Ajout quelques lemmes; noms des variables liees

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@4845 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 235 insertions, 155 deletions

diff --git a/theories/ZArith/Zorder.v b/theories/ZArith/Zorder.v
index 905466b62..e2cc554b3 100644
--- a/theories/ZArith/Zorder.v
+++ b/theories/ZArith/Zorder.v
@@ -27,7 +27,7 @@ Implicit Variable Type x,y,z:Z.
 
 (** Trichotomy *)
 
-Theorem Ztrichotomy_inf : (m,n:Z) {Zlt m n} + {m=n} + {Zgt m n}.
+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.
@@ -37,7 +37,7 @@ Unfold Zgt Zlt; Intros m n; Assert H:=(refl_equal ? (Zcompare m n)).
     | Right ]; Reflexivity.
 Qed.
 
-Theorem Ztrichotomy : (m,n:Z) (Zlt m n) \/ m=n \/ (Zgt m n).
+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.
@@ -65,7 +65,7 @@ Intros H1 H2; Elim (Dcompare (Zcompare x y));
     | Auto]].
 Qed.
 
-Theorem dec_Zle: (x,y:Z) (decidable (Zle x y)).
+Theorem dec_Zle: (x,y:Z) (decidable `x<=y`).
 Proof.
 Intros x y; Unfold decidable Zle ; Elim (Zcompare x y); [
     Left; Discriminate
@@ -73,13 +73,13 @@ Intros x y; Unfold decidable Zle ; Elim (Zcompare x y); [
   | Right; Unfold not ; Intros H; Apply H; Trivial with arith].
 Qed.
 
-Theorem dec_Zgt: (x,y:Z) (decidable (Zgt x y)).
+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 (Zge x y)).
+Theorem dec_Zge: (x,y:Z) (decidable `x>=y`).
 Proof.
 Intros x y; Unfold decidable Zge ; Elim (Zcompare x y); [
   Left; Discriminate
@@ -87,13 +87,13 @@ Intros x y; Unfold decidable Zge ; Elim (Zcompare x y); [
 | Left; Discriminate]. 
 Qed.
 
-Theorem dec_Zlt: (x,y:Z) (decidable (Zlt x y)).
+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 -> (Zlt x y) \/ (Zlt y x).
+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]
@@ -103,70 +103,68 @@ Qed.
 
 (** Relating strict and large orders *)
 
-Lemma Zgt_lt : (m,n:Z) (Zgt m n) -> (Zlt n m).
+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) (Zlt m n) -> (Zgt n m).
+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) (Zge m n) -> (Zle n m).
+Lemma Zge_le : (m,n:Z) `m>=n` -> `n<=m`.
 Proof.
-Intros m n; Change ~(Zlt m n)-> ~(Zgt n m);
+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) (Zle m n) -> (Zge n m).
+Lemma Zle_ge : (m,n:Z) `m<=n` -> `n>=m`.
 Proof.
-Intros m n; Change ~(Zgt m n)-> ~(Zlt n m);
+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)(Zle n m) -> ~(Zgt n m).
+Lemma Zle_not_gt     : (n,m:Z)`n<=m` -> ~`n>m`.
 Proof.
 Trivial.
 Qed.
 
-Lemma Zgt_not_le     : (n,m:Z)(Zgt n m) -> ~(Zle n m).
+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)(Zle n m) -> ~(Zlt m n).
+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)(Zlt n m) -> ~(Zle m n).
+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.
 
-Theorem not_Zge : (x,y:Z) ~(Zge x y) -> (Zlt x y).
+Theorem 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.
  
-Theorem not_Zlt : (x,y:Z) ~(Zlt x y) -> (Zge x y).
+Theorem not_Zlt : (x,y:Z) ~`x<y` -> `x>=y`.
 Proof.
 Unfold Zlt Zge; Auto with arith.
 Qed.
 
-Lemma not_Zgt     : (n,m:Z)~(Zgt n m) -> (Zle n m).
+Lemma not_Zgt : (x,y:Z)~`x>y` -> `x<=y`.
 Proof.
 Trivial.
 Qed.
 
-V7only [Notation Znot_gt_le := not_Zgt.].
-
-Theorem not_Zle : (x,y:Z) ~(Zle x y) -> (Zgt x y).
+Theorem 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].
@@ -181,63 +179,63 @@ Qed.
 
