Numbers: change definition of divide (compat with Znumtheory)

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14237 85f007b7-540e-0410-9357-904b9bb8a0f7

4 files changed, 29 insertions, 31 deletions

diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v
index f72c1b343..e1802dbee 100644
--- a/theories/Numbers/Integer/Abstract/ZDivEucl.v
+++ b/theories/Numbers/Integer/Abstract/ZDivEucl.v
@@ -604,11 +604,9 @@ Lemma mod_divides : forall a b, b~=0 ->
  (a mod b == 0 <-> (b|a)).
 Proof.
 intros a b Hb. split.
-intros Hab. exists (a/b). rewrite (div_mod a b Hb) at 2.
- rewrite Hab; now nzsimpl.
-intros (c,Hc).
-rewrite <- Hc, mul_comm.
-now apply mod_mul.
+intros Hab. exists (a/b). rewrite mul_comm.
+ rewrite (div_mod a b Hb) at 1. rewrite Hab; now nzsimpl.
+intros (c,Hc). rewrite Hc. now apply mod_mul.
 Qed.
 
 End ZEuclidProp.
diff --git a/theories/Numbers/Integer/Abstract/ZGcd.v b/theories/Numbers/Integer/Abstract/ZGcd.v
index 87a95e9d7..404fc0c43 100644
--- a/theories/Numbers/Integer/Abstract/ZGcd.v
+++ b/theories/Numbers/Integer/Abstract/ZGcd.v
@@ -21,7 +21,7 @@ Module Type ZGcdProp
 
 Lemma divide_opp_l : forall n m, (-n | m) <-> (n | m).
 Proof.
- intros n m. split; intros (p,Hp); exists (-p); rewrite <- Hp.
+ intros n m. split; intros (p,Hp); exists (-p); rewrite Hp.
   now rewrite mul_opp_l, mul_opp_r.
   now rewrite mul_opp_opp.
 Qed.
@@ -29,8 +29,8 @@ Qed.
 Lemma divide_opp_r : forall n m, (n | -m) <-> (n | m).
 Proof.
  intros n m. split; intros (p,Hp); exists (-p).
-  now rewrite mul_opp_r, Hp, opp_involutive.
-  now rewrite <- Hp, mul_opp_r.
+  now rewrite mul_opp_l, <- Hp, opp_involutive.
+  now rewrite Hp, mul_opp_l.
 Qed.
 
 Lemma divide_abs_l : forall n m, (abs n | m) <-> (n | m).
@@ -53,7 +53,7 @@ Qed.
 
 Lemma divide_1_r : forall n, (n | 1) -> n==1 \/ n==-1.
 Proof.
- intros n (m,Hm). now apply eq_mul_1 with m.
+ intros n (m,H). rewrite mul_comm in H. now apply eq_mul_1 with m.
 Qed.
 
 Lemma divide_antisym_abs : forall n m,
@@ -210,11 +210,11 @@ Proof.
  apply gcd_unique.
  apply mul_nonneg_nonneg; trivial using gcd_nonneg, abs_nonneg.
  destruct (gcd_divide_l n m) as (q,Hq).
- rewrite <- Hq at 2. rewrite <- (abs_sgn p) at 2.
- rewrite mul_shuffle1. apply divide_factor_l.
+ rewrite Hq at 2. rewrite mul_assoc. apply mul_divide_mono_r.
+ rewrite <- (abs_sgn p) at 2. rewrite <- mul_assoc. apply divide_factor_l.
  destruct (gcd_divide_r n m) as (q,Hq).
- rewrite <- Hq at 2. rewrite <- (abs_sgn p) at 2.
- rewrite mul_shuffle1. apply divide_factor_l.
+ rewrite Hq at 2. rewrite mul_assoc. apply mul_divide_mono_r.
+ rewrite <- (abs_sgn p) at 2. rewrite <- mul_assoc. apply divide_factor_l.
  intros q H H'.
  destruct (gcd_bezout n m (gcd n m) (eq_refl _)) as (a & b & EQ).
  rewrite <- EQ, <- sgn_abs, mul_add_distr_l. apply divide_add_r.
@@ -257,15 +257,15 @@ Proof.
  apply le_lteq in G; destruct G as [G|G].
  destruct (gcd_divide_l n m) as (q,Hq).
  exists (gcd n m). exists q.
- split. easy.
+ split. now rewrite mul_comm.
  split. apply gcd_divide_r.
  destruct (gcd_divide_r n m) as (r,Hr).
- rewrite <- Hr in H. rewrite <- Hq in H at 1.
- rewrite <- mul_assoc in H. apply mul_divide_cancel_l in H; [|order].
+ rewrite Hr in H. rewrite Hq in H at 1.
+ rewrite mul_shuffle0 in H. apply mul_divide_cancel_r in H; [|order].
  apply gauss with r; trivial.
- apply mul_cancel_l with (gcd n m); [order|].
- rewrite mul_1_r.
- rewrite <- gcd_mul_mono_l_nonneg, Hq, Hr; order.
+ apply mul_cancel_r with (gcd n m); [order|].
+ rewrite mul_1_l.
+ rewrite <- gcd_mul_mono_r_nonneg, <- Hq, <- Hr; order.
  symmetry in G. apply gcd_eq_0 in G. destruct G as (Hn',_); order.
 Qed.
 
