (sigTT ? [delta:posreal]``delta<=b-a``/\(x,y:R)``a<=x<=b``->``a<=y<=b``->``(Rabsolu (x-y)) < delta``->``(Rabsolu ((f x)-(f y))) < eps``). +Intro f; Intros; Pose E := [l:R]``0<l<=b-a``/\(x,y:R)``a <= x <= b``->``a <= y <= b``->``(Rabsolu (x-y)) < l``->``(Rabsolu ((f x)-(f y))) < eps``; Assert H1 : (bound E). +Unfold bound; Exists ``b-a``; Unfold is_upper_bound; Intros; Unfold E in H1; Elim H1; Clear H1; Intros H1 _; Elim H1; Intros; Assumption. +Assert H2 : (EXT x:R | (E x)). +Assert H2 := (Heine f [x:R]``a<=x<=b`` (compact_P3 a b) H0 eps); Elim H2; Intros; Exists (Rmin x ``b-a``); Unfold E; Split; [Split; [Unfold Rmin; Case (total_order_Rle x ``b-a``); Intro; [Apply (cond_pos x) | Apply Rlt_Rminus; Assumption] | Apply Rmin_r] | Intros; Apply H3; Try Assumption; Apply Rlt_le_trans with (Rmin x ``b-a``); [Assumption | Apply Rmin_l]]. +Assert H3 := (complet E H1 H2); Elim H3; Intros; Cut ``0<x<=b-a``. +Intro; Elim H4; Clear H4; Intros; Apply existTT with (mkposreal ? H4); Split. +Apply H5. +Unfold is_lub in p; Elim p; Intros; Unfold is_upper_bound in H6; Pose D := ``(Rabsolu (x0-y))``; Elim (classic (EXT y:R | ``D<y``/\(E y))); Intro. +Elim H11; Intros; Elim H12; Clear H12; Intros; Unfold E in H13; Elim H13; Intros; Apply H15; Assumption. +Assert H12 := (not_ex_all_not ? [y:R]``D < y``/\(E y) H11); Assert H13 : (is_upper_bound E D). +Unfold is_upper_bound; Intros; Assert H14 := (H12 x1); Elim (not_and_or ``D<x1`` (E x1) H14); Intro. +Case (total_order_Rle x1 D); Intro. +Assumption. +Elim H15; Auto with real. +Elim H15; Assumption. +Assert H14 := (H7 ? H13); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H14 H10)). +Unfold is_lub in p; Unfold is_upper_bound in p; Elim p; Clear p; Intros; Split. +Elim H2; Intros; Assert H7 := (H4 ? H6); Unfold E in H6; Elim H6; Clear H6; Intros H6 _; Elim H6; Intros; Apply Rlt_le_trans with x0; Assumption. +Apply H5; Intros; Unfold E in H6; Elim H6; Clear H6; Intros H6 _; Elim H6; Intros; Assumption. +Qed. + +Lemma Heine_cor2 : (f:(R->R); a,b:R) ((x:R)``a <= x <= b``->(continuity_pt f x))->(eps:posreal)(sigTT posreal [delta:posreal]((x,y:R)``a <= x <= b``->``a <= y <= b``->``(Rabsolu (x-y)) < delta``->``(Rabsolu ((f x)-(f y))) < eps``)). +Intro f; Intros; Case (total_order_T a b); Intro. +Elim s; Intro. +Assert H0 := (Heine_cor1 a0 H eps); Elim H0; Intros; Apply existTT with x; Elim p; Intros; Apply H2; Assumption. +Apply existTT with (mkposreal ? Rlt_R0_R1); Intros; Assert H3 : x==y; [Elim H0; Elim H1; Intros; Rewrite b0 in H3; Rewrite b0 in H5; Apply Rle_antisym; Apply Rle_trans with b; Assumption | Rewrite H3; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos eps)]. +Apply existTT with (mkposreal ? Rlt_R0_R1); Intros; Elim H0; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (Rle_trans ? ? ? H3 H4) r)). +Qed. + +Lemma SubEqui_P1 : (a,b:R;del:posreal;h:``a<b``) (pos_Rl (SubEqui del h) O)==a. +Intros; Unfold SubEqui; Case (maxN del h); Intros; Reflexivity. +Qed. + +Lemma SubEqui_P2 : (a,b:R;del:posreal;h:``a<b``) (pos_Rl (SubEqui del h) (pred (Rlength (SubEqui del h))))==b. +Intros; Unfold SubEqui; Case (maxN del h); Intros; Clear a0; Cut (x:nat)(a:R)(del:posreal)(pos_Rl (SubEquiN (S x) a b del) (pred (Rlength (SubEquiN (S x) a b del)))) == b; [Intro; Apply H | Induction x0; [Intros; Reflexivity | Intros; Change (pos_Rl (SubEquiN (S n) ``a0+del0`` b del0) (pred (Rlength (SubEquiN (S n) ``a0+del0`` b del0))))==b; Apply H]]. +Qed. + +Lemma SubEqui_P3 : (N:nat;a,b:R;del:posreal) (Rlength (SubEquiN N a b del))=(S N). +Induction N; Intros; [Reflexivity | Simpl; Rewrite H; Reflexivity]. +Qed. + +Lemma SubEqui_P4 : (N:nat;a,b:R;del:posreal;i:nat) (lt i (S N)) -> (pos_Rl (SubEquiN (S N) a b del) i)==``a+(INR i)*del``. +Induction N; [Intros; Inversion H; [Simpl; Ring | Elim (le_Sn_O ? H1)] | Intros; Induction i; [Simpl; Ring | Change (pos_Rl (SubEquiN (S n) ``a+del`` b del) i)==``a+(INR (S i))*del``; Rewrite H; [Rewrite S_INR; Ring | Apply lt_S_n; Apply H0]]]. +Qed. + +Lemma SubEqui_P5 : (a,b:R;del:posreal;h:``a<b``) (Rlength (SubEqui del h))=(S (S (max_N del h))). +Intros; Unfold SubEqui; Apply SubEqui_P3. +Qed. + +Lemma SubEqui_P6 : (a,b:R;del:posreal;h:``a<b``;i:nat) (lt i (S (max_N del h))) -> (pos_Rl (SubEqui del h) i)==``a+(INR i)*del``. +Intros; Unfold SubEqui; Apply SubEqui_P4; Assumption. +Qed. + +Lemma SubEqui_P7 : (a,b:R;del:posreal;h:``a<b``) (ordered_Rlist (SubEqui del h)). +Intros; Unfold ordered_Rlist; Intros; Rewrite SubEqui_P5 in H; Simpl in H; Inversion H. +Rewrite (SubEqui_P6 3!del 4!h 5!(max_N del h)). +Replace (S (max_N del h)) with (pred (Rlength (SubEqui del h))). +Rewrite SubEqui_P2; Unfold max_N; Case (maxN del h); Intros; Left; Elim a0; Intros; Assumption. +Rewrite SubEqui_P5; Reflexivity. +Apply lt_n_Sn. +Repeat Rewrite SubEqui_P6. +3:Assumption. +2:Apply le_lt_n_Sm; Assumption. +Apply Rle_compatibility; Rewrite S_INR; Rewrite Rmult_Rplus_distrl; Pattern 1 ``(INR i)*del``; Rewrite <- Rplus_Or; Apply Rle_compatibility; Rewrite Rmult_1l; Left; Apply (cond_pos del). +Qed. + +Lemma SubEqui_P8 : (a,b:R;del:posreal;h:``a<b``;i:nat) (lt i (Rlength (SubEqui del h))) -> ``a<=(pos_Rl (SubEqui del h) i)<=b``. +Intros; Split. +Pattern 1 a; Rewrite <- (SubEqui_P1 del h); Apply RList_P5. +Apply SubEqui_P7. +Elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); Intros; Apply H1; Exists i; Split; [Reflexivity | Assumption]. +Pattern 2 b; Rewrite <- (SubEqui_P2 del h); Apply RList_P7; [Apply SubEqui_P7 | Elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); Intros; Apply H1; Exists i; Split; [Reflexivity | Assumption]]. +Qed. + +Lemma SubEqui_P9 : (a,b:R;del:posreal;f:R->R;h:``a<b``) (sigTT ? [g:(StepFun a b)](g b)==(f b)/\(i:nat)(lt i (pred (Rlength (SubEqui del h))))->(constant_D_eq g (co_interval (pos_Rl (SubEqui del h) i) (pos_Rl (SubEqui del h) (S i))) (f (pos_Rl (SubEqui del h) i)))). +Intros; Apply StepFun_P38; [Apply SubEqui_P7 | Apply SubEqui_P1 | Apply SubEqui_P2]. +Qed. + +Lemma RiemannInt_P6 : (f:R->R;a,b:R) ``a<b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> (Riemann_integrable f a b). +Intros; Unfold Riemann_integrable; Intro; Assert H1 : ``0<eps/(2*(b-a))``. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Apply Rlt_Rminus; Assumption]]. +Assert H2 : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Left; Assumption]. +Assert H3 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Left; Assumption]. +Elim (Heine_cor2 H0 (mkposreal ? H1)); Intros del H4; Elim (SubEqui_P9 del f H); Intros phi [H5 H6]; Split with phi; Split with (mkStepFun (StepFun_P4 a b ``eps/(2*(b-a))``)); Split. +2:Rewrite StepFun_P18; Unfold Rdiv; Rewrite Rinv_Rmult. +2:Do 2 Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +2:Rewrite Rmult_1r; Rewrite Rabsolu_right. +2:Apply Rlt_monotony_contra with ``2``. +2:Sup0. +2:Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +2:Rewrite Rmult_1l; Pattern 1 (pos eps); Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Apply (cond_pos eps). +2:DiscrR. +2:Apply Rle_sym1; Left; Apply Rmult_lt_pos. +2:Apply (cond_pos eps). +2:Apply Rlt_Rinv; Sup0. +2:Apply Rminus_eq_contra; Red; Intro; Clear H6; Rewrite H7 in H; Elim (Rlt_antirefl ? H). +2:DiscrR. +2:Apply Rminus_eq_contra; Red; Intro; Clear H6; Rewrite H7 in H; Elim (Rlt_antirefl ? H). +Intros; Rewrite H2 in H7; Rewrite H3 in H7; Simpl; Unfold fct_cte; Cut (t:R)``a<=t<=b``->t==b\/(EX i:nat | (lt i (pred (Rlength (SubEqui del H))))/\(co_interval (pos_Rl (SubEqui del H) i) (pos_Rl (SubEqui del H) (S i)) t)). +Intro; Elim (H8 ? H7); Intro. +Rewrite H9; Rewrite H5; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. +Elim H9; Clear H9; Intros I [H9 H10]; Assert H11 := (H6 I H9 t H10); Rewrite H11; Left; Apply H4. +Assumption. +Apply SubEqui_P8; Apply lt_trans with (pred (Rlength (SubEqui del H))). +Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H9; Elim (lt_n_O ? H9). +Unfold co_interval in H10; Elim H10; Clear H10; Intros; Rewrite Rabsolu_right. +Rewrite SubEqui_P5 in H9; Simpl in H9; Inversion H9. +Apply Rlt_anti_compatibility with (pos_Rl (SubEqui del H) (max_N del H)). +Replace ``(pos_Rl (SubEqui del H) (max_N del H))+(t-(pos_Rl (SubEqui del H) (max_N del H)))`` with t; [Idtac | Ring]; Apply Rlt_le_trans with b. +Rewrite H14 in H12; Assert H13 : (S (max_N del H))=(pred (Rlength (SubEqui del H))). +Rewrite SubEqui_P5; Reflexivity. +Rewrite H13 in H12; Rewrite SubEqui_P2 in H12; Apply H12. +Rewrite SubEqui_P6. +2:Apply lt_n_Sn. +Unfold max_N; Case (maxN del H); Intros; Elim a0; Clear a0; Intros _ H13; Replace ``a+(INR x)*del+del`` with ``a+(INR (S x))*del``; [Assumption | Rewrite S_INR; Ring]. +Apply Rlt_anti_compatibility with (pos_Rl (SubEqui del H) I); Replace ``(pos_Rl (SubEqui del H) I)+(t-(pos_Rl (SubEqui del H) I))`` with t; [Idtac | Ring]; Replace ``(pos_Rl (SubEqui del H) I)+del`` with (pos_Rl (SubEqui del H) (S I)). +Assumption. +Repeat Rewrite SubEqui_P6. +Rewrite S_INR; Ring. +Assumption. +Apply le_lt_n_Sm; Assumption. +Apply Rge_minus; Apply Rle_sym1; Assumption. +Intros; Clear H0 H1 H4 phi H5 H6 t H7; Case (Req_EM t0 b); Intro. +Left; Assumption. +Right; Pose I := [j:nat]``a+(INR j)*del<=t0``; Assert H1 : (EX n:nat | (I n)). +Exists O; Unfold I; Rewrite Rmult_Ol; Rewrite Rplus_Or; Elim H8; Intros; Assumption. +Assert H4 : (Nbound I). +Unfold Nbound; Exists (S (max_N del H)); Intros; Unfold max_N; Case (maxN del H); Intros; Elim a0; Clear a0; Intros _ H5; Apply INR_le; Apply Rle_monotony_contra with (pos del). +Apply (cond_pos del). +Apply Rle_anti_compatibility with a; Do 2 Rewrite (Rmult_sym del); Apply Rle_trans with t0; Unfold I in H4; Try Assumption; Apply Rle_trans with b; Try Assumption; Elim H8; Intros; Assumption. +Elim (Nzorn H1 H4); Intros N [H5 H6]; Assert H7 : (lt N (S (max_N del H))). +Unfold max_N; Case (maxN del H); Intros; Apply INR_lt; Apply Rlt_monotony_contra with (pos del). +Apply (cond_pos del). +Apply Rlt_anti_compatibility with a; Do 2 Rewrite (Rmult_sym del); Apply Rle_lt_trans with t0; Unfold I in H5; Try Assumption; Elim a0; Intros; Apply Rlt_le_trans with b; Try Assumption; Elim H8; Intros. +Elim H11; Intro. +Assumption. +Elim H0; Assumption. +Exists N; Split. +Rewrite SubEqui_P5; Simpl; Assumption. +Unfold co_interval; Split. +Rewrite SubEqui_P6. +Apply H5. +Assumption. +Inversion H7. +Replace (S (max_N del H)) with (pred (Rlength (SubEqui del H))). +Rewrite (SubEqui_P2 del H); Elim H8; Intros. +Elim H11; Intro. +Assumption. +Elim H0; Assumption. +Rewrite SubEqui_P5; Reflexivity. +Rewrite SubEqui_P6. +Case (total_order_Rle ``a+(INR (S N))*del`` t0); Intro. +Assert H11 := (H6 (S N) r); Elim (le_Sn_n ? H11). +Auto with real. +Apply le_lt_n_Sm; Assumption. +Qed. + +Lemma RiemannInt_P7 : (f:R->R;a:R) (Riemann_integrable f a a). +Unfold Riemann_integrable; Intro f; Intros; Split with (mkStepFun (StepFun_P4 a a (f a))); Split with (mkStepFun (StepFun_P4 a a R0)); Split. +Intros; Simpl; Unfold fct_cte; Replace t with a. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Right; Reflexivity. +Generalize H; Unfold Rmin Rmax; Case (total_order_Rle a a); Intros; Elim H0; Intros; Apply Rle_antisym; Assumption. +Rewrite StepFun_P18; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Apply (cond_pos eps). +Qed. + +Lemma continuity_implies_RiemannInt : (f:R->R;a,b:R) ``a<=b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> (Riemann_integrable f a b). +Intros; Case (total_order_T a b); Intro; [Elim s; Intro; [Apply RiemannInt_P6; Assumption | Rewrite b0; Apply RiemannInt_P7] | Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r))]. +Qed. + +Lemma RiemannInt_P8 : (f:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable f b a)) ``(RiemannInt pr1)==-(RiemannInt pr2)``. +Intro f; Intros; EApply UL_sequence. +Unfold RiemannInt; Case (RiemannInt_exists pr1 5!RinvN RinvN_cv); Intros; Apply u. +Unfold RiemannInt; Case (RiemannInt_exists pr2 5!RinvN RinvN_cv); Intros; Cut (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr1 n)] t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). +Cut (EXT psi2:nat->(StepFun b a) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr2 n)] t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). +Intros; Elim H; Clear H; Intros psi2 H; Elim H0; Clear H0; Intros psi1 H0; Assert H1 := RinvN_cv; Unfold Un_cv; Intros; Assert H3 : ``0<eps/3``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Unfold Un_cv in H1; Elim (H1 ? H3); Clear H1; Intros N0 H1; Unfold R_dist in H1; Simpl in H1; Assert H4 : (n:nat)(ge n N0)->``(RinvN n)<eps/3``. +Intros; Assert H5 := (H1 ? H4); Replace (pos (RinvN n)) with ``(Rabsolu (/((INR n)+1)-0))``; [Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Left; Apply (cond_pos (RinvN n))]. +Clear H1; Unfold Un_cv in u; Elim (u ? H3); Clear u; Intros N1 H1; Exists (max N0 N1); Intros; Unfold R_dist; Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)])))+(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x))``. +Rewrite <- (Rabsolu_Ropp ``(RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x``); Replace ``(RiemannInt_SF [(phi_sequence RinvN pr1 n)])- -x`` with ``((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))+ -((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x)``; [Apply Rabsolu_triang | Ring]. +Replace eps with ``2*eps/3+eps/3``. +Apply Rplus_lt. +Rewrite (StepFun_P39 (phi_sequence RinvN pr2 n)); Replace ``(RiemannInt_SF [(phi_sequence RinvN pr1 n)])+ -(RiemannInt_SF (mkStepFun (StepFun_P6 (pre [(phi_sequence RinvN pr2 n)]))))`` with ``(RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(-1)*(RiemannInt_SF (mkStepFun (StepFun_P6 (pre [(phi_sequence RinvN pr2 n)]))))``; [Idtac | Ring]; Rewrite <- StepFun_P30. +Case (total_order_Rle a b); Intro. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (phi_sequence RinvN pr1 n) (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n))))))))). +Apply StepFun_P34; Assumption. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 ``1`` (psi1 n) (mkStepFun (StepFun_P6 (pre (psi2 n))))))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu (([(phi_sequence RinvN pr1 n)] x0)-(f x0)))+(Rabsolu ((f x0)-([(phi_sequence RinvN pr2 n)] x0)))``. +Replace ``([(phi_sequence RinvN pr1 n)] x0)+ -1*([(phi_sequence RinvN pr2 n)] x0)`` with ``(([(phi_sequence RinvN pr1 n)] x0)-(f x0))+((f x0)-([(phi_sequence RinvN pr2 n)] x0))``; [Apply Rabsolu_triang | Ring]. +Assert H7 : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Assert H8 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Apply Rplus_le. +Elim (H0 n); Intros; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H9; Rewrite H7; Rewrite H8. +Elim H6; Intros; Split; Left; Assumption. +Elim (H n); Intros; Apply H9; Rewrite H7; Rewrite H8. +Elim H6; Intros; Split; Left; Assumption. +Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Apply Rplus_lt. +Elim (H0 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))); [Apply Rle_Rabsolu | Apply Rlt_trans with (pos (RinvN n)); [Assumption | Apply H4; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Assumption]]]. +Elim (H n); Intros; Rewrite <- (Ropp_Ropp (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (psi2 n)))))); Rewrite <- StepFun_P39; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))); [Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu | Apply Rlt_trans with (pos (RinvN n)); [Assumption | Apply H4; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Assumption]]]. +Assert Hyp : ``b<=a``. +Auto with real. +Rewrite StepFun_P39; Rewrite Rabsolu_Ropp; Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P6 (StepFun_P28 ``-1`` (phi_sequence RinvN pr1 n) (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n)))))))))). +Apply StepFun_P34; Assumption. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 ``1`` (mkStepFun (StepFun_P6 (pre (psi1 n)))) (psi2 n)))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu (([(phi_sequence RinvN pr1 n)] x0)-(f x0)))+(Rabsolu ((f x0)-([(phi_sequence RinvN pr2 n)] x0)))``. +Replace ``([(phi_sequence RinvN pr1 n)] x0)+ -1*([(phi_sequence RinvN pr2 n)] x0)`` with ``(([(phi_sequence RinvN pr1 n)] x0)-(f x0))+((f x0)-([(phi_sequence RinvN pr2 n)] x0))``; [Apply Rabsolu_triang | Ring]. +Assert H7 : (Rmin a b)==b. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Elim n0; Assumption | Reflexivity]. +Assert H8 : (Rmax a b)==a. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Elim n0; Assumption | Reflexivity]. +Apply Rplus_le. +Elim (H0 n); Intros; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H9; Rewrite H7; Rewrite H8. +Elim H6; Intros; Split; Left; Assumption. +Elim (H n); Intros; Apply H9; Rewrite H7; Rewrite H8; Elim H6; Intros; Split; Left; Assumption. +Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Apply Rplus_lt. +Elim (H0 n); Intros; Rewrite <- (Ropp_Ropp (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (psi1 n)))))); Rewrite <- StepFun_P39; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))); [Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu | Apply Rlt_trans with (pos (RinvN n)); [Assumption | Apply H4; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Assumption]]]. +Elim (H n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))); [Apply Rle_Rabsolu | Apply Rlt_trans with (pos (RinvN n)); [Assumption | Apply H4; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Assumption]]]. +Unfold R_dist in H1; Apply H1; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_r | Assumption]. +Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Rewrite Rmin_sym; Rewrite RmaxSym; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n)). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)). +Qed. + +Lemma RiemannInt_P9 : (f:R->R;a:R;pr:(Riemann_integrable f a a)) ``(RiemannInt pr)==0``. +Intros; Assert H := (RiemannInt_P8 pr pr); Apply r_Rmult_mult with ``2``; [Rewrite Rmult_Or; Rewrite double; Pattern 2 (RiemannInt pr); Rewrite H; Apply Rplus_Ropp_r | DiscrR]. +Qed. + +Lemma Req_EM_T :(r1,r2:R) (sumboolT (r1==r2) ``r1<>r2``). +Intros; Elim (total_order_T r1 r2);Intros; [Elim a;Intro; [Right; Red; Intro; Rewrite H in a0; Elim (Rlt_antirefl ``r2`` a0) | Left;Assumption] | Right; Red; Intro; Rewrite H in b; Elim (Rlt_antirefl ``r2`` b)]. +Qed. + +(* L1([a,b]) is a vectorial space *) +Lemma RiemannInt_P10 : (f,g:R->R;a,b,l:R) (Riemann_integrable f a b) -> (Riemann_integrable g a b) -> (Riemann_integrable [x:R]``(f x)+l*(g x)`` a b). +Unfold Riemann_integrable; Intros f g; Intros; Case (Req_EM_T l R0); Intro. +Elim (X eps); Intros; Split with x; Elim p; Intros; Split with x0; Elim p0; Intros; Split; Try Assumption; Rewrite e; Intros; Rewrite Rmult_Ol; Rewrite Rplus_Or; Apply H; Assumption. +Assert H : ``0<eps/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Sup0]. +Assert H0 : ``0<eps/(2*(Rabsolu l))``. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Apply Rabsolu_pos_lt; Assumption]]. +Elim (X (mkposreal ? H)); Intros; Elim (X0 (mkposreal ? H0)); Intros; Split with (mkStepFun (StepFun_P28 l x x0)); Elim p0; Elim p; Intros; Split with (mkStepFun (StepFun_P28 (Rabsolu l) x1 x2)); Elim p1; Elim p2; Clear p1 p2 p0 p X X0; Intros; Split. +Intros; Simpl; Apply Rle_trans with ``(Rabsolu ((f t)-(x t)))+(Rabsolu (l*((g t)-(x0 t))))``. +Replace ``(f t)+l*(g t)-((x t)+l*(x0 t))`` with ``((f t)-(x t))+ l*((g t)-(x0 t))``; [Apply Rabsolu_triang | Ring]. +Apply Rplus_le; [Apply H3; Assumption | Rewrite Rabsolu_mult; Apply Rle_monotony; [Apply Rabsolu_pos | Apply H1; Assumption]]. +Rewrite StepFun_P30; Apply Rle_lt_trans with ``(Rabsolu (RiemannInt_SF x1))+(Rabsolu ((Rabsolu l)*(RiemannInt_SF x2)))``. +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Apply H4. +Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Apply Rlt_monotony_contra with ``/(Rabsolu l)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1l; Replace ``/(Rabsolu l)*eps/2`` with ``eps/(2*(Rabsolu l))``; [Apply H2 | Unfold Rdiv; Rewrite Rinv_Rmult; [Ring | DiscrR | Apply Rabsolu_no_R0; Assumption]] | Apply Rabsolu_no_R0; Assumption]. +Qed. + +Lemma RiemannInt_P11 : (f:R->R;a,b,l:R;un:nat->posreal;phi1,phi2,psi1,psi2:nat->(StepFun a b)) (Un_cv un R0) -> ((n:nat)((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(phi1 n t)))<=(psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n)))<(un n)``) -> ((n:nat)((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(phi2 n t)))<=(psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n)))<(un n)``) -> (Un_cv [N:nat](RiemannInt_SF (phi1 N)) l) -> (Un_cv [N:nat](RiemannInt_SF (phi2 N)) l). +Unfold Un_cv; Intro f; Intros; Intros. +Case (total_order_Rle a b); Intro Hyp. +Assert H4 : ``0<eps/3``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H ? H4); Clear H; Intros N0 H. +Elim (H2 ? H4); Clear H2; Intros N1 H2. +Pose N := (max N0 N1); Exists N; Intros; Unfold R_dist. +Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n))))+(Rabsolu ((RiemannInt_SF (phi1 n))-l))``. +Replace ``(RiemannInt_SF (phi2 n))-l`` with ``((RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n)))+((RiemannInt_SF (phi1 n))-l)``; [Apply Rabsolu_triang | Ring]. +Replace ``eps`` with ``2*eps/3+eps/3``. +Apply Rplus_lt. +Replace ``(RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n))`` with ``(RiemannInt_SF (phi2 n))+(-1)*(RiemannInt_SF (phi1 n))``; [Idtac | Ring]. +Rewrite <- StepFun_P30. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (phi2 n) (phi1 n)))))). +Apply StepFun_P34; Assumption. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 R1 (psi1 n) (psi2 n)))). +Apply StepFun_P37; Try Assumption; Intros; Simpl; Rewrite Rmult_1l. +Apply Rle_trans with ``(Rabsolu ((phi2 n x)-(f x)))+(Rabsolu ((f x)-(phi1 n x)))``. +Replace ``(phi2 n x)+-1*(phi1 n x)`` with ``((phi2 n x)-(f x))+((f x)-(phi1 n x))``; [Apply Rabsolu_triang | Ring]. +Rewrite (Rplus_sym (psi1 n x)); Apply Rplus_le. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Elim (H1 n); Intros; Apply H7. +Assert H10 : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Assert H11 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Rewrite H10; Rewrite H11; Elim H6; Intros; Split; Left; Assumption. +Elim (H0 n); Intros; Apply H7; Assert H10 : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Assert H11 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Rewrite H10; Rewrite H11; Elim H6; Intros; Split; Left; Assumption. +Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Apply Rplus_lt. +Apply Rlt_trans with (pos (un n)). +Elim (H0 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))). +Apply Rle_Rabsolu. +Assumption. +Replace (pos (un n)) with (R_dist (un n) R0). +Apply H; Unfold ge; Apply le_trans with N; Try Assumption. +Unfold N; Apply le_max_l. +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right. +Apply Rle_sym1; Left; Apply (cond_pos (un n)). +Apply Rlt_trans with (pos (un n)). +Elim (H1 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))). +Apply Rle_Rabsolu; Assumption. +Assumption. +Replace (pos (un n)) with (R_dist (un n) R0). +Apply H; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_l. +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (un n)). +Unfold R_dist in H2; Apply H2; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_r. +Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Assert H4 : ``0<eps/3``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H ? H4); Clear H; Intros N0 H. +Elim (H2 ? H4); Clear H2; Intros N1 H2. +Pose N := (max N0 N1); Exists N; Intros; Unfold R_dist. +Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n))))+(Rabsolu ((RiemannInt_SF (phi1 n))-l))``. +Replace ``(RiemannInt_SF (phi2 n))-l`` with ``((RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n)))+((RiemannInt_SF (phi1 n))-l)``; [Apply Rabsolu_triang | Ring]. +Assert Hyp_b : ``b<=a``. +Auto with real. +Replace ``eps`` with ``2*eps/3+eps/3``. +Apply Rplus_lt. +Replace ``(RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n))`` with ``(RiemannInt_SF (phi2 n))+(-1)*(RiemannInt_SF (phi1 n))``; [Idtac | Ring]. +Rewrite <- StepFun_P30. +Rewrite StepFun_P39. +Rewrite Rabsolu_Ropp. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 ``-1`` (phi2 n) (phi1 n))))))))). +Apply StepFun_P34; Try Assumption. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 R1 (psi1 n) (psi2 n))))))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Rewrite Rmult_1l. +Apply Rle_trans with ``(Rabsolu ((phi2 n x)-(f x)))+(Rabsolu ((f x)-(phi1 n x)))``. +Replace ``(phi2 n x)+-1*(phi1 n x)`` with ``((phi2 n x)-(f x))+((f x)-(phi1 n x))``; [Apply Rabsolu_triang | Ring]. +Rewrite (Rplus_sym (psi1 n x)); Apply Rplus_le. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Elim (H1 n); Intros; Apply H7. +Assert H10 : (Rmin a b)==b. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Elim Hyp; Assumption | Reflexivity]. +Assert H11 : (Rmax a b)==a. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Elim Hyp; Assumption | Reflexivity]. +Rewrite H10; Rewrite H11; Elim H6; Intros; Split; Left; Assumption. +Elim (H0 n); Intros; Apply H7; Assert H10 : (Rmin a b)==b. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Elim Hyp; Assumption | Reflexivity]. +Assert H11 : (Rmax a b)==a. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Elim Hyp; Assumption | Reflexivity]. +Rewrite H10; Rewrite H11; Elim H6; Intros; Split; Left; Assumption. +Rewrite <- (Ropp_Ropp (RiemannInt_SF + (mkStepFun + (StepFun_P6 (pre (mkStepFun (StepFun_P28 R1 (psi1 n) (psi2 n)))))))). +Rewrite <- StepFun_P39. +Rewrite StepFun_P30. +Rewrite Rmult_1l; Rewrite double. +Rewrite Ropp_distr1; Apply Rplus_lt. +Apply Rlt_trans with (pos (un n)). +Elim (H0 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Assumption. +Replace (pos (un n)) with (R_dist (un n) R0). +Apply H; Unfold ge; Apply le_trans with N; Try Assumption. +Unfold N; Apply le_max_l. +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right. +Apply Rle_sym1; Left; Apply (cond_pos (un n)). +Apply Rlt_trans with (pos (un n)). +Elim (H1 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu; Assumption. +Assumption. +Replace (pos (un n)) with (R_dist (un n) R0). +Apply H; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_l. +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (un n)). +Unfold R_dist in H2; Apply H2; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_r. +Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Qed. + +Lemma RiemannInt_P12 : (f,g:R->R;a,b,l:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable g a b);pr3:(Riemann_integrable [x:R]``(f x)+l*(g x)`` a b)) ``a<=b`` -> ``(RiemannInt pr3)==(RiemannInt pr1)+l*(RiemannInt pr2)``. +Intro f; Intros; Case (Req_EM l R0); Intro. +Pattern 2 l; Rewrite H0; Rewrite Rmult_Ol; Rewrite Rplus_Or; Unfold RiemannInt; Case (RiemannInt_exists pr3 5!RinvN RinvN_cv); Case (RiemannInt_exists pr1 5!RinvN RinvN_cv); Intros; EApply UL_sequence; [Apply u0 | Pose psi1 := [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Pose psi2 := [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr3 n)); Apply RiemannInt_P11 with f RinvN (phi_sequence RinvN pr1) psi1 psi2; [Apply RinvN_cv | Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)) | Intro; Assert H1 : ((t:R) ``(Rmin a b) <= t``/\``t <= (Rmax a b)`` -> (Rle (Rabsolu (Rminus ``(f t)+l*(g t)`` (phi_sequence RinvN pr3 n t))) (psi2 n t))) /\ ``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``; [Apply (projT2 ? ? (phi_sequence_prop RinvN pr3 n)) | Elim H1; Intros; Split; Try Assumption; Intros; Replace (f t) with ``(f t)+l*(g t)``; [Apply H2; Assumption | Rewrite H0; Ring]] | Assumption]]. +EApply UL_sequence. +Unfold RiemannInt; Case (RiemannInt_exists pr3 5!RinvN RinvN_cv); Intros; Apply u. +Unfold Un_cv; Intros; Unfold RiemannInt; Case (RiemannInt_exists pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists pr2 5!RinvN RinvN_cv); Unfold Un_cv; Intros; Assert H2 : ``0<eps/5``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (u0 ? H2); Clear u0; Intros N0 H3; Assert H4 := RinvN_cv; Unfold Un_cv in H4; Elim (H4 ? H2); Clear H4 H2; Intros N1 H4; Assert H5 : ``0<eps/(5*(Rabsolu l))``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Apply Rabsolu_pos_lt; Assumption]]. +Elim (u ? H5); Clear u; Intros N2 H6; Assert H7 := RinvN_cv; Unfold Un_cv in H7; Elim (H7 ? H5); Clear H7 H5; Intros N3 H5; Unfold R_dist in H3 H4 H5 H6; Pose N := (max (max N0 N1) (max N2 N3)). +Assert H7 : (n:nat) (ge n N1)->``(RinvN n)< eps/5``. +Intros; Replace (pos (RinvN n)) with ``(Rabsolu ((RinvN n)-0))``; [Unfold RinvN; Apply H4; Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Left; Apply (cond_pos (RinvN n))]. +Clear H4; Assert H4 := H7; Clear H7; Assert H7 : (n:nat) (ge n N3)->``(RinvN n)< eps/(5*(Rabsolu l))``. +Intros; Replace (pos (RinvN n)) with ``(Rabsolu ((RinvN n)-0))``; [Unfold RinvN; Apply H5; Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Left; Apply (cond_pos (RinvN n))]. +Clear H5; Assert H5 := H7; Clear H7; Exists N; Intros; Unfold R_dist. +Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))))+(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0))+(Rabsolu l)*(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x))``. +Apply Rle_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))))+(Rabsolu (((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0)+l*((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x)))``. +Replace ``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])-(x0+l*x)`` with ``(((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))))+(((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0)+l*((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x))``; [Apply Rabsolu_triang | Ring]. +Rewrite Rplus_assoc; Apply Rle_compatibility; Rewrite <- Rabsolu_mult; Replace ``(RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0+l*((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x)`` with ``((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0)+(l*((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x))``; [Apply Rabsolu_triang | Ring]. +Replace eps with ``3*eps/5+eps/5+eps/5``. +Repeat Apply Rplus_lt. +Assert H7 : (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr1 n)] t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n0)). +Assert H8 : (EXT psi2:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((g t)-([(phi_sequence RinvN pr2 n)] t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n0)). +Assert H9 : (EXT psi3:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu (((f t)+l*(g t))-([(phi_sequence RinvN pr3 n)] t)))<= (psi3 n t)``)/\``(Rabsolu (RiemannInt_SF (psi3 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr3 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr3 n0)). +Elim H7; Clear H7; Intros psi1 H7; Elim H8; Clear H8; Intros psi2 H8; Elim H9; Clear H9; Intros psi3 H9; Replace ``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))`` with ``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])+(-1)*((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))``; [Idtac | Ring]; Do 2 Rewrite <- StepFun_P30; Assert H10 : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Assert H11 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Rewrite H10 in H7; Rewrite H10 in H8; Rewrite H10 in H9; Rewrite H11 in H7; Rewrite H11 in H8; Rewrite H11 in H9; Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (phi_sequence RinvN pr3 n) (mkStepFun (StepFun_P28 l (phi_sequence RinvN pr1 n) (phi_sequence RinvN pr2 n)))))))). +Apply StepFun_P34; Assumption. +Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 R1 (psi3 n) (mkStepFun (StepFun_P28 (Rabsolu l) (psi1 n) (psi2 n)))))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Rewrite Rmult_1l. +Apply Rle_trans with ``(Rabsolu (([(phi_sequence RinvN pr3 n)] x1)-((f x1)+l*(g x1))))+(Rabsolu (((f x1)+l*(g x1))+ -1*(([(phi_sequence RinvN pr1 n)] x1)+l*([(phi_sequence RinvN pr2 n)] x1))))``. +Replace ``([(phi_sequence RinvN pr3 n)] x1)+ -1*(([(phi_sequence RinvN pr1 n)] x1)+l*([(phi_sequence RinvN pr2 n)] x1))`` with ``(([(phi_sequence RinvN pr3 n)] x1)-((f x1)+l*(g x1)))+(((f x1)+l*(g x1))+ -1*(([(phi_sequence RinvN pr1 n)] x1)+l*([(phi_sequence RinvN pr2 n)] x1)))``; [Apply Rabsolu_triang | Ring]. +Rewrite Rplus_assoc; Apply Rplus_le. +Elim (H9 n); Intros; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H13. +Elim H12; Intros; Split; Left; Assumption. +Apply Rle_trans with ``(Rabsolu ((f x1)-([(phi_sequence RinvN pr1 n)] x1)))+(Rabsolu l)*(Rabsolu ((g x1)-([(phi_sequence RinvN pr2 n)] x1)))``. +Rewrite <- Rabsolu_mult; Replace ``((f x1)+(l*(g x1)+ -1*(([(phi_sequence RinvN pr1 n)] x1)+l*([(phi_sequence RinvN pr2 n)] x1))))`` with ``((f x1)-([(phi_sequence RinvN pr1 n)] x1))+l*((g x1)-([(phi_sequence RinvN pr2 n)] x1))``; [Apply Rabsolu_triang | Ring]. +Apply Rplus_le. +Elim (H7 n); Intros; Apply H13. +Elim H12; Intros; Split; Left; Assumption. +Apply Rle_monotony; [Apply Rabsolu_pos | Elim (H8 n); Intros; Apply H13; Elim H12; Intros; Split; Left; Assumption]. +Do 2 Rewrite StepFun_P30; Rewrite Rmult_1l; Replace ``3*eps/5`` with ``eps/5+(eps/5+eps/5)``; [Repeat Apply Rplus_lt | Ring]. +Apply Rlt_trans with (pos (RinvN n)); [Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi3 n))); [Apply Rle_Rabsolu | Elim (H9 n); Intros; Assumption] | Apply H4; Unfold ge; Apply le_trans with N; [Apply le_trans with (max N0 N1); [Apply le_max_r | Unfold N; Apply le_max_l] | Assumption]]. +Apply Rlt_trans with (pos (RinvN n)); [Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))); [Apply Rle_Rabsolu | Elim (H7 n); Intros; Assumption] | Apply H4; Unfold ge; Apply le_trans with N; [Apply le_trans with (max N0 N1); [Apply le_max_r | Unfold N; Apply le_max_l] | Assumption]]. +Apply Rlt_monotony_contra with ``/(Rabsolu l)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Replace ``/(Rabsolu l)*eps/5`` with ``eps/(5*(Rabsolu l))``. +Apply Rlt_trans with (pos (RinvN n)); [Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))); [Apply Rle_Rabsolu | Elim (H8 n); Intros; Assumption] | Apply H5; Unfold ge; Apply le_trans with N; [Apply le_trans with (max N2 N3); [Apply le_max_r | Unfold N; Apply le_max_r] | Assumption]]. +Unfold Rdiv; Rewrite Rinv_Rmult; [Ring | DiscrR | Apply Rabsolu_no_R0; Assumption]. +Apply Rabsolu_no_R0; Assumption. +Apply H3; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Apply le_trans with N; [Unfold N; Apply le_max_l | Assumption]]. +Apply Rlt_monotony_contra with ``/(Rabsolu l)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Replace ``/(Rabsolu l)*eps/5`` with ``eps/(5*(Rabsolu l))``. +Apply H6; Unfold ge; Apply le_trans with (max N2 N3); [Apply le_max_l | Apply le_trans with N; [Unfold N; Apply le_max_r | Assumption]]. +Unfold Rdiv; Rewrite Rinv_Rmult; [Ring | DiscrR | Apply Rabsolu_no_R0; Assumption]. +Apply Rabsolu_no_R0; Assumption. +Apply r_Rmult_mult with ``5``; [Unfold Rdiv; Do 2 Rewrite Rmult_Rplus_distr; Do 3 Rewrite (Rmult_sym ``5``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Qed. + +Lemma RiemannInt_P13 : (f,g:R->R;a,b,l:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable g a b);pr3:(Riemann_integrable [x:R]``(f x)+l*(g x)`` a b)) ``(RiemannInt pr3)==(RiemannInt pr1)+l*(RiemannInt pr2)``. +Intros; Case (total_order_Rle a b); Intro; [Apply RiemannInt_P12; Assumption | Assert H : ``b<=a``; [Auto with real | Replace (RiemannInt pr3) with (Ropp (RiemannInt (RiemannInt_P1 pr3))); [Idtac | Symmetry; Apply RiemannInt_P8]; Replace (RiemannInt pr2) with (Ropp (RiemannInt (RiemannInt_P1 pr2))); [Idtac | Symmetry; Apply RiemannInt_P8]; Replace (RiemannInt pr1) with (Ropp (RiemannInt (RiemannInt_P1 pr1))); [Idtac | Symmetry; Apply RiemannInt_P8]; Rewrite (RiemannInt_P12 (RiemannInt_P1 pr1) (RiemannInt_P1 pr2) (RiemannInt_P1 pr3) H); Ring]]. +Qed. + +Lemma RiemannInt_P14 : (a,b,c:R) (Riemann_integrable (fct_cte c) a b). +Unfold Riemann_integrable; Intros; Split with (mkStepFun (StepFun_P4 a b c)); Split with (mkStepFun (StepFun_P4 a b R0)); Split; [Intros; Simpl; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Unfold fct_cte; Right; Reflexivity | Rewrite StepFun_P18; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Apply (cond_pos eps)]. +Qed. + +Lemma RiemannInt_P15 : (a,b,c:R;pr:(Riemann_integrable (fct_cte c) a b)) ``(RiemannInt pr)==c*(b-a)``. +Intros; Unfold RiemannInt; Case (RiemannInt_exists 1!(fct_cte c) 2!a 3!b pr 5!RinvN RinvN_cv); Intros; EApply UL_sequence. +Apply u. +Pose phi1 := [N:nat](phi_sequence RinvN 2!(fct_cte c) 3!a 4!b pr N); Change (Un_cv [N:nat](RiemannInt_SF (phi1 N)) ``c*(b-a)``); Pose f := (fct_cte c); Assert H1 : (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr n)] t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr n)). +Elim H1; Clear H1; Intros psi1 H1; Pose phi2 := [n:nat](mkStepFun (StepFun_P4 a b c)); Pose psi2 := [n:nat](mkStepFun (StepFun_P4 a b R0)); Apply RiemannInt_P11 with f RinvN phi2 psi2 psi1; Try Assumption. +Apply RinvN_cv. +Intro; Split. +Intros; Unfold f; Simpl; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Unfold fct_cte; Right; Reflexivity. +Unfold psi2; Rewrite StepFun_P18; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Apply (cond_pos (RinvN n)). +Unfold Un_cv; Intros; Split with O; Intros; Unfold R_dist; Unfold phi2; Rewrite StepFun_P18; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H. +Qed. + +Lemma RiemannInt_P16 : (f:R->R;a,b:R) (Riemann_integrable f a b) -> (Riemann_integrable [x:R](Rabsolu (f x)) a b). +Unfold Riemann_integrable; Intro f; Intros; Elim (X eps); Clear X; Intros phi [psi [H H0]]; Split with (mkStepFun (StepFun_P32 phi)); Split with psi; Split; Try Assumption; Intros; Simpl; Apply Rle_trans with ``(Rabsolu ((f t)-(phi t)))``; [Apply Rabsolu_triang_inv2 | Apply H; Assumption]. +Qed. + +Lemma Rle_cv_lim : (Un,Vn:nat->R;l1,l2:R) ((n:nat)``(Un n)<=(Vn n)``) -> (Un_cv Un l1) -> (Un_cv Vn l2) -> ``l1<=l2``. +Intros; Case (total_order_Rle l1 l2); Intro. +Assumption. +Assert H2 : ``l2<l1``. +Auto with real. +Clear n; Assert H3 : ``0<(l1-l2)/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply Rlt_Rminus; Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H1 ? H3); Elim (H0 ? H3); Clear H0 H1; Unfold R_dist; Intros; Pose N := (max x x0); Cut ``(Vn N)<(Un N)``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (H N) H4)). +Apply Rlt_trans with ``(l1+l2)/2``. +Apply Rlt_anti_compatibility with ``-l2``; Replace ``-l2+(l1+l2)/2`` with ``(l1-l2)/2``. +Rewrite Rplus_sym; Apply Rle_lt_trans with ``(Rabsolu ((Vn N)-l2))``. +Apply Rle_Rabsolu. +Apply H1; Unfold ge; Unfold N; Apply le_max_r. +Apply r_Rmult_mult with ``2``; [Unfold Rdiv; Do 2 Rewrite -> (Rmult_sym ``2``); Rewrite (Rmult_Rplus_distrl ``-l2`` ``(l1+l2)*/2`` ``2``); Repeat Rewrite -> Rmult_assoc; Rewrite <- Rinv_l_sym; [ Ring | DiscrR ] | DiscrR]. +Apply Ropp_Rlt; Apply Rlt_anti_compatibility with l1; Replace ``l1+ -((l1+l2)/2)`` with ``(l1-l2)/2``. +Apply Rle_lt_trans with ``(Rabsolu ((Un N)-l1))``. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply Rle_Rabsolu. +Apply H0; Unfold ge; Unfold N; Apply le_max_l. +Apply r_Rmult_mult with ``2``; [Unfold Rdiv; Do 2 Rewrite -> (Rmult_sym ``2``); Rewrite (Rmult_Rplus_distrl ``l1`` ``-((l1+l2)*/2)`` ``2``); Rewrite <- Ropp_mul1; Repeat Rewrite -> Rmult_assoc; Rewrite <- Rinv_l_sym; [ Ring | DiscrR ] | DiscrR]. +Qed. + +Lemma RiemannInt_P17 : (f:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable [x:R](Rabsolu (f x)) a b)) ``a<=b`` -> ``(Rabsolu (RiemannInt pr1))<=(RiemannInt pr2)``. +Intro f; Intros; Unfold RiemannInt; Case (RiemannInt_exists 1!f 2!a 3!b pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists 1!([x0:R](Rabsolu (f x0))) 2!a 3!b pr2 5!RinvN RinvN_cv); Intros; Pose phi1 := (phi_sequence RinvN pr1); Pose phi2 := [N:nat](mkStepFun (StepFun_P32 (phi1 N))); Apply Rle_cv_lim with [N:nat](Rabsolu (RiemannInt_SF (phi1 N))) [N:nat](RiemannInt_SF (phi2 N)). +Intro; Unfold phi2; Apply StepFun_P34; Assumption. +Fold phi1 in u0; Apply (continuity_seq Rabsolu [N:nat](RiemannInt_SF (phi1 N)) x0); Try Assumption. +Apply continuity_Rabsolu. +Pose phi3 := (phi_sequence RinvN pr2); Assert H0 : (EXT psi3:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((Rabsolu (f t))-((phi3 n) t)))<= (psi3 n t)``)/\``(Rabsolu (RiemannInt_SF (psi3 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n)). +Assert H1 : (EXT psi2:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((Rabsolu (f t))-((phi2 n) t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). +Assert H1 : (EXT psi2:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-((phi1 n) t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)). +Elim H1; Clear H1; Intros psi2 H1; Split with psi2; Intros; Elim (H1 n); Clear H1; Intros; Split; Try Assumption. +Intros; Unfold phi2; Simpl; Apply Rle_trans with ``(Rabsolu ((f t)-((phi1 n) t)))``. +Apply Rabsolu_triang_inv2. +Apply H1; Assumption. +Elim H0; Clear H0; Intros psi3 H0; Elim H1; Clear H1; Intros psi2 H1; Apply RiemannInt_P11 with [x:R](Rabsolu (f x)) RinvN phi3 psi3 psi2; Try Assumption; Apply RinvN_cv. +Qed. + +Lemma RiemannInt_P18 : (f,g:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable g a b)) ``a<=b`` -> ((x:R)``a<x<b``->``(f x)==(g x)``) -> ``(RiemannInt pr1)==(RiemannInt pr2)``. +Intro f; Intros; Unfold RiemannInt; Case (RiemannInt_exists 1!f 2!a 3!b pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists 1!g 2!a 3!b pr2 5!RinvN RinvN_cv); Intros; EApply UL_sequence. +Apply u0. +Pose phi1 := [N:nat](phi_sequence RinvN 2!f 3!a 4!b pr1 N); Change (Un_cv [N:nat](RiemannInt_SF (phi1 N)) x); Assert H1 : (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-((phi1 n) t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)). +Elim H1; Clear H1; Intros psi1 H1; Pose phi2 := [N:nat](phi_sequence RinvN 2!g 3!a 4!b pr2 N). +Pose phi2_aux := [N:nat][x:R](Cases (Req_EM_T x a) of + | (leftT _) => (f a) + | (rightT _) => (Cases (Req_EM_T x b) of + | (leftT _) => (f b) + | (rightT _) => (phi2 N x) end) end). +Cut (N:nat)(IsStepFun (phi2_aux N) a b). +Intro; Pose phi2_m := [N:nat](mkStepFun (X N)). +Assert H2 : (EXT psi2:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((g t)-((phi2 n) t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n)). +Elim H2; Clear H2; Intros psi2 H2; Apply RiemannInt_P11 with f RinvN phi2_m psi2 psi1; Try Assumption. +Apply RinvN_cv. +Intro; Elim (H2 n); Intros; Split; Try Assumption. +Intros; Unfold phi2_m; Simpl; Unfold phi2_aux; Case (Req_EM_T t a); Case (Req_EM_T t b); Intros. +Rewrite e0; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rle_trans with ``(Rabsolu ((g t)-((phi2 n) t)))``. +Apply Rabsolu_pos. +Pattern 3 a; Rewrite <- e0; Apply H3; Assumption. +Rewrite e; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rle_trans with ``(Rabsolu ((g t)-((phi2 n) t)))``. +Apply Rabsolu_pos. +Pattern 3 a; Rewrite <- e; Apply H3; Assumption. +Rewrite e; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rle_trans with ``(Rabsolu ((g t)-((phi2 n) t)))``. +Apply Rabsolu_pos. +Pattern 3 b; Rewrite <- e; Apply H3; Assumption. +Replace (f t) with (g t). +Apply H3; Assumption. +Symmetry; Apply H0; Elim H5; Clear H5; Intros. +Assert H7 : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n2; Assumption]. +Assert H8 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n2; Assumption]. +Rewrite H7 in H5; Rewrite H8 in H6; Split. +Elim H5; Intro; [Assumption | Elim n1; Symmetry; Assumption]. +Elim H6; Intro; [Assumption | Elim n0; Assumption]. +Cut (N:nat)(RiemannInt_SF (phi2_m N))==(RiemannInt_SF (phi2 N)). +Intro; Unfold Un_cv; Intros; Elim (u ? H4); Intros; Exists x1; Intros; Rewrite (H3 n); Apply H5; Assumption. +Intro; Apply Rle_antisym. +Apply StepFun_P37; Try Assumption. +Intros; Unfold phi2_m; Simpl; Unfold phi2_aux; Case (Req_EM_T x1 a); Case (Req_EM_T x1 b); Intros. +Elim H3; Intros; Rewrite e0 in H4; Elim (Rlt_antirefl ? H4). +Elim H3; Intros; Rewrite e in H4; Elim (Rlt_antirefl ? H4). +Elim H3; Intros; Rewrite e in H5; Elim (Rlt_antirefl ? H5). +Right; Reflexivity. +Apply StepFun_P37; Try Assumption. +Intros; Unfold phi2_m; Simpl; Unfold phi2_aux; Case (Req_EM_T x1 a); Case (Req_EM_T x1 b); Intros. +Elim H3; Intros; Rewrite e0 in H4; Elim (Rlt_antirefl ? H4). +Elim H3; Intros; Rewrite e in H4; Elim (Rlt_antirefl ? H4). +Elim H3; Intros; Rewrite e in H5; Elim (Rlt_antirefl ? H5). +Right; Reflexivity. +Intro; Assert H2 := (pre (phi2 N)); Unfold IsStepFun in H2; Unfold is_subdivision in H2; Elim H2; Clear H2; Intros l [lf H2]; Split with l; Split with lf; Unfold adapted_couple in H2; Decompose [and] H2; Clear H2; Unfold adapted_couple; Repeat Split; Try Assumption. +Intros; Assert H9 := (H8 i H2); Unfold constant_D_eq open_interval in H9; Unfold constant_D_eq open_interval; Intros; Rewrite <- (H9 x1 H7); Assert H10 : ``a<=(pos_Rl l i)``. +Replace a with (Rmin a b). +Rewrite <- H5; Elim (RList_P6 l); Intros; Apply H10. +Assumption. +Apply le_O_n. +Apply lt_trans with (pred (Rlength l)); [Assumption | Apply lt_pred_n_n]. +Apply neq_O_lt; Intro; Rewrite <- H12 in H6; Discriminate. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Assert H11 : ``(pos_Rl l (S i))<=b``. +Replace b with (Rmax a b). +Rewrite <- H4; Elim (RList_P6 l); Intros; Apply H11. +Assumption. +Apply lt_le_S; Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Intro; Rewrite <- H13 in H6; Discriminate. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Elim H7; Clear H7; Intros; Unfold phi2_aux; Case (Req_EM_T x1 a); Case (Req_EM_T x1 b); Intros. +Rewrite e in H12; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H11 H12)). +Rewrite e in H7; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H10 H7)). +Rewrite e in H12; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H11 H12)). +Reflexivity. +Qed. + +Lemma RiemannInt_P19 : (f,g:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable g a b)) ``a<=b`` -> ((x:R)``a<x<b``->``(f x)<=(g x)``) -> ``(RiemannInt pr1)<=(RiemannInt pr2)``. +Intro f; Intros; Apply Rle_anti_compatibility with ``-(RiemannInt pr1)``; Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Apply Rle_trans with (Rabsolu (RiemannInt (RiemannInt_P10 ``-1`` pr2 pr1))). +Apply Rabsolu_pos. +Replace ``(RiemannInt pr2)+ -(RiemannInt pr1)`` with (RiemannInt (RiemannInt_P16 (RiemannInt_P10 ``-1`` pr2 pr1))). +Apply (RiemannInt_P17 (RiemannInt_P10 ``-1`` pr2 pr1) (RiemannInt_P16 (RiemannInt_P10 ``-1`` pr2 pr1))); Assumption. +Replace ``(RiemannInt pr2)+-(RiemannInt pr1)`` with (RiemannInt (RiemannInt_P10 ``-1`` pr2 pr1)). +Apply RiemannInt_P18; Try Assumption. +Intros; Apply Rabsolu_right. +Apply Rle_sym1; Apply Rle_anti_compatibility with (f x); Rewrite Rplus_Or; Replace ``(f x)+((g x)+ -1*(f x))`` with (g x); [Apply H0; Assumption | Ring]. +Rewrite (RiemannInt_P12 pr2 pr1 (RiemannInt_P10 ``-1`` pr2 pr1)); [Ring | Assumption]. +Qed. + +Lemma FTC_P1 : (f:R->R;a,b:R) ``a<=b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> ((x:R)``a<=x``->``x<=b``->(Riemann_integrable f a x)). +Intros; Apply continuity_implies_RiemannInt; [Assumption | Intros; Apply H0; Elim H3; Intros; Split; Assumption Orelse Apply Rle_trans with x; Assumption]. +Qed. +V7only [Notation FTC_P2 := Rle_refl.]. + +Definition primitive [f:R->R;a,b:R;h:``a<=b``;pr:((x:R)``a<=x``->``x<=b``->(Riemann_integrable f a x))] : R->R := [x:R] Cases (total_order_Rle a x) of + | (leftT r) => Cases (total_order_Rle x b) of + | (leftT r0) => (RiemannInt (pr x r r0)) + | (rightT _) => ``(f b)*(x-b)+(RiemannInt (pr b h (FTC_P2 b)))`` end + | (rightT _) => ``(f a)*(x-a)`` end. + +Lemma RiemannInt_P20 : (f:R->R;a,b:R;h:``a<=b``;pr:((x:R)``a<=x``->``x<=b``->(Riemann_integrable f a x));pr0:(Riemann_integrable f a b)) ``(RiemannInt pr0)==(primitive h pr b)-(primitive h pr a)``. +Intros; Replace (primitive h pr a) with R0. +Replace (RiemannInt pr0) with (primitive h pr b). +Ring. +Unfold primitive; Case (total_order_Rle a b); Case (total_order_Rle b b); Intros; [Apply RiemannInt_P5 | Elim n; Right; Reflexivity | Elim n; Assumption | Elim n0; Assumption]. +Symmetry; Unfold primitive; Case (total_order_Rle a a); Case (total_order_Rle a b); Intros; [Apply RiemannInt_P9 | Elim n; Assumption | Elim n; Right; Reflexivity | Elim n0; Right; Reflexivity]. +Qed. + +Lemma RiemannInt_P21 : (f:R->R;a,b,c:R) ``a<=b``-> ``b<=c`` -> (Riemann_integrable f a b) -> (Riemann_integrable f b c) -> (Riemann_integrable f a c). +Unfold Riemann_integrable; Intros f a b c Hyp1 Hyp2 X X0 eps. +Assert H : ``0<eps/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Sup0]. +Elim (X (mkposreal ? H)); Clear X; Intros phi1 [psi1 H1]; Elim (X0 (mkposreal ? H)); Clear X0; Intros phi2 [psi2 H2]. +Pose phi3 := [x:R] Cases (total_order_Rle a x) of + | (leftT _) => Cases (total_order_Rle x b) of + | (leftT _) => (phi1 x) + | (rightT _) => (phi2 x) end + | (rightT _) => R0 end. +Pose psi3 := [x:R] Cases (total_order_Rle a x) of + | (leftT _) => Cases (total_order_Rle x b) of + | (leftT _) => (psi1 x) + | (rightT _) => (psi2 x) end + | (rightT _) => R0 end. +Cut (IsStepFun phi3 a c). +Intro; Cut (IsStepFun psi3 a b). +Intro; Cut (IsStepFun psi3 b c). +Intro; Cut (IsStepFun psi3 a c). +Intro; Split with (mkStepFun X); Split with (mkStepFun X2); Simpl; Split. +Intros; Unfold phi3 psi3; Case (total_order_Rle t b); Case (total_order_Rle a t); Intros. +Elim H1; Intros; Apply H3. +Replace (Rmin a b) with a. +Replace (Rmax a b) with b. +Split; Assumption. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Elim n; Replace a with (Rmin a c). +Elim H0; Intros; Assumption. +Unfold Rmin; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. +Elim H2; Intros; Apply H3. +Replace (Rmax b c) with (Rmax a c). +Elim H0; Intros; Split; Try Assumption. +Replace (Rmin b c) with b. +Auto with real. +Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n0; Assumption]. +Unfold Rmax; Case (total_order_Rle a c); Case (total_order_Rle b c); Intros; Try (Elim n0; Assumption Orelse Elim n0; Apply Rle_trans with b; Assumption). +Reflexivity. +Elim n; Replace a with (Rmin a c). +Elim H0; Intros; Assumption. +Unfold Rmin; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n1; Apply Rle_trans with b; Assumption]. +Rewrite <- (StepFun_P43 X0 X1 X2). +Apply Rle_lt_trans with ``(Rabsolu (RiemannInt_SF (mkStepFun X0)))+(Rabsolu (RiemannInt_SF (mkStepFun X1)))``. +Apply Rabsolu_triang. +Rewrite (double_var eps); Replace (RiemannInt_SF (mkStepFun X0)) with (RiemannInt_SF psi1). +Replace (RiemannInt_SF (mkStepFun X1)) with (RiemannInt_SF psi2). +Apply Rplus_lt. +Elim H1; Intros; Assumption. +Elim H2; Intros; Assumption. +Apply Rle_antisym. +Apply StepFun_P37; Try Assumption. +Simpl; Intros; Unfold psi3; Elim H0; Clear H0; Intros; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H0)) | Right; Reflexivity | Elim n; Apply Rle_trans with b; [Assumption | Left; Assumption] | Elim n0; Apply Rle_trans with b; [Assumption | Left; Assumption]]. +Apply StepFun_P37; Try Assumption. +Simpl; Intros; Unfold psi3; Elim H0; Clear H0; Intros; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H0)) | Right; Reflexivity | Elim n; Apply Rle_trans with b; [Assumption | Left; Assumption] | Elim n0; Apply Rle_trans with b; [Assumption | Left; Assumption]]. +Apply Rle_antisym. +Apply StepFun_P37; Try Assumption. +Simpl; Intros; Unfold psi3; Elim H0; Clear H0; Intros; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Right; Reflexivity | Elim n; Left; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. +Apply StepFun_P37; Try Assumption. +Simpl; Intros; Unfold psi3; Elim H0; Clear H0; Intros; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Right; Reflexivity | Elim n; Left; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. +Apply StepFun_P46 with b; Assumption. +Assert H3 := (pre psi2); Unfold IsStepFun in H3; Unfold is_subdivision in H3; Elim H3; Clear H3; Intros l1 [lf1 H3]; Split with l1; Split with lf1; Unfold adapted_couple in H3; Decompose [and] H3; Clear H3; Unfold adapted_couple; Repeat Split; Try Assumption. +Intros; Assert H9 := (H8 i H3); Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9; Intros; Rewrite <- (H9 x H7); Unfold psi3; Assert H10 : ``b<x``. +Apply Rle_lt_trans with (pos_Rl l1 i). +Replace b with (Rmin b c). +Rewrite <- H5; Elim (RList_P6 l1); Intros; Apply H10; Try Assumption. +Apply le_O_n. +Apply lt_trans with (pred (Rlength l1)); Try Assumption; Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H6; Discriminate. +Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n; Assumption]. +Elim H7; Intros; Assumption. +Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H10)) | Reflexivity | Elim n; Apply Rle_trans with b; [Assumption | Left; Assumption] | Elim n0; Apply Rle_trans with b; [Assumption | Left; Assumption]]. +Assert H3 := (pre psi1); Unfold IsStepFun in H3; Unfold is_subdivision in H3; Elim H3; Clear H3; Intros l1 [lf1 H3]; Split with l1; Split with lf1; Unfold adapted_couple in H3; Decompose [and] H3; Clear H3; Unfold adapted_couple; Repeat Split; Try Assumption. +Intros; Assert H9 := (H8 i H3); Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9; Intros; Rewrite <- (H9 x H7); Unfold psi3; Assert H10 : ``x<=b``. +Apply Rle_trans with (pos_Rl l1 (S i)). +Elim H7; Intros; Left; Assumption. +Replace b with (Rmax a b). +Rewrite <- H4; Elim (RList_P6 l1); Intros; Apply H10; Try Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H6; Discriminate. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Assert H11 : ``a<=x``. +Apply Rle_trans with (pos_Rl l1 i). +Replace a with (Rmin a b). +Rewrite <- H5; Elim (RList_P6 l1); Intros; Apply H11; Try Assumption. +Apply le_O_n. +Apply lt_trans with (pred (Rlength l1)); Try Assumption; Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H6; Discriminate. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Left; Elim H7; Intros; Assumption. +Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; Reflexivity Orelse Elim n; Assumption. +Apply StepFun_P46 with b. +Assert H3 := (pre phi1); Unfold IsStepFun in H3; Unfold is_subdivision in H3; Elim H3; Clear H3; Intros l1 [lf1 H3]; Split with l1; Split with lf1; Unfold adapted_couple in H3; Decompose [and] H3; Clear H3; Unfold adapted_couple; Repeat Split; Try Assumption. +Intros; Assert H9 := (H8 i H3); Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9; Intros; Rewrite <- (H9 x H7); Unfold psi3; Assert H10 : ``x<=b``. +Apply Rle_trans with (pos_Rl l1 (S i)). +Elim H7; Intros; Left; Assumption. +Replace b with (Rmax a b). +Rewrite <- H4; Elim (RList_P6 l1); Intros; Apply H10; Try Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H6; Discriminate. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Assert H11 : ``a<=x``. +Apply Rle_trans with (pos_Rl l1 i). +Replace a with (Rmin a b). +Rewrite <- H5; Elim (RList_P6 l1); Intros; Apply H11; Try Assumption. +Apply le_O_n. +Apply lt_trans with (pred (Rlength l1)); Try Assumption; Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H6; Discriminate. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Left; Elim H7; Intros; Assumption. +Unfold phi3; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; Reflexivity Orelse Elim n; Assumption. +Assert H3 := (pre phi2); Unfold IsStepFun in H3; Unfold is_subdivision in H3; Elim H3; Clear H3; Intros l1 [lf1 H3]; Split with l1; Split with lf1; Unfold adapted_couple in H3; Decompose [and] H3; Clear H3; Unfold adapted_couple; Repeat Split; Try Assumption. +Intros; Assert H9 := (H8 i H3); Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9; Intros; Rewrite <- (H9 x H7); Unfold psi3; Assert H10 : ``b<x``. +Apply Rle_lt_trans with (pos_Rl l1 i). +Replace b with (Rmin b c). +Rewrite <- H5; Elim (RList_P6 l1); Intros; Apply H10; Try Assumption. +Apply le_O_n. +Apply lt_trans with (pred (Rlength l1)); Try Assumption; Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H6; Discriminate. +Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n; Assumption]. +Elim H7; Intros; Assumption. +Unfold phi3; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H10)) | Reflexivity | Elim n; Apply Rle_trans with b; [Assumption | Left; Assumption] | Elim n0; Apply Rle_trans with b; [Assumption | Left; Assumption]]. +Qed. + +Lemma RiemannInt_P22 : (f:R->R;a,b,c:R) (Riemann_integrable f a b) -> ``a<=c<=b`` -> (Riemann_integrable f a c). +Unfold Riemann_integrable; Intros; Elim (X eps); Clear X; Intros phi [psi H0]; Elim H; Elim H0; Clear H H0; Intros; Assert H3 : (IsStepFun phi a c). +Apply StepFun_P44 with b. +Apply (pre phi). +Split; Assumption. +Assert H4 : (IsStepFun psi a c). +Apply StepFun_P44 with b. +Apply (pre psi). +Split; Assumption. +Split with (mkStepFun H3); Split with (mkStepFun H4); Split. +Simpl; Intros; Apply H. +Replace (Rmin a b) with (Rmin a c). +Elim H5; Intros; Split; Try Assumption. +Apply Rle_trans with (Rmax a c); Try Assumption. +Replace (Rmax a b) with b. +Replace (Rmax a c) with c. +Assumption. +Unfold Rmax; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n; Assumption]. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Unfold Rmin; Case (total_order_Rle a c); Case (total_order_Rle a b); Intros; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption | Elim n; Assumption | Elim n0; Assumption]. +Rewrite Rabsolu_right. +Assert H5 : (IsStepFun psi c b). +Apply StepFun_P46 with a. +Apply StepFun_P6; Assumption. +Apply (pre psi). +Replace (RiemannInt_SF (mkStepFun H4)) with ``(RiemannInt_SF psi)-(RiemannInt_SF (mkStepFun H5))``. +Apply Rle_lt_trans with (RiemannInt_SF psi). +Unfold Rminus; Pattern 2 (RiemannInt_SF psi); Rewrite <- Rplus_Or; Apply Rle_compatibility; Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Replace R0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b R0))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Unfold fct_cte; Apply Rle_trans with ``(Rabsolu ((f x)-(phi x)))``. +Apply Rabsolu_pos. +Apply H. +Replace (Rmin a b) with a. +Replace (Rmax a b) with b. +Elim H6; Intros; Split; Left. +Apply Rle_lt_trans with c; Assumption. +Assumption. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Rewrite StepFun_P18; Ring. +Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi)). +Apply Rle_Rabsolu. +Assumption. +Assert H6 : (IsStepFun psi a b). +Apply (pre psi). +Replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)). +Rewrite <- (StepFun_P43 H4 H5 H6); Ring. +Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. +EApply StepFun_P17. +Apply StepFun_P1. +Simpl; Apply StepFun_P1. +Apply eq_Ropp; EApply StepFun_P17. +Apply StepFun_P1. +Simpl; Apply StepFun_P1. +Apply Rle_sym1; Replace R0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c R0))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Unfold fct_cte; Apply Rle_trans with ``(Rabsolu ((f x)-(phi x)))``. +Apply Rabsolu_pos. +Apply H. +Replace (Rmin a b) with a. +Replace (Rmax a b) with b. +Elim H5; Intros; Split; Left. +Assumption. +Apply Rlt_le_trans with c; Assumption. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Rewrite StepFun_P18; Ring. +Qed. + +Lemma RiemannInt_P23 : (f:R->R;a,b,c:R) (Riemann_integrable f a b) -> ``a<=c<=b`` -> (Riemann_integrable f c b). +Unfold Riemann_integrable; Intros; Elim (X eps); Clear X; Intros phi [psi H0]; Elim H; Elim H0; Clear H H0; Intros; Assert H3 : (IsStepFun phi c b). +Apply StepFun_P45 with a. +Apply (pre phi). +Split; Assumption. +Assert H4 : (IsStepFun psi c b). +Apply StepFun_P45 with a. +Apply (pre psi). +Split; Assumption. +Split with (mkStepFun H3); Split with (mkStepFun H4); Split. +Simpl; Intros; Apply H. +Replace (Rmax a b) with (Rmax c b). +Elim H5; Intros; Split; Try Assumption. +Apply Rle_trans with (Rmin c b); Try Assumption. +Replace (Rmin a b) with a. +Replace (Rmin c b) with c. +Assumption. +Unfold Rmin; Case (total_order_Rle c b); Intro; [Reflexivity | Elim n; Assumption]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Unfold Rmax; Case (total_order_Rle c b); Case (total_order_Rle a b); Intros; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption | Elim n; Assumption | Elim n0; Assumption]. +Rewrite Rabsolu_right. +Assert H5 : (IsStepFun psi a c). +Apply StepFun_P46 with b. +Apply (pre psi). +Apply StepFun_P6; Assumption. +Replace (RiemannInt_SF (mkStepFun H4)) with ``(RiemannInt_SF psi)-(RiemannInt_SF (mkStepFun H5))``. +Apply Rle_lt_trans with (RiemannInt_SF psi). +Unfold Rminus; Pattern 2 (RiemannInt_SF psi); Rewrite <- Rplus_Or; Apply Rle_compatibility; Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Replace R0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c R0))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Unfold fct_cte; Apply Rle_trans with ``(Rabsolu ((f x)-(phi x)))``. +Apply Rabsolu_pos. +Apply H. +Replace (Rmin a b) with a. +Replace (Rmax a b) with b. +Elim H6; Intros; Split; Left. +Assumption. +Apply Rlt_le_trans with c; Assumption. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Rewrite StepFun_P18; Ring. +Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi)). +Apply Rle_Rabsolu. +Assumption. +Assert H6 : (IsStepFun psi a b). +Apply (pre psi). +Replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)). +Rewrite <- (StepFun_P43 H5 H4 H6); Ring. +Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. +EApply StepFun_P17. +Apply StepFun_P1. +Simpl; Apply StepFun_P1. +Apply eq_Ropp; EApply StepFun_P17. +Apply StepFun_P1. +Simpl; Apply StepFun_P1. +Apply Rle_sym1; Replace R0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b R0))). +Apply StepFun_P37; Try Assumption. +Intros; Simpl; Unfold fct_cte; Apply Rle_trans with ``(Rabsolu ((f x)-(phi x)))``. +Apply Rabsolu_pos. +Apply H. +Replace (Rmin a b) with a. +Replace (Rmax a b) with b. +Elim H5; Intros; Split; Left. +Apply Rle_lt_trans with c; Assumption. +Assumption. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. +Rewrite StepFun_P18; Ring. +Qed. + +Lemma RiemannInt_P24 : (f:R->R;a,b,c:R) (Riemann_integrable f a b) -> (Riemann_integrable f b c) -> (Riemann_integrable f a c). +Intros; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros. +Apply RiemannInt_P21 with b; Assumption. +Case (total_order_Rle a c); Intro. +Apply RiemannInt_P22 with b; Try Assumption. +Split; [Assumption | Auto with real]. +Apply RiemannInt_P1; Apply RiemannInt_P22 with b. +Apply RiemannInt_P1; Assumption. +Split; Auto with real. +Case (total_order_Rle a c); Intro. +Apply RiemannInt_P23 with b; Try Assumption. +Split; Auto with real. +Apply RiemannInt_P1; Apply RiemannInt_P23 with b. +Apply RiemannInt_P1; Assumption. +Split; [Assumption | Auto with real]. +Apply RiemannInt_P1; Apply RiemannInt_P21 with b; Auto with real Orelse Apply RiemannInt_P1; Assumption. +Qed. + +Lemma RiemannInt_P25 : (f:R->R;a,b,c:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable f b c);pr3:(Riemann_integrable f a c)) ``a<=b``->``b<=c``->``(RiemannInt pr1)+(RiemannInt pr2)==(RiemannInt pr3)``. +Intros f a b c pr1 pr2 pr3 Hyp1 Hyp2; Unfold RiemannInt; Case (RiemannInt_exists 1!f 2!a 3!b pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists 1!f 2!b 3!c pr2 5!RinvN RinvN_cv); Case (RiemannInt_exists 1!f 2!a 3!c pr3 5!RinvN RinvN_cv); Intros; Symmetry; EApply UL_sequence. +Apply u. +Unfold Un_cv; Intros; Assert H0 : ``0<eps/3``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (u1 ? H0); Clear u1; Intros N1 H1; Elim (u0 ? H0); Clear u0; Intros N2 H2; Cut (Un_cv [n:nat]``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))`` R0). +Intro; Elim (H3 ? H0); Clear H3; Intros N3 H3; Pose N0 := (max (max N1 N2) N3); Exists N0; Intros; Unfold R_dist; Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))))+(Rabsolu (((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))-(x1+x0)))``. +Replace ``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])-(x1+x0)`` with ``((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)])))+(((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))-(x1+x0))``; [Apply Rabsolu_triang | Ring]. +Replace eps with ``eps/3+eps/3+eps/3``. +Rewrite Rplus_assoc; Apply Rplus_lt. +Unfold R_dist in H3; Cut (ge n N3). +Intro; Assert H6 := (H3 ? H5); Unfold Rminus in H6; Rewrite Ropp_O in H6; Rewrite Rplus_Or in H6; Apply H6. +Unfold ge; Apply le_trans with N0; [Unfold N0; Apply le_max_r | Assumption]. +Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x1))+(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x0))``. +Replace ``((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))-(x1+x0)`` with ``((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x1)+((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x0)``; [Apply Rabsolu_triang | Ring]. +Apply Rplus_lt. +Unfold R_dist in H1; Apply H1. +Unfold ge; Apply le_trans with N0; [Apply le_trans with (max N1 N2); [Apply le_max_l | Unfold N0; Apply le_max_l] | Assumption]. +Unfold R_dist in H2; Apply H2. +Unfold ge; Apply le_trans with N0; [Apply le_trans with (max N1 N2); [Apply le_max_r | Unfold N0; Apply le_max_l] | Assumption]. +Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Repeat Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Clear x u x0 x1 eps H H0 N1 H1 N2 H2; Assert H1 : (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr1 n)] t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)). +Assert H2 : (EXT psi2:nat->(StepFun b c) | (n:nat) ((t:R)``(Rmin b c) <= t``/\``t <= (Rmax b c)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr2 n)] t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n)). +Assert H3 : (EXT psi3:nat->(StepFun a c) | (n:nat) ((t:R)``(Rmin a c) <= t``/\``t <= (Rmax a c)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr3 n)] t)))<= (psi3 n t)``)/\``(Rabsolu (RiemannInt_SF (psi3 n))) < (RinvN n)``). +Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr3 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr3 n)). +Elim H1; Clear H1; Intros psi1 H1; Elim H2; Clear H2; Intros psi2 H2; Elim H3; Clear H3; Intros psi3 H3; Assert H := RinvN_cv; Unfold Un_cv; Intros; Assert H4 : ``0<eps/3``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H ? H4); Clear H; Intros N0 H; Assert H5 : (n:nat)(ge n N0)->``(RinvN n)<eps/3``. +Intros; Replace (pos (RinvN n)) with ``(R_dist (mkposreal (/((INR n)+1)) (RinvN_pos n)) 0)``. +Apply H; Assumption. +Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (RinvN n)). +Exists N0; Intros; Elim (H1 n); Elim (H2 n); Elim (H3 n); Clear H1 H2 H3; Intros; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Pose phi1 := (phi_sequence RinvN pr1 n); Fold phi1 in H8; Pose phi2 := (phi_sequence RinvN pr2 n); Fold phi2 in H3; Pose phi3 := (phi_sequence RinvN pr3 n); Fold phi2 in H1; Assert H10 : (IsStepFun phi3 a b). +Apply StepFun_P44 with c. +Apply (pre phi3). +Split; Assumption. +Assert H11 : (IsStepFun (psi3 n) a b). +Apply StepFun_P44 with c. +Apply (pre (psi3 n)). +Split; Assumption. +Assert H12 : (IsStepFun phi3 b c). +Apply StepFun_P45 with a. +Apply (pre phi3). +Split; Assumption. +Assert H13 : (IsStepFun (psi3 n) b c). +Apply StepFun_P45 with a. +Apply (pre (psi3 n)). +Split; Assumption. +Replace (RiemannInt_SF phi3) with ``(RiemannInt_SF (mkStepFun H10))+(RiemannInt_SF (mkStepFun H12))``. +Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF (mkStepFun H10))-(RiemannInt_SF phi1)))+(Rabsolu ((RiemannInt_SF (mkStepFun H12))-(RiemannInt_SF phi2)))``. +Replace ``(RiemannInt_SF (mkStepFun H10))+(RiemannInt_SF (mkStepFun H12))+ -((RiemannInt_SF phi1)+(RiemannInt_SF phi2))`` with ``((RiemannInt_SF (mkStepFun H10))-(RiemannInt_SF phi1))+((RiemannInt_SF (mkStepFun H12))-(RiemannInt_SF phi2))``; [Apply Rabsolu_triang | Ring]. +Replace ``(RiemannInt_SF (mkStepFun H10))-(RiemannInt_SF phi1)`` with (RiemannInt_SF (mkStepFun (StepFun_P28 ``-1`` (mkStepFun H10) phi1))). +Replace ``(RiemannInt_SF (mkStepFun H12))-(RiemannInt_SF phi2)`` with (RiemannInt_SF (mkStepFun (StepFun_P28 ``-1`` (mkStepFun H12) phi2))). +Apply Rle_lt_trans with ``(RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))))+(RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2)))))``. +Apply Rle_trans with ``(Rabsolu (RiemannInt_SF (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1))))+(RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2)))))``. +Apply Rle_compatibility. +Apply StepFun_P34; Try Assumption. +Do 2 Rewrite <- (Rplus_sym (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (mkStepFun H12) phi2)))))); Apply Rle_compatibility; Apply StepFun_P34; Try Assumption. +Apply Rle_lt_trans with ``(RiemannInt_SF (mkStepFun (StepFun_P28 R1 (mkStepFun H11) (psi1 n))))+(RiemannInt_SF (mkStepFun (StepFun_P28 R1 (mkStepFun H13) (psi2 n))))``. +Apply Rle_trans with ``(RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))))+(RiemannInt_SF (mkStepFun (StepFun_P28 R1 (mkStepFun H13) (psi2 n))))``. +Apply Rle_compatibility; Apply StepFun_P37; Try Assumption. +Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu ((f x)-(phi3 x)))+(Rabsolu ((f x)-(phi2 x)))``. +Rewrite <- (Rabsolu_Ropp ``(f x)-(phi3 x)``); Rewrite Ropp_distr2; Replace ``(phi3 x)+ -1*(phi2 x)`` with ``((phi3 x)-(f x))+((f x)-(phi2 x))``; [Apply Rabsolu_triang | Ring]. +Apply Rplus_le. +Fold phi3 in H1; Apply H1. +Elim H14; Intros; Split. +Replace (Rmin a c) with a. +Apply Rle_trans with b; Try Assumption. +Left; Assumption. +Unfold Rmin; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. +Replace (Rmax a c) with c. +Left; Assumption. +Unfold Rmax; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. +Apply H3. +Elim H14; Intros; Split. +Replace (Rmin b c) with b. +Left; Assumption. +Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n0; Assumption]. +Replace (Rmax b c) with c. +Left; Assumption. +Unfold Rmax; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n0; Assumption]. +Do 2 Rewrite <- (Rplus_sym ``(RiemannInt_SF (mkStepFun (StepFun_P28 R1 (mkStepFun H13) (psi2 n))))``); Apply Rle_compatibility; Apply StepFun_P37; Try Assumption. +Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu ((f x)-(phi3 x)))+(Rabsolu ((f x)-(phi1 x)))``. +Rewrite <- (Rabsolu_Ropp ``(f x)-(phi3 x)``); Rewrite Ropp_distr2; Replace ``(phi3 x)+ -1*(phi1 x)`` with ``((phi3 x)-(f x))+((f x)-(phi1 x))``; [Apply Rabsolu_triang | Ring]. +Apply Rplus_le. +Apply H1. +Elim H14; Intros; Split. +Replace (Rmin a c) with a. +Left; Assumption. +Unfold Rmin; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. +Replace (Rmax a c) with c. +Apply Rle_trans with b. +Left; Assumption. +Assumption. +Unfold Rmax; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. +Apply H8. +Elim H14; Intros; Split. +Replace (Rmin a b) with a. +Left; Assumption. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Replace (Rmax a b) with b. +Left; Assumption. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. +Do 2 Rewrite StepFun_P30. +Do 2 Rewrite Rmult_1l; Replace ``(RiemannInt_SF (mkStepFun H11))+(RiemannInt_SF (psi1 n))+((RiemannInt_SF (mkStepFun H13))+(RiemannInt_SF (psi2 n)))`` with ``(RiemannInt_SF (psi3 n))+(RiemannInt_SF (psi1 n))+(RiemannInt_SF (psi2 n))``. +Replace eps with ``eps/3+eps/3+eps/3``. +Repeat Rewrite Rplus_assoc; Repeat Apply Rplus_lt. +Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi3 n))). +Apply Rle_Rabsolu. +Apply Rlt_trans with (pos (RinvN n)). +Assumption. +Apply H5; Assumption. +Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))). +Apply Rle_Rabsolu. +Apply Rlt_trans with (pos (RinvN n)). +Assumption. +Apply H5; Assumption. +Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))). +Apply Rle_Rabsolu. +Apply Rlt_trans with (pos (RinvN n)). +Assumption. +Apply H5; Assumption. +Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Repeat Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Replace (RiemannInt_SF (psi3 n)) with (RiemannInt_SF (mkStepFun (pre (psi3 n)))). +Rewrite <- (StepFun_P43 H11 H13 (pre (psi3 n))); Ring. +Reflexivity. +Rewrite StepFun_P30; Ring. +Rewrite StepFun_P30; Ring. +Apply (StepFun_P43 H10 H12 (pre phi3)). +Qed. + +Lemma RiemannInt_P26 : (f:R->R;a,b,c:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable f b c);pr3:(Riemann_integrable f a c)) ``(RiemannInt pr1)+(RiemannInt pr2)==(RiemannInt pr3)``. +Intros; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros. +Apply RiemannInt_P25; Assumption. +Case (total_order_Rle a c); Intro. +Assert H : ``c<=b``. +Auto with real. +Rewrite <- (RiemannInt_P25 pr3 (RiemannInt_P1 pr2) pr1 r0 H); Rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); Ring. +Assert H : ``c<=a``. +Auto with real. +Rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); Rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr3) pr1 (RiemannInt_P1 pr2) H r); Rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); Ring. +Assert H : ``b<=a``. +Auto with real. +Case (total_order_Rle a c); Intro. +Rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr1) pr3 pr2 H r0); Rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); Ring. +Assert H0 : ``c<=a``. +Auto with real. +Rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); Rewrite <- (RiemannInt_P25 pr2 (RiemannInt_P1 pr3) (RiemannInt_P1 pr1) r H0); Rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); Ring. +Rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); Rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); Rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); Rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr2) (RiemannInt_P1 pr1) (RiemannInt_P1 pr3)); [Ring | Auto with real | Auto with real]. +Qed. + +Lemma RiemannInt_P27 : (f:R->R;a,b,x:R;h:``a<=b``;C0:((x:R)``a<=x<=b``->(continuity_pt f x))) ``a<x<b`` -> (derivable_pt_lim (primitive h (FTC_P1 h C0)) x (f x)). +Intro f; Intros; Elim H; Clear H; Intros; Assert H1 : (continuity_pt f x). +Apply C0; Split; Left; Assumption. +Unfold derivable_pt_lim; Intros; Assert Hyp : ``0<eps/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H1 ? Hyp); Unfold dist D_x no_cond; Simpl; Unfold R_dist; Intros; Pose del := (Rmin x0 (Rmin ``b-x`` ``x-a``)); Assert H4 : ``0<del``. +Unfold del; Unfold Rmin; Case (total_order_Rle ``b-x`` ``x-a``); Intro. +Case (total_order_Rle x0 ``b-x``); Intro; [Elim H3; Intros; Assumption | Apply Rlt_Rminus; Assumption]. +Case (total_order_Rle x0 ``x-a``); Intro; [Elim H3; Intros; Assumption | Apply Rlt_Rminus; Assumption]. +Split with (mkposreal ? H4); Intros; Assert H7 : (Riemann_integrable f x ``x+h0``). +Case (total_order_Rle x ``x+h0``); Intro. +Apply continuity_implies_RiemannInt; Try Assumption. +Intros; Apply C0; Elim H7; Intros; Split. +Apply Rle_trans with x; [Left; Assumption | Assumption]. +Apply Rle_trans with ``x+h0``. +Assumption. +Left; Apply Rlt_le_trans with ``x+del``. +Apply Rlt_compatibility; Apply Rle_lt_trans with (Rabsolu h0); [Apply Rle_Rabsolu | Apply H6]. +Unfold del; Apply Rle_trans with ``x+(Rmin (b-x) (x-a))``. +Apply Rle_compatibility; Apply Rmin_r. +Pattern 2 b; Replace b with ``x+(b-x)``; [Apply Rle_compatibility; Apply Rmin_l | Ring]. +Apply RiemannInt_P1; Apply continuity_implies_RiemannInt; Auto with real. +Intros; Apply C0; Elim H7; Intros; Split. +Apply Rle_trans with ``x+h0``. +Left; Apply Rle_lt_trans with ``x-del``. +Unfold del; Apply Rle_trans with ``x-(Rmin (b-x) (x-a))``. +Pattern 1 a; Replace a with ``x+(a-x)``; [Idtac | Ring]. +Unfold Rminus; Apply Rle_compatibility; Apply Ropp_Rle. +Rewrite Ropp_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Rewrite (Rplus_sym x); Apply Rmin_r. +Unfold Rminus; Apply Rle_compatibility; Apply Ropp_Rle. +Do 2 Rewrite Ropp_Ropp; Apply Rmin_r. +Unfold Rminus; Apply Rlt_compatibility; Apply Ropp_Rlt. +Rewrite Ropp_Ropp; Apply Rle_lt_trans with (Rabsolu h0); [Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu | Apply H6]. +Assumption. +Apply Rle_trans with x; [Assumption | Left; Assumption]. +Replace ``(primitive h (FTC_P1 h C0) (x+h0))-(primitive h (FTC_P1 h C0) x)`` with (RiemannInt H7). +Replace (f x) with ``(RiemannInt (RiemannInt_P14 x (x+h0) (f x)))/h0``. +Replace ``(RiemannInt H7)/h0-(RiemannInt (RiemannInt_P14 x (x+h0) (f x)))/h0`` with ``((RiemannInt H7)-(RiemannInt (RiemannInt_P14 x (x+h0) (f x))))/h0``. +Replace ``(RiemannInt H7)-(RiemannInt (RiemannInt_P14 x (x+h0) (f x)))`` with (RiemannInt (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))). +Unfold Rdiv; Rewrite Rabsolu_mult; Case (total_order_Rle x ``x+h0``); Intro. +Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P16 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x+h0) (f x)))))*(Rabsolu (/h0))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. +Apply Rabsolu_pos. +Apply (RiemannInt_P17 (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x))) (RiemannInt_P16 (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x))))); Assumption. +Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P14 x (x+h0) (eps/2)))*(Rabsolu (/h0))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. +Apply Rabsolu_pos. +Apply RiemannInt_P19; Try Assumption. +Intros; Replace ``(f x1)+ -1*(fct_cte (f x) x1)`` with ``(f x1)-(f x)``. +Unfold fct_cte; Case (Req_EM x x1); Intro. +Rewrite H9; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. +Elim H3; Intros; Left; Apply H11. +Repeat Split. +Assumption. +Rewrite Rabsolu_right. +Apply Rlt_anti_compatibility with x; Replace ``x+(x1-x)`` with x1; [Idtac | Ring]. +Apply Rlt_le_trans with ``x+h0``. +Elim H8; Intros; Assumption. +Apply Rle_compatibility; Apply Rle_trans with del. +Left; Apply Rle_lt_trans with (Rabsolu h0); [Apply Rle_Rabsolu | Assumption]. +Unfold del; Apply Rmin_l. +Apply Rge_minus; Apply Rle_sym1; Left; Elim H8; Intros; Assumption. +Unfold fct_cte; Ring. +Rewrite RiemannInt_P15. +Rewrite Rmult_assoc; Replace ``(x+h0-x)*(Rabsolu (/h0))`` with R1. +Rewrite Rmult_1r; Unfold Rdiv; Apply Rlt_monotony_contra with ``2``; [Sup0 | Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Pattern 1 eps; Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Rewrite Rabsolu_right. +Replace ``x+h0-x`` with h0; [Idtac | Ring]. +Apply Rinv_r_sym. +Assumption. +Apply Rle_sym1; Left; Apply Rlt_Rinv. +Elim r; Intro. +Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Assumption. +Elim H5; Symmetry; Apply r_Rplus_plus with x; Rewrite Rplus_Or; Assumption. +Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P16 (RiemannInt_P1 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x+h0) (f x))))))*(Rabsolu (/h0))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. +Apply Rabsolu_pos. +Replace (RiemannInt (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))) with ``-(RiemannInt (RiemannInt_P1 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x+h0) (f x)))))``. +Rewrite Rabsolu_Ropp; Apply (RiemannInt_P17 (RiemannInt_P1 (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))) (RiemannInt_P16 (RiemannInt_P1 (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))))); Auto with real. +Symmetry; Apply RiemannInt_P8. +Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P14 (x+h0) x (eps/2)))*(Rabsolu (/h0))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. +Apply Rabsolu_pos. +Apply RiemannInt_P19. +Auto with real. +Intros; Replace ``(f x1)+ -1*(fct_cte (f x) x1)`` with ``(f x1)-(f x)``. +Unfold fct_cte; Case (Req_EM x x1); Intro. +Rewrite H9; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. +Elim H3; Intros; Left; Apply H11. +Repeat Split. +Assumption. +Rewrite Rabsolu_left. +Apply Rlt_anti_compatibility with ``x1-x0``; Replace ``x1-x0+x0`` with x1; [Idtac | Ring]. +Replace ``x1-x0+ -(x1-x)`` with ``x-x0``; [Idtac | Ring]. +Apply Rle_lt_trans with ``x+h0``. +Unfold Rminus; Apply Rle_compatibility; Apply Ropp_Rle. +Rewrite Ropp_Ropp; Apply Rle_trans with (Rabsolu h0). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Apply Rle_trans with del; [Left; Assumption | Unfold del; Apply Rmin_l]. +Elim H8; Intros; Assumption. +Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Replace ``x+(x1-x)`` with x1; [Elim H8; Intros; Assumption | Ring]. +Unfold fct_cte; Ring. +Rewrite RiemannInt_P15. +Rewrite Rmult_assoc; Replace ``(x-(x+h0))*(Rabsolu (/h0))`` with R1. +Rewrite Rmult_1r; Unfold Rdiv; Apply Rlt_monotony_contra with ``2``; [Sup0 | Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Pattern 1 eps; Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Rewrite Rabsolu_left. +Replace ``x-(x+h0)`` with ``-h0``; [Idtac | Ring]. +Rewrite Ropp_mul1; Rewrite Ropp_mul3; Rewrite Ropp_Ropp; Apply Rinv_r_sym. +Assumption. +Apply Rlt_Rinv2. +Assert H8 : ``x+h0<x``. +Auto with real. +Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Assumption. +Rewrite (RiemannInt_P13 H7 (RiemannInt_P14 x ``x+h0`` (f x)) (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))). +Ring. +Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring. +Rewrite RiemannInt_P15; Apply r_Rmult_mult with h0; [Unfold Rdiv; Rewrite -> (Rmult_sym h0); Repeat Rewrite -> Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | Assumption] | Assumption]. +Cut ``a<=x+h0``. +Cut ``x+h0<=b``. +Intros; Unfold primitive. +Case (total_order_Rle a ``x+h0``); Case (total_order_Rle ``x+h0`` b); Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; Try (Elim n; Assumption Orelse Left; Assumption). +Rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r0 r) H7 (FTC_P1 h C0 r2 r1)); Ring. +Apply Rle_anti_compatibility with ``-x``; Replace ``-x+(x+h0)`` with h0; [Idtac | Ring]. +Rewrite Rplus_sym; Apply Rle_trans with (Rabsolu h0). +Apply Rle_Rabsolu. +Apply Rle_trans with del; [Left; Assumption | Unfold del; Apply Rle_trans with ``(Rmin (b-x) (x-a))``; [Apply Rmin_r | Apply Rmin_l]]. +Apply Ropp_Rle; Apply Rle_anti_compatibility with ``x``; Replace ``x+-(x+h0)`` with ``-h0``; [Idtac | Ring]. +Apply Rle_trans with (Rabsolu h0); [Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu | Apply Rle_trans with del; [Left; Assumption | Unfold del; Apply Rle_trans with ``(Rmin (b-x) (x-a))``; Apply Rmin_r]]. +Qed. + +Lemma RiemannInt_P28 : (f:R->R;a,b,x:R;h:``a<=b``;C0:((x:R)``a<=x<=b``->(continuity_pt f x))) ``a<=x<=b`` -> (derivable_pt_lim (primitive h (FTC_P1 h C0)) x (f x)). +Intro f; Intros; Elim h; Intro. +Elim H; Clear H; Intros; Elim H; Intro. +Elim H1; Intro. +Apply RiemannInt_P27; Split; Assumption. +Pose f_b := [x:R]``(f b)*(x-b)+(RiemannInt [(FTC_P1 h C0 h (FTC_P2 b))])``; Rewrite H3. +Assert H4 : (derivable_pt_lim f_b b (f b)). +Unfold f_b; Pattern 2 (f b); Replace (f b) with ``(f b)+0``. +Change (derivable_pt_lim (plus_fct (mult_fct (fct_cte (f b)) (minus_fct id (fct_cte b))) (fct_cte (RiemannInt (FTC_P1 h C0 h (FTC_P2 b))))) b ``(f b)+0``). +Apply derivable_pt_lim_plus. +Pattern 2 (f b); Replace (f b) with ``0*((minus_fct id (fct_cte b)) b)+((fct_cte (f b)) b)*1``. +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_const. +Replace R1 with ``1-0``; [Idtac | Ring]. +Apply derivable_pt_lim_minus. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_const. +Unfold fct_cte; Ring. +Apply derivable_pt_lim_const. +Ring. +Unfold derivable_pt_lim; Intros; Elim (H4 ? H5); Intros; Assert H7 : (continuity_pt f b). +Apply C0; Split; [Left; Assumption | Right; Reflexivity]. +Assert H8 : ``0<eps/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H7 ? H8); Unfold D_x no_cond dist; Simpl; Unfold R_dist; Intros; Pose del := (Rmin x0 (Rmin x1 ``b-a``)); Assert H10 : ``0<del``. +Unfold del; Unfold Rmin; Case (total_order_Rle x1 ``b-a``); Intros. +Case (total_order_Rle x0 x1); Intro; [Apply (cond_pos x0) | Elim H9; Intros; Assumption]. +Case (total_order_Rle x0 ``b-a``); Intro; [Apply (cond_pos x0) | Apply Rlt_Rminus; Assumption]. +Split with (mkposreal ? H10); Intros; Case (case_Rabsolu h0); Intro. +Assert H14 : ``b+h0<b``. +Pattern 2 b; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. +Assert H13 : (Riemann_integrable f ``b+h0`` b). +Apply continuity_implies_RiemannInt. +Left; Assumption. +Intros; Apply C0; Elim H13; Intros; Split; Try Assumption. +Apply Rle_trans with ``b+h0``; Try Assumption. +Apply Rle_anti_compatibility with ``-a-h0``. +Replace ``-a-h0+a`` with ``-h0``; [Idtac | Ring]. +Replace ``-a-h0+(b+h0)`` with ``b-a``; [Idtac | Ring]. +Apply Rle_trans with del. +Apply Rle_trans with (Rabsolu h0). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Left; Assumption. +Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); Apply Rmin_r. +Replace ``[(primitive h (FTC_P1 h C0) (b+h0))]-[(primitive h (FTC_P1 h C0) b)]`` with ``-(RiemannInt H13)``. +Replace (f b) with ``-[(RiemannInt (RiemannInt_P14 (b+h0) b (f b)))]/h0``. +Rewrite <- Rabsolu_Ropp; Unfold Rminus; Unfold Rdiv; Rewrite Ropp_mul1; Rewrite Ropp_distr1; Repeat Rewrite Ropp_Ropp; Replace ``(RiemannInt H13)*/h0+ -(RiemannInt (RiemannInt_P14 (b+h0) b (f b)))*/h0`` with ``((RiemannInt H13)-(RiemannInt (RiemannInt_P14 (b+h0) b (f b))))/h0``. +Replace ``(RiemannInt H13)-(RiemannInt (RiemannInt_P14 (b+h0) b (f b)))`` with (RiemannInt (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 ``b+h0`` b (f b)))). +Unfold Rdiv; Rewrite Rabsolu_mult; Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P16 (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b+h0) b (f b)))))*(Rabsolu (/h0))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. +Apply Rabsolu_pos. +Apply (RiemannInt_P17 (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 ``b+h0`` b (f b))) (RiemannInt_P16 (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 ``b+h0`` b (f b))))); Left; Assumption. +Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P14 (b+h0) b (eps/2)))*(Rabsolu (/h0))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. +Apply Rabsolu_pos. +Apply RiemannInt_P19. +Left; Assumption. +Intros; Replace ``(f x2)+ -1*(fct_cte (f b) x2)`` with ``(f x2)-(f b)``. +Unfold fct_cte; Case (Req_EM b x2); Intro. +Rewrite H16; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. +Elim H9; Intros; Left; Apply H18. +Repeat Split. +Assumption. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Rewrite Rabsolu_right. +Apply Rlt_anti_compatibility with ``x2-x1``; Replace ``x2-x1+(b-x2)`` with ``b-x1``; [Idtac | Ring]. +Replace ``x2-x1+x1`` with x2; [Idtac | Ring]. +Apply Rlt_le_trans with ``b+h0``. +2:Elim H15; Intros; Left; Assumption. +Unfold Rminus; Apply Rlt_compatibility; Apply Ropp_Rlt; Rewrite Ropp_Ropp; Apply Rle_lt_trans with (Rabsolu h0). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Apply Rlt_le_trans with del; [Assumption | Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); [Apply Rmin_r | Apply Rmin_l]]. +Apply Rle_sym1; Left; Apply Rlt_Rminus; Elim H15; Intros; Assumption. +Unfold fct_cte; Ring. +Rewrite RiemannInt_P15. +Rewrite Rmult_assoc; Replace ``(b-(b+h0))*(Rabsolu (/h0))`` with R1. +Rewrite Rmult_1r; Unfold Rdiv; Apply Rlt_monotony_contra with ``2``; [Sup0 | Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Pattern 1 eps; Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Rewrite Rabsolu_left. +Apply r_Rmult_mult with h0; [Do 2 Rewrite (Rmult_sym h0); Rewrite Rmult_assoc; Rewrite Ropp_mul1; Rewrite <- Rinv_l_sym; [ Ring | Assumption ] | Assumption]. +Apply Rlt_Rinv2; Assumption. +Rewrite (RiemannInt_P13 H13 (RiemannInt_P14 ``b+h0`` b (f b)) (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 ``b+h0`` b (f b)))); Ring. +Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring. +Rewrite RiemannInt_P15. +Rewrite <- Ropp_mul1; Apply r_Rmult_mult with h0; [Repeat Rewrite (Rmult_sym h0); Unfold Rdiv; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | Assumption] | Assumption]. +Cut ``a<=b+h0``. +Cut ``b+h0<=b``. +Intros; Unfold primitive; Case (total_order_Rle a ``b+h0``); Case (total_order_Rle ``b+h0`` b); Case (total_order_Rle a b); Case (total_order_Rle b b); Intros; Try (Elim n; Right; Reflexivity) Orelse (Elim n; Left; Assumption). +Rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r3 r2) H13 (FTC_P1 h C0 r1 r0)); Ring. +Elim n; Assumption. +Left; Assumption. +Apply Rle_anti_compatibility with ``-a-h0``. +Replace ``-a-h0+a`` with ``-h0``; [Idtac | Ring]. +Replace ``-a-h0+(b+h0)`` with ``b-a``; [Idtac | Ring]. +Apply Rle_trans with del. +Apply Rle_trans with (Rabsolu h0). +Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. +Left; Assumption. +Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); Apply Rmin_r. +Cut (primitive h (FTC_P1 h C0) b)==(f_b b). +Intro; Cut (primitive h (FTC_P1 h C0) ``b+h0``)==(f_b ``b+h0``). +Intro; Rewrite H13; Rewrite H14; Apply H6. +Assumption. +Apply Rlt_le_trans with del; [Assumption | Unfold del; Apply Rmin_l]. +Assert H14 : ``b<b+h0``. +Pattern 1 b; Rewrite <- Rplus_Or; Apply Rlt_compatibility. +Assert H14 := (Rle_sym2 ? ? r); Elim H14; Intro. +Assumption. +Elim H11; Symmetry; Assumption. +Unfold primitive; Case (total_order_Rle a ``b+h0``); Case (total_order_Rle ``b+h0`` b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 H14)) | Unfold f_b; Reflexivity | Elim n; Left; Apply Rlt_trans with b; Assumption | Elim n0; Left; Apply Rlt_trans with b; Assumption]. +Unfold f_b; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rmult_Or; Rewrite Rplus_Ol; Unfold primitive; Case (total_order_Rle a b); Case (total_order_Rle b b); Intros; [Apply RiemannInt_P5 | Elim n; Right; Reflexivity | Elim n; Left; Assumption | Elim n; Right; Reflexivity]. +(*****) +Pose f_a := [x:R]``(f a)*(x-a)``; Rewrite <- H2; Assert H3 : (derivable_pt_lim f_a a (f a)). +Unfold f_a; Change (derivable_pt_lim (mult_fct (fct_cte (f a)) (minus_fct id (fct_cte a))) a (f a)); Pattern 2 (f a); Replace (f a) with ``0*((minus_fct id (fct_cte a)) a)+((fct_cte (f a)) a)*1``. +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_const. +Replace R1 with ``1-0``; [Idtac | Ring]. +Apply derivable_pt_lim_minus. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_const. +Unfold fct_cte; Ring. +Unfold derivable_pt_lim; Intros; Elim (H3 ? H4); Intros. +Assert H6 : (continuity_pt f a). +Apply C0; Split; [Right; Reflexivity | Left; Assumption]. +Assert H7 : ``0<eps/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Elim (H6 ? H7); Unfold D_x no_cond dist; Simpl; Unfold R_dist; Intros. +Pose del := (Rmin x0 (Rmin x1 ``b-a``)). +Assert H9 : ``0<del``. +Unfold del; Unfold Rmin. +Case (total_order_Rle x1 ``b-a``); Intros. +Case (total_order_Rle x0 x1); Intro. +Apply (cond_pos x0). +Elim H8; Intros; Assumption. +Case (total_order_Rle x0 ``b-a``); Intro. +Apply (cond_pos x0). +Apply Rlt_Rminus; Assumption. +Split with (mkposreal ? H9). +Intros; Case (case_Rabsolu h0); Intro. +Assert H12 : ``a+h0<a``. +Pattern 2 a; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. +Unfold primitive. +Case (total_order_Rle a ``a+h0``); Case (total_order_Rle ``a+h0`` b); Case (total_order_Rle a a); Case (total_order_Rle a b); Intros; Try (Elim n; Left; Assumption) Orelse (Elim n; Right; Reflexivity). +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r3 H12)). +Elim n; Left; Apply Rlt_trans with a; Assumption. +Rewrite RiemannInt_P9; Replace R0 with (f_a a). +Replace ``(f a)*(a+h0-a)`` with (f_a ``a+h0``). +Apply H5; Try Assumption. +Apply Rlt_le_trans with del; [Assumption | Unfold del; Apply Rmin_l]. +Unfold f_a; Ring. +Unfold f_a; Ring. +Elim n; Left; Apply Rlt_trans with a; Assumption. +Assert H12 : ``a<a+h0``. +Pattern 1 a; Rewrite <- Rplus_Or; Apply Rlt_compatibility. +Assert H12 := (Rle_sym2 ? ? r); Elim H12; Intro. +Assumption. +Elim H10; Symmetry; Assumption. +Assert H13 : (Riemann_integrable f a ``a+h0``). +Apply continuity_implies_RiemannInt. +Left; Assumption. +Intros; Apply C0; Elim H13; Intros; Split; Try Assumption. +Apply Rle_trans with ``a+h0``; Try Assumption. +Apply Rle_anti_compatibility with ``-b-h0``. +Replace ``-b-h0+b`` with ``-h0``; [Idtac | Ring]. +Replace ``-b-h0+(a+h0)`` with ``a-b``; [Idtac | Ring]. +Apply Ropp_Rle; Rewrite Ropp_Ropp; Rewrite Ropp_distr2; Apply Rle_trans with del. +Apply Rle_trans with (Rabsolu h0); [Apply Rle_Rabsolu | Left; Assumption]. +Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); Apply Rmin_r. +Replace ``(primitive h (FTC_P1 h C0) (a+h0))-(primitive h (FTC_P1 h C0) a)`` with ``(RiemannInt H13)``. +Replace (f a) with ``(RiemannInt (RiemannInt_P14 a (a+h0) (f a)))/h0``. +Replace ``(RiemannInt H13)/h0-(RiemannInt (RiemannInt_P14 a (a+h0) (f a)))/h0`` with ``((RiemannInt H13)-(RiemannInt (RiemannInt_P14 a (a+h0) (f a))))/h0``. +Replace ``(RiemannInt H13)-(RiemannInt (RiemannInt_P14 a (a+h0) (f a)))`` with (RiemannInt (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 a ``a+h0`` (f a)))). +Unfold Rdiv; Rewrite Rabsolu_mult; Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P16 (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a+h0) (f a)))))*(Rabsolu (/h0))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. +Apply Rabsolu_pos. +Apply (RiemannInt_P17 (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 a ``a+h0`` (f a))) (RiemannInt_P16 (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 a ``a+h0`` (f a))))); Left; Assumption. +Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P14 a (a+h0) (eps/2)))*(Rabsolu (/h0))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. +Apply Rabsolu_pos. +Apply RiemannInt_P19. +Left; Assumption. +Intros; Replace ``(f x2)+ -1*(fct_cte (f a) x2)`` with ``(f x2)-(f a)``. +Unfold fct_cte; Case (Req_EM a x2); Intro. +Rewrite H15; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. +Elim H8; Intros; Left; Apply H17; Repeat Split. +Assumption. +Rewrite Rabsolu_right. +Apply Rlt_anti_compatibility with a; Replace ``a+(x2-a)`` with x2; [Idtac | Ring]. +Apply Rlt_le_trans with ``a+h0``. +Elim H14; Intros; Assumption. +Apply Rle_compatibility; Left; Apply Rle_lt_trans with (Rabsolu h0). +Apply Rle_Rabsolu. +Apply Rlt_le_trans with del; [Assumption | Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); [Apply Rmin_r | Apply Rmin_l]]. +Apply Rle_sym1; Left; Apply Rlt_Rminus; Elim H14; Intros; Assumption. +Unfold fct_cte; Ring. +Rewrite RiemannInt_P15. +Rewrite Rmult_assoc; Replace ``((a+h0)-a)*(Rabsolu (/h0))`` with R1. +Rewrite Rmult_1r; Unfold Rdiv; Apply Rlt_monotony_contra with ``2``; [Sup0 | Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Pattern 1 eps; Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Rewrite Rabsolu_right. +Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Rewrite <- Rinv_r_sym; [ Reflexivity | Assumption ]. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Assert H14 := (Rle_sym2 ? ? r); Elim H14; Intro. +Assumption. +Elim H10; Symmetry; Assumption. +Rewrite (RiemannInt_P13 H13 (RiemannInt_P14 a ``a+h0`` (f a)) (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 a ``a+h0`` (f a)))); Ring. +Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring. +Rewrite RiemannInt_P15. +Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [ Ring | Assumption ]. +Cut ``a<=a+h0``. +Cut ``a+h0<=b``. +Intros; Unfold primitive; Case (total_order_Rle a ``a+h0``); Case (total_order_Rle ``a+h0`` b); Case (total_order_Rle a a); Case (total_order_Rle a b); Intros; Try (Elim n; Right; Reflexivity) Orelse (Elim n; Left; Assumption). +Rewrite RiemannInt_P9; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply RiemannInt_P5. +Elim n; Assumption. +Elim n; Assumption. +2:Left; Assumption. +Apply Rle_anti_compatibility with ``-a``; Replace ``-a+(a+h0)`` with h0; [Idtac | Ring]. +Rewrite Rplus_sym; Apply Rle_trans with del; [Apply Rle_trans with (Rabsolu h0); [Apply Rle_Rabsolu | Left; Assumption] | Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); Apply Rmin_r]. +(*****) +Assert H1 : x==a. +Rewrite <- H0 in H; Elim H; Intros; Apply Rle_antisym; Assumption. +Pose f_a := [x:R]``(f a)*(x-a)``. +Assert H2 : (derivable_pt_lim f_a a (f a)). +Unfold f_a; Change (derivable_pt_lim (mult_fct (fct_cte (f a)) (minus_fct id (fct_cte a))) a (f a)); Pattern 2 (f a); Replace (f a) with ``0*((minus_fct id (fct_cte a)) a)+((fct_cte (f a)) a)*1``. +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_const. +Replace R1 with ``1-0``; [Idtac | Ring]. +Apply derivable_pt_lim_minus. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_const. +Unfold fct_cte; Ring. +Pose f_b := [x:R]``(f b)*(x-b)+(RiemannInt (FTC_P1 h C0 b h (FTC_P2 b)))``. +Assert H3 : (derivable_pt_lim f_b b (f b)). +Unfold f_b; Pattern 2 (f b); Replace (f b) with ``(f b)+0``. +Change (derivable_pt_lim (plus_fct (mult_fct (fct_cte (f b)) (minus_fct id (fct_cte b))) (fct_cte (RiemannInt (FTC_P1 h C0 h (FTC_P2 b))))) b ``(f b)+0``). +Apply derivable_pt_lim_plus. +Pattern 2 (f b); Replace (f b) with ``0*((minus_fct id (fct_cte b)) b)+((fct_cte (f b)) b)*1``. +Apply derivable_pt_lim_mult. +Apply derivable_pt_lim_const. +Replace R1 with ``1-0``; [Idtac | Ring]. +Apply derivable_pt_lim_minus. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_const. +Unfold fct_cte; Ring. +Apply derivable_pt_lim_const. +Ring. +Unfold derivable_pt_lim; Intros; Elim (H2 ? H4); Intros; Elim (H3 ? H4); Intros; Pose del := (Rmin x0 x1). +Assert H7 : ``0<del``. +Unfold del; Unfold Rmin; Case (total_order_Rle x0 x1); Intro. +Apply (cond_pos x0). +Apply (cond_pos x1). +Split with (mkposreal ? H7); Intros; Case (case_Rabsolu h0); Intro. +Assert H10 : ``a+h0<a``. +Pattern 2 a; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. +Rewrite H1; Unfold primitive; Case (total_order_Rle a ``a+h0``); Case (total_order_Rle ``a+h0`` b); Case (total_order_Rle a a); Case (total_order_Rle a b); Intros; Try (Elim n; Right; Assumption Orelse Reflexivity). +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r3 H10)). +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r2 H10)). +Rewrite RiemannInt_P9; Replace R0 with (f_a a). +Replace ``(f a)*(a+h0-a)`` with (f_a ``a+h0``). +Apply H5; Try Assumption. +Apply Rlt_le_trans with del; Try Assumption. +Unfold del; Apply Rmin_l. +Unfold f_a; Ring. +Unfold f_a; Ring. +Elim n; Rewrite <- H0; Left; Assumption. +Assert H10 : ``a<a+h0``. +Pattern 1 a; Rewrite <- Rplus_Or; Apply Rlt_compatibility. +Assert H10 := (Rle_sym2 ? ? r); Elim H10; Intro. +Assumption. +Elim H8; Symmetry; Assumption. +Rewrite H0 in H1; Rewrite H1; Unfold primitive; Case (total_order_Rle a ``b+h0``); Case (total_order_Rle ``b+h0`` b); Case (total_order_Rle a b); Case (total_order_Rle b b); Intros; Try (Elim n; Right; Assumption Orelse Reflexivity). +Rewrite H0 in H10; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r2 H10)). +Repeat Rewrite RiemannInt_P9. +Replace (RiemannInt (FTC_P1 h C0 r1 r0)) with (f_b b). +Fold (f_b ``b+h0``). +Apply H6; Try Assumption. +Apply Rlt_le_trans with del; Try Assumption. +Unfold del; Apply Rmin_r. +Unfold f_b; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rmult_Or; Rewrite Rplus_Ol; Apply RiemannInt_P5. +Elim n; Rewrite <- H0; Left; Assumption. +Elim n0; Rewrite <- H0; Left; Assumption. +Qed. + +Lemma RiemannInt_P29 : (f:R->R;a,b;h:``a<=b``;C0:((x:R)``a<=x<=b``->(continuity_pt f x))) (antiderivative f (primitive h (FTC_P1 h C0)) a b). +Intro f; Intros; Unfold antiderivative; Split; Try Assumption; Intros; Assert H0 := (RiemannInt_P28 h C0 H); Assert H1 : (derivable_pt (primitive h (FTC_P1 h C0)) x); [Unfold derivable_pt; Split with (f x); Apply H0 | Split with H1; Symmetry; Apply derive_pt_eq_0; Apply H0]. +Qed. + +Lemma RiemannInt_P30 : (f:R->R;a,b:R) ``a<=b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> (sigTT ? [g:R->R](antiderivative f g a b)). +Intros; Split with (primitive H (FTC_P1 H H0)); Apply RiemannInt_P29. +Qed. + +Record C1_fun : Type := mkC1 { +c1 :> R->R; +diff0 : (derivable c1); +cont1 : (continuity (derive c1 diff0)) }. + +Lemma RiemannInt_P31 : (f:C1_fun;a,b:R) ``a<=b`` -> (antiderivative (derive f (diff0 f)) f a b). +Intro f; Intros; Unfold antiderivative; Split; Try Assumption; Intros; Split with (diff0 f x); Reflexivity. +Qed. + +Lemma RiemannInt_P32 : (f:C1_fun;a,b:R) (Riemann_integrable (derive f (diff0 f)) a b). +Intro f; Intros; Case (total_order_Rle a b); Intro; [Apply continuity_implies_RiemannInt; Try Assumption; Intros; Apply (cont1 f) | Assert H : ``b<=a``; [Auto with real | Apply RiemannInt_P1; Apply continuity_implies_RiemannInt; Try Assumption; Intros; Apply (cont1 f)]]. +Qed. + +Lemma RiemannInt_P33 : (f:C1_fun;a,b:R;pr:(Riemann_integrable (derive f (diff0 f)) a b)) ``a<=b`` -> (RiemannInt pr)==``(f b)-(f a)``. +Intro f; Intros; Assert H0 : (x:R)``a<=x<=b``->(continuity_pt (derive f (diff0 f)) x). +Intros; Apply (cont1 f). +Rewrite (RiemannInt_P20 H (FTC_P1 H H0) pr); Assert H1 := (RiemannInt_P29 H H0); Assert H2 := (RiemannInt_P31 f H); Elim (antiderivative_Ucte (derive f (diff0 f)) ? ? ? ? H1 H2); Intros C H3; Repeat Rewrite H3; [Ring | Split; [Right; Reflexivity | Assumption] | Split; [Assumption | Right; Reflexivity]]. +Qed. + +Lemma FTC_Riemann : (f:C1_fun;a,b:R;pr:(Riemann_integrable (derive f (diff0 f)) a b)) (RiemannInt pr)==``(f b)-(f a)``. +Intro f; Intros; Case (total_order_Rle a b); Intro; [Apply RiemannInt_P33; Assumption | Assert H : ``b<=a``; [Auto with real | Assert H0 := (RiemannInt_P1 pr); Rewrite (RiemannInt_P8 pr H0); Rewrite (RiemannInt_P33 H0 H); Ring]]. +Qed. diff --git a/theories7/Reals/RiemannInt_SF.v b/theories7/Reals/RiemannInt_SF.v new file mode 100644 index 00000000..3e2cc457 --- /dev/null +++ b/theories7/Reals/RiemannInt_SF.v @@ -0,0 +1,1400 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: RiemannInt_SF.v,v 1.2.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis. +Require Classical_Prop. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Implicit Arguments On. + +(**************************************************) +(* Each bounded subset of N has a maximal element *) +(**************************************************) + +Definition Nbound [I:nat->Prop] : Prop := (EX n:nat | (i:nat)(I i)->(le i n)). + +Lemma IZN_var:(z:Z)(`0<=z`)->{ n:nat | z=(INZ n)}. +Intros; Apply inject_nat_complete_inf; Assumption. +Qed. + +Lemma Nzorn : (I:nat->Prop) (EX n:nat | (I n)) -> (Nbound I) -> (sigTT ? [n:nat](I n)/\(i:nat)(I i)->(le i n)). +Intros I H H0; Pose E := [x:R](EX i:nat | (I i)/\(INR i)==x); Assert H1 : (bound E). +Unfold Nbound in H0; Elim H0; Intros N H1; Unfold bound; Exists (INR N); Unfold is_upper_bound; Intros; Unfold E in H2; Elim H2; Intros; Elim H3; Intros; Rewrite <- H5; Apply le_INR; Apply H1; Assumption. +Assert H2 : (EXT x:R | (E x)). +Elim H; Intros; Exists (INR x); Unfold E; Exists x; Split; [Assumption | Reflexivity]. +Assert H3 := (complet E H1 H2); Elim H3; Intros; Unfold is_lub in p; Elim p; Clear p; Intros; Unfold is_upper_bound in H4 H5; Assert H6 : ``0<=x``. +Elim H2; Intros; Unfold E in H6; Elim H6; Intros; Elim H7; Intros; Apply Rle_trans with x0; [Rewrite <- H9; Change ``(INR O)<=(INR x1)``; Apply le_INR; Apply le_O_n | Apply H4; Assumption]. +Assert H7 := (archimed x); Elim H7; Clear H7; Intros; Assert H9 : ``x<=(IZR (up x))-1``. +Apply H5; Intros; Assert H10 := (H4 ? H9); Unfold E in H9; Elim H9; Intros; Elim H11; Intros; Rewrite <- H13; Apply Rle_anti_compatibility with R1; Replace ``1+((IZR (up x))-1)`` with (IZR (up x)); [Idtac | Ring]; Replace ``1+(INR x1)`` with (INR (S x1)); [Idtac | Rewrite S_INR; Ring]. +Assert H14 : `0<=(up x)`. +Apply le_IZR; Apply Rle_trans with x; [Apply H6 | Left; Assumption]. +Assert H15 := (IZN ? H14); Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- INR_IZR_INZ; Apply le_INR; Apply lt_le_S; Apply INR_lt; Rewrite H13; Apply Rle_lt_trans with x; [Assumption | Rewrite INR_IZR_INZ; Rewrite <- H15; Assumption]. +Assert H10 : ``x==(IZR (up x))-1``. +Apply Rle_antisym; [Assumption | Apply Rle_anti_compatibility with ``-x+1``; Replace `` -x+1+((IZR (up x))-1)`` with ``(IZR (up x))-x``; [Idtac | Ring]; Replace ``-x+1+x`` with R1; [Assumption | Ring]]. +Assert H11 : `0<=(up x)`. +Apply le_IZR; Apply Rle_trans with x; [Apply H6 | Left; Assumption]. +Assert H12 := (IZN_var H11); Elim H12; Clear H12; Intros; Assert H13 : (E x). +Elim (classic (E x)); Intro; Try Assumption. +Cut ((y:R)(E y)->``y<=x-1``). +Intro; Assert H14 := (H5 ? H13); Cut ``x-1<x``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H14 H15)). +Apply Rminus_lt; Replace ``x-1-x`` with ``-1``; [Idtac | Ring]; Rewrite <- Ropp_O; Apply Rlt_Ropp; Apply Rlt_R0_R1. +Intros; Assert H14 := (H4 ? H13); Elim H14; Intro; Unfold E in H13; Elim H13; Intros; Elim H16; Intros; Apply Rle_anti_compatibility with R1. +Replace ``1+(x-1)`` with x; [Idtac | Ring]; Rewrite <- H18; Replace ``1+(INR x1)`` with (INR (S x1)); [Idtac | Rewrite S_INR; Ring]. +Cut x==(INR (pred x0)). +Intro; Rewrite H19; Apply le_INR; Apply lt_le_S; Apply INR_lt; Rewrite H18; Rewrite <- H19; Assumption. +Rewrite H10; Rewrite p; Rewrite <- INR_IZR_INZ; Replace R1 with (INR (S O)); [Idtac | Reflexivity]; Rewrite <- minus_INR. +Replace (minus x0 (S O)) with (pred x0); [Reflexivity | Case x0; [Reflexivity | Intro; Simpl; Apply minus_n_O]]. +Induction x0; [Rewrite p in H7; Rewrite <- INR_IZR_INZ in H7; Simpl in H7; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H6 H7)) | Apply le_n_S; Apply le_O_n]. +Rewrite H15 in H13; Elim H12; Assumption. +Split with (pred x0); Unfold E in H13; Elim H13; Intros; Elim H12; Intros; Rewrite H10 in H15; Rewrite p in H15; Rewrite <- INR_IZR_INZ in H15; Assert H16 : ``(INR x0)==(INR x1)+1``. +Rewrite H15; Ring. +Rewrite <- S_INR in H16; Assert H17 := (INR_eq ? ? H16); Rewrite H17; Simpl; Split. +Assumption. +Intros; Apply INR_le; Rewrite H15; Rewrite <- H15; Elim H12; Intros; Rewrite H20; Apply H4; Unfold E; Exists i; Split; [Assumption | Reflexivity]. +Qed. + +(*******************************************) +(* Step functions *) +(*******************************************) + +Definition open_interval [a,b:R] : R->Prop := [x:R]``a<x<b``. +Definition co_interval [a,b:R] : R->Prop := [x:R]``a<=x<b``. + +Definition adapted_couple [f:R->R;a,b:R;l,lf:Rlist] : Prop := (ordered_Rlist l)/\``(pos_Rl l O)==(Rmin a b)``/\``(pos_Rl l (pred (Rlength l)))==(Rmax a b)``/\(Rlength l)=(S (Rlength lf))/\(i:nat)(lt i (pred (Rlength l)))->(constant_D_eq f (open_interval (pos_Rl l i) (pos_Rl l (S i))) (pos_Rl lf i)). + +Definition adapted_couple_opt [f:R->R;a,b:R;l,lf:Rlist] := (adapted_couple f a b l lf)/\((i:nat)(lt i (pred (Rlength lf)))->(``(pos_Rl lf i)<>(pos_Rl lf (S i))``\/``(f (pos_Rl l (S i)))<>(pos_Rl lf i)``))/\((i:nat)(lt i (pred (Rlength l)))->``(pos_Rl l i)<>(pos_Rl l (S i))``). + +Definition is_subdivision [f:R->R;a,b:R;l:Rlist] : Type := (sigTT ? [l0:Rlist](adapted_couple f a b l l0)). + +Definition IsStepFun [f:R->R;a,b:R] : Type := (SigT ? [l:Rlist](is_subdivision f a b l)). + +(* Class of step functions *) +Record StepFun [a,b:R] : Type := mkStepFun { + fe:> R->R; + pre:(IsStepFun fe a b)}. + +Definition subdivision [a,b:R;f:(StepFun a b)] : Rlist := (projT1 ? ? (pre f)). + +Definition subdivision_val [a,b:R;f:(StepFun a b)] : Rlist := Cases (projT2 ? ? (pre f)) of (existTT a b) => a end. + +Fixpoint Int_SF [l:Rlist] : Rlist -> R := +[k:Rlist] Cases l of +| nil => R0 +| (cons a l') => Cases k of + | nil => R0 + | (cons x nil) => R0 + | (cons x (cons y k')) => ``a*(y-x)+(Int_SF l' (cons y k'))`` + end +end. + +(* Integral of step functions *) +Definition RiemannInt_SF [a,b:R;f:(StepFun a b)] : R := +Cases (total_order_Rle a b) of + (leftT _) => (Int_SF (subdivision_val f) (subdivision f)) +| (rightT _) => ``-(Int_SF (subdivision_val f) (subdivision f))`` +end. + +(********************************) +(* Properties of step functions *) +(********************************) + +Lemma StepFun_P1 : (a,b:R;f:(StepFun a b)) (adapted_couple f a b (subdivision f) (subdivision_val f)). +Intros a b f; Unfold subdivision_val; Case (projT2 Rlist ([l:Rlist](is_subdivision f a b l)) (pre f)); Intros; Apply a0. +Qed. + +Lemma StepFun_P2 : (a,b:R;f:R->R;l,lf:Rlist) (adapted_couple f a b l lf) -> (adapted_couple f b a l lf). +Unfold adapted_couple; Intros; Decompose [and] H; Clear H; Repeat Split; Try Assumption. +Rewrite H2; Unfold Rmin; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. +Apply Rle_antisym; Assumption. +Apply Rle_antisym; Auto with real. +Rewrite H1; Unfold Rmax; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. +Apply Rle_antisym; Assumption. +Apply Rle_antisym; Auto with real. +Qed. + +Lemma StepFun_P3 : (a,b,c:R) ``a<=b`` -> (adapted_couple (fct_cte c) a b (cons a (cons b nil)) (cons c nil)). +Intros; Unfold adapted_couple; Repeat Split. +Unfold ordered_Rlist; Intros; Simpl in H0; Inversion H0; [Simpl; Assumption | Elim (le_Sn_O ? H2)]. +Simpl; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Simpl; Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Unfold constant_D_eq open_interval; Intros; Simpl in H0; Inversion H0; [Reflexivity | Elim (le_Sn_O ? H3)]. +Qed. + +Lemma StepFun_P4 : (a,b,c:R) (IsStepFun (fct_cte c) a b). +Intros; Unfold IsStepFun; Case (total_order_Rle a b); Intro. +Apply Specif.existT with (cons a (cons b nil)); Unfold is_subdivision; Apply existTT with (cons c nil); Apply (StepFun_P3 c r). +Apply Specif.existT with (cons b (cons a nil)); Unfold is_subdivision; Apply existTT with (cons c nil); Apply StepFun_P2; Apply StepFun_P3; Auto with real. +Qed. + +Lemma StepFun_P5 : (a,b:R;f:R->R;l:Rlist) (is_subdivision f a b l) -> (is_subdivision f b a l). +Unfold is_subdivision; Intros; Elim X; Intros; Exists x; Unfold adapted_couple in p; Decompose [and] p; Clear p; Unfold adapted_couple; Repeat Split; Try Assumption. +Rewrite H1; Unfold Rmin; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. +Apply Rle_antisym; Assumption. +Apply Rle_antisym; Auto with real. +Rewrite H0; Unfold Rmax; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. +Apply Rle_antisym; Assumption. +Apply Rle_antisym; Auto with real. +Qed. + +Lemma StepFun_P6 : (f:R->R;a,b:R) (IsStepFun f a b) -> (IsStepFun f b a). +Unfold IsStepFun; Intros; Elim X; Intros; Apply Specif.existT with x; Apply StepFun_P5; Assumption. +Qed. + +Lemma StepFun_P7 : (a,b,r1,r2,r3:R;f:R->R;l,lf:Rlist) ``a<=b`` -> (adapted_couple f a b (cons r1 (cons r2 l)) (cons r3 lf)) -> (adapted_couple f r2 b (cons r2 l) lf). +Unfold adapted_couple; Intros; Decompose [and] H0; Clear H0; Assert H5 : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Assert H7 : ``r2<=b``. +Rewrite H5 in H2; Rewrite <- H2; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity]. +Repeat Split. +Apply RList_P4 with r1; Assumption. +Rewrite H5 in H2; Unfold Rmin; Case (total_order_Rle r2 b); Intro; [Reflexivity | Elim n; Assumption]. +Unfold Rmax; Case (total_order_Rle r2 b); Intro; [Rewrite H5 in H2; Rewrite <- H2; Reflexivity | Elim n; Assumption]. +Simpl in H4; Simpl; Apply INR_eq; Apply r_Rplus_plus with R1; Do 2 Rewrite (Rplus_sym R1); Do 2 Rewrite <- S_INR; Rewrite H4; Reflexivity. +Intros; Unfold constant_D_eq open_interval; Intros; Unfold constant_D_eq open_interval in H6; Assert H9 : (lt (S i) (pred (Rlength (cons r1 (cons r2 l))))). +Simpl; Simpl in H0; Apply lt_n_S; Assumption. +Assert H10 := (H6 ? H9); Apply H10; Assumption. +Qed. + +Lemma StepFun_P8 : (f:R->R;l1,lf1:Rlist;a,b:R) (adapted_couple f a b l1 lf1) -> a==b -> (Int_SF lf1 l1)==R0. +Induction l1. +Intros; Induction lf1; Reflexivity. +Induction r0. +Intros; Induction lf1. +Reflexivity. +Unfold adapted_couple in H0; Decompose [and] H0; Clear H0; Simpl in H5; Discriminate. +Intros; Induction lf1. +Reflexivity. +Simpl; Cut r==r1. +Intro; Rewrite H3; Rewrite (H0 lf1 r b). +Ring. +Rewrite H3; Apply StepFun_P7 with a r r3; [Right; Assumption | Assumption]. +Clear H H0 Hreclf1 r0; Unfold adapted_couple in H1; Decompose [and] H1; Intros; Simpl in H4; Rewrite H4; Unfold Rmin; Case (total_order_Rle a b); Intro; [Assumption | Reflexivity]. +Unfold adapted_couple in H1; Decompose [and] H1; Intros; Apply Rle_antisym. +Apply (H3 O); Simpl; Apply lt_O_Sn. +Simpl in H5; Rewrite H2 in H5; Rewrite H5; Replace (Rmin b b) with (Rmax a b); [Rewrite <- H4; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity] | Unfold Rmin Rmax; Case (total_order_Rle b b); Case (total_order_Rle a b); Intros; Try Assumption Orelse Reflexivity]. +Qed. + +Lemma StepFun_P9 : (a,b:R;f:R->R;l,lf:Rlist) (adapted_couple f a b l lf) -> ``a<>b`` -> (le (2) (Rlength l)). +Intros; Unfold adapted_couple in H; Decompose [and] H; Clear H; Induction l; [Simpl in H4; Discriminate | Induction l; [Simpl in H3; Simpl in H2; Generalize H3; Generalize H2; Unfold Rmin Rmax; Case (total_order_Rle a b); Intros; Elim H0; Rewrite <- H5; Rewrite <- H7; Reflexivity | Simpl; Do 2 Apply le_n_S; Apply le_O_n]]. +Qed. + +Lemma StepFun_P10 : (f:R->R;l,lf:Rlist;a,b:R) ``a<=b`` -> (adapted_couple f a b l lf) -> (EXT l':Rlist | (EXT lf':Rlist | (adapted_couple_opt f a b l' lf'))). +Induction l. +Intros; Unfold adapted_couple in H0; Decompose [and] H0; Simpl in H4; Discriminate. +Intros; Case (Req_EM a b); Intro. +Exists (cons a nil); Exists nil; Unfold adapted_couple_opt; Unfold adapted_couple; Unfold ordered_Rlist; Repeat Split; Try (Intros; Simpl in H3; Elim (lt_n_O ? H3)). +Simpl; Rewrite <- H2; Unfold Rmin; Case (total_order_Rle a a); Intro; Reflexivity. +Simpl; Rewrite <- H2; Unfold Rmax; Case (total_order_Rle a a); Intro; Reflexivity. +Elim (RList_P20 ? (StepFun_P9 H1 H2)); Intros t1 [t2 [t3 H3]]; Induction lf. +Unfold adapted_couple in H1; Decompose [and] H1; Rewrite H3 in H7; Simpl in H7; Discriminate. +Clear Hreclf; Assert H4 : (adapted_couple f t2 b r0 lf). +Rewrite H3 in H1; Assert H4 := (RList_P21 ? ? H3); Simpl in H4; Rewrite H4; EApply StepFun_P7; [Apply H0 | Apply H1]. +Cut ``t2<=b``. +Intro; Assert H6 := (H ? ? ? H5 H4); Case (Req_EM t1 t2); Intro Hyp_eq. +Replace a with t2. +Apply H6. +Rewrite <- Hyp_eq; Rewrite H3 in H1; Unfold adapted_couple in H1; Decompose [and] H1; Clear H1; Simpl in H9; Rewrite H9; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Elim H6; Clear H6; Intros l' [lf' H6]; Case (Req_EM t2 b); Intro. +Exists (cons a (cons b nil)); Exists (cons r1 nil); Unfold adapted_couple_opt; Unfold adapted_couple; Repeat Split. +Unfold ordered_Rlist; Intros; Simpl in H8; Inversion H8; [Simpl; Assumption | Elim (le_Sn_O ? H10)]. +Simpl; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Simpl; Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Intros; Simpl in H8; Inversion H8. +Unfold constant_D_eq open_interval; Intros; Simpl; Simpl in H9; Rewrite H3 in H1; Unfold adapted_couple in H1; Decompose [and] H1; Apply (H16 O). +Simpl; Apply lt_O_Sn. +Unfold open_interval; Simpl; Rewrite H7; Simpl in H13; Rewrite H13; Unfold Rmin; Case (total_order_Rle a b); Intro; [Assumption | Elim n; Assumption]. +Elim (le_Sn_O ? H10). +Intros; Simpl in H8; Elim (lt_n_O ? H8). +Intros; Simpl in H8; Inversion H8; [Simpl; Assumption | Elim (le_Sn_O ? H10)]. +Assert Hyp_min : (Rmin t2 b)==t2. +Unfold Rmin; Case (total_order_Rle t2 b); Intro; [Reflexivity | Elim n; Assumption]. +Unfold adapted_couple in H6; Elim H6; Clear H6; Intros; Elim (RList_P20 ? (StepFun_P9 H6 H7)); Intros s1 [s2 [s3 H9]]; Induction lf'. +Unfold adapted_couple in H6; Decompose [and] H6; Rewrite H9 in H13; Simpl in H13; Discriminate. +Clear Hreclf'; Case (Req_EM r1 r2); Intro. +Case (Req_EM (f t2) r1); Intro. +Exists (cons t1 (cons s2 s3)); Exists (cons r1 lf'); Rewrite H3 in H1; Rewrite H9 in H6; Unfold adapted_couple in H6 H1; Decompose [and] H1; Decompose [and] H6; Clear H1 H6; Unfold adapted_couple_opt; Unfold adapted_couple; Repeat Split. +Unfold ordered_Rlist; Intros; Simpl in H1; Induction i. +Simpl; Apply Rle_trans with s1. +Replace s1 with t2. +Apply (H12 O). +Simpl; Apply lt_O_Sn. +Simpl in H19; Rewrite H19; Symmetry; Apply Hyp_min. +Apply (H16 O); Simpl; Apply lt_O_Sn. +Change ``(pos_Rl (cons s2 s3) i)<=(pos_Rl (cons s2 s3) (S i))``; Apply (H16 (S i)); Simpl; Assumption. +Simpl; Simpl in H14; Rewrite H14; Reflexivity. +Simpl; Simpl in H18; Rewrite H18; Unfold Rmax; Case (total_order_Rle a b); Case (total_order_Rle t2 b); Intros; Reflexivity Orelse Elim n; Assumption. +Simpl; Simpl in H20; Apply H20. +Intros; Simpl in H1; Unfold constant_D_eq open_interval; Intros; Induction i. +Simpl; Simpl in H6; Case (total_order_T x t2); Intro. +Elim s; Intro. +Apply (H17 O); [Simpl; Apply lt_O_Sn | Unfold open_interval; Simpl; Elim H6; Intros; Split; Assumption]. +Rewrite b0; Assumption. +Rewrite H10; Apply (H22 O); [Simpl; Apply lt_O_Sn | Unfold open_interval; Simpl; Replace s1 with t2; [Elim H6; Intros; Split; Assumption | Simpl in H19; Rewrite H19; Rewrite Hyp_min; Reflexivity]]. +Simpl; Simpl in H6; Apply (H22 (S i)); [Simpl; Assumption | Unfold open_interval; Simpl; Apply H6]. +Intros; Simpl in H1; Rewrite H10; Change ``(pos_Rl (cons r2 lf') i)<>(pos_Rl (cons r2 lf') (S i))``\/``(f (pos_Rl (cons s1 (cons s2 s3)) (S i)))<>(pos_Rl (cons r2 lf') i)``; Rewrite <- H9; Elim H8; Intros; Apply H6; Simpl; Apply H1. +Intros; Induction i. +Simpl; Red; Intro; Elim Hyp_eq; Apply Rle_antisym. +Apply (H12 O); Simpl; Apply lt_O_Sn. +Rewrite <- Hyp_min; Rewrite H6; Simpl in H19; Rewrite <- H19; Apply (H16 O); Simpl; Apply lt_O_Sn. +Elim H8; Intros; Rewrite H9 in H21; Apply (H21 (S i)); Simpl; Simpl in H1; Apply H1. +Exists (cons t1 l'); Exists (cons r1 (cons r2 lf')); Rewrite H9 in H6; Rewrite H3 in H1; Unfold adapted_couple in H1 H6; Decompose [and] H6; Decompose [and] H1; Clear H6 H1; Unfold adapted_couple_opt; Unfold adapted_couple; Repeat Split. +Rewrite H9; Unfold ordered_Rlist; Intros; Simpl in H1; Induction i. +Simpl; Replace s1 with t2. +Apply (H16 O); Simpl; Apply lt_O_Sn. +Simpl in H14; Rewrite H14; Rewrite Hyp_min; Reflexivity. +Change ``(pos_Rl (cons s1 (cons s2 s3)) i)<=(pos_Rl (cons s1 (cons s2 s3)) (S i))``; Apply (H12 i); Simpl; Apply lt_S_n; Assumption. +Simpl; Simpl in H19; Apply H19. +Rewrite H9; Simpl; Simpl in H13; Rewrite H13; Unfold Rmax; Case (total_order_Rle t2 b); Case (total_order_Rle a b); Intros; Reflexivity Orelse Elim n; Assumption. +Rewrite H9; Simpl; Simpl in H15; Rewrite H15; Reflexivity. +Intros; Simpl in H1; Unfold constant_D_eq open_interval; Intros; Induction i. +Simpl; Rewrite H9 in H6; Simpl in H6; Apply (H22 O). +Simpl; Apply lt_O_Sn. +Unfold open_interval; Simpl. +Replace t2 with s1. +Assumption. +Simpl in H14; Rewrite H14; Rewrite Hyp_min; Reflexivity. +Change (f x)==(pos_Rl (cons r2 lf') i); Clear Hreci; Apply (H17 i). +Simpl; Rewrite H9 in H1; Simpl in H1; Apply lt_S_n; Apply H1. +Rewrite H9 in H6; Unfold open_interval; Apply H6. +Intros; Simpl in H1; Induction i. +Simpl; Rewrite H9; Right; Simpl; Replace s1 with t2. +Assumption. +Simpl in H14; Rewrite H14; Rewrite Hyp_min; Reflexivity. +Elim H8; Intros; Apply (H6 i). +Simpl; Apply lt_S_n; Apply H1. +Intros; Rewrite H9; Induction i. +Simpl; Red; Intro; Elim Hyp_eq; Apply Rle_antisym. +Apply (H16 O); Simpl; Apply lt_O_Sn. +Rewrite <- Hyp_min; Rewrite H6; Simpl in H14; Rewrite <- H14; Right; Reflexivity. +Elim H8; Intros; Rewrite <- H9; Apply (H21 i); Rewrite H9; Rewrite H9 in H1; Simpl; Simpl in H1; Apply lt_S_n; Apply H1. +Exists (cons t1 l'); Exists (cons r1 (cons r2 lf')); Rewrite H9 in H6; Rewrite H3 in H1; Unfold adapted_couple in H1 H6; Decompose [and] H6; Decompose [and] H1; Clear H6 H1; Unfold adapted_couple_opt; Unfold adapted_couple; Repeat Split. +Rewrite H9; Unfold ordered_Rlist; Intros; Simpl in H1; Induction i. +Simpl; Replace s1 with t2. +Apply (H15 O); Simpl; Apply lt_O_Sn. +Simpl in H13; Rewrite H13; Rewrite Hyp_min; Reflexivity. +Change ``(pos_Rl (cons s1 (cons s2 s3)) i)<=(pos_Rl (cons s1 (cons s2 s3)) (S i))``; Apply (H11 i); Simpl; Apply lt_S_n; Assumption. +Simpl; Simpl in H18; Apply H18. +Rewrite H9; Simpl; Simpl in H12; Rewrite H12; Unfold Rmax; Case (total_order_Rle t2 b); Case (total_order_Rle a b); Intros; Reflexivity Orelse Elim n; Assumption. +Rewrite H9; Simpl; Simpl in H14; Rewrite H14; Reflexivity. +Intros; Simpl in H1; Unfold constant_D_eq open_interval; Intros; Induction i. +Simpl; Rewrite H9 in H6; Simpl in H6; Apply (H21 O). +Simpl; Apply lt_O_Sn. +Unfold open_interval; Simpl; Replace t2 with s1. +Assumption. +Simpl in H13; Rewrite H13; Rewrite Hyp_min; Reflexivity. +Change (f x)==(pos_Rl (cons r2 lf') i); Clear Hreci; Apply (H16 i). +Simpl; Rewrite H9 in H1; Simpl in H1; Apply lt_S_n; Apply H1. +Rewrite H9 in H6; Unfold open_interval; Apply H6. +Intros; Simpl in H1; Induction i. +Simpl; Left; Assumption. +Elim H8; Intros; Apply (H6 i). +Simpl; Apply lt_S_n; Apply H1. +Intros; Rewrite H9; Induction i. +Simpl; Red; Intro; Elim Hyp_eq; Apply Rle_antisym. +Apply (H15 O); Simpl; Apply lt_O_Sn. +Rewrite <- Hyp_min; Rewrite H6; Simpl in H13; Rewrite <- H13; Right; Reflexivity. +Elim H8; Intros; Rewrite <- H9; Apply (H20 i); Rewrite H9; Rewrite H9 in H1; Simpl; Simpl in H1; Apply lt_S_n; Apply H1. +Rewrite H3 in H1; Clear H4; Unfold adapted_couple in H1; Decompose [and] H1; Clear H1; Clear H H7 H9; Cut (Rmax a b)==b; [Intro; Rewrite H in H5; Rewrite <- H5; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity] | Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]]. +Qed. + +Lemma StepFun_P11 : (a,b,r,r1,r3,s1,s2,r4:R;r2,lf1,s3,lf2:Rlist;f:R->R) ``a<b`` -> (adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1)) -> (adapted_couple_opt f a b (cons s1 (cons s2 s3)) (cons r4 lf2)) -> ``r1<=s2``. +Intros; Unfold adapted_couple_opt in H1; Elim H1; Clear H1; Intros; Unfold adapted_couple in H0 H1; Decompose [and] H0; Decompose [and] H1; Clear H0 H1; Assert H12 : r==s1. +Simpl in H10; Simpl in H5; Rewrite H10; Rewrite H5; Reflexivity. +Assert H14 := (H3 O (lt_O_Sn ?)); Simpl in H14; Elim H14; Intro. +Assert H15 := (H7 O (lt_O_Sn ?)); Simpl in H15; Elim H15; Intro. +Rewrite <- H12 in H1; Case (total_order_Rle r1 s2); Intro; Try Assumption. +Assert H16 : ``s2<r1``; Auto with real. +Induction s3. +Simpl in H9; Rewrite H9 in H16; Cut ``r1<=(Rmax a b)``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H17 H16)). +Rewrite <- H4; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity]. +Clear Hrecs3; Induction lf2. +Simpl in H11; Discriminate. +Clear Hreclf2; Assert H17 : r3==r4. +Pose x := ``(r+s2)/2``; Assert H17 := (H8 O (lt_O_Sn ?)); Assert H18 := (H13 O (lt_O_Sn ?)); Unfold constant_D_eq open_interval in H17 H18; Simpl in H17; Simpl in H18; Rewrite <- (H17 x). +Rewrite <- (H18 x). +Reflexivity. +Rewrite <- H12; Unfold x; Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym r); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Unfold x; Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Apply Rlt_trans with s2; [Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym r); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]] | Assumption]. +Assert H18 : (f s2)==r3. +Apply (H8 O); [Simpl; Apply lt_O_Sn | Unfold open_interval; Simpl; Split; Assumption]. +Assert H19 : r3 == r5. +Assert H19 := (H7 (S O)); Simpl in H19; Assert H20 := (H19 (lt_n_S ? ? (lt_O_Sn ?))); Elim H20; Intro. +Pose x := ``(s2+(Rmin r1 r0))/2``; Assert H22 := (H8 O); Assert H23 := (H13 (S O)); Simpl in H22; Simpl in H23; Rewrite <- (H22 (lt_O_Sn ?) x). +Rewrite <- (H23 (lt_n_S ? ? (lt_O_Sn ?)) x). +Reflexivity. +Unfold open_interval; Simpl; Unfold x; Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Unfold Rmin; Case (total_order_Rle r1 r0); Intro; Assumption | DiscrR]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_le_trans with ``r0+(Rmin r1 r0)``; [Do 2 Rewrite <- (Rplus_sym (Rmin r1 r0)); Apply Rlt_compatibility; Assumption | Apply Rle_compatibility; Apply Rmin_r] | DiscrR]]. +Unfold open_interval; Simpl; Unfold x; Split. +Apply Rlt_trans with s2; [Assumption | Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Unfold Rmin; Case (total_order_Rle r1 r0); Intro; Assumption | DiscrR]]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_le_trans with ``r1+(Rmin r1 r0)``; [Do 2 Rewrite <- (Rplus_sym (Rmin r1 r0)); Apply Rlt_compatibility; Assumption | Apply Rle_compatibility; Apply Rmin_l] | DiscrR]]. +Elim H2; Clear H2; Intros; Assert H23 := (H22 (S O)); Simpl in H23; Assert H24 := (H23 (lt_n_S ? ? (lt_O_Sn ?))); Elim H24; Assumption. +Elim H2; Intros; Assert H22 := (H20 O); Simpl in H22; Assert H23 := (H22 (lt_O_Sn ?)); Elim H23; Intro; [Elim H24; Rewrite <- H17; Rewrite <- H19; Reflexivity | Elim H24; Rewrite <- H17; Assumption]. +Elim H2; Clear H2; Intros; Assert H17 := (H16 O); Simpl in H17; Elim (H17 (lt_O_Sn ?)); Assumption. +Rewrite <- H0; Rewrite H12; Apply (H7 O); Simpl; Apply lt_O_Sn. +Qed. + +Lemma StepFun_P12 : (a,b:R;f:R->R;l,lf:Rlist) (adapted_couple_opt f a b l lf) -> (adapted_couple_opt f b a l lf). +Unfold adapted_couple_opt; Unfold adapted_couple; Intros; Decompose [and] H; Clear H; Repeat Split; Try Assumption. +Rewrite H0; Unfold Rmin; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. +Apply Rle_antisym; Assumption. +Apply Rle_antisym; Auto with real. +Rewrite H3; Unfold Rmax; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. +Apply Rle_antisym; Assumption. +Apply Rle_antisym; Auto with real. +Qed. + +Lemma StepFun_P13 : (a,b,r,r1,r3,s1,s2,r4:R;r2,lf1,s3,lf2:Rlist;f:R->R) ``a<>b`` -> (adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1)) -> (adapted_couple_opt f a b (cons s1 (cons s2 s3)) (cons r4 lf2)) -> ``r1<=s2``. +Intros; Case (total_order_T a b); Intro. +Elim s; Intro. +EApply StepFun_P11; [Apply a0 | Apply H0 | Apply H1]. +Elim H; Assumption. +EApply StepFun_P11; [Apply r0 | Apply StepFun_P2; Apply H0 | Apply StepFun_P12; Apply H1]. +Qed. + +Lemma StepFun_P14 : (f:R->R;l1,l2,lf1,lf2:Rlist;a,b:R) ``a<=b`` -> (adapted_couple f a b l1 lf1) -> (adapted_couple_opt f a b l2 lf2) -> (Int_SF lf1 l1)==(Int_SF lf2 l2). +Induction l1. +Intros l2 lf1 lf2 a b Hyp H H0; Unfold adapted_couple in H; Decompose [and] H; Clear H H0 H2 H3 H1 H6; Simpl in H4; Discriminate. +Induction r0. +Intros; Case (Req_EM a b); Intro. +Unfold adapted_couple_opt in H2; Elim H2; Intros; Rewrite (StepFun_P8 H4 H3); Rewrite (StepFun_P8 H1 H3); Reflexivity. +Assert H4 := (StepFun_P9 H1 H3); Simpl in H4; Elim (le_Sn_O ? (le_S_n ? ? H4)). +Intros; Clear H; Unfold adapted_couple_opt in H3; Elim H3; Clear H3; Intros; Case (Req_EM a b); Intro. +Rewrite (StepFun_P8 H2 H4); Rewrite (StepFun_P8 H H4); Reflexivity. +Assert Hyp_min : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Assert Hyp_max : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Elim (RList_P20 ? (StepFun_P9 H H4)); Intros s1 [s2 [s3 H5]]; Rewrite H5 in H; Rewrite H5; Induction lf1. +Unfold adapted_couple in H2; Decompose [and] H2; Clear H H2 H4 H5 H3 H6 H8 H7 H11; Simpl in H9; Discriminate. +Clear Hreclf1; Induction lf2. +Unfold adapted_couple in H; Decompose [and] H; Clear H H2 H4 H5 H3 H6 H8 H7 H11; Simpl in H9; Discriminate. +Clear Hreclf2; Assert H6 : r==s1. +Unfold adapted_couple in H H2; Decompose [and] H; Decompose [and] H2; Clear H H2; Simpl in H13; Simpl in H8; Rewrite H13; Rewrite H8; Reflexivity. +Assert H7 : r3==r4\/r==r1. +Case (Req_EM r r1); Intro. +Right; Assumption. +Left; Cut ``r1<=s2``. +Intro; Unfold adapted_couple in H2 H; Decompose [and] H; Decompose [and] H2; Clear H H2; Pose x := ``(r+r1)/2``; Assert H18 := (H14 O); Assert H20 := (H19 O); Unfold constant_D_eq open_interval in H18 H20; Simpl in H18; Simpl in H20; Rewrite <- (H18 (lt_O_Sn ?) x). +Rewrite <- (H20 (lt_O_Sn ?) x). +Reflexivity. +Assert H21 := (H13 O (lt_O_Sn ?)); Simpl in H21; Elim H21; Intro; [Idtac | Elim H7; Assumption]; Unfold x; Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Apply H | DiscrR]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite <- (Rplus_sym r1); Rewrite double; Apply Rlt_compatibility; Apply H | DiscrR]]. +Rewrite <- H6; Assert H21 := (H13 O (lt_O_Sn ?)); Simpl in H21; Elim H21; Intro; [Idtac | Elim H7; Assumption]; Unfold x; Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Apply H | DiscrR]]. +Apply Rlt_le_trans with r1; [Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite <- (Rplus_sym r1); Rewrite double; Apply Rlt_compatibility; Apply H | DiscrR]] | Assumption]. +EApply StepFun_P13. +Apply H4. +Apply H2. +Unfold adapted_couple_opt; Split. +Apply H. +Rewrite H5 in H3; Apply H3. +Assert H8 : ``r1<=s2``. +EApply StepFun_P13. +Apply H4. +Apply H2. +Unfold adapted_couple_opt; Split. +Apply H. +Rewrite H5 in H3; Apply H3. +Elim H7; Intro. +Simpl; Elim H8; Intro. +Replace ``r4*(s2-s1)`` with ``r3*(r1-r)+r3*(s2-r1)``; [Idtac | Rewrite H9; Rewrite H6; Ring]. +Rewrite Rplus_assoc; Apply Rplus_plus_r; Change (Int_SF lf1 (cons r1 r2))==(Int_SF (cons r3 lf2) (cons r1 (cons s2 s3))); Apply H0 with r1 b. +Unfold adapted_couple in H2; Decompose [and] H2; Clear H2; Replace b with (Rmax a b). +Rewrite <- H12; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity]. +EApply StepFun_P7. +Apply H1. +Apply H2. +Unfold adapted_couple_opt; Split. +Apply StepFun_P7 with a a r3. +Apply H1. +Unfold adapted_couple in H2 H; Decompose [and] H2; Decompose [and] H; Clear H H2; Assert H20 : r==a. +Simpl in H13; Rewrite H13; Apply Hyp_min. +Unfold adapted_couple; Repeat Split. +Unfold ordered_Rlist; Intros; Simpl in H; Induction i. +Simpl; Rewrite <- H20; Apply (H11 O). +Simpl; Apply lt_O_Sn. +Induction i. +Simpl; Assumption. +Change ``(pos_Rl (cons s2 s3) i)<=(pos_Rl (cons s2 s3) (S i))``; Apply (H15 (S i)); Simpl; Apply lt_S_n; Assumption. +Simpl; Symmetry; Apply Hyp_min. +Rewrite <- H17; Reflexivity. +Simpl in H19; Simpl; Rewrite H19; Reflexivity. +Intros; Simpl in H; Unfold constant_D_eq open_interval; Intros; Induction i. +Simpl; Apply (H16 O). +Simpl; Apply lt_O_Sn. +Simpl in H2; Rewrite <- H20 in H2; Unfold open_interval; Simpl; Apply H2. +Clear Hreci; Induction i. +Simpl; Simpl in H2; Rewrite H9; Apply (H21 O). +Simpl; Apply lt_O_Sn. +Unfold open_interval; Simpl; Elim H2; Intros; Split. +Apply Rle_lt_trans with r1; Try Assumption; Rewrite <- H6; Apply (H11 O); Simpl; Apply lt_O_Sn. +Assumption. +Clear Hreci; Simpl; Apply (H21 (S i)). +Simpl; Apply lt_S_n; Assumption. +Unfold open_interval; Apply H2. +Elim H3; Clear H3; Intros; Split. +Rewrite H9; Change (i:nat) (lt i (pred (Rlength (cons r4 lf2)))) ->``(pos_Rl (cons r4 lf2) i)<>(pos_Rl (cons r4 lf2) (S i))``\/``(f (pos_Rl (cons s1 (cons s2 s3)) (S i)))<>(pos_Rl (cons r4 lf2) i)``; Rewrite <- H5; Apply H3. +Rewrite H5 in H11; Intros; Simpl in H12; Induction i. +Simpl; Red; Intro; Rewrite H13 in H10; Elim (Rlt_antirefl ? H10). +Clear Hreci; Apply (H11 (S i)); Simpl; Apply H12. +Rewrite H9; Rewrite H10; Rewrite H6; Apply Rplus_plus_r; Rewrite <- H10; Apply H0 with r1 b. +Unfold adapted_couple in H2; Decompose [and] H2; Clear H2; Replace b with (Rmax a b). +Rewrite <- H12; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity]. +EApply StepFun_P7. +Apply H1. +Apply H2. +Unfold adapted_couple_opt; Split. +Apply StepFun_P7 with a a r3. +Apply H1. +Unfold adapted_couple in H2 H; Decompose [and] H2; Decompose [and] H; Clear H H2; Assert H20 : r==a. +Simpl in H13; Rewrite H13; Apply Hyp_min. +Unfold adapted_couple; Repeat Split. +Unfold ordered_Rlist; Intros; Simpl in H; Induction i. +Simpl; Rewrite <- H20; Apply (H11 O); Simpl; Apply lt_O_Sn. +Rewrite H10; Apply (H15 (S i)); Simpl; Assumption. +Simpl; Symmetry; Apply Hyp_min. +Rewrite <- H17; Rewrite H10; Reflexivity. +Simpl in H19; Simpl; Apply H19. +Intros; Simpl in H; Unfold constant_D_eq open_interval; Intros; Induction i. +Simpl; Apply (H16 O). +Simpl; Apply lt_O_Sn. +Simpl in H2; Rewrite <- H20 in H2; Unfold open_interval; Simpl; Apply H2. +Clear Hreci; Simpl; Apply (H21 (S i)). +Simpl; Assumption. +Rewrite <- H10; Unfold open_interval; Apply H2. +Elim H3; Clear H3; Intros; Split. +Rewrite H5 in H3; Intros; Apply (H3 (S i)). +Simpl; Replace (Rlength lf2) with (S (pred (Rlength lf2))). +Apply lt_n_S; Apply H12. +Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H12; Elim (lt_n_O ? H12). +Intros; Simpl in H12; Rewrite H10; Rewrite H5 in H11; Apply (H11 (S i)); Simpl; Apply lt_n_S; Apply H12. +Simpl; Rewrite H9; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rmult_Or; Rewrite Rplus_Ol; Change (Int_SF lf1 (cons r1 r2))==(Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))); EApply H0. +Apply H1. +2: Rewrite H5 in H3; Unfold adapted_couple_opt; Split; Assumption. +Assert H10 : r==a. +Unfold adapted_couple in H2; Decompose [and] H2; Clear H2; Simpl in H12; Rewrite H12; Apply Hyp_min. +Rewrite <- H9; Rewrite H10; Apply StepFun_P7 with a r r3; [Apply H1 | Pattern 2 a; Rewrite <- H10; Pattern 2 r; Rewrite H9; Apply H2]. +Qed. + +Lemma StepFun_P15 : (f:R->R;l1,l2,lf1,lf2:Rlist;a,b:R) (adapted_couple f a b l1 lf1) -> (adapted_couple_opt f a b l2 lf2) -> (Int_SF lf1 l1)==(Int_SF lf2 l2). +Intros; Case (total_order_Rle a b); Intro; [Apply (StepFun_P14 r H H0) | Assert H1 : ``b<=a``; [Auto with real | EApply StepFun_P14; [Apply H1 | Apply StepFun_P2; Apply H | Apply StepFun_P12; Apply H0]]]. +Qed. + +Lemma StepFun_P16 : (f:R->R;l,lf:Rlist;a,b:R) (adapted_couple f a b l lf) -> (EXT l':Rlist | (EXT lf':Rlist | (adapted_couple_opt f a b l' lf'))). +Intros; Case (total_order_Rle a b); Intro; [Apply (StepFun_P10 r H) | Assert H1 : ``b<=a``; [Auto with real | Assert H2 := (StepFun_P10 H1 (StepFun_P2 H)); Elim H2; Intros l' [lf' H3]; Exists l'; Exists lf'; Apply StepFun_P12; Assumption]]. +Qed. + +Lemma StepFun_P17 : (f:R->R;l1,l2,lf1,lf2:Rlist;a,b:R) (adapted_couple f a b l1 lf1) -> (adapted_couple f a b l2 lf2) -> (Int_SF lf1 l1)==(Int_SF lf2 l2). +Intros; Elim (StepFun_P16 H); Intros l' [lf' H1]; Rewrite (StepFun_P15 H H1); Rewrite (StepFun_P15 H0 H1); Reflexivity. +Qed. + +Lemma StepFun_P18 : (a,b,c:R) (RiemannInt_SF (mkStepFun (StepFun_P4 a b c)))==``c*(b-a)``. +Intros; Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. +Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c))) (subdivision (mkStepFun (StepFun_P4 a b c)))) with (Int_SF (cons c nil) (cons a (cons b nil))); [Simpl; Ring | Apply StepFun_P17 with (fct_cte c) a b; [Apply StepFun_P3; Assumption | Apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c)))]]. +Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c))) (subdivision (mkStepFun (StepFun_P4 a b c)))) with (Int_SF (cons c nil) (cons b (cons a nil))); [Simpl; Ring | Apply StepFun_P17 with (fct_cte c) a b; [Apply StepFun_P2; Apply StepFun_P3; Auto with real | Apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c)))]]. +Qed. + +Lemma StepFun_P19 : (l1:Rlist;f,g:R->R;l:R) (Int_SF (FF l1 [x:R]``(f x)+l*(g x)``) l1)==``(Int_SF (FF l1 f) l1)+l*(Int_SF (FF l1 g) l1)``. +Intros; Induction l1; [Simpl; Ring | Induction l1; Simpl; [Ring | Simpl in Hrecl1; Rewrite Hrecl1; Ring]]. +Qed. + +Lemma StepFun_P20 : (l:Rlist;f:R->R) (lt O (Rlength l)) -> (Rlength l)=(S (Rlength (FF l f))). +Intros l f H; NewInduction l; [Elim (lt_n_n ? H) | Simpl; Rewrite RList_P18; Rewrite RList_P14; Reflexivity]. +Qed. + +Lemma StepFun_P21 : (a,b:R;f:R->R;l:Rlist) (is_subdivision f a b l) -> (adapted_couple f a b l (FF l f)). +Intros; Unfold adapted_couple; Unfold is_subdivision in X; Unfold adapted_couple in X; Elim X; Clear X; Intros; Decompose [and] p; Clear p; Repeat Split; Try Assumption. +Apply StepFun_P20; Rewrite H2; Apply lt_O_Sn. +Intros; Assert H5 := (H4 ? H3); Unfold constant_D_eq open_interval in H5; Unfold constant_D_eq open_interval; Intros; Induction l. +Discriminate. +Unfold FF; Rewrite RList_P12. +Simpl; Change (f x0)==(f (pos_Rl (mid_Rlist (cons r l) r) (S i))); Rewrite RList_P13; Try Assumption; Rewrite (H5 x0 H6); Rewrite H5. +Reflexivity. +Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Elim H6; Intros; Apply Rlt_trans with x0; Assumption | DiscrR]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Rewrite (Rplus_sym (pos_Rl (cons r l) i)); Apply Rlt_compatibility; Elim H6; Intros; Apply Rlt_trans with x0; Assumption | DiscrR]]. +Rewrite RList_P14; Simpl in H3; Apply H3. +Qed. + +Lemma StepFun_P22 : (a,b:R;f,g:R->R;lf,lg:Rlist) ``a<=b`` -> (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision f a b (cons_ORlist lf lg)). +Unfold is_subdivision; Intros a b f g lf lg Hyp X X0; Elim X; Elim X0; Clear X X0; Intros lg0 p lf0 p0; Assert Hyp_min : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Assert Hyp_max : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Apply existTT with (FF (cons_ORlist lf lg) f); Unfold adapted_couple in p p0; Decompose [and] p; Decompose [and] p0; Clear p p0; Rewrite Hyp_min in H6; Rewrite Hyp_min in H1; Rewrite Hyp_max in H0; Rewrite Hyp_max in H5; Unfold adapted_couple; Repeat Split. +Apply RList_P2; Assumption. +Rewrite Hyp_min; Symmetry; Apply Rle_antisym. +Induction lf. +Simpl; Right; Symmetry; Assumption. +Assert H10 : (In (pos_Rl (cons_ORlist (cons r lf) lg) (0)) (cons_ORlist (cons r lf) lg)). +Elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros _ H10; Apply H10; Exists O; Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_O_Sn]. +Elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H12 _; Assert H13 := (H12 H10); Elim H13; Intro. +Elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H11 _; Assert H14 := (H11 H8); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H6; Elim (RList_P6 (cons r lf)); Intros; Apply H17; [Assumption | Apply le_O_n | Assumption]. +Elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H11 _; Assert H14 := (H11 H8); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H1; Elim (RList_P6 lg); Intros; Apply H17; [Assumption | Apply le_O_n | Assumption]. +Induction lf. +Simpl; Right; Assumption. +Assert H8 : (In a (cons_ORlist (cons r lf) lg)). +Elim (RList_P9 (cons r lf) lg a); Intros; Apply H10; Left; Elim (RList_P3 (cons r lf) a); Intros; Apply H12; Exists O; Split; [Symmetry; Assumption | Simpl; Apply lt_O_Sn]. +Apply RList_P5; [Apply RList_P2; Assumption | Assumption]. +Rewrite Hyp_max; Apply Rle_antisym. +Induction lf. +Simpl; Right; Assumption. +Assert H8 : (In (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg)))) (cons_ORlist (cons r lf) lg)). +Elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros _ H10; Apply H10; Exists (pred (Rlength (cons_ORlist (cons r lf) lg))); Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_n_Sn]. +Elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H10 _. +Assert H11 := (H10 H8); Elim H11; Intro. +Elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H13 _; Assert H14 := (H13 H12); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H5; Elim (RList_P6 (cons r lf)); Intros; Apply H17; [Assumption | Simpl; Simpl in H14; Apply lt_n_Sm_le; Assumption | Simpl; Apply lt_n_Sn]. +Elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H13 _; Assert H14 := (H13 H12); Elim H14; Intros; Elim H15; Clear H15; Intros. +Rewrite H15; Assert H17 : (Rlength lg)=(S (pred (Rlength lg))). +Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H17 in H16; Elim (lt_n_O ? H16). +Rewrite <- H0; Elim (RList_P6 lg); Intros; Apply H18; [Assumption | Rewrite H17 in H16; Apply lt_n_Sm_le; Assumption | Apply lt_pred_n_n; Rewrite H17; Apply lt_O_Sn]. +Induction lf. +Simpl; Right; Symmetry; Assumption. +Assert H8 : (In b (cons_ORlist (cons r lf) lg)). +Elim (RList_P9 (cons r lf) lg b); Intros; Apply H10; Left; Elim (RList_P3 (cons r lf) b); Intros; Apply H12; Exists (pred (Rlength (cons r lf))); Split; [Symmetry; Assumption | Simpl; Apply lt_n_Sn]. +Apply RList_P7; [Apply RList_P2; Assumption | Assumption]. +Apply StepFun_P20; Rewrite RList_P11; Rewrite H2; Rewrite H7; Simpl; Apply lt_O_Sn. +Intros; Unfold constant_D_eq open_interval; Intros; Cut (EXT l:R | (constant_D_eq f (open_interval (pos_Rl (cons_ORlist lf lg) i) (pos_Rl (cons_ORlist lf lg) (S i))) l)). +Intros; Elim H11; Clear H11; Intros; Assert H12 := H11; Assert Hyp_cons : (EXT r:R | (EXT r0:Rlist | (cons_ORlist lf lg)==(cons r r0))). +Apply RList_P19; Red; Intro; Rewrite H13 in H8; Elim (lt_n_O ? H8). +Elim Hyp_cons; Clear Hyp_cons; Intros r [r0 Hyp_cons]; Rewrite Hyp_cons; Unfold FF; Rewrite RList_P12. +Change (f x)==(f (pos_Rl (mid_Rlist (cons r r0) r) (S i))); Rewrite <- Hyp_cons; Rewrite RList_P13. +Assert H13 := (RList_P2 ? ? H ? H8); Elim H13; Intro. +Unfold constant_D_eq open_interval in H11 H12; Rewrite (H11 x H10); Assert H15 : ``(pos_Rl (cons_ORlist lf lg) i)<((pos_Rl (cons_ORlist lf lg) i)+(pos_Rl (cons_ORlist lf lg) (S i)))/2<(pos_Rl (cons_ORlist lf lg) (S i))``. +Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Rewrite (Rplus_sym (pos_Rl (cons_ORlist lf lg) i)); Apply Rlt_compatibility; Assumption | DiscrR]]. +Rewrite (H11 ? H15); Reflexivity. +Elim H10; Intros; Rewrite H14 in H15; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H16 H15)). +Apply H8. +Rewrite RList_P14; Rewrite Hyp_cons in H8; Simpl in H8; Apply H8. +Assert H11 : ``a<b``. +Apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i). +Rewrite <- H6; Rewrite <- (RList_P15 lf lg). +Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H11. +Apply RList_P2; Assumption. +Apply le_O_n. +Apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); [Assumption | Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H8; Elim (lt_n_O ? H8)]. +Assumption. +Assumption. +Rewrite H1; Assumption. +Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). +Elim H10; Intros; Apply Rlt_trans with x; Assumption. +Rewrite <- H5; Rewrite <- (RList_P16 lf lg); Try Assumption. +Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H11. +Apply RList_P2; Assumption. +Apply lt_n_Sm_le; Apply lt_n_S; Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H8; Elim (lt_n_O ? H8). +Rewrite H0; Assumption. +Pose I := [j:nat]``(pos_Rl lf j)<=(pos_Rl (cons_ORlist lf lg) i)``/\(lt j (Rlength lf)); Assert H12 : (Nbound I). +Unfold Nbound; Exists (Rlength lf); Intros; Unfold I in H12; Elim H12; Intros; Apply lt_le_weak; Assumption. +Assert H13 : (EX n:nat | (I n)). +Exists O; Unfold I; Split. +Apply Rle_trans with (pos_Rl (cons_ORlist lf lg) O). +Right; Symmetry. +Apply RList_P15; Try Assumption; Rewrite H1; Assumption. +Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H13. +Apply RList_P2; Assumption. +Apply le_O_n. +Apply lt_trans with (pred (Rlength (cons_ORlist lf lg))). +Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H15 in H8; Elim (lt_n_O ? H8). +Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H5; Rewrite <- H6 in H11; Rewrite <- H5 in H11; Elim (Rlt_antirefl ? H11). +Assert H14 := (Nzorn H13 H12); Elim H14; Clear H14; Intros x0 H14; Exists (pos_Rl lf0 x0); Unfold constant_D_eq open_interval; Intros; Assert H16 := (H9 x0); Assert H17 : (lt x0 (pred (Rlength lf))). +Elim H14; Clear H14; Intros; Unfold I in H14; Elim H14; Clear H14; Intros; Apply lt_S_n; Replace (S (pred (Rlength lf))) with (Rlength lf). +Inversion H18. +2:Apply lt_n_S; Assumption. +Cut x0=(pred (Rlength lf)). +Intro; Rewrite H19 in H14; Rewrite H5 in H14; Cut ``(pos_Rl (cons_ORlist lf lg) i)<b``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H14 H21)). +Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). +Elim H10; Intros; Apply Rlt_trans with x; Assumption. +Rewrite <- H5; Apply Rle_trans with (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))). +Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H21. +Apply RList_P2; Assumption. +Apply lt_n_Sm_le; Apply lt_n_S; Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H23 in H8; Elim (lt_n_O ? H8). +Right; Apply RList_P16; Try Assumption; Rewrite H0; Assumption. +Rewrite <- H20; Reflexivity. +Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H19 in H18; Elim (lt_n_O ? H18). +Assert H18 := (H16 H17); Unfold constant_D_eq open_interval in H18; Rewrite (H18 x1). +Reflexivity. +Elim H15; Clear H15; Intros; Elim H14; Clear H14; Intros; Unfold I in H14; Elim H14; Clear H14; Intros; Split. +Apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); Assumption. +Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); Try Assumption. +Assert H22 : (lt (S x0) (Rlength lf)). +Replace (Rlength lf) with (S (pred (Rlength lf))); [Apply lt_n_S; Assumption | Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H22 in H21; Elim (lt_n_O ? H21)]. +Elim (total_order_Rle (pos_Rl lf (S x0)) (pos_Rl (cons_ORlist lf lg) i)); Intro. +Assert H23 : (le (S x0) x0). +Apply H20; Unfold I; Split; Assumption. +Elim (le_Sn_n ? H23). +Assert H23 : ``(pos_Rl (cons_ORlist lf lg) i)<(pos_Rl lf (S x0))``. +Auto with real. +Clear b0; Apply RList_P17; Try Assumption. +Apply RList_P2; Assumption. +Elim (RList_P9 lf lg (pos_Rl lf (S x0))); Intros; Apply H25; Left; Elim (RList_P3 lf (pos_Rl lf (S x0))); Intros; Apply H27; Exists (S x0); Split; [Reflexivity | Apply H22]. +Qed. + +Lemma StepFun_P23 : (a,b:R;f,g:R->R;lf,lg:Rlist) (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision f a b (cons_ORlist lf lg)). +Intros; Case (total_order_Rle a b); Intro; [Apply StepFun_P22 with g; Assumption | Apply StepFun_P5; Apply StepFun_P22 with g; [Auto with real | Apply StepFun_P5; Assumption | Apply StepFun_P5; Assumption]]. +Qed. + +Lemma StepFun_P24 : (a,b:R;f,g:R->R;lf,lg:Rlist) ``a<=b`` -> (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision g a b (cons_ORlist lf lg)). +Unfold is_subdivision; Intros a b f g lf lg Hyp X X0; Elim X; Elim X0; Clear X X0; Intros lg0 p lf0 p0; Assert Hyp_min : (Rmin a b)==a. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Assert Hyp_max : (Rmax a b)==b. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Apply existTT with (FF (cons_ORlist lf lg) g); Unfold adapted_couple in p p0; Decompose [and] p; Decompose [and] p0; Clear p p0; Rewrite Hyp_min in H1; Rewrite Hyp_min in H6; Rewrite Hyp_max in H0; Rewrite Hyp_max in H5; Unfold adapted_couple; Repeat Split. +Apply RList_P2; Assumption. +Rewrite Hyp_min; Symmetry; Apply Rle_antisym. +Induction lf. +Simpl; Right; Symmetry; Assumption. +Assert H10 : (In (pos_Rl (cons_ORlist (cons r lf) lg) (0)) (cons_ORlist (cons r lf) lg)). +Elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros _ H10; Apply H10; Exists O; Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_O_Sn]. +Elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H12 _; Assert H13 := (H12 H10); Elim H13; Intro. +Elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H11 _; Assert H14 := (H11 H8); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H6; Elim (RList_P6 (cons r lf)); Intros; Apply H17; [Assumption | Apply le_O_n | Assumption]. +Elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H11 _; Assert H14 := (H11 H8); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H1; Elim (RList_P6 lg); Intros; Apply H17; [Assumption | Apply le_O_n | Assumption]. +Induction lf. +Simpl; Right; Assumption. +Assert H8 : (In a (cons_ORlist (cons r lf) lg)). +Elim (RList_P9 (cons r lf) lg a); Intros; Apply H10; Left; Elim (RList_P3 (cons r lf) a); Intros; Apply H12; Exists O; Split; [Symmetry; Assumption | Simpl; Apply lt_O_Sn]. +Apply RList_P5; [Apply RList_P2; Assumption | Assumption]. +Rewrite Hyp_max; Apply Rle_antisym. +Induction lf. +Simpl; Right; Assumption. +Assert H8 : (In (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg)))) (cons_ORlist (cons r lf) lg)). +Elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros _ H10; Apply H10; Exists (pred (Rlength (cons_ORlist (cons r lf) lg))); Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_n_Sn]. +Elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H10 _; Assert H11 := (H10 H8); Elim H11; Intro. +Elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H13 _; Assert H14 := (H13 H12); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H5; Elim (RList_P6 (cons r lf)); Intros; Apply H17; [Assumption | Simpl; Simpl in H14; Apply lt_n_Sm_le; Assumption | Simpl; Apply lt_n_Sn]. +Elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H13 _; Assert H14 := (H13 H12); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Assert H17 : (Rlength lg)=(S (pred (Rlength lg))). +Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H17 in H16; Elim (lt_n_O ? H16). +Rewrite <- H0; Elim (RList_P6 lg); Intros; Apply H18; [Assumption | Rewrite H17 in H16; Apply lt_n_Sm_le; Assumption | Apply lt_pred_n_n; Rewrite H17; Apply lt_O_Sn]. +Induction lf. +Simpl; Right; Symmetry; Assumption. +Assert H8 : (In b (cons_ORlist (cons r lf) lg)). +Elim (RList_P9 (cons r lf) lg b); Intros; Apply H10; Left; Elim (RList_P3 (cons r lf) b); Intros; Apply H12; Exists (pred (Rlength (cons r lf))); Split; [Symmetry; Assumption | Simpl; Apply lt_n_Sn]. +Apply RList_P7; [Apply RList_P2; Assumption | Assumption]. +Apply StepFun_P20; Rewrite RList_P11; Rewrite H7; Rewrite H2; Simpl; Apply lt_O_Sn. +Unfold constant_D_eq open_interval; Intros; Cut (EXT l:R | (constant_D_eq g (open_interval (pos_Rl (cons_ORlist lf lg) i) (pos_Rl (cons_ORlist lf lg) (S i))) l)). +Intros; Elim H11; Clear H11; Intros; Assert H12 := H11; Assert Hyp_cons : (EXT r:R | (EXT r0:Rlist | (cons_ORlist lf lg)==(cons r r0))). +Apply RList_P19; Red; Intro; Rewrite H13 in H8; Elim (lt_n_O ? H8). +Elim Hyp_cons; Clear Hyp_cons; Intros r [r0 Hyp_cons]; Rewrite Hyp_cons; Unfold FF; Rewrite RList_P12. +Change (g x)==(g (pos_Rl (mid_Rlist (cons r r0) r) (S i))); Rewrite <- Hyp_cons; Rewrite RList_P13. +Assert H13 := (RList_P2 ? ? H ? H8); Elim H13; Intro. +Unfold constant_D_eq open_interval in H11 H12; Rewrite (H11 x H10); Assert H15 : ``(pos_Rl (cons_ORlist lf lg) i)<((pos_Rl (cons_ORlist lf lg) i)+(pos_Rl (cons_ORlist lf lg) (S i)))/2<(pos_Rl (cons_ORlist lf lg) (S i))``. +Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Rewrite (Rplus_sym (pos_Rl (cons_ORlist lf lg) i)); Apply Rlt_compatibility; Assumption | DiscrR]]. +Rewrite (H11 ? H15); Reflexivity. +Elim H10; Intros; Rewrite H14 in H15; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H16 H15)). +Apply H8. +Rewrite RList_P14; Rewrite Hyp_cons in H8; Simpl in H8; Apply H8. +Assert H11 : ``a<b``. +Apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i). +Rewrite <- H6; Rewrite <- (RList_P15 lf lg); Try Assumption. +Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H11. +Apply RList_P2; Assumption. +Apply le_O_n. +Apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); [Assumption | Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H8; Elim (lt_n_O ? H8)]. +Rewrite H1; Assumption. +Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). +Elim H10; Intros; Apply Rlt_trans with x; Assumption. +Rewrite <- H5; Rewrite <- (RList_P16 lf lg); Try Assumption. +Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H11. +Apply RList_P2; Assumption. +Apply lt_n_Sm_le; Apply lt_n_S; Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H8; Elim (lt_n_O ? H8). +Rewrite H0; Assumption. +Pose I := [j:nat]``(pos_Rl lg j)<=(pos_Rl (cons_ORlist lf lg) i)``/\(lt j (Rlength lg)); Assert H12 : (Nbound I). +Unfold Nbound; Exists (Rlength lg); Intros; Unfold I in H12; Elim H12; Intros; Apply lt_le_weak; Assumption. +Assert H13 : (EX n:nat | (I n)). +Exists O; Unfold I; Split. +Apply Rle_trans with (pos_Rl (cons_ORlist lf lg) O). +Right; Symmetry; Rewrite H1; Rewrite <- H6; Apply RList_P15; Try Assumption; Rewrite H1; Assumption. +Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H13; [Apply RList_P2; Assumption | Apply le_O_n | Apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); [Assumption | Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H15 in H8; Elim (lt_n_O ? H8)]]. +Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H0; Rewrite <- H1 in H11; Rewrite <- H0 in H11; Elim (Rlt_antirefl ? H11). +Assert H14 := (Nzorn H13 H12); Elim H14; Clear H14; Intros x0 H14; Exists (pos_Rl lg0 x0); Unfold constant_D_eq open_interval; Intros; Assert H16 := (H4 x0); Assert H17 : (lt x0 (pred (Rlength lg))). +Elim H14; Clear H14; Intros; Unfold I in H14; Elim H14; Clear H14; Intros; Apply lt_S_n; Replace (S (pred (Rlength lg))) with (Rlength lg). +Inversion H18. +2:Apply lt_n_S; Assumption. +Cut x0=(pred (Rlength lg)). +Intro; Rewrite H19 in H14; Rewrite H0 in H14; Cut ``(pos_Rl (cons_ORlist lf lg) i)<b``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H14 H21)). +Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). +Elim H10; Intros; Apply Rlt_trans with x; Assumption. +Rewrite <- H0; Apply Rle_trans with (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))). +Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H21. +Apply RList_P2; Assumption. +Apply lt_n_Sm_le; Apply lt_n_S; Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H23 in H8; Elim (lt_n_O ? H8). +Right; Rewrite H0; Rewrite <- H5; Apply RList_P16; Try Assumption. +Rewrite H0; Assumption. +Rewrite <- H20; Reflexivity. +Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H19 in H18; Elim (lt_n_O ? H18). +Assert H18 := (H16 H17); Unfold constant_D_eq open_interval in H18; Rewrite (H18 x1). +Reflexivity. +Elim H15; Clear H15; Intros; Elim H14; Clear H14; Intros; Unfold I in H14; Elim H14; Clear H14; Intros; Split. +Apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); Assumption. +Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); Try Assumption. +Assert H22 : (lt (S x0) (Rlength lg)). +Replace (Rlength lg) with (S (pred (Rlength lg))). +Apply lt_n_S; Assumption. +Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H22 in H21; Elim (lt_n_O ? H21). +Elim (total_order_Rle (pos_Rl lg (S x0)) (pos_Rl (cons_ORlist lf lg) i)); Intro. +Assert H23 : (le (S x0) x0); [Apply H20; Unfold I; Split; Assumption | Elim (le_Sn_n ? H23)]. +Assert H23 : ``(pos_Rl (cons_ORlist lf lg) i)<(pos_Rl lg (S x0))``. +Auto with real. +Clear b0; Apply RList_P17; Try Assumption; [Apply RList_P2; Assumption | Elim (RList_P9 lf lg (pos_Rl lg (S x0))); Intros; Apply H25; Right; Elim (RList_P3 lg (pos_Rl lg (S x0))); Intros; Apply H27; Exists (S x0); Split; [Reflexivity | Apply H22]]. +Qed. + +Lemma StepFun_P25 : (a,b:R;f,g:R->R;lf,lg:Rlist) (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision g a b (cons_ORlist lf lg)). +Intros a b f g lf lg H H0; Case (total_order_Rle a b); Intro; [Apply StepFun_P24 with f; Assumption | Apply StepFun_P5; Apply StepFun_P24 with f; [Auto with real | Apply StepFun_P5; Assumption | Apply StepFun_P5; Assumption]]. +Qed. + +Lemma StepFun_P26 : (a,b,l:R;f,g:R->R;l1:Rlist) (is_subdivision f a b l1) -> (is_subdivision g a b l1) -> (is_subdivision [x:R]``(f x)+l*(g x)`` a b l1). +Intros a b l f g l1; Unfold is_subdivision; Intros; Elim X; Elim X0; Intros; Clear X X0; Unfold adapted_couple in p p0; Decompose [and] p; Decompose [and] p0; Clear p p0; Apply existTT with (FF l1 [x:R]``(f x)+l*(g x)``); Unfold adapted_couple; Repeat Split; Try Assumption. +Apply StepFun_P20; Apply neq_O_lt; Red; Intro; Rewrite <- H8 in H7; Discriminate. +Intros; Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9 H4; Intros; Rewrite (H9 ? H8 ? H10); Rewrite (H4 ? H8 ? H10); Assert H11 : ~l1==nil. +Red; Intro; Rewrite H11 in H8; Elim (lt_n_O ? H8). +Assert H12 := (RList_P19 ? H11); Elim H12; Clear H12; Intros r [r0 H12]; Rewrite H12; Unfold FF; Change ``(pos_Rl x0 i)+l*(pos_Rl x i)`` == (pos_Rl (app_Rlist (mid_Rlist (cons r r0) r) [x2:R]``(f x2)+l*(g x2)``) (S i)); Rewrite RList_P12. +Rewrite RList_P13. +Rewrite <- H12; Rewrite (H9 ? H8); Try Rewrite (H4 ? H8); Reflexivity Orelse (Elim H10; Clear H10; Intros; Split; [Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Apply Rlt_trans with x1; Assumption | DiscrR]] | Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Rewrite (Rplus_sym (pos_Rl l1 i)); Apply Rlt_compatibility; Apply Rlt_trans with x1; Assumption | DiscrR]]]). +Rewrite <- H12; Assumption. +Rewrite RList_P14; Simpl; Rewrite H12 in H8; Simpl in H8; Apply lt_n_S; Apply H8. +Qed. + +Lemma StepFun_P27 : (a,b,l:R;f,g:R->R;lf,lg:Rlist) (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision [x:R]``(f x)+l*(g x)`` a b (cons_ORlist lf lg)). +Intros a b l f g lf lg H H0; Apply StepFun_P26; [Apply StepFun_P23 with g; Assumption | Apply StepFun_P25 with f; Assumption]. +Qed. + +(* The set of step functions on [a,b] is a vectorial space *) +Lemma StepFun_P28 : (a,b,l:R;f,g:(StepFun a b)) (IsStepFun [x:R]``(f x)+l*(g x)`` a b). +Intros a b l f g; Unfold IsStepFun; Assert H := (pre f); Assert H0 := (pre g); Unfold IsStepFun in H H0; Elim H; Elim H0; Intros; Apply Specif.existT with (cons_ORlist x0 x); Apply StepFun_P27; Assumption. +Qed. + +Lemma StepFun_P29 : (a,b:R;f:(StepFun a b)) (is_subdivision f a b (subdivision f)). +Intros a b f; Unfold is_subdivision; Apply existTT with (subdivision_val f); Apply StepFun_P1. +Qed. + +Lemma StepFun_P30 : (a,b,l:R;f,g:(StepFun a b)) ``(RiemannInt_SF (mkStepFun (StepFun_P28 l f g)))==(RiemannInt_SF f)+l*(RiemannInt_SF g)``. +Intros a b l f g; Unfold RiemannInt_SF; Case (total_order_Rle a b); (Intro; Replace ``(Int_SF (subdivision_val (mkStepFun (StepFun_P28 l f g))) (subdivision (mkStepFun (StepFun_P28 l f g))))`` with (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) [x:R]``(f x)+l*(g x)``) (cons_ORlist (subdivision f) (subdivision g))); [Rewrite StepFun_P19; Replace (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) f) (cons_ORlist (subdivision f) (subdivision g))) with (Int_SF (subdivision_val f) (subdivision f)); [Replace (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) g) (cons_ORlist (subdivision f) (subdivision g))) with (Int_SF (subdivision_val g) (subdivision g)); [Ring | Apply StepFun_P17 with (fe g) a b; [Apply StepFun_P1 | Apply StepFun_P21; Apply StepFun_P25 with (fe f); Apply StepFun_P29]] | Apply StepFun_P17 with (fe f) a b; [Apply StepFun_P1 | Apply StepFun_P21; Apply StepFun_P23 with (fe g); Apply StepFun_P29]] | Apply StepFun_P17 with [x:R]``(f x)+l*(g x)`` a b; [Apply StepFun_P21; Apply StepFun_P27; Apply StepFun_P29 | Apply (StepFun_P1 (mkStepFun (StepFun_P28 l f g)))]]). +Qed. + +Lemma StepFun_P31 : (a,b:R;f:R->R;l,lf:Rlist) (adapted_couple f a b l lf) -> (adapted_couple [x:R](Rabsolu (f x)) a b l (app_Rlist lf Rabsolu)). +Unfold adapted_couple; Intros; Decompose [and] H; Clear H; Repeat Split; Try Assumption. +Symmetry; Rewrite H3; Rewrite RList_P18; Reflexivity. +Intros; Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H5; Intros; Rewrite (H5 ? H ? H4); Rewrite RList_P12; [Reflexivity | Rewrite H3 in H; Simpl in H; Apply H]. +Qed. + +Lemma StepFun_P32 : (a,b:R;f:(StepFun a b)) (IsStepFun [x:R](Rabsolu (f x)) a b). +Intros a b f; Unfold IsStepFun; Apply Specif.existT with (subdivision f); Unfold is_subdivision; Apply existTT with (app_Rlist (subdivision_val f) Rabsolu); Apply StepFun_P31; Apply StepFun_P1. +Qed. + +Lemma StepFun_P33 : (l2,l1:Rlist) (ordered_Rlist l1) -> ``(Rabsolu (Int_SF l2 l1))<=(Int_SF (app_Rlist l2 Rabsolu) l1)``. +Induction l2; Intros. +Simpl; Rewrite Rabsolu_R0; Right; Reflexivity. +Simpl; Induction l1. +Rewrite Rabsolu_R0; Right; Reflexivity. +Induction l1. +Rewrite Rabsolu_R0; Right; Reflexivity. +Apply Rle_trans with ``(Rabsolu (r*(r2-r1)))+(Rabsolu (Int_SF r0 (cons r2 l1)))``. +Apply Rabsolu_triang. +Rewrite Rabsolu_mult; Rewrite (Rabsolu_right ``r2-r1``); [Apply Rle_compatibility; Apply H; Apply RList_P4 with r1; Assumption | Apply Rge_minus; Apply Rle_sym1; Apply (H0 O); Simpl; Apply lt_O_Sn]. +Qed. + +Lemma StepFun_P34 : (a,b:R;f:(StepFun a b)) ``a<=b`` -> ``(Rabsolu (RiemannInt_SF f))<=(RiemannInt_SF (mkStepFun (StepFun_P32 f)))``. +Intros; Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. +Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P32 f))) (subdivision (mkStepFun (StepFun_P32 f)))) with (Int_SF (app_Rlist (subdivision_val f) Rabsolu) (subdivision f)). +Apply StepFun_P33; Assert H0 := (StepFun_P29 f); Unfold is_subdivision in H0; Elim H0; Intros; Unfold adapted_couple in p; Decompose [and] p; Assumption. +Apply StepFun_P17 with [x:R](Rabsolu (f x)) a b; [Apply StepFun_P31; Apply StepFun_P1 | Apply (StepFun_P1 (mkStepFun (StepFun_P32 f)))]. +Elim n; Assumption. +Qed. + +Lemma StepFun_P35 : (l:Rlist;a,b:R;f,g:R->R) (ordered_Rlist l) -> (pos_Rl l O)==a -> (pos_Rl l (pred (Rlength l)))==b -> ((x:R)``a<x<b``->``(f x)<=(g x)``) -> ``(Int_SF (FF l f) l)<=(Int_SF (FF l g) l)``. +Induction l; Intros. +Right; Reflexivity. +Simpl; Induction r0. +Right; Reflexivity. +Simpl; Apply Rplus_le. +Case (Req_EM r r0); Intro. +Rewrite H4; Right; Ring. +Do 2 Rewrite <- (Rmult_sym ``r0-r``); Apply Rle_monotony. +Apply Rle_sym2; Apply Rge_minus; Apply Rle_sym1; Apply (H0 O); Simpl; Apply lt_O_Sn. +Apply H3; Split. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Assert H5 : r==a. +Apply H1. +Rewrite H5; Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility. +Assert H6 := (H0 O (lt_O_Sn ?)). +Simpl in H6. +Elim H6; Intro. +Rewrite H5 in H7; Apply H7. +Elim H4; Assumption. +DiscrR. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite double; Assert H5 : ``r0<=b``. +Replace b with (pos_Rl (cons r (cons r0 r1)) (pred (Rlength (cons r (cons r0 r1))))). +Replace r0 with (pos_Rl (cons r (cons r0 r1)) (S O)). +Elim (RList_P6 (cons r (cons r0 r1))); Intros; Apply H5. +Assumption. +Simpl; Apply le_n_S. +Apply le_O_n. +Simpl; Apply lt_n_Sn. +Reflexivity. +Apply Rle_lt_trans with ``r+b``. +Apply Rle_compatibility; Assumption. +Rewrite (Rplus_sym r); Apply Rlt_compatibility. +Apply Rlt_le_trans with r0. +Assert H6 := (H0 O (lt_O_Sn ?)). +Simpl in H6. +Elim H6; Intro. +Apply H7. +Elim H4; Assumption. +Assumption. +DiscrR. +Simpl in H; Apply H with r0 b. +Apply RList_P4 with r; Assumption. +Reflexivity. +Rewrite <- H2; Reflexivity. +Intros; Apply H3; Elim H4; Intros; Split; Try Assumption. +Apply Rle_lt_trans with r0; Try Assumption. +Rewrite <- H1. +Simpl; Apply (H0 O); Simpl; Apply lt_O_Sn. +Qed. + +Lemma StepFun_P36 : (a,b:R;f,g:(StepFun a b);l:Rlist) ``a<=b`` -> (is_subdivision f a b l) -> (is_subdivision g a b l) -> ((x:R)``a<x<b``->``(f x)<=(g x)``) -> ``(RiemannInt_SF f) <= (RiemannInt_SF g)``. +Intros; Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. +Replace (Int_SF (subdivision_val f) (subdivision f)) with (Int_SF (FF l f) l). +Replace (Int_SF (subdivision_val g) (subdivision g)) with (Int_SF (FF l g) l). +Unfold is_subdivision in X; Elim X; Clear X; Intros; Unfold adapted_couple in p; Decompose [and] p; Clear p; Assert H5 : (Rmin a b)==a; [Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption] | Assert H7 : (Rmax a b)==b; [Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption] | Rewrite H5 in H3; Rewrite H7 in H2; EApply StepFun_P35 with a b; Assumption]]. +Apply StepFun_P17 with (fe g) a b; [Apply StepFun_P21; Assumption | Apply StepFun_P1]. +Apply StepFun_P17 with (fe f) a b; [Apply StepFun_P21; Assumption | Apply StepFun_P1]. +Elim n; Assumption. +Qed. + +Lemma StepFun_P37 : (a,b:R;f,g:(StepFun a b)) ``a<=b`` -> ((x:R)``a<x<b``->``(f x)<=(g x)``) -> ``(RiemannInt_SF f) <= (RiemannInt_SF g)``. +Intros; EApply StepFun_P36; Try Assumption. +EApply StepFun_P25; Apply StepFun_P29. +EApply StepFun_P23; Apply StepFun_P29. +Qed. + +Lemma StepFun_P38 : (l:Rlist;a,b:R;f:R->R) (ordered_Rlist l) -> (pos_Rl l O)==a -> (pos_Rl l (pred (Rlength l)))==b -> (sigTT ? [g:(StepFun a b)](g b)==(f b)/\(i:nat)(lt i (pred (Rlength l)))->(constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i))) (f (pos_Rl l i)))). +Intros l a b f; Generalize a; Clear a; NewInduction l. +Intros a H H0 H1; Simpl in H0; Simpl in H1; Exists (mkStepFun (StepFun_P4 a b (f b))); Split. +Reflexivity. +Intros; Elim (lt_n_O ? H2). +Intros; NewDestruct l as [|r1 l]. +Simpl in H1; Simpl in H0; Exists (mkStepFun (StepFun_P4 a b (f b))); Split. +Reflexivity. +Intros i H2; Elim (lt_n_O ? H2). +Intros; Assert H2 : (ordered_Rlist (cons r1 l)). +Apply RList_P4 with r; Assumption. +Assert H3 : (pos_Rl (cons r1 l) O)==r1. +Reflexivity. +Assert H4 : (pos_Rl (cons r1 l) (pred (Rlength (cons r1 l))))==b. +Rewrite <- H1; Reflexivity. +Elim (IHl r1 H2 H3 H4); Intros g [H5 H6]. +Pose g' := [x:R]Cases (total_order_Rle r1 x) of + | (leftT _) => (g x) + | (rightT _) => (f a) end. +Assert H7 : ``r1<=b``. +Rewrite <- H4; Apply RList_P7; [Assumption | Left; Reflexivity]. +Assert H8 : (IsStepFun g' a b). +Unfold IsStepFun; Assert H8 := (pre g); Unfold IsStepFun in H8; Elim H8; Intros lg H9; Unfold is_subdivision in H9; Elim H9; Clear H9; Intros lg2 H9; Split with (cons a lg); Unfold is_subdivision; Split with (cons (f a) lg2); Unfold adapted_couple in H9; Decompose [and] H9; Clear H9; Unfold adapted_couple; Repeat Split. +Unfold ordered_Rlist; Intros; Simpl in H9; Induction i. +Simpl; Rewrite H12; Replace (Rmin r1 b) with r1. +Simpl in H0; Rewrite <- H0; Apply (H O); Simpl; Apply lt_O_Sn. +Unfold Rmin; Case (total_order_Rle r1 b); Intro; [Reflexivity | Elim n; Assumption]. +Apply (H10 i); Apply lt_S_n. +Replace (S (pred (Rlength lg))) with (Rlength lg). +Apply H9. +Apply S_pred with O; Apply neq_O_lt; Intro; Rewrite <- H14 in H9; Elim (lt_n_O ? H9). +Simpl; Assert H14 : ``a<=b``. +Rewrite <- H1; Simpl in H0; Rewrite <- H0; Apply RList_P7; [Assumption | Left; Reflexivity]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Assert H14 : ``a<=b``. +Rewrite <- H1; Simpl in H0; Rewrite <- H0; Apply RList_P7; [Assumption | Left; Reflexivity]. +Replace (Rmax a b) with (Rmax r1 b). +Rewrite <- H11; Induction lg. +Simpl in H13; Discriminate. +Reflexivity. +Unfold Rmax; Case (total_order_Rle a b); Case (total_order_Rle r1 b); Intros; Reflexivity Orelse Elim n; Assumption. +Simpl; Rewrite H13; Reflexivity. +Intros; Simpl in H9; Induction i. +Unfold constant_D_eq open_interval; Simpl; Intros; Assert H16 : (Rmin r1 b)==r1. +Unfold Rmin; Case (total_order_Rle r1 b); Intro; [Reflexivity | Elim n; Assumption]. +Rewrite H16 in H12; Rewrite H12 in H14; Elim H14; Clear H14; Intros _ H14; Unfold g'; Case (total_order_Rle r1 x); Intro r3. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r3 H14)). +Reflexivity. +Change (constant_D_eq g' (open_interval (pos_Rl lg i) (pos_Rl lg (S i))) (pos_Rl lg2 i)); Clear Hreci; Assert H16 := (H15 i); Assert H17 : (lt i (pred (Rlength lg))). +Apply lt_S_n. +Replace (S (pred (Rlength lg))) with (Rlength lg). +Assumption. +Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H14 in H9; Elim (lt_n_O ? H9). +Assert H18 := (H16 H17); Unfold constant_D_eq open_interval in H18; Unfold constant_D_eq open_interval; Intros; Assert H19 := (H18 ? H14); Rewrite <- H19; Unfold g'; Case (total_order_Rle r1 x); Intro. +Reflexivity. +Elim n; Replace r1 with (Rmin r1 b). +Rewrite <- H12; Elim H14; Clear H14; Intros H14 _; Left; Apply Rle_lt_trans with (pos_Rl lg i); Try Assumption. +Apply RList_P5. +Assumption. +Elim (RList_P3 lg (pos_Rl lg i)); Intros; Apply H21; Exists i; Split. +Reflexivity. +Apply lt_trans with (pred (Rlength lg)); Try Assumption. +Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H22 in H17; Elim (lt_n_O ? H17). +Unfold Rmin; Case (total_order_Rle r1 b); Intro; [Reflexivity | Elim n0; Assumption]. +Exists (mkStepFun H8); Split. +Simpl; Unfold g'; Case (total_order_Rle r1 b); Intro. +Assumption. +Elim n; Assumption. +Intros; Simpl in H9; Induction i. +Unfold constant_D_eq co_interval; Simpl; Intros; Simpl in H0; Rewrite H0; Elim H10; Clear H10; Intros; Unfold g'; Case (total_order_Rle r1 x); Intro r3. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r3 H11)). +Reflexivity. +Clear Hreci; Change (constant_D_eq (mkStepFun H8) (co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i))) (f (pos_Rl (cons r1 l) i))); Assert H10 := (H6 i); Assert H11 : (lt i (pred (Rlength (cons r1 l)))). +Simpl; Apply lt_S_n; Assumption. +Assert H12 := (H10 H11); Unfold constant_D_eq co_interval in H12; Unfold constant_D_eq co_interval; Intros; Rewrite <- (H12 ? H13); Simpl; Unfold g'; Case (total_order_Rle r1 x); Intro. +Reflexivity. +Elim n; Elim H13; Clear H13; Intros; Apply Rle_trans with (pos_Rl (cons r1 l) i); Try Assumption; Change ``(pos_Rl (cons r1 l) O)<=(pos_Rl (cons r1 l) i)``; Elim (RList_P6 (cons r1 l)); Intros; Apply H15; [Assumption | Apply le_O_n | Simpl; Apply lt_trans with (Rlength l); [Apply lt_S_n; Assumption | Apply lt_n_Sn]]. +Qed. + +Lemma StepFun_P39 : (a,b:R;f:(StepFun a b)) (RiemannInt_SF f)==(Ropp (RiemannInt_SF (mkStepFun (StepFun_P6 (pre f))))). +Intros; Unfold RiemannInt_SF; Case (total_order_Rle a b); Case (total_order_Rle b a); Intros. +Assert H : (adapted_couple f a b (subdivision f) (subdivision_val f)); [Apply StepFun_P1 | Assert H0 : (adapted_couple (mkStepFun (StepFun_P6 (pre f))) b a (subdivision (mkStepFun (StepFun_P6 (pre f)))) (subdivision_val (mkStepFun (StepFun_P6 (pre f))))); [Apply StepFun_P1 | Assert H1 : a==b; [Apply Rle_antisym; Assumption | Rewrite (StepFun_P8 H H1); Assert H2 : b==a; [Symmetry; Apply H1 | Rewrite (StepFun_P8 H0 H2); Ring]]]]. +Rewrite Ropp_Ropp; EApply StepFun_P17; [Apply StepFun_P1 | Apply StepFun_P2; Pose H := (StepFun_P6 (pre f)); Unfold IsStepFun in H; Elim H; Intros; Unfold is_subdivision; Elim p; Intros; Apply p0]. +Apply eq_Ropp; EApply StepFun_P17; [Apply StepFun_P1 | Apply StepFun_P2; Pose H := (StepFun_P6 (pre f)); Unfold IsStepFun in H; Elim H; Intros; Unfold is_subdivision; Elim p; Intros; Apply p0]. +Assert H : ``a<b``; [Auto with real | Assert H0 : ``b<a``; [Auto with real | Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H H0))]]. +Qed. + +Lemma StepFun_P40 : (f:R->R;a,b,c:R;l1,l2,lf1,lf2:Rlist) ``a<b`` -> ``b<c`` -> (adapted_couple f a b l1 lf1) -> (adapted_couple f b c l2 lf2) -> (adapted_couple f a c (cons_Rlist l1 l2) (FF (cons_Rlist l1 l2) f)). +Intros f a b c l1 l2 lf1 lf2 H H0 H1 H2; Unfold adapted_couple in H1 H2; Unfold adapted_couple; Decompose [and] H1; Decompose [and] H2; Clear H1 H2; Repeat Split. +Apply RList_P25; Try Assumption. +Rewrite H10; Rewrite H4; Unfold Rmin Rmax; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros; (Right; Reflexivity) Orelse (Elim n; Left; Assumption). +Rewrite RList_P22. +Rewrite H5; Unfold Rmin Rmax; Case (total_order_Rle a b); Case (total_order_Rle a c); Intros; [Reflexivity | Elim n; Apply Rle_trans with b; Left; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. +Red; Intro; Rewrite H1 in H6; Discriminate. +Rewrite RList_P24. +Rewrite H9; Unfold Rmin Rmax; Case (total_order_Rle b c); Case (total_order_Rle a c); Intros; [Reflexivity | Elim n; Apply Rle_trans with b; Left; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. +Red; Intro; Rewrite H1 in H11; Discriminate. +Apply StepFun_P20. +Rewrite RList_P23; Apply neq_O_lt; Red; Intro. +Assert H2 : (plus (Rlength l1) (Rlength l2))=O. +Symmetry; Apply H1. +Elim (plus_is_O ? ? H2); Intros; Rewrite H12 in H6; Discriminate. +Unfold constant_D_eq open_interval; Intros; Elim (le_or_lt (S (S i)) (Rlength l1)); Intro. +Assert H14 : (pos_Rl (cons_Rlist l1 l2) i) == (pos_Rl l1 i). +Apply RList_P26; Apply lt_S_n; Apply le_lt_n_Sm; Apply le_S_n; Apply le_trans with (Rlength l1); [Assumption | Apply le_n_Sn]. +Assert H15 : (pos_Rl (cons_Rlist l1 l2) (S i))==(pos_Rl l1 (S i)). +Apply RList_P26; Apply lt_S_n; Apply le_lt_n_Sm; Assumption. +Rewrite H14 in H2; Rewrite H15 in H2; Assert H16 : (le (2) (Rlength l1)). +Apply le_trans with (S (S i)); [Repeat Apply le_n_S; Apply le_O_n | Assumption]. +Elim (RList_P20 ? H16); Intros r1 [r2 [r3 H17]]; Rewrite H17; Change (f x)==(pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i); Rewrite RList_P12. +Induction i. +Simpl; Assert H18 := (H8 O); Unfold constant_D_eq open_interval in H18; Assert H19 : (lt O (pred (Rlength l1))). +Rewrite H17; Simpl; Apply lt_O_Sn. +Assert H20 := (H18 H19); Repeat Rewrite H20. +Reflexivity. +Assert H21 : ``r1<=r2``. +Rewrite H17 in H3; Apply (H3 O). +Simpl; Apply lt_O_Sn. +Elim H21; Intro. +Split. +Rewrite H17; Simpl; Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Rewrite H17; Simpl; Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym r1); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Elim H2; Intros; Rewrite H17 in H23; Rewrite H17 in H24; Simpl in H24; Simpl in H23; Rewrite H22 in H23; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H23 H24)). +Assumption. +Clear Hreci; Rewrite RList_P13. +Rewrite H17 in H14; Rewrite H17 in H15; Change (pos_Rl (cons_Rlist (cons r2 r3) l2) i)== (pos_Rl (cons r1 (cons r2 r3)) (S i)) in H14; Rewrite H14; Change (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i))==(pos_Rl (cons r1 (cons r2 r3)) (S (S i))) in H15; Rewrite H15; Assert H18 := (H8 (S i)); Unfold constant_D_eq open_interval in H18; Assert H19 : (lt (S i) (pred (Rlength l1))). +Apply lt_pred; Apply lt_S_n; Apply le_lt_n_Sm; Assumption. +Assert H20 := (H18 H19); Repeat Rewrite H20. +Reflexivity. +Rewrite <- H17; Assert H21 : ``(pos_Rl l1 (S i))<=(pos_Rl l1 (S (S i)))``. +Apply (H3 (S i)); Apply lt_pred; Apply lt_S_n; Apply le_lt_n_Sm; Assumption. +Elim H21; Intro. +Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym (pos_Rl l1 (S i))); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Elim H2; Intros; Rewrite H22 in H23; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H23 H24)). +Assumption. +Simpl; Rewrite H17 in H1; Simpl in H1; Apply lt_S_n; Assumption. +Rewrite RList_P14; Rewrite H17 in H1; Simpl in H1; Apply H1. +Inversion H12. +Assert H16 : (pos_Rl (cons_Rlist l1 l2) (S i))==b. +Rewrite RList_P29. +Rewrite H15; Rewrite <- minus_n_n; Rewrite H10; Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n; Left; Assumption]. +Rewrite H15; Apply le_n. +Induction l1. +Simpl in H15; Discriminate. +Clear Hrecl1; Simpl in H1; Simpl; Apply lt_n_S; Assumption. +Assert H17 : (pos_Rl (cons_Rlist l1 l2) i)==b. +Rewrite RList_P26. +Replace i with (pred (Rlength l1)); [Rewrite H4; Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Left; Assumption] | Rewrite H15; Reflexivity]. +Rewrite H15; Apply lt_n_Sn. +Rewrite H16 in H2; Rewrite H17 in H2; Elim H2; Intros; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H14 H18)). +Assert H16 : (pos_Rl (cons_Rlist l1 l2) i) == (pos_Rl l2 (minus i (Rlength l1))). +Apply RList_P29. +Apply le_S_n; Assumption. +Apply lt_le_trans with (pred (Rlength (cons_Rlist l1 l2))); [Assumption | Apply le_pred_n]. +Assert H17 : (pos_Rl (cons_Rlist l1 l2) (S i))==(pos_Rl l2 (S (minus i (Rlength l1)))). +Replace (S (minus i (Rlength l1))) with (minus (S i) (Rlength l1)). +Apply RList_P29. +Apply le_S_n; Apply le_trans with (S i); [Assumption | Apply le_n_Sn]. +Induction l1. +Simpl in H6; Discriminate. +Clear Hrecl1; Simpl in H1; Simpl; Apply lt_n_S; Assumption. +Symmetry; Apply minus_Sn_m; Apply le_S_n; Assumption. +Assert H18 : (le (2) (Rlength l1)). +Clear f c l2 lf2 H0 H3 H8 H7 H10 H9 H11 H13 i H1 x H2 H12 m H14 H15 H16 H17; Induction l1. +Discriminate. +Clear Hrecl1; Induction l1. +Simpl in H5; Simpl in H4; Assert H0 : ``(Rmin a b)<(Rmax a b)``. +Unfold Rmin Rmax; Case (total_order_Rle a b); Intro; [Assumption | Elim n; Left; Assumption]. +Rewrite <- H5 in H0; Rewrite <- H4 in H0; Elim (Rlt_antirefl ? H0). +Clear Hrecl1; Simpl; Repeat Apply le_n_S; Apply le_O_n. +Elim (RList_P20 ? H18); Intros r1 [r2 [r3 H19]]; Rewrite H19; Change (f x)==(pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i); Rewrite RList_P12. +Induction i. +Assert H20 := (le_S_n ? ? H15); Assert H21 := (le_trans ? ? ? H18 H20); Elim (le_Sn_O ? H21). +Clear Hreci; Rewrite RList_P13. +Rewrite H19 in H16; Rewrite H19 in H17; Change (pos_Rl (cons_Rlist (cons r2 r3) l2) i)== (pos_Rl l2 (minus (S i) (Rlength (cons r1 (cons r2 r3))))) in H16; Rewrite H16; Change (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i))== (pos_Rl l2 (S (minus (S i) (Rlength (cons r1 (cons r2 r3)))))) in H17; Rewrite H17; Assert H20 := (H13 (minus (S i) (Rlength l1))); Unfold constant_D_eq open_interval in H20; Assert H21 : (lt (minus (S i) (Rlength l1)) (pred (Rlength l2))). +Apply lt_pred; Rewrite minus_Sn_m. +Apply simpl_lt_plus_l with (Rlength l1); Rewrite <- le_plus_minus. +Rewrite H19 in H1; Simpl in H1; Rewrite H19; Simpl; Rewrite RList_P23 in H1; Apply lt_n_S; Assumption. +Apply le_trans with (S i); [Apply le_S_n; Assumption | Apply le_n_Sn]. +Apply le_S_n; Assumption. +Assert H22 := (H20 H21); Repeat Rewrite H22. +Reflexivity. +Rewrite <- H19; Assert H23 : ``(pos_Rl l2 (minus (S i) (Rlength l1)))<=(pos_Rl l2 (S (minus (S i) (Rlength l1))))``. +Apply H7; Apply lt_pred. +Rewrite minus_Sn_m. +Apply simpl_lt_plus_l with (Rlength l1); Rewrite <- le_plus_minus. +Rewrite H19 in H1; Simpl in H1; Rewrite H19; Simpl; Rewrite RList_P23 in H1; Apply lt_n_S; Assumption. +Apply le_trans with (S i); [Apply le_S_n; Assumption | Apply le_n_Sn]. +Apply le_S_n; Assumption. +Elim H23; Intro. +Split. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym (pos_Rl l2 (minus (S i) (Rlength l1)))); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. +Rewrite <- H19 in H16; Rewrite <- H19 in H17; Elim H2; Intros; Rewrite H19 in H25; Rewrite H19 in H26; Simpl in H25; Simpl in H16; Rewrite H16 in H25; Simpl in H26; Simpl in H17; Rewrite H17 in H26; Simpl in H24; Rewrite H24 in H25; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H25 H26)). +Assert H23 : (pos_Rl (cons_Rlist l1 l2) (S i))==(pos_Rl l2 (minus (S i) (Rlength l1))). +Rewrite H19; Simpl; Simpl in H16; Apply H16. +Assert H24 : (pos_Rl (cons_Rlist l1 l2) (S (S i)))==(pos_Rl l2 (S (minus (S i) (Rlength l1)))). +Rewrite H19; Simpl; Simpl in H17; Apply H17. +Rewrite <- H23; Rewrite <- H24; Assumption. +Simpl; Rewrite H19 in H1; Simpl in H1; Apply lt_S_n; Assumption. +Rewrite RList_P14; Rewrite H19 in H1; Simpl in H1; Simpl; Apply H1. +Qed. + +Lemma StepFun_P41 : (f:R->R;a,b,c:R) ``a<=b``->``b<=c``->(IsStepFun f a b) -> (IsStepFun f b c) -> (IsStepFun f a c). +Unfold IsStepFun; Unfold is_subdivision; Intros; Elim X; Clear X; Intros l1 [lf1 H1]; Elim X0; Clear X0; Intros l2 [lf2 H2]; Case (total_order_T a b); Intro. +Elim s; Intro. +Case (total_order_T b c); Intro. +Elim s0; Intro. +Split with (cons_Rlist l1 l2); Split with (FF (cons_Rlist l1 l2) f); Apply StepFun_P40 with b lf1 lf2; Assumption. +Split with l1; Split with lf1; Rewrite b0 in H1; Assumption. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 r)). +Split with l2; Split with lf2; Rewrite <- b0 in H2; Assumption. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Qed. + +Lemma StepFun_P42 : (l1,l2:Rlist;f:R->R) (pos_Rl l1 (pred (Rlength l1)))==(pos_Rl l2 O) -> ``(Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)) == (Int_SF (FF l1 f) l1) + (Int_SF (FF l2 f) l2)``. +Intros l1 l2 f; NewInduction l1 as [|r l1 IHl1]; Intros H; [ Simpl; Ring | NewDestruct l1; [Simpl in H; Simpl; NewDestruct l2; [Simpl; Ring | Simpl; Simpl in H; Rewrite H; Ring] | Simpl; Rewrite Rplus_assoc; Apply Rplus_plus_r; Apply IHl1; Rewrite <- H; Reflexivity]]. +Qed. + +Lemma StepFun_P43 : (f:R->R;a,b,c:R;pr1:(IsStepFun f a b);pr2:(IsStepFun f b c);pr3:(IsStepFun f a c)) ``(RiemannInt_SF (mkStepFun pr1))+(RiemannInt_SF (mkStepFun pr2))==(RiemannInt_SF (mkStepFun pr3))``. +Intros f; Intros; Assert H1 : (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f a b l l0))). +Apply pr1. +Assert H2 : (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f b c l l0))). +Apply pr2. +Assert H3 : (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f a c l l0))). +Apply pr3. +Elim H1; Clear H1; Intros l1 [lf1 H1]; Elim H2; Clear H2; Intros l2 [lf2 H2]; Elim H3; Clear H3; Intros l3 [lf3 H3]. +Replace (RiemannInt_SF (mkStepFun pr1)) with (Cases (total_order_Rle a b) of (leftT _) => (Int_SF lf1 l1) | (rightT _) => ``-(Int_SF lf1 l1)`` end). +Replace (RiemannInt_SF (mkStepFun pr2)) with (Cases (total_order_Rle b c) of (leftT _) => (Int_SF lf2 l2) | (rightT _) => ``-(Int_SF lf2 l2)`` end). +Replace (RiemannInt_SF (mkStepFun pr3)) with (Cases (total_order_Rle a c) of (leftT _) => (Int_SF lf3 l3) | (rightT _) => ``-(Int_SF lf3 l3)`` end). +Case (total_order_Rle a b); Case (total_order_Rle b c); Case (total_order_Rle a c); Intros. +Elim r1; Intro. +Elim r0; Intro. +Replace (Int_SF lf3 l3) with (Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)). +Replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). +Replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). +Symmetry; Apply StepFun_P42. +Unfold adapted_couple in H1 H2; Decompose [and] H1; Decompose [and] H2; Clear H1 H2; Rewrite H11; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros; Reflexivity Orelse Elim n; Assumption. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf2; Apply H2; Assumption | Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf1; Apply H1 | Assumption]. +EApply StepFun_P17; [Apply (StepFun_P40 H H0 H1 H2) | Apply H3]. +Replace (Int_SF lf2 l2) with R0. +Rewrite Rplus_Or; EApply StepFun_P17; [Apply H1 | Rewrite <- H0 in H3; Apply H3]. +Symmetry; EApply StepFun_P8; [Apply H2 | Assumption]. +Replace (Int_SF lf1 l1) with R0. +Rewrite Rplus_Ol; EApply StepFun_P17; [Apply H2 | Rewrite H in H3; Apply H3]. +Symmetry; EApply StepFun_P8; [Apply H1 | Assumption]. +Elim n; Apply Rle_trans with b; Assumption. +Apply r_Rplus_plus with (Int_SF lf2 l2); Replace ``(Int_SF lf2 l2)+((Int_SF lf1 l1)+ -(Int_SF lf2 l2))`` with (Int_SF lf1 l1); [Idtac | Ring]. +Assert H : ``c<b``. +Auto with real. +Elim r; Intro. +Rewrite Rplus_sym; Replace (Int_SF lf1 l1) with (Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2)). +Replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). +Replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). +Apply StepFun_P42. +Unfold adapted_couple in H2 H3; Decompose [and] H2; Decompose [and] H3; Clear H3 H2; Rewrite H10; Rewrite H6; Unfold Rmax Rmin; Case (total_order_Rle a c); Case (total_order_Rle b c); Intros; [Elim n; Assumption | Reflexivity | Elim n0; Assumption | Elim n1; Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf2; Apply H2 | Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf3; Apply H3 | Assumption]. +EApply StepFun_P17; [Apply (StepFun_P40 H0 H H3 (StepFun_P2 H2)) | Apply H1]. +Replace (Int_SF lf3 l3) with R0. +Rewrite Rplus_Or; EApply StepFun_P17; [Apply H1 | Apply StepFun_P2; Rewrite <- H0 in H2; Apply H2]. +Symmetry; EApply StepFun_P8; [Apply H3 | Assumption]. +Replace (Int_SF lf2 l2) with ``(Int_SF lf3 l3)+(Int_SF lf1 l1)``. +Ring. +Elim r; Intro. +Replace (Int_SF lf2 l2) with (Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)). +Replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). +Replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). +Symmetry; Apply StepFun_P42. +Unfold adapted_couple in H1 H3; Decompose [and] H1; Decompose [and] H3; Clear H3 H1; Rewrite H9; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a c); Case (total_order_Rle a b); Intros; [Elim n; Assumption | Elim n1; Assumption | Reflexivity | Elim n1; Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf1; Apply H1 | Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf3; Apply H3 | Assumption]. +EApply StepFun_P17. +Assert H0 : ``c<a``. +Auto with real. +Apply (StepFun_P40 H0 H (StepFun_P2 H3) H1). +Apply StepFun_P2; Apply H2. +Replace (Int_SF lf1 l1) with R0. +Rewrite Rplus_Or; EApply StepFun_P17; [Apply H3 | Rewrite <- H in H2; Apply H2]. +Symmetry; EApply StepFun_P8; [Apply H1 | Assumption]. +Assert H : ``b<a``. +Auto with real. +Replace (Int_SF lf2 l2) with ``(Int_SF lf3 l3)+(Int_SF lf1 l1)``. +Ring. +Rewrite Rplus_sym; Elim r; Intro. +Replace (Int_SF lf2 l2) with (Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)). +Replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). +Replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). +Symmetry; Apply StepFun_P42. +Unfold adapted_couple in H1 H3; Decompose [and] H1; Decompose [and] H3; Clear H3 H1; Rewrite H11; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a c); Case (total_order_Rle a b); Intros; [Elim n; Assumption | Reflexivity | Elim n0; Assumption | Elim n1; Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf1; Apply H1 | Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf3; Apply H3 | Assumption]. +EApply StepFun_P17. +Apply (StepFun_P40 H H0 (StepFun_P2 H1) H3). +Apply H2. +Replace (Int_SF lf3 l3) with R0. +Rewrite Rplus_Or; EApply StepFun_P17; [Apply H1 | Rewrite <- H0 in H2; Apply StepFun_P2; Apply H2]. +Symmetry; EApply StepFun_P8; [Apply H3 | Assumption]. +Assert H : ``c<a``. +Auto with real. +Replace (Int_SF lf1 l1) with ``(Int_SF lf2 l2)+(Int_SF lf3 l3)``. +Ring. +Elim r; Intro. +Replace (Int_SF lf1 l1) with (Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)). +Replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). +Replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). +Symmetry; Apply StepFun_P42. +Unfold adapted_couple in H2 H3; Decompose [and] H2; Decompose [and] H3; Clear H3 H2; Rewrite H11; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a c); Case (total_order_Rle b c); Intros; [Elim n; Assumption | Elim n1; Assumption | Reflexivity | Elim n1; Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf2; Apply H2 | Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf3; Apply H3 | Assumption]. +EApply StepFun_P17. +Apply (StepFun_P40 H0 H H2 (StepFun_P2 H3)). +Apply StepFun_P2; Apply H1. +Replace (Int_SF lf2 l2) with R0. +Rewrite Rplus_Ol; EApply StepFun_P17; [Apply H3 | Rewrite H0 in H1; Apply H1]. +Symmetry; EApply StepFun_P8; [Apply H2 | Assumption]. +Elim n; Apply Rle_trans with a; Try Assumption. +Auto with real. +Assert H : ``c<b``. +Auto with real. +Assert H0 : ``b<a``. +Auto with real. +Replace (Int_SF lf3 l3) with ``(Int_SF lf2 l2)+(Int_SF lf1 l1)``. +Ring. +Replace (Int_SF lf3 l3) with (Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)). +Replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). +Replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). +Symmetry; Apply StepFun_P42. +Unfold adapted_couple in H2 H1; Decompose [and] H2; Decompose [and] H1; Clear H1 H2; Rewrite H11; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros; [Elim n1; Assumption | Elim n1; Assumption | Elim n0; Assumption | Reflexivity]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf2; Apply H2 | Assumption]. +EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf1; Apply H1 | Assumption]. +EApply StepFun_P17. +Apply (StepFun_P40 H H0 (StepFun_P2 H2) (StepFun_P2 H1)). +Apply StepFun_P2; Apply H3. +Unfold RiemannInt_SF; Case (total_order_Rle a c); Intro. +EApply StepFun_P17. +Apply H3. +Change (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun 1!a 2!c 3!f pr3)) (subdivision_val (mkStepFun 1!a 2!c 3!f pr3))); Apply StepFun_P1. +Apply eq_Ropp; EApply StepFun_P17. +Apply H3. +Change (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun 1!a 2!c 3!f pr3)) (subdivision_val (mkStepFun 1!a 2!c 3!f pr3))); Apply StepFun_P1. +Unfold RiemannInt_SF; Case (total_order_Rle b c); Intro. +EApply StepFun_P17. +Apply H2. +Change (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun 1!b 2!c 3!f pr2)) (subdivision_val (mkStepFun 1!b 2!c 3!f pr2))); Apply StepFun_P1. +Apply eq_Ropp; EApply StepFun_P17. +Apply H2. +Change (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun 1!b 2!c 3!f pr2)) (subdivision_val (mkStepFun 1!b 2!c 3!f pr2))); Apply StepFun_P1. +Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. +EApply StepFun_P17. +Apply H1. +Change (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun 1!a 2!b 3!f pr1)) (subdivision_val (mkStepFun 1!a 2!b 3!f pr1))); Apply StepFun_P1. +Apply eq_Ropp; EApply StepFun_P17. +Apply H1. +Change (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun 1!a 2!b 3!f pr1)) (subdivision_val (mkStepFun 1!a 2!b 3!f pr1))); Apply StepFun_P1. +Qed. + +Lemma StepFun_P44 : (f:R->R;a,b,c:R) (IsStepFun f a b) -> ``a<=c<=b`` -> (IsStepFun f a c). +Intros f; Intros; Assert H0 : ``a<=b``. +Elim H; Intros; Apply Rle_trans with c; Assumption. +Elim H; Clear H; Intros; Unfold IsStepFun in X; Unfold is_subdivision in X; Elim X; Clear X; Intros l1 [lf1 H2]; Cut (l1,lf1:Rlist;a,b,c:R;f:R->R) (adapted_couple f a b l1 lf1) -> ``a<=c<=b`` -> (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f a c l l0))). +Intros; Unfold IsStepFun; Unfold is_subdivision; EApply X. +Apply H2. +Split; Assumption. +Clear f a b c H0 H H1 H2 l1 lf1; Induction l1. +Intros; Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H4; Discriminate. +Induction r0. +Intros; Assert H1 : ``a==b``. +Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H3; Simpl in H2; Assert H7 : ``a<=b``. +Elim H0; Intros; Apply Rle_trans with c; Assumption. +Replace a with (Rmin a b). +Pattern 2 b; Replace b with (Rmax a b). +Rewrite <- H2; Rewrite H3; Reflexivity. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Split with (cons r nil); Split with lf1; Assert H2 : ``c==b``. +Rewrite H1 in H0; Elim H0; Intros; Apply Rle_antisym; Assumption. +Rewrite H2; Assumption. +Intros; Clear X; Induction lf1. +Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H4; Discriminate. +Clear Hreclf1; Assert H1 : (sumboolT ``c<=r1`` ``r1<c``). +Case (total_order_Rle c r1); Intro; [Left; Assumption | Right; Auto with real]. +Elim H1; Intro. +Split with (cons r (cons c nil)); Split with (cons r3 nil); Unfold adapted_couple in H; Decompose [and] H; Clear H; Assert H6 : ``r==a``. +Simpl in H4; Rewrite H4; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Elim H0; Intros; Apply Rle_trans with c; Assumption]. +Elim H0; Clear H0; Intros; Unfold adapted_couple; Repeat Split. +Rewrite H6; Unfold ordered_Rlist; Intros; Simpl in H8; Inversion H8; [Simpl; Assumption | Elim (le_Sn_O ? H10)]. +Simpl; Unfold Rmin; Case (total_order_Rle a c); Intro; [Assumption | Elim n; Assumption]. +Simpl; Unfold Rmax; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n; Assumption]. +Unfold constant_D_eq open_interval; Intros; Simpl in H8; Inversion H8. +Simpl; Assert H10 := (H7 O); Assert H12 : (lt (0) (pred (Rlength (cons r (cons r1 r2))))). +Simpl; Apply lt_O_Sn. +Apply (H10 H12); Unfold open_interval; Simpl; Rewrite H11 in H9; Simpl in H9; Elim H9; Clear H9; Intros; Split; Try Assumption. +Apply Rlt_le_trans with c; Assumption. +Elim (le_Sn_O ? H11). +Cut (adapted_couple f r1 b (cons r1 r2) lf1). +Cut ``r1<=c<=b``. +Intros. +Elim (X0 ? ? ? ? ? H3 H2); Intros l1' [lf1' H4]; Split with (cons r l1'); Split with (cons r3 lf1'); Unfold adapted_couple in H H4; Decompose [and] H; Decompose [and] H4; Clear H H4 X0; Assert H14 : ``a<=b``. +Elim H0; Intros; Apply Rle_trans with c; Assumption. +Assert H16 : ``r==a``. +Simpl in H7; Rewrite H7; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Induction l1'. +Simpl in H13; Discriminate. +Clear Hrecl1'; Unfold adapted_couple; Repeat Split. +Unfold ordered_Rlist; Intros; Simpl in H; Induction i. +Simpl; Replace r4 with r1. +Apply (H5 O). +Simpl; Apply lt_O_Sn. +Simpl in H12; Rewrite H12; Unfold Rmin; Case (total_order_Rle r1 c); Intro; [Reflexivity | Elim n; Left; Assumption]. +Apply (H9 i); Simpl; Apply lt_S_n; Assumption. +Simpl; Unfold Rmin; Case (total_order_Rle a c); Intro; [Assumption | Elim n; Elim H0; Intros; Assumption]. +Replace (Rmax a c) with (Rmax r1 c). +Rewrite <- H11; Reflexivity. +Unfold Rmax; Case (total_order_Rle r1 c); Case (total_order_Rle a c); Intros; [Reflexivity | Elim n; Elim H0; Intros; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. +Simpl; Simpl in H13; Rewrite H13; Reflexivity. +Intros; Simpl in H; Unfold constant_D_eq open_interval; Intros; Induction i. +Simpl; Assert H17 := (H10 O); Assert H18 : (lt (0) (pred (Rlength (cons r (cons r1 r2))))). +Simpl; Apply lt_O_Sn. +Apply (H17 H18); Unfold open_interval; Simpl; Simpl in H4; Elim H4; Clear H4; Intros; Split; Try Assumption; Replace r1 with r4. +Assumption. +Simpl in H12; Rewrite H12; Unfold Rmin; Case (total_order_Rle r1 c); Intro; [Reflexivity | Elim n; Left; Assumption]. +Clear Hreci; Simpl; Apply H15. +Simpl; Apply lt_S_n; Assumption. +Unfold open_interval; Apply H4. +Split. +Left; Assumption. +Elim H0; Intros; Assumption. +EApply StepFun_P7; [Elim H0; Intros; Apply Rle_trans with c; [Apply H2 | Apply H3] | Apply H]. +Qed. + +Lemma StepFun_P45 : (f:R->R;a,b,c:R) (IsStepFun f a b) -> ``a<=c<=b`` -> (IsStepFun f c b). +Intros f; Intros; Assert H0 : ``a<=b``. +Elim H; Intros; Apply Rle_trans with c; Assumption. +Elim H; Clear H; Intros; Unfold IsStepFun in X; Unfold is_subdivision in X; Elim X; Clear X; Intros l1 [lf1 H2]; Cut (l1,lf1:Rlist;a,b,c:R;f:R->R) (adapted_couple f a b l1 lf1) -> ``a<=c<=b`` -> (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f c b l l0))). +Intros; Unfold IsStepFun; Unfold is_subdivision; EApply X; [Apply H2 | Split; Assumption]. +Clear f a b c H0 H H1 H2 l1 lf1; Induction l1. +Intros; Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H4; Discriminate. +Induction r0. +Intros; Assert H1 : ``a==b``. +Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H3; Simpl in H2; Assert H7 : ``a<=b``. +Elim H0; Intros; Apply Rle_trans with c; Assumption. +Replace a with (Rmin a b). +Pattern 2 b; Replace b with (Rmax a b). +Rewrite <- H2; Rewrite H3; Reflexivity. +Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. +Split with (cons r nil); Split with lf1; Assert H2 : ``c==b``. +Rewrite H1 in H0; Elim H0; Intros; Apply Rle_antisym; Assumption. +Rewrite <- H2 in H1; Rewrite <- H1; Assumption. +Intros; Clear X; Induction lf1. +Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H4; Discriminate. +Clear Hreclf1; Assert H1 : (sumboolT ``c<=r1`` ``r1<c``). +Case (total_order_Rle c r1); Intro; [Left; Assumption | Right; Auto with real]. +Elim H1; Intro. +Split with (cons c (cons r1 r2)); Split with (cons r3 lf1); Unfold adapted_couple in H; Decompose [and] H; Clear H; Unfold adapted_couple; Repeat Split. +Unfold ordered_Rlist; Intros; Simpl in H; Induction i. +Simpl; Assumption. +Clear Hreci; Apply (H2 (S i)); Simpl; Assumption. +Simpl; Unfold Rmin; Case (total_order_Rle c b); Intro; [Reflexivity | Elim n; Elim H0; Intros; Assumption]. +Replace (Rmax c b) with (Rmax a b). +Rewrite <- H3; Reflexivity. +Unfold Rmax; Case (total_order_Rle a b); Case (total_order_Rle c b); Intros; [Reflexivity | Elim n; Elim H0; Intros; Assumption | Elim n; Elim H0; Intros; Apply Rle_trans with c; Assumption | Elim n0; Elim H0; Intros; Apply Rle_trans with c; Assumption]. +Simpl; Simpl in H5; Apply H5. +Intros; Simpl in H; Induction i. +Unfold constant_D_eq open_interval; Intros; Simpl; Apply (H7 O). +Simpl; Apply lt_O_Sn. +Unfold open_interval; Simpl; Simpl in H6; Elim H6; Clear H6; Intros; Split; Try Assumption; Apply Rle_lt_trans with c; Try Assumption; Replace r with a. +Elim H0; Intros; Assumption. +Simpl in H4; Rewrite H4; Unfold Rmin; Case (total_order_Rle a b); Intros; [Reflexivity | Elim n; Elim H0; Intros; Apply Rle_trans with c; Assumption]. +Clear Hreci; Apply (H7 (S i)); Simpl; Assumption. +Cut (adapted_couple f r1 b (cons r1 r2) lf1). +Cut ``r1<=c<=b``. +Intros; Elim (X0 ? ? ? ? ? H3 H2); Intros l1' [lf1' H4]; Split with l1'; Split with lf1'; Assumption. +Split; [Left; Assumption | Elim H0; Intros; Assumption]. +EApply StepFun_P7; [Elim H0; Intros; Apply Rle_trans with c; [Apply H2 | Apply H3] | Apply H]. +Qed. + +Lemma StepFun_P46 : (f:R->R;a,b,c:R) (IsStepFun f a b) -> (IsStepFun f b c) -> (IsStepFun f a c). +Intros f; Intros; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros. +Apply StepFun_P41 with b; Assumption. +Case (total_order_Rle a c); Intro. +Apply StepFun_P44 with b; Try Assumption. +Split; [Assumption | Auto with real]. +Apply StepFun_P6; Apply StepFun_P44 with b. +Apply StepFun_P6; Assumption. +Split; Auto with real. +Case (total_order_Rle a c); Intro. +Apply StepFun_P45 with b; Try Assumption. +Split; Auto with real. +Apply StepFun_P6; Apply StepFun_P45 with b. +Apply StepFun_P6; Assumption. +Split; [Assumption | Auto with real]. +Apply StepFun_P6; Apply StepFun_P41 with b; Auto with real Orelse Apply StepFun_P6; Assumption. +Qed. diff --git a/theories7/Reals/Rlimit.v b/theories7/Reals/Rlimit.v new file mode 100644 index 00000000..3308b2e3 --- /dev/null +++ b/theories7/Reals/Rlimit.v @@ -0,0 +1,539 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rlimit.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +(*********************************************************) +(* Definition of the limit *) +(* *) +(*********************************************************) + +Require Rbase. +Require Rfunctions. +Require Classical_Prop. +Require Fourier. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(*******************************) +(* Calculus *) +(*******************************) +(*********) +Lemma eps2_Rgt_R0:(eps:R)(Rgt eps R0)-> + (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). +Intros;Fourier. +Qed. + +(*********) +Lemma eps2:(eps:R)(Rplus (Rmult eps (Rinv (Rplus R1 R1))) + (Rmult eps (Rinv (Rplus R1 R1))))==eps. +Intro esp. +Assert H := (double_var esp). +Unfold Rdiv in H. +Symmetry; Exact H. +Qed. + +(*********) +Lemma eps4:(eps:R) + (Rplus (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1) ))) + (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1) ))))== + (Rmult eps (Rinv (Rplus R1 R1))). +Intro eps. +Replace ``2+2`` with ``2*2``. +Pattern 3 eps; Rewrite double_var. +Rewrite (Rmult_Rplus_distrl ``eps/2`` ``eps/2`` ``/2``). +Unfold Rdiv. +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_Rmult. +Reflexivity. +DiscrR. +DiscrR. +Ring. +Qed. + +(*********) +Lemma Rlt_eps2_eps:(eps:R)(Rgt eps R0)-> + (Rlt (Rmult eps (Rinv (Rplus R1 R1))) eps). +Intros. +Pattern 2 eps; Rewrite <- Rmult_1r. +Repeat Rewrite (Rmult_sym eps). +Apply Rlt_monotony_r. +Exact H. +Apply Rlt_monotony_contra with ``2``. +Fourier. +Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. +Fourier. +DiscrR. +Qed. + +(*********) +Lemma Rlt_eps4_eps:(eps:R)(Rgt eps R0)-> + (Rlt (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1)))) eps). +Intros. +Replace ``2+2`` with ``4``. +Pattern 2 eps; Rewrite <- Rmult_1r. +Repeat Rewrite (Rmult_sym eps). +Apply Rlt_monotony_r. +Exact H. +Apply Rlt_monotony_contra with ``4``. +Replace ``4`` with ``2*2``. +Apply Rmult_lt_pos; Fourier. +Ring. +Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. +Fourier. +DiscrR. +Ring. +Qed. + +(*********) +Lemma prop_eps:(r:R)((eps:R)(Rgt eps R0)->(Rlt r eps))->(Rle r R0). +Intros;Elim (total_order r R0); Intro. +Apply Rlt_le; Assumption. +Elim H0; Intro. +Apply eq_Rle; Assumption. +Clear H0;Generalize (H r H1); Intro;Generalize (Rlt_antirefl r); + Intro;ElimType False; Auto. +Qed. + +(*********) +Definition mul_factor := [l,l':R](Rinv (Rplus R1 (Rplus (Rabsolu l) + (Rabsolu l')))). + +(*********) +Lemma mul_factor_wd : (l,l':R) + ~(Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))==R0. +Intros;Rewrite (Rplus_sym R1 (Rplus (Rabsolu l) (Rabsolu l'))); + Apply tech_Rplus. +Cut (Rle (Rabsolu (Rplus l l')) (Rplus (Rabsolu l) (Rabsolu l'))). +Cut (Rle R0 (Rabsolu (Rplus l l'))). +Exact (Rle_trans ? ? ?). +Exact (Rabsolu_pos (Rplus l l')). +Exact (Rabsolu_triang ? ?). +Exact Rlt_R0_R1. +Qed. + +(*********) +Lemma mul_factor_gt:(eps:R)(l,l':R)(Rgt eps R0)-> + (Rgt (Rmult eps (mul_factor l l')) R0). +Intros;Unfold Rgt;Rewrite <- (Rmult_Or eps);Apply Rlt_monotony. +Assumption. +Unfold mul_factor;Apply Rlt_Rinv; + Cut (Rle R1 (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))). +Cut (Rlt R0 R1). +Exact (Rlt_le_trans ? ? ?). +Exact Rlt_R0_R1. +Replace (Rle R1 (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))) + with (Rle (Rplus R1 R0) (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))). +Apply Rle_compatibility. +Cut (Rle (Rabsolu (Rplus l l')) (Rplus (Rabsolu l) (Rabsolu l'))). +Cut (Rle R0 (Rabsolu (Rplus l l'))). +Exact (Rle_trans ? ? ?). +Exact (Rabsolu_pos ?). +Exact (Rabsolu_triang ? ?). +Rewrite (proj1 ? ? (Rplus_ne R1));Trivial. +Qed. + +(*********) +Lemma mul_factor_gt_f:(eps:R)(l,l':R)(Rgt eps R0)-> + (Rgt (Rmin R1 (Rmult eps (mul_factor l l'))) R0). +Intros;Apply Rmin_Rgt_r;Split. +Exact Rlt_R0_R1. +Exact (mul_factor_gt eps l l' H). +Qed. + + +(*******************************) +(* Metric space *) +(*******************************) + +(*********) +Record Metric_Space:Type:= { + Base:Type; + dist:Base->Base->R; + dist_pos:(x,y:Base)(Rge (dist x y) R0); + dist_sym:(x,y:Base)(dist x y)==(dist y x); + dist_refl:(x,y:Base)((dist x y)==R0<->x==y); + dist_tri:(x,y,z:Base)(Rle (dist x y) + (Rplus (dist x z) (dist z y))) }. + +(*******************************) +(* Limit in Metric space *) +(*******************************) + +(*********) +Definition limit_in:= + [X:Metric_Space; X':Metric_Space; f:(Base X)->(Base X'); + D:(Base X)->Prop; x0:(Base X); l:(Base X')] + (eps:R)(Rgt eps R0)-> + (EXT alp:R | (Rgt alp R0)/\(x:(Base X))(D x)/\ + (Rlt (dist X x x0) alp)-> + (Rlt (dist X' (f x) l) eps)). + +(*******************************) +(* R is a metric space *) +(*******************************) + +(*********) +Definition R_met:Metric_Space:=(Build_Metric_Space R R_dist + R_dist_pos R_dist_sym R_dist_refl R_dist_tri). + +(*******************************) +(* Limit 1 arg *) +(*******************************) +(*********) +Definition Dgf:=[Df,Dg:R->Prop][f:R->R][x:R](Df x)/\(Dg (f x)). + +(*********) +Definition limit1_in:(R->R)->(R->Prop)->R->R->Prop:= + [f:R->R; D:R->Prop; l:R; x0:R](limit_in R_met R_met f D x0 l). + +(*********) +Lemma tech_limit:(f:R->R)(D:R->Prop)(l:R)(x0:R)(D x0)-> + (limit1_in f D l x0)->l==(f x0). +Intros f D l x0 H H0. +Case (Rabsolu_pos (Rminus (f x0) l)); Intros H1. +Absurd (Rlt (dist R_met (f x0) l) (dist R_met (f x0) l)). +Apply Rlt_antirefl. +Case (H0 (dist R_met (f x0) l)); Auto. +Intros alpha1 (H2, H3); Apply H3; Auto; Split; Auto. +Case (dist_refl R_met x0 x0); Intros Hr1 Hr2; Rewrite Hr2; Auto. +Case (dist_refl R_met (f x0) l); Intros Hr1 Hr2; Apply sym_eqT; Auto. +Qed. + +(*********) +Lemma tech_limit_contr:(f:R->R)(D:R->Prop)(l:R)(x0:R)(D x0)->~l==(f x0) + ->~(limit1_in f D l x0). +Intros;Generalize (tech_limit f D l x0);Tauto. +Qed. + +(*********) +Lemma lim_x:(D:R->Prop)(x0:R)(limit1_in [x:R]x D x0 x0). +Unfold limit1_in; Unfold limit_in; Simpl; Intros;Split with eps; + Split; Auto;Intros;Elim H0; Intros; Auto. +Qed. + +(*********) +Lemma limit_plus:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) + (limit1_in f D l x0)->(limit1_in g D l' x0)-> + (limit1_in [x:R](Rplus (f x) (g x)) D (Rplus l l') x0). +Intros;Unfold limit1_in; Unfold limit_in; Simpl; Intros; + Elim (H (Rmult eps (Rinv (Rplus R1 R1))) (eps2_Rgt_R0 eps H1)); + Elim (H0 (Rmult eps (Rinv (Rplus R1 R1))) (eps2_Rgt_R0 eps H1)); + Simpl;Clear H H0; Intros; Elim H; Elim H0; Clear H H0; Intros; + Split with (Rmin x1 x); Split. +Exact (Rmin_Rgt_r x1 x R0 (conj ? ? H H2)). +Intros;Elim H4; Clear H4; Intros; + Cut (Rlt (Rplus (R_dist (f x2) l) (R_dist (g x2) l')) eps). + Cut (Rle (R_dist (Rplus (f x2) (g x2)) (Rplus l l')) + (Rplus (R_dist (f x2) l) (R_dist (g x2) l'))). +Exact (Rle_lt_trans ? ? ?). +Exact (R_dist_plus ? ? ? ?). +Elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); Clear H5; Intros. +Generalize (H3 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H4 H6)); + Generalize (H0 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H4 H5)); + Intros; + Replace eps + with (Rplus (Rmult eps (Rinv (Rplus R1 R1))) + (Rmult eps (Rinv (Rplus R1 R1)))). +Exact (Rplus_lt ? ? ? ? H7 H8). +Exact (eps2 eps). +Qed. + +(*********) +Lemma limit_Ropp:(f:R->R)(D:R->Prop)(l:R)(x0:R) + (limit1_in f D l x0)->(limit1_in [x:R](Ropp (f x)) D (Ropp l) x0). +Unfold limit1_in;Unfold limit_in;Simpl;Intros;Elim (H eps H0);Clear H; + Intros;Elim H;Clear H;Intros;Split with x;Split;Auto;Intros; + Generalize (H1 x1 H2);Clear H1;Intro;Unfold R_dist;Unfold Rminus; + Rewrite (Ropp_Ropp l);Rewrite (Rplus_sym (Ropp (f x1)) l); + Fold (Rminus l (f x1));Fold (R_dist l (f x1));Rewrite R_dist_sym; + Assumption. +Qed. + +(*********) +Lemma limit_minus:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) + (limit1_in f D l x0)->(limit1_in g D l' x0)-> + (limit1_in [x:R](Rminus (f x) (g x)) D (Rminus l l') x0). +Intros;Unfold Rminus;Generalize (limit_Ropp g D l' x0 H0);Intro; + Exact (limit_plus f [x:R](Ropp (g x)) D l (Ropp l') x0 H H1). +Qed. + +(*********) +Lemma limit_free:(f:R->R)(D:R->Prop)(x:R)(x0:R) + (limit1_in [h:R](f x) D (f x) x0). +Unfold limit1_in;Unfold limit_in;Simpl;Intros;Split with eps;Split; + Auto;Intros;Elim (R_dist_refl (f x) (f x));Intros a b; + Rewrite (b (refl_eqT R (f x)));Unfold Rgt in H;Assumption. +Qed. + +(*********) +Lemma limit_mul:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) + (limit1_in f D l x0)->(limit1_in g D l' x0)-> + (limit1_in [x:R](Rmult (f x) (g x)) D (Rmult l l') x0). +Intros;Unfold limit1_in; Unfold limit_in; Simpl; Intros; + Elim (H (Rmin R1 (Rmult eps (mul_factor l l'))) + (mul_factor_gt_f eps l l' H1)); + Elim (H0 (Rmult eps (mul_factor l l')) (mul_factor_gt eps l l' H1)); + Clear H H0; Simpl; Intros; Elim H; Elim H0; Clear H H0; Intros; + Split with (Rmin x1 x); Split. +Exact (Rmin_Rgt_r x1 x R0 (conj ? ? H H2)). +Intros; Elim H4; Clear H4; Intros;Unfold R_dist; + Replace (Rminus (Rmult (f x2) (g x2)) (Rmult l l')) with + (Rplus (Rmult (f x2) (Rminus (g x2) l')) (Rmult l' (Rminus (f x2) l))). +Cut (Rlt (Rplus (Rabsolu (Rmult (f x2) (Rminus (g x2) l'))) (Rabsolu (Rmult l' + (Rminus (f x2) l)))) eps). +Cut (Rle (Rabsolu (Rplus (Rmult (f x2) (Rminus (g x2) l')) (Rmult l' (Rminus + (f x2) l)))) (Rplus (Rabsolu (Rmult (f x2) (Rminus (g x2) l'))) (Rabsolu + (Rmult l' (Rminus (f x2) l))))). +Exact (Rle_lt_trans ? ? ?). +Exact (Rabsolu_triang ? ?). +Rewrite (Rabsolu_mult (f x2) (Rminus (g x2) l')); + Rewrite (Rabsolu_mult l' (Rminus (f x2) l)); + Cut (Rle (Rplus (Rmult (Rplus R1 (Rabsolu l)) (Rmult eps (mul_factor l l'))) + (Rmult (Rabsolu l') (Rmult eps (mul_factor l l')))) eps). +Cut (Rlt (Rplus (Rmult (Rabsolu (f x2)) (Rabsolu (Rminus (g x2) l'))) (Rmult + (Rabsolu l') (Rabsolu (Rminus (f x2) l)))) (Rplus (Rmult (Rplus R1 (Rabsolu + l)) (Rmult eps (mul_factor l l'))) (Rmult (Rabsolu l') (Rmult eps + (mul_factor l l'))))). +Exact (Rlt_le_trans ? ? ?). +Elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); Clear H5; Intros; + Generalize (H0 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H4 H5));Intro; + Generalize (Rmin_Rgt_l ? ? ? H7);Intro;Elim H8;Intros;Clear H0 H8; + Apply Rplus_lt_le_lt. +Apply Rmult_lt_0. +Apply Rle_sym1. +Exact (Rabsolu_pos (Rminus (g x2) l')). +Rewrite (Rplus_sym R1 (Rabsolu l));Unfold Rgt;Apply Rlt_r_plus_R1; + Exact (Rabsolu_pos l). +Unfold R_dist in H9; + Apply (Rlt_anti_compatibility (Ropp (Rabsolu l)) (Rabsolu (f x2)) + (Rplus R1 (Rabsolu l))). +Rewrite <- (Rplus_assoc (Ropp (Rabsolu l)) R1 (Rabsolu l)); + Rewrite (Rplus_sym (Ropp (Rabsolu l)) R1); + Rewrite (Rplus_assoc R1 (Ropp (Rabsolu l)) (Rabsolu l)); + Rewrite (Rplus_Ropp_l (Rabsolu l)); + Rewrite (proj1 ? ? (Rplus_ne R1)); + Rewrite (Rplus_sym (Ropp (Rabsolu l)) (Rabsolu (f x2))); + Generalize H9; +Cut (Rle (Rminus (Rabsolu (f x2)) (Rabsolu l)) (Rabsolu (Rminus (f x2) l))). +Exact (Rle_lt_trans ? ? ?). +Exact (Rabsolu_triang_inv ? ?). +Generalize (H3 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H4 H6));Trivial. +Apply Rle_monotony. +Exact (Rabsolu_pos l'). +Unfold Rle;Left;Assumption. +Rewrite (Rmult_sym (Rplus R1 (Rabsolu l)) (Rmult eps (mul_factor l l'))); + Rewrite (Rmult_sym (Rabsolu l') (Rmult eps (mul_factor l l'))); + Rewrite <- (Rmult_Rplus_distr + (Rmult eps (mul_factor l l')) + (Rplus R1 (Rabsolu l)) + (Rabsolu l')); + Rewrite (Rmult_assoc eps (mul_factor l l') (Rplus (Rplus R1 (Rabsolu l)) + (Rabsolu l'))); + Rewrite (Rplus_assoc R1 (Rabsolu l) (Rabsolu l'));Unfold mul_factor; + Rewrite (Rinv_l (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l'))) + (mul_factor_wd l l')); + Rewrite (proj1 ? ? (Rmult_ne eps));Apply eq_Rle;Trivial. +Ring. +Qed. + +(*********) +Definition adhDa:(R->Prop)->R->Prop:=[D:R->Prop][a:R] + (alp:R)(Rgt alp R0)->(EXT x:R | (D x)/\(Rlt (R_dist x a) alp)). + +(*********) +Lemma single_limit:(f:R->R)(D:R->Prop)(l:R)(l':R)(x0:R) + (adhDa D x0)->(limit1_in f D l x0)->(limit1_in f D l' x0)->l==l'. +Unfold limit1_in; Unfold limit_in; Intros. +Cut (eps:R)(Rgt eps R0)->(Rlt (dist R_met l l') + (Rmult (Rplus R1 R1) eps)). +Clear H0 H1;Unfold dist; Unfold R_met; Unfold R_dist; + Unfold Rabsolu;Case (case_Rabsolu (Rminus l l')); Intros. +Cut (eps:R)(Rgt eps R0)->(Rlt (Ropp (Rminus l l')) eps). +Intro;Generalize (prop_eps (Ropp (Rminus l l')) H1);Intro; + Generalize (Rlt_RoppO (Rminus l l') r); Intro;Unfold Rgt in H3; + Generalize (Rle_not (Ropp (Rminus l l')) R0 H3); Intro; + ElimType False; Auto. +Intros;Cut (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). +Intro;Generalize (H0 (Rmult eps (Rinv (Rplus R1 R1))) H2); + Rewrite (Rmult_sym eps (Rinv (Rplus R1 R1))); + Rewrite <- (Rmult_assoc (Rplus R1 R1) (Rinv (Rplus R1 R1)) eps); + Rewrite (Rinv_r (Rplus R1 R1)). +Elim (Rmult_ne eps);Intros a b;Rewrite b;Clear a b;Trivial. +Apply (imp_not_Req (Rplus R1 R1) R0);Right;Generalize Rlt_R0_R1;Intro; + Unfold Rgt;Generalize (Rlt_compatibility R1 R0 R1 H3);Intro; + Elim (Rplus_ne R1);Intros a b;Rewrite a in H4;Clear a b; + Apply (Rlt_trans R0 R1 (Rplus R1 R1) H3 H4). +Unfold Rgt;Unfold Rgt in H1; + Rewrite (Rmult_sym eps(Rinv (Rplus R1 R1))); + Rewrite <-(Rmult_Or (Rinv (Rplus R1 R1))); + Apply (Rlt_monotony (Rinv (Rplus R1 R1)) R0 eps);Auto. +Apply (Rlt_Rinv (Rplus R1 R1));Cut (Rlt R1 (Rplus R1 R1)). +Intro;Apply (Rlt_trans R0 R1 (Rplus R1 R1) Rlt_R0_R1 H2). +Generalize (Rlt_compatibility R1 R0 R1 Rlt_R0_R1);Elim (Rplus_ne R1); + Intros a b;Rewrite a;Clear a b;Trivial. +(**) +Cut (eps:R)(Rgt eps R0)->(Rlt (Rminus l l') eps). +Intro;Generalize (prop_eps (Rminus l l') H1);Intro; + Elim (Rle_le_eq (Rminus l l') R0);Intros a b;Clear b; + Apply (Rminus_eq l l');Apply a;Split. +Assumption. +Apply (Rle_sym2 R0 (Rminus l l') r). +Intros;Cut (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). +Intro;Generalize (H0 (Rmult eps (Rinv (Rplus R1 R1))) H2); + Rewrite (Rmult_sym eps (Rinv (Rplus R1 R1))); + Rewrite <- (Rmult_assoc (Rplus R1 R1) (Rinv (Rplus R1 R1)) eps); + Rewrite (Rinv_r (Rplus R1 R1)). +Elim (Rmult_ne eps);Intros a b;Rewrite b;Clear a b;Trivial. +Apply (imp_not_Req (Rplus R1 R1) R0);Right;Generalize Rlt_R0_R1;Intro; + Unfold Rgt;Generalize (Rlt_compatibility R1 R0 R1 H3);Intro; + Elim (Rplus_ne R1);Intros a b;Rewrite a in H4;Clear a b; + Apply (Rlt_trans R0 R1 (Rplus R1 R1) H3 H4). +Unfold Rgt;Unfold Rgt in H1; + Rewrite (Rmult_sym eps(Rinv (Rplus R1 R1))); + Rewrite <-(Rmult_Or (Rinv (Rplus R1 R1))); + Apply (Rlt_monotony (Rinv (Rplus R1 R1)) R0 eps);Auto. +Apply (Rlt_Rinv (Rplus R1 R1));Cut (Rlt R1 (Rplus R1 R1)). +Intro;Apply (Rlt_trans R0 R1 (Rplus R1 R1) Rlt_R0_R1 H2). +Generalize (Rlt_compatibility R1 R0 R1 Rlt_R0_R1);Elim (Rplus_ne R1); + Intros a b;Rewrite a;Clear a b;Trivial. +(**) +Intros;Unfold adhDa in H;Elim (H0 eps H2);Intros;Elim (H1 eps H2); + Intros;Clear H0 H1;Elim H3;Elim H4;Clear H3 H4;Intros; + Simpl;Simpl in H1 H4;Generalize (Rmin_Rgt x x1 R0);Intro;Elim H5; + Intros;Clear H5; + Elim (H (Rmin x x1) (H7 (conj (Rgt x R0) (Rgt x1 R0) H3 H0))); + Intros; Elim H5;Intros;Clear H5 H H6 H7; + Generalize (Rmin_Rgt x x1 (R_dist x2 x0));Intro;Elim H; + Intros;Clear H H6;Unfold Rgt in H5;Elim (H5 H9);Intros;Clear H5 H9; + Generalize (H1 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H8 H6)); + Generalize (H4 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H8 H)); + Clear H8 H H6 H1 H4 H0 H3;Intros; + Generalize (Rplus_lt (R_dist (f x2) l) eps (R_dist (f x2) l') eps + H H0); Unfold R_dist;Intros; + Rewrite (Rabsolu_minus_sym (f x2) l) in H1; + Rewrite (Rmult_sym (Rplus R1 R1) eps);Rewrite (Rmult_Rplus_distr eps R1 R1); + Elim (Rmult_ne eps);Intros a b;Rewrite a;Clear a b; + Generalize (R_dist_tri l l' (f x2));Unfold R_dist;Intros; + Apply (Rle_lt_trans (Rabsolu (Rminus l l')) + (Rplus (Rabsolu (Rminus l (f x2))) (Rabsolu (Rminus (f x2) l'))) + (Rplus eps eps) H3 H1). +Qed. + +(*********) +Lemma limit_comp:(f,g:R->R)(Df,Dg:R->Prop)(l,l':R)(x0:R) + (limit1_in f Df l x0)->(limit1_in g Dg l' l)-> + (limit1_in [x:R](g (f x)) (Dgf Df Dg f) l' x0). +Unfold limit1_in limit_in Dgf;Simpl. +Intros f g Df Dg l l' x0 Hf Hg eps eps_pos. +Elim (Hg eps eps_pos). +Intros alpg lg. +Elim (Hf alpg). +2: Tauto. +Intros alpf lf. +Exists alpf. +Intuition. +Qed. + +(*********) + +Lemma limit_inv : (f:R->R)(D:R->Prop)(l:R)(x0:R) (limit1_in f D l x0)->~(l==R0)->(limit1_in [x:R](Rinv (f x)) D (Rinv l) x0). +Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H ``(Rabsolu l)/2``). +Intros delta1 H2; Elim (H ``eps*((Rsqr l)/2)``). +Intros delta2 H3; Elim H2; Elim H3; Intros; Exists (Rmin delta1 delta2); Split. +Unfold Rmin; Case (total_order_Rle delta1 delta2); Intro; Assumption. +Intro; Generalize (H5 x); Clear H5; Intro H5; Generalize (H7 x); Clear H7; Intro H7; Intro H10; Elim H10; Intros; Cut (D x)/\``(Rabsolu (x-x0))<delta1``. +Cut (D x)/\``(Rabsolu (x-x0))<delta2``. +Intros; Generalize (H5 H11); Clear H5; Intro H5; Generalize (H7 H12); Clear H7; Intro H7; Generalize (Rabsolu_triang_inv l (f x)); Intro; Rewrite Rabsolu_minus_sym in H7; Generalize (Rle_lt_trans ``(Rabsolu l)-(Rabsolu (f x))`` ``(Rabsolu (l-(f x)))`` ``(Rabsolu l)/2`` H13 H7); Intro; Generalize (Rlt_compatibility ``(Rabsolu (f x))-(Rabsolu l)/2`` ``(Rabsolu l)-(Rabsolu (f x))`` ``(Rabsolu l)/2`` H14); Replace ``(Rabsolu (f x))-(Rabsolu l)/2+((Rabsolu l)-(Rabsolu (f x)))`` with ``(Rabsolu l)/2``. +Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Intro; Cut ~``(f x)==0``. +Intro; Replace ``/(f x)+ -/l`` with ``(l-(f x))*/(l*(f x))``. +Rewrite Rabsolu_mult; Rewrite Rabsolu_Rinv. +Cut ``/(Rabsolu (l*(f x)))<2/(Rsqr l)``. +Intro; Rewrite Rabsolu_minus_sym in H5; Cut ``0<=/(Rabsolu (l*(f x)))``. +Intro; Generalize (Rmult_lt2 ``(Rabsolu (l-(f x)))`` ``eps*(Rsqr l)/2`` ``/(Rabsolu (l*(f x)))`` ``2/(Rsqr l)`` (Rabsolu_pos ``l-(f x)``) H18 H5 H17); Replace ``eps*(Rsqr l)/2*2/(Rsqr l)`` with ``eps``. +Intro; Assumption. +Unfold Rdiv; Unfold Rsqr; Rewrite Rinv_Rmult. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym l). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite (Rmult_sym l). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Reflexivity. +DiscrR. +Exact H0. +Exact H0. +Exact H0. +Exact H0. +Left; Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply prod_neq_R0; Assumption. +Rewrite Rmult_sym; Rewrite Rabsolu_mult; Rewrite Rinv_Rmult. +Rewrite (Rsqr_abs l); Unfold Rsqr; Unfold Rdiv; Rewrite Rinv_Rmult. +Repeat Rewrite <- Rmult_assoc; Apply Rlt_monotony_r. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Apply Rlt_monotony_contra with ``(Rabsolu (f x))*(Rabsolu l)*/2``. +Repeat Apply Rmult_lt_pos. +Apply Rabsolu_pos_lt; Assumption. +Apply Rabsolu_pos_lt; Assumption. +Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H17; Generalize (lt_INR_0 (2) (neq_O_lt (2) H17)); Unfold INR; Intro H18; Assumption | Discriminate]. +Replace ``(Rabsolu (f x))*(Rabsolu l)*/2*/(Rabsolu (f x))`` with ``(Rabsolu l)/2``. +Replace ``(Rabsolu (f x))*(Rabsolu l)*/2*(2*/(Rabsolu l))`` with ``(Rabsolu (f x))``. +Assumption. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym (Rabsolu l)). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Reflexivity. +DiscrR. +Apply Rabsolu_no_R0. +Assumption. +Unfold Rdiv. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym (Rabsolu (f x))). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Reflexivity. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Apply prod_neq_R0; Assumption. +Rewrite (Rinv_Rmult ? ? H0 H16). +Unfold Rminus; Rewrite Rmult_Rplus_distrl. +Rewrite <- Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l. +Rewrite Ropp_mul1. +Rewrite (Rmult_sym (f x)). +Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Reflexivity. +Assumption. +Assumption. +Red; Intro; Rewrite H16 in H15; Rewrite Rabsolu_R0 in H15; Cut ``0<(Rabsolu l)/2``. +Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(Rabsolu l)/2`` ``0`` H17 H15)). +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rabsolu_pos_lt; Assumption. +Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H17; Generalize (lt_INR_0 (2) (neq_O_lt (2) H17)); Unfold INR; Intro; Assumption | Discriminate]. +Pattern 3 (Rabsolu l); Rewrite double_var. +Ring. +Split; [Assumption | Apply Rlt_le_trans with (Rmin delta1 delta2); [Assumption | Apply Rmin_r]]. +Split; [Assumption | Apply Rlt_le_trans with (Rmin delta1 delta2); [Assumption | Apply Rmin_l]]. +Change ``0<eps*(Rsqr l)/2``; Unfold Rdiv; Repeat Rewrite Rmult_assoc; Repeat Apply Rmult_lt_pos. +Assumption. +Apply Rsqr_pos_lt; Assumption. +Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H3; Generalize (lt_INR_0 (2) (neq_O_lt (2) H3)); Unfold INR; Intro; Assumption | Discriminate]. +Change ``0<(Rabsolu l)/2``; Unfold Rdiv; Apply Rmult_lt_pos; [Apply Rabsolu_pos_lt; Assumption | Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H3; Generalize (lt_INR_0 (2) (neq_O_lt (2) H3)); Unfold INR; Intro; Assumption | Discriminate]]. +Qed. diff --git a/theories7/Reals/Rpower.v b/theories7/Reals/Rpower.v new file mode 100644 index 00000000..0acfa8d2 --- /dev/null +++ b/theories7/Reals/Rpower.v @@ -0,0 +1,560 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rpower.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) +(*i Due to L.Thery i*) + +(************************************************************) +(* Definitions of log and Rpower : R->R->R; main properties *) +(************************************************************) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis1. +Require Exp_prop. +Require Rsqrt_def. +Require R_sqrt. +Require MVT. +Require Ranalysis4. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Lemma P_Rmin: (P : R -> Prop) (x, y : R) (P x) -> (P y) -> (P (Rmin x y)). +Intros P x y H1 H2; Unfold Rmin; Case (total_order_Rle x y); Intro; Assumption. +Qed. + +Lemma exp_le_3 : ``(exp 1)<=3``. +Assert exp_1 : ``(exp 1)<>0``. +Assert H0 := (exp_pos R1); Red; Intro; Rewrite H in H0; Elim (Rlt_antirefl ? H0). +Apply Rle_monotony_contra with ``/(exp 1)``. +Apply Rlt_Rinv; Apply exp_pos. +Rewrite <- Rinv_l_sym. +Apply Rle_monotony_contra with ``/3``. +Apply Rlt_Rinv; Sup0. +Rewrite Rmult_1r; Rewrite <- (Rmult_sym ``3``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Replace ``/(exp 1)`` with ``(exp (-1))``. +Unfold exp; Case (exist_exp ``-1``); Intros; Simpl; Unfold exp_in in e; Assert H := (alternated_series_ineq [i:nat]``/(INR (fact i))`` x (S O)). +Cut ``(sum_f_R0 (tg_alt [([i:nat]``/(INR (fact i))``)]) (S (mult (S (S O)) (S O)))) <= x <= (sum_f_R0 (tg_alt [([i:nat]``/(INR (fact i))``)]) (mult (S (S O)) (S O)))``. +Intro; Elim H0; Clear H0; Intros H0 _; Simpl in H0; Unfold tg_alt in H0; Simpl in H0. +Replace ``/3`` with ``1*/1+ -1*1*/1+ -1*( -1*1)*/2+ -1*( -1*( -1*1))*/(2+1+1+1+1)``. +Apply H0. +Repeat Rewrite Rinv_R1; Repeat Rewrite Rmult_1r; Rewrite Ropp_mul1; Rewrite Rmult_1l; Rewrite Ropp_Ropp; Rewrite Rplus_Ropp_r; Rewrite Rmult_1r; Rewrite Rplus_Ol; Rewrite Rmult_1l; Apply r_Rmult_mult with ``6``. +Rewrite Rmult_Rplus_distr; Replace ``2+1+1+1+1`` with ``6``. +Rewrite <- (Rmult_sym ``/6``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Replace ``6`` with ``2*3``. +Do 2 Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Rewrite (Rmult_sym ``3``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Ring. +DiscrR. +DiscrR. +Ring. +DiscrR. +Ring. +DiscrR. +Apply H. +Unfold Un_decreasing; Intros; Apply Rle_monotony_contra with ``(INR (fact n))``. +Apply INR_fact_lt_0. +Apply Rle_monotony_contra with ``(INR (fact (S n)))``. +Apply INR_fact_lt_0. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Apply le_INR; Apply fact_growing; Apply le_n_Sn. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Assert H0 := (cv_speed_pow_fact R1); Unfold Un_cv; Unfold Un_cv in H0; Intros; Elim (H0 ? H1); Intros; Exists x0; Intros; Unfold R_dist in H2; Unfold R_dist; Replace ``/(INR (fact n))`` with ``(pow 1 n)/(INR (fact n))``. +Apply (H2 ? H3). +Unfold Rdiv; Rewrite pow1; Rewrite Rmult_1l; Reflexivity. +Unfold infinit_sum in e; Unfold Un_cv tg_alt; Intros; Elim (e ? H0); Intros; Exists x0; Intros; Replace (sum_f_R0 ([i:nat]``(pow ( -1) i)*/(INR (fact i))``) n) with (sum_f_R0 ([i:nat]``/(INR (fact i))*(pow ( -1) i)``) n). +Apply (H1 ? H2). +Apply sum_eq; Intros; Apply Rmult_sym. +Apply r_Rmult_mult with ``(exp 1)``. +Rewrite <- exp_plus; Rewrite Rplus_Ropp_r; Rewrite exp_0; Rewrite <- Rinv_r_sym. +Reflexivity. +Assumption. +Assumption. +DiscrR. +Assumption. +Qed. + +(******************************************************************) +(* Properties of Exp *) +(******************************************************************) + +Theorem exp_increasing: (x, y : R) ``x<y`` -> ``(exp x)<(exp y)``. +Intros x y H. +Assert H0 : (derivable exp). +Apply derivable_exp. +Assert H1 := (positive_derivative ? H0). +Unfold strict_increasing in H1. +Apply H1. +Intro. +Replace (derive_pt exp x0 (H0 x0)) with (exp x0). +Apply exp_pos. +Symmetry; Apply derive_pt_eq_0. +Apply (derivable_pt_lim_exp x0). +Apply H. +Qed. + +Theorem exp_lt_inv: (x, y : R) ``(exp x)<(exp y)`` -> ``x<y``. +Intros x y H; Case (total_order x y); [Intros H1 | Intros [H1|H1]]. +Assumption. +Rewrite H1 in H; Elim (Rlt_antirefl ? H). +Assert H2 := (exp_increasing ? ? H1). +Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H H2)). +Qed. + +Lemma exp_ineq1 : (x:R) ``0<x`` -> ``1+x < (exp x)``. +Intros; Apply Rlt_anti_compatibility with ``-(exp 0)``; Rewrite <- (Rplus_sym (exp x)); Assert H0 := (MVT_cor1 exp R0 x derivable_exp H); Elim H0; Intros; Elim H1; Intros; Unfold Rminus in H2; Rewrite H2; Rewrite Ropp_O; Rewrite Rplus_Or; Replace (derive_pt exp x0 (derivable_exp x0)) with (exp x0). +Rewrite exp_0; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Pattern 1 x; Rewrite <- Rmult_1r; Rewrite (Rmult_sym (exp x0)); Apply Rlt_monotony. +Apply H. +Rewrite <- exp_0; Apply exp_increasing; Elim H3; Intros; Assumption. +Symmetry; Apply derive_pt_eq_0; Apply derivable_pt_lim_exp. +Qed. + +Lemma ln_exists1 : (y:R) ``0<y``->``1<=y``->(sigTT R [z:R]``y==(exp z)``). +Intros; Pose f := [x:R]``(exp x)-y``; Cut ``(f 0)<=0``. +Intro; Cut (continuity f). +Intro; Cut ``0<=(f y)``. +Intro; Cut ``(f 0)*(f y)<=0``. +Intro; Assert X := (IVT_cor f R0 y H2 (Rlt_le ? ? H) H4); Elim X; Intros t H5; Apply existTT with t; Elim H5; Intros; Unfold f in H7; Apply Rminus_eq_right; Exact H7. +Pattern 2 R0; Rewrite <- (Rmult_Or (f y)); Rewrite (Rmult_sym (f R0)); Apply Rle_monotony; Assumption. +Unfold f; Apply Rle_anti_compatibility with y; Left; Apply Rlt_trans with ``1+y``. +Rewrite <- (Rplus_sym y); Apply Rlt_compatibility; Apply Rlt_R0_R1. +Replace ``y+((exp y)-y)`` with (exp y); [Apply (exp_ineq1 y H) | Ring]. +Unfold f; Change (continuity (minus_fct exp (fct_cte y))); Apply continuity_minus; [Apply derivable_continuous; Apply derivable_exp | Apply derivable_continuous; Apply derivable_const]. +Unfold f; Rewrite exp_0; Apply Rle_anti_compatibility with y; Rewrite Rplus_Or; Replace ``y+(1-y)`` with R1; [Apply H0 | Ring]. +Qed. + +(**********) +Lemma ln_exists : (y:R) ``0<y`` -> (sigTT R [z:R]``y==(exp z)``). +Intros; Case (total_order_Rle R1 y); Intro. +Apply (ln_exists1 ? H r). +Assert H0 : ``1<=/y``. +Apply Rle_monotony_contra with y. +Apply H. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Left; Apply (not_Rle ? ? n). +Red; Intro; Rewrite H0 in H; Elim (Rlt_antirefl ? H). +Assert H1 : ``0</y``. +Apply Rlt_Rinv; Apply H. +Assert H2 := (ln_exists1 ? H1 H0); Elim H2; Intros; Apply existTT with ``-x``; Apply r_Rmult_mult with ``(exp x)/y``. +Unfold Rdiv; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite <- (Rmult_sym ``/y``); Rewrite Rmult_assoc; Rewrite <- exp_plus; Rewrite Rplus_Ropp_r; Rewrite exp_0; Rewrite Rmult_1r; Symmetry; Apply p. +Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H). +Unfold Rdiv; Apply prod_neq_R0. +Assert H3 := (exp_pos x); Red; Intro; Rewrite H4 in H3; Elim (Rlt_antirefl ? H3). +Apply Rinv_neq_R0; Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H). +Qed. + +(* Definition of log R+* -> R *) +Definition Rln [y:posreal] : R := Cases (ln_exists (pos y) (RIneq.cond_pos y)) of (existTT a b) => a end. + +(* Extension on R *) +Definition ln : R->R := [x:R](Cases (total_order_Rlt R0 x) of + (leftT a) => (Rln (mkposreal x a)) + | (rightT a) => R0 end). + +Lemma exp_ln : (x : R) ``0<x`` -> (exp (ln x)) == x. +Intros; Unfold ln; Case (total_order_Rlt R0 x); Intro. +Unfold Rln; Case (ln_exists (mkposreal x r) (RIneq.cond_pos (mkposreal x r))); Intros. +Simpl in e; Symmetry; Apply e. +Elim n; Apply H. +Qed. + +Theorem exp_inv: (x, y : R) (exp x) == (exp y) -> x == y. +Intros x y H; Case (total_order x y); [Intros H1 | Intros [H1|H1]]; Auto; Assert H2 := (exp_increasing ? ? H1); Rewrite H in H2; Elim (Rlt_antirefl ? H2). +Qed. + +Theorem exp_Ropp: (x : R) ``(exp (-x)) == /(exp x)``. +Intros x; Assert H : ``(exp x)<>0``. +Assert H := (exp_pos x); Red; Intro; Rewrite H0 in H; Elim (Rlt_antirefl ? H). +Apply r_Rmult_mult with r := (exp x). +Rewrite <- exp_plus; Rewrite Rplus_Ropp_r; Rewrite exp_0. +Apply Rinv_r_sym. +Apply H. +Apply H. +Qed. + +(******************************************************************) +(* Properties of Ln *) +(******************************************************************) + +Theorem ln_increasing: + (x, y : R) ``0<x`` -> ``x<y`` -> ``(ln x) < (ln y)``. +Intros x y H H0; Apply exp_lt_inv. +Repeat Rewrite exp_ln. +Apply H0. +Apply Rlt_trans with x; Assumption. +Apply H. +Qed. + +Theorem ln_exp: (x : R) (ln (exp x)) == x. +Intros x; Apply exp_inv. +Apply exp_ln. +Apply exp_pos. +Qed. + +Theorem ln_1: ``(ln 1) == 0``. +Rewrite <- exp_0; Rewrite ln_exp; Reflexivity. +Qed. + +Theorem ln_lt_inv: + (x, y : R) ``0<x`` -> ``0<y`` -> ``(ln x)<(ln y)`` -> ``x<y``. +Intros x y H H0 H1; Rewrite <- (exp_ln x); Try Rewrite <- (exp_ln y). +Apply exp_increasing; Apply H1. +Assumption. +Assumption. +Qed. + +Theorem ln_inv: (x, y : R) ``0<x`` -> ``0<y`` -> (ln x) == (ln y) -> x == y. +Intros x y H H0 H'0; Case (total_order x y); [Intros H1 | Intros [H1|H1]]; Auto. +Assert H2 := (ln_increasing ? ? H H1); Rewrite H'0 in H2; Elim (Rlt_antirefl ? H2). +Assert H2 := (ln_increasing ? ? H0 H1); Rewrite H'0 in H2; Elim (Rlt_antirefl ? H2). +Qed. + +Theorem ln_mult: (x, y : R) ``0<x`` -> ``0<y`` -> ``(ln (x*y)) == (ln x)+(ln y)``. +Intros x y H H0; Apply exp_inv. +Rewrite exp_plus. +Repeat Rewrite exp_ln. +Reflexivity. +Assumption. +Assumption. +Apply Rmult_lt_pos; Assumption. +Qed. + +Theorem ln_Rinv: (x : R) ``0<x`` -> ``(ln (/x)) == -(ln x)``. +Intros x H; Apply exp_inv; Repeat (Rewrite exp_ln Orelse Rewrite exp_Ropp). +Reflexivity. +Assumption. +Apply Rlt_Rinv; Assumption. +Qed. + +Theorem ln_continue: + (y : R) ``0<y`` -> (continue_in ln [x : R] (Rlt R0 x) y). +Intros y H. +Unfold continue_in limit1_in limit_in; Intros eps Heps. +Cut (Rlt R1 (exp eps)); [Intros H1 | Idtac]. +Cut (Rlt (exp (Ropp eps)) R1); [Intros H2 | Idtac]. +Exists + (Rmin (Rmult y (Rminus (exp eps) R1)) (Rmult y (Rminus R1 (exp (Ropp eps))))); + Split. +Red; Apply P_Rmin. +Apply Rmult_lt_pos. +Assumption. +Apply Rlt_anti_compatibility with R1. +Rewrite Rplus_Or; Replace ``(1+((exp eps)-1))`` with (exp eps); [Apply H1 | Ring]. +Apply Rmult_lt_pos. +Assumption. +Apply Rlt_anti_compatibility with ``(exp (-eps))``. +Rewrite Rplus_Or; Replace ``(exp ( -eps))+(1-(exp ( -eps)))`` with R1; [Apply H2 | Ring]. +Unfold dist R_met R_dist; Simpl. +Intros x ((H3, H4), H5). +Cut (Rmult y (Rmult x (Rinv y))) == x. +Intro Hxyy. +Replace (Rminus (ln x) (ln y)) with (ln (Rmult x (Rinv y))). +Case (total_order x y); [Intros Hxy | Intros [Hxy|Hxy]]. +Rewrite Rabsolu_left. +Apply Ropp_Rlt; Rewrite Ropp_Ropp. +Apply exp_lt_inv. +Rewrite exp_ln. +Apply Rlt_monotony_contra with z := y. +Apply H. +Rewrite Hxyy. +Apply Ropp_Rlt. +Apply Rlt_anti_compatibility with r := y. +Replace (Rplus y (Ropp (Rmult y (exp (Ropp eps))))) + with (Rmult y (Rminus R1 (exp (Ropp eps)))); [Idtac | Ring]. +Replace (Rplus y (Ropp x)) with (Rabsolu (Rminus x y)); [Idtac | Ring]. +Apply Rlt_le_trans with 1 := H5; Apply Rmin_r. +Rewrite Rabsolu_left; [Ring | Idtac]. +Apply (Rlt_minus ? ? Hxy). +Apply Rmult_lt_pos; [Apply H3 | Apply (Rlt_Rinv ? H)]. +Rewrite <- ln_1. +Apply ln_increasing. +Apply Rmult_lt_pos; [Apply H3 | Apply (Rlt_Rinv ? H)]. +Apply Rlt_monotony_contra with z := y. +Apply H. +Rewrite Hxyy; Rewrite Rmult_1r; Apply Hxy. +Rewrite Hxy; Rewrite Rinv_r. +Rewrite ln_1; Rewrite Rabsolu_R0; Apply Heps. +Red; Intro; Rewrite H0 in H; Elim (Rlt_antirefl ? H). +Rewrite Rabsolu_right. +Apply exp_lt_inv. +Rewrite exp_ln. +Apply Rlt_monotony_contra with z := y. +Apply H. +Rewrite Hxyy. +Apply Rlt_anti_compatibility with r := (Ropp y). +Replace (Rplus (Ropp y) (Rmult y (exp eps))) + with (Rmult y (Rminus (exp eps) R1)); [Idtac | Ring]. +Replace (Rplus (Ropp y) x) with (Rabsolu (Rminus x y)); [Idtac | Ring]. +Apply Rlt_le_trans with 1 := H5; Apply Rmin_l. +Rewrite Rabsolu_right; [Ring | Idtac]. +Left; Apply (Rgt_minus ? ? Hxy). +Apply Rmult_lt_pos; [Apply H3 | Apply (Rlt_Rinv ? H)]. +Rewrite <- ln_1. +Apply Rgt_ge; Red; Apply ln_increasing. +Apply Rlt_R0_R1. +Apply Rlt_monotony_contra with z := y. +Apply H. +Rewrite Hxyy; Rewrite Rmult_1r; Apply Hxy. +Rewrite ln_mult. +Rewrite ln_Rinv. +Ring. +Assumption. +Assumption. +Apply Rlt_Rinv; Assumption. +Rewrite (Rmult_sym x); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Ring. +Red; Intro; Rewrite H0 in H; Elim (Rlt_antirefl ? H). +Apply Rlt_monotony_contra with (exp eps). +Apply exp_pos. +Rewrite <- exp_plus; Rewrite Rmult_1r; Rewrite Rplus_Ropp_r; Rewrite exp_0; Apply H1. +Rewrite <- exp_0. +Apply exp_increasing; Apply Heps. +Qed. + +(******************************************************************) +(* Definition of Rpower *) +(******************************************************************) + +Definition Rpower := [x : R] [y : R] ``(exp (y*(ln x)))``. + +Infix Local "^R" Rpower (at level 2, left associativity) : R_scope. + +(******************************************************************) +(* Properties of Rpower *) +(******************************************************************) + +Theorem Rpower_plus: + (x, y, z : R) ``(Rpower z (x+y)) == (Rpower z x)*(Rpower z y)``. +Intros x y z; Unfold Rpower. +Rewrite Rmult_Rplus_distrl; Rewrite exp_plus; Auto. +Qed. + +Theorem Rpower_mult: + (x, y, z : R) ``(Rpower (Rpower x y) z) == (Rpower x (y*z))``. +Intros x y z; Unfold Rpower. +Rewrite ln_exp. +Replace (Rmult z (Rmult y (ln x))) with (Rmult (Rmult y z) (ln x)). +Reflexivity. +Ring. +Qed. + +Theorem Rpower_O: (x : R) ``0<x`` -> ``(Rpower x 0) == 1``. +Intros x H; Unfold Rpower. +Rewrite Rmult_Ol; Apply exp_0. +Qed. + +Theorem Rpower_1: (x : R) ``0<x`` -> ``(Rpower x 1) == x``. +Intros x H; Unfold Rpower. +Rewrite Rmult_1l; Apply exp_ln; Apply H. +Qed. + +Theorem Rpower_pow: + (n : nat) (x : R) ``0<x`` -> (Rpower x (INR n)) == (pow x n). +Intros n; Elim n; Simpl; Auto; Fold INR. +Intros x H; Apply Rpower_O; Auto. +Intros n1; Case n1. +Intros H x H0; Simpl; Rewrite Rmult_1r; Apply Rpower_1; Auto. +Intros n0 H x H0; Rewrite Rpower_plus; Rewrite H; Try Rewrite Rpower_1; Try Apply Rmult_sym Orelse Assumption. +Qed. + +Theorem Rpower_lt: (x, y, z : R) ``1<x`` -> ``0<=y`` -> ``y<z`` -> ``(Rpower x y) < (Rpower x z)``. +Intros x y z H H0 H1. +Unfold Rpower. +Apply exp_increasing. +Apply Rlt_monotony_r. +Rewrite <- ln_1; Apply ln_increasing. +Apply Rlt_R0_R1. +Apply H. +Apply H1. +Qed. + +Theorem Rpower_sqrt: (x : R) ``0<x`` -> ``(Rpower x (/2)) == (sqrt x)``. +Intros x H. +Apply ln_inv. +Unfold Rpower; Apply exp_pos. +Apply sqrt_lt_R0; Apply H. +Apply r_Rmult_mult with (INR (S (S O))). +Apply exp_inv. +Fold Rpower. +Cut (Rpower (Rpower x (Rinv (Rplus R1 R1))) (INR (S (S O)))) == (Rpower (sqrt x) (INR (S (S O)))). +Unfold Rpower; Auto. +Rewrite Rpower_mult. +Rewrite Rinv_l. +Replace R1 with (INR (S O)); Auto. +Repeat Rewrite Rpower_pow; Simpl. +Pattern 1 x; Rewrite <- (sqrt_sqrt x (Rlt_le ? ? H)). +Ring. +Apply sqrt_lt_R0; Apply H. +Apply H. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Qed. + +Theorem Rpower_Ropp: (x, y : R) ``(Rpower x (-y)) == /(Rpower x y)``. +Unfold Rpower. +Intros x y; Rewrite Ropp_mul1. +Apply exp_Ropp. +Qed. + +Theorem Rle_Rpower: (e,n,m : R) ``1<e`` -> ``0<=n`` -> ``n<=m`` -> ``(Rpower e n)<=(Rpower e m)``. +Intros e n m H H0 H1; Case H1. +Intros H2; Left; Apply Rpower_lt; Assumption. +Intros H2; Rewrite H2; Right; Reflexivity. +Qed. + +Theorem ln_lt_2: ``/2<(ln 2)``. +Apply Rlt_monotony_contra with z := (Rplus R1 R1). +Sup0. +Rewrite Rinv_r. +Apply exp_lt_inv. +Apply Rle_lt_trans with 1 := exp_le_3. +Change (Rlt (Rplus R1 (Rplus R1 R1)) (Rpower (Rplus R1 R1) (Rplus R1 R1))). +Repeat Rewrite Rpower_plus; Repeat Rewrite Rpower_1. +Repeat Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_Rplus_distr; + Repeat Rewrite Rmult_1l. +Pattern 1 ``3``; Rewrite <- Rplus_Or; Replace ``2+2`` with ``3+1``; [Apply Rlt_compatibility; Apply Rlt_R0_R1 | Ring]. +Sup0. +DiscrR. +Qed. + +(**************************************) +(* Differentiability of Ln and Rpower *) +(**************************************) + +Theorem limit1_ext: (f, g : R -> R)(D : R -> Prop)(l, x : R) ((x : R) (D x) -> (f x) == (g x)) -> (limit1_in f D l x) -> (limit1_in g D l x). +Intros f g D l x H; Unfold limit1_in limit_in. +Intros H0 eps H1; Case (H0 eps); Auto. +Intros x0 (H2, H3); Exists x0; Split; Auto. +Intros x1 (H4, H5); Rewrite <- H; Auto. +Qed. + +Theorem limit1_imp: (f : R -> R)(D, D1 : R -> Prop)(l, x : R) ((x : R) (D1 x) -> (D x)) -> (limit1_in f D l x) -> (limit1_in f D1 l x). +Intros f D D1 l x H; Unfold limit1_in limit_in. +Intros H0 eps H1; Case (H0 eps H1); Auto. +Intros alpha (H2, H3); Exists alpha; Split; Auto. +Intros d (H4, H5); Apply H3; Split; Auto. +Qed. + +Theorem Rinv_Rdiv: (x, y : R) ``x<>0`` -> ``y<>0`` -> ``/(x/y) == y/x``. +Intros x y H1 H2; Unfold Rdiv; Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Apply Rmult_sym. +Assumption. +Assumption. +Apply Rinv_neq_R0; Assumption. +Qed. + +Theorem Dln: (y : R) ``0<y`` -> (D_in ln Rinv [x:R]``0<x`` y). +Intros y Hy; Unfold D_in. +Apply limit1_ext with f := [x : R](Rinv (Rdiv (Rminus (exp (ln x)) (exp (ln y))) (Rminus (ln x) (ln y)))). +Intros x (HD1, HD2); Repeat Rewrite exp_ln. +Unfold Rdiv; Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Apply Rmult_sym. +Apply Rminus_eq_contra. +Red; Intros H2; Case HD2. +Symmetry; Apply (ln_inv ? ? HD1 Hy H2). +Apply Rminus_eq_contra; Apply (not_sym ? ? HD2). +Apply Rinv_neq_R0; Apply Rminus_eq_contra; Red; Intros H2; Case HD2; Apply ln_inv; Auto. +Assumption. +Assumption. +Apply limit_inv with f := [x : R] (Rdiv (Rminus (exp (ln x)) (exp (ln y))) (Rminus (ln x) (ln y))). +Apply limit1_imp with f := [x : R] ([x : R] (Rdiv (Rminus (exp x) (exp (ln y))) (Rminus x (ln y))) (ln x)) D := (Dgf (D_x [x : R] (Rlt R0 x) y) (D_x [x : R] True (ln y)) ln). +Intros x (H1, H2); Split. +Split; Auto. +Split; Auto. +Red; Intros H3; Case H2; Apply ln_inv; Auto. +Apply limit_comp with l := (ln y) g := [x : R] (Rdiv (Rminus (exp x) (exp (ln y))) (Rminus x (ln y))) f := ln. +Apply ln_continue; Auto. +Assert H0 := (derivable_pt_lim_exp (ln y)); Unfold derivable_pt_lim in H0; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H0 ? H); Intros; Exists (pos x); Split. +Apply (RIneq.cond_pos x). +Intros; Pattern 3 y; Rewrite <- exp_ln. +Pattern 1 x0; Replace x0 with ``(ln y)+(x0-(ln y))``; [Idtac | Ring]. +Apply H1. +Elim H2; Intros H3 _; Unfold D_x in H3; Elim H3; Clear H3; Intros _ H3; Apply Rminus_eq_contra; Apply not_sym; Apply H3. +Elim H2; Clear H2; Intros _ H2; Apply H2. +Assumption. +Red; Intro; Rewrite H in Hy; Elim (Rlt_antirefl ? Hy). +Qed. + +Lemma derivable_pt_lim_ln : (x:R) ``0<x`` -> (derivable_pt_lim ln x ``/x``). +Intros; Assert H0 := (Dln x H); Unfold D_in in H0; Unfold limit1_in in H0; Unfold limit_in in H0; Simpl in H0; Unfold R_dist in H0; Unfold derivable_pt_lim; Intros; Elim (H0 ? H1); Intros; Elim H2; Clear H2; Intros; Pose alp := (Rmin x0 ``x/2``); Assert H4 : ``0<alp``. +Unfold alp; Unfold Rmin; Case (total_order_Rle x0 ``x/2``); Intro. +Apply H2. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Exists (mkposreal ? H4); Intros; Pattern 2 h; Replace h with ``(x+h)-x``; [Idtac | Ring]. +Apply H3; Split. +Unfold D_x; Split. +Case (case_Rabsolu h); Intro. +Assert H7 : ``(Rabsolu h)<x/2``. +Apply Rlt_le_trans with alp. +Apply H6. +Unfold alp; Apply Rmin_r. +Apply Rlt_trans with ``x/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Rewrite Rabsolu_left in H7. +Apply Rlt_anti_compatibility with ``-h-x/2``. +Replace ``-h-x/2+x/2`` with ``-h``; [Idtac | Ring]. +Pattern 2 x; Rewrite double_var. +Replace ``-h-x/2+(x/2+x/2+h)`` with ``x/2``; [Apply H7 | Ring]. +Apply r. +Apply gt0_plus_ge0_is_gt0; [Assumption | Apply Rle_sym2; Apply r]. +Apply not_sym; Apply Rminus_not_eq; Replace ``x+h-x`` with h; [Apply H5 | Ring]. +Replace ``x+h-x`` with h; [Apply Rlt_le_trans with alp; [Apply H6 | Unfold alp; Apply Rmin_l] | Ring]. +Qed. + +Theorem D_in_imp: (f, g : R -> R)(D, D1 : R -> Prop)(x : R) ((x : R) (D1 x) -> (D x)) -> (D_in f g D x) -> (D_in f g D1 x). +Intros f g D D1 x H; Unfold D_in. +Intros H0; Apply limit1_imp with D := (D_x D x); Auto. +Intros x1 (H1, H2); Split; Auto. +Qed. + +Theorem D_in_ext: (f, g, h : R -> R)(D : R -> Prop) (x : R) (f x) == (g x) -> (D_in h f D x) -> (D_in h g D x). +Intros f g h D x H; Unfold D_in. +Rewrite H; Auto. +Qed. + +Theorem Dpower: (y, z : R) ``0<y`` -> (D_in [x:R](Rpower x z) [x:R](Rmult z (Rpower x (Rminus z R1))) [x:R]``0<x`` y). +Intros y z H; Apply D_in_imp with D := (Dgf [x : R] (Rlt R0 x) [x : R] True ln). +Intros x H0; Repeat Split. +Assumption. +Apply D_in_ext with f := [x : R] (Rmult (Rinv x) (Rmult z (exp (Rmult z (ln x))))). +Unfold Rminus; Rewrite Rpower_plus; Rewrite Rpower_Ropp; Rewrite (Rpower_1 ? H); Ring. +Apply Dcomp with f := ln g := [x : R] (exp (Rmult z x)) df := Rinv dg := [x : R] (Rmult z (exp (Rmult z x))). +Apply (Dln ? H). +Apply D_in_imp with D := (Dgf [x : R] True [x : R] True [x : R] (Rmult z x)). +Intros x H1; Repeat Split; Auto. +Apply (Dcomp [_ : R] True [_ : R] True [x : ?] z exp [x : R] (Rmult z x) exp); Simpl. +Apply D_in_ext with f := [x : R] (Rmult z R1). +Apply Rmult_1r. +Apply (Dmult_const [x : ?] True [x : ?] x [x : ?] R1); Apply Dx. +Assert H0 := (derivable_pt_lim_D_in exp exp ``z*(ln y)``); Elim H0; Clear H0; Intros _ H0; Apply H0; Apply derivable_pt_lim_exp. +Qed. + +Theorem derivable_pt_lim_power: (x, y : R) (Rlt R0 x) -> (derivable_pt_lim [x : ?] (Rpower x y) x (Rmult y (Rpower x (Rminus y R1)))). +Intros x y H. +Unfold Rminus; Rewrite Rpower_plus. +Rewrite Rpower_Ropp. +Rewrite Rpower_1; Auto. +Rewrite <- Rmult_assoc. +Unfold Rpower. +Apply derivable_pt_lim_comp with f1 := ln f2 := [x : ?] (exp (Rmult y x)). +Apply derivable_pt_lim_ln; Assumption. +Rewrite (Rmult_sym y). +Apply derivable_pt_lim_comp with f1 := [x : ?] (Rmult y x) f2 := exp. +Pattern 2 y; Replace y with (Rplus (Rmult R0 (ln x)) (Rmult y R1)). +Apply derivable_pt_lim_mult with f1 := [x : R] y f2 := [x : R] x. +Apply derivable_pt_lim_const with a := y. +Apply derivable_pt_lim_id. +Ring. +Apply derivable_pt_lim_exp. +Qed. diff --git a/theories7/Reals/Rprod.v b/theories7/Reals/Rprod.v new file mode 100644 index 00000000..a524a915 --- /dev/null +++ b/theories7/Reals/Rprod.v @@ -0,0 +1,164 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rprod.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Compare. +Require Rbase. +Require Rfunctions. +Require Rseries. +Require PartSum. +Require Binomial. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(* TT Ak; 1<=k<=N *) +Fixpoint prod_f_SO [An:nat->R;N:nat] : R := Cases N of + O => R1 +| (S p) => ``(prod_f_SO An p)*(An (S p))`` +end. + +(**********) +Lemma prod_SO_split : (An:nat->R;n,k:nat) (le k n) -> (prod_f_SO An n)==(Rmult (prod_f_SO An k) (prod_f_SO [l:nat](An (plus k l)) (minus n k))). +Intros; Induction n. +Cut k=O; [Intro; Rewrite H0; Simpl; Ring | Inversion H; Reflexivity]. +Cut k=(S n)\/(le k n). +Intro; Elim H0; Intro. +Rewrite H1; Simpl; Rewrite <- minus_n_n; Simpl; Ring. +Replace (minus (S n) k) with (S (minus n k)). +Simpl; Replace (plus k (S (minus n k))) with (S n). +Rewrite Hrecn; [Ring | Assumption]. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite S_INR; Rewrite minus_INR; [Ring | Assumption]. +Apply INR_eq; Rewrite S_INR; Repeat Rewrite minus_INR. +Rewrite S_INR; Ring. +Apply le_trans with n; [Assumption | Apply le_n_Sn]. +Assumption. +Inversion H; [Left; Reflexivity | Right; Assumption]. +Qed. + +(**********) +Lemma prod_SO_pos : (An:nat->R;N:nat) ((n:nat)(le n N)->``0<=(An n)``) -> ``0<=(prod_f_SO An N)``. +Intros; Induction N. +Simpl; Left; Apply Rlt_R0_R1. +Simpl; Apply Rmult_le_pos. +Apply HrecN; Intros; Apply H; Apply le_trans with N; [Assumption | Apply le_n_Sn]. +Apply H; Apply le_n. +Qed. + +(**********) +Lemma prod_SO_Rle : (An,Bn:nat->R;N:nat) ((n:nat)(le n N)->``0<=(An n)<=(Bn n)``) -> ``(prod_f_SO An N)<=(prod_f_SO Bn N)``. +Intros; Induction N. +Right; Reflexivity. +Simpl; Apply Rle_trans with ``(prod_f_SO An N)*(Bn (S N))``. +Apply Rle_monotony. +Apply prod_SO_pos; Intros; Elim (H n (le_trans ? ? ? H0 (le_n_Sn N))); Intros; Assumption. +Elim (H (S N) (le_n (S N))); Intros; Assumption. +Do 2 Rewrite <- (Rmult_sym (Bn (S N))); Apply Rle_monotony. +Elim (H (S N) (le_n (S N))); Intros. +Apply Rle_trans with (An (S N)); Assumption. +Apply HrecN; Intros; Elim (H n (le_trans ? ? ? H0 (le_n_Sn N))); Intros; Split; Assumption. +Qed. + +(* Application to factorial *) +Lemma fact_prodSO : (n:nat) (INR (fact n))==(prod_f_SO [k:nat](INR k) n). +Intro; Induction n. +Reflexivity. +Change (INR (mult (S n) (fact n)))==(prod_f_SO ([k:nat](INR k)) (S n)). +Rewrite mult_INR; Rewrite Rmult_sym; Rewrite Hrecn; Reflexivity. +Qed. + +Lemma le_n_2n : (n:nat) (le n (mult (2) n)). +Induction n. +Replace (mult (2) (O)) with O; [Apply le_n | Ring]. +Intros; Replace (mult (2) (S n0)) with (S (S (mult (2) n0))). +Apply le_n_S; Apply le_S; Assumption. +Replace (S (S (mult (2) n0))) with (plus (mult (2) n0) (2)); [Idtac | Ring]. +Replace (S n0) with (plus n0 (1)); [Idtac | Ring]. +Ring. +Qed. + +(* We prove that (N!)²<=(2N-k)!*k! forall k in [|O;2N|] *) +Lemma RfactN_fact2N_factk : (N,k:nat) (le k (mult (2) N)) -> ``(Rsqr (INR (fact N)))<=(INR (fact (minus (mult (S (S O)) N) k)))*(INR (fact k))``. +Intros; Unfold Rsqr; Repeat Rewrite fact_prodSO. +Cut (le k N)\/(le N k). +Intro; Elim H0; Intro. +Rewrite (prod_SO_split [l:nat](INR l) (minus (mult (2) N) k) N). +Rewrite Rmult_assoc; Apply Rle_monotony. +Apply prod_SO_pos; Intros; Apply pos_INR. +Replace (minus (minus (mult (2) N) k) N) with (minus N k). +Rewrite Rmult_sym; Rewrite (prod_SO_split [l:nat](INR l) N k). +Apply Rle_monotony. +Apply prod_SO_pos; Intros; Apply pos_INR. +Apply prod_SO_Rle; Intros; Split. +Apply pos_INR. +Apply le_INR; Apply le_reg_r; Assumption. +Assumption. +Apply INR_eq; Repeat Rewrite minus_INR. +Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply le_trans with N; [Assumption | Apply le_n_2n]. +Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. +Replace (mult (2) N) with (plus N N); [Idtac | Ring]. +Apply le_reg_r; Assumption. +Assumption. +Assumption. +Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. +Replace (mult (2) N) with (plus N N); [Idtac | Ring]. +Apply le_reg_r; Assumption. +Assumption. +Rewrite <- (Rmult_sym (prod_f_SO [l:nat](INR l) k)); Rewrite (prod_SO_split [l:nat](INR l) k N). +Rewrite Rmult_assoc; Apply Rle_monotony. +Apply prod_SO_pos; Intros; Apply pos_INR. +Rewrite Rmult_sym; Rewrite (prod_SO_split [l:nat](INR l) N (minus (mult (2) N) k)). +Apply Rle_monotony. +Apply prod_SO_pos; Intros; Apply pos_INR. +Replace (minus N (minus (mult (2) N) k)) with (minus k N). +Apply prod_SO_Rle; Intros; Split. +Apply pos_INR. +Apply le_INR; Apply le_reg_r. +Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. +Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. +Assumption. +Apply INR_eq; Repeat Rewrite minus_INR. +Rewrite mult_INR; Do 2 Rewrite S_INR; Ring. +Assumption. +Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. +Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. +Assumption. +Assumption. +Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. +Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. +Assumption. +Assumption. +Elim (le_dec k N); Intro; [Left; Assumption | Right; Assumption]. +Qed. + +(**********) +Lemma INR_fact_lt_0 : (n:nat) ``0<(INR (fact n))``. +Intro; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Elim (fact_neq_0 n); Symmetry; Assumption. +Qed. + +(* We have the following inequality : (C 2N k) <= (C 2N N) forall k in [|O;2N|] *) +Lemma C_maj : (N,k:nat) (le k (mult (2) N)) -> ``(C (mult (S (S O)) N) k)<=(C (mult (S (S O)) N) N)``. +Intros; Unfold C; Unfold Rdiv; Apply Rle_monotony. +Apply pos_INR. +Replace (minus (mult (2) N) N) with N. +Apply Rle_monotony_contra with ``((INR (fact N))*(INR (fact N)))``. +Apply Rmult_lt_pos; Apply INR_fact_lt_0. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_sym; Apply Rle_monotony_contra with ``((INR (fact k))* + (INR (fact (minus (mult (S (S O)) N) k))))``. +Apply Rmult_lt_pos; Apply INR_fact_lt_0. +Rewrite Rmult_1r; Rewrite <- mult_INR; Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite mult_INR; Rewrite (Rmult_sym (INR (fact k))); Replace ``(INR (fact N))*(INR (fact N))`` with (Rsqr (INR (fact N))). +Apply RfactN_fact2N_factk. +Assumption. +Reflexivity. +Rewrite mult_INR; Apply prod_neq_R0; Apply INR_fact_neq_0. +Apply prod_neq_R0; Apply INR_fact_neq_0. +Apply INR_eq; Rewrite minus_INR; [Rewrite mult_INR; Do 2 Rewrite S_INR; Ring | Apply le_n_2n]. +Qed. diff --git a/theories7/Reals/Rseries.v b/theories7/Reals/Rseries.v new file mode 100644 index 00000000..a38099dd --- /dev/null +++ b/theories7/Reals/Rseries.v @@ -0,0 +1,279 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rseries.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Classical. +Require Compare. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Implicit Variable Type r:R. + +(* classical is needed for [Un_cv_crit] *) +(*********************************************************) +(* Definition of sequence and properties *) +(* *) +(*********************************************************) + +Section sequence. + +(*********) +Variable Un:nat->R. + +(*********) +Fixpoint Rmax_N [N:nat]:R:= + Cases N of + O => (Un O) + |(S n) => (Rmax (Un (S n)) (Rmax_N n)) + end. + +(*********) +Definition EUn:R->Prop:=[r:R](Ex [i:nat] (r==(Un i))). + +(*********) +Definition Un_cv:R->Prop:=[l:R] + (eps:R)(Rgt eps R0)->(Ex[N:nat](n:nat)(ge n N)-> + (Rlt (R_dist (Un n) l) eps)). + +(*********) +Definition Cauchy_crit:Prop:=(eps:R)(Rgt eps R0)-> + (Ex[N:nat] (n,m:nat)(ge n N)->(ge m N)-> + (Rlt (R_dist (Un n) (Un m)) eps)). + +(*********) +Definition Un_growing:Prop:=(n:nat)(Rle (Un n) (Un (S n))). + +(*********) +Lemma EUn_noempty:(ExT [r:R] (EUn r)). +Unfold EUn;Split with (Un O);Split with O;Trivial. +Qed. + +(*********) +Lemma Un_in_EUn:(n:nat)(EUn (Un n)). +Intro;Unfold EUn;Split with n;Trivial. +Qed. + +(*********) +Lemma Un_bound_imp:(x:R)((n:nat)(Rle (Un n) x))->(is_upper_bound EUn x). +Intros;Unfold is_upper_bound;Intros;Unfold EUn in H0;Elim H0;Clear H0; + Intros;Generalize (H x1);Intro;Rewrite <- H0 in H1;Trivial. +Qed. + +(*********) +Lemma growing_prop:(n,m:nat)Un_growing->(ge n m)->(Rge (Un n) (Un m)). +Double Induction n m;Intros. +Unfold Rge;Right;Trivial. +ElimType False;Unfold ge in H1;Generalize (le_Sn_O n0);Intro;Auto. +Cut (ge n0 (0)). +Generalize H0;Intros;Unfold Un_growing in H0; + Apply (Rge_trans (Un (S n0)) (Un n0) (Un (0)) + (Rle_sym1 (Un n0) (Un (S n0)) (H0 n0)) (H O H2 H3)). +Elim n0;Auto. +Elim (lt_eq_lt_dec n1 n0);Intro y. +Elim y;Clear y;Intro y. +Unfold ge in H2;Generalize (le_not_lt n0 n1 (le_S_n n0 n1 H2));Intro; + ElimType False;Auto. +Rewrite y;Unfold Rge;Right;Trivial. +Unfold ge in H0;Generalize (H0 (S n0) H1 (lt_le_S n0 n1 y));Intro; + Unfold Un_growing in H1; + Apply (Rge_trans (Un (S n1)) (Un n1) (Un (S n0)) + (Rle_sym1 (Un n1) (Un (S n1)) (H1 n1)) H3). +Qed. + + +(* classical is needed: [not_all_not_ex] *) +(*********) +Lemma Un_cv_crit:Un_growing->(bound EUn)->(ExT [l:R] (Un_cv l)). +Unfold Un_growing Un_cv;Intros; + Generalize (complet_weak EUn H0 EUn_noempty);Intro; + Elim H1;Clear H1;Intros;Split with x;Intros; + Unfold is_lub in H1;Unfold bound in H0;Unfold is_upper_bound in H0 H1; + Elim H0;Clear H0;Intros;Elim H1;Clear H1;Intros; + Generalize (H3 x0 H0);Intro;Cut (n:nat)(Rle (Un n) x);Intro. +Cut (Ex [N:nat] (Rlt (Rminus x eps) (Un N))). +Intro;Elim H6;Clear H6;Intros;Split with x1. +Intros;Unfold R_dist;Apply (Rabsolu_def1 (Rminus (Un n) x) eps). +Unfold Rgt in H2; + Apply (Rle_lt_trans (Rminus (Un n) x) R0 eps + (Rle_minus (Un n) x (H5 n)) H2). +Fold Un_growing in H;Generalize (growing_prop n x1 H H7);Intro; + Generalize (Rlt_le_trans (Rminus x eps) (Un x1) (Un n) H6 + (Rle_sym2 (Un x1) (Un n) H8));Intro; + Generalize (Rlt_compatibility (Ropp x) (Rminus x eps) (Un n) H9); + Unfold Rminus;Rewrite <-(Rplus_assoc (Ropp x) x (Ropp eps)); + Rewrite (Rplus_sym (Ropp x) (Un n));Fold (Rminus (Un n) x); + Rewrite Rplus_Ropp_l;Rewrite (let (H1,H2)=(Rplus_ne (Ropp eps)) in H2); + Trivial. +Cut ~((N:nat)(Rge (Rminus x eps) (Un N))). +Intro;Apply (not_all_not_ex nat ([N:nat](Rlt (Rminus x eps) (Un N)))); + Red;Intro;Red in H6;Elim H6;Clear H6;Intro; + Apply (Rlt_not_ge (Rminus x eps) (Un N) (H7 N)). +Red;Intro;Cut (N:nat)(Rle (Un N) (Rminus x eps)). +Intro;Generalize (Un_bound_imp (Rminus x eps) H7);Intro; + Unfold is_upper_bound in H8;Generalize (H3 (Rminus x eps) H8);Intro; + Generalize (Rle_minus x (Rminus x eps) H9);Unfold Rminus; + Rewrite Ropp_distr1;Rewrite <- Rplus_assoc;Rewrite Rplus_Ropp_r; + Rewrite (let (H1,H2)=(Rplus_ne (Ropp (Ropp eps))) in H2); + Rewrite Ropp_Ropp;Intro;Unfold Rgt in H2; + Generalize (Rle_not eps R0 H2);Intro;Auto. +Intro;Elim (H6 N);Intro;Unfold Rle. +Left;Unfold Rgt in H7;Assumption. +Right;Auto. +Apply (H1 (Un n) (Un_in_EUn n)). +Qed. + +(*********) +Lemma finite_greater:(N:nat)(ExT [M:R] (n:nat)(le n N)->(Rle (Un n) M)). +Intro;Induction N. +Split with (Un O);Intros;Rewrite (le_n_O_eq n H); + Apply (eq_Rle (Un (n)) (Un (n)) (refl_eqT R (Un (n)))). +Elim HrecN;Clear HrecN;Intros;Split with (Rmax (Un (S N)) x);Intros; + Elim (Rmax_Rle (Un (S N)) x (Un n));Intros;Clear H1;Inversion H0. +Rewrite <-H1;Rewrite <-H1 in H2; + Apply (H2 (or_introl (Rle (Un n) (Un n)) (Rle (Un n) x) + (eq_Rle (Un n) (Un n) (refl_eqT R (Un n))))). +Apply (H2 (or_intror (Rle (Un n) (Un (S N))) (Rle (Un n) x) + (H n H3))). +Qed. + +(*********) +Lemma cauchy_bound:Cauchy_crit->(bound EUn). +Unfold Cauchy_crit bound;Intros;Unfold is_upper_bound; + Unfold Rgt in H;Elim (H R1 Rlt_R0_R1);Clear H;Intros; + Generalize (H x);Intro;Generalize (le_dec x);Intro; + Elim (finite_greater x);Intros;Split with (Rmax x0 (Rplus (Un x) R1)); + Clear H;Intros;Unfold EUn in H;Elim H;Clear H;Intros;Elim (H1 x2); + Clear H1;Intro y. +Unfold ge in H0;Generalize (H0 x2 (le_n x) y);Clear H0;Intro; + Rewrite <- H in H0;Unfold R_dist in H0; + Elim (Rabsolu_def2 (Rminus (Un x) x1) R1 H0);Clear H0;Intros; + Elim (Rmax_Rle x0 (Rplus (Un x) R1) x1);Intros;Apply H4;Clear H3 H4; + Right;Clear H H0 y;Apply (Rlt_le x1 (Rplus (Un x) R1)); + Generalize (Rlt_minus (Ropp R1) (Rminus (Un x) x1) H1);Clear H1; + Intro;Apply (Rminus_lt x1 (Rplus (Un x) R1)); + Cut (Rminus (Ropp R1) (Rminus (Un x) x1))== + (Rminus x1 (Rplus (Un x) R1));[Intro;Rewrite H0 in H;Assumption|Ring]. +Generalize (H2 x2 y);Clear H2 H0;Intro;Rewrite<-H in H0; + Elim (Rmax_Rle x0 (Rplus (Un x) R1) x1);Intros;Clear H1;Apply H2; + Left;Assumption. +Qed. + +End sequence. + +(*****************************************************************) +(* Definition of Power Series and properties *) +(* *) +(*****************************************************************) + +Section Isequence. + +(*********) +Variable An:nat->R. + +(*********) +Definition Pser:R->R->Prop:=[x,l:R] + (infinit_sum [n:nat](Rmult (An n) (pow x n)) l). + +End Isequence. + +Lemma GP_infinite: + (x:R) (Rlt (Rabsolu x) R1) + -> (Pser ([n:nat] R1) x (Rinv(Rminus R1 x))). +Intros;Unfold Pser; Unfold infinit_sum;Intros;Elim (Req_EM x R0). +Intros;Exists O; Intros;Rewrite H1;Rewrite minus_R0;Rewrite Rinv_R1; + Cut (sum_f_R0 [n0:nat](Rmult R1 (pow R0 n0)) n)==R1. +Intros; Rewrite H3;Rewrite R_dist_eq;Auto. +Elim n; Simpl. +Ring. +Intros;Rewrite H3;Ring. +Intro;Cut (Rlt R0 + (Rmult eps (Rmult (Rabsolu (Rminus R1 x)) + (Rabsolu (Rinv x))))). +Intro;Elim (pow_lt_1_zero x H + (Rmult eps (Rmult (Rabsolu (Rminus R1 x)) + (Rabsolu (Rinv x)))) + H2);Intro N; Intros;Exists N; Intros; + Cut (sum_f_R0 [n0:nat](Rmult R1 (pow x n0)) n)== + (sum_f_R0 [n0:nat](pow x n0) n). +Intros; Rewrite H5;Apply (Rlt_monotony_rev + (Rabsolu (Rminus R1 x)) + (R_dist (sum_f_R0 [n0:nat](pow x n0) n) + (Rinv (Rminus R1 x))) + eps). +Apply Rabsolu_pos_lt. +Apply Rminus_eq_contra. +Apply imp_not_Req. +Right; Unfold Rgt. +Apply (Rle_lt_trans x (Rabsolu x) R1). +Apply Rle_Rabsolu. +Assumption. +Unfold R_dist; Rewrite <- Rabsolu_mult. +Rewrite Rminus_distr. +Cut (Rmult (Rminus R1 x) (sum_f_R0 [n0:nat](pow x n0) n))== + (Ropp (Rmult(sum_f_R0 [n0:nat](pow x n0) n) + (Rminus x R1))). +Intro; Rewrite H6. +Rewrite GP_finite. +Rewrite Rinv_r. +Cut (Rminus (Ropp (Rminus (pow x (plus n (1))) R1)) R1)== + (Ropp (pow x (plus n (1)))). +Intro; Rewrite H7. +Rewrite Rabsolu_Ropp;Cut (plus n (S O))=(S n);Auto. +Intro H8;Rewrite H8;Simpl;Rewrite Rabsolu_mult; + Apply (Rlt_le_trans (Rmult (Rabsolu x) (Rabsolu (pow x n))) + (Rmult (Rabsolu x) + (Rmult eps + (Rmult (Rabsolu (Rminus R1 x)) + (Rabsolu (Rinv x))))) + (Rmult (Rabsolu (Rminus R1 x)) eps)). +Apply Rlt_monotony. +Apply Rabsolu_pos_lt. +Assumption. +Auto. +Cut (Rmult (Rabsolu x) + (Rmult eps (Rmult (Rabsolu (Rminus R1 x)) + (Rabsolu (Rinv x)))))== + (Rmult (Rmult (Rabsolu x) (Rabsolu (Rinv x))) + (Rmult eps (Rabsolu (Rminus R1 x)))). +Clear H8;Intros; Rewrite H8;Rewrite <- Rabsolu_mult;Rewrite Rinv_r. +Rewrite Rabsolu_R1;Cut (Rmult R1 (Rmult eps (Rabsolu (Rminus R1 x))))== + (Rmult (Rabsolu (Rminus R1 x)) eps). +Intros; Rewrite H9;Unfold Rle; Right; Reflexivity. +Ring. +Assumption. +Ring. +Ring. +Ring. +Apply Rminus_eq_contra. +Apply imp_not_Req. +Right; Unfold Rgt. +Apply (Rle_lt_trans x (Rabsolu x) R1). +Apply Rle_Rabsolu. +Assumption. +Ring; Ring. +Elim n; Simpl. +Ring. +Intros; Rewrite H5. +Ring. +Apply Rmult_lt_pos. +Auto. +Apply Rmult_lt_pos. +Apply Rabsolu_pos_lt. +Apply Rminus_eq_contra. +Apply imp_not_Req. +Right; Unfold Rgt. +Apply (Rle_lt_trans x (Rabsolu x) R1). +Apply Rle_Rabsolu. +Assumption. +Apply Rabsolu_pos_lt. +Apply Rinv_neq_R0. +Assumption. +Qed. diff --git a/theories7/Reals/Rsigma.v b/theories7/Reals/Rsigma.v new file mode 100644 index 00000000..f9e8e92b --- /dev/null +++ b/theories7/Reals/Rsigma.v @@ -0,0 +1,117 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rsigma.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require PartSum. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Set Implicit Arguments. + +Section Sigma. + +Variable f : nat->R. + +Definition sigma [low,high:nat] : R := (sum_f_R0 [k:nat](f (plus low k)) (minus high low)). + +Theorem sigma_split : (low,high,k:nat) (le low k)->(lt k high)->``(sigma low high)==(sigma low k)+(sigma (S k) high)``. +Intros; Induction k. +Cut low = O. +Intro; Rewrite H1; Unfold sigma; Rewrite <- minus_n_n; Rewrite <- minus_n_O; Simpl; Replace (minus high (S O)) with (pred high). +Apply (decomp_sum [k:nat](f k)). +Assumption. +Apply pred_of_minus. +Inversion H; Reflexivity. +Cut (le low k)\/low=(S k). +Intro; Elim H1; Intro. +Replace (sigma low (S k)) with ``(sigma low k)+(f (S k))``. +Rewrite Rplus_assoc; Replace ``(f (S k))+(sigma (S (S k)) high)`` with (sigma (S k) high). +Apply Hreck. +Assumption. +Apply lt_trans with (S k); [Apply lt_n_Sn | Assumption]. +Unfold sigma; Replace (minus high (S (S k))) with (pred (minus high (S k))). +Pattern 3 (S k); Replace (S k) with (plus (S k) O); [Idtac | Ring]. +Replace (sum_f_R0 [k0:nat](f (plus (S (S k)) k0)) (pred (minus high (S k)))) with (sum_f_R0 [k0:nat](f (plus (S k) (S k0))) (pred (minus high (S k)))). +Apply (decomp_sum [i:nat](f (plus (S k) i))). +Apply lt_minus_O_lt; Assumption. +Apply sum_eq; Intros; Replace (plus (S k) (S i)) with (plus (S (S k)) i). +Reflexivity. +Apply INR_eq; Do 2 Rewrite plus_INR; Do 3 Rewrite S_INR; Ring. +Replace (minus high (S (S k))) with (minus (minus high (S k)) (S O)). +Apply pred_of_minus. +Apply INR_eq; Repeat Rewrite minus_INR. +Do 4 Rewrite S_INR; Ring. +Apply lt_le_S; Assumption. +Apply lt_le_weak; Assumption. +Apply lt_le_S; Apply lt_minus_O_lt; Assumption. +Unfold sigma; Replace (minus (S k) low) with (S (minus k low)). +Pattern 1 (S k); Replace (S k) with (plus low (S (minus k low))). +Symmetry; Apply (tech5 [i:nat](f (plus low i))). +Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite minus_INR. +Ring. +Assumption. +Apply minus_Sn_m; Assumption. +Rewrite <- H2; Unfold sigma; Rewrite <- minus_n_n; Simpl; Replace (minus high (S low)) with (pred (minus high low)). +Replace (sum_f_R0 [k0:nat](f (S (plus low k0))) (pred (minus high low))) with (sum_f_R0 [k0:nat](f (plus low (S k0))) (pred (minus high low))). +Apply (decomp_sum [k0:nat](f (plus low k0))). +Apply lt_minus_O_lt. +Apply le_lt_trans with (S k); [Rewrite H2; Apply le_n | Assumption]. +Apply sum_eq; Intros; Replace (S (plus low i)) with (plus low (S i)). +Reflexivity. +Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. +Replace (minus high (S low)) with (minus (minus high low) (S O)). +Apply pred_of_minus. +Apply INR_eq; Repeat Rewrite minus_INR. +Do 2 Rewrite S_INR; Ring. +Apply lt_le_S; Rewrite H2; Assumption. +Rewrite H2; Apply lt_le_weak; Assumption. +Apply lt_le_S; Apply lt_minus_O_lt; Rewrite H2; Assumption. +Inversion H; [ + Right; Reflexivity +| Left; Assumption]. +Qed. + +Theorem sigma_diff : (low,high,k:nat) (le low k) -> (lt k high )->``(sigma low high)-(sigma low k)==(sigma (S k) high)``. +Intros low high k H1 H2; Symmetry; Rewrite -> (sigma_split H1 H2); Ring. +Qed. + +Theorem sigma_diff_neg : (low,high,k:nat) (le low k) -> (lt k high)-> ``(sigma low k)-(sigma low high)==-(sigma (S k) high)``. +Intros low high k H1 H2; Rewrite -> (sigma_split H1 H2); Ring. +Qed. + +Theorem sigma_first : (low,high:nat) (lt low high) -> ``(sigma low high)==(f low)+(sigma (S low) high)``. +Intros low high H1; Generalize (lt_le_S low high H1); Intro H2; Generalize (lt_le_weak low high H1); Intro H3; Replace ``(f low)`` with ``(sigma low low)``. +Apply sigma_split. +Apply le_n. +Assumption. +Unfold sigma; Rewrite <- minus_n_n. +Simpl. +Replace (plus low O) with low; [Reflexivity | Ring]. +Qed. + +Theorem sigma_last : (low,high:nat) (lt low high) -> ``(sigma low high)==(f high)+(sigma low (pred high))``. +Intros low high H1; Generalize (lt_le_S low high H1); Intro H2; Generalize (lt_le_weak low high H1); Intro H3; Replace ``(f high)`` with ``(sigma high high)``. +Rewrite Rplus_sym; Cut high = (S (pred high)). +Intro; Pattern 3 high; Rewrite H. +Apply sigma_split. +Apply le_S_n; Rewrite <- H; Apply lt_le_S; Assumption. +Apply lt_pred_n_n; Apply le_lt_trans with low; [Apply le_O_n | Assumption]. +Apply S_pred with O; Apply le_lt_trans with low; [Apply le_O_n | Assumption]. +Unfold sigma; Rewrite <- minus_n_n; Simpl; Replace (plus high O) with high; [Reflexivity | Ring]. +Qed. + +Theorem sigma_eq_arg : (low:nat) (sigma low low)==(f low). +Intro; Unfold sigma; Rewrite <- minus_n_n. +Simpl; Replace (plus low O) with low; [Reflexivity | Ring]. +Qed. + +End Sigma. diff --git a/theories7/Reals/Rsqrt_def.v b/theories7/Reals/Rsqrt_def.v new file mode 100644 index 00000000..17367dce --- /dev/null +++ b/theories7/Reals/Rsqrt_def.v @@ -0,0 +1,688 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rsqrt_def.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Sumbool. +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Ranalysis1. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Fixpoint Dichotomy_lb [x,y:R;P:R->bool;N:nat] : R := +Cases N of + O => x +| (S n) => let down = (Dichotomy_lb x y P n) in let up = (Dichotomy_ub x y P n) in let z = ``(down+up)/2`` in if (P z) then down else z +end +with Dichotomy_ub [x,y:R;P:R->bool;N:nat] : R := +Cases N of + O => y +| (S n) => let down = (Dichotomy_lb x y P n) in let up = (Dichotomy_ub x y P n) in let z = ``(down+up)/2`` in if (P z) then z else up +end. + +Definition dicho_lb [x,y:R;P:R->bool] : nat->R := [N:nat](Dichotomy_lb x y P N). +Definition dicho_up [x,y:R;P:R->bool] : nat->R := [N:nat](Dichotomy_ub x y P N). + +(**********) +Lemma dicho_comp : (x,y:R;P:R->bool;n:nat) ``x<=y`` -> ``(dicho_lb x y P n)<=(dicho_up x y P n)``. +Intros. +Induction n. +Simpl; Assumption. +Simpl. +Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). +Unfold Rdiv; Apply Rle_monotony_contra with ``2``. +Sup0. +Pattern 1 ``2``; Rewrite Rmult_sym. +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. +Rewrite Rmult_1r. +Rewrite double. +Apply Rle_compatibility. +Assumption. +Unfold Rdiv; Apply Rle_monotony_contra with ``2``. +Sup0. +Pattern 3 ``2``; Rewrite Rmult_sym. +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. +Rewrite Rmult_1r. +Rewrite double. +Rewrite <- (Rplus_sym (Dichotomy_ub x y P n)). +Apply Rle_compatibility. +Assumption. +Qed. + +Lemma dicho_lb_growing : (x,y:R;P:R->bool) ``x<=y`` -> (Un_growing (dicho_lb x y P)). +Intros. +Unfold Un_growing. +Intro. +Simpl. +Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). +Right; Reflexivity. +Unfold Rdiv; Apply Rle_monotony_contra with ``2``. +Sup0. +Pattern 1 ``2``; Rewrite Rmult_sym. +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. +Rewrite Rmult_1r. +Rewrite double. +Apply Rle_compatibility. +Replace (Dichotomy_ub x y P n) with (dicho_up x y P n); [Apply dicho_comp; Assumption | Reflexivity]. +Qed. + +Lemma dicho_up_decreasing : (x,y:R;P:R->bool) ``x<=y`` -> (Un_decreasing (dicho_up x y P)). +Intros. +Unfold Un_decreasing. +Intro. +Simpl. +Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). +Unfold Rdiv; Apply Rle_monotony_contra with ``2``. +Sup0. +Pattern 3 ``2``; Rewrite Rmult_sym. +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. +Rewrite Rmult_1r. +Rewrite double. +Replace (Dichotomy_ub x y P n) with (dicho_up x y P n); [Idtac | Reflexivity]. +Replace (Dichotomy_lb x y P n) with (dicho_lb x y P n); [Idtac | Reflexivity]. +Rewrite <- (Rplus_sym ``(dicho_up x y P n)``). +Apply Rle_compatibility. +Apply dicho_comp; Assumption. +Right; Reflexivity. +Qed. + +Lemma dicho_lb_maj_y : (x,y:R;P:R->bool) ``x<=y`` -> (n:nat)``(dicho_lb x y P n)<=y``. +Intros. +Induction n. +Simpl; Assumption. +Simpl. +Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). +Assumption. +Unfold Rdiv; Apply Rle_monotony_contra with ``2``. +Sup0. +Pattern 3 ``2``; Rewrite Rmult_sym. +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1r | DiscrR]. +Rewrite double; Apply Rplus_le. +Assumption. +Pattern 2 y; Replace y with (Dichotomy_ub x y P O); [Idtac | Reflexivity]. +Apply decreasing_prop. +Assert H0 := (dicho_up_decreasing x y P H). +Assumption. +Apply le_O_n. +Qed. + +Lemma dicho_lb_maj : (x,y:R;P:R->bool) ``x<=y`` -> (has_ub (dicho_lb x y P)). +Intros. +Cut (n:nat)``(dicho_lb x y P n)<=y``. +Intro. +Unfold has_ub. +Unfold bound. +Exists y. +Unfold is_upper_bound. +Intros. +Elim H1; Intros. +Rewrite H2; Apply H0. +Apply dicho_lb_maj_y; Assumption. +Qed. + +Lemma dicho_up_min_x : (x,y:R;P:R->bool) ``x<=y`` -> (n:nat)``x<=(dicho_up x y P n)``. +Intros. +Induction n. +Simpl; Assumption. +Simpl. +Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). +Unfold Rdiv; Apply Rle_monotony_contra with ``2``. +Sup0. +Pattern 1 ``2``; Rewrite Rmult_sym. +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1r | DiscrR]. +Rewrite double; Apply Rplus_le. +Pattern 1 x; Replace x with (Dichotomy_lb x y P O); [Idtac | Reflexivity]. +Apply tech9. +Assert H0 := (dicho_lb_growing x y P H). +Assumption. +Apply le_O_n. +Assumption. +Assumption. +Qed. + +Lemma dicho_up_min : (x,y:R;P:R->bool) ``x<=y`` -> (has_lb (dicho_up x y P)). +Intros. +Cut (n:nat)``x<=(dicho_up x y P n)``. +Intro. +Unfold has_lb. +Unfold bound. +Exists ``-x``. +Unfold is_upper_bound. +Intros. +Elim H1; Intros. +Rewrite H2. +Unfold opp_seq. +Apply Rle_Ropp1. +Apply H0. +Apply dicho_up_min_x; Assumption. +Qed. + +Lemma dicho_lb_cv : (x,y:R;P:R->bool) ``x<=y`` -> (sigTT R [l:R](Un_cv (dicho_lb x y P) l)). +Intros. +Apply growing_cv. +Apply dicho_lb_growing; Assumption. +Apply dicho_lb_maj; Assumption. +Qed. + +Lemma dicho_up_cv : (x,y:R;P:R->bool) ``x<=y`` -> (sigTT R [l:R](Un_cv (dicho_up x y P) l)). +Intros. +Apply decreasing_cv. +Apply dicho_up_decreasing; Assumption. +Apply dicho_up_min; Assumption. +Qed. + +Lemma dicho_lb_dicho_up : (x,y:R;P:R->bool;n:nat) ``x<=y`` -> ``(dicho_up x y P n)-(dicho_lb x y P n)==(y-x)/(pow 2 n)``. +Intros. +Induction n. +Simpl. +Unfold Rdiv; Rewrite Rinv_R1; Ring. +Simpl. +Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). +Unfold Rdiv. +Replace ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))*/2- + (Dichotomy_lb x y P n)`` with ``((dicho_up x y P n)-(dicho_lb x y P n))/2``. +Unfold Rdiv; Rewrite Hrecn. +Unfold Rdiv. +Rewrite Rinv_Rmult. +Ring. +DiscrR. +Apply pow_nonzero; DiscrR. +Pattern 2 (Dichotomy_lb x y P n); Rewrite (double_var (Dichotomy_lb x y P n)); Unfold dicho_up dicho_lb Rminus Rdiv; Ring. +Replace ``(Dichotomy_ub x y P n)-((Dichotomy_lb x y P n)+ + (Dichotomy_ub x y P n))/2`` with ``((dicho_up x y P n)-(dicho_lb x y P n))/2``. +Unfold Rdiv; Rewrite Hrecn. +Unfold Rdiv. +Rewrite Rinv_Rmult. +Ring. +DiscrR. +Apply pow_nonzero; DiscrR. +Pattern 1 (Dichotomy_ub x y P n); Rewrite (double_var (Dichotomy_ub x y P n)); Unfold dicho_up dicho_lb Rminus Rdiv; Ring. +Qed. + +Definition pow_2_n := [n:nat](pow ``2`` n). + +Lemma pow_2_n_neq_R0 : (n:nat) ``(pow_2_n n)<>0``. +Intro. +Unfold pow_2_n. +Apply pow_nonzero. +DiscrR. +Qed. + +Lemma pow_2_n_growing : (Un_growing pow_2_n). +Unfold Un_growing. +Intro. +Replace (S n) with (plus n (1)); [Unfold pow_2_n; Rewrite pow_add | Ring]. +Pattern 1 (pow ``2`` n); Rewrite <- Rmult_1r. +Apply Rle_monotony. +Left; Apply pow_lt; Sup0. +Simpl. +Rewrite Rmult_1r. +Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply Rlt_R0_R1. +Qed. + +Lemma pow_2_n_infty : (cv_infty pow_2_n). +Cut (N:nat)``(INR N)<=(pow 2 N)``. +Intros. +Unfold cv_infty. +Intro. +Case (total_order_T R0 M); Intro. +Elim s; Intro. +Pose N := (up M). +Cut `0<=N`. +Intro. +Elim (IZN N H0); Intros N0 H1. +Exists N0. +Intros. +Apply Rlt_le_trans with (INR N0). +Rewrite INR_IZR_INZ. +Rewrite <- H1. +Unfold N. +Assert H3 := (archimed M). +Elim H3; Intros; Assumption. +Apply Rle_trans with (pow_2_n N0). +Unfold pow_2_n; Apply H. +Apply Rle_sym2. +Apply growing_prop. +Apply pow_2_n_growing. +Assumption. +Apply le_IZR. +Unfold N. +Simpl. +Assert H0 := (archimed M); Elim H0; Intros. +Left; Apply Rlt_trans with M; Assumption. +Exists O; Intros. +Rewrite <- b. +Unfold pow_2_n; Apply pow_lt; Sup0. +Exists O; Intros. +Apply Rlt_trans with R0. +Assumption. +Unfold pow_2_n; Apply pow_lt; Sup0. +Induction N. +Simpl. +Left; Apply Rlt_R0_R1. +Intros. +Pattern 2 (S n); Replace (S n) with (plus n (1)); [Idtac | Ring]. +Rewrite S_INR; Rewrite pow_add. +Simpl. +Rewrite Rmult_1r. +Apply Rle_trans with ``(pow 2 n)``. +Rewrite <- (Rplus_sym R1). +Rewrite <- (Rmult_1r (INR n)). +Apply (poly n R1). +Apply Rlt_R0_R1. +Pattern 1 (pow ``2`` n); Rewrite <- Rplus_Or. +Rewrite <- (Rmult_sym ``2``). +Rewrite double. +Apply Rle_compatibility. +Left; Apply pow_lt; Sup0. +Qed. + +Lemma cv_dicho : (x,y,l1,l2:R;P:R->bool) ``x<=y`` -> (Un_cv (dicho_lb x y P) l1) -> (Un_cv (dicho_up x y P) l2) -> l1==l2. +Intros. +Assert H2 := (CV_minus ? ? ? ? H0 H1). +Cut (Un_cv [i:nat]``(dicho_lb x y P i)-(dicho_up x y P i)`` R0). +Intro. +Assert H4 := (UL_sequence ? ? ? H2 H3). +Symmetry; Apply Rminus_eq_right; Assumption. +Unfold Un_cv; Unfold R_dist. +Intros. +Assert H4 := (cv_infty_cv_R0 pow_2_n pow_2_n_neq_R0 pow_2_n_infty). +Case (total_order_T x y); Intro. +Elim s; Intro. +Unfold Un_cv in H4; Unfold R_dist in H4. +Cut ``0<y-x``. +Intro Hyp. +Cut ``0<eps/(y-x)``. +Intro. +Elim (H4 ``eps/(y-x)`` H5); Intros N H6. +Exists N; Intros. +Replace ``(dicho_lb x y P n)-(dicho_up x y P n)-0`` with ``(dicho_lb x y P n)-(dicho_up x y P n)``; [Idtac | Ring]. +Rewrite <- Rabsolu_Ropp. +Rewrite Ropp_distr3. +Rewrite dicho_lb_dicho_up. +Unfold Rdiv; Rewrite Rabsolu_mult. +Rewrite (Rabsolu_right ``y-x``). +Apply Rlt_monotony_contra with ``/(y-x)``. +Apply Rlt_Rinv; Assumption. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Replace ``/(pow 2 n)`` with ``/(pow 2 n)-0``; [Unfold pow_2_n Rdiv in H6; Rewrite <- (Rmult_sym eps); Apply H6; Assumption | Ring]. +Red; Intro; Rewrite H8 in Hyp; Elim (Rlt_antirefl ? Hyp). +Apply Rle_sym1. +Apply Rle_anti_compatibility with x; Rewrite Rplus_Or. +Replace ``x+(y-x)`` with y; [Assumption | Ring]. +Assumption. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Assumption]. +Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or. +Replace ``x+(y-x)`` with y; [Assumption | Ring]. +Exists O; Intros. +Replace ``(dicho_lb x y P n)-(dicho_up x y P n)-0`` with ``(dicho_lb x y P n)-(dicho_up x y P n)``; [Idtac | Ring]. +Rewrite <- Rabsolu_Ropp. +Rewrite Ropp_distr3. +Rewrite dicho_lb_dicho_up. +Rewrite b. +Unfold Rminus Rdiv; Rewrite Rplus_Ropp_r; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Assumption. +Assumption. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Qed. + +Definition cond_positivity [x:R] : bool := Cases (total_order_Rle R0 x) of + (leftT _) => true +| (rightT _) => false end. + +(* Sequential caracterisation of continuity *) +Lemma continuity_seq : (f:R->R;Un:nat->R;l:R) (continuity_pt f l) -> (Un_cv Un l) -> (Un_cv [i:nat](f (Un i)) (f l)). +Unfold continuity_pt Un_cv; Unfold continue_in. +Unfold limit1_in. +Unfold limit_in. +Unfold dist. +Simpl. +Unfold R_dist. +Intros. +Elim (H eps H1); Intros alp H2. +Elim H2; Intros. +Elim (H0 alp H3); Intros N H5. +Exists N; Intros. +Case (Req_EM (Un n) l); Intro. +Rewrite H7; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Apply H4. +Split. +Unfold D_x no_cond. +Split. +Trivial. +Apply not_sym; Assumption. +Apply H5; Assumption. +Qed. + +Lemma dicho_lb_car : (x,y:R;P:R->bool;n:nat) (P x)=false -> (P (dicho_lb x y P n))=false. +Intros. +Induction n. +Simpl. +Assumption. +Simpl. +Assert X := (sumbool_of_bool (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``)). +Elim X; Intro. +Rewrite a. +Unfold dicho_lb in Hrecn; Assumption. +Rewrite b. +Assumption. +Qed. + +Lemma dicho_up_car : (x,y:R;P:R->bool;n:nat) (P y)=true -> (P (dicho_up x y P n))=true. +Intros. +Induction n. +Simpl. +Assumption. +Simpl. +Assert X := (sumbool_of_bool (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``)). +Elim X; Intro. +Rewrite a. +Unfold dicho_lb in Hrecn; Assumption. +Rewrite b. +Assumption. +Qed. + +(* Intermediate Value Theorem *) +Lemma IVT : (f:R->R;x,y:R) (continuity f) -> ``x<y`` -> ``(f x)<0`` -> ``0<(f y)`` -> (sigTT R [z:R]``x<=z<=y``/\``(f z)==0``). +Intros. +Cut ``x<=y``. +Intro. +Generalize (dicho_lb_cv x y [z:R](cond_positivity (f z)) H3). +Generalize (dicho_up_cv x y [z:R](cond_positivity (f z)) H3). +Intros. +Elim X; Intros. +Elim X0; Intros. +Assert H4 := (cv_dicho ? ? ? ? ? H3 p0 p). +Rewrite H4 in p0. +Apply existTT with x0. +Split. +Split. +Apply Rle_trans with (dicho_lb x y [z:R](cond_positivity (f z)) O). +Simpl. +Right; Reflexivity. +Apply growing_ineq. +Apply dicho_lb_growing; Assumption. +Assumption. +Apply Rle_trans with (dicho_up x y [z:R](cond_positivity (f z)) O). +Apply decreasing_ineq. +Apply dicho_up_decreasing; Assumption. +Assumption. +Right; Reflexivity. +2:Left; Assumption. +Pose Vn := [n:nat](dicho_lb x y [z:R](cond_positivity (f z)) n). +Pose Wn := [n:nat](dicho_up x y [z:R](cond_positivity (f z)) n). +Cut ((n:nat)``(f (Vn n))<=0``)->``(f x0)<=0``. +Cut ((n:nat)``0<=(f (Wn n))``)->``0<=(f x0)``. +Intros. +Cut (n:nat)``(f (Vn n))<=0``. +Cut (n:nat)``0<=(f (Wn n))``. +Intros. +Assert H9 := (H6 H8). +Assert H10 := (H5 H7). +Apply Rle_antisym; Assumption. +Intro. +Unfold Wn. +Cut (z:R) (cond_positivity z)=true <-> ``0<=z``. +Intro. +Assert H8 := (dicho_up_car x y [z:R](cond_positivity (f z)) n). +Elim (H7 (f (dicho_up x y [z:R](cond_positivity (f z)) n))); Intros. +Apply H9. +Apply H8. +Elim (H7 (f y)); Intros. +Apply H12. +Left; Assumption. +Intro. +Unfold cond_positivity. +Case (total_order_Rle R0 z); Intro. +Split. +Intro; Assumption. +Intro; Reflexivity. +Split. +Intro; Elim diff_false_true; Assumption. +Intro. +Elim n0; Assumption. +Unfold Vn. +Cut (z:R) (cond_positivity z)=false <-> ``z<0``. +Intros. +Assert H8 := (dicho_lb_car x y [z:R](cond_positivity (f z)) n). +Left. +Elim (H7 (f (dicho_lb x y [z:R](cond_positivity (f z)) n))); Intros. +Apply H9. +Apply H8. +Elim (H7 (f x)); Intros. +Apply H12. +Assumption. +Intro. +Unfold cond_positivity. +Case (total_order_Rle R0 z); Intro. +Split. +Intro; Elim diff_true_false; Assumption. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H7)). +Split. +Intro; Auto with real. +Intro; Reflexivity. +Cut (Un_cv Wn x0). +Intros. +Assert H7 := (continuity_seq f Wn x0 (H x0) H5). +Case (total_order_T R0 (f x0)); Intro. +Elim s; Intro. +Left; Assumption. +Rewrite <- b; Right; Reflexivity. +Unfold Un_cv in H7; Unfold R_dist in H7. +Cut ``0< -(f x0)``. +Intro. +Elim (H7 ``-(f x0)`` H8); Intros. +Cut (ge x2 x2); [Intro | Unfold ge; Apply le_n]. +Assert H11 := (H9 x2 H10). +Rewrite Rabsolu_right in H11. +Pattern 1 ``-(f x0)`` in H11; Rewrite <- Rplus_Or in H11. +Unfold Rminus in H11; Rewrite (Rplus_sym (f (Wn x2))) in H11. +Assert H12 := (Rlt_anti_compatibility ? ? ? H11). +Assert H13 := (H6 x2). +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H13 H12)). +Apply Rle_sym1; Left; Unfold Rminus; Apply ge0_plus_gt0_is_gt0. +Apply H6. +Exact H8. +Apply Rgt_RO_Ropp; Assumption. +Unfold Wn; Assumption. +Cut (Un_cv Vn x0). +Intros. +Assert H7 := (continuity_seq f Vn x0 (H x0) H5). +Case (total_order_T R0 (f x0)); Intro. +Elim s; Intro. +Unfold Un_cv in H7; Unfold R_dist in H7. +Elim (H7 ``(f x0)`` a); Intros. +Cut (ge x2 x2); [Intro | Unfold ge; Apply le_n]. +Assert H10 := (H8 x2 H9). +Rewrite Rabsolu_left in H10. +Pattern 2 ``(f x0)`` in H10; Rewrite <- Rplus_Or in H10. +Rewrite Ropp_distr3 in H10. +Unfold Rminus in H10. +Assert H11 := (Rlt_anti_compatibility ? ? ? H10). +Assert H12 := (H6 x2). +Cut ``0<(f (Vn x2))``. +Intro. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H13 H12)). +Rewrite <- (Ropp_Ropp (f (Vn x2))). +Apply Rgt_RO_Ropp; Assumption. +Apply Rlt_anti_compatibility with ``(f x0)-(f (Vn x2))``. +Rewrite Rplus_Or; Replace ``(f x0)-(f (Vn x2))+((f (Vn x2))-(f x0))`` with R0; [Unfold Rminus; Apply gt0_plus_ge0_is_gt0 | Ring]. +Assumption. +Apply Rge_RO_Ropp; Apply Rle_sym1; Apply H6. +Right; Rewrite <- b; Reflexivity. +Left; Assumption. +Unfold Vn; Assumption. +Qed. + +Lemma IVT_cor : (f:R->R;x,y:R) (continuity f) -> ``x<=y`` -> ``(f x)*(f y)<=0`` -> (sigTT R [z:R]``x<=z<=y``/\``(f z)==0``). +Intros. +Case (total_order_T R0 (f x)); Intro. +Case (total_order_T R0 (f y)); Intro. +Elim s; Intro. +Elim s0; Intro. +Cut ``0<(f x)*(f y)``; [Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 H2)) | Apply Rmult_lt_pos; Assumption]. +Exists y. +Split. +Split; [Assumption | Right; Reflexivity]. +Symmetry; Exact b. +Exists x. +Split. +Split; [Right; Reflexivity | Assumption]. +Symmetry; Exact b. +Elim s; Intro. +Cut ``x<y``. +Intro. +Assert H3 := (IVT (opp_fct f) x y (continuity_opp f H) H2). +Cut ``(opp_fct f x)<0``. +Cut ``0<(opp_fct f y)``. +Intros. +Elim (H3 H5 H4); Intros. +Apply existTT with x0. +Elim p; Intros. +Split. +Assumption. +Unfold opp_fct in H7. +Rewrite <- (Ropp_Ropp (f x0)). +Apply eq_RoppO; Assumption. +Unfold opp_fct; Apply Rgt_RO_Ropp; Assumption. +Unfold opp_fct. +Apply Rlt_anti_compatibility with (f x); Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Assumption. +Inversion H0. +Assumption. +Rewrite H2 in a. +Elim (Rlt_antirefl ? (Rlt_trans ? ? ? r a)). +Apply existTT with x. +Split. +Split; [Right; Reflexivity | Assumption]. +Symmetry; Assumption. +Case (total_order_T R0 (f y)); Intro. +Elim s; Intro. +Cut ``x<y``. +Intro. +Apply IVT; Assumption. +Inversion H0. +Assumption. +Rewrite H2 in r. +Elim (Rlt_antirefl ? (Rlt_trans ? ? ? r a)). +Apply existTT with y. +Split. +Split; [Assumption | Right; Reflexivity]. +Symmetry; Assumption. +Cut ``0<(f x)*(f y)``. +Intro. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H2 H1)). +Rewrite <- Ropp_mul2; Apply Rmult_lt_pos; Apply Rgt_RO_Ropp; Assumption. +Qed. + +(* We can now define the square root function as the reciprocal transformation of the square root function *) +Lemma Rsqrt_exists : (y:R) ``0<=y`` -> (sigTT R [z:R]``0<=z``/\``y==(Rsqr z)``). +Intros. +Pose f := [x:R]``(Rsqr x)-y``. +Cut ``(f 0)<=0``. +Intro. +Cut (continuity f). +Intro. +Case (total_order_T y R1); Intro. +Elim s; Intro. +Cut ``0<=(f 1)``. +Intro. +Cut ``(f 0)*(f 1)<=0``. +Intro. +Assert X := (IVT_cor f R0 R1 H1 (Rlt_le ? ? Rlt_R0_R1) H3). +Elim X; Intros t H4. +Apply existTT with t. +Elim H4; Intros. +Split. +Elim H5; Intros; Assumption. +Unfold f in H6. +Apply Rminus_eq_right; Exact H6. +Rewrite Rmult_sym; Pattern 2 R0; Rewrite <- (Rmult_Or (f R1)). +Apply Rle_monotony; Assumption. +Unfold f. +Rewrite Rsqr_1. +Apply Rle_anti_compatibility with y. +Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Left; Assumption. +Apply existTT with R1. +Split. +Left; Apply Rlt_R0_R1. +Rewrite b; Symmetry; Apply Rsqr_1. +Cut ``0<=(f y)``. +Intro. +Cut ``(f 0)*(f y)<=0``. +Intro. +Assert X := (IVT_cor f R0 y H1 H H3). +Elim X; Intros t H4. +Apply existTT with t. +Elim H4; Intros. +Split. +Elim H5; Intros; Assumption. +Unfold f in H6. +Apply Rminus_eq_right; Exact H6. +Rewrite Rmult_sym; Pattern 2 R0; Rewrite <- (Rmult_Or (f y)). +Apply Rle_monotony; Assumption. +Unfold f. +Apply Rle_anti_compatibility with y. +Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. +Pattern 1 y; Rewrite <- Rmult_1r. +Unfold Rsqr; Apply Rle_monotony. +Assumption. +Left; Exact r. +Replace f with (minus_fct Rsqr (fct_cte y)). +Apply continuity_minus. +Apply derivable_continuous; Apply derivable_Rsqr. +Apply derivable_continuous; Apply derivable_const. +Reflexivity. +Unfold f; Rewrite Rsqr_O. +Unfold Rminus; Rewrite Rplus_Ol. +Apply Rle_sym2. +Apply Rle_RO_Ropp; Assumption. +Qed. + +(* Definition of the square root: R+->R *) +Definition Rsqrt [y:nonnegreal] : R := Cases (Rsqrt_exists (nonneg y) (cond_nonneg y)) of (existTT a b) => a end. + +(**********) +Lemma Rsqrt_positivity : (x:nonnegreal) ``0<=(Rsqrt x)``. +Intro. +Assert X := (Rsqrt_exists (nonneg x) (cond_nonneg x)). +Elim X; Intros. +Cut x0==(Rsqrt x). +Intros. +Elim p; Intros. +Rewrite H in H0; Assumption. +Unfold Rsqrt. +Case (Rsqrt_exists x (cond_nonneg x)). +Intros. +Elim p; Elim a; Intros. +Apply Rsqr_inj. +Assumption. +Assumption. +Rewrite <- H0; Rewrite <- H2; Reflexivity. +Qed. + +(**********) +Lemma Rsqrt_Rsqrt : (x:nonnegreal) ``(Rsqrt x)*(Rsqrt x)==x``. +Intros. +Assert X := (Rsqrt_exists (nonneg x) (cond_nonneg x)). +Elim X; Intros. +Cut x0==(Rsqrt x). +Intros. +Rewrite <- H. +Elim p; Intros. +Rewrite H1; Reflexivity. +Unfold Rsqrt. +Case (Rsqrt_exists x (cond_nonneg x)). +Intros. +Elim p; Elim a; Intros. +Apply Rsqr_inj. +Assumption. +Assumption. +Rewrite <- H0; Rewrite <- H2; Reflexivity. +Qed. diff --git a/theories7/Reals/Rsyntax.v b/theories7/Reals/Rsyntax.v new file mode 100644 index 00000000..7b1b6266 --- /dev/null +++ b/theories7/Reals/Rsyntax.v @@ -0,0 +1,236 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Rsyntax.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Export Rdefinitions. + +Axiom NRplus : R->R. +Axiom NRmult : R->R. + +V7only[ +Grammar rnatural ident := + nat_id [ prim:var($id) ] -> [$id] + +with rnegnumber : constr := + neg_expr [ "-" rnumber ($c) ] -> [ (Ropp $c) ] + +with rnumber := + +with rformula : constr := + form_expr [ rexpr($p) ] -> [ $p ] +(* | form_eq [ rexpr($p) "==" rexpr($c) ] -> [ (eqT R $p $c) ] *) +| form_eq [ rexpr($p) "==" rexpr($c) ] -> [ (eqT ? $p $c) ] +| form_eq2 [ rexpr($p) "=" rexpr($c) ] -> [ (eqT ? $p $c) ] +| form_le [ rexpr($p) "<=" rexpr($c) ] -> [ (Rle $p $c) ] +| form_lt [ rexpr($p) "<" rexpr($c) ] -> [ (Rlt $p $c) ] +| form_ge [ rexpr($p) ">=" rexpr($c) ] -> [ (Rge $p $c) ] +| form_gt [ rexpr($p) ">" rexpr($c) ] -> [ (Rgt $p $c) ] +(* +| form_eq_eq [ rexpr($p) "==" rexpr($c) "==" rexpr($c1) ] + -> [ (eqT R $p $c)/\(eqT R $c $c1) ] +*) +| form_eq_eq [ rexpr($p) "==" rexpr($c) "==" rexpr($c1) ] + -> [ (eqT ? $p $c)/\(eqT ? $c $c1) ] +| form_le_le [ rexpr($p) "<=" rexpr($c) "<=" rexpr($c1) ] + -> [ (Rle $p $c)/\(Rle $c $c1) ] +| form_le_lt [ rexpr($p) "<=" rexpr($c) "<" rexpr($c1) ] + -> [ (Rle $p $c)/\(Rlt $c $c1) ] +| form_lt_le [ rexpr($p) "<" rexpr($c) "<=" rexpr($c1) ] + -> [ (Rlt $p $c)/\(Rle $c $c1) ] +| form_lt_lt [ rexpr($p) "<" rexpr($c) "<" rexpr($c1) ] + -> [ (Rlt $p $c)/\(Rlt $c $c1) ] +| form_neq [ rexpr($p) "<>" rexpr($c) ] -> [ ~(eqT ? $p $c) ] + +with rexpr : constr := + expr_plus [ rexpr($p) "+" rexpr($c) ] -> [ (Rplus $p $c) ] +| expr_minus [ rexpr($p) "-" rexpr($c) ] -> [ (Rminus $p $c) ] +| rexpr2 [ rexpr2($e) ] -> [ $e ] + +with rexpr2 : constr := + expr_mult [ rexpr2($p) "*" rexpr2($c) ] -> [ (Rmult $p $c) ] +| rexpr0 [ rexpr0($e) ] -> [ $e ] + + +with rexpr0 : constr := + expr_id [ constr:global($c) ] -> [ $c ] +| expr_com [ "[" constr:constr($c) "]" ] -> [ $c ] +| expr_appl [ "(" rapplication($a) ")" ] -> [ $a ] +| expr_num [ rnumber($s) ] -> [ $s ] +| expr_negnum [ "-" rnegnumber($n) ] -> [ $n ] +| expr_div [ rexpr0($p) "/" rexpr0($c) ] -> [ (Rdiv $p $c) ] +| expr_opp [ "-" rexpr0($c) ] -> [ (Ropp $c) ] +| expr_inv [ "/" rexpr0($c) ] -> [ (Rinv $c) ] +| expr_meta [ meta($m) ] -> [ $m ] + +with meta := +| rimpl [ "?" ] -> [ ? ] +| rmeta0 [ "?" "0" ] -> [ ?0 ] +| rmeta1 [ "?" "1" ] -> [ ?1 ] +| rmeta2 [ "?" "2" ] -> [ ?2 ] +| rmeta3 [ "?" "3" ] -> [ ?3 ] +| rmeta4 [ "?" "4" ] -> [ ?4 ] +| rmeta5 [ "?" "5" ] -> [ ?5 ] + +with rapplication : constr := + apply [ rapplication($p) rexpr($c1) ] -> [ ($p $c1) ] +| pair [ rexpr($p) "," rexpr($c) ] -> [ ($p, $c) ] +| appl0 [ rexpr($a) ] -> [ $a ]. + +Grammar constr constr0 := + r_in_com [ "``" rnatural:rformula($c) "``" ] -> [ $c ]. + +Grammar constr atomic_pattern := + r_in_pattern [ "``" rnatural:rnumber($c) "``" ] -> [ $c ]. + +(*i* pp **) + +Syntax constr + level 0: + Rle [ (Rle $n1 $n2) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "<= " (REXPR $n2) "``"]] + | Rlt [ (Rlt $n1 $n2) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "< "(REXPR $n2) "``" ]] + | Rge [ (Rge $n1 $n2) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] ">= "(REXPR $n2) "``" ]] + | Rgt [ (Rgt $n1 $n2) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "> "(REXPR $n2) "``" ]] + | Req [ (eqT R $n1 $n2) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "= "(REXPR $n2)"``"]] + | Rneq [ ~(eqT R $n1 $n2) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "<> "(REXPR $n2) "``"]] + | Rle_Rle [ (Rle $n1 $n2)/\(Rle $n2 $n3) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "<= " (REXPR $n2) + [1 0] "<= " (REXPR $n3) "``"]] + | Rle_Rlt [ (Rle $n1 $n2)/\(Rlt $n2 $n3) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "<= "(REXPR $n2) + [1 0] "< " (REXPR $n3) "``"]] + | Rlt_Rle [ (Rlt $n1 $n2)/\(Rle $n2 $n3) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "< " (REXPR $n2) + [1 0] "<= " (REXPR $n3) "``"]] + | Rlt_Rlt [ (Rlt $n1 $n2)/\(Rlt $n2 $n3) ] -> + [[<hov 0> "``" (REXPR $n1) [1 0] "< " (REXPR $n2) + [1 0] "< " (REXPR $n3) "``"]] + | Rzero [ R0 ] -> [ "``0``" ] + | Rone [ R1 ] -> [ "``1``" ] + ; + + level 7: + Rplus [ (Rplus $n1 $n2) ] + -> [ [<hov 0> "``"(REXPR $n1):E "+" [0 0] (REXPR $n2):L "``"] ] + | Rodd_outside [(Rplus R1 $r)] -> [ $r:"r_printer_odd_outside"] + | Rminus [ (Rminus $n1 $n2) ] + -> [ [<hov 0> "``"(REXPR $n1):E "-" [0 0] (REXPR $n2):L "``"] ] + ; + + level 6: + Rmult [ (Rmult $n1 $n2) ] + -> [ [<hov 0> "``"(REXPR $n1):E "*" [0 0] (REXPR $n2):L "``"] ] + | Reven_outside [ (Rmult (Rplus R1 R1) $r) ] -> [ $r:"r_printer_even_outside"] + | Rdiv [ (Rdiv $n1 $n2) ] + -> [ [<hov 0> "``"(REXPR $n1):E "/" [0 0] (REXPR $n2):L "``"] ] + ; + + level 8: + Ropp [(Ropp $n1)] -> [ [<hov 0> "``" "-"(REXPR $n1):E "``"] ] + | Rinv [(Rinv $n1)] -> [ [<hov 0> "``" "/"(REXPR $n1):E "``"] ] + ; + + level 0: + rescape_inside [<< (REXPR $r) >>] -> [ "[" $r:E "]" ] + ; + + level 4: + Rappl_inside [<<(REXPR (APPLIST $h ($LIST $t)))>>] + -> [ [<hov 0> "("(REXPR $h):E [1 0] (RAPPLINSIDETAIL ($LIST $t)):E ")"] ] + | Rappl_inside_tail [<<(RAPPLINSIDETAIL $h ($LIST $t))>>] + -> [(REXPR $h):E [1 0] (RAPPLINSIDETAIL ($LIST $t)):E] + | Rappl_inside_one [<<(RAPPLINSIDETAIL $e)>>] ->[(REXPR $e):E] + | rpair_inside [<<(REXPR <<(pair $s1 $s2 $r1 $r2)>>)>>] + -> [ [<hov 0> "("(REXPR $r1):E "," [1 0] (REXPR $r2):E ")"] ] + ; + + level 3: + rvar_inside [<<(REXPR ($VAR $i))>>] -> [$i] + | rsecvar_inside [<<(REXPR (SECVAR $i))>>] -> [(SECVAR $i)] + | rconst_inside [<<(REXPR (CONST $c))>>] -> [(CONST $c)] + | rmutind_inside [<<(REXPR (MUTIND $i $n))>>] + -> [(MUTIND $i $n)] + | rmutconstruct_inside [<<(REXPR (MUTCONSTRUCT $c1 $c2 $c3))>>] + -> [ (MUTCONSTRUCT $c1 $c2 $c3) ] + | rimplicit_head_inside [<<(REXPR (XTRA "!" $c))>>] -> [ $c ] + | rimplicit_arg_inside [<<(REXPR (XTRA "!" $n $c))>>] -> [ ] + + ; + + + level 7: + Rplus_inside + [<<(REXPR <<(Rplus $n1 $n2)>>)>>] + -> [ (REXPR $n1):E "+" [0 0] (REXPR $n2):L ] + | Rminus_inside + [<<(REXPR <<(Rminus $n1 $n2)>>)>>] + -> [ (REXPR $n1):E "-" [0 0] (REXPR $n2):L ] + | NRplus_inside + [<<(REXPR <<(NRplus $r)>>)>>] -> [ "(" "1" "+" (REXPR $r):L ")"] + ; + + level 6: + Rmult_inside + [<<(REXPR <<(Rmult $n1 $n2)>>)>>] + -> [ (REXPR $n1):E "*" (REXPR $n2):L ] + | NRmult_inside + [<<(REXPR <<(NRmult $r)>>)>>] -> [ "(" "2" "*" (REXPR $r):L ")"] + ; + + level 5: + Ropp_inside [<<(REXPR <<(Ropp $n1)>>)>>] -> [ " -" (REXPR $n1):E ] + | Rinv_inside [<<(REXPR <<(Rinv $n1)>>)>>] -> [ "/" (REXPR $n1):E ] + | Rdiv_inside + [<<(REXPR <<(Rdiv $n1 $n2)>>)>>] + -> [ (REXPR $n1):E "/" [0 0] (REXPR $n2):L ] + ; + + level 0: + Rzero_inside [<<(REXPR <<R0>>)>>] -> ["0"] + | Rone_inside [<<(REXPR <<R1>>)>>] -> ["1"] + | Rodd_inside [<<(REXPR <<(Rplus R1 $r)>>)>>] -> [ $r:"r_printer_odd" ] + | Reven_inside [<<(REXPR <<(Rmult (Rplus R1 R1) $r)>>)>>] -> [ $r:"r_printer_even" ] +. + +(* For parsing/printing based on scopes *) +Module R_scope. + +Infix "<=" Rle (at level 5, no associativity) : R_scope V8only. +Infix "<" Rlt (at level 5, no associativity) : R_scope V8only. +Infix ">=" Rge (at level 5, no associativity) : R_scope V8only. +Infix ">" Rgt (at level 5, no associativity) : R_scope V8only. +Infix "+" Rplus (at level 4) : R_scope V8only. +Infix "-" Rminus (at level 4) : R_scope V8only. +Infix "*" Rmult (at level 3) : R_scope V8only. +Infix "/" Rdiv (at level 3) : R_scope V8only. +Notation "- x" := (Ropp x) (at level 0) : R_scope V8only. +Notation "x == y == z" := (eqT R x y)/\(eqT R y z) + (at level 5, y at level 4, no associtivity): R_scope. +Notation "x <= y <= z" := (Rle x y)/\(Rle y z) + (at level 5, y at level 4) : R_scope + V8only. +Notation "x <= y < z" := (Rle x y)/\(Rlt y z) + (at level 5, y at level 4) : R_scope + V8only. +Notation "x < y < z" := (Rlt x y)/\(Rlt y z) + (at level 5, y at level 4) : R_scope + V8only. +Notation "x < y <= z" := (Rlt x y)/\(Rle y z) + (at level 5, y at level 4) : R_scope + V8only. +Notation "/ x" := (Rinv x) (at level 0): R_scope + V8only. + +Open Local Scope R_scope. +End R_scope. +]. diff --git a/theories7/Reals/Rtopology.v b/theories7/Reals/Rtopology.v new file mode 100644 index 00000000..f2ae19b9 --- /dev/null +++ b/theories7/Reals/Rtopology.v @@ -0,0 +1,1178 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rtopology.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +Require RList. +Require Classical_Prop. +Require Classical_Pred_Type. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Definition included [D1,D2:R->Prop] : Prop := (x:R)(D1 x)->(D2 x). +Definition disc [x:R;delta:posreal] : R->Prop := [y:R]``(Rabsolu (y-x))<delta``. +Definition neighbourhood [V:R->Prop;x:R] : Prop := (EXT delta:posreal | (included (disc x delta) V)). +Definition open_set [D:R->Prop] : Prop := (x:R) (D x)->(neighbourhood D x). +Definition complementary [D:R->Prop] : R->Prop := [c:R]~(D c). +Definition closed_set [D:R->Prop] : Prop := (open_set (complementary D)). +Definition intersection_domain [D1,D2:R->Prop] : R->Prop := [c:R](D1 c)/\(D2 c). +Definition union_domain [D1,D2:R->Prop] : R->Prop := [c:R](D1 c)\/(D2 c). +Definition interior [D:R->Prop] : R->Prop := [x:R](neighbourhood D x). + +Lemma interior_P1 : (D:R->Prop) (included (interior D) D). +Intros; Unfold included; Unfold interior; Intros; Unfold neighbourhood in H; Elim H; Intros; Unfold included in H0; Apply H0; Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos x0). +Qed. + +Lemma interior_P2 : (D:R->Prop) (open_set D) -> (included D (interior D)). +Intros; Unfold open_set in H; Unfold included; Intros; Assert H1 := (H ? H0); Unfold interior; Apply H1. +Qed. + +Definition point_adherent [D:R->Prop;x:R] : Prop := (V:R->Prop) (neighbourhood V x) -> (EXT y:R | (intersection_domain V D y)). +Definition adherence [D:R->Prop] : R->Prop := [x:R](point_adherent D x). + +Lemma adherence_P1 : (D:R->Prop) (included D (adherence D)). +Intro; Unfold included; Intros; Unfold adherence; Unfold point_adherent; Intros; Exists x; Unfold intersection_domain; Split. +Unfold neighbourhood in H0; Elim H0; Intros; Unfold included in H1; Apply H1; Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos x0). +Apply H. +Qed. + +Lemma included_trans : (D1,D2,D3:R->Prop) (included D1 D2) -> (included D2 D3) -> (included D1 D3). +Unfold included; Intros; Apply H0; Apply H; Apply H1. +Qed. + +Lemma interior_P3 : (D:R->Prop) (open_set (interior D)). +Intro; Unfold open_set interior; Unfold neighbourhood; Intros; Elim H; Intros. +Exists x0; Unfold included; Intros. +Pose del := ``x0-(Rabsolu (x-x1))``. +Cut ``0<del``. +Intro; Exists (mkposreal del H2); Intros. +Cut (included (disc x1 (mkposreal del H2)) (disc x x0)). +Intro; Assert H5 := (included_trans ? ? ? H4 H0). +Apply H5; Apply H3. +Unfold included; Unfold disc; Intros. +Apply Rle_lt_trans with ``(Rabsolu (x3-x1))+(Rabsolu (x1-x))``. +Replace ``x3-x`` with ``(x3-x1)+(x1-x)``; [Apply Rabsolu_triang | Ring]. +Replace (pos x0) with ``del+(Rabsolu (x1-x))``. +Do 2 Rewrite <- (Rplus_sym (Rabsolu ``x1-x``)); Apply Rlt_compatibility; Apply H4. +Unfold del; Rewrite <- (Rabsolu_Ropp ``x-x1``); Rewrite Ropp_distr2; Ring. +Unfold del; Apply Rlt_anti_compatibility with ``(Rabsolu (x-x1))``; Rewrite Rplus_Or; Replace ``(Rabsolu (x-x1))+(x0-(Rabsolu (x-x1)))`` with (pos x0); [Idtac | Ring]. +Unfold disc in H1; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H1. +Qed. + +Lemma complementary_P1 : (D:R->Prop) ~(EXT y:R | (intersection_domain D (complementary D) y)). +Intro; Red; Intro; Elim H; Intros; Unfold intersection_domain complementary in H0; Elim H0; Intros; Elim H2; Assumption. +Qed. + +Lemma adherence_P2 : (D:R->Prop) (closed_set D) -> (included (adherence D) D). +Unfold closed_set; Unfold open_set complementary; Intros; Unfold included adherence; Intros; Assert H1 := (classic (D x)); Elim H1; Intro. +Assumption. +Assert H3 := (H ? H2); Assert H4 := (H0 ? H3); Elim H4; Intros; Unfold intersection_domain in H5; Elim H5; Intros; Elim H6; Assumption. +Qed. + +Lemma adherence_P3 : (D:R->Prop) (closed_set (adherence D)). +Intro; Unfold closed_set adherence; Unfold open_set complementary point_adherent; Intros; Pose P := [V:R->Prop](neighbourhood V x)->(EXT y:R | (intersection_domain V D y)); Assert H0 := (not_all_ex_not ? P H); Elim H0; Intros V0 H1; Unfold P in H1; Assert H2 := (imply_to_and ? ? H1); Unfold neighbourhood; Elim H2; Intros; Unfold neighbourhood in H3; Elim H3; Intros; Exists x0; Unfold included; Intros; Red; Intro. +Assert H8 := (H7 V0); Cut (EXT delta:posreal | (x:R)(disc x1 delta x)->(V0 x)). +Intro; Assert H10 := (H8 H9); Elim H4; Assumption. +Cut ``0<x0-(Rabsolu (x-x1))``. +Intro; Pose del := (mkposreal ? H9); Exists del; Intros; Unfold included in H5; Apply H5; Unfold disc; Apply Rle_lt_trans with ``(Rabsolu (x2-x1))+(Rabsolu (x1-x))``. +Replace ``x2-x`` with ``(x2-x1)+(x1-x)``; [Apply Rabsolu_triang | Ring]. +Replace (pos x0) with ``del+(Rabsolu (x1-x))``. +Do 2 Rewrite <- (Rplus_sym ``(Rabsolu (x1-x))``); Apply Rlt_compatibility; Apply H10. +Unfold del; Simpl; Rewrite <- (Rabsolu_Ropp ``x-x1``); Rewrite Ropp_distr2; Ring. +Apply Rlt_anti_compatibility with ``(Rabsolu (x-x1))``; Rewrite Rplus_Or; Replace ``(Rabsolu (x-x1))+(x0-(Rabsolu (x-x1)))`` with (pos x0); [Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H6 | Ring]. +Qed. + +Definition eq_Dom [D1,D2:R->Prop] : Prop := (included D1 D2)/\(included D2 D1). + +Infix "=_D" eq_Dom (at level 5, no associativity). + +Lemma open_set_P1 : (D:R->Prop) (open_set D) <-> D =_D (interior D). +Intro; Split. +Intro; Unfold eq_Dom; Split. +Apply interior_P2; Assumption. +Apply interior_P1. +Intro; Unfold eq_Dom in H; Elim H; Clear H; Intros; Unfold open_set; Intros; Unfold included interior in H; Unfold included in H0; Apply (H ? H1). +Qed. + +Lemma closed_set_P1 : (D:R->Prop) (closed_set D) <-> D =_D (adherence D). +Intro; Split. +Intro; Unfold eq_Dom; Split. +Apply adherence_P1. +Apply adherence_P2; Assumption. +Unfold eq_Dom; Unfold included; Intros; Assert H0 := (adherence_P3 D); Unfold closed_set in H0; Unfold closed_set; Unfold open_set; Unfold open_set in H0; Intros; Assert H2 : (complementary (adherence D) x). +Unfold complementary; Unfold complementary in H1; Red; Intro; Elim H; Clear H; Intros _ H; Elim H1; Apply (H ? H2). +Assert H3 := (H0 ? H2); Unfold neighbourhood; Unfold neighbourhood in H3; Elim H3; Intros; Exists x0; Unfold included; Unfold included in H4; Intros; Assert H6 := (H4 ? H5); Unfold complementary in H6; Unfold complementary; Red; Intro; Elim H; Clear H; Intros H _; Elim H6; Apply (H ? H7). +Qed. + +Lemma neighbourhood_P1 : (D1,D2:R->Prop;x:R) (included D1 D2) -> (neighbourhood D1 x) -> (neighbourhood D2 x). +Unfold included neighbourhood; Intros; Elim H0; Intros; Exists x0; Intros; Unfold included; Unfold included in H1; Intros; Apply (H ? (H1 ? H2)). +Qed. + +Lemma open_set_P2 : (D1,D2:R->Prop) (open_set D1) -> (open_set D2) -> (open_set (union_domain D1 D2)). +Unfold open_set; Intros; Unfold union_domain in H1; Elim H1; Intro. +Apply neighbourhood_P1 with D1. +Unfold included union_domain; Tauto. +Apply H; Assumption. +Apply neighbourhood_P1 with D2. +Unfold included union_domain; Tauto. +Apply H0; Assumption. +Qed. + +Lemma open_set_P3 : (D1,D2:R->Prop) (open_set D1) -> (open_set D2) -> (open_set (intersection_domain D1 D2)). +Unfold open_set; Intros; Unfold intersection_domain in H1; Elim H1; Intros. +Assert H4 := (H ? H2); Assert H5 := (H0 ? H3); Unfold intersection_domain; Unfold neighbourhood in H4 H5; Elim H4; Clear H; Intros del1 H; Elim H5; Clear H0; Intros del2 H0; Cut ``0<(Rmin del1 del2)``. +Intro; Pose del := (mkposreal ? H6). +Exists del; Unfold included; Intros; Unfold included in H H0; Unfold disc in H H0 H7. +Split. +Apply H; Apply Rlt_le_trans with (pos del). +Apply H7. +Unfold del; Simpl; Apply Rmin_l. +Apply H0; Apply Rlt_le_trans with (pos del). +Apply H7. +Unfold del; Simpl; Apply Rmin_r. +Unfold Rmin; Case (total_order_Rle del1 del2); Intro. +Apply (cond_pos del1). +Apply (cond_pos del2). +Qed. + +Lemma open_set_P4 : (open_set [x:R]False). +Unfold open_set; Intros; Elim H. +Qed. + +Lemma open_set_P5 : (open_set [x:R]True). +Unfold open_set; Intros; Unfold neighbourhood. +Exists (mkposreal R1 Rlt_R0_R1); Unfold included; Intros; Trivial. +Qed. + +Lemma disc_P1 : (x:R;del:posreal) (open_set (disc x del)). +Intros; Assert H := (open_set_P1 (disc x del)). +Elim H; Intros; Apply H1. +Unfold eq_Dom; Split. +Unfold included interior disc; Intros; Cut ``0<del-(Rabsolu (x-x0))``. +Intro; Pose del2 := (mkposreal ? H3). +Exists del2; Unfold included; Intros. +Apply Rle_lt_trans with ``(Rabsolu (x1-x0))+(Rabsolu (x0 -x))``. +Replace ``x1-x`` with ``(x1-x0)+(x0-x)``; [Apply Rabsolu_triang | Ring]. +Replace (pos del) with ``del2 + (Rabsolu (x0-x))``. +Do 2 Rewrite <- (Rplus_sym ``(Rabsolu (x0-x))``); Apply Rlt_compatibility. +Apply H4. +Unfold del2; Simpl; Rewrite <- (Rabsolu_Ropp ``x-x0``); Rewrite Ropp_distr2; Ring. +Apply Rlt_anti_compatibility with ``(Rabsolu (x-x0))``; Rewrite Rplus_Or; Replace ``(Rabsolu (x-x0))+(del-(Rabsolu (x-x0)))`` with (pos del); [Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H2 | Ring]. +Apply interior_P1. +Qed. + +Lemma continuity_P1 : (f:R->R;x:R) (continuity_pt f x) <-> (W:R->Prop)(neighbourhood W (f x)) -> (EXT V:R->Prop | (neighbourhood V x) /\ ((y:R)(V y)->(W (f y)))). +Intros; Split. +Intros; Unfold neighbourhood in H0. +Elim H0; Intros del1 H1. +Unfold continuity_pt in H; Unfold continue_in in H; Unfold limit1_in in H; Unfold limit_in in H; Simpl in H; Unfold R_dist in H. +Assert H2 := (H del1 (cond_pos del1)). +Elim H2; Intros del2 H3. +Elim H3; Intros. +Exists (disc x (mkposreal del2 H4)). +Intros; Unfold included in H1; Split. +Unfold neighbourhood disc. +Exists (mkposreal del2 H4). +Unfold included; Intros; Assumption. +Intros; Apply H1; Unfold disc; Case (Req_EM y x); Intro. +Rewrite H7; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos del1). +Apply H5; Split. +Unfold D_x no_cond; Split. +Trivial. +Apply not_sym; Apply H7. +Unfold disc in H6; Apply H6. +Intros; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Intros. +Assert H1 := (H (disc (f x) (mkposreal eps H0))). +Cut (neighbourhood (disc (f x) (mkposreal eps H0)) (f x)). +Intro; Assert H3 := (H1 H2). +Elim H3; Intros D H4; Elim H4; Intros; Unfold neighbourhood in H5; Elim H5; Intros del1 H7. +Exists (pos del1); Split. +Apply (cond_pos del1). +Intros; Elim H8; Intros; Simpl in H10; Unfold R_dist in H10; Simpl; Unfold R_dist; Apply (H6 ? (H7 ? H10)). +Unfold neighbourhood disc; Exists (mkposreal eps H0); Unfold included; Intros; Assumption. +Qed. + +Definition image_rec [f:R->R;D:R->Prop] : R->Prop := [x:R](D (f x)). + +(**********) +Lemma continuity_P2 : (f:R->R;D:R->Prop) (continuity f) -> (open_set D) -> (open_set (image_rec f D)). +Intros; Unfold open_set in H0; Unfold open_set; Intros; Assert H2 := (continuity_P1 f x); Elim H2; Intros H3 _; Assert H4 := (H3 (H x)); Unfold neighbourhood image_rec; Unfold image_rec in H1; Assert H5 := (H4 D (H0 (f x) H1)); Elim H5; Intros V0 H6; Elim H6; Intros; Unfold neighbourhood in H7; Elim H7; Intros del H9; Exists del; Unfold included in H9; Unfold included; Intros; Apply (H8 ? (H9 ? H10)). +Qed. + +(**********) +Lemma continuity_P3 : (f:R->R) (continuity f) <-> (D:R->Prop) (open_set D)->(open_set (image_rec f D)). +Intros; Split. +Intros; Apply continuity_P2; Assumption. +Intros; Unfold continuity; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Cut (open_set (disc (f x) (mkposreal ? H0))). +Intro; Assert H2 := (H ? H1). +Unfold open_set image_rec in H2; Cut (disc (f x) (mkposreal ? H0) (f x)). +Intro; Assert H4 := (H2 ? H3). +Unfold neighbourhood in H4; Elim H4; Intros del H5. +Exists (pos del); Split. +Apply (cond_pos del). +Intros; Unfold included in H5; Apply H5; Elim H6; Intros; Apply H8. +Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H0. +Apply disc_P1. +Qed. + +(**********) +Theorem Rsepare : (x,y:R) ``x<>y``->(EXT V:R->Prop | (EXT W:R->Prop | (neighbourhood V x)/\(neighbourhood W y)/\~(EXT y:R | (intersection_domain V W y)))). +Intros x y Hsep; Pose D := ``(Rabsolu (x-y))``. +Cut ``0<D/2``. +Intro; Exists (disc x (mkposreal ? H)). +Exists (disc y (mkposreal ? H)); Split. +Unfold neighbourhood; Exists (mkposreal ? H); Unfold included; Tauto. +Split. +Unfold neighbourhood; Exists (mkposreal ? H); Unfold included; Tauto. +Red; Intro; Elim H0; Intros; Unfold intersection_domain in H1; Elim H1; Intros. +Cut ``D<D``. +Intro; Elim (Rlt_antirefl ? H4). +Change ``(Rabsolu (x-y))<D``; Apply Rle_lt_trans with ``(Rabsolu (x-x0))+(Rabsolu (x0-y))``. +Replace ``x-y`` with ``(x-x0)+(x0-y)``; [Apply Rabsolu_triang | Ring]. +Rewrite (double_var D); Apply Rplus_lt. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H2. +Apply H3. +Unfold Rdiv; Apply Rmult_lt_pos. +Unfold D; Apply Rabsolu_pos_lt; Apply (Rminus_eq_contra ? ? Hsep). +Apply Rlt_Rinv; Sup0. +Qed. + +Record family : Type := mkfamily { + ind : R->Prop; + f :> R->R->Prop; + cond_fam : (x:R)(EXT y:R|(f x y))->(ind x) }. + +Definition family_open_set [f:family] : Prop := (x:R) (open_set (f x)). + +Definition domain_finite [D:R->Prop] : Prop := (EXT l:Rlist | (x:R)(D x)<->(In x l)). + +Definition family_finite [f:family] : Prop := (domain_finite (ind f)). + +Definition covering [D:R->Prop;f:family] : Prop := (x:R) (D x)->(EXT y:R | (f y x)). + +Definition covering_open_set [D:R->Prop;f:family] : Prop := (covering D f)/\(family_open_set f). + +Definition covering_finite [D:R->Prop;f:family] : Prop := (covering D f)/\(family_finite f). + +Lemma restriction_family : (f:family;D:R->Prop) (x:R)(EXT y:R|([z1:R][z2:R](f z1 z2)/\(D z1) x y))->(intersection_domain (ind f) D x). +Intros; Elim H; Intros; Unfold intersection_domain; Elim H0; Intros; Split. +Apply (cond_fam f0); Exists x0; Assumption. +Assumption. +Qed. + +Definition subfamily [f:family;D:R->Prop] : family := (mkfamily (intersection_domain (ind f) D) [x:R][y:R](f x y)/\(D x) (restriction_family f D)). + +Definition compact [X:R->Prop] : Prop := (f:family) (covering_open_set X f) -> (EXT D:R->Prop | (covering_finite X (subfamily f D))). + +(**********) +Lemma family_P1 : (f:family;D:R->Prop) (family_open_set f) -> (family_open_set (subfamily f D)). +Unfold family_open_set; Intros; Unfold subfamily; Simpl; Assert H0 := (classic (D x)). +Elim H0; Intro. +Cut (open_set (f0 x))->(open_set [y:R](f0 x y)/\(D x)). +Intro; Apply H2; Apply H. +Unfold open_set; Unfold neighbourhood; Intros; Elim H3; Intros; Assert H6 := (H2 ? H4); Elim H6; Intros; Exists x1; Unfold included; Intros; Split. +Apply (H7 ? H8). +Assumption. +Cut (open_set [y:R]False) -> (open_set [y:R](f0 x y)/\(D x)). +Intro; Apply H2; Apply open_set_P4. +Unfold open_set; Unfold neighbourhood; Intros; Elim H3; Intros; Elim H1; Assumption. +Qed. + +Definition bounded [D:R->Prop] : Prop := (EXT m:R | (EXT M:R | (x:R)(D x)->``m<=x<=M``)). + +Lemma open_set_P6 : (D1,D2:R->Prop) (open_set D1) -> D1 =_D D2 -> (open_set D2). +Unfold open_set; Unfold neighbourhood; Intros. +Unfold eq_Dom in H0; Elim H0; Intros. +Assert H4 := (H ? (H3 ? H1)). +Elim H4; Intros. +Exists x0; Apply included_trans with D1; Assumption. +Qed. + +(**********) +Lemma compact_P1 : (X:R->Prop) (compact X) -> (bounded X). +Intros; Unfold compact in H; Pose D := [x:R]True; Pose g := [x:R][y:R]``(Rabsolu y)<x``; Cut (x:R)(EXT y|(g x y))->True; [Intro | Intro; Trivial]. +Pose f0 := (mkfamily D g H0); Assert H1 := (H f0); Cut (covering_open_set X f0). +Intro; Assert H3 := (H1 H2); Elim H3; Intros D' H4; Unfold covering_finite in H4; Elim H4; Intros; Unfold family_finite in H6; Unfold domain_finite in H6; Elim H6; Intros l H7; Unfold bounded; Pose r := (MaxRlist l). +Exists ``-r``; Exists r; Intros. +Unfold covering in H5; Assert H9 := (H5 ? H8); Elim H9; Intros; Unfold subfamily in H10; Simpl in H10; Elim H10; Intros; Assert H13 := (H7 x0); Simpl in H13; Cut (intersection_domain D D' x0). +Elim H13; Clear H13; Intros. +Assert H16 := (H13 H15); Unfold g in H11; Split. +Cut ``x0<=r``. +Intro; Cut ``(Rabsolu x)<r``. +Intro; Assert H19 := (Rabsolu_def2 x r H18); Elim H19; Intros; Left; Assumption. +Apply Rlt_le_trans with x0; Assumption. +Apply (MaxRlist_P1 l x0 H16). +Cut ``x0<=r``. +Intro; Apply Rle_trans with (Rabsolu x). +Apply Rle_Rabsolu. +Apply Rle_trans with x0. +Left; Apply H11. +Assumption. +Apply (MaxRlist_P1 l x0 H16). +Unfold intersection_domain D; Tauto. +Unfold covering_open_set; Split. +Unfold covering; Intros; Simpl; Exists ``(Rabsolu x)+1``; Unfold g; Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. +Unfold family_open_set; Intro; Case (total_order R0 x); Intro. +Apply open_set_P6 with (disc R0 (mkposreal ? H2)). +Apply disc_P1. +Unfold eq_Dom; Unfold f0; Simpl; Unfold g disc; Split. +Unfold included; Intros; Unfold Rminus in H3; Rewrite Ropp_O in H3; Rewrite Rplus_Or in H3; Apply H3. +Unfold included; Intros; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply H3. +Apply open_set_P6 with [x:R]False. +Apply open_set_P4. +Unfold eq_Dom; Split. +Unfold included; Intros; Elim H3. +Unfold included f0; Simpl; Unfold g; Intros; Elim H2; Intro; [Rewrite <- H4 in H3; Assert H5 := (Rabsolu_pos x0); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H5 H3)) | Assert H6 := (Rabsolu_pos x0); Assert H7 := (Rlt_trans ? ? ? H3 H4); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H6 H7))]. +Qed. + +(**********) +Lemma compact_P2 : (X:R->Prop) (compact X) -> (closed_set X). +Intros; Assert H0 := (closed_set_P1 X); Elim H0; Clear H0; Intros _ H0; Apply H0; Clear H0. +Unfold eq_Dom; Split. +Apply adherence_P1. +Unfold included; Unfold adherence; Unfold point_adherent; Intros; Unfold compact in H; Assert H1 := (classic (X x)); Elim H1; Clear H1; Intro. +Assumption. +Cut (y:R)(X y)->``0<(Rabsolu (y-x))/2``. +Intro; Pose D := X; Pose g := [y:R][z:R]``(Rabsolu (y-z))<(Rabsolu (y-x))/2``/\(D y); Cut (x:R)(EXT y|(g x y))->(D x). +Intro; Pose f0 := (mkfamily D g H3); Assert H4 := (H f0); Cut (covering_open_set X f0). +Intro; Assert H6 := (H4 H5); Elim H6; Clear H6; Intros D' H6. +Unfold covering_finite in H6; Decompose [and] H6; Unfold covering subfamily in H7; Simpl in H7; Unfold family_finite subfamily in H8; Simpl in H8; Unfold domain_finite in H8; Elim H8; Clear H8; Intros l H8; Pose alp := (MinRlist (AbsList l x)); Cut ``0<alp``. +Intro; Assert H10 := (H0 (disc x (mkposreal ? H9))); Cut (neighbourhood (disc x (mkposreal alp H9)) x). +Intro; Assert H12 := (H10 H11); Elim H12; Clear H12; Intros y H12; Unfold intersection_domain in H12; Elim H12; Clear H12; Intros; Assert H14 := (H7 ? H13); Elim H14; Clear H14; Intros y0 H14; Elim H14; Clear H14; Intros; Unfold g in H14; Elim H14; Clear H14; Intros; Unfold disc in H12; Simpl in H12; Cut ``alp<=(Rabsolu (y0-x))/2``. +Intro; Assert H18 := (Rlt_le_trans ? ? ? H12 H17); Cut ``(Rabsolu (y0-x))<(Rabsolu (y0-x))``. +Intro; Elim (Rlt_antirefl ? H19). +Apply Rle_lt_trans with ``(Rabsolu (y0-y))+(Rabsolu (y-x))``. +Replace ``y0-x`` with ``(y0-y)+(y-x)``; [Apply Rabsolu_triang | Ring]. +Rewrite (double_var ``(Rabsolu (y0-x))``); Apply Rplus_lt; Assumption. +Apply (MinRlist_P1 (AbsList l x) ``(Rabsolu (y0-x))/2``); Apply AbsList_P1; Elim (H8 y0); Clear H8; Intros; Apply H8; Unfold intersection_domain; Split; Assumption. +Assert H11 := (disc_P1 x (mkposreal alp H9)); Unfold open_set in H11; Apply H11. +Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H9. +Unfold alp; Apply MinRlist_P2; Intros; Assert H10 := (AbsList_P2 ? ? ? H9); Elim H10; Clear H10; Intros z H10; Elim H10; Clear H10; Intros; Rewrite H11; Apply H2; Elim (H8 z); Clear H8; Intros; Assert H13 := (H12 H10); Unfold intersection_domain D in H13; Elim H13; Clear H13; Intros; Assumption. +Unfold covering_open_set; Split. +Unfold covering; Intros; Exists x0; Simpl; Unfold g; Split. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Unfold Rminus in H2; Apply (H2 ? H5). +Apply H5. +Unfold family_open_set; Intro; Simpl; Unfold g; Elim (classic (D x0)); Intro. +Apply open_set_P6 with (disc x0 (mkposreal ? (H2 ? H5))). +Apply disc_P1. +Unfold eq_Dom; Split. +Unfold included disc; Simpl; Intros; Split. +Rewrite <- (Rabsolu_Ropp ``x0-x1``); Rewrite Ropp_distr2; Apply H6. +Apply H5. +Unfold included disc; Simpl; Intros; Elim H6; Intros; Rewrite <- (Rabsolu_Ropp ``x1-x0``); Rewrite Ropp_distr2; Apply H7. +Apply open_set_P6 with [z:R]False. +Apply open_set_P4. +Unfold eq_Dom; Split. +Unfold included; Intros; Elim H6. +Unfold included; Intros; Elim H6; Intros; Elim H5; Assumption. +Intros; Elim H3; Intros; Unfold g in H4; Elim H4; Clear H4; Intros _ H4; Apply H4. +Intros; Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rabsolu_pos_lt; Apply Rminus_eq_contra; Red; Intro; Rewrite H3 in H2; Elim H1; Apply H2. +Apply Rlt_Rinv; Sup0. +Qed. + +(**********) +Lemma compact_EMP : (compact [_:R]False). +Unfold compact; Intros; Exists [x:R]False; Unfold covering_finite; Split. +Unfold covering; Intros; Elim H0. +Unfold family_finite; Unfold domain_finite; Exists nil; Intro. +Split. +Simpl; Unfold intersection_domain; Intros; Elim H0. +Elim H0; Clear H0; Intros _ H0; Elim H0. +Simpl; Intro; Elim H0. +Qed. + +Lemma compact_eqDom : (X1,X2:R->Prop) (compact X1) -> X1 =_D X2 -> (compact X2). +Unfold compact; Intros; Unfold eq_Dom in H0; Elim H0; Clear H0; Unfold included; Intros; Assert H3 : (covering_open_set X1 f0). +Unfold covering_open_set; Unfold covering_open_set in H1; Elim H1; Clear H1; Intros; Split. +Unfold covering in H1; Unfold covering; Intros; Apply (H1 ? (H0 ? H4)). +Apply H3. +Elim (H ? H3); Intros D H4; Exists D; Unfold covering_finite; Unfold covering_finite in H4; Elim H4; Intros; Split. +Unfold covering in H5; Unfold covering; Intros; Apply (H5 ? (H2 ? H7)). +Apply H6. +Qed. + +(* Borel-Lebesgue's lemma *) +Lemma compact_P3 : (a,b:R) (compact [c:R]``a<=c<=b``). +Intros; Case (total_order_Rle a b); Intro. +Unfold compact; Intros; Pose A := [x:R]``a<=x<=b``/\(EXT D:R->Prop | (covering_finite [c:R]``a <= c <= x`` (subfamily f0 D))); Cut (A a). +Intro; Cut (bound A). +Intro; Cut (EXT a0:R | (A a0)). +Intro; Assert H3 := (complet A H1 H2); Elim H3; Clear H3; Intros m H3; Unfold is_lub in H3; Cut ``a<=m<=b``. +Intro; Unfold covering_open_set in H; Elim H; Clear H; Intros; Unfold covering in H; Assert H6 := (H m H4); Elim H6; Clear H6; Intros y0 H6; Unfold family_open_set in H5; Assert H7 := (H5 y0); Unfold open_set in H7; Assert H8 := (H7 m H6); Unfold neighbourhood in H8; Elim H8; Clear H8; Intros eps H8; Cut (EXT x:R | (A x)/\``m-eps<x<=m``). +Intro; Elim H9; Clear H9; Intros x H9; Elim H9; Clear H9; Intros; Case (Req_EM m b); Intro. +Rewrite H11 in H10; Rewrite H11 in H8; Unfold A in H9; Elim H9; Clear H9; Intros; Elim H12; Clear H12; Intros Dx H12; Pose Db := [x:R](Dx x)\/x==y0; Exists Db; Unfold covering_finite; Split. +Unfold covering; Unfold covering_finite in H12; Elim H12; Clear H12; Intros; Unfold covering in H12; Case (total_order_Rle x0 x); Intro. +Cut ``a<=x0<=x``. +Intro; Assert H16 := (H12 x0 H15); Elim H16; Clear H16; Intros; Exists x1; Simpl in H16; Simpl; Unfold Db; Elim H16; Clear H16; Intros; Split; [Apply H16 | Left; Apply H17]. +Split. +Elim H14; Intros; Assumption. +Assumption. +Exists y0; Simpl; Split. +Apply H8; Unfold disc; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Rewrite Rabsolu_right. +Apply Rlt_trans with ``b-x``. +Unfold Rminus; Apply Rlt_compatibility; Apply Rlt_Ropp; Auto with real. +Elim H10; Intros H15 _; Apply Rlt_anti_compatibility with ``x-eps``; Replace ``x-eps+(b-x)`` with ``b-eps``; [Replace ``x-eps+eps`` with x; [Apply H15 | Ring] | Ring]. +Apply Rge_minus; Apply Rle_sym1; Elim H14; Intros _ H15; Apply H15. +Unfold Db; Right; Reflexivity. +Unfold family_finite; Unfold domain_finite; Unfold covering_finite in H12; Elim H12; Clear H12; Intros; Unfold family_finite in H13; Unfold domain_finite in H13; Elim H13; Clear H13; Intros l H13; Exists (cons y0 l); Intro; Split. +Intro; Simpl in H14; Unfold intersection_domain in H14; Elim (H13 x0); Clear H13; Intros; Case (Req_EM x0 y0); Intro. +Simpl; Left; Apply H16. +Simpl; Right; Apply H13. +Simpl; Unfold intersection_domain; Unfold Db in H14; Decompose [and or] H14. +Split; Assumption. +Elim H16; Assumption. +Intro; Simpl in H14; Elim H14; Intro; Simpl; Unfold intersection_domain. +Split. +Apply (cond_fam f0); Rewrite H15; Exists m; Apply H6. +Unfold Db; Right; Assumption. +Simpl; Unfold intersection_domain; Elim (H13 x0). +Intros _ H16; Assert H17 := (H16 H15); Simpl in H17; Unfold intersection_domain in H17; Split. +Elim H17; Intros; Assumption. +Unfold Db; Left; Elim H17; Intros; Assumption. +Pose m' := (Rmin ``m+eps/2`` b); Cut (A m'). +Intro; Elim H3; Intros; Unfold is_upper_bound in H13; Assert H15 := (H13 m' H12); Cut ``m<m'``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H15 H16)). +Unfold m'; Unfold Rmin; Case (total_order_Rle ``m+eps/2`` b); Intro. +Pattern 1 m; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Sup0]. +Elim H4; Intros. +Elim H17; Intro. +Assumption. +Elim H11; Assumption. +Unfold A; Split. +Split. +Apply Rle_trans with m. +Elim H4; Intros; Assumption. +Unfold m'; Unfold Rmin; Case (total_order_Rle ``m+eps/2`` b); Intro. +Pattern 1 m; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Sup0]. +Elim H4; Intros. +Elim H13; Intro. +Assumption. +Elim H11; Assumption. +Unfold m'; Apply Rmin_r. +Unfold A in H9; Elim H9; Clear H9; Intros; Elim H12; Clear H12; Intros Dx H12; Pose Db := [x:R](Dx x)\/x==y0; Exists Db; Unfold covering_finite; Split. +Unfold covering; Unfold covering_finite in H12; Elim H12; Clear H12; Intros; Unfold covering in H12; Case (total_order_Rle x0 x); Intro. +Cut ``a<=x0<=x``. +Intro; Assert H16 := (H12 x0 H15); Elim H16; Clear H16; Intros; Exists x1; Simpl in H16; Simpl; Unfold Db. +Elim H16; Clear H16; Intros; Split; [Apply H16 | Left; Apply H17]. +Elim H14; Intros; Split; Assumption. +Exists y0; Simpl; Split. +Apply H8; Unfold disc; Unfold Rabsolu; Case (case_Rabsolu ``x0-m``); Intro. +Rewrite Ropp_distr2; Apply Rlt_trans with ``m-x``. +Unfold Rminus; Apply Rlt_compatibility; Apply Rlt_Ropp; Auto with real. +Apply Rlt_anti_compatibility with ``x-eps``; Replace ``x-eps+(m-x)`` with ``m-eps``. +Replace ``x-eps+eps`` with x. +Elim H10; Intros; Assumption. +Ring. +Ring. +Apply Rle_lt_trans with ``m'-m``. +Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-m``); Apply Rle_compatibility; Elim H14; Intros; Assumption. +Apply Rlt_anti_compatibility with m; Replace ``m+(m'-m)`` with m'. +Apply Rle_lt_trans with ``m+eps/2``. +Unfold m'; Apply Rmin_l. +Apply Rlt_compatibility; Apply Rlt_monotony_contra with ``2``. +Sup0. +Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Pattern 1 (pos eps); Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Apply (cond_pos eps). +DiscrR. +Ring. +Unfold Db; Right; Reflexivity. +Unfold family_finite; Unfold domain_finite; Unfold covering_finite in H12; Elim H12; Clear H12; Intros; Unfold family_finite in H13; Unfold domain_finite in H13; Elim H13; Clear H13; Intros l H13; Exists (cons y0 l); Intro; Split. +Intro; Simpl in H14; Unfold intersection_domain in H14; Elim (H13 x0); Clear H13; Intros; Case (Req_EM x0 y0); Intro. +Simpl; Left; Apply H16. +Simpl; Right; Apply H13; Simpl; Unfold intersection_domain; Unfold Db in H14; Decompose [and or] H14. +Split; Assumption. +Elim H16; Assumption. +Intro; Simpl in H14; Elim H14; Intro; Simpl; Unfold intersection_domain. +Split. +Apply (cond_fam f0); Rewrite H15; Exists m; Apply H6. +Unfold Db; Right; Assumption. +Elim (H13 x0); Intros _ H16. +Assert H17 := (H16 H15). +Simpl in H17. +Unfold intersection_domain in H17. +Split. +Elim H17; Intros; Assumption. +Unfold Db; Left; Elim H17; Intros; Assumption. +Elim (classic (EXT x:R | (A x)/\``m-eps < x <= m``)); Intro. +Assumption. +Elim H3; Intros; Cut (is_upper_bound A ``m-eps``). +Intro; Assert H13 := (H11 ? H12); Cut ``m-eps<m``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H13 H14)). +Pattern 2 m; Rewrite <- Rplus_Or; Unfold Rminus; Apply Rlt_compatibility; Apply Ropp_Rlt; Rewrite Ropp_Ropp; Rewrite Ropp_O; Apply (cond_pos eps). +Pose P := [n:R](A n)/\``m-eps<n<=m``; Assert H12 := (not_ex_all_not ? P H9); Unfold P in H12; Unfold is_upper_bound; Intros; Assert H14 := (not_and_or ? ? (H12 x)); Elim H14; Intro. +Elim H15; Apply H13. +Elim (not_and_or ? ? H15); Intro. +Case (total_order_Rle x ``m-eps``); Intro. +Assumption. +Elim H16; Auto with real. +Unfold is_upper_bound in H10; Assert H17 := (H10 x H13); Elim H16; Apply H17. +Elim H3; Clear H3; Intros. +Unfold is_upper_bound in H3. +Split. +Apply (H3 ? H0). +Apply (H4 b); Unfold is_upper_bound; Intros; Unfold A in H5; Elim H5; Clear H5; Intros H5 _; Elim H5; Clear H5; Intros _ H5; Apply H5. +Exists a; Apply H0. +Unfold bound; Exists b; Unfold is_upper_bound; Intros; Unfold A in H1; Elim H1; Clear H1; Intros H1 _; Elim H1; Clear H1; Intros _ H1; Apply H1. +Unfold A; Split. +Split; [Right; Reflexivity | Apply r]. +Unfold covering_open_set in H; Elim H; Clear H; Intros; Unfold covering in H; Cut ``a<=a<=b``. +Intro; Elim (H ? H1); Intros y0 H2; Pose D':=[x:R]x==y0; Exists D'; Unfold covering_finite; Split. +Unfold covering; Simpl; Intros; Cut x==a. +Intro; Exists y0; Split. +Rewrite H4; Apply H2. +Unfold D'; Reflexivity. +Elim H3; Intros; Apply Rle_antisym; Assumption. +Unfold family_finite; Unfold domain_finite; Exists (cons y0 nil); Intro; Split. +Simpl; Unfold intersection_domain; Intro; Elim H3; Clear H3; Intros; Unfold D' in H4; Left; Apply H4. +Simpl; Unfold intersection_domain; Intro; Elim H3; Intro. +Split; [Rewrite H4; Apply (cond_fam f0); Exists a; Apply H2 | Apply H4]. +Elim H4. +Split; [Right; Reflexivity | Apply r]. +Apply compact_eqDom with [c:R]False. +Apply compact_EMP. +Unfold eq_Dom; Split. +Unfold included; Intros; Elim H. +Unfold included; Intros; Elim H; Clear H; Intros; Assert H1 := (Rle_trans ? ? ? H H0); Elim n; Apply H1. +Qed. + +Lemma compact_P4 : (X,F:R->Prop) (compact X) -> (closed_set F) -> (included F X) -> (compact F). +Unfold compact; Intros; Elim (classic (EXT z:R | (F z))); Intro Hyp_F_NE. +Pose D := (ind f0); Pose g := (f f0); Unfold closed_set in H0. +Pose g' := [x:R][y:R](f0 x y)\/((complementary F y)/\(D x)). +Pose D' := D. +Cut (x:R)(EXT y:R | (g' x y))->(D' x). +Intro; Pose f' := (mkfamily D' g' H3); Cut (covering_open_set X f'). +Intro; Elim (H ? H4); Intros DX H5; Exists DX. +Unfold covering_finite; Unfold covering_finite in H5; Elim H5; Clear H5; Intros. +Split. +Unfold covering; Unfold covering in H5; Intros. +Elim (H5 ? (H1 ? H7)); Intros y0 H8; Exists y0; Simpl in H8; Simpl; Elim H8; Clear H8; Intros. +Split. +Unfold g' in H8; Elim H8; Intro. +Apply H10. +Elim H10; Intros H11 _; Unfold complementary in H11; Elim H11; Apply H7. +Apply H9. +Unfold family_finite; Unfold domain_finite; Unfold family_finite in H6; Unfold domain_finite in H6; Elim H6; Clear H6; Intros l H6; Exists l; Intro; Assert H7 := (H6 x); Elim H7; Clear H7; Intros. +Split. +Intro; Apply H7; Simpl; Unfold intersection_domain; Simpl in H9; Unfold intersection_domain in H9; Unfold D'; Apply H9. +Intro; Assert H10 := (H8 H9); Simpl in H10; Unfold intersection_domain in H10; Simpl; Unfold intersection_domain; Unfold D' in H10; Apply H10. +Unfold covering_open_set; Unfold covering_open_set in H2; Elim H2; Clear H2; Intros. +Split. +Unfold covering; Unfold covering in H2; Intros. +Elim (classic (F x)); Intro. +Elim (H2 ? H6); Intros y0 H7; Exists y0; Simpl; Unfold g'; Left; Assumption. +Cut (EXT z:R | (D z)). +Intro; Elim H7; Clear H7; Intros x0 H7; Exists x0; Simpl; Unfold g'; Right. +Split. +Unfold complementary; Apply H6. +Apply H7. +Elim Hyp_F_NE; Intros z0 H7. +Assert H8 := (H2 ? H7). +Elim H8; Clear H8; Intros t H8; Exists t; Apply (cond_fam f0); Exists z0; Apply H8. +Unfold family_open_set; Intro; Simpl; Unfold g'; Elim (classic (D x)); Intro. +Apply open_set_P6 with (union_domain (f0 x) (complementary F)). +Apply open_set_P2. +Unfold family_open_set in H4; Apply H4. +Apply H0. +Unfold eq_Dom; Split. +Unfold included union_domain complementary; Intros. +Elim H6; Intro; [Left; Apply H7 | Right; Split; Assumption]. +Unfold included union_domain complementary; Intros. +Elim H6; Intro; [Left; Apply H7 | Right; Elim H7; Intros; Apply H8]. +Apply open_set_P6 with (f0 x). +Unfold family_open_set in H4; Apply H4. +Unfold eq_Dom; Split. +Unfold included complementary; Intros; Left; Apply H6. +Unfold included complementary; Intros. +Elim H6; Intro. +Apply H7. +Elim H7; Intros _ H8; Elim H5; Apply H8. +Intros; Elim H3; Intros y0 H4; Unfold g' in H4; Elim H4; Intro. +Apply (cond_fam f0); Exists y0; Apply H5. +Elim H5; Clear H5; Intros _ H5; Apply H5. +(* Cas ou F est l'ensemble vide *) +Cut (compact F). +Intro; Apply (H3 f0 H2). +Apply compact_eqDom with [_:R]False. +Apply compact_EMP. +Unfold eq_Dom; Split. +Unfold included; Intros; Elim H3. +Assert H3 := (not_ex_all_not ? ? Hyp_F_NE); Unfold included; Intros; Elim (H3 x); Apply H4. +Qed. + +(**********) +Lemma compact_P5 : (X:R->Prop) (closed_set X)->(bounded X)->(compact X). +Intros; Unfold bounded in H0. +Elim H0; Clear H0; Intros m H0. +Elim H0; Clear H0; Intros M H0. +Assert H1 := (compact_P3 m M). +Apply (compact_P4 [c:R]``m<=c<=M`` X H1 H H0). +Qed. + +(**********) +Lemma compact_carac : (X:R->Prop) (compact X)<->(closed_set X)/\(bounded X). +Intro; Split. +Intro; Split; [Apply (compact_P2 ? H) | Apply (compact_P1 ? H)]. +Intro; Elim H; Clear H; Intros; Apply (compact_P5 ? H H0). +Qed. + +Definition image_dir [f:R->R;D:R->Prop] : R->Prop := [x:R](EXT y:R | x==(f y)/\(D y)). + +(**********) +Lemma continuity_compact : (f:R->R;X:R->Prop) ((x:R)(continuity_pt f x)) -> (compact X) -> (compact (image_dir f X)). +Unfold compact; Intros; Unfold covering_open_set in H1. +Elim H1; Clear H1; Intros. +Pose D := (ind f1). +Pose g := [x:R][y:R](image_rec f0 (f1 x) y). +Cut (x:R)(EXT y:R | (g x y))->(D x). +Intro; Pose f' := (mkfamily D g H3). +Cut (covering_open_set X f'). +Intro; Elim (H0 f' H4); Intros D' H5; Exists D'. +Unfold covering_finite in H5; Elim H5; Clear H5; Intros; Unfold covering_finite; Split. +Unfold covering image_dir; Simpl; Unfold covering in H5; Intros; Elim H7; Intros y H8; Elim H8; Intros; Assert H11 := (H5 ? H10); Simpl in H11; Elim H11; Intros z H12; Exists z; Unfold g in H12; Unfold image_rec in H12; Rewrite H9; Apply H12. +Unfold family_finite in H6; Unfold domain_finite in H6; Unfold family_finite; Unfold domain_finite; Elim H6; Intros l H7; Exists l; Intro; Elim (H7 x); Intros; Split; Intro. +Apply H8; Simpl in H10; Simpl; Apply H10. +Apply (H9 H10). +Unfold covering_open_set; Split. +Unfold covering; Intros; Simpl; Unfold covering in H1; Unfold image_dir in H1; Unfold g; Unfold image_rec; Apply H1. +Exists x; Split; [Reflexivity | Apply H4]. +Unfold family_open_set; Unfold family_open_set in H2; Intro; Simpl; Unfold g; Cut ([y:R](image_rec f0 (f1 x) y))==(image_rec f0 (f1 x)). +Intro; Rewrite H4. +Apply (continuity_P2 f0 (f1 x) H (H2 x)). +Reflexivity. +Intros; Apply (cond_fam f1); Unfold g in H3; Unfold image_rec in H3; Elim H3; Intros; Exists (f0 x0); Apply H4. +Qed. + +Lemma Rlt_Rminus : (a,b:R) ``a<b`` -> ``0<b-a``. +Intros; Apply Rlt_anti_compatibility with a; Rewrite Rplus_Or; Replace ``a+(b-a)`` with b; [Assumption | Ring]. +Qed. + +Lemma prolongement_C0 : (f:R->R;a,b:R) ``a<=b`` -> ((c:R)``a<=c<=b``->(continuity_pt f c)) -> (EXT g:R->R | (continuity g)/\((c:R)``a<=c<=b``->(g c)==(f c))). +Intros; Elim H; Intro. +Pose h := [x:R](Cases (total_order_Rle x a) of + (leftT _) => (f0 a) +| (rightT _) => (Cases (total_order_Rle x b) of + (leftT _) => (f0 x) + | (rightT _) => (f0 b) end) end). +Assert H2 : ``0<b-a``. +Apply Rlt_Rminus; Assumption. +Exists h; Split. +Unfold continuity; Intro; Case (total_order x a); Intro. +Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Exists ``a-x``; Split. +Change ``0<a-x``; Apply Rlt_Rminus; Assumption. +Intros; Elim H5; Clear H5; Intros _ H5; Unfold h. +Case (total_order_Rle x a); Intro. +Case (total_order_Rle x0 a); Intro. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Elim n; Left; Apply Rlt_anti_compatibility with ``-x``; Do 2 Rewrite (Rplus_sym ``-x``); Apply Rle_lt_trans with ``(Rabsolu (x0-x))``. +Apply Rle_Rabsolu. +Assumption. +Elim n; Left; Assumption. +Elim H3; Intro. +Assert H5 : ``a<=a<=b``. +Split; [Right; Reflexivity | Left; Assumption]. +Assert H6 := (H0 ? H5); Unfold continuity_pt in H6; Unfold continue_in in H6; Unfold limit1_in in H6; Unfold limit_in in H6; Simpl in H6; Unfold R_dist in H6; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H6 ? H7); Intros; Exists (Rmin x0 ``b-a``); Split. +Unfold Rmin; Case (total_order_Rle x0 ``b-a``); Intro. +Elim H8; Intros; Assumption. +Change ``0<b-a``; Apply Rlt_Rminus; Assumption. +Intros; Elim H9; Clear H9; Intros _ H9; Cut ``x1<b``. +Intro; Unfold h; Case (total_order_Rle x a); Intro. +Case (total_order_Rle x1 a); Intro. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Case (total_order_Rle x1 b); Intro. +Elim H8; Intros; Apply H12; Split. +Unfold D_x no_cond; Split. +Trivial. +Red; Intro; Elim n; Right; Symmetry; Assumption. +Apply Rlt_le_trans with (Rmin x0 ``b-a``). +Rewrite H4 in H9; Apply H9. +Apply Rmin_l. +Elim n0; Left; Assumption. +Elim n; Right; Assumption. +Apply Rlt_anti_compatibility with ``-a``; Do 2 Rewrite (Rplus_sym ``-a``); Rewrite H4 in H9; Apply Rle_lt_trans with ``(Rabsolu (x1-a))``. +Apply Rle_Rabsolu. +Apply Rlt_le_trans with ``(Rmin x0 (b-a))``. +Assumption. +Apply Rmin_r. +Case (total_order x b); Intro. +Assert H6 : ``a<=x<=b``. +Split; Left; Assumption. +Assert H7 := (H0 ? H6); Unfold continuity_pt in H7; Unfold continue_in in H7; Unfold limit1_in in H7; Unfold limit_in in H7; Simpl in H7; Unfold R_dist in H7; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H7 ? H8); Intros; Elim H9; Clear H9; Intros. +Assert H11 : ``0<x-a``. +Apply Rlt_Rminus; Assumption. +Assert H12 : ``0<b-x``. +Apply Rlt_Rminus; Assumption. +Exists (Rmin x0 (Rmin ``x-a`` ``b-x``)); Split. +Unfold Rmin; Case (total_order_Rle ``x-a`` ``b-x``); Intro. +Case (total_order_Rle x0 ``x-a``); Intro. +Assumption. +Assumption. +Case (total_order_Rle x0 ``b-x``); Intro. +Assumption. +Assumption. +Intros; Elim H13; Clear H13; Intros; Cut ``a<x1<b``. +Intro; Elim H15; Clear H15; Intros; Unfold h; Case (total_order_Rle x a); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H4)). +Case (total_order_Rle x b); Intro. +Case (total_order_Rle x1 a); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 H15)). +Case (total_order_Rle x1 b); Intro. +Apply H10; Split. +Assumption. +Apply Rlt_le_trans with ``(Rmin x0 (Rmin (x-a) (b-x)))``. +Assumption. +Apply Rmin_l. +Elim n1; Left; Assumption. +Elim n0; Left; Assumption. +Split. +Apply Ropp_Rlt; Apply Rlt_anti_compatibility with x; Apply Rle_lt_trans with ``(Rabsolu (x1-x))``. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply Rle_Rabsolu. +Apply Rlt_le_trans with ``(Rmin x0 (Rmin (x-a) (b-x)))``. +Assumption. +Apply Rle_trans with ``(Rmin (x-a) (b-x))``. +Apply Rmin_r. +Apply Rmin_l. +Apply Rlt_anti_compatibility with ``-x``; Do 2 Rewrite (Rplus_sym ``-x``); Apply Rle_lt_trans with ``(Rabsolu (x1-x))``. +Apply Rle_Rabsolu. +Apply Rlt_le_trans with ``(Rmin x0 (Rmin (x-a) (b-x)))``. +Assumption. +Apply Rle_trans with ``(Rmin (x-a) (b-x))``; Apply Rmin_r. +Elim H5; Intro. +Assert H7 : ``a<=b<=b``. +Split; [Left; Assumption | Right; Reflexivity]. +Assert H8 := (H0 ? H7); Unfold continuity_pt in H8; Unfold continue_in in H8; Unfold limit1_in in H8; Unfold limit_in in H8; Simpl in H8; Unfold R_dist in H8; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H8 ? H9); Intros; Exists (Rmin x0 ``b-a``); Split. +Unfold Rmin; Case (total_order_Rle x0 ``b-a``); Intro. +Elim H10; Intros; Assumption. +Change ``0<b-a``; Apply Rlt_Rminus; Assumption. +Intros; Elim H11; Clear H11; Intros _ H11; Cut ``a<x1``. +Intro; Unfold h; Case (total_order_Rle x a); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H4)). +Case (total_order_Rle x1 a); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H12)). +Case (total_order_Rle x b); Intro. +Case (total_order_Rle x1 b); Intro. +Rewrite H6; Elim H10; Intros; Elim r0; Intro. +Apply H14; Split. +Unfold D_x no_cond; Split. +Trivial. +Red; Intro; Rewrite <- H16 in H15; Elim (Rlt_antirefl ? H15). +Rewrite H6 in H11; Apply Rlt_le_trans with ``(Rmin x0 (b-a))``. +Apply H11. +Apply Rmin_l. +Rewrite H15; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Rewrite H6; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Elim n1; Right; Assumption. +Rewrite H6 in H11; Apply Ropp_Rlt; Apply Rlt_anti_compatibility with b; Apply Rle_lt_trans with ``(Rabsolu (x1-b))``. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply Rle_Rabsolu. +Apply Rlt_le_trans with ``(Rmin x0 (b-a))``. +Assumption. +Apply Rmin_r. +Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Exists ``x-b``; Split. +Change ``0<x-b``; Apply Rlt_Rminus; Assumption. +Intros; Elim H8; Clear H8; Intros. +Assert H10 : ``b<x0``. +Apply Ropp_Rlt; Apply Rlt_anti_compatibility with x; Apply Rle_lt_trans with ``(Rabsolu (x0-x))``. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply Rle_Rabsolu. +Assumption. +Unfold h; Case (total_order_Rle x a); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H4)). +Case (total_order_Rle x b); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H6)). +Case (total_order_Rle x0 a); Intro. +Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H1 (Rlt_le_trans ? ? ? H10 r))). +Case (total_order_Rle x0 b); Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H10)). +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Intros; Elim H3; Intros; Unfold h; Case (total_order_Rle c a); Intro. +Elim r; Intro. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 H6)). +Rewrite H6; Reflexivity. +Case (total_order_Rle c b); Intro. +Reflexivity. +Elim n0; Assumption. +Exists [_:R](f0 a); Split. +Apply derivable_continuous; Apply (derivable_const (f0 a)). +Intros; Elim H2; Intros; Rewrite H1 in H3; Cut b==c. +Intro; Rewrite <- H5; Rewrite H1; Reflexivity. +Apply Rle_antisym; Assumption. +Qed. + +(**********) +Lemma continuity_ab_maj : (f:R->R;a,b:R) ``a<=b`` -> ((c:R)``a<=c<=b``->(continuity_pt f c)) -> (EXT Mx : R | ((c:R)``a<=c<=b``->``(f c)<=(f Mx)``)/\``a<=Mx<=b``). +Intros; Cut (EXT g:R->R | (continuity g)/\((c:R)``a<=c<=b``->(g c)==(f0 c))). +Intro HypProl. +Elim HypProl; Intros g Hcont_eq. +Elim Hcont_eq; Clear Hcont_eq; Intros Hcont Heq. +Assert H1 := (compact_P3 a b). +Assert H2 := (continuity_compact g [c:R]``a<=c<=b`` Hcont H1). +Assert H3 := (compact_P2 ? H2). +Assert H4 := (compact_P1 ? H2). +Cut (bound (image_dir g [c:R]``a <= c <= b``)). +Cut (ExT [x:R] ((image_dir g [c:R]``a <= c <= b``) x)). +Intros; Assert H7 := (complet ? H6 H5). +Elim H7; Clear H7; Intros M H7; Cut (image_dir g [c:R]``a <= c <= b`` M). +Intro; Unfold image_dir in H8; Elim H8; Clear H8; Intros Mxx H8; Elim H8; Clear H8; Intros; Exists Mxx; Split. +Intros; Rewrite <- (Heq c H10); Rewrite <- (Heq Mxx H9); Intros; Rewrite <- H8; Unfold is_lub in H7; Elim H7; Clear H7; Intros H7 _; Unfold is_upper_bound in H7; Apply H7; Unfold image_dir; Exists c; Split; [Reflexivity | Apply H10]. +Apply H9. +Elim (classic (image_dir g [c:R]``a <= c <= b`` M)); Intro. +Assumption. +Cut (EXT eps:posreal | (y:R)~(intersection_domain (disc M eps) (image_dir g [c:R]``a <= c <= b``) y)). +Intro; Elim H9; Clear H9; Intros eps H9; Unfold is_lub in H7; Elim H7; Clear H7; Intros; Cut (is_upper_bound (image_dir g [c:R]``a <= c <= b``) ``M-eps``). +Intro; Assert H12 := (H10 ? H11); Cut ``M-eps<M``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H12 H13)). +Pattern 2 M; Rewrite <- Rplus_Or; Unfold Rminus; Apply Rlt_compatibility; Apply Ropp_Rlt; Rewrite Ropp_O; Rewrite Ropp_Ropp; Apply (cond_pos eps). +Unfold is_upper_bound image_dir; Intros; Cut ``x<=M``. +Intro; Case (total_order_Rle x ``M-eps``); Intro. +Apply r. +Elim (H9 x); Unfold intersection_domain disc image_dir; Split. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Rewrite Rabsolu_right. +Apply Rlt_anti_compatibility with ``x-eps``; Replace ``x-eps+(M-x)`` with ``M-eps``. +Replace ``x-eps+eps`` with x. +Auto with real. +Ring. +Ring. +Apply Rge_minus; Apply Rle_sym1; Apply H12. +Apply H11. +Apply H7; Apply H11. +Cut (EXT V:R->Prop | (neighbourhood V M)/\((y:R)~(intersection_domain V (image_dir g [c:R]``a <= c <= b``) y))). +Intro; Elim H9; Intros V H10; Elim H10; Clear H10; Intros. +Unfold neighbourhood in H10; Elim H10; Intros del H12; Exists del; Intros; Red; Intro; Elim (H11 y). +Unfold intersection_domain; Unfold intersection_domain in H13; Elim H13; Clear H13; Intros; Split. +Apply (H12 ? H13). +Apply H14. +Cut ~(point_adherent (image_dir g [c:R]``a <= c <= b``) M). +Intro; Unfold point_adherent in H9. +Assert H10 := (not_all_ex_not ? [V:R->Prop](neighbourhood V M) + ->(EXT y:R | + (intersection_domain V + (image_dir g [c:R]``a <= c <= b``) y)) H9). +Elim H10; Intros V0 H11; Exists V0; Assert H12 := (imply_to_and ? ? H11); Elim H12; Clear H12; Intros. +Split. +Apply H12. +Apply (not_ex_all_not ? ? H13). +Red; Intro; Cut (adherence (image_dir g [c:R]``a <= c <= b``) M). +Intro; Elim (closed_set_P1 (image_dir g [c:R]``a <= c <= b``)); Intros H11 _; Assert H12 := (H11 H3). +Elim H8. +Unfold eq_Dom in H12; Elim H12; Clear H12; Intros. +Apply (H13 ? H10). +Apply H9. +Exists (g a); Unfold image_dir; Exists a; Split. +Reflexivity. +Split; [Right; Reflexivity | Apply H]. +Unfold bound; Unfold bounded in H4; Elim H4; Clear H4; Intros m H4; Elim H4; Clear H4; Intros M H4; Exists M; Unfold is_upper_bound; Intros; Elim (H4 ? H5); Intros _ H6; Apply H6. +Apply prolongement_C0; Assumption. +Qed. + +(**********) +Lemma continuity_ab_min : (f:(R->R); a,b:R) ``a <= b``->((c:R)``a<=c<=b``->(continuity_pt f c))->(EXT mx:R | ((c:R)``a <= c <= b``->``(f mx) <= (f c)``)/\``a <= mx <= b``). +Intros. +Cut ((c:R)``a<=c<=b``->(continuity_pt (opp_fct f0) c)). +Intro; Assert H2 := (continuity_ab_maj (opp_fct f0) a b H H1); Elim H2; Intros x0 H3; Exists x0; Intros; Split. +Intros; Rewrite <- (Ropp_Ropp (f0 x0)); Rewrite <- (Ropp_Ropp (f0 c)); Apply Rle_Ropp1; Elim H3; Intros; Unfold opp_fct in H5; Apply H5; Apply H4. +Elim H3; Intros; Assumption. +Intros. +Assert H2 := (H0 ? H1). +Apply (continuity_pt_opp ? ? H2). +Qed. + + +(********************************************************) +(* Proof of Bolzano-Weierstrass theorem *) +(********************************************************) + +Definition ValAdh [un:nat->R;x:R] : Prop := (V:R->Prop;N:nat) (neighbourhood V x) -> (EX p:nat | (le N p)/\(V (un p))). + +Definition intersection_family [f:family] : R->Prop := [x:R](y:R)(ind f y)->(f y x). + +Lemma ValAdh_un_exists : (un:nat->R) let D=[x:R](EX n:nat | x==(INR n)) in let f=[x:R](adherence [y:R](EX p:nat | y==(un p)/\``x<=(INR p)``)/\(D x)) in ((x:R)(EXT y:R | (f x y))->(D x)). +Intros; Elim H; Intros; Unfold f in H0; Unfold adherence in H0; Unfold point_adherent in H0; Assert H1 : (neighbourhood (disc x0 (mkposreal ? Rlt_R0_R1)) x0). +Unfold neighbourhood disc; Exists (mkposreal ? Rlt_R0_R1); Unfold included; Trivial. +Elim (H0 ? H1); Intros; Unfold intersection_domain in H2; Elim H2; Intros; Elim H4; Intros; Apply H6. +Qed. + +Definition ValAdh_un [un:nat->R] : R->Prop := let D=[x:R](EX n:nat | x==(INR n)) in let f=[x:R](adherence [y:R](EX p:nat | y==(un p)/\``x<=(INR p)``)/\(D x)) in (intersection_family (mkfamily D f (ValAdh_un_exists un))). + +Lemma ValAdh_un_prop : (un:nat->R;x:R) (ValAdh un x) <-> (ValAdh_un un x). +Intros; Split; Intro. +Unfold ValAdh in H; Unfold ValAdh_un; Unfold intersection_family; Simpl; Intros; Elim H0; Intros N H1; Unfold adherence; Unfold point_adherent; Intros; Elim (H V N H2); Intros; Exists (un x0); Unfold intersection_domain; Elim H3; Clear H3; Intros; Split. +Assumption. +Split. +Exists x0; Split; [Reflexivity | Rewrite H1; Apply (le_INR ? ? H3)]. +Exists N; Assumption. +Unfold ValAdh; Intros; Unfold ValAdh_un in H; Unfold intersection_family in H; Simpl in H; Assert H1 : (adherence [y0:R](EX p:nat | ``y0 == (un p)``/\``(INR N) <= (INR p)``)/\(EX n:nat | ``(INR N) == (INR n)``) x). +Apply H; Exists N; Reflexivity. +Unfold adherence in H1; Unfold point_adherent in H1; Assert H2 := (H1 ? H0); Elim H2; Intros; Unfold intersection_domain in H3; Elim H3; Clear H3; Intros; Elim H4; Clear H4; Intros; Elim H4; Clear H4; Intros; Elim H4; Clear H4; Intros; Exists x1; Split. +Apply (INR_le ? ? H6). +Rewrite H4 in H3; Apply H3. +Qed. + +Lemma adherence_P4 : (F,G:R->Prop) (included F G) -> (included (adherence F) (adherence G)). +Unfold adherence included; Unfold point_adherent; Intros; Elim (H0 ? H1); Unfold intersection_domain; Intros; Elim H2; Clear H2; Intros; Exists x0; Split; [Assumption | Apply (H ? H3)]. +Qed. + +Definition family_closed_set [f:family] : Prop := (x:R) (closed_set (f x)). + +Definition intersection_vide_in [D:R->Prop;f:family] : Prop := ((x:R)((ind f x)->(included (f x) D))/\~(EXT y:R | (intersection_family f y))). + +Definition intersection_vide_finite_in [D:R->Prop;f:family] : Prop := (intersection_vide_in D f)/\(family_finite f). + +(**********) +Lemma compact_P6 : (X:R->Prop) (compact X) -> (EXT z:R | (X z)) -> ((g:family) (family_closed_set g) -> (intersection_vide_in X g) -> (EXT D:R->Prop | (intersection_vide_finite_in X (subfamily g D)))). +Intros X H Hyp g H0 H1. +Pose D' := (ind g). +Pose f' := [x:R][y:R](complementary (g x) y)/\(D' x). +Assert H2 : (x:R)(EXT y:R|(f' x y))->(D' x). +Intros; Elim H2; Intros; Unfold f' in H3; Elim H3; Intros; Assumption. +Pose f0 := (mkfamily D' f' H2). +Unfold compact in H; Assert H3 : (covering_open_set X f0). +Unfold covering_open_set; Split. +Unfold covering; Intros; Unfold intersection_vide_in in H1; Elim (H1 x); Intros; Unfold intersection_family in H5; Assert H6 := (not_ex_all_not ? [y:R](y0:R)(ind g y0)->(g y0 y) H5 x); Assert H7 := (not_all_ex_not ? [y0:R](ind g y0)->(g y0 x) H6); Elim H7; Intros; Exists x0; Elim (imply_to_and ? ? H8); Intros; Unfold f0; Simpl; Unfold f'; Split; [Apply H10 | Apply H9]. +Unfold family_open_set; Intro; Elim (classic (D' x)); Intro. +Apply open_set_P6 with (complementary (g x)). +Unfold family_closed_set in H0; Unfold closed_set in H0; Apply H0. +Unfold f0; Simpl; Unfold f'; Unfold eq_Dom; Split. +Unfold included; Intros; Split; [Apply H4 | Apply H3]. +Unfold included; Intros; Elim H4; Intros; Assumption. +Apply open_set_P6 with [_:R]False. +Apply open_set_P4. +Unfold eq_Dom; Unfold included; Split; Intros; [Elim H4 | Simpl in H4; Unfold f' in H4; Elim H4; Intros; Elim H3; Assumption]. +Elim (H ? H3); Intros SF H4; Exists SF; Unfold intersection_vide_finite_in; Split. +Unfold intersection_vide_in; Simpl; Intros; Split. +Intros; Unfold included; Intros; Unfold intersection_vide_in in H1; Elim (H1 x); Intros; Elim H6; Intros; Apply H7. +Unfold intersection_domain in H5; Elim H5; Intros; Assumption. +Assumption. +Elim (classic (EXT y:R | (intersection_domain (ind g) SF y))); Intro Hyp'. +Red; Intro; Elim H5; Intros; Unfold intersection_family in H6; Simpl in H6. +Cut (X x0). +Intro; Unfold covering_finite in H4; Elim H4; Clear H4; Intros H4 _; Unfold covering in H4; Elim (H4 x0 H7); Intros; Simpl in H8; Unfold intersection_domain in H6; Cut (ind g x1)/\(SF x1). +Intro; Assert H10 := (H6 x1 H9); Elim H10; Clear H10; Intros H10 _; Elim H8; Clear H8; Intros H8 _; Unfold f' in H8; Unfold complementary in H8; Elim H8; Clear H8; Intros H8 _; Elim H8; Assumption. +Split. +Apply (cond_fam f0). +Exists x0; Elim H8; Intros; Assumption. +Elim H8; Intros; Assumption. +Unfold intersection_vide_in in H1; Elim Hyp'; Intros; Assert H8 := (H6 ? H7); Elim H8; Intros; Cut (ind g x1). +Intro; Elim (H1 x1); Intros; Apply H12. +Apply H11. +Apply H9. +Apply (cond_fam g); Exists x0; Assumption. +Unfold covering_finite in H4; Elim H4; Clear H4; Intros H4 _; Cut (EXT z:R | (X z)). +Intro; Elim H5; Clear H5; Intros; Unfold covering in H4; Elim (H4 x0 H5); Intros; Simpl in H6; Elim Hyp'; Exists x1; Elim H6; Intros; Unfold intersection_domain; Split. +Apply (cond_fam f0); Exists x0; Apply H7. +Apply H8. +Apply Hyp. +Unfold covering_finite in H4; Elim H4; Clear H4; Intros; Unfold family_finite in H5; Unfold domain_finite in H5; Unfold family_finite; Unfold domain_finite; Elim H5; Clear H5; Intros l H5; Exists l; Intro; Elim (H5 x); Intros; Split; Intro; [Apply H6; Simpl; Simpl in H8; Apply H8 | Apply (H7 H8)]. +Qed. + +Theorem Bolzano_Weierstrass : (un:nat->R;X:R->Prop) (compact X) -> ((n:nat)(X (un n))) -> (EXT l:R | (ValAdh un l)). +Intros; Cut (EXT l:R | (ValAdh_un un l)). +Intro; Elim H1; Intros; Exists x; Elim (ValAdh_un_prop un x); Intros; Apply (H4 H2). +Assert H1 : (EXT z:R | (X z)). +Exists (un O); Apply H0. +Pose D:=[x:R](EX n:nat | x==(INR n)). +Pose g:=[x:R](adherence [y:R](EX p:nat | y==(un p)/\``x<=(INR p)``)/\(D x)). +Assert H2 : (x:R)(EXT y:R | (g x y))->(D x). +Intros; Elim H2; Intros; Unfold g in H3; Unfold adherence in H3; Unfold point_adherent in H3. +Assert H4 : (neighbourhood (disc x0 (mkposreal ? Rlt_R0_R1)) x0). +Unfold neighbourhood; Exists (mkposreal ? Rlt_R0_R1); Unfold included; Trivial. +Elim (H3 ? H4); Intros; Unfold intersection_domain in H5; Decompose [and] H5; Assumption. +Pose f0 := (mkfamily D g H2). +Assert H3 := (compact_P6 X H H1 f0). +Elim (classic (EXT l:R | (ValAdh_un un l))); Intro. +Assumption. +Cut (family_closed_set f0). +Intro; Cut (intersection_vide_in X f0). +Intro; Assert H7 := (H3 H5 H6). +Elim H7; Intros SF H8; Unfold intersection_vide_finite_in in H8; Elim H8; Clear H8; Intros; Unfold intersection_vide_in in H8; Elim (H8 R0); Intros _ H10; Elim H10; Unfold family_finite in H9; Unfold domain_finite in H9; Elim H9; Clear H9; Intros l H9; Pose r := (MaxRlist l); Cut (D r). +Intro; Unfold D in H11; Elim H11; Intros; Exists (un x); Unfold intersection_family; Simpl; Unfold intersection_domain; Intros; Split. +Unfold g; Apply adherence_P1; Split. +Exists x; Split; [Reflexivity | Rewrite <- H12; Unfold r; Apply MaxRlist_P1; Elim (H9 y); Intros; Apply H14; Simpl; Apply H13]. +Elim H13; Intros; Assumption. +Elim H13; Intros; Assumption. +Elim (H9 r); Intros. +Simpl in H12; Unfold intersection_domain in H12; Cut (In r l). +Intro; Elim (H12 H13); Intros; Assumption. +Unfold r; Apply MaxRlist_P2; Cut (EXT z:R | (intersection_domain (ind f0) SF z)). +Intro; Elim H13; Intros; Elim (H9 x); Intros; Simpl in H15; Assert H17 := (H15 H14); Exists x; Apply H17. +Elim (classic (EXT z:R | (intersection_domain (ind f0) SF z))); Intro. +Assumption. +Elim (H8 R0); Intros _ H14; Elim H1; Intros; Assert H16 := (not_ex_all_not ? [y:R](intersection_family (subfamily f0 SF) y) H14); Assert H17 := (not_ex_all_not ? [z:R](intersection_domain (ind f0) SF z) H13); Assert H18 := (H16 x); Unfold intersection_family in H18; Simpl in H18; Assert H19 := (not_all_ex_not ? [y:R](intersection_domain D SF y)->(g y x)/\(SF y) H18); Elim H19; Intros; Assert H21 := (imply_to_and ? ? H20); Elim (H17 x0); Elim H21; Intros; Assumption. +Unfold intersection_vide_in; Intros; Split. +Intro; Simpl in H6; Unfold f0; Simpl; Unfold g; Apply included_trans with (adherence X). +Apply adherence_P4. +Unfold included; Intros; Elim H7; Intros; Elim H8; Intros; Elim H10; Intros; Rewrite H11; Apply H0. +Apply adherence_P2; Apply compact_P2; Assumption. +Apply H4. +Unfold family_closed_set; Unfold f0; Simpl; Unfold g; Intro; Apply adherence_P3. +Qed. + +(********************************************************) +(* Proof of Heine's theorem *) +(********************************************************) + +Definition uniform_continuity [f:R->R;X:R->Prop] : Prop := (eps:posreal)(EXT delta:posreal | (x,y:R) (X x)->(X y)->``(Rabsolu (x-y))<delta`` ->``(Rabsolu ((f x)-(f y)))<eps``). + +Lemma is_lub_u : (E:R->Prop;x,y:R) (is_lub E x) -> (is_lub E y) -> x==y. +Unfold is_lub; Intros; Elim H; Elim H0; Intros; Apply Rle_antisym; [Apply (H4 ? H1) | Apply (H2 ? H3)]. +Qed. + +Lemma domain_P1 : (X:R->Prop) ~(EXT y:R | (X y))\/(EXT y:R | (X y)/\((x:R)(X x)->x==y))\/(EXT x:R | (EXT y:R | (X x)/\(X y)/\``x<>y``)). +Intro; Elim (classic (EXT y:R | (X y))); Intro. +Right; Elim H; Intros; Elim (classic (EXT y:R | (X y)/\``y<>x``)); Intro. +Right; Elim H1; Intros; Elim H2; Intros; Exists x; Exists x0; Intros. +Split; [Assumption | Split; [Assumption | Apply not_sym; Assumption]]. +Left; Exists x; Split. +Assumption. +Intros; Case (Req_EM x0 x); Intro. +Assumption. +Elim H1; Exists x0; Split; Assumption. +Left; Assumption. +Qed. + +Theorem Heine : (f:R->R;X:R->Prop) (compact X) -> ((x:R)(X x)->(continuity_pt f x)) -> (uniform_continuity f X). +Intros f0 X H0 H; Elim (domain_P1 X); Intro Hyp. +(* X est vide *) +Unfold uniform_continuity; Intros; Exists (mkposreal ? Rlt_R0_R1); Intros; Elim Hyp; Exists x; Assumption. +Elim Hyp; Clear Hyp; Intro Hyp. +(* X possède un seul élément *) +Unfold uniform_continuity; Intros; Exists (mkposreal ? Rlt_R0_R1); Intros; Elim Hyp; Clear Hyp; Intros; Elim H4; Clear H4; Intros; Assert H6 := (H5 ? H1); Assert H7 := (H5 ? H2); Rewrite H6; Rewrite H7; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos eps). +(* X possède au moins deux éléments distincts *) +Assert X_enc : (EXT m:R | (EXT M:R | ((x:R)(X x)->``m<=x<=M``)/\``m<M``)). +Assert H1 := (compact_P1 X H0); Unfold bounded in H1; Elim H1; Intros; Elim H2; Intros; Exists x; Exists x0; Split. +Apply H3. +Elim Hyp; Intros; Elim H4; Intros; Decompose [and] H5; Assert H10 := (H3 ? H6); Assert H11 := (H3 ? H8); Elim H10; Intros; Elim H11; Intros; Case (total_order_T x x0); Intro. +Elim s; Intro. +Assumption. +Rewrite b in H13; Rewrite b in H7; Elim H9; Apply Rle_antisym; Apply Rle_trans with x0; Assumption. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (Rle_trans ? ? ? H13 H14) r)). +Elim X_enc; Clear X_enc; Intros m X_enc; Elim X_enc; Clear X_enc; Intros M X_enc; Elim X_enc; Clear X_enc Hyp; Intros X_enc Hyp; Unfold uniform_continuity; Intro; Assert H1 : (t:posreal)``0<t/2``. +Intro; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos t) | Apply Rlt_Rinv; Sup0]. +Pose g := [x:R][y:R](X x)/\(EXT del:posreal | ((z:R) ``(Rabsolu (z-x))<del``->``(Rabsolu ((f0 z)-(f0 x)))<eps/2``)/\(is_lub [zeta:R]``0<zeta<=M-m``/\((z:R) ``(Rabsolu (z-x))<zeta``->``(Rabsolu ((f0 z)-(f0 x)))<eps/2``) del)/\(disc x (mkposreal ``del/2`` (H1 del)) y)). +Assert H2 : (x:R)(EXT y:R | (g x y))->(X x). +Intros; Elim H2; Intros; Unfold g in H3; Elim H3; Clear H3; Intros H3 _; Apply H3. +Pose f' := (mkfamily X g H2); Unfold compact in H0; Assert H3 : (covering_open_set X f'). +Unfold covering_open_set; Split. +Unfold covering; Intros; Exists x; Simpl; Unfold g; Split. +Assumption. +Assert H4 := (H ? H3); Unfold continuity_pt in H4; Unfold continue_in in H4; Unfold limit1_in in H4; Unfold limit_in in H4; Simpl in H4; Unfold R_dist in H4; Elim (H4 ``eps/2`` (H1 eps)); Intros; Pose E:=[zeta:R]``0<zeta <= M-m``/\((z:R)``(Rabsolu (z-x)) < zeta``->``(Rabsolu ((f0 z)-(f0 x))) < eps/2``); Assert H6 : (bound E). +Unfold bound; Exists ``M-m``; Unfold is_upper_bound; Unfold E; Intros; Elim H6; Clear H6; Intros H6 _; Elim H6; Clear H6; Intros _ H6; Apply H6. +Assert H7 : (EXT x:R | (E x)). +Elim H5; Clear H5; Intros; Exists (Rmin x0 ``M-m``); Unfold E; Intros; Split. +Split. +Unfold Rmin; Case (total_order_Rle x0 ``M-m``); Intro. +Apply H5. +Apply Rlt_Rminus; Apply Hyp. +Apply Rmin_r. +Intros; Case (Req_EM x z); Intro. +Rewrite H9; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (H1 eps). +Apply H7; Split. +Unfold D_x no_cond; Split; [Trivial | Assumption]. +Apply Rlt_le_trans with (Rmin x0 ``M-m``); [Apply H8 | Apply Rmin_l]. +Assert H8 := (complet ? H6 H7); Elim H8; Clear H8; Intros; Cut ``0<x1<=(M-m)``. +Intro; Elim H8; Clear H8; Intros; Exists (mkposreal ? H8); Split. +Intros; Cut (EXT alp:R | ``(Rabsolu (z-x))<alp<=x1``/\(E alp)). +Intros; Elim H11; Intros; Elim H12; Clear H12; Intros; Unfold E in H13; Elim H13; Intros; Apply H15. +Elim H12; Intros; Assumption. +Elim (classic (EXT alp:R | ``(Rabsolu (z-x)) < alp <= x1``/\(E alp))); Intro. +Assumption. +Assert H12 := (not_ex_all_not ? [alp:R]``(Rabsolu (z-x)) < alp <= x1``/\(E alp) H11); Unfold is_lub in p; Elim p; Intros; Cut (is_upper_bound E ``(Rabsolu (z-x))``). +Intro; Assert H16 := (H14 ? H15); Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H10 H16)). +Unfold is_upper_bound; Intros; Unfold is_upper_bound in H13; Assert H16 := (H13 ? H15); Case (total_order_Rle x2 ``(Rabsolu (z-x))``); Intro. +Assumption. +Elim (H12 x2); Split; [Split; [Auto with real | Assumption] | Assumption]. +Split. +Apply p. +Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Simpl; Unfold Rdiv; Apply Rmult_lt_pos; [Apply H8 | Apply Rlt_Rinv; Sup0]. +Elim H7; Intros; Unfold E in H8; Elim H8; Intros H9 _; Elim H9; Intros H10 _; Unfold is_lub in p; Elim p; Intros; Unfold is_upper_bound in H12; Unfold is_upper_bound in H11; Split. +Apply Rlt_le_trans with x2; [Assumption | Apply (H11 ? H8)]. +Apply H12; Intros; Unfold E in H13; Elim H13; Intros; Elim H14; Intros; Assumption. +Unfold family_open_set; Intro; Simpl; Elim (classic (X x)); Intro. +Unfold g; Unfold open_set; Intros; Elim H4; Clear H4; Intros _ H4; Elim H4; Clear H4; Intros; Elim H4; Clear H4; Intros; Unfold neighbourhood; Case (Req_EM x x0); Intro. +Exists (mkposreal ? (H1 x1)); Rewrite <- H6; Unfold included; Intros; Split. +Assumption. +Exists x1; Split. +Apply H4. +Split. +Elim H5; Intros; Apply H8. +Apply H7. +Pose d := ``x1/2-(Rabsolu (x0-x))``; Assert H7 : ``0<d``. +Unfold d; Apply Rlt_Rminus; Elim H5; Clear H5; Intros; Unfold disc in H7; Apply H7. +Exists (mkposreal ? H7); Unfold included; Intros; Split. +Assumption. +Exists x1; Split. +Apply H4. +Elim H5; Intros; Split. +Assumption. +Unfold disc in H8; Simpl in H8; Unfold disc; Simpl; Unfold disc in H10; Simpl in H10; Apply Rle_lt_trans with ``(Rabsolu (x2-x0))+(Rabsolu (x0-x))``. +Replace ``x2-x`` with ``(x2-x0)+(x0-x)``; [Apply Rabsolu_triang | Ring]. +Replace ``x1/2`` with ``d+(Rabsolu (x0-x))``; [Idtac | Unfold d; Ring]. +Do 2 Rewrite <- (Rplus_sym ``(Rabsolu (x0-x))``); Apply Rlt_compatibility; Apply H8. +Apply open_set_P6 with [_:R]False. +Apply open_set_P4. +Unfold eq_Dom; Unfold included; Intros; Split. +Intros; Elim H4. +Intros; Unfold g in H4; Elim H4; Clear H4; Intros H4 _; Elim H3; Apply H4. +Elim (H0 ? H3); Intros DF H4; Unfold covering_finite in H4; Elim H4; Clear H4; Intros; Unfold family_finite in H5; Unfold domain_finite in H5; Unfold covering in H4; Simpl in H4; Simpl in H5; Elim H5; Clear H5; Intros l H5; Unfold intersection_domain in H5; Cut (x:R)(In x l)->(EXT del:R | ``0<del``/\((z:R)``(Rabsolu (z-x)) < del``->``(Rabsolu ((f0 z)-(f0 x))) < eps/2``)/\(included (g x) [z:R]``(Rabsolu (z-x))<del/2``)). +Intros; Assert H7 := (Rlist_P1 l [x:R][del:R]``0<del``/\((z:R)``(Rabsolu (z-x)) < del``->``(Rabsolu ((f0 z)-(f0 x))) < eps/2``)/\(included (g x) [z:R]``(Rabsolu (z-x))<del/2``) H6); Elim H7; Clear H7; Intros l' H7; Elim H7; Clear H7; Intros; Pose D := (MinRlist l'); Cut ``0<D/2``. +Intro; Exists (mkposreal ? H9); Intros; Assert H13 := (H4 ? H10); Elim H13; Clear H13; Intros xi H13; Assert H14 : (In xi l). +Unfold g in H13; Decompose [and] H13; Elim (H5 xi); Intros; Apply H14; Split; Assumption. +Elim (pos_Rl_P2 l xi); Intros H15 _; Elim (H15 H14); Intros i H16; Elim H16; Intros; Apply Rle_lt_trans with ``(Rabsolu ((f0 x)-(f0 xi)))+(Rabsolu ((f0 xi)-(f0 y)))``. +Replace ``(f0 x)-(f0 y)`` with ``((f0 x)-(f0 xi))+((f0 xi)-(f0 y))``; [Apply Rabsolu_triang | Ring]. +Rewrite (double_var eps); Apply Rplus_lt. +Assert H19 := (H8 i H17); Elim H19; Clear H19; Intros; Rewrite <- H18 in H20; Elim H20; Clear H20; Intros; Apply H20; Unfold included in H21; Apply Rlt_trans with ``(pos_Rl l' i)/2``. +Apply H21. +Elim H13; Clear H13; Intros; Assumption. +Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. +Sup0. +Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Pattern 1 (pos_Rl l' i); Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Apply H19. +DiscrR. +Assert H19 := (H8 i H17); Elim H19; Clear H19; Intros; Rewrite <- H18 in H20; Elim H20; Clear H20; Intros; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H20; Unfold included in H21; Elim H13; Intros; Assert H24 := (H21 x H22); Apply Rle_lt_trans with ``(Rabsolu (y-x))+(Rabsolu (x-xi))``. +Replace ``y-xi`` with ``(y-x)+(x-xi)``; [Apply Rabsolu_triang | Ring]. +Rewrite (double_var (pos_Rl l' i)); Apply Rplus_lt. +Apply Rlt_le_trans with ``D/2``. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H12. +Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/2``); Apply Rle_monotony. +Left; Apply Rlt_Rinv; Sup0. +Unfold D; Apply MinRlist_P1; Elim (pos_Rl_P2 l' (pos_Rl l' i)); Intros; Apply H26; Exists i; Split; [Rewrite <- H7; Assumption | Reflexivity]. +Assumption. +Unfold Rdiv; Apply Rmult_lt_pos; [Unfold D; Apply MinRlist_P2; Intros; Elim (pos_Rl_P2 l' y); Intros; Elim (H10 H9); Intros; Elim H12; Intros; Rewrite H14; Rewrite <- H7 in H13; Elim (H8 x H13); Intros; Apply H15 | Apply Rlt_Rinv; Sup0]. +Intros; Elim (H5 x); Intros; Elim (H8 H6); Intros; Pose E:=[zeta:R]``0<zeta <= M-m``/\((z:R)``(Rabsolu (z-x)) < zeta``->``(Rabsolu ((f0 z)-(f0 x))) < eps/2``); Assert H11 : (bound E). +Unfold bound; Exists ``M-m``; Unfold is_upper_bound; Unfold E; Intros; Elim H11; Clear H11; Intros H11 _; Elim H11; Clear H11; Intros _ H11; Apply H11. +Assert H12 : (EXT x:R | (E x)). +Assert H13 := (H ? H9); Unfold continuity_pt in H13; Unfold continue_in in H13; Unfold limit1_in in H13; Unfold limit_in in H13; Simpl in H13; Unfold R_dist in H13; Elim (H13 ? (H1 eps)); Intros; Elim H12; Clear H12; Intros; Exists (Rmin x0 ``M-m``); Unfold E; Intros; Split. +Split; [Unfold Rmin; Case (total_order_Rle x0 ``M-m``); Intro; [Apply H12 | Apply Rlt_Rminus; Apply Hyp] | Apply Rmin_r]. +Intros; Case (Req_EM x z); Intro. +Rewrite H16; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (H1 eps). +Apply H14; Split; [Unfold D_x no_cond; Split; [Trivial | Assumption] | Apply Rlt_le_trans with (Rmin x0 ``M-m``); [Apply H15 | Apply Rmin_l]]. +Assert H13 := (complet ? H11 H12); Elim H13; Clear H13; Intros; Cut ``0<x0<=M-m``. +Intro; Elim H13; Clear H13; Intros; Exists x0; Split. +Assumption. +Split. +Intros; Cut (EXT alp:R | ``(Rabsolu (z-x))<alp<=x0``/\(E alp)). +Intros; Elim H16; Intros; Elim H17; Clear H17; Intros; Unfold E in H18; Elim H18; Intros; Apply H20; Elim H17; Intros; Assumption. +Elim (classic (EXT alp:R | ``(Rabsolu (z-x)) < alp <= x0``/\(E alp))); Intro. +Assumption. +Assert H17 := (not_ex_all_not ? [alp:R]``(Rabsolu (z-x)) < alp <= x0``/\(E alp) H16); Unfold is_lub in p; Elim p; Intros; Cut (is_upper_bound E ``(Rabsolu (z-x))``). +Intro; Assert H21 := (H19 ? H20); Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H15 H21)). +Unfold is_upper_bound; Intros; Unfold is_upper_bound in H18; Assert H21 := (H18 ? H20); Case (total_order_Rle x1 ``(Rabsolu (z-x))``); Intro. +Assumption. +Elim (H17 x1); Split. +Split; [Auto with real | Assumption]. +Assumption. +Unfold included g; Intros; Elim H15; Intros; Elim H17; Intros; Decompose [and] H18; Cut x0==x2. +Intro; Rewrite H20; Apply H22. +Unfold E in p; EApply is_lub_u. +Apply p. +Apply H21. +Elim H12; Intros; Unfold E in H13; Elim H13; Intros H14 _; Elim H14; Intros H15 _; Unfold is_lub in p; Elim p; Intros; Unfold is_upper_bound in H16; Unfold is_upper_bound in H17; Split. +Apply Rlt_le_trans with x1; [Assumption | Apply (H16 ? H13)]. +Apply H17; Intros; Unfold E in H18; Elim H18; Intros; Elim H19; Intros; Assumption. +Qed. diff --git a/theories7/Reals/Rtrigo.v b/theories7/Reals/Rtrigo.v new file mode 100644 index 00000000..2b19a00a --- /dev/null +++ b/theories7/Reals/Rtrigo.v @@ -0,0 +1,1111 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rtrigo.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Export Rtrigo_fun. +Require Export Rtrigo_def. +Require Export Rtrigo_alt. +Require Export Cos_rel. +Require Export Cos_plus. +Require ZArith_base. +Require Zcomplements. +Require Classical_Prop. +V7only [Import nat_scope. Import Z_scope. Import R_scope.]. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +(** sin_PI2 is the only remaining axiom **) +Axiom sin_PI2 : ``(sin (PI/2))==1``. + +(**********) +Lemma PI_neq0 : ~``PI==0``. +Red; Intro; Assert H0 := PI_RGT_0; Rewrite H in H0; Elim (Rlt_antirefl ? H0). +Qed. + +(**********) +Lemma cos_minus : (x,y:R) ``(cos (x-y))==(cos x)*(cos y)+(sin x)*(sin y)``. +Intros; Unfold Rminus; Rewrite cos_plus. +Rewrite <- cos_sym; Rewrite sin_antisym; Ring. +Qed. + +(**********) +Lemma sin2_cos2 : (x:R) ``(Rsqr (sin x)) + (Rsqr (cos x))==1``. +Intro; Unfold Rsqr; Rewrite Rplus_sym; Rewrite <- (cos_minus x x); Unfold Rminus; Rewrite Rplus_Ropp_r; Apply cos_0. +Qed. + +Lemma cos2 : (x:R) ``(Rsqr (cos x))==1-(Rsqr (sin x))``. +Intro x; Generalize (sin2_cos2 x); Intro H1; Rewrite <- H1; Unfold Rminus; Rewrite <- (Rplus_sym (Rsqr (cos x))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Symmetry; Apply Rplus_Or. +Qed. + +(**********) +Lemma cos_PI2 : ``(cos (PI/2))==0``. +Apply Rsqr_eq_0; Rewrite cos2; Rewrite sin_PI2; Rewrite Rsqr_1; Unfold Rminus; Apply Rplus_Ropp_r. +Qed. + +(**********) +Lemma cos_PI : ``(cos PI)==-1``. +Replace ``PI`` with ``PI/2+PI/2``. +Rewrite cos_plus. +Rewrite sin_PI2; Rewrite cos_PI2. +Ring. +Symmetry; Apply double_var. +Qed. + +Lemma sin_PI : ``(sin PI)==0``. +Assert H := (sin2_cos2 PI). +Rewrite cos_PI in H. +Rewrite <- Rsqr_neg in H. +Rewrite Rsqr_1 in H. +Cut (Rsqr (sin PI))==R0. +Intro; Apply (Rsqr_eq_0 ? H0). +Apply r_Rplus_plus with R1. +Rewrite Rplus_Or; Rewrite Rplus_sym; Exact H. +Qed. + +(**********) +Lemma neg_cos : (x:R) ``(cos (x+PI))==-(cos x)``. +Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. +Qed. + +(**********) +Lemma sin_cos : (x:R) ``(sin x)==-(cos (PI/2+x))``. +Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. +Qed. + +(**********) +Lemma sin_plus : (x,y:R) ``(sin (x+y))==(sin x)*(cos y)+(cos x)*(sin y)``. +Intros. +Rewrite (sin_cos ``x+y``). +Replace ``PI/2+(x+y)`` with ``(PI/2+x)+y``; [Rewrite cos_plus | Ring]. +Rewrite (sin_cos ``PI/2+x``). +Replace ``PI/2+(PI/2+x)`` with ``x+PI``. +Rewrite neg_cos. +Replace (cos ``PI/2+x``) with ``-(sin x)``. +Ring. +Rewrite sin_cos; Rewrite Ropp_Ropp; Reflexivity. +Pattern 1 PI; Rewrite (double_var PI); Ring. +Qed. + +Lemma sin_minus : (x,y:R) ``(sin (x-y))==(sin x)*(cos y)-(cos x)*(sin y)``. +Intros; Unfold Rminus; Rewrite sin_plus. +Rewrite <- cos_sym; Rewrite sin_antisym; Ring. +Qed. + +(**********) +Definition tan [x:R] : R := ``(sin x)/(cos x)``. + +Lemma tan_plus : (x,y:R) ~``(cos x)==0`` -> ~``(cos y)==0`` -> ~``(cos (x+y))==0`` -> ~``1-(tan x)*(tan y)==0`` -> ``(tan (x+y))==((tan x)+(tan y))/(1-(tan x)*(tan y))``. +Intros; Unfold tan; Rewrite sin_plus; Rewrite cos_plus; Unfold Rdiv; Replace ``((cos x)*(cos y)-(sin x)*(sin y))`` with ``((cos x)*(cos y))*(1-(sin x)*/(cos x)*((sin y)*/(cos y)))``. +Rewrite Rinv_Rmult. +Repeat Rewrite <- Rmult_assoc; Replace ``((sin x)*(cos y)+(cos x)*(sin y))*/((cos x)*(cos y))`` with ``((sin x)*/(cos x)+(sin y)*/(cos y))``. +Reflexivity. +Rewrite Rmult_Rplus_distrl; Rewrite Rinv_Rmult. +Repeat Rewrite Rmult_assoc; Repeat Rewrite (Rmult_sym ``(sin x)``); Repeat Rewrite <- Rmult_assoc. +Repeat Rewrite Rinv_r_simpl_m; [Reflexivity | Assumption | Assumption]. +Assumption. +Assumption. +Apply prod_neq_R0; Assumption. +Assumption. +Unfold Rminus; Rewrite Rmult_Rplus_distr; Rewrite Rmult_1r; Apply Rplus_plus_r; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``(sin x)``); Rewrite (Rmult_sym ``(cos y)``); Rewrite <- Ropp_mul3; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite (Rmult_sym (sin x)); Rewrite <- Ropp_mul3; Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r; Rewrite (Rmult_sym ``/(cos y)``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Apply Rmult_1r. +Assumption. +Assumption. +Qed. + +(*******************************************************) +(* Some properties of cos, sin and tan *) +(*******************************************************) + +Lemma sin2 : (x:R) ``(Rsqr (sin x))==1-(Rsqr (cos x))``. +Intro x; Generalize (cos2 x); Intro H1; Rewrite -> H1. +Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Symmetry; Apply Ropp_Ropp. +Qed. + +Lemma sin_2a : (x:R) ``(sin (2*x))==2*(sin x)*(cos x)``. +Intro x; Rewrite double; Rewrite sin_plus. +Rewrite <- (Rmult_sym (sin x)); Symmetry; Rewrite Rmult_assoc; Apply double. +Qed. + +Lemma cos_2a : (x:R) ``(cos (2*x))==(cos x)*(cos x)-(sin x)*(sin x)``. +Intro x; Rewrite double; Apply cos_plus. +Qed. + +Lemma cos_2a_cos : (x:R) ``(cos (2*x))==2*(cos x)*(cos x)-1``. +Intro x; Rewrite double; Unfold Rminus; Rewrite Rmult_assoc; Rewrite cos_plus; Generalize (sin2_cos2 x); Rewrite double; Intro H1; Rewrite <- H1; SqRing. +Qed. + +Lemma cos_2a_sin : (x:R) ``(cos (2*x))==1-2*(sin x)*(sin x)``. +Intro x; Rewrite Rmult_assoc; Unfold Rminus; Repeat Rewrite double. +Generalize (sin2_cos2 x); Intro H1; Rewrite <- H1; Rewrite cos_plus; SqRing. +Qed. + +Lemma tan_2a : (x:R) ~``(cos x)==0`` -> ~``(cos (2*x))==0`` -> ~``1-(tan x)*(tan x)==0`` ->``(tan (2*x))==(2*(tan x))/(1-(tan x)*(tan x))``. +Repeat Rewrite double; Intros; Repeat Rewrite double; Rewrite double in H0; Apply tan_plus; Assumption. +Qed. + +Lemma sin_neg : (x:R) ``(sin (-x))==-(sin x)``. +Apply sin_antisym. +Qed. + +Lemma cos_neg : (x:R) ``(cos (-x))==(cos x)``. +Intro; Symmetry; Apply cos_sym. +Qed. + +Lemma tan_0 : ``(tan 0)==0``. +Unfold tan; Rewrite -> sin_0; Rewrite -> cos_0. +Unfold Rdiv; Apply Rmult_Ol. +Qed. + +Lemma tan_neg : (x:R) ``(tan (-x))==-(tan x)``. +Intros x; Unfold tan; Rewrite sin_neg; Rewrite cos_neg; Unfold Rdiv. +Apply Ropp_mul1. +Qed. + +Lemma tan_minus : (x,y:R) ~``(cos x)==0`` -> ~``(cos y)==0`` -> ~``(cos (x-y))==0`` -> ~``1+(tan x)*(tan y)==0`` -> ``(tan (x-y))==((tan x)-(tan y))/(1+(tan x)*(tan y))``. +Intros; Unfold Rminus; Rewrite tan_plus. +Rewrite tan_neg; Unfold Rminus; Rewrite <- Ropp_mul1; Rewrite Ropp_mul2; Reflexivity. +Assumption. +Rewrite cos_neg; Assumption. +Assumption. +Rewrite tan_neg; Unfold Rminus; Rewrite <- Ropp_mul1; Rewrite Ropp_mul2; Assumption. +Qed. + +Lemma cos_3PI2 : ``(cos (3*(PI/2)))==0``. +Replace ``3*(PI/2)`` with ``PI+(PI/2)``. +Rewrite -> cos_plus; Rewrite -> sin_PI; Rewrite -> cos_PI2; Ring. +Pattern 1 PI; Rewrite (double_var PI). +Ring. +Qed. + +Lemma sin_2PI : ``(sin (2*PI))==0``. +Rewrite -> sin_2a; Rewrite -> sin_PI; Ring. +Qed. + +Lemma cos_2PI : ``(cos (2*PI))==1``. +Rewrite -> cos_2a; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. +Qed. + +Lemma neg_sin : (x:R) ``(sin (x+PI))==-(sin x)``. +Intro x; Rewrite -> sin_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. +Qed. + +Lemma sin_PI_x : (x:R) ``(sin (PI-x))==(sin x)``. +Intro x; Rewrite -> sin_minus; Rewrite -> sin_PI; Rewrite -> cos_PI; Rewrite Rmult_Ol; Unfold Rminus; Rewrite Rplus_Ol; Rewrite Ropp_mul1; Rewrite Ropp_Ropp; Apply Rmult_1l. +Qed. + +Lemma sin_period : (x:R)(k:nat) ``(sin (x+2*(INR k)*PI))==(sin x)``. +Intros x k; Induction k. +Cut ``x+2*(INR O)*PI==x``; [Intro; Rewrite H; Reflexivity | Ring]. +Replace ``x+2*(INR (S k))*PI`` with ``(x+2*(INR k)*PI)+(2*PI)``; [Rewrite -> sin_plus; Rewrite -> sin_2PI; Rewrite -> cos_2PI; Ring; Apply Hreck | Rewrite -> S_INR; Ring]. +Qed. + +Lemma cos_period : (x:R)(k:nat) ``(cos (x+2*(INR k)*PI))==(cos x)``. +Intros x k; Induction k. +Cut ``x+2*(INR O)*PI==x``; [Intro; Rewrite H; Reflexivity | Ring]. +Replace ``x+2*(INR (S k))*PI`` with ``(x+2*(INR k)*PI)+(2*PI)``; [Rewrite -> cos_plus; Rewrite -> sin_2PI; Rewrite -> cos_2PI; Ring; Apply Hreck | Rewrite -> S_INR; Ring]. +Qed. + +Lemma sin_shift : (x:R) ``(sin (PI/2-x))==(cos x)``. +Intro x; Rewrite -> sin_minus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. +Qed. + +Lemma cos_shift : (x:R) ``(cos (PI/2-x))==(sin x)``. +Intro x; Rewrite -> cos_minus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. +Qed. + +Lemma cos_sin : (x:R) ``(cos x)==(sin (PI/2+x))``. +Intro x; Rewrite -> sin_plus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. +Qed. + +Lemma PI2_RGT_0 : ``0<PI/2``. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup]. +Qed. + +Lemma SIN_bound : (x:R) ``-1<=(sin x)<=1``. +Intro; Case (total_order_Rle ``-1`` (sin x)); Intro. +Case (total_order_Rle (sin x) ``1``); Intro. +Split; Assumption. +Cut ``1<(sin x)``. +Intro; Generalize (Rsqr_incrst_1 ``1`` (sin x) H (Rlt_le ``0`` ``1`` Rlt_R0_R1) (Rlt_le ``0`` (sin x) (Rlt_trans ``0`` ``1`` (sin x) Rlt_R0_R1 H))); Rewrite Rsqr_1; Intro; Rewrite sin2 in H0; Unfold Rminus in H0; Generalize (Rlt_compatibility ``-1`` ``1`` ``1+ -(Rsqr (cos x))`` H0); Repeat Rewrite <- Rplus_assoc; Repeat Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Intro; Rewrite <- Ropp_O in H1; Generalize (Rlt_Ropp ``-0`` ``-(Rsqr (cos x))`` H1); Repeat Rewrite Ropp_Ropp; Intro; Generalize (pos_Rsqr (cos x)); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` (Rsqr (cos x)) ``0`` H3 H2)). +Auto with real. +Cut ``(sin x)< -1``. +Intro; Generalize (Rlt_Ropp (sin x) ``-1`` H); Rewrite Ropp_Ropp; Clear H; Intro; Generalize (Rsqr_incrst_1 ``1`` ``-(sin x)`` H (Rlt_le ``0`` ``1`` Rlt_R0_R1) (Rlt_le ``0`` ``-(sin x)`` (Rlt_trans ``0`` ``1`` ``-(sin x)`` Rlt_R0_R1 H))); Rewrite Rsqr_1; Intro; Rewrite <- Rsqr_neg in H0; Rewrite sin2 in H0; Unfold Rminus in H0; Generalize (Rlt_compatibility ``-1`` ``1`` ``1+ -(Rsqr (cos x))`` H0); Repeat Rewrite <- Rplus_assoc; Repeat Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Intro; Rewrite <- Ropp_O in H1; Generalize (Rlt_Ropp ``-0`` ``-(Rsqr (cos x))`` H1); Repeat Rewrite Ropp_Ropp; Intro; Generalize (pos_Rsqr (cos x)); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` (Rsqr (cos x)) ``0`` H3 H2)). +Auto with real. +Qed. + +Lemma COS_bound : (x:R) ``-1<=(cos x)<=1``. +Intro; Rewrite <- sin_shift; Apply SIN_bound. +Qed. + +Lemma cos_sin_0 : (x:R) ~(``(cos x)==0``/\``(sin x)==0``). +Intro; Red; Intro; Elim H; Intros; Generalize (sin2_cos2 x); Intro; Rewrite H0 in H2; Rewrite H1 in H2; Repeat Rewrite Rsqr_O in H2; Rewrite Rplus_Or in H2; Generalize Rlt_R0_R1; Intro; Rewrite <- H2 in H3; Elim (Rlt_antirefl ``0`` H3). +Qed. + +Lemma cos_sin_0_var : (x:R) ~``(cos x)==0``\/~``(sin x)==0``. +Intro; Apply not_and_or; Apply cos_sin_0. +Qed. + +(*****************************************************************) +(* Using series definitions of cos and sin *) +(*****************************************************************) + +Definition sin_lb [a:R] : R := (sin_approx a (3)). +Definition sin_ub [a:R] : R := (sin_approx a (4)). +Definition cos_lb [a:R] : R := (cos_approx a (3)). +Definition cos_ub [a:R] : R := (cos_approx a (4)). + +Lemma sin_lb_gt_0 : (a:R) ``0<a``->``a<=PI/2``->``0<(sin_lb a)``. +Intros. +Unfold sin_lb; Unfold sin_approx; Unfold sin_term. +Pose Un := [i:nat]``(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))``. +Replace (sum_f_R0 [i:nat] ``(pow ( -1) i)*(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))`` (S (S (S O)))) with (sum_f_R0 [i:nat]``(pow (-1) i)*(Un i)`` (3)); [Idtac | Apply sum_eq; Intros; Unfold Un; Reflexivity]. +Cut (n:nat)``(Un (S n))<(Un n)``. +Intro; Simpl. +Repeat Rewrite Rmult_1l; Repeat Rewrite Rmult_1r; Replace ``-1*(Un (S O))`` with ``-(Un (S O))``; [Idtac | Ring]; Replace ``-1* -1*(Un (S (S O)))`` with ``(Un (S (S O)))``; [Idtac | Ring]; Replace ``-1*( -1* -1)*(Un (S (S (S O))))`` with ``-(Un (S (S (S O))))``; [Idtac | Ring]; Replace ``(Un O)+ -(Un (S O))+(Un (S (S O)))+ -(Un (S (S (S O))))`` with ``((Un O)-(Un (S O)))+((Un (S (S O)))-(Un (S (S (S O)))))``; [Idtac | Ring]. +Apply gt0_plus_gt0_is_gt0. +Unfold Rminus; Apply Rlt_anti_compatibility with (Un (S O)); Rewrite Rplus_Or; Rewrite (Rplus_sym (Un (S O))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H1. +Unfold Rminus; Apply Rlt_anti_compatibility with (Un (S (S (S O)))); Rewrite Rplus_Or; Rewrite (Rplus_sym (Un (S (S (S O))))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H1. +Intro; Unfold Un. +Cut (plus (mult (2) (S n)) (S O)) = (plus (plus (mult (2) n) (S O)) (2)). +Intro; Rewrite H1. +Rewrite pow_add; Unfold Rdiv; Rewrite Rmult_assoc; Apply Rlt_monotony. +Apply pow_lt; Assumption. +Rewrite <- H1; Apply Rlt_monotony_contra with (INR (fact (plus (mult (S (S O)) n) (S O)))). +Apply lt_INR_0; Apply neq_O_lt. +Assert H2 := (fact_neq_0 (plus (mult (2) n) (1))). +Red; Intro; Elim H2; Symmetry; Assumption. +Rewrite <- Rinv_r_sym. +Apply Rlt_monotony_contra with (INR (fact (plus (mult (S (S O)) (S n)) (S O)))). +Apply lt_INR_0; Apply neq_O_lt. +Assert H2 := (fact_neq_0 (plus (mult (2) (S n)) (1))). +Red; Intro; Elim H2; Symmetry; Assumption. +Rewrite (Rmult_sym (INR (fact (plus (mult (S (S O)) (S n)) (S O))))); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Do 2 Rewrite Rmult_1r; Apply Rle_lt_trans with ``(INR (fact (plus (mult (S (S O)) n) (S O))))*4``. +Apply Rle_monotony. +Replace R0 with (INR O); [Idtac | Reflexivity]; Apply le_INR; Apply le_O_n. +Simpl; Rewrite Rmult_1r; Replace ``4`` with ``(Rsqr 2)``; [Idtac | SqRing]; Replace ``a*a`` with (Rsqr a); [Idtac | Reflexivity]; Apply Rsqr_incr_1. +Apply Rle_trans with ``PI/2``; [Assumption | Unfold Rdiv; Apply Rle_monotony_contra with ``2``; [ Sup0 | Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m; [Replace ``2*2`` with ``4``; [Apply PI_4 | Ring] | DiscrR]]]. +Left; Assumption. +Left; Sup0. +Rewrite H1; Replace (plus (plus (mult (S (S O)) n) (S O)) (S (S O))) with (S (S (plus (mult (S (S O)) n) (S O)))). +Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR. +Repeat Rewrite <- Rmult_assoc. +Rewrite <- (Rmult_sym (INR (fact (plus (mult (S (S O)) n) (S O))))). +Rewrite Rmult_assoc. +Apply Rlt_monotony. +Apply lt_INR_0; Apply neq_O_lt. +Assert H2 := (fact_neq_0 (plus (mult (2) n) (1))). +Red; Intro; Elim H2; Symmetry; Assumption. +Do 2 Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Pose x := (INR n); Unfold INR. +Replace ``(2*x+1+1+1)*(2*x+1+1)`` with ``4*x*x+10*x+6``; [Idtac | Ring]. +Apply Rlt_anti_compatibility with ``-4``; Rewrite Rplus_Ropp_l; Replace ``-4+(4*x*x+10*x+6)`` with ``(4*x*x+10*x)+2``; [Idtac | Ring]. +Apply ge0_plus_gt0_is_gt0. +Cut ``0<=x``. +Intro; Apply ge0_plus_ge0_is_ge0; Repeat Apply Rmult_le_pos; Assumption Orelse Left; Sup. +Unfold x; Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Sup0. +Apply INR_eq; Do 2 Rewrite S_INR; Do 3 Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_eq; Do 3 Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Qed. + +Lemma SIN : (a:R) ``0<=a`` -> ``a<=PI`` -> ``(sin_lb a)<=(sin a)<=(sin_ub a)``. +Intros; Unfold sin_lb sin_ub; Apply (sin_bound a (S O) H H0). +Qed. + +Lemma COS : (a:R) ``-PI/2<=a`` -> ``a<=PI/2`` -> ``(cos_lb a)<=(cos a)<=(cos_ub a)``. +Intros; Unfold cos_lb cos_ub; Apply (cos_bound a (S O) H H0). +Qed. + +(**********) +Lemma _PI2_RLT_0 : ``-(PI/2)<0``. +Rewrite <- Ropp_O; Apply Rlt_Ropp1; Apply PI2_RGT_0. +Qed. + +Lemma PI4_RLT_PI2 : ``PI/4<PI/2``. +Unfold Rdiv; Apply Rlt_monotony. +Apply PI_RGT_0. +Apply Rinv_lt. +Apply Rmult_lt_pos; Sup0. +Pattern 1 ``2``; Rewrite <- Rplus_Or. +Replace ``4`` with ``2+2``; [Apply Rlt_compatibility; Sup0 | Ring]. +Qed. + +Lemma PI2_Rlt_PI : ``PI/2<PI``. +Unfold Rdiv; Pattern 2 PI; Rewrite <- Rmult_1r. +Apply Rlt_monotony. +Apply PI_RGT_0. +Pattern 3 R1; Rewrite <- Rinv_R1; Apply Rinv_lt. +Rewrite Rmult_1l; Sup0. +Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. +Qed. + +(********************************************) +(* Increasing and decreasing of COS and SIN *) +(********************************************) +Theorem sin_gt_0 : (x:R) ``0<x`` -> ``x<PI`` -> ``0<(sin x)``. +Intros; Elim (SIN x (Rlt_le R0 x H) (Rlt_le x PI H0)); Intros H1 _; Case (total_order x ``PI/2``); Intro H2. +Apply Rlt_le_trans with (sin_lb x). +Apply sin_lb_gt_0; [Assumption | Left; Assumption]. +Assumption. +Elim H2; Intro H3. +Rewrite H3; Rewrite sin_PI2; Apply Rlt_R0_R1. +Rewrite <- sin_PI_x; Generalize (Rgt_Ropp x ``PI/2`` H3); Intro H4; Generalize (Rlt_compatibility PI (Ropp x) (Ropp ``PI/2``) H4). +Replace ``PI+(-x)`` with ``PI-x``. +Replace ``PI+ -(PI/2)`` with ``PI/2``. +Intro H5; Generalize (Rlt_Ropp x PI H0); Intro H6; Change ``-PI < -x`` in H6; Generalize (Rlt_compatibility PI (Ropp PI) (Ropp x) H6). +Rewrite Rplus_Ropp_r. +Replace ``PI+ -x`` with ``PI-x``. +Intro H7; Elim (SIN ``PI-x`` (Rlt_le R0 ``PI-x`` H7) (Rlt_le ``PI-x`` PI (Rlt_trans ``PI-x`` ``PI/2`` ``PI`` H5 PI2_Rlt_PI))); Intros H8 _; Generalize (sin_lb_gt_0 ``PI-x`` H7 (Rlt_le ``PI-x`` ``PI/2`` H5)); Intro H9; Apply (Rlt_le_trans ``0`` ``(sin_lb (PI-x))`` ``(sin (PI-x))`` H9 H8). +Reflexivity. +Pattern 2 PI; Rewrite double_var; Ring. +Reflexivity. +Qed. + +Theorem cos_gt_0 : (x:R) ``-(PI/2)<x`` -> ``x<PI/2`` -> ``0<(cos x)``. +Intros; Rewrite cos_sin; Generalize (Rlt_compatibility ``PI/2`` ``-(PI/2)`` x H). +Rewrite Rplus_Ropp_r; Intro H1; Generalize (Rlt_compatibility ``PI/2`` x ``PI/2`` H0); Rewrite <- double_var; Intro H2; Apply (sin_gt_0 ``PI/2+x`` H1 H2). +Qed. + +Lemma sin_ge_0 : (x:R) ``0<=x`` -> ``x<=PI`` -> ``0<=(sin x)``. +Intros x H1 H2; Elim H1; Intro H3; [ Elim H2; Intro H4; [ Left ; Apply (sin_gt_0 x H3 H4) | Rewrite H4; Right; Symmetry; Apply sin_PI ] | Rewrite <- H3; Right; Symmetry; Apply sin_0]. +Qed. + +Lemma cos_ge_0 : (x:R) ``-(PI/2)<=x`` -> ``x<=PI/2`` -> ``0<=(cos x)``. +Intros x H1 H2; Elim H1; Intro H3; [ Elim H2; Intro H4; [ Left ; Apply (cos_gt_0 x H3 H4) | Rewrite H4; Right; Symmetry; Apply cos_PI2 ] | Rewrite <- H3; Rewrite cos_neg; Right; Symmetry; Apply cos_PI2 ]. +Qed. + +Lemma sin_le_0 : (x:R) ``PI<=x`` -> ``x<=2*PI`` -> ``(sin x)<=0``. +Intros x H1 H2; Apply Rle_sym2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (sin x)); Apply Rle_Ropp; Rewrite <- neg_sin; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``; [Rewrite -> (sin_period (Rminus x PI) (S O)); Apply sin_ge_0; [Replace ``x-PI`` with ``x+(-PI)``; [Rewrite Rplus_sym; Replace ``0`` with ``(-PI)+PI``; [Apply Rle_compatibility; Assumption | Ring] | Ring] | Replace ``x-PI`` with ``x+(-PI)``; Rewrite Rplus_sym; [Pattern 2 PI; Replace ``PI`` with ``(-PI)+2*PI``; [Apply Rle_compatibility; Assumption | Ring] | Ring]] |Unfold INR; Ring]. +Qed. + +Lemma cos_le_0 : (x:R) ``PI/2<=x``->``x<=3*(PI/2)``->``(cos x)<=0``. +Intros x H1 H2; Apply Rle_sym2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (cos x)); Apply Rle_Ropp; Rewrite <- neg_cos; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``. +Rewrite cos_period; Apply cos_ge_0. +Replace ``-(PI/2)`` with ``-PI+(PI/2)``. +Unfold Rminus; Rewrite (Rplus_sym x); Apply Rle_compatibility; Assumption. +Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. +Unfold Rminus; Rewrite Rplus_sym; Replace ``PI/2`` with ``(-PI)+3*(PI/2)``. +Apply Rle_compatibility; Assumption. +Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. +Unfold INR; Ring. +Qed. + +Lemma sin_lt_0 : (x:R) ``PI<x`` -> ``x<2*PI`` -> ``(sin x)<0``. +Intros x H1 H2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (sin x)); Apply Rlt_Ropp; Rewrite <- neg_sin; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``; [Rewrite -> (sin_period (Rminus x PI) (S O)); Apply sin_gt_0; [Replace ``x-PI`` with ``x+(-PI)``; [Rewrite Rplus_sym; Replace ``0`` with ``(-PI)+PI``; [Apply Rlt_compatibility; Assumption | Ring] | Ring] | Replace ``x-PI`` with ``x+(-PI)``; Rewrite Rplus_sym; [Pattern 2 PI; Replace ``PI`` with ``(-PI)+2*PI``; [Apply Rlt_compatibility; Assumption | Ring] | Ring]] |Unfold INR; Ring]. +Qed. + +Lemma sin_lt_0_var : (x:R) ``-PI<x`` -> ``x<0`` -> ``(sin x)<0``. +Intros; Generalize (Rlt_compatibility ``2*PI`` ``-PI`` x H); Replace ``2*PI+(-PI)`` with ``PI``; [Intro H1; Rewrite Rplus_sym in H1; Generalize (Rlt_compatibility ``2*PI`` x ``0`` H0); Intro H2; Rewrite (Rplus_sym ``2*PI``) in H2; Rewrite <- (Rplus_sym R0) in H2; Rewrite Rplus_Ol in H2; Rewrite <- (sin_period x (1)); Unfold INR; Replace ``2*1*PI`` with ``2*PI``; [Apply (sin_lt_0 ``x+2*PI`` H1 H2) | Ring] | Ring]. +Qed. + +Lemma cos_lt_0 : (x:R) ``PI/2<x`` -> ``x<3*(PI/2)``-> ``(cos x)<0``. +Intros x H1 H2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (cos x)); Apply Rlt_Ropp; Rewrite <- neg_cos; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``. +Rewrite cos_period; Apply cos_gt_0. +Replace ``-(PI/2)`` with ``-PI+(PI/2)``. +Unfold Rminus; Rewrite (Rplus_sym x); Apply Rlt_compatibility; Assumption. +Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. +Unfold Rminus; Rewrite Rplus_sym; Replace ``PI/2`` with ``(-PI)+3*(PI/2)``. +Apply Rlt_compatibility; Assumption. +Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. +Unfold INR; Ring. +Qed. + +Lemma tan_gt_0 : (x:R) ``0<x`` -> ``x<PI/2`` -> ``0<(tan x)``. +Intros x H1 H2; Unfold tan; Generalize _PI2_RLT_0; Generalize (Rlt_trans R0 x ``PI/2`` H1 H2); Intros; Generalize (Rlt_trans ``-(PI/2)`` R0 x H0 H1); Intro H5; Generalize (Rlt_trans x ``PI/2`` PI H2 PI2_Rlt_PI); Intro H7; Unfold Rdiv; Apply Rmult_lt_pos. +Apply sin_gt_0; Assumption. +Apply Rlt_Rinv; Apply cos_gt_0; Assumption. +Qed. + +Lemma tan_lt_0 : (x:R) ``-(PI/2)<x``->``x<0``->``(tan x)<0``. +Intros x H1 H2; Unfold tan; Generalize (cos_gt_0 x H1 (Rlt_trans x ``0`` ``PI/2`` H2 PI2_RGT_0)); Intro H3; Rewrite <- Ropp_O; Replace ``(sin x)/(cos x)`` with ``- ((-(sin x))/(cos x))``. +Rewrite <- sin_neg; Apply Rgt_Ropp; Change ``0<(sin (-x))/(cos x)``; Unfold Rdiv; Apply Rmult_lt_pos. +Apply sin_gt_0. +Rewrite <- Ropp_O; Apply Rgt_Ropp; Assumption. +Apply Rlt_trans with ``PI/2``. +Rewrite <- (Ropp_Ropp ``PI/2``); Apply Rgt_Ropp; Assumption. +Apply PI2_Rlt_PI. +Apply Rlt_Rinv; Assumption. +Unfold Rdiv; Ring. +Qed. + +Lemma cos_ge_0_3PI2 : (x:R) ``3*(PI/2)<=x``->``x<=2*PI``->``0<=(cos x)``. +Intros; Rewrite <- cos_neg; Rewrite <- (cos_period ``-x`` (1)); Unfold INR; Replace ``-x+2*1*PI`` with ``2*PI-x``. +Generalize (Rle_Ropp x ``2*PI`` H0); Intro H1; Generalize (Rle_sym2 ``-(2*PI)`` ``-x`` H1); Clear H1; Intro H1; Generalize (Rle_compatibility ``2*PI`` ``-(2*PI)`` ``-x`` H1). +Rewrite Rplus_Ropp_r. +Intro H2; Generalize (Rle_Ropp ``3*(PI/2)`` x H); Intro H3; Generalize (Rle_sym2 ``-x`` ``-(3*(PI/2))`` H3); Clear H3; Intro H3; Generalize (Rle_compatibility ``2*PI`` ``-x`` ``-(3*(PI/2))`` H3). +Replace ``2*PI+ -(3*PI/2)`` with ``PI/2``. +Intro H4; Apply (cos_ge_0 ``2*PI-x`` (Rlt_le ``-(PI/2)`` ``2*PI-x`` (Rlt_le_trans ``-(PI/2)`` ``0`` ``2*PI-x`` _PI2_RLT_0 H2)) H4). +Rewrite double; Pattern 2 3 PI; Rewrite double_var; Ring. +Ring. +Qed. + +Lemma form1 : (p,q:R) ``(cos p)+(cos q)==2*(cos ((p-q)/2))*(cos ((p+q)/2))``. +Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. +Rewrite <- (cos_neg q); Replace``-q`` with ``(p-q)/2-(p+q)/2``. +Rewrite cos_plus; Rewrite cos_minus; Ring. +Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. +Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. +Qed. + +Lemma form2 : (p,q:R) ``(cos p)-(cos q)==-2*(sin ((p-q)/2))*(sin ((p+q)/2))``. +Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. +Rewrite <- (cos_neg q); Replace``-q`` with ``(p-q)/2-(p+q)/2``. +Rewrite cos_plus; Rewrite cos_minus; Ring. +Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. +Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. +Qed. + +Lemma form3 : (p,q:R) ``(sin p)+(sin q)==2*(cos ((p-q)/2))*(sin ((p+q)/2))``. +Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. +Pattern 3 q; Replace ``q`` with ``(p+q)/2-(p-q)/2``. +Rewrite sin_plus; Rewrite sin_minus; Ring. +Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. +Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. +Qed. + +Lemma form4 : (p,q:R) ``(sin p)-(sin q)==2*(cos ((p+q)/2))*(sin ((p-q)/2))``. +Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. +Pattern 3 q; Replace ``q`` with ``(p+q)/2-(p-q)/2``. +Rewrite sin_plus; Rewrite sin_minus; Ring. +Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. +Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. + +Qed. + +Lemma sin_increasing_0 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``(sin x)<(sin y)``->``x<y``. +Intros; Cut ``(sin ((x-y)/2))<0``. +Intro H4; Case (total_order ``(x-y)/2`` ``0``); Intro H5. +Assert Hyp : ``0<2``. +Sup0. +Generalize (Rlt_monotony ``2`` ``(x-y)/2`` ``0`` Hyp H5). +Unfold Rdiv. +Rewrite <- Rmult_assoc. +Rewrite Rinv_r_simpl_m. +Rewrite Rmult_Or. +Clear H5; Intro H5; Apply Rminus_lt; Assumption. +DiscrR. +Elim H5; Intro H6. +Rewrite H6 in H4; Rewrite sin_0 in H4; Elim (Rlt_antirefl ``0`` H4). +Change ``0<(x-y)/2`` in H6; Generalize (Rle_Ropp ``-(PI/2)`` y H1). +Rewrite Ropp_Ropp. +Intro H7; Generalize (Rle_sym2 ``-y`` ``PI/2`` H7); Clear H7; Intro H7; Generalize (Rplus_le x ``PI/2`` ``-y`` ``PI/2`` H0 H7). +Rewrite <- double_var. +Intro H8. +Assert Hyp : ``0<2``. +Sup0. +Generalize (Rle_monotony ``(Rinv 2)`` ``x-y`` PI (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H8). +Repeat Rewrite (Rmult_sym ``/2``). +Intro H9; Generalize (sin_gt_0 ``(x-y)/2`` H6 (Rle_lt_trans ``(x-y)/2`` ``PI/2`` PI H9 PI2_Rlt_PI)); Intro H10; Elim (Rlt_antirefl ``(sin ((x-y)/2))`` (Rlt_trans ``(sin ((x-y)/2))`` ``0`` ``(sin ((x-y)/2))`` H4 H10)). +Generalize (Rlt_minus (sin x) (sin y) H3); Clear H3; Intro H3; Rewrite form4 in H3; Generalize (Rplus_le x ``PI/2`` y ``PI/2`` H0 H2). +Rewrite <- double_var. +Assert Hyp : ``0<2``. +Sup0. +Intro H4; Generalize (Rle_monotony ``(Rinv 2)`` ``x+y`` PI (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H4). +Repeat Rewrite (Rmult_sym ``/2``). +Clear H4; Intro H4; Generalize (Rplus_le ``-(PI/2)`` x ``-(PI/2)`` y H H1); Replace ``-(PI/2)+(-(PI/2))`` with ``-PI``. +Intro H5; Generalize (Rle_monotony ``(Rinv 2)`` ``-PI`` ``x+y`` (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H5). +Replace ``/2*(x+y)`` with ``(x+y)/2``. +Replace ``/2*(-PI)`` with ``-(PI/2)``. +Clear H5; Intro H5; Elim H4; Intro H40. +Elim H5; Intro H50. +Generalize (cos_gt_0 ``(x+y)/2`` H50 H40); Intro H6; Generalize (Rlt_monotony ``2`` ``0`` ``(cos ((x+y)/2))`` Hyp H6). +Rewrite Rmult_Or. +Clear H6; Intro H6; Case (case_Rabsolu ``(sin ((x-y)/2))``); Intro H7. +Assumption. +Generalize (Rle_sym2 ``0`` ``(sin ((x-y)/2))`` H7); Clear H7; Intro H7; Generalize (Rmult_le_pos ``2*(cos ((x+y)/2))`` ``(sin ((x-y)/2))`` (Rlt_le ``0`` ``2*(cos ((x+y)/2))`` H6) H7); Intro H8; Generalize (Rle_lt_trans ``0`` ``2*(cos ((x+y)/2))*(sin ((x-y)/2))`` ``0`` H8 H3); Intro H9; Elim (Rlt_antirefl ``0`` H9). +Rewrite <- H50 in H3; Rewrite cos_neg in H3; Rewrite cos_PI2 in H3; Rewrite Rmult_Or in H3; Rewrite Rmult_Ol in H3; Elim (Rlt_antirefl ``0`` H3). +Unfold Rdiv in H3. +Rewrite H40 in H3; Assert H50 := cos_PI2; Unfold Rdiv in H50; Rewrite H50 in H3; Rewrite Rmult_Or in H3; Rewrite Rmult_Ol in H3; Elim (Rlt_antirefl ``0`` H3). +Unfold Rdiv. +Rewrite <- Ropp_mul1. +Apply Rmult_sym. +Unfold Rdiv; Apply Rmult_sym. +Pattern 1 PI; Rewrite double_var. +Rewrite Ropp_distr1. +Reflexivity. +Qed. + +Lemma sin_increasing_1 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``x<y``->``(sin x)<(sin y)``. +Intros; Generalize (Rlt_compatibility ``x`` ``x`` ``y`` H3); Intro H4; Generalize (Rplus_le ``-(PI/2)`` x ``-(PI/2)`` x H H); Replace ``-(PI/2)+ (-(PI/2))`` with ``-PI``. +Assert Hyp : ``0<2``. +Sup0. +Intro H5; Generalize (Rle_lt_trans ``-PI`` ``x+x`` ``x+y`` H5 H4); Intro H6; Generalize (Rlt_monotony ``(Rinv 2)`` ``-PI`` ``x+y`` (Rlt_Rinv ``2`` Hyp) H6); Replace ``/2*(-PI)`` with ``-(PI/2)``. +Replace ``/2*(x+y)`` with ``(x+y)/2``. +Clear H4 H5 H6; Intro H4; Generalize (Rlt_compatibility ``y`` ``x`` ``y`` H3); Intro H5; Rewrite Rplus_sym in H5; Generalize (Rplus_le y ``PI/2`` y ``PI/2`` H2 H2). +Rewrite <- double_var. +Intro H6; Generalize (Rlt_le_trans ``x+y`` ``y+y`` PI H5 H6); Intro H7; Generalize (Rlt_monotony ``(Rinv 2)`` ``x+y`` PI (Rlt_Rinv ``2`` Hyp) H7); Replace ``/2*PI`` with ``PI/2``. +Replace ``/2*(x+y)`` with ``(x+y)/2``. +Clear H5 H6 H7; Intro H5; Generalize (Rle_Ropp ``-(PI/2)`` y H1); Rewrite Ropp_Ropp; Clear H1; Intro H1; Generalize (Rle_sym2 ``-y`` ``PI/2`` H1); Clear H1; Intro H1; Generalize (Rle_Ropp y ``PI/2`` H2); Clear H2; Intro H2; Generalize (Rle_sym2 ``-(PI/2)`` ``-y`` H2); Clear H2; Intro H2; Generalize (Rlt_compatibility ``-y`` x y H3); Replace ``-y+x`` with ``x-y``. +Rewrite Rplus_Ropp_l. +Intro H6; Generalize (Rlt_monotony ``(Rinv 2)`` ``x-y`` ``0`` (Rlt_Rinv ``2`` Hyp) H6); Rewrite Rmult_Or; Replace ``/2*(x-y)`` with ``(x-y)/2``. +Clear H6; Intro H6; Generalize (Rplus_le ``-(PI/2)`` x ``-(PI/2)`` ``-y`` H H2); Replace ``-(PI/2)+ (-(PI/2))`` with ``-PI``. +Replace `` x+ -y`` with ``x-y``. +Intro H7; Generalize (Rle_monotony ``(Rinv 2)`` ``-PI`` ``x-y`` (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H7); Replace ``/2*(-PI)`` with ``-(PI/2)``. +Replace ``/2*(x-y)`` with ``(x-y)/2``. +Clear H7; Intro H7; Clear H H0 H1 H2; Apply Rminus_lt; Rewrite form4; Generalize (cos_gt_0 ``(x+y)/2`` H4 H5); Intro H8; Generalize (Rmult_lt_pos ``2`` ``(cos ((x+y)/2))`` Hyp H8); Clear H8; Intro H8; Cut ``-PI< -(PI/2)``. +Intro H9; Generalize (sin_lt_0_var ``(x-y)/2`` (Rlt_le_trans ``-PI`` ``-(PI/2)`` ``(x-y)/2`` H9 H7) H6); Intro H10; Generalize (Rlt_anti_monotony ``(sin ((x-y)/2))`` ``0`` ``2*(cos ((x+y)/2))`` H10 H8); Intro H11; Rewrite Rmult_Or in H11; Rewrite Rmult_sym; Assumption. +Apply Rlt_Ropp; Apply PI2_Rlt_PI. +Unfold Rdiv; Apply Rmult_sym. +Unfold Rdiv; Rewrite <- Ropp_mul1; Apply Rmult_sym. +Reflexivity. +Pattern 1 PI; Rewrite double_var. +Rewrite Ropp_distr1. +Reflexivity. +Unfold Rdiv; Apply Rmult_sym. +Unfold Rminus; Apply Rplus_sym. +Unfold Rdiv; Apply Rmult_sym. +Unfold Rdiv; Apply Rmult_sym. +Unfold Rdiv; Apply Rmult_sym. +Unfold Rdiv. +Rewrite <- Ropp_mul1. +Apply Rmult_sym. +Pattern 1 PI; Rewrite double_var. +Rewrite Ropp_distr1. +Reflexivity. +Qed. + +Lemma sin_decreasing_0 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``(sin x)<(sin y)`` -> ``y<x``. +Intros; Rewrite <- (sin_PI_x x) in H3; Rewrite <- (sin_PI_x y) in H3; Generalize (Rlt_Ropp ``(sin (PI-x))`` ``(sin (PI-y))`` H3); Repeat Rewrite <- sin_neg; Generalize (Rle_compatibility ``-PI`` x ``3*(PI/2)`` H); Generalize (Rle_compatibility ``-PI`` ``PI/2`` x H0); Generalize (Rle_compatibility ``-PI`` y ``3*(PI/2)`` H1); Generalize (Rle_compatibility ``-PI`` ``PI/2`` y H2); Replace ``-PI+x`` with ``x-PI``. +Replace ``-PI+PI/2`` with ``-(PI/2)``. +Replace ``-PI+y`` with ``y-PI``. +Replace ``-PI+3*(PI/2)`` with ``PI/2``. +Replace ``-(PI-x)`` with ``x-PI``. +Replace ``-(PI-y)`` with ``y-PI``. +Intros; Change ``(sin (y-PI))<(sin (x-PI))`` in H8; Apply Rlt_anti_compatibility with ``-PI``; Rewrite Rplus_sym; Replace ``y+ (-PI)`` with ``y-PI``. +Rewrite Rplus_sym; Replace ``x+ (-PI)`` with ``x-PI``. +Apply (sin_increasing_0 ``y-PI`` ``x-PI`` H4 H5 H6 H7 H8). +Reflexivity. +Reflexivity. +Unfold Rminus; Rewrite Ropp_distr1. +Rewrite Ropp_Ropp. +Apply Rplus_sym. +Unfold Rminus; Rewrite Ropp_distr1. +Rewrite Ropp_Ropp. +Apply Rplus_sym. +Pattern 2 PI; Rewrite double_var. +Rewrite Ropp_distr1. +Ring. +Unfold Rminus; Apply Rplus_sym. +Pattern 2 PI; Rewrite double_var. +Rewrite Ropp_distr1. +Ring. +Unfold Rminus; Apply Rplus_sym. +Qed. + +Lemma sin_decreasing_1 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``x<y`` -> ``(sin y)<(sin x)``. +Intros; Rewrite <- (sin_PI_x x); Rewrite <- (sin_PI_x y); Generalize (Rle_compatibility ``-PI`` x ``3*(PI/2)`` H); Generalize (Rle_compatibility ``-PI`` ``PI/2`` x H0); Generalize (Rle_compatibility ``-PI`` y ``3*(PI/2)`` H1); Generalize (Rle_compatibility ``-PI`` ``PI/2`` y H2); Generalize (Rlt_compatibility ``-PI`` x y H3); Replace ``-PI+PI/2`` with ``-(PI/2)``. +Replace ``-PI+y`` with ``y-PI``. +Replace ``-PI+3*(PI/2)`` with ``PI/2``. +Replace ``-PI+x`` with ``x-PI``. +Intros; Apply Ropp_Rlt; Repeat Rewrite <- sin_neg; Replace ``-(PI-x)`` with ``x-PI``. +Replace ``-(PI-y)`` with ``y-PI``. +Apply (sin_increasing_1 ``x-PI`` ``y-PI`` H7 H8 H5 H6 H4). +Unfold Rminus; Rewrite Ropp_distr1. +Rewrite Ropp_Ropp. +Apply Rplus_sym. +Unfold Rminus; Rewrite Ropp_distr1. +Rewrite Ropp_Ropp. +Apply Rplus_sym. +Unfold Rminus; Apply Rplus_sym. +Pattern 2 PI; Rewrite double_var; Ring. +Unfold Rminus; Apply Rplus_sym. +Pattern 2 PI; Rewrite double_var; Ring. +Qed. + +Lemma cos_increasing_0 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``(cos x)<(cos y)`` -> ``x<y``. +Intros x y H1 H2 H3 H4; Rewrite <- (cos_neg x); Rewrite <- (cos_neg y); Rewrite <- (cos_period ``-x`` (1)); Rewrite <- (cos_period ``-y`` (1)); Unfold INR; Replace ``-x+2*1*PI`` with ``PI/2-(x-3*(PI/2))``. +Replace ``-y+2*1*PI`` with ``PI/2-(y-3*(PI/2))``. +Repeat Rewrite cos_shift; Intro H5; Generalize (Rle_compatibility ``-3*(PI/2)`` PI x H1); Generalize (Rle_compatibility ``-3*(PI/2)`` x ``2*PI`` H2); Generalize (Rle_compatibility ``-3*(PI/2)`` PI y H3); Generalize (Rle_compatibility ``-3*(PI/2)`` y ``2*PI`` H4). +Replace ``-3*(PI/2)+y`` with ``y-3*(PI/2)``. +Replace ``-3*(PI/2)+x`` with ``x-3*(PI/2)``. +Replace ``-3*(PI/2)+2*PI`` with ``PI/2``. +Replace ``-3*PI/2+PI`` with ``-(PI/2)``. +Clear H1 H2 H3 H4; Intros H1 H2 H3 H4; Apply Rlt_anti_compatibility with ``-3*(PI/2)``; Replace ``-3*PI/2+x`` with ``x-3*(PI/2)``. +Replace ``-3*PI/2+y`` with ``y-3*(PI/2)``. +Apply (sin_increasing_0 ``x-3*(PI/2)`` ``y-3*(PI/2)`` H4 H3 H2 H1 H5). +Unfold Rminus. +Rewrite Ropp_mul1. +Apply Rplus_sym. +Unfold Rminus. +Rewrite Ropp_mul1. +Apply Rplus_sym. +Pattern 3 PI; Rewrite double_var. +Ring. +Rewrite double; Pattern 3 4 PI; Rewrite double_var. +Ring. +Unfold Rminus. +Rewrite Ropp_mul1. +Apply Rplus_sym. +Unfold Rminus. +Rewrite Ropp_mul1. +Apply Rplus_sym. +Rewrite Rmult_1r. +Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. +Ring. +Rewrite Rmult_1r. +Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. +Ring. +Qed. + +Lemma cos_increasing_1 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``x<y`` -> ``(cos x)<(cos y)``. +Intros x y H1 H2 H3 H4 H5; Generalize (Rle_compatibility ``-3*(PI/2)`` PI x H1); Generalize (Rle_compatibility ``-3*(PI/2)`` x ``2*PI`` H2); Generalize (Rle_compatibility ``-3*(PI/2)`` PI y H3); Generalize (Rle_compatibility ``-3*(PI/2)`` y ``2*PI`` H4); Generalize (Rlt_compatibility ``-3*(PI/2)`` x y H5); Rewrite <- (cos_neg x); Rewrite <- (cos_neg y); Rewrite <- (cos_period ``-x`` (1)); Rewrite <- (cos_period ``-y`` (1)); Unfold INR; Replace ``-3*(PI/2)+x`` with ``x-3*(PI/2)``. +Replace ``-3*(PI/2)+y`` with ``y-3*(PI/2)``. +Replace ``-3*(PI/2)+PI`` with ``-(PI/2)``. +Replace ``-3*(PI/2)+2*PI`` with ``PI/2``. +Clear H1 H2 H3 H4 H5; Intros H1 H2 H3 H4 H5; Replace ``-x+2*1*PI`` with ``(PI/2)-(x-3*(PI/2))``. +Replace ``-y+2*1*PI`` with ``(PI/2)-(y-3*(PI/2))``. +Repeat Rewrite cos_shift; Apply (sin_increasing_1 ``x-3*(PI/2)`` ``y-3*(PI/2)`` H5 H4 H3 H2 H1). +Rewrite Rmult_1r. +Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. +Ring. +Rewrite Rmult_1r. +Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. +Ring. +Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. +Ring. +Pattern 3 PI; Rewrite double_var; Ring. +Unfold Rminus. +Rewrite <- Ropp_mul1. +Apply Rplus_sym. +Unfold Rminus. +Rewrite <- Ropp_mul1. +Apply Rplus_sym. +Qed. + +Lemma cos_decreasing_0 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``(cos x)<(cos y)``->``y<x``. +Intros; Generalize (Rlt_Ropp (cos x) (cos y) H3); Repeat Rewrite <- neg_cos; Intro H4; Change ``(cos (y+PI))<(cos (x+PI))`` in H4; Rewrite (Rplus_sym x) in H4; Rewrite (Rplus_sym y) in H4; Generalize (Rle_compatibility PI ``0`` x H); Generalize (Rle_compatibility PI x PI H0); Generalize (Rle_compatibility PI ``0`` y H1); Generalize (Rle_compatibility PI y PI H2); Rewrite Rplus_Or. +Rewrite <- double. +Clear H H0 H1 H2 H3; Intros; Apply Rlt_anti_compatibility with ``PI``; Apply (cos_increasing_0 ``PI+y`` ``PI+x`` H0 H H2 H1 H4). +Qed. + +Lemma cos_decreasing_1 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``x<y``->``(cos y)<(cos x)``. +Intros; Apply Ropp_Rlt; Repeat Rewrite <- neg_cos; Rewrite (Rplus_sym x); Rewrite (Rplus_sym y); Generalize (Rle_compatibility PI ``0`` x H); Generalize (Rle_compatibility PI x PI H0); Generalize (Rle_compatibility PI ``0`` y H1); Generalize (Rle_compatibility PI y PI H2); Rewrite Rplus_Or. +Rewrite <- double. +Generalize (Rlt_compatibility PI x y H3); Clear H H0 H1 H2 H3; Intros; Apply (cos_increasing_1 ``PI+x`` ``PI+y`` H3 H2 H1 H0 H). +Qed. + +Lemma tan_diff : (x,y:R) ~``(cos x)==0``->~``(cos y)==0``->``(tan x)-(tan y)==(sin (x-y))/((cos x)*(cos y))``. +Intros; Unfold tan;Rewrite sin_minus. +Unfold Rdiv. +Unfold Rminus. +Rewrite Rmult_Rplus_distrl. +Rewrite Rinv_Rmult. +Repeat Rewrite (Rmult_sym (sin x)). +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym (cos y)). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite (Rmult_sym (sin x)). +Apply Rplus_plus_r. +Rewrite <- Ropp_mul1. +Rewrite <- Ropp_mul3. +Rewrite (Rmult_sym ``/(cos x)``). +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym (cos x)). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Reflexivity. +Assumption. +Assumption. +Assumption. +Assumption. +Qed. + +Lemma tan_increasing_0 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``(tan x)<(tan y)``->``x<y``. +Intros; Generalize PI4_RLT_PI2; Intro H4; Generalize (Rlt_Ropp ``PI/4`` ``PI/2`` H4); Intro H5; Change ``-(PI/2)< -(PI/4)`` in H5; Generalize (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)); Intro HP1; Generalize (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)); Intro HP2; Generalize (not_sym ``0`` (cos x) (Rlt_not_eq ``0`` (cos x) (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)))); Intro H6; Generalize (not_sym ``0`` (cos y) (Rlt_not_eq ``0`` (cos y) (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)))); Intro H7; Generalize (tan_diff x y H6 H7); Intro H8; Generalize (Rlt_minus (tan x) (tan y) H3); Clear H3; Intro H3; Rewrite H8 in H3; Cut ``(sin (x-y))<0``. +Intro H9; Generalize (Rle_Ropp ``-(PI/4)`` y H1); Rewrite Ropp_Ropp; Intro H10; Generalize (Rle_sym2 ``-y`` ``PI/4`` H10); Clear H10; Intro H10; Generalize (Rle_Ropp y ``PI/4`` H2); Intro H11; Generalize (Rle_sym2 ``-(PI/4)`` ``-y`` H11); Clear H11; Intro H11; Generalize (Rplus_le ``-(PI/4)`` x ``-(PI/4)`` ``-y`` H H11); Generalize (Rplus_le x ``PI/4`` ``-y`` ``PI/4`` H0 H10); Replace ``x+ -y`` with ``x-y``. +Replace ``PI/4+PI/4`` with ``PI/2``. +Replace ``-(PI/4)+ -(PI/4)`` with ``-(PI/2)``. +Intros; Case (total_order ``0`` ``x-y``); Intro H14. +Generalize (sin_gt_0 ``x-y`` H14 (Rle_lt_trans ``x-y`` ``PI/2`` PI H12 PI2_Rlt_PI)); Intro H15; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(sin (x-y))`` ``0`` H15 H9)). +Elim H14; Intro H15. +Rewrite <- H15 in H9; Rewrite -> sin_0 in H9; Elim (Rlt_antirefl ``0`` H9). +Apply Rminus_lt; Assumption. +Pattern 1 PI; Rewrite double_var. +Unfold Rdiv. +Rewrite Rmult_Rplus_distrl. +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_Rmult. +Rewrite Ropp_distr1. +Replace ``2*2`` with ``4``. +Reflexivity. +Ring. +DiscrR. +DiscrR. +Pattern 1 PI; Rewrite double_var. +Unfold Rdiv. +Rewrite Rmult_Rplus_distrl. +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_Rmult. +Replace ``2*2`` with ``4``. +Reflexivity. +Ring. +DiscrR. +DiscrR. +Reflexivity. +Case (case_Rabsolu ``(sin (x-y))``); Intro H9. +Assumption. +Generalize (Rle_sym2 ``0`` ``(sin (x-y))`` H9); Clear H9; Intro H9; Generalize (Rlt_Rinv (cos x) HP1); Intro H10; Generalize (Rlt_Rinv (cos y) HP2); Intro H11; Generalize (Rmult_lt_pos (Rinv (cos x)) (Rinv (cos y)) H10 H11); Replace ``/(cos x)*/(cos y)`` with ``/((cos x)*(cos y))``. +Intro H12; Generalize (Rmult_le_pos ``(sin (x-y))`` ``/((cos x)*(cos y))`` H9 (Rlt_le ``0`` ``/((cos x)*(cos y))`` H12)); Intro H13; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``(sin (x-y))*/((cos x)*(cos y))`` ``0`` H13 H3)). +Rewrite Rinv_Rmult. +Reflexivity. +Assumption. +Assumption. +Qed. + +Lemma tan_increasing_1 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``x<y``->``(tan x)<(tan y)``. +Intros; Apply Rminus_lt; Generalize PI4_RLT_PI2; Intro H4; Generalize (Rlt_Ropp ``PI/4`` ``PI/2`` H4); Intro H5; Change ``-(PI/2)< -(PI/4)`` in H5; Generalize (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)); Intro HP1; Generalize (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)); Intro HP2; Generalize (not_sym ``0`` (cos x) (Rlt_not_eq ``0`` (cos x) (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)))); Intro H6; Generalize (not_sym ``0`` (cos y) (Rlt_not_eq ``0`` (cos y) (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)))); Intro H7; Rewrite (tan_diff x y H6 H7); Generalize (Rlt_Rinv (cos x) HP1); Intro H10; Generalize (Rlt_Rinv (cos y) HP2); Intro H11; Generalize (Rmult_lt_pos (Rinv (cos x)) (Rinv (cos y)) H10 H11); Replace ``/(cos x)*/(cos y)`` with ``/((cos x)*(cos y))``. +Clear H10 H11; Intro H8; Generalize (Rle_Ropp y ``PI/4`` H2); Intro H11; Generalize (Rle_sym2 ``-(PI/4)`` ``-y`` H11); Clear H11; Intro H11; Generalize (Rplus_le ``-(PI/4)`` x ``-(PI/4)`` ``-y`` H H11); Replace ``x+ -y`` with ``x-y``. +Replace ``-(PI/4)+ -(PI/4)`` with ``-(PI/2)``. +Clear H11; Intro H9; Generalize (Rlt_minus x y H3); Clear H3; Intro H3; Clear H H0 H1 H2 H4 H5 HP1 HP2; Generalize PI2_Rlt_PI; Intro H1; Generalize (Rlt_Ropp ``PI/2`` PI H1); Clear H1; Intro H1; Generalize (sin_lt_0_var ``x-y`` (Rlt_le_trans ``-PI`` ``-(PI/2)`` ``x-y`` H1 H9) H3); Intro H2; Generalize (Rlt_anti_monotony ``(sin (x-y))`` ``0`` ``/((cos x)*(cos y))`` H2 H8); Rewrite Rmult_Or; Intro H4; Assumption. +Pattern 1 PI; Rewrite double_var. +Unfold Rdiv. +Rewrite Rmult_Rplus_distrl. +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_Rmult. +Replace ``2*2`` with ``4``. +Rewrite Ropp_distr1. +Reflexivity. +Ring. +DiscrR. +DiscrR. +Reflexivity. +Apply Rinv_Rmult; Assumption. +Qed. + +Lemma sin_incr_0 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``(sin x)<=(sin y)``->``x<=y``. +Intros; Case (total_order (sin x) (sin y)); Intro H4; [Left; Apply (sin_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (sin_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (sin y) H8)]] | Elim (Rlt_antirefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5))]]. +Qed. + +Lemma sin_incr_1 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``x<=y``->``(sin x)<=(sin y)``. +Intros; Case (total_order x y); Intro H4; [Left; Apply (sin_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (sin x) (sin y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (sin_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. +Qed. + +Lemma sin_decr_0 : (x,y:R) ``x<=3*(PI/2)``->``PI/2<=x``->``y<=3*(PI/2)``->``PI/2<=y``-> ``(sin x)<=(sin y)`` -> ``y<=x``. +Intros; Case (total_order (sin x) (sin y)); Intro H4; [Left; Apply (sin_decreasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Generalize (sin_decreasing_1 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (sin y) H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5))]]. +Qed. + +Lemma sin_decr_1 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``x<=y`` -> ``(sin y)<=(sin x)``. +Intros; Case (total_order x y); Intro H4; [Left; Apply (sin_decreasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (sin x) (sin y)); Intro H6; [Generalize (sin_decreasing_0 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. +Qed. + +Lemma cos_incr_0 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``(cos x)<=(cos y)`` -> ``x<=y``. +Intros; Case (total_order (cos x) (cos y)); Intro H4; [Left; Apply (cos_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (cos_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (cos y) H8)]] | Elim (Rlt_antirefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5))]]. +Qed. + +Lemma cos_incr_1 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``x<=y`` -> ``(cos x)<=(cos y)``. +Intros; Case (total_order x y); Intro H4; [Left; Apply (cos_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (cos x) (cos y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (cos_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. +Qed. + +Lemma cos_decr_0 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``(cos x)<=(cos y)`` -> ``y<=x``. +Intros; Case (total_order (cos x) (cos y)); Intro H4; [Left; Apply (cos_decreasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Generalize (cos_decreasing_1 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (cos y) H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5))]]. +Qed. + +Lemma cos_decr_1 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``x<=y``->``(cos y)<=(cos x)``. +Intros; Case (total_order x y); Intro H4; [Left; Apply (cos_decreasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (cos x) (cos y)); Intro H6; [Generalize (cos_decreasing_0 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. +Qed. + +Lemma tan_incr_0 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``(tan x)<=(tan y)``->``x<=y``. +Intros; Case (total_order (tan x) (tan y)); Intro H4; [Left; Apply (tan_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (tan_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (tan y) H8)]] | Elim (Rlt_antirefl (tan x) (Rle_lt_trans (tan x) (tan y) (tan x) H3 H5))]]. +Qed. + +Lemma tan_incr_1 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``x<=y``->``(tan x)<=(tan y)``. +Intros; Case (total_order x y); Intro H4; [Left; Apply (tan_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (tan x) (tan y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (tan_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. +Qed. + +(**********) +Lemma sin_eq_0_1 : (x:R) (EXT k:Z | x==(Rmult (IZR k) PI)) -> (sin x)==R0. +Intros. +Elim H; Intros. +Apply (Zcase_sign x0). +Intro. +Rewrite H1 in H0. +Simpl in H0. +Rewrite H0; Rewrite Rmult_Ol; Apply sin_0. +Intro. +Cut `0<=x0`. +Intro. +Elim (IZN x0 H2); Intros. +Rewrite H3 in H0. +Rewrite <- INR_IZR_INZ in H0. +Rewrite H0. +Elim (even_odd_cor x1); Intros. +Elim H4; Intro. +Rewrite H5. +Rewrite mult_INR. +Simpl. +Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). +Rewrite sin_period. +Apply sin_0. +Rewrite H5. +Rewrite S_INR; Rewrite mult_INR. +Simpl. +Rewrite Rmult_Rplus_distrl. +Rewrite Rmult_1l; Rewrite sin_plus. +Rewrite sin_PI. +Rewrite Rmult_Or. +Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). +Rewrite sin_period. +Rewrite sin_0; Ring. +Apply le_IZR. +Left; Apply IZR_lt. +Assert H2 := Zgt_iff_lt. +Elim (H2 x0 `0`); Intros. +Apply H3; Assumption. +Intro. +Rewrite H0. +Replace ``(sin ((IZR x0)*PI))`` with ``-(sin (-(IZR x0)*PI))``. +Cut `0<=-x0`. +Intro. +Rewrite <- Ropp_Ropp_IZR. +Elim (IZN `-x0` H2); Intros. +Rewrite H3. +Rewrite <- INR_IZR_INZ. +Elim (even_odd_cor x1); Intros. +Elim H4; Intro. +Rewrite H5. +Rewrite mult_INR. +Simpl. +Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). +Rewrite sin_period. +Rewrite sin_0; Ring. +Rewrite H5. +Rewrite S_INR; Rewrite mult_INR. +Simpl. +Rewrite Rmult_Rplus_distrl. +Rewrite Rmult_1l; Rewrite sin_plus. +Rewrite sin_PI. +Rewrite Rmult_Or. +Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). +Rewrite sin_period. +Rewrite sin_0; Ring. +Apply le_IZR. +Apply Rle_anti_compatibility with ``(IZR x0)``. +Rewrite Rplus_Or. +Rewrite Ropp_Ropp_IZR. +Rewrite Rplus_Ropp_r. +Left; Replace R0 with (IZR `0`); [Apply IZR_lt | Reflexivity]. +Assumption. +Rewrite <- sin_neg. +Rewrite Ropp_mul1. +Rewrite Ropp_Ropp. +Reflexivity. +Qed. + +Lemma sin_eq_0_0 : (x:R) (sin x)==R0 -> (EXT k:Z | x==(Rmult (IZR k) PI)). +Intros. +Assert H0 := (euclidian_division x PI PI_neq0). +Elim H0; Intros q H1. +Elim H1; Intros r H2. +Exists q. +Cut r==R0. +Intro. +Elim H2; Intros H4 _; Rewrite H4; Rewrite H3. +Apply Rplus_Or. +Elim H2; Intros. +Rewrite H3 in H. +Rewrite sin_plus in H. +Cut ``(sin ((IZR q)*PI))==0``. +Intro. +Rewrite H5 in H. +Rewrite Rmult_Ol in H. +Rewrite Rplus_Ol in H. +Assert H6 := (without_div_Od ? ? H). +Elim H6; Intro. +Assert H8 := (sin2_cos2 ``(IZR q)*PI``). +Rewrite H5 in H8; Rewrite H7 in H8. +Rewrite Rsqr_O in H8. +Rewrite Rplus_Or in H8. +Elim R1_neq_R0; Symmetry; Assumption. +Cut r==R0\/``0<r<PI``. +Intro; Elim H8; Intro. +Assumption. +Elim H9; Intros. +Assert H12 := (sin_gt_0 ? H10 H11). +Rewrite H7 in H12; Elim (Rlt_antirefl ? H12). +Rewrite Rabsolu_right in H4. +Elim H4; Intros. +Case (total_order R0 r); Intro. +Right; Split; Assumption. +Elim H10; Intro. +Left; Symmetry; Assumption. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H8 H11)). +Apply Rle_sym1. +Left; Apply PI_RGT_0. +Apply sin_eq_0_1. +Exists q; Reflexivity. +Qed. + +Lemma cos_eq_0_0 : (x:R) (cos x)==R0 -> (EXT k : Z | ``x==(IZR k)*PI+PI/2``). +Intros x H; Rewrite -> cos_sin in H; Generalize (sin_eq_0_0 (Rplus (Rdiv PI (INR (2))) x) H); Intro H2; Elim H2; Intros x0 H3; Exists (Zminus x0 (inject_nat (S O))); Rewrite <- Z_R_minus; Ring; Rewrite Rmult_sym; Rewrite <- H3; Unfold INR. +Rewrite (double_var ``-PI``); Unfold Rdiv; Ring. +Qed. + +Lemma cos_eq_0_1 : (x:R) (EXT k : Z | ``x==(IZR k)*PI+PI/2``) -> ``(cos x)==0``. +Intros x H1; Rewrite cos_sin; Elim H1; Intros x0 H2; Rewrite H2; Replace ``PI/2+((IZR x0)*PI+PI/2)`` with ``(IZR x0)*PI+PI``. +Rewrite neg_sin; Rewrite <- Ropp_O. +Apply eq_Ropp; Apply sin_eq_0_1; Exists x0; Reflexivity. +Pattern 2 PI; Rewrite (double_var PI); Ring. +Qed. + +Lemma sin_eq_O_2PI_0 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``(sin x)==0`` -> ``x==0``\/``x==PI``\/``x==2*PI``. +Intros; Generalize (sin_eq_0_0 x H1); Intro. +Elim H2; Intros k0 H3. +Case (total_order PI x); Intro. +Rewrite H3 in H4; Rewrite H3 in H0. +Right; Right. +Generalize (Rlt_monotony_r ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv ``PI`` PI_RGT_0) H4); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``(IZR k0)*PI`` ``2*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H0); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. +Repeat Rewrite Rmult_1r; Intro; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H5); Rewrite <- plus_IZR. +Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``. +Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``2`` H6); Rewrite <- plus_IZR. +Replace ``(IZR (NEG (xO xH)))+2`` with ``0``. +Intro; Cut ``-1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``. +Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H9); Intro. +Cut k0=`2`. +Intro; Rewrite H11 in H3; Rewrite H3; Simpl. +Reflexivity. +Rewrite <- (Zplus_inverse_l `2`) in H10; Generalize (Zsimpl_plus_l `-2` k0 `2` H10); Intro; Assumption. +Split. +Assumption. +Apply Rle_lt_trans with ``0``. +Assumption. +Apply Rlt_R0_R1. +Simpl; Ring. +Simpl; Ring. +Apply PI_neq0. +Apply PI_neq0. +Elim H4; Intro. +Right; Left. +Symmetry; Assumption. +Left. +Rewrite H3 in H5; Rewrite H3 in H; Generalize (Rlt_monotony_r ``/PI`` ``(IZR k0)*PI`` PI (Rlt_Rinv ``PI`` PI_RGT_0) H5); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``0`` ``(IZR k0)*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Rewrite Rmult_Ol; Intro. +Cut ``-1 < (IZR (k0)) < 1``. +Intro; Generalize (one_IZR_lt1 k0 H8); Intro; Rewrite H9 in H3; Rewrite H3; Simpl; Apply Rmult_Ol. +Split. +Apply Rlt_le_trans with ``0``. +Rewrite <- Ropp_O; Apply Rgt_Ropp; Apply Rlt_R0_R1. +Assumption. +Assumption. +Apply PI_neq0. +Apply PI_neq0. +Qed. + +Lemma sin_eq_O_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==0``\/``x==PI``\/``x==2*PI`` -> ``(sin x)==0``. +Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> sin_0; Reflexivity | Elim H4; Intro H5; [Rewrite H5; Rewrite -> sin_PI; Reflexivity | Rewrite H5; Rewrite -> sin_2PI; Reflexivity]]. +Qed. + +Lemma cos_eq_0_2PI_0 : (x:R) ``R0<=x`` -> ``x<=2*PI`` -> ``(cos x)==0`` -> ``x==(PI/2)``\/``x==3*(PI/2)``. +Intros; Case (total_order x ``3*(PI/2)``); Intro. +Rewrite cos_sin in H1. +Cut ``0<=PI/2+x``. +Cut ``PI/2+x<=2*PI``. +Intros; Generalize (sin_eq_O_2PI_0 ``PI/2+x`` H4 H3 H1); Intros. +Decompose [or] H5. +Generalize (Rle_compatibility ``PI/2`` ``0`` x H); Rewrite Rplus_Or; Rewrite H6; Intro. +Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``PI/2`` ``0`` PI2_RGT_0 H7)). +Left. +Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` PI H7). +Replace ``-(PI/2)+(PI/2+x)`` with x. +Replace ``-(PI/2)+PI`` with ``PI/2``. +Intro; Assumption. +Pattern 3 PI; Rewrite (double_var PI); Ring. +Ring. +Right. +Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` ``2*PI`` H7). +Replace ``-(PI/2)+(PI/2+x)`` with x. +Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``. +Intro; Assumption. +Rewrite double; Pattern 3 4 PI; Rewrite (double_var PI); Ring. +Ring. +Left; Replace ``2*PI`` with ``PI/2+3*(PI/2)``. +Apply Rlt_compatibility; Assumption. +Rewrite (double PI); Pattern 3 4 PI; Rewrite (double_var PI); Ring. +Apply ge0_plus_ge0_is_ge0. +Left; Unfold Rdiv; Apply Rmult_lt_pos. +Apply PI_RGT_0. +Apply Rlt_Rinv; Sup0. +Assumption. +Elim H2; Intro. +Right; Assumption. +Generalize (cos_eq_0_0 x H1); Intro; Elim H4; Intros k0 H5. +Rewrite H5 in H3; Rewrite H5 in H0; Generalize (Rlt_compatibility ``-(PI/2)`` ``3*PI/2`` ``(IZR k0)*PI+PI/2`` H3); Generalize (Rle_compatibility ``-(PI/2)`` ``(IZR k0)*PI+PI/2`` ``2*PI`` H0). +Replace ``-(PI/2)+3*PI/2`` with PI. +Replace ``-(PI/2)+((IZR k0)*PI+PI/2)`` with ``(IZR k0)*PI``. +Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``. +Intros; Generalize (Rlt_monotony ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv PI PI_RGT_0) H7); Generalize (Rle_monotony ``/PI`` ``(IZR k0)*PI`` ``3*(PI/2)`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv PI PI_RGT_0)) H6). +Replace ``/PI*((IZR k0)*PI)`` with (IZR k0). +Replace ``/PI*(3*PI/2)`` with ``3*/2``. +Rewrite <- Rinv_l_sym. +Intros; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H9); Rewrite <- plus_IZR. +Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``. +Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``3*/2`` H8); Rewrite <- plus_IZR. +Replace ``(IZR (NEG (xO xH)))+2`` with ``0``. +Intro; Cut `` -1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``. +Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H12); Intro. +Cut k0=`2`. +Intro; Rewrite H14 in H8. +Assert Hyp : ``0<2``. +Sup0. +Generalize (Rle_monotony ``2`` ``(IZR (POS (xO xH)))`` ``3*/2`` (Rlt_le ``0`` ``2`` Hyp) H8); Simpl. +Replace ``2*2`` with ``4``. +Replace ``2*(3*/2)`` with ``3``. +Intro; Cut ``3<4``. +Intro; Elim (Rlt_antirefl ``3`` (Rlt_le_trans ``3`` ``4`` ``3`` H16 H15)). +Generalize (Rlt_compatibility ``3`` ``0`` ``1`` Rlt_R0_R1); Rewrite Rplus_Or. +Replace ``3+1`` with ``4``. +Intro; Assumption. +Ring. +Symmetry; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. +DiscrR. +Ring. +Rewrite <- (Zplus_inverse_l `2`) in H13; Generalize (Zsimpl_plus_l `-2` k0 `2` H13); Intro; Assumption. +Split. +Assumption. +Apply Rle_lt_trans with ``(IZR (NEG (xO xH)))+3*/2``. +Assumption. +Simpl; Replace ``-2+3*/2`` with ``-(1*/2)``. +Apply Rlt_trans with ``0``. +Rewrite <- Ropp_O; Apply Rlt_Ropp. +Apply Rmult_lt_pos; [Apply Rlt_R0_R1 | Apply Rlt_Rinv; Sup0]. +Apply Rlt_R0_R1. +Rewrite Rmult_1l; Apply r_Rmult_mult with ``2``. +Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym. +Rewrite Rmult_Rplus_distr; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. +Ring. +DiscrR. +DiscrR. +DiscrR. +Simpl; Ring. +Simpl; Ring. +Apply PI_neq0. +Unfold Rdiv; Pattern 1 ``3``; Rewrite (Rmult_sym ``3``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Apply Rmult_sym. +Apply PI_neq0. +Symmetry; Rewrite (Rmult_sym ``/PI``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Apply Rmult_1r. +Apply PI_neq0. +Rewrite double; Pattern 3 4 PI; Rewrite double_var; Ring. +Ring. +Pattern 1 PI; Rewrite double_var; Ring. +Qed. + +Lemma cos_eq_0_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==PI/2``\/``x==3*(PI/2)`` -> ``(cos x)==0``. +Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> cos_PI2; Reflexivity | Rewrite H4; Rewrite -> cos_3PI2; Reflexivity ]. +Qed. diff --git a/theories7/Reals/Rtrigo_alt.v b/theories7/Reals/Rtrigo_alt.v new file mode 100644 index 00000000..db0e2fea --- /dev/null +++ b/theories7/Reals/Rtrigo_alt.v @@ -0,0 +1,294 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rtrigo_alt.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo_def. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(*****************************************************************) +(* Using series definitions of cos and sin *) +(*****************************************************************) + +Definition sin_term [a:R] : nat->R := [i:nat] ``(pow (-1) i)*(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))``. + +Definition cos_term [a:R] : nat->R := [i:nat] ``(pow (-1) i)*(pow a (mult (S (S O)) i))/(INR (fact (mult (S (S O)) i)))``. + +Definition sin_approx [a:R;n:nat] : R := (sum_f_R0 (sin_term a) n). + +Definition cos_approx [a:R;n:nat] : R := (sum_f_R0 (cos_term a) n). + +(**********) +Lemma PI_4 : ``PI<=4``. +Assert H0 := (PI_ineq O). +Elim H0; Clear H0; Intros _ H0. +Unfold tg_alt PI_tg in H0; Simpl in H0. +Rewrite Rinv_R1 in H0; Rewrite Rmult_1r in H0; Unfold Rdiv in H0. +Apply Rle_monotony_contra with ``/4``. +Apply Rlt_Rinv; Sup0. +Rewrite <- Rinv_l_sym; [Rewrite Rmult_sym; Assumption | DiscrR]. +Qed. + +(**********) +Theorem sin_bound : (a:R; n:nat) ``0 <= a``->``a <= PI``->``(sin_approx a (plus (mult (S (S O)) n) (S O))) <= (sin a)<= (sin_approx a (mult (S (S O)) (plus n (S O))))``. +Intros; Case (Req_EM a R0); Intro Hyp_a. +Rewrite Hyp_a; Rewrite sin_0; Split; Right; Unfold sin_approx; Apply sum_eq_R0 Orelse (Symmetry; Apply sum_eq_R0); Intros; Unfold sin_term; Rewrite pow_add; Simpl; Unfold Rdiv; Rewrite Rmult_Ol; Ring. +Unfold sin_approx; Cut ``0<a``. +Intro Hyp_a_pos. +Rewrite (decomp_sum (sin_term a) (plus (mult (S (S O)) n) (S O))). +Rewrite (decomp_sum (sin_term a) (mult (S (S O)) (plus n (S O)))). +Replace (sin_term a O) with a. +Cut (Rle (sum_f_R0 [i:nat](sin_term a (S i)) (pred (plus (mult (S (S O)) n) (S O)))) ``(sin a)-a``)/\(Rle ``(sin a)-a`` (sum_f_R0 [i:nat](sin_term a (S i)) (pred (mult (S (S O)) (plus n (S O)))))) -> (Rle (Rplus a (sum_f_R0 [i:nat](sin_term a (S i)) (pred (plus (mult (S (S O)) n) (S O))))) (sin a))/\(Rle (sin a) (Rplus a (sum_f_R0 [i:nat](sin_term a (S i)) (pred (mult (S (S O)) (plus n (S O))))))). +Intro; Apply H1. +Pose Un := [n:nat]``(pow a (plus (mult (S (S O)) (S n)) (S O)))/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))``. +Replace (pred (plus (mult (S (S O)) n) (S O))) with (mult (S (S O)) n). +Replace (pred (mult (S (S O)) (plus n (S O)))) with (S (mult (S (S O)) n)). +Replace (sum_f_R0 [i:nat](sin_term a (S i)) (mult (S (S O)) n)) with ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``. +Replace (sum_f_R0 [i:nat](sin_term a (S i)) (S (mult (S (S O)) n))) with ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``. +Cut ``(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))<=a-(sin a)<=(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``->`` -(sum_f_R0 (tg_alt Un) (mult (S (S O)) n)) <= (sin a)-a <= -(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``. +Intro; Apply H2. +Apply alternated_series_ineq. +Unfold Un_decreasing Un; Intro; Cut (plus (mult (S (S O)) (S (S n0))) (S O))=(S (S (plus (mult (S (S O)) (S n0)) (S O)))). +Intro; Rewrite H3. +Replace ``(pow a (S (S (plus (mult (S (S O)) (S n0)) (S O)))))`` with ``(pow a (plus (mult (S (S O)) (S n0)) (S O)))*(a*a)``. +Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply pow_lt; Assumption. +Apply Rle_monotony_contra with ``(INR (fact (S (S (plus (mult (S (S O)) (S n0)) (S O))))))``. +Rewrite <- H3; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H5 := (sym_eq ? ? ? H4); Elim (fact_neq_0 ? H5). +Rewrite <- H3; Rewrite (Rmult_sym ``(INR (fact (plus (mult (S (S O)) (S (S n0))) (S O))))``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite H3; Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r. +Do 2 Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Simpl; Replace ``((0+1+1)*((INR n0)+1)+(0+1)+1+1)*((0+1+1)*((INR n0)+1)+(0+1)+1)`` with ``4*(INR n0)*(INR n0)+18*(INR n0)+20``; [Idtac | Ring]. +Apply Rle_trans with ``20``. +Apply Rle_trans with ``16``. +Replace ``16`` with ``(Rsqr 4)``; [Idtac | SqRing]. +Replace ``a*a`` with (Rsqr a); [Idtac | Reflexivity]. +Apply Rsqr_incr_1. +Apply Rle_trans with PI; [Assumption | Apply PI_4]. +Assumption. +Left; Sup0. +Rewrite <- (Rplus_Or ``16``); Replace ``20`` with ``16+4``; [Apply Rle_compatibility; Left; Sup0 | Ring]. +Rewrite <- (Rplus_sym ``20``); Pattern 1 ``20``; Rewrite <- Rplus_Or; Apply Rle_compatibility. +Apply ge0_plus_ge0_is_ge0. +Repeat Apply Rmult_le_pos. +Left; Sup0. +Left; Sup0. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Apply Rmult_le_pos. +Left; Sup0. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Assert H3 := (cv_speed_pow_fact a); Unfold Un; Unfold Un_cv in H3; Unfold R_dist in H3; Unfold Un_cv; Unfold R_dist; Intros; Elim (H3 eps H4); Intros N H5. +Exists N; Intros; Apply H5. +Replace (plus (mult (2) (S n0)) (1)) with (S (mult (2) (S n0))). +Unfold ge; Apply le_trans with (mult (2) (S n0)). +Apply le_trans with (mult (2) (S N)). +Apply le_trans with (mult (2) N). +Apply le_n_2n. +Apply mult_le; Apply le_n_Sn. +Apply mult_le; Apply le_n_S; Assumption. +Apply le_n_Sn. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Reflexivity. +Assert X := (exist_sin (Rsqr a)); Elim X; Intros. +Cut ``x==(sin a)/a``. +Intro; Rewrite H3 in p; Unfold sin_in in p; Unfold infinit_sum in p; Unfold R_dist in p; Unfold Un_cv; Unfold R_dist; Intros. +Cut ``0<eps/(Rabsolu a)``. +Intro; Elim (p ? H5); Intros N H6. +Exists N; Intros. +Replace (sum_f_R0 (tg_alt Un) n0) with (Rmult a (Rminus R1 (sum_f_R0 [i:nat]``(sin_n i)*(pow (Rsqr a) i)`` (S n0)))). +Unfold Rminus; Rewrite Rmult_Rplus_distr; Rewrite Rmult_1r; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Repeat Rewrite Rplus_assoc; Rewrite (Rplus_sym a); Rewrite (Rplus_sym ``-a``); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rlt_monotony_contra with ``/(Rabsolu a)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Pattern 1 ``/(Rabsolu a)``; Rewrite <- (Rabsolu_Rinv a Hyp_a). +Rewrite <- Rabsolu_mult; Rewrite Rmult_Rplus_distr; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1l | Assumption]; Rewrite (Rmult_sym ``/a``); Rewrite (Rmult_sym ``/(Rabsolu a)``); Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Unfold Rminus Rdiv in H6; Apply H6; Unfold ge; Apply le_trans with n0; [Exact H7 | Apply le_n_Sn]. +Rewrite (decomp_sum [i:nat]``(sin_n i)*(pow (Rsqr a) i)`` (S n0)). +Replace (sin_n O) with R1. +Simpl; Rewrite Rmult_1r; Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Rewrite Ropp_mul3; Rewrite <- Ropp_mul1; Rewrite scal_sum; Apply sum_eq. +Intros; Unfold sin_n Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-(pow (-1) i)``. +Replace ``(pow a (plus (mult (S (S O)) (S i)) (S O)))`` with ``(Rsqr a)*(pow (Rsqr a) i)*a``. +Unfold Rdiv; Ring. +Rewrite pow_add; Rewrite pow_Rsqr; Simpl; Ring. +Simpl; Ring. +Unfold sin_n; Unfold Rdiv; Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. +Apply lt_O_Sn. +Unfold Rdiv; Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Unfold sin; Case (exist_sin (Rsqr a)). +Intros; Cut x==x0. +Intro; Rewrite H3; Unfold Rdiv. +Symmetry; Apply Rinv_r_simpl_m; Assumption. +Unfold sin_in in p; Unfold sin_in in s; EApply unicity_sum. +Apply p. +Apply s. +Intros; Elim H2; Intros. +Replace ``(sin a)-a`` with ``-(a-(sin a))``; [Idtac | Ring]. +Split; Apply Rle_Ropp1; Assumption. +Replace ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))`` with ``-1*(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``; [Rewrite scal_sum | Ring]. +Apply sum_eq; Intros; Unfold sin_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. +Unfold Rdiv; Ring. +Reflexivity. +Replace ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))`` with ``-1*(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``; [Rewrite scal_sum | Ring]. +Apply sum_eq; Intros. +Unfold sin_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. +Unfold Rdiv; Ring. +Reflexivity. +Replace (mult (2) (plus n (1))) with (S (S (mult (2) n))). +Reflexivity. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)). +Reflexivity. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Intro; Elim H1; Intros. +Split. +Apply Rle_anti_compatibility with ``-a``. +Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-a``); Apply H2. +Apply Rle_anti_compatibility with ``-a``. +Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-a``); Apply H3. +Unfold sin_term; Simpl; Unfold Rdiv; Rewrite Rinv_R1; Ring. +Replace (mult (2) (plus n (1))) with (S (S (mult (2) n))). +Apply lt_O_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)). +Apply lt_O_Sn. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Inversion H; [Assumption | Elim Hyp_a; Symmetry; Assumption]. +Qed. + +(**********) +Lemma cos_bound : (a:R; n:nat) `` -PI/2 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``. +Cut ((a:R; n:nat) ``0 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``) -> ((a:R; n:nat) `` -PI/2 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``). +Intros H a n; Apply H. +Intros; Unfold cos_approx. +Rewrite (decomp_sum (cos_term a0) (plus (mult (S (S O)) n0) (S O))). +Rewrite (decomp_sum (cos_term a0) (mult (S (S O)) (plus n0 (S O)))). +Replace (cos_term a0 O) with R1. +Cut (Rle (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (plus (mult (S (S O)) n0) (S O)))) ``(cos a0)-1``)/\(Rle ``(cos a0)-1`` (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (mult (S (S O)) (plus n0 (S O)))))) -> (Rle (Rplus R1 (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (plus (mult (S (S O)) n0) (S O))))) (cos a0))/\(Rle (cos a0) (Rplus R1 (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (mult (S (S O)) (plus n0 (S O))))))). +Intro; Apply H2. +Pose Un := [n:nat]``(pow a0 (mult (S (S O)) (S n)))/(INR (fact (mult (S (S O)) (S n))))``. +Replace (pred (plus (mult (S (S O)) n0) (S O))) with (mult (S (S O)) n0). +Replace (pred (mult (S (S O)) (plus n0 (S O)))) with (S (mult (S (S O)) n0)). +Replace (sum_f_R0 [i:nat](cos_term a0 (S i)) (mult (S (S O)) n0)) with ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``. +Replace (sum_f_R0 [i:nat](cos_term a0 (S i)) (S (mult (S (S O)) n0))) with ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``. +Cut ``(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))<=1-(cos a0)<=(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``->`` -(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0)) <= (cos a0)-1 <= -(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``. +Intro; Apply H3. +Apply alternated_series_ineq. +Unfold Un_decreasing; Intro; Unfold Un. +Cut (mult (S (S O)) (S (S n1)))=(S (S (mult (S (S O)) (S n1)))). +Intro; Rewrite H4; Replace ``(pow a0 (S (S (mult (S (S O)) (S n1)))))`` with ``(pow a0 (mult (S (S O)) (S n1)))*(a0*a0)``. +Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. +Apply pow_le; Assumption. +Apply Rle_monotony_contra with ``(INR (fact (S (S (mult (S (S O)) (S n1))))))``. +Rewrite <- H4; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H6 := (sym_eq ? ? ? H5); Elim (fact_neq_0 ? H6). +Rewrite <- H4; Rewrite (Rmult_sym ``(INR (fact (mult (S (S O)) (S (S n1)))))``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite H4; Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Do 2 Rewrite S_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Simpl; Replace ``((0+1+1)*((INR n1)+1)+1+1)*((0+1+1)*((INR n1)+1)+1)`` with ``4*(INR n1)*(INR n1)+14*(INR n1)+12``; [Idtac | Ring]. +Apply Rle_trans with ``12``. +Apply Rle_trans with ``4``. +Replace ``4`` with ``(Rsqr 2)``; [Idtac | SqRing]. +Replace ``a0*a0`` with (Rsqr a0); [Idtac | Reflexivity]. +Apply Rsqr_incr_1. +Apply Rle_trans with ``PI/2``. +Assumption. +Unfold Rdiv; Apply Rle_monotony_contra with ``2``. +Sup0. +Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. +Replace ``2*2`` with ``4``; [Apply PI_4 | Ring]. +DiscrR. +Assumption. +Left; Sup0. +Pattern 1 ``4``; Rewrite <- Rplus_Or; Replace ``12`` with ``4+8``; [Apply Rle_compatibility; Left; Sup0 | Ring]. +Rewrite <- (Rplus_sym ``12``); Pattern 1 ``12``; Rewrite <- Rplus_Or; Apply Rle_compatibility. +Apply ge0_plus_ge0_is_ge0. +Repeat Apply Rmult_le_pos. +Left; Sup0. +Left; Sup0. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Apply Rmult_le_pos. +Left; Sup0. +Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Assert H4 := (cv_speed_pow_fact a0); Unfold Un; Unfold Un_cv in H4; Unfold R_dist in H4; Unfold Un_cv; Unfold R_dist; Intros; Elim (H4 eps H5); Intros N H6; Exists N; Intros. +Apply H6; Unfold ge; Apply le_trans with (mult (2) (S N)). +Apply le_trans with (mult (2) N). +Apply le_n_2n. +Apply mult_le; Apply le_n_Sn. +Apply mult_le; Apply le_n_S; Assumption. +Assert X := (exist_cos (Rsqr a0)); Elim X; Intros. +Cut ``x==(cos a0)``. +Intro; Rewrite H4 in p; Unfold cos_in in p; Unfold infinit_sum in p; Unfold R_dist in p; Unfold Un_cv; Unfold R_dist; Intros. +Elim (p ? H5); Intros N H6. +Exists N; Intros. +Replace (sum_f_R0 (tg_alt Un) n1) with (Rminus R1 (sum_f_R0 [i:nat]``(cos_n i)*(pow (Rsqr a0) i)`` (S n1))). +Unfold Rminus; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Repeat Rewrite Rplus_assoc; Rewrite (Rplus_sym R1); Rewrite (Rplus_sym ``-1``); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Unfold Rminus in H6; Apply H6. +Unfold ge; Apply le_trans with n1. +Exact H7. +Apply le_n_Sn. +Rewrite (decomp_sum [i:nat]``(cos_n i)*(pow (Rsqr a0) i)`` (S n1)). +Replace (cos_n O) with R1. +Simpl; Rewrite Rmult_1r; Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Replace (Ropp (sum_f_R0 [i:nat]``(cos_n (S i))*((Rsqr a0)*(pow (Rsqr a0) i))`` n1)) with (Rmult ``-1`` (sum_f_R0 [i:nat]``(cos_n (S i))*((Rsqr a0)*(pow (Rsqr a0) i))`` n1)); [Idtac | Ring]; Rewrite scal_sum; Apply sum_eq; Intros; Unfold cos_n Un tg_alt. +Replace ``(pow (-1) (S i))`` with ``-(pow (-1) i)``. +Replace ``(pow a0 (mult (S (S O)) (S i)))`` with ``(Rsqr a0)*(pow (Rsqr a0) i)``. +Unfold Rdiv; Ring. +Rewrite pow_Rsqr; Reflexivity. +Simpl; Ring. +Unfold cos_n; Unfold Rdiv; Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. +Apply lt_O_Sn. +Unfold cos; Case (exist_cos (Rsqr a0)); Intros; Unfold cos_in in p; Unfold cos_in in c; EApply unicity_sum. +Apply p. +Apply c. +Intros; Elim H3; Intros; Replace ``(cos a0)-1`` with ``-(1-(cos a0))``; [Idtac | Ring]. +Split; Apply Rle_Ropp1; Assumption. +Replace ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))`` with ``-1*(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``; [Rewrite scal_sum | Ring]. +Apply sum_eq; Intros; Unfold cos_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. +Unfold Rdiv; Ring. +Reflexivity. +Replace ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))`` with ``-1*(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``; [Rewrite scal_sum | Ring]; Apply sum_eq; Intros; Unfold cos_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. +Unfold Rdiv; Ring. +Reflexivity. +Replace (mult (2) (plus n0 (1))) with (S (S (mult (2) n0))). +Reflexivity. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) n0) (1)) with (S (mult (2) n0)). +Reflexivity. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Intro; Elim H2; Intros; Split. +Apply Rle_anti_compatibility with ``-1``. +Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-1``); Apply H3. +Apply Rle_anti_compatibility with ``-1``. +Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-1``); Apply H4. +Unfold cos_term; Simpl; Unfold Rdiv; Rewrite Rinv_R1; Ring. +Replace (mult (2) (plus n0 (1))) with (S (S (mult (2) n0))). +Apply lt_O_Sn. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. +Replace (plus (mult (2) n0) (1)) with (S (mult (2) n0)). +Apply lt_O_Sn. +Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Intros; Case (total_order_T R0 a); Intro. +Elim s; Intro. +Apply H; [Left; Assumption | Assumption]. +Apply H; [Right; Assumption | Assumption]. +Cut ``0< -a``. +Intro; Cut (x:R;n:nat) (cos_approx x n)==(cos_approx ``-x`` n). +Intro; Rewrite H3; Rewrite (H3 a (mult (S (S O)) (plus n (S O)))); Rewrite cos_sym; Apply H. +Left; Assumption. +Rewrite <- (Ropp_Ropp ``PI/2``); Apply Rle_Ropp1; Unfold Rdiv; Unfold Rdiv in H0; Rewrite <- Ropp_mul1; Exact H0. +Intros; Unfold cos_approx; Apply sum_eq; Intros; Unfold cos_term; Do 2 Rewrite pow_Rsqr; Rewrite Rsqr_neg; Unfold Rdiv; Reflexivity. +Apply Rgt_RO_Ropp; Assumption. +Qed. diff --git a/theories7/Reals/Rtrigo_calc.v b/theories7/Reals/Rtrigo_calc.v new file mode 100644 index 00000000..ab181106 --- /dev/null +++ b/theories7/Reals/Rtrigo_calc.v @@ -0,0 +1,350 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rtrigo_calc.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require R_sqrt. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Lemma tan_PI : ``(tan PI)==0``. +Unfold tan; Rewrite sin_PI; Rewrite cos_PI; Unfold Rdiv; Apply Rmult_Ol. +Qed. + +Lemma sin_3PI2 : ``(sin (3*(PI/2)))==(-1)``. +Replace ``3*(PI/2)`` with ``PI+(PI/2)``. +Rewrite sin_plus; Rewrite sin_PI; Rewrite cos_PI; Rewrite sin_PI2; Ring. +Pattern 1 PI; Rewrite (double_var PI); Ring. +Qed. + +Lemma tan_2PI : ``(tan (2*PI))==0``. +Unfold tan; Rewrite sin_2PI; Unfold Rdiv; Apply Rmult_Ol. +Qed. + +Lemma sin_cos_PI4 : ``(sin (PI/4)) == (cos (PI/4))``. +Proof with Trivial. +Rewrite cos_sin. +Replace ``PI/2+PI/4`` with ``-(PI/4)+PI``. +Rewrite neg_sin; Rewrite sin_neg; Ring. +Cut ``PI==PI/2+PI/2``; [Intro | Apply double_var]. +Pattern 2 3 PI; Rewrite H; Pattern 2 3 PI; Rewrite H. +Assert H0 : ``2<>0``; [DiscrR | Unfold Rdiv; Rewrite Rinv_Rmult; Try Ring]. +Qed. + +Lemma sin_PI3_cos_PI6 : ``(sin (PI/3))==(cos (PI/6))``. +Proof with Trivial. +Replace ``PI/6`` with ``(PI/2)-(PI/3)``. +Rewrite cos_shift. +Assert H0 : ``6<>0``; [DiscrR | Idtac]. +Assert H1 : ``3<>0``; [DiscrR | Idtac]. +Assert H2 : ``2<>0``; [DiscrR | Idtac]. +Apply r_Rmult_mult with ``6``. +Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``6``). +Unfold Rdiv; Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite (Rmult_sym ``/3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Pattern 2 PI; Rewrite (Rmult_sym PI); Repeat Rewrite Rmult_1r; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Ring. +Qed. + +Lemma sin_PI6_cos_PI3 : ``(cos (PI/3))==(sin (PI/6))``. +Proof with Trivial. +Replace ``PI/6`` with ``(PI/2)-(PI/3)``. +Rewrite sin_shift. +Assert H0 : ``6<>0``; [DiscrR | Idtac]. +Assert H1 : ``3<>0``; [DiscrR | Idtac]. +Assert H2 : ``2<>0``; [DiscrR | Idtac]. +Apply r_Rmult_mult with ``6``. +Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``6``). +Unfold Rdiv; Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite (Rmult_sym ``/3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Pattern 2 PI; Rewrite (Rmult_sym PI); Repeat Rewrite Rmult_1r; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Ring. +Qed. + +Lemma PI6_RGT_0 : ``0<PI/6``. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. +Qed. + +Lemma PI6_RLT_PI2 : ``PI/6<PI/2``. +Unfold Rdiv; Apply Rlt_monotony. +Apply PI_RGT_0. +Apply Rinv_lt; Sup. +Qed. + +Lemma sin_PI6 : ``(sin (PI/6))==1/2``. +Proof with Trivial. +Assert H : ``2<>0``; [DiscrR | Idtac]. +Apply r_Rmult_mult with ``2*(cos (PI/6))``. +Replace ``2*(cos (PI/6))*(sin (PI/6))`` with ``2*(sin (PI/6))*(cos (PI/6))``. +Rewrite <- sin_2a; Replace ``2*(PI/6)`` with ``PI/3``. +Rewrite sin_PI3_cos_PI6. +Unfold Rdiv; Rewrite Rmult_1l; Rewrite Rmult_assoc; Pattern 2 ``2``; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Unfold Rdiv; Rewrite Rinv_Rmult. +Rewrite (Rmult_sym ``/2``); Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +DiscrR. +Ring. +Apply prod_neq_R0. +Cut ``0<(cos (PI/6))``; [Intro H1; Auto with real | Apply cos_gt_0; [Apply (Rlt_trans ``-(PI/2)`` ``0`` ``PI/6`` _PI2_RLT_0 PI6_RGT_0) | Apply PI6_RLT_PI2]]. +Qed. + +Lemma sqrt2_neq_0 : ~``(sqrt 2)==0``. +Assert Hyp:``0<2``; [Sup0 | Generalize (Rlt_le ``0`` ``2`` Hyp); Intro H1; Red; Intro H2; Generalize (sqrt_eq_0 ``2`` H1 H2); Intro H; Absurd ``2==0``; [ DiscrR | Assumption]]. +Qed. + +Lemma R1_sqrt2_neq_0 : ~``1/(sqrt 2)==0``. +Generalize (Rinv_neq_R0 ``(sqrt 2)`` sqrt2_neq_0); Intro H; Generalize (prod_neq_R0 ``1`` ``(Rinv (sqrt 2))`` R1_neq_R0 H); Intro H0; Assumption. +Qed. + +Lemma sqrt3_2_neq_0 : ~``2*(sqrt 3)==0``. +Apply prod_neq_R0; [DiscrR | Assert Hyp:``0<3``; [Sup0 | Generalize (Rlt_le ``0`` ``3`` Hyp); Intro H1; Red; Intro H2; Generalize (sqrt_eq_0 ``3`` H1 H2); Intro H; Absurd ``3==0``; [ DiscrR | Assumption]]]. +Qed. + +Lemma Rlt_sqrt2_0 : ``0<(sqrt 2)``. +Assert Hyp:``0<2``; [Sup0 | Generalize (sqrt_positivity ``2`` (Rlt_le ``0`` ``2`` Hyp)); Intro H1; Elim H1; Intro H2; [Assumption | Absurd ``0 == (sqrt 2)``; [Apply not_sym; Apply sqrt2_neq_0 | Assumption]]]. +Qed. + +Lemma Rlt_sqrt3_0 : ``0<(sqrt 3)``. +Cut ~(O=(1)); [Intro H0; Assert Hyp:``0<2``; [Sup0 | Generalize (Rlt_le ``0`` ``2`` Hyp); Intro H1; Assert Hyp2:``0<3``; [Sup0 | Generalize (Rlt_le ``0`` ``3`` Hyp2); Intro H2; Generalize (lt_INR_0 (1) (neq_O_lt (1) H0)); Unfold INR; Intro H3; Generalize (Rlt_compatibility ``2`` ``0`` ``1`` H3); Rewrite Rplus_sym; Rewrite Rplus_Ol; Replace ``2+1`` with ``3``; [Intro H4; Generalize (sqrt_lt_1 ``2`` ``3`` H1 H2 H4); Clear H3; Intro H3; Apply (Rlt_trans ``0`` ``(sqrt 2)`` ``(sqrt 3)`` Rlt_sqrt2_0 H3) | Ring]]] | Discriminate]. +Qed. + +Lemma PI4_RGT_0 : ``0<PI/4``. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. +Qed. + +Lemma cos_PI4 : ``(cos (PI/4))==1/(sqrt 2)``. +Proof with Trivial. +Apply Rsqr_inj. +Apply cos_ge_0. +Left; Apply (Rlt_trans ``-(PI/2)`` R0 ``PI/4`` _PI2_RLT_0 PI4_RGT_0). +Left; Apply PI4_RLT_PI2. +Left; Apply (Rmult_lt_pos R1 ``(Rinv (sqrt 2))``). +Sup. +Apply Rlt_Rinv; Apply Rlt_sqrt2_0. +Rewrite Rsqr_div. +Rewrite Rsqr_1; Rewrite Rsqr_sqrt. +Assert H : ``2<>0``; [DiscrR | Idtac]. +Unfold Rsqr; Pattern 1 ``(cos (PI/4))``; Rewrite <- sin_cos_PI4; Replace ``(sin (PI/4))*(cos (PI/4))`` with ``(1/2)*(2*(sin (PI/4))*(cos (PI/4)))``. +Rewrite <- sin_2a; Replace ``2*(PI/4)`` with ``PI/2``. +Rewrite sin_PI2. +Apply Rmult_1r. +Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rinv_Rmult. +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Unfold Rdiv; Rewrite Rmult_1l; Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Left; Sup. +Apply sqrt2_neq_0. +Qed. + +Lemma sin_PI4 : ``(sin (PI/4))==1/(sqrt 2)``. +Rewrite sin_cos_PI4; Apply cos_PI4. +Qed. + +Lemma tan_PI4 : ``(tan (PI/4))==1``. +Unfold tan; Rewrite sin_cos_PI4. +Unfold Rdiv; Apply Rinv_r. +Change ``(cos (PI/4))<>0``; Rewrite cos_PI4; Apply R1_sqrt2_neq_0. +Qed. + +Lemma cos3PI4 : ``(cos (3*(PI/4)))==-1/(sqrt 2)``. +Proof with Trivial. +Replace ``3*(PI/4)`` with ``(PI/2)-(-(PI/4))``. +Rewrite cos_shift; Rewrite sin_neg; Rewrite sin_PI4. +Unfold Rdiv; Rewrite Ropp_mul1. +Unfold Rminus; Rewrite Ropp_Ropp; Pattern 1 PI; Rewrite double_var; Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_Rmult; [Ring | DiscrR | DiscrR]. +Qed. + +Lemma sin3PI4 : ``(sin (3*(PI/4)))==1/(sqrt 2)``. +Proof with Trivial. +Replace ``3*(PI/4)`` with ``(PI/2)-(-(PI/4))``. +Rewrite sin_shift; Rewrite cos_neg; Rewrite cos_PI4. +Unfold Rminus; Rewrite Ropp_Ropp; Pattern 1 PI; Rewrite double_var; Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_Rmult; [Ring | DiscrR | DiscrR]. +Qed. + +Lemma cos_PI6 : ``(cos (PI/6))==(sqrt 3)/2``. +Proof with Trivial. +Apply Rsqr_inj. +Apply cos_ge_0. +Left; Apply (Rlt_trans ``-(PI/2)`` R0 ``PI/6`` _PI2_RLT_0 PI6_RGT_0). +Left; Apply PI6_RLT_PI2. +Left; Apply (Rmult_lt_pos ``(sqrt 3)`` ``(Rinv 2)``). +Apply Rlt_sqrt3_0. +Apply Rlt_Rinv; Sup0. +Assert H : ``2<>0``; [DiscrR | Idtac]. +Assert H1 : ``4<>0``; [Apply prod_neq_R0 | Idtac]. +Rewrite Rsqr_div. +Rewrite cos2; Unfold Rsqr; Rewrite sin_PI6; Rewrite sqrt_def. +Unfold Rdiv; Rewrite Rmult_1l; Apply r_Rmult_mult with ``4``. +Rewrite Rminus_distr; Rewrite (Rmult_sym ``3``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite Rmult_1r. +Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- Rinv_r_sym. +Ring. +Left; Sup0. +Qed. + +Lemma tan_PI6 : ``(tan (PI/6))==1/(sqrt 3)``. +Unfold tan; Rewrite sin_PI6; Rewrite cos_PI6; Unfold Rdiv; Repeat Rewrite Rmult_1l; Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Rewrite (Rmult_sym ``/2``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Apply Rmult_1r. +DiscrR. +DiscrR. +Red; Intro; Assert H1 := Rlt_sqrt3_0; Rewrite H in H1; Elim (Rlt_antirefl ``0`` H1). +Apply Rinv_neq_R0; DiscrR. +Qed. + +Lemma sin_PI3 : ``(sin (PI/3))==(sqrt 3)/2``. +Rewrite sin_PI3_cos_PI6; Apply cos_PI6. +Qed. + +Lemma cos_PI3 : ``(cos (PI/3))==1/2``. +Rewrite sin_PI6_cos_PI3; Apply sin_PI6. +Qed. + +Lemma tan_PI3 : ``(tan (PI/3))==(sqrt 3)``. +Unfold tan; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Rewrite Rmult_1l; Rewrite Rinv_Rinv. +Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Apply Rmult_1r. +DiscrR. +DiscrR. +Qed. + +Lemma sin_2PI3 : ``(sin (2*(PI/3)))==(sqrt 3)/2``. +Rewrite double; Rewrite sin_plus; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Repeat Rewrite Rmult_1l; Rewrite (Rmult_sym ``/2``); Repeat Rewrite <- Rmult_assoc; Rewrite double_var; Reflexivity. +Qed. + +Lemma cos_2PI3 : ``(cos (2*(PI/3)))==-1/2``. +Proof with Trivial. +Assert H : ``2<>0``; [DiscrR | Idtac]. +Assert H0 : ``4<>0``; [Apply prod_neq_R0 | Idtac]. +Rewrite double; Rewrite cos_plus; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Rewrite Rmult_1l; Apply r_Rmult_mult with ``4``. +Rewrite Rminus_distr; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``). +Repeat Rewrite Rmult_assoc; Rewrite <- (Rinv_l_sym). +Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. +Pattern 4 ``2``; Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite Ropp_mul3; Rewrite Rmult_1r. +Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite (Rmult_sym ``2``); Rewrite (Rmult_sym ``/2``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Rewrite sqrt_def. +Ring. +Left; Sup. +Qed. + +Lemma tan_2PI3 : ``(tan (2*(PI/3)))==-(sqrt 3)``. +Proof with Trivial. +Assert H : ``2<>0``; [DiscrR | Idtac]. +Unfold tan; Rewrite sin_2PI3; Rewrite cos_2PI3; Unfold Rdiv; Rewrite Ropp_mul1; Rewrite Rmult_1l; Rewrite <- Ropp_Rinv. +Rewrite Rinv_Rinv. +Rewrite Rmult_assoc; Rewrite Ropp_mul3; Rewrite <- Rinv_l_sym. +Ring. +Apply Rinv_neq_R0. +Qed. + +Lemma cos_5PI4 : ``(cos (5*(PI/4)))==-1/(sqrt 2)``. +Proof with Trivial. +Replace ``5*(PI/4)`` with ``(PI/4)+(PI)``. +Rewrite neg_cos; Rewrite cos_PI4; Unfold Rdiv; Rewrite Ropp_mul1. +Pattern 2 PI; Rewrite double_var; Pattern 2 3 PI; Rewrite double_var; Assert H : ``2<>0``; [DiscrR | Unfold Rdiv; Repeat Rewrite Rinv_Rmult; Try Ring]. +Qed. + +Lemma sin_5PI4 : ``(sin (5*(PI/4)))==-1/(sqrt 2)``. +Proof with Trivial. +Replace ``5*(PI/4)`` with ``(PI/4)+(PI)``. +Rewrite neg_sin; Rewrite sin_PI4; Unfold Rdiv; Rewrite Ropp_mul1. +Pattern 2 PI; Rewrite double_var; Pattern 2 3 PI; Rewrite double_var; Assert H : ``2<>0``; [DiscrR | Unfold Rdiv; Repeat Rewrite Rinv_Rmult; Try Ring]. +Qed. + +Lemma sin_cos5PI4 : ``(cos (5*(PI/4)))==(sin (5*(PI/4)))``. +Rewrite cos_5PI4; Rewrite sin_5PI4; Reflexivity. +Qed. + +Lemma Rgt_3PI2_0 : ``0<3*(PI/2)``. +Apply Rmult_lt_pos; [Sup0 | Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]]. +Qed. + +Lemma Rgt_2PI_0 : ``0<2*PI``. +Apply Rmult_lt_pos; [Sup0 | Apply PI_RGT_0]. +Qed. + +Lemma Rlt_PI_3PI2 : ``PI<3*(PI/2)``. +Generalize PI2_RGT_0; Intro H1; Generalize (Rlt_compatibility PI ``0`` ``PI/2`` H1); Replace ``PI+(PI/2)`` with ``3*(PI/2)``. +Rewrite Rplus_Or; Intro H2; Assumption. +Pattern 2 PI; Rewrite double_var; Ring. +Qed. + +Lemma Rlt_3PI2_2PI : ``3*(PI/2)<2*PI``. +Generalize PI2_RGT_0; Intro H1; Generalize (Rlt_compatibility ``3*(PI/2)`` ``0`` ``PI/2`` H1); Replace ``3*(PI/2)+(PI/2)`` with ``2*PI``. +Rewrite Rplus_Or; Intro H2; Assumption. +Rewrite double; Pattern 1 2 PI; Rewrite double_var; Ring. +Qed. + +(***************************************************************) +(* Radian -> Degree | Degree -> Radian *) +(***************************************************************) + +Definition plat : R := ``180``. +Definition toRad [x:R] : R := ``x*PI*/plat``. +Definition toDeg [x:R] : R := ``x*plat*/PI``. + +Lemma rad_deg : (x:R) (toRad (toDeg x))==x. +Intro; Unfold toRad toDeg; Replace ``x*plat*/PI*PI*/plat`` with ``x*(plat*/plat)*(PI*/PI)``; [Idtac | Ring]. +Repeat Rewrite <- Rinv_r_sym. +Ring. +Apply PI_neq0. +Unfold plat; DiscrR. +Qed. + +Lemma toRad_inj : (x,y:R) (toRad x)==(toRad y) -> x==y. +Intros; Unfold toRad in H; Apply r_Rmult_mult with PI. +Rewrite <- (Rmult_sym x); Rewrite <- (Rmult_sym y). +Apply r_Rmult_mult with ``/plat``. +Rewrite <- (Rmult_sym ``x*PI``); Rewrite <- (Rmult_sym ``y*PI``); Assumption. +Apply Rinv_neq_R0; Unfold plat; DiscrR. +Apply PI_neq0. +Qed. + +Lemma deg_rad : (x:R) (toDeg (toRad x))==x. +Intro x; Apply toRad_inj; Rewrite -> (rad_deg (toRad x)); Reflexivity. +Qed. + +Definition sind [x:R] : R := (sin (toRad x)). +Definition cosd [x:R] : R := (cos (toRad x)). +Definition tand [x:R] : R := (tan (toRad x)). + +Lemma Rsqr_sin_cos_d_one : (x:R) ``(Rsqr (sind x))+(Rsqr (cosd x))==1``. +Intro x; Unfold sind; Unfold cosd; Apply sin2_cos2. +Qed. + +(***************************************************) +(* Other properties *) +(***************************************************) + +Lemma sin_lb_ge_0 : (a:R) ``0<=a``->``a<=PI/2``->``0<=(sin_lb a)``. +Intros; Case (total_order R0 a); Intro. +Left; Apply sin_lb_gt_0; Assumption. +Elim H1; Intro. +Rewrite <- H2; Unfold sin_lb; Unfold sin_approx; Unfold sum_f_R0; Unfold sin_term; Repeat Rewrite pow_ne_zero. +Unfold Rdiv; Repeat Rewrite Rmult_Ol; Repeat Rewrite Rmult_Or; Repeat Rewrite Rplus_Or; Right; Reflexivity. +Discriminate. +Discriminate. +Discriminate. +Discriminate. +Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` a ``0`` H H2)). +Qed. diff --git a/theories7/Reals/Rtrigo_def.v b/theories7/Reals/Rtrigo_def.v new file mode 100644 index 00000000..0897416b --- /dev/null +++ b/theories7/Reals/Rtrigo_def.v @@ -0,0 +1,357 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rtrigo_def.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo_fun. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(*****************************) +(* Definition of exponential *) +(*****************************) +Definition exp_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``/(INR (fact i))*(pow x i)`` l). + +Lemma exp_cof_no_R0 : (n:nat) ``/(INR (fact n))<>0``. +Intro. +Apply Rinv_neq_R0. +Apply INR_fact_neq_0. +Qed. + +Lemma exist_exp : (x:R)(SigT R [l:R](exp_in x l)). +Intro; Generalize (Alembert_C3 [n:nat](Rinv (INR (fact n))) x exp_cof_no_R0 Alembert_exp). +Unfold Pser exp_in. +Trivial. +Defined. + +Definition exp : R -> R := [x:R](projT1 ? ? (exist_exp x)). + +Lemma pow_i : (i:nat) (lt O i) -> (pow R0 i)==R0. +Intros; Apply pow_ne_zero. +Red; Intro; Rewrite H0 in H; Elim (lt_n_n ? H). +Qed. + +(*i Calculus of $e^0$ *) +Lemma exist_exp0 : (SigT R [l:R](exp_in R0 l)). +Apply Specif.existT with R1. +Unfold exp_in; Unfold infinit_sum; Intros. +Exists O. +Intros; Replace (sum_f_R0 ([i:nat]``/(INR (fact i))*(pow R0 i)``) n) with R1. +Unfold R_dist; Replace ``1-1`` with R0; [Rewrite Rabsolu_R0; Assumption | Ring]. +Induction n. +Simpl; Rewrite Rinv_R1; Ring. +Rewrite tech5. +Rewrite <- Hrecn. +Simpl. +Ring. +Unfold ge; Apply le_O_n. +Defined. + +Lemma exp_0 : ``(exp 0)==1``. +Cut (exp_in R0 (exp R0)). +Cut (exp_in R0 R1). +Unfold exp_in; Intros; EApply unicity_sum. +Apply H0. +Apply H. +Exact (projT2 ? ? exist_exp0). +Exact (projT2 ? ? (exist_exp R0)). +Qed. + +(**************************************) +(* Definition of hyperbolic functions *) +(**************************************) +Definition cosh : R->R := [x:R]``((exp x)+(exp (-x)))/2``. +Definition sinh : R->R := [x:R]``((exp x)-(exp (-x)))/2``. +Definition tanh : R->R := [x:R]``(sinh x)/(cosh x)``. + +Lemma cosh_0 : ``(cosh 0)==1``. +Unfold cosh; Rewrite Ropp_O; Rewrite exp_0. +Unfold Rdiv; Rewrite <- Rinv_r_sym; [Reflexivity | DiscrR]. +Qed. + +Lemma sinh_0 : ``(sinh 0)==0``. +Unfold sinh; Rewrite Ropp_O; Rewrite exp_0. +Unfold Rminus Rdiv; Rewrite Rplus_Ropp_r; Apply Rmult_Ol. +Qed. + +Definition cos_n [n:nat] : R := ``(pow (-1) n)/(INR (fact (mult (S (S O)) n)))``. + +Lemma simpl_cos_n : (n:nat) (Rdiv (cos_n (S n)) (cos_n n))==(Ropp (Rinv (INR (mult (mult (2) (S n)) (plus (mult (2) n) (1)))))). +Intro; Unfold cos_n; Replace (S n) with (plus n (1)); [Idtac | Ring]. +Rewrite pow_add; Unfold Rdiv; Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Replace ``(pow ( -1) n)*(pow ( -1) (S O))*/(INR (fact (mult (S (S O)) (plus n (S O)))))*(/(pow ( -1) n)*(INR (fact (mult (S (S O)) n))))`` with ``((pow ( -1) n)*/(pow ( -1) n))*/(INR (fact (mult (S (S O)) (plus n (S O)))))*(INR (fact (mult (S (S O)) n)))*(pow (-1) (S O))``; [Idtac | Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Unfold pow; Rewrite Rmult_1r. +Replace (mult (S (S O)) (plus n (S O))) with (S (S (mult (S (S O)) n))); [Idtac | Ring]. +Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rinv_Rmult; Try (Apply not_O_INR; Discriminate). +Rewrite <- (Rmult_sym ``-1``). +Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Replace (S (mult (S (S O)) n)) with (plus (mult (S (S O)) n) (S O)); [Idtac | Ring]. +Rewrite mult_INR; Rewrite Rinv_Rmult. +Ring. +Apply not_O_INR; Discriminate. +Replace (plus (mult (S (S O)) n) (S O)) with (S (mult (S (S O)) n)); [Apply not_O_INR; Discriminate | Ring]. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. +Apply pow_nonzero; DiscrR. +Apply INR_fact_neq_0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Qed. + +Lemma archimed_cor1 : (eps:R) ``0<eps`` -> (EX N : nat | ``/(INR N) < eps``/\(lt O N)). +Intros; Cut ``/eps < (IZR (up (/eps)))``. +Intro; Cut `0<=(up (Rinv eps))`. +Intro; Assert H2 := (IZN ? H1); Elim H2; Intros; Exists (max x (1)). +Split. +Cut ``0<(IZR (INZ x))``. +Intro; Rewrite INR_IZR_INZ; Apply Rle_lt_trans with ``/(IZR (INZ x))``. +Apply Rle_monotony_contra with (IZR (INZ x)). +Assumption. +Rewrite <- Rinv_r_sym; [Idtac | Red; Intro; Rewrite H5 in H4; Elim (Rlt_antirefl ? H4)]. +Apply Rle_monotony_contra with (IZR (INZ (max x (1)))). +Apply Rlt_le_trans with (IZR (INZ x)). +Assumption. +Repeat Rewrite <- INR_IZR_INZ; Apply le_INR; Apply le_max_l. +Rewrite Rmult_1r; Rewrite (Rmult_sym (IZR (INZ (max x (S O))))); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Repeat Rewrite <- INR_IZR_INZ; Apply le_INR; Apply le_max_l. +Rewrite <- INR_IZR_INZ; Apply not_O_INR. +Red; Intro;Assert H6 := (le_max_r x (1)); Cut (lt O (1)); [Intro | Apply lt_O_Sn]; Assert H8 := (lt_le_trans ? ? ? H7 H6); Rewrite H5 in H8; Elim (lt_n_n ? H8). +Pattern 1 eps; Rewrite <- Rinv_Rinv. +Apply Rinv_lt. +Apply Rmult_lt_pos; [Apply Rlt_Rinv; Assumption | Assumption]. +Rewrite H3 in H0; Assumption. +Red; Intro; Rewrite H5 in H; Elim (Rlt_antirefl ? H). +Apply Rlt_trans with ``/eps``. +Apply Rlt_Rinv; Assumption. +Rewrite H3 in H0; Assumption. +Apply lt_le_trans with (1); [Apply lt_O_Sn | Apply le_max_r]. +Apply le_IZR; Replace (IZR `0`) with R0; [Idtac | Reflexivity]; Left; Apply Rlt_trans with ``/eps``; [Apply Rlt_Rinv; Assumption | Assumption]. +Assert H0 := (archimed ``/eps``). +Elim H0; Intros; Assumption. +Qed. + +Lemma Alembert_cos : (Un_cv [n:nat]``(Rabsolu (cos_n (S n))/(cos_n n))`` R0). +Unfold Un_cv; Intros. +Assert H0 := (archimed_cor1 eps H). +Elim H0; Intros; Exists x. +Intros; Rewrite simpl_cos_n; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Rewrite Rabsolu_Ropp; Rewrite Rabsolu_right. +Rewrite mult_INR; Rewrite Rinv_Rmult. +Cut ``/(INR (mult (S (S O)) (S n)))<1``. +Intro; Cut ``/(INR (plus (mult (S (S O)) n) (S O)))<eps``. +Intro; Rewrite <- (Rmult_1l eps). +Apply Rmult_lt; Try Assumption. +Change ``0</(INR (plus (mult (S (S O)) n) (S O)))``; Apply Rlt_Rinv; Apply lt_INR_0. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. +Apply Rlt_R0_R1. +Cut (lt x (plus (mult (2) n) (1))). +Intro; Assert H5 := (lt_INR ? ? H4). +Apply Rlt_trans with ``/(INR x)``. +Apply Rinv_lt. +Apply Rmult_lt_pos. +Apply lt_INR_0. +Elim H1; Intros; Assumption. +Apply lt_INR_0; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. +Assumption. +Elim H1; Intros; Assumption. +Apply lt_le_trans with (S n). +Unfold ge in H2; Apply le_lt_n_Sm; Assumption. +Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Idtac | Ring]. +Apply le_n_S; Apply le_n_2n. +Apply Rlt_monotony_contra with (INR (mult (S (S O)) (S n))). +Apply lt_INR_0; Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Apply lt_O_Sn. +Replace (S n) with (plus n (1)); [Idtac | Ring]. +Ring. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Replace R1 with (INR (1)); [Apply lt_INR | Reflexivity]. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Apply lt_n_S; Apply lt_O_Sn. +Replace (S n) with (plus n (1)); [Ring | Ring]. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Replace (plus (mult (S (S O)) n) (S O)) with (S (mult (2) n)); [Apply not_O_INR; Discriminate | Ring]. +Apply Rle_sym1; Left; Apply Rlt_Rinv. +Apply lt_INR_0. +Replace (mult (mult (2) (S n)) (plus (mult (2) n) (1))) with (S (S (plus (mult (4) (mult n n)) (mult (6) n)))). +Apply lt_O_Sn. +Apply INR_eq. +Repeat Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite mult_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Replace (INR O) with R0; [Ring | Reflexivity]. +Qed. + +Lemma cosn_no_R0 : (n:nat)``(cos_n n)<>0``. +Intro; Unfold cos_n; Unfold Rdiv; Apply prod_neq_R0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0. +Apply INR_fact_neq_0. +Qed. + +(**********) +Definition cos_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``(cos_n i)*(pow x i)`` l). + +(**********) +Lemma exist_cos : (x:R)(SigT R [l:R](cos_in x l)). +Intro; Generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos). +Unfold Pser cos_in; Trivial. +Qed. + +(* Definition of cosinus *) +(*************************) +Definition cos : R -> R := [x:R](Cases (exist_cos (Rsqr x)) of (Specif.existT a b) => a end). + + +Definition sin_n [n:nat] : R := ``(pow (-1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))``. + +Lemma simpl_sin_n : (n:nat) (Rdiv (sin_n (S n)) (sin_n n))==(Ropp (Rinv (INR (mult (plus (mult (2) (S n)) (1)) (mult (2) (S n)))))). +Intro; Unfold sin_n; Replace (S n) with (plus n (1)); [Idtac | Ring]. +Rewrite pow_add; Unfold Rdiv; Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Replace ``(pow ( -1) n)*(pow ( -1) (S O))*/(INR (fact (plus (mult (S (S O)) (plus n (S O))) (S O))))*(/(pow ( -1) n)*(INR (fact (plus (mult (S (S O)) n) (S O)))))`` with ``((pow ( -1) n)*/(pow ( -1) n))*/(INR (fact (plus (mult (S (S O)) (plus n (S O))) (S O))))*(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow (-1) (S O))``; [Idtac | Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Unfold pow; Rewrite Rmult_1r; Replace (plus (mult (S (S O)) (plus n (S O))) (S O)) with (S (S (plus (mult (S (S O)) n) (S O)))). +Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. +Rewrite <- (Rmult_sym ``-1``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Replace (S (plus (mult (S (S O)) n) (S O))) with (mult (S (S O)) (plus n (S O))). +Repeat Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. +Ring. +Apply not_O_INR; Discriminate. +Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. +Apply not_O_INR; Discriminate. +Apply prod_neq_R0. +Apply not_O_INR; Discriminate. +Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. +Apply not_O_INR; Discriminate. +Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. +Rewrite mult_plus_distr_r; Cut (n:nat) (S n)=(plus n (1)). +Intros; Rewrite (H (plus (mult (2) n) (1))). +Ring. +Intros; Ring. +Apply INR_fact_neq_0. +Apply not_O_INR; Discriminate. +Apply INR_fact_neq_0. +Apply not_O_INR; Discriminate. +Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. +Cut (n:nat) (S (S n))=(plus n (2)); [Intros; Rewrite (H (plus (mult (2) n) (1))); Ring | Intros; Ring]. +Apply pow_nonzero; DiscrR. +Apply INR_fact_neq_0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Qed. + +Lemma Alembert_sin : (Un_cv [n:nat]``(Rabsolu (sin_n (S n))/(sin_n n))`` R0). +Unfold Un_cv; Intros; Assert H0 := (archimed_cor1 eps H). +Elim H0; Intros; Exists x. +Intros; Rewrite simpl_sin_n; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Rewrite Rabsolu_Ropp; Rewrite Rabsolu_right. +Rewrite mult_INR; Rewrite Rinv_Rmult. +Cut ``/(INR (mult (S (S O)) (S n)))<1``. +Intro; Cut ``/(INR (plus (mult (S (S O)) (S n)) (S O)))<eps``. +Intro; Rewrite <- (Rmult_1l eps); Rewrite (Rmult_sym ``/(INR (plus (mult (S (S O)) (S n)) (S O)))``); Apply Rmult_lt; Try Assumption. +Change ``0</(INR (plus (mult (S (S O)) (S n)) (S O)))``; Apply Rlt_Rinv; Apply lt_INR_0; Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Apply lt_O_Sn | Ring]. +Apply Rlt_R0_R1. +Cut (lt x (plus (mult (2) (S n)) (1))). +Intro; Assert H5 := (lt_INR ? ? H4); Apply Rlt_trans with ``/(INR x)``. +Apply Rinv_lt. +Apply Rmult_lt_pos. +Apply lt_INR_0; Elim H1; Intros; Assumption. +Apply lt_INR_0; Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Apply lt_O_Sn | Ring]. +Assumption. +Elim H1; Intros; Assumption. +Apply lt_le_trans with (S n). +Unfold ge in H2; Apply le_lt_n_Sm; Assumption. +Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Idtac | Ring]. +Apply le_S; Apply le_n_2n. +Apply Rlt_monotony_contra with (INR (mult (S (S O)) (S n))). +Apply lt_INR_0; Replace (mult (2) (S n)) with (S (S (mult (2) n))); [Apply lt_O_Sn | Replace (S n) with (plus n (1)); [Idtac | Ring]; Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Replace R1 with (INR (1)); [Apply lt_INR | Reflexivity]. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Apply lt_n_S; Apply lt_O_Sn. +Replace (S n) with (plus n (1)); [Ring | Ring]. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Apply not_O_INR; Discriminate. +Left; Change ``0</(INR (mult (plus (mult (S (S O)) (S n)) (S O)) (mult (S (S O)) (S n))))``; Apply Rlt_Rinv. +Apply lt_INR_0. +Replace (mult (plus (mult (2) (S n)) (1)) (mult (2) (S n))) with (S (S (S (S (S (S (plus (mult (4) (mult n n)) (mult (10) n)))))))). +Apply lt_O_Sn. +Apply INR_eq; Repeat Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite mult_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Replace (INR O) with R0; [Ring | Reflexivity]. +Qed. + +Lemma sin_no_R0 : (n:nat)``(sin_n n)<>0``. +Intro; Unfold sin_n; Unfold Rdiv; Apply prod_neq_R0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Qed. + +(**********) +Definition sin_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``(sin_n i)*(pow x i)`` l). + +(**********) +Lemma exist_sin : (x:R)(SigT R [l:R](sin_in x l)). +Intro; Generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin). +Unfold Pser sin_n; Trivial. +Qed. + +(***********************) +(* Definition of sinus *) +Definition sin : R -> R := [x:R](Cases (exist_sin (Rsqr x)) of (Specif.existT a b) => ``x*a`` end). + +(*********************************************) +(* PROPERTIES *) +(*********************************************) + +Lemma cos_sym : (x:R) ``(cos x)==(cos (-x))``. +Intros; Unfold cos; Replace ``(Rsqr (-x))`` with (Rsqr x). +Reflexivity. +Apply Rsqr_neg. +Qed. + +Lemma sin_antisym : (x:R)``(sin (-x))==-(sin x)``. +Intro; Unfold sin; Replace ``(Rsqr (-x))`` with (Rsqr x); [Idtac | Apply Rsqr_neg]. +Case (exist_sin (Rsqr x)); Intros; Ring. +Qed. + +Lemma sin_0 : ``(sin 0)==0``. +Unfold sin; Case (exist_sin (Rsqr R0)). +Intros; Ring. +Qed. + +Lemma exist_cos0 : (SigT R [l:R](cos_in R0 l)). +Apply Specif.existT with R1. +Unfold cos_in; Unfold infinit_sum; Intros; Exists O. +Intros. +Unfold R_dist. +Induction n. +Unfold cos_n; Simpl. +Unfold Rdiv; Rewrite Rinv_R1. +Do 2 Rewrite Rmult_1r. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Rewrite tech5. +Replace ``(cos_n (S n))*(pow 0 (S n))`` with R0. +Rewrite Rplus_Or. +Apply Hrecn; Unfold ge; Apply le_O_n. +Simpl; Ring. +Defined. + +(* Calculus of (cos 0) *) +Lemma cos_0 : ``(cos 0)==1``. +Cut (cos_in R0 (cos R0)). +Cut (cos_in R0 R1). +Unfold cos_in; Intros; EApply unicity_sum. +Apply H0. +Apply H. +Exact (projT2 ? ? exist_cos0). +Assert H := (projT2 ? ? (exist_cos (Rsqr R0))); Unfold cos; Pattern 1 R0; Replace R0 with (Rsqr R0); [Exact H | Apply Rsqr_O]. +Qed. diff --git a/theories7/Reals/Rtrigo_fun.v b/theories7/Reals/Rtrigo_fun.v new file mode 100644 index 00000000..bc72c0e1 --- /dev/null +++ b/theories7/Reals/Rtrigo_fun.v @@ -0,0 +1,118 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rtrigo_fun.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(*****************************************************************) +(* To define transcendental functions *) +(* *) +(*****************************************************************) +(*****************************************************************) +(* For exponential function *) +(* *) +(*****************************************************************) + +(*********) +Lemma Alembert_exp:(Un_cv + [n:nat](Rabsolu (Rmult (Rinv (INR (fact (S n)))) + (Rinv (Rinv (INR (fact n)))))) R0). +Unfold Un_cv;Intros;Elim (total_order_Rgt eps R1);Intro. +Split with O;Intros;Rewrite (simpl_fact n);Unfold R_dist; + Rewrite (minus_R0 (Rabsolu (Rinv (INR (S n))))); + Rewrite (Rabsolu_Rabsolu (Rinv (INR (S n)))); + Cut (Rgt (Rinv (INR (S n))) R0). +Intro; Rewrite (Rabsolu_pos_eq (Rinv (INR (S n)))). +Cut (Rlt (Rminus (Rinv eps) R1) R0). +Intro;Generalize (Rlt_le_trans (Rminus (Rinv eps) R1) R0 (INR n) H2 + (pos_INR n));Clear H2;Intro; + Unfold Rminus in H2;Generalize (Rlt_compatibility R1 + (Rplus (Rinv eps) (Ropp R1)) (INR n) H2); + Replace (Rplus R1 (Rplus (Rinv eps) (Ropp R1))) with (Rinv eps); + [Clear H2;Intro|Ring]. +Rewrite (Rplus_sym R1 (INR n)) in H2;Rewrite <-(S_INR n) in H2; + Generalize (Rmult_gt (Rinv (INR (S n))) eps H1 H);Intro; + Unfold Rgt in H3; + Generalize (Rlt_monotony (Rmult (Rinv (INR (S n))) eps) (Rinv eps) + (INR (S n)) H3 H2);Intro; + Rewrite (Rmult_assoc (Rinv (INR (S n))) eps (Rinv eps)) in H4; + Rewrite (Rinv_r eps (imp_not_Req eps R0 + (or_intror (Rlt eps R0) (Rgt eps R0) H))) + in H4;Rewrite (let (H1,H2)=(Rmult_ne (Rinv (INR (S n)))) in H1) + in H4;Rewrite (Rmult_sym (Rinv (INR (S n)))) in H4; + Rewrite (Rmult_assoc eps (Rinv (INR (S n))) (INR (S n))) in H4; + Rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) + (sym_not_equal nat O (S n) (O_S n)))) in H4; + Rewrite (let (H1,H2)=(Rmult_ne eps) in H1) in H4;Assumption. +Apply Rlt_minus;Unfold Rgt in a;Rewrite <- Rinv_R1; + Apply (Rinv_lt R1 eps);Auto; + Rewrite (let (H1,H2)=(Rmult_ne eps) in H2);Unfold Rgt in H;Assumption. +Unfold Rgt in H1;Apply Rlt_le;Assumption. +Unfold Rgt;Apply Rlt_Rinv; Apply lt_INR_0;Apply lt_O_Sn. +(**) +Cut `0<=(up (Rminus (Rinv eps) R1))`. +Intro;Elim (IZN (up (Rminus (Rinv eps) R1)) H0);Intros; + Split with x;Intros;Rewrite (simpl_fact n);Unfold R_dist; + Rewrite (minus_R0 (Rabsolu (Rinv (INR (S n))))); + Rewrite (Rabsolu_Rabsolu (Rinv (INR (S n)))); + Cut (Rgt (Rinv (INR (S n))) R0). +Intro; Rewrite (Rabsolu_pos_eq (Rinv (INR (S n)))). +Cut (Rlt (Rminus (Rinv eps) R1) (INR x)). +Intro;Generalize (Rlt_le_trans (Rminus (Rinv eps) R1) (INR x) (INR n) + H4 (le_INR x n ([n,m:nat; H:(ge m n)]H x n H2))); + Clear H4;Intro;Unfold Rminus in H4;Generalize (Rlt_compatibility R1 + (Rplus (Rinv eps) (Ropp R1)) (INR n) H4); + Replace (Rplus R1 (Rplus (Rinv eps) (Ropp R1))) with (Rinv eps); + [Clear H4;Intro|Ring]. +Rewrite (Rplus_sym R1 (INR n)) in H4;Rewrite <-(S_INR n) in H4; + Generalize (Rmult_gt (Rinv (INR (S n))) eps H3 H);Intro; + Unfold Rgt in H5; + Generalize (Rlt_monotony (Rmult (Rinv (INR (S n))) eps) (Rinv eps) + (INR (S n)) H5 H4);Intro; + Rewrite (Rmult_assoc (Rinv (INR (S n))) eps (Rinv eps)) in H6; + Rewrite (Rinv_r eps (imp_not_Req eps R0 + (or_intror (Rlt eps R0) (Rgt eps R0) H))) + in H6;Rewrite (let (H1,H2)=(Rmult_ne (Rinv (INR (S n)))) in H1) + in H6;Rewrite (Rmult_sym (Rinv (INR (S n)))) in H6; + Rewrite (Rmult_assoc eps (Rinv (INR (S n))) (INR (S n))) in H6; + Rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) + (sym_not_equal nat O (S n) (O_S n)))) in H6; + Rewrite (let (H1,H2)=(Rmult_ne eps) in H1) in H6;Assumption. +Cut (IZR (up (Rminus (Rinv eps) R1)))==(IZR (INZ x)); + [Intro|Rewrite H1;Trivial]. +Elim (archimed (Rminus (Rinv eps) R1));Intros;Clear H6; + Unfold Rgt in H5;Rewrite H4 in H5;Rewrite INR_IZR_INZ;Assumption. +Unfold Rgt in H1;Apply Rlt_le;Assumption. +Unfold Rgt;Apply Rlt_Rinv; Apply lt_INR_0;Apply lt_O_Sn. +Apply (le_O_IZR (up (Rminus (Rinv eps) R1))); + Apply (Rle_trans R0 (Rminus (Rinv eps) R1) + (IZR (up (Rminus (Rinv eps) R1)))). +Generalize (Rgt_not_le eps R1 b);Clear b;Unfold Rle;Intro;Elim H0; + Clear H0;Intro. +Left;Unfold Rgt in H; + Generalize (Rlt_monotony (Rinv eps) eps R1 (Rlt_Rinv eps H) H0); + Rewrite (Rinv_l eps (sym_not_eqT R R0 eps + (imp_not_Req R0 eps (or_introl (Rlt R0 eps) (Rgt R0 eps) H)))); + Rewrite (let (H1,H2)=(Rmult_ne (Rinv eps)) in H1);Intro; + Fold (Rgt (Rminus (Rinv eps) R1) R0);Apply Rgt_minus;Unfold Rgt; + Assumption. +Right;Rewrite H0;Rewrite Rinv_R1;Apply sym_eqT;Apply eq_Rminus;Auto. +Elim (archimed (Rminus (Rinv eps) R1));Intros;Clear H1; + Unfold Rgt in H0;Apply Rlt_le;Assumption. +Qed. + + + + + + diff --git a/theories7/Reals/Rtrigo_reg.v b/theories7/Reals/Rtrigo_reg.v new file mode 100644 index 00000000..02e40caf --- /dev/null +++ b/theories7/Reals/Rtrigo_reg.v @@ -0,0 +1,497 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rtrigo_reg.