 (** Antisymmetry *)
 
-Lemma Zle_antisym : (n,m:Z)(Zle n m)->(Zle m n)->n=m.
+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 (Zgt m n); [ Apply Zle_not_gt | Apply Zlt_gt]; Assumption.
+  Absurd `m>n`; [ Apply Zle_not_gt | Apply Zlt_gt]; Assumption.
   Assumption.
-  Absurd (Zgt n m); [ Apply Zle_not_gt | Idtac]; Assumption.
+  Absurd `n>m`; [ Apply Zle_not_gt | Idtac]; Assumption.
 Qed.
 
 (** Asymmetry *)
 
-Lemma Zgt_not_sym    : (n,m:Z)(Zgt n m) -> ~(Zgt m n).
+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)(Zlt n m) -> ~(Zlt m n).
+Lemma Zlt_not_sym : (n,m:Z)`n<m` -> ~`m<n`.
 Proof.
 Intros n m H H1;
-Assert H2:(Zgt m n). Apply Zlt_gt; Assumption.
-Assert H3: (Zgt n m). Apply Zlt_gt; Assumption.
+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)~(Zgt n n).
+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)~(Zlt n n).
+Lemma Zlt_n_n : (n:Z)~`n<n`.
 Proof.
 Intros n H; Apply (Zlt_not_sym n n H H).
 Qed.
 
 (** Large = strict or equal *)
 
-Lemma Zle_lt_or_eq : (n,m:Z)(Zle n m)->((Zlt n m) \/ n=m).
+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 (Zgt n m); [Apply Zle_not_gt|Idtac]; Assumption ].
+| Absurd `n>m`; [Apply Zle_not_gt|Idtac]; Assumption ].
 Qed.
 
-Lemma Zlt_le_weak : (n,m:Z)(Zlt n m)->(Zle n m).
+Lemma Zlt_le_weak : (n,m:Z)`n<m`->`n<=m`.
 Proof.
-Intros n m Hlt; Apply Znot_gt_le; Apply Zgt_not_sym; Apply Zlt_gt; Assumption.
+Intros n m Hlt; Apply not_Zgt; Apply Zgt_not_sym; Apply Zlt_gt; Assumption.
 Qed.
 
 (** Dichotomy *)
 
-Lemma Zle_or_lt : (n,m:Z)(Zle n m)\/(Zlt m n).
+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 Znot_gt_le; Intro Hgt; Assert Hgt':=(Zlt_gt ? ? Hlt);
+  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 ].
@@ -245,12 +243,12 @@ Qed.
 
 (** Transitivity of strict orders *)
 
-Lemma Zgt_trans      : (n,m,p:Z)(Zgt n m)->(Zgt m p)->(Zgt n p).
+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)(Zlt n m)->(Zlt m p)->(Zlt n p).
+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.
@@ -258,26 +256,26 @@ Qed.
 
 (** Mixed transitivity *)
 
-Lemma Zle_gt_trans : (n,m,p:Z)(Zle m n)->(Zgt m p)->(Zgt n p).
+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)(Zgt n m)->(Zle p m)->(Zgt n p).
+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)(Zlt n m)->(Zle m p)->(Zlt n p).
+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)(Zle n m)->(Zlt m p)->(Zlt n p).
+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 ].
@@ -285,14 +283,14 @@ Qed.
 
 (** Transitivity of large orders *)
 
-Lemma Zle_trans : (n,m,p:Z)(Zle n m)->(Zle m p)->(Zle n p).
+Lemma Zle_trans : (n,m,p:Z)`n<=m`->`m<=p`->`n<=p`.
 Proof.
-Intros n m p H1 H2; Apply Znot_gt_le.
+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) (Zge n m) -> (Zge m p) -> (Zge n p).
+Lemma Zge_trans : (n, m, p : Z) `n>=m` -> `m>=p` -> `n>=p`.
 Proof.
 Intros n m p H1 H2.
 Apply Zle_ge.
@@ -301,27 +299,27 @@ Qed.
 
 (** Compatibility of successor wrt to order *)
 
-Lemma Zle_n_S : (n,m:Z) (Zle m n) -> (Zle (Zs m) (Zs n)).
+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)(Zgt m n) -> (Zgt (Zs m) (Zs n)).
+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.
 
