ZArith + other : favor the use of modern names instead of compat notations

 - For instance, refl_equal --> eq_refl
 - Npos, Zpos, Zneg now admit more uniform qualified aliases
   N.pos, Z.pos, Z.neg.
 - A new module BinInt.Pos2Z with results about injections from
   positive to Z
 - A result about Z.pow pushed in the generic layer
 - Zmult_le_compat_{r,l} --> Z.mul_le_mono_nonneg_{r,l}
 - Using tactic Z.le_elim instead of Zle_lt_or_eq
 - Some cleanup in ring, field, micromega
   (use of "Equivalence", "Proper" ...)
 - Some adaptions in QArith (for instance changed Qpower.Qpower_decomp)
 - In ZMake and ZMake, functor parameters are now named NN and ZZ
   instead of N and Z for avoiding confusions

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15515 85f007b7-540e-0410-9357-904b9bb8a0f7

6 files changed, 29 insertions, 29 deletions

diff --git a/theories/Logic/ClassicalFacts.v b/theories/Logic/ClassicalFacts.v
index bcec657aad..7d5cec74d5 100644
--- a/theories/Logic/ClassicalFacts.v
+++ b/theories/Logic/ClassicalFacts.v
@@ -191,7 +191,7 @@ Section Proof_irrelevance_gen.
     intros Ext Ind.
     case (ext_prop_fixpoint Ext bool true); intros G Gfix.
     set (neg := fun b:bool => bool_elim bool false true b).
-    generalize (refl_equal (G neg)).
+    generalize (eq_refl (G neg)).
     pattern (G neg) at 1 in |- *.
     apply Ind with (b := G neg); intro Heq.
     rewrite (bool_elim_redl bool false true).
@@ -228,9 +228,9 @@ Section Proof_irrelevance_Prop_Ext_CC.
   Definition FalseP : BoolP := fun C c1 c2 => c2.
   Definition BoolP_elim C c1 c2 (b:BoolP) := b C c1 c2.
   Definition BoolP_elim_redl (C:Prop) (c1 c2:C) :
-    c1 = BoolP_elim C c1 c2 TrueP := refl_equal c1.
+    c1 = BoolP_elim C c1 c2 TrueP := eq_refl c1.
   Definition BoolP_elim_redr (C:Prop) (c1 c2:C) :
-    c2 = BoolP_elim C c1 c2 FalseP := refl_equal c2.
+    c2 = BoolP_elim C c1 c2 FalseP := eq_refl c2.
 
   Definition BoolP_dep_induction :=
     forall P:BoolP -> Prop, P TrueP -> P FalseP -> forall b:BoolP, P b.
@@ -263,9 +263,9 @@ Section Proof_irrelevance_CIC.
     | trueP : boolP
     | falseP : boolP.
   Definition boolP_elim_redl (C:Prop) (c1 c2:C) :
-    c1 = boolP_ind C c1 c2 trueP := refl_equal c1.
+    c1 = boolP_ind C c1 c2 trueP := eq_refl c1.
   Definition boolP_elim_redr (C:Prop) (c1 c2:C) :
-    c2 = boolP_ind C c1 c2 falseP := refl_equal c2.
+    c2 = boolP_ind C c1 c2 falseP := eq_refl c2.
   Scheme boolP_indd := Induction for boolP Sort Prop.
 
   Lemma ext_prop_dep_proof_irrel_cic : prop_extensionality -> proof_irrelevance.
@@ -392,9 +392,9 @@ Section Proof_irrelevance_CCI.
   Hypothesis em : forall A:Prop, A \/ ~ A.
 
   Definition or_elim_redl (A B C:Prop) (f:A -> C) (g:B -> C)
-    (a:A) : f a = or_ind f g (or_introl B a) := refl_equal (f a).
+    (a:A) : f a = or_ind f g (or_introl B a) := eq_refl (f a).
   Definition or_elim_redr (A B C:Prop) (f:A -> C) (g:B -> C)
-    (b:B) : g b = or_ind f g (or_intror A b) := refl_equal (g b).
+    (b:B) : g b = or_ind f g (or_intror A b) := eq_refl (g b).
   Scheme or_indd := Induction for or Sort Prop.
 
   Theorem proof_irrelevance_cci : forall (B:Prop) (b1 b2:B), b1 = b2.
diff --git a/theories/Logic/Classical_Prop.v b/theories/Logic/Classical_Prop.v
index d2b35da298..5d7764e7e1 100644
--- a/theories/Logic/Classical_Prop.v
+++ b/theories/Logic/Classical_Prop.v
@@ -112,7 +112,7 @@ Module Eq_rect_eq.
 Lemma eq_rect_eq :
   forall (U:Type) (p:U) (Q:U -> Type) (x:Q p) (h:p = p), x = eq_rect p Q x p h.
 Proof.
-intros; rewrite proof_irrelevance with (p1:=h) (p2:=refl_equal p); reflexivity.
+intros; rewrite proof_irrelevance with (p1:=h) (p2:=eq_refl p); reflexivity.
 Qed.
 