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis1. +Require PSeries_reg. +V7only [Import nat_scope. Import Z_scope. Import R_scope.]. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +Lemma CVN_R_cos : (fn:nat->R->R) (fn == [N:nat][x:R]``(pow (-1) N)/(INR (fact (mult (S (S O)) N)))*(pow x (mult (S (S O)) N))``) -> (CVN_R fn). +Unfold CVN_R; Intros. +Cut (r::R)<>``0``. +Intro hyp_r; Unfold CVN_r. +Apply Specif.existT with [n:nat]``/(INR (fact (mult (S (S O)) n)))*(pow r (mult (S (S O)) n))``. +Cut (SigT ? [l:R](Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu ``/(INR (fact (mult (S (S O)) k)))*(pow r (mult (S (S O)) k))``) n) l)). +Intro; Elim X; Intros. +Apply existTT with x. +Split. +Apply p. +Intros; Rewrite H; Unfold Rdiv; Do 2 Rewrite Rabsolu_mult. +Rewrite pow_1_abs; Rewrite Rmult_1l. +Cut ``0</(INR (fact (mult (S (S O)) n)))``. +Intro; Rewrite (Rabsolu_right ? (Rle_sym1 ? ? (Rlt_le ? ? H1))). +Apply Rle_monotony. +Left; Apply H1. +Rewrite <- Pow_Rabsolu; Apply pow_maj_Rabs. +Rewrite Rabsolu_Rabsolu. +Unfold Boule in H0; Rewrite minus_R0 in H0. +Left; Apply H0. +Apply Rlt_Rinv; Apply INR_fact_lt_0. +Apply Alembert_C2. +Intro; Apply Rabsolu_no_R0. +Apply prod_neq_R0. +Apply Rinv_neq_R0. +Apply INR_fact_neq_0. +Apply pow_nonzero; Assumption. +Assert H0 := Alembert_cos. +Unfold cos_n in H0; Unfold Un_cv in H0; Unfold Un_cv; Intros. +Cut ``0<eps/(Rsqr r)``. +Intro; Elim (H0 ? H2); Intros N0 H3. +Exists N0; Intros. +Unfold R_dist; Assert H5 := (H3 ? H4). +Unfold R_dist in H5; Replace ``(Rabsolu ((Rabsolu (/(INR (fact (mult (S (S O)) (S n))))*(pow r (mult (S (S O)) (S n)))))/(Rabsolu (/(INR (fact (mult (S (S O)) n)))*(pow r (mult (S (S O)) n))))))`` with ``(Rsqr r)*(Rabsolu ((pow ( -1) (S n))/(INR (fact (mult (S (S O)) (S n))))/((pow ( -1) n)/(INR (fact (mult (S (S O)) n))))))``. +Apply Rlt_monotony_contra with ``/(Rsqr r)``. +Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. +Pattern 1 ``/(Rsqr r)``; Replace ``/(Rsqr r)`` with ``(Rabsolu (/(Rsqr r)))``. +Rewrite <- Rabsolu_mult; Rewrite Rminus_distr; Rewrite Rmult_Or; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Apply H5. +Unfold Rsqr; Apply prod_neq_R0; Assumption. +Rewrite Rabsolu_Rinv. +Rewrite Rabsolu_right. +Reflexivity. +Apply Rle_sym1; Apply pos_Rsqr. +Unfold Rsqr; Apply prod_neq_R0; Assumption. +Rewrite (Rmult_sym (Rsqr r)); Unfold Rdiv; Repeat Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite pow_1_abs; Rewrite Rmult_1l; Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r. +Rewrite Rabsolu_Rinv. +Rewrite Rabsolu_mult; Rewrite (pow_1_abs n); Rewrite Rmult_1l; Rewrite <- Rabsolu_Rinv. +Rewrite Rinv_Rinv. +Rewrite Rinv_Rmult. +Rewrite Rabsolu_Rinv. +Rewrite Rinv_Rinv. +Rewrite (Rmult_sym ``(Rabsolu (Rabsolu (pow r (mult (S (S O)) (S n)))))``); Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite Rmult_assoc; Apply Rmult_mult_r. +Rewrite Rabsolu_Rinv. +Do 2 Rewrite Rabsolu_Rabsolu; Repeat Rewrite Rabsolu_right. +Replace ``(pow r (mult (S (S O)) (S n)))`` with ``(pow r (mult (S (S O)) n))*r*r``. +Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Unfold Rsqr; Ring. +Apply pow_nonzero; Assumption. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). +Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply Rabsolu_no_R0; Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply Rabsolu_no_R0; Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply INR_fact_neq_0. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply prod_neq_R0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply H1. +Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. +Assert H0 := (cond_pos r); Red; Intro; Rewrite H1 in H0; Elim (Rlt_antirefl ? H0). +Qed. + +(**********) +Lemma continuity_cos : (continuity cos). +Pose fn := [N:nat][x:R]``(pow (-1) N)/(INR (fact (mult (S (S O)) N)))*(pow x (mult (S (S O)) N))``. +Cut (CVN_R fn). +Intro; Cut (x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l)). +Intro cv; Cut ((n:nat)(continuity (fn n))). +Intro; Cut (x:R)(cos x)==(SFL fn cv x). +Intro; Cut (continuity (SFL fn cv))->(continuity cos). +Intro; Apply H1. +Apply SFL_continuity; Assumption. +Unfold continuity; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Elim (H1 x ? H2); Intros. +Exists x0; Intros. +Elim H3; Intros. +Split. +Apply H4. +Intros; Rewrite (H0 x); Rewrite (H0 x1); Apply H5; Apply H6. +Intro; Unfold cos SFL. +Case (cv x); Case (exist_cos (Rsqr x)); Intros. +Symmetry; EApply UL_sequence. +Apply u. +Unfold cos_in in c; Unfold infinit_sum in c; Unfold Un_cv; Intros. +Elim (c ? H0); Intros N0 H1. +Exists N0; Intros. +Unfold R_dist in H1; Unfold R_dist SP. +Replace (sum_f_R0 [k:nat](fn k x) n) with (sum_f_R0 [i:nat]``(cos_n i)*(pow (Rsqr x) i)`` n). +Apply H1; Assumption. +Apply sum_eq; Intros. +Unfold cos_n fn; Apply Rmult_mult_r. +Unfold Rsqr; Rewrite pow_sqr; Reflexivity. +Intro; Unfold fn; Replace [x:R]``(pow ( -1) n)/(INR (fact (mult (S (S O)) n)))*(pow x (mult (S (S O)) n))`` with (mult_fct (fct_cte ``(pow ( -1) n)/(INR (fact (mult (S (S O)) n)))``) (pow_fct (mult (S (S O)) n))); [Idtac | Reflexivity]. +Apply continuity_mult. +Apply derivable_continuous; Apply derivable_const. +Apply derivable_continuous; Apply (derivable_pow (mult (2) n)). +Apply CVN_R_CVS; Apply X. +Apply CVN_R_cos; Unfold fn; Reflexivity. +Qed. + +(**********) +Lemma continuity_sin : (continuity sin). +Unfold continuity; Intro. +Assert H0 := (continuity_cos ``PI/2-x``). +Unfold continuity_pt in H0; Unfold continue_in in H0; Unfold limit1_in in H0; Unfold limit_in in H0; Simpl in H0; Unfold R_dist in H0; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Elim (H0 ? H); Intros. +Exists x0; Intros. +Elim H1; Intros. +Split. +Assumption. +Intros; Rewrite <- (cos_shift x); Rewrite <- (cos_shift x1); Apply H3. +Elim H4; Intros. +Split. +Unfold D_x no_cond; Split. +Trivial. +Red; Intro; Unfold D_x no_cond in H5; Elim H5; Intros _ H8; Elim H8; Rewrite <- (Ropp_Ropp x); Rewrite <- (Ropp_Ropp x1); Apply eq_Ropp; Apply r_Rplus_plus with ``PI/2``; Apply H7. +Replace ``PI/2-x1-(PI/2-x)`` with ``x-x1``; [Idtac | Ring]; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr3; Apply H6. +Qed. + +Lemma CVN_R_sin : (fn:nat->R->R) (fn == [N:nat][x:R]``(pow ( -1) N)/(INR (fact (plus (mult (S (S O)) N) (S O))))*(pow x (mult (S (S O)) N))``) -> (CVN_R fn). +Unfold CVN_R; Unfold CVN_r; Intros fn H r. +Apply Specif.existT with [n:nat]``/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow r (mult (S (S O)) n))``. +Cut (SigT ? [l:R](Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu ``/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow r (mult (S (S O)) k))``) n) l)). +Intro; Elim X; Intros. +Apply existTT with x. +Split. +Apply p. +Intros; Rewrite H; Unfold Rdiv; Do 2 Rewrite Rabsolu_mult; Rewrite pow_1_abs; Rewrite Rmult_1l. +Cut ``0</(INR (fact (plus (mult (S (S O)) n) (S O))))``. +Intro; Rewrite (Rabsolu_right ? (Rle_sym1 ? ? (Rlt_le ? ? H1))). +Apply Rle_monotony. +Left; Apply H1. +Rewrite <- Pow_Rabsolu; Apply pow_maj_Rabs. +Rewrite Rabsolu_Rabsolu; Unfold Boule in H0; Rewrite minus_R0 in H0; Left; Apply H0. +Apply Rlt_Rinv; Apply INR_fact_lt_0. +Cut (r::R)<>``0``. +Intro; Apply Alembert_C2. +Intro; Apply Rabsolu_no_R0. +Apply prod_neq_R0. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply pow_nonzero; Assumption. +Assert H1 := Alembert_sin. +Unfold sin_n in H1; Unfold Un_cv in H1; Unfold Un_cv; Intros. +Cut ``0<eps/(Rsqr r)``. +Intro; Elim (H1 ? H3); Intros N0 H4. +Exists N0; Intros. +Unfold R_dist; Assert H6 := (H4 ? H5). +Unfold R_dist in H5; Replace ``(Rabsolu ((Rabsolu (/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))*(pow r (mult (S (S O)) (S n)))))/(Rabsolu (/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow r (mult (S (S O)) n))))))`` with ``(Rsqr r)*(Rabsolu ((pow ( -1) (S n))/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))/((pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O)))))))``. +Apply Rlt_monotony_contra with ``/(Rsqr r)``. +Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. +Pattern 1 ``/(Rsqr r)``; Rewrite <- (Rabsolu_right ``/(Rsqr r)``). +Rewrite <- Rabsolu_mult. +Rewrite Rminus_distr. +Rewrite Rmult_Or; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps). +Apply H6. +Unfold Rsqr; Apply prod_neq_R0; Assumption. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. +Unfold Rdiv; Rewrite (Rmult_sym (Rsqr r)); Repeat Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite pow_1_abs. +Rewrite Rmult_1l. +Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r. +Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Rewrite Rabsolu_mult. +Rewrite Rabsolu_Rinv. +Rewrite pow_1_abs; Rewrite Rinv_R1; Rewrite Rmult_1l. +Rewrite Rinv_Rmult. +Rewrite <- Rabsolu_Rinv. +Rewrite Rinv_Rinv. +Rewrite Rabsolu_mult. +Do 2 Rewrite Rabsolu_Rabsolu. +Rewrite (Rmult_sym ``(Rabsolu (pow r (mult (S (S O)) (S n))))``). +Rewrite Rmult_assoc; Apply Rmult_mult_r. +Rewrite Rabsolu_Rinv. +Rewrite Rabsolu_Rabsolu. +Repeat Rewrite Rabsolu_right. +Replace ``(pow r (mult (S (S O)) (S n)))`` with ``(pow r (mult (S (S O)) n))*r*r``. +Do 2 Rewrite <- Rmult_assoc. +Rewrite <- Rinv_l_sym. +Unfold Rsqr; Ring. +Apply pow_nonzero; Assumption. +Replace (mult (2) (S n)) with (S (S (mult (2) n))). +Simpl; Ring. +Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). +Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply INR_fact_neq_0. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply Rabsolu_no_R0; Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. +Apply pow_nonzero; DiscrR. +Apply INR_fact_neq_0. +Apply pow_nonzero; DiscrR. +Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption]. +Assert H0 := (cond_pos r); Red; Intro; Rewrite H1 in H0; Elim (Rlt_antirefl ? H0). +Qed. + +(* (sin h)/h -> 1 when h -> 0 *) +Lemma derivable_pt_lim_sin_0 : (derivable_pt_lim sin R0 R1). +Unfold derivable_pt_lim; Intros. +Pose fn := [N:nat][x:R]``(pow ( -1) N)/(INR (fact (plus (mult (S (S O)) N) (S O))))*(pow x (mult (S (S O)) N))``. +Cut (CVN_R fn). +Intro; Cut (x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l)). +Intro cv. +Pose r := (mkposreal ? Rlt_R0_R1). +Cut (CVN_r fn r). +Intro; Cut ((n:nat; y:R)(Boule ``0`` r y)->(continuity_pt (fn n) y)). +Intro; Cut (Boule R0 r R0). +Intro; Assert H2 := (SFL_continuity_pt ? cv ? X0 H0 ? H1). +Unfold continuity_pt in H2; Unfold continue_in in H2; Unfold limit1_in in H2; Unfold limit_in in H2; Simpl in H2; Unfold R_dist in H2. +Elim (H2 ? H); Intros alp H3. +Elim H3; Intros. +Exists (mkposreal ? H4). +Simpl; Intros. +Rewrite sin_0; Rewrite Rplus_Ol; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. +Cut ``(Rabsolu ((SFL fn cv h)-(SFL fn cv 0))) < eps``. +Intro; Cut (SFL fn cv R0)==R1. +Intro; Cut (SFL fn cv h)==``(sin h)/h``. +Intro; Rewrite H9 in H8; Rewrite H10 in H8. +Apply H8. +Unfold SFL sin. +Case (cv h); Intros. +Case (exist_sin (Rsqr h)); Intros. +Unfold Rdiv; Rewrite (Rinv_r_simpl_m h x0 H6). +EApply UL_sequence. +Apply u. +Unfold sin_in in s; Unfold sin_n infinit_sum in s; Unfold SP fn Un_cv; Intros. +Elim (s ? H10); Intros N0 H11. +Exists N0; Intros. +Unfold R_dist; Unfold R_dist in H11. +Replace (sum_f_R0 [k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow h (mult (S (S O)) k))`` n) with (sum_f_R0 [i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (Rsqr h) i)`` n). +Apply H11; Assumption. +Apply sum_eq; Intros; Apply Rmult_mult_r; Unfold Rsqr; Rewrite pow_sqr; Reflexivity. +Unfold SFL sin. +Case (cv R0); Intros. +EApply UL_sequence. +Apply u. +Unfold SP fn; Unfold Un_cv; Intros; Exists (S O); Intros. +Unfold R_dist; Replace (sum_f_R0 [k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow 0 (mult (S (S O)) k))`` n) with R1. +Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Rewrite decomp_sum. +Simpl; Rewrite Rmult_1r; Unfold Rdiv; Rewrite Rinv_R1; Rewrite Rmult_1r; Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rplus_plus_r. +Symmetry; Apply sum_eq_R0; Intros. +Rewrite Rmult_Ol; Rewrite Rmult_Or; Reflexivity. +Unfold ge in H10; Apply lt_le_trans with (1); [Apply lt_n_Sn | Apply H10]. +Apply H5. +Split. +Unfold D_x no_cond; Split. +Trivial. +Apply not_sym; Apply H6. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply H7. +Unfold Boule; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_R0; Apply (cond_pos r). +Intros; Unfold fn; Replace [x:R]``(pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow x (mult (S (S O)) n))`` with (mult_fct (fct_cte ``(pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))``) (pow_fct (mult (S (S O)) n))); [Idtac | Reflexivity]. +Apply continuity_pt_mult. +Apply derivable_continuous_pt. +Apply derivable_pt_const. +Apply derivable_continuous_pt. +Apply (derivable_pt_pow (mult (2) n) y). +Apply (X r). +Apply (CVN_R_CVS ? X). +Apply CVN_R_sin; Unfold fn; Reflexivity. +Qed. + +(* ((cos h)-1)/h -> 0 when h -> 0 *) +Lemma derivable_pt_lim_cos_0 : (derivable_pt_lim cos ``0`` ``0``). +Unfold derivable_pt_lim; Intros. +Assert H0 := derivable_pt_lim_sin_0. +Unfold derivable_pt_lim in H0. +Cut ``0<eps/2``. +Intro; Elim (H0 ? H1); Intros del H2. +Cut (continuity_pt sin ``0``). +Intro; Unfold continuity_pt in H3; Unfold continue_in in H3; Unfold limit1_in in H3; Unfold limit_in in H3; Simpl in H3; Unfold R_dist in H3. +Cut ``0<eps/2``; [Intro | Assumption]. +Elim (H3 ? H4); Intros del_c H5. +Cut ``0<(Rmin del del_c)``. +Intro; Pose delta := (mkposreal ? H6). +Exists delta; Intros. +Rewrite Rplus_Ol; Replace ``((cos h)-(cos 0))`` with ``-2*(Rsqr (sin (h/2)))``. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. +Unfold Rdiv; Do 2 Rewrite Ropp_mul1. +Rewrite Rabsolu_Ropp. +Replace ``2*(Rsqr (sin (h*/2)))*/h`` with ``(sin (h/2))*((sin (h/2))/(h/2)-1)+(sin (h/2))``. +Apply Rle_lt_trans with ``(Rabsolu ((sin (h/2))*((sin (h/2))/(h/2)-1)))+(Rabsolu ((sin (h/2))))``. +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Apply Rle_lt_trans with ``(Rabsolu ((sin (h/2))/(h/2)-1))``. +Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu ((sin (h/2))/(h/2)-1))``; Rewrite <- Rmult_1r; Apply Rle_monotony. +Apply Rabsolu_pos. +Assert H9 := (SIN_bound ``h/2``). +Unfold Rabsolu; Case (case_Rabsolu ``(sin (h/2))``); Intro. +Pattern 3 R1; Rewrite <- (Ropp_Ropp ``1``). +Apply Rle_Ropp1. +Elim H9; Intros; Assumption. +Elim H9; Intros; Assumption. +Cut ``(Rabsolu (h/2))<del``. +Intro; Cut ``h/2<>0``. +Intro; Assert H11 := (H2 ? H10 H9). +Rewrite Rplus_Ol in H11; Rewrite sin_0 in H11. +Rewrite minus_R0 in H11; Apply H11. +Unfold Rdiv; Apply prod_neq_R0. +Apply H7. +Apply Rinv_neq_R0; DiscrR. +Apply Rlt_trans with ``del/2``. +Unfold Rdiv; Rewrite Rabsolu_mult. +Rewrite (Rabsolu_right ``/2``). +Do 2 Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony. +Apply Rlt_Rinv; Sup0. +Apply Rlt_le_trans with (pos delta). +Apply H8. +Unfold delta; Simpl; Apply Rmin_l. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Sup0. +Rewrite <- (Rplus_Or ``del/2``); Pattern 1 del; Rewrite (double_var del); Apply Rlt_compatibility; Unfold Rdiv; Apply Rmult_lt_pos. +Apply (cond_pos del). +Apply Rlt_Rinv; Sup0. +Elim H5; Intros; Assert H11 := (H10 ``h/2``). +Rewrite sin_0 in H11; Do 2 Rewrite minus_R0 in H11. +Apply H11. +Split. +Unfold D_x no_cond; Split. +Trivial. +Apply not_sym; Unfold Rdiv; Apply prod_neq_R0. +Apply H7. +Apply Rinv_neq_R0; DiscrR. +Apply Rlt_trans with ``del_c/2``. +Unfold Rdiv; Rewrite Rabsolu_mult. +Rewrite (Rabsolu_right ``/2``). +Do 2 Rewrite <- (Rmult_sym ``/2``). +Apply Rlt_monotony. +Apply Rlt_Rinv; Sup0. +Apply Rlt_le_trans with (pos delta). +Apply H8. +Unfold delta; Simpl; Apply Rmin_r. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Sup0. +Rewrite <- (Rplus_Or ``del_c/2``); Pattern 2 del_c; Rewrite (double_var del_c); Apply Rlt_compatibility. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply H9. +Apply Rlt_Rinv; Sup0. +Rewrite Rminus_distr; Rewrite Rmult_1r; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Rewrite (Rmult_sym ``2``); Unfold Rdiv Rsqr. +Repeat Rewrite Rmult_assoc. +Repeat Apply Rmult_mult_r. +Rewrite Rinv_Rmult. +Rewrite Rinv_Rinv. +Apply Rmult_sym. +DiscrR. +Apply H7. +Apply Rinv_neq_R0; DiscrR. +Pattern 2 h; Replace h with ``2*(h/2)``. +Rewrite (cos_2a_sin ``h/2``). +Rewrite cos_0; Unfold Rsqr; Ring. +Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. +DiscrR. +Unfold Rmin; Case (total_order_Rle del del_c); Intro. +Apply (cond_pos del). +Elim H5; Intros; Assumption. +Apply continuity_sin. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Qed. + +(**********) +Theorem derivable_pt_lim_sin : (x:R)(derivable_pt_lim sin x (cos x)). +Intro; Assert H0 := derivable_pt_lim_sin_0. +Assert H := derivable_pt_lim_cos_0. +Unfold derivable_pt_lim in H0 H. +Unfold derivable_pt_lim; Intros. +Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Apply H1 | Apply Rlt_Rinv; Sup0]]. +Elim (H0 ? H2); Intros alp1 H3. +Elim (H ? H2); Intros alp2 H4. +Pose alp := (Rmin alp1 alp2). +Cut ``0<alp``. +Intro; Exists (mkposreal ? H5); Intros. +Replace ``((sin (x+h))-(sin x))/h-(cos x)`` with ``(sin x)*((cos h)-1)/h+(cos x)*((sin h)/h-1)``. +Apply Rle_lt_trans with ``(Rabsolu ((sin x)*((cos h)-1)/h))+(Rabsolu ((cos x)*((sin h)/h-1)))``. +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Apply Rle_lt_trans with ``(Rabsolu ((cos h)-1)/h)``. +Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu (((cos h)-1)/h))``; Rewrite <- Rmult_1r; Apply Rle_monotony. +Apply Rabsolu_pos. +Assert H8 := (SIN_bound x); Elim H8; Intros. +Unfold Rabsolu; Case (case_Rabsolu (sin x)); Intro. +Rewrite <- (Ropp_Ropp R1). +Apply Rle_Ropp1; Assumption. +Assumption. +Cut ``(Rabsolu h)<alp2``. +Intro; Assert H9 := (H4 ? H6 H8). +Rewrite cos_0 in H9; Rewrite Rplus_Ol in H9; Rewrite minus_R0 in H9; Apply H9. +Apply Rlt_le_trans with alp. +Apply H7. +Unfold alp; Apply Rmin_r. +Apply Rle_lt_trans with ``(Rabsolu ((sin h)/h-1))``. +Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu ((sin h)/h-1))``; Rewrite <- Rmult_1r; Apply Rle_monotony. +Apply Rabsolu_pos. +Assert H8 := (COS_bound x); Elim H8; Intros. +Unfold Rabsolu; Case (case_Rabsolu (cos x)); Intro. +Rewrite <- (Ropp_Ropp R1); Apply Rle_Ropp1; Assumption. +Assumption. +Cut ``(Rabsolu h)<alp1``. +Intro; Assert H9 := (H3 ? H6 H8). +Rewrite sin_0 in H9; Rewrite Rplus_Ol in H9; Rewrite minus_R0 in H9; Apply H9. +Apply Rlt_le_trans with alp. +Apply H7. +Unfold alp; Apply Rmin_l. +Rewrite sin_plus; Unfold Rminus Rdiv; Repeat Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_Rplus_distr; Repeat Rewrite Rmult_assoc; Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Rewrite (Rplus_sym ``(sin x)*( -1*/h)``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. +Rewrite Ropp_mul3; Rewrite Ropp_mul1; Rewrite Rmult_1r; Rewrite Rmult_1l; Rewrite Ropp_mul3; Rewrite <- Ropp_mul1; Apply Rplus_sym. +Unfold alp; Unfold Rmin; Case (total_order_Rle alp1 alp2); Intro. +Apply (cond_pos alp1). +Apply (cond_pos alp2). +Qed. + +Lemma derivable_pt_lim_cos : (x:R) (derivable_pt_lim cos x ``-(sin x)``). +Intro; Cut (h:R)``(sin (h+PI/2))``==(cos h). +Intro; Replace ``-(sin x)`` with (Rmult (cos ``x+PI/2``) (Rplus R1 R0)). +Generalize (derivable_pt_lim_comp (plus_fct id (fct_cte ``PI/2``)) sin); Intros. +Cut (derivable_pt_lim (plus_fct id (fct_cte ``PI/2``)) x ``1+0``). +Cut (derivable_pt_lim sin (plus_fct id (fct_cte ``PI/2``) x) ``(cos (x+PI/2))``). +Intros; Generalize (H0 ? ? ? H2 H1); Replace (comp sin (plus_fct id (fct_cte ``PI/2``))) with [x:R]``(sin (x+PI/2))``; [Idtac | Reflexivity]. +Unfold derivable_pt_lim; Intros. +Elim (H3 eps H4); Intros. +Exists x0. +Intros; Rewrite <- (H ``x+h``); Rewrite <- (H x); Apply H5; Assumption. +Apply derivable_pt_lim_sin. +Apply derivable_pt_lim_plus. +Apply derivable_pt_lim_id. +Apply derivable_pt_lim_const. +Rewrite sin_cos; Rewrite <- (Rplus_sym x); Ring. +Intro; Rewrite cos_sin; Rewrite Rplus_sym; Reflexivity. +Qed. + +Lemma derivable_pt_sin : (x:R) (derivable_pt sin x). +Unfold derivable_pt; Intro. +Apply Specif.existT with (cos x). +Apply derivable_pt_lim_sin. +Qed. + +Lemma derivable_pt_cos : (x:R) (derivable_pt cos x). +Unfold derivable_pt; Intro. +Apply Specif.existT with ``-(sin x)``. +Apply derivable_pt_lim_cos. +Qed. + +Lemma derivable_sin : (derivable sin). +Unfold derivable; Intro; Apply derivable_pt_sin. +Qed. + +Lemma derivable_cos : (derivable cos). +Unfold derivable; Intro; Apply derivable_pt_cos. +Qed. + +Lemma derive_pt_sin : (x:R) ``(derive_pt sin x (derivable_pt_sin ?))==(cos x)``. +Intros; Apply derive_pt_eq_0. +Apply derivable_pt_lim_sin. +Qed. + +Lemma derive_pt_cos : (x:R) ``(derive_pt cos x (derivable_pt_cos ?))==-(sin x)``. +Intros; Apply derive_pt_eq_0. +Apply derivable_pt_lim_cos. +Qed. diff --git a/theories7/Reals/SeqProp.v b/theories7/Reals/SeqProp.v new file mode 100644 index 00000000..b34fa339 --- /dev/null +++ b/theories7/Reals/SeqProp.v @@ -0,0 +1,1089 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: SeqProp.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require Classical. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Definition Un_decreasing [Un:nat->R] : Prop := (n:nat) (Rle (Un (S n)) (Un n)). +Definition opp_seq [Un:nat->R] : nat->R := [n:nat]``-(Un n)``. +Definition has_ub [Un:nat->R] : Prop := (bound (EUn Un)). +Definition has_lb [Un:nat->R] : Prop := (bound (EUn (opp_seq Un))). + +(**********) +Lemma growing_cv : (Un:nat->R) (Un_growing Un) -> (has_ub Un) -> (sigTT R [l:R](Un_cv Un l)). +Unfold Un_growing Un_cv;Intros; + NewDestruct (complet (EUn Un) H0 (EUn_noempty Un)) as [x [H2 H3]]. + Exists x;Intros eps H1. + Unfold is_upper_bound in H2 H3. +Assert H5:(n:nat)(Rle (Un n) x). + Intro n; Apply (H2 (Un n) (Un_in_EUn Un n)). +Cut (Ex [N:nat] (Rlt (Rminus x eps) (Un N))). +Intro H6;NewDestruct H6 as [N H6];Exists N. +Intros n H7;Unfold R_dist;Apply (Rabsolu_def1 (Rminus (Un n) x) eps). +Unfold Rgt in H1. + Apply (Rle_lt_trans (Rminus (Un n) x) R0 eps + (Rle_minus (Un n) x (H5 n)) H1). +Fold Un_growing in H;Generalize (growing_prop Un n N H H7);Intro H8. + Generalize (Rlt_le_trans (Rminus x eps) (Un N) (Un n) H6 + (Rle_sym2 (Un N) (Un n) H8));Intro H9; + Generalize (Rlt_compatibility (Ropp x) (Rminus x eps) (Un n) H9); + Unfold Rminus;Rewrite <-(Rplus_assoc (Ropp x) x (Ropp eps)); + Rewrite (Rplus_sym (Ropp x) (Un n));Fold (Rminus (Un n) x); + Rewrite Rplus_Ropp_l;Rewrite (let (H1,H2)=(Rplus_ne (Ropp eps)) in H2); + Trivial. +Cut ~((N:nat)(Rle (Un N) (Rminus x eps))). +Intro H6;Apply (not_all_not_ex nat ([N:nat](Rlt (Rminus x eps) (Un N)))). + Intro H7; Apply H6; Intro N; Apply Rnot_lt_le; Apply H7. +Intro H7;Generalize (Un_bound_imp Un (Rminus x eps) H7);Intro H8; + Unfold is_upper_bound in H8;Generalize (H3 (Rminus x eps) H8); + Apply Rlt_le_not; Apply tech_Rgt_minus; Exact H1. +Qed. + +Lemma decreasing_growing : (Un:nat->R) (Un_decreasing Un) -> (Un_growing (opp_seq Un)). +Intro. +Unfold Un_growing opp_seq Un_decreasing. +Intros. +Apply Rle_Ropp1. +Apply H. +Qed. + +Lemma decreasing_cv : (Un:nat->R) (Un_decreasing Un) -> (has_lb Un) -> (sigTT R [l:R](Un_cv Un l)). +Intros. +Cut (sigTT R [l:R](Un_cv (opp_seq Un) l)) -> (sigTT R [l:R](Un_cv Un l)). +Intro. +Apply X. +Apply growing_cv. +Apply decreasing_growing; Assumption. +Exact H0. +Intro. +Elim X; Intros. +Apply existTT with ``-x``. +Unfold Un_cv in p. +Unfold R_dist in p. +Unfold opp_seq in p. +Unfold Un_cv. +Unfold R_dist. +Intros. +Elim (p eps H1); Intros. +Exists x0; Intros. +Assert H4 := (H2 n H3). +Rewrite <- Rabsolu_Ropp. +Replace ``-((Un n)- -x)`` with ``-(Un n)-x``; [Assumption | Ring]. +Qed. + +(***********) +Lemma maj_sup : (Un:nat->R) (has_ub Un) -> (sigTT R [l:R](is_lub (EUn Un) l)). +Intros. +Unfold has_ub in H. +Apply complet. +Assumption. +Exists (Un O). +Unfold EUn. +Exists O; Reflexivity. +Qed. + +(**********) +Lemma min_inf : (Un:nat->R) (has_lb Un) -> (sigTT R [l:R](is_lub (EUn (opp_seq Un)) l)). +Intros; Unfold has_lb in H. +Apply complet. +Assumption. +Exists ``-(Un O)``. +Exists O. +Reflexivity. +Qed. + +Definition majorant [Un:nat->R;pr:(has_ub Un)] : R := Cases (maj_sup Un pr) of (existTT a b) => a end. + +Definition minorant [Un:nat->R;pr:(has_lb Un)] : R := Cases (min_inf Un pr) of (existTT a b) => ``-a`` end. + +Lemma maj_ss : (Un:nat->R;k:nat) (has_ub Un) -> (has_ub [i:nat](Un (plus k i))). +Intros. +Unfold has_ub in H. +Unfold bound in H. +Elim H; Intros. +Unfold is_upper_bound in H0. +Unfold has_ub. +Exists x. +Unfold is_upper_bound. +Intros. +Apply H0. +Elim H1; Intros. +Exists (plus k x1); Assumption. +Qed. + +Lemma min_ss : (Un:nat->R;k:nat) (has_lb Un) -> (has_lb [i:nat](Un (plus k i))). +Intros. +Unfold has_lb in H. +Unfold bound in H. +Elim H; Intros. +Unfold is_upper_bound in H0. +Unfold has_lb. +Exists x. +Unfold is_upper_bound. +Intros. +Apply H0. +Elim H1; Intros. +Exists (plus k x1); Assumption. +Qed. + +Definition sequence_majorant [Un:nat->R;pr:(has_ub Un)] : nat -> R := [i:nat](majorant [k:nat](Un (plus i k)) (maj_ss Un i pr)). + +Definition sequence_minorant [Un:nat->R;pr:(has_lb Un)] : nat -> R := [i:nat](minorant [k:nat](Un (plus i k)) (min_ss Un i pr)). + +Lemma Wn_decreasing : (Un:nat->R;pr:(has_ub Un)) (Un_decreasing (sequence_majorant Un pr)). +Intros. +Unfold Un_decreasing. +Intro. +Unfold sequence_majorant. +Assert H := (maj_sup [k:nat](Un (plus (S n) k)) (maj_ss Un (S n) pr)). +Assert H0 := (maj_sup [k:nat](Un (plus n k)) (maj_ss Un n pr)). +Elim H; Intros. +Elim H0; Intros. +Cut (majorant ([k:nat](Un (plus (S n) k))) (maj_ss Un (S n) pr)) == x; [Intro Maj1; Rewrite Maj1 | Idtac]. +Cut (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr)) == x0; [Intro Maj2; Rewrite Maj2 | Idtac]. +Unfold is_lub in p. +Unfold is_lub in p0. +Elim p; Intros. +Apply H2. +Elim p0; Intros. +Unfold is_upper_bound. +Intros. +Unfold is_upper_bound in H3. +Apply H3. +Elim H5; Intros. +Exists (plus (1) x2). +Replace (plus n (plus (S O) x2)) with (plus (S n) x2). +Assumption. +Replace (S n) with (plus (1) n); [Ring | Ring]. +Cut (is_lub (EUn [k:nat](Un (plus n k))) (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr))). +Intro. +Unfold is_lub in p0; Unfold is_lub in H1. +Elim p0; Intros; Elim H1; Intros. +Assert H6 := (H5 x0 H2). +Assert H7 := (H3 (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr)) H4). +Apply Rle_antisym; Assumption. +Unfold majorant. +Case (maj_sup [k:nat](Un (plus n k)) (maj_ss Un n pr)). +Trivial. +Cut (is_lub (EUn [k:nat](Un (plus (S n) k))) (majorant ([k:nat](Un (plus (S n) k))) (maj_ss Un (S n) pr))). +Intro. +Unfold is_lub in p; Unfold is_lub in H1. +Elim p; Intros; Elim H1; Intros. +Assert H6 := (H5 x H2). +Assert H7 := (H3 (majorant ([k:nat](Un (plus (S n) k))) (maj_ss Un (S n) pr)) H4). +Apply Rle_antisym; Assumption. +Unfold majorant. +Case (maj_sup [k:nat](Un (plus (S n) k)) (maj_ss Un (S n) pr)). +Trivial. +Qed. + +Lemma Vn_growing : (Un:nat->R;pr:(has_lb Un)) (Un_growing (sequence_minorant Un pr)). +Intros. +Unfold Un_growing. +Intro. +Unfold sequence_minorant. +Assert H := (min_inf [k:nat](Un (plus (S n) k)) (min_ss Un (S n) pr)). +Assert H0 := (min_inf [k:nat](Un (plus n k)) (min_ss Un n pr)). +Elim H; Intros. +Elim H0; Intros. +Cut (minorant ([k:nat](Un (plus (S n) k))) (min_ss Un (S n) pr)) == ``-x``; [Intro Maj1; Rewrite Maj1 | Idtac]. +Cut (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr)) == ``-x0``; [Intro Maj2; Rewrite Maj2 | Idtac]. +Unfold is_lub in p. +Unfold is_lub in p0. +Elim p; Intros. +Apply Rle_Ropp1. +Apply H2. +Elim p0; Intros. +Unfold is_upper_bound. +Intros. +Unfold is_upper_bound in H3. +Apply H3. +Elim H5; Intros. +Exists (plus (1) x2). +Unfold opp_seq in H6. +Unfold opp_seq. +Replace (plus n (plus (S O) x2)) with (plus (S n) x2). +Assumption. +Replace (S n) with (plus (1) n); [Ring | Ring]. +Cut (is_lub (EUn (opp_seq [k:nat](Un (plus n k)))) (Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr)))). +Intro. +Unfold is_lub in p0; Unfold is_lub in H1. +Elim p0; Intros; Elim H1; Intros. +Assert H6 := (H5 x0 H2). +Assert H7 := (H3 (Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr))) H4). +Rewrite <- (Ropp_Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr))). +Apply eq_Ropp; Apply Rle_antisym; Assumption. +Unfold minorant. +Case (min_inf [k:nat](Un (plus n k)) (min_ss Un n pr)). +Intro; Rewrite Ropp_Ropp. +Trivial. +Cut (is_lub (EUn (opp_seq [k:nat](Un (plus (S n) k)))) (Ropp (minorant ([k:nat](Un (plus (S n) k))) (min_ss Un (S n) pr)))). +Intro. +Unfold is_lub in p; Unfold is_lub in H1. +Elim p; Intros; Elim H1; Intros. +Assert H6 := (H5 x H2). +Assert H7 := (H3 (Ropp (minorant ([k:nat](Un (plus (S n) k))) (min_ss Un (S n) pr))) H4). +Rewrite <- (Ropp_Ropp (minorant ([k:nat](Un (plus (S n) k))) (min_ss Un (S n) pr))). +Apply eq_Ropp; Apply Rle_antisym; Assumption. +Unfold minorant. +Case (min_inf [k:nat](Un (plus (S n) k)) (min_ss Un (S n) pr)). +Intro; Rewrite Ropp_Ropp. +Trivial. +Qed. + +(**********) +Lemma Vn_Un_Wn_order : (Un:nat->R;pr1:(has_ub Un);pr2:(has_lb Un)) (n:nat) ``((sequence_minorant Un pr2) n)<=(Un n)<=((sequence_majorant Un pr1) n)``. +Intros. +Split. +Unfold sequence_minorant. +Cut (sigTT R [l:R](is_lub (EUn (opp_seq [i:nat](Un (plus n i)))) l)). +Intro. +Elim X; Intros. +Replace (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr2)) with ``-x``. +Unfold is_lub in p. +Elim p; Intros. +Unfold is_upper_bound in H. +Rewrite <- (Ropp_Ropp (Un n)). +Apply Rle_Ropp1. +Apply H. +Exists O. +Unfold opp_seq. +Replace (plus n O) with n; [Reflexivity | Ring]. +Cut (is_lub (EUn (opp_seq [k:nat](Un (plus n k)))) (Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr2)))). +Intro. +Unfold is_lub in p; Unfold is_lub in H. +Elim p; Intros; Elim H; Intros. +Assert H4 := (H3 x H0). +Assert H5 := (H1 (Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr2))) H2). +Rewrite <- (Ropp_Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr2))). +Apply eq_Ropp; Apply Rle_antisym; Assumption. +Unfold minorant. +Case (min_inf [k:nat](Un (plus n k)) (min_ss Un n pr2)). +Intro; Rewrite Ropp_Ropp. +Trivial. +Apply min_inf. +Apply min_ss; Assumption. +Unfold sequence_majorant. +Cut (sigTT R [l:R](is_lub (EUn [i:nat](Un (plus n i))) l)). +Intro. +Elim X; Intros. +Replace (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr1)) with ``x``. +Unfold is_lub in p. +Elim p; Intros. +Unfold is_upper_bound in H. +Apply H. +Exists O. +Replace (plus n O) with n; [Reflexivity | Ring]. +Cut (is_lub (EUn [k:nat](Un (plus n k))) (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr1))). +Intro. +Unfold is_lub in p; Unfold is_lub in H. +Elim p; Intros; Elim H; Intros. +Assert H4 := (H3 x H0). +Assert H5 := (H1 (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr1)) H2). +Apply Rle_antisym; Assumption. +Unfold majorant. +Case (maj_sup [k:nat](Un (plus n k)) (maj_ss Un n pr1)). +Intro; Trivial. +Apply maj_sup. +Apply maj_ss; Assumption. +Qed. + +Lemma min_maj : (Un:nat->R;pr1:(has_ub Un);pr2:(has_lb Un)) (has_ub (sequence_minorant Un pr2)). +Intros. +Assert H := (Vn_Un_Wn_order Un pr1 pr2). +Unfold has_ub. +Unfold bound. +Unfold has_ub in pr1. +Unfold bound in pr1. +Elim pr1; Intros. +Exists x. +Unfold is_upper_bound. +Intros. +Unfold is_upper_bound in H0. +Elim H1; Intros. +Rewrite H2. +Apply Rle_trans with (Un x1). +Assert H3 := (H x1); Elim H3; Intros; Assumption. +Apply H0. +Exists x1; Reflexivity. +Qed. + +Lemma maj_min : (Un:nat->R;pr1:(has_ub Un);pr2:(has_lb Un)) (has_lb (sequence_majorant Un pr1)). +Intros. +Assert H := (Vn_Un_Wn_order Un pr1 pr2). +Unfold has_lb. +Unfold bound. +Unfold has_lb in pr2. +Unfold bound in pr2. +Elim pr2; Intros. +Exists x. +Unfold is_upper_bound. +Intros. +Unfold is_upper_bound in H0. +Elim H1; Intros. +Rewrite H2. +Apply Rle_trans with ((opp_seq Un) x1). +Assert H3 := (H x1); Elim H3; Intros. +Unfold opp_seq; Apply Rle_Ropp1. +Assumption. +Apply H0. +Exists x1; Reflexivity. +Qed. + +(**********) +Lemma cauchy_maj : (Un:nat->R) (Cauchy_crit Un) -> (has_ub Un). +Intros. +Unfold has_ub. +Apply cauchy_bound. +Assumption. +Qed. + +(**********) +Lemma cauchy_opp : (Un:nat->R) (Cauchy_crit Un) -> (Cauchy_crit (opp_seq Un)). +Intro. +Unfold Cauchy_crit. +Unfold R_dist. +Intros. +Elim (H eps H0); Intros. +Exists x; Intros. +Unfold opp_seq. +Rewrite <- Rabsolu_Ropp. +Replace ``-( -(Un n)- -(Un m))`` with ``(Un n)-(Un m)``; [Apply H1; Assumption | Ring]. +Qed. + +(**********) +Lemma cauchy_min : (Un:nat->R) (Cauchy_crit Un) -> (has_lb Un). +Intros. +Unfold has_lb. +Assert H0 := (cauchy_opp ? H). +Apply cauchy_bound. +Assumption. +Qed. + +(**********) +Lemma maj_cv : (Un:nat->R;pr:(Cauchy_crit Un)) (sigTT R [l:R](Un_cv (sequence_majorant Un (cauchy_maj Un pr)) l)). +Intros. +Apply decreasing_cv. +Apply Wn_decreasing. +Apply maj_min. +Apply cauchy_min. +Assumption. +Qed. + +(**********) +Lemma min_cv : (Un:nat->R;pr:(Cauchy_crit Un)) (sigTT R [l:R](Un_cv (sequence_minorant Un (cauchy_min Un pr)) l)). +Intros. +Apply growing_cv. +Apply Vn_growing. +Apply min_maj. +Apply cauchy_maj. +Assumption. +Qed. + +Lemma cond_eq : (x,y:R) ((eps:R)``0<eps``->``(Rabsolu (x-y))<eps``) -> x==y. +Intros. +Case (total_order_T x y); Intro. +Elim s; Intro. +Cut ``0<y-x``. +Intro. +Assert H1 := (H ``y-x`` H0). +Rewrite <- Rabsolu_Ropp in H1. +Cut ``-(x-y)==y-x``; [Intro; Rewrite H2 in H1 | Ring]. +Rewrite Rabsolu_right in H1. +Elim (Rlt_antirefl ? H1). +Left; Assumption. +Apply Rlt_anti_compatibility with x. +Rewrite Rplus_Or; Replace ``x+(y-x)`` with y; [Assumption | Ring]. +Assumption. +Cut ``0<x-y``. +Intro. +Assert H1 := (H ``x-y`` H0). +Rewrite Rabsolu_right in H1. +Elim (Rlt_antirefl ? H1). +Left; Assumption. +Apply Rlt_anti_compatibility with y. +Rewrite Rplus_Or; Replace ``y+(x-y)`` with x; [Assumption | Ring]. +Qed. + +Lemma not_Rlt : (r1,r2:R)~(``r1<r2``)->``r1>=r2``. +Intros r1 r2 ; Generalize (total_order r1 r2) ; Unfold Rge. +Tauto. +Qed. + +(**********) +Lemma approx_maj : (Un:nat->R;pr:(has_ub Un)) (eps:R) ``0<eps`` -> (EX k : nat | ``(Rabsolu ((majorant Un pr)-(Un k))) < eps``). +Intros. +Pose P := [k:nat]``(Rabsolu ((majorant Un pr)-(Un k))) < eps``. +Unfold P. +Cut (EX k:nat | (P k)) -> (EX k:nat | ``(Rabsolu ((majorant Un pr)-(Un k))) < eps``). +Intros. +Apply H0. +Apply not_all_not_ex. +Red; Intro. +2:Unfold P; Trivial. +Unfold P in H1. +Cut (n:nat)``(Rabsolu ((majorant Un pr)-(Un n))) >= eps``. +Intro. +Cut (is_lub (EUn Un) (majorant Un pr)). +Intro. +Unfold is_lub in H3. +Unfold is_upper_bound in H3. +Elim H3; Intros. +Cut (n:nat)``eps<=(majorant Un pr)-(Un n)``. +Intro. +Cut (n:nat)``(Un n)<=(majorant Un pr)-eps``. +Intro. +Cut ((x:R)(EUn Un x)->``x <= (majorant Un pr)-eps``). +Intro. +Assert H9 := (H5 ``(majorant Un pr)-eps`` H8). +Cut ``eps<=0``. +Intro. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H H10)). +Apply Rle_anti_compatibility with ``(majorant Un pr)-eps``. +Rewrite Rplus_Or. +Replace ``(majorant Un pr)-eps+eps`` with (majorant Un pr); [Assumption | Ring]. +Intros. +Unfold EUn in H8. +Elim H8; Intros. +Rewrite H9; Apply H7. +Intro. +Assert H7 := (H6 n). +Apply Rle_anti_compatibility with ``eps-(Un n)``. +Replace ``eps-(Un n)+(Un n)`` with ``eps``. +Replace ``eps-(Un n)+((majorant Un pr)-eps)`` with ``(majorant Un pr)-(Un n)``. +Assumption. +Ring. +Ring. +Intro. +Assert H6 := (H2 n). +Rewrite Rabsolu_right in H6. +Apply Rle_sym2. +Assumption. +Apply Rle_sym1. +Apply Rle_anti_compatibility with (Un n). +Rewrite Rplus_Or; Replace ``(Un n)+((majorant Un pr)-(Un n))`` with (majorant Un pr); [Apply H4 | Ring]. +Exists n; Reflexivity. +Unfold majorant. +Case (maj_sup Un pr). +Trivial. +Intro. +Assert H2 := (H1 n). +Apply not_Rlt; Assumption. +Qed. + +(**********) +Lemma approx_min : (Un:nat->R;pr:(has_lb Un)) (eps:R) ``0<eps`` -> (EX k :nat | ``(Rabsolu ((minorant Un pr)-(Un k))) < eps``). +Intros. +Pose P := [k:nat]``(Rabsolu ((minorant Un pr)-(Un k))) < eps``. +Unfold P. +Cut (EX k:nat | (P k)) -> (EX k:nat | ``(Rabsolu ((minorant Un pr)-(Un k))) < eps``). +Intros. +Apply H0. +Apply not_all_not_ex. +Red; Intro. +2:Unfold P; Trivial. +Unfold P in H1. +Cut (n:nat)``(Rabsolu ((minorant Un pr)-(Un n))) >= eps``. +Intro. +Cut (is_lub (EUn (opp_seq Un)) ``-(minorant Un pr)``). +Intro. +Unfold is_lub in H3. +Unfold is_upper_bound in H3. +Elim H3; Intros. +Cut (n:nat)``eps<=(Un n)-(minorant Un pr)``. +Intro. +Cut (n:nat)``((opp_seq Un) n)<=-(minorant Un pr)-eps``. +Intro. +Cut ((x:R)(EUn (opp_seq Un) x)->``x <= -(minorant Un pr)-eps``). +Intro. +Assert H9 := (H5 ``-(minorant Un pr)-eps`` H8). +Cut ``eps<=0``. +Intro. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H H10)). +Apply Rle_anti_compatibility with ``-(minorant Un pr)-eps``. +Rewrite Rplus_Or. +Replace ``-(minorant Un pr)-eps+eps`` with ``-(minorant Un pr)``; [Assumption | Ring]. +Intros. +Unfold EUn in H8. +Elim H8; Intros. +Rewrite H9; Apply H7. +Intro. +Assert H7 := (H6 n). +Unfold opp_seq. +Apply Rle_anti_compatibility with ``eps+(Un n)``. +Replace ``eps+(Un n)+ -(Un n)`` with ``eps``. +Replace ``eps+(Un n)+(-(minorant Un pr)-eps)`` with ``(Un n)-(minorant Un pr)``. +Assumption. +Ring. +Ring. +Intro. +Assert H6 := (H2 n). +Rewrite Rabsolu_left1 in H6. +Apply Rle_sym2. +Replace ``(Un n)-(minorant Un pr)`` with `` -((minorant Un pr)-(Un n))``; [Assumption | Ring]. +Apply Rle_anti_compatibility with ``-(minorant Un pr)``. +Rewrite Rplus_Or; Replace ``-(minorant Un pr)+((minorant Un pr)-(Un n))`` with ``-(Un n)``. +Apply H4. +Exists n; Reflexivity. +Ring. +Unfold minorant. +Case (min_inf Un pr). +Intro. +Rewrite Ropp_Ropp. +Trivial. +Intro. +Assert H2 := (H1 n). +Apply not_Rlt; Assumption. +Qed. + +(* Unicity of limit for convergent sequences *) +Lemma UL_sequence : (Un:nat->R;l1,l2:R) (Un_cv Un l1) -> (Un_cv Un l2) -> l1==l2. +Intros Un l1 l2; Unfold Un_cv; Unfold R_dist; Intros. +Apply cond_eq. +Intros; Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. +Elim (H ``eps/2`` H2); Intros. +Elim (H0 ``eps/2`` H2); Intros. +Pose N := (max x x0). +Apply Rle_lt_trans with ``(Rabsolu (l1 -(Un N)))+(Rabsolu ((Un N)-l2))``. +Replace ``l1-l2`` with ``(l1-(Un N))+((Un N)-l2)``; [Apply Rabsolu_triang | Ring]. +Rewrite (double_var eps); Apply Rplus_lt. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H3; Unfold ge N; Apply le_max_l. +Apply H4; Unfold ge N; Apply le_max_r. +Qed. + +(**********) +Lemma CV_plus : (An,Bn:nat->R;l1,l2:R) (Un_cv An l1) -> (Un_cv Bn l2) -> (Un_cv [i:nat]``(An i)+(Bn i)`` ``l1+l2``). +Unfold Un_cv; Unfold R_dist; Intros. +Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. +Elim (H ``eps/2`` H2); Intros. +Elim (H0 ``eps/2`` H2); Intros. +Pose N := (max x x0). +Exists N; Intros. +Replace ``(An n)+(Bn n)-(l1+l2)`` with ``((An n)-l1)+((Bn n)-l2)``; [Idtac | Ring]. +Apply Rle_lt_trans with ``(Rabsolu ((An n)-l1))+(Rabsolu ((Bn n)-l2))``. +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Apply H3; Unfold ge; Apply le_trans with N; [Unfold N; Apply le_max_l | Assumption]. +Apply H4; Unfold ge; Apply le_trans with N; [Unfold N; Apply le_max_r | Assumption]. +Qed. + +(**********) +Lemma cv_cvabs : (Un:nat->R;l:R) (Un_cv Un l) -> (Un_cv [i:nat](Rabsolu (Un i)) (Rabsolu l)). +Unfold Un_cv; Unfold R_dist; Intros. +Elim (H eps H0); Intros. +Exists x; Intros. +Apply Rle_lt_trans with ``(Rabsolu ((Un n)-l))``. +Apply Rabsolu_triang_inv2. +Apply H1; Assumption. +Qed. + +(**********) +Lemma CV_Cauchy : (Un:nat->R) (sigTT R [l:R](Un_cv Un l)) -> (Cauchy_crit Un). +Intros; Elim X; Intros. +Unfold Cauchy_crit; Intros. +Unfold Un_cv in p; Unfold R_dist in p. +Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. +Elim (p ``eps/2`` H0); Intros. +Exists x0; Intros. +Unfold R_dist; Apply Rle_lt_trans with ``(Rabsolu ((Un n)-x))+(Rabsolu (x-(Un m)))``. +Replace ``(Un n)-(Un m)`` with ``((Un n)-x)+(x-(Un m))``; [Apply Rabsolu_triang | Ring]. +Rewrite (double_var eps); Apply Rplus_lt. +Apply H1; Assumption. +Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H1; Assumption. +Qed. + +(**********) +Lemma maj_by_pos : (Un:nat->R) (sigTT R [l:R](Un_cv Un l)) -> (EXT l:R | ``0<l``/\((n:nat)``(Rabsolu (Un n))<=l``)). +Intros; Elim X; Intros. +Cut (sigTT R [l:R](Un_cv [k:nat](Rabsolu (Un k)) l)). +Intro. +Assert H := (CV_Cauchy [k:nat](Rabsolu (Un k)) X0). +Assert H0 := (cauchy_bound [k:nat](Rabsolu (Un k)) H). +Elim H0; Intros. +Exists ``x0+1``. +Cut ``0<=x0``. +Intro. +Split. +Apply ge0_plus_gt0_is_gt0; [Assumption | Apply Rlt_R0_R1]. +Intros. +Apply Rle_trans with x0. +Unfold is_upper_bound in H1. +Apply H1. +Exists n; Reflexivity. +Pattern 1 x0; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply Rlt_R0_R1. +Apply Rle_trans with (Rabsolu (Un O)). +Apply Rabsolu_pos. +Unfold is_upper_bound in H1. +Apply H1. +Exists O; Reflexivity. +Apply existTT with (Rabsolu x). +Apply cv_cvabs; Assumption. +Qed. + +(**********) +Lemma CV_mult : (An,Bn:nat->R;l1,l2:R) (Un_cv An l1) -> (Un_cv Bn l2) -> (Un_cv [i:nat]``(An i)*(Bn i)`` ``l1*l2``). +Intros. +Cut (sigTT R [l:R](Un_cv An l)). +Intro. +Assert H1 := (maj_by_pos An X). +Elim H1; Intros M H2. +Elim H2; Intros. +Unfold Un_cv; Unfold R_dist; Intros. +Cut ``0<eps/(2*M)``. +Intro. +Case (Req_EM l2 R0); Intro. +Unfold Un_cv in H0; Unfold R_dist in H0. +Elim (H0 ``eps/(2*M)`` H6); Intros. +Exists x; Intros. +Apply Rle_lt_trans with ``(Rabsolu ((An n)*(Bn n)-(An n)*l2))+(Rabsolu ((An n)*l2-l1*l2))``. +Replace ``(An n)*(Bn n)-l1*l2`` with ``((An n)*(Bn n)-(An n)*l2)+((An n)*l2-l1*l2)``; [Apply Rabsolu_triang | Ring]. +Replace ``(Rabsolu ((An n)*(Bn n)-(An n)*l2))`` with ``(Rabsolu (An n))*(Rabsolu ((Bn n)-l2))``. +Replace ``(Rabsolu ((An n)*l2-l1*l2))`` with R0. +Rewrite Rplus_Or. +Apply Rle_lt_trans with ``M*(Rabsolu ((Bn n)-l2))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu ((Bn n)-l2))``). +Apply Rle_monotony. +Apply Rabsolu_pos. +Apply H4. +Apply Rlt_monotony_contra with ``/M``. +Apply Rlt_Rinv; Apply H3. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite (Rmult_sym ``/M``). +Apply Rlt_trans with ``eps/(2*M)``. +Apply H8; Assumption. +Unfold Rdiv; Rewrite Rinv_Rmult. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Replace ``2*(eps*(/2*/M))`` with ``(2*/2)*(eps*/M)``; [Idtac | Ring]. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite double. +Pattern 1 ``eps*/M``; Rewrite <- Rplus_Or. +Apply Rlt_compatibility; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Assumption]. +DiscrR. +DiscrR. +Red; Intro; Rewrite H10 in H3; Elim (Rlt_antirefl ? H3). +Red; Intro; Rewrite H10 in H3; Elim (Rlt_antirefl ? H3). +Rewrite H7; Do 2 Rewrite Rmult_Or; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Reflexivity. +Replace ``(An n)*(Bn n)-(An n)*l2`` with ``(An n)*((Bn n)-l2)``; [Idtac | Ring]. +Symmetry; Apply Rabsolu_mult. +Cut ``0<eps/(2*(Rabsolu l2))``. +Intro. +Unfold Un_cv in H; Unfold R_dist in H; Unfold Un_cv in H0; Unfold R_dist in H0. +Elim (H ``eps/(2*(Rabsolu l2))`` H8); Intros N1 H9. +Elim (H0 ``eps/(2*M)`` H6); Intros N2 H10. +Pose N := (max N1 N2). +Exists N; Intros. +Apply Rle_lt_trans with ``(Rabsolu ((An n)*(Bn n)-(An n)*l2))+(Rabsolu ((An n)*l2-l1*l2))``. +Replace ``(An n)*(Bn n)-l1*l2`` with ``((An n)*(Bn n)-(An n)*l2)+((An n)*l2-l1*l2)``; [Apply Rabsolu_triang | Ring]. +Replace ``(Rabsolu ((An n)*(Bn n)-(An n)*l2))`` with ``(Rabsolu (An n))*(Rabsolu ((Bn n)-l2))``. +Replace ``(Rabsolu ((An n)*l2-l1*l2))`` with ``(Rabsolu l2)*(Rabsolu ((An n)-l1))``. +Rewrite (double_var eps); Apply Rplus_lt. +Apply Rle_lt_trans with ``M*(Rabsolu ((Bn n)-l2))``. +Do 2 Rewrite <- (Rmult_sym ``(Rabsolu ((Bn n)-l2))``). +Apply Rle_monotony. +Apply Rabsolu_pos. +Apply H4. +Apply Rlt_monotony_contra with ``/M``. +Apply Rlt_Rinv; Apply H3. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite (Rmult_sym ``/M``). +Apply Rlt_le_trans with ``eps/(2*M)``. +Apply H10. +Unfold ge; Apply le_trans with N. +Unfold N; Apply le_max_r. +Assumption. +Unfold Rdiv; Rewrite Rinv_Rmult. +Right; Ring. +DiscrR. +Red; Intro; Rewrite H12 in H3; Elim (Rlt_antirefl ? H3). +Red; Intro; Rewrite H12 in H3; Elim (Rlt_antirefl ? H3). +Apply Rlt_monotony_contra with ``/(Rabsolu l2)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Apply Rlt_le_trans with ``eps/(2*(Rabsolu l2))``. +Apply H9. +Unfold ge; Apply le_trans with N. +Unfold N; Apply le_max_l. +Assumption. +Unfold Rdiv; Right; Rewrite Rinv_Rmult. +Ring. +DiscrR. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Replace ``(An n)*l2-l1*l2`` with ``l2*((An n)-l1)``; [Symmetry; Apply Rabsolu_mult | Ring]. +Replace ``(An n)*(Bn n)-(An n)*l2`` with ``(An n)*((Bn n)-l2)``; [Symmetry; Apply Rabsolu_mult | Ring]. +Unfold Rdiv; Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Apply Rabsolu_pos_lt; Assumption]. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Assumption]]. +Apply existTT with l1; Assumption. +Qed. + +Lemma tech9 : (Un:nat->R) (Un_growing Un) -> ((m,n:nat)(le m n)->``(Un m)<=(Un n)``). +Intros; Unfold Un_growing in H. +Induction n. +Induction m. +Right; Reflexivity. +Elim (le_Sn_O ? H0). +Cut (le m n)\/m=(S n). +Intro; Elim H1; Intro. +Apply Rle_trans with (Un n). +Apply Hrecn; Assumption. +Apply H. +Rewrite H2; Right; Reflexivity. +Inversion H0. +Right; Reflexivity. +Left; Assumption. +Qed. + +Lemma tech10 : (Un:nat->R;x:R) (Un_growing Un) -> (is_lub (EUn Un) x) -> (Un_cv Un x). +Intros; Cut (bound (EUn Un)). +Intro; Assert H2 := (Un_cv_crit ? H H1). +Elim H2; Intros. +Case (total_order_T x x0); Intro. +Elim s; Intro. +Cut (n:nat)``(Un n)<=x``. +Intro; Unfold Un_cv in H3; Cut ``0<x0-x``. +Intro; Elim (H3 ``x0-x`` H5); Intros. +Cut (ge x1 x1). +Intro; Assert H8 := (H6 x1 H7). +Unfold R_dist in H8; Rewrite Rabsolu_left1 in H8. +Rewrite Ropp_distr2 in H8; Unfold Rminus in H8. +Assert H9 := (Rlt_anti_compatibility ``x0`` ? ? H8). +Assert H10 := (Ropp_Rlt ? ? H9). +Assert H11 := (H4 x1). +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H10 H11)). +Apply Rle_minus; Apply Rle_trans with x. +Apply H4. +Left; Assumption. +Unfold ge; Apply le_n. +Apply Rgt_minus; Assumption. +Intro; Unfold is_lub in H0; Unfold is_upper_bound in H0; Elim H0; Intros. +Apply H4; Unfold EUn; Exists n; Reflexivity. +Rewrite b; Assumption. +Cut ((n:nat)``(Un n)<=x0``). +Intro; Unfold is_lub in H0; Unfold is_upper_bound in H0; Elim H0; Intros. +Cut (y:R)(EUn Un y)->``y<=x0``. +Intro; Assert H8 := (H6 ? H7). +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H8 r)). +Unfold EUn; Intros; Elim H7; Intros. +Rewrite H8; Apply H4. +Intro; Case (total_order_Rle (Un n) x0); Intro. +Assumption. +Cut (n0:nat)(le n n0) -> ``x0<(Un n0)``. +Intro; Unfold Un_cv in H3; Cut ``0<(Un n)-x0``. +Intro; Elim (H3 ``(Un n)-x0`` H5); Intros. +Cut (ge (max n x1) x1). +Intro; Assert H8 := (H6 (max n x1) H7). +Unfold R_dist in H8. +Rewrite Rabsolu_right in H8. +Unfold Rminus in H8; Do 2 Rewrite <- (Rplus_sym ``-x0``) in H8. +Assert H9 := (Rlt_anti_compatibility ? ? ? H8). +Cut ``(Un n)<=(Un (max n x1))``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H10 H9)). +Apply tech9; [Assumption | Apply le_max_l]. +Apply Rge_trans with ``(Un n)-x0``. +Unfold Rminus; Apply Rle_sym1; Do 2 Rewrite <- (Rplus_sym ``-x0``); Apply Rle_compatibility. +Apply tech9; [Assumption | Apply le_max_l]. +Left; Assumption. +Unfold ge; Apply le_max_r. +Apply Rlt_anti_compatibility with x0. +Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym x0); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H4; Apply le_n. +Intros; Apply Rlt_le_trans with (Un n). +Case (total_order_Rlt_Rle x0 (Un n)); Intro. +Assumption. +Elim n0; Assumption. +Apply tech9; Assumption. +Unfold bound; Exists x; Unfold is_lub in H0; Elim H0; Intros; Assumption. +Qed. + +Lemma tech13 : (An:nat->R;k:R) ``0<=k<1`` -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (EXT k0 : R | ``k<k0<1`` /\ (EX N:nat | (n:nat) (le N n)->``(Rabsolu ((An (S n))/(An n)))<k0``)). +Intros; Exists ``k+(1-k)/2``. +Split. +Split. +Pattern 1 k; Rewrite <- Rplus_Or; Apply Rlt_compatibility. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rlt_anti_compatibility with k; Rewrite Rplus_Or; Replace ``k+(1-k)`` with R1; [Elim H; Intros; Assumption | Ring]. +Apply Rlt_Rinv; Sup0. +Apply Rlt_monotony_contra with ``2``. +Sup0. +Unfold Rdiv; Rewrite Rmult_1r; Rewrite Rmult_Rplus_distr; Pattern 1 ``2``; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1r; Replace ``2*k+(1-k)`` with ``1+k``; [Idtac | Ring]. +Elim H; Intros. +Apply Rlt_compatibility; Assumption. +Unfold Un_cv in H0; Cut ``0<(1-k)/2``. +Intro; Elim (H0 ``(1-k)/2`` H1); Intros. +Exists x; Intros. +Assert H4 := (H2 n H3). +Unfold R_dist in H4; Rewrite <- Rabsolu_Rabsolu; Replace ``(Rabsolu ((An (S n))/(An n)))`` with ``((Rabsolu ((An (S n))/(An n)))-k)+k``; [Idtac | Ring]; Apply Rle_lt_trans with ``(Rabsolu ((Rabsolu ((An (S n))/(An n)))-k))+(Rabsolu k)``. +Apply Rabsolu_triang. +Rewrite (Rabsolu_right k). +Apply Rlt_anti_compatibility with ``-k``; Rewrite <- (Rplus_sym k); Repeat Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Repeat Rewrite Rplus_Ol; Apply H4. +Apply Rle_sym1; Elim H; Intros; Assumption. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rlt_anti_compatibility with k; Rewrite Rplus_Or; Elim H; Intros; Replace ``k+(1-k)`` with R1; [Assumption | Ring]. +Apply Rlt_Rinv; Sup0. +Qed. + +(**********) +Lemma growing_ineq : (Un:nat->R;l:R) (Un_growing Un) -> (Un_cv Un l) -> ((n:nat)``(Un n)<=l``). +Intros; Case (total_order_T (Un n) l); Intro. +Elim s; Intro. +Left; Assumption. +Right; Assumption. +Cut ``0<(Un n)-l``. +Intro; Unfold Un_cv in H0; Unfold R_dist in H0. +Elim (H0 ``(Un n)-l`` H1); Intros N1 H2. +Pose N := (max n N1). +Cut ``(Un n)-l<=(Un N)-l``. +Intro; Cut ``(Un N)-l<(Un n)-l``. +Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H3 H4)). +Apply Rle_lt_trans with ``(Rabsolu ((Un N)-l))``. +Apply Rle_Rabsolu. +Apply H2. +Unfold ge N; Apply le_max_r. +Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-l``); Apply Rle_compatibility. +Apply tech9. +Assumption. +Unfold N; Apply le_max_l. +Apply Rlt_anti_compatibility with l. +Rewrite Rplus_Or. +Replace ``l+((Un n)-l)`` with (Un n); [Assumption | Ring]. +Qed. + +(* Un->l => (-Un) -> (-l) *) +Lemma CV_opp : (An:nat->R;l:R) (Un_cv An l) -> (Un_cv (opp_seq An) ``-l``). +Intros An l. +Unfold Un_cv; Unfold R_dist; Intros. +Elim (H eps H0); Intros. +Exists x; Intros. +Unfold opp_seq; Replace ``-(An n)- (-l)`` with ``-((An n)-l)``; [Rewrite Rabsolu_Ropp | Ring]. +Apply H1; Assumption. +Qed. + +(**********) +Lemma decreasing_ineq : (Un:nat->R;l:R) (Un_decreasing Un) -> (Un_cv Un l) -> ((n:nat)``l<=(Un n)``). +Intros. +Assert H1 := (decreasing_growing ? H). +Assert H2 := (CV_opp ? ? H0). +Assert H3 := (growing_ineq ? ? H1 H2). +Apply Ropp_Rle. +Unfold opp_seq in H3; Apply H3. +Qed. + +(**********) +Lemma CV_minus : (An,Bn:nat->R;l1,l2:R) (Un_cv An l1) -> (Un_cv Bn l2) -> (Un_cv [i:nat]``(An i)-(Bn i)`` ``l1-l2``). +Intros. +Replace [i:nat]``(An i)-(Bn i)`` with [i:nat]``(An i)+((opp_seq Bn) i)``. +Unfold Rminus; Apply CV_plus. +Assumption. +Apply CV_opp; Assumption. +Unfold Rminus opp_seq; Reflexivity. +Qed. + +(* Un -> +oo *) +Definition cv_infty [Un:nat->R] : Prop := (M:R)(EXT N:nat | (n:nat) (le N n) -> ``M<(Un n)``). + +(* Un -> +oo => /Un -> O *) +Lemma cv_infty_cv_R0 : (Un:nat->R) ((n:nat)``(Un n)<>0``) -> (cv_infty Un) -> (Un_cv [n:nat]``/(Un n)`` R0). +Unfold cv_infty Un_cv; Unfold R_dist; Intros. +Elim (H0 ``/eps``); Intros N0 H2. +Exists N0; Intros. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite (Rabsolu_Rinv ? (H n)). +Apply Rlt_monotony_contra with (Rabsolu (Un n)). +Apply Rabsolu_pos_lt; Apply H. +Rewrite <- Rinv_r_sym. +Apply Rlt_monotony_contra with ``/eps``. +Apply Rlt_Rinv; Assumption. +Rewrite Rmult_1r; Rewrite (Rmult_sym ``/eps``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Apply Rlt_le_trans with (Un n). +Apply H2; Assumption. +Apply Rle_Rabsolu. +Red; Intro; Rewrite H4 in H1; Elim (Rlt_antirefl ? H1). +Apply Rabsolu_no_R0; Apply H. +Qed. + +(**********) +Lemma decreasing_prop : (Un:nat->R;m,n:nat) (Un_decreasing Un) -> (le m n) -> ``(Un n)<=(Un m)``. +Unfold Un_decreasing; Intros. +Induction n. +Induction m. +Right; Reflexivity. +Elim (le_Sn_O ? H0). +Cut (le m n)\/m=(S n). +Intro; Elim H1; Intro. +Apply Rle_trans with (Un n). +Apply H. +Apply Hrecn; Assumption. +Rewrite H2; Right; Reflexivity. +Inversion H0; [Right; Reflexivity | Left; Assumption]. +Qed. + +(* |x|^n/n! -> 0 *) +Lemma cv_speed_pow_fact : (x:R) (Un_cv [n:nat]``(pow x n)/(INR (fact n))`` R0). +Intro; Cut (Un_cv [n:nat]``(pow (Rabsolu x) n)/(INR (fact n))`` R0) -> (Un_cv [n:nat]``(pow x n)/(INR (fact n))`` ``0``). +Intro; Apply H. +Unfold Un_cv; Unfold R_dist; Intros; Case (Req_EM x R0); Intro. +Exists (S O); Intros. +Rewrite H1; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_R0; Rewrite pow_ne_zero; [Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Assumption | Red; Intro; Rewrite H3 in H2; Elim (le_Sn_n ? H2)]. +Assert H2 := (Rabsolu_pos_lt x H1); Pose M := (up (Rabsolu x)); Cut `0<=M`. +Intro; Elim (IZN M H3); Intros M_nat H4. +Pose Un := [n:nat]``(pow (Rabsolu x) (plus M_nat n))/(INR (fact (plus M_nat n)))``. +Cut (Un_cv Un R0); Unfold Un_cv; Unfold R_dist; Intros. +Elim (H5 eps H0); Intros N H6. +Exists (plus M_nat N); Intros; Cut (EX p:nat | (ge p N)/\n=(plus M_nat p)). +Intro; Elim H8; Intros p H9. +Elim H9; Intros; Rewrite H11; Unfold Un in H6; Apply H6; Assumption. +Exists (minus n M_nat). +Split. +Unfold ge; Apply simpl_le_plus_l with M_nat; Rewrite <- le_plus_minus. +Assumption. +Apply le_trans with (plus M_nat N). +Apply le_plus_l. +Assumption. +Apply le_plus_minus; Apply le_trans with (plus M_nat N); [Apply le_plus_l | Assumption]. +Pose Vn := [n:nat]``(Rabsolu x)*(Un O)/(INR (S n))``. +Cut (le (1) M_nat). +Intro; Cut (n:nat)``0<(Un n)``. +Intro; Cut (Un_decreasing Un). +Intro; Cut (n:nat)``(Un (S n))<=(Vn n)``. +Intro; Cut (Un_cv Vn R0). +Unfold Un_cv; Unfold R_dist; Intros. +Elim (H10 eps0 H5); Intros N1 H11. +Exists (S N1); Intros. +Cut (n:nat)``0<(Vn n)``. +Intro; Apply Rle_lt_trans with ``(Rabsolu ((Vn (pred n))-0))``. +Repeat Rewrite Rabsolu_right. +Unfold Rminus; Rewrite Ropp_O; Do 2 Rewrite Rplus_Or; Replace n with (S (pred n)). +Apply H9. +Inversion H12; Simpl; Reflexivity. +Apply Rle_sym1; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Left; Apply H13. +Apply Rle_sym1; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Left; Apply H7. +Apply H11; Unfold ge; Apply le_S_n; Replace (S (pred n)) with n; [Unfold ge in H12; Exact H12 | Inversion H12; Simpl; Reflexivity]. +Intro; Apply Rlt_le_trans with (Un (S n0)); [Apply H7 | Apply H9]. +Cut (cv_infty [n:nat](INR (S n))). +Intro; Cut (Un_cv [n:nat]``/(INR (S n))`` R0). +Unfold Un_cv R_dist; Intros; Unfold Vn. +Cut ``0<eps1/((Rabsolu x)*(Un O))``. +Intro; Elim (H11 ? H13); Intros N H14. +Exists N; Intros; Replace ``(Rabsolu x)*(Un O)/(INR (S n))-0`` with ``((Rabsolu x)*(Un O))*(/(INR (S n))-0)``; [Idtac | Unfold Rdiv; Ring]. +Rewrite Rabsolu_mult; Apply Rlt_monotony_contra with ``/(Rabsolu ((Rabsolu x)*(Un O)))``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt. +Apply prod_neq_R0. +Apply Rabsolu_no_R0; Assumption. +Assert H16 := (H7 O); Red; Intro; Rewrite H17 in H16; Elim (Rlt_antirefl ? H16). +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l. +Replace ``/(Rabsolu ((Rabsolu x)*(Un O)))*eps1`` with ``eps1/((Rabsolu x)*(Un O))``. +Apply H14; Assumption. +Unfold Rdiv; Rewrite (Rabsolu_right ``(Rabsolu x)*(Un O)``). +Apply Rmult_sym. +Apply Rle_sym1; Apply Rmult_le_pos. +Apply Rabsolu_pos. +Left; Apply H7. +Apply Rabsolu_no_R0. +Apply prod_neq_R0; [Apply Rabsolu_no_R0; Assumption | Assert H16 := (H7 O); Red; Intro; Rewrite H17 in H16; Elim (Rlt_antirefl ? H16)]. +Unfold Rdiv; Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv; Apply Rmult_lt_pos. +Apply Rabsolu_pos_lt; Assumption. +Apply H7. +Apply (cv_infty_cv_R0 [n:nat]``(INR (S n))``). +Intro; Apply not_O_INR; Discriminate. +Assumption. +Unfold cv_infty; Intro; Case (total_order_T M0 R0); Intro. +Elim s; Intro. +Exists O; Intros. +Apply Rlt_trans with R0; [Assumption | Apply lt_INR_0; Apply lt_O_Sn]. +Exists O; Intros; Rewrite b; Apply lt_INR_0; Apply lt_O_Sn. +Pose M0_z := (up M0). +Assert H10 := (archimed M0). +Cut `0<=M0_z`. +Intro; Elim (IZN ? H11); Intros M0_nat H12. +Exists M0_nat; Intros. +Apply Rlt_le_trans with (IZR M0_z). +Elim H10; Intros; Assumption. +Rewrite H12; Rewrite <- INR_IZR_INZ; Apply le_INR. +Apply le_trans with n; [Assumption | Apply le_n_Sn]. +Apply le_IZR; Left; Simpl; Unfold M0_z; Apply Rlt_trans with M0; [Assumption | Elim H10; Intros; Assumption]. +Intro; Apply Rle_trans with ``(Rabsolu x)*(Un n)*/(INR (S n))``. +Unfold Un; Replace (plus M_nat (S n)) with (plus (plus M_nat n) (1)). +Rewrite pow_add; Replace (pow (Rabsolu x) (S O)) with (Rabsolu x); [Idtac | Simpl; Ring]. +Unfold Rdiv; Rewrite <- (Rmult_sym (Rabsolu x)); Repeat Rewrite Rmult_assoc; Repeat Apply Rle_monotony. +Apply Rabsolu_pos. +Left; Apply pow_lt; Assumption. +Replace (plus (plus M_nat n) (S O)) with (S (plus M_nat n)). +Rewrite fact_simpl; Rewrite mult_sym; Rewrite mult_INR; Rewrite Rinv_Rmult. +Apply Rle_monotony. +Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H10 := (sym_eq ? ? ? H9); Elim (fact_neq_0 ? H10). +Left; Apply Rinv_lt. +Apply Rmult_lt_pos; Apply lt_INR_0; Apply lt_O_Sn. +Apply lt_INR; Apply lt_n_S. +Pattern 1 n; Replace n with (plus O n); [Idtac | Reflexivity]. +Apply lt_reg_r. +Apply lt_le_trans with (S O); [Apply lt_O_Sn | Assumption]. +Apply INR_fact_neq_0. +Apply not_O_INR; Discriminate. +Apply INR_eq; Rewrite S_INR; Do 3 Rewrite plus_INR; Reflexivity. +Apply INR_eq; Do 3 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. +Unfold Vn; Rewrite Rmult_assoc; Unfold Rdiv; Rewrite (Rmult_sym (Un O)); Rewrite (Rmult_sym (Un n)). +Repeat Apply Rle_monotony. +Apply Rabsolu_pos. +Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply lt_O_Sn. +Apply decreasing_prop; [Assumption | Apply le_O_n]. +Unfold Un_decreasing; Intro; Unfold Un. +Replace (plus M_nat (S n)) with (plus (plus M_nat n) (1)). +Rewrite pow_add; Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. +Left; Apply pow_lt; Assumption. +Replace (pow (Rabsolu x) (S O)) with (Rabsolu x); [Idtac | Simpl; Ring]. +Replace (plus (plus M_nat n) (S O)) with (S (plus M_nat n)). +Apply Rle_monotony_contra with (INR (fact (S (plus M_nat n)))). +Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H9 := (sym_eq ? ? ? H8); Elim (fact_neq_0 ? H9). +Rewrite (Rmult_sym (Rabsolu x)); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l. +Rewrite fact_simpl; Rewrite mult_INR; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Apply Rle_trans with (INR M_nat). +Left; Rewrite INR_IZR_INZ. +Rewrite <- H4; Assert H8 := (archimed (Rabsolu x)); Elim H8; Intros; Assumption. +Apply le_INR; Apply le_trans with (S M_nat); [Apply le_n_Sn | Apply le_n_S; Apply le_plus_l]. +Apply INR_fact_neq_0. +Apply INR_fact_neq_0. +Apply INR_eq; Rewrite S_INR; Do 3 Rewrite plus_INR; Reflexivity. +Apply INR_eq; Do 3 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. +Intro; Unfold Un; Unfold Rdiv; Apply Rmult_lt_pos. +Apply pow_lt; Assumption. +Apply Rlt_Rinv; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H8 := (sym_eq ? ? ? H7); Elim (fact_neq_0 ? H8). +Clear Un Vn; Apply INR_le; Simpl. +Induction M_nat. +Assert H6 := (archimed (Rabsolu x)); Fold M in H6; Elim H6; Intros. +Rewrite H4 in H7; Rewrite <- INR_IZR_INZ in H7. +Simpl in H7; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H2 H7)). +Replace R1 with (INR (S O)); [Apply le_INR | Reflexivity]; Apply le_n_S; Apply le_O_n. +Apply le_IZR; Simpl; Left; Apply Rlt_trans with (Rabsolu x). +Assumption. +Elim (archimed (Rabsolu x)); Intros; Assumption. +Unfold Un_cv; Unfold R_dist; Intros; Elim (H eps H0); Intros. +Exists x0; Intros; Apply Rle_lt_trans with ``(Rabsolu ((pow (Rabsolu x) n)/(INR (fact n))-0))``. +Unfold Rminus; Rewrite Ropp_O; Do 2 Rewrite Rplus_Or; Rewrite (Rabsolu_right ``(pow (Rabsolu x) n)/(INR (fact n))``). +Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite (Rabsolu_right ``/(INR (fact n))``). +Rewrite Pow_Rabsolu; Right; Reflexivity. +Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H4 := (sym_eq ? ? ? H3); Elim (fact_neq_0 ? H4). +Apply Rle_sym1; Unfold Rdiv; Apply Rmult_le_pos. +Case (Req_EM x R0); Intro. +Rewrite H3; Rewrite Rabsolu_R0. +Induction n; [Simpl; Left; Apply Rlt_R0_R1 | Simpl; Rewrite Rmult_Ol; Right; Reflexivity]. +Left; Apply pow_lt; Apply Rabsolu_pos_lt; Assumption. +Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H4 := (sym_eq ? ? ? H3); Elim (fact_neq_0 ? H4). +Apply H1; Assumption. +Qed. diff --git a/theories7/Reals/SeqSeries.v b/theories7/Reals/SeqSeries.v new file mode 100644 index 00000000..dd93c304 --- /dev/null +++ b/theories7/Reals/SeqSeries.v @@ -0,0 +1,307 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: SeqSeries.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Max. +Require Export Rseries. +Require Export SeqProp. +Require Export Rcomplete. +Require Export PartSum. +Require Export AltSeries. +Require Export Binomial. +Require Export Rsigma. +Require Export Rprod. +Require Export Cauchy_prod. +Require Export Alembert. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(**********) +Lemma sum_maj1 : (fn:nat->R->R;An:nat->R;x,l1,l2:R;N:nat) (Un_cv [n:nat](SP fn n x) l1) -> (Un_cv [n:nat](sum_f_R0 An n) l2) -> ((n:nat)``(Rabsolu (fn n x))<=(An n)``) -> ``(Rabsolu (l1-(SP fn N x)))<=l2-(sum_f_R0 An N)``. +Intros; Cut (sigTT R [l:R](Un_cv [n:nat](sum_f_R0 [l:nat](fn (plus (S N) l) x) n) l)). +Intro; Cut (sigTT R [l:R](Un_cv [n:nat](sum_f_R0 [l:nat](An (plus (S N) l)) n) l)). +Intro; Elim X; Intros l1N H2. +Elim X0; Intros l2N H3. +Cut ``l1-(SP fn N x)==l1N``. +Intro; Cut ``l2-(sum_f_R0 An N)==l2N``. +Intro; Rewrite H4; Rewrite H5. +Apply sum_cv_maj with [l:nat](An (plus (S N) l)) [l:nat][x:R](fn (plus (S N) l) x) x. +Unfold SP; Apply H2. +Apply H3. +Intros; Apply H1. +Symmetry; EApply UL_sequence. +Apply H3. +Unfold Un_cv in H0; Unfold Un_cv; Intros; Elim (H0 eps H5); Intros N0 H6. +Unfold R_dist in H6; Exists N0; Intros. +Unfold R_dist; Replace (Rminus (sum_f_R0 [l:nat](An (plus (S N) l)) n) (Rminus l2 (sum_f_R0 An N))) with (Rminus (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) l2); [Idtac | Ring]. +Replace (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) with (sum_f_R0 An (S (plus N n))). +Apply H6; Unfold ge; Apply le_trans with n. +Apply H7. +Apply le_trans with (plus N n). +Apply le_plus_r. +Apply le_n_Sn. +Cut (le O N). +Cut (lt N (S (plus N n))). +Intros; Assert H10 := (sigma_split An H9 H8). +Unfold sigma in H10. +Do 2 Rewrite <- minus_n_O in H10. +Replace (sum_f_R0 An (S (plus N n))) with (sum_f_R0 [k:nat](An (plus (0) k)) (S (plus N n))). +Replace (sum_f_R0 An N) with (sum_f_R0 [k:nat](An (plus (0) k)) N). +Cut (minus (S (plus N n)) (S N))=n. +Intro; Rewrite H11 in H10. +Apply H10. +Apply INR_eq; Rewrite minus_INR. +Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_n_S; Apply le_plus_l. +Apply sum_eq; Intros. +Reflexivity. +Apply sum_eq; Intros. +Reflexivity. +Apply le_lt_n_Sm; Apply le_plus_l. +Apply le_O_n. +Symmetry; EApply UL_sequence. +Apply H2. +Unfold Un_cv in H; Unfold Un_cv; Intros. +Elim (H eps H4); Intros N0 H5. +Unfold R_dist in H5; Exists N0; Intros. +Unfold R_dist SP; Replace (Rminus (sum_f_R0 [l:nat](fn (plus (S N) l) x) n) (Rminus l1 (sum_f_R0 [k:nat](fn k x) N))) with (Rminus (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) l1); [Idtac | Ring]. +Replace (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) with (sum_f_R0 [k:nat](fn k x) (S (plus N n))). +Unfold SP in H5; Apply H5; Unfold ge; Apply le_trans with n. +Apply H6. +Apply le_trans with (plus N n). +Apply le_plus_r. +Apply le_n_Sn. +Cut (le O N). +Cut (lt N (S (plus N n))). +Intros; Assert H9 := (sigma_split [k:nat](fn k x) H8 H7). +Unfold sigma in H9. +Do 2 Rewrite <- minus_n_O in H9. +Replace (sum_f_R0 [k:nat](fn k x) (S (plus N n))) with (sum_f_R0 [k:nat](fn (plus (0) k) x) (S (plus N n))). +Replace (sum_f_R0 [k:nat](fn k x) N) with (sum_f_R0 [k:nat](fn (plus (0) k) x) N). +Cut (minus (S (plus N n)) (S N))=n. +Intro; Rewrite H10 in H9. +Apply H9. +Apply INR_eq; Rewrite minus_INR. +Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_n_S; Apply le_plus_l. +Apply sum_eq; Intros. +Reflexivity. +Apply sum_eq; Intros. +Reflexivity. +Apply le_lt_n_Sm. +Apply le_plus_l. +Apply le_O_n. +Apply existTT with ``l2-(sum_f_R0 An N)``. +Unfold Un_cv in H0; Unfold Un_cv; Intros. +Elim (H0 eps H2); Intros N0 H3. +Unfold R_dist in H3; Exists N0; Intros. +Unfold R_dist; Replace (Rminus (sum_f_R0 [l:nat](An (plus (S N) l)) n) (Rminus l2 (sum_f_R0 An N))) with (Rminus (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) l2); [Idtac | Ring]. +Replace (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) with (sum_f_R0 An (S (plus N n))). +Apply H3; Unfold ge; Apply le_trans with n. +Apply H4. +Apply le_trans with (plus N n). +Apply le_plus_r. +Apply le_n_Sn. +Cut (le O N). +Cut (lt N (S (plus N n))). +Intros; Assert H7 := (sigma_split An H6 H5). +Unfold sigma in H7. +Do 2 Rewrite <- minus_n_O in H7. +Replace (sum_f_R0 An (S (plus N n))) with (sum_f_R0 [k:nat](An (plus (0) k)) (S (plus N n))). +Replace (sum_f_R0 An N) with (sum_f_R0 [k:nat](An (plus (0) k)) N). +Cut (minus (S (plus N n)) (S N))=n. +Intro; Rewrite H8 in H7. +Apply H7. +Apply INR_eq; Rewrite minus_INR. +Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_n_S; Apply le_plus_l. +Apply sum_eq; Intros. +Reflexivity. +Apply sum_eq; Intros. +Reflexivity. +Apply le_lt_n_Sm. +Apply le_plus_l. +Apply le_O_n. +Apply existTT with ``l1-(SP fn N x)``. +Unfold Un_cv in H; Unfold Un_cv; Intros. +Elim (H eps H2); Intros N0 H3. +Unfold R_dist in H3; Exists N0; Intros. +Unfold R_dist SP. +Replace (Rminus (sum_f_R0 [l:nat](fn (plus (S N) l) x) n) (Rminus l1 (sum_f_R0 [k:nat](fn k x) N))) with (Rminus (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) l1); [Idtac | Ring]. +Replace (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) with (sum_f_R0 [k:nat](fn k x) (S (plus N n))). +Unfold SP in H3; Apply H3. +Unfold ge; Apply le_trans with n. +Apply H4. +Apply le_trans with (plus N n). +Apply le_plus_r. +Apply le_n_Sn. +Cut (le O N). +Cut (lt N (S (plus N n))). +Intros; Assert H7 := (sigma_split [k:nat](fn k x) H6 H5). +Unfold sigma in H7. +Do 2 Rewrite <- minus_n_O in H7. +Replace (sum_f_R0 [k:nat](fn k x) (S (plus N n))) with (sum_f_R0 [k:nat](fn (plus (0) k) x) (S (plus N n))). +Replace (sum_f_R0 [k:nat](fn k x) N) with (sum_f_R0 [k:nat](fn (plus (0) k) x) N). +Cut (minus (S (plus N n)) (S N))=n. +Intro; Rewrite H8 in H7. +Apply H7. +Apply INR_eq; Rewrite minus_INR. +Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. +Apply le_n_S; Apply le_plus_l. +Apply sum_eq; Intros. +Reflexivity. +Apply sum_eq; Intros. +Reflexivity. +Apply le_lt_n_Sm. +Apply le_plus_l. +Apply le_O_n. +Qed. + +(* Comparaison of convergence for series *) +Lemma Rseries_CV_comp : (An,Bn:nat->R) ((n:nat)``0<=(An n)<=(Bn n)``) -> (sigTT ? [l:R](Un_cv [N:nat](sum_f_R0 Bn N) l)) -> (sigTT ? [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). +Intros; Apply cv_cauchy_2. +Assert H0 := (cv_cauchy_1 ? X). +Unfold Cauchy_crit_series; Unfold Cauchy_crit. +Intros; Elim (H0 eps H1); Intros. +Exists x; Intros. +Cut (Rle (R_dist (sum_f_R0 An n) (sum_f_R0 An m)) (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m))). +Intro; Apply Rle_lt_trans with (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)). +Assumption. +Apply H2; Assumption. +Assert H5 := (lt_eq_lt_dec n m). +Elim H5; Intro. +Elim a; Intro. +Rewrite (tech2 An n m); [Idtac | Assumption]. +Rewrite (tech2 Bn n m); [Idtac | Assumption]. +Unfold R_dist; Unfold Rminus; Do 2 Rewrite Ropp_distr1; Do 2 Rewrite <- Rplus_assoc; Do 2 Rewrite Rplus_Ropp_r; Do 2 Rewrite Rplus_Ol; Do 2 Rewrite Rabsolu_Ropp; Repeat Rewrite Rabsolu_right. +Apply sum_Rle; Intros. +Elim (H (plus (S n) n0)); Intros. +Apply H8. +Apply Rle_sym1; Apply cond_pos_sum; Intro. +Elim (H (plus (S n) n0)); Intros. +Apply Rle_trans with (An (plus (S n) n0)); Assumption. +Apply Rle_sym1; Apply cond_pos_sum; Intro. +Elim (H (plus (S n) n0)); Intros; Assumption. +Rewrite b; Unfold R_dist; Unfold Rminus; Do 2 Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Right; Reflexivity. +Rewrite (tech2 An m n); [Idtac | Assumption]. +Rewrite (tech2 Bn m n); [Idtac | Assumption]. +Unfold R_dist; Unfold Rminus; Do 2 Rewrite Rplus_assoc; Rewrite (Rplus_sym (sum_f_R0 An m)); Rewrite (Rplus_sym (sum_f_R0 Bn m)); Do 2 Rewrite Rplus_assoc; Do 2 Rewrite Rplus_Ropp_l; Do 2 Rewrite Rplus_Or; Repeat Rewrite Rabsolu_right. +Apply sum_Rle; Intros. +Elim (H (plus (S m) n0)); Intros; Apply H8. +Apply Rle_sym1; Apply cond_pos_sum; Intro. +Elim (H (plus (S m) n0)); Intros. +Apply Rle_trans with (An (plus (S m) n0)); Assumption. +Apply Rle_sym1. +Apply cond_pos_sum; Intro. +Elim (H (plus (S m) n0)); Intros; Assumption. +Qed. + +(* Cesaro's theorem *) +Lemma Cesaro : (An,Bn:nat->R;l:R) (Un_cv Bn l) -> ((n:nat)``0<(An n)``) -> (cv_infty [n:nat](sum_f_R0 An n)) -> (Un_cv [n:nat](Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 An n)) l). +Proof with Trivial. +Unfold Un_cv; Intros; Assert H3 : (n:nat)``0<(sum_f_R0 An n)``. +Intro; Apply tech1. +Assert H4 : (n:nat) ``(sum_f_R0 An n)<>0``. +Intro; Red; Intro; Assert H5 := (H3 n); Rewrite H4 in H5; Elim (Rlt_antirefl ? H5). +Assert H5 := (cv_infty_cv_R0 ? H4 H1); Assert H6 : ``0<eps/2``. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rlt_Rinv; Sup. +Elim (H ? H6); Clear H; Intros N1 H; Pose C := (Rabsolu (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` N1)); Assert H7 : (EX N:nat | (n:nat) (le N n) -> ``C/(sum_f_R0 An n)<eps/2``). +Case (Req_EM C R0); Intro. +Exists O; Intros. +Rewrite H7; Unfold Rdiv; Rewrite Rmult_Ol; Apply Rmult_lt_pos. +Apply Rlt_Rinv; Sup. +Assert H8 : ``0<eps/(2*(Rabsolu C))``. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rlt_Rinv; Apply Rmult_lt_pos. +Sup. +Apply Rabsolu_pos_lt. +Elim (H5 ? H8); Intros; Exists x; Intros; Assert H11 := (H9 ? H10); Unfold R_dist in H11; Unfold Rminus in H11; Rewrite Ropp_O in H11; Rewrite Rplus_Or in H11. +Apply Rle_lt_trans with (Rabsolu ``C/(sum_f_R0 An n)``). +Apply Rle_Rabsolu. +Unfold Rdiv; Rewrite Rabsolu_mult; Apply Rlt_monotony_contra with ``/(Rabsolu C)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Replace ``/(Rabsolu C)*(eps*/2)`` with ``eps/(2*(Rabsolu C))``. +Unfold Rdiv; Rewrite Rinv_Rmult. +Ring. +DiscrR. +Apply Rabsolu_no_R0. +Apply Rabsolu_no_R0. +Elim H7; Clear H7; Intros N2 H7; Pose N := (max N1 N2); Exists (S N); Intros; Unfold R_dist; Replace (Rminus (Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 An n)) l) with (Rdiv (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` n) (sum_f_R0 An n)). +Assert H9 : (lt N1 n). +Apply lt_le_trans with (S N). +Apply le_lt_n_Sm; Unfold N; Apply le_max_l. +Rewrite (tech2 [k:nat]``(An k)*((Bn k)-l)`` ? ? H9); Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Apply Rle_lt_trans with (Rplus (Rabsolu (Rdiv (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` N1) (sum_f_R0 An n))) (Rabsolu (Rdiv (sum_f_R0 [i:nat]``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)`` (minus n (S N1))) (sum_f_R0 An n)))). +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Unfold Rdiv; Rewrite Rabsolu_mult; Fold C; Rewrite Rabsolu_right. +Apply (H7 n); Apply le_trans with (S N). +Apply le_trans with N; [Unfold N; Apply le_max_r | Apply le_n_Sn]. +Apply Rle_sym1; Left; Apply Rlt_Rinv. + +Unfold R_dist in H; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite (Rabsolu_right ``/(sum_f_R0 An n)``). +Apply Rle_lt_trans with (Rmult (sum_f_R0 [i:nat](Rabsolu ``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)``) (minus n (S N1))) ``/(sum_f_R0 An n)``). +Do 2 Rewrite <- (Rmult_sym ``/(sum_f_R0 An n)``); Apply Rle_monotony. +Left; Apply Rlt_Rinv. +Apply (sum_Rabsolu [i:nat]``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)`` (minus n (S N1))). +Apply Rle_lt_trans with (Rmult (sum_f_R0 [i:nat]``(An (plus (S N1) i))*eps/2`` (minus n (S N1))) ``/(sum_f_R0 An n)``). +Do 2 Rewrite <- (Rmult_sym ``/(sum_f_R0 An n)``); Apply Rle_monotony. +Left; Apply Rlt_Rinv. +Apply sum_Rle; Intros; Rewrite Rabsolu_mult; Pattern 2 (An (plus (S N1) n0)); Rewrite <- (Rabsolu_right (An (plus (S N1) n0))). +Apply Rle_monotony. +Apply Rabsolu_pos. +Left; Apply H; Unfold ge; Apply le_trans with (S N1); [Apply le_n_Sn | Apply le_plus_l]. +Apply Rle_sym1; Left. +Rewrite <- (scal_sum [i:nat](An (plus (S N1) i)) (minus n (S N1)) ``eps/2``); Unfold Rdiv; Repeat Rewrite Rmult_assoc; Apply Rlt_monotony. +Pattern 2 ``/2``; Rewrite <- Rmult_1r; Apply Rlt_monotony. +Apply Rlt_Rinv; Sup. +Rewrite Rmult_sym; Apply Rlt_monotony_contra with (sum_f_R0 An n). +Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite Rmult_1r; Rewrite (tech2 An N1 n). +Rewrite Rplus_sym; Pattern 1 (sum_f_R0 [i:nat](An (plus (S N1) i)) (minus n (S N1))); Rewrite <- Rplus_Or; Apply Rlt_compatibility. +Apply Rle_sym1; Left; Apply Rlt_Rinv. +Replace (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` n) with (Rplus (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 [k:nat]``(An k)*-l`` n)). +Rewrite <- (scal_sum An n ``-l``); Field. +Rewrite <- plus_sum; Apply sum_eq; Intros; Ring. +Qed. + +Lemma Cesaro_1 : (An:nat->R;l:R) (Un_cv An l) -> (Un_cv [n:nat]``(sum_f_R0 An (pred n))/(INR n)`` l). +Proof with Trivial. +Intros Bn l H; Pose An := [_:nat]R1. +Assert H0 : (n:nat) ``0<(An n)``. +Intro; Unfold An; Apply Rlt_R0_R1. +Assert H1 : (n:nat)``0<(sum_f_R0 An n)``. +Intro; Apply tech1. +Assert H2 : (cv_infty [n:nat](sum_f_R0 An n)). +Unfold cv_infty; Intro; Case (total_order_Rle M R0); Intro. +Exists O; Intros; Apply Rle_lt_trans with R0. +Assert H2 : ``0<M``. +Auto with real. +Clear n; Pose m := (up M); Elim (archimed M); Intros; Assert H5 : `0<=m`. +Apply le_IZR; Unfold m; Simpl; Left; Apply Rlt_trans with M. +Elim (IZN ? H5); Intros; Exists x; Intros; Unfold An; Rewrite sum_cte; Rewrite Rmult_1l; Apply Rlt_trans with (IZR (up M)). +Apply Rle_lt_trans with (INR x). +Rewrite INR_IZR_INZ; Fold m; Rewrite <- H6; Right. +Apply lt_INR; Apply le_lt_n_Sm. +Assert H3 := (Cesaro ? ? ? H H0 H2). +Unfold Un_cv; Unfold Un_cv in H3; Intros; Elim (H3 ? H4); Intros; Exists (S x); Intros; Unfold R_dist; Unfold R_dist in H5; Apply Rle_lt_trans with (Rabsolu (Rminus (Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` (pred n)) (sum_f_R0 An (pred n))) l)). +Right; Replace ``(sum_f_R0 Bn (pred n))/(INR n)-l`` with (Rminus (Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` (pred n)) (sum_f_R0 An (pred n))) l). +Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-l``); Apply Rplus_plus_r. +Unfold An; Replace (sum_f_R0 [k:nat]``1*(Bn k)`` (pred n)) with (sum_f_R0 Bn (pred n)). +Rewrite sum_cte; Rewrite Rmult_1l; Replace (S (pred n)) with n. +Apply S_pred with O; Apply lt_le_trans with (S x). +Apply lt_O_Sn. +Apply sum_eq; Intros; Ring. +Apply H5; Unfold ge; Apply le_S_n; Replace (S (pred n)) with n. +Apply S_pred with O; Apply lt_le_trans with (S x). +Apply lt_O_Sn. +Qed. diff --git a/theories7/Reals/SplitAbsolu.v b/theories7/Reals/SplitAbsolu.v new file mode 100644 index 00000000..30580a0c --- /dev/null +++ b/theories7/Reals/SplitAbsolu.v @@ -0,0 +1,22 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: SplitAbsolu.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbasic_fun. + +Recursive Tactic Definition SplitAbs := + Match Context With + | [ |- [(case_Rabsolu ?1)] ] -> + Case (case_Rabsolu ?1); Try SplitAbs. + + +Recursive Tactic Definition SplitAbsolu := + Match Context With + | [ id:[(Rabsolu ?)] |- ? ] -> Generalize id; Clear id;Try SplitAbsolu + | [ |- [(Rabsolu ?1)] ] -> Unfold Rabsolu; Try SplitAbs;Intros. diff --git a/theories7/Reals/SplitRmult.v b/theories7/Reals/SplitRmult.v new file mode 100644 index 00000000..392675c3 --- /dev/null +++ b/theories7/Reals/SplitRmult.v @@ -0,0 +1,19 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: SplitRmult.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +(*i Lemma mult_non_zero :(r1,r2:R)``r1<>0`` /\ ``r2<>0`` -> ``r1*r2<>0``. i*) + + +Require Rbase. + +Recursive Tactic Definition SplitRmult := + Match Context With + | [ |- ~(Rmult ?1 ?2)==R0 ] -> Apply mult_non_zero; Split;Try SplitRmult. + diff --git a/theories7/Reals/Sqrt_reg.v b/theories7/Reals/Sqrt_reg.v new file mode 100644 index 00000000..d2068e5d --- /dev/null +++ b/theories7/Reals/Sqrt_reg.v @@ -0,0 +1,297 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Sqrt_reg.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +Require R_sqrt. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(**********) +Lemma sqrt_var_maj : (h:R) ``(Rabsolu h) <= 1`` -> ``(Rabsolu ((sqrt (1+h))-1))<=(Rabsolu h)``. +Intros; Cut ``0<=1+h``. +Intro; Apply Rle_trans with ``(Rabsolu ((sqrt (Rsqr (1+h)))-1))``. +Case (total_order_T h R0); Intro. +Elim s; Intro. +Repeat Rewrite Rabsolu_left. +Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-1``). +Do 2 Rewrite Ropp_distr1;Rewrite Ropp_Ropp; Apply Rle_compatibility. +Apply Rle_Ropp1; Apply sqrt_le_1. +Apply pos_Rsqr. +Apply H0. +Pattern 2 ``1+h``; Rewrite <- Rmult_1r; Unfold Rsqr; Apply Rle_monotony. +Apply H0. +Pattern 2 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Assumption. +Apply Rlt_anti_compatibility with R1; Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. +Pattern 2 R1; Rewrite <- sqrt_1; Apply sqrt_lt_1. +Apply pos_Rsqr. +Left; Apply Rlt_R0_R1. +Pattern 2 R1; Rewrite <- Rsqr_1; Apply Rsqr_incrst_1. +Pattern 2 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. +Apply H0. +Left; Apply Rlt_R0_R1. +Apply Rlt_anti_compatibility with R1; Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. +Pattern 2 R1; Rewrite <- sqrt_1; Apply sqrt_lt_1. +Apply H0. +Left; Apply Rlt_R0_R1. +Pattern 2 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. +Rewrite b; Rewrite Rplus_Or; Rewrite Rsqr_1; Rewrite sqrt_1; Right; Reflexivity. +Repeat Rewrite Rabsolu_right. +Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-1``); Apply Rle_compatibility. +Apply sqrt_le_1. +Apply H0. +Apply pos_Rsqr. +Pattern 1 ``1+h``; Rewrite <- Rmult_1r; Unfold Rsqr; Apply Rle_monotony. +Apply H0. +Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Assumption. +Apply Rle_sym1; Apply Rle_anti_compatibility with R1. +Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. +Pattern 1 R1; Rewrite <- sqrt_1; Apply sqrt_le_1. +Left; Apply Rlt_R0_R1. +Apply pos_Rsqr. +Pattern 1 R1; Rewrite <- Rsqr_1; Apply Rsqr_incr_1. +Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_R0_R1. +Apply H0. +Apply Rle_sym1; Left; Apply Rlt_anti_compatibility with R1. +Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. +Pattern 1 R1; Rewrite <- sqrt_1; Apply sqrt_lt_1. +Left; Apply Rlt_R0_R1. +Apply H0. +Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. +Rewrite sqrt_Rsqr. +Replace ``(1+h)-1`` with h; [Right; Reflexivity | Ring]. +Apply H0. +Case (total_order_T h R0); Intro. +Elim s; Intro. +Rewrite (Rabsolu_left h a) in H. +Apply Rle_anti_compatibility with ``-h``. +Rewrite Rplus_Or; Rewrite Rplus_sym; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Exact H. +Left; Rewrite b; Rewrite Rplus_Or; Apply Rlt_R0_R1. +Left; Apply gt0_plus_gt0_is_gt0. +Apply Rlt_R0_R1. +Apply r. +Qed. + +(* sqrt is continuous in 1 *) +Lemma sqrt_continuity_pt_R1 : (continuity_pt sqrt R1). +Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Pose alpha := (Rmin eps R1). +Exists alpha; Intros. +Split. +Unfold alpha; Unfold Rmin; Case (total_order_Rle eps R1); Intro. +Assumption. +Apply Rlt_R0_R1. +Intros; Elim H0; Intros. +Rewrite sqrt_1; Replace x with ``1+(x-1)``; [Idtac | Ring]; Apply Rle_lt_trans with ``(Rabsolu (x-1))``. +Apply sqrt_var_maj. +Apply Rle_trans with alpha. +Left; Apply H2. +Unfold alpha; Apply Rmin_r. +Apply Rlt_le_trans with alpha; [Apply H2 | Unfold alpha; Apply Rmin_l]. +Qed. + +(* sqrt is continuous forall x>0 *) +Lemma sqrt_continuity_pt : (x:R) ``0<x`` -> (continuity_pt sqrt x). +Intros; Generalize sqrt_continuity_pt_R1. +Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. +Cut ``0<eps/(sqrt x)``. +Intro; Elim (H0 ? H2); Intros alp_1 H3. +Elim H3; Intros. +Pose alpha := ``alp_1*x``. +Exists (Rmin alpha x); Intros. +Split. +Change ``0<(Rmin alpha x)``; Unfold Rmin; Case (total_order_Rle alpha x); Intro. +Unfold alpha; Apply Rmult_lt_pos; Assumption. +Apply H. +Intros; Replace x0 with ``x+(x0-x)``; [Idtac | Ring]; Replace ``(sqrt (x+(x0-x)))-(sqrt x)`` with ``(sqrt x)*((sqrt (1+(x0-x)/x))-(sqrt 1))``. +Rewrite Rabsolu_mult; Rewrite (Rabsolu_right (sqrt x)). +Apply Rlt_monotony_contra with ``/(sqrt x)``. +Apply Rlt_Rinv; Apply sqrt_lt_R0; Assumption. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Rewrite Rmult_sym. +Unfold Rdiv in H5. +Case (Req_EM x x0); Intro. +Rewrite H7; Unfold Rminus Rdiv; Rewrite Rplus_Ropp_r; Rewrite Rmult_Ol; Rewrite Rplus_Or; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0. +Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv; Rewrite <- H7; Apply sqrt_lt_R0; Assumption. +Apply H5. +Split. +Unfold D_x no_cond. +Split. +Trivial. +Red; Intro. +Cut ``(x0-x)*/x==0``. +Intro. +Elim (without_div_Od ? ? H9); Intro. +Elim H7. +Apply (Rminus_eq_right ? ? H10). +Assert H11 := (without_div_Oi1 ? x H10). +Rewrite <- Rinv_l_sym in H11. +Elim R1_neq_R0; Exact H11. +Red; Intro; Rewrite H12 in H; Elim (Rlt_antirefl ? H). +Symmetry; Apply r_Rplus_plus with R1; Rewrite Rplus_Or; Unfold Rdiv in H8; Exact H8. +Unfold Rminus; Rewrite Rplus_sym; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Elim H6; Intros. +Unfold Rdiv; Rewrite Rabsolu_mult. +Rewrite Rabsolu_Rinv. +Rewrite (Rabsolu_right x). +Rewrite Rmult_sym; Apply Rlt_monotony_contra with x. +Apply H. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite Rmult_sym; Fold alpha. +Apply Rlt_le_trans with (Rmin alpha x). +Apply H9. +Apply Rmin_l. +Red; Intro; Rewrite H10 in H; Elim (Rlt_antirefl ? H). +Apply Rle_sym1; Left; Apply H. +Red; Intro; Rewrite H10 in H; Elim (Rlt_antirefl ? H). +Assert H7 := (sqrt_lt_R0 x H). +Red; Intro; Rewrite H8 in H7; Elim (Rlt_antirefl ? H7). +Apply Rle_sym1; Apply sqrt_positivity. +Left; Apply H. +Unfold Rminus; Rewrite Rmult_Rplus_distr; Rewrite Ropp_mul3; Repeat Rewrite <- sqrt_times. +Rewrite Rmult_1r; Rewrite Rmult_Rplus_distr; Rewrite Rmult_1r; Unfold Rdiv; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Reflexivity. +Red; Intro; Rewrite H7 in H; Elim (Rlt_antirefl ? H). +Left; Apply H. +Left; Apply Rlt_R0_R1. +Left; Apply H. +Elim H6; Intros. +Case (case_Rabsolu ``x0-x``); Intro. +Rewrite (Rabsolu_left ``x0-x`` r) in H8. +Rewrite Rplus_sym. +Apply Rle_anti_compatibility with ``-((x0-x)/x)``. +Rewrite Rplus_Or; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Unfold Rdiv; Rewrite <- Ropp_mul1. +Apply Rle_monotony_contra with x. +Apply H. +Rewrite Rmult_1r; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Left; Apply Rlt_le_trans with (Rmin alpha x). +Apply H8. +Apply Rmin_r. +Red; Intro; Rewrite H9 in H; Elim (Rlt_antirefl ? H). +Apply ge0_plus_ge0_is_ge0. +Left; Apply Rlt_R0_R1. +Unfold Rdiv; Apply Rmult_le_pos. +Apply Rle_sym2; Exact r. +Left; Apply Rlt_Rinv; Apply H. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply H1. +Apply Rlt_Rinv; Apply sqrt_lt_R0; Apply H. +Qed. + +(* sqrt is derivable for all x>0 *) +Lemma derivable_pt_lim_sqrt : (x:R) ``0<x`` -> (derivable_pt_lim sqrt x ``/(2*(sqrt x))``). +Intros; Pose g := [h:R]``(sqrt x)+(sqrt (x+h))``. +Cut (continuity_pt g R0). +Intro; Cut ``(g 0)<>0``. +Intro; Assert H2 := (continuity_pt_inv g R0 H0 H1). +Unfold derivable_pt_lim; Intros; Unfold continuity_pt in H2; Unfold continue_in in H2; Unfold limit1_in in H2; Unfold limit_in in H2; Simpl in H2; Unfold R_dist in H2. +Elim (H2 eps H3); Intros alpha H4. +Elim H4; Intros. +Pose alpha1 := (Rmin alpha x). +Cut ``0<alpha1``. +Intro; Exists (mkposreal alpha1 H7); Intros. +Replace ``((sqrt (x+h))-(sqrt x))/h`` with ``/((sqrt x)+(sqrt (x+h)))``. +Unfold inv_fct g in H6; Replace ``2*(sqrt x)`` with ``(sqrt x)+(sqrt (x+0))``. +Apply H6. +Split. +Unfold D_x no_cond. +Split. +Trivial. +Apply not_sym; Exact H8. +Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rlt_le_trans with alpha1. +Exact H9. +Unfold alpha1; Apply Rmin_l. +Rewrite Rplus_Or; Ring. +Cut ``0<=x+h``. +Intro; Cut ``0<(sqrt x)+(sqrt (x+h))``. +Intro; Apply r_Rmult_mult with ``((sqrt x)+(sqrt (x+h)))``. +Rewrite <- Rinv_r_sym. +Rewrite Rplus_sym; Unfold Rdiv; Rewrite <- Rmult_assoc; Rewrite Rsqr_plus_minus; Repeat Rewrite Rsqr_sqrt. +Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Rewrite <- Rinv_r_sym. +Reflexivity. +Apply H8. +Left; Apply H. +Assumption. +Red; Intro; Rewrite H12 in H11; Elim (Rlt_antirefl ? H11). +Red; Intro; Rewrite H12 in H11; Elim (Rlt_antirefl ? H11). +Apply gt0_plus_ge0_is_gt0. +Apply sqrt_lt_R0; Apply H. +Apply sqrt_positivity; Apply H10. +Case (case_Rabsolu h); Intro. +Rewrite (Rabsolu_left h r) in H9. +Apply Rle_anti_compatibility with ``-h``. +Rewrite Rplus_Or; Rewrite Rplus_sym; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Left; Apply Rlt_le_trans with alpha1. +Apply H9. +Unfold alpha1; Apply Rmin_r. +Apply ge0_plus_ge0_is_ge0. +Left; Assumption. +Apply Rle_sym2; Apply r. +Unfold alpha1; Unfold Rmin; Case (total_order_Rle alpha x); Intro. +Apply H5. +Apply H. +Unfold g; Rewrite Rplus_Or. +Cut ``0<(sqrt x)+(sqrt x)``. +Intro; Red; Intro; Rewrite H2 in H1; Elim (Rlt_antirefl ? H1). +Apply gt0_plus_gt0_is_gt0; Apply sqrt_lt_R0; Apply H. +Replace g with (plus_fct (fct_cte (sqrt x)) (comp sqrt (plus_fct (fct_cte x) id))); [Idtac | Reflexivity]. +Apply continuity_pt_plus. +Apply continuity_pt_const; Unfold constant fct_cte; Intro; Reflexivity. +Apply continuity_pt_comp. +Apply continuity_pt_plus. +Apply continuity_pt_const; Unfold constant fct_cte; Intro; Reflexivity. +Apply derivable_continuous_pt; Apply derivable_pt_id. +Apply sqrt_continuity_pt. +Unfold plus_fct fct_cte id; Rewrite Rplus_Or; Apply H. +Qed. + +(**********) +Lemma derivable_pt_sqrt : (x:R) ``0<x`` -> (derivable_pt sqrt x). +Unfold derivable_pt; Intros. +Apply Specif.existT with ``/(2*(sqrt x))``. +Apply derivable_pt_lim_sqrt; Assumption. +Qed. + +(**********) +Lemma derive_pt_sqrt : (x:R;pr:``0<x``) ``(derive_pt sqrt x (derivable_pt_sqrt ? pr)) == /(2*(sqrt x))``. +Intros. +Apply derive_pt_eq_0. +Apply derivable_pt_lim_sqrt; Assumption. +Qed. + +(* We show that sqrt is continuous for all x>=0 *) +(* Remark : by definition of sqrt (as extension of Rsqrt on |R), *) +(* we could also show that sqrt is continuous for all x *) +Lemma continuity_pt_sqrt : (x:R) ``0<=x`` -> (continuity_pt sqrt x). +Intros; Case (total_order R0 x); Intro. +Apply (sqrt_continuity_pt x H0). +Elim H0; Intro. +Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Exists (Rsqr eps); Intros. +Split. +Change ``0<(Rsqr eps)``; Apply Rsqr_pos_lt. +Red; Intro; Rewrite H3 in H2; Elim (Rlt_antirefl ? H2). +Intros; Elim H3; Intros. +Rewrite <- H1; Rewrite sqrt_0; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite <- H1 in H5; Unfold Rminus in H5; Rewrite Ropp_O in H5; Rewrite Rplus_Or in H5. +Case (case_Rabsolu x0); Intro. +Unfold sqrt; Case (case_Rabsolu x0); Intro. +Rewrite Rabsolu_R0; Apply H2. +Assert H6 := (Rle_sym2 ? ? r0); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H6 r)). +Rewrite Rabsolu_right. +Apply Rsqr_incrst_0. +Rewrite Rsqr_sqrt. +Rewrite (Rabsolu_right x0 r) in H5; Apply H5. +Apply Rle_sym2; Exact r. +Apply sqrt_positivity; Apply Rle_sym2; Exact r. +Left; Exact H2. +Apply Rle_sym1; Apply sqrt_positivity; Apply Rle_sym2; Exact r. +Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H1 H)). +Qed. diff --git a/theories7/Relations/Newman.v b/theories7/Relations/Newman.v new file mode 100755 index 00000000..c53db971 --- /dev/null +++ b/theories7/Relations/Newman.v @@ -0,0 +1,115 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Newman.v,v 1.1.2.1 2004/07/16 19:31:37 herbelin Exp $ i*) + +Require Rstar. + +Section Newman. + +Variable A: Type. +Variable R: A->A->Prop. + +Local Rstar := (Rstar A R). +Local Rstar_reflexive := (Rstar_reflexive A R). +Local Rstar_transitive := (Rstar_transitive A R). +Local Rstar_Rstar' := (Rstar_Rstar' A R). + +Definition coherence := [x:A][y:A] (exT2 ? (Rstar x) (Rstar y)). + +Theorem coherence_intro : (x:A)(y:A)(z:A)(Rstar x z)->(Rstar y z)->(coherence x y). +Proof [x:A][y:A][z:A][h1:(Rstar x z)][h2:(Rstar y z)] + (exT_intro2 A (Rstar x) (Rstar y) z h1 h2). + +(** A very simple case of coherence : *) + +Lemma Rstar_coherence : (x:A)(y:A)(Rstar x y)->(coherence x y). + Proof [x:A][y:A][h:(Rstar x y)](coherence_intro x y y h (Rstar_reflexive y)). + +(** coherence is symmetric *) +Lemma coherence_sym: (x:A)(y:A)(coherence x y)->(coherence y x). + Proof [x:A][y:A][h:(coherence x y)] + (exT2_ind A (Rstar x) (Rstar y) (coherence y x) + [w:A][h1:(Rstar x w)][h2:(Rstar y w)] + (coherence_intro y x w h2 h1) h). + +Definition confluence := + [x:A](y:A)(z:A)(Rstar x y)->(Rstar x z)->(coherence y z). + +Definition local_confluence := + [x:A](y:A)(z:A)(R x y)->(R x z)->(coherence y z). + +Definition noetherian := + (x:A)(P:A->Prop)((y:A)((z:A)(R y z)->(P z))->(P y))->(P x). + +Section Newman_section. + +(** The general hypotheses of the theorem *) + +Hypothesis Hyp1:noetherian. +Hypothesis Hyp2:(x:A)(local_confluence x). + +(** The induction hypothesis *) + +Section Induct. + Variable x:A. + Hypothesis hyp_ind:(u:A)(R x u)->(confluence u). + +(** Confluence in [x] *) + + Variables y,z:A. + Hypothesis h1:(Rstar x y). + Hypothesis h2:(Rstar x z). + +(** particular case [x->u] and [u->*y] *) +Section Newman_. + Variable u:A. + Hypothesis t1:(R x u). + Hypothesis t2:(Rstar u y). + +(** In the usual diagram, we assume also [x->v] and [v->*z] *) + +Theorem Diagram : (v:A)(u1:(R x v))(u2:(Rstar v z))(coherence y z). + +Proof (* We draw the diagram ! *) + [v:A][u1:(R x v)][u2:(Rstar v z)] + (exT2_ind A (Rstar u) (Rstar v) (* local confluence in x for u,v *) + (coherence y z) (* gives w, u->*w and v->*w *) + ([w:A][s1:(Rstar u w)][s2:(Rstar v w)] + (exT2_ind A (Rstar y) (Rstar w) (* confluence in u => coherence(y,w) *) + (coherence y z) (* gives a, y->*a and z->*a *) + ([a:A][v1:(Rstar y a)][v2:(Rstar w a)] + (exT2_ind A (Rstar a) (Rstar z) (* confluence in v => coherence(a,z) *) + (coherence y z) (* gives b, a->*b and z->*b *) + ([b:A][w1:(Rstar a b)][w2:(Rstar z b)] + (coherence_intro y z b (Rstar_transitive y a b v1 w1) w2)) + (hyp_ind v u1 a z (Rstar_transitive v w a s2 v2) u2))) + (hyp_ind u t1 y w t2 s1))) + (Hyp2 x u v t1 u1)). + +Theorem caseRxy : (coherence y z). +Proof (Rstar_Rstar' x z h2 + ([v:A][w:A](coherence y w)) + (coherence_sym x y (Rstar_coherence x y h1)) (*i case x=z i*) + Diagram). (*i case x->v->*z i*) +End Newman_. + +Theorem Ind_proof : (coherence y z). +Proof (Rstar_Rstar' x y h1 ([u:A][v:A](coherence v z)) + (Rstar_coherence x z h2) (*i case x=y i*) + caseRxy). (*i case x->u->*z i*) +End Induct. + +Theorem Newman : (x:A)(confluence x). +Proof [x:A](Hyp1 x confluence Ind_proof). + +End Newman_section. + + +End Newman. + diff --git a/theories7/Relations/Operators_Properties.v b/theories7/Relations/Operators_Properties.v new file mode 100755 index 00000000..4f1818bc --- /dev/null +++ b/theories7/Relations/Operators_Properties.v @@ -0,0 +1,98 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Operators_Properties.v,v 1.1.2.1 2004/07/16 19:31:37 herbelin Exp $ i*) + +(****************************************************************************) +(* Bruno Barras *) +(****************************************************************************) + +Require Relation_Definitions. +Require Relation_Operators. + + +Section Properties. + + Variable A: Set. + Variable R: (relation A). + + Local incl : (relation A)->(relation A)->Prop := + [R1,R2: (relation A)] (x,y:A) (R1 x y) -> (R2 x y). + +Section Clos_Refl_Trans. + + Lemma clos_rt_is_preorder: (preorder A (clos_refl_trans A R)). +Apply Build_preorder. +Exact (rt_refl A R). + +Exact (rt_trans A R). +Qed. + + + +Lemma clos_rt_idempotent: + (incl (clos_refl_trans A (clos_refl_trans A R)) + (clos_refl_trans A R)). +Red. +NewInduction 1; Auto with sets. +Intros. +Apply rt_trans with y; Auto with sets. +Qed. + + Lemma clos_refl_trans_ind_left: (A:Set)(R:A->A->Prop)(M:A)(P:A->Prop) + (P M) + ->((P0,N:A) + (clos_refl_trans A R M P0)->(P P0)->(R P0 N)->(P N)) + ->(a:A)(clos_refl_trans A R M a)->(P a). +Intros. +Generalize H H0 . +Clear H H0. +Elim H1; Intros; Auto with sets. +Apply H2 with x; Auto with sets. + +Apply H3. +Apply H0; Auto with sets. + +Intros. +Apply H5 with P0; Auto with sets. +Apply rt_trans with y; Auto with sets. +Qed. + + +End Clos_Refl_Trans. + + +Section Clos_Refl_Sym_Trans. + + Lemma clos_rt_clos_rst: (inclusion A (clos_refl_trans A R) + (clos_refl_sym_trans A R)). +Red. +NewInduction 1; Auto with sets. +Apply rst_trans with y; Auto with sets. +Qed. + + Lemma clos_rst_is_equiv: (equivalence A (clos_refl_sym_trans A R)). +Apply Build_equivalence. +Exact (rst_refl A R). + +Exact (rst_trans A R). + +Exact (rst_sym A R). +Qed. + + Lemma clos_rst_idempotent: + (incl (clos_refl_sym_trans A (clos_refl_sym_trans A R)) + (clos_refl_sym_trans A R)). +Red. +NewInduction 1; Auto with sets. +Apply rst_trans with y; Auto with sets. +Qed. + +End Clos_Refl_Sym_Trans. + +End Properties. diff --git a/theories7/Relations/Relation_Definitions.v b/theories7/Relations/Relation_Definitions.v new file mode 100755 index 00000000..1e38e753 --- /dev/null +++ b/theories7/Relations/Relation_Definitions.v @@ -0,0 +1,83 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Relation_Definitions.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +Section Relation_Definition. + + Variable A: Type. + + Definition relation := A -> A -> Prop. + + Variable R: relation. + + +Section General_Properties_of_Relations. + + Definition reflexive : Prop := (x: A) (R x x). + Definition transitive : Prop := (x,y,z: A) (R x y) -> (R y z) -> (R x z). + Definition symmetric : Prop := (x,y: A) (R x y) -> (R y x). + Definition antisymmetric : Prop := (x,y: A) (R x y) -> (R y x) -> x=y. + + (* for compatibility with Equivalence in ../PROGRAMS/ALG/ *) + Definition equiv := reflexive /\ transitive /\ symmetric. + +End General_Properties_of_Relations. + + + +Section Sets_of_Relations. + + Record preorder : Prop := { + preord_refl : reflexive; + preord_trans : transitive }. + + Record order : Prop := { + ord_refl : reflexive; + ord_trans : transitive; + ord_antisym : antisymmetric }. + + Record equivalence : Prop := { + equiv_refl : reflexive; + equiv_trans : transitive; + equiv_sym : symmetric }. + + Record PER : Prop := { + per_sym : symmetric; + per_trans : transitive }. + +End Sets_of_Relations. + + + +Section Relations_of_Relations. + + Definition inclusion : relation -> relation -> Prop := + [R1,R2: relation] (x,y:A) (R1 x y) -> (R2 x y). + + Definition same_relation : relation -> relation -> Prop := + [R1,R2: relation] (inclusion R1 R2) /\ (inclusion R2 R1). + + Definition commut : relation -> relation -> Prop := + [R1,R2:relation] (x,y:A) (R1 y x) -> (z:A) (R2 z y) + -> (EX y':A |(R2 y' x) & (R1 z y')). + +End Relations_of_Relations. + + +End Relation_Definition. + +Hints Unfold reflexive transitive antisymmetric symmetric : sets v62. + +Hints Resolve Build_preorder Build_order Build_equivalence + Build_PER preord_refl preord_trans + ord_refl ord_trans ord_antisym + equiv_refl equiv_trans equiv_sym + per_sym per_trans : sets v62. + +Hints Unfold inclusion same_relation commut : sets v62. diff --git a/theories7/Relations/Relation_Operators.v b/theories7/Relations/Relation_Operators.v new file mode 100755 index 00000000..14c2ae30 --- /dev/null +++ b/theories7/Relations/Relation_Operators.v @@ -0,0 +1,157 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Relation_Operators.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +(****************************************************************************) +(* Bruno Barras, Cristina Cornes *) +(* *) +(* Some of these definitons were taken from : *) +(* Constructing Recursion Operators in Type Theory *) +(* L. Paulson JSC (1986) 2, 325-355 *) +(****************************************************************************) + +Require Relation_Definitions. +Require PolyList. +Require PolyListSyntax. + +(** Some operators to build relations *) + +Section Transitive_Closure. + Variable A: Set. + Variable R: (relation A). + + Inductive clos_trans : A->A->Prop := + t_step: (x,y:A)(R x y)->(clos_trans x y) + | t_trans: (x,y,z:A)(clos_trans x y)->(clos_trans y z)->(clos_trans x z). +End Transitive_Closure. + + +Section Reflexive_Transitive_Closure. + Variable A: Set. + Variable R: (relation A). + + Inductive clos_refl_trans: (relation A) := + rt_step: (x,y:A)(R x y)->(clos_refl_trans x y) + | rt_refl: (x:A)(clos_refl_trans x x) + | rt_trans: (x,y,z: A)(clos_refl_trans x y)->(clos_refl_trans y z) + ->(clos_refl_trans x z). +End Reflexive_Transitive_Closure. + + +Section Reflexive_Symetric_Transitive_Closure. + Variable A: Set. + Variable R: (relation A). + + Inductive clos_refl_sym_trans: (relation A) := + rst_step: (x,y:A)(R x y)->(clos_refl_sym_trans x y) + | rst_refl: (x:A)(clos_refl_sym_trans x x) + | rst_sym: (x,y:A)(clos_refl_sym_trans x y)->(clos_refl_sym_trans y x) + | rst_trans: (x,y,z:A)(clos_refl_sym_trans x y)->(clos_refl_sym_trans y z) + ->(clos_refl_sym_trans x z). +End Reflexive_Symetric_Transitive_Closure. + + +Section Transposee. + Variable A: Set. + Variable R: (relation A). + + Definition transp := [x,y:A](R y x). +End Transposee. + + +Section Union. + Variable A: Set. + Variable R1,R2: (relation A). + + Definition union := [x,y:A](R1 x y)\/(R2 x y). +End Union. + + +Section Disjoint_Union. +Variable A,B:Set. +Variable leA: A->A->Prop. +Variable leB: B->B->Prop. + +Inductive le_AsB : A+B->A+B->Prop := + le_aa: (x,y:A) (leA x y) -> (le_AsB (inl A B x) (inl A B y)) +| le_ab: (x:A)(y:B) (le_AsB (inl A B x) (inr A B y)) +| le_bb: (x,y:B) (leB x y) -> (le_AsB (inr A B x) (inr A B y)). + +End Disjoint_Union. + + + +Section Lexicographic_Product. +(* Lexicographic order on dependent pairs *) + +Variable A:Set. +Variable B:A->Set. +Variable leA: A->A->Prop. +Variable leB: (x:A)(B x)->(B x)->Prop. + +Inductive lexprod : (sigS A B) -> (sigS A B) ->Prop := + left_lex : (x,x':A)(y:(B x)) (y':(B x')) + (leA x x') ->(lexprod (existS A B x y) (existS A B x' y')) +| right_lex : (x:A) (y,y':(B x)) + (leB x y y') -> (lexprod (existS A B x y) (existS A B x y')). +End Lexicographic_Product. + + +Section Symmetric_Product. + Variable A:Set. + Variable B:Set. + Variable leA: A->A->Prop. + Variable leB: B->B->Prop. + + Inductive symprod : (A*B) -> (A*B) ->Prop := + left_sym : (x,x':A)(leA x x')->(y:B)(symprod (x,y) (x',y)) + | right_sym : (y,y':B)(leB y y')->(x:A)(symprod (x,y) (x,y')). + +End Symmetric_Product. + + +Section Swap. + Variable A:Set. + Variable R:A->A->Prop. + + Inductive swapprod: (A*A)->(A*A)->Prop := + sp_noswap: (x,x':A*A)(symprod A A R R x x')->(swapprod x x') + | sp_swap: (x,y:A)(p:A*A)(symprod A A R R (x,y) p)->(swapprod (y,x) p). +End Swap. + + +Section Lexicographic_Exponentiation. + +Variable A : Set. +Variable leA : A->A->Prop. +Local Nil := (nil A). +Local List := (list A). + +Inductive Ltl : List->List->Prop := + Lt_nil: (a:A)(x:List)(Ltl Nil (cons a x)) +| Lt_hd : (a,b:A) (leA a b)-> (x,y:(list A))(Ltl (cons a x) (cons b y)) +| Lt_tl : (a:A)(x,y:List)(Ltl x y) -> (Ltl (cons a x) (cons a y)). + + +Inductive Desc : List->Prop := + d_nil : (Desc Nil) +| d_one : (x:A)(Desc (cons x Nil)) +| d_conc : (x,y:A)(l:List)(leA x y) + -> (Desc l^(cons y Nil))->(Desc (l^(cons y Nil))^(cons x Nil)). + +Definition Pow :Set := (sig List Desc). + +Definition lex_exp : Pow -> Pow ->Prop := + [a,b:Pow](Ltl (proj1_sig List Desc a) (proj1_sig List Desc b)). + +End Lexicographic_Exponentiation. + +Hints Unfold transp union : sets v62. +Hints Resolve t_step rt_step rt_refl rst_step rst_refl : sets v62. +Hints Immediate rst_sym : sets v62. diff --git a/theories7/Relations/Relations.v b/theories7/Relations/Relations.v new file mode 100755 index 00000000..694d0eec --- /dev/null +++ b/theories7/Relations/Relations.v @@ -0,0 +1,28 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Relations.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +Require Export Relation_Definitions. +Require Export Relation_Operators. +Require Export Operators_Properties. + +Lemma inverse_image_of_equivalence : (A,B:Set)(f:A->B) + (r:(relation B))(equivalence B r)->(equivalence A [x,y:A](r (f x) (f y))). +Intros; Split; Elim H; Red; Auto. +Intros _ equiv_trans _ x y z H0 H1; Apply equiv_trans with (f y); Assumption. +Qed. + +Lemma inverse_image_of_eq : (A,B:Set)(f:A->B) + (equivalence A [x,y:A](f x)=(f y)). +Split; Red; +[ (* reflexivity *) Reflexivity +| (* transitivity *) Intros; Transitivity (f y); Assumption +| (* symmetry *) Intros; Symmetry; Assumption +]. +Qed. diff --git a/theories7/Relations/Rstar.v b/theories7/Relations/Rstar.v new file mode 100755 index 00000000..3747b45e --- /dev/null +++ b/theories7/Relations/Rstar.v @@ -0,0 +1,78 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rstar.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +(** Properties of a binary relation [R] on type [A] *) + +Section Rstar. + +Variable A : Type. +Variable R : A->A->Prop. + +(** Definition of the reflexive-transitive closure [R*] of [R] *) +(** Smallest reflexive [P] containing [R o P] *) + +Definition Rstar := [x,y:A](P:A->A->Prop) + ((u:A)(P u u))->((u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)) -> (P x y). + +Theorem Rstar_reflexive: (x:A)(Rstar x x). + Proof [x:A][P:A->A->Prop] + [h1:(u:A)(P u u)][h2:(u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)] + (h1 x). + +Theorem Rstar_R: (x:A)(y:A)(z:A)(R x y)->(Rstar y z)->(Rstar x z). + Proof [x:A][y:A][z:A][t1:(R x y)][t2:(Rstar y z)] + [P:A->A->Prop] + [h1:(u:A)(P u u)][h2:(u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)] + (h2 x y z t1 (t2 P h1 h2)). + +(** We conclude with transitivity of [Rstar] : *) + +Theorem Rstar_transitive: (x:A)(y:A)(z:A)(Rstar x y)->(Rstar y z)->(Rstar x z). + Proof [x:A][y:A][z:A][h:(Rstar x y)] + (h ([u:A][v:A](Rstar v z)->(Rstar u z)) + ([u:A][t:(Rstar u z)]t) + ([u:A][v:A][w:A][t1:(R u v)][t2:(Rstar w z)->(Rstar v z)] + [t3:(Rstar w z)](Rstar_R u v z t1 (t2 t3)))). + +(** Another characterization of [R*] *) +(** Smallest reflexive [P] containing [R o R*] *) + +Definition Rstar' := [x:A][y:A](P:A->A->Prop) + ((P x x))->((u:A)(R x u)->(Rstar u y)->(P x y)) -> (P x y). + +Theorem Rstar'_reflexive: (x:A)(Rstar' x x). + Proof [x:A][P:A->A->Prop][h:(P x x)][h':(u:A)(R x u)->(Rstar u x)->(P x x)]h. + +Theorem Rstar'_R: (x:A)(y:A)(z:A)(R x z)->(Rstar z y)->(Rstar' x y). + Proof [x:A][y:A][z:A][t1:(R x z)][t2:(Rstar z y)] + [P:A->A->Prop][h1:(P x x)] + [h2:(u:A)(R x u)->(Rstar u y)->(P x y)](h2 z t1 t2). + +(** Equivalence of the two definitions: *) + +Theorem Rstar'_Rstar: (x:A)(y:A)(Rstar' x y)->(Rstar x y). + Proof [x:A][y:A][h:(Rstar' x y)] + (h Rstar (Rstar_reflexive x) ([u:A](Rstar_R x u y))). + +Theorem Rstar_Rstar': (x:A)(y:A)(Rstar x y)->(Rstar' x y). + Proof [x:A][y:A][h:(Rstar x y)](h Rstar' ([u:A](Rstar'_reflexive u)) + ([u:A][v:A][w:A][h1:(R u v)][h2:(Rstar' v w)] + (Rstar'_R u w v h1 (Rstar'_Rstar v w h2)))). + + +(** Property of Commutativity of two relations *) + +Definition commut := [A:Set][R1,R2:A->A->Prop] + (x,y:A)(R1 y x)->(z:A)(R2 z y) + ->(EX y':A |(R2 y' x) & (R1 z y')). + + +End Rstar. + diff --git a/theories7/Setoids/Setoid.v b/theories7/Setoids/Setoid.v new file mode 100644 index 00000000..f8176f60 --- /dev/null +++ b/theories7/Setoids/Setoid.v @@ -0,0 +1,73 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Setoid.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $: i*) + +Section Setoid. + +Variable A : Type. +Variable Aeq : A -> A -> Prop. + +Record Setoid_Theory : Prop := +{ Seq_refl : (x:A) (Aeq x x); + Seq_sym : (x,y:A) (Aeq x y) -> (Aeq y x); + Seq_trans : (x,y,z:A) (Aeq x y) -> (Aeq y z) -> (Aeq x z) +}. + +End Setoid. + +Definition Prop_S : (Setoid_Theory Prop iff). +Split; [Exact iff_refl | Exact iff_sym | Exact iff_trans]. +Qed. + +Add Setoid Prop iff Prop_S. + +Hint prop_set : setoid := Resolve (Seq_refl Prop iff Prop_S). +Hint prop_set : setoid := Resolve (Seq_sym Prop iff Prop_S). +Hint prop_set : setoid := Resolve (Seq_trans Prop iff Prop_S). + +Add Morphism or : or_ext. +Intros. +Inversion H1. +Left. +Inversion H. +Apply (H3 H2). + +Right. +Inversion H0. +Apply (H3 H2). +Qed. + +Add Morphism and : and_ext. +Intros. +Inversion H1. +Split. +Inversion H. +Apply (H4 H2). + +Inversion H0. +Apply (H4 H3). +Qed. + +Add Morphism not : not_ext. +Red ; Intros. +Apply H0. +Inversion H. +Apply (H3 H1). +Qed. + +Definition fleche [A,B:Prop] := A -> B. + +Add Morphism fleche : fleche_ext. +Unfold fleche. +Intros. +Inversion H0. +Inversion H. +Apply (H3 (H1 (H6 H2))). +Qed. + diff --git a/theories7/Sets/Classical_sets.v b/theories7/Sets/Classical_sets.v new file mode 100755 index 00000000..a6928ffd --- /dev/null +++ b/theories7/Sets/Classical_sets.v @@ -0,0 +1,133 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Classical_sets.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +Require Export Ensembles. +Require Export Constructive_sets. +Require Export Classical_Type. + +(* Hints Unfold not . *) + +Section Ensembles_classical. +Variable U: Type. + +Lemma not_included_empty_Inhabited: + (A: (Ensemble U)) ~ (Included U A (Empty_set U)) -> (Inhabited U A). +Proof. +Intros A NI. +Elim (not_all_ex_not U [x:U]~(In U A x)). +Intros x H; Apply Inhabited_intro with x. +Apply NNPP; Auto with sets. +Red; Intro. +Apply NI; Red. +Intros x H'; Elim (H x); Trivial with sets. +Qed. +Hints Resolve not_included_empty_Inhabited. + +Lemma not_empty_Inhabited: + (A: (Ensemble U)) ~ A == (Empty_set U) -> (Inhabited U A). +Proof. +Intros; Apply not_included_empty_Inhabited. +Red; Auto with sets. +Qed. + +Lemma Inhabited_Setminus : +(X, Y: (Ensemble U)) (Included U X Y) -> ~ (Included U Y X) -> + (Inhabited U (Setminus U Y X)). +Proof. +Intros X Y I NI. +Elim (not_all_ex_not U [x:U](In U Y x)->(In U X x) NI). +Intros x YX. +Apply Inhabited_intro with x. +Apply Setminus_intro. +Apply not_imply_elim with (In U X x); Trivial with sets. +Auto with sets. +Qed. +Hints Resolve Inhabited_Setminus. + +Lemma Strict_super_set_contains_new_element: + (X, Y: (Ensemble U)) (Included U X Y) -> ~ X == Y -> + (Inhabited U (Setminus U Y X)). +Proof. +Auto 7 with sets. +Qed. +Hints Resolve Strict_super_set_contains_new_element. + +Lemma Subtract_intro: + (A: (Ensemble U)) (x, y: U) (In U A y) -> ~ x == y -> + (In U (Subtract U A x) y). +Proof. +Unfold 1 Subtract; Auto with sets. +Qed. +Hints Resolve Subtract_intro. + +Lemma Subtract_inv: + (A: (Ensemble U)) (x, y: U) (In U (Subtract U A x) y) -> + (In U A y) /\ ~ x == y. +Proof. +Intros A x y H'; Elim H'; Auto with sets. +Qed. + +Lemma Included_Strict_Included: + (X, Y: (Ensemble U)) (Included U X Y) -> (Strict_Included U X Y) \/ X == Y. +Proof. +Intros X Y H'; Try Assumption. +Elim (classic X == Y); Auto with sets. +Qed. + +Lemma Strict_Included_inv: + (X, Y: (Ensemble U)) (Strict_Included U X Y) -> + (Included U X Y) /\ (Inhabited U (Setminus U Y X)). +Proof. +Intros X Y H'; Red in H'. +Split; [Tauto | Idtac]. +Elim H'; Intros H'0 H'1; Try Exact H'1; Clear H'. +Apply Strict_super_set_contains_new_element; Auto with sets. +Qed. + +Lemma not_SIncl_empty: + (X: (Ensemble U)) ~ (Strict_Included U X (Empty_set U)). +Proof. +Intro X; Red; Intro H'; Try Exact H'. +LApply (Strict_Included_inv X (Empty_set U)); Auto with sets. +Intro H'0; Elim H'0; Intros H'1 H'2; Elim H'2; Clear H'0. +Intros x H'0; Elim H'0. +Intro H'3; Elim H'3. +Qed. + +Lemma Complement_Complement : + (A: (Ensemble U)) (Complement U (Complement U A)) == A. +Proof. +Unfold Complement; Intros; Apply Extensionality_Ensembles; Auto with sets. +Red; Split; Auto with sets. +Red; Intros; Apply NNPP; Auto with sets. +Qed. + +End Ensembles_classical. + +Hints Resolve Strict_super_set_contains_new_element Subtract_intro + not_SIncl_empty : sets v62. diff --git a/theories7/Sets/Constructive_sets.v b/theories7/Sets/Constructive_sets.v new file mode 100755 index 00000000..35c88e9d --- /dev/null +++ b/theories7/Sets/Constructive_sets.v @@ -0,0 +1,162 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Constructive_sets.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +Require Export Ensembles. + +Section Ensembles_facts. +Variable U: Type. + +Lemma Extension: (B, C: (Ensemble U)) B == C -> (Same_set U B C). +Proof. +Intros B C H'; Rewrite H'; Auto with sets. +Qed. + +Lemma Noone_in_empty: (x: U) ~ (In U (Empty_set U) x). +Proof. +Red; NewDestruct 1. +Qed. +Hints Resolve Noone_in_empty. + +Lemma Included_Empty: (A: (Ensemble U))(Included U (Empty_set U) A). +Proof. +Intro; Red. +Intros x H; Elim (Noone_in_empty x); Auto with sets. +Qed. +Hints Resolve Included_Empty. + +Lemma Add_intro1: + (A: (Ensemble U)) (x, y: U) (In U A y) -> (In U (Add U A x) y). +Proof. +Unfold 1 Add; Auto with sets. +Qed. +Hints Resolve Add_intro1. + +Lemma Add_intro2: (A: (Ensemble U)) (x: U) (In U (Add U A x) x). +Proof. +Unfold 1 Add; Auto with sets. +Qed. +Hints Resolve Add_intro2. + +Lemma Inhabited_add: (A: (Ensemble U)) (x: U) (Inhabited U (Add U A x)). +Proof. +Intros A x. +Apply Inhabited_intro with x := x; Auto with sets. +Qed. +Hints Resolve Inhabited_add. + +Lemma Inhabited_not_empty: + (X: (Ensemble U)) (Inhabited U X) -> ~ X == (Empty_set U). +Proof. +Intros X H'; Elim H'. +Intros x H'0; Red; Intro H'1. +Absurd (In U X x); Auto with sets. +Rewrite H'1; Auto with sets. +Qed. +Hints Resolve Inhabited_not_empty. + +Lemma Add_not_Empty : + (A: (Ensemble U)) (x: U) ~ (Add U A x) == (Empty_set U). +Proof. +Auto with sets. +Qed. +Hints Resolve Add_not_Empty. + +Lemma not_Empty_Add : + (A: (Ensemble U)) (x: U) ~ (Empty_set U) == (Add U A x). +Proof. +Intros; Red; Intro H; Generalize (Add_not_Empty A x); Auto with sets. +Qed. +Hints Resolve not_Empty_Add. + +Lemma Singleton_inv: (x, y: U) (In U (Singleton U x) y) -> x == y. +Proof. +Intros x y H'; Elim H'; Trivial with sets. +Qed. +Hints Resolve Singleton_inv. + +Lemma Singleton_intro: (x, y: U) x == y -> (In U (Singleton U x) y). +Proof. +Intros x y H'; Rewrite H'; Trivial with sets. +Qed. +Hints Resolve Singleton_intro. + +Lemma Union_inv: (B, C: (Ensemble U)) (x: U) + (In U (Union U B C) x) -> (In U B x) \/ (In U C x). +Proof. +Intros B C x H'; Elim H'; Auto with sets. +Qed. + +Lemma Add_inv: + (A: (Ensemble U)) (x, y: U) (In U (Add U A x) y) -> (In U A y) \/ x == y. +Proof. +Intros A x y H'; Elim H'; Auto with sets. +Qed. + +Lemma Intersection_inv: + (B, C: (Ensemble U)) (x: U) (In U (Intersection U B C) x) -> + (In U B x) /\ (In U C x). +Proof. +Intros B C x H'; Elim H'; Auto with sets. +Qed. +Hints Resolve Intersection_inv. + +Lemma Couple_inv: (x, y, z: U) (In U (Couple U x y) z) -> z == x \/ z == y. +Proof. +Intros x y z H'; Elim H'; Auto with sets. +Qed. +Hints Resolve Couple_inv. + +Lemma Setminus_intro: + (A, B: (Ensemble U)) (x: U) (In U A x) -> ~ (In U B x) -> + (In U (Setminus U A B) x). +Proof. +Unfold 1 Setminus; Red; Auto with sets. +Qed. +Hints Resolve Setminus_intro. + +Lemma Strict_Included_intro: + (X, Y: (Ensemble U)) (Included U X Y) /\ ~ X == Y -> + (Strict_Included U X Y). +Proof. +Auto with sets. +Qed. +Hints Resolve Strict_Included_intro. + +Lemma Strict_Included_strict: (X: (Ensemble U)) ~ (Strict_Included U X X). +Proof. +Intro X; Red; Intro H'; Elim H'. +Intros H'0 H'1; Elim H'1; Auto with sets. +Qed. +Hints Resolve Strict_Included_strict. + +End Ensembles_facts. + +Hints Resolve Singleton_inv Singleton_intro Add_intro1 Add_intro2 + Intersection_inv Couple_inv Setminus_intro Strict_Included_intro + Strict_Included_strict Noone_in_empty Inhabited_not_empty + Add_not_Empty not_Empty_Add Inhabited_add Included_Empty : sets v62. diff --git a/theories7/Sets/Cpo.v b/theories7/Sets/Cpo.v new file mode 100755 index 00000000..2fe46be6 --- /dev/null +++ b/theories7/Sets/Cpo.v @@ -0,0 +1,107 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Cpo.v,v 1.1.2.1 2004/07/16 19:31:38 herbelin Exp $ i*) + +Require Export Ensembles. +Require Export Relations_1. +Require Export Partial_Order. + +Section Bounds. +Variable U: Type. +Variable D: (PO U). + +Local C := (Carrier_of U D). + +Local R := (Rel_of U D). + +Inductive Upper_Bound [B:(Ensemble U); x:U]: Prop := + Upper_Bound_definition: + (In U C x) -> ((y: U) (In U B y) -> (R y x)) -> (Upper_Bound B x). + +Inductive Lower_Bound [B:(Ensemble U); x:U]: Prop := + Lower_Bound_definition: + (In U C x) -> ((y: U) (In U B y) -> (R x y)) -> (Lower_Bound B x). + +Inductive Lub [B:(Ensemble U); x:U]: Prop := + Lub_definition: + (Upper_Bound B x) -> ((y: U) (Upper_Bound B y) -> (R x y)) -> (Lub B x). + +Inductive Glb [B:(Ensemble U); x:U]: Prop := + Glb_definition: + (Lower_Bound B x) -> ((y: U) (Lower_Bound B y) -> (R y x)) -> (Glb B x). + +Inductive Bottom [bot:U]: Prop := + Bottom_definition: + (In U C bot) -> ((y: U) (In U C y) -> (R bot y)) -> (Bottom bot). + +Inductive Totally_ordered [B:(Ensemble U)]: Prop := + Totally_ordered_definition: + ((Included U B C) -> + (x: U) (y: U) (Included U (Couple U x y) B) -> (R x y) \/ (R y x)) -> + (Totally_ordered B). + +Definition Compatible : (Relation U) := + [x: U] [y: U] (In U C x) -> (In U C y) -> + (EXT z | (In U C z) /\ (Upper_Bound (Couple U x y) z)). + +Inductive Directed [X:(Ensemble U)]: Prop := + Definition_of_Directed: + (Included U X C) -> + (Inhabited U X) -> + ((x1: U) (x2: U) (Included U (Couple U x1 x2) X) -> + (EXT x3 | (In U X x3) /\ (Upper_Bound (Couple U x1 x2) x3))) -> + (Directed X). + +Inductive Complete : Prop := + Definition_of_Complete: + ((EXT bot | (Bottom bot))) -> + ((X: (Ensemble U)) (Directed X) -> (EXT bsup | (Lub X bsup))) -> + Complete. + +Inductive Conditionally_complete : Prop := + Definition_of_Conditionally_complete: + ((X: (Ensemble U)) + (Included U X C) -> (EXT maj | (Upper_Bound X maj)) -> + (EXT bsup | (Lub X bsup))) -> Conditionally_complete. +End Bounds. +Hints Resolve Totally_ordered_definition Upper_Bound_definition + Lower_Bound_definition Lub_definition Glb_definition + Bottom_definition Definition_of_Complete + Definition_of_Complete Definition_of_Conditionally_complete. + +Section Specific_orders. +Variable U: Type. + +Record Cpo : Type := Definition_of_cpo { + PO_of_cpo: (PO U); + Cpo_cond: (Complete U PO_of_cpo) }. + +Record Chain : Type := Definition_of_chain { + PO_of_chain: (PO U); + Chain_cond: (Totally_ordered U PO_of_chain (Carrier_of U PO_of_chain)) }. + +End Specific_orders. diff --git a/theories7/Sets/Ensembles.v b/theories7/Sets/Ensembles.v new file mode 100755 index 00000000..c3a044c0 --- /dev/null +++ b/theories7/Sets/Ensembles.v @@ -0,0 +1,108 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Ensembles.