 (** Simplification of successor wrt to order *)
 
-Lemma Zgt_S_n        : (n,p:Z)(Zgt (Zs p) (Zs n))->(Zgt p n).
+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) (Zle (Zs m) (Zs n)) -> (Zle m n).
+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));
@@ -330,41 +328,39 @@ Qed.
 
 (** Compatibility of addition wrt to order *)
 
-Lemma Zgt_reg_l      
-	: (n,m,p:Z)(Zgt n m)->(Zgt (Zplus p n) (Zplus p m)).
+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)(Zgt n m)->(Zgt (Zplus n p) (Zplus m p)).
+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)(Zle n m)->(Zle (Zplus p n) (Zplus p m)).
+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) (Zle n m)->(Zle (Zplus n p) (Zplus m p)).
+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)(Zlt n m)->(Zlt (Zplus p n) (Zplus p m)).
+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)(Zlt n m)->(Zlt (Zplus n p) (Zplus m p)).
+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) (Zlt a b)->(Zle c d)->(Zlt (Zplus a c) (Zplus b d)).
+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).
@@ -372,8 +368,7 @@ Apply  Zlt_reg_r; Trivial.
 Apply  Zle_reg_l; Trivial.
 Qed.
 
-Lemma Zle_lt_reg :
- (a,b,c,d:Z) (Zle a b)->(Zlt c d)->(Zlt (Zplus a c) (Zplus b d)).
+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).
@@ -381,8 +376,7 @@ Apply  Zle_reg_r; Trivial.
 Apply  Zlt_reg_l; Trivial.
 Qed.
 
-Lemma Zle_plus_plus : 
- (n,m,p,q:Z) (Zle n m)->(Zle p q)->(Zle (Zplus n p) (Zplus m q)).
+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 ].
@@ -390,8 +384,7 @@ Qed.
 
 V7only [Set Implicit Arguments.].
 
-Lemma Zlt_Zplus : 
-  (x1,x2,y1,y2:Z)`x1 < x2` -> `y1 < y2` -> `x1 + y1 < x2 + y2`.
+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.
 
@@ -399,49 +392,44 @@ V7only [Unset Implicit Arguments.].
 
 (** Compatibility of addition wrt to being positive *)
 
-Theorem Zle_0_plus : 
-  (x,y:Z) (Zle ZERO x) -> (Zle ZERO y) -> (Zle ZERO (Zplus x y)).
+Theorem 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)(Zgt (Zplus p n) (Zplus p m))->(Zgt n m).
+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)(Zgt (Zplus n p) (Zplus m p))->(Zgt n m).
+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 : (p,n,m:Z)(Zle (Zplus p n) (Zplus p m))->(Zle n m).
+Lemma Zsimpl_le_plus_l : (p,n,m:Z)`p+n<=p+m`->`n<=m`.
 Proof.
 Intros p n m; Unfold Zle not ;Intros H1 H2;Apply H1; 
 Rewrite (Zcompare_Zplus_compatible n m p); Assumption.
 Qed.
  
-Lemma Zsimpl_le_plus_r : (p,n,m:Z)(Zle (Zplus n p) (Zplus m p))->(Zle n m).
+Lemma Zsimpl_le_plus_r : (p,n,m:Z)`n+p<=m+p`->`n<=m`.
 Proof.
 Intros p n m 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)(Zlt (Zplus p n) (Zplus p m))->(Zlt n m).
+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)(Zlt (Zplus n p) (Zplus m p))->(Zlt n m).
+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.
@@ -449,12 +437,12 @@ Qed.
  
 (** Order, predecessor and successor *)
 
-Lemma Zgt_Sn_n : (n:Z)(Zgt (Zs n) n).
+Lemma Zgt_Sn_n : (n:Z)`(Zs n)>n`.
 Proof.
 Exact Zcompare_Zs_SUPERIEUR.
 Qed.
 
-Lemma Zgt_le_S       : (n,p:Z)(Zgt p n)->(Zle (Zs n) p).
+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; [
@@ -462,26 +450,31 @@ Unfold not ;Intros H3; Unfold not in H1; Apply H1; [
 | Elim (Zcompare_ANTISYM (Zplus n (POS xH)) p);Intros H4 H5;Apply H4;Exact H3].
 Qed.
 