diff --git a/theories/Numbers/Integer/Abstract/ZLcm.v b/theories/Numbers/Integer/Abstract/ZLcm.v
index 052d68ab6..06af04d16 100644
--- a/theories/Numbers/Integer/Abstract/ZLcm.v
+++ b/theories/Numbers/Integer/Abstract/ZLcm.v
@@ -86,17 +86,17 @@ Qed.
 Lemma mod_divide : forall a b, b~=0 -> (a mod b == 0 <-> (b|a)).
 Proof.
  intros a b Hb. split.
- intros Hab. exists (a/b). rewrite (div_mod a b Hb) at 2.
-  rewrite Hab; now nzsimpl.
- intros (c,Hc). rewrite <- Hc, mul_comm. now apply mod_mul.
+ intros Hab. exists (a/b). rewrite mul_comm.
+ rewrite (div_mod a b Hb) at 1. rewrite Hab; now nzsimpl.
+ intros (c,Hc). rewrite Hc. now apply mod_mul.
 Qed.
 
 Lemma rem_divide : forall a b, b~=0 -> (a rem b == 0 <-> (b|a)).
 Proof.
  intros a b Hb. split.
- intros Hab. exists (a÷b). rewrite (quot_rem a b Hb) at 2.
-  rewrite Hab; now nzsimpl.
- intros (c,Hc). rewrite <- Hc, mul_comm. now apply rem_mul.
+ intros Hab. exists (a÷b). rewrite mul_comm.
+ rewrite (quot_rem a b Hb) at 1. rewrite Hab; now nzsimpl.
+ intros (c,Hc). rewrite Hc. now apply rem_mul.
 Qed.
 
 Lemma rem_mod_eq_0 : forall a b, b~=0 -> (a rem b == 0 <-> a mod b == 0).
@@ -248,7 +248,7 @@ Qed.
 Lemma divide_div : forall a b c, a~=0 -> (a|b) -> (b|c) -> (b/a|c/a).
 Proof.
  intros a b c Ha Hb (c',Hc). exists c'.
- now rewrite mul_comm, <- divide_div_mul_exact, mul_comm, Hc.
+ now rewrite <- divide_div_mul_exact, <- Hc.
 Qed.
 
 Lemma lcm_least : forall a b c,
@@ -262,14 +262,14 @@ Proof.
  set (g:=gcd a b) in *.
  assert (Ha' := divide_div g a c NEQ Ga Ha).
  assert (Hb' := divide_div g b c NEQ Gb Hb).
- destruct Ha' as (a',Ha'). rewrite <- Ha' in Hb'.
+ destruct Ha' as (a',Ha'). rewrite Ha', mul_comm in Hb'.
  apply gauss in Hb'; [|apply gcd_div_gcd; unfold g; trivial using gcd_comm].
  destruct Hb' as (b',Hb').
  exists b'.
- rewrite <- mul_assoc, Hb'.
+ rewrite mul_shuffle3, <- Hb'.
  rewrite (proj2 (div_exact c g NEQ)).
- rewrite <- Ha', mul_assoc. f_equiv.
- apply div_exact; trivial.
+ rewrite Ha', mul_shuffle3, (mul_comm a a'). f_equiv.
+ symmetry. apply div_exact; trivial.
  apply mod_divide; trivial.
  apply mod_divide; trivial. transitivity a; trivial.
 Qed.
diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
index e3fc512e7..44dd2c593 100644
--- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
+++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
@@ -409,7 +409,7 @@ Qed.
 
 (** Gcd *)
 
-Definition divide n m := exists p, n*p == m.
+Definition divide n m := exists p, m == p*n.
 Local Notation "( x | y )" := (divide x y) (at level 0).
 
 Lemma spec_divide : forall n m, (n|m) <-> Z.divide [n] [m].