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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. + |