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Section Ensembles. +Variable U: Type. + +Definition Ensemble := U -> Prop. + +Definition In : Ensemble -> U -> Prop := [A: Ensemble] [x: U] (A x). + +Definition Included : Ensemble -> Ensemble -> Prop := + [B, C: Ensemble] (x: U) (In B x) -> (In C x). + +Inductive Empty_set : Ensemble := + . + +Inductive Full_set : Ensemble := + Full_intro: (x: U) (In Full_set x). + +(** NB: The following definition builds-in equality of elements in [U] as + Leibniz equality. + + This may have to be changed if we replace [U] by a Setoid on [U] + with its own equality [eqs], with + [In_singleton: (y: U)(eqs x y) -> (In (Singleton x) y)]. *) + +Inductive Singleton [x:U] : Ensemble := + In_singleton: (In (Singleton x) x). + +Inductive Union [B, C: Ensemble] : Ensemble := + Union_introl: (x: U) (In B x) -> (In (Union B C) x) + | Union_intror: (x: U) (In C x) -> (In (Union B C) x). + +Definition Add : Ensemble -> U -> Ensemble := + [B: Ensemble] [x: U] (Union B (Singleton x)). + +Inductive Intersection [B, C:Ensemble] : Ensemble := + Intersection_intro: + (x: U) (In B x) -> (In C x) -> (In (Intersection B C) x). + +Inductive Couple [x,y:U] : Ensemble := + Couple_l: (In (Couple x y) x) + | Couple_r: (In (Couple x y) y). + +Inductive Triple[x, y, z:U] : Ensemble := + Triple_l: (In (Triple x y z) x) + | Triple_m: (In (Triple x y z) y) + | Triple_r: (In (Triple x y z) z). + +Definition Complement : Ensemble -> Ensemble := + [A: Ensemble] [x: U] ~ (In A x). + +Definition Setminus : Ensemble -> Ensemble -> Ensemble := + [B: Ensemble] [C: Ensemble] [x: U] (In B x) /\ ~ (In C x). + +Definition Subtract : Ensemble -> U -> Ensemble := + [B: Ensemble] [x: U] (Setminus B (Singleton x)). + +Inductive Disjoint [B, C:Ensemble] : Prop := + Disjoint_intro: ((x: U) ~ (In (Intersection B C) x)) -> (Disjoint B C). + +Inductive Inhabited [B:Ensemble] : Prop := + Inhabited_intro: (x: U) (In B x) -> (Inhabited B). + +Definition Strict_Included : Ensemble -> Ensemble -> Prop := + [B, C: Ensemble] (Included B C) /\ ~ B == C. + +Definition Same_set : Ensemble -> Ensemble -> Prop := + [B, C: Ensemble] (Included B C) /\ (Included C B). + +(** Extensionality Axiom *) + +Axiom Extensionality_Ensembles: + (A,B: Ensemble) (Same_set A B) -> A == B. +Hints Resolve Extensionality_Ensembles. + +End Ensembles. + +Hints Unfold In Included Same_set Strict_Included Add Setminus Subtract : sets v62. + +Hints Resolve Union_introl Union_intror Intersection_intro In_singleton Couple_l + Couple_r Triple_l Triple_m Triple_r Disjoint_intro + Extensionality_Ensembles : sets v62. diff --git a/theories7/Sets/Finite_sets.v b/theories7/Sets/Finite_sets.v new file mode 100755 index 00000000..fb53994d --- /dev/null +++ b/theories7/Sets/Finite_sets.v @@ -0,0 +1,74 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Finite_sets.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Ensembles. + +Section Ensembles_finis. +Variable U: Type. + +Inductive Finite : (Ensemble U) -> Prop := + Empty_is_finite: (Finite (Empty_set U)) + | Union_is_finite: + (A: (Ensemble U)) (Finite A) -> + (x: U) ~ (In U A x) -> (Finite (Add U A x)). + +Inductive cardinal : (Ensemble U) -> nat -> Prop := + card_empty: (cardinal (Empty_set U) O) + | card_add: + (A: (Ensemble U)) (n: nat) (cardinal A n) -> + (x: U) ~ (In U A x) -> (cardinal (Add U A x) (S n)). + +End Ensembles_finis. + +Hints Resolve Empty_is_finite Union_is_finite : sets v62. +Hints Resolve card_empty card_add : sets v62. + +Require Constructive_sets. + +Section Ensembles_finis_facts. +Variable U: Type. + +Lemma cardinal_invert : + (X: (Ensemble U)) (p:nat)(cardinal U X p) -> Case p of + X == (Empty_set U) + [n:nat] (EXT A | (EXT x | + X == (Add U A x) /\ ~ (In U A x) /\ (cardinal U A n))) end. +Proof. +NewInduction 1; Simpl; Auto. +Exists A; Exists x; Auto. +Qed. + +Lemma cardinal_elim : + (X: (Ensemble U)) (p:nat)(cardinal U X p) -> Case p of + X == (Empty_set U) + [n:nat](Inhabited U X) end. +Proof. +Intros X p C; Elim C; Simpl; Trivial with sets. +Qed. + +End Ensembles_finis_facts. diff --git a/theories7/Sets/Finite_sets_facts.v b/theories7/Sets/Finite_sets_facts.v new file mode 100755 index 00000000..63d4d2ad --- /dev/null +++ b/theories7/Sets/Finite_sets_facts.v @@ -0,0 +1,345 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Finite_sets_facts.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Export Finite_sets. +Require Export Constructive_sets. +Require Export Classical_Type. +Require Export Classical_sets. +Require Export Powerset. +Require Export Powerset_facts. +Require Export Powerset_Classical_facts. +Require Export Gt. +Require Export Lt. + +Section Finite_sets_facts. +Variable U: Type. + +Lemma finite_cardinal : + (X: (Ensemble U)) (Finite U X) -> (EX n:nat |(cardinal U X n)). +Proof. +NewInduction 1 as [|A _ [n H]]. +Exists O; Auto with sets. +Exists (S n); Auto with sets. +Qed. + +Lemma cardinal_finite: + (X: (Ensemble U)) (n: nat) (cardinal U X n) -> (Finite U X). +Proof. +NewInduction 1; Auto with sets. +Qed. + +Theorem Add_preserves_Finite: + (X: (Ensemble U)) (x: U) (Finite U X) -> (Finite U (Add U X x)). +Proof. +Intros X x H'. +Elim (classic (In U X x)); Intro H'0; Auto with sets. +Rewrite (Non_disjoint_union U X x); Auto with sets. +Qed. +Hints Resolve Add_preserves_Finite. + +Theorem Singleton_is_finite: (x: U) (Finite U (Singleton U x)). +Proof. +Intro x; Rewrite <- (Empty_set_zero U (Singleton U x)). +Change (Finite U (Add U (Empty_set U) x)); Auto with sets. +Qed. +Hints Resolve Singleton_is_finite. + +Theorem Union_preserves_Finite: + (X, Y: (Ensemble U)) (Finite U X) -> (Finite U Y) -> + (Finite U (Union U X Y)). +Proof. +Intros X Y H'; Elim H'. +Rewrite (Empty_set_zero U Y); Auto with sets. +Intros A H'0 H'1 x H'2 H'3. +Rewrite (Union_commutative U (Add U A x) Y). +Rewrite <- (Union_add U Y A x). +Rewrite (Union_commutative U Y A); Auto with sets. +Qed. + +Lemma Finite_downward_closed: + (A: (Ensemble U)) (Finite U A) -> + (X: (Ensemble U)) (Included U X A) -> (Finite U X). +Proof. +Intros A H'; Elim H'; Auto with sets. +Intros X H'0. +Rewrite (less_than_empty U X H'0); Auto with sets. +Intros; Elim Included_Add with U X A0 x; Auto with sets. +NewDestruct 1 as [A' [H5 H6]]. +Rewrite H5; Auto with sets. +Qed. + +Lemma Intersection_preserves_finite: + (A: (Ensemble U)) (Finite U A) -> + (X: (Ensemble U)) (Finite U (Intersection U X A)). +Proof. +Intros A H' X; Apply Finite_downward_closed with A; Auto with sets. +Qed. + +Lemma cardinalO_empty: + (X: (Ensemble U)) (cardinal U X O) -> X == (Empty_set U). +Proof. +Intros X H; Apply (cardinal_invert U X O); Trivial with sets. +Qed. +Hints Resolve cardinalO_empty. + +Lemma inh_card_gt_O: + (X: (Ensemble U)) (Inhabited U X) -> (n: nat) (cardinal U X n) -> (gt n O). +Proof. +NewInduction 1 as [x H']. +Intros n H'0. +Elim (gt_O_eq n); Auto with sets. +Intro H'1; Generalize H'; Generalize H'0. +Rewrite <- H'1; Intro H'2. +Rewrite (cardinalO_empty X); Auto with sets. +Intro H'3; Elim H'3. +Qed. + +Lemma card_soustr_1: + (X: (Ensemble U)) (n: nat) (cardinal U X n) -> + (x: U) (In U X x) -> (cardinal U (Subtract U X x) (pred n)). +Proof. +Intros X n H'; Elim H'. +Intros x H'0; Elim H'0. +Clear H' n X. +Intros X n H' H'0 x H'1 x0 H'2. +Elim (classic (In U X x0)). +Intro H'4; Rewrite (add_soustr_xy U X x x0). +Elim (classic x == x0). +Intro H'5. +Absurd (In U X x0); Auto with sets. +Rewrite <- H'5; Auto with sets. +Intro H'3; Try Assumption. +Cut (S (pred n)) = (pred (S n)). +Intro H'5; Rewrite <- H'5. +Apply card_add; Auto with sets. +Red; Intro H'6; Elim H'6. +Intros H'7 H'8; Try Assumption. +Elim H'1; Auto with sets. +Unfold 2 pred; Symmetry. +Apply S_pred with m := O. +Change (gt n O). +Apply inh_card_gt_O with X := X; Auto with sets. +Apply Inhabited_intro with x := x0; Auto with sets. +Red; Intro H'3. +Apply H'1. +Elim H'3; Auto with sets. +Rewrite H'3; Auto with sets. +Elim (classic x == x0). +Intro H'3; Rewrite <- H'3. +Cut (Subtract U (Add U X x) x) == X; Auto with sets. +Intro H'4; Rewrite H'4; Auto with sets. +Intros H'3 H'4; Try Assumption. +Absurd (In U (Add U X x) x0); Auto with sets. +Red; Intro H'5; Try Exact H'5. +LApply (Add_inv U X x x0); Tauto. +Qed. + +Lemma cardinal_is_functional: + (X: (Ensemble U)) (c1: nat) (cardinal U X c1) -> + (Y: (Ensemble U)) (c2: nat) (cardinal U Y c2) -> X == Y -> + c1 = c2. +Proof. +Intros X c1 H'; Elim H'. +Intros Y c2 H'0; Elim H'0; Auto with sets. +Intros A n H'1 H'2 x H'3 H'5. +Elim (not_Empty_Add U A x); Auto with sets. +Clear H' c1 X. +Intros X n H' H'0 x H'1 Y c2 H'2. +Elim H'2. +Intro H'3. +Elim (not_Empty_Add U X x); Auto with sets. +Clear H'2 c2 Y. +Intros X0 c2 H'2 H'3 x0 H'4 H'5. +Elim (classic (In U X0 x)). +Intro H'6; Apply f_equal with nat. +Apply H'0 with Y := (Subtract U (Add U X0 x0) x). +ElimType (pred (S c2)) = c2; Auto with sets. +Apply card_soustr_1; Auto with sets. +Rewrite <- H'5. +Apply Sub_Add_new; Auto with sets. +Elim (classic x == x0). +Intros H'6 H'7; Apply f_equal with nat. +Apply H'0 with Y := X0; Auto with sets. +Apply Simplify_add with x := x; Auto with sets. +Pattern 2 x; Rewrite H'6; Auto with sets. +Intros H'6 H'7. +Absurd (Add U X x) == (Add U X0 x0); Auto with sets. +Clear H'0 H' H'3 n H'5 H'4 H'2 H'1 c2. +Red; Intro H'. +LApply (Extension U (Add U X x) (Add U X0 x0)); Auto with sets. +Clear H'. +Intro H'; Red in H'. +Elim H'; Intros H'0 H'1; Red in H'0; Clear H' H'1. +Absurd (In U (Add U X0 x0) x); Auto with sets. +LApply (Add_inv U X0 x0 x); [ Intuition | Apply (H'0 x); Apply Add_intro2 ]. +Qed. + +Lemma cardinal_Empty : (m:nat)(cardinal U (Empty_set U) m) -> O = m. +Proof. +Intros m Cm; Generalize (cardinal_invert U (Empty_set U) m Cm). +Elim m; Auto with sets. +Intros; Elim H0; Intros; Elim H1; Intros; Elim H2; Intros. +Elim (not_Empty_Add U x x0 H3). +Qed. + +Lemma cardinal_unicity : + (X: (Ensemble U)) (n: nat) (cardinal U X n) -> + (m: nat) (cardinal U X m) -> n = m. +Proof. +Intros; Apply cardinal_is_functional with X X; Auto with sets. +Qed. + +Lemma card_Add_gen: + (A: (Ensemble U)) + (x: U) (n, n': nat) (cardinal U A n) -> (cardinal U (Add U A x) n') -> + (le n' (S n)). +Proof. +Intros A x n n' H'. +Elim (classic (In U A x)). +Intro H'0. +Rewrite (Non_disjoint_union U A x H'0). +Intro H'1; Cut n = n'. +Intro E; Rewrite E; Auto with sets. +Apply cardinal_unicity with A; Auto with sets. +Intros H'0 H'1. +Cut n'=(S n). +Intro E; Rewrite E; Auto with sets. +Apply cardinal_unicity with (Add U A x); Auto with sets. +Qed. + +Lemma incl_st_card_lt: + (X: (Ensemble U)) (c1: nat) (cardinal U X c1) -> + (Y: (Ensemble U)) (c2: nat) (cardinal U Y c2) -> (Strict_Included U X Y) -> + (gt c2 c1). +Proof. +Intros X c1 H'; Elim H'. +Intros Y c2 H'0; Elim H'0; Auto with sets arith. +Intro H'1. +Elim (Strict_Included_strict U (Empty_set U)); Auto with sets arith. +Clear H' c1 X. +Intros X n H' H'0 x H'1 Y c2 H'2. +Elim H'2. +Intro H'3; Elim (not_SIncl_empty U (Add U X x)); Auto with sets arith. +Clear H'2 c2 Y. +Intros X0 c2 H'2 H'3 x0 H'4 H'5; Elim (classic (In U X0 x)). +Intro H'6; Apply gt_n_S. +Apply H'0 with Y := (Subtract U (Add U X0 x0) x). +ElimType (pred (S c2)) = c2; Auto with sets arith. +Apply card_soustr_1; Auto with sets arith. +Apply incl_st_add_soustr; Auto with sets arith. +Elim (classic x == x0). +Intros H'6 H'7; Apply gt_n_S. +Apply H'0 with Y := X0; Auto with sets arith. +Apply sincl_add_x with x := x0. +Rewrite <- H'6; Auto with sets arith. +Pattern 1 x0; Rewrite <- H'6; Trivial with sets arith. +Intros H'6 H'7; Red in H'5. +Elim H'5; Intros H'8 H'9; Try Exact H'8; Clear H'5. +Red in H'8. +Generalize (H'8 x). +Intro H'5; LApply H'5; Auto with sets arith. +Intro H; Elim Add_inv with U X0 x0 x; Auto with sets arith. +Intro; Absurd (In U X0 x); Auto with sets arith. +Intro; Absurd x==x0; Auto with sets arith. +Qed. + +Lemma incl_card_le: + (X,Y: (Ensemble U)) (n,m: nat) (cardinal U X n) -> (cardinal U Y m) -> + (Included U X Y) -> (le n m). +Proof. +Intros; +Elim Included_Strict_Included with U X Y; Auto with sets arith; Intro. +Cut (gt m n); Auto with sets arith. +Apply incl_st_card_lt with X := X Y := Y; Auto with sets arith. +Generalize H0; Rewrite <- H2; Intro. +Cut n=m. +Intro E; Rewrite E; Auto with sets arith. +Apply cardinal_unicity with X; Auto with sets arith. +Qed. + +Lemma G_aux: + (P:(Ensemble U) ->Prop) + ((X:(Ensemble U)) + (Finite U X) -> ((Y:(Ensemble U)) (Strict_Included U Y X) ->(P Y)) ->(P X)) -> + (P (Empty_set U)). +Proof. +Intros P H'; Try Assumption. +Apply H'; Auto with sets. +Clear H'; Auto with sets. +Intros Y H'; Try Assumption. +Red in H'. +Elim H'; Intros H'0 H'1; Try Exact H'1; Clear H'. +LApply (less_than_empty U Y); [Intro H'3; Try Exact H'3 | Assumption]. +Elim H'1; Auto with sets. +Qed. + +Hints Unfold not. + +Lemma Generalized_induction_on_finite_sets: + (P:(Ensemble U) ->Prop) + ((X:(Ensemble U)) + (Finite U X) -> ((Y:(Ensemble U)) (Strict_Included U Y X) ->(P Y)) ->(P X)) -> + (X:(Ensemble U)) (Finite U X) ->(P X). +Proof. +Intros P H'0 X H'1. +Generalize P H'0; Clear H'0 P. +Elim H'1. +Intros P H'0. +Apply G_aux; Auto with sets. +Clear H'1 X. +Intros A H' H'0 x H'1 P H'3. +Cut (Y:(Ensemble U)) (Included U Y (Add U A x)) ->(P Y); Auto with sets. +Generalize H'1. +Apply H'0. +Intros X K H'5 L Y H'6; Apply H'3; Auto with sets. +Apply Finite_downward_closed with A := (Add U X x); Auto with sets. +Intros Y0 H'7. +Elim (Strict_inclusion_is_transitive_with_inclusion U Y0 Y (Add U X x)); Auto with sets. +Intros H'2 H'4. +Elim (Included_Add U Y0 X x); + [Intro H'14 | + Intro H'14; Elim H'14; Intros A' E; Elim E; Intros H'15 H'16; Clear E H'14 | + Idtac]; Auto with sets. +Elim (Included_Strict_Included U Y0 X); Auto with sets. +Intro H'9; Apply H'5 with Y := Y0; Auto with sets. +Intro H'9; Rewrite H'9. +Apply H'3; Auto with sets. +Intros Y1 H'8; Elim H'8. +Intros H'10 H'11; Apply H'5 with Y := Y1; Auto with sets. +Elim (Included_Strict_Included U A' X); Auto with sets. +Intro H'8; Apply H'5 with Y := A'; Auto with sets. +Rewrite <- H'15; Auto with sets. +Intro H'8. +Elim H'7. +Intros H'9 H'10; Apply H'10 Orelse Elim H'10; Try Assumption. +Generalize H'6. +Rewrite <- H'8. +Rewrite <- H'15; Auto with sets. +Qed. + +End Finite_sets_facts. diff --git a/theories7/Sets/Image.v b/theories7/Sets/Image.v new file mode 100755 index 00000000..0794a3bb --- /dev/null +++ b/theories7/Sets/Image.v @@ -0,0 +1,199 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Image.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Export Finite_sets. +Require Export Constructive_sets. +Require Export Classical_Type. +Require Export Classical_sets. +Require Export Powerset. +Require Export Powerset_facts. +Require Export Powerset_Classical_facts. +Require Export Gt. +Require Export Lt. +Require Export Le. +Require Export Finite_sets_facts. + +Section Image. +Variables U, V: Type. + +Inductive Im [X:(Ensemble U); f:U -> V]: (Ensemble V) := + Im_intro: (x: U) (In ? X x) -> (y: V) y == (f x) -> (In ? (Im X f) y). + +Lemma Im_def: + (X: (Ensemble U)) (f: U -> V) (x: U) (In ? X x) -> (In ? (Im X f) (f x)). +Proof. +Intros X f x H'; Try Assumption. +Apply Im_intro with x := x; Auto with sets. +Qed. +Hints Resolve Im_def. + +Lemma Im_add: + (X: (Ensemble U)) (x: U) (f: U -> V) + (Im (Add ? X x) f) == (Add ? (Im X f) (f x)). +Proof. +Intros X x f. +Apply Extensionality_Ensembles. +Split; Red; Intros x0 H'. +Elim H'; Intros. +Rewrite H0. +Elim Add_inv with U X x x1; Auto with sets. +NewDestruct 1; Auto with sets. +Elim Add_inv with V (Im X f) (f x) x0; Auto with sets. +NewDestruct 1 as [x0 H y H0]. +Rewrite H0; Auto with sets. +NewDestruct 1; Auto with sets. +Qed. + +Lemma image_empty: (f: U -> V) (Im (Empty_set U) f) == (Empty_set V). +Proof. +Intro f; Try Assumption. +Apply Extensionality_Ensembles. +Split; Auto with sets. +Red. +Intros x H'; Elim H'. +Intros x0 H'0; Elim H'0; Auto with sets. +Qed. +Hints Resolve image_empty. + +Lemma finite_image: + (X: (Ensemble U)) (f: U -> V) (Finite ? X) -> (Finite ? (Im X f)). +Proof. +Intros X f H'; Elim H'. +Rewrite (image_empty f); Auto with sets. +Intros A H'0 H'1 x H'2; Clear H' X. +Rewrite (Im_add A x f); Auto with sets. +Apply Add_preserves_Finite; Auto with sets. +Qed. +Hints Resolve finite_image. + +Lemma Im_inv: + (X: (Ensemble U)) (f: U -> V) (y: V) (In ? (Im X f) y) -> + (exT ? [x: U] (In ? X x) /\ (f x) == y). +Proof. +Intros X f y H'; Elim H'. +Intros x H'0 y0 H'1; Rewrite H'1. +Exists x; Auto with sets. +Qed. + +Definition injective := [f: U -> V] (x, y: U) (f x) == (f y) -> x == y. + +Lemma not_injective_elim: + (f: U -> V) ~ (injective f) -> + (EXT x | (EXT y | (f x) == (f y) /\ ~ x == y)). +Proof. +Unfold injective; Intros f H. +Cut (EXT x | ~ ((y: U) (f x) == (f y) -> x == y)). +2: Apply not_all_ex_not with P:=[x:U](y: U) (f x) == (f y) -> x == y; + Trivial with sets. +NewDestruct 1 as [x C]; Exists x. +Cut (EXT y | ~((f x)==(f y)->x==y)). +2: Apply not_all_ex_not with P:=[y:U](f x)==(f y)->x==y; Trivial with sets. +NewDestruct 1 as [y D]; Exists y. +Apply imply_to_and; Trivial with sets. +Qed. + +Lemma cardinal_Im_intro: + (A: (Ensemble U)) (f: U -> V) (n: nat) (cardinal ? A n) -> + (EX p: nat | (cardinal ? (Im A f) p)). +Proof. +Intros. +Apply finite_cardinal; Apply finite_image. +Apply cardinal_finite with n; Trivial with sets. +Qed. + +Lemma In_Image_elim: + (A: (Ensemble U)) (f: U -> V) (injective f) -> + (x: U) (In ? (Im A f) (f x)) -> (In ? A x). +Proof. +Intros. +Elim Im_inv with A f (f x); Trivial with sets. +Intros z C; Elim C; Intros InAz E. +Elim (H z x E); Trivial with sets. +Qed. + +Lemma injective_preserves_cardinal: + (A: (Ensemble U)) (f: U -> V) (n: nat) (injective f) -> (cardinal ? A n) -> + (n': nat) (cardinal ? (Im A f) n') -> n' = n. +Proof. +NewInduction 2 as [|A n H'0 H'1 x H'2]; Auto with sets. +Rewrite (image_empty f). +Intros n' CE. +Apply cardinal_unicity with V (Empty_set V); Auto with sets. +Intro n'. +Rewrite (Im_add A x f). +Intro H'3. +Elim cardinal_Im_intro with A f n; Trivial with sets. +Intros i CI. +LApply (H'1 i); Trivial with sets. +Cut ~ (In ? (Im A f) (f x)). +Intros H0 H1. +Apply cardinal_unicity with V (Add ? (Im A f) (f x)); Trivial with sets. +Apply card_add; Auto with sets. +Rewrite <- H1; Trivial with sets. +Red; Intro; Apply H'2. +Apply In_Image_elim with f; Trivial with sets. +Qed. + +Lemma cardinal_decreases: + (A: (Ensemble U)) (f: U -> V) (n: nat) (cardinal U A n) -> + (n': nat) (cardinal V (Im A f) n') -> (le n' n). +Proof. +NewInduction 1 as [|A n H'0 H'1 x H'2]; Auto with sets. +Rewrite (image_empty f); Intros. +Cut n' = O. +Intro E; Rewrite E; Trivial with sets. +Apply cardinal_unicity with V (Empty_set V); Auto with sets. +Intro n'. +Rewrite (Im_add A x f). +Elim cardinal_Im_intro with A f n; Trivial with sets. +Intros p C H'3. +Apply le_trans with (S p). +Apply card_Add_gen with V (Im A f) (f x); Trivial with sets. +Apply le_n_S; Auto with sets. +Qed. + +Theorem Pigeonhole: + (A: (Ensemble U)) (f: U -> V) (n: nat) (cardinal U A n) -> + (n': nat) (cardinal V (Im A f) n') -> (lt n' n) -> ~ (injective f). +Proof. +Unfold not; Intros A f n CAn n' CIfn' ltn'n I. +Cut n' = n. +Intro E; Generalize ltn'n; Rewrite E; Exact (lt_n_n n). +Apply injective_preserves_cardinal with A := A f := f n := n; Trivial with sets. +Qed. + +Lemma Pigeonhole_principle: + (A: (Ensemble U)) (f: U -> V) (n: nat) (cardinal ? A n) -> + (n': nat) (cardinal ? (Im A f) n') -> (lt n' n) -> + (EXT x | (EXT y | (f x) == (f y) /\ ~ x == y)). +Proof. +Intros; Apply not_injective_elim. +Apply Pigeonhole with A n n'; Trivial with sets. +Qed. +End Image. +Hints Resolve Im_def image_empty finite_image : sets v62. diff --git a/theories7/Sets/Infinite_sets.v b/theories7/Sets/Infinite_sets.v new file mode 100755 index 00000000..bf423753 --- /dev/null +++ b/theories7/Sets/Infinite_sets.v @@ -0,0 +1,232 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Infinite_sets.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Export Finite_sets. +Require Export Constructive_sets. +Require Export Classical_Type. +Require Export Classical_sets. +Require Export Powerset. +Require Export Powerset_facts. +Require Export Powerset_Classical_facts. +Require Export Gt. +Require Export Lt. +Require Export Le. +Require Export Finite_sets_facts. +Require Export Image. + +Section Approx. +Variable U: Type. + +Inductive Approximant [A, X:(Ensemble U)] : Prop := + Defn_of_Approximant: (Finite U X) -> (Included U X A) -> (Approximant A X). +End Approx. + +Hints Resolve Defn_of_Approximant. + +Section Infinite_sets. +Variable U: Type. + +Lemma make_new_approximant: + (A: (Ensemble U)) (X: (Ensemble U)) ~ (Finite U A) -> (Approximant U A X) -> + (Inhabited U (Setminus U A X)). +Proof. +Intros A X H' H'0. +Elim H'0; Intros H'1 H'2. +Apply Strict_super_set_contains_new_element; Auto with sets. +Red; Intro H'3; Apply H'. +Rewrite <- H'3; Auto with sets. +Qed. + +Lemma approximants_grow: + (A: (Ensemble U)) (X: (Ensemble U)) ~ (Finite U A) -> + (n: nat) (cardinal U X n) -> (Included U X A) -> + (EXT Y | (cardinal U Y (S n)) /\ (Included U Y A)). +Proof. +Intros A X H' n H'0; Elim H'0; Auto with sets. +Intro H'1. +Cut (Inhabited U (Setminus U A (Empty_set U))). +Intro H'2; Elim H'2. +Intros x H'3. +Exists (Add U (Empty_set U) x); Auto with sets. +Split. +Apply card_add; Auto with sets. +Cut (In U A x). +Intro H'4; Red; Auto with sets. +Intros x0 H'5; Elim H'5; Auto with sets. +Intros x1 H'6; Elim H'6; Auto with sets. +Elim H'3; Auto with sets. +Apply make_new_approximant; Auto with sets. +Intros A0 n0 H'1 H'2 x H'3 H'5. +LApply H'2; [Intro H'6; Elim H'6; Clear H'2 | Clear H'2]; Auto with sets. +Intros x0 H'2; Try Assumption. +Elim H'2; Intros H'7 H'8; Try Exact H'8; Clear H'2. +Elim (make_new_approximant A x0); Auto with sets. +Intros x1 H'2; Try Assumption. +Exists (Add U x0 x1); Auto with sets. +Split. +Apply card_add; Auto with sets. +Elim H'2; Auto with sets. +Red. +Intros x2 H'9; Elim H'9; Auto with sets. +Intros x3 H'10; Elim H'10; Auto with sets. +Elim H'2; Auto with sets. +Auto with sets. +Apply Defn_of_Approximant; Auto with sets. +Apply cardinal_finite with n := (S n0); Auto with sets. +Qed. + +Lemma approximants_grow': + (A: (Ensemble U)) (X: (Ensemble U)) ~ (Finite U A) -> + (n: nat) (cardinal U X n) -> (Approximant U A X) -> + (EXT Y | (cardinal U Y (S n)) /\ (Approximant U A Y)). +Proof. +Intros A X H' n H'0 H'1; Try Assumption. +Elim H'1. +Intros H'2 H'3. +ElimType (EXT Y | (cardinal U Y (S n)) /\ (Included U Y A)). +Intros x H'4; Elim H'4; Intros H'5 H'6; Try Exact H'5; Clear H'4. +Exists x; Auto with sets. +Split; [Auto with sets | Idtac]. +Apply Defn_of_Approximant; Auto with sets. +Apply cardinal_finite with n := (S n); Auto with sets. +Apply approximants_grow with X := X; Auto with sets. +Qed. + +Lemma approximant_can_be_any_size: + (A: (Ensemble U)) (X: (Ensemble U)) ~ (Finite U A) -> + (n: nat) (EXT Y | (cardinal U Y n) /\ (Approximant U A Y)). +Proof. +Intros A H' H'0 n; Elim n. +Exists (Empty_set U); Auto with sets. +Intros n0 H'1; Elim H'1. +Intros x H'2. +Apply approximants_grow' with X := x; Tauto. +Qed. + +Variable V: Type. + +Theorem Image_set_continuous: + (A: (Ensemble U)) + (f: U -> V) (X: (Ensemble V)) (Finite V X) -> (Included V X (Im U V A f)) -> + (EX n | + (EXT Y | ((cardinal U Y n) /\ (Included U Y A)) /\ (Im U V Y f) == X)). +Proof. +Intros A f X H'; Elim H'. +Intro H'0; Exists O. +Exists (Empty_set U); Auto with sets. +Intros A0 H'0 H'1 x H'2 H'3; Try Assumption. +LApply H'1; + [Intro H'4; Elim H'4; Intros n E; Elim E; Clear H'4 H'1 | Clear H'1]; Auto with sets. +Intros x0 H'1; Try Assumption. +Exists (S n); Try Assumption. +Elim H'1; Intros H'4 H'5; Elim H'4; Intros H'6 H'7; Try Exact H'6; Clear H'4 H'1. +Clear E. +Generalize H'2. +Rewrite <- H'5. +Intro H'1; Try Assumption. +Red in H'3. +Generalize (H'3 x). +Intro H'4; LApply H'4; [Intro H'8; Try Exact H'8; Clear H'4 | Clear H'4]; Auto with sets. +Specialize 5 Im_inv with U := U V:=V X := A f := f y := x; Intro H'11; + LApply H'11; [Intro H'13; Elim H'11; Clear H'11 | Clear H'11]; Auto with sets. +Intros x1 H'4; Try Assumption. +Apply exT_intro with x := (Add U x0 x1). +Split; [Split; [Try Assumption | Idtac] | Idtac]. +Apply card_add; Auto with sets. +Red; Intro H'9; Try Exact H'9. +Apply H'1. +Elim H'4; Intros H'10 H'11; Rewrite <- H'11; Clear H'4; Auto with sets. +Elim H'4; Intros H'9 H'10; Try Exact H'9; Clear H'4; Auto with sets. +Red; Auto with sets. +Intros x2 H'4; Elim H'4; Auto with sets. +Intros x3 H'11; Elim H'11; Auto with sets. +Elim H'4; Intros H'9 H'10; Rewrite <- H'10; Clear H'4; Auto with sets. +Apply Im_add; Auto with sets. +Qed. + +Theorem Image_set_continuous': + (A: (Ensemble U)) + (f: U -> V) (X: (Ensemble V)) (Approximant V (Im U V A f) X) -> + (EXT Y | (Approximant U A Y) /\ (Im U V Y f) == X). +Proof. +Intros A f X H'; Try Assumption. +Cut (EX n | (EXT Y | + ((cardinal U Y n) /\ (Included U Y A)) /\ (Im U V Y f) == X)). +Intro H'0; Elim H'0; Intros n E; Elim E; Clear H'0. +Intros x H'0; Try Assumption. +Elim H'0; Intros H'1 H'2; Elim H'1; Intros H'3 H'4; Try Exact H'3; + Clear H'1 H'0; Auto with sets. +Exists x. +Split; [Idtac | Try Assumption]. +Apply Defn_of_Approximant; Auto with sets. +Apply cardinal_finite with n := n; Auto with sets. +Apply Image_set_continuous; Auto with sets. +Elim H'; Auto with sets. +Elim H'; Auto with sets. +Qed. + +Theorem Pigeonhole_bis: + (A: (Ensemble U)) (f: U -> V) ~ (Finite U A) -> (Finite V (Im U V A f)) -> + ~ (injective U V f). +Proof. +Intros A f H'0 H'1; Try Assumption. +Elim (Image_set_continuous' A f (Im U V A f)); Auto with sets. +Intros x H'2; Elim H'2; Intros H'3 H'4; Try Exact H'3; Clear H'2. +Elim (make_new_approximant A x); Auto with sets. +Intros x0 H'2; Elim H'2. +Intros H'5 H'6. +Elim (finite_cardinal V (Im U V A f)); Auto with sets. +Intros n E. +Elim (finite_cardinal U x); Auto with sets. +Intros n0 E0. +Apply Pigeonhole with A := (Add U x x0) n := (S n0) n' := n. +Apply card_add; Auto with sets. +Rewrite (Im_add U V x x0 f); Auto with sets. +Cut (In V (Im U V x f) (f x0)). +Intro H'8. +Rewrite (Non_disjoint_union V (Im U V x f) (f x0)); Auto with sets. +Rewrite H'4; Auto with sets. +Elim (Extension V (Im U V x f) (Im U V A f)); Auto with sets. +Apply le_lt_n_Sm. +Apply cardinal_decreases with U := U V := V A := x f := f; Auto with sets. +Rewrite H'4; Auto with sets. +Elim H'3; Auto with sets. +Qed. + +Theorem Pigeonhole_ter: + (A: (Ensemble U)) + (f: U -> V) (n: nat) (injective U V f) -> (Finite V (Im U V A f)) -> + (Finite U A). +Proof. +Intros A f H' H'0 H'1. +Apply NNPP. +Red; Intro H'2. +Elim (Pigeonhole_bis A f); Auto with sets. +Qed. + +End Infinite_sets. diff --git a/theories7/Sets/Integers.v b/theories7/Sets/Integers.v new file mode 100755 index 00000000..7dee371f --- /dev/null +++ b/theories7/Sets/Integers.v @@ -0,0 +1,166 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Integers.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Export Finite_sets. +Require Export Constructive_sets. +Require Export Classical_Type. +Require Export Classical_sets. +Require Export Powerset. +Require Export Powerset_facts. +Require Export Powerset_Classical_facts. +Require Export Gt. +Require Export Lt. +Require Export Le. +Require Export Finite_sets_facts. +Require Export Image. +Require Export Infinite_sets. +Require Export Compare_dec. +Require Export Relations_1. +Require Export Partial_Order. +Require Export Cpo. + +Section Integers_sect. + +Inductive Integers : (Ensemble nat) := + Integers_defn: (x: nat) (In nat Integers x). +Hints Resolve Integers_defn. + +Lemma le_reflexive: (Reflexive nat le). +Proof. +Red; Auto with arith. +Qed. + +Lemma le_antisym: (Antisymmetric nat le). +Proof. +Red; Intros x y H H';Rewrite (le_antisym x y);Auto. +Qed. + +Lemma le_trans: (Transitive nat le). +Proof. +Red; Intros; Apply le_trans with y;Auto. +Qed. +Hints Resolve le_reflexive le_antisym le_trans. + +Lemma le_Order: (Order nat le). +Proof. +Auto with sets arith. +Qed. +Hints Resolve le_Order. + +Lemma triv_nat: (n: nat) (In nat Integers n). +Proof. +Auto with sets arith. +Qed. +Hints Resolve triv_nat. + +Definition nat_po: (PO nat). +Apply Definition_of_PO with Carrier_of := Integers Rel_of := le; Auto with sets arith. +Apply Inhabited_intro with x := O; Auto with sets arith. +Defined. +Hints Unfold nat_po. + +Lemma le_total_order: (Totally_ordered nat nat_po Integers). +Proof. +Apply Totally_ordered_definition. +Simpl. +Intros H' x y H'0. +Specialize 2 le_or_lt with n := x m := y; Intro H'2; Elim H'2. +Intro H'1; Left; Auto with sets arith. +Intro H'1; Right. +Cut (le y x); Auto with sets arith. +Qed. +Hints Resolve le_total_order. + +Lemma Finite_subset_has_lub: + (X: (Ensemble nat)) (Finite nat X) -> + (EXT m: nat | (Upper_Bound nat nat_po X m)). +Proof. +Intros X H'; Elim H'. +Exists O. +Apply Upper_Bound_definition; Auto with sets arith. +Intros y H'0; Elim H'0; Auto with sets arith. +Intros A H'0 H'1 x H'2; Try Assumption. +Elim H'1; Intros x0 H'3; Clear H'1. +Elim le_total_order. +Simpl. +Intro H'1; Try Assumption. +LApply H'1; [Intro H'4; Idtac | Try Assumption]; Auto with sets arith. +Generalize (H'4 x0 x). +Clear H'4. +Clear H'1. +Intro H'1; LApply H'1; + [Intro H'4; Elim H'4; + [Intro H'5; Try Exact H'5; Clear H'4 H'1 | Intro H'5; Clear H'4 H'1] | + Clear H'1]. +Exists x. +Apply Upper_Bound_definition; Auto with sets arith; Simpl. +Intros y H'1; Elim H'1. +Generalize le_trans. +Intro H'4; Red in H'4. +Intros x1 H'6; Try Assumption. +Apply H'4 with y := x0; Auto with sets arith. +Elim H'3; Simpl; Auto with sets arith. +Intros x1 H'4; Elim H'4; Auto with sets arith. +Exists x0. +Apply Upper_Bound_definition; Auto with sets arith; Simpl. +Intros y H'1; Elim H'1. +Intros x1 H'4; Try Assumption. +Elim H'3; Simpl; Auto with sets arith. +Intros x1 H'4; Elim H'4; Auto with sets arith. +Red. +Intros x1 H'1; Elim H'1; Auto with sets arith. +Qed. + +Lemma Integers_has_no_ub: ~ (EXT m:nat | (Upper_Bound nat nat_po Integers m)). +Proof. +Red; Intro H'; Elim H'. +Intros x H'0. +Elim H'0; Intros H'1 H'2. +Cut (In nat Integers (S x)). +Intro H'3. +Specialize 1 H'2 with y := (S x); Intro H'4; LApply H'4; + [Intro H'5; Clear H'4 | Try Assumption; Clear H'4]. +Simpl in H'5. +Absurd (le (S x) x); Auto with arith. +Auto with sets arith. +Qed. + +Lemma Integers_infinite: ~ (Finite nat Integers). +Proof. +Generalize Integers_has_no_ub. +Intro H'; Red; Intro H'0; Try Exact H'0. +Apply H'. +Apply Finite_subset_has_lub; Auto with sets arith. +Qed. + +End Integers_sect. + + + + + diff --git a/theories7/Sets/Multiset.v b/theories7/Sets/Multiset.v new file mode 100755 index 00000000..b5d5edf7 --- /dev/null +++ b/theories7/Sets/Multiset.v @@ -0,0 +1,186 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Multiset.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +(* G. Huet 1-9-95 *) + +Require Permut. + +Set Implicit Arguments. + +Section multiset_defs. + +Variable A : Set. +Variable eqA : A -> A -> Prop. +Hypothesis Aeq_dec : (x,y:A){(eqA x y)}+{~(eqA x y)}. + +Inductive multiset : Set := + Bag : (A->nat) -> multiset. + +Definition EmptyBag := (Bag [a:A]O). +Definition SingletonBag := [a:A] + (Bag [a':A]Cases (Aeq_dec a a') of + (left _) => (S O) + | (right _) => O + end + ). + +Definition multiplicity : multiset -> A -> nat := + [m:multiset][a:A]let (f) = m in (f a). + +(** multiset equality *) +Definition meq := [m1,m2:multiset] + (a:A)(multiplicity m1 a)=(multiplicity m2 a). + +Hints Unfold meq multiplicity. + +Lemma meq_refl : (x:multiset)(meq x x). +Proof. +NewDestruct x; Auto. +Qed. +Hints Resolve meq_refl. + +Lemma meq_trans : (x,y,z:multiset)(meq x y)->(meq y z)->(meq x z). +Proof. +Unfold meq. +NewDestruct x; NewDestruct y; NewDestruct z. +Intros; Rewrite H; Auto. +Qed. + +Lemma meq_sym : (x,y:multiset)(meq x y)->(meq y x). +Proof. +Unfold meq. +NewDestruct x; NewDestruct y; Auto. +Qed. +Hints Immediate meq_sym. + +(** multiset union *) +Definition munion := [m1,m2:multiset] + (Bag [a:A](plus (multiplicity m1 a)(multiplicity m2 a))). + +Lemma munion_empty_left : + (x:multiset)(meq x (munion EmptyBag x)). +Proof. +Unfold meq; Unfold munion; Simpl; Auto. +Qed. +Hints Resolve munion_empty_left. + +Lemma munion_empty_right : + (x:multiset)(meq x (munion x EmptyBag)). +Proof. +Unfold meq; Unfold munion; Simpl; Auto. +Qed. + + +Require Plus. (* comm. and ass. of plus *) + +Lemma munion_comm : (x,y:multiset)(meq (munion x y) (munion y x)). +Proof. +Unfold meq; Unfold multiplicity; Unfold munion. +NewDestruct x; NewDestruct y; Auto with arith. +Qed. +Hints Resolve munion_comm. + +Lemma munion_ass : + (x,y,z:multiset)(meq (munion (munion x y) z) (munion x (munion y z))). +Proof. +Unfold meq; Unfold munion; Unfold multiplicity. +NewDestruct x; NewDestruct y; NewDestruct z; Auto with arith. +Qed. +Hints Resolve munion_ass. + +Lemma meq_left : (x,y,z:multiset)(meq x y)->(meq (munion x z) (munion y z)). +Proof. +Unfold meq; Unfold munion; Unfold multiplicity. +NewDestruct x; NewDestruct y; NewDestruct z. +Intros; Elim H; Auto with arith. +Qed. +Hints Resolve meq_left. + +Lemma meq_right : (x,y,z:multiset)(meq x y)->(meq (munion z x) (munion z y)). +Proof. +Unfold meq; Unfold munion; Unfold multiplicity. +NewDestruct x; NewDestruct y; NewDestruct z. +Intros; Elim H; Auto. +Qed. +Hints Resolve meq_right. + + +(** Here we should make multiset an abstract datatype, by hiding [Bag], + [munion], [multiplicity]; all further properties are proved abstractly *) + +Lemma munion_rotate : + (x,y,z:multiset)(meq (munion x (munion y z)) (munion z (munion x y))). +Proof. +Intros; Apply (op_rotate multiset munion meq); Auto. +Exact meq_trans. +Qed. + +Lemma meq_congr : (x,y,z,t:multiset)(meq x y)->(meq z t)-> + (meq (munion x z) (munion y t)). +Proof. +Intros; Apply (cong_congr multiset munion meq); Auto. +Exact meq_trans. +Qed. + +Lemma munion_perm_left : + (x,y,z:multiset)(meq (munion x (munion y z)) (munion y (munion x z))). +Proof. +Intros; Apply (perm_left multiset munion meq); Auto. +Exact meq_trans. +Qed. + +Lemma multiset_twist1 : (x,y,z,t:multiset) + (meq (munion x (munion (munion y z) t)) (munion (munion y (munion x t)) z)). +Proof. +Intros; Apply (twist multiset munion meq); Auto. +Exact meq_trans. +Qed. + +Lemma multiset_twist2 : (x,y,z,t:multiset) + (meq (munion x (munion (munion y z) t)) (munion (munion y (munion x z)) t)). +Proof. +Intros; Apply meq_trans with (munion (munion x (munion y z)) t). +Apply meq_sym; Apply munion_ass. +Apply meq_left; Apply munion_perm_left. +Qed. + +(** specific for treesort *) + +Lemma treesort_twist1 : (x,y,z,t,u:multiset) (meq u (munion y z)) -> + (meq (munion x (munion u t)) (munion (munion y (munion x t)) z)). +Proof. +Intros; Apply meq_trans with (munion x (munion (munion y z) t)). +Apply meq_right; Apply meq_left; Trivial. +Apply multiset_twist1. +Qed. + +Lemma treesort_twist2 : (x,y,z,t,u:multiset) (meq u (munion y z)) -> + (meq (munion x (munion u t)) (munion (munion y (munion x z)) t)). +Proof. +Intros; Apply meq_trans with (munion x (munion (munion y z) t)). +Apply meq_right; Apply meq_left; Trivial. +Apply multiset_twist2. +Qed. + + +(*i theory of minter to do similarly +Require Min. +(* multiset intersection *) +Definition minter := [m1,m2:multiset] + (Bag [a:A](min (multiplicity m1 a)(multiplicity m2 a))). +i*) + +End multiset_defs. + +Unset Implicit Arguments. + +Hints Unfold meq multiplicity : v62 datatypes. +Hints Resolve munion_empty_right munion_comm munion_ass meq_left meq_right munion_empty_left : v62 datatypes. +Hints Immediate meq_sym : v62 datatypes. diff --git a/theories7/Sets/Partial_Order.v b/theories7/Sets/Partial_Order.v new file mode 100755 index 00000000..759cf4e2 --- /dev/null +++ b/theories7/Sets/Partial_Order.v @@ -0,0 +1,100 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Partial_Order.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Export Ensembles. +Require Export Relations_1. + +Section Partial_orders. +Variable U: Type. + +Definition Carrier := (Ensemble U). + +Definition Rel := (Relation U). + +Record PO : Type := Definition_of_PO { + Carrier_of: (Ensemble U); + Rel_of: (Relation U); + PO_cond1: (Inhabited U Carrier_of); + PO_cond2: (Order U Rel_of) }. +Variable p: PO. + +Definition Strict_Rel_of : Rel := [x, y: U] (Rel_of p x y) /\ ~ x == y. + +Inductive covers [y, x:U]: Prop := + Definition_of_covers: + (Strict_Rel_of x y) -> + ~ (EXT z | (Strict_Rel_of x z) /\ (Strict_Rel_of z y)) -> + (covers y x). + +End Partial_orders. + +Hints Unfold Carrier_of Rel_of Strict_Rel_of : sets v62. +Hints Resolve Definition_of_covers : sets v62. + + +Section Partial_order_facts. +Variable U:Type. +Variable D:(PO U). + +Lemma Strict_Rel_Transitive_with_Rel: + (x:U) (y:U) (z:U) (Strict_Rel_of U D x y) -> (Rel_of U D y z) -> + (Strict_Rel_of U D x z). +Unfold 1 Strict_Rel_of. +Red. +Elim D; Simpl. +Intros C R H' H'0; Elim H'0. +Intros H'1 H'2 H'3 x y z H'4 H'5; Split. +Apply H'2 with y := y; Tauto. +Red; Intro H'6. +Elim H'4; Intros H'7 H'8; Apply H'8; Clear H'4. +Apply H'3; Auto. +Rewrite H'6; Tauto. +Qed. + +Lemma Strict_Rel_Transitive_with_Rel_left: + (x:U) (y:U) (z:U) (Rel_of U D x y) -> (Strict_Rel_of U D y z) -> + (Strict_Rel_of U D x z). +Unfold 1 Strict_Rel_of. +Red. +Elim D; Simpl. +Intros C R H' H'0; Elim H'0. +Intros H'1 H'2 H'3 x y z H'4 H'5; Split. +Apply H'2 with y := y; Tauto. +Red; Intro H'6. +Elim H'5; Intros H'7 H'8; Apply H'8; Clear H'5. +Apply H'3; Auto. +Rewrite <- H'6; Auto. +Qed. + +Lemma Strict_Rel_Transitive: (Transitive U (Strict_Rel_of U D)). +Red. +Intros x y z H' H'0. +Apply Strict_Rel_Transitive_with_Rel with y := y; + [ Intuition | Unfold Strict_Rel_of in H' H'0; Intuition ]. +Qed. +End Partial_order_facts. diff --git a/theories7/Sets/Permut.v b/theories7/Sets/Permut.v new file mode 100755 index 00000000..2d0413a8 --- /dev/null +++ b/theories7/Sets/Permut.v @@ -0,0 +1,91 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Permut.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +(* G. Huet 1-9-95 *) + +(** We consider a Set [U], given with a commutative-associative operator [op], + and a congruence [cong]; we show permutation lemmas *) + +Section Axiomatisation. + +Variable U: Set. + +Variable op: U -> U -> U. + +Variable cong : U -> U -> Prop. + +Hypothesis op_comm : (x,y:U)(cong (op x y) (op y x)). +Hypothesis op_ass : (x,y,z:U)(cong (op (op x y) z) (op x (op y z))). + +Hypothesis cong_left : (x,y,z:U)(cong x y)->(cong (op x z) (op y z)). +Hypothesis cong_right : (x,y,z:U)(cong x y)->(cong (op z x) (op z y)). +Hypothesis cong_trans : (x,y,z:U)(cong x y)->(cong y z)->(cong x z). +Hypothesis cong_sym : (x,y:U)(cong x y)->(cong y x). + +(** Remark. we do not need: [Hypothesis cong_refl : (x:U)(cong x x)]. *) + +Lemma cong_congr : + (x,y,z,t:U)(cong x y)->(cong z t)->(cong (op x z) (op y t)). +Proof. +Intros; Apply cong_trans with (op y z). +Apply cong_left; Trivial. +Apply cong_right; Trivial. +Qed. + +Lemma comm_right : (x,y,z:U)(cong (op x (op y z)) (op x (op z y))). +Proof. +Intros; Apply cong_right; Apply op_comm. +Qed. + +Lemma comm_left : (x,y,z:U)(cong (op (op x y) z) (op (op y x) z)). +Proof. +Intros; Apply cong_left; Apply op_comm. +Qed. + +Lemma perm_right : (x,y,z:U)(cong (op (op x y) z) (op (op x z) y)). +Proof. +Intros. +Apply cong_trans with (op x (op y z)). +Apply op_ass. +Apply cong_trans with (op x (op z y)). +Apply cong_right; Apply op_comm. +Apply cong_sym; Apply op_ass. +Qed. + +Lemma perm_left : (x,y,z:U)(cong (op x (op y z)) (op y (op x z))). +Proof. +Intros. +Apply cong_trans with (op (op x y) z). +Apply cong_sym; Apply op_ass. +Apply cong_trans with (op (op y x) z). +Apply cong_left; Apply op_comm. +Apply op_ass. +Qed. + +Lemma op_rotate : (x,y,z,t:U)(cong (op x (op y z)) (op z (op x y))). +Proof. +Intros; Apply cong_trans with (op (op x y) z). +Apply cong_sym; Apply op_ass. +Apply op_comm. +Qed. + +(* Needed for treesort ... *) +Lemma twist : (x,y,z,t:U) + (cong (op x (op (op y z) t)) (op (op y (op x t)) z)). +Proof. +Intros. +Apply cong_trans with (op x (op (op y t) z)). +Apply cong_right; Apply perm_right. +Apply cong_trans with (op (op x (op y t)) z). +Apply cong_sym; Apply op_ass. +Apply cong_left; Apply perm_left. +Qed. + +End Axiomatisation. diff --git a/theories7/Sets/Powerset.v b/theories7/Sets/Powerset.v new file mode 100755 index 00000000..b1fa892c --- /dev/null +++ b/theories7/Sets/Powerset.v @@ -0,0 +1,188 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Powerset.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Export Ensembles. +Require Export Relations_1. +Require Export Relations_1_facts. +Require Export Partial_Order. +Require Export Cpo. + +Section The_power_set_partial_order. +Variable U: Type. + +Inductive Power_set [A:(Ensemble U)]: (Ensemble (Ensemble U)) := + Definition_of_Power_set: + (X: (Ensemble U)) (Included U X A) -> (In (Ensemble U) (Power_set A) X). +Hints Resolve Definition_of_Power_set. + +Theorem Empty_set_minimal: (X: (Ensemble U)) (Included U (Empty_set U) X). +Intro X; Red. +Intros x H'; Elim H'. +Qed. +Hints Resolve Empty_set_minimal. + +Theorem Power_set_Inhabited: + (X: (Ensemble U)) (Inhabited (Ensemble U) (Power_set X)). +Intro X. +Apply Inhabited_intro with (Empty_set U); Auto with sets. +Qed. +Hints Resolve Power_set_Inhabited. + +Theorem Inclusion_is_an_order: (Order (Ensemble U) (Included U)). +Auto 6 with sets. +Qed. +Hints Resolve Inclusion_is_an_order. + +Theorem Inclusion_is_transitive: (Transitive (Ensemble U) (Included U)). +Elim Inclusion_is_an_order; Auto with sets. +Qed. +Hints Resolve Inclusion_is_transitive. + +Definition Power_set_PO: (Ensemble U) -> (PO (Ensemble U)). +Intro A; Try Assumption. +Apply Definition_of_PO with (Power_set A) (Included U); Auto with sets. +Defined. +Hints Unfold Power_set_PO. + +Theorem Strict_Rel_is_Strict_Included: + (same_relation + (Ensemble U) (Strict_Included U) + (Strict_Rel_of (Ensemble U) (Power_set_PO (Full_set U)))). +Auto with sets. +Qed. +Hints Resolve Strict_Rel_Transitive Strict_Rel_is_Strict_Included. + +Lemma Strict_inclusion_is_transitive_with_inclusion: + (x, y, z:(Ensemble U)) (Strict_Included U x y) -> (Included U y z) -> + (Strict_Included U x z). +Intros x y z H' H'0; Try Assumption. +Elim Strict_Rel_is_Strict_Included. +Unfold contains. +Intros H'1 H'2; Try Assumption. +Apply H'1. +Apply Strict_Rel_Transitive_with_Rel with y := y; Auto with sets. +Qed. + +Lemma Strict_inclusion_is_transitive_with_inclusion_left: + (x, y, z:(Ensemble U)) (Included U x y) -> (Strict_Included U y z) -> + (Strict_Included U x z). +Intros x y z H' H'0; Try Assumption. +Elim Strict_Rel_is_Strict_Included. +Unfold contains. +Intros H'1 H'2; Try Assumption. +Apply H'1. +Apply Strict_Rel_Transitive_with_Rel_left with y := y; Auto with sets. +Qed. + +Lemma Strict_inclusion_is_transitive: + (Transitive (Ensemble U) (Strict_Included U)). +Apply cong_transitive_same_relation + with R := (Strict_Rel_of (Ensemble U) (Power_set_PO (Full_set U))); Auto with sets. +Qed. + +Theorem Empty_set_is_Bottom: + (A: (Ensemble U)) (Bottom (Ensemble U) (Power_set_PO A) (Empty_set U)). +Intro A; Apply Bottom_definition; Simpl; Auto with sets. +Qed. +Hints Resolve Empty_set_is_Bottom. + +Theorem Union_minimal: + (a, b, X: (Ensemble U)) (Included U a X) -> (Included U b X) -> + (Included U (Union U a b) X). +Intros a b X H' H'0; Red. +Intros x H'1; Elim H'1; Auto with sets. +Qed. +Hints Resolve Union_minimal. + +Theorem Intersection_maximal: + (a, b, X: (Ensemble U)) (Included U X a) -> (Included U X b) -> + (Included U X (Intersection U a b)). +Auto with sets. +Qed. + +Theorem Union_increases_l: (a, b: (Ensemble U)) (Included U a (Union U a b)). +Auto with sets. +Qed. + +Theorem Union_increases_r: (a, b: (Ensemble U)) (Included U b (Union U a b)). +Auto with sets. +Qed. + +Theorem Intersection_decreases_l: + (a, b: (Ensemble U)) (Included U (Intersection U a b) a). +Intros a b; Red. +Intros x H'; Elim H'; Auto with sets. +Qed. + +Theorem Intersection_decreases_r: + (a, b: (Ensemble U)) (Included U (Intersection U a b) b). +Intros a b; Red. +Intros x H'; Elim H'; Auto with sets. +Qed. +Hints Resolve Union_increases_l Union_increases_r Intersection_decreases_l + Intersection_decreases_r. + +Theorem Union_is_Lub: + (A: (Ensemble U)) (a, b: (Ensemble U)) (Included U a A) -> (Included U b A) -> + (Lub (Ensemble U) (Power_set_PO A) (Couple (Ensemble U) a b) (Union U a b)). +Intros A a b H' H'0. +Apply Lub_definition; Simpl. +Apply Upper_Bound_definition; Simpl; Auto with sets. +Intros y H'1; Elim H'1; Auto with sets. +Intros y H'1; Elim H'1; Simpl; Auto with sets. +Qed. + +Theorem Intersection_is_Glb: + (A: (Ensemble U)) (a, b: (Ensemble U)) (Included U a A) -> (Included U b A) -> + (Glb + (Ensemble U) + (Power_set_PO A) + (Couple (Ensemble U) a b) + (Intersection U a b)). +Intros A a b H' H'0. +Apply Glb_definition; Simpl. +Apply Lower_Bound_definition; Simpl; Auto with sets. +Apply Definition_of_Power_set. +Generalize Inclusion_is_transitive; Intro IT; Red in IT; Apply IT with a; Auto with sets. +Intros y H'1; Elim H'1; Auto with sets. +Intros y H'1; Elim H'1; Simpl; Auto with sets. +Qed. + +End The_power_set_partial_order. + +Hints Resolve Empty_set_minimal : sets v62. +Hints Resolve Power_set_Inhabited : sets v62. +Hints Resolve Inclusion_is_an_order : sets v62. +Hints Resolve Inclusion_is_transitive : sets v62. +Hints Resolve Union_minimal : sets v62. +Hints Resolve Union_increases_l : sets v62. +Hints Resolve Union_increases_r : sets v62. +Hints Resolve Intersection_decreases_l : sets v62. +Hints Resolve Intersection_decreases_r : sets v62. +Hints Resolve Empty_set_is_Bottom : sets v62. +Hints Resolve Strict_inclusion_is_transitive : sets v62. diff --git a/theories7/Sets/Powerset_Classical_facts.v b/theories7/Sets/Powerset_Classical_facts.v new file mode 100755 index 00000000..1a51c562 --- /dev/null +++ b/theories7/Sets/Powerset_Classical_facts.v @@ -0,0 +1,338 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Powerset_Classical_facts.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Export Ensembles. +Require Export Constructive_sets. +Require Export Relations_1. +Require Export Relations_1_facts. +Require Export Partial_Order. +Require Export Cpo. +Require Export Powerset. +Require Export Powerset_facts. +Require Export Classical_Type. +Require Export Classical_sets. + +Section Sets_as_an_algebra. + +Variable U: Type. + +Lemma sincl_add_x: + (A, B: (Ensemble U)) + (x: U) ~ (In U A x) -> (Strict_Included U (Add U A x) (Add U B x)) -> + (Strict_Included U A B). +Proof. +Intros A B x H' H'0; Red. +LApply (Strict_Included_inv U (Add U A x) (Add U B x)); Auto with sets. +Clear H'0; Intro H'0; Split. +Apply incl_add_x with x := x; Tauto. +Elim H'0; Intros H'1 H'2; Elim H'2; Clear H'0 H'2. +Intros x0 H'0. +Red; Intro H'2. +Elim H'0; Clear H'0. +Rewrite <- H'2; Auto with sets. +Qed. + +Lemma incl_soustr_in: + (X: (Ensemble U)) (x: U) (In U X x) -> (Included U (Subtract U X x) X). +Proof. +Intros X x H'; Red. +Intros x0 H'0; Elim H'0; Auto with sets. +Qed. +Hints Resolve incl_soustr_in : sets v62. + +Lemma incl_soustr: + (X, Y: (Ensemble U)) (x: U) (Included U X Y) -> + (Included U (Subtract U X x) (Subtract U Y x)). +Proof. +Intros X Y x H'; Red. +Intros x0 H'0; Elim H'0. +Intros H'1 H'2. +Apply Subtract_intro; Auto with sets. +Qed. +Hints Resolve incl_soustr : sets v62. + + +Lemma incl_soustr_add_l: + (X: (Ensemble U)) (x: U) (Included U (Subtract U (Add U X x) x) X). +Proof. +Intros X x; Red. +Intros x0 H'; Elim H'; Auto with sets. +Intro H'0; Elim H'0; Auto with sets. +Intros t H'1 H'2; Elim H'2; Auto with sets. +Qed. +Hints Resolve incl_soustr_add_l : sets v62. + +Lemma incl_soustr_add_r: + (X: (Ensemble U)) (x: U) ~ (In U X x) -> + (Included U X (Subtract U (Add U X x) x)). +Proof. +Intros X x H'; Red. +Intros x0 H'0; Try Assumption. +Apply Subtract_intro; Auto with sets. +Red; Intro H'1; Apply H'; Rewrite H'1; Auto with sets. +Qed. +Hints Resolve incl_soustr_add_r : sets v62. + +Lemma add_soustr_2: + (X: (Ensemble U)) (x: U) (In U X x) -> + (Included U X (Add U (Subtract U X x) x)). +Proof. +Intros X x H'; Red. +Intros x0 H'0; Try Assumption. +Elim (classic x == x0); Intro K; Auto with sets. +Elim K; Auto with sets. +Qed. + +Lemma add_soustr_1: + (X: (Ensemble U)) (x: U) (In U X x) -> + (Included U (Add U (Subtract U X x) x) X). +Proof. +Intros X x H'; Red. +Intros x0 H'0; Elim H'0; Auto with sets. +Intros y H'1; Elim H'1; Auto with sets. +Intros t H'1; Try Assumption. +Rewrite <- (Singleton_inv U x t); Auto with sets. +Qed. +Hints Resolve add_soustr_1 add_soustr_2 : sets v62. + +Lemma add_soustr_xy: + (X: (Ensemble U)) (x, y: U) ~ x == y -> + (Subtract U (Add U X x) y) == (Add U (Subtract U X y) x). +Proof. +Intros X x y H'; Apply Extensionality_Ensembles. +Split; Red. +Intros x0 H'0; Elim H'0; Auto with sets. +Intro H'1; Elim H'1. +Intros u H'2 H'3; Try Assumption. +Apply Add_intro1. +Apply Subtract_intro; Auto with sets. +Intros t H'2 H'3; Try Assumption. +Elim (Singleton_inv U x t); Auto with sets. +Intros u H'2; Try Assumption. +Elim (Add_inv U (Subtract U X y) x u); Auto with sets. +Intro H'0; Elim H'0; Auto with sets. +Intro H'0; Rewrite <- H'0; Auto with sets. +Qed. +Hints Resolve add_soustr_xy : sets v62. + +Lemma incl_st_add_soustr: + (X, Y: (Ensemble U)) (x: U) ~ (In U X x) -> + (Strict_Included U (Add U X x) Y) -> + (Strict_Included U X (Subtract U Y x)). +Proof. +Intros X Y x H' H'0; Apply sincl_add_x with x := x; Auto with sets. +Split. +Elim H'0. +Intros H'1 H'2. +Generalize (Inclusion_is_transitive U). +Intro H'4; Red in H'4. +Apply H'4 with y := Y; Auto with sets. +Red in H'0. +Elim H'0; Intros H'1 H'2; Try Exact H'1; Clear H'0. (* PB *) +Red; Intro H'0; Apply H'2. +Rewrite H'0; Auto 8 with sets. +Qed. + +Lemma Sub_Add_new: + (X: (Ensemble U)) (x: U) ~ (In U X x) -> X == (Subtract U (Add U X x) x). +Proof. +Auto with sets. +Qed. + +Lemma Simplify_add: + (X, X0 : (Ensemble U)) (x: U) + ~ (In U X x) -> ~ (In U X0 x) -> (Add U X x) == (Add U X0 x) -> X == X0. +Proof. +Intros X X0 x H' H'0 H'1; Try Assumption. +Rewrite (Sub_Add_new X x); Auto with sets. +Rewrite (Sub_Add_new X0 x); Auto with sets. +Rewrite H'1; Auto with sets. +Qed. + +Lemma Included_Add: + (X, A: (Ensemble U)) (x: U) (Included U X (Add U A x)) -> + (Included U X A) \/ + (EXT A' | X == (Add U A' x) /\ (Included U A' A)). +Proof. +Intros X A x H'0; Try Assumption. +Elim (classic (In U X x)). +Intro H'1; Right; Try Assumption. +Exists (Subtract U X x). +Split; Auto with sets. +Red in H'0. +Red. +Intros x0 H'2; Try Assumption. +LApply (Subtract_inv U X x x0); Auto with sets. +Intro H'3; Elim H'3; Intros K K'; Clear H'3. +LApply (H'0 x0); Auto with sets. +Intro H'3; Try Assumption. +LApply (Add_inv U A x x0); Auto with sets. +Intro H'4; Elim H'4; + [Intro H'5; Try Exact H'5; Clear H'4 | Intro H'5; Clear H'4]. +Elim K'; Auto with sets. +Intro H'1; Left; Try Assumption. +Red in H'0. +Red. +Intros x0 H'2; Try Assumption. +LApply (H'0 x0); Auto with sets. +Intro H'3; Try Assumption. +LApply (Add_inv U A x x0); Auto with sets. +Intro H'4; Elim H'4; + [Intro H'5; Try Exact H'5; Clear H'4 | Intro H'5; Clear H'4]. +Absurd (In U X x0); Auto with sets. +Rewrite <- H'5; Auto with sets. +Qed. + +Lemma setcover_inv: + (A: (Ensemble U)) + (x, y: (Ensemble U)) (covers (Ensemble U) (Power_set_PO U A) y x) -> + (Strict_Included U x y) /\ + ((z: (Ensemble U)) (Included U x z) -> (Included U z y) -> x == z \/ z == y). +Proof. +Intros A x y H'; Elim H'. +Unfold Strict_Rel_of; Simpl. +Intros H'0 H'1; Split; [Auto with sets | Idtac]. +Intros z H'2 H'3; Try Assumption. +Elim (classic x == z); Auto with sets. +Intro H'4; Right; Try Assumption. +Elim (classic z == y); Auto with sets. +Intro H'5; Try Assumption. +Elim H'1. +Exists z; Auto with sets. +Qed. + +Theorem Add_covers: + (A: (Ensemble U)) (a: (Ensemble U)) (Included U a A) -> + (x: U) (In U A x) -> ~ (In U a x) -> + (covers (Ensemble U) (Power_set_PO U A) (Add U a x) a). +Proof. +Intros A a H' x H'0 H'1; Try Assumption. +Apply setcover_intro; Auto with sets. +Red. +Split; [Idtac | Red; Intro H'2; Try Exact H'2]; Auto with sets. +Apply H'1. +Rewrite H'2; Auto with sets. +Red; Intro H'2; Elim H'2; Clear H'2. +Intros z H'2; Elim H'2; Intros H'3 H'4; Try Exact H'3; Clear H'2. +LApply (Strict_Included_inv U a z); Auto with sets; Clear H'3. +Intro H'2; Elim H'2; Intros H'3 H'5; Elim H'5; Clear H'2 H'5. +Intros x0 H'2; Elim H'2. +Intros H'5 H'6; Try Assumption. +Generalize H'4; Intro K. +Red in H'4. +Elim H'4; Intros H'8 H'9; Red in H'8; Clear H'4. +LApply (H'8 x0); Auto with sets. +Intro H'7; Try Assumption. +Elim (Add_inv U a x x0); Auto with sets. +Intro H'15. +Cut (Included U (Add U a x) z). +Intro H'10; Try Assumption. +Red in K. +Elim K; Intros H'11 H'12; Apply H'12; Clear K; Auto with sets. +Rewrite H'15. +Red. +Intros x1 H'10; Elim H'10; Auto with sets. +Intros x2 H'11; Elim H'11; Auto with sets. +Qed. + +Theorem covers_Add: + (A: (Ensemble U)) + (a, a': (Ensemble U)) + (Included U a A) -> + (Included U a' A) -> (covers (Ensemble U) (Power_set_PO U A) a' a) -> + (EXT x | a' == (Add U a x) /\ ((In U A x) /\ ~ (In U a x))). +Proof. +Intros A a a' H' H'0 H'1; Try Assumption. +Elim (setcover_inv A a a'); Auto with sets. +Intros H'6 H'7. +Clear H'1. +Elim (Strict_Included_inv U a a'); Auto with sets. +Intros H'5 H'8; Elim H'8. +Intros x H'1; Elim H'1. +Intros H'2 H'3; Try Assumption. +Exists x. +Split; [Try Assumption | Idtac]. +Clear H'8 H'1. +Elim (H'7 (Add U a x)); Auto with sets. +Intro H'1. +Absurd a ==(Add U a x); Auto with sets. +Red; Intro H'8; Try Exact H'8. +Apply H'3. +Rewrite H'8; Auto with sets. +Auto with sets. +Red. +Intros x0 H'1; Elim H'1; Auto with sets. +Intros x1 H'8; Elim H'8; Auto with sets. +Split; [Idtac | Try Assumption]. +Red in H'0; Auto with sets. +Qed. + +Theorem covers_is_Add: + (A: (Ensemble U)) + (a, a': (Ensemble U)) (Included U a A) -> (Included U a' A) -> + (iff + (covers (Ensemble U) (Power_set_PO U A) a' a) + (EXT x | a' == (Add U a x) /\ ((In U A x) /\ ~ (In U a x)))). +Proof. +Intros A a a' H' H'0; Split; Intro K. +Apply covers_Add with A := A; Auto with sets. +Elim K. +Intros x H'1; Elim H'1; Intros H'2 H'3; Rewrite H'2; Clear H'1. +Apply Add_covers; Intuition. +Qed. + +Theorem Singleton_atomic: + (x:U) (A:(Ensemble U)) (In U A x) -> + (covers (Ensemble U) (Power_set_PO U A) (Singleton U x) (Empty_set U)). +Intros x A H'. +Rewrite <- (Empty_set_zero' U x). +Apply Add_covers; Auto with sets. +Qed. + +Lemma less_than_singleton: + (X:(Ensemble U)) (x:U) (Strict_Included U X (Singleton U x)) -> + X ==(Empty_set U). +Intros X x H'; Try Assumption. +Red in H'. +LApply (Singleton_atomic x (Full_set U)); + [Intro H'2; Try Exact H'2 | Apply Full_intro]. +Elim H'; Intros H'0 H'1; Try Exact H'1; Clear H'. +Elim (setcover_inv (Full_set U) (Empty_set U) (Singleton U x)); + [Intros H'6 H'7; Try Exact H'7 | Idtac]; Auto with sets. +Elim (H'7 X); [Intro H'5; Try Exact H'5 | Intro H'5 | Idtac | Idtac]; Auto with sets. +Elim H'1; Auto with sets. +Qed. + +End Sets_as_an_algebra. + +Hints Resolve incl_soustr_in : sets v62. +Hints Resolve incl_soustr : sets v62. +Hints Resolve incl_soustr_add_l : sets v62. +Hints Resolve incl_soustr_add_r : sets v62. +Hints Resolve add_soustr_1 add_soustr_2 : sets v62. +Hints Resolve add_soustr_xy : sets v62. diff --git a/theories7/Sets/Powerset_facts.v b/theories7/Sets/Powerset_facts.v new file mode 100755 index 00000000..fbe7d93e --- /dev/null +++ b/theories7/Sets/Powerset_facts.v @@ -0,0 +1,276 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Powerset_facts.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Require Export Ensembles. +Require Export Constructive_sets. +Require Export Relations_1. +Require Export Relations_1_facts. +Require Export Partial_Order. +Require Export Cpo. +Require Export Powerset. + +Section Sets_as_an_algebra. +Variable U: Type. +Hints Unfold not. + +Theorem Empty_set_zero : + (X: (Ensemble U)) (Union U (Empty_set U) X) == X. +Proof. +Auto 6 with sets. +Qed. +Hints Resolve Empty_set_zero. + +Theorem Empty_set_zero' : + (x: U) (Add U (Empty_set U) x) == (Singleton U x). +Proof. +Unfold 1 Add; Auto with sets. +Qed. +Hints Resolve Empty_set_zero'. + +Lemma less_than_empty : + (X: (Ensemble U)) (Included U X (Empty_set U)) -> X == (Empty_set U). +Proof. +Auto with sets. +Qed. +Hints Resolve less_than_empty. + +Theorem Union_commutative : + (A,B: (Ensemble U)) (Union U A B) == (Union U B A). +Proof. +Auto with sets. +Qed. + +Theorem Union_associative : + (A, B, C: (Ensemble U)) + (Union U (Union U A B) C) == (Union U A (Union U B C)). +Proof. +Auto 9 with sets. +Qed. +Hints Resolve Union_associative. + +Theorem Union_idempotent : (A: (Ensemble U)) (Union U A A) == A. +Proof. +Auto 7 with sets. +Qed. + +Lemma Union_absorbs : + (A, B: (Ensemble U)) (Included U B A) -> (Union U A B) == A. +Proof. +Auto 7 with sets. +Qed. + +Theorem Couple_as_union: + (x, y: U) (Union U (Singleton U x) (Singleton U y)) == (Couple U x y). +Proof. +Intros x y; Apply Extensionality_Ensembles; Split; Red. +Intros x0 H'; Elim H'; (Intros x1 H'0; Elim H'0; Auto with sets). +Intros x0 H'; Elim H'; Auto with sets. +Qed. + +Theorem Triple_as_union : + (x, y, z: U) + (Union U (Union U (Singleton U x) (Singleton U y)) (Singleton U z)) == + (Triple U x y z). +Proof. +Intros x y z; Apply Extensionality_Ensembles; Split; Red. +Intros x0 H'; Elim H'. +Intros x1 H'0; Elim H'0; (Intros x2 H'1; Elim H'1; Auto with sets). +Intros x1 H'0; Elim H'0; Auto with sets. +Intros x0 H'; Elim H'; Auto with sets. +Qed. + +Theorem Triple_as_Couple : (x, y: U) (Couple U x y) == (Triple U x x y). +Proof. +Intros x y. +Rewrite <- (Couple_as_union x y). +Rewrite <- (Union_idempotent (Singleton U x)). +Apply Triple_as_union. +Qed. + +Theorem Triple_as_Couple_Singleton : + (x, y, z: U) (Triple U x y z) == (Union U (Couple U x y) (Singleton U z)). +Proof. +Intros x y z. +Rewrite <- (Triple_as_union x y z). +Rewrite <- (Couple_as_union x y); Auto with sets. +Qed. + +Theorem Intersection_commutative : + (A,B: (Ensemble U)) (Intersection U A B) == (Intersection U B A). +Proof. +Intros A B. +Apply Extensionality_Ensembles. +Split; Red; Intros x H'; Elim H'; Auto with sets. +Qed. + +Theorem Distributivity : + (A, B, C: (Ensemble U)) + (Intersection U A (Union U B C)) == + (Union U (Intersection U A B) (Intersection U A C)). +Proof. +Intros A B C. +Apply Extensionality_Ensembles. +Split; Red; Intros x H'. +Elim H'. +Intros x0 H'0 H'1; Generalize H'0. +Elim H'1; Auto with sets. +Elim H'; Intros x0 H'0; Elim H'0; Auto with sets. +Qed. + +Theorem Distributivity' : + (A, B, C: (Ensemble U)) + (Union U A (Intersection U B C)) == + (Intersection U (Union U A B) (Union U A C)). +Proof. +Intros A B C. +Apply Extensionality_Ensembles. +Split; Red; Intros x H'. +Elim H'; Auto with sets. +Intros x0 H'0; Elim H'0; Auto with sets. +Elim H'. +Intros x0 H'0; Elim H'0; Auto with sets. +Intros x1 H'1 H'2; Try Exact H'2. +Generalize H'1. +Elim H'2; Auto with sets. +Qed. + +Theorem Union_add : + (A, B: (Ensemble U)) (x: U) + (Add U (Union U A B) x) == (Union U A (Add U B x)). +Proof. +Unfold Add; Auto with sets. +Qed. +Hints Resolve Union_add. + +Theorem Non_disjoint_union : + (X: (Ensemble U)) (x: U) (In U X x) -> (Add U X x) == X. +Intros X x H'; Unfold Add. +Apply Extensionality_Ensembles; Red. +Split; Red; Auto with sets. +Intros x0 H'0; Elim H'0; Auto with sets. +Intros t H'1; Elim H'1; Auto with sets. +Qed. + +Theorem Non_disjoint_union' : + (X: (Ensemble U)) (x: U) ~ (In U X x) -> (Subtract U X x) == X. +Proof. +Intros X x H'; Unfold Subtract. +Apply Extensionality_Ensembles. +Split; Red; Auto with sets. +Intros x0 H'0; Elim H'0; Auto with sets. +Intros x0 H'0; Apply Setminus_intro; Auto with sets. +Red; Intro H'1; Elim H'1. +LApply (Singleton_inv U x x0); Auto with sets. +Intro H'4; Apply H'; Rewrite H'4; Auto with sets. +Qed. + +Lemma singlx : (x, y: U) (In U (Add U (Empty_set U) x) y) -> x == y. +Proof. +Intro x; Rewrite (Empty_set_zero' x); Auto with sets. +Qed. +Hints Resolve singlx. + +Lemma incl_add : + (A, B: (Ensemble U)) (x: U) (Included U A B) -> + (Included U (Add U A x) (Add U B x)). +Proof. +Intros A B x H'; Red; Auto with sets. +Intros x0 H'0. +LApply (Add_inv U A x x0); Auto with sets. +Intro H'1; Elim H'1; + [Intro H'2; Clear H'1 | Intro H'2; Rewrite <- H'2; Clear H'1]; Auto with sets. +Qed. +Hints Resolve incl_add. + +Lemma incl_add_x : + (A, B: (Ensemble U)) + (x: U) ~ (In U A x) -> (Included U (Add U A x) (Add U B x)) -> + (Included U A B). +Proof. +Unfold Included. +Intros A B x H' H'0 x0 H'1. +LApply (H'0 x0); Auto with sets. +Intro H'2; LApply (Add_inv U B x x0); Auto with sets. +Intro H'3; Elim H'3; + [Intro H'4; Try Exact H'4; Clear H'3 | Intro H'4; Clear H'3]. +Absurd (In U A x0); Auto with sets. +Rewrite <- H'4; Auto with sets. +Qed. + +Lemma Add_commutative : + (A: (Ensemble U)) (x, y: U) (Add U (Add U A x) y) == (Add U (Add U A y) x). +Proof. +Intros A x y. +Unfold Add. +Rewrite (Union_associative A (Singleton U x) (Singleton U y)). +Rewrite (Union_commutative (Singleton U x) (Singleton U y)). +Rewrite <- (Union_associative A (Singleton U y) (Singleton U x)); Auto with sets. +Qed. + +Lemma Add_commutative' : + (A: (Ensemble U)) (x, y, z: U) + (Add U (Add U (Add U A x) y) z) == (Add U (Add U (Add U A z) x) y). +Proof. +Intros A x y z. +Rewrite (Add_commutative (Add U A x) y z). +Rewrite (Add_commutative A x z); Auto with sets. +Qed. + +Lemma Add_distributes : + (A, B: (Ensemble U)) (x, y: U) (Included U B A) -> + (Add U (Add U A x) y) == (Union U (Add U A x) (Add U B y)). +Proof. +Intros A B x y H'; Try Assumption. +Rewrite <- (Union_add (Add U A x) B y). +Unfold 4 Add. +Rewrite (Union_commutative A (Singleton U x)). +Rewrite Union_associative. +Rewrite (Union_absorbs A B H'). +Rewrite (Union_commutative (Singleton U x) A). +Auto with sets. +Qed. + +Lemma setcover_intro : + (U: Type) + (A: (Ensemble U)) + (x, y: (Ensemble U)) + (Strict_Included U x y) -> + ~ (EXT z | (Strict_Included U x z) + /\ (Strict_Included U z y)) -> + (covers (Ensemble U) (Power_set_PO U A) y x). +Proof. +Intros; Apply Definition_of_covers; Auto with sets. +Qed. +Hints Resolve setcover_intro. + +End Sets_as_an_algebra. + +Hints Resolve Empty_set_zero Empty_set_zero' Union_associative Union_add + singlx incl_add : sets v62. + + diff --git a/theories7/Sets/Relations_1.v b/theories7/Sets/Relations_1.v new file mode 100755 index 00000000..d4ed823b --- /dev/null +++ b/theories7/Sets/Relations_1.v @@ -0,0 +1,67 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Relations_1.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*) + +Section Relations_1. + Variable U: Type. + + Definition Relation := U -> U -> Prop. + Variable R: Relation. + + Definition Reflexive : Prop := (x: U) (R x x). + + Definition Transitive : Prop := (x,y,z: U) (R x y) -> (R y z) -> (R x z). + + Definition Symmetric : Prop := (x,y: U) (R x y) -> (R y x). + + Definition Antisymmetric : Prop := + (x: U) (y: U) (R x y) -> (R y x) -> x == y. + + Definition contains : Relation -> Relation -> Prop := + [R,R': Relation] (x: U) (y: U) (R' x y) -> (R x y). + + Definition same_relation : Relation -> Relation -> Prop := + [R,R': Relation] (contains R R') /\ (contains R' R). + + Inductive Preorder : Prop := + Definition_of_preorder: Reflexive -> Transitive -> Preorder. + + Inductive Order : Prop := + Definition_of_order: Reflexive -> Transitive -> Antisymmetric -> Order. + + Inductive Equivalence : Prop := + Definition_of_equivalence: + Reflexive -> Transitive -> Symmetric -> Equivalence. + + Inductive PER : Prop := + Definition_of_PER: Symmetric -> Transitive -> PER. + +End Relations_1. +Hints Unfold Reflexive Transitive Antisymmetric Symmetric contains + same_relation : sets v62. +Hints Resolve Definition_of_preorder Definition_of_order + Definition_of_equivalence Definition_of_PER : sets v62. diff --git a/theories7/Sets/Relations_1_facts.v b/theories7/Sets/Relations_1_facts.v new file mode 100755 index 00000000..cf73ce8b --- /dev/null +++ b/theories7/Sets/Relations_1_facts.v @@ -0,0 +1,109 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Relations_1_facts.v,v 1.1.2.1 2004/07/16 19:31:40 herbelin Exp $ i*) + +Require Export Relations_1. + +Definition Complement : (U: Type) (Relation U) -> (Relation U) := + [U: Type] [R: (Relation U)] [x,y: U] ~ (R x y). + +Theorem Rsym_imp_notRsym: (U: Type) (R: (Relation U)) (Symmetric U R) -> + (Symmetric U (Complement U R)). +Proof. +Unfold Symmetric Complement. +Intros U R H' x y H'0; Red; Intro H'1; Apply H'0; Auto with sets. +Qed. + +Theorem Equiv_from_preorder : + (U: Type) (R: (Relation U)) (Preorder U R) -> + (Equivalence U [x,y: U] (R x y) /\ (R y x)). +Proof. +Intros U R H'; Elim H'; Intros H'0 H'1. +Apply Definition_of_equivalence. +Red in H'0; Auto 10 with sets. +2:Red; Intros x y h; Elim h; Intros H'3 H'4; Auto 10 with sets. +Red in H'1; Red; Auto 10 with sets. +Intros x y z h; Elim h; Intros H'3 H'4; Clear h. +Intro h; Elim h; Intros H'5 H'6; Clear h. +Split; Apply H'1 with y; Auto 10 with sets. +Qed. +Hints Resolve Equiv_from_preorder. + +Theorem Equiv_from_order : + (U: Type) (R: (Relation U)) (Order U R) -> + (Equivalence U [x,y: U] (R x y) /\ (R y x)). +Proof. +Intros U R H'; Elim H'; Auto 10 with sets. +Qed. +Hints Resolve Equiv_from_order. + +Theorem contains_is_preorder : + (U: Type) (Preorder (Relation U) (contains U)). +Proof. +Auto 10 with sets. +Qed. +Hints Resolve contains_is_preorder. + +Theorem same_relation_is_equivalence : + (U: Type) (Equivalence (Relation U) (same_relation U)). +Proof. +Unfold 1 same_relation; Auto 10 with sets. +Qed. +Hints Resolve same_relation_is_equivalence. + +Theorem cong_reflexive_same_relation: + (U:Type) (R, R':(Relation U)) (same_relation U R R') -> (Reflexive U R) -> + (Reflexive U R'). +Proof. +Unfold same_relation; Intuition. +Qed. + +Theorem cong_symmetric_same_relation: + (U:Type) (R, R':(Relation U)) (same_relation U R R') -> (Symmetric U R) -> + (Symmetric U R'). +Proof. + Compute;Intros;Elim H;Intros;Clear H;Apply (H3 y x (H0 x y (H2 x y H1))). +(*Intuition.*) +Qed. + +Theorem cong_antisymmetric_same_relation: + (U:Type) (R, R':(Relation U)) (same_relation U R R') -> + (Antisymmetric U R) -> (Antisymmetric U R'). +Proof. + Compute;Intros;Elim H;Intros;Clear H;Apply (H0 x y (H3 x y H1) (H3 y x H2)). +(*Intuition.*) +Qed. + +Theorem cong_transitive_same_relation: + (U:Type) (R, R':(Relation U)) (same_relation U R R') -> (Transitive U R) -> + (Transitive U R'). +Proof. +Intros U R R' H' H'0; Red. +Elim H'. +Intros H'1 H'2 x y z H'3 H'4; Apply H'2. +Apply H'0 with y; Auto with sets. +Qed. diff --git a/theories7/Sets/Relations_2.v b/theories7/Sets/Relations_2.v new file mode 100755 index 00000000..92a1236e --- /dev/null +++ b/theories7/Sets/Relations_2.v @@ -0,0 +1,56 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Relations_2.v,v 1.1.2.1 2004/07/16 19:31:40 herbelin Exp $ i*) + +Require Export Relations_1. + +Section Relations_2. +Variable U: Type. +Variable R: (Relation U). + +Inductive Rstar : (Relation U) := + Rstar_0: (x: U) (Rstar x x) + | Rstar_n: (x, y, z: U) (R x y) -> (Rstar y z) -> (Rstar x z). + +Inductive Rstar1 : (Relation U) := + Rstar1_0: (x: U) (Rstar1 x x) + | Rstar1_1: (x: U) (y: U) (R x y) -> (Rstar1 x y) + | Rstar1_n: (x, y, z: U) (Rstar1 x y) -> (Rstar1 y z) -> (Rstar1 x z). + +Inductive Rplus : (Relation U) := + Rplus_0: (x, y: U) (R x y) -> (Rplus x y) + | Rplus_n: (x, y, z: U) (R x y) -> (Rplus y z) -> (Rplus x z). + +Definition Strongly_confluent : Prop := + (x, a, b: U) (R x a) -> (R x b) -> (exT U [z: U] (R a z) /\ (R b z)). + +End Relations_2. + +Hints Resolve Rstar_0 : sets v62. +Hints Resolve Rstar1_0 : sets v62. +Hints Resolve Rstar1_1 : sets v62. +Hints Resolve Rplus_0 : sets v62. diff --git a/theories7/Sets/Relations_2_facts.v b/theories7/Sets/Relations_2_facts.v new file mode 100755 index 00000000..b82438eb --- /dev/null +++ b/theories7/Sets/Relations_2_facts.v @@ -0,0 +1,151 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Relations_2_facts.v,v 1.1.2.1 2004/07/16 19:31:40 herbelin Exp $ i*) + +Require Export Relations_1. +Require Export Relations_1_facts. +Require Export Relations_2. + +Theorem Rstar_reflexive : + (U: Type) (R: (Relation U)) (Reflexive U (Rstar U R)). +Proof. +Auto with sets. +Qed. + +Theorem Rplus_contains_R : + (U: Type) (R: (Relation U)) (contains U (Rplus U R) R). +Proof. +Auto with sets. +Qed. + +Theorem Rstar_contains_R : + (U: Type) (R: (Relation U)) (contains U (Rstar U R) R). +Proof. +Intros U R; Red; Intros x y H'; Apply Rstar_n with y; Auto with sets. +Qed. + +Theorem Rstar_contains_Rplus : + (U: Type) (R: (Relation U)) (contains U (Rstar U R) (Rplus U R)). +Proof. +Intros U R; Red. +Intros x y H'; Elim H'. +Generalize Rstar_contains_R; Intro T; Red in T; Auto with sets. +Intros x0 y0 z H'0 H'1 H'2; Apply Rstar_n with y0; Auto with sets. +Qed. + +Theorem Rstar_transitive : + (U: Type) (R: (Relation U)) (Transitive U (Rstar U R)). +Proof. +Intros U R; Red. +Intros x y z H'; Elim H'; Auto with sets. +Intros x0 y0 z0 H'0 H'1 H'2 H'3; Apply Rstar_n with y0; Auto with sets. +Qed. + +Theorem Rstar_cases : + (U: Type) (R: (Relation U)) (x, y: U) (Rstar U R x y) -> + x == y \/ (EXT u | (R x u) /\ (Rstar U R u y)). +Proof. +Intros U R x y H'; Elim H'; Auto with sets. +Intros x0 y0 z H'0 H'1 H'2; Right; Exists y0; Auto with sets. +Qed. + +Theorem Rstar_equiv_Rstar1 : + (U: Type) (R: (Relation U)) (same_relation U (Rstar U R) (Rstar1 U R)). +Proof. +Generalize Rstar_contains_R; Intro T; Red in T. +Intros U R; Unfold same_relation contains. +Split; Intros x y H'; Elim H'; Auto with sets. +Generalize Rstar_transitive; Intro T1; Red in T1. +Intros x0 y0 z H'0 H'1 H'2 H'3; Apply T1 with y0; Auto with sets. +Intros x0 y0 z H'0 H'1 H'2; Apply Rstar1_n with y0; Auto with sets. +Qed. + +Theorem Rsym_imp_Rstarsym : + (U: Type) (R: (Relation U)) (Symmetric U R) -> (Symmetric U (Rstar U R)). +Proof. +Intros U R H'; Red. +Intros x y H'0; Elim H'0; Auto with sets. +Intros x0 y0 z H'1 H'2 H'3. +Generalize Rstar_transitive; Intro T1; Red in T1. +Apply T1 with y0; Auto with sets. +Apply Rstar_n with x0; Auto with sets. +Qed. + +Theorem Sstar_contains_Rstar : + (U: Type) (R, S: (Relation U)) (contains U (Rstar U S) R) -> + (contains U (Rstar U S) (Rstar U R)). +Proof. +Unfold contains. +Intros U R S H' x y H'0; Elim H'0; Auto with sets. +Generalize Rstar_transitive; Intro T1; Red in T1. +Intros x0 y0 z H'1 H'2 H'3; Apply T1 with y0; Auto with sets. +Qed. + +Theorem star_monotone : + (U: Type) (R, S: (Relation U)) (contains U S R) -> + (contains U (Rstar U S) (Rstar U R)). +Proof. +Intros U R S H'. +Apply Sstar_contains_Rstar; Auto with sets. +Generalize (Rstar_contains_R U S); Auto with sets. +Qed. + +Theorem RstarRplus_RRstar : + (U: Type) (R: (Relation U)) (x, y, z: U) + (Rstar U R x y) -> (Rplus U R y z) -> + (EXT u | (R x u) /\ (Rstar U R u z)). +Proof. +Generalize Rstar_contains_Rplus; Intro T; Red in T. +Generalize Rstar_transitive; Intro T1; Red in T1. +Intros U R x y z H'; Elim H'. +Intros x0 H'0; Elim H'0. +Intros x1 y0 H'1; Exists y0; Auto with sets. +Intros x1 y0 z0 H'1 H'2 H'3; Exists y0; Auto with sets. +Intros x0 y0 z0 H'0 H'1 H'2 H'3; Exists y0. +Split; [Try Assumption | Idtac]. +Apply T1 with z0; Auto with sets. +Qed. + +Theorem Lemma1 : + (U: Type) (R: (Relation U)) (Strongly_confluent U R) -> + (x, b: U) (Rstar U R x b) -> + (a: U) (R x a) -> (EXT z | (Rstar U R a z) /\ (R b z)). +Proof. +Intros U R H' x b H'0; Elim H'0. +Intros x0 a H'1; Exists a; Auto with sets. +Intros x0 y z H'1 H'2 H'3 a H'4. +Red in H'. +Specialize 3 H' with x := x0 a := a b := y; Intro H'7; LApply H'7; + [Intro H'8; LApply H'8; + [Intro H'9; Try Exact H'9; Clear H'8 H'7 | Clear H'8 H'7] | Clear H'7]; Auto with sets. +Elim H'9. +Intros t H'5; Elim H'5; Intros H'6 H'7; Try Exact H'6; Clear H'5. +Elim (H'3 t); Auto with sets. +Intros z1 H'5; Elim H'5; Intros H'8 H'10; Try Exact H'8; Clear H'5. +Exists z1; Split; [Idtac | Assumption]. +Apply Rstar_n with t; Auto with sets. +Qed. diff --git a/theories7/Sets/Relations_3.v b/theories7/Sets/Relations_3.v new file mode 100755 index 00000000..092fc534 --- /dev/null +++ b/theories7/Sets/Relations_3.v @@ -0,0 +1,63 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Relations_3.v,v 1.1.2.1 2004/07/16 19:31:40 herbelin Exp $ i*) + +Require Export Relations_1. +Require Export Relations_2. + +Section Relations_3. + Variable U: Type. + Variable R: (Relation U). + + Definition coherent : U -> U -> Prop := + [x,y: U] (EXT z | (Rstar U R x z) /\ (Rstar U R y z)). + + Definition locally_confluent : U -> Prop := + [x: U] (y,z: U) (R x y) -> (R x z) -> (coherent y z). + + Definition Locally_confluent : Prop := (x: U) (locally_confluent x). + + Definition confluent : U -> Prop := + [x: U] (y,z: U) (Rstar U R x y) -> (Rstar U R x z) -> (coherent y z). + + Definition Confluent : Prop := (x: U) (confluent x). + + Inductive noetherian : U -> Prop := + definition_of_noetherian: + (x: U) ((y: U) (R x y) -> (noetherian y)) -> (noetherian x). + + Definition Noetherian : Prop := (x: U) (noetherian x). + +End Relations_3. +Hints Unfold coherent : sets v62. +Hints Unfold locally_confluent : sets v62. +Hints Unfold confluent : sets v62. +Hints Unfold Confluent : sets v62. +Hints Resolve definition_of_noetherian : sets v62. +Hints Unfold Noetherian : sets v62. + + diff --git a/theories7/Sets/Relations_3_facts.v b/theories7/Sets/Relations_3_facts.v new file mode 100755 index 00000000..822f550a --- /dev/null +++ b/theories7/Sets/Relations_3_facts.v @@ -0,0 +1,157 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(****************************************************************************) +(* *) +(* Naive Set Theory in Coq *) +(* *) +(* INRIA INRIA *) +(* Rocquencourt Sophia-Antipolis *) +(* *) +(* Coq V6.1 *) +(* *) +(* Gilles Kahn *) +(* Gerard Huet *) +(* *) +(* *) +(* *) +(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks *) +(* to the Newton Institute for providing an exceptional work environment *) +(* in Summer 1995. Several developments by E. Ledinot were an inspiration. *) +(****************************************************************************) + +(*i $Id: Relations_3_facts.v,v 1.1.2.1 2004/07/16 19:31:40 herbelin Exp $ i*) + +Require Export Relations_1. +Require Export Relations_1_facts. +Require Export Relations_2. +Require Export Relations_2_facts. +Require Export Relations_3. + +Theorem Rstar_imp_coherent : + (U: Type) (R: (Relation U)) (x: U) (y: U) (Rstar U R x y) -> + (coherent U R x y). +Proof. +Intros U R x y H'; Red. +Exists y; Auto with sets. +Qed. +Hints Resolve Rstar_imp_coherent. + +Theorem coherent_symmetric : + (U: Type) (R: (Relation U)) (Symmetric U (coherent U R)). +Proof. +Unfold 1 coherent. +Intros U R; Red. +Intros x y H'; Elim H'. +Intros z H'0; Exists z; Tauto. +Qed. + +Theorem Strong_confluence : + (U: Type) (R: (Relation U)) (Strongly_confluent U R) -> (Confluent U R). +Proof. +Intros U R H'; Red. +Intro x; Red; Intros a b H'0. +Unfold 1 coherent. +Generalize b; Clear b. +Elim H'0; Clear H'0. +Intros x0 b H'1; Exists b; Auto with sets. +Intros x0 y z H'1 H'2 H'3 b H'4. +Generalize (Lemma1 U R); Intro h; LApply h; + [Intro H'0; Generalize (H'0 x0 b); Intro h0; LApply h0; + [Intro H'5; Generalize (H'5 y); Intro h1; LApply h1; + [Intro h2; Elim h2; Intros z0 h3; Elim h3; Intros H'6 H'7; + Clear h h0 h1 h2 h3 | Clear h h0 h1] | Clear h h0] | Clear h]; Auto with sets. +Generalize (H'3 z0); Intro h; LApply h; + [Intro h0; Elim h0; Intros z1 h1; Elim h1; Intros H'8 H'9; Clear h h0 h1 | + Clear h]; Auto with sets. +Exists z1; Split; Auto with sets. +Apply Rstar_n with z0; Auto with sets. +Qed. + +Theorem Strong_confluence_direct : + (U: Type) (R: (Relation U)) (Strongly_confluent U R) -> (Confluent U R). +Proof. +Intros U R H'; Red. +Intro x; Red; Intros a b H'0. +Unfold 1 coherent. +Generalize b; Clear b. +Elim H'0; Clear H'0. +Intros x0 b H'1; Exists b; Auto with sets. +Intros x0 y z H'1 H'2 H'3 b H'4. +Cut (exT U [t: U] (Rstar U R y t) /\ (R b t)). +Intro h; Elim h; Intros t h0; Elim h0; Intros H'0 H'5; Clear h h0. +Generalize (H'3 t); Intro h; LApply h; + [Intro h0; Elim h0; Intros z0 h1; Elim h1; Intros H'6 H'7; Clear h h0 h1 | + Clear h]; Auto with sets. +Exists z0; Split; [Assumption | Idtac]. +Apply Rstar_n with t; Auto with sets. +Generalize H'1; Generalize y; Clear H'1. +Elim H'4. +Intros x1 y0 H'0; Exists y0; Auto with sets. +Intros x1 y0 z0 H'0 H'1 H'5 y1 H'6. +Red in H'. +Generalize (H' x1 y0 y1); Intro h; LApply h; + [Intro H'7; LApply H'7; + [Intro h0; Elim h0; Intros z1 h1; Elim h1; Intros H'8 H'9; Clear h H'7 h0 h1 | + Clear h] | Clear h]; Auto with sets. +Generalize (H'5 z1); Intro h; LApply h; + [Intro h0; Elim h0; Intros t h1; Elim h1; Intros H'7 H'10; Clear h h0 h1 | + Clear h]; Auto with sets. +Exists t; Split; Auto with sets. +Apply Rstar_n with z1; Auto with sets. +Qed. + +Theorem Noetherian_contains_Noetherian : + (U: Type) (R, R': (Relation U)) (Noetherian U R) -> (contains U R R') -> + (Noetherian U R'). +Proof. +Unfold 2 Noetherian. +Intros U R R' H' H'0 x. +Elim (H' x); Auto with sets. +Qed. + +Theorem Newman : + (U: Type) (R: (Relation U)) (Noetherian U R) -> (Locally_confluent U R) -> + (Confluent U R). +Proof. +Intros U R H' H'0; Red; Intro x. +Elim (H' x); Unfold confluent. +Intros x0 H'1 H'2 y z H'3 H'4. +Generalize (Rstar_cases U R x0 y); Intro h; LApply h; + [Intro h0; Elim h0; + [Clear h h0; Intro h1 | + Intro h1; Elim h1; Intros u h2; Elim h2; Intros H'5 H'6; Clear h h0 h1 h2] | + Clear h]; Auto with sets. +Elim h1; Auto with sets. +Generalize (Rstar_cases U R x0 z); Intro h; LApply h; + [Intro h0; Elim h0; + [Clear h h0; Intro h1 | + Intro h1; Elim h1; Intros v h2; Elim h2; Intros H'7 H'8; Clear h h0 h1 h2] | + Clear h]; Auto with sets. +Elim h1; Generalize coherent_symmetric; Intro t; Red in t; Auto with sets. +Unfold Locally_confluent locally_confluent coherent in H'0. +Generalize (H'0 x0 u v); Intro h; LApply h; + [Intro H'9; LApply H'9; + [Intro h0; Elim h0; Intros t h1; Elim h1; Intros H'10 H'11; + Clear h H'9 h0 h1 | Clear h] | Clear h]; Auto with sets. +Clear H'0. +Unfold 1 coherent in H'2. +Generalize (H'2 u); Intro h; LApply h; + [Intro H'0; Generalize (H'0 y t); Intro h0; LApply h0; + [Intro H'9; LApply H'9; + [Intro h1; Elim h1; Intros y1 h2; Elim h2; Intros H'12 H'13; + Clear h h0 H'9 h1 h2 | Clear h h0] | Clear h h0] | Clear h]; Auto with sets. +Generalize Rstar_transitive; Intro T; Red in T. +Generalize (H'2 v); Intro h; LApply h; + [Intro H'9; Generalize (H'9 y1 z); Intro h0; LApply h0; + [Intro H'14; LApply H'14; + [Intro h1; Elim h1; Intros z1 h2; Elim h2; Intros H'15 H'16; + Clear h h0 H'14 h1 h2 | Clear h h0] | Clear h h0] | Clear h]; Auto with sets. +Red; (Exists z1; Split); Auto with sets. +Apply T with y1; Auto with sets. +Apply T with t; Auto with sets. +Qed. diff --git a/theories7/Sets/Uniset.v b/theories7/Sets/Uniset.v new file mode 100644 index 00000000..33880214 --- /dev/null +++ b/theories7/Sets/Uniset.v @@ -0,0 +1,212 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Uniset.v,v 1.1.2.1 2004/07/16 19:31:40 herbelin Exp $ i*) + +(** Sets as characteristic functions *) + +(* G. Huet 1-9-95 *) +(* Updated Papageno 12/98 *) + +Require Bool. + +Set Implicit Arguments. + +Section defs. + +Variable A : Set. +Variable eqA : A -> A -> Prop. +Hypothesis eqA_dec : (x,y:A){(eqA x y)}+{~(eqA x y)}. + +Inductive uniset : Set := + Charac : (A->bool) -> uniset. + +Definition charac : uniset -> A -> bool := + [s:uniset][a:A]Case s of [f:A->bool](f a) end. + +Definition Emptyset := (Charac [a:A]false). + +Definition Fullset := (Charac [a:A]true). + +Definition Singleton := [a:A](Charac [a':A] + Case (eqA_dec a a') of + [h:(eqA a a')] true + [h: ~(eqA a a')] false end). + +Definition In : uniset -> A -> Prop := + [s:uniset][a:A](charac s a)=true. +Hints Unfold In. + +(** uniset inclusion *) +Definition incl := [s1,s2:uniset] + (a:A)(leb (charac s1 a) (charac s2 a)). +Hints Unfold incl. + +(** uniset equality *) +Definition seq := [s1,s2:uniset] + (a:A)(charac s1 a) = (charac s2 a). +Hints Unfold seq. + +Lemma leb_refl : (b:bool)(leb b b). +Proof. +NewDestruct b; Simpl; Auto. +Qed. +Hints Resolve leb_refl. + +Lemma incl_left : (s1,s2:uniset)(seq s1 s2)->(incl s1 s2). +Proof. +Unfold incl; Intros s1 s2 E a; Elim (E a); Auto. +Qed. + +Lemma incl_right : (s1,s2:uniset)(seq s1 s2)->(incl s2 s1). +Proof. +Unfold incl; Intros s1 s2 E a; Elim (E a); Auto. +Qed. + +Lemma seq_refl : (x:uniset)(seq x x). +Proof. +NewDestruct x; Unfold seq; Auto. +Qed. +Hints Resolve seq_refl. + +Lemma seq_trans : (x,y,z:uniset)(seq x y)->(seq y z)->(seq x z). +Proof. +Unfold seq. +NewDestruct x; NewDestruct y; NewDestruct z; Simpl; Intros. +Rewrite H; Auto. +Qed. + +Lemma seq_sym : (x,y:uniset)(seq x y)->(seq y x). +Proof. +Unfold seq. +NewDestruct x; NewDestruct y; Simpl; Auto. +Qed. + +(** uniset union *) +Definition union := [m1,m2:uniset] + (Charac [a:A](orb (charac m1 a)(charac m2 a))). + +Lemma union_empty_left : + (x:uniset)(seq x (union Emptyset x)). +Proof. +Unfold seq; Unfold union; Simpl; Auto. +Qed. +Hints Resolve union_empty_left. + +Lemma union_empty_right : + (x:uniset)(seq x (union x Emptyset)). +Proof. +Unfold seq; Unfold union; Simpl. +Intros x a; Rewrite (orb_b_false (charac x a)); Auto. +Qed. +Hints Resolve union_empty_right. + +Lemma union_comm : (x,y:uniset)(seq (union x y) (union y x)). +Proof. +Unfold seq; Unfold charac; Unfold union. +NewDestruct x; NewDestruct y; Auto with bool. +Qed. +Hints Resolve union_comm. + +Lemma union_ass : + (x,y,z:uniset)(seq (union (union x y) z) (union x (union y z))). +Proof. +Unfold seq; Unfold union; Unfold charac. +NewDestruct x; NewDestruct y; NewDestruct z; Auto with bool. +Qed. +Hints Resolve union_ass. + +Lemma seq_left : (x,y,z:uniset)(seq x y)->(seq (union x z) (union y z)). +Proof. +Unfold seq; Unfold union; Unfold charac. +NewDestruct x; NewDestruct y; NewDestruct z. +Intros; Elim H; Auto. +Qed. +Hints Resolve seq_left. + +Lemma seq_right : (x,y,z:uniset)(seq x y)->(seq (union z x) (union z y)). +Proof. +Unfold seq; Unfold union; Unfold charac. +NewDestruct x; NewDestruct y; NewDestruct z. +Intros; Elim H; Auto. +Qed. +Hints Resolve seq_right. + + +(** All the proofs that follow duplicate [Multiset_of_A] *) + +(** Here we should make uniset an abstract datatype, by hiding [Charac], + [union], [charac]; all further properties are proved abstractly *) + +Require Permut. + +Lemma union_rotate : + (x,y,z:uniset)(seq (union x (union y z)) (union z (union x y))). +Proof. +Intros; Apply (op_rotate uniset union seq); Auto. +Exact seq_trans. +Qed. + +Lemma seq_congr : (x,y,z,t:uniset)(seq x y)->(seq z t)-> + (seq (union x z) (union y t)). +Proof. +Intros; Apply (cong_congr uniset union seq); Auto. +Exact seq_trans. +Qed. + +Lemma union_perm_left : + (x,y,z:uniset)(seq (union x (union y z)) (union y (union x z))). +Proof. +Intros; Apply (perm_left uniset union seq); Auto. +Exact seq_trans. +Qed. + +Lemma uniset_twist1 : (x,y,z,t:uniset) + (seq (union x (union (union y z) t)) (union (union y (union x t)) z)). +Proof. +Intros; Apply (twist uniset union seq); Auto. +Exact seq_trans. +Qed. + +Lemma uniset_twist2 : (x,y,z,t:uniset) + (seq (union x (union (union y z) t)) (union (union y (union x z)) t)). +Proof. +Intros; Apply seq_trans with (union (union x (union y z)) t). +Apply seq_sym; Apply union_ass. +Apply seq_left; Apply union_perm_left. +Qed. + +(** specific for treesort *) + +Lemma treesort_twist1 : (x,y,z,t,u:uniset) (seq u (union y z)) -> + (seq (union x (union u t)) (union (union y (union x t)) z)). +Proof. +Intros; Apply seq_trans with (union x (union (union y z) t)). +Apply seq_right; Apply seq_left; Trivial. +Apply uniset_twist1. +Qed. + +Lemma treesort_twist2 : (x,y,z,t,u:uniset) (seq u (union y z)) -> + (seq (union x (union u t)) (union (union y (union x z)) t)). +Proof. +Intros; Apply seq_trans with (union x (union (union y z) t)). +Apply seq_right; Apply seq_left; Trivial. +Apply uniset_twist2. +Qed. + + +(*i theory of minter to do similarly +Require Min. +(* uniset intersection *) +Definition minter := [m1,m2:uniset] + (Charac [a:A](andb (charac m1 a)(charac m2 a))). +i*) + +End defs. + +Unset Implicit Arguments. diff --git a/theories7/Sorting/Heap.v b/theories7/Sorting/Heap.v new file mode 100644 index 00000000..63e7f324 --- /dev/null +++ b/theories7/Sorting/Heap.v @@ -0,0 +1,223 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Heap.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*) + +(** A development of Treesort on Heap trees *) + +(* G. Huet 1-9-95 uses Multiset *) + +Require PolyList. +Require Multiset. +Require Permutation. +Require Relations. +Require Sorting. + + +Section defs. + +Variable A : Set. +Variable leA : (relation A). +Variable eqA : (relation A). + +Local gtA := [x,y:A]~(leA x y). + +Hypothesis leA_dec : (x,y:A){(leA x y)}+{(leA y x)}. +Hypothesis eqA_dec : (x,y:A){(eqA x y)}+{~(eqA x y)}. +Hypothesis leA_refl : (x,y:A) (eqA x y) -> (leA x y). +Hypothesis leA_trans : (x,y,z:A) (leA x y) -> (leA y z) -> (leA x z). +Hypothesis leA_antisym : (x,y:A)(leA x y) -> (leA y x) -> (eqA x y). + +Hints Resolve leA_refl. +Hints Immediate eqA_dec leA_dec leA_antisym. + +Local emptyBag := (EmptyBag A). +Local singletonBag := (SingletonBag eqA_dec). + +Inductive Tree : Set := + Tree_Leaf : Tree + | Tree_Node : A -> Tree -> Tree -> Tree. + +(** [a] is lower than a Tree [T] if [T] is a Leaf + or [T] is a Node holding [b>a] *) + +Definition leA_Tree := [a:A; t:Tree] + Cases t of + Tree_Leaf => True + | (Tree_Node b T1 T2) => (leA a b) + end. + +Lemma leA_Tree_Leaf : (a:A)(leA_Tree a Tree_Leaf). +Proof. +Simpl; Auto with datatypes. +Qed. + +Lemma leA_Tree_Node : (a,b:A)(G,D:Tree)(leA a b) -> + (leA_Tree a (Tree_Node b G D)). +Proof. +Simpl; Auto with datatypes. +Qed. + +Hints Resolve leA_Tree_Leaf leA_Tree_Node. + + +(** The heap property *) + +Inductive is_heap : Tree -> Prop := + nil_is_heap : (is_heap Tree_Leaf) + | node_is_heap : (a:A)(T1,T2:Tree) + (leA_Tree a T1) -> + (leA_Tree a T2) -> + (is_heap T1) -> (is_heap T2) -> + (is_heap (Tree_Node a T1 T2)). + +Hint constr_is_heap := Constructors is_heap. + +Lemma invert_heap : (a:A)(T1,T2:Tree)(is_heap (Tree_Node a T1 T2))-> + (leA_Tree a T1) /\ (leA_Tree a T2) /\ + (is_heap T1) /\ (is_heap T2). +Proof. +Intros; Inversion H; Auto with datatypes. +Qed. + +(* This lemma ought to be generated automatically by the Inversion tools *) +Lemma is_heap_rec : (P:Tree->Set) + (P Tree_Leaf)-> + ((a:A) + (T1:Tree) + (T2:Tree) + (leA_Tree a T1)-> + (leA_Tree a T2)-> + (is_heap T1)-> + (P T1)->(is_heap T2)->(P T2)->(P (Tree_Node a T1 T2))) + -> (T:Tree)(is_heap T) -> (P T). +Proof. +Induction T; Auto with datatypes. +Intros a G PG D PD PN. +Elim (invert_heap a G D); Auto with datatypes. +Intros H1 H2; Elim H2; Intros H3 H4; Elim H4; Intros. +Apply H0; Auto with datatypes. +Qed. + +Lemma low_trans : + (T:Tree)(a,b:A)(leA a b) -> (leA_Tree b T) -> (leA_Tree a T). +Proof. +Induction T; Auto with datatypes. +Intros; Simpl; Apply leA_trans with b; Auto with datatypes. +Qed. + +(** contents of a tree as a multiset *) + +(** Nota Bene : In what follows the definition of SingletonBag + in not used. Actually, we could just take as postulate: + [Parameter SingletonBag : A->multiset]. *) + +Fixpoint contents [t:Tree] : (multiset A) := + Cases t of + Tree_Leaf => emptyBag + | (Tree_Node a t1 t2) => (munion (contents t1) + (munion (contents t2) (singletonBag a))) +end. + + +(** equivalence of two trees is equality of corresponding multisets *) + +Definition equiv_Tree := [t1,t2:Tree](meq (contents t1) (contents t2)). + + +(** specification of heap insertion *) + +Inductive insert_spec [a:A; T:Tree] : Set := + insert_exist : (T1:Tree)(is_heap T1) -> + (meq (contents T1) (munion (contents T) (singletonBag a))) -> + ((b:A)(leA b a)->(leA_Tree b T)->(leA_Tree b T1)) -> + (insert_spec a T). + + +Lemma insert : (T:Tree)(is_heap T) -> (a:A)(insert_spec a T). +Proof. +Induction 1; Intros. +Apply insert_exist with (Tree_Node a Tree_Leaf Tree_Leaf); Auto with datatypes. +Simpl; Unfold meq munion; Auto with datatypes. +Elim (leA_dec a a0); Intros. +Elim (H3 a0); Intros. +Apply insert_exist with (Tree_Node a T2 T0); Auto with datatypes. +Simpl; Apply treesort_twist1; Trivial with datatypes. +Elim (H3 a); Intros T3 HeapT3 ConT3 LeA. +Apply insert_exist with (Tree_Node a0 T2 T3); Auto with datatypes. +Apply node_is_heap; Auto with datatypes. +Apply low_trans with a; Auto with datatypes. +Apply LeA; Auto with datatypes. +Apply low_trans with a; Auto with datatypes. +Simpl; Apply treesort_twist2; Trivial with datatypes. +Qed. + +(** building a heap from a list *) + +Inductive build_heap [l:(list A)] : Set := + heap_exist : (T:Tree)(is_heap T) -> + (meq (list_contents eqA_dec l)(contents T)) -> + (build_heap l). + +Lemma list_to_heap : (l:(list A))(build_heap l). +Proof. +Induction l. +Apply (heap_exist (nil A) Tree_Leaf); Auto with datatypes. +Simpl; Unfold meq; Auto with datatypes. +Induction 1. +Intros T i m; Elim (insert T i a). +Intros; Apply heap_exist with T1; Simpl; Auto with datatypes. +Apply meq_trans with (munion (contents T) (singletonBag a)). +Apply meq_trans with (munion (singletonBag a) (contents T)). +Apply meq_right; Trivial with datatypes. +Apply munion_comm. +Apply meq_sym; Trivial with datatypes. +Qed. + + +(** building the sorted list *) + +Inductive flat_spec [T:Tree] : Set := + flat_exist : (l:(list A))(sort leA l) -> + ((a:A)(leA_Tree a T)->(lelistA leA a l)) -> + (meq (contents T) (list_contents eqA_dec l)) -> + (flat_spec T). + +Lemma heap_to_list : (T:Tree)(is_heap T) -> (flat_spec T). +Proof. + Intros T h; Elim h; Intros. + Apply flat_exist with (nil A); Auto with datatypes. + Elim H2; Intros l1 s1 i1 m1; Elim H4; Intros l2 s2 i2 m2. + Elim (merge leA_dec eqA_dec s1 s2); Intros. + Apply flat_exist with (cons a l); Simpl; Auto with datatypes. + Apply meq_trans with + (munion (list_contents eqA_dec l1) (munion (list_contents eqA_dec l2) + (singletonBag a))). + Apply meq_congr; Auto with datatypes. + Apply meq_trans with + (munion (singletonBag a) (munion (list_contents eqA_dec l1) + (list_contents eqA_dec l2))). + Apply munion_rotate. + Apply meq_right; Apply meq_sym; Trivial with datatypes. +Qed. + +(** specification of treesort *) + +Theorem treesort : (l:(list A)) + {m:(list A) | (sort leA m) & (permutation eqA_dec l m)}. +Proof. + Intro l; Unfold permutation. + Elim (list_to_heap l). + Intros. + Elim (heap_to_list T); Auto with datatypes. + Intros. + Exists l0; Auto with datatypes. + Apply meq_trans with (contents T); Trivial with datatypes. +Qed. + +End defs. diff --git a/theories7/Sorting/Permutation.v b/theories7/Sorting/Permutation.v new file mode 100644 index 00000000..46b8da00 --- /dev/null +++ b/theories7/Sorting/Permutation.v @@ -0,0 +1,111 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Permutation.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*) + +Require Relations. +Require PolyList. +Require Multiset. + +Set Implicit Arguments. + +Section defs. + +Variable A : Set. +Variable leA : (relation A). +Variable eqA : (relation A). + +Local gtA := [x,y:A]~(leA x y). + +Hypothesis leA_dec : (x,y:A){(leA x y)}+{~(leA x y)}. +Hypothesis eqA_dec : (x,y:A){(eqA x y)}+{~(eqA x y)}. +Hypothesis leA_refl : (x,y:A) (eqA x y) -> (leA x y). +Hypothesis leA_trans : (x,y,z:A) (leA x y) -> (leA y z) -> (leA x z). +Hypothesis leA_antisym : (x,y:A)(leA x y) -> (leA y x) -> (eqA x y). + +Hints Resolve leA_refl : default. +Hints Immediate eqA_dec leA_dec leA_antisym : default. + +Local emptyBag := (EmptyBag A). +Local singletonBag := (SingletonBag eqA_dec). + +(** contents of a list *) + +Fixpoint list_contents [l:(list A)] : (multiset A) := + Cases l of + nil => emptyBag + | (cons a l) => (munion (singletonBag a) (list_contents l)) + end. + +Lemma list_contents_app : (l,m:(list A)) + (meq (list_contents (app l m)) (munion (list_contents l) (list_contents m))). +Proof. +Induction l; Simpl; Auto with datatypes. +Intros. +Apply meq_trans with + (munion (singletonBag a) (munion (list_contents l0) (list_contents m))); Auto with datatypes. +Qed. +Hints Resolve list_contents_app. + +Definition permutation := [l,m:(list A)](meq (list_contents l) (list_contents m)). + +Lemma permut_refl : (l:(list A))(permutation l l). +Proof. +Unfold permutation; Auto with datatypes. +Qed. +Hints Resolve permut_refl. + +Lemma permut_tran : (l,m,n:(list A)) + (permutation l m) -> (permutation m n) -> (permutation l n). +Proof. +Unfold permutation; Intros. +Apply meq_trans with (list_contents m); Auto with datatypes. +Qed. + +Lemma permut_right : (l,m:(list A)) + (permutation l m) -> (a:A)(permutation (cons a l) (cons a m)). +Proof. +Unfold permutation; Simpl; Auto with datatypes. +Qed. +Hints Resolve permut_right. + +Lemma permut_app : (l,l',m,m':(list A)) + (permutation l l') -> (permutation m m') -> + (permutation (app l m) (app l' m')). +Proof. +Unfold permutation; Intros. +Apply meq_trans with (munion (list_contents l) (list_contents m)); Auto with datatypes. +Apply meq_trans with (munion (list_contents l') (list_contents m')); Auto with datatypes. +Apply meq_trans with (munion (list_contents l') (list_contents m)); Auto with datatypes. +Qed. +Hints Resolve permut_app. + +Lemma permut_cons : (l,m:(list A))(permutation l m) -> + (a:A)(permutation (cons a l) (cons a m)). +Proof. +Intros l m H a. +Change (permutation (app (cons a (nil A)) l) (app (cons a (nil A)) m)). +Apply permut_app; Auto with datatypes. +Qed. +Hints Resolve permut_cons. + +Lemma permut_middle : (l,m:(list A)) + (a:A)(permutation (cons a (app l m)) (app l (cons a m))). +Proof. +Unfold permutation. +Induction l; Simpl; Auto with datatypes. +Intros. +Apply meq_trans with (munion (singletonBag a) + (munion (singletonBag a0) (list_contents (app l0 m)))); Auto with datatypes. +Apply munion_perm_left; Auto with datatypes. +Qed. +Hints Resolve permut_middle. + +End defs. +Unset Implicit Arguments. + diff --git a/theories7/Sorting/Sorting.v b/theories7/Sorting/Sorting.v new file mode 100644 index 00000000..a6e38976 --- /dev/null +++ b/theories7/Sorting/Sorting.v @@ -0,0 +1,117 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Sorting.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*) + +Require PolyList. +Require Multiset. +Require Permutation. +Require Relations. + +Set Implicit Arguments. + +Section defs. + +Variable A : Set. +Variable leA : (relation A). +Variable eqA : (relation A). + +Local gtA := [x,y:A]~(leA x y). + +Hypothesis leA_dec : (x,y:A){(leA x y)}+{(leA y x)}. +Hypothesis eqA_dec : (x,y:A){(eqA x y)}+{~(eqA x y)}. +Hypothesis leA_refl : (x,y:A) (eqA x y) -> (leA x y). +Hypothesis leA_trans : (x,y,z:A) (leA x y) -> (leA y z) -> (leA x z). +Hypothesis leA_antisym : (x,y:A)(leA x y) -> (leA y x) -> (eqA x y). + +Hints Resolve leA_refl. +Hints Immediate eqA_dec leA_dec leA_antisym. + +Local emptyBag := (EmptyBag A). +Local singletonBag := (SingletonBag eqA_dec). + +(** [lelistA] *) + +Inductive lelistA [a:A] : (list A) -> Prop := + nil_leA : (lelistA a (nil A)) + | cons_leA : (b:A)(l:(list A))(leA a b)->(lelistA a (cons b l)). +Hint constr_lelistA := Constructors lelistA. + +Lemma lelistA_inv : (a,b:A)(l:(list A)) + (lelistA a (cons b l)) -> (leA a b). +Proof. + Intros; Inversion H; Trivial with datatypes. +Qed. + +(** definition for a list to be sorted *) + +Inductive sort : (list A) -> Prop := + nil_sort : (sort (nil A)) + | cons_sort : (a:A)(l:(list A))(sort l) -> (lelistA a l) -> (sort (cons a l)). +Hint constr_sort := Constructors sort. + +Lemma sort_inv : (a:A)(l:(list A))(sort (cons a l))->(sort l) /\ (lelistA a l). +Proof. +Intros; Inversion H; Auto with datatypes. +Qed. + +Lemma sort_rec : (P:(list A)->Set) + (P (nil A)) -> + ((a:A)(l:(list A))(sort l)->(P l)->(lelistA a l)->(P (cons a l))) -> + (y:(list A))(sort y) -> (P y). +Proof. +Induction y; Auto with datatypes. +Intros; Elim (!sort_inv a l); Auto with datatypes. +Qed. + +(** merging two sorted lists *) + +Inductive merge_lem [l1:(list A);l2:(list A)] : Set := + merge_exist : (l:(list A))(sort l) -> + (meq (list_contents eqA_dec l) + (munion (list_contents eqA_dec l1) (list_contents eqA_dec l2))) -> + ((a:A)(lelistA a l1)->(lelistA a l2)->(lelistA a l)) -> + (merge_lem l1 l2). + +Lemma merge : (l1:(list A))(sort l1)->(l2:(list A))(sort l2)->(merge_lem l1 l2). +Proof. + Induction 1; Intros. + Apply merge_exist with l2; Auto with datatypes. + Elim H3; Intros. + Apply merge_exist with (cons a l); Simpl; Auto with datatypes. + Elim (leA_dec a a0); Intros. + +(* 1 (leA a a0) *) + Cut (merge_lem l (cons a0 l0)); Auto with datatypes. + Intros (l3, l3sorted, l3contents, Hrec). + Apply merge_exist with (cons a l3); Simpl; Auto with datatypes. + Apply meq_trans with (munion (singletonBag a) + (munion (list_contents eqA_dec l) + (list_contents eqA_dec (cons a0 l0)))). + Apply meq_right; Trivial with datatypes. + Apply meq_sym; Apply munion_ass. + Intros; Apply cons_leA. + Apply lelistA_inv with l; Trivial with datatypes. + +(* 2 (leA a0 a) *) + Elim H5; Simpl; Intros. + Apply merge_exist with (cons a0 l3); Simpl; Auto with datatypes. + Apply meq_trans with (munion (singletonBag a0) + (munion (munion (singletonBag a) + (list_contents eqA_dec l)) + (list_contents eqA_dec l0))). + Apply meq_right; Trivial with datatypes. + Apply munion_perm_left. + Intros; Apply cons_leA; Apply lelistA_inv with l0; Trivial with datatypes. +Qed. + +End defs. + +Unset Implicit Arguments. +Hint constr_sort : datatypes v62 := Constructors sort. +Hint constr_lelistA : datatypes v62 := Constructors lelistA. diff --git a/theories7/Wellfounded/Disjoint_Union.v b/theories7/Wellfounded/Disjoint_Union.v new file mode 100644 index 00000000..04930170 --- /dev/null +++ b/theories7/Wellfounded/Disjoint_Union.v @@ -0,0 +1,56 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Disjoint_Union.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*) + +(** Author: Cristina Cornes + From : Constructing Recursion Operators in Type Theory + L. Paulson JSC (1986) 2, 325-355 *) + +Require Relation_Operators. + +Section Wf_Disjoint_Union. +Variable A,B:Set. +Variable leA: A->A->Prop. +Variable leB: B->B->Prop. + +Notation Le_AsB := (le_AsB A B leA leB). + +Lemma acc_A_sum: (x:A)(Acc A leA x)->(Acc A+B Le_AsB (inl A B x)). +Proof. + NewInduction 1. + Apply Acc_intro;Intros y H2. + Inversion_clear H2. + Auto with sets. +Qed. + +Lemma acc_B_sum: (well_founded A leA) ->(x:B)(Acc B leB x) + ->(Acc A+B Le_AsB (inr A B x)). +Proof. + NewInduction 2. + Apply Acc_intro;Intros y H3. + Inversion_clear H3;Auto with sets. + Apply acc_A_sum;Auto with sets. +Qed. + + +Lemma wf_disjoint_sum: + (well_founded A leA) + -> (well_founded B leB) -> (well_founded A+B Le_AsB). +Proof. + Intros. + Unfold well_founded . + NewDestruct a as [a|b]. + Apply (acc_A_sum a). + Apply (H a). + + Apply (acc_B_sum H b). + Apply (H0 b). +Qed. + +End Wf_Disjoint_Union. diff --git a/theories7/Wellfounded/Inclusion.v b/theories7/Wellfounded/Inclusion.v new file mode 100644 index 00000000..6a515333 --- /dev/null +++ b/theories7/Wellfounded/Inclusion.v @@ -0,0 +1,33 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Inclusion.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*) + +(** Author: Bruno Barras *) + +Require Relation_Definitions. + +Section WfInclusion. + Variable A:Set. + Variable R1,R2:A->A->Prop. + + Lemma Acc_incl: (inclusion A R1 R2)->(z:A)(Acc A R2 z)->(Acc A R1 z). + Proof. + NewInduction 2. + Apply Acc_intro;Auto with sets. + Qed. + + Hints Resolve Acc_incl. + + Theorem wf_incl: + (inclusion A R1 R2)->(well_founded A R2)->(well_founded A R1). + Proof. + Unfold well_founded ;Auto with sets. + Qed. + +End WfInclusion. diff --git a/theories7/Wellfounded/Inverse_Image.v b/theories7/Wellfounded/Inverse_Image.v new file mode 100644 index 00000000..6c9c3e65 --- /dev/null +++ b/theories7/Wellfounded/Inverse_Image.v @@ -0,0 +1,58 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Inverse_Image.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*) + +(** Author: Bruno Barras *) + +Section Inverse_Image. + + Variables A,B:Set. + Variable R : B->B->Prop. + Variable f:A->B. + + Local Rof : A->A->Prop := [x,y:A](R (f x) (f y)). + + Remark Acc_lemma : (y:B)(Acc B R y)->(x:A)(y=(f x))->(Acc A Rof x). + NewInduction 1 as [y _ IHAcc]; Intros x H. + Apply Acc_intro; Intros y0 H1. + Apply (IHAcc (f y0)); Try Trivial. + Rewrite H; Trivial. + Qed. + + Lemma Acc_inverse_image : (x:A)(Acc B R (f x)) -> (Acc A Rof x). + Intros; Apply (Acc_lemma (f x)); Trivial. + Qed. + + Theorem wf_inverse_image: (well_founded B R)->(well_founded A Rof). + Red; Intros; Apply Acc_inverse_image; Auto. + Qed. + + Variable F : A -> B -> Prop. + Local RoF : A -> A -> Prop := [x,y] + (EX b : B | (F x b) & (c:B)(F y c)->(R b c)). + +Lemma Acc_inverse_rel : + (b:B)(Acc B R b)->(x:A)(F x b)->(Acc A RoF x). +NewInduction 1 as [x _ IHAcc]; Intros x0 H2. +Constructor; Intros y H3. +NewDestruct H3. +Apply (IHAcc x1); Auto. +Save. + + +Theorem wf_inverse_rel : + (well_founded B R)->(well_founded A RoF). + Red; Constructor; Intros. + Case H0; Intros. + Apply (Acc_inverse_rel x); Auto. +Save. + +End Inverse_Image. + + diff --git a/theories7/Wellfounded/Lexicographic_Exponentiation.v b/theories7/Wellfounded/Lexicographic_Exponentiation.v new file mode 100644 index 00000000..17f6d650 --- /dev/null +++ b/theories7/Wellfounded/Lexicographic_Exponentiation.v @@ -0,0 +1,386 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Lexicographic_Exponentiation.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*) + +(** Author: Cristina Cornes + + From : Constructing Recursion Operators in Type Theory + L. Paulson JSC (1986) 2, 325-355 *) + +Require Eqdep. +Require PolyList. +Require PolyListSyntax. +Require Relation_Operators. +Require Transitive_Closure. + +Section Wf_Lexicographic_Exponentiation. +Variable A:Set. +Variable leA: A->A->Prop. + +Notation Power := (Pow A leA). +Notation Lex_Exp := (lex_exp A leA). +Notation ltl := (Ltl A leA). +Notation Descl := (Desc A leA). + +Notation List := (list A). +Notation Nil := (nil A). +(* useless but symmetric *) +Notation Cons := (cons 1!A). +Notation "<< x , y >>" := (exist List Descl x y) (at level 0) + V8only (at level 0, x,y at level 100). + +V7only[ +Syntax constr + level 1: + List [ (list A) ] -> ["List"] + | Nil [ (nil A) ] -> ["Nil"] + | Cons [ (cons A) ] -> ["Cons"] + ; + level 10: + Cons2 [ (cons A $e $l) ] -> ["Cons " $e:L " " $l:L ]. + +Syntax constr + level 1: + pair_sig [ (exist (list A) Desc $e $d) ] -> ["<<" $e:L "," $d:L ">>"]. +]. +Hints Resolve d_one d_nil t_step. + +Lemma left_prefix : (x,y,z:List)(ltl x^y z)-> (ltl x z). +Proof. + Induction x. + Induction z. + Simpl;Intros H. + Inversion_clear H. + Simpl;Intros;Apply (Lt_nil A leA). + Intros a l HInd. + Simpl. + Intros. + Inversion_clear H. + Apply (Lt_hd A leA);Auto with sets. + Apply (Lt_tl A leA). + Apply (HInd y y0);Auto with sets. +Qed. + + +Lemma right_prefix : + (x,y,z:List)(ltl x y^z)-> (ltl x y) \/ (EX y':List | x=(y^y') /\ (ltl y' z)). +Proof. + Intros x y;Generalize x. + Elim y;Simpl. + Right. + Exists x0 ;Auto with sets. + Intros. + Inversion H0. + Left;Apply (Lt_nil A leA). + Left;Apply (Lt_hd A leA);Auto with sets. + Generalize (H x1 z H3) . + Induction 1. + Left;Apply (Lt_tl A leA);Auto with sets. + Induction 1. + Induction 1;Intros. + Rewrite -> H8. + Right;Exists x2 ;Auto with sets. +Qed. + + + +Lemma desc_prefix: (x:List)(a:A)(Descl x^(Cons a Nil))->(Descl x). +Proof. + Intros. + Inversion H. + Generalize (app_cons_not_nil H1); Induction 1. + Cut (x^(Cons a Nil))=(Cons x0 Nil); Auto with sets. + Intro. + Generalize (app_eq_unit H0) . + Induction 1; Induction 1; Intros. + Rewrite -> H4; Auto with sets. + Discriminate H5. + Generalize (app_inj_tail H0) . + Induction 1; Intros. + Rewrite <- H4; Auto with sets. +Qed. + +Lemma desc_tail: (x:List)(a,b:A) + (Descl (Cons b (x^(Cons a Nil))))-> (clos_trans A leA a b). +Proof. + Intro. + Apply rev_ind with A:=A + P:=[x:List](a,b:A) + (Descl (Cons b (x^(Cons a Nil))))-> (clos_trans A leA a b). + Intros. + + Inversion H. + Cut (Cons b (Cons a Nil))= ((Nil^(Cons b Nil))^ (Cons a Nil)); Auto with sets; Intro. + Generalize H0. + Intro. + Generalize (app_inj_tail 2!(l^(Cons y Nil)) 3!(Nil^(Cons b Nil)) H4); + Induction 1. + Intros. + + Generalize (app_inj_tail H6); Induction 1; Intros. + Generalize H1. + Rewrite <- H10; Rewrite <- H7; Intro. + Apply (t_step A leA); Auto with sets. + + + + Intros. + Inversion H0. + Generalize (app_cons_not_nil H3); Intro. + Elim H1. + + Generalize H0. + Generalize (app_comm_cons (l^(Cons x0 Nil)) (Cons a Nil) b); Induction 1. + Intro. + Generalize (desc_prefix (Cons b (l^(Cons x0 Nil))) a H5); Intro. + Generalize (H x0 b H6). + Intro. + Apply t_trans with A:=A y:=x0; Auto with sets. + + Apply t_step. + Generalize H1. + Rewrite -> H4; Intro. + + Generalize (app_inj_tail H8); Induction 1. + Intros. + Generalize H2; Generalize (app_comm_cons l (Cons x0 Nil) b). + Intro. + Generalize H10. + Rewrite ->H12; Intro. + Generalize (app_inj_tail H13); Induction 1. + Intros. + Rewrite <- H11; Rewrite <- H16; Auto with sets. +Qed. + + +Lemma dist_aux : (z:List)(Descl z)->(x,y:List)z=(x^y)->(Descl x)/\ (Descl y). +Proof. + Intros z D. + Elim D. + Intros. + Cut (x^y)=Nil;Auto with sets; Intro. + Generalize (app_eq_nil H0) ; Induction 1. + Intros. + Rewrite -> H2;Rewrite -> H3; Split;Apply d_nil. + + Intros. + Cut (x0^y)=(Cons x Nil); Auto with sets. + Intros E. + Generalize (app_eq_unit E); Induction 1. + Induction 1;Intros. + Rewrite -> H2;Rewrite -> H3; Split. + Apply d_nil. + + Apply d_one. + + Induction 1; Intros. + Rewrite -> H2;Rewrite -> H3; Split. + Apply d_one. + + Apply d_nil. + + Do 5 Intro. + Intros Hind. + Do 2 Intro. + Generalize x0 . + Apply rev_ind with A:=A + P:=[y0:List] + (x0:List) + ((l^(Cons y Nil))^(Cons x Nil))=(x0^y0)->(Descl x0)/\(Descl y0). + + Intro. + Generalize (app_nil_end x1) ; Induction 1; Induction 1. + Split. Apply d_conc; Auto with sets. + + Apply d_nil. + + Do 3 Intro. + Generalize x1 . + Apply rev_ind with + A:=A + P:=[l0:List] + (x1:A) + (x0:List) + ((l^(Cons y Nil))^(Cons x Nil))=(x0^(l0^(Cons x1 Nil))) + ->(Descl x0)/\(Descl (l0^(Cons x1 Nil))). + + + Simpl. + Split. + Generalize (app_inj_tail H2) ;Induction 1. + Induction 1;Auto with sets. + + Apply d_one. + Do 5 Intro. + Generalize (app_ass x4 (l1^(Cons x2 Nil)) (Cons x3 Nil)) . + Induction 1. + Generalize (app_ass x4 l1 (Cons x2 Nil)) ;Induction 1. + Intro E. + Generalize (app_inj_tail E) . + Induction 1;Intros. + Generalize (app_inj_tail H6) ;Induction 1;Intros. + Rewrite <- H7; Rewrite <- H10; Generalize H6. + Generalize (app_ass x4 l1 (Cons x2 Nil)); Intro E1. + Rewrite -> E1. + Intro. + Generalize (Hind x4 (l1^(Cons x2 Nil)) H11) . + Induction 1;Split. + Auto with sets. + + Generalize H14. + Rewrite <- H10; Intro. + Apply d_conc;Auto with sets. +Qed. + + + +Lemma dist_Desc_concat : (x,y:List)(Descl x^y)->(Descl x)/\(Descl y). +Proof. + Intros. + Apply (dist_aux (x^y) H x y); Auto with sets. +Qed. + + +Lemma desc_end:(a,b:A)(x:List) + (Descl x^(Cons a Nil)) /\ (ltl x^(Cons a Nil) (Cons b Nil)) + -> (clos_trans A leA a b). + +Proof. + Intros a b x. + Case x. + Simpl. + Induction 1. + Intros. + Inversion H1;Auto with sets. + Inversion H3. + + Induction 1. + Generalize (app_comm_cons l (Cons a Nil) a0). + Intros E; Rewrite <- E; Intros. + Generalize (desc_tail l a a0 H0); Intro. + Inversion H1. + Apply t_trans with y:=a0 ;Auto with sets. + + Inversion H4. +Qed. + + + + +Lemma ltl_unit: (x:List)(a,b:A) + (Descl (x^(Cons a Nil))) -> (ltl x^(Cons a Nil) (Cons b Nil)) + -> (ltl x (Cons b Nil)). +Proof. + Intro. + Case x. + Intros;Apply (Lt_nil A leA). + + Simpl;Intros. + Inversion_clear H0. + Apply (Lt_hd A leA a b);Auto with sets. + + Inversion_clear H1. +Qed. + + +Lemma acc_app: + (x1,x2:List)(y1:(Descl x1^x2)) + (Acc Power Lex_Exp (exist List Descl (x1^x2) y1)) + ->(x:List) + (y:(Descl x)) + (ltl x (x1^x2))->(Acc Power Lex_Exp (exist List Descl x y)). +Proof. + Intros. + Apply (Acc_inv Power Lex_Exp (exist List Descl (x1^x2) y1)). + Auto with sets. + + Unfold lex_exp ;Simpl;Auto with sets. +Qed. + + +Theorem wf_lex_exp : + (well_founded A leA)->(well_founded Power Lex_Exp). +Proof. + Unfold 2 well_founded . + Induction a;Intros x y. + Apply Acc_intro. + Induction y0. + Unfold 1 lex_exp ;Simpl. + Apply rev_ind with A:=A P:=[x:List] + (x0:List) + (y:(Descl x0)) + (ltl x0 x) + ->(Acc Power Lex_Exp (exist List Descl x0 y)) . + Intros. + Inversion_clear H0. + + Intro. + Generalize (well_founded_ind A (clos_trans A leA) (wf_clos_trans A leA H)). + Intros GR. + Apply GR with P:=[x0:A] + (l:List) + ((x1:List) + (y:(Descl x1)) + (ltl x1 l) + ->(Acc Power Lex_Exp (exist List Descl x1 y))) + ->(x1:List) + (y:(Descl x1)) + (ltl x1 (l^(Cons x0 Nil))) + ->(Acc Power Lex_Exp (exist List Descl x1 y)) . + Intro;Intros HInd; Intros. + Generalize (right_prefix x2 l (Cons x1 Nil) H1) . + Induction 1. + Intro; Apply (H0 x2 y1 H3). + + Induction 1. + Intro;Induction 1. + Clear H4 H2. + Intro;Generalize y1 ;Clear y1. + Rewrite -> H2. + Apply rev_ind with A:=A P:=[x3:List] + (y1:(Descl (l^x3))) + (ltl x3 (Cons x1 Nil)) + ->(Acc Power Lex_Exp + (exist List Descl (l^x3) y1)) . + Intros. + Generalize (app_nil_end l) ;Intros Heq. + Generalize y1 . + Clear y1. + Rewrite <- Heq. + Intro. + Apply Acc_intro. + Induction y2. + Unfold 1 lex_exp . + Simpl;Intros x4 y3. Intros. + Apply (H0 x4 y3);Auto with sets. + + Intros. + Generalize (dist_Desc_concat l (l0^(Cons x4 Nil)) y1) . + Induction 1. + Intros. + Generalize (desc_end x4 x1 l0 (conj ? ? H8 H5)) ; Intros. + Generalize y1 . + Rewrite <- (app_ass l l0 (Cons x4 Nil)); Intro. + Generalize (HInd x4 H9 (l^l0)) ; Intros HInd2. + Generalize (ltl_unit l0 x4 x1 H8 H5); Intro. + Generalize (dist_Desc_concat (l^l0) (Cons x4 Nil) y2) . + Induction 1;Intros. + Generalize (H4 H12 H10); Intro. + Generalize (Acc_inv Power Lex_Exp (exist List Descl (l^l0) H12) H14) . + Generalize (acc_app l l0 H12 H14). + Intros f g. + Generalize (HInd2 f);Intro. + Apply Acc_intro. + Induction y3. + Unfold 1 lex_exp ;Simpl; Intros. + Apply H15;Auto with sets. +Qed. + + +End Wf_Lexicographic_Exponentiation. diff --git a/theories7/Wellfounded/Lexicographic_Product.v b/theories7/Wellfounded/Lexicographic_Product.v new file mode 100644 index 00000000..f31d8c3f --- /dev/null +++ b/theories7/Wellfounded/Lexicographic_Product.v @@ -0,0 +1,191 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Lexicographic_Product.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +(** Authors: Bruno Barras, Cristina Cornes *) + +Require Eqdep. +Require Relation_Operators. +Require Transitive_Closure. + +(** From : Constructing Recursion Operators in Type Theory + L. Paulson JSC (1986) 2, 325-355 *) + +Section WfLexicographic_Product. +Variable A:Set. +Variable B:A->Set. +Variable leA: A->A->Prop. +Variable leB: (x:A)(B x)->(B x)->Prop. + +Notation LexProd := (lexprod A B leA leB). + +Hints Resolve t_step Acc_clos_trans wf_clos_trans. + +Lemma acc_A_B_lexprod : (x:A)(Acc A leA x) + ->((x0:A)(clos_trans A leA x0 x)->(well_founded (B x0) (leB x0))) + ->(y:(B x))(Acc (B x) (leB x) y) + ->(Acc (sigS A B) LexProd (existS A B x y)). +Proof. + NewInduction 1 as [x _ IHAcc]; Intros H2 y. + NewInduction 1 as [x0 H IHAcc0];Intros. + Apply Acc_intro. + NewDestruct y as [x2 y1]; Intro H6. + Simple Inversion H6; Intro. + Cut (leA x2 x);Intros. + Apply IHAcc;Auto with sets. + Intros. + Apply H2. + Apply t_trans with x2 ;Auto with sets. + + Red in H2. + Apply H2. + Auto with sets. + + Injection H1. + NewDestruct 2. + Injection H3. + NewDestruct 2;Auto with sets. + + Rewrite <- H1. + Injection H3; Intros _ Hx1. + Subst x1. + Apply IHAcc0. + Elim inj_pair2 with A B x y' x0; Assumption. +Qed. + +Theorem wf_lexprod: + (well_founded A leA) ->((x:A) (well_founded (B x) (leB x))) + -> (well_founded (sigS A B) LexProd). +Proof. + Intros wfA wfB;Unfold well_founded . + NewDestruct a. + Apply acc_A_B_lexprod;Auto with sets;Intros. + Red in wfB. + Auto with sets. +Qed. + + +End WfLexicographic_Product. + + +Section Wf_Symmetric_Product. + Variable A:Set. + Variable B:Set. + Variable leA: A->A->Prop. + Variable leB: B->B->Prop. + + Notation Symprod := (symprod A B leA leB). + +(*i + Local sig_prod:= + [x:A*B]<{_:A&B}>Case x of [a:A][b:B](existS A [_:A]B a b) end. + +Lemma incl_sym_lexprod: (included (A*B) Symprod + (R_o_f (A*B) {_:A&B} sig_prod (lexprod A [_:A]B leA [_:A]leB))). +Proof. + Red. + Induction x. + (Induction y1;Intros). + Red. + Unfold sig_prod . + Inversion_clear H. + (Apply left_lex;Auto with sets). + + (Apply right_lex;Auto with sets). +Qed. +i*) + + Lemma Acc_symprod: (x:A)(Acc A leA x)->(y:B)(Acc B leB y) + ->(Acc (A*B) Symprod (x,y)). + Proof. + NewInduction 1 as [x _ IHAcc]; Intros y H2. + NewInduction H2 as [x1 H3 IHAcc1]. + Apply Acc_intro;Intros y H5. + Inversion_clear H5;Auto with sets. + Apply IHAcc; Auto. + Apply Acc_intro;Trivial. +Qed. + + +Lemma wf_symprod: (well_founded A leA)->(well_founded B leB) + ->(well_founded (A*B) Symprod). +Proof. + Red. + NewDestruct a. + Apply Acc_symprod;Auto with sets. +Qed. + +End Wf_Symmetric_Product. + + +Section Swap. + + Variable A:Set. + Variable R:A->A->Prop. + + Notation SwapProd :=(swapprod A R). + + + Lemma swap_Acc: (x,y:A)(Acc A*A SwapProd (x,y))->(Acc A*A SwapProd (y,x)). +Proof. + Intros. + Inversion_clear H. + Apply Acc_intro. + NewDestruct y0;Intros. + Inversion_clear H;Inversion_clear H1;Apply H0. + Apply sp_swap. + Apply right_sym;Auto with sets. + + Apply sp_swap. + Apply left_sym;Auto with sets. + + Apply sp_noswap. + Apply right_sym;Auto with sets. + + Apply sp_noswap. + Apply left_sym;Auto with sets. +Qed. + + + Lemma Acc_swapprod: (x,y:A)(Acc A R x)->(Acc A R y) + ->(Acc A*A SwapProd (x,y)). +Proof. + NewInduction 1 as [x0 _ IHAcc0];Intros H2. + Cut (y0:A)(R y0 x0)->(Acc ? SwapProd (y0,y)). + Clear IHAcc0. + NewInduction H2 as [x1 _ IHAcc1]; Intros H4. + Cut (y:A)(R y x1)->(Acc ? SwapProd (x0,y)). + Clear IHAcc1. + Intro. + Apply Acc_intro. + NewDestruct y; Intro H5. + Inversion_clear H5. + Inversion_clear H0;Auto with sets. + + Apply swap_Acc. + Inversion_clear H0;Auto with sets. + + Intros. + Apply IHAcc1;Auto with sets;Intros. + Apply Acc_inv with (y0,x1) ;Auto with sets. + Apply sp_noswap. + Apply right_sym;Auto with sets. + + Auto with sets. +Qed. + + + Lemma wf_swapprod: (well_founded A R)->(well_founded A*A SwapProd). +Proof. + Red. + NewDestruct a;Intros. + Apply Acc_swapprod;Auto with sets. +Qed. + +End Swap. diff --git a/theories7/Wellfounded/Transitive_Closure.v b/theories7/Wellfounded/Transitive_Closure.v new file mode 100644 index 00000000..4d6cbe28 --- /dev/null +++ b/theories7/Wellfounded/Transitive_Closure.v @@ -0,0 +1,47 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Transitive_Closure.