-Lemma Zgt_S_le       : (n,p:Z)(Zgt (Zs p) n)->(Zle n p).
+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 Zle_S_gt : (n,m:Z) (Zle (Zs n) m) -> (Zgt m n).
+V7only [ (* Relevance confirmed from Zextensions *) ].
+Lemma Zlt_S_le: (n,p:Z)`n < (Zs p)`->`n <= p`.
+Proof.
+Intros; 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.
 
-Lemma Zle_gt_S       : (n,p:Z)(Zle n p)->(Zgt (Zs p) n).
+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 Zgt_pred       
-	: (n,p:Z)(Zgt p (Zs n))->(Zgt (Zpred p) n).
+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));
@@ -489,7 +482,7 @@ Rewrite (Zplus_sym p); Rewrite Zplus_assoc; Rewrite [x:Z](Zplus_sym x n);
 Simpl; Assumption.
 Qed.
 
-Lemma Zlt_ZERO_pred_le_ZERO : (n:Z) (Zlt ZERO n) -> (Zle ZERO (Zpred n)).
+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.
@@ -507,55 +500,67 @@ V7only [Unset Implicit Arguments.].
 
 (** Special cases of ordered integers *)
 
-Lemma Zle_n_Sn : (n:Z)(Zle n (Zs n)).
+V7only [ (* Relevance confirmed from Zdivides *) ].
+Theorem Z_O_1: `0<1`.
+Proof.
+Red; Simpl; Auto; Intros; Red; Intros; Discriminate.
+Qed.
+
+V7only [ (* Relevance confirmed from Zdivides *) ].
+Theorem Zle_NEG_POS: (p,q:positive) `(NEG p)<=(POS q)`.
+Proof.
+Intros p; Red; Simpl; Red; Intros H; Discriminate.
+Qed.
+
+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.
 
-Lemma Zle_pred_n : (n:Z)(Zle (Zpred n) n).
+Lemma Zle_pred_n : (n:Z)`(Zpred n)<=n`.
 Proof.
 Intros n;Pattern 2 n ;Rewrite Zs_pred; Apply Zle_n_Sn.
 Qed.
 
-Lemma POS_gt_ZERO : (p:positive) (Zgt (POS p) ZERO).
+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) (Zle ZERO (POS p)).
+Lemma ZERO_le_POS : (p:positive) `0<=(POS p)`.
 Intro; Unfold Zle; Discriminate.
 Qed.
 
-Lemma NEG_lt_ZERO : (p:positive)(Zlt (NEG p) ZERO).
+Lemma NEG_lt_ZERO : (p:positive)`(NEG p)<0`.
 Unfold Zlt; Trivial.
 Qed.
 
 (** Weakening equality within order *)
  
-Lemma Zlt_not_eq : (x,y:Z)(Zlt x y) -> ~x=y.
+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.
 
-Lemma Zle_refl : (n,m:Z) n=m -> (Zle n m).
+Lemma Zle_refl : (n,m:Z) n=m -> `n<=m`.
 Proof.
 Intros; Rewrite H; Apply Zle_n.
 Qed.
 
 (** Transitivity using successor *)
 
-Lemma Zgt_trans_S  : (n,m,p:Z)(Zgt (Zs n) m)->(Zgt m p)->(Zgt n p).
+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.
 
-Lemma Zgt_S        : (n,m:Z)(Zgt (Zs n) m)->((Zgt n m)\/(m=n)).
+Lemma Zgt_S        : (n,m:Z)`(Zs n)>m`->(`n>m`\/(m=n)).
 Proof.
 Intros n m H.
-Assert Hle : (Zle m n).
+Assert Hle : `m<=n`.
   Apply Zgt_S_le; Assumption.
 NewDestruct (Zle_lt_or_eq ? ? Hle) as [Hlt|Heq].
   Left; Apply Zlt_gt; Assumption.  
@@ -565,63 +570,63 @@ Qed.
 Hints Resolve Zle_n Zle_n_Sn Zle_trans Zle_n_S : zarith.
 Hints Immediate Zle_refl : zarith.
 
-Lemma Zle_trans_S : (n,m:Z)(Zle (Zs n) m)->(Zle n m).
+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.
 
-Lemma Zle_Sn_n : (n:Z)~(Zle (Zs n) n).
+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)(Zlt n (Zs n)).
+Lemma Zlt_n_Sn : (n:Z)`n<(Zs n)`.
 Proof.
 Intro n; Apply Zgt_lt; Apply Zgt_Sn_n.
 Qed.
 