 End Eq_rect_eq.
diff --git a/theories/Logic/Diaconescu.v b/theories/Logic/Diaconescu.v
index 8569e55ec3..d1327af3e9 100644
--- a/theories/Logic/Diaconescu.v
+++ b/theories/Logic/Diaconescu.v
@@ -188,8 +188,8 @@ Lemma projT1_injective : a1=a2 -> a1'=a2'.
 Proof.
   intro Heq ; unfold a1', a2', A'.
   rewrite Heq.
-  replace (or_introl (a2=a2) (refl_equal a2))
-     with (or_intror (a2=a2) (refl_equal a2)).
+  replace (or_introl (a2=a2) (eq_refl a2))
+     with (or_intror (a2=a2) (eq_refl a2)).
   reflexivity.
   apply proof_irrelevance.
 Qed.
diff --git a/theories/Logic/EqdepFacts.v b/theories/Logic/EqdepFacts.v
index d84cd82405..4d0560bd7f 100644
--- a/theories/Logic/EqdepFacts.v
+++ b/theories/Logic/EqdepFacts.v
@@ -101,7 +101,7 @@ Section Dependent_Equality.
     forall (p q:U) (x:P p) (y:P q), eq_dep p x q y -> eq_dep1 p x q y.
   Proof.
     destruct 1.
-    apply eq_dep1_intro with (refl_equal p).
+    apply eq_dep1_intro with (eq_refl p).
     simpl; trivial.
   Qed.
 
@@ -121,7 +121,7 @@ Proof.
   apply eq_dep_intro.
 Qed.
 
-Notation eq_sigS_eq_dep := eq_sigT_eq_dep (only parsing). (* Compatibility *)
+Notation eq_sigS_eq_dep := eq_sigT_eq_dep (compat "8.2"). (* Compatibility *)
 
 Lemma eq_dep_eq_sigT :
   forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q),
@@ -250,12 +250,12 @@ Section Equivalences.
   (** Uniqueness of Reflexive Identity Proofs *)
 
   Definition UIP_refl_ :=
-    forall (x:U) (p:x = x), p = refl_equal x.
+    forall (x:U) (p:x = x), p = eq_refl x.
 
   (** Streicher's axiom K *)
 
   Definition Streicher_K_ :=
-    forall (x:U) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p.
+    forall (x:U) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.
 
   (** Injectivity of Dependent Equality is a consequence of *)
   (** Invariance by Substitution of Reflexive Equality Proof *)
@@ -389,14 +389,14 @@ Proof (eq_dep_eq__UIP U eq_dep_eq).
 
 (** Uniqueness of Reflexive Identity Proofs is a direct instance of UIP *)
 
-Lemma UIP_refl : forall (x:U) (p:x = x), p = refl_equal x.
+Lemma UIP_refl : forall (x:U) (p:x = x), p = eq_refl x.
 Proof (UIP__UIP_refl U UIP).
 
 (** Streicher's axiom K is a direct consequence of Uniqueness of
     Reflexive Identity Proofs *)
 
 Lemma Streicher_K :
-  forall (x:U) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p.
+  forall (x:U) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.
 Proof (UIP_refl__Streicher_K U UIP_refl).
 
 End Axioms.
diff --git a/theories/Logic/Eqdep_dec.v b/theories/Logic/Eqdep_dec.v
index 59088aa75f..2ed5d428c2 100644
--- a/theories/Logic/Eqdep_dec.v
+++ b/theories/Logic/Eqdep_dec.v
@@ -9,7 +9,7 @@
 (* Created by Bruno Barras, Jan 1998 *)
 (* Made a module instance for EqdepFacts by Hugo Herbelin, Mar 2006 *)
 
-(** We prove that there is only one proof of [x=x], i.e [refl_equal x].
+(** We prove that there is only one proof of [x=x], i.e [eq_refl x].
     This holds if the equality upon the set of [x] is decidable.
     A corollary of this theorem is the equality of the right projections
     of two equal dependent pairs.
@@ -43,7 +43,7 @@ Section EqdepDec.
   Let comp (x y y':A) (eq1:x = y) (eq2:x = y') : y = y' :=
     eq_ind _ (fun a => a = y') eq2 _ eq1.
 
-  Remark trans_sym_eq : forall (x y:A) (u:x = y), comp u u = refl_equal y.
+  Remark trans_sym_eq : forall (x y:A) (u:x = y), comp u u = eq_refl y.
   Proof.
     intros.
     case u; trivial.
@@ -69,7 +69,7 @@ Section EqdepDec.
   Qed.
 
 
-  Let nu_inv (y:A) (v:x = y) : x = y := comp (nu (refl_equal x)) v.
+  Let nu_inv (y:A) (v:x = y) : x = y := comp (nu (eq_refl x)) v.
 
 
   Remark nu_left_inv : forall (y:A) (u:x = y), nu_inv (nu u) = u.
@@ -90,10 +90,10 @@ Section EqdepDec.
   Qed.
 
   Theorem K_dec :
-    forall P:x = x -> Prop, P (refl_