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +(** Author: Bruno Barras *) + +Require Relation_Definitions. +Require Relation_Operators. + +Section Wf_Transitive_Closure. + Variable A: Set. + Variable R: (relation A). + + Notation trans_clos := (clos_trans A R). + + Lemma incl_clos_trans: (inclusion A R trans_clos). + Red;Auto with sets. + Qed. + + Lemma Acc_clos_trans: (x:A)(Acc A R x)->(Acc A trans_clos x). + NewInduction 1 as [x0 _ H1]. + Apply Acc_intro. + Intros y H2. + NewInduction H2;Auto with sets. + Apply Acc_inv with y ;Auto with sets. + Qed. + + Hints Resolve Acc_clos_trans. + + Lemma Acc_inv_trans: (x,y:A)(trans_clos y x)->(Acc A R x)->(Acc A R y). + Proof. + NewInduction 1 as [|x y];Auto with sets. + Intro; Apply Acc_inv with y; Assumption. + Qed. + + Theorem wf_clos_trans: (well_founded A R) ->(well_founded A trans_clos). + Proof. + Unfold well_founded;Auto with sets. + Qed. + +End Wf_Transitive_Closure. diff --git a/theories7/Wellfounded/Union.v b/theories7/Wellfounded/Union.v new file mode 100644 index 00000000..9b31f72d --- /dev/null +++ b/theories7/Wellfounded/Union.v @@ -0,0 +1,74 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Union.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +(** Author: Bruno Barras *) + +Require Relation_Operators. +Require Relation_Definitions. +Require Transitive_Closure. + +Section WfUnion. + Variable A: Set. + Variable R1,R2: (relation A). + + Notation Union := (union A R1 R2). + + Hints Resolve Acc_clos_trans wf_clos_trans. + +Remark strip_commut: + (commut A R1 R2)->(x,y:A)(clos_trans A R1 y x)->(z:A)(R2 z y) + ->(EX y':A | (R2 y' x) & (clos_trans A R1 z y')). +Proof. + NewInduction 2 as [x y|x y z H0 IH1 H1 IH2]; Intros. + Elim H with y x z ;Auto with sets;Intros x0 H2 H3. + Exists x0;Auto with sets. + + Elim IH1 with z0 ;Auto with sets;Intros. + Elim IH2 with x0 ;Auto with sets;Intros. + Exists x1;Auto with sets. + Apply t_trans with x0; Auto with sets. +Qed. + + + Lemma Acc_union: (commut A R1 R2)->((x:A)(Acc A R2 x)->(Acc A R1 x)) + ->(a:A)(Acc A R2 a)->(Acc A Union a). +Proof. + NewInduction 3 as [x H1 H2]. + Apply Acc_intro;Intros. + Elim H3;Intros;Auto with sets. + Cut (clos_trans A R1 y x);Auto with sets. + ElimType (Acc A (clos_trans A R1) y);Intros. + Apply Acc_intro;Intros. + Elim H8;Intros. + Apply H6;Auto with sets. + Apply t_trans with x0 ;Auto with sets. + + Elim strip_commut with x x0 y0 ;Auto with sets;Intros. + Apply Acc_inv_trans with x1 ;Auto with sets. + Unfold union . + Elim H11;Auto with sets;Intros. + Apply t_trans with y1 ;Auto with sets. + + Apply (Acc_clos_trans A). + Apply Acc_inv with x ;Auto with sets. + Apply H0. + Apply Acc_intro;Auto with sets. +Qed. + + + Theorem wf_union: (commut A R1 R2)->(well_founded A R1)->(well_founded A R2) + ->(well_founded A Union). +Proof. + Unfold well_founded . + Intros. + Apply Acc_union;Auto with sets. +Qed. + +End WfUnion. diff --git a/theories7/Wellfounded/Well_Ordering.v b/theories7/Wellfounded/Well_Ordering.v new file mode 100644 index 00000000..5c2b2405 --- /dev/null +++ b/theories7/Wellfounded/Well_Ordering.v @@ -0,0 +1,72 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Well_Ordering.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +(** Author: Cristina Cornes. + From: Constructing Recursion Operators in Type Theory + L. Paulson JSC (1986) 2, 325-355 *) + +Require Eqdep. + +Section WellOrdering. +Variable A:Set. +Variable B:A->Set. + +Inductive WO : Set := + sup : (a:A)(f:(B a)->WO)WO. + + +Inductive le_WO : WO->WO->Prop := + le_sup : (a:A)(f:(B a)->WO)(v:(B a)) (le_WO (f v) (sup a f)). + + +Theorem wf_WO : (well_founded WO le_WO ). +Proof. + Unfold well_founded ;Intro. + Apply Acc_intro. + Elim a. + Intros. + Inversion H0. + Apply Acc_intro. + Generalize H4 ;Generalize H1 ;Generalize f0 ;Generalize v. + Rewrite -> H3. + Intros. + Apply (H v0 y0). + Cut (eq ? f f1). + Intros E;Rewrite -> E;Auto. + Symmetry. + Apply (inj_pair2 A [a0:A](B a0)->WO a0 f1 f H5). +Qed. + +End WellOrdering. + + +Section Characterisation_wf_relations. + +(** Wellfounded relations are the inverse image of wellordering types *) +(* in course of development *) + + +Variable A:Set. +Variable leA:A->A->Prop. + +Definition B:= [a:A] {x:A | (leA x a)}. + +Definition wof: (well_founded A leA)-> A-> (WO A B). +Proof. + Intros. + Apply (well_founded_induction A leA H [a:A](WO A B));Auto. + Intros. + Apply (sup A B x). + Unfold 1 B . + NewDestruct 1 as [x0]. + Apply (H1 x0);Auto. +Qed. + +End Characterisation_wf_relations. diff --git a/theories7/Wellfounded/Wellfounded.v b/theories7/Wellfounded/Wellfounded.v new file mode 100644 index 00000000..d1a8dd01 --- /dev/null +++ b/theories7/Wellfounded/Wellfounded.v @@ -0,0 +1,20 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Wellfounded.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +Require Export Disjoint_Union. +Require Export Inclusion. +Require Export Inverse_Image. +Require Export Lexicographic_Exponentiation. +Require Export Lexicographic_Product. +Require Export Transitive_Closure. +Require Export Union. +Require Export Well_Ordering. + + diff --git a/theories7/ZArith/BinInt.v b/theories7/ZArith/BinInt.v new file mode 100644 index 00000000..9071896b --- /dev/null +++ b/theories7/ZArith/BinInt.v @@ -0,0 +1,1005 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: BinInt.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +(***********************************************************) +(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) +(***********************************************************) + +Require Export BinPos. +Require Export Pnat. +Require BinNat. +Require Plus. +Require Mult. +(**********************************************************************) +(** Binary integer numbers *) + +Inductive Z : Set := + ZERO : Z | POS : positive -> Z | NEG : positive -> Z. + +(** Declare Scope Z_scope with Key Z *) +Delimits Scope Z_scope with Z. + +(** Automatically open scope positive_scope for the constructors of Z *) +Bind Scope Z_scope with Z. +Arguments Scope POS [ positive_scope ]. +Arguments Scope NEG [ positive_scope ]. + +(** Subtraction of positive into Z *) + +Definition Zdouble_plus_one [x:Z] := + Cases x of + | ZERO => (POS xH) + | (POS p) => (POS (xI p)) + | (NEG p) => (NEG (double_moins_un p)) + end. + +Definition Zdouble_minus_one [x:Z] := + Cases x of + | ZERO => (NEG xH) + | (NEG p) => (NEG (xI p)) + | (POS p) => (POS (double_moins_un p)) + end. + +Definition Zdouble [x:Z] := + Cases x of + | ZERO => ZERO + | (POS p) => (POS (xO p)) + | (NEG p) => (NEG (xO p)) + end. + +Fixpoint ZPminus [x,y:positive] : Z := + Cases x y of + | (xI x') (xI y') => (Zdouble (ZPminus x' y')) + | (xI x') (xO y') => (Zdouble_plus_one (ZPminus x' y')) + | (xI x') xH => (POS (xO x')) + | (xO x') (xI y') => (Zdouble_minus_one (ZPminus x' y')) + | (xO x') (xO y') => (Zdouble (ZPminus x' y')) + | (xO x') xH => (POS (double_moins_un x')) + | xH (xI y') => (NEG (xO y')) + | xH (xO y') => (NEG (double_moins_un y')) + | xH xH => ZERO + end. + +(** Addition on integers *) + +Definition Zplus := [x,y:Z] + Cases x y of + ZERO y => y + | x ZERO => x + | (POS x') (POS y') => (POS (add x' y')) + | (POS x') (NEG y') => + Cases (compare x' y' EGAL) of + | EGAL => ZERO + | INFERIEUR => (NEG (true_sub y' x')) + | SUPERIEUR => (POS (true_sub x' y')) + end + | (NEG x') (POS y') => + Cases (compare x' y' EGAL) of + | EGAL => ZERO + | INFERIEUR => (POS (true_sub y' x')) + | SUPERIEUR => (NEG (true_sub x' y')) + end + | (NEG x') (NEG y') => (NEG (add x' y')) + end. + +V8Infix "+" Zplus : Z_scope. + +(** Opposite *) + +Definition Zopp := [x:Z] + Cases x of + ZERO => ZERO + | (POS x) => (NEG x) + | (NEG x) => (POS x) + end. + +V8Notation "- x" := (Zopp x) : Z_scope. + +(** Successor on integers *) + +Definition Zs := [x:Z](Zplus x (POS xH)). + +(** Predecessor on integers *) + +Definition Zpred := [x:Z](Zplus x (NEG xH)). + +(** Subtraction on integers *) + +Definition Zminus := [m,n:Z](Zplus m (Zopp n)). + +V8Infix "-" Zminus : Z_scope. + +(** Multiplication on integers *) + +Definition Zmult := [x,y:Z] + Cases x y of + | ZERO _ => ZERO + | _ ZERO => ZERO + | (POS x') (POS y') => (POS (times x' y')) + | (POS x') (NEG y') => (NEG (times x' y')) + | (NEG x') (POS y') => (NEG (times x' y')) + | (NEG x') (NEG y') => (POS (times x' y')) + end. + +V8Infix "*" Zmult : Z_scope. + +(** Comparison of integers *) + +Definition Zcompare := [x,y:Z] + Cases x y of + | ZERO ZERO => EGAL + | ZERO (POS y') => INFERIEUR + | ZERO (NEG y') => SUPERIEUR + | (POS x') ZERO => SUPERIEUR + | (POS x') (POS y') => (compare x' y' EGAL) + | (POS x') (NEG y') => SUPERIEUR + | (NEG x') ZERO => INFERIEUR + | (NEG x') (POS y') => INFERIEUR + | (NEG x') (NEG y') => (Op (compare x' y' EGAL)) + end. + +V8Infix "?=" Zcompare (at level 70, no associativity) : Z_scope. + +Tactic Definition ElimCompare com1 com2:= + Case (Dcompare (Zcompare com1 com2)); [ Idtac | + Let x = FreshId "H" In Intro x; Case x; Clear x ]. + +(** Sign function *) + +Definition Zsgn [z:Z] : Z := + Cases z of + ZERO => ZERO + | (POS p) => (POS xH) + | (NEG p) => (NEG xH) + end. + +(** Direct, easier to handle variants of successor and addition *) + +Definition Zsucc' [x:Z] := + Cases x of + | ZERO => (POS xH) + | (POS x') => (POS (add_un x')) + | (NEG x') => (ZPminus xH x') + end. + +Definition Zpred' [x:Z] := + Cases x of + | ZERO => (NEG xH) + | (POS x') => (ZPminus x' xH) + | (NEG x') => (NEG (add_un x')) + end. + +Definition Zplus' := [x,y:Z] + Cases x y of + ZERO y => y + | x ZERO => x + | (POS x') (POS y') => (POS (add x' y')) + | (POS x') (NEG y') => (ZPminus x' y') + | (NEG x') (POS y') => (ZPminus y' x') + | (NEG x') (NEG y') => (NEG (add x' y')) + end. + +Open Local Scope Z_scope. + +(**********************************************************************) +(** Inductive specification of Z *) + +Theorem Zind : (P:(Z ->Prop)) + (P ZERO) -> ((x:Z)(P x) ->(P (Zsucc' x))) -> ((x:Z)(P x) ->(P (Zpred' x))) -> + (z:Z)(P z). +Proof. +Intros P H0 Hs Hp z; NewDestruct z. + Assumption. + Apply Pind with P:=[p](P (POS p)). + Change (P (Zsucc' ZERO)); Apply Hs; Apply H0. + Intro n; Exact (Hs (POS n)). + Apply Pind with P:=[p](P (NEG p)). + Change (P (Zpred' ZERO)); Apply Hp; Apply H0. + Intro n; Exact (Hp (NEG n)). +Qed. + +(**********************************************************************) +(** Properties of opposite on binary integer numbers *) + +Theorem Zopp_NEG : (x:positive) (Zopp (NEG x)) = (POS x). +Proof. +Reflexivity. +Qed. + +(** [opp] is involutive *) + +Theorem Zopp_Zopp: (x:Z) (Zopp (Zopp x)) = x. +Proof. +Intro x; NewDestruct x; Reflexivity. +Qed. + +(** Injectivity of the opposite *) + +Theorem Zopp_intro : (x,y:Z) (Zopp x) = (Zopp y) -> x = y. +Proof. +Intros x y;Case x;Case y;Simpl;Intros; [ + Trivial | Discriminate H | Discriminate H | Discriminate H +| Simplify_eq H; Intro E; Rewrite E; Trivial +| Discriminate H | Discriminate H | Discriminate H +| Simplify_eq H; Intro E; Rewrite E; Trivial ]. +Qed. + +(**********************************************************************) +(* Properties of the direct definition of successor and predecessor *) + +Lemma Zpred'_succ' : (x:Z)(Zpred' (Zsucc' x))=x. +Proof. +Intro x; NewDestruct x; Simpl. + Reflexivity. +NewDestruct p; Simpl; Try Rewrite double_moins_un_add_un_xI; Reflexivity. +NewDestruct p; Simpl; Try Rewrite is_double_moins_un; Reflexivity. +Qed. + +Lemma Zsucc'_discr : (x:Z)x<>(Zsucc' x). +Proof. +Intro x; NewDestruct x; Simpl. + Discriminate. + Injection; Apply add_un_discr. + NewDestruct p; Simpl. + Discriminate. + Intro H; Symmetry in H; Injection H; Apply double_moins_un_xO_discr. + Discriminate. +Qed. + +(**********************************************************************) +(** Other properties of binary integer numbers *) + +Lemma ZL0 : (S (S O))=(plus (S O) (S O)). +Proof. +Reflexivity. +Qed. + +(**********************************************************************) +(** Properties of the addition on integers *) + +(** zero is left neutral for addition *) + +Theorem Zero_left: (x:Z) (Zplus ZERO x) = x. +Proof. +Intro x; NewDestruct x; Reflexivity. +Qed. + +(** zero is right neutral for addition *) + +Theorem Zero_right: (x:Z) (Zplus x ZERO) = x. +Proof. +Intro x; NewDestruct x; Reflexivity. +Qed. + +(** addition is commutative *) + +Theorem Zplus_sym: (x,y:Z) (Zplus x y) = (Zplus y x). +Proof. +Intro x;NewInduction x as [|p|p];Intro y; NewDestruct y as [|q|q];Simpl;Try Reflexivity. + Rewrite add_sym; Reflexivity. + Rewrite ZC4; NewDestruct (compare q p EGAL); Reflexivity. + Rewrite ZC4; NewDestruct (compare q p EGAL); Reflexivity. + Rewrite add_sym; Reflexivity. +Qed. + +(** opposite distributes over addition *) + +Theorem Zopp_Zplus: + (x,y:Z) (Zopp (Zplus x y)) = (Zplus (Zopp x) (Zopp y)). +Proof. +Intro x; NewDestruct x as [|p|p]; Intro y; NewDestruct y as [|q|q]; Simpl; + Reflexivity Orelse NewDestruct (compare p q EGAL); Reflexivity. +Qed. + +(** opposite is inverse for addition *) + +Theorem Zplus_inverse_r: (x:Z) (Zplus x (Zopp x)) = ZERO. +Proof. +Intro x; NewDestruct x as [|p|p]; Simpl; [ + Reflexivity +| Rewrite (convert_compare_EGAL p); Reflexivity +| Rewrite (convert_compare_EGAL p); Reflexivity ]. +Qed. + +Theorem Zplus_inverse_l: (x:Z) (Zplus (Zopp x) x) = ZERO. +Proof. +Intro; Rewrite Zplus_sym; Apply Zplus_inverse_r. +Qed. + +Hints Local Resolve Zero_left Zero_right. + +(** addition is associative *) + +Lemma weak_assoc : + (x,y:positive)(z:Z) (Zplus (POS x) (Zplus (POS y) z))= + (Zplus (Zplus (POS x) (POS y)) z). +Proof. +Intros x y z';Case z'; [ + Auto with arith +| Intros z;Simpl; Rewrite add_assoc;Auto with arith +| Intros z; Simpl; ElimPcompare y z; + Intros E0;Rewrite E0; + ElimPcompare '(add x y) 'z;Intros E1;Rewrite E1; [ + Absurd (compare (add x y) z EGAL)=EGAL; [ (* Case 1 *) + Rewrite convert_compare_SUPERIEUR; [ + Discriminate + | Rewrite convert_add; Rewrite (compare_convert_EGAL y z E0); + Elim (ZL4 x);Intros k E2;Rewrite E2; Simpl; Unfold gt lt; Apply le_n_S; + Apply le_plus_r ] + | Assumption ] + | Absurd (compare (add x y) z EGAL)=INFERIEUR; [ (* Case 2 *) + Rewrite convert_compare_SUPERIEUR; [ + Discriminate + | Rewrite convert_add; Rewrite (compare_convert_EGAL y z E0); + Elim (ZL4 x);Intros k E2;Rewrite E2; Simpl; Unfold gt lt; Apply le_n_S; + Apply le_plus_r] + | Assumption ] + | Rewrite (compare_convert_EGAL y z E0); (* Case 3 *) + Elim (sub_pos_SUPERIEUR (add x z) z);[ + Intros t H; Elim H;Intros H1 H2;Elim H2;Intros H3 H4; + Unfold true_sub; Rewrite H1; Cut x=t; [ + Intros E;Rewrite E;Auto with arith + | Apply simpl_add_r with z:=z; Rewrite <- H3; Rewrite add_sym; Trivial with arith ] + | Pattern 1 z; Rewrite <- (compare_convert_EGAL y z E0); Assumption ] + | Elim (sub_pos_SUPERIEUR z y); [ (* Case 4 *) + Intros k H;Elim H;Intros H1 H2;Elim H2;Intros H3 H4; Unfold 1 true_sub; + Rewrite H1; Cut x=k; [ + Intros E;Rewrite E; Rewrite (convert_compare_EGAL k); Trivial with arith + | Apply simpl_add_r with z:=y; Rewrite (add_sym k y); Rewrite H3; + Apply compare_convert_EGAL; Assumption ] + | Apply ZC2;Assumption] + | Elim (sub_pos_SUPERIEUR z y); [ (* Case 5 *) + Intros k H;Elim H;Intros H1 H2;Elim H2;Intros H3 H4; + Unfold 1 3 5 true_sub; Rewrite H1; + Cut (compare x k EGAL)=INFERIEUR; [ + Intros E2;Rewrite E2; Elim (sub_pos_SUPERIEUR k x); [ + Intros i H5;Elim H5;Intros H6 H7;Elim H7;Intros H8 H9; + Elim (sub_pos_SUPERIEUR z (add x y)); [ + Intros j H10;Elim H10;Intros H11 H12;Elim H12;Intros H13 H14; + Unfold true_sub ;Rewrite H6;Rewrite H11; Cut i=j; [ + Intros E;Rewrite E;Auto with arith + | Apply (simpl_add_l (add x y)); Rewrite H13; + Rewrite (add_sym x y); Rewrite <- add_assoc; Rewrite H8; + Assumption ] + | Apply ZC2; Assumption] + | Apply ZC2;Assumption] + | Apply convert_compare_INFERIEUR; + Apply simpl_lt_plus_l with p:=(convert y); + Do 2 Rewrite <- convert_add; Apply compare_convert_INFERIEUR; + Rewrite H3; Rewrite add_sym; Assumption ] + | Apply ZC2; Assumption ] + | Elim (sub_pos_SUPERIEUR z y); [ (* Case 6 *) + Intros k H;Elim H;Intros H1 H2;Elim H2;Intros H3 H4; + Elim (sub_pos_SUPERIEUR (add x y) z); [ + Intros i H5;Elim H5;Intros H6 H7;Elim H7;Intros H8 H9; + Unfold true_sub; Rewrite H1;Rewrite H6; + Cut (compare x k EGAL)=SUPERIEUR; [ + Intros H10;Elim (sub_pos_SUPERIEUR x k H10); + Intros j H11;Elim H11;Intros H12 H13;Elim H13;Intros H14 H15; + Rewrite H10; Rewrite H12; Cut i=j; [ + Intros H16;Rewrite H16;Auto with arith + | Apply (simpl_add_l (add z k)); Rewrite <- (add_assoc z k j); + Rewrite H14; Rewrite (add_sym z k); Rewrite <- add_assoc; + Rewrite H8; Rewrite (add_sym x y); Rewrite add_assoc; + Rewrite (add_sym k y); Rewrite H3; Trivial with arith] + | Apply convert_compare_SUPERIEUR; Unfold lt gt; + Apply simpl_lt_plus_l with p:=(convert y); + Do 2 Rewrite <- convert_add; Apply compare_convert_INFERIEUR; + Rewrite H3; Rewrite add_sym; Apply ZC1; Assumption ] + | Assumption ] + | Apply ZC2;Assumption ] + | Absurd (compare (add x y) z EGAL)=EGAL; [ (* Case 7 *) + Rewrite convert_compare_SUPERIEUR; [ + Discriminate + | Rewrite convert_add; Unfold gt;Apply lt_le_trans with m:=(convert y);[ + Apply compare_convert_INFERIEUR; Apply ZC1; Assumption + | Apply le_plus_r]] + | Assumption ] + | Absurd (compare (add x y) z EGAL)=INFERIEUR; [ (* Case 8 *) + Rewrite convert_compare_SUPERIEUR; [ + Discriminate + | Unfold gt; Apply lt_le_trans with m:=(convert y);[ + Exact (compare_convert_SUPERIEUR y z E0) + | Rewrite convert_add; Apply le_plus_r]] + | Assumption ] + | Elim sub_pos_SUPERIEUR with 1:=E0;Intros k H1; (* Case 9 *) + Elim sub_pos_SUPERIEUR with 1:=E1; Intros i H2;Elim H1;Intros H3 H4; + Elim H4;Intros H5 H6; Elim H2;Intros H7 H8;Elim H8;Intros H9 H10; + Unfold true_sub ;Rewrite H3;Rewrite H7; Cut (add x k)=i; [ + Intros E;Rewrite E;Auto with arith + | Apply (simpl_add_l z);Rewrite (add_sym x k); + Rewrite add_assoc; Rewrite H5;Rewrite H9; + Rewrite add_sym; Trivial with arith ]]]. +Qed. + +Hints Local Resolve weak_assoc. + +Theorem Zplus_assoc : + (n,m,p:Z) (Zplus n (Zplus m p))= (Zplus (Zplus n m) p). +Proof. +Intros x y z;Case x;Case y;Case z;Auto with arith; Intros; [ + Rewrite (Zplus_sym (NEG p0)); Rewrite weak_assoc; + Rewrite (Zplus_sym (Zplus (POS p1) (NEG p0))); Rewrite weak_assoc; + Rewrite (Zplus_sym (POS p1)); Trivial with arith +| Apply Zopp_intro; Do 4 Rewrite Zopp_Zplus; + Do 2 Rewrite Zopp_NEG; Rewrite Zplus_sym; Rewrite <- weak_assoc; + Rewrite (Zplus_sym (Zopp (POS p1))); + Rewrite (Zplus_sym (Zplus (POS p0) (Zopp (POS p1)))); + Rewrite (weak_assoc p); Rewrite weak_assoc; Rewrite (Zplus_sym (POS p0)); + Trivial with arith +| Rewrite Zplus_sym; Rewrite (Zplus_sym (POS p0) (POS p)); + Rewrite <- weak_assoc; Rewrite Zplus_sym; Rewrite (Zplus_sym (POS p0)); + Trivial with arith +| Apply Zopp_intro; Do 4 Rewrite Zopp_Zplus; + Do 2 Rewrite Zopp_NEG; Rewrite (Zplus_sym (Zopp (POS p0))); + Rewrite weak_assoc; Rewrite (Zplus_sym (Zplus (POS p1) (Zopp (POS p0)))); + Rewrite weak_assoc;Rewrite (Zplus_sym (POS p)); Trivial with arith +| Apply Zopp_intro; Do 4 Rewrite Zopp_Zplus; Do 2 Rewrite Zopp_NEG; + Apply weak_assoc +| Apply Zopp_intro; Do 4 Rewrite Zopp_Zplus; Do 2 Rewrite Zopp_NEG; + Apply weak_assoc] +. +Qed. + +V7only [Notation Zplus_assoc_l := Zplus_assoc.]. + +Lemma Zplus_assoc_r : (n,m,p:Z)(Zplus (Zplus n m) p) =(Zplus n (Zplus m p)). +Proof. +Intros; Symmetry; Apply Zplus_assoc. +Qed. + +(** Associativity mixed with commutativity *) + +Theorem Zplus_permute : (n,m,p:Z) (Zplus n (Zplus m p))=(Zplus m (Zplus n p)). +Proof. +Intros n m p; +Rewrite Zplus_sym;Rewrite <- Zplus_assoc; Rewrite (Zplus_sym p n); Trivial with arith. +Qed. + +(** addition simplifies *) + +Theorem Zsimpl_plus_l : (n,m,p:Z)(Zplus n m)=(Zplus n p)->m=p. +Intros n m p H; Cut (Zplus (Zopp n) (Zplus n m))=(Zplus (Zopp n) (Zplus n p));[ + Do 2 Rewrite -> Zplus_assoc; Rewrite -> (Zplus_sym (Zopp n) n); + Rewrite -> Zplus_inverse_r;Simpl; Trivial with arith +| Rewrite -> H; Trivial with arith ]. +Qed. + +(** addition and successor permutes *) + +Lemma Zplus_S_n: (x,y:Z) (Zplus (Zs x) y) = (Zs (Zplus x y)). +Proof. +Intros x y; Unfold Zs; Rewrite (Zplus_sym (Zplus x y)); Rewrite Zplus_assoc; +Rewrite (Zplus_sym (POS xH)); Trivial with arith. +Qed. + +Lemma Zplus_n_Sm : (n,m:Z) (Zs (Zplus n m))=(Zplus n (Zs m)). +Proof. +Intros n m; Unfold Zs; Rewrite Zplus_assoc; Trivial with arith. +Qed. + +Lemma Zplus_Snm_nSm : (n,m:Z)(Zplus (Zs n) m)=(Zplus n (Zs m)). +Proof. +Unfold Zs ;Intros n m; Rewrite <- Zplus_assoc; Rewrite (Zplus_sym (POS xH)); +Trivial with arith. +Qed. + +(** Misc properties, usually redundant or non natural *) + +Lemma Zplus_n_O : (n:Z) n=(Zplus n ZERO). +Proof. +Symmetry; Apply Zero_right. +Qed. + +Lemma Zplus_unit_left : (n,m:Z) (Zplus n ZERO)=m -> n=m. +Proof. +Intros n m; Rewrite Zero_right; Intro; Assumption. +Qed. + +Lemma Zplus_unit_right : (n,m:Z) n=(Zplus m ZERO) -> n=m. +Proof. +Intros n m; Rewrite Zero_right; Intro; Assumption. +Qed. + +Lemma Zplus_simpl : (x,y,z,t:Z) x=y -> z=t -> (Zplus x z)=(Zplus y t). +Proof. +Intros; Rewrite H; Rewrite H0; Reflexivity. +Qed. + +Lemma Zplus_Zopp_expand : (x,y,z:Z) + (Zplus x (Zopp y))=(Zplus (Zplus x (Zopp z)) (Zplus z (Zopp y))). +Proof. +Intros x y z. +Rewrite <- (Zplus_assoc x). +Rewrite (Zplus_assoc (Zopp z)). +Rewrite Zplus_inverse_l. +Reflexivity. +Qed. + +(**********************************************************************) +(** Properties of successor and predecessor on binary integer numbers *) + +Theorem Zn_Sn : (x:Z) ~ x=(Zs x). +Proof. +Intros n;Cut ~ZERO=(POS xH);[ + Unfold not ;Intros H1 H2;Apply H1;Apply (Zsimpl_plus_l n);Rewrite Zero_right; + Exact H2 +| Discriminate ]. +Qed. + +Theorem add_un_Zs : (x:positive) (POS (add_un x)) = (Zs (POS x)). +Proof. +Intro; Rewrite -> ZL12; Unfold Zs; Simpl; Trivial with arith. +Qed. + +(** successor and predecessor are inverse functions *) + +Theorem Zs_pred : (n:Z) n=(Zs (Zpred n)). +Proof. +Intros n; Unfold Zs Zpred ;Rewrite <- Zplus_assoc; Simpl; Rewrite Zero_right; +Trivial with arith. +Qed. + +Hints Immediate Zs_pred : zarith. + +Theorem Zpred_Sn : (x:Z) x=(Zpred (Zs x)). +Proof. +Intros m; Unfold Zpred Zs; Rewrite <- Zplus_assoc; Simpl; +Rewrite Zplus_sym; Auto with arith. +Qed. + +Theorem Zeq_add_S : (n,m:Z) (Zs n)=(Zs m) -> n=m. +Proof. +Intros n m H. +Change (Zplus (Zplus (NEG xH) (POS xH)) n)= + (Zplus (Zplus (NEG xH) (POS xH)) m); +Do 2 Rewrite <- Zplus_assoc; Do 2 Rewrite (Zplus_sym (POS xH)); +Unfold Zs in H;Rewrite H; Trivial with arith. +Qed. + +(** Misc properties, usually redundant or non natural *) + +Lemma Zeq_S : (n,m:Z) n=m -> (Zs n)=(Zs m). +Proof. +Intros n m H; Rewrite H; Reflexivity. +Qed. + +Lemma Znot_eq_S : (n,m:Z) ~(n=m) -> ~((Zs n)=(Zs m)). +Proof. +Unfold not ;Intros n m H1 H2;Apply H1;Apply Zeq_add_S; Assumption. +Qed. + +(**********************************************************************) +(** Properties of subtraction on binary integer numbers *) + +Lemma Zminus_0_r : (x:Z) (Zminus x ZERO)=x. +Proof. +Intro; Unfold Zminus; Simpl;Rewrite Zero_right; Trivial with arith. +Qed. + +Lemma Zminus_n_O : (x:Z) x=(Zminus x ZERO). +Proof. +Intro; Symmetry; Apply Zminus_0_r. +Qed. + +Lemma Zminus_diag : (n:Z)(Zminus n n)=ZERO. +Proof. +Intro; Unfold Zminus; Rewrite Zplus_inverse_r; Trivial with arith. +Qed. + +Lemma Zminus_n_n : (n:Z)(ZERO=(Zminus n n)). +Proof. +Intro; Symmetry; Apply Zminus_diag. +Qed. + +Lemma Zplus_minus : (x,y,z:Z)(x=(Zplus y z))->(z=(Zminus x y)). +Proof. +Intros n m p H;Unfold Zminus;Apply (Zsimpl_plus_l m); +Rewrite (Zplus_sym m (Zplus n (Zopp m))); Rewrite <- Zplus_assoc; +Rewrite Zplus_inverse_l; Rewrite Zero_right; Rewrite H; Trivial with arith. +Qed. + +Lemma Zminus_plus : (x,y:Z)(Zminus (Zplus x y) x)=y. +Proof. +Intros n m;Unfold Zminus ;Rewrite -> (Zplus_sym n m);Rewrite <- Zplus_assoc; +Rewrite -> Zplus_inverse_r; Apply Zero_right. +Qed. + +Lemma Zle_plus_minus : (n,m:Z) (Zplus n (Zminus m n))=m. +Proof. +Unfold Zminus; Intros n m; Rewrite Zplus_permute; Rewrite Zplus_inverse_r; +Apply Zero_right. +Qed. + +Lemma Zminus_Sn_m : (n,m:Z)((Zs (Zminus n m))=(Zminus (Zs n) m)). +Proof. +Intros n m;Unfold Zminus Zs; Rewrite (Zplus_sym n (Zopp m)); +Rewrite <- Zplus_assoc;Apply Zplus_sym. +Qed. + +Lemma Zminus_plus_simpl_l : + (x,y,z:Z)(Zminus (Zplus z x) (Zplus z y))=(Zminus x y). +Proof. +Intros n m p;Unfold Zminus; Rewrite Zopp_Zplus; Rewrite Zplus_assoc; +Rewrite (Zplus_sym p); Rewrite <- (Zplus_assoc n p); Rewrite Zplus_inverse_r; +Rewrite Zero_right; Trivial with arith. +Qed. + +Lemma Zminus_plus_simpl : + (x,y,z:Z)((Zminus x y)=(Zminus (Zplus z x) (Zplus z y))). +Proof. +Intros; Symmetry; Apply Zminus_plus_simpl_l. +Qed. + +Lemma Zminus_Zplus_compatible : + (x,y,z:Z) (Zminus (Zplus x z) (Zplus y z)) = (Zminus x y). +Intros x y n. +Unfold Zminus. +Rewrite -> Zopp_Zplus. +Rewrite -> (Zplus_sym (Zopp y) (Zopp n)). +Rewrite -> Zplus_assoc. +Rewrite <- (Zplus_assoc x n (Zopp n)). +Rewrite -> (Zplus_inverse_r n). +Rewrite <- Zplus_n_O. +Reflexivity. +Qed. + +(** Misc redundant properties *) + +V7only [Set Implicit Arguments.]. + +Lemma Zeq_Zminus : (x,y:Z)x=y -> (Zminus x y)=ZERO. +Proof. +Intros x y H; Rewrite H; Symmetry; Apply Zminus_n_n. +Qed. + +Lemma Zminus_Zeq : (x,y:Z)(Zminus x y)=ZERO -> x=y. +Proof. +Intros x y H; Rewrite <- (Zle_plus_minus y x); Rewrite H; Apply Zero_right. +Qed. + +V7only [Unset Implicit Arguments.]. + +(**********************************************************************) +(** Properties of multiplication on binary integer numbers *) + +(** One is neutral for multiplication *) + +Theorem Zmult_1_n : (n:Z)(Zmult (POS xH) n)=n. +Proof. +Intro x; NewDestruct x; Reflexivity. +Qed. +V7only [Notation Zmult_one := Zmult_1_n.]. + +Theorem Zmult_n_1 : (n:Z)(Zmult n (POS xH))=n. +Proof. +Intro x; NewDestruct x; Simpl; Try Rewrite times_x_1; Reflexivity. +Qed. + +(** Zero property of multiplication *) + +Theorem Zero_mult_left: (x:Z) (Zmult ZERO x) = ZERO. +Proof. +Intro x; NewDestruct x; Reflexivity. +Qed. + +Theorem Zero_mult_right: (x:Z) (Zmult x ZERO) = ZERO. +Proof. +Intro x; NewDestruct x; Reflexivity. +Qed. + +Hints Local Resolve Zero_mult_left Zero_mult_right. + +Lemma Zmult_n_O : (n:Z) ZERO=(Zmult n ZERO). +Proof. +Intro x; NewDestruct x; Reflexivity. +Qed. + +(** Commutativity of multiplication *) + +Theorem Zmult_sym : (x,y:Z) (Zmult x y) = (Zmult y x). +Proof. +Intros x y; NewDestruct x as [|p|p]; NewDestruct y as [|q|q]; Simpl; + Try Rewrite (times_sym p q); Reflexivity. +Qed. + +(** Associativity of multiplication *) + +Theorem Zmult_assoc : + (x,y,z:Z) (Zmult x (Zmult y z))= (Zmult (Zmult x y) z). +Proof. +Intros x y z; NewDestruct x; NewDestruct y; NewDestruct z; Simpl; + Try Rewrite times_assoc; Reflexivity. +Qed. +V7only [Notation Zmult_assoc_l := Zmult_assoc.]. + +Lemma Zmult_assoc_r : (n,m,p:Z)((Zmult (Zmult n m) p) = (Zmult n (Zmult m p))). +Proof. +Intros n m p; Rewrite Zmult_assoc; Trivial with arith. +Qed. + +(** Associativity mixed with commutativity *) + +Theorem Zmult_permute : (n,m,p:Z)(Zmult n (Zmult m p)) = (Zmult m (Zmult n p)). +Proof. +Intros x y z; Rewrite -> (Zmult_assoc y x z); Rewrite -> (Zmult_sym y x). +Apply Zmult_assoc. +Qed. + +(** Z is integral *) + +Theorem Zmult_eq: (x,y:Z) ~(x=ZERO) -> (Zmult y x) = ZERO -> y = ZERO. +Proof. +Intros x y; NewDestruct x as [|p|p]. + Intro H; Absurd ZERO=ZERO; Trivial. + Intros _ H; NewDestruct y as [|q|q]; Reflexivity Orelse Discriminate. + Intros _ H; NewDestruct y as [|q|q]; Reflexivity Orelse Discriminate. +Qed. + +V7only [Set Implicit Arguments.]. + +Theorem Zmult_zero : (x,y:Z)(Zmult x y)=ZERO -> x=ZERO \/ y=ZERO. +Proof. +Intros x y; NewDestruct x; NewDestruct y; Auto; Simpl; Intro H; Discriminate H. +Qed. + +V7only [Unset Implicit Arguments.]. + +Lemma Zmult_1_inversion_l : + (x,y:Z) (Zmult x y)=(POS xH) -> x=(POS xH) \/ x=(NEG xH). +Proof. +Intros x y; NewDestruct x as [|p|p]; Intro; [ Discriminate | Left | Right ]; + (NewDestruct y as [|q|q]; Try Discriminate; + Simpl in H; Injection H; Clear H; Intro H; + Rewrite times_one_inversion_l with 1:=H; Reflexivity). +Qed. + +(** Multiplication and Opposite *) + +Theorem Zopp_Zmult_l : (x,y:Z)(Zopp (Zmult x y)) = (Zmult (Zopp x) y). +Proof. +Intros x y; NewDestruct x; NewDestruct y; Reflexivity. +Qed. + +Theorem Zopp_Zmult_r : (x,y:Z)(Zopp (Zmult x y)) = (Zmult x (Zopp y)). +Intros x y; Rewrite (Zmult_sym x y); Rewrite Zopp_Zmult_l; Apply Zmult_sym. +Qed. + +Lemma Zopp_Zmult: (x,y:Z) (Zmult (Zopp x) y) = (Zopp (Zmult x y)). +Proof. +Intros x y; Symmetry; Apply Zopp_Zmult_l. +Qed. + +Theorem Zmult_Zopp_left : (x,y:Z)(Zmult (Zopp x) y) = (Zmult x (Zopp y)). +Intros x y; Rewrite Zopp_Zmult; Rewrite Zopp_Zmult_r; Trivial with arith. +Qed. + +Theorem Zmult_Zopp_Zopp: (x,y:Z) (Zmult (Zopp x) (Zopp y)) = (Zmult x y). +Proof. +Intros x y; NewDestruct x; NewDestruct y; Reflexivity. +Qed. + +Theorem Zopp_one : (x:Z)(Zopp x)=(Zmult x (NEG xH)). +Intro x; NewInduction x; Intros; Rewrite Zmult_sym; Auto with arith. +Qed. + +(** Distributivity of multiplication over addition *) + +Lemma weak_Zmult_plus_distr_r: + (x:positive)(y,z:Z) + (Zmult (POS x) (Zplus y z)) = (Zplus (Zmult (POS x) y) (Zmult (POS x) z)). +Proof. +Intros x y' z';Case y';Case z';Auto with arith;Intros y z; + (Simpl; Rewrite times_add_distr; Trivial with arith) +Orelse + (Simpl; ElimPcompare z y; Intros E0;Rewrite E0; [ + Rewrite (compare_convert_EGAL z y E0); + Rewrite (convert_compare_EGAL (times x y)); Trivial with arith + | Cut (compare (times x z) (times x y) EGAL)=INFERIEUR; [ + Intros E;Rewrite E; Rewrite times_true_sub_distr; [ + Trivial with arith + | Apply ZC2;Assumption ] + | Apply convert_compare_INFERIEUR;Do 2 Rewrite times_convert; + Elim (ZL4 x);Intros h H1;Rewrite H1;Apply lt_mult_left; + Exact (compare_convert_INFERIEUR z y E0)] + | Cut (compare (times x z) (times x y) EGAL)=SUPERIEUR; [ + Intros E;Rewrite E; Rewrite times_true_sub_distr; Auto with arith + | Apply convert_compare_SUPERIEUR; Unfold gt; Do 2 Rewrite times_convert; + Elim (ZL4 x);Intros h H1;Rewrite H1;Apply lt_mult_left; + Exact (compare_convert_SUPERIEUR z y E0) ]]). +Qed. + +Theorem Zmult_plus_distr_r: + (x,y,z:Z) (Zmult x (Zplus y z)) = (Zplus (Zmult x y) (Zmult x z)). +Proof. +Intros x y z; Case x; [ + Auto with arith +| Intros x';Apply weak_Zmult_plus_distr_r +| Intros p; Apply Zopp_intro; Rewrite Zopp_Zplus; + Do 3 Rewrite <- Zopp_Zmult; Rewrite Zopp_NEG; + Apply weak_Zmult_plus_distr_r ]. +Qed. + +Theorem Zmult_plus_distr_l : + (n,m,p:Z)((Zmult (Zplus n m) p)=(Zplus (Zmult n p) (Zmult m p))). +Proof. +Intros n m p;Rewrite Zmult_sym;Rewrite Zmult_plus_distr_r; +Do 2 Rewrite -> (Zmult_sym p); Trivial with arith. +Qed. + +(** Distributivity of multiplication over subtraction *) + +Lemma Zmult_Zminus_distr_l : + (x,y,z:Z)((Zmult (Zminus x y) z)=(Zminus (Zmult x z) (Zmult y z))). +Proof. +Intros x y z; Unfold Zminus. +Rewrite <- Zopp_Zmult. +Apply Zmult_plus_distr_l. +Qed. + +V7only [Notation Zmult_minus_distr := Zmult_Zminus_distr_l.]. + +Lemma Zmult_Zminus_distr_r : + (x,y,z:Z)(Zmult z (Zminus x y)) = (Zminus (Zmult z x) (Zmult z y)). +Proof. +Intros x y z; Rewrite (Zmult_sym z (Zminus x y)). +Rewrite (Zmult_sym z x). +Rewrite (Zmult_sym z y). +Apply Zmult_Zminus_distr_l. +Qed. + +(** Simplification of multiplication for non-zero integers *) +V7only [Set Implicit Arguments.]. + +Lemma Zmult_reg_left : (x,y,z:Z) z<>ZERO -> (Zmult z x)=(Zmult z y) -> x=y. +Proof. +Intros x y z H H0. +Generalize (Zeq_Zminus H0). +Intro. +Apply Zminus_Zeq. +Rewrite <- Zmult_Zminus_distr_r in H1. +Clear H0; NewDestruct (Zmult_zero H1). +Contradiction. +Trivial. +Qed. + +Lemma Zmult_reg_right : (x,y,z:Z) z<>ZERO -> (Zmult x z)=(Zmult y z) -> x=y. +Proof. +Intros x y z Hz. +Rewrite (Zmult_sym x z). +Rewrite (Zmult_sym y z). +Intro; Apply Zmult_reg_left with z; Assumption. +Qed. +V7only [Unset Implicit Arguments.]. + +(** Addition and multiplication by 2 *) + +Lemma Zplus_Zmult_2 : (x:Z) (Zplus x x) = (Zmult x (POS (xO xH))). +Proof. +Intros x; Pattern 1 2 x ; Rewrite <- (Zmult_n_1 x); +Rewrite <- Zmult_plus_distr_r; Reflexivity. +Qed. + +(** Multiplication and successor *) + +Lemma Zmult_succ_r : (n,m:Z) (Zmult n (Zs m))=(Zplus (Zmult n m) n). +Proof. +Intros n m;Unfold Zs; Rewrite Zmult_plus_distr_r; +Rewrite (Zmult_sym n (POS xH));Rewrite Zmult_one; Trivial with arith. +Qed. + +Lemma Zmult_n_Sm : (n,m:Z) (Zplus (Zmult n m) n)=(Zmult n (Zs m)). +Proof. +Intros; Symmetry; Apply Zmult_succ_r. +Qed. + +Lemma Zmult_succ_l : (n,m:Z) (Zmult (Zs n) m)=(Zplus (Zmult n m) m). +Proof. +Intros n m; Unfold Zs; Rewrite Zmult_plus_distr_l; Rewrite Zmult_1_n; +Trivial with arith. +Qed. + +Lemma Zmult_Sm_n : (n,m:Z) (Zplus (Zmult n m) m)=(Zmult (Zs n) m). +Proof. +Intros; Symmetry; Apply Zmult_succ_l. +Qed. + +(** Misc redundant properties *) + +Lemma Z_eq_mult: + (x,y:Z) y = ZERO -> (Zmult y x) = ZERO. +Intros x y H; Rewrite H; Auto with arith. +Qed. + +(**********************************************************************) +(** Relating binary positive numbers and binary integers *) + +Lemma POS_xI : (p:positive) (POS (xI p))=(Zplus (Zmult (POS (xO xH)) (POS p)) (POS xH)). +Proof. +Intro; Apply refl_equal. +Qed. + +Lemma POS_xO : (p:positive) (POS (xO p))=(Zmult (POS (xO xH)) (POS p)). +Proof. +Intro; Apply refl_equal. +Qed. + +Lemma NEG_xI : (p:positive) (NEG (xI p))=(Zminus (Zmult (POS (xO xH)) (NEG p)) (POS xH)). +Proof. +Intro; Apply refl_equal. +Qed. + +Lemma NEG_xO : (p:positive) (NEG (xO p))=(Zmult (POS (xO xH)) (NEG p)). +Proof. +Reflexivity. +Qed. + +Lemma POS_add : (p,p':positive)(POS (add p p'))=(Zplus (POS p) (POS p')). +Proof. +Intros p p'; NewDestruct p; NewDestruct p'; Reflexivity. +Qed. + +Lemma NEG_add : (p,p':positive)(NEG (add p p'))=(Zplus (NEG p) (NEG p')). +Proof. +Intros p p'; NewDestruct p; NewDestruct p'; Reflexivity. +Qed. + +(**********************************************************************) +(** Order relations *) + +Definition Zlt := [x,y:Z](Zcompare x y) = INFERIEUR. +Definition Zgt := [x,y:Z](Zcompare x y) = SUPERIEUR. +Definition Zle := [x,y:Z]~(Zcompare x y) = SUPERIEUR. +Definition Zge := [x,y:Z]~(Zcompare x y) = INFERIEUR. +Definition Zne := [x,y:Z] ~(x=y). + +V8Infix "<=" Zle : Z_scope. +V8Infix "<" Zlt : Z_scope. +V8Infix ">=" Zge : Z_scope. +V8Infix ">" Zgt : Z_scope. + +V8Notation "x <= y <= z" := (Zle x y)/\(Zle y z) :Z_scope. +V8Notation "x <= y < z" := (Zle x y)/\(Zlt y z) :Z_scope. +V8Notation "x < y < z" := (Zlt x y)/\(Zlt y z) :Z_scope. +V8Notation "x < y <= z" := (Zlt x y)/\(Zle y z) :Z_scope. + +(**********************************************************************) +(** Absolute value on integers *) + +Definition absolu [x:Z] : nat := + Cases x of + ZERO => O + | (POS p) => (convert p) + | (NEG p) => (convert p) + end. + +Definition Zabs [z:Z] : Z := + Cases z of + ZERO => ZERO + | (POS p) => (POS p) + | (NEG p) => (POS p) + end. + +(**********************************************************************) +(** From [nat] to [Z] *) + +Definition inject_nat := + [x:nat]Cases x of + O => ZERO + | (S y) => (POS (anti_convert y)) + end. + +Require BinNat. + +Definition entier_of_Z := + [z:Z]Cases z of ZERO => Nul | (POS p) => (Pos p) | (NEG p) => (Pos p) end. + +Definition Z_of_entier := + [x:entier]Cases x of Nul => ZERO | (Pos p) => (POS p) end. diff --git a/theories7/ZArith/Wf_Z.v b/theories7/ZArith/Wf_Z.v new file mode 100644 index 00000000..e6cf4610 --- /dev/null +++ b/theories7/ZArith/Wf_Z.v @@ -0,0 +1,194 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Wf_Z.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +Require BinInt. +Require Zcompare. +Require Zorder. +Require Znat. +Require Zmisc. +Require Zsyntax. +Require Wf_nat. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +(** Our purpose is to write an induction shema for {0,1,2,...} + similar to the [nat] schema (Theorem [Natlike_rec]). For that the + following implications will be used : +<< + (n:nat)(Q n)==(n:nat)(P (inject_nat n)) ===> (x:Z)`x > 0) -> (P x) + + /\ + || + || + + (Q O) (n:nat)(Q n)->(Q (S n)) <=== (P 0) (x:Z) (P x) -> (P (Zs x)) + + <=== (inject_nat (S n))=(Zs (inject_nat n)) + + <=== inject_nat_complete +>> + Then the diagram will be closed and the theorem proved. *) + +Lemma inject_nat_complete : + (x:Z)`0 <= x` -> (EX n:nat | x=(inject_nat n)). +Intro x; NewDestruct x; Intros; +[ Exists O; Auto with arith +| Specialize (ZL4 p); Intros Hp; Elim Hp; Intros; + Exists (S x); Intros; Simpl; + Specialize (bij1 x); Intro Hx0; + Rewrite <- H0 in Hx0; + Apply f_equal with f:=POS; + Apply convert_intro; Auto with arith +| Absurd `0 <= (NEG p)`; + [ Unfold Zle; Simpl; Do 2 (Unfold not); Auto with arith + | Assumption] +]. +Qed. + +Lemma ZL4_inf: (y:positive) { h:nat | (convert y)=(S h) }. +Intro y; NewInduction y as [p H|p H1|]; [ + Elim H; Intros x H1; Exists (plus (S x) (S x)); + Unfold convert ;Simpl; Rewrite ZL0; Rewrite ZL2; Unfold convert in H1; + Rewrite H1; Auto with arith +| Elim H1;Intros x H2; Exists (plus x (S x)); Unfold convert; + Simpl; Rewrite ZL0; Rewrite ZL2;Unfold convert in H2; Rewrite H2; Auto with arith +| Exists O ;Auto with arith]. +Qed. + +Lemma inject_nat_complete_inf : + (x:Z)`0 <= x` -> { n:nat | (x=(inject_nat n)) }. +Intro x; NewDestruct x; Intros; +[ Exists O; Auto with arith +| Specialize (ZL4_inf p); Intros Hp; Elim Hp; Intros x0 H0; + Exists (S x0); Intros; Simpl; + Specialize (bij1 x0); Intro Hx0; + Rewrite <- H0 in Hx0; + Apply f_equal with f:=POS; + Apply convert_intro; Auto with arith +| Absurd `0 <= (NEG p)`; + [ Unfold Zle; Simpl; Do 2 (Unfold not); Auto with arith + | Assumption] +]. +Qed. + +Lemma inject_nat_prop : + (P:Z->Prop)((n:nat)(P (inject_nat n))) -> + (x:Z) `0 <= x` -> (P x). +Intros P H x H0. +Specialize (inject_nat_complete x H0). +Intros Hn; Elim Hn; Intros. +Rewrite -> H1; Apply H. +Qed. + +Lemma inject_nat_set : + (P:Z->Set)((n:nat)(P (inject_nat n))) -> + (x:Z) `0 <= x` -> (P x). +Intros P H x H0. +Specialize (inject_nat_complete_inf x H0). +Intros Hn; Elim Hn; Intros. +Rewrite -> p; Apply H. +Qed. + +Lemma natlike_ind : (P:Z->Prop) (P `0`) -> + ((x:Z)(`0 <= x` -> (P x) -> (P (Zs x)))) -> + (x:Z) `0 <= x` -> (P x). +Intros P H H0 x H1; Apply inject_nat_prop; +[ Induction n; + [ Simpl; Assumption + | Intros; Rewrite -> (inj_S n0); + Exact (H0 (inject_nat n0) (ZERO_le_inj n0) H2) ] +| Assumption]. +Qed. + +Lemma natlike_rec : (P:Z->Set) (P `0`) -> + ((x:Z)(`0 <= x` -> (P x) -> (P (Zs x)))) -> + (x:Z) `0 <= x` -> (P x). +Intros P H H0 x H1; Apply inject_nat_set; +[ Induction n; + [ Simpl; Assumption + | Intros; Rewrite -> (inj_S n0); + Exact (H0 (inject_nat n0) (ZERO_le_inj n0) H2) ] +| Assumption]. +Qed. + +Section Efficient_Rec. + +(** [natlike_rec2] is the same as [natlike_rec], but with a different proof, designed + to give a better extracted term. *) + +Local R := [a,b:Z]`0<=a`/\`a<b`. + +Local R_wf : (well_founded Z R). +Proof. +LetTac f := [z]Cases z of (POS p) => (convert p) | ZERO => O | (NEG _) => O end. +Apply well_founded_lt_compat with f. +Unfold R f; Clear f R. +Intros x y; Case x; Intros; Elim H; Clear H. +Case y; Intros; Apply compare_convert_O Orelse Inversion H0. +Case y; Intros; Apply compare_convert_INFERIEUR Orelse Inversion H0; Auto. +Intros; Elim H; Auto. +Qed. + +Lemma natlike_rec2 : (P:Z->Type)(P `0`) -> + ((z:Z)`0<=z` -> (P z) -> (P (Zs z))) -> (z:Z)`0<=z` -> (P z). +Proof. +Intros P Ho Hrec z; Pattern z; Apply (well_founded_induction_type Z R R_wf). +Intro x; Case x. +Trivial. +Intros. +Assert `0<=(Zpred (POS p))`. +Apply Zlt_ZERO_pred_le_ZERO; Unfold Zlt; Simpl; Trivial. +Rewrite Zs_pred. +Apply Hrec. +Auto. +Apply X; Auto; Unfold R; Intuition; Apply Zlt_pred_n_n. +Intros; Elim H; Simpl; Trivial. +Qed. + +(** A variant of the previous using [Zpred] instead of [Zs]. *) + +Lemma natlike_rec3 : (P:Z->Type)(P `0`) -> + ((z:Z)`0<z` -> (P (Zpred z)) -> (P z)) -> (z:Z)`0<=z` -> (P z). +Proof. +Intros P Ho Hrec z; Pattern z; Apply (well_founded_induction_type Z R R_wf). +Intro x; Case x. +Trivial. +Intros; Apply Hrec. +Unfold Zlt; Trivial. +Assert `0<=(Zpred (POS p))`. +Apply Zlt_ZERO_pred_le_ZERO; Unfold Zlt; Simpl; Trivial. +Apply X; Auto; Unfold R; Intuition; Apply Zlt_pred_n_n. +Intros; Elim H; Simpl; Trivial. +Qed. + +(** A more general induction principal using [Zlt]. *) + +Lemma Z_lt_rec : (P:Z->Type) + ((x:Z)((y:Z)`0 <= y < x`->(P y))->(P x)) -> (x:Z)`0 <= x`->(P x). +Proof. +Intros P Hrec z; Pattern z; Apply (well_founded_induction_type Z R R_wf). +Intro x; Case x; Intros. +Apply Hrec; Intros. +Assert H2: `0<0`. + Apply Zle_lt_trans with y; Intuition. +Inversion H2. +Firstorder. +Unfold Zle Zcompare in H; Elim H; Auto. +Defined. + +Lemma Z_lt_induction : + (P:Z->Prop) + ((x:Z)((y:Z)`0 <= y < x`->(P y))->(P x)) + -> (x:Z)`0 <= x`->(P x). +Proof. +Exact Z_lt_rec. +Qed. + +End Efficient_Rec. diff --git a/theories7/ZArith/ZArith.v b/theories7/ZArith/ZArith.v new file mode 100644 index 00000000..e1746433 --- /dev/null +++ b/theories7/ZArith/ZArith.v @@ -0,0 +1,22 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ZArith.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +(** Library for manipulating integers based on binary encoding *) + +Require Export ZArith_base. + +(** Extra modules using [Omega] or [Ring]. *) + +Require Export Zcomplements. +Require Export Zsqrt. +Require Export Zpower. +Require Export Zdiv. +Require Export Zlogarithm. +Require Export Zbool. diff --git a/theories7/ZArith/ZArith_base.v b/theories7/ZArith/ZArith_base.v new file mode 100644 index 00000000..7f2863d6 --- /dev/null +++ b/theories7/ZArith/ZArith_base.v @@ -0,0 +1,39 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(* $Id: ZArith_base.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ *) + +(** Library for manipulating integers based on binary encoding. + These are the basic modules, required by [Omega] and [Ring] for instance. + The full library is [ZArith]. *) + +V7only [ +Require Export fast_integer. +Require Export zarith_aux. +]. +Require Export BinPos. +Require Export BinNat. +Require Export BinInt. +Require Export Zcompare. +Require Export Zorder. +Require Export Zeven. +Require Export Zmin. +Require Export Zabs. +Require Export Znat. +Require Export auxiliary. +Require Export Zsyntax. +Require Export ZArith_dec. +Require Export Zbool. +Require Export Zmisc. +Require Export Wf_Z. + +Hints Resolve Zle_n Zplus_sym Zplus_assoc Zmult_sym Zmult_assoc + Zero_left Zero_right Zmult_one Zplus_inverse_l Zplus_inverse_r + Zmult_plus_distr_l Zmult_plus_distr_r : zarith. + +Require Export Zhints. diff --git a/theories7/ZArith/ZArith_dec.v b/theories7/ZArith/ZArith_dec.v new file mode 100644 index 00000000..985f7601 --- /dev/null +++ b/theories7/ZArith/ZArith_dec.v @@ -0,0 +1,243 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: ZArith_dec.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +Require Sumbool. + +Require BinInt. +Require Zorder. +Require Zcompare. +Require Zsyntax. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +Lemma Dcompare_inf : (r:relation) {r=EGAL} + {r=INFERIEUR} + {r=SUPERIEUR}. +Proof. +Induction r; Auto with arith. +Defined. + +Lemma Zcompare_rec : + (P:Set)(x,y:Z) + ((Zcompare x y)=EGAL -> P) -> + ((Zcompare x y)=INFERIEUR -> P) -> + ((Zcompare x y)=SUPERIEUR -> P) -> + P. +Proof. +Intros P x y H1 H2 H3. +Elim (Dcompare_inf (Zcompare x y)). +Intro H. Elim H; Auto with arith. Auto with arith. +Defined. + +Section decidability. + +Variables x,y : Z. + +(** Decidability of equality on binary integers *) + +Definition Z_eq_dec : {x=y}+{~x=y}. +Proof. +Apply Zcompare_rec with x:=x y:=y. +Intro. Left. Elim (Zcompare_EGAL x y); Auto with arith. +Intro H3. Right. Elim (Zcompare_EGAL x y). Intros H1 H2. Unfold not. Intro H4. + Rewrite (H2 H4) in H3. Discriminate H3. +Intro H3. Right. Elim (Zcompare_EGAL x y). Intros H1 H2. Unfold not. Intro H4. + Rewrite (H2 H4) in H3. Discriminate H3. +Defined. + +(** Decidability of order on binary integers *) + +Definition Z_lt_dec : {(Zlt x y)}+{~(Zlt x y)}. +Proof. +Unfold Zlt. +Apply Zcompare_rec with x:=x y:=y; Intro H. +Right. Rewrite H. Discriminate. +Left; Assumption. +Right. Rewrite H. Discriminate. +Defined. + +Definition Z_le_dec : {(Zle x y)}+{~(Zle x y)}. +Proof. +Unfold Zle. +Apply Zcompare_rec with x:=x y:=y; Intro H. +Left. Rewrite H. Discriminate. +Left. Rewrite H. Discriminate. +Right. Tauto. +Defined. + +Definition Z_gt_dec : {(Zgt x y)}+{~(Zgt x y)}. +Proof. +Unfold Zgt. +Apply Zcompare_rec with x:=x y:=y; Intro H. +Right. Rewrite H. Discriminate. +Right. Rewrite H. Discriminate. +Left; Assumption. +Defined. + +Definition Z_ge_dec : {(Zge x y)}+{~(Zge x y)}. +Proof. +Unfold Zge. +Apply Zcompare_rec with x:=x y:=y; Intro H. +Left. Rewrite H. Discriminate. +Right. Tauto. +Left. Rewrite H. Discriminate. +Defined. + +Definition Z_lt_ge_dec : {`x < y`}+{`x >= y`}. +Proof. +Exact Z_lt_dec. +Defined. + +V7only [ (* From Zextensions *) ]. +Lemma Z_lt_le_dec: {`x < y`}+{`y <= x`}. +Proof. +Intros. +Elim Z_lt_ge_dec. +Intros; Left; Assumption. +Intros; Right; Apply Zge_le; Assumption. +Qed. + +Definition Z_le_gt_dec : {`x <= y`}+{`x > y`}. +Proof. +Elim Z_le_dec; Auto with arith. +Intro. Right. Apply not_Zle; Auto with arith. +Defined. + +Definition Z_gt_le_dec : {`x > y`}+{`x <= y`}. +Proof. +Exact Z_gt_dec. +Defined. + +Definition Z_ge_lt_dec : {`x >= y`}+{`x < y`}. +Proof. +Elim Z_ge_dec; Auto with arith. +Intro. Right. Apply not_Zge; Auto with arith. +Defined. + +Definition Z_le_lt_eq_dec : `x <= y` -> {`x < y`}+{x=y}. +Proof. +Intro H. +Apply Zcompare_rec with x:=x y:=y. +Intro. Right. Elim (Zcompare_EGAL x y); Auto with arith. +Intro. Left. Elim (Zcompare_EGAL x y); Auto with arith. +Intro H1. Absurd `x > y`; Auto with arith. +Defined. + +End decidability. + +(** Cotransitivity of order on binary integers *) + +Lemma Zlt_cotrans:(n,m:Z)`n<m`->(p:Z){`n<p`}+{`p<m`}. +Proof. + Intros x y H z. + Case (Z_lt_ge_dec x z). + Intro. + Left. + Assumption. + Intro. + Right. + Apply Zle_lt_trans with m:=x. + Apply Zge_le. + Assumption. + Assumption. +Defined. + +Lemma Zlt_cotrans_pos:(x,y:Z)`0<x+y`->{`0<x`}+{`0<y`}. +Proof. + Intros x y H. + Case (Zlt_cotrans `0` `x+y` H x). + Intro. + Left. + Assumption. + Intro. + Right. + Apply Zsimpl_lt_plus_l with p:=`x`. + Rewrite Zero_right. + Assumption. +Defined. + +Lemma Zlt_cotrans_neg:(x,y:Z)`x+y<0`->{`x<0`}+{`y<0`}. +Proof. + Intros x y H; + Case (Zlt_cotrans `x+y` `0` H x); + Intro Hxy; + [ Right; + Apply Zsimpl_lt_plus_l with p:=`x`; + Rewrite Zero_right + | Left + ]; + Assumption. +Defined. + +Lemma not_Zeq_inf:(x,y:Z)`x<>y`->{`x<y`}+{`y<x`}. +Proof. + Intros x y H. + Case Z_lt_ge_dec with x y. + Intro. + Left. + Assumption. + Intro H0. + Generalize (Zge_le ? ? H0). + Intro. + Case (Z_le_lt_eq_dec ? ? H1). + Intro. + Right. + Assumption. + Intro. + Apply False_rec. + Apply H. + Symmetry. + Assumption. +Defined. + +Lemma Z_dec:(x,y:Z){`x<y`}+{`x>y`}+{`x=y`}. +Proof. + Intros x y. + Case (Z_lt_ge_dec x y). + Intro H. + Left. + Left. + Assumption. + Intro H. + Generalize (Zge_le ? ? H). + Intro H0. + Case (Z_le_lt_eq_dec y x H0). + Intro H1. + Left. + Right. + Apply Zlt_gt. + Assumption. + Intro. + Right. + Symmetry. + Assumption. +Defined. + + +Lemma Z_dec':(x,y:Z){`x<y`}+{`y<x`}+{`x=y`}. +Proof. + Intros x y. + Case (Z_eq_dec x y); + Intro H; + [ Right; + Assumption + | Left; + Apply (not_Zeq_inf ?? H) + ]. +Defined. + + + +Definition Z_zerop : (x:Z){(`x = 0`)}+{(`x <> 0`)}. +Proof. +Exact [x:Z](Z_eq_dec x ZERO). +Defined. + +Definition Z_notzerop := [x:Z](sumbool_not ? ? (Z_zerop x)). + +Definition Z_noteq_dec := [x,y:Z](sumbool_not ? ? (Z_eq_dec x y)). diff --git a/theories7/ZArith/Zabs.v b/theories7/ZArith/Zabs.v new file mode 100644 index 00000000..57778cae --- /dev/null +++ b/theories7/ZArith/Zabs.v @@ -0,0 +1,138 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Zabs.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *) + +Require Arith. +Require BinPos. +Require BinInt. +Require Zorder. +Require Zsyntax. +Require ZArith_dec. + +V7only [Import nat_scope.]. +Open Local Scope Z_scope. + +(**********************************************************************) +(** Properties of absolute value *) + +Lemma Zabs_eq : (x:Z) (Zle ZERO x) -> (Zabs x)=x. +Intro x; NewDestruct x; Auto with arith. +Compute; Intros; Absurd SUPERIEUR=SUPERIEUR; Trivial with arith. +Qed. + +Lemma Zabs_non_eq : (x:Z) (Zle x ZERO) -> (Zabs x)=(Zopp x). +Proof. +Intro x; NewDestruct x; Auto with arith. +Compute; Intros; Absurd SUPERIEUR=SUPERIEUR; Trivial with arith. +Qed. + +V7only [ (* From Zdivides *) ]. +Theorem Zabs_Zopp: (z : Z) (Zabs (Zopp z)) = (Zabs z). +Proof. +Intros z; Case z; Simpl; Auto. +Qed. + +(** Proving a property of the absolute value by cases *) + +Lemma Zabs_ind : + (P:Z->Prop)(x:Z)(`x >= 0` -> (P x)) -> (`x <= 0` -> (P `-x`)) -> + (P `|x|`). +Proof. +Intros P x H H0; Elim (Z_lt_ge_dec x `0`); Intro. +Assert `x<=0`. Apply Zlt_le_weak; Assumption. +Rewrite Zabs_non_eq. Apply H0. Assumption. Assumption. +Rewrite Zabs_eq. Apply H; Assumption. Apply Zge_le. Assumption. +Save. + +V7only [ (* From Zdivides *) ]. +Theorem Zabs_intro: (P : ?) (z : Z) (P (Zopp z)) -> (P z) -> (P (Zabs z)). +Intros P z; Case z; Simpl; Auto. +Qed. + +Definition Zabs_dec : (x:Z){x=(Zabs x)}+{x=(Zopp (Zabs x))}. +Proof. +Intro x; NewDestruct x;Auto with arith. +Defined. + +Lemma Zabs_pos : (x:Z)(Zle ZERO (Zabs x)). +Intro x; NewDestruct x;Auto with arith; Compute; Intros H;Inversion H. +Qed. + +V7only [ (* From Zdivides *) ]. +Theorem Zabs_eq_case: + (z1, z2 : Z) (Zabs z1) = (Zabs z2) -> z1 = z2 \/ z1 = (Zopp z2). +Proof. +Intros z1 z2; Case z1; Case z2; Simpl; Auto; Try (Intros; Discriminate); + Intros p1 p2 H1; Injection H1; (Intros H2; Rewrite H2); Auto. +Qed. + +(** Triangular inequality *) + +Hints Local Resolve Zle_NEG_POS :zarith. + +V7only [ (* From Zdivides *) ]. +Theorem Zabs_triangle: + (z1, z2 : Z) (Zle (Zabs (Zplus z1 z2)) (Zplus (Zabs z1) (Zabs z2))). +Proof. +Intros z1 z2; Case z1; Case z2; Try (Simpl; Auto with zarith; Fail). +Intros p1 p2; + Apply Zabs_intro + with P := [x : ?] (Zle x (Zplus (Zabs (POS p2)) (Zabs (NEG p1)))); + Try Rewrite Zopp_Zplus; Auto with zarith. +Apply Zle_plus_plus; Simpl; Auto with zarith. +Apply Zle_plus_plus; Simpl; Auto with zarith. +Intros p1 p2; + Apply Zabs_intro + with P := [x : ?] (Zle x (Zplus (Zabs (POS p2)) (Zabs (NEG p1)))); + Try Rewrite Zopp_Zplus; Auto with zarith. +Apply Zle_plus_plus; Simpl; Auto with zarith. +Apply Zle_plus_plus; Simpl; Auto with zarith. +Qed. + +(** Absolute value and multiplication *) + +Lemma Zsgn_Zabs: (x:Z)(Zmult x (Zsgn x))=(Zabs x). +Proof. +Intro x; NewDestruct x; Rewrite Zmult_sym; Auto with arith. +Qed. + +Lemma Zabs_Zsgn: (x:Z)(Zmult (Zabs x) (Zsgn x))=x. +Proof. +Intro x; NewDestruct x; Rewrite Zmult_sym; Auto with arith. +Qed. + +V7only [ (* From Zdivides *) ]. +Theorem Zabs_Zmult: + (z1, z2 : Z) (Zabs (Zmult z1 z2)) = (Zmult (Zabs z1) (Zabs z2)). +Proof. +Intros z1 z2; Case z1; Case z2; Simpl; Auto. +Qed. + +(** absolute value in nat is compatible with order *) + +Lemma absolu_lt : (x,y:Z) (Zle ZERO x)/\(Zlt x y) -> (lt (absolu x) (absolu y)). +Proof. +Intros x y. Case x; Simpl. Case y; Simpl. + +Intro. Absurd (Zlt ZERO ZERO). Compute. Intro H0. Discriminate H0. Intuition. +Intros. Elim (ZL4 p). Intros. Rewrite H0. Auto with arith. +Intros. Elim (ZL4 p). Intros. Rewrite H0. Auto with arith. + +Case y; Simpl. +Intros. Absurd (Zlt (POS p) ZERO). Compute. Intro H0. Discriminate H0. Intuition. +Intros. Change (gt (convert p) (convert p0)). +Apply compare_convert_SUPERIEUR. +Elim H; Auto with arith. Intro. Exact (ZC2 p0 p). + +Intros. Absurd (Zlt (POS p0) (NEG p)). +Compute. Intro H0. Discriminate H0. Intuition. + +Intros. Absurd (Zle ZERO (NEG p)). Compute. Auto with arith. Intuition. +Qed. diff --git a/theories7/ZArith/Zbinary.v b/theories7/ZArith/Zbinary.v new file mode 100644 index 00000000..c3efbe1e --- /dev/null +++ b/theories7/ZArith/Zbinary.v @@ -0,0 +1,425 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zbinary.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) + +(** Bit vectors interpreted as integers. + Contribution by Jean Duprat (ENS Lyon). *) + +Require Bvector. +Require ZArith. +Require Export Zpower. +Require Omega. + +(* +L'évaluation des vecteurs de booléens se font à la fois en binaire et +en complément à deux. Le nombre appartient à Z. +On utilise donc Omega pour faire les calculs dans Z. +De plus, on utilise les fonctions 2^n où n est un naturel, ici la longueur. + two_power_nat = [n:nat](POS (shift_nat n xH)) + : nat->Z + two_power_nat_S + : (n:nat)`(two_power_nat (S n)) = 2*(two_power_nat n)` + Z_lt_ge_dec + : (x,y:Z){`x < y`}+{`x >= y`} +*) + + +Section VALUE_OF_BOOLEAN_VECTORS. + +(* +Les calculs sont effectués dans la convention positive usuelle. +Les valeurs correspondent soit à l'écriture binaire (nat), +soit au complément à deux (int). +On effectue le calcul suivant le schéma de Horner. +Le complément à deux n'a de sens que sur les vecteurs de taille +supérieure ou égale à un, le bit de signe étant évalué négativement. +*) + +Definition bit_value [b:bool] : Z := +Cases b of + | true => `1` + | false => `0` +end. + +Lemma binary_value : (n:nat) (Bvector n) -> Z. +Proof. + Induction n; Intros. + Exact `0`. + + Inversion H0. + Exact (Zplus (bit_value a) (Zmult `2` (H H2))). +Defined. + +Lemma two_compl_value : (n:nat) (Bvector (S n)) -> Z. +Proof. + Induction n; Intros. + Inversion H. + Exact (Zopp (bit_value a)). + + Inversion H0. + Exact (Zplus (bit_value a) (Zmult `2` (H H2))). +Defined. + +(* +Coq < Eval Compute in (binary_value (3) (Bcons true (2) (Bcons false (1) (Bcons true (0) Bnil)))). + = `5` + : Z +*) + +(* +Coq < Eval Compute in (two_compl_value (3) (Bcons true (3) (Bcons false (2) (Bcons true (1) (Bcons true (0) Bnil))))). + = `-3` + : Z +*) + +End VALUE_OF_BOOLEAN_VECTORS. + +Section ENCODING_VALUE. + +(* +On calcule la valeur binaire selon un schema de Horner. +Le calcul s'arrete à la longueur du vecteur sans vérification. +On definit une fonction Zmod2 calquee sur Zdiv2 mais donnant le quotient +de la division z=2q+r avec 0<=r<=1. +La valeur en complément à deux est calculée selon un schema de Horner +avec Zmod2, le paramètre est la taille moins un. +*) + +Definition Zmod2 := [z:Z] Cases z of + | ZERO => `0` + | ((POS p)) => Cases p of + | (xI q) => (POS q) + | (xO q) => (POS q) + | xH => `0` + end + | ((NEG p)) => Cases p of + | (xI q) => `(NEG q) - 1` + | (xO q) => (NEG q) + | xH => `-1` + end + end. + +V7only [ +Notation double_moins_un_add_un := + [p](sym_eq ? ? ? (double_moins_un_add_un_xI p)). +]. + +Lemma Zmod2_twice : (z:Z) + `z = (2*(Zmod2 z) + (bit_value (Zodd_bool z)))`. +Proof. + NewDestruct z; Simpl. + Trivial. + + NewDestruct p; Simpl; Trivial. + + NewDestruct p; Simpl. + NewDestruct p as [p|p|]; Simpl. + Rewrite <- (double_moins_un_add_un_xI p); Trivial. + + Trivial. + + Trivial. + + Trivial. + + Trivial. +Save. + +Lemma Z_to_binary : (n:nat) Z -> (Bvector n). +Proof. + Induction n; Intros. + Exact Bnil. + + Exact (Bcons (Zodd_bool H0) n0 (H (Zdiv2 H0))). +Defined. + +(* +Eval Compute in (Z_to_binary (5) `5`). + = (Vcons bool true (4) + (Vcons bool false (3) + (Vcons bool true (2) + (Vcons bool false (1) (Vcons bool false (0) (Vnil bool)))))) + : (Bvector (5)) +*) + +Lemma Z_to_two_compl : (n:nat) Z -> (Bvector (S n)). +Proof. + Induction n; Intros. + Exact (Bcons (Zodd_bool H) (0) Bnil). + + Exact (Bcons (Zodd_bool H0) (S n0) (H (Zmod2 H0))). +Defined. + +(* +Eval Compute in (Z_to_two_compl (3) `0`). + = (Vcons bool false (3) + (Vcons bool false (2) + (Vcons bool false (1) (Vcons bool false (0) (Vnil bool))))) + : (vector bool (4)) + +Eval Compute in (Z_to_two_compl (3) `5`). + = (Vcons bool true (3) + (Vcons bool false (2) + (Vcons bool true (1) (Vcons bool false (0) (Vnil bool))))) + : (vector bool (4)) + +Eval Compute in (Z_to_two_compl (3) `-5`). + = (Vcons bool true (3) + (Vcons bool true (2) + (Vcons bool false (1) (Vcons bool true (0) (Vnil bool))))) + : (vector bool (4)) +*) + +End ENCODING_VALUE. + +Section Z_BRIC_A_BRAC. + +(* +Bibliotheque de lemmes utiles dans la section suivante. +Utilise largement ZArith. +Meriterait d'etre reecrite. +*) + +Lemma binary_value_Sn : (n:nat) (b:bool) (bv : (Bvector n)) + (binary_value (S n) (Vcons bool b n bv))=`(bit_value b) + 2*(binary_value n bv)`. +Proof. + Intros; Auto. +Save. + +Lemma Z_to_binary_Sn : (n:nat) (b:bool) (z:Z) + `z>=0`-> + (Z_to_binary (S n) `(bit_value b) + 2*z`)=(Bcons b n (Z_to_binary n z)). +Proof. + NewDestruct b; NewDestruct z; Simpl; Auto. + Intro H; Elim H; Trivial. +Save. + +Lemma binary_value_pos : (n:nat) (bv:(Bvector n)) + `(binary_value n bv) >= 0`. +Proof. + NewInduction bv as [|a n v IHbv]; Simpl. + Omega. + + NewDestruct a; NewDestruct (binary_value n v); Simpl; Auto. + Auto with zarith. +Save. + +V7only [Notation add_un_double_moins_un_xO := is_double_moins_un.]. + +Lemma two_compl_value_Sn : (n:nat) (bv : (Bvector (S n))) (b:bool) + (two_compl_value (S n) (Bcons b (S n) bv)) = + `(bit_value b) + 2*(two_compl_value n bv)`. +Proof. + Intros; Auto. +Save. + +Lemma Z_to_two_compl_Sn : (n:nat) (b:bool) (z:Z) + (Z_to_two_compl (S n) `(bit_value b) + 2*z`) = + (Bcons b (S n) (Z_to_two_compl n z)). +Proof. + NewDestruct b; NewDestruct z as [|p|p]; Auto. + NewDestruct p as [p|p|]; Auto. + NewDestruct p as [p|p|]; Simpl; Auto. + Intros; Rewrite (add_un_double_moins_un_xO p); Trivial. +Save. + +Lemma Z_to_binary_Sn_z : (n:nat) (z:Z) + (Z_to_binary (S n) z)=(Bcons (Zodd_bool z) n (Z_to_binary n (Zdiv2 z))). +Proof. + Intros; Auto. +Save. + +Lemma Z_div2_value : (z:Z) + ` z>=0 `-> + `(bit_value (Zodd_bool z))+2*(Zdiv2 z) = z`. +Proof. + NewDestruct z as [|p|p]; Auto. + NewDestruct p; Auto. + Intro H; Elim H; Trivial. +Save. + +Lemma Zdiv2_pos : (z:Z) + ` z >= 0 ` -> + `(Zdiv2 z) >= 0 `. +Proof. + NewDestruct z as [|p|p]. + Auto. + + NewDestruct p; Auto. + Simpl; Intros; Omega. + + Intro H; Elim H; Trivial. + +Save. + +Lemma Zdiv2_two_power_nat : (z:Z) (n:nat) + ` z >= 0 ` -> + ` z < (two_power_nat (S n)) ` -> + `(Zdiv2 z) < (two_power_nat n) `. +Proof. + Intros. + Cut (Zlt (Zmult `2` (Zdiv2 z)) (Zmult `2` (two_power_nat n))); Intros. + Omega. + + Rewrite <- two_power_nat_S. + NewDestruct (Zeven_odd_dec z); Intros. + Rewrite <- Zeven_div2; Auto. + + Generalize (Zodd_div2 z H z0); Omega. +Save. + +(* + +Lemma Z_minus_one_or_zero : (z:Z) + `z >= -1` -> + `z < 1` -> + {`z=-1`} + {`z=0`}. +Proof. + NewDestruct z; Auto. + NewDestruct p; Auto. + Tauto. + + Tauto. + + Intros. + Right; Omega. + + NewDestruct p. + Tauto. + + Tauto. + + Intros; Left; Omega. +Save. +*) + +Lemma Z_to_two_compl_Sn_z : (n:nat) (z:Z) + (Z_to_two_compl (S n) z)=(Bcons (Zodd_bool z) (S n) (Z_to_two_compl n (Zmod2 z))). +Proof. + Intros; Auto. +Save. + +Lemma Zeven_bit_value : (z:Z) + (Zeven z) -> + `(bit_value (Zodd_bool z))=0`. +Proof. + NewDestruct z; Unfold bit_value; Auto. + NewDestruct p; Tauto Orelse (Intro H; Elim H). + NewDestruct p; Tauto Orelse (Intro H; Elim H). +Save. + +Lemma Zodd_bit_value : (z:Z) + (Zodd z) -> + `(bit_value (Zodd_bool z))=1`. +Proof. + NewDestruct z; Unfold bit_value; Auto. + Intros; Elim H. + NewDestruct p; Tauto Orelse (Intros; Elim H). + NewDestruct p; Tauto Orelse (Intros; Elim H). +Save. + +Lemma Zge_minus_two_power_nat_S : (n:nat) (z:Z) + `z >= (-(two_power_nat (S n)))`-> + `(Zmod2 z) >= (-(two_power_nat n))`. +Proof. + Intros n z; Rewrite (two_power_nat_S n). + Generalize (Zmod2_twice z). + NewDestruct (Zeven_odd_dec z) as [H|H]. + Rewrite (Zeven_bit_value z H); Intros; Omega. + + Rewrite (Zodd_bit_value z H); Intros; Omega. +Save. + +Lemma Zlt_two_power_nat_S : (n:nat) (z:Z) + `z < (two_power_nat (S n))`-> + `(Zmod2 z) < (two_power_nat n)`. +Proof. + Intros n z; Rewrite (two_power_nat_S n). + Generalize (Zmod2_twice z). + NewDestruct (Zeven_odd_dec z) as [H|H]. + Rewrite (Zeven_bit_value z H); Intros; Omega. + + Rewrite (Zodd_bit_value z H); Intros; Omega. +Save. + +End Z_BRIC_A_BRAC. + +Section COHERENT_VALUE. + +(* +On vérifie que dans l'intervalle de définition les fonctions sont +réciproques l'une de l'autre. +Elles utilisent les lemmes du bric-a-brac. +*) + +Lemma binary_to_Z_to_binary : (n:nat) (bv : (Bvector n)) + (Z_to_binary n (binary_value n bv))=bv. +Proof. + NewInduction bv as [|a n bv IHbv]. + Auto. + + Rewrite binary_value_Sn. + Rewrite Z_to_binary_Sn. + Rewrite IHbv; Trivial. + + Apply binary_value_pos. +Save. + +Lemma two_compl_to_Z_to_two_compl : (n:nat) (bv : (Bvector n)) (b:bool) + (Z_to_two_compl n (two_compl_value n (Bcons b n bv)))= + (Bcons b n bv). +Proof. + NewInduction bv as [|a n bv IHbv]; Intro b. + NewDestruct b; Auto. + + Rewrite two_compl_value_Sn. + Rewrite Z_to_two_compl_Sn. + Rewrite IHbv; Trivial. +Save. + +Lemma Z_to_binary_to_Z : (n:nat) (z : Z) + `z >= 0 `-> + `z < (two_power_nat n) `-> + (binary_value n (Z_to_binary n z))=z. +Proof. + NewInduction n as [|n IHn]. + Unfold two_power_nat shift_nat; Simpl; Intros; Omega. + + Intros; Rewrite Z_to_binary_Sn_z. + Rewrite binary_value_Sn. + Rewrite IHn. + Apply Z_div2_value; Auto. + + Apply Zdiv2_pos; Trivial. + + Apply Zdiv2_two_power_nat; Trivial. +Save. + +Lemma Z_to_two_compl_to_Z : (n:nat) (z : Z) + `z >= -(two_power_nat n) `-> + `z < (two_power_nat n) `-> + (two_compl_value n (Z_to_two_compl n z))=z. +Proof. + NewInduction n as [|n IHn]. + Unfold two_power_nat shift_nat; Simpl; Intros. + Assert `z=-1`\/`z=0`. Omega. +Intuition; Subst z; Trivial. + + Intros; Rewrite Z_to_two_compl_Sn_z. + Rewrite two_compl_value_Sn. + Rewrite IHn. + Generalize (Zmod2_twice z); Omega. + + Apply Zge_minus_two_power_nat_S; Auto. + + Apply Zlt_two_power_nat_S; Auto. +Save. + +End COHERENT_VALUE. + diff --git a/theories7/ZArith/Zbool.v b/theories7/ZArith/Zbool.v new file mode 100644 index 00000000..258a485d --- /dev/null +++ b/theories7/ZArith/Zbool.v @@ -0,0 +1,158 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(* $Id: Zbool.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ *) + +Require BinInt. +Require Zeven. +Require Zorder. +Require Zcompare. +Require ZArith_dec. +Require Zsyntax. +Require Sumbool. + +(** The decidability of equality and order relations over + type [Z] give some boolean functions with the adequate specification. *) + +Definition Z_lt_ge_bool := [x,y:Z](bool_of_sumbool (Z_lt_ge_dec x y)). +Definition Z_ge_lt_bool := [x,y:Z](bool_of_sumbool (Z_ge_lt_dec x y)). + +Definition Z_le_gt_bool := [x,y:Z](bool_of_sumbool (Z_le_gt_dec x y)). +Definition Z_gt_le_bool := [x,y:Z](bool_of_sumbool (Z_gt_le_dec x y)). + +Definition Z_eq_bool := [x,y:Z](bool_of_sumbool (Z_eq_dec x y)). +Definition Z_noteq_bool := [x,y:Z](bool_of_sumbool (Z_noteq_dec x y)). + +Definition Zeven_odd_bool := [x:Z](bool_of_sumbool (Zeven_odd_dec x)). + +(**********************************************************************) +(** Boolean comparisons of binary integers *) + +Definition Zle_bool := + [x,y:Z]Cases `x ?= y` of SUPERIEUR => false | _ => true end. +Definition Zge_bool := + [x,y:Z]Cases `x ?= y` of INFERIEUR => false | _ => true end. +Definition Zlt_bool := + [x,y:Z]Cases `x ?= y` of INFERIEUR => true | _ => false end. +Definition Zgt_bool := + [x,y:Z]Cases ` x ?= y` of SUPERIEUR => true | _ => false end. +Definition Zeq_bool := + [x,y:Z]Cases `x ?= y` of EGAL => true | _ => false end. +Definition Zneq_bool := + [x,y:Z]Cases `x ?= y` of EGAL => false | _ => true end. + +Lemma Zle_cases : (x,y:Z)if (Zle_bool x y) then `x<=y` else `x>y`. +Proof. +Intros x y; Unfold Zle_bool Zle Zgt. +Case (Zcompare x y); Auto; Discriminate. +Qed. + +Lemma Zlt_cases : (x,y:Z)if (Zlt_bool x y) then `x<y` else `x>=y`. +Proof. +Intros x y; Unfold Zlt_bool Zlt Zge. +Case (Zcompare x y); Auto; Discriminate. +Qed. + +Lemma Zge_cases : (x,y:Z)if (Zge_bool x y) then `x>=y` else `x<y`. +Proof. +Intros x y; Unfold Zge_bool Zge Zlt. +Case (Zcompare x y); Auto; Discriminate. +Qed. + +Lemma Zgt_cases : (x,y:Z)if (Zgt_bool x y) then `x>y` else `x<=y`. +Proof. +Intros x y; Unfold Zgt_bool Zgt Zle. +Case (Zcompare x y); Auto; Discriminate. +Qed. + +(** Lemmas on [Zle_bool] used in contrib/graphs *) + +Lemma Zle_bool_imp_le : (x,y:Z) (Zle_bool x y)=true -> (Zle x y). +Proof. + Unfold Zle_bool Zle. Intros x y. Unfold not. + Case (Zcompare x y); Intros; Discriminate. +Qed. + +Lemma Zle_imp_le_bool : (x,y:Z) (Zle x y) -> (Zle_bool x y)=true. +Proof. + Unfold Zle Zle_bool. Intros x y. Case (Zcompare x y); Trivial. Intro. Elim (H (refl_equal ? ?)). +Qed. + +Lemma Zle_bool_refl : (x:Z) (Zle_bool x x)=true. +Proof. + Intro. Apply Zle_imp_le_bool. Apply Zle_refl. Reflexivity. +Qed. + +Lemma Zle_bool_antisym : (x,y:Z) (Zle_bool x y)=true -> (Zle_bool y x)=true -> x=y. +Proof. + Intros. Apply Zle_antisym. Apply Zle_bool_imp_le. Assumption. + Apply Zle_bool_imp_le. Assumption. +Qed. + +Lemma Zle_bool_trans : (x,y,z:Z) (Zle_bool x y)=true -> (Zle_bool y z)=true -> (Zle_bool x z)=true. +Proof. + Intros x y z; Intros. Apply Zle_imp_le_bool. Apply Zle_trans with m:=y. Apply Zle_bool_imp_le. Assumption. + Apply Zle_bool_imp_le. Assumption. +Qed. + +Definition Zle_bool_total : (x,y:Z) {(Zle_bool x y)=true}+{(Zle_bool y x)=true}. +Proof. + Intros x y; Intros. Unfold Zle_bool. Cut (Zcompare x y)=SUPERIEUR<->(Zcompare y x)=INFERIEUR. + Case (Zcompare x y). Left . Reflexivity. + Left . Reflexivity. + Right . Rewrite (proj1 ? ? H (refl_equal ? ?)). Reflexivity. + Apply Zcompare_ANTISYM. +Defined. + +Lemma Zle_bool_plus_mono : (x,y,z,t:Z) (Zle_bool x y)=true -> (Zle_bool z t)=true -> + (Zle_bool (Zplus x z) (Zplus y t))=true. +Proof. + Intros. Apply Zle_imp_le_bool. Apply Zle_plus_plus. Apply Zle_bool_imp_le. Assumption. + Apply Zle_bool_imp_le. Assumption. +Qed. + +Lemma Zone_pos : (Zle_bool `1` `0`)=false. +Proof. + Reflexivity. +Qed. + +Lemma Zone_min_pos : (x:Z) (Zle_bool x `0`)=false -> (Zle_bool `1` x)=true. +Proof. + Intros x; Intros. Apply Zle_imp_le_bool. Change (Zle (Zs ZERO) x). Apply Zgt_le_S. Generalize H. + Unfold Zle_bool Zgt. Case (Zcompare x ZERO). Intro H0. Discriminate H0. + Intro H0. Discriminate H0. + Reflexivity. +Qed. + + + Lemma Zle_is_le_bool : (x,y:Z) (Zle x y) <-> (Zle_bool x y)=true. + Proof. + Intros. Split. Intro. Apply Zle_imp_le_bool. Assumption. + Intro. Apply Zle_bool_imp_le. Assumption. + Qed. + + Lemma Zge_is_le_bool : (x,y:Z) (Zge x y) <-> (Zle_bool y x)=true. + Proof. + Intros. Split. Intro. Apply Zle_imp_le_bool. Apply Zge_le. Assumption. + Intro. Apply Zle_ge. Apply Zle_bool_imp_le. Assumption. + Qed. + + Lemma Zlt_is_le_bool : (x,y:Z) (Zlt x y) <-> (Zle_bool x `y-1`)=true. + Proof. + Intros x y. Split. Intro. Apply Zle_imp_le_bool. Apply Zlt_n_Sm_le. Rewrite (Zs_pred y) in H. + Assumption. + Intro. Rewrite (Zs_pred y). Apply Zle_lt_n_Sm. Apply Zle_bool_imp_le. Assumption. + Qed. + + Lemma Zgt_is_le_bool : (x,y:Z) (Zgt x y) <-> (Zle_bool y `x-1`)=true. + Proof. + Intros x y. Apply iff_trans with `y < x`. Split. Exact (Zgt_lt x y). + Exact (Zlt_gt y x). + Exact (Zlt_is_le_bool y x). + Qed. + diff --git a/theories7/ZArith/Zcompare.v b/theories7/ZArith/Zcompare.v new file mode 100644 index 00000000..fd11ae9b --- /dev/null +++ b/theories7/ZArith/Zcompare.v @@ -0,0 +1,480 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $$ i*) + +Require Export BinPos. +Require Export BinInt. +Require Zsyntax. +Require Lt. +Require Gt. +Require Plus. +Require Mult. + +Open Local Scope Z_scope. + +(**********************************************************************) +(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) +(**********************************************************************) + +(**********************************************************************) +(** Comparison on integers *) + +Lemma Zcompare_x_x : (x:Z) (Zcompare x x) = EGAL. +Proof. +Intro x; NewDestruct x as [|p|p]; Simpl; [ Reflexivity | Apply convert_compare_EGAL + | Rewrite convert_compare_EGAL; Reflexivity ]. +Qed. + +Lemma Zcompare_EGAL_eq : (x,y:Z) (Zcompare x y) = EGAL -> x = y. +Proof. +Intros x y; NewDestruct x as [|x'|x'];NewDestruct y as [|y'|y'];Simpl;Intro H; Reflexivity Orelse Try Discriminate H; [ + Rewrite (compare_convert_EGAL x' y' H); Reflexivity + | Rewrite (compare_convert_EGAL x' y'); [ + Reflexivity + | NewDestruct (compare x' y' EGAL); + Reflexivity Orelse Discriminate]]. +Qed. + +Lemma Zcompare_EGAL : (x,y:Z) (Zcompare x y) = EGAL <-> x = y. +Proof. +Intros x y;Split; Intro E; [ Apply Zcompare_EGAL_eq; Assumption + | Rewrite E; Apply Zcompare_x_x ]. +Qed. + +Lemma Zcompare_antisym : + (x,y:Z)(Op (Zcompare x y)) = (Zcompare y x). +Proof. +Intros x y; NewDestruct x; NewDestruct y; Simpl; + Reflexivity Orelse Discriminate H Orelse + Rewrite Pcompare_antisym; Reflexivity. +Qed. + +Lemma Zcompare_ANTISYM : + (x,y:Z) (Zcompare x y) = SUPERIEUR <-> (Zcompare y x) = INFERIEUR. +Proof. +Intros x y; Split; Intro H; [ + Change INFERIEUR with (Op SUPERIEUR); + Rewrite <- Zcompare_antisym; Rewrite H; Reflexivity +| Change SUPERIEUR with (Op INFERIEUR); + Rewrite <- Zcompare_antisym; Rewrite H; Reflexivity ]. +Qed. + +(** Transitivity of comparison *) + +Lemma Zcompare_trans_SUPERIEUR : + (x,y,z:Z) (Zcompare x y) = SUPERIEUR -> + (Zcompare y z) = SUPERIEUR -> + (Zcompare x z) = SUPERIEUR. +Proof. +Intros x y z;Case x;Case y;Case z; Simpl; +Try (Intros; Discriminate H Orelse Discriminate H0); +Auto with arith; [ + Intros p q r H H0;Apply convert_compare_SUPERIEUR; Unfold gt; + Apply lt_trans with m:=(convert q); + Apply compare_convert_INFERIEUR;Apply ZC1;Assumption +| Intros p q r; Do 3 Rewrite <- ZC4; Intros H H0; + Apply convert_compare_SUPERIEUR;Unfold gt;Apply lt_trans with m:=(convert q); + Apply compare_convert_INFERIEUR;Apply ZC1;Assumption ]. +Qed. + +(** Comparison and opposite *) + +Lemma Zcompare_Zopp : + (x,y:Z) (Zcompare x y) = (Zcompare (Zopp y) (Zopp x)). +Proof. +(Intros x y;Case x;Case y;Simpl;Auto with arith); +Intros;Rewrite <- ZC4;Trivial with arith. +Qed. + +Hints Local Resolve convert_compare_EGAL. + +(** Comparison first-order specification *) + +Lemma SUPERIEUR_POS : + (x,y:Z) (Zcompare x y) = SUPERIEUR -> + (EX h:positive |(Zplus x (Zopp y)) = (POS h)). +Proof. +Intros x y;Case x;Case y; [ + Simpl; Intros H; Discriminate H +| Simpl; Intros p H; Discriminate H +| Intros p H; Exists p; Simpl; Auto with arith +| Intros p H; Exists p; Simpl; Auto with arith +| Intros q p H; Exists (true_sub p q); Unfold Zplus Zopp; + Unfold Zcompare in H; Rewrite H; Trivial with arith +| Intros q p H; Exists (add p q); Simpl; Trivial with arith +| Simpl; Intros p H; Discriminate H +| Simpl; Intros q p H; Discriminate H +| Unfold Zcompare; Intros q p; Rewrite <- ZC4; Intros H; Exists (true_sub q p); + Simpl; Rewrite (ZC1 q p H); Trivial with arith]. +Qed. + +(** Comparison and addition *) + +Lemma weaken_Zcompare_Zplus_compatible : + ((n,m:Z) (p:positive) + (Zcompare (Zplus (POS p) n) (Zplus (POS p) m)) = (Zcompare n m)) -> + (x,y,z:Z) (Zcompare (Zplus z x) (Zplus z y)) = (Zcompare x y). +Proof. +Intros H x y z; NewDestruct z; [ + Reflexivity +| Apply H +| Rewrite (Zcompare_Zopp x y); Rewrite Zcompare_Zopp; + Do 2 Rewrite Zopp_Zplus; Rewrite Zopp_NEG; Apply H ]. +Qed. + +Hints Local Resolve ZC4. + +Lemma weak_Zcompare_Zplus_compatible : + (x,y:Z) (z:positive) + (Zcompare (Zplus (POS z) x) (Zplus (POS z) y)) = (Zcompare x y). +Proof. +Intros x y z;Case x;Case y;Simpl;Auto with arith; [ + Intros p;Apply convert_compare_INFERIEUR; Apply ZL17 +| Intros p;ElimPcompare z p;Intros E;Rewrite E;Auto with arith; + Apply convert_compare_SUPERIEUR; Rewrite true_sub_convert; [ Unfold gt ; + Apply ZL16 | Assumption ] +| Intros p;ElimPcompare z p; + Intros E;Auto with arith; Apply convert_compare_SUPERIEUR; + Unfold gt;Apply ZL17 +| Intros p q; + ElimPcompare q p; + Intros E;Rewrite E;[ + Rewrite (compare_convert_EGAL q p E); Apply convert_compare_EGAL + | Apply convert_compare_INFERIEUR;Do 2 Rewrite convert_add;Apply lt_reg_l; + Apply compare_convert_INFERIEUR with 1:=E + | Apply convert_compare_SUPERIEUR;Unfold gt ;Do 2 Rewrite convert_add; + Apply lt_reg_l;Exact (compare_convert_SUPERIEUR q p E) ] +| Intros p q; + ElimPcompare z p; + Intros E;Rewrite E;Auto with arith; + Apply convert_compare_SUPERIEUR; Rewrite true_sub_convert; [ + Unfold gt; Apply lt_trans with m:=(convert z); [Apply ZL16 | Apply ZL17] + | Assumption ] +| Intros p;ElimPcompare z p;Intros E;Rewrite E;Auto with arith; Simpl; + Apply convert_compare_INFERIEUR;Rewrite true_sub_convert;[Apply ZL16| + Assumption] +| Intros p q; + ElimPcompare z q; + Intros E;Rewrite E;Auto with arith; Simpl;Apply convert_compare_INFERIEUR; + Rewrite true_sub_convert;[ + Apply lt_trans with m:=(convert z) ;[Apply ZL16|Apply ZL17] + | Assumption] +| Intros p q; ElimPcompare z q; Intros E0;Rewrite E0; + ElimPcompare z p; Intros E1;Rewrite E1; ElimPcompare q p; + Intros E2;Rewrite E2;Auto with arith; [ + Absurd (compare q p EGAL)=INFERIEUR; [ + Rewrite <- (compare_convert_EGAL z q E0); + Rewrite <- (compare_convert_EGAL z p E1); + Rewrite (convert_compare_EGAL z); Discriminate + | Assumption ] + | Absurd (compare q p EGAL)=SUPERIEUR; [ + Rewrite <- (compare_convert_EGAL z q E0); + Rewrite <- (compare_convert_EGAL z p E1); + Rewrite (convert_compare_EGAL z);Discriminate + | Assumption] + | Absurd (compare z p EGAL)=INFERIEUR; [ + Rewrite (compare_convert_EGAL z q E0); + Rewrite <- (compare_convert_EGAL q p E2); + Rewrite (convert_compare_EGAL q);Discriminate + | Assumption ] + | Absurd (compare z p EGAL)=INFERIEUR; [ + Rewrite (compare_convert_EGAL z q E0); Rewrite E2;Discriminate + | Assumption] + | Absurd (compare z p EGAL)=SUPERIEUR;[ + Rewrite (compare_convert_EGAL z q E0); + Rewrite <- (compare_convert_EGAL q p E2); + Rewrite (convert_compare_EGAL q);Discriminate + | Assumption] + | Absurd (compare z p EGAL)=SUPERIEUR;[ + Rewrite (compare_convert_EGAL z q E0);Rewrite E2;Discriminate + | Assumption] + | Absurd (compare z q EGAL)=INFERIEUR;[ + Rewrite (compare_convert_EGAL z p E1); + Rewrite (compare_convert_EGAL q p E2); + Rewrite (convert_compare_EGAL p); Discriminate + | Assumption] + | Absurd (compare p q EGAL)=SUPERIEUR; [ + Rewrite <- (compare_convert_EGAL z p E1); + Rewrite E0; Discriminate + | Apply ZC2;Assumption ] + | Simpl; Rewrite (compare_convert_EGAL q p E2); + Rewrite (convert_compare_EGAL (true_sub p z)); Auto with arith + | Simpl; Rewrite <- ZC4; Apply convert_compare_SUPERIEUR; + Rewrite true_sub_convert; [ + Rewrite true_sub_convert; [ + Unfold gt; Apply simpl_lt_plus_l with p:=(convert z); + Rewrite le_plus_minus_r; [ + Rewrite le_plus_minus_r; [ + Apply compare_convert_INFERIEUR;Assumption + | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Assumption ] + | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Assumption ] + | Apply ZC2;Assumption ] + | Apply ZC2;Assumption ] + | Simpl; Rewrite <- ZC4; Apply convert_compare_INFERIEUR; + Rewrite true_sub_convert; [ + Rewrite true_sub_convert; [ + Apply simpl_lt_plus_l with p:=(convert z); + Rewrite le_plus_minus_r; [ + Rewrite le_plus_minus_r; [ + Apply compare_convert_INFERIEUR;Apply ZC1;Assumption + | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Assumption ] + | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Assumption ] + | Apply ZC2;Assumption] + | Apply ZC2;Assumption ] + | Absurd (compare z q EGAL)=INFERIEUR; [ + Rewrite (compare_convert_EGAL q p E2);Rewrite E1;Discriminate + | Assumption ] + | Absurd (compare q p EGAL)=INFERIEUR; [ + Cut (compare q p EGAL)=SUPERIEUR; [ + Intros E;Rewrite E;Discriminate + | Apply convert_compare_SUPERIEUR; Unfold gt; + Apply lt_trans with m:=(convert z); [ + Apply compare_convert_INFERIEUR;Apply ZC1;Assumption + | Apply compare_convert_INFERIEUR;Assumption ]] + | Assumption ] + | Absurd (compare z q EGAL)=SUPERIEUR; [ + Rewrite (compare_convert_EGAL z p E1); + Rewrite (compare_convert_EGAL q p E2); + Rewrite (convert_compare_EGAL p); Discriminate + | Assumption ] + | Absurd (compare z q EGAL)=SUPERIEUR; [ + Rewrite (compare_convert_EGAL z p E1); + Rewrite ZC1; [Discriminate | Assumption ] + | Assumption ] + | Absurd (compare z q EGAL)=SUPERIEUR; [ + Rewrite (compare_convert_EGAL q p E2); Rewrite E1; Discriminate + | Assumption ] + | Absurd (compare q p EGAL)=SUPERIEUR; [ + Rewrite ZC1; [ + Discriminate + | Apply convert_compare_SUPERIEUR; Unfold gt; + Apply lt_trans with m:=(convert z); [ + Apply compare_convert_INFERIEUR;Apply ZC1;Assumption + | Apply compare_convert_INFERIEUR;Assumption ]] + | Assumption ] + | Simpl; Rewrite (compare_convert_EGAL q p E2); Apply convert_compare_EGAL + | Simpl; Apply convert_compare_SUPERIEUR; Unfold gt; + Rewrite true_sub_convert; [ + Rewrite true_sub_convert; [ + Apply simpl_lt_plus_l with p:=(convert p); Rewrite le_plus_minus_r; [ + Rewrite plus_sym; Apply simpl_lt_plus_l with p:=(convert q); + Rewrite plus_assoc_l; Rewrite le_plus_minus_r; [ + Rewrite (plus_sym (convert q)); Apply lt_reg_l; + Apply compare_convert_INFERIEUR;Assumption + | Apply lt_le_weak; Apply compare_convert_INFERIEUR; + Apply ZC1;Assumption ] + | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Apply ZC1; + Assumption ] + | Assumption ] + | Assumption ] + | Simpl; Apply convert_compare_INFERIEUR; Rewrite true_sub_convert; [ + Rewrite true_sub_convert; [ + Apply simpl_lt_plus_l with p:=(convert q); Rewrite le_plus_minus_r; [ + Rewrite plus_sym; Apply simpl_lt_plus_l with p:=(convert p); + Rewrite plus_assoc_l; Rewrite le_plus_minus_r; [ + Rewrite (plus_sym (convert p)); Apply lt_reg_l; + Apply compare_convert_INFERIEUR;Apply ZC1;Assumption + | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Apply ZC1; + Assumption ] + | Apply lt_le_weak;Apply compare_convert_INFERIEUR;Apply ZC1;Assumption] + | Assumption] + | Assumption]]]. +Qed. + +Lemma Zcompare_Zplus_compatible : + (x,y,z:Z) (Zcompare (Zplus z x) (Zplus z y)) = (Zcompare x y). +Proof. +Exact (weaken_Zcompare_Zplus_compatible weak_Zcompare_Zplus_compatible). +Qed. + +Lemma Zcompare_Zplus_compatible2 : + (r:relation)(x,y,z,t:Z) + (Zcompare x y) = r -> (Zcompare z t) = r -> + (Zcompare (Zplus x z) (Zplus y t)) = r. +Proof. +Intros r x y z t; Case r; [ + Intros H1 H2; Elim (Zcompare_EGAL x y); Elim (Zcompare_EGAL z t); + Intros H3 H4 H5 H6; Rewrite H3; [ + Rewrite H5; [ Elim (Zcompare_EGAL (Zplus y t) (Zplus y t)); Auto with arith | Auto with arith ] + | Auto with arith ] +| Intros H1 H2; Elim (Zcompare_ANTISYM (Zplus y t) (Zplus x z)); + Intros H3 H4; Apply H3; + Apply Zcompare_trans_SUPERIEUR with y:=(Zplus y z) ; [ + Rewrite Zcompare_Zplus_compatible; + Elim (Zcompare_ANTISYM t z); Auto with arith + | Do 2 Rewrite <- (Zplus_sym z); + Rewrite Zcompare_Zplus_compatible; + Elim (Zcompare_ANTISYM y x); Auto with arith] +| Intros H1 H2; + Apply Zcompare_trans_SUPERIEUR with y:=(Zplus x t) ; [ + Rewrite Zcompare_Zplus_compatible; Assumption + | Do 2 Rewrite <- (Zplus_sym t); + Rewrite Zcompare_Zplus_compatible; Assumption]]. +Qed. + +Lemma Zcompare_Zs_SUPERIEUR : (x:Z)(Zcompare (Zs x) x)=SUPERIEUR. +Proof. +Intro x; Unfold Zs; Pattern 2 x; Rewrite <- (Zero_right x); +Rewrite Zcompare_Zplus_compatible;Reflexivity. +Qed. + +Lemma Zcompare_et_un: + (x,y:Z) (Zcompare x y)=SUPERIEUR <-> + ~(Zcompare x (Zplus y (POS xH)))=INFERIEUR. +Proof. +Intros x y; Split; [ + Intro H; (ElimCompare 'x '(Zplus y (POS xH)));[ + Intro H1; Rewrite H1; Discriminate + | Intros H1; Elim SUPERIEUR_POS with 1:=H; Intros h H2; + Absurd (gt (convert h) O) /\ (lt (convert h) (S O)); [ + Unfold not ;Intros H3;Elim H3;Intros H4 H5; Absurd (gt (convert h) O); [ + Unfold gt ;Apply le_not_lt; Apply le_S_n; Exact H5 + | Assumption] + | Split; [ + Elim (ZL4 h); Intros i H3;Rewrite H3; Apply gt_Sn_O + | Change (lt (convert h) (convert xH)); + Apply compare_convert_INFERIEUR; + Change (Zcompare (POS h) (POS xH))=INFERIEUR; + Rewrite <- H2; Rewrite <- [m,n:Z](Zcompare_Zplus_compatible m n y); + Rewrite (Zplus_sym x);Rewrite Zplus_assoc; Rewrite Zplus_inverse_r; + Simpl; Exact H1 ]] + | Intros H1;Rewrite -> H1;Discriminate ] +| Intros H; (ElimCompare 'x '(Zplus y (POS xH))); [ + Intros H1;Elim (Zcompare_EGAL x (Zplus y (POS xH))); Intros H2 H3; + Rewrite (H2 H1); Exact (Zcompare_Zs_SUPERIEUR y) + | Intros H1;Absurd (Zcompare x (Zplus y (POS xH)))=INFERIEUR;Assumption + | Intros H1; Apply Zcompare_trans_SUPERIEUR with y:=(Zs y); + [ Exact H1 | Exact (Zcompare_Zs_SUPERIEUR y)]]]. +Qed. + +(** Successor and comparison *) + +Lemma Zcompare_n_S : (n,m:Z)(Zcompare (Zs n) (Zs m)) = (Zcompare n m). +Proof. +Intros n m;Unfold Zs ;Do 2 Rewrite -> [t:Z](Zplus_sym t (POS xH)); +Rewrite -> Zcompare_Zplus_compatible;Auto with arith. +Qed. + +(** Multiplication and comparison *) + +Lemma Zcompare_Zmult_compatible : + (x:positive)(y,z:Z) + (Zcompare (Zmult (POS x) y) (Zmult (POS x) z)) = (Zcompare y z). +Proof. +Intros x; NewInduction x as [p H|p H|]; [ + Intros y z; + Cut (POS (xI p))=(Zplus (Zplus (POS p) (POS p)) (POS xH)); [ + Intros E; Rewrite E; Do 4 Rewrite Zmult_plus_distr_l; + Do 2 Rewrite Zmult_one; + Apply Zcompare_Zplus_compatible2; [ + Apply Zcompare_Zplus_compatible2; Apply H + | Trivial with arith] + | Simpl; Rewrite (add_x_x p); Trivial with arith] +| Intros y z; Cut (POS (xO p))=(Zplus (POS p) (POS p)); [ + Intros E; Rewrite E; Do 2 Rewrite Zmult_plus_distr_l; + Apply Zcompare_Zplus_compatible2; Apply H + | Simpl; Rewrite (add_x_x p); Trivial with arith] + | Intros y z; Do 2 Rewrite Zmult_one; Trivial with arith]. +Qed. + + +(** Reverting [x ?= y] to trichotomy *) + +Lemma rename : (A:Set)(P:A->Prop)(x:A) ((y:A)(x=y)->(P y)) -> (P x). +Proof. +Auto with arith. +Qed. + +Lemma Zcompare_elim : + (c1,c2,c3:Prop)(x,y:Z) + ((x=y) -> c1) ->(`x<y` -> c2) ->(`x>y`-> c3) + -> Cases (Zcompare x y) of EGAL => c1 | INFERIEUR => c2 | SUPERIEUR => c3 end. +Proof. +Intros c1 c2 c3 x y; Intros. +Apply rename with x:=(Zcompare x y); Intro r; Elim r; +[ Intro; Apply H; Apply (Zcompare_EGAL_eq x y); Assumption +| Unfold Zlt in H0; Assumption +| Unfold Zgt in H1; Assumption ]. +Qed. + +Lemma Zcompare_eq_case : + (c1,c2,c3:Prop)(x,y:Z) c1 -> x=y -> + Cases (Zcompare x y) of EGAL => c1 | INFERIEUR => c2 | SUPERIEUR => c3 end. +Proof. +Intros c1 c2 c3 x y; Intros. +Rewrite H0; Rewrite (Zcompare_x_x). +Assumption. +Qed. + +(** Decompose an egality between two [?=] relations into 3 implications *) + +Lemma Zcompare_egal_dec : + (x1,y1,x2,y2:Z) + (`x1<y1`->`x2<y2`) + ->((Zcompare x1 y1)=EGAL -> (Zcompare x2 y2)=EGAL) + ->(`x1>y1`->`x2>y2`)->(Zcompare x1 y1)=(Zcompare x2 y2). +Proof. +Intros x1 y1 x2 y2. +Unfold Zgt; Unfold Zlt; +Case (Zcompare x1 y1); Case (Zcompare x2 y2); Auto with arith; Symmetry; Auto with arith. +Qed. + +(** Relating [x ?= y] to [Zle], [Zlt], [Zge] or [Zgt] *) + +Lemma Zle_Zcompare : + (x,y:Z)`x<=y` -> + Cases (Zcompare x y) of EGAL => True | INFERIEUR => True | SUPERIEUR => False end. +Proof. +Intros x y; Unfold Zle; Elim (Zcompare x y); Auto with arith. +Qed. + +Lemma Zlt_Zcompare : + (x,y:Z)`x<y` -> + Cases (Zcompare x y) of EGAL => False | INFERIEUR => True | SUPERIEUR => False end. +Proof. +Intros x y; Unfold Zlt; Elim (Zcompare x y); Intros; Discriminate Orelse Trivial with arith. +Qed. + +Lemma Zge_Zcompare : + (x,y:Z)`x>=y`-> + Cases (Zcompare x y) of EGAL => True | INFERIEUR => False | SUPERIEUR => True end. +Proof. +Intros x y; Unfold Zge; Elim (Zcompare x y); Auto with arith. +Qed. + +Lemma Zgt_Zcompare : + (x,y:Z)`x>y` -> + Cases (Zcompare x y) of EGAL => False | INFERIEUR => False | SUPERIEUR => True end. +Proof. +Intros x y; Unfold Zgt; Elim (Zcompare x y); Intros; Discriminate Orelse Trivial with arith. +Qed. + +(**********************************************************************) +(* Other properties *) + +V7only [Set Implicit Arguments.]. + +Lemma Zcompare_Zmult_left : (x,y,z:Z)`z>0` -> `x ?= y`=`z*x ?= z*y`. +Proof. +Intros x y z H; NewDestruct z. + Discriminate H. + Rewrite Zcompare_Zmult_compatible; Reflexivity. + Discriminate H. +Qed. + +Lemma Zcompare_Zmult_right : (x,y,z:Z)` z>0` -> `x ?= y`=`x*z ?= y*z`. +Proof. +Intros x y z H; +Rewrite (Zmult_sym x z); +Rewrite (Zmult_sym y z); +Apply Zcompare_Zmult_left; Assumption. +Qed. + +V7only [Unset Implicit Arguments.]. + diff --git a/theories7/ZArith/Zcomplements.v b/theories7/ZArith/Zcomplements.v new file mode 100644 index 00000000..72d837b6 --- /dev/null +++ b/theories7/ZArith/Zcomplements.v @@ -0,0 +1,212 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zcomplements.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +Require ZArithRing. +Require ZArith_base. +Require Omega. +Require Wf_nat. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +V7only [Set Implicit Arguments.]. + +(**********************************************************************) +(** About parity *) + +Lemma two_or_two_plus_one : (x:Z) { y:Z | `x = 2*y`}+{ y:Z | `x = 2*y+1`}. +Proof. +Intro x; NewDestruct x. +Left ; Split with ZERO; Reflexivity. + +NewDestruct p. +Right ; Split with (POS p); Reflexivity. + +Left ; Split with (POS p); Reflexivity. + +Right ; Split with ZERO; Reflexivity. + +NewDestruct p. +Right ; Split with (NEG (add xH p)). +Rewrite NEG_xI. +Rewrite NEG_add. +Omega. + +Left ; Split with (NEG p); Reflexivity. + +Right ; Split with `-1`; Reflexivity. +Qed. + +(**********************************************************************) +(** The biggest power of 2 that is stricly less than [a] + + Easy to compute: replace all "1" of the binary representation by + "0", except the first "1" (or the first one :-) *) + +Fixpoint floor_pos [a : positive] : positive := + Cases a of + | xH => xH + | (xO a') => (xO (floor_pos a')) + | (xI b') => (xO (floor_pos b')) + end. + +Definition floor := [a:positive](POS (floor_pos a)). + +Lemma floor_gt0 : (x:positive) `(floor x) > 0`. +Proof. +Intro. +Compute. +Trivial. +Qed. + +Lemma floor_ok : (a:positive) + `(floor a) <= (POS a) < 2*(floor a)`. +Proof. +Unfold floor. +Intro a; NewInduction a as [p|p|]. + +Simpl. +Repeat Rewrite POS_xI. +Rewrite (POS_xO (xO (floor_pos p))). +Rewrite (POS_xO (floor_pos p)). +Omega. + +Simpl. +Repeat Rewrite POS_xI. +Rewrite (POS_xO (xO (floor_pos p))). +Rewrite (POS_xO (floor_pos p)). +Rewrite (POS_xO p). +Omega. + +Simpl; Omega. +Qed. + +(**********************************************************************) +(** Two more induction principles over [Z]. *) + +Theorem Z_lt_abs_rec : (P: Z -> Set) + ((n: Z) ((m: Z) `|m|<|n|` -> (P m)) -> (P n)) -> (p: Z) (P p). +Proof. +Intros P HP p. +LetTac Q:=[z]`0<=z`->(P z)*(P `-z`). +Cut (Q `|p|`);[Intros|Apply (Z_lt_rec Q);Auto with zarith]. +Elim (Zabs_dec p);Intro eq;Rewrite eq;Elim H;Auto with zarith. +Unfold Q;Clear Q;Intros. +Apply pair;Apply HP. +Rewrite Zabs_eq;Auto;Intros. +Elim (H `|m|`);Intros;Auto with zarith. +Elim (Zabs_dec m);Intro eq;Rewrite eq;Trivial. +Rewrite Zabs_non_eq;Auto with zarith. +Rewrite Zopp_Zopp;Intros. +Elim (H `|m|`);Intros;Auto with zarith. +Elim (Zabs_dec m);Intro eq;Rewrite eq;Trivial. +Qed. + +Theorem Z_lt_abs_induction : (P: Z -> Prop) + ((n: Z) ((m: Z) `|m|<|n|` -> (P m)) -> (P n)) -> (p: Z) (P p). +Proof. +Intros P HP p. +LetTac Q:=[z]`0<=z`->(P z) /\ (P `-z`). +Cut (Q `|p|`);[Intros|Apply (Z_lt_induction Q);Auto with zarith]. +Elim (Zabs_dec p);Intro eq;Rewrite eq;Elim H;Auto with zarith. +Unfold Q;Clear Q;Intros. +Split;Apply HP. +Rewrite Zabs_eq;Auto;Intros. +Elim (H `|m|`);Intros;Auto with zarith. +Elim (Zabs_dec m);Intro eq;Rewrite eq;Trivial. +Rewrite Zabs_non_eq;Auto with zarith. +Rewrite Zopp_Zopp;Intros. +Elim (H `|m|`);Intros;Auto with zarith. +Elim (Zabs_dec m);Intro eq;Rewrite eq;Trivial. +Qed. +V7only [Unset Implicit Arguments.]. + +(** To do case analysis over the sign of [z] *) + +Lemma Zcase_sign : (x:Z)(P:Prop) + (`x=0` -> P) -> + (`x>0` -> P) -> + (`x<0` -> P) -> P. +Proof. +Intros x P Hzero Hpos Hneg. +Induction x. +Apply Hzero; Trivial. +Apply Hpos; Apply POS_gt_ZERO. +Apply Hneg; Apply NEG_lt_ZERO. +Save. + +Lemma sqr_pos : (x:Z)`x*x >= 0`. +Proof. +Intro x. +Apply (Zcase_sign x `x*x >= 0`). +Intros H; Rewrite H; Omega. +Intros H; Replace `0` with `0*0`. +Apply Zge_Zmult_pos_compat; Omega. +Omega. +Intros H; Replace `0` with `0*0`. +Replace `x*x` with `(-x)*(-x)`. +Apply Zge_Zmult_pos_compat; Omega. +Ring. +Omega. +Save. + +(**********************************************************************) +(** A list length in Z, tail recursive. *) + +Require PolyList. + +Fixpoint Zlength_aux [acc: Z; A:Set; l:(list A)] : Z := Cases l of + nil => acc + | (cons _ l) => (Zlength_aux (Zs acc) A l) +end. + +Definition Zlength := (Zlength_aux 0). +Implicits Zlength [1]. + +Section Zlength_properties. + +Variable A:Set. + +Implicit Variable Type l:(list A). + +Lemma Zlength_correct : (l:(list A))(Zlength l)=(inject_nat (length l)). +Proof. +Assert (l:(list A))(acc:Z)(Zlength_aux acc A l)=acc+(inject_nat (length l)). +Induction l. +Simpl; Auto with zarith. +Intros; Simpl (length (cons a l0)); Rewrite inj_S. +Simpl; Rewrite H; Auto with zarith. +Unfold Zlength; Intros; Rewrite H; Auto. +Qed. + +Lemma Zlength_nil : (Zlength 1!A (nil A))=0. +Proof. +Auto. +Qed. + +Lemma Zlength_cons : (x:A)(l:(list A))(Zlength (cons x l))=(Zs (Zlength l)). +Proof. +Intros; Do 2 Rewrite Zlength_correct. +Simpl (length (cons x l)); Rewrite inj_S; Auto. +Qed. + +Lemma Zlength_nil_inv : (l:(list A))(Zlength l)=0 -> l=(nil ?). +Proof. +Intro l; Rewrite Zlength_correct. +Case l; Auto. +Intros x l'; Simpl (length (cons x l')). +Rewrite inj_S. +Intros; ElimType False; Generalize (ZERO_le_inj (length l')); Omega. +Qed. + +End Zlength_properties. + +Implicits Zlength_correct [1]. +Implicits Zlength_cons [1]. +Implicits Zlength_nil_inv [1]. diff --git a/theories7/ZArith/Zdiv.v b/theories7/ZArith/Zdiv.v new file mode 100644 index 00000000..84d53931 --- /dev/null +++ b/theories7/ZArith/Zdiv.v @@ -0,0 +1,432 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zdiv.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +(* Contribution by Claude Marché and Xavier Urbain *) + +(** + +Euclidean Division + +Defines first of function that allows Coq to normalize. +Then only after proves the main required property. + +*) + +Require Export ZArith_base. +Require Zbool. +Require Omega. +Require ZArithRing. +Require Zcomplements. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +(** + + Euclidean division of a positive by a integer + (that is supposed to be positive). + + total function than returns an arbitrary value when + divisor is not positive + +*) + +Fixpoint Zdiv_eucl_POS [a:positive] : Z -> Z*Z := [b:Z] + Cases a of + | xH => if `(Zge_bool b 2)` then `(0,1)` else `(1,0)` + | (xO a') => + let (q,r) = (Zdiv_eucl_POS a' b) in + [r':=`2*r`] if `(Zgt_bool b r')` then `(2*q,r')` else `(2*q+1,r'-b)` + | (xI a') => + let (q,r) = (Zdiv_eucl_POS a' b) in + [r':=`2*r+1`] if `(Zgt_bool b r')` then `(2*q,r')` else `(2*q+1,r'-b)` + end. + + +(** + + Euclidean division of integers. + + Total function than returns (0,0) when dividing by 0. + +*) + +(* + + The pseudo-code is: + + if b = 0 : (0,0) + + if b <> 0 and a = 0 : (0,0) + + if b > 0 and a < 0 : let (q,r) = div_eucl_pos (-a) b in + if r = 0 then (-q,0) else (-(q+1),b-r) + + if b < 0 and a < 0 : let (q,r) = div_eucl (-a) (-b) in (q,-r) + + if b < 0 and a > 0 : let (q,r) = div_eucl a (-b) in + if r = 0 then (-q,0) else (-(q+1),b+r) + + In other word, when b is non-zero, q is chosen to be the greatest integer + smaller or equal to a/b. And sgn(r)=sgn(b) and |r| < |b|. + +*) + +Definition Zdiv_eucl [a,b:Z] : Z*Z := + Cases a b of + | ZERO _ => `(0,0)` + | _ ZERO => `(0,0)` + | (POS a') (POS _) => (Zdiv_eucl_POS a' b) + | (NEG a') (POS _) => + let (q,r) = (Zdiv_eucl_POS a' b) in + Cases r of + | ZERO => `(-q,0)` + | _ => `(-(q+1),b-r)` + end + | (NEG a') (NEG b') => + let (q,r) = (Zdiv_eucl_POS a' (POS b')) in `(q,-r)` + | (POS a') (NEG b') => + let (q,r) = (Zdiv_eucl_POS a' (POS b')) in + Cases r of + | ZERO => `(-q,0)` + | _ => `(-(q+1),b+r)` + end + end. + + +(** Division and modulo are projections of [Zdiv_eucl] *) + +Definition Zdiv [a,b:Z] : Z := let (q,_) = (Zdiv_eucl a b) in q. + +Definition Zmod [a,b:Z] : Z := let (_,r) = (Zdiv_eucl a b) in r. + +(* Tests: + +Eval Compute in `(Zdiv_eucl 7 3)`. + +Eval Compute in `(Zdiv_eucl (-7) 3)`. + +Eval Compute in `(Zdiv_eucl 7 (-3))`. + +Eval Compute in `(Zdiv_eucl (-7) (-3))`. + +*) + + +(** + + Main division theorem. + + First a lemma for positive + +*) + +Lemma Z_div_mod_POS : (b:Z)`b > 0` -> (a:positive) + let (q,r)=(Zdiv_eucl_POS a b) in `(POS a) = b*q + r`/\`0<=r<b`. +Proof. +Induction a; Unfold Zdiv_eucl_POS; Fold Zdiv_eucl_POS. + +Intro p; Case (Zdiv_eucl_POS p b); Intros q r (H0,H1). +Generalize (Zgt_cases b `2*r+1`). +Case (Zgt_bool b `2*r+1`); +(Rewrite POS_xI; Rewrite H0; Split ; [ Ring | Omega ]). + +Intros p; Case (Zdiv_eucl_POS p b); Intros q r (H0,H1). +Generalize (Zgt_cases b `2*r`). +Case (Zgt_bool b `2*r`); + Rewrite POS_xO; Change (POS (xO p)) with `2*(POS p)`; + Rewrite H0; (Split; [Ring | Omega]). + +Generalize (Zge_cases b `2`). +Case (Zge_bool b `2`); (Intros; Split; [Ring | Omega ]). +Omega. +Qed. + + +Theorem Z_div_mod : (a,b:Z)`b > 0` -> + let (q,r) = (Zdiv_eucl a b) in `a = b*q + r` /\ `0<=r<b`. +Proof. +Intros a b; Case a; Case b; Try (Simpl; Intros; Omega). +Unfold Zdiv_eucl; Intros; Apply Z_div_mod_POS; Trivial. + +Intros; Discriminate. + +Intros. +Generalize (Z_div_mod_POS (POS p) H p0). +Unfold Zdiv_eucl. +Case (Zdiv_eucl_POS p0 (POS p)). +Intros z z0. +Case z0. + +Intros [H1 H2]. +Split; Trivial. +Replace (NEG p0) with `-(POS p0)`; [ Rewrite H1; Ring | Trivial ]. + +Intros p1 [H1 H2]. +Split; Trivial. +Replace (NEG p0) with `-(POS p0)`; [ Rewrite H1; Ring | Trivial ]. +Generalize (POS_gt_ZERO p1); Omega. + +Intros p1 [H1 H2]. +Split; Trivial. +Replace (NEG p0) with `-(POS p0)`; [ Rewrite H1; Ring | Trivial ]. +Generalize (NEG_lt_ZERO p1); Omega. + +Intros; Discriminate. +Qed. + +(** Existence theorems *) + +Theorem Zdiv_eucl_exist : (b:Z)`b > 0` -> (a:Z) + { qr:Z*Z | let (q,r)=qr in `a=b*q+r` /\ `0 <= r < b` }. +Proof. +Intros b Hb a. +Exists (Zdiv_eucl a b). +Exact (Z_div_mod a b Hb). +Qed. + +Implicits Zdiv_eucl_exist. + +Theorem Zdiv_eucl_extended : (b:Z)`b <> 0` -> (a:Z) + { qr:Z*Z | let (q,r)=qr in `a=b*q+r` /\ `0 <= r < |b|` }. +Proof. +Intros b Hb a. +Elim (Z_le_gt_dec `0` b);Intro Hb'. +Cut `b>0`;[Intro Hb''|Omega]. +Rewrite Zabs_eq;[Apply Zdiv_eucl_exist;Assumption|Assumption]. +Cut `-b>0`;[Intro Hb''|Omega]. +Elim (Zdiv_eucl_exist Hb'' a);Intros qr. +Elim qr;Intros q r Hqr. +Exists (pair ? ? `-q` r). +Elim Hqr;Intros. +Split. +Rewrite <- Zmult_Zopp_left;Assumption. +Rewrite Zabs_non_eq;[Assumption|Omega]. +Qed. + +Implicits Zdiv_eucl_extended. + +(** Auxiliary lemmas about [Zdiv] and [Zmod] *) + +Lemma Z_div_mod_eq : (a,b:Z)`b > 0` -> `a = b * (Zdiv a b) + (Zmod a b)`. +Proof. +Unfold Zdiv Zmod. +Intros a b Hb. +Generalize (Z_div_mod a b Hb). +Case (Zdiv_eucl); Tauto. +Save. + +Lemma Z_mod_lt : (a,b:Z)`b > 0` -> `0 <= (Zmod a b) < b`. +Proof. +Unfold Zmod. +Intros a b Hb. +Generalize (Z_div_mod a b Hb). +Case (Zdiv_eucl a b); Tauto. +Save. + +Lemma Z_div_POS_ge0 : (b:Z)(a:positive) + let (q,_) = (Zdiv_eucl_POS a b) in `q >= 0`. +Proof. +Induction a; Unfold Zdiv_eucl_POS; Fold Zdiv_eucl_POS. +Intro p; Case (Zdiv_eucl_POS p b). +Intros; Case (Zgt_bool b `2*z0+1`); Intros; Omega. +Intro p; Case (Zdiv_eucl_POS p b). +Intros; Case (Zgt_bool b `2*z0`); Intros; Omega. +Case (Zge_bool b `2`); Simpl; Omega. +Save. + +Lemma Z_div_ge0 : (a,b:Z)`b > 0` -> `a >= 0` -> `(Zdiv a b) >= 0`. +Proof. +Intros a b Hb; Unfold Zdiv Zdiv_eucl; Case a; Simpl; Intros. +Case b; Simpl; Trivial. +Generalize Hb; Case b; Try Trivial. +Auto with zarith. +Intros p0 Hp0; Generalize (Z_div_POS_ge0 (POS p0) p). +Case (Zdiv_eucl_POS p (POS p0)); Simpl; Tauto. +Intros; Discriminate. +Elim H; Trivial. +Save. + +Lemma Z_div_lt : (a,b:Z)`b >= 2` -> `a > 0` -> `(Zdiv a b) < a`. +Proof. +Intros. Cut `b > 0`; [Intro Hb | Omega]. +Generalize (Z_div_mod a b Hb). +Cut `a >= 0`; [Intro Ha | Omega]. +Generalize (Z_div_ge0 a b Hb Ha). +Unfold Zdiv; Case (Zdiv_eucl a b); Intros q r H1 [H2 H3]. +Cut `a >= 2*q` -> `q < a`; [ Intro h; Apply h; Clear h | Intros; Omega ]. +Apply Zge_trans with `b*q`. +Omega. +Auto with zarith. +Save. + +(** Syntax *) + +V7only[ +Grammar znatural expr2 : constr := + expr_div [ expr2($p) "/" expr2($c) ] -> [ (Zdiv $p $c) ] +| expr_mod [ expr2($p) "%" expr2($c) ] -> [ (Zmod $p $c) ] +. + +Syntax constr + level 6: + Zdiv [ (Zdiv $n1 $n2) ] + -> [ [<hov 0> "`"(ZEXPR $n1):E "/" [0 0] (ZEXPR $n2):L "`"] ] + | Zmod [ (Zmod $n1 $n2) ] + -> [ [<hov 0> "`"(ZEXPR $n1):E "%" [0 0] (ZEXPR $n2):L "`"] ] + | Zdiv_inside + [ << (ZEXPR <<(Zdiv $n1 $n2)>>) >> ] + -> [ (ZEXPR $n1):E "/" [0 0] (ZEXPR $n2):L ] + | Zmod_inside + [ << (ZEXPR <<(Zmod $n1 $n2)>>) >> ] + -> [ (ZEXPR $n1):E " %" [1 0] (ZEXPR $n2):L ] +. +]. + + +Infix 3 "/" Zdiv (no associativity) : Z_scope V8only. +Infix 3 "mod" Zmod (no associativity) : Z_scope. + +(** Other lemmas (now using the syntax for [Zdiv] and [Zmod]). *) + +Lemma Z_div_ge : (a,b,c:Z)`c > 0`->`a >= b`->`a/c >= b/c`. +Proof. +Intros a b c cPos aGeb. +Generalize (Z_div_mod_eq a c cPos). +Generalize (Z_mod_lt a c cPos). +Generalize (Z_div_mod_eq b c cPos). +Generalize (Z_mod_lt b c cPos). +Intros. +Elim (Z_ge_lt_dec `a/c` `b/c`); Trivial. +Intro. +Absurd `b-a >= 1`. +Omega. +Rewrite -> H0. +Rewrite -> H2. +Assert `c*(b/c)+b % c-(c*(a/c)+a % c) = c*(b/c - a/c) + b % c - a % c`. +Ring. +Rewrite H3. +Assert `c*(b/c-a/c) >= c*1`. +Apply Zge_Zmult_pos_left. +Omega. +Omega. +Assert `c*1=c`. +Ring. +Omega. +Save. + +Lemma Z_mod_plus : (a,b,c:Z)`c > 0`->`(a+b*c) % c = a % c`. +Proof. +Intros a b c cPos. +Generalize (Z_div_mod_eq a c cPos). +Generalize (Z_mod_lt a c cPos). +Generalize (Z_div_mod_eq `a+b*c` c cPos). +Generalize (Z_mod_lt `a+b*c` c cPos). +Intros. + +Assert `(a+b*c) % c - a % c = c*(b+a/c - (a+b*c)/c)`. +Replace `(a+b*c) % c` with `a+b*c - c*((a+b*c)/c)`. +Replace `a % c` with `a - c*(a/c)`. +Ring. +Omega. +Omega. +LetTac q := `b+a/c-(a+b*c)/c`. +Apply (Zcase_sign q); Intros. +Assert `c*q=0`. +Rewrite H4; Ring. +Rewrite H5 in H3. +Omega. + +Assert `c*q >= c`. +Pattern 2 c; Replace c with `c*1`. +Apply Zge_Zmult_pos_left; Omega. +Ring. +Omega. + +Assert `c*q <= -c`. +Replace `-c` with `c*(-1)`. +Apply Zle_Zmult_pos_left; Omega. +Ring. +Omega. +Save. + +Lemma Z_div_plus : (a,b,c:Z)`c > 0`->`(a+b*c)/c = a/c+b`. +Proof. +Intros a b c cPos. +Generalize (Z_div_mod_eq a c cPos). +Generalize (Z_mod_lt a c cPos). +Generalize (Z_div_mod_eq `a+b*c` c cPos). +Generalize (Z_mod_lt `a+b*c` c cPos). +Intros. +Apply Zmult_reg_left with c. Omega. +Replace `c*((a+b*c)/c)` with `a+b*c-(a+b*c) % c`. +Rewrite (Z_mod_plus a b c cPos). +Pattern 1 a; Rewrite H2. +Ring. +Pattern 1 `a+b*c`; Rewrite H0. +Ring. +Save. + +Lemma Z_div_mult : (a,b:Z)`b > 0`->`(a*b)/b = a`. +Intros; Replace `a*b` with `0+a*b`; Auto. +Rewrite Z_div_plus; Auto. +Save. + +Lemma Z_mult_div_ge : (a,b:Z)`b>0`->`b*(a/b) <= a`. +Proof. +Intros a b bPos. +Generalize (Z_div_mod_eq `a` ? bPos); Intros. +Generalize (Z_mod_lt `a` ? bPos); Intros. +Pattern 2 a; Rewrite H. +Omega. +Save. + +Lemma Z_mod_same : (a:Z)`a>0`->`a % a=0`. +Proof. +Intros a aPos. +Generalize (Z_mod_plus `0` `1` a aPos). +Replace `0+1*a` with `a`. +Intros. +Rewrite H. +Compute. +Trivial. +Ring. +Save. + +Lemma Z_div_same : (a:Z)`a>0`->`a/a=1`. +Proof. +Intros a aPos. +Generalize (Z_div_plus `0` `1` a aPos). +Replace `0+1*a` with `a`. +Intros. +Rewrite H. +Compute. +Trivial. +Ring. +Save. + +Lemma Z_div_exact_1 : (a,b:Z)`b>0` -> `a = b*(a/b)` -> `a%b = 0`. +Intros a b Hb; Generalize (Z_div_mod a b Hb); Unfold Zmod Zdiv. +Case (Zdiv_eucl a b); Intros q r; Omega. +Save. + +Lemma Z_div_exact_2 : (a,b:Z)`b>0` -> `a%b = 0` -> `a = b*(a/b)`. +Intros a b Hb; Generalize (Z_div_mod a b Hb); Unfold Zmod Zdiv. +Case (Zdiv_eucl a b); Intros q r; Omega. +Save. + +Lemma Z_mod_zero_opp : (a,b:Z)`b>0` -> `a%b = 0` -> `(-a)%b = 0`. +Intros a b Hb. +Intros. +Rewrite Z_div_exact_2 with a b; Auto. +Replace `-(b*(a/b))` with `0+(-(a/b))*b`. +Rewrite Z_mod_plus; Auto. +Ring. +Save. + diff --git a/theories7/ZArith/Zeven.v b/theories7/ZArith/Zeven.v new file mode 100644 index 00000000..04b3ec09 --- /dev/null +++ b/theories7/ZArith/Zeven.v @@ -0,0 +1,184 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zeven.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +Require BinInt. +Require Zsyntax. + +(**********************************************************************) +(** About parity: even and odd predicates on Z, division by 2 on Z *) + +(**********************************************************************) +(** [Zeven], [Zodd], [Zdiv2] and their related properties *) + +Definition Zeven := + [z:Z]Cases z of ZERO => True + | (POS (xO _)) => True + | (NEG (xO _)) => True + | _ => False + end. + +Definition Zodd := + [z:Z]Cases z of (POS xH) => True + | (NEG xH) => True + | (POS (xI _)) => True + | (NEG (xI _)) => True + | _ => False + end. + +Definition Zeven_bool := + [z:Z]Cases z of ZERO => true + | (POS (xO _)) => true + | (NEG (xO _)) => true + | _ => false + end. + +Definition Zodd_bool := + [z:Z]Cases z of ZERO => false + | (POS (xO _)) => false + | (NEG (xO _)) => false + | _ => true + end. + +Definition Zeven_odd_dec : (z:Z) { (Zeven z) }+{ (Zodd z) }. +Proof. + Intro z. Case z; + [ Left; Compute; Trivial + | Intro p; Case p; Intros; + (Right; Compute; Exact I) Orelse (Left; Compute; Exact I) + | Intro p; Case p; Intros; + (Right; Compute; Exact I) Orelse (Left; Compute; Exact I) ]. +Defined. + +Definition Zeven_dec : (z:Z) { (Zeven z) }+{ ~(Zeven z) }. +Proof. + Intro z. Case z; + [ Left; Compute; Trivial + | Intro p; Case p; Intros; + (Left; Compute; Exact I) Orelse (Right; Compute; Trivial) + | Intro p; Case p; Intros; + (Left; Compute; Exact I) Orelse (Right; Compute; Trivial) ]. +Defined. + +Definition Zodd_dec : (z:Z) { (Zodd z) }+{ ~(Zodd z) }. +Proof. + Intro z. Case z; + [ Right; Compute; Trivial + | Intro p; Case p; Intros; + (Left; Compute; Exact I) Orelse (Right; Compute; Trivial) + | Intro p; Case p; Intros; + (Left; Compute; Exact I) Orelse (Right; Compute; Trivial) ]. +Defined. + +Lemma Zeven_not_Zodd : (z:Z)(Zeven z) -> ~(Zodd z). +Proof. + Intro z; NewDestruct z; [ Idtac | NewDestruct p | NewDestruct p ]; Compute; Trivial. +Qed. + +Lemma Zodd_not_Zeven : (z:Z)(Zodd z) -> ~(Zeven z). +Proof. + Intro z; NewDestruct z; [ Idtac | NewDestruct p | NewDestruct p ]; Compute; Trivial. +Qed. + +Lemma Zeven_Sn : (z:Z)(Zodd z) -> (Zeven (Zs z)). +Proof. + Intro z; NewDestruct z; Unfold Zs; [ Idtac | NewDestruct p | NewDestruct p ]; Simpl; Trivial. + Unfold double_moins_un; Case p; Simpl; Auto. +Qed. + +Lemma Zodd_Sn : (z:Z)(Zeven z) -> (Zodd (Zs z)). +Proof. + Intro z; NewDestruct z; Unfold Zs; [ Idtac | NewDestruct p | NewDestruct p ]; Simpl; Trivial. + Unfold double_moins_un; Case p; Simpl; Auto. +Qed. + +Lemma Zeven_pred : (z:Z)(Zodd z) -> (Zeven (Zpred z)). +Proof. + Intro z; NewDestruct z; Unfold Zpred; [ Idtac | NewDestruct p | NewDestruct p ]; Simpl; Trivial. + Unfold double_moins_un; Case p; Simpl; Auto. +Qed. + +Lemma Zodd_pred : (z:Z)(Zeven z) -> (Zodd (Zpred z)). +Proof. + Intro z; NewDestruct z; Unfold Zpred; [ Idtac | NewDestruct p | NewDestruct p ]; Simpl; Trivial. + Unfold double_moins_un; Case p; Simpl; Auto. +Qed. + +Hints Unfold Zeven Zodd : zarith. + +(**********************************************************************) +(** [Zdiv2] is defined on all [Z], but notice that for odd negative + integers it is not the euclidean quotient: in that case we have [n = + 2*(n/2)-1] *) + +Definition Zdiv2 := + [z:Z]Cases z of ZERO => ZERO + | (POS xH) => ZERO + | (POS p) => (POS (Zdiv2_pos p)) + | (NEG xH) => ZERO + | (NEG p) => (NEG (Zdiv2_pos p)) + end. + +Lemma Zeven_div2 : (x:Z) (Zeven x) -> `x = 2*(Zdiv2 x)`. +Proof. +Intro x; NewDestruct x. +Auto with arith. +NewDestruct p; Auto with arith. +Intros. Absurd (Zeven (POS (xI p))); Red; Auto with arith. +Intros. Absurd (Zeven `1`); Red; Auto with arith. +NewDestruct p; Auto with arith. +Intros. Absurd (Zeven (NEG (xI p))); Red; Auto with arith. +Intros. Absurd (Zeven `-1`); Red; Auto with arith. +Qed. + +Lemma Zodd_div2 : (x:Z) `x >= 0` -> (Zodd x) -> `x = 2*(Zdiv2 x)+1`. +Proof. +Intro x; NewDestruct x. +Intros. Absurd (Zodd `0`); Red; Auto with arith. +NewDestruct p; Auto with arith. +Intros. Absurd (Zodd (POS (xO p))); Red; Auto with arith. +Intros. Absurd `(NEG p) >= 0`; Red; Auto with arith. +Qed. + +Lemma Zodd_div2_neg : (x:Z) `x <= 0` -> (Zodd x) -> `x = 2*(Zdiv2 x)-1`. +Proof. +Intro x; NewDestruct x. +Intros. Absurd (Zodd `0`); Red; Auto with arith. +Intros. Absurd `(NEG p) >= 0`; Red; Auto with arith. +NewDestruct p; Auto with arith. +Intros. Absurd (Zodd (NEG (xO p))); Red; Auto with arith. +Qed. + +Lemma Z_modulo_2 : (x:Z) { y:Z | `x=2*y` }+{ y:Z | `x=2*y+1` }. +Proof. +Intros x. +Elim (Zeven_odd_dec x); Intro. +Left. Split with (Zdiv2 x). Exact (Zeven_div2 x a). +Right. Generalize b; Clear b; Case x. +Intro b; Inversion b. +Intro p; Split with (Zdiv2 (POS p)). Apply (Zodd_div2 (POS p)); Trivial. +Unfold Zge Zcompare; Simpl; Discriminate. +Intro p; Split with (Zdiv2 (Zpred (NEG p))). +Pattern 1 (NEG p); Rewrite (Zs_pred (NEG p)). +Pattern 1 (Zpred (NEG p)); Rewrite (Zeven_div2 (Zpred (NEG p))). +Reflexivity. +Apply Zeven_pred; Assumption. +Qed. + +Lemma Zsplit2 : (x:Z) { p : Z*Z | let (x1,x2)=p in (`x=x1+x2` /\ (x1=x2 \/ `x2=x1+1`)) }. +Proof. +Intros x. +Elim (Z_modulo_2 x); Intros (y,Hy); Rewrite Zmult_sym in Hy; Rewrite <- Zplus_Zmult_2 in Hy. +Exists (y,y); Split. +Assumption. +Left; Reflexivity. +Exists (y,`y+1`); Split. +Rewrite Zplus_assoc; Assumption. +Right; Reflexivity. +Qed. diff --git a/theories7/ZArith/Zhints.v b/theories7/ZArith/Zhints.v new file mode 100644 index 00000000..01860d18 --- /dev/null +++ b/theories7/ZArith/Zhints.v @@ -0,0 +1,387 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zhints.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +(** This file centralizes the lemmas about [Z], classifying them + according to the way they can be used in automatic search *) + +(*i*) + +(* Lemmas which clearly leads to simplification during proof search are *) +(* declared as Hints. A definite status (Hint or not) for the other lemmas *) +(* remains to be given *) + +(* Structure of the file *) +(* - simplification lemmas (only those are declared as Hints) *) +(* - reversible lemmas relating operators *) +(* - useful Bottom-up lemmas *) +(* - irreversible lemmas with meta-variables *) +(* - unclear or too specific lemmas *) +(* - lemmas to be used as rewrite rules *) + +(* Lemmas involving positive and compare are not taken into account *) + +Require BinInt. +Require Zorder. +Require Zmin. +Require Zabs. +Require Zcompare. +Require Znat. +Require auxiliary. +Require Zsyntax. +Require Zmisc. +Require Wf_Z. + +(**********************************************************************) +(* Simplification lemmas *) +(* No subgoal or smaller subgoals *) + +Hints Resolve + (* A) Reversible simplification lemmas (no loss of information) *) + (* Should clearly declared as hints *) + + (* Lemmas ending by eq *) + Zeq_S (* :(n,m:Z)`n = m`->`(Zs n) = (Zs m)` *) + + (* Lemmas ending by Zgt *) + Zgt_n_S (* :(n,m:Z)`m > n`->`(Zs m) > (Zs n)` *) + Zgt_Sn_n (* :(n:Z)`(Zs n) > n` *) + POS_gt_ZERO (* :(p:positive)`(POS p) > 0` *) + Zgt_reg_l (* :(n,m,p:Z)`n > m`->`p+n > p+m` *) + Zgt_reg_r (* :(n,m,p:Z)`n > m`->`n+p > m+p` *) + + (* Lemmas ending by Zlt *) + Zlt_n_Sn (* :(n:Z)`n < (Zs n)` *) + Zlt_n_S (* :(n,m:Z)`n < m`->`(Zs n) < (Zs m)` *) + Zlt_pred_n_n (* :(n:Z)`(Zpred n) < n` *) + Zlt_reg_l (* :(n,m,p:Z)`n < m`->`p+n < p+m` *) + Zlt_reg_r (* :(n,m,p:Z)`n < m`->`n+p < m+p` *) + + (* Lemmas ending by Zle *) + ZERO_le_inj (* :(n:nat)`0 <= (inject_nat n)` *) + ZERO_le_POS (* :(p:positive)`0 <= (POS p)` *) + Zle_n (* :(n:Z)`n <= n` *) + Zle_n_Sn (* :(n:Z)`n <= (Zs n)` *) + Zle_n_S (* :(n,m:Z)`m <= n`->`(Zs m) <= (Zs n)` *) + Zle_pred_n (* :(n:Z)`(Zpred n) <= n` *) + Zle_min_l (* :(n,m:Z)`(Zmin n m) <= n` *) + Zle_min_r (* :(n,m:Z)`(Zmin n m) <= m` *) + Zle_reg_l (* :(n,m,p:Z)`n <= m`->`p+n <= p+m` *) + Zle_reg_r (* :(a,b,c:Z)`a <= b`->`a+c <= b+c` *) + Zabs_pos (* :(x:Z)`0 <= |x|` *) + + (* B) Irreversible simplification lemmas : Probably to be declared as *) + (* hints, when no other simplification is possible *) + + (* Lemmas ending by eq *) + Z_eq_mult (* :(x,y:Z)`y = 0`->`y*x = 0` *) + Zplus_simpl (* :(n,m,p,q:Z)`n = m`->`p = q`->`n+p = m+q` *) + + (* Lemmas ending by Zge *) + Zge_Zmult_pos_right (* :(a,b,c:Z)`a >= b`->`c >= 0`->`a*c >= b*c` *) + Zge_Zmult_pos_left (* :(a,b,c:Z)`a >= b`->`c >= 0`->`c*a >= c*b` *) + Zge_Zmult_pos_compat (* : + (a,b,c,d:Z)`a >= c`->`b >= d`->`c >= 0`->`d >= 0`->`a*b >= c*d` *) + + (* Lemmas ending by Zlt *) + Zgt_ZERO_mult (* :(a,b:Z)`a > 0`->`b > 0`->`a*b > 0` *) + Zlt_S (* :(n,m:Z)`n < m`->`n < (Zs m)` *) + + (* Lemmas ending by Zle *) + Zle_ZERO_mult (* :(x,y:Z)`0 <= x`->`0 <= y`->`0 <= x*y` *) + Zle_Zmult_pos_right (* :(a,b,c:Z)`a <= b`->`0 <= c`->`a*c <= b*c` *) + Zle_Zmult_pos_left (* :(a,b,c:Z)`a <= b`->`0 <= c`->`c*a <= c*b` *) + OMEGA2 (* :(x,y:Z)`0 <= x`->`0 <= y`->`0 <= x+y` *) + Zle_le_S (* :(x,y:Z)`x <= y`->`x <= (Zs y)` *) + Zle_plus_plus (* :(n,m,p,q:Z)`n <= m`->`p <= q`->`n+p <= m+q` *) + +: zarith. + +(**********************************************************************) +(* Reversible lemmas relating operators *) +(* Probably to be declared as hints but need to define precedences *) + +(* A) Conversion between comparisons/predicates and arithmetic operators + +(* Lemmas ending by eq *) +Zegal_left: (x,y:Z)`x = y`->`x+(-y) = 0` +Zabs_eq: (x:Z)`0 <= x`->`|x| = x` +Zeven_div2: (x:Z)(Zeven x)->`x = 2*(Zdiv2 x)` +Zodd_div2: (x:Z)`x >= 0`->(Zodd x)->`x = 2*(Zdiv2 x)+1` + +(* Lemmas ending by Zgt *) +Zgt_left_rev: (x,y:Z)`x+(-y) > 0`->`x > y` +Zgt_left_gt: (x,y:Z)`x > y`->`x+(-y) > 0` + +(* Lemmas ending by Zlt *) +Zlt_left_rev: (x,y:Z)`0 < y+(-x)`->`x < y` +Zlt_left_lt: (x,y:Z)`x < y`->`0 < y+(-x)` +Zlt_O_minus_lt: (n,m:Z)`0 < n-m`->`m < n` + +(* Lemmas ending by Zle *) +Zle_left: (x,y:Z)`x <= y`->`0 <= y+(-x)` +Zle_left_rev: (x,y:Z)`0 <= y+(-x)`->`x <= y` +Zlt_left: (x,y:Z)`x < y`->`0 <= y+(-1)+(-x)` +Zge_left: (x,y:Z)`x >= y`->`0 <= x+(-y)` +Zgt_left: (x,y:Z)`x > y`->`0 <= x+(-1)+(-y)` + +(* B) Conversion between nat comparisons and Z comparisons *) + +(* Lemmas ending by eq *) +inj_eq: (x,y:nat)x=y->`(inject_nat x) = (inject_nat y)` + +(* Lemmas ending by Zge *) +inj_ge: (x,y:nat)(ge x y)->`(inject_nat x) >= (inject_nat y)` + +(* Lemmas ending by Zgt *) +inj_gt: (x,y:nat)(gt x y)->`(inject_nat x) > (inject_nat y)` + +(* Lemmas ending by Zlt *) +inj_lt: (x,y:nat)(lt x y)->`(inject_nat x) < (inject_nat y)` + +(* Lemmas ending by Zle *) +inj_le: (x,y:nat)(le x y)->`(inject_nat x) <= (inject_nat y)` + +(* C) Conversion between comparisons *) + +(* Lemmas ending by Zge *) +not_Zlt: (x,y:Z)~`x < y`->`x >= y` +Zle_ge: (m,n:Z)`m <= n`->`n >= m` + +(* Lemmas ending by Zgt *) +Zle_gt_S: (n,p:Z)`n <= p`->`(Zs p) > n` +not_Zle: (x,y:Z)~`x <= y`->`x > y` +Zlt_gt: (m,n:Z)`m < n`->`n > m` +Zle_S_gt: (n,m:Z)`(Zs n) <= m`->`m > n` + +(* Lemmas ending by Zlt *) +not_Zge: (x,y:Z)~`x >= y`->`x < y` +Zgt_lt: (m,n:Z)`m > n`->`n < m` +Zle_lt_n_Sm: (n,m:Z)`n <= m`->`n < (Zs m)` + +(* Lemmas ending by Zle *) +Zlt_ZERO_pred_le_ZERO: (x:Z)`0 < x`->`0 <= (Zpred x)` +not_Zgt: (x,y:Z)~`x > y`->`x <= y` +Zgt_le_S: (n,p:Z)`p > n`->`(Zs n) <= p` +Zgt_S_le: (n,p:Z)`(Zs p) > n`->`n <= p` +Zge_le: (m,n:Z)`m >= n`->`n <= m` +Zlt_le_S: (n,p:Z)`n < p`->`(Zs n) <= p` +Zlt_n_Sm_le: (n,m:Z)`n < (Zs m)`->`n <= m` +Zlt_le_weak: (n,m:Z)`n < m`->`n <= m` +Zle_refl: (n,m:Z)`n = m`->`n <= m` + +(* D) Irreversible simplification involving several comparaisons, *) +(* useful with clear precedences *) + +(* Lemmas ending by Zlt *) +Zlt_le_reg :(a,b,c,d:Z)`a < b`->`c <= d`->`a+c < b+d` +Zle_lt_reg : (a,b,c,d:Z)`a <= b`->`c < d`->`a+c < b+d` + +(* D) What is decreasing here ? *) + +(* Lemmas ending by eq *) +Zplus_minus: (n,m,p:Z)`n = m+p`->`p = n-m` + +(* Lemmas ending by Zgt *) +Zgt_pred: (n,p:Z)`p > (Zs n)`->`(Zpred p) > n` + +(* Lemmas ending by Zlt *) +Zlt_pred: (n,p:Z)`(Zs n) < p`->`n < (Zpred p)` + +*) + +(**********************************************************************) +(* Useful Bottom-up lemmas *) + +(* A) Bottom-up simplification: should be used + +(* Lemmas ending by eq *) +Zeq_add_S: (n,m:Z)`(Zs n) = (Zs m)`->`n = m` +Zsimpl_plus_l: (n,m,p:Z)`n+m = n+p`->`m = p` +Zplus_unit_left: (n,m:Z)`n+0 = m`->`n = m` +Zplus_unit_right: (n,m:Z)`n = m+0`->`n = m` + +(* Lemmas ending by Zgt *) +Zsimpl_gt_plus_l: (n,m,p:Z)`p+n > p+m`->`n > m` +Zsimpl_gt_plus_r: (n,m,p:Z)`n+p > m+p`->`n > m` +Zgt_S_n: (n,p:Z)`(Zs p) > (Zs n)`->`p > n` + +(* Lemmas ending by Zlt *) +Zsimpl_lt_plus_l: (n,m,p:Z)`p+n < p+m`->`n < m` +Zsimpl_lt_plus_r: (n,m,p:Z)`n+p < m+p`->`n < m` +Zlt_S_n: (n,m:Z)`(Zs n) < (Zs m)`->`n < m` + +(* Lemmas ending by Zle *) +Zsimpl_le_plus_l: (p,n,m:Z)`p+n <= p+m`->`n <= m` +Zsimpl_le_plus_r: (p,n,m:Z)`n+p <= m+p`->`n <= m` +Zle_S_n: (n,m:Z)`(Zs m) <= (Zs n)`->`m <= n` + +(* B) Bottom-up irreversible (syntactic) simplification *) + +(* Lemmas ending by Zle *) +Zle_trans_S: (n,m:Z)`(Zs n) <= m`->`n <= m` + +(* C) Other unclearly simplifying lemmas *) + +(* Lemmas ending by Zeq *) +Zmult_eq: (x,y:Z)`x <> 0`->`y*x = 0`->`y = 0` + +(* Lemmas ending by Zgt *) +Zmult_gt: (x,y:Z)`x > 0`->`x*y > 0`->`y > 0` + +(* Lemmas ending by Zlt *) +pZmult_lt: (x,y:Z)`x > 0`->`0 < y*x`->`0 < y` + +(* Lemmas ending by Zle *) +Zmult_le: (x,y:Z)`x > 0`->`0 <= y*x`->`0 <= y` +OMEGA1: (x,y:Z)`x = y`->`0 <= x`->`0 <= y` +*) + +(**********************************************************************) +(* Irreversible lemmas with meta-variables *) +(* To be used by EAuto + +Hints Immediate +(* Lemmas ending by eq *) +Zle_antisym: (n,m:Z)`n <= m`->`m <= n`->`n = m` + +(* Lemmas ending by Zge *) +Zge_trans: (n,m,p:Z)`n >= m`->`m >= p`->`n >= p` + +(* Lemmas ending by Zgt *) +Zgt_trans: (n,m,p:Z)`n > m`->`m > p`->`n > p` +Zgt_trans_S: (n,m,p:Z)`(Zs n) > m`->`m > p`->`n > p` +Zle_gt_trans: (n,m,p:Z)`m <= n`->`m > p`->`n > p` +Zgt_le_trans: (n,m,p:Z)`n > m`->`p <= m`->`n > p` + +(* Lemmas ending by Zlt *) +Zlt_trans: (n,m,p:Z)`n < m`->`m < p`->`n < p` +Zlt_le_trans: (n,m,p:Z)`n < m`->`m <= p`->`n < p` +Zle_lt_trans: (n,m,p:Z)`n <= m`->`m < p`->`n < p` + +(* Lemmas ending by Zle *) +Zle_trans: (n,m,p:Z)`n <= m`->`m <= p`->`n <= p` +*) + +(**********************************************************************) +(* Unclear or too specific lemmas *) +(* Not to be used ?? *) + +(* A) Irreversible and too specific (not enough regular) + +(* Lemmas ending by Zle *) +Zle_mult: (x,y:Z)`x > 0`->`0 <= y`->`0 <= y*x` +Zle_mult_approx: (x,y,z:Z)`x > 0`->`z > 0`->`0 <= y`->`0 <= y*x+z` +OMEGA6: (x,y,z:Z)`0 <= x`->`y = 0`->`0 <= x+y*z` +OMEGA7: (x,y,z,t:Z)`z > 0`->`t > 0`->`0 <= x`->`0 <= y`->`0 <= x*z+y*t` + + +(* B) Expansion and too specific ? *) + +(* Lemmas ending by Zge *) +Zge_mult_simpl: (a,b,c:Z)`c > 0`->`a*c >= b*c`->`a >= b` + +(* Lemmas ending by Zgt *) +Zgt_mult_simpl: (a,b,c:Z)`c > 0`->`a*c > b*c`->`a > b` +Zgt_square_simpl: (x,y:Z)`x >= 0`->`y >= 0`->`x*x > y*y`->`x > y` + +(* Lemmas ending by Zle *) +Zle_mult_simpl: (a,b,c:Z)`c > 0`->`a*c <= b*c`->`a <= b` +Zmult_le_approx: (x,y,z:Z)`x > 0`->`x > z`->`0 <= y*x+z`->`0 <= y` + +(* C) Reversible but too specific ? *) + +(* Lemmas ending by Zlt *) +Zlt_minus: (n,m:Z)`0 < m`->`n-m < n` +*) + +(**********************************************************************) +(* Lemmas to be used as rewrite rules *) +(* but can also be used as hints + +(* Left-to-right simplification lemmas (a symbol disappears) *) + +Zcompare_n_S: (n,m:Z)(Zcompare (Zs n) (Zs m))=(Zcompare n m) +Zmin_n_n: (n:Z)`(Zmin n n) = n` +Zmult_1_n: (n:Z)`1*n = n` +Zmult_n_1: (n:Z)`n*1 = n` +Zminus_plus: (n,m:Z)`n+m-n = m` +Zle_plus_minus: (n,m:Z)`n+(m-n) = m` +Zopp_Zopp: (x:Z)`(-(-x)) = x` +Zero_left: (x:Z)`0+x = x` +Zero_right: (x:Z)`x+0 = x` +Zplus_inverse_r: (x:Z)`x+(-x) = 0` +Zplus_inverse_l: (x:Z)`(-x)+x = 0` +Zopp_intro: (x,y:Z)`(-x) = (-y)`->`x = y` +Zmult_one: (x:Z)`1*x = x` +Zero_mult_left: (x:Z)`0*x = 0` +Zero_mult_right: (x:Z)`x*0 = 0` +Zmult_Zopp_Zopp: (x,y:Z)`(-x)*(-y) = x*y` + +(* Right-to-left simplification lemmas (a symbol disappears) *) + +Zpred_Sn: (m:Z)`m = (Zpred (Zs m))` +Zs_pred: (n:Z)`n = (Zs (Zpred n))` +Zplus_n_O: (n:Z)`n = n+0` +Zmult_n_O: (n:Z)`0 = n*0` +Zminus_n_O: (n:Z)`n = n-0` +Zminus_n_n: (n:Z)`0 = n-n` +Zred_factor6: (x:Z)`x = x+0` +Zred_factor0: (x:Z)`x = x*1` + +(* Unclear orientation (no symbol disappears) *) + +Zplus_n_Sm: (n,m:Z)`(Zs (n+m)) = n+(Zs m)` +Zmult_n_Sm: (n,m:Z)`n*m+n = n*(Zs m)` +Zmin_SS: (n,m:Z)`(Zs (Zmin n m)) = (Zmin (Zs n) (Zs m))` +Zplus_assoc_l: (n,m,p:Z)`n+(m+p) = n+m+p` +Zplus_assoc_r: (n,m,p:Z)`n+m+p = n+(m+p)` +Zplus_permute: (n,m,p:Z)`n+(m+p) = m+(n+p)` +Zplus_Snm_nSm: (n,m:Z)`(Zs n)+m = n+(Zs m)` +Zminus_plus_simpl: (n,m,p:Z)`n-m = p+n-(p+m)` +Zminus_Sn_m: (n,m:Z)`(Zs (n-m)) = (Zs n)-m` +Zmult_plus_distr_l: (n,m,p:Z)`(n+m)*p = n*p+m*p` +Zmult_minus_distr: (n,m,p:Z)`(n-m)*p = n*p-m*p` +Zmult_assoc_r: (n,m,p:Z)`n*m*p = n*(m*p)` +Zmult_assoc_l: (n,m,p:Z)`n*(m*p) = n*m*p` +Zmult_permute: (n,m,p:Z)`n*(m*p) = m*(n*p)` +Zmult_Sm_n: (n,m:Z)`n*m+m = (Zs n)*m` +Zmult_Zplus_distr: (x,y,z:Z)`x*(y+z) = x*y+x*z` +Zmult_plus_distr: (n,m,p:Z)`(n+m)*p = n*p+m*p` +Zopp_Zplus: (x,y:Z)`(-(x+y)) = (-x)+(-y)` +Zplus_sym: (x,y:Z)`x+y = y+x` +Zplus_assoc: (x,y,z:Z)`x+(y+z) = x+y+z` +Zmult_sym: (x,y:Z)`x*y = y*x` +Zmult_assoc: (x,y,z:Z)`x*(y*z) = x*y*z` +Zopp_Zmult: (x,y:Z)`(-x)*y = (-(x*y))` +Zplus_S_n: (x,y:Z)`(Zs x)+y = (Zs (x+y))` +Zopp_one: (x:Z)`(-x) = x*(-1)` +Zopp_Zmult_r: (x,y:Z)`(-(x*y)) = x*(-y)` +Zmult_Zopp_left: (x,y:Z)`(-x)*y = x*(-y)` +Zopp_Zmult_l: (x,y:Z)`(-(x*y)) = (-x)*y` +Zred_factor1: (x:Z)`x+x = x*2` +Zred_factor2: (x,y:Z)`x+x*y = x*(1+y)` +Zred_factor3: (x,y:Z)`x*y+x = x*(1+y)` +Zred_factor4: (x,y,z:Z)`x*y+x*z = x*(y+z)` +Zminus_Zplus_compatible: (x,y,n:Z)`x+n-(y+n) = x-y` +Zmin_plus: (x,y,n:Z)`(Zmin (x+n) (y+n)) = (Zmin x y)+n` + +(* nat <-> Z *) +inj_S: (y:nat)`(inject_nat (S y)) = (Zs (inject_nat y))` +inj_plus: (x,y:nat)`(inject_nat (plus x y)) = (inject_nat x)+(inject_nat y)` +inj_mult: (x,y:nat)`(inject_nat (mult x y)) = (inject_nat x)*(inject_nat y)` +inj_minus1: + (x,y:nat)(le y x)->`(inject_nat (minus x y)) = (inject_nat x)-(inject_nat y)` +inj_minus2: (x,y:nat)(gt y x)->`(inject_nat (minus x y)) = 0` + +(* Too specific ? *) +Zred_factor5: (x,y:Z)`x*0+y = y` +*) + +(*i*) diff --git a/theories7/ZArith/Zlogarithm.v b/theories7/ZArith/Zlogarithm.v new file mode 100644 index 00000000..dc850738 --- /dev/null +++ b/theories7/ZArith/Zlogarithm.v @@ -0,0 +1,272 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zlogarithm.