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diff --git a/theories7/ZArith/Zorder.v b/theories7/ZArith/Zorder.v deleted file mode 100644 index d49a0800..00000000 --- a/theories7/ZArith/Zorder.v +++ /dev/null @@ -1,969 +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: 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. |