-Lemma Zlt_S : (n,m:Z)(Zlt n m)->(Zlt n (Zs m)).
+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 Zlt_n_S : (n,m:Z)(Zlt n m)->(Zlt (Zs n) (Zs m)).
+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.
 
-Lemma Zlt_S_n : (n,m:Z)(Zlt (Zs n) (Zs m))->(Zlt n m).
+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.
 
-Lemma Zlt_pred : (n,p:Z)(Zlt (Zs n) p)->(Zlt n (Zpred p)).
+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.
 
-Lemma Zlt_pred_n_n : (n:Z)(Zlt (Zpred n) n).
+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.
  
-Lemma Zlt_le_S : (n,p:Z)(Zlt n p)->(Zle (Zs n) p).
+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 Zlt_n_Sm_le : (n,m:Z)(Zlt n (Zs m))->(Zle n m).
+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_lt_n_Sm : (n,m:Z)(Zle n m)->(Zlt n (Zs m)).
+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 Zle_le_S : (x,y:Z)(Zle x y)->(Zle x (Zs y)).
+Lemma Zle_le_S : (x,y:Z)`x<=y`->`x<=(Zs y)`.
 Proof.
 Intros.
 Apply Zle_trans with y; Trivial with zarith.
@@ -633,9 +638,7 @@ Hints Resolve Zle_le_S : zarith.
 
 V7only [Set Implicit Arguments.].
 
-Lemma Zle_Zmult_pos_right : 
-	(a,b,c : Z) 
-	(Zle a b) -> (Zle ZERO c) -> (Zle (Zmult a c) (Zmult b c)).
+Lemma Zle_Zmult_pos_right : (a,b,c : Z) `a<=b` -> `0<=c` -> `a*c<=b*c`.
 Proof.
 Intros; NewDestruct c.
   Do 2 Rewrite Zero_mult_right; Assumption.
@@ -644,9 +647,7 @@ Intros; NewDestruct c.
   Unfold Zle in H0; Contradiction H0; Reflexivity.
 Qed.
 
-Lemma Zle_Zmult_pos_left : 
-	(a,b,c : Z) 
-	(Zle a b) -> (Zle ZERO c) -> (Zle (Zmult c a) (Zmult c b)).
+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.
@@ -661,6 +662,12 @@ Intros; NewDestruct z.
   Discriminate H.
 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; Apply Zlt_Zmult_right; Try Apply Zlt_gt; Assumption.
+Save.
+
 Lemma Zle_Zmult_right : (x,y,z:Z)`z>0` -> `x <= y` -> `x*z <= y*z`.
 Proof.
 Intros x y z Hz Hxy.
@@ -671,6 +678,12 @@ 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; Apply Zle_Zmult_right; Try Apply Zlt_gt; Assumption.
+Qed.
+
 Lemma Zgt_Zmult_right : (x,y,z:Z)`z>0` -> `x > y` -> `x*z > y*z`.
 Proof.
 Intros; Apply Zlt_gt; Apply Zlt_Zmult_right; 
@@ -684,6 +697,14 @@ 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;
+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;
@@ -691,26 +712,20 @@ 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) 
-	(Zge a b) -> (Zge c ZERO) -> (Zge (Zmult a c) (Zmult b c)).
+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) 
-	(Zge a b) -> (Zge c ZERO) -> (Zge (Zmult c a) (Zmult c b)).
+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) 
-	(Zge a c) -> (Zge b d) -> (Zge c ZERO) -> (Zge d ZERO) 
-	-> (Zge (Zmult a b) (Zmult c d)).
+  (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).
@@ -719,6 +734,18 @@ 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`.
@@ -730,7 +757,14 @@ Intros; NewDestruct z.
   Discriminate H.
 Qed.
 
-Lemma Zle_Zmult_right2 :  (x,y,z:Z)`z>0` -> `x*z <= y*z` -> `x <= y`.
+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).
@@ -741,21 +775,25 @@ Apply Zmult_reg_right with z.
   Intro. Rewrite H0 in Hz. Contradiction (Zgt_antirefl `0`).
 Assumption.
 Qed.
-V7only [Notation Zle_mult_simpl := Zle_Zmult_right2.
-(* Zle_mult_simpl 
- : (a,b,c:Z)(Zgt c ZERO)->(Zle (Zmult a c) (Zmult b c))->(Zle a b). *)
+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 ; 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) (Zgt c ZERO)->(Zge (Zmult a c) (Zmult b c))->(Zge a b).
+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) (Zgt c ZERO)->(Zgt (Zmult a c) (Zmult b c))->(Zgt a b).
+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.
@@ -763,8 +801,7 @@ Qed.
 
