diff options
Diffstat (limited to 'theories7/NArith')
| -rw-r--r-- | theories7/NArith/BinNat.v | 205 | ||||
| -rw-r--r-- | theories7/NArith/BinPos.v | 894 | ||||
| -rw-r--r-- | theories7/NArith/NArith.v | 14 | ||||
| -rw-r--r-- | theories7/NArith/Pnat.v | 472 |
4 files changed, 0 insertions, 1585 deletions
diff --git a/theories7/NArith/BinNat.v b/theories7/NArith/BinNat.v deleted file mode 100644 index 5e04e22e..00000000 --- a/theories7/NArith/BinNat.v +++ /dev/null @@ -1,205 +0,0 @@ -(************************************************************************) -(* 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 deleted file mode 100644 index ae61587d..00000000 --- a/theories7/NArith/BinPos.v +++ /dev/null @@ -1,894 +0,0 @@ -(************************************************************************) -(* 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 deleted file mode 100644 index d924ae2e..00000000 --- a/theories7/NArith/NArith.v +++ /dev/null @@ -1,14 +0,0 @@ -(************************************************************************) -(* 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 deleted file mode 100644 index d62661ed..00000000 --- a/theories7/NArith/Pnat.v +++ /dev/null @@ -1,472 +0,0 @@ -(************************************************************************) -(* 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. - |