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +(**********************************************************************) +(** The integer logarithms with base 2. + + There are three logarithms, + depending on the rounding of the real 2-based logarithm: + - [Log_inf]: [y = (Log_inf x) iff 2^y <= x < 2^(y+1)] + i.e. [Log_inf x] is the biggest integer that is smaller than [Log x] + - [Log_sup]: [y = (Log_sup x) iff 2^(y-1) < x <= 2^y] + i.e. [Log_inf x] is the smallest integer that is bigger than [Log x] + - [Log_nearest]: [y= (Log_nearest x) iff 2^(y-1/2) < x <= 2^(y+1/2)] + i.e. [Log_nearest x] is the integer nearest from [Log x] *) + +Require ZArith_base. +Require Omega. +Require Zcomplements. +Require Zpower. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +Section Log_pos. (* Log of positive integers *) + +(** First we build [log_inf] and [log_sup] *) + +Fixpoint log_inf [p:positive] : Z := + Cases p of + xH => `0` (* 1 *) + | (xO q) => (Zs (log_inf q)) (* 2n *) + | (xI q) => (Zs (log_inf q)) (* 2n+1 *) + end. +Fixpoint log_sup [p:positive] : Z := + Cases p of + xH => `0` (* 1 *) + | (xO n) => (Zs (log_sup n)) (* 2n *) + | (xI n) => (Zs (Zs (log_inf n))) (* 2n+1 *) + end. + +Hints Unfold log_inf log_sup. + +(** Then we give the specifications of [log_inf] and [log_sup] + and prove their validity *) + +(*i Hints Resolve ZERO_le_S : zarith. i*) +Hints Resolve Zle_trans : zarith. + +Theorem log_inf_correct : (x:positive) ` 0 <= (log_inf x)` /\ + ` (two_p (log_inf x)) <= (POS x) < (two_p (Zs (log_inf x)))`. +Induction x; Intros; Simpl; +[ Elim H; Intros Hp HR; Clear H; Split; + [ Auto with zarith + | Conditional (Apply Zle_le_S; Trivial) Rewrite two_p_S with x:=(Zs (log_inf p)); + Conditional Trivial Rewrite two_p_S; + Conditional Trivial Rewrite two_p_S in HR; + Rewrite (POS_xI p); Omega ] +| Elim H; Intros Hp HR; Clear H; Split; + [ Auto with zarith + | Conditional (Apply Zle_le_S; Trivial) Rewrite two_p_S with x:=(Zs (log_inf p)); + Conditional Trivial Rewrite two_p_S; + Conditional Trivial Rewrite two_p_S in HR; + Rewrite (POS_xO p); Omega ] +| Unfold two_power_pos; Unfold shift_pos; Simpl; Omega +]. +Qed. + +Definition log_inf_correct1 := + [p:positive](proj1 ? ? (log_inf_correct p)). +Definition log_inf_correct2 := + [p:positive](proj2 ? ? (log_inf_correct p)). + +Opaque log_inf_correct1 log_inf_correct2. + +Hints Resolve log_inf_correct1 log_inf_correct2 : zarith. + +Lemma log_sup_correct1 : (p:positive)` 0 <= (log_sup p)`. +Induction p; Intros; Simpl; Auto with zarith. +Qed. + +(** For every [p], either [p] is a power of two and [(log_inf p)=(log_sup p)] + either [(log_sup p)=(log_inf p)+1] *) + +Theorem log_sup_log_inf : (p:positive) + IF (POS p)=(two_p (log_inf p)) + then (POS p)=(two_p (log_sup p)) + else ` (log_sup p)=(Zs (log_inf p))`. + +Induction p; Intros; +[ Elim H; Right; Simpl; + Rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); + Rewrite POS_xI; Unfold Zs; Omega +| Elim H; Clear H; Intro Hif; + [ Left; Simpl; + Rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); + Rewrite (two_p_S (log_sup p0) (log_sup_correct1 p0)); + Rewrite <- (proj1 ? ? Hif); Rewrite <- (proj2 ? ? Hif); + Auto + | Right; Simpl; + Rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); + Rewrite POS_xO; Unfold Zs; Omega ] +| Left; Auto ]. +Qed. + +Theorem log_sup_correct2 : (x:positive) + ` (two_p (Zpred (log_sup x))) < (POS x) <= (two_p (log_sup x))`. + +Intro. +Elim (log_sup_log_inf x). +(* x is a power of two and [log_sup = log_inf] *) +Intros (E1,E2); Rewrite E2. +Split ; [ Apply two_p_pred; Apply log_sup_correct1 | Apply Zle_n ]. +Intros (E1,E2); Rewrite E2. +Rewrite <- (Zpred_Sn (log_inf x)). +Generalize (log_inf_correct2 x); Omega. +Qed. + +Lemma log_inf_le_log_sup : + (p:positive) `(log_inf p) <= (log_sup p)`. +Induction p; Simpl; Intros; Omega. +Qed. + +Lemma log_sup_le_Slog_inf : + (p:positive) `(log_sup p) <= (Zs (log_inf p))`. +Induction p; Simpl; Intros; Omega. +Qed. + +(** Now it's possible to specify and build the [Log] rounded to the nearest *) + +Fixpoint log_near[x:positive] : Z := + Cases x of + xH => `0` + | (xO xH) => `1` + | (xI xH) => `2` + | (xO y) => (Zs (log_near y)) + | (xI y) => (Zs (log_near y)) + end. + +Theorem log_near_correct1 : (p:positive)` 0 <= (log_near p)`. +Induction p; Simpl; Intros; +[Elim p0; Auto with zarith | Elim p0; Auto with zarith | Trivial with zarith ]. +Intros; Apply Zle_le_S. +Generalize H0; Elim p1; Intros; Simpl; + [ Assumption | Assumption | Apply ZERO_le_POS ]. +Intros; Apply Zle_le_S. +Generalize H0; Elim p1; Intros; Simpl; + [ Assumption | Assumption | Apply ZERO_le_POS ]. +Qed. + +Theorem log_near_correct2: (p:positive) + (log_near p)=(log_inf p) +\/(log_near p)=(log_sup p). +Induction p. +Intros p0 [Einf|Esup]. +Simpl. Rewrite Einf. +Case p0; [Left | Left | Right]; Reflexivity. +Simpl; Rewrite Esup. +Elim (log_sup_log_inf p0). +Generalize (log_inf_le_log_sup p0). +Generalize (log_sup_le_Slog_inf p0). +Case p0; Auto with zarith. +Intros; Omega. +Case p0; Intros; Auto with zarith. +Intros p0 [Einf|Esup]. +Simpl. +Repeat Rewrite Einf. +Case p0; Intros; Auto with zarith. +Simpl. +Repeat Rewrite Esup. +Case p0; Intros; Auto with zarith. +Auto. +Qed. + +(*i****************** +Theorem log_near_correct: (p:positive) + `| (two_p (log_near p)) - (POS p) | <= (POS p)-(two_p (log_inf p))` + /\`| (two_p (log_near p)) - (POS p) | <= (two_p (log_sup p))-(POS p)`. +Intro. +Induction p. +Intros p0 [(Einf1,Einf2)|(Esup1,Esup2)]. +Unfold log_near log_inf log_sup. Fold log_near log_inf log_sup. +Rewrite Einf1. +Repeat Rewrite two_p_S. +Case p0; [Left | Left | Right]. + +Split. +Simpl. +Rewrite E1; Case p0; Try Reflexivity. +Compute. +Unfold log_near; Fold log_near. +Unfold log_inf; Fold log_inf. +Repeat Rewrite E1. +Split. +**********************************i*) + +End Log_pos. + +Section divers. + +(** Number of significative digits. *) + +Definition N_digits := + [x:Z]Cases x of + (POS p) => (log_inf p) + | (NEG p) => (log_inf p) + | ZERO => `0` + end. + +Lemma ZERO_le_N_digits : (x:Z) ` 0 <= (N_digits x)`. +Induction x; Simpl; +[ Apply Zle_n +| Exact log_inf_correct1 +| Exact log_inf_correct1]. +Qed. + +Lemma log_inf_shift_nat : + (n:nat)(log_inf (shift_nat n xH))=(inject_nat n). +Induction n; Intros; +[ Try Trivial +| Rewrite -> inj_S; Rewrite <- H; Reflexivity]. +Qed. + +Lemma log_sup_shift_nat : + (n:nat)(log_sup (shift_nat n xH))=(inject_nat n). +Induction n; Intros; +[ Try Trivial +| Rewrite -> inj_S; Rewrite <- H; Reflexivity]. +Qed. + +(** [Is_power p] means that p is a power of two *) +Fixpoint Is_power[p:positive] : Prop := + Cases p of + xH => True + | (xO q) => (Is_power q) + | (xI q) => False + end. + +Lemma Is_power_correct : + (p:positive) (Is_power p) <-> (Ex [y:nat](p=(shift_nat y xH))). + +Split; +[ Elim p; + [ Simpl; Tauto + | Simpl; Intros; Generalize (H H0); Intro H1; Elim H1; Intros y0 Hy0; + Exists (S y0); Rewrite Hy0; Reflexivity + | Intro; Exists O; Reflexivity] +| Intros; Elim H; Intros; Rewrite -> H0; Elim x; Intros; Simpl; Trivial]. +Qed. + +Lemma Is_power_or : (p:positive) (Is_power p)\/~(Is_power p). +Induction p; +[ Intros; Right; Simpl; Tauto +| Intros; Elim H; + [ Intros; Left; Simpl; Exact H0 + | Intros; Right; Simpl; Exact H0] +| Left; Simpl; Trivial]. +Qed. + +End divers. + + + + + + + diff --git a/theories7/ZArith/Zmin.v b/theories7/ZArith/Zmin.v new file mode 100644 index 00000000..753fe461 --- /dev/null +++ b/theories7/ZArith/Zmin.v @@ -0,0 +1,102 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Zmin.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *) + +Require Arith. +Require BinInt. +Require Zcompare. +Require Zorder. + +Open Local Scope Z_scope. + +(**********************************************************************) +(** Minimum on binary integer numbers *) + +Definition Zmin := [n,m:Z] + <Z>Cases (Zcompare n m) of + EGAL => n + | INFERIEUR => n + | SUPERIEUR => m + end. + +(** Properties of minimum on binary integer numbers *) + +Lemma Zmin_SS : (n,m:Z)((Zs (Zmin n m))=(Zmin (Zs n) (Zs m))). +Proof. +Intros n m;Unfold Zmin; Rewrite (Zcompare_n_S n m); +(ElimCompare 'n 'm);Intros E;Rewrite E;Auto with arith. +Qed. + +Lemma Zle_min_l : (n,m:Z)(Zle (Zmin n m) n). +Proof. +Intros n m;Unfold Zmin ; (ElimCompare 'n 'm);Intros E;Rewrite -> E; + [ Apply Zle_n | Apply Zle_n | Apply Zlt_le_weak; Apply Zgt_lt;Exact E ]. +Qed. + +Lemma Zle_min_r : (n,m:Z)(Zle (Zmin n m) m). +Proof. +Intros n m;Unfold Zmin ; (ElimCompare 'n 'm);Intros E;Rewrite -> E;[ + Unfold Zle ;Rewrite -> E;Discriminate +| Unfold Zle ;Rewrite -> E;Discriminate +| Apply Zle_n ]. +Qed. + +Lemma Zmin_case : (n,m:Z)(P:Z->Set)(P n)->(P m)->(P (Zmin n m)). +Proof. +Intros n m P H1 H2; Unfold Zmin; Case (Zcompare n m);Auto with arith. +Qed. + +Lemma Zmin_or : (n,m:Z)(Zmin n m)=n \/ (Zmin n m)=m. +Proof. +Unfold Zmin; Intros; Elim (Zcompare n m); Auto. +Qed. + +Lemma Zmin_n_n : (n:Z) (Zmin n n)=n. +Proof. +Unfold Zmin; Intros; Elim (Zcompare n n); Auto. +Qed. + +Lemma Zmin_plus : + (x,y,n:Z)(Zmin (Zplus x n) (Zplus y n))=(Zplus (Zmin x y) n). +Proof. +Intros x y n; Unfold Zmin. +Rewrite (Zplus_sym x n); +Rewrite (Zplus_sym y n); +Rewrite (Zcompare_Zplus_compatible x y n). +Case (Zcompare x y); Apply Zplus_sym. +Qed. + +(**********************************************************************) +(** Maximum of two binary integer numbers *) +V7only [ (* From Zdivides *) ]. + +Definition Zmax := + [a, b : ?] Cases (Zcompare a b) of INFERIEUR => b | _ => a end. + +(** Properties of maximum on binary integer numbers *) + +Tactic Definition CaseEq name := +Generalize (refl_equal ? name); Pattern -1 name; Case name. + +Theorem Zmax1: (a, b : ?) (Zle a (Zmax a b)). +Proof. +Intros a b; Unfold Zmax; (CaseEq '(Zcompare a b)); Simpl; Auto with zarith. +Unfold Zle; Intros H; Rewrite H; Red; Intros; Discriminate. +Qed. + +Theorem Zmax2: (a, b : ?) (Zle b (Zmax a b)). +Proof. +Intros a b; Unfold Zmax; (CaseEq '(Zcompare a b)); Simpl; Auto with zarith. +Intros H; + (Case (Zle_or_lt b a); Auto; Unfold Zlt; Rewrite H; Intros; Discriminate). +Intros H; + (Case (Zle_or_lt b a); Auto; Unfold Zlt; Rewrite H; Intros; Discriminate). +Qed. + diff --git a/theories7/ZArith/Zmisc.v b/theories7/ZArith/Zmisc.v new file mode 100644 index 00000000..bd89ec66 --- /dev/null +++ b/theories7/ZArith/Zmisc.v @@ -0,0 +1,188 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zmisc.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +Require BinInt. +Require Zcompare. +Require Zorder. +Require Zsyntax. +Require Bool. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +(**********************************************************************) +(** Iterators *) + +(** [n]th iteration of the function [f] *) +Fixpoint iter_nat[n:nat] : (A:Set)(f:A->A)A->A := + [A:Set][f:A->A][x:A] + Cases n of + O => x + | (S n') => (f (iter_nat n' A f x)) + end. + +Fixpoint iter_pos[n:positive] : (A:Set)(f:A->A)A->A := + [A:Set][f:A->A][x:A] + Cases n of + xH => (f x) + | (xO n') => (iter_pos n' A f (iter_pos n' A f x)) + | (xI n') => (f (iter_pos n' A f (iter_pos n' A f x))) + end. + +Definition iter := + [n:Z][A:Set][f:A->A][x:A]Cases n of + ZERO => x + | (POS p) => (iter_pos p A f x) + | (NEG p) => x + end. + +Theorem iter_nat_plus : + (n,m:nat)(A:Set)(f:A->A)(x:A) + (iter_nat (plus n m) A f x)=(iter_nat n A f (iter_nat m A f x)). +Proof. +Induction n; +[ Simpl; Auto with arith +| Intros; Simpl; Apply f_equal with f:=f; Apply H +]. +Qed. + +Theorem iter_convert : (n:positive)(A:Set)(f:A->A)(x:A) + (iter_pos n A f x) = (iter_nat (convert n) A f x). +Proof. +Intro n; NewInduction n as [p H|p H|]; +[ Intros; Simpl; Rewrite -> (H A f x); + Rewrite -> (H A f (iter_nat (convert p) A f x)); + Rewrite -> (ZL6 p); Symmetry; Apply f_equal with f:=f; + Apply iter_nat_plus +| Intros; Unfold convert; Simpl; Rewrite -> (H A f x); + Rewrite -> (H A f (iter_nat (convert p) A f x)); + Rewrite -> (ZL6 p); Symmetry; + Apply iter_nat_plus +| Simpl; Auto with arith +]. +Qed. + +Theorem iter_pos_add : + (n,m:positive)(A:Set)(f:A->A)(x:A) + (iter_pos (add n m) A f x)=(iter_pos n A f (iter_pos m A f x)). +Proof. +Intros n m; Intros. +Rewrite -> (iter_convert m A f x). +Rewrite -> (iter_convert n A f (iter_nat (convert m) A f x)). +Rewrite -> (iter_convert (add n m) A f x). +Rewrite -> (convert_add n m). +Apply iter_nat_plus. +Qed. + +(** Preservation of invariants : if [f : A->A] preserves the invariant [Inv], + then the iterates of [f] also preserve it. *) + +Theorem iter_nat_invariant : + (n:nat)(A:Set)(f:A->A)(Inv:A->Prop) + ((x:A)(Inv x)->(Inv (f x)))->(x:A)(Inv x)->(Inv (iter_nat n A f x)). +Proof. +Induction n; Intros; +[ Trivial with arith +| Simpl; Apply H0 with x:=(iter_nat n0 A f x); Apply H; Trivial with arith]. +Qed. + +Theorem iter_pos_invariant : + (n:positive)(A:Set)(f:A->A)(Inv:A->Prop) + ((x:A)(Inv x)->(Inv (f x)))->(x:A)(Inv x)->(Inv (iter_pos n A f x)). +Proof. +Intros; Rewrite iter_convert; Apply iter_nat_invariant; Trivial with arith. +Qed. + +V7only [ +(* Compatibility *) +Require Zbool. +Require Zeven. +Require Zabs. +Require Zmin. +Notation rename := rename. +Notation POS_xI := POS_xI. +Notation POS_xO := POS_xO. +Notation NEG_xI := NEG_xI. +Notation NEG_xO := NEG_xO. +Notation POS_add := POS_add. +Notation NEG_add := NEG_add. +Notation Zle_cases := Zle_cases. +Notation Zlt_cases := Zlt_cases. +Notation Zge_cases := Zge_cases. +Notation Zgt_cases := Zgt_cases. +Notation POS_gt_ZERO := POS_gt_ZERO. +Notation ZERO_le_POS := ZERO_le_POS. +Notation Zlt_ZERO_pred_le_ZERO := Zlt_ZERO_pred_le_ZERO. +Notation NEG_lt_ZERO := NEG_lt_ZERO. +Notation Zeven_not_Zodd := Zeven_not_Zodd. +Notation Zodd_not_Zeven := Zodd_not_Zeven. +Notation Zeven_Sn := Zeven_Sn. +Notation Zodd_Sn := Zodd_Sn. +Notation Zeven_pred := Zeven_pred. +Notation Zodd_pred := Zodd_pred. +Notation Zeven_div2 := Zeven_div2. +Notation Zodd_div2 := Zodd_div2. +Notation Zodd_div2_neg := Zodd_div2_neg. +Notation Z_modulo_2 := Z_modulo_2. +Notation Zsplit2 := Zsplit2. +Notation Zminus_Zplus_compatible := Zminus_Zplus_compatible. +Notation Zcompare_egal_dec := Zcompare_egal_dec. +Notation Zcompare_elim := Zcompare_elim. +Notation Zcompare_x_x := Zcompare_x_x. +Notation Zlt_not_eq := Zlt_not_eq. +Notation Zcompare_eq_case := Zcompare_eq_case. +Notation Zle_Zcompare := Zle_Zcompare. +Notation Zlt_Zcompare := Zlt_Zcompare. +Notation Zge_Zcompare := Zge_Zcompare. +Notation Zgt_Zcompare := Zgt_Zcompare. +Notation Zmin_plus := Zmin_plus. +Notation absolu_lt := absolu_lt. +Notation Zle_bool_imp_le := Zle_bool_imp_le. +Notation Zle_imp_le_bool := Zle_imp_le_bool. +Notation Zle_bool_refl := Zle_bool_refl. +Notation Zle_bool_antisym := Zle_bool_antisym. +Notation Zle_bool_trans := Zle_bool_trans. +Notation Zle_bool_plus_mono := Zle_bool_plus_mono. +Notation Zone_pos := Zone_pos. +Notation Zone_min_pos := Zone_min_pos. +Notation Zle_is_le_bool := Zle_is_le_bool. +Notation Zge_is_le_bool := Zge_is_le_bool. +Notation Zlt_is_le_bool := Zlt_is_le_bool. +Notation Zgt_is_le_bool := Zgt_is_le_bool. +Notation Zle_plus_swap := Zle_plus_swap. +Notation Zge_iff_le := Zge_iff_le. +Notation Zlt_plus_swap := Zlt_plus_swap. +Notation Zgt_iff_lt := Zgt_iff_lt. +Notation Zeq_plus_swap := Zeq_plus_swap. +(* Definitions *) +Notation entier_of_Z := entier_of_Z. +Notation Z_of_entier := Z_of_entier. +Notation Zle_bool := Zle_bool. +Notation Zge_bool := Zge_bool. +Notation Zlt_bool := Zlt_bool. +Notation Zgt_bool := Zgt_bool. +Notation Zeq_bool := Zeq_bool. +Notation Zneq_bool := Zneq_bool. +Notation Zeven := Zeven. +Notation Zodd := Zodd. +Notation Zeven_bool := Zeven_bool. +Notation Zodd_bool := Zodd_bool. +Notation Zeven_odd_dec := Zeven_odd_dec. +Notation Zeven_dec := Zeven_dec. +Notation Zodd_dec := Zodd_dec. +Notation Zdiv2_pos := Zdiv2_pos. +Notation Zdiv2 := Zdiv2. +Notation Zle_bool_total := Zle_bool_total. +Export Zbool. +Export Zeven. +Export Zabs. +Export Zmin. +Export Zorder. +Export Zcompare. +]. diff --git a/theories7/ZArith/Znat.v b/theories7/ZArith/Znat.v new file mode 100644 index 00000000..99d1422f --- /dev/null +++ b/theories7/ZArith/Znat.v @@ -0,0 +1,138 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Znat.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) + +Require Export Arith. +Require BinPos. +Require BinInt. +Require Zcompare. +Require Zorder. +Require Decidable. +Require Peano_dec. +Require Export Compare_dec. + +Open Local Scope Z_scope. + +Definition neq := [x,y:nat] ~(x=y). + +(**********************************************************************) +(** Properties of the injection from nat into Z *) + +Theorem inj_S : (y:nat) (inject_nat (S y)) = (Zs (inject_nat y)). +Proof. +Intro y; NewInduction y as [|n H]; [ + Unfold Zs ; Simpl; Trivial with arith +| Change (POS (add_un (anti_convert n)))=(Zs (inject_nat (S n))); + Rewrite add_un_Zs; Trivial with arith]. +Qed. + +Theorem inj_plus : + (x,y:nat) (inject_nat (plus x y)) = (Zplus (inject_nat x) (inject_nat y)). +Proof. +Intro x; NewInduction x as [|n H]; Intro y; NewDestruct y as [|m]; [ + Simpl; Trivial with arith +| Simpl; Trivial with arith +| Simpl; Rewrite <- plus_n_O; Trivial with arith +| Change (inject_nat (S (plus n (S m))))= + (Zplus (inject_nat (S n)) (inject_nat (S m))); + Rewrite inj_S; Rewrite H; Do 2 Rewrite inj_S; Rewrite Zplus_S_n; Trivial with arith]. +Qed. + +Theorem inj_mult : + (x,y:nat) (inject_nat (mult x y)) = (Zmult (inject_nat x) (inject_nat y)). +Proof. +Intro x; NewInduction x as [|n H]; [ + Simpl; Trivial with arith +| Intro y; Rewrite -> inj_S; Rewrite <- Zmult_Sm_n; + Rewrite <- H;Rewrite <- inj_plus; Simpl; Rewrite plus_sym; Trivial with arith]. +Qed. + +Theorem inj_neq: + (x,y:nat) (neq x y) -> (Zne (inject_nat x) (inject_nat y)). +Proof. +Unfold neq Zne not ; Intros x y H1 H2; Apply H1; Generalize H2; +Case x; Case y; Intros; [ + Auto with arith +| Discriminate H0 +| Discriminate H0 +| Simpl in H0; Injection H0; Do 2 Rewrite <- bij1; Intros E; Rewrite E; Auto with arith]. +Qed. + +Theorem inj_le: + (x,y:nat) (le x y) -> (Zle (inject_nat x) (inject_nat y)). +Proof. +Intros x y; Intros H; Elim H; [ + Unfold Zle ; Elim (Zcompare_EGAL (inject_nat x) (inject_nat x)); + Intros H1 H2; Rewrite H2; [ Discriminate | Trivial with arith] +| Intros m H1 H2; Apply Zle_trans with (inject_nat m); + [Assumption | Rewrite inj_S; Apply Zle_n_Sn]]. +Qed. + +Theorem inj_lt: (x,y:nat) (lt x y) -> (Zlt (inject_nat x) (inject_nat y)). +Proof. +Intros x y H; Apply Zgt_lt; Apply Zle_S_gt; Rewrite <- inj_S; Apply inj_le; +Exact H. +Qed. + +Theorem inj_gt: (x,y:nat) (gt x y) -> (Zgt (inject_nat x) (inject_nat y)). +Proof. +Intros x y H; Apply Zlt_gt; Apply inj_lt; Exact H. +Qed. + +Theorem inj_ge: (x,y:nat) (ge x y) -> (Zge (inject_nat x) (inject_nat y)). +Proof. +Intros x y H; Apply Zle_ge; Apply inj_le; Apply H. +Qed. + +Theorem inj_eq: (x,y:nat) x=y -> (inject_nat x) = (inject_nat y). +Proof. +Intros x y H; Rewrite H; Trivial with arith. +Qed. + +Theorem intro_Z : + (x:nat) (EX y:Z | (inject_nat x)=y /\ + (Zle ZERO (Zplus (Zmult y (POS xH)) ZERO))). +Proof. +Intros x; Exists (inject_nat x); Split; [ + Trivial with arith +| Rewrite Zmult_sym; Rewrite Zmult_one; Rewrite Zero_right; + Unfold Zle ; Elim x; Intros;Simpl; Discriminate ]. +Qed. + +Theorem inj_minus1 : + (x,y:nat) (le y x) -> + (inject_nat (minus x y)) = (Zminus (inject_nat x) (inject_nat y)). +Proof. +Intros x y H; Apply (Zsimpl_plus_l (inject_nat y)); Unfold Zminus ; +Rewrite Zplus_permute; Rewrite Zplus_inverse_r; Rewrite <- inj_plus; +Rewrite <- (le_plus_minus y x H);Rewrite Zero_right; Trivial with arith. +Qed. + +Theorem inj_minus2: (x,y:nat) (gt y x) -> (inject_nat (minus x y)) = ZERO. +Proof. +Intros x y H; Rewrite inj_minus_aux; [ Trivial with arith | Apply gt_not_le; Assumption]. +Qed. + +V7only [ (* From Zdivides *) ]. +Theorem POS_inject: (x : positive) (POS x) = (inject_nat (convert x)). +Proof. +Intros x; Elim x; Simpl; Auto. +Intros p H; Rewrite ZL6. +Apply f_equal with f := POS. +Apply convert_intro. +Rewrite bij1; Unfold convert; Simpl. +Rewrite ZL6; Auto. +Intros p H; Unfold convert; Simpl. +Rewrite ZL6; Simpl. +Rewrite inj_plus; Repeat Rewrite <- H. +Rewrite POS_xO; Simpl; Rewrite add_x_x; Reflexivity. +Qed. + diff --git a/theories7/ZArith/Znumtheory.v b/theories7/ZArith/Znumtheory.v new file mode 100644 index 00000000..b8e5f300 --- /dev/null +++ b/theories7/ZArith/Znumtheory.v @@ -0,0 +1,629 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Znumtheory.v,v 1.3.2.1 2004/07/16 19:31:43 herbelin Exp $ i*) + +Require ZArith_base. +Require ZArithRing. +Require Zcomplements. +Require Zdiv. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +(** This file contains some notions of number theory upon Z numbers: + - a divisibility predicate [Zdivide] + - a gcd predicate [gcd] + - Euclid algorithm [euclid] + - an efficient [Zgcd] function + - a relatively prime predicate [rel_prime] + - a prime predicate [prime] +*) + +(** * Divisibility *) + +Inductive Zdivide [a,b:Z] : Prop := + Zdivide_intro : (q:Z) `b = q * a` -> (Zdivide a b). + +(** Syntax for divisibility *) + +Notation "( a | b )" := (Zdivide a b) + (at level 0, a,b at level 10) : Z_scope + V8only "( a | b )" (at level 0). + +(** Results concerning divisibility*) + +Lemma Zdivide_refl : (a:Z) (a|a). +Proof. +Intros; Apply Zdivide_intro with `1`; Ring. +Save. + +Lemma Zone_divide : (a:Z) (1|a). +Proof. +Intros; Apply Zdivide_intro with `a`; Ring. +Save. + +Lemma Zdivide_0 : (a:Z) (a|0). +Proof. +Intros; Apply Zdivide_intro with `0`; Ring. +Save. + +Hints Resolve Zdivide_refl Zone_divide Zdivide_0 : zarith. + +Lemma Zdivide_mult_left : (a,b,c:Z) (a|b) -> (`c*a`|`c*b`). +Proof. +Induction 1; Intros; Apply Zdivide_intro with q. +Rewrite H0; Ring. +Save. + +Lemma Zdivide_mult_right : (a,b,c:Z) (a|b) -> (`a*c`|`b*c`). +Proof. +Intros a b c; Rewrite (Zmult_sym a c); Rewrite (Zmult_sym b c). +Apply Zdivide_mult_left; Trivial. +Save. + +Hints Resolve Zdivide_mult_left Zdivide_mult_right : zarith. + +Lemma Zdivide_plus : (a,b,c:Z) (a|b) -> (a|c) -> (a|`b+c`). +Proof. +Induction 1; Intros q Hq; Induction 1; Intros q' Hq'. +Apply Zdivide_intro with `q+q'`. +Rewrite Hq; Rewrite Hq'; Ring. +Save. + +Lemma Zdivide_opp : (a,b:Z) (a|b) -> (a|`-b`). +Proof. +Induction 1; Intros; Apply Zdivide_intro with `-q`. +Rewrite H0; Ring. +Save. + +Lemma Zdivide_opp_rev : (a,b:Z) (a|`-b`) -> (a| b). +Proof. +Intros; Replace b with `-(-b)`. Apply Zdivide_opp; Trivial. Ring. +Save. + +Lemma Zdivide_opp_left : (a,b:Z) (a|b) -> (`-a`|b). +Proof. +Induction 1; Intros; Apply Zdivide_intro with `-q`. +Rewrite H0; Ring. +Save. + +Lemma Zdivide_opp_left_rev : (a,b:Z) (`-a`|b) -> (a|b). +Proof. +Intros; Replace a with `-(-a)`. Apply Zdivide_opp_left; Trivial. Ring. +Save. + +Lemma Zdivide_minus : (a,b,c:Z) (a|b) -> (a|c) -> (a|`b-c`). +Proof. +Induction 1; Intros q Hq; Induction 1; Intros q' Hq'. +Apply Zdivide_intro with `q-q'`. +Rewrite Hq; Rewrite Hq'; Ring. +Save. + +Lemma Zdivide_left : (a,b,c:Z) (a|b) -> (a|`b*c`). +Proof. +Induction 1; Intros q Hq; Apply Zdivide_intro with `q*c`. +Rewrite Hq; Ring. +Save. + +Lemma Zdivide_right : (a,b,c:Z) (a|c) -> (a|`b*c`). +Proof. +Induction 1; Intros q Hq; Apply Zdivide_intro with `q*b`. +Rewrite Hq; Ring. +Save. + +Lemma Zdivide_a_ab : (a,b:Z) (a|`a*b`). +Proof. +Intros; Apply Zdivide_intro with b; Ring. +Save. + +Lemma Zdivide_a_ba : (a,b:Z) (a|`b*a`). +Proof. +Intros; Apply Zdivide_intro with b; Ring. +Save. + +Hints Resolve Zdivide_plus Zdivide_opp Zdivide_opp_rev + Zdivide_opp_left Zdivide_opp_left_rev + Zdivide_minus Zdivide_left Zdivide_right + Zdivide_a_ab Zdivide_a_ba : zarith. + +(** Auxiliary result. *) + +Lemma Zmult_one : + (x,y:Z) `x>=0` -> `x*y=1` -> `x=1`. +Proof. +Intros x y H H0; NewDestruct (Zmult_1_inversion_l ? ? H0) as [Hpos|Hneg]. + Assumption. + Rewrite Hneg in H; Simpl in H. + Contradiction (Zle_not_lt `0` `-1`). + Apply Zge_le; Assumption. + Apply NEG_lt_ZERO. +Save. + +(** Only [1] and [-1] divide [1]. *) + +Lemma Zdivide_1 : (x:Z) (x|1) -> `x=1` \/ `x=-1`. +Proof. +Induction 1; Intros. +Elim (Z_lt_ge_dec `0` x); [Left|Right]. +Apply Zmult_one with q; Auto with zarith; Rewrite H0; Ring. +Assert `(-x) = 1`; Auto with zarith. +Apply Zmult_one with (-q); Auto with zarith; Rewrite H0; Ring. +Save. + +(** If [a] divides [b] and [b] divides [a] then [a] is [b] or [-b]. *) + +Lemma Zdivide_antisym : (a,b:Z) (a|b) -> (b|a) -> `a=b` \/ `a=-b`. +Proof. +Induction 1; Intros. +Inversion H1. +Rewrite H0 in H2; Clear H H1. +Case (Z_zerop a); Intro. +Left; Rewrite H0; Rewrite e; Ring. +Assert Hqq0: `q0*q = 1`. +Apply Zmult_reg_left with a. +Assumption. +Ring. +Pattern 2 a; Rewrite H2; Ring. +Assert (q|1). +Rewrite <- Hqq0; Auto with zarith. +Elim (Zdivide_1 q H); Intros. +Rewrite H1 in H0; Left; Omega. +Rewrite H1 in H0; Right; Omega. +Save. + +(** If [a] divides [b] and [b<>0] then [|a| <= |b|]. *) + +Lemma Zdivide_bounds : (a,b:Z) (a|b) -> `b<>0` -> `|a| <= |b|`. +Proof. +Induction 1; Intros. +Assert `|b|=|q|*|a|`. + Subst; Apply Zabs_Zmult. +Rewrite H2. +Assert H3 := (Zabs_pos q). +Assert H4 := (Zabs_pos a). +Assert `|q|*|a|>=1*|a|`; Auto with zarith. +Apply Zge_Zmult_pos_compat; Auto with zarith. +Elim (Z_lt_ge_dec `|q|` `1`); [ Intros | Auto with zarith ]. +Assert `|q|=0`. + Omega. +Assert `q=0`. + Rewrite <- (Zabs_Zsgn q). +Rewrite H5; Auto with zarith. +Subst q; Omega. +Save. + +(** * Greatest common divisor (gcd). *) + +(** There is no unicity of the gcd; hence we define the predicate [gcd a b d] + expressing that [d] is a gcd of [a] and [b]. + (We show later that the [gcd] is actually unique if we discard its sign.) *) + +Inductive gcd [a,b,d:Z] : Prop := + gcd_intro : + (d|a) -> (d|b) -> ((x:Z) (x|a) -> (x|b) -> (x|d)) -> (gcd a b d). + +(** Trivial properties of [gcd] *) + +Lemma gcd_sym : (a,b,d:Z)(gcd a b d) -> (gcd b a d). +Proof. +Induction 1; Constructor; Intuition. +Save. + +Lemma gcd_0 : (a:Z)(gcd a `0` a). +Proof. +Constructor; Auto with zarith. +Save. + +Lemma gcd_minus :(a,b,d:Z)(gcd a `-b` d) -> (gcd b a d). +Proof. +Induction 1; Constructor; Intuition. +Save. + +Lemma gcd_opp :(a,b,d:Z)(gcd a b d) -> (gcd b a `-d`). +Proof. +Induction 1; Constructor; Intuition. +Save. + +Hints Resolve gcd_sym gcd_0 gcd_minus gcd_opp : zarith. + +(** * Extended Euclid algorithm. *) + +(** Euclid's algorithm to compute the [gcd] mainly relies on + the following property. *) + +Lemma gcd_for_euclid : + (a,b,d,q:Z) (gcd b `a-q*b` d) -> (gcd a b d). +Proof. +Induction 1; Constructor; Intuition. +Replace a with `a-q*b+q*b`. Auto with zarith. Ring. +Save. + +Lemma gcd_for_euclid2 : + (b,d,q,r:Z) (gcd r b d) -> (gcd b `b*q+r` d). +Proof. +Induction 1; Constructor; Intuition. +Apply H2; Auto. +Replace r with `b*q+r-b*q`. Auto with zarith. Ring. +Save. + +(** We implement the extended version of Euclid's algorithm, + i.e. the one computing Bezout's coefficients as it computes + the [gcd]. We follow the algorithm given in Knuth's + "Art of Computer Programming", vol 2, page 325. *) + +Section extended_euclid_algorithm. + +Variable a,b : Z. + +(** The specification of Euclid's algorithm is the existence of + [u], [v] and [d] such that [ua+vb=d] and [(gcd a b d)]. *) + +Inductive Euclid : Set := + Euclid_intro : + (u,v,d:Z) `u*a+v*b=d` -> (gcd a b d) -> Euclid. + +(** The recursive part of Euclid's algorithm uses well-founded + recursion of non-negative integers. It maintains 6 integers + [u1,u2,u3,v1,v2,v3] such that the following invariant holds: + [u1*a+u2*b=u3] and [v1*a+v2*b=v3] and [gcd(u2,v3)=gcd(a,b)]. + *) + +Lemma euclid_rec : + (v3:Z) `0 <= v3` -> (u1,u2,u3,v1,v2:Z) `u1*a+u2*b=u3` -> `v1*a+v2*b=v3` -> + ((d:Z)(gcd u3 v3 d) -> (gcd a b d)) -> Euclid. +Proof. +Intros v3 Hv3; Generalize Hv3; Pattern v3. +Apply Z_lt_rec. +Clear v3 Hv3; Intros. +Elim (Z_zerop x); Intro. +Apply Euclid_intro with u:=u1 v:=u2 d:=u3. +Assumption. +Apply H2. +Rewrite a0; Auto with zarith. +LetTac q := (Zdiv u3 x). +Assert Hq: `0 <= u3-q*x < x`. +Replace `u3-q*x` with `u3%x`. +Apply Z_mod_lt; Omega. +Assert xpos : `x > 0`. Omega. +Generalize (Z_div_mod_eq u3 x xpos). +Unfold q. +Intro eq; Pattern 2 u3; Rewrite eq; Ring. +Apply (H `u3-q*x` Hq (proj1 ? ? Hq) v1 v2 x `u1-q*v1` `u2-q*v2`). +Tauto. +Replace `(u1-q*v1)*a+(u2-q*v2)*b` with `(u1*a+u2*b)-q*(v1*a+v2*b)`. +Rewrite H0; Rewrite H1; Trivial. +Ring. +Intros; Apply H2. +Apply gcd_for_euclid with q; Assumption. +Assumption. +Save. + +(** We get Euclid's algorithm by applying [euclid_rec] on + [1,0,a,0,1,b] when [b>=0] and [1,0,a,0,-1,-b] when [b<0]. *) + +Lemma euclid : Euclid. +Proof. +Case (Z_le_gt_dec `0` b); Intro. +Intros; Apply euclid_rec with u1:=`1` u2:=`0` u3:=a + v1:=`0` v2:=`1` v3:=b; +Auto with zarith; Ring. +Intros; Apply euclid_rec with u1:=`1` u2:=`0` u3:=a + v1:=`0` v2:=`-1` v3:=`-b`; +Auto with zarith; Try Ring. +Save. + +End extended_euclid_algorithm. + +Theorem gcd_uniqueness_apart_sign : + (a,b,d,d':Z) (gcd a b d) -> (gcd a b d') -> `d = d'` \/ `d = -d'`. +Proof. +Induction 1. +Intros H1 H2 H3; Induction 1; Intros. +Generalize (H3 d' H4 H5); Intro Hd'd. +Generalize (H6 d H1 H2); Intro Hdd'. +Exact (Zdivide_antisym d d' Hdd' Hd'd). +Save. + +(** * Bezout's coefficients *) + +Inductive Bezout [a,b,d:Z] : Prop := + Bezout_intro : (u,v:Z) `u*a + v*b = d` -> (Bezout a b d). + +(** Existence of Bezout's coefficients for the [gcd] of [a] and [b] *) + +Lemma gcd_bezout : (a,b,d:Z) (gcd a b d) -> (Bezout a b d). +Proof. +Intros a b d Hgcd. +Elim (euclid a b); Intros u v d0 e g. +Generalize (gcd_uniqueness_apart_sign a b d d0 Hgcd g). +Intro H; Elim H; Clear H; Intros. +Apply Bezout_intro with u v. +Rewrite H; Assumption. +Apply Bezout_intro with `-u` `-v`. +Rewrite H; Rewrite <- e; Ring. +Save. + +(** gcd of [ca] and [cb] is [c gcd(a,b)]. *) + +Lemma gcd_mult : (a,b,c,d:Z) (gcd a b d) -> (gcd `c*a` `c*b` `c*d`). +Proof. +Intros a b c d; Induction 1; Constructor; Intuition. +Elim (gcd_bezout a b d H); Intros. +Elim H3; Intros. +Elim H4; Intros. +Apply Zdivide_intro with `u*q+v*q0`. +Rewrite <- H5. +Replace `c*(u*a+v*b)` with `u*(c*a)+v*(c*b)`. +Rewrite H6; Rewrite H7; Ring. +Ring. +Save. + +(** We could obtain a [Zgcd] function via [euclid]. But we propose + here a more direct version of a [Zgcd], with better extraction + (no bezout coeffs). *) + +Definition Zgcd_pos : (a:Z)`0<=a` -> (b:Z) + { g:Z | `0<=a` -> (gcd a b g) /\ `g>=0` }. +Proof. +Intros a Ha. +Apply (Z_lt_rec [a:Z](b:Z) { g:Z | `0<=a` -> (gcd a b g) /\`g>=0` }); Try Assumption. +Intro x; Case x. +Intros _ b; Exists (Zabs b). + Elim (Z_le_lt_eq_dec ? ? (Zabs_pos b)). + Intros H0; Split. + Apply Zabs_ind. + Intros; Apply gcd_sym; Apply gcd_0; Auto. + Intros; Apply gcd_opp; Apply gcd_0; Auto. + Auto with zarith. + + Intros H0; Rewrite <- H0. + Rewrite <- (Zabs_Zsgn b); Rewrite <- H0; Simpl. + Split; [Apply gcd_0|Idtac];Auto with zarith. + +Intros p Hrec b. +Generalize (Z_div_mod b (POS p)). +Case (Zdiv_eucl b (POS p)); Intros q r Hqr. +Elim Hqr; Clear Hqr; Intros; Auto with zarith. +Elim (Hrec r H0 (POS p)); Intros g Hgkl. +Inversion_clear H0. +Elim (Hgkl H1); Clear Hgkl; Intros H3 H4. +Exists g; Intros. +Split; Auto. +Rewrite H. +Apply gcd_for_euclid2; Auto. + +Intros p Hrec b. +Exists `0`; Intros. +Elim H; Auto. +Defined. + +Definition Zgcd_spec : (a,b:Z){ g : Z | (gcd a b g) /\ `g>=0` }. +Proof. +Intros a; Case (Z_gt_le_dec `0` a). +Intros; Assert `0 <= -a`. +Omega. +Elim (Zgcd_pos `-a` H b); Intros g Hgkl. +Exists g. +Intuition. +Intros Ha b; Elim (Zgcd_pos a Ha b); Intros g; Exists g; Intuition. +Defined. + +Definition Zgcd := [a,b:Z](let (g,_) = (Zgcd_spec a b) in g). + +Lemma Zgcd_is_pos : (a,b:Z)`(Zgcd a b) >=0`. +Intros a b; Unfold Zgcd; Case (Zgcd_spec a b); Tauto. +Qed. + +Lemma Zgcd_is_gcd : (a,b:Z)(gcd a b (Zgcd a b)). +Intros a b; Unfold Zgcd; Case (Zgcd_spec a b); Tauto. +Qed. + +(** * Relative primality *) + +Definition rel_prime [a,b:Z] : Prop := (gcd a b `1`). + +(** Bezout's theorem: [a] and [b] are relatively prime if and + only if there exist [u] and [v] such that [ua+vb = 1]. *) + +Lemma rel_prime_bezout : + (a,b:Z) (rel_prime a b) -> (Bezout a b `1`). +Proof. +Intros a b; Exact (gcd_bezout a b `1`). +Save. + +Lemma bezout_rel_prime : + (a,b:Z) (Bezout a b `1`) -> (rel_prime a b). +Proof. +Induction 1; Constructor; Auto with zarith. +Intros. Rewrite <- H0; Auto with zarith. +Save. + +(** Gauss's theorem: if [a] divides [bc] and if [a] and [b] are + relatively prime, then [a] divides [c]. *) + +Theorem Gauss : (a,b,c:Z) (a |`b*c`) -> (rel_prime a b) -> (a | c). +Proof. +Intros. Elim (rel_prime_bezout a b H0); Intros. +Replace c with `c*1`; [ Idtac | Ring ]. +Rewrite <- H1. +Replace `c*(u*a+v*b)` with `(c*u)*a + v*(b*c)`; [ EAuto with zarith | Ring ]. +Save. + +(** If [a] is relatively prime to [b] and [c], then it is to [bc] *) + +Lemma rel_prime_mult : + (a,b,c:Z) (rel_prime a b) -> (rel_prime a c) -> (rel_prime a `b*c`). +Proof. +Intros a b c Hb Hc. +Elim (rel_prime_bezout a b Hb); Intros. +Elim (rel_prime_bezout a c Hc); Intros. +Apply bezout_rel_prime. +Apply Bezout_intro with u:=`u*u0*a+v0*c*u+u0*v*b` v:=`v*v0`. +Rewrite <- H. +Replace `u*a+v*b` with `(u*a+v*b) * 1`; [ Idtac | Ring ]. +Rewrite <- H0. +Ring. +Save. + +Lemma rel_prime_cross_prod : + (a,b,c,d:Z) (rel_prime a b) -> (rel_prime c d) -> `b>0` -> `d>0` -> + `a*d = b*c` -> (a=c /\ b=d). +Proof. +Intros a b c d; Intros. +Elim (Zdivide_antisym b d). +Split; Auto with zarith. +Rewrite H4 in H3. +Rewrite Zmult_sym in H3. +Apply Zmult_reg_left with d; Auto with zarith. +Intros; Omega. +Apply Gauss with a. +Rewrite H3. +Auto with zarith. +Red; Auto with zarith. +Apply Gauss with c. +Rewrite Zmult_sym. +Rewrite <- H3. +Auto with zarith. +Red; Auto with zarith. +Save. + +(** After factorization by a gcd, the original numbers are relatively prime. *) + +Lemma gcd_rel_prime : + (a,b,g:Z)`b>0` -> `g>=0`-> (gcd a b g) -> (rel_prime `a/g` `b/g`). +Intros a b g; Intros. +Assert `g <> 0`. + Intro. + Elim H1; Intros. + Elim H4; Intros. + Rewrite H2 in H6; Subst b; Omega. +Unfold rel_prime. +Elim (Zgcd_spec `a/g` `b/g`); Intros g' (H3,H4). +Assert H5 := (gcd_mult ? ? g ? H3). +Rewrite <- Z_div_exact_2 in H5; Auto with zarith. +Rewrite <- Z_div_exact_2 in H5; Auto with zarith. +Elim (gcd_uniqueness_apart_sign ? ? ? ? H1 H5). +Intros; Rewrite (!Zmult_reg_left `1` g' g); Auto with zarith. +Intros; Rewrite (!Zmult_reg_left `1` `-g'` g); Auto with zarith. +Pattern 1 g; Rewrite H6; Ring. + +Elim H1; Intros. +Elim H7; Intros. +Rewrite H9. +Replace `q*g` with `0+q*g`. +Rewrite Z_mod_plus. +Compute; Auto. +Omega. +Ring. + +Elim H1; Intros. +Elim H6; Intros. +Rewrite H9. +Replace `q*g` with `0+q*g`. +Rewrite Z_mod_plus. +Compute; Auto. +Omega. +Ring. +Save. + +(** * Primality *) + +Inductive prime [p:Z] : Prop := + prime_intro : + `1 < p` -> ((n:Z) `1 <= n < p` -> (rel_prime n p)) -> (prime p). + +(** The sole divisors of a prime number [p] are [-1], [1], [p] and [-p]. *) + +Lemma prime_divisors : + (p:Z) (prime p) -> + (a:Z) (a|p) -> `a = -1` \/ `a = 1` \/ a = p \/ `a = -p`. +Proof. +Induction 1; Intros. +Assert `a = (-p)`\/`-p<a< -1`\/`a = -1`\/`a=0`\/`a = 1`\/`1<a<p`\/`a = p`. +Assert `|a| <= |p|`. Apply Zdivide_bounds; [ Assumption | Omega ]. +Generalize H3. +Pattern `|a|`; Apply Zabs_ind; Pattern `|p|`; Apply Zabs_ind; Intros; Omega. +Intuition Idtac. +(* -p < a < -1 *) +Absurd (rel_prime `-a` p); Intuition. +Inversion H3. +Assert (`-a` | `-a`); Auto with zarith. +Assert (`-a` | p); Auto with zarith. +Generalize (H8 `-a` H9 H10); Intuition Idtac. +Generalize (Zdivide_1 `-a` H11); Intuition. +(* a = 0 *) +Inversion H2. Subst a; Omega. +(* 1 < a < p *) +Absurd (rel_prime a p); Intuition. +Inversion H3. +Assert (a | a); Auto with zarith. +Assert (a | p); Auto with zarith. +Generalize (H8 a H9 H10); Intuition Idtac. +Generalize (Zdivide_1 a H11); Intuition. +Save. + +(** A prime number is relatively prime with any number it does not divide *) + +Lemma prime_rel_prime : + (p:Z) (prime p) -> (a:Z) ~ (p|a) -> (rel_prime p a). +Proof. +Induction 1; Intros. +Constructor; Intuition. +Elim (prime_divisors p H x H3); Intuition; Subst; Auto with zarith. +Absurd (p | a); Auto with zarith. +Absurd (p | a); Intuition. +Save. + +Hints Resolve prime_rel_prime : zarith. + +(** [Zdivide] can be expressed using [Zmod]. *) + +Lemma Zmod_Zdivide : (a,b:Z) `b>0` -> `a%b = 0` -> (b|a). +Intros a b H H0. +Apply Zdivide_intro with `(a/b)`. +Pattern 1 a; Rewrite (Z_div_mod_eq a b H). +Rewrite H0; Ring. +Save. + +Lemma Zdivide_Zmod : (a,b:Z) `b>0` -> (b|a) -> `a%b = 0`. +Intros a b; Destruct 2; Intros; Subst. +Change `q*b` with `0+q*b`. +Rewrite Z_mod_plus; Auto. +Save. + +(** [Zdivide] is hence decidable *) + +Lemma Zdivide_dec : (a,b:Z) { (a|b) } + { ~ (a|b) }. +Proof. +Intros a b; Elim (Ztrichotomy_inf a `0`). +(* a<0 *) +Intros H; Elim H; Intros. +Case (Z_eq_dec `b%(-a)` `0`). +Left; Apply Zdivide_opp_left_rev; Apply Zmod_Zdivide; Auto with zarith. +Intro H1; Right; Intro; Elim H1; Apply Zdivide_Zmod; Auto with zarith. +(* a=0 *) +Case (Z_eq_dec b `0`); Intro. +Left; Subst; Auto with zarith. +Right; Subst; Intro H0; Inversion H0; Omega. +(* a>0 *) +Intro H; Case (Z_eq_dec `b%a` `0`). +Left; Apply Zmod_Zdivide; Auto with zarith. +Intro H1; Right; Intro; Elim H1; Apply Zdivide_Zmod; Auto with zarith. +Save. + +(** If a prime [p] divides [ab] then it divides either [a] or [b] *) + +Lemma prime_mult : + (p:Z) (prime p) -> (a,b:Z) (p | `a*b`) -> (p | a) \/ (p | b). +Proof. +Intro p; Induction 1; Intros. +Case (Zdivide_dec p a); Intuition. +Right; Apply Gauss with a; Auto with zarith. +Save. + + diff --git a/theories7/ZArith/Zorder.v b/theories7/ZArith/Zorder.v new file mode 100644 index 00000000..d49a0800 --- /dev/null +++ b/theories7/ZArith/Zorder.v @@ -0,0 +1,969 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: Zorder.v,v 1.1.2.1 2004/07/16 19:31:44 herbelin Exp $ i*) + +(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *) + +Require BinPos. +Require BinInt. +Require Arith. +Require Decidable. +Require Zsyntax. +Require Zcompare. + +V7only [Import nat_scope.]. +Open Local Scope Z_scope. + +Implicit Variable Type x,y,z:Z. + +(**********************************************************************) +(** Properties of the order relations on binary integers *) + +(** Trichotomy *) + +Theorem Ztrichotomy_inf : (m,n:Z) {`m<n`} + {m=n} + {`m>n`}. +Proof. +Unfold Zgt Zlt; Intros m n; Assert H:=(refl_equal ? (Zcompare m n)). + LetTac x := (Zcompare m n) in 2 H Goal. + NewDestruct x; + [Left; Right;Rewrite Zcompare_EGAL_eq with 1:=H + | Left; Left + | Right ]; Reflexivity. +Qed. + +Theorem Ztrichotomy : (m,n:Z) `m<n` \/ m=n \/ `m>n`. +Proof. + Intros m n; NewDestruct (Ztrichotomy_inf m n) as [[Hlt|Heq]|Hgt]; + [Left | Right; Left |Right; Right]; Assumption. +Qed. + +(**********************************************************************) +(** Decidability of equality and order on Z *) + +Theorem dec_eq: (x,y:Z) (decidable (x=y)). +Proof. +Intros x y; Unfold decidable ; Elim (Zcompare_EGAL x y); +Intros H1 H2; Elim (Dcompare (Zcompare x y)); [ + Tauto + | Intros H3; Right; Unfold not ; Intros H4; + Elim H3; Rewrite (H2 H4); Intros H5; Discriminate H5]. +Qed. + +Theorem dec_Zne: (x,y:Z) (decidable (Zne x y)). +Proof. +Intros x y; Unfold decidable Zne ; Elim (Zcompare_EGAL x y). +Intros H1 H2; Elim (Dcompare (Zcompare x y)); + [ Right; Rewrite H1; Auto + | Left; Unfold not; Intro; Absurd (Zcompare x y)=EGAL; + [ Elim H; Intros HR; Rewrite HR; Discriminate + | Auto]]. +Qed. + +Theorem dec_Zle: (x,y:Z) (decidable `x<=y`). +Proof. +Intros x y; Unfold decidable Zle ; Elim (Zcompare x y); [ + Left; Discriminate + | Left; Discriminate + | Right; Unfold not ; Intros H; Apply H; Trivial with arith]. +Qed. + +Theorem dec_Zgt: (x,y:Z) (decidable `x>y`). +Proof. +Intros x y; Unfold decidable Zgt ; Elim (Zcompare x y); + [ Right; Discriminate | Right; Discriminate | Auto with arith]. +Qed. + +Theorem dec_Zge: (x,y:Z) (decidable `x>=y`). +Proof. +Intros x y; Unfold decidable Zge ; Elim (Zcompare x y); [ + Left; Discriminate +| Right; Unfold not ; Intros H; Apply H; Trivial with arith +| Left; Discriminate]. +Qed. + +Theorem dec_Zlt: (x,y:Z) (decidable `x<y`). +Proof. +Intros x y; Unfold decidable Zlt ; Elim (Zcompare x y); + [ Right; Discriminate | Auto with arith | Right; Discriminate]. +Qed. + +Theorem not_Zeq : (x,y:Z) ~ x=y -> `x<y` \/ `y<x`. +Proof. +Intros x y; Elim (Dcompare (Zcompare x y)); [ + Intros H1 H2; Absurd x=y; [ Assumption | Elim (Zcompare_EGAL x y); Auto with arith] +| Unfold Zlt ; Intros H; Elim H; Intros H1; + [Auto with arith | Right; Elim (Zcompare_ANTISYM x y); Auto with arith]]. +Qed. + +(** Relating strict and large orders *) + +Lemma Zgt_lt : (m,n:Z) `m>n` -> `n<m`. +Proof. +Unfold Zgt Zlt ;Intros m n H; Elim (Zcompare_ANTISYM m n); Auto with arith. +Qed. + +Lemma Zlt_gt : (m,n:Z) `m<n` -> `n>m`. +Proof. +Unfold Zgt Zlt ;Intros m n H; Elim (Zcompare_ANTISYM n m); Auto with arith. +Qed. + +Lemma Zge_le : (m,n:Z) `m>=n` -> `n<=m`. +Proof. +Intros m n; Change ~`m<n`-> ~`n>m`; +Unfold not; Intros H1 H2; Apply H1; Apply Zgt_lt; Assumption. +Qed. + +Lemma Zle_ge : (m,n:Z) `m<=n` -> `n>=m`. +Proof. +Intros m n; Change ~`m>n`-> ~`n<m`; +Unfold not; Intros H1 H2; Apply H1; Apply Zlt_gt; Assumption. +Qed. + +Lemma Zle_not_gt : (n,m:Z)`n<=m` -> ~`n>m`. +Proof. +Trivial. +Qed. + +Lemma Zgt_not_le : (n,m:Z)`n>m` -> ~`n<=m`. +Proof. +Intros n m H1 H2; Apply H2; Assumption. +Qed. + +Lemma Zle_not_lt : (n,m:Z)`n<=m` -> ~`m<n`. +Proof. +Intros n m H1 H2. +Assert H3:=(Zlt_gt ? ? H2). +Apply Zle_not_gt with n m; Assumption. +Qed. + +Lemma Zlt_not_le : (n,m:Z)`n<m` -> ~`m<=n`. +Proof. +Intros n m H1 H2. +Apply Zle_not_lt with m n; Assumption. +Qed. + +Lemma not_Zge : (x,y:Z) ~`x>=y` -> `x<y`. +Proof. +Unfold Zge Zlt ; Intros x y H; Apply dec_not_not; + [ Exact (dec_Zlt x y) | Assumption]. +Qed. + +Lemma not_Zlt : (x,y:Z) ~`x<y` -> `x>=y`. +Proof. +Unfold Zlt Zge; Auto with arith. +Qed. + +Lemma not_Zgt : (x,y:Z)~`x>y` -> `x<=y`. +Proof. +Trivial. +Qed. + +Lemma not_Zle : (x,y:Z) ~`x<=y` -> `x>y`. +Proof. +Unfold Zle Zgt ; Intros x y H; Apply dec_not_not; + [ Exact (dec_Zgt x y) | Assumption]. +Qed. + +Lemma Zge_iff_le : (x,y:Z) `x>=y` <-> `y<=x`. +Proof. + Intros x y; Intros. Split. Intro. Apply Zge_le. Assumption. + Intro. Apply Zle_ge. Assumption. +Qed. + +Lemma Zgt_iff_lt : (x,y:Z) `x>y` <-> `y<x`. +Proof. + Intros x y. Split. Intro. Apply Zgt_lt. Assumption. + Intro. Apply Zlt_gt. Assumption. +Qed. + +(** Reflexivity *) + +Lemma Zle_n : (n:Z) (Zle n n). +Proof. +Intros n; Unfold Zle; Rewrite (Zcompare_x_x n); Discriminate. +Qed. + +Lemma Zle_refl : (n,m:Z) n=m -> `n<=m`. +Proof. +Intros; Rewrite H; Apply Zle_n. +Qed. + +Hints Resolve Zle_n : zarith. + +(** Antisymmetry *) + +Lemma Zle_antisym : (n,m:Z)`n<=m`->`m<=n`->n=m. +Proof. +Intros n m H1 H2; NewDestruct (Ztrichotomy n m) as [Hlt|[Heq|Hgt]]. + Absurd `m>n`; [ Apply Zle_not_gt | Apply Zlt_gt]; Assumption. + Assumption. + Absurd `n>m`; [ Apply Zle_not_gt | Idtac]; Assumption. +Qed. + +(** Asymmetry *) + +Lemma Zgt_not_sym : (n,m:Z)`n>m` -> ~`m>n`. +Proof. +Unfold Zgt ;Intros n m H; Elim (Zcompare_ANTISYM n m); Intros H1 H2; +Rewrite -> H1; [ Discriminate | Assumption ]. +Qed. + +Lemma Zlt_not_sym : (n,m:Z)`n<m` -> ~`m<n`. +Proof. +Intros n m H H1; +Assert H2:`m>n`. Apply Zlt_gt; Assumption. +Assert H3: `n>m`. Apply Zlt_gt; Assumption. +Apply Zgt_not_sym with m n; Assumption. +Qed. + +(** Irreflexivity *) + +Lemma Zgt_antirefl : (n:Z)~`n>n`. +Proof. +Intros n H; Apply (Zgt_not_sym n n H H). +Qed. + +Lemma Zlt_n_n : (n:Z)~`n<n`. +Proof. +Intros n H; Apply (Zlt_not_sym n n H H). +Qed. + +Lemma Zlt_not_eq : (x,y:Z)`x<y` -> ~x=y. +Proof. +Unfold not; Intros x y H H0. +Rewrite H0 in H. +Apply (Zlt_n_n ? H). +Qed. + +(** Large = strict or equal *) + +Lemma Zlt_le_weak : (n,m:Z)`n<m`->`n<=m`. +Proof. +Intros n m Hlt; Apply not_Zgt; Apply Zgt_not_sym; Apply Zlt_gt; Assumption. +Qed. + +Lemma Zle_lt_or_eq : (n,m:Z)`n<=m`->(`n<m` \/ n=m). +Proof. +Intros n m H; NewDestruct (Ztrichotomy n m) as [Hlt|[Heq|Hgt]]; [ + Left; Assumption +| Right; Assumption +| Absurd `n>m`; [Apply Zle_not_gt|Idtac]; Assumption ]. +Qed. + +(** Dichotomy *) + +Lemma Zle_or_lt : (n,m:Z)`n<=m`\/`m<n`. +Proof. +Intros n m; NewDestruct (Ztrichotomy n m) as [Hlt|[Heq|Hgt]]; [ + Left; Apply not_Zgt; Intro Hgt; Assert Hgt':=(Zlt_gt ? ? Hlt); + Apply Zgt_not_sym with m n; Assumption +| Left; Rewrite Heq; Apply Zle_n +| Right; Apply Zgt_lt; Assumption ]. +Qed. + +(** Transitivity of strict orders *) + +Lemma Zgt_trans : (n,m,p:Z)`n>m`->`m>p`->`n>p`. +Proof. +Exact Zcompare_trans_SUPERIEUR. +Qed. + +Lemma Zlt_trans : (n,m,p:Z)`n<m`->`m<p`->`n<p`. +Proof. +Intros n m p H1 H2; Apply Zgt_lt; Apply Zgt_trans with m:= m; +Apply Zlt_gt; Assumption. +Qed. + +(** Mixed transitivity *) + +Lemma Zle_gt_trans : (n,m,p:Z)`m<=n`->`m>p`->`n>p`. +Proof. +Intros n m p H1 H2; NewDestruct (Zle_lt_or_eq m n H1) as [Hlt|Heq]; [ + Apply Zgt_trans with m; [Apply Zlt_gt; Assumption | Assumption ] +| Rewrite <- Heq; Assumption ]. +Qed. + +Lemma Zgt_le_trans : (n,m,p:Z)`n>m`->`p<=m`->`n>p`. +Proof. +Intros n m p H1 H2; NewDestruct (Zle_lt_or_eq p m H2) as [Hlt|Heq]; [ + Apply Zgt_trans with m; [Assumption|Apply Zlt_gt; Assumption] +| Rewrite Heq; Assumption ]. +Qed. + +Lemma Zlt_le_trans : (n,m,p:Z)`n<m`->`m<=p`->`n<p`. +Intros n m p H1 H2;Apply Zgt_lt;Apply Zle_gt_trans with m:=m; + [ Assumption | Apply Zlt_gt;Assumption ]. +Qed. + +Lemma Zle_lt_trans : (n,m,p:Z)`n<=m`->`m<p`->`n<p`. +Proof. +Intros n m p H1 H2;Apply Zgt_lt;Apply Zgt_le_trans with m:=m; + [ Apply Zlt_gt;Assumption | Assumption ]. +Qed. + +(** Transitivity of large orders *) + +Lemma Zle_trans : (n,m,p:Z)`n<=m`->`m<=p`->`n<=p`. +Proof. +Intros n m p H1 H2; Apply not_Zgt. +Intro Hgt; Apply Zle_not_gt with n m. Assumption. +Exact (Zgt_le_trans n p m Hgt H2). +Qed. + +Lemma Zge_trans : (n, m, p : Z) `n>=m` -> `m>=p` -> `n>=p`. +Proof. +Intros n m p H1 H2. +Apply Zle_ge. +Apply Zle_trans with m; Apply Zge_le; Trivial. +Qed. + +Hints Resolve Zle_trans : zarith. + +(** Compatibility of successor wrt to order *) + +Lemma Zle_n_S : (n,m:Z) `m<=n` -> `(Zs m)<=(Zs n)`. +Proof. +Unfold Zle not ;Intros m n H1 H2; Apply H1; +Rewrite <- (Zcompare_Zplus_compatible n m (POS xH)); +Do 2 Rewrite (Zplus_sym (POS xH)); Exact H2. +Qed. + +Lemma Zgt_n_S : (n,m:Z)`m>n` -> `(Zs m)>(Zs n)`. +Proof. +Unfold Zgt; Intros n m H; Rewrite Zcompare_n_S; Auto with arith. +Qed. + +Lemma Zlt_n_S : (n,m:Z)`n<m`->`(Zs n)<(Zs m)`. +Proof. +Intros n m H;Apply Zgt_lt;Apply Zgt_n_S;Apply Zlt_gt; Assumption. +Qed. + +Hints Resolve Zle_n_S : zarith. + +(** Simplification of successor wrt to order *) + +Lemma Zgt_S_n : (n,p:Z)`(Zs p)>(Zs n)`->`p>n`. +Proof. +Unfold Zs Zgt;Intros n p;Do 2 Rewrite -> [m:Z](Zplus_sym m (POS xH)); +Rewrite -> (Zcompare_Zplus_compatible p n (POS xH));Trivial with arith. +Qed. + +Lemma Zle_S_n : (n,m:Z) `(Zs m)<=(Zs n)` -> `m<=n`. +Proof. +Unfold Zle not ;Intros m n H1 H2;Apply H1; +Unfold Zs ;Do 2 Rewrite <- (Zplus_sym (POS xH)); +Rewrite -> (Zcompare_Zplus_compatible n m (POS xH));Assumption. +Qed. + +Lemma Zlt_S_n : (n,m:Z)`(Zs n)<(Zs m)`->`n<m`. +Proof. +Intros n m H;Apply Zgt_lt;Apply Zgt_S_n;Apply Zlt_gt; Assumption. +Qed. + +(** Compatibility of addition wrt to order *) + +Lemma Zgt_reg_l : (n,m,p:Z)`n>m`->`p+n>p+m`. +Proof. +Unfold Zgt; Intros n m p H; Rewrite (Zcompare_Zplus_compatible n m p); +Assumption. +Qed. + +Lemma Zgt_reg_r : (n,m,p:Z)`n>m`->`n+p>m+p`. +Proof. +Intros n m p H; Rewrite (Zplus_sym n p); Rewrite (Zplus_sym m p); Apply Zgt_reg_l; Trivial. +Qed. + +Lemma Zle_reg_l : (n,m,p:Z)`n<=m`->`p+n<=p+m`. +Proof. +Intros n m p; Unfold Zle not ;Intros H1 H2;Apply H1; +Rewrite <- (Zcompare_Zplus_compatible n m p); Assumption. +Qed. + +Lemma Zle_reg_r : (n,m,p:Z) `n<=m`->`n+p<=m+p`. +Proof. +Intros a b c;Do 2 Rewrite [n:Z](Zplus_sym n c); Exact (Zle_reg_l a b c). +Qed. + +Lemma Zlt_reg_l : (n,m,p:Z)`n<m`->`p+n<p+m`. +Proof. +Unfold Zlt ;Intros n m p; Rewrite Zcompare_Zplus_compatible;Trivial with arith. +Qed. + +Lemma Zlt_reg_r : (n,m,p:Z)`n<m`->`n+p<m+p`. +Proof. +Intros n m p H; Rewrite (Zplus_sym n p); Rewrite (Zplus_sym m p); Apply Zlt_reg_l; Trivial. +Qed. + +Lemma Zlt_le_reg : (a,b,c,d:Z) `a<b`->`c<=d`->`a+c<b+d`. +Proof. +Intros a b c d H0 H1. +Apply Zlt_le_trans with (Zplus b c). +Apply Zlt_reg_r; Trivial. +Apply Zle_reg_l; Trivial. +Qed. + +Lemma Zle_lt_reg : (a,b,c,d:Z) `a<=b`->`c<d`->`a+c<b+d`. +Proof. +Intros a b c d H0 H1. +Apply Zle_lt_trans with (Zplus b c). +Apply Zle_reg_r; Trivial. +Apply Zlt_reg_l; Trivial. +Qed. + +Lemma Zle_plus_plus : (n,m,p,q:Z) `n<=m`->(Zle p q)->`n+p<=m+q`. +Proof. +Intros n m p q; Intros H1 H2;Apply Zle_trans with m:=(Zplus n q); [ + Apply Zle_reg_l;Assumption | Apply Zle_reg_r;Assumption ]. +Qed. + +V7only [Set Implicit Arguments.]. + +Lemma Zlt_Zplus : (x1,x2,y1,y2:Z)`x1 < x2` -> `y1 < y2` -> `x1 + y1 < x2 + y2`. +Intros; Apply Zle_lt_reg. Apply Zlt_le_weak; Assumption. Assumption. +Qed. + +V7only [Unset Implicit Arguments.]. + +(** Compatibility of addition wrt to being positive *) + +Lemma Zle_0_plus : (x,y:Z) `0<=x` -> `0<=y` -> `0<=x+y`. +Proof. +Intros x y H1 H2;Rewrite <- (Zero_left ZERO); Apply Zle_plus_plus; Assumption. +Qed. + +(** Simplification of addition wrt to order *) + +Lemma Zsimpl_gt_plus_l : (n,m,p:Z)`p+n>p+m`->`n>m`. +Proof. +Unfold Zgt; Intros n m p H; + Rewrite <- (Zcompare_Zplus_compatible n m p); Assumption. +Qed. + +Lemma Zsimpl_gt_plus_r : (n,m,p:Z)`n+p>m+p`->`n>m`. +Proof. +Intros n m p H; Apply Zsimpl_gt_plus_l with p. +Rewrite (Zplus_sym p n); Rewrite (Zplus_sym p m); Trivial. +Qed. + +Lemma Zsimpl_le_plus_l : (n,m,p:Z)`p+n<=p+m`->`n<=m`. +Proof. +Intros n m p; Unfold Zle not ;Intros H1 H2;Apply H1; +Rewrite (Zcompare_Zplus_compatible n m p); Assumption. +Qed. + +Lemma Zsimpl_le_plus_r : (n,m,p:Z)`n+p<=m+p`->`n<=m`. +Proof. +Intros n m p H; Apply Zsimpl_le_plus_l with p. +Rewrite (Zplus_sym p n); Rewrite (Zplus_sym p m); Trivial. +Qed. + +Lemma Zsimpl_lt_plus_l : (n,m,p:Z)`p+n<p+m`->`n<m`. +Proof. +Unfold Zlt ;Intros n m p; + Rewrite Zcompare_Zplus_compatible;Trivial with arith. +Qed. + +Lemma Zsimpl_lt_plus_r : (n,m,p:Z)`n+p<m+p`->`n<m`. +Proof. +Intros n m p H; Apply Zsimpl_lt_plus_l with p. +Rewrite (Zplus_sym p n); Rewrite (Zplus_sym p m); Trivial. +Qed. + +(** Special base instances of order *) + +Lemma Zgt_Sn_n : (n:Z)`(Zs n)>n`. +Proof. +Exact Zcompare_Zs_SUPERIEUR. +Qed. + +Lemma Zle_Sn_n : (n:Z)~`(Zs n)<=n`. +Proof. +Intros n; Apply Zgt_not_le; Apply Zgt_Sn_n. +Qed. + +Lemma Zlt_n_Sn : (n:Z)`n<(Zs n)`. +Proof. +Intro n; Apply Zgt_lt; Apply Zgt_Sn_n. +Qed. + +Lemma Zlt_pred_n_n : (n:Z)`(Zpred n)<n`. +Proof. +Intros n; Apply Zlt_S_n; Rewrite <- Zs_pred; Apply Zlt_n_Sn. +Qed. + +(** Relating strict and large order using successor or predecessor *) + +Lemma Zgt_le_S : (n,p:Z)`p>n`->`(Zs n)<=p`. +Proof. +Unfold Zgt Zle; Intros n p H; Elim (Zcompare_et_un p n); Intros H1 H2; +Unfold not ;Intros H3; Unfold not in H1; Apply H1; [ + Assumption +| Elim (Zcompare_ANTISYM (Zplus n (POS xH)) p);Intros H4 H5;Apply H4;Exact H3]. +Qed. + +Lemma Zle_gt_S : (n,p:Z)`n<=p`->`(Zs p)>n`. +Proof. +Intros n p H; Apply Zgt_le_trans with p. + Apply Zgt_Sn_n. + Assumption. +Qed. + +Lemma Zle_lt_n_Sm : (n,m:Z)`n<=m`->`n<(Zs m)`. +Proof. +Intros n m H; Apply Zgt_lt; Apply Zle_gt_S; Assumption. +Qed. + +Lemma Zlt_le_S : (n,p:Z)`n<p`->`(Zs n)<=p`. +Proof. +Intros n p H; Apply Zgt_le_S; Apply Zlt_gt; Assumption. +Qed. + +Lemma Zgt_S_le : (n,p:Z)`(Zs p)>n`->`n<=p`. +Proof. +Intros n p H;Apply Zle_S_n; Apply Zgt_le_S; Assumption. +Qed. + +Lemma Zlt_n_Sm_le : (n,m:Z)`n<(Zs m)`->`n<=m`. +Proof. +Intros n m H; Apply Zgt_S_le; Apply Zlt_gt; Assumption. +Qed. + +Lemma Zle_S_gt : (n,m:Z) `(Zs n)<=m` -> `m>n`. +Proof. +Intros n m H;Apply Zle_gt_trans with m:=(Zs n); + [ Assumption | Apply Zgt_Sn_n ]. +Qed. + +(** Weakening order *) + +Lemma Zle_n_Sn : (n:Z)`n<=(Zs n)`. +Proof. +Intros n; Apply Zgt_S_le;Apply Zgt_trans with m:=(Zs n) ;Apply Zgt_Sn_n. +Qed. + +Hints Resolve Zle_n_Sn : zarith. + +Lemma Zle_pred_n : (n:Z)`(Zpred n)<=n`. +Proof. +Intros n;Pattern 2 n ;Rewrite Zs_pred; Apply Zle_n_Sn. +Qed. + +Lemma Zlt_S : (n,m:Z)`n<m`->`n<(Zs m)`. +Intros n m H;Apply Zgt_lt; Apply Zgt_trans with m:=m; [ + Apply Zgt_Sn_n +| Apply Zlt_gt; Assumption ]. +Qed. + +Lemma Zle_le_S : (x,y:Z)`x<=y`->`x<=(Zs y)`. +Proof. +Intros x y H. +Apply Zle_trans with y; Trivial with zarith. +Qed. + +Lemma Zle_trans_S : (n,m:Z)`(Zs n)<=m`->`n<=m`. +Proof. +Intros n m H;Apply Zle_trans with m:=(Zs n); [ Apply Zle_n_Sn | Assumption ]. +Qed. + +Hints Resolve Zle_le_S : zarith. + +(** Relating order wrt successor and order wrt predecessor *) + +Lemma Zgt_pred : (n,p:Z)`p>(Zs n)`->`(Zpred p)>n`. +Proof. +Unfold Zgt Zs Zpred ;Intros n p H; +Rewrite <- [x,y:Z](Zcompare_Zplus_compatible x y (POS xH)); +Rewrite (Zplus_sym p); Rewrite Zplus_assoc; Rewrite [x:Z](Zplus_sym x n); +Simpl; Assumption. +Qed. + +Lemma Zlt_pred : (n,p:Z)`(Zs n)<p`->`n<(Zpred p)`. +Proof. +Intros n p H;Apply Zlt_S_n; Rewrite <- Zs_pred; Assumption. +Qed. + +(** Relating strict order and large order on positive *) + +Lemma Zlt_ZERO_pred_le_ZERO : (n:Z) `0<n` -> `0<=(Zpred n)`. +Intros x H. +Rewrite (Zs_pred x) in H. +Apply Zgt_S_le. +Apply Zlt_gt. +Assumption. +Qed. + +V7only [Set Implicit Arguments.]. + +Lemma Zgt0_le_pred : (y:Z) `y > 0` -> `0 <= (Zpred y)`. +Intros; Apply Zlt_ZERO_pred_le_ZERO; Apply Zgt_lt. Assumption. +Qed. + +V7only [Unset Implicit Arguments.]. + +(** Special cases of ordered integers *) + +V7only [ (* Relevance confirmed from Zdivides *) ]. +Lemma Z_O_1: `0<1`. +Proof. +Change `0<(Zs 0)`. Apply Zlt_n_Sn. +Qed. + +Lemma Zle_0_1: `0<=1`. +Proof. +Change `0<=(Zs 0)`. Apply Zle_n_Sn. +Qed. + +V7only [ (* Relevance confirmed from Zdivides *) ]. +Lemma Zle_NEG_POS: (p,q:positive) `(NEG p)<=(POS q)`. +Proof. +Intros p; Red; Simpl; Red; Intros H; Discriminate. +Qed. + +Lemma POS_gt_ZERO : (p:positive) `(POS p)>0`. +Unfold Zgt; Trivial. +Qed. + + (* weaker but useful (in [Zpower] for instance) *) +Lemma ZERO_le_POS : (p:positive) `0<=(POS p)`. +Intro; Unfold Zle; Discriminate. +Qed. + +Lemma NEG_lt_ZERO : (p:positive)`(NEG p)<0`. +Unfold Zlt; Trivial. +Qed. + +Lemma ZERO_le_inj : + (n:nat) `0 <= (inject_nat n)`. +Induction n; Simpl; Intros; +[ Apply Zle_n +| Unfold Zle; Simpl; Discriminate]. +Qed. + +Hints Immediate Zle_refl : zarith. + +(** Transitivity using successor *) + +Lemma Zgt_trans_S : (n,m,p:Z)`(Zs n)>m`->`m>p`->`n>p`. +Proof. +Intros n m p H1 H2;Apply Zle_gt_trans with m:=m; + [ Apply Zgt_S_le; Assumption | Assumption ]. +Qed. + +(** Derived lemma *) + +Lemma Zgt_S : (n,m:Z)`(Zs n)>m`->(`n>m`\/(m=n)). +Proof. +Intros n m H. +Assert Hle : `m<=n`. + Apply Zgt_S_le; Assumption. +NewDestruct (Zle_lt_or_eq ? ? Hle) as [Hlt|Heq]. + Left; Apply Zlt_gt; Assumption. + Right; Assumption. +Qed. + +(** Compatibility of multiplication by a positive wrt to order *) + +V7only [Set Implicit Arguments.]. + +Lemma Zle_Zmult_pos_right : (a,b,c : Z) `a<=b` -> `0<=c` -> `a*c<=b*c`. +Proof. +Intros a b c H H0; NewDestruct c. + Do 2 Rewrite Zero_mult_right; Assumption. + Rewrite (Zmult_sym a); Rewrite (Zmult_sym b). + Unfold Zle; Rewrite Zcompare_Zmult_compatible; Assumption. + Unfold Zle in H0; Contradiction H0; Reflexivity. +Qed. + +Lemma Zle_Zmult_pos_left : (a,b,c : Z) `a<=b` -> `0<=c` -> `c*a<=c*b`. +Proof. +Intros a b c H1 H2; Rewrite (Zmult_sym c a);Rewrite (Zmult_sym c b). +Apply Zle_Zmult_pos_right; Trivial. +Qed. + +V7only [ (* Relevance confirmed from Zextensions *) ]. +Lemma Zmult_lt_compat_r : (x,y,z:Z)`0<z` -> `x < y` -> `x*z < y*z`. +Proof. +Intros x y z H H0; NewDestruct z. + Contradiction (Zlt_n_n `0`). + Rewrite (Zmult_sym x); Rewrite (Zmult_sym y). + Unfold Zlt; Rewrite Zcompare_Zmult_compatible; Assumption. + Discriminate H. +Save. + +Lemma Zgt_Zmult_right : (x,y,z:Z)`z>0` -> `x > y` -> `x*z > y*z`. +Proof. +Intros x y z; Intros; Apply Zlt_gt; Apply Zmult_lt_compat_r; + Apply Zgt_lt; Assumption. +Qed. + +Lemma Zlt_Zmult_right : (x,y,z:Z)`z>0` -> `x < y` -> `x*z < y*z`. +Proof. +Intros x y z; Intros; Apply Zmult_lt_compat_r; + [Apply Zgt_lt; Assumption | Assumption]. +Qed. + +Lemma Zle_Zmult_right : (x,y,z:Z)`z>0` -> `x <= y` -> `x*z <= y*z`. +Proof. +Intros x y z Hz Hxy. +Elim (Zle_lt_or_eq x y Hxy). +Intros; Apply Zlt_le_weak. +Apply Zlt_Zmult_right; Trivial. +Intros; Apply Zle_refl. +Rewrite H; Trivial. +Qed. + +V7only [ (* Relevance confirmed from Zextensions *) ]. +Lemma Zmult_lt_0_le_compat_r : (x,y,z:Z)`0 < z`->`x <= y`->`x*z <= y*z`. +Proof. +Intros x y z; Intros; Apply Zle_Zmult_right; Try Apply Zlt_gt; Assumption. +Qed. + +Lemma Zlt_Zmult_left : (x,y,z:Z)`z>0` -> `x < y` -> `z*x < z*y`. +Proof. +Intros x y z; Intros. +Rewrite (Zmult_sym z x); Rewrite (Zmult_sym z y); +Apply Zlt_Zmult_right; Assumption. +Qed. + +V7only [ (* Relevance confirmed from Zextensions *) ]. +Lemma Zmult_lt_compat_l : (x,y,z:Z)`0<z` -> `x < y` -> `z*x < z*y`. +Proof. +Intros x y z; Intros. +Rewrite (Zmult_sym z x); Rewrite (Zmult_sym z y); +Apply Zlt_Zmult_right; Try Apply Zlt_gt; Assumption. +Save. + +Lemma Zgt_Zmult_left : (x,y,z:Z)`z>0` -> `x > y` -> `z*x > z*y`. +Proof. +Intros x y z; Intros; +Rewrite (Zmult_sym z x); Rewrite (Zmult_sym z y); +Apply Zgt_Zmult_right; Assumption. +Qed. + +Lemma Zge_Zmult_pos_right : (a,b,c : Z) `a>=b` -> `c>=0` -> `a*c>=b*c`. +Proof. +Intros a b c H1 H2; Apply Zle_ge. +Apply Zle_Zmult_pos_right; Apply Zge_le; Trivial. +Qed. + +Lemma Zge_Zmult_pos_left : (a,b,c : Z) `a>=b` -> `c>=0` -> `c*a>=c*b`. +Proof. +Intros a b c H1 H2; Apply Zle_ge. +Apply Zle_Zmult_pos_left; Apply Zge_le; Trivial. +Qed. + +Lemma Zge_Zmult_pos_compat : + (a,b,c,d : Z) `a>=c` -> `b>=d` -> `c>=0` -> `d>=0` -> `a*b>=c*d`. +Proof. +Intros a b c d H0 H1 H2 H3. +Apply Zge_trans with (Zmult a d). +Apply Zge_Zmult_pos_left; Trivial. +Apply Zge_trans with c; Trivial. +Apply Zge_Zmult_pos_right; Trivial. +Qed. + +V7only [ (* Relevance confirmed from Zextensions *) ]. +Lemma Zmult_le_compat: (a, b, c, d : Z) + `a<=c` -> `b<=d` -> `0<=a` -> `0<=b` -> `a*b<=c*d`. +Proof. +Intros a b c d H0 H1 H2 H3. +Apply Zle_trans with (Zmult c b). +Apply Zle_Zmult_pos_right; Assumption. +Apply Zle_Zmult_pos_left. +Assumption. +Apply Zle_trans with a; Assumption. +Qed. + +(** Simplification of multiplication by a positive wrt to being positive *) + +Lemma Zlt_Zmult_right2 : (x,y,z:Z)`z>0` -> `x*z < y*z` -> `x < y`. +Proof. +Intros x y z; Intros; NewDestruct z. + Contradiction (Zgt_antirefl `0`). + Rewrite (Zmult_sym x) in H0; Rewrite (Zmult_sym y) in H0. + Unfold Zlt in H0; Rewrite Zcompare_Zmult_compatible in H0; Assumption. + Discriminate H. +Qed. + +V7only [ (* Relevance confirmed from Zextensions *) ]. +Lemma Zmult_lt_reg_r : (a, b, c : Z) `0<c` -> `a*c<b*c` -> `a<b`. +Proof. +Intros a b c H0 H1. +Apply Zlt_Zmult_right2 with c; Try Apply Zlt_gt; Assumption. +Qed. + +Lemma Zle_mult_simpl : (a,b,c:Z)`c>0`->`a*c<=b*c`->`a<=b`. +Proof. +Intros x y z Hz Hxy. +Elim (Zle_lt_or_eq `x*z` `y*z` Hxy). +Intros; Apply Zlt_le_weak. +Apply Zlt_Zmult_right2 with z; Trivial. +Intros; Apply Zle_refl. +Apply Zmult_reg_right with z. + Intro. Rewrite H0 in Hz. Contradiction (Zgt_antirefl `0`). +Assumption. +Qed. +V7only [Notation Zle_Zmult_right2 := Zle_mult_simpl. +(* Zle_Zmult_right2 : (x,y,z:Z)`z>0` -> `x*z <= y*z` -> `x <= y`. *) +]. + +V7only [ (* Relevance confirmed from Zextensions *) ]. +Lemma Zmult_lt_0_le_reg_r: (x,y,z:Z)`0 <z`->`x*z <= y*z`->`x <= y`. +Intros x y z; Intros ; Apply Zle_mult_simpl with z. +Try Apply Zlt_gt; Assumption. +Assumption. +Qed. + +V7only [Unset Implicit Arguments.]. + +Lemma Zge_mult_simpl : (a,b,c:Z) `c>0`->`a*c>=b*c`->`a>=b`. +Intros a b c H1 H2; Apply Zle_ge; Apply Zle_mult_simpl with c; Trivial. +Apply Zge_le; Trivial. +Qed. + +Lemma Zgt_mult_simpl : (a,b,c:Z) `c>0`->`a*c>b*c`->`a>b`. +Intros a b c H1 H2; Apply Zlt_gt; Apply Zlt_Zmult_right2 with c; Trivial. +Apply Zgt_lt; Trivial. +Qed. + + +(** Compatibility of multiplication by a positive wrt to being positive *) + +Lemma Zle_ZERO_mult : (x,y:Z) `0<=x` -> `0<=y` -> `0<=x*y`. +Proof. +Intros x y; Case x. +Intros; Rewrite Zero_mult_left; Trivial. +Intros p H1; Unfold Zle. + Pattern 2 ZERO ; Rewrite <- (Zero_mult_right (POS p)). + Rewrite Zcompare_Zmult_compatible; Trivial. +Intros p H1 H2; Absurd (Zgt ZERO (NEG p)); Trivial. +Unfold Zgt; Simpl; Auto with zarith. +Qed. + +Lemma Zgt_ZERO_mult: (a,b:Z) `a>0`->`b>0`->`a*b>0`. +Proof. +Intros x y; Case x. +Intros H; Discriminate H. +Intros p H1; Unfold Zgt; +Pattern 2 ZERO ; Rewrite <- (Zero_mult_right (POS p)). + Rewrite Zcompare_Zmult_compatible; Trivial. +Intros p H; Discriminate H. +Qed. + +V7only [ (* Relevance confirmed from Zextensions *) ]. +Lemma Zmult_lt_O_compat : (a, b : Z) `0<a` -> `0<b` -> `0<a*b`. +Intros a b apos bpos. +Apply Zgt_lt. +Apply Zgt_ZERO_mult; Try Apply Zlt_gt; Assumption. +Qed. + +Lemma Zle_mult: (x,y:Z) `x>0` -> `0<=y` -> `0<=(Zmult y x)`. +Proof. +Intros x y H1 H2; Apply Zle_ZERO_mult; Trivial. +Apply Zlt_le_weak; Apply Zgt_lt; Trivial. +Qed. + +(** Simplification of multiplication by a positive wrt to being positive *) + +Lemma Zmult_le: (x,y:Z) `x>0` -> `0<=(Zmult y x)` -> `0<=y`. +Proof. +Intros x y; Case x; [ + Simpl; Unfold Zgt ; Simpl; Intros H; Discriminate H +| Intros p H1; Unfold Zle; Rewrite -> Zmult_sym; + Pattern 1 ZERO ; Rewrite <- (Zero_mult_right (POS p)); + Rewrite Zcompare_Zmult_compatible; Auto with arith +| Intros p; Unfold Zgt ; Simpl; Intros H; Discriminate H]. +Qed. + +Lemma Zmult_lt: (x,y:Z) `x>0` -> `0<y*x` -> `0<y`. +Proof. +Intros x y; Case x; [ + Simpl; Unfold Zgt ; Simpl; Intros H; Discriminate H +| Intros p H1; Unfold Zlt; Rewrite -> Zmult_sym; + Pattern 1 ZERO ; Rewrite <- (Zero_mult_right (POS p)); + Rewrite Zcompare_Zmult_compatible; Auto with arith +| Intros p; Unfold Zgt ; Simpl; Intros H; Discriminate H]. +Qed. + +V7only [ (* Relevance confirmed from Zextensions *) ]. +Lemma Zmult_lt_0_reg_r : (x,y:Z)`0 < x`->`0 < y*x`->`0 < y`. +Proof. +Intros x y; Intros; EApply Zmult_lt with x ; Try Apply Zlt_gt; Assumption. +Qed. + +Lemma Zmult_gt: (x,y:Z) `x>0` -> `x*y>0` -> `y>0`. +Proof. +Intros x y; Case x. + Intros H; Discriminate H. + Intros p H1; Unfold Zgt. + Pattern 1 ZERO ; Rewrite <- (Zero_mult_right (POS p)). + Rewrite Zcompare_Zmult_compatible; Trivial. +Intros p H; Discriminate H. +Qed. + +(** Simplification of square wrt order *) + +Lemma Zgt_square_simpl: (x, y : Z) `x>=0` -> `y>=0` -> `x*x>y*y` -> `x>y`. +Proof. +Intros x y H0 H1 H2. +Case (dec_Zlt y x). +Intro; Apply Zlt_gt; Trivial. +Intros H3; Cut (Zge y x). +Intros H. +Elim Zgt_not_le with 1 := H2. +Apply Zge_le. +Apply Zge_Zmult_pos_compat; Auto. +Apply not_Zlt; Trivial. +Qed. + +Lemma Zlt_square_simpl: (x,y:Z) `0<=x` -> `0<=y` -> `y*y<x*x` -> `y<x`. +Proof. +Intros x y H0 H1 H2. +Apply Zgt_lt. +Apply Zgt_square_simpl; Try Apply Zle_ge; Try Apply Zlt_gt; Assumption. +Qed. + +(** Equivalence between inequalities *) + +Lemma Zle_plus_swap : (x,y,z:Z) `x+z<=y` <-> `x<=y-z`. +Proof. + Intros x y z; Intros. Split. Intro. Rewrite <- (Zero_right x). Rewrite <- (Zplus_inverse_r z). + Rewrite Zplus_assoc_l. Exact (Zle_reg_r ? ? ? H). + Intro. Rewrite <- (Zero_right y). Rewrite <- (Zplus_inverse_l z). Rewrite Zplus_assoc_l. + Apply Zle_reg_r. Assumption. +Qed. + +Lemma Zlt_plus_swap : (x,y,z:Z) `x+z<y` <-> `x<y-z`. +Proof. + Intros x y z; Intros. Split. Intro. Unfold Zminus. Rewrite Zplus_sym. Rewrite <- (Zero_left x). + Rewrite <- (Zplus_inverse_l z). Rewrite Zplus_assoc_r. Apply Zlt_reg_l. Rewrite Zplus_sym. + Assumption. + Intro. Rewrite Zplus_sym. Rewrite <- (Zero_left y). Rewrite <- (Zplus_inverse_r z). + Rewrite Zplus_assoc_r. Apply Zlt_reg_l. Rewrite Zplus_sym. Assumption. +Qed. + +Lemma Zeq_plus_swap : (x,y,z:Z)`x+z=y` <-> `x=y-z`. +Proof. +Intros x y z; Intros. Split. Intro. Apply Zplus_minus. Symmetry. Rewrite Zplus_sym. + Assumption. +Intro. Rewrite H. Unfold Zminus. Rewrite Zplus_assoc_r. + Rewrite Zplus_inverse_l. Apply Zero_right. +Qed. + +Lemma Zlt_minus : (n,m:Z)`0<m`->`n-m<n`. +Proof. +Intros n m H; Apply Zsimpl_lt_plus_l with p:=m; Rewrite Zle_plus_minus; +Pattern 1 n ;Rewrite <- (Zero_right n); Rewrite (Zplus_sym m n); +Apply Zlt_reg_l; Assumption. +Qed. + +Lemma Zlt_O_minus_lt : (n,m:Z)`0<n-m`->`m<n`. +Proof. +Intros n m H; Apply Zsimpl_lt_plus_l with p:=(Zopp m); Rewrite Zplus_inverse_l; +Rewrite Zplus_sym;Exact H. +Qed. diff --git a/theories7/ZArith/Zpower.v b/theories7/ZArith/Zpower.v new file mode 100644 index 00000000..97c2b3c9 --- /dev/null +++ b/theories7/ZArith/Zpower.v @@ -0,0 +1,394 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zpower.v,v 1.2.2.1 2004/07/16 19:31:44 herbelin Exp $ i*) + +Require ZArith_base. +Require Omega. +Require Zcomplements. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +Section section1. + +(** [Zpower_nat z n] is the n-th power of [z] when [n] is an unary + integer (type [nat]) and [z] a signed integer (type [Z]) *) + +Definition Zpower_nat := + [z:Z][n:nat] (iter_nat n Z ([x:Z]` z * x `) `1`). + +(** [Zpower_nat_is_exp] says [Zpower_nat] is a morphism for + [plus : nat->nat] and [Zmult : Z->Z] *) + +Lemma Zpower_nat_is_exp : + (n,m:nat)(z:Z) + `(Zpower_nat z (plus n m)) = (Zpower_nat z n)*(Zpower_nat z m)`. + +Intros; Elim n; +[ Simpl; Elim (Zpower_nat z m); Auto with zarith +| Unfold Zpower_nat; Intros; Simpl; Rewrite H; + Apply Zmult_assoc]. +Qed. + +(** [Zpower_pos z n] is the n-th power of [z] when [n] is an binary + integer (type [positive]) and [z] a signed integer (type [Z]) *) + +Definition Zpower_pos := + [z:Z][n:positive] (iter_pos n Z ([x:Z]`z * x`) `1`). + +(** This theorem shows that powers of unary and binary integers + are the same thing, modulo the function convert : [positive -> nat] *) + +Theorem Zpower_pos_nat : + (z:Z)(p:positive)(Zpower_pos z p) = (Zpower_nat z (convert p)). + +Intros; Unfold Zpower_pos; Unfold Zpower_nat; Apply iter_convert. +Qed. + +(** Using the theorem [Zpower_pos_nat] and the lemma [Zpower_nat_is_exp] we + deduce that the function [[n:positive](Zpower_pos z n)] is a morphism + for [add : positive->positive] and [Zmult : Z->Z] *) + +Theorem Zpower_pos_is_exp : + (n,m:positive)(z:Z) + ` (Zpower_pos z (add n m)) = (Zpower_pos z n)*(Zpower_pos z m)`. + +Intros. +Rewrite -> (Zpower_pos_nat z n). +Rewrite -> (Zpower_pos_nat z m). +Rewrite -> (Zpower_pos_nat z (add n m)). +Rewrite -> (convert_add n m). +Apply Zpower_nat_is_exp. +Qed. + +Definition Zpower := + [x,y:Z]Cases y of + (POS p) => (Zpower_pos x p) + | ZERO => `1` + | (NEG p) => `0` + end. + +V8Infix "^" Zpower : Z_scope. + +Hints Immediate Zpower_nat_is_exp : zarith. +Hints Immediate Zpower_pos_is_exp : zarith. +Hints Unfold Zpower_pos : zarith. +Hints Unfold Zpower_nat : zarith. + +Lemma Zpower_exp : (x:Z)(n,m:Z) + `n >= 0` -> `m >= 0` -> `(Zpower x (n+m))=(Zpower x n)*(Zpower x m)`. +NewDestruct n; NewDestruct m; Auto with zarith. +Simpl; Intros; Apply Zred_factor0. +Simpl; Auto with zarith. +Intros; Compute in H0; Absurd INFERIEUR=INFERIEUR; Auto with zarith. +Intros; Compute in H0; Absurd INFERIEUR=INFERIEUR; Auto with zarith. +Qed. + +End section1. + +(* Exporting notation "^" *) + +V8Infix "^" Zpower : Z_scope. + +Hints Immediate Zpower_nat_is_exp : zarith. +Hints Immediate Zpower_pos_is_exp : zarith. +Hints Unfold Zpower_pos : zarith. +Hints Unfold Zpower_nat : zarith. + +Section Powers_of_2. + +(** For the powers of two, that will be widely used, a more direct + calculus is possible. We will also prove some properties such + as [(x:positive) x < 2^x] that are true for all integers bigger + than 2 but more difficult to prove and useless. *) + +(** [shift n m] computes [2^n * m], or [m] shifted by [n] positions *) + +Definition shift_nat := + [n:nat][z:positive](iter_nat n positive xO z). +Definition shift_pos := + [n:positive][z:positive](iter_pos n positive xO z). +Definition shift := + [n:Z][z:positive] + Cases n of + ZERO => z + | (POS p) => (iter_pos p positive xO z) + | (NEG p) => z + end. + +Definition two_power_nat := [n:nat] (POS (shift_nat n xH)). +Definition two_power_pos := [x:positive] (POS (shift_pos x xH)). + +Lemma two_power_nat_S : + (n:nat)` (two_power_nat (S n)) = 2*(two_power_nat n)`. +Intro; Simpl; Apply refl_equal. +Qed. + +Lemma shift_nat_plus : + (n,m:nat)(x:positive) + (shift_nat (plus n m) x)=(shift_nat n (shift_nat m x)). + +Intros; Unfold shift_nat; Apply iter_nat_plus. +Qed. + +Theorem shift_nat_correct : + (n:nat)(x:positive)(POS (shift_nat n x))=`(Zpower_nat 2 n)*(POS x)`. + +Unfold shift_nat; Induction n; +[ Simpl; Trivial with zarith +| Intros; Replace (Zpower_nat `2` (S n0)) with `2 * (Zpower_nat 2 n0)`; +[ Rewrite <- Zmult_assoc; Rewrite <- (H x); Simpl; Reflexivity +| Auto with zarith ] +]. +Qed. + +Theorem two_power_nat_correct : + (n:nat)(two_power_nat n)=(Zpower_nat `2` n). + +Intro n. +Unfold two_power_nat. +Rewrite -> (shift_nat_correct n). +Omega. +Qed. + +(** Second we show that [two_power_pos] and [two_power_nat] are the same *) +Lemma shift_pos_nat : (p:positive)(x:positive) + (shift_pos p x)=(shift_nat (convert p) x). + +Unfold shift_pos. +Unfold shift_nat. +Intros; Apply iter_convert. +Qed. + +Lemma two_power_pos_nat : + (p:positive) (two_power_pos p)=(two_power_nat (convert p)). + +Intro; Unfold two_power_pos; Unfold two_power_nat. +Apply f_equal with f:=POS. +Apply shift_pos_nat. +Qed. + +(** Then we deduce that [two_power_pos] is also correct *) + +Theorem shift_pos_correct : + (p,x:positive) ` (POS (shift_pos p x)) = (Zpower_pos 2 p) * (POS x)`. + +Intros. +Rewrite -> (shift_pos_nat p x). +Rewrite -> (Zpower_pos_nat `2` p). +Apply shift_nat_correct. +Qed. + +Theorem two_power_pos_correct : + (x:positive) (two_power_pos x)=(Zpower_pos `2` x). + +Intro. +Rewrite -> two_power_pos_nat. +Rewrite -> Zpower_pos_nat. +Apply two_power_nat_correct. +Qed. + +(** Some consequences *) + +Theorem two_power_pos_is_exp : + (x,y:positive) (two_power_pos (add x y)) + =(Zmult (two_power_pos x) (two_power_pos y)). +Intros. +Rewrite -> (two_power_pos_correct (add x y)). +Rewrite -> (two_power_pos_correct x). +Rewrite -> (two_power_pos_correct y). +Apply Zpower_pos_is_exp. +Qed. + +(** The exponentiation [z -> 2^z] for [z] a signed integer. + For convenience, we assume that [2^z = 0] for all [z < 0] + We could also define a inductive type [Log_result] with + 3 contructors [ Zero | Pos positive -> | minus_infty] + but it's more complexe and not so useful. *) + +Definition two_p := + [x:Z]Cases x of + ZERO => `1` + | (POS y) => (two_power_pos y) + | (NEG y) => `0` + end. + +Theorem two_p_is_exp : + (x,y:Z) ` 0 <= x` -> ` 0 <= y` -> + ` (two_p (x+y)) = (two_p x)*(two_p y)`. +Induction x; +[ Induction y; Simpl; Auto with zarith +| Induction y; + [ Unfold two_p; Rewrite -> (Zmult_sym (two_power_pos p) `1`); + Rewrite -> (Zmult_one (two_power_pos p)); Auto with zarith + | Unfold Zplus; Unfold two_p; + Intros; Apply two_power_pos_is_exp + | Intros; Unfold Zle in H0; Unfold Zcompare in H0; + Absurd SUPERIEUR=SUPERIEUR; Trivial with zarith + ] +| Induction y; + [ Simpl; Auto with zarith + | Intros; Unfold Zle in H; Unfold Zcompare in H; + Absurd (SUPERIEUR=SUPERIEUR); Trivial with zarith + | Intros; Unfold Zle in H; Unfold Zcompare in H; + Absurd (SUPERIEUR=SUPERIEUR); Trivial with zarith + ] +]. +Qed. + +Lemma two_p_gt_ZERO : (x:Z) ` 0 <= x` -> ` (two_p x) > 0`. +Induction x; Intros; +[ Simpl; Omega +| Simpl; Unfold two_power_pos; Apply POS_gt_ZERO +| Absurd ` 0 <= (NEG p)`; + [ Simpl; Unfold Zle; Unfold Zcompare; + Do 2 Unfold not; Auto with zarith + | Assumption ] +]. +Qed. + +Lemma two_p_S : (x:Z) ` 0 <= x` -> + `(two_p (Zs x)) = 2 * (two_p x)`. +Intros; Unfold Zs. +Rewrite (two_p_is_exp x `1` H (ZERO_le_POS xH)). +Apply Zmult_sym. +Qed. + +Lemma two_p_pred : + (x:Z)` 0 <= x` -> ` (two_p (Zpred x)) < (two_p x)`. +Intros; Apply natlike_ind +with P:=[x:Z]` (two_p (Zpred x)) < (two_p x)`; +[ Simpl; Unfold Zlt; Auto with zarith +| Intros; Elim (Zle_lt_or_eq `0` x0 H0); + [ Intros; + Replace (two_p (Zpred (Zs x0))) + with (two_p (Zs (Zpred x0))); + [ Rewrite -> (two_p_S (Zpred x0)); + [ Rewrite -> (two_p_S x0); + [ Omega + | Assumption] + | Apply Zlt_ZERO_pred_le_ZERO; Assumption] + | Rewrite <- (Zs_pred x0); Rewrite <- (Zpred_Sn x0); Trivial with zarith] + | Intro Hx0; Rewrite <- Hx0; Simpl; Unfold Zlt; Auto with zarith] +| Assumption]. +Qed. + +Lemma Zlt_lt_double : (x,y:Z) ` 0 <= x < y` -> ` x < 2*y`. +Intros; Omega. Qed. + +End Powers_of_2. + +Hints Resolve two_p_gt_ZERO : zarith. +Hints Immediate two_p_pred two_p_S : zarith. + +Section power_div_with_rest. + +(** Division by a power of two. + To [n:Z] and [p:positive], [q],[r] are associated such that + [n = 2^p.q + r] and [0 <= r < 2^p] *) + +(** Invariant: [d*q + r = d'*q + r /\ d' = 2*d /\ 0<= r < d /\ 0 <= r' < d'] *) +Definition Zdiv_rest_aux := + [qrd:(Z*Z)*Z] + let (qr,d)=qrd in let (q,r)=qr in + (Cases q of + ZERO => ` (0, r)` + | (POS xH) => ` (0, d + r)` + | (POS (xI n)) => ` ((POS n), d + r)` + | (POS (xO n)) => ` ((POS n), r)` + | (NEG xH) => ` (-1, d + r)` + | (NEG (xI n)) => ` ((NEG n) - 1, d + r)` + | (NEG (xO n)) => ` ((NEG n), r)` + end, ` 2*d`). + +Definition Zdiv_rest := + [x:Z][p:positive]let (qr,d)=(iter_pos p ? Zdiv_rest_aux ((x,`0`),`1`)) in qr. + +Lemma Zdiv_rest_correct1 : + (x:Z)(p:positive) + let (qr,d)=(iter_pos p ? Zdiv_rest_aux ((x,`0`),`1`)) in d=(two_power_pos p). + +Intros x p; +Rewrite (iter_convert p ? Zdiv_rest_aux ((x,`0`),`1`)); +Rewrite (two_power_pos_nat p); +Elim (convert p); Simpl; +[ Trivial with zarith +| Intro n; Rewrite (two_power_nat_S n); + Unfold 2 Zdiv_rest_aux; + Elim (iter_nat n (Z*Z)*Z Zdiv_rest_aux ((x,`0`),`1`)); + NewDestruct a; Intros; Apply f_equal with f:=[z:Z]`2*z`; Assumption ]. +Qed. + +Lemma Zdiv_rest_correct2 : + (x:Z)(p:positive) + let (qr,d)=(iter_pos p ? Zdiv_rest_aux ((x,`0`),`1`)) in + let (q,r)=qr in + ` x=q*d + r` /\ ` 0 <= r < d`. + +Intros; Apply iter_pos_invariant with + f:=Zdiv_rest_aux + Inv:=[qrd:(Z*Z)*Z]let (qr,d)=qrd in let (q,r)=qr in + ` x=q*d + r` /\ ` 0 <= r < d`; +[ Intro x0; Elim x0; Intro y0; Elim y0; + Intros q r d; Unfold Zdiv_rest_aux; + Elim q; + [ Omega + | NewDestruct p0; + [ Rewrite POS_xI; Intro; Elim H; Intros; Split; + [ Rewrite H0; Rewrite Zplus_assoc; + Rewrite Zmult_plus_distr_l; + Rewrite Zmult_1_n; Rewrite Zmult_assoc; + Rewrite (Zmult_sym (POS p0) `2`); Apply refl_equal + | Omega ] + | Rewrite POS_xO; Intro; Elim H; Intros; Split; + [ Rewrite H0; + Rewrite Zmult_assoc; Rewrite (Zmult_sym (POS p0) `2`); + Apply refl_equal + | Omega ] + | Omega ] + | NewDestruct p0; + [ Rewrite NEG_xI; Unfold Zminus; Intro; Elim H; Intros; Split; + [ Rewrite H0; Rewrite Zplus_assoc; + Apply f_equal with f:=[z:Z]`z+r`; + Do 2 (Rewrite Zmult_plus_distr_l); + Rewrite Zmult_assoc; + Rewrite (Zmult_sym (NEG p0) `2`); + Rewrite <- Zplus_assoc; + Apply f_equal with f:=[z:Z]`2 * (NEG p0) * d + z`; + Omega + | Omega ] + | Rewrite NEG_xO; Unfold Zminus; Intro; Elim H; Intros; Split; + [ Rewrite H0; + Rewrite Zmult_assoc; Rewrite (Zmult_sym (NEG p0) `2`); + Apply refl_equal + | Omega ] + | Omega ] ] +| Omega]. +Qed. + +Inductive Set Zdiv_rest_proofs[x:Z; p:positive] := + Zdiv_rest_proof : (q:Z)(r:Z) + `x = q * (two_power_pos p) + r` + -> `0 <= r` + -> `r < (two_power_pos p)` + -> (Zdiv_rest_proofs x p). + +Lemma Zdiv_rest_correct : + (x:Z)(p:positive)(Zdiv_rest_proofs x p). +Intros x p. +Generalize (Zdiv_rest_correct1 x p); Generalize (Zdiv_rest_correct2 x p). +Elim (iter_pos p (Z*Z)*Z Zdiv_rest_aux ((x,`0`),`1`)). +Induction a. +Intros. +Elim H; Intros H1 H2; Clear H. +Rewrite -> H0 in H1; Rewrite -> H0 in H2; +Elim H2; Intros; +Apply Zdiv_rest_proof with q:=a0 r:=b; Assumption. +Qed. + +End power_div_with_rest. diff --git a/theories7/ZArith/Zsqrt.v b/theories7/ZArith/Zsqrt.v new file mode 100644 index 00000000..72a2e9cf --- /dev/null +++ b/theories7/ZArith/Zsqrt.v @@ -0,0 +1,136 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(* $Id: Zsqrt.v,v 1.1.2.1 2004/07/16 19:31:44 herbelin Exp $ *) + +Require Omega. +Require Export ZArith_base. +Require Export ZArithRing. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +(**********************************************************************) +(** Definition and properties of square root on Z *) + +(** The following tactic replaces all instances of (POS (xI ...)) by + `2*(POS ...)+1` , but only when ... is not made only with xO, XI, or xH. *) +Tactic Definition compute_POS := + Match Context With + | [|- [(POS (xI ?1))]] -> + (Match ?1 With + | [[xH]] -> Fail + | _ -> Rewrite (POS_xI ?1)) + | [|- [(POS (xO ?1))]] -> + (Match ?1 With + | [[xH]] -> Fail + | _ -> Rewrite (POS_xO ?1)). + +Inductive sqrt_data [n : Z] : Set := + c_sqrt: (s, r :Z)`n=s*s+r`->`0<=r<=2*s`->(sqrt_data n) . + +Definition sqrtrempos: (p : positive) (sqrt_data (POS p)). +Refine (Fix sqrtrempos { + sqrtrempos [p : positive] : (sqrt_data (POS p)) := + <[p : ?] (sqrt_data (POS p))> Cases p of + xH => (c_sqrt `1` `1` `0` ? ?) + | (xO xH) => (c_sqrt `2` `1` `1` ? ?) + | (xI xH) => (c_sqrt `3` `1` `2` ? ?) + | (xO (xO p')) => + Cases (sqrtrempos p') of + (c_sqrt s' r' Heq Hint) => + Cases (Z_le_gt_dec `4*s'+1` `4*r'`) of + (left Hle) => + (c_sqrt (POS (xO (xO p'))) `2*s'+1` `4*r'-(4*s'+1)` ? ?) + | (right Hgt) => + (c_sqrt (POS (xO (xO p'))) `2*s'` `4*r'` ? ?) + end + end + | (xO (xI p')) => + Cases (sqrtrempos p') of + (c_sqrt s' r' Heq Hint) => + Cases + (Z_le_gt_dec `4*s'+1` `4*r'+2`) of + (left Hle) => + (c_sqrt + (POS (xO (xI p'))) `2*s'+1` `4*r'+2-(4*s'+1)` ? ?) + | (right Hgt) => + (c_sqrt (POS (xO (xI p'))) `2*s'` `4*r'+2` ? ?) + end + end + | (xI (xO p')) => + Cases (sqrtrempos p') of + (c_sqrt s' r' Heq Hint) => + Cases + (Z_le_gt_dec `4*s'+1` `4*r'+1`) of + (left Hle) => + (c_sqrt + (POS (xI (xO p'))) `2*s'+1` `4*r'+1-(4*s'+1)` ? ?) + | (right Hgt) => + (c_sqrt (POS (xI (xO p'))) `2*s'` `4*r'+1` ? ?) + end + end + | (xI (xI p')) => + Cases (sqrtrempos p') of + (c_sqrt s' r' Heq Hint) => + Cases + (Z_le_gt_dec `4*s'+1` `4*r'+3`) of + (left Hle) => + (c_sqrt + (POS (xI (xI p'))) `2*s'+1` `4*r'+3-(4*s'+1)` ? ?) + | (right Hgt) => + (c_sqrt (POS (xI (xI p'))) `2*s'` `4*r'+3` ? ?) + end + end + end + }); Clear sqrtrempos; Repeat compute_POS; + Try (Try Rewrite Heq; Ring; Fail); Try Omega. +Defined. + +(** Define with integer input, but with a strong (readable) specification. *) +Definition Zsqrt : (x:Z)`0<=x`->{s:Z & {r:Z | x=`s*s+r` /\ `s*s<=x<(s+1)*(s+1)`}}. +Refine [x] + <[x:Z]`0<=x`->{s:Z & {r:Z | x=`s*s+r` /\ `s*s<=x<(s+1)*(s+1)`}}>Cases x of + (POS p) => [h]Cases (sqrtrempos p) of + (c_sqrt s r Heq Hint) => + (existS ? [s:Z]{r:Z | `(POS p)=s*s+r` /\ + `s*s<=(POS p)<(s+1)*(s+1)`} + s + (exist Z [r:Z]((POS p)=`s*s+r` /\ `s*s<=(POS p)<(s+1)*(s+1)`) + r ?)) + end + | (NEG p) => [h](False_rec + {s:Z & {r:Z | + (NEG p)=`s*s+r` /\ `s*s<=(NEG p)<(s+1)*(s+1)`}} + (h (refl_equal ? SUPERIEUR))) + | ZERO => [h](existS ? [s:Z]{r:Z | `0=s*s+r` /\ `s*s<=0<(s+1)*(s+1)`} + `0` (exist Z [r:Z](`0=0*0+r`/\`0*0<=0<(0+1)*(0+1)`) + `0` ?)) + end;Try Omega. +Split;[Omega|Rewrite Heq;Ring `(s+1)*(s+1)`;Omega]. +Defined. + +(** Define a function of type Z->Z that computes the integer square root, + but only for positive numbers, and 0 for others. *) +Definition Zsqrt_plain : Z->Z := + [x]Cases x of + (POS p)=>Cases (Zsqrt (POS p) (ZERO_le_POS p)) of (existS s _) => s end + |(NEG p)=>`0` + |ZERO=>`0` + end. + +(** A basic theorem about Zsqrt_plain *) +Theorem Zsqrt_interval :(x:Z)`0<=x`-> + `(Zsqrt_plain x)*(Zsqrt_plain x)<= x < ((Zsqrt_plain x)+1)*((Zsqrt_plain x)+1)`. +Intros x;Case x. +Unfold Zsqrt_plain;Omega. +Intros p;Unfold Zsqrt_plain;Case (Zsqrt (POS p) (ZERO_le_POS p)). +Intros s (r,(Heq,Hint)) Hle;Assumption. +Intros p Hle;Elim Hle;Auto. +Qed. + + diff --git a/theories7/ZArith/Zsyntax.v b/theories7/ZArith/Zsyntax.v new file mode 100644 index 00000000..3c7f3a57 --- /dev/null +++ b/theories7/ZArith/Zsyntax.v @@ -0,0 +1,278 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zsyntax.v,v 1.1.2.1 2004/07/16 19:31:44 herbelin Exp $ i*) + +Require Export BinInt. + +V7only[ + +Grammar znatural ident := + nat_id [ prim:var($id) ] -> [$id] + +with number := + +with negnumber := + +with formula : constr := + form_expr [ expr($p) ] -> [$p] +(*| form_eq [ expr($p) "=" expr($c) ] -> [ (eq Z $p $c) ]*) +| form_eq [ expr($p) "=" expr($c) ] -> [ (Coq.Init.Logic.eq ? $p $c) ] +| form_le [ expr($p) "<=" expr($c) ] -> [ (Zle $p $c) ] +| form_lt [ expr($p) "<" expr($c) ] -> [ (Zlt $p $c) ] +| form_ge [ expr($p) ">=" expr($c) ] -> [ (Zge $p $c) ] +| form_gt [ expr($p) ">" expr($c) ] -> [ (Zgt $p $c) ] +(*| form_eq_eq [ expr($p) "=" expr($c) "=" expr($c1) ] + -> [ (eq Z $p $c)/\(eq Z $c $c1) ]*) +| form_eq_eq [ expr($p) "=" expr($c) "=" expr($c1) ] + -> [ (Coq.Init.Logic.eq ? $p $c)/\(Coq.Init.Logic.eq ? $c $c1) ] +| form_le_le [ expr($p) "<=" expr($c) "<=" expr($c1) ] + -> [ (Zle $p $c)/\(Zle $c $c1) ] +| form_le_lt [ expr($p) "<=" expr($c) "<" expr($c1) ] + -> [ (Zle $p $c)/\(Zlt $c $c1) ] +| form_lt_le [ expr($p) "<" expr($c) "<=" expr($c1) ] + -> [ (Zlt $p $c)/\(Zle $c $c1) ] +| form_lt_lt [ expr($p) "<" expr($c) "<" expr($c1) ] + -> [ (Zlt $p $c)/\(Zlt $c $c1) ] +(*| form_neq [ expr($p) "<>" expr($c) ] -> [ ~(Coq.Init.Logic.eq Z $p $c) ]*) +| form_neq [ expr($p) "<>" expr($c) ] -> [ ~(Coq.Init.Logic.eq ? $p $c) ] +| form_comp [ expr($p) "?=" expr($c) ] -> [ (Zcompare $p $c) ] + +with expr : constr := + expr_plus [ expr($p) "+" expr($c) ] -> [ (Zplus $p $c) ] +| expr_minus [ expr($p) "-" expr($c) ] -> [ (Zminus $p $c) ] +| expr2 [ expr2($e) ] -> [$e] + +with expr2 : constr := + expr_mult [ expr2($p) "*" expr2($c) ] -> [ (Zmult $p $c) ] +| expr1 [ expr1($e) ] -> [$e] + +with expr1 : constr := + expr_abs [ "|" expr($c) "|" ] -> [ (Zabs $c) ] +| expr0 [ expr0($e) ] -> [$e] + +with expr0 : constr := + expr_id [ constr:global($c) ] -> [ $c ] +| expr_com [ "[" constr:constr($c) "]" ] -> [$c] +| expr_appl [ "(" application($a) ")" ] -> [$a] +| expr_num [ number($s) ] -> [$s ] +| expr_negnum [ "-" negnumber($n) ] -> [ $n ] +| expr_inv [ "-" expr0($c) ] -> [ (Zopp $c) ] +| expr_meta [ zmeta($m) ] -> [ $m ] + +with zmeta := +| rimpl [ "?" ] -> [ ? ] +| rmeta0 [ "?" "0" ] -> [ ?0 ] +| rmeta1 [ "?" "1" ] -> [ ?1 ] +| rmeta2 [ "?" "2" ] -> [ ?2 ] +| rmeta3 [ "?" "3" ] -> [ ?3 ] +| rmeta4 [ "?" "4" ] -> [ ?4 ] +| rmeta5 [ "?" "5" ] -> [ ?5 ] + +with application : constr := + apply [ application($p) expr($c1) ] -> [ ($p $c1) ] +| apply_inject_nat [ "inject_nat" constr:constr($c1) ] -> [ (inject_nat $c1) ] +| pair [ expr($p) "," expr($c) ] -> [ ($p, $c) ] +| appl0 [ expr($a) ] -> [$a] +. + +Grammar constr constr0 := + z_in_com [ "`" znatural:formula($c) "`" ] -> [$c]. + +Grammar constr pattern := + z_in_pattern [ "`" prim:bigint($c) "`" ] -> [ 'Z: $c ' ]. + +(* The symbols "`" "`" must be printed just once at the top of the expressions, + to avoid printings like |``x` + `y`` < `45`| + for |x + y < 45|. + So when a Z-expression is to be printed, its sub-expresssions are + enclosed into an ast (ZEXPR \$subexpr), which is printed like \$subexpr + but without symbols "`" "`" around. + + There is just one problem: NEG and Zopp have the same printing rules. + If Zopp is opaque, we may not be able to solve a goal like + ` -5 = -5 ` by reflexivity. (In fact, this precise Goal is solved + by the Reflexivity tactic, but more complex problems may arise + + SOLUTION : Print (Zopp 5) for constants and -x for variables *) + +Syntax constr + level 0: + Zle [ (Zle $n1 $n2) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "<= " (ZEXPR $n2) "`"]] + | Zlt [ (Zlt $n1 $n2) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "< " (ZEXPR $n2) "`" ]] + | Zge [ (Zge $n1 $n2) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] ">= " (ZEXPR $n2) "`" ]] + | Zgt [ (Zgt $n1 $n2) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "> " (ZEXPR $n2) "`" ]] + | Zcompare [<<(Zcompare $n1 $n2)>>] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "?= " (ZEXPR $n2) "`" ]] + | Zeq [ (eq Z $n1 $n2) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "= " (ZEXPR $n2)"`"]] + | Zneq [ ~(eq Z $n1 $n2) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "<> " (ZEXPR $n2) "`"]] + | Zle_Zle [ (Zle $n1 $n2)/\(Zle $n2 $n3) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "<= " (ZEXPR $n2) + [1 0] "<= " (ZEXPR $n3) "`"]] + | Zle_Zlt [ (Zle $n1 $n2)/\(Zlt $n2 $n3) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "<= " (ZEXPR $n2) + [1 0] "< " (ZEXPR $n3) "`"]] + | Zlt_Zle [ (Zlt $n1 $n2)/\(Zle $n2 $n3) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "< " (ZEXPR $n2) + [1 0] "<= " (ZEXPR $n3) "`"]] + | Zlt_Zlt [ (Zlt $n1 $n2)/\(Zlt $n2 $n3) ] -> + [[<hov 0> "`" (ZEXPR $n1) [1 0] "< " (ZEXPR $n2) + [1 0] "< " (ZEXPR $n3) "`"]] + | ZZero_v7 [ ZERO ] -> [ "`0`" ] + | ZPos_v7 [ (POS $r) ] -> [$r:"positive_printer":9] + | ZNeg_v7 [ (NEG $r) ] -> [$r:"negative_printer":9] + ; + + level 7: + Zplus [ (Zplus $n1 $n2) ] + -> [ [<hov 0> "`" (ZEXPR $n1):E "+" [0 0] (ZEXPR $n2):L "`"] ] + | Zminus [ (Zminus $n1 $n2) ] + -> [ [<hov 0> "`" (ZEXPR $n1):E "-" [0 0] (ZEXPR $n2):L "`"] ] + ; + + level 6: + Zmult [ (Zmult $n1 $n2) ] + -> [ [<hov 0> "`" (ZEXPR $n1):E "*" [0 0] (ZEXPR $n2):L "`"] ] + ; + + level 8: + Zopp [ (Zopp $n1) ] -> [ [<hov 0> "`" "-" (ZEXPR $n1):E "`"] ] + | Zopp_POS [ (Zopp (POS $r)) ] -> + [ [<hov 0> "`(" "Zopp" [1 0] $r:"positive_printer_inside" ")`"] ] + | Zopp_ZERO [ (Zopp ZERO) ] -> [ [<hov 0> "`(" "Zopp" [1 0] "0" ")`"] ] + | Zopp_NEG [ (Zopp (NEG $r)) ] -> + [ [<hov 0> "`(" "Zopp" [1 0] "(" $r:"negative_printer_inside" "))`"] ] + ; + + level 4: + Zabs [ (Zabs $n1) ] -> [ [<hov 0> "`|" (ZEXPR $n1):E "|`"] ] + ; + + level 0: + escape_inside [ << (ZEXPR $r) >> ] -> [ "[" $r:E "]" ] + ; + + level 4: + Zappl_inside [ << (ZEXPR (APPLIST $h ($LIST $t))) >> ] + -> [ [<hov 0> "("(ZEXPR $h):E [1 0] (ZAPPLINSIDETAIL ($LIST $t)):E ")"] ] + | Zappl_inject_nat [ << (ZEXPR (APPLIST <<inject_nat>> $n)) >> ] + -> [ [<hov 0> "(inject_nat" [1 1] $n:L ")"] ] + | Zappl_inside_tail [ << (ZAPPLINSIDETAIL $h ($LIST $t)) >> ] + -> [(ZEXPR $h):E [1 0] (ZAPPLINSIDETAIL ($LIST $t)):E] + | Zappl_inside_one [ << (ZAPPLINSIDETAIL $e) >> ] ->[(ZEXPR $e):E] + | pair_inside [ << (ZEXPR <<(pair $s1 $s2 $z1 $z2)>>) >> ] + -> [ [<hov 0> "("(ZEXPR $z1):E "," [1 0] (ZEXPR $z2):E ")"] ] + ; + + level 3: + var_inside [ << (ZEXPR ($VAR $i)) >> ] -> [$i] + | secvar_inside [ << (ZEXPR (SECVAR $i)) >> ] -> [(SECVAR $i)] + | const_inside [ << (ZEXPR (CONST $c)) >> ] -> [(CONST $c)] + | mutind_inside [ << (ZEXPR (MUTIND $i $n)) >> ] + -> [(MUTIND $i $n)] + | mutconstruct_inside [ << (ZEXPR (MUTCONSTRUCT $c1 $c2 $c3)) >> ] + -> [ (MUTCONSTRUCT $c1 $c2 $c3) ] + + | O_inside [ << (ZEXPR << O >>) >> ] -> [ "O" ] (* To shunt Arith printer *) + + (* Added by JCF, 9/3/98; updated HH, 11/9/01 *) + | implicit_head_inside [ << (ZEXPR (APPLISTEXPL ($LIST $c))) >> ] + -> [ (APPLIST ($LIST $c)) ] + | implicit_arg_inside [ << (ZEXPR (EXPL "!" $n $c)) >> ] -> [ ] + + ; + + level 7: + Zplus_inside + [ << (ZEXPR <<(Zplus $n1 $n2)>>) >> ] + -> [ (ZEXPR $n1):E "+" [0 0] (ZEXPR $n2):L ] + | Zminus_inside + [ << (ZEXPR <<(Zminus $n1 $n2)>>) >> ] + -> [ (ZEXPR $n1):E "-" [0 0] (ZEXPR $n2):L ] + ; + + level 6: + Zmult_inside + [ << (ZEXPR <<(Zmult $n1 $n2)>>) >> ] + -> [ (ZEXPR $n1):E "*" [0 0] (ZEXPR $n2):L ] + ; + + level 5: + Zopp_inside [ << (ZEXPR <<(Zopp $n1)>>) >> ] -> [ "(-" (ZEXPR $n1):E ")" ] + ; + + level 10: + Zopp_POS_inside [ << (ZEXPR <<(Zopp (POS $r))>>) >> ] -> + [ [<hov 0> "Zopp" [1 0] $r:"positive_printer_inside" ] ] + | Zopp_ZERO_inside [ << (ZEXPR <<(Zopp ZERO)>>) >> ] -> + [ [<hov 0> "Zopp" [1 0] "0"] ] + | Zopp_NEG_inside [ << (ZEXPR <<(Zopp (NEG $r))>>) >> ] -> + [ [<hov 0> "Zopp" [1 0] $r:"negative_printer_inside" ] ] + ; + + level 4: + Zabs_inside [ << (ZEXPR <<(Zabs $n1)>>) >> ] -> [ "|" (ZEXPR $n1) "|"] + ; + + level 0: + ZZero_inside [ << (ZEXPR <<ZERO>>) >> ] -> ["0"] + | ZPos_inside [ << (ZEXPR <<(POS $p)>>) >>] -> + [$p:"positive_printer_inside":9] + | ZNeg_inside [ << (ZEXPR <<(NEG $p)>>) >>] -> + [$p:"negative_printer_inside":9] +. +]. + +V7only[ +(* For parsing/printing based on scopes *) +Module Z_scope. + +Infix LEFTA 4 "+" Zplus : Z_scope. +Infix LEFTA 4 "-" Zminus : Z_scope. +Infix LEFTA 3 "*" Zmult : Z_scope. +Notation "- x" := (Zopp x) (at level 0): Z_scope V8only. +Infix NONA 5 "<=" Zle : Z_scope. +Infix NONA 5 "<" Zlt : Z_scope. +Infix NONA 5 ">=" Zge : Z_scope. +Infix NONA 5 ">" Zgt : Z_scope. +Infix NONA 5 "?=" Zcompare : Z_scope. +Notation "x <= y <= z" := (Zle x y)/\(Zle y z) + (at level 5, y at level 4):Z_scope + V8only (at level 70, y at next level). +Notation "x <= y < z" := (Zle x y)/\(Zlt y z) + (at level 5, y at level 4):Z_scope + V8only (at level 70, y at next level). +Notation "x < y < z" := (Zlt x y)/\(Zlt y z) + (at level 5, y at level 4):Z_scope + V8only (at level 70, y at next level). +Notation "x < y <= z" := (Zlt x y)/\(Zle y z) + (at level 5, y at level 4):Z_scope + V8only (at level 70, y at next level). +Notation "x = y = z" := x=y/\y=z : Z_scope + V8only (at level 70, y at next level). + +(* Now a polymorphic notation +Notation "x <> y" := ~(eq Z x y) (at level 5, no associativity) : Z_scope. +*) + +(* Notation "| x |" (Zabs x) : Z_scope.(* "|" conflicts with THENS *)*) + +(* Overwrite the printing of "`x = y`" *) +Syntax constr level 0: + Zeq [ (eq Z $n1 $n2) ] -> [[<hov 0> $n1 [1 0] "= " $n2 ]]. + +Open Scope Z_scope. + +End Z_scope. +]. diff --git a/theories7/ZArith/Zwf.v b/theories7/ZArith/Zwf.v new file mode 100644 index 00000000..c2e6ca2a --- /dev/null +++ b/theories7/ZArith/Zwf.v @@ -0,0 +1,96 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(* $Id: Zwf.v,v 1.1.2.1 2004/07/16 19:31:44 herbelin Exp $ *) + +Require ZArith_base. +Require Export Wf_nat. +Require Omega. +V7only [Import Z_scope.]. +Open Local Scope Z_scope. + +(** Well-founded relations on Z. *) + +(** We define the following family of relations on [Z x Z]: + + [x (Zwf c) y] iff [x < y & c <= y] + *) + +Definition Zwf := [c:Z][x,y:Z] `c <= y` /\ `x < y`. + +(** and we prove that [(Zwf c)] is well founded *) + +Section wf_proof. + +Variable c : Z. + +(** The proof of well-foundness is classic: we do the proof by induction + on a measure in nat, which is here [|x-c|] *) + +Local f := [z:Z](absolu (Zminus z c)). + +Lemma Zwf_well_founded : (well_founded Z (Zwf c)). +Red; Intros. +Assert (n:nat)(a:Z)(lt (f a) n)\/(`a<c`) -> (Acc Z (Zwf c) a). +Clear a; Induction n; Intros. +(** n= 0 *) +Case H; Intros. +Case (lt_n_O (f a)); Auto. +Apply Acc_intro; Unfold Zwf; Intros. +Assert False;Omega Orelse Contradiction. +(** inductive case *) +Case H0; Clear H0; Intro; Auto. +Apply Acc_intro; Intros. +Apply H. +Unfold Zwf in H1. +Case (Zle_or_lt c y); Intro; Auto with zarith. +Left. +Red in H0. +Apply lt_le_trans with (f a); Auto with arith. +Unfold f. +Apply absolu_lt; Omega. +Apply (H (S (f a))); Auto. +Save. + +End wf_proof. + +Hints Resolve Zwf_well_founded : datatypes v62. + + +(** We also define the other family of relations: + + [x (Zwf_up c) y] iff [y < x <= c] + *) + +Definition Zwf_up := [c:Z][x,y:Z] `y < x <= c`. + +(** and we prove that [(Zwf_up c)] is well founded *) + +Section wf_proof_up. + +Variable c : Z. + +(** The proof of well-foundness is classic: we do the proof by induction + on a measure in nat, which is here [|c-x|] *) + +Local f := [z:Z](absolu (Zminus c z)). + +Lemma Zwf_up_well_founded : (well_founded Z (Zwf_up c)). +Proof. +Apply well_founded_lt_compat with f:=f. +Unfold Zwf_up f. +Intros. +Apply absolu_lt. +Unfold Zminus. Split. +Apply Zle_left; Intuition. +Apply Zlt_reg_l; Unfold Zlt; Rewrite <- Zcompare_Zopp; Intuition. +Save. + +End wf_proof_up. + +Hints Resolve Zwf_up_well_founded : datatypes v62. diff --git a/theories7/ZArith/auxiliary.v b/theories7/ZArith/auxiliary.v new file mode 100644 index 00000000..8db2c852 --- /dev/null +++ b/theories7/ZArith/auxiliary.v @@ -0,0 +1,219 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: auxiliary.v,v 1.1.2.1 2004/07/16 19:31:44 herbelin Exp $ i*) + +(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) + +Require Export Arith. +Require BinInt. +Require Zorder. +Require Decidable. +Require Peano_dec. +Require Export Compare_dec. + +Open Local Scope Z_scope. + +(**********************************************************************) +(** Moving terms from one side to the other of an inequality *) + +Theorem Zne_left : (x,y:Z) (Zne x y) -> (Zne (Zplus x (Zopp y)) ZERO). +Proof. +Intros x y; Unfold Zne; Unfold not; Intros H1 H2; Apply H1; +Apply Zsimpl_plus_l with (Zopp y); Rewrite Zplus_inverse_l; Rewrite Zplus_sym; +Trivial with arith. +Qed. + +Theorem Zegal_left : (x,y:Z) (x=y) -> (Zplus x (Zopp y)) = ZERO. +Proof. +Intros x y H; +Apply (Zsimpl_plus_l y);Rewrite -> Zplus_permute; +Rewrite -> Zplus_inverse_r;Do 2 Rewrite -> Zero_right;Assumption. +Qed. + +Theorem Zle_left : (x,y:Z) (Zle x y) -> (Zle ZERO (Zplus y (Zopp x))). +Proof. +Intros x y H; Replace ZERO with (Zplus x (Zopp x)). +Apply Zle_reg_r; Trivial. +Apply Zplus_inverse_r. +Qed. + +Theorem Zle_left_rev : (x,y:Z) (Zle ZERO (Zplus y (Zopp x))) + -> (Zle x y). +Proof. +Intros x y H; Apply Zsimpl_le_plus_r with (Zopp x). +Rewrite Zplus_inverse_r; Trivial. +Qed. + +Theorem Zlt_left_rev : (x,y:Z) (Zlt ZERO (Zplus y (Zopp x))) + -> (Zlt x y). +Proof. +Intros x y H; Apply Zsimpl_lt_plus_r with (Zopp x). +Rewrite Zplus_inverse_r; Trivial. +Qed. + +Theorem Zlt_left : + (x,y:Z) (Zlt x y) -> (Zle ZERO (Zplus (Zplus y (NEG xH)) (Zopp x))). +Proof. +Intros x y H; Apply Zle_left; Apply Zle_S_n; +Change (Zle (Zs x) (Zs (Zpred y))); Rewrite <- Zs_pred; Apply Zlt_le_S; +Assumption. +Qed. + +Theorem Zlt_left_lt : + (x,y:Z) (Zlt x y) -> (Zlt ZERO (Zplus y (Zopp x))). +Proof. +Intros x y H; Replace ZERO with (Zplus x (Zopp x)). +Apply Zlt_reg_r; Trivial. +Apply Zplus_inverse_r. +Qed. + +Theorem Zge_left : (x,y:Z) (Zge x y) -> (Zle ZERO (Zplus x (Zopp y))). +Proof. +Intros x y H; Apply Zle_left; Apply Zge_le; Assumption. +Qed. + +Theorem Zgt_left : + (x,y:Z) (Zgt x y) -> (Zle ZERO (Zplus (Zplus x (NEG xH)) (Zopp y))). +Proof. +Intros x y H; Apply Zlt_left; Apply Zgt_lt; Assumption. +Qed. + +Theorem Zgt_left_gt : + (x,y:Z) (Zgt x y) -> (Zgt (Zplus x (Zopp y)) ZERO). +Proof. +Intros x y H; Replace ZERO with (Zplus y (Zopp y)). +Apply Zgt_reg_r; Trivial. +Apply Zplus_inverse_r. +Qed. + +Theorem Zgt_left_rev : (x,y:Z) (Zgt (Zplus x (Zopp y)) ZERO) + -> (Zgt x y). +Proof. +Intros x y H; Apply Zsimpl_gt_plus_r with (Zopp y). +Rewrite Zplus_inverse_r; Trivial. +Qed. + +(**********************************************************************) +(** Factorization lemmas *) + +Theorem Zred_factor0 : (x:Z) x = (Zmult x (POS xH)). +Intro x; Rewrite (Zmult_n_1 x); Reflexivity. +Qed. + +Theorem Zred_factor1 : (x:Z) (Zplus x x) = (Zmult x (POS (xO xH))). +Proof. +Exact Zplus_Zmult_2. +Qed. + +Theorem Zred_factor2 : + (x,y:Z) (Zplus x (Zmult x y)) = (Zmult x (Zplus (POS xH) y)). + +Intros x y; Pattern 1 x ; Rewrite <- (Zmult_n_1 x); +Rewrite <- Zmult_plus_distr_r; Trivial with arith. +Qed. + +Theorem Zred_factor3 : + (x,y:Z) (Zplus (Zmult x y) x) = (Zmult x (Zplus (POS xH) y)). + +Intros x y; Pattern 2 x ; Rewrite <- (Zmult_n_1 x); +Rewrite <- Zmult_plus_distr_r; Rewrite Zplus_sym; Trivial with arith. +Qed. +Theorem Zred_factor4 : + (x,y,z:Z) (Zplus (Zmult x y) (Zmult x z)) = (Zmult x (Zplus y z)). +Intros x y z; Symmetry; Apply Zmult_plus_distr_r. +Qed. + +Theorem Zred_factor5 : (x,y:Z) (Zplus (Zmult x ZERO) y) = y. + +Intros x y; Rewrite <- Zmult_n_O;Auto with arith. +Qed. + +Theorem Zred_factor6 : (x:Z) x = (Zplus x ZERO). + +Intro; Rewrite Zero_right; Trivial with arith. +Qed. + +Theorem Zle_mult_approx: + (x,y,z:Z) (Zgt x ZERO) -> (Zgt z ZERO) -> (Zle ZERO y) -> + (Zle ZERO (Zplus (Zmult y x) z)). + +Intros x y z H1 H2 H3; Apply Zle_trans with m:=(Zmult y x) ; [ + Apply Zle_mult; Assumption +| Pattern 1 (Zmult y x) ; Rewrite <- Zero_right; Apply Zle_reg_l; + Apply Zlt_le_weak; Apply Zgt_lt; Assumption]. +Qed. + +Theorem Zmult_le_approx: + (x,y,z:Z) (Zgt x ZERO) -> (Zgt x z) -> + (Zle ZERO (Zplus (Zmult y x) z)) -> (Zle ZERO y). + +Intros x y z H1 H2 H3; Apply Zlt_n_Sm_le; Apply Zmult_lt with x; [ + Assumption + | Apply Zle_lt_trans with 1:=H3 ; Rewrite <- Zmult_Sm_n; + Apply Zlt_reg_l; Apply Zgt_lt; Assumption]. + +Qed. + +V7only [ +(* Compatibility *) +Require Znat. +Require Zcompare. +Notation neq := neq. +Notation Zne := Zne. +Notation OMEGA2 := Zle_0_plus. +Notation add_un_Zs := add_un_Zs. +Notation inj_S := inj_S. +Notation Zplus_S_n := Zplus_S_n. +Notation inj_plus := inj_plus. +Notation inj_mult := inj_mult. +Notation inj_neq := inj_neq. +Notation inj_le := inj_le. +Notation inj_lt := inj_lt. +Notation inj_gt := inj_gt. +Notation inj_ge := inj_ge. +Notation inj_eq := inj_eq. +Notation intro_Z := intro_Z. +Notation inj_minus1 := inj_minus1. +Notation inj_minus2 := inj_minus2. +Notation dec_eq := dec_eq. +Notation dec_Zne := dec_Zne. +Notation dec_Zle := dec_Zle. +Notation dec_Zgt := dec_Zgt. +Notation dec_Zge := dec_Zge. +Notation dec_Zlt := dec_Zlt. +Notation dec_eq_nat := dec_eq_nat. +Notation not_Zge := not_Zge. +Notation not_Zlt := not_Zlt. +Notation not_Zle := not_Zle. +Notation not_Zgt := not_Zgt. +Notation not_Zeq := not_Zeq. +Notation Zopp_one := Zopp_one. +Notation Zopp_Zmult_r := Zopp_Zmult_r. +Notation Zmult_Zopp_left := Zmult_Zopp_left. +Notation Zopp_Zmult_l := Zopp_Zmult_l. +Notation Zcompare_Zplus_compatible2 := Zcompare_Zplus_compatible2. +Notation Zcompare_Zmult_compatible := Zcompare_Zmult_compatible. +Notation Zmult_eq := Zmult_eq. +Notation Z_eq_mult := Z_eq_mult. +Notation Zmult_le := Zmult_le. +Notation Zle_ZERO_mult := Zle_ZERO_mult. +Notation Zgt_ZERO_mult := Zgt_ZERO_mult. +Notation Zle_mult := Zle_mult. +Notation Zmult_lt := Zmult_lt. +Notation Zmult_gt := Zmult_gt. +Notation Zle_Zmult_pos_right := Zle_Zmult_pos_right. +Notation Zle_Zmult_pos_left := Zle_Zmult_pos_left. +Notation Zge_Zmult_pos_right := Zge_Zmult_pos_right. +Notation Zge_Zmult_pos_left := Zge_Zmult_pos_left. +Notation Zge_Zmult_pos_compat := Zge_Zmult_pos_compat. +Notation Zle_mult_simpl := Zle_mult_simpl. +Notation Zge_mult_simpl := Zge_mult_simpl. +Notation Zgt_mult_simpl := Zgt_mult_simpl. +Notation Zgt_square_simpl := Zgt_square_simpl. +]. diff --git a/theories7/ZArith/fast_integer.v b/theories7/ZArith/fast_integer.v new file mode 100644 index 00000000..7e3fe306 --- /dev/null +++ b/theories7/ZArith/fast_integer.v @@ -0,0 +1,191 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: fast_integer.v,v 1.1.2.1 2004/07/16 19:31:44 herbelin Exp $ i*) + +(***********************************************************) +(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) +(***********************************************************) + +Require BinPos. +Require BinNat. +Require BinInt. +Require Zcompare. +Require Mult. + +V7only [ +(* Defs and ppties on positive, entier and Z, previously in fast_integer *) +(* For v7 compatibility *) +Notation positive := positive. +Notation xO := xO. +Notation xI := xI. +Notation xH := xH. +Notation add_un := add_un. +Notation add := add. +Notation convert := convert. +Notation convert_add_un := convert_add_un. +Notation cvt_carry := cvt_carry. +Notation convert_add := convert_add. +Notation positive_to_nat := positive_to_nat. +Notation anti_convert := anti_convert. +Notation double_moins_un := double_moins_un. +Notation sub_un := sub_un. +Notation positive_mask := positive_mask. +Notation Un_suivi_de_mask := Un_suivi_de_mask. +Notation Zero_suivi_de_mask := Zero_suivi_de_mask. +Notation double_moins_deux := double_moins_deux. +Notation sub_pos := sub_pos. +Notation true_sub := true_sub. +Notation times := times. +Notation relation := relation. +Notation SUPERIEUR := SUPERIEUR. +Notation INFERIEUR := INFERIEUR. +Notation EGAL := EGAL. +Notation Op := Op. +Notation compare := compare. +Notation compare_convert1 := compare_convert1. +Notation compare_convert_EGAL := compare_convert_EGAL. +Notation ZLSI := ZLSI. +Notation ZLIS := ZLIS. +Notation ZLII := ZLII. +Notation ZLSS := ZLSS. +Notation Dcompare := Dcompare. +Notation convert_compare_EGAL := convert_compare_EGAL. +Notation ZL0 := ZL0. +Notation ZL11 := ZL11. +Notation xI_add_un_xO := xI_add_un_xO. +Notation is_double_moins_un := is_double_moins_un. +Notation double_moins_un_add_un_xI := double_moins_un_add_un_xI. +Notation ZL1 := ZL1. +Notation add_un_not_un := add_un_not_un. +Notation sub_add_one := sub_add_one. +Notation add_sub_one := add_sub_one. +Notation add_un_inj := add_un_inj. +Notation ZL12 := ZL12. +Notation ZL12bis := ZL12bis. +Notation ZL13 := ZL13. +Notation add_sym := add_sym. +Notation ZL14 := ZL14. +Notation ZL14bis := ZL14bis. +Notation ZL15 := ZL15. +Notation add_no_neutral := add_no_neutral. +Notation add_carry_not_add_un := add_carry_not_add_un. +Notation add_carry_add := add_carry_add. +Notation simpl_add_r := simpl_add_r. +Notation simpl_add_carry_r := simpl_add_carry_r. +Notation simpl_add_l := simpl_add_l. +Notation simpl_add_carry_l := simpl_add_carry_l. +Notation add_assoc := add_assoc. +Notation add_xI_double_moins_un := add_xI_double_moins_un. +Notation add_x_x := add_x_x. +Notation ZS := ZS. +Notation US := US. +Notation USH := USH. +Notation ZSH := ZSH. +Notation sub_pos_x_x := sub_pos_x_x. +Notation ZL10 := ZL10. +Notation sub_pos_SUPERIEUR := sub_pos_SUPERIEUR. +Notation sub_add := sub_add. +Notation convert_add_carry := convert_add_carry. +Notation add_verif := add_verif. +Notation ZL2 := ZL2. +Notation ZL6 := ZL6. +Notation positive_to_nat_mult := positive_to_nat_mult. +Notation times_convert := times_convert. +Notation compare_positive_to_nat_O := compare_positive_to_nat_O. +Notation compare_convert_O := compare_convert_O. +Notation convert_xH := convert_xH. +Notation convert_xO := convert_xO. +Notation convert_xI := convert_xI. +Notation bij1 := bij1. +Notation ZL3 := ZL3. +Notation ZL4 := ZL4. +Notation ZL5 := ZL5. +Notation bij2 := bij2. +Notation bij3 := bij3. +Notation ZL7 := ZL7. +Notation ZL8 := ZL8. +Notation compare_convert_INFERIEUR := compare_convert_INFERIEUR. +Notation compare_convert_SUPERIEUR := compare_convert_SUPERIEUR. +Notation convert_compare_INFERIEUR := convert_compare_INFERIEUR. +Notation convert_compare_SUPERIEUR := convert_compare_SUPERIEUR. +Notation ZC1 := ZC1. +Notation ZC2 := ZC2. +Notation ZC3 := ZC3. +Notation ZC4 := ZC4. +Notation true_sub_convert := true_sub_convert. +Notation convert_intro := convert_intro. +Notation ZL16 := ZL16. +Notation ZL17 := ZL17. +Notation compare_true_sub_right := compare_true_sub_right. +Notation compare_true_sub_left := compare_true_sub_left. +Notation times_x_ := times_x_1. +Notation times_x_double := times_x_double. +Notation times_x_double_plus_one := times_x_double_plus_one. +Notation times_sym := times_sym. +Notation times_add_distr := times_add_distr. +Notation times_add_distr_l := times_add_distr_l. +Notation times_assoc := times_assoc. +Notation times_true_sub_distr := times_true_sub_distr. +Notation times_discr_xO_xI := times_discr_xO_xI. +Notation times_discr_xO := times_discr_xO. +Notation simpl_times_r := simpl_times_r. +Notation simpl_times_l := simpl_times_l. +Notation iterate_add := iterate_add. +Notation entier := entier. +Notation Nul := Nul. +Notation Pos := Pos. +Notation Un_suivi_de := Un_suivi_de. +Notation Zero_suivi_de := Zero_suivi_de. +Notation times1 := + [x:positive;_:positive->positive;y:positive](times x y). +Notation times1_convert := + [x,y:positive;_:positive->positive](times_convert x y). + +Notation Z := Z. +Notation POS := POS. +Notation NEG := NEG. +Notation ZERO := ZERO. +Notation Zero_left := Zero_left. +Notation Zopp_Zopp := Zopp_Zopp. +Notation Zero_right := Zero_right. +Notation Zplus_inverse_r := Zplus_inverse_r. +Notation Zopp_Zplus := Zopp_Zplus. +Notation Zplus_sym := Zplus_sym. +Notation Zplus_inverse_l := Zplus_inverse_l. +Notation Zopp_intro := Zopp_intro. +Notation Zopp_NEG := Zopp_NEG. +Notation weak_assoc := weak_assoc. +Notation Zplus_assoc := Zplus_assoc. +Notation Zplus_simpl := Zplus_simpl. +Notation Zmult_sym := Zmult_sym. +Notation Zmult_assoc := Zmult_assoc. +Notation Zmult_one := Zmult_one. +Notation lt_mult_left := lt_mult_left. (* Mult*) +Notation Zero_mult_left := Zero_mult_left. +Notation Zero_mult_right := Zero_mult_right. +Notation Zopp_Zmult := Zopp_Zmult. +Notation Zmult_Zopp_Zopp := Zmult_Zopp_Zopp. +Notation weak_Zmult_plus_distr_r := weak_Zmult_plus_distr_r. +Notation Zmult_plus_distr_r := Zmult_plus_distr_r. +Notation Zcompare_EGAL := Zcompare_EGAL. +Notation Zcompare_ANTISYM := Zcompare_ANTISYM. +Notation le_minus := le_minus. +Notation Zcompare_Zopp := Zcompare_Zopp. +Notation weaken_Zcompare_Zplus_compatible := weaken_Zcompare_Zplus_compatible. +Notation weak_Zcompare_Zplus_compatible := weak_Zcompare_Zplus_compatible. +Notation Zcompare_Zplus_compatible := Zcompare_Zplus_compatible. +Notation Zcompare_trans_SUPERIEUR := Zcompare_trans_SUPERIEUR. +Notation SUPERIEUR_POS := SUPERIEUR_POS. +Export Datatypes. +Export BinPos. +Export BinNat. +Export BinInt. +Export Zcompare. +Export Mult. +]. diff --git a/theories7/ZArith/zarith_aux.v b/theories7/ZArith/zarith_aux.v new file mode 100644 index 00000000..cd67d46b --- /dev/null +++ b/theories7/ZArith/zarith_aux.v @@ -0,0 +1,163 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(*i $Id: zarith_aux.v,v 1.2.2.1 2004/07/16 19:31:44 herbelin Exp $ i*) + +Require Export BinInt. +Require Export Zcompare. +Require Export Zorder. +Require Export Zmin. +Require Export Zabs. + +V7only [ +Notation Zlt := Zlt. +Notation Zgt := Zgt. +Notation Zle := Zle. +Notation Zge := Zge. +Notation Zsgn := Zsgn. +Notation absolu := absolu. +Notation Zabs := Zabs. +Notation Zabs_eq := Zabs_eq. +Notation Zabs_non_eq := Zabs_non_eq. +Notation Zabs_dec := Zabs_dec. +Notation Zabs_pos := Zabs_pos. +Notation Zsgn_Zabs := Zsgn_Zabs. +Notation Zabs_Zsgn := Zabs_Zsgn. +Notation inject_nat := inject_nat. +Notation Zs := Zs. +Notation Zpred := Zpred. +Notation Zgt_Sn_n := Zgt_Sn_n. +Notation Zle_gt_trans := Zle_gt_trans. +Notation Zgt_le_trans := Zgt_le_trans. +Notation Zle_S_gt := Zle_S_gt. +Notation Zcompare_n_S := Zcompare_n_S. +Notation Zgt_n_S := Zgt_n_S. +Notation Zle_not_gt := Zle_not_gt. +Notation Zgt_antirefl := Zgt_antirefl. +Notation Zgt_not_sym := Zgt_not_sym. +Notation Zgt_not_le := Zgt_not_le. +Notation Zgt_trans := Zgt_trans. +Notation Zle_gt_S := Zle_gt_S. +Notation Zgt_pred := Zgt_pred. +Notation Zsimpl_gt_plus_l := Zsimpl_gt_plus_l. +Notation Zsimpl_gt_plus_r := Zsimpl_gt_plus_r. +Notation Zgt_reg_l := Zgt_reg_l. +Notation Zgt_reg_r := Zgt_reg_r. +Notation Zcompare_et_un := Zcompare_et_un. +Notation Zgt_S_n := Zgt_S_n. +Notation Zle_S_n := Zle_S_n. +Notation Zgt_le_S := Zgt_le_S. +Notation Zgt_S_le := Zgt_S_le. +Notation Zgt_S := Zgt_S. +Notation Zgt_trans_S := Zgt_trans_S. +Notation Zeq_S := Zeq_S. +Notation Zpred_Sn := Zpred_Sn. +Notation Zeq_add_S := Zeq_add_S. +Notation Znot_eq_S := Znot_eq_S. +Notation Zsimpl_plus_l := Zsimpl_plus_l. +Notation Zn_Sn := Zn_Sn. +Notation Zplus_n_O := Zplus_n_O. +Notation Zplus_unit_left := Zplus_unit_left. +Notation Zplus_unit_right := Zplus_unit_right. +Notation Zplus_n_Sm := Zplus_n_Sm. +Notation Zmult_n_O := Zmult_n_O. +Notation Zmult_n_Sm := Zmult_n_Sm. +Notation Zle_n := Zle_n. +Notation Zle_refl := Zle_refl. +Notation Zle_trans := Zle_trans. +Notation Zle_n_Sn := Zle_n_Sn. +Notation Zle_n_S := Zle_n_S. +Notation Zs_pred := Zs_pred. (* BinInt *) +Notation Zle_pred_n := Zle_pred_n. +Notation Zle_trans_S := Zle_trans_S. +Notation Zle_Sn_n := Zle_Sn_n. +Notation Zle_antisym := Zle_antisym. +Notation Zgt_lt := Zgt_lt. +Notation Zlt_gt := Zlt_gt. +Notation Zge_le := Zge_le. +Notation Zle_ge := Zle_ge. +Notation Zge_trans := Zge_trans. +Notation Zlt_n_Sn := Zlt_n_Sn. +Notation Zlt_S := Zlt_S. +Notation Zlt_n_S := Zlt_n_S. +Notation Zlt_S_n := Zlt_S_n. +Notation Zlt_n_n := Zlt_n_n. +Notation Zlt_pred := Zlt_pred. +Notation Zlt_pred_n_n := Zlt_pred_n_n. +Notation Zlt_le_S := Zlt_le_S. +Notation Zlt_n_Sm_le := Zlt_n_Sm_le. +Notation Zle_lt_n_Sm := Zle_lt_n_Sm. +Notation Zlt_le_weak := Zlt_le_weak. +Notation Zlt_trans := Zlt_trans. +Notation Zlt_le_trans := Zlt_le_trans. +Notation Zle_lt_trans := Zle_lt_trans. +Notation Zle_lt_or_eq := Zle_lt_or_eq. +Notation Zle_or_lt := Zle_or_lt. +Notation Zle_not_lt := Zle_not_lt. +Notation Zlt_not_le := Zlt_not_le. +Notation Zlt_not_sym := Zlt_not_sym. +Notation Zle_le_S := Zle_le_S. +Notation Zmin := Zmin. +Notation Zmin_SS := Zmin_SS. +Notation Zle_min_l := Zle_min_l. +Notation Zle_min_r := Zle_min_r. +Notation Zmin_case := Zmin_case. +Notation Zmin_or := Zmin_or. +Notation Zmin_n_n := Zmin_n_n. +Notation Zplus_assoc_l := Zplus_assoc_l. +Notation Zplus_assoc_r := Zplus_assoc_r. +Notation Zplus_permute := Zplus_permute. +Notation Zsimpl_le_plus_l := Zsimpl_le_plus_l. +Notation "'Zsimpl_le_plus_l' c" := [a,b:Z](Zsimpl_le_plus_l a b c) + (at level 10, c at next level). +Notation "'Zsimpl_le_plus_l' c a" := [b:Z](Zsimpl_le_plus_l a b c) + (at level 10, a, c at next level). +Notation "'Zsimpl_le_plus_l' c a b" := (Zsimpl_le_plus_l a b c) + (at level 10, a, b, c at next level). +Notation Zsimpl_le_plus_r := Zsimpl_le_plus_r. +Notation "'Zsimpl_le_plus_r' c" := [a,b:Z](Zsimpl_le_plus_r a b c) + (at level 10, c at next level). +Notation "'Zsimpl_le_plus_r' c a" := [b:Z](Zsimpl_le_plus_r a b c) + (at level 10, a, c at next level). +Notation "'Zsimpl_le_plus_r' c a b" := (Zsimpl_le_plus_r a b c) + (at level 10, a, b, c at next level). +Notation Zle_reg_l := Zle_reg_l. +Notation Zle_reg_r := Zle_reg_r. +Notation Zle_plus_plus := Zle_plus_plus. +Notation Zplus_Snm_nSm := Zplus_Snm_nSm. +Notation Zsimpl_lt_plus_l := Zsimpl_lt_plus_l. +Notation Zsimpl_lt_plus_r := Zsimpl_lt_plus_r. +Notation Zlt_reg_l := Zlt_reg_l. +Notation Zlt_reg_r := Zlt_reg_r. +Notation Zlt_le_reg := Zlt_le_reg. +Notation Zle_lt_reg := Zle_lt_reg. +Notation Zminus := Zminus. +Notation Zminus_plus_simpl := Zminus_plus_simpl. +Notation Zminus_n_O := Zminus_n_O. +Notation Zminus_n_n := Zminus_n_n. +Notation Zplus_minus := Zplus_minus. +Notation Zminus_plus := Zminus_plus. +Notation Zle_plus_minus := Zle_plus_minus. +Notation Zminus_Sn_m := Zminus_Sn_m. +Notation Zlt_minus := Zlt_minus. +Notation Zlt_O_minus_lt := Zlt_O_minus_lt. +Notation Zmult_plus_distr_l := Zmult_plus_distr_l. +Notation Zmult_plus_distr := BinInt.Zmult_plus_distr_l. +Notation Zmult_minus_distr := Zmult_minus_distr. +Notation Zmult_assoc_r := Zmult_assoc_r. +Notation Zmult_assoc_l := Zmult_assoc_l. +Notation Zmult_permute := Zmult_permute. +Notation Zmult_1_n := Zmult_1_n. +Notation Zmult_n_1 := Zmult_n_1. +Notation Zmult_Sm_n := Zmult_Sm_n. +Notation Zmult_Zplus_distr := Zmult_plus_distr_r. +Export BinInt. +Export Zorder. +Export Zmin. +Export Zabs. +Export Zcompare. +]. |