 (** Compatibility of multiplication by a positive wrt to being positive *)
 
-Theorem Zle_ZERO_mult : 
-	 (x,y:Z) (Zle ZERO x) -> (Zle ZERO y) -> (Zle ZERO (Zmult x y)).
+Theorem 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.
@@ -775,8 +812,7 @@ Intros p H1 H2; Absurd (Zgt ZERO (NEG p)); Trivial.
 Unfold Zgt; Simpl; Auto with zarith.
 Qed.
 
-Lemma Zgt_ZERO_mult: (a,b:Z) (Zgt a ZERO)->(Zgt b ZERO)
-	->(Zgt (Zmult a b) ZERO).
+Lemma Zgt_ZERO_mult: (a,b:Z) `a>0`->`b>0`->`a*b>0`.
 Proof.
 Intros x y; Case x.
 Intros H; Discriminate H.
@@ -786,8 +822,14 @@ Pattern 2 ZERO ; Rewrite <- (Zero_mult_right (POS p)).
 Intros p H; Discriminate H.
 Qed.
 
-Theorem Zle_mult:
-  (x,y:Z) (Zgt x ZERO) -> (Zle ZERO y) -> (Zle ZERO (Zmult y x)).
+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.
+
+Theorem 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.
@@ -795,8 +837,7 @@ Qed.
 
 (** Simplification of multiplication by a positive wrt to being positive *)
 
-Theorem Zmult_le:
-  (x,y:Z) (Zgt x ZERO) -> (Zle ZERO (Zmult y x)) -> (Zle ZERO y).
+Theorem 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
@@ -806,8 +847,7 @@ Intros x y; Case x; [
 | Intros p; Unfold Zgt ; Simpl; Intros H; Discriminate H].
 Qed.
 
-Theorem Zmult_lt:
-  (x,y:Z) (Zgt x ZERO) -> (Zlt ZERO (Zmult y x)) -> (Zlt ZERO y).
+Theorem 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
@@ -817,8 +857,13 @@ Intros x y; Case x; [
 | Intros p; Unfold Zgt ; Simpl; Intros H; Discriminate H].
 Qed.
 
-Theorem Zmult_gt:
-  (x,y:Z) (Zgt x ZERO) -> (Zgt (Zmult x y) ZERO) -> (Zgt y ZERO).
+V7only [ (* Relevance confirmed from Zextensions *) ].
+Lemma Zmult_lt_0_reg_r : (x,y:Z)`0 < x`->`0 < y*x`->`0 < y`.
+Proof.
+Intros ; EApply Zmult_lt with x ; Try Apply Zlt_gt; Assumption.
+Qed.
+
+Theorem Zmult_gt: (x,y:Z) `x>0` -> `x*y>0` -> `y>0`.
 Proof.
 Intros x y; Case x.
  Intros H; Discriminate H.
@@ -828,9 +873,31 @@ Intros x y; Case x.
 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` -> (Zgt 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 (used in contrib/graph) *)
 
-Lemma Zle_plus_swap : (x,y,z:Z) (Zle (Zplus x z) y) <-> (Zle x (Zminus y z)).
+Lemma Zle_plus_swap : (x,y,z:Z) `x+z<=y` <-> `x<=y-z`.
 Proof.
     Intros. Split. Intro. Rewrite <- (Zero_right x). Rewrite <- (Zplus_inverse_r z).
     Rewrite Zplus_assoc_l. Exact (Zle_reg_r ? ? ? H).
@@ -838,13 +905,13 @@ Proof.
     Apply Zle_reg_r. Assumption.
 Qed.
 
-Lemma Zge_iff_le : (x,y:Z) (Zge x y) <-> (Zle y x).
+Lemma Zge_iff_le : (x,y:Z) `x>=y` <-> `y<=x`.
 Proof.
     Intros. Split. Intro. Apply Zge_le. Assumption.
     Intro. Apply Zle_ge. Assumption.
 Qed.
 
-Lemma Zlt_plus_swap : (x,y,z:Z) (Zlt (Zplus x z) y) <-> (Zlt x (Zminus y z)).
+Lemma Zlt_plus_swap : (x,y,z:Z) `x+z<y` <-> `x<y-z`.
 Proof.
     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.
@@ -853,13 +920,13 @@ Proof.
     Rewrite Zplus_assoc_r. Apply Zlt_reg_l. Rewrite Zplus_sym. Assumption.
 Qed.
 
-Lemma Zgt_iff_lt : (x,y:Z) (Zgt x y) <-> (Zlt y x).
+Lemma Zgt_iff_lt : (x,y:Z) `x>y` <-> `y<x`.
 Proof.
     Intros. Split. Intro. Apply Zgt_lt. Assumption.
     Intro. Apply Zlt_gt. Assumption.
 Qed.
 
-Lemma Zeq_plus_swap : (x,y,z:Z) (Zplus x z)=y <-> x=(Zminus y z).
+Lemma Zeq_plus_swap : (x,y,z:Z)`x+z=y` <-> `x=y-z`.
 Proof.
 Intros. Split. Intro. Apply Zplus_minus. Symmetry. Rewrite Zplus_sym. 
   Assumption.
@@ -867,6 +934,19 @@ 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.
+
 (** Reverting [x ?= y] to trichotomy *)
 
 Lemma rename : (A:Set)(P:A->Prop)(x:A) ((y:A)(x=y)->(P y)) -> (P x).
@@ -876,7 +956,7 @@ Qed.
 
 Theorem Zcompare_elim :
   (c1,c2,c3:Prop)(x,y:Z)
-    ((x=y) -> c1) ->((Zlt x y) -> c2) ->((Zgt x y)-> c3)
+    ((x=y) -> c1) ->(`x<y` -> c2) ->(`x>y`-> c3)
      -> Cases (Zcompare x y) of EGAL => c1 | INFERIEUR => c2 | SUPERIEUR => c3 end.
 Proof.
 Intros.
@@ -899,9 +979,9 @@ Qed.
 
 Theorem Zcompare_egal_dec :
    (x1,y1,x2,y2:Z)
-    ((Zlt x1 y1)->(Zlt x2 y2))
+    (`x1<y1`->`x2<y2`)
      ->((Zcompare x1 y1)=EGAL -> (Zcompare x2 y2)=EGAL)
-        ->((Zgt x1 y1)->(Zgt x2 y2))->(Zcompare x1 y1)=(Zcompare x2 y2).
+        ->(`x1>y1`->`x2>y2`)->(Zcompare x1 y1)=(Zcompare x2 y2).
 Proof.
 Intros x1 y1 x2 y2.
 Unfold Zgt; Unfold Zlt;
@@ -911,28 +991,28 @@ Qed.
 (** Relating [x ?= y] to [Zle], [Zlt], [Zge] or [Zgt] *)
 
 Lemma Zle_Zcompare :
-  (x,y:Z)(Zle x y) ->
+  (x,y:Z)`x<=y` ->
   Cases (Zcompare x y) of EGAL => True | INFERIEUR => True | SUPERIEUR => False end.
 Proof.
 Intros x y; Unfold Zle; Elim (Zcompare x y); Auto with arith.
 Qed.
 
 Lemma Zlt_Zcompare :
-  (x,y:Z)(Zlt x y)  ->
+  (x,y:Z)`x<y`  ->
   Cases (Zcompare x y) of EGAL => False | INFERIEUR => True | SUPERIEUR => False end.
 Proof.
 Intros x y; Unfold Zlt; Elim (Zcompare x y); Intros; Discriminate Orelse Trivial with arith.
 Qed.
 
 Lemma Zge_Zcompare :
-  (x,y:Z)(Zge x y)->
+  (x,y:Z)`x>=y`->
   Cases (Zcompare x y) of EGAL => True | INFERIEUR => False | SUPERIEUR => True end.
 Proof.
 Intros x y; Unfold Zge; Elim (Zcompare x y); Auto with arith. 
 Qed.
 
 Lemma Zgt_Zcompare :
-  (x,y:Z)(Zgt x y) ->
+  (x,y:Z)`x>y` ->
   Cases (Zcompare x y) of EGAL => False | INFERIEUR => False | SUPERIEUR => True end.
 Proof.
 Intros x y; Unfold Zgt; Elim (Zcompare x y); Intros; Discriminate Orelse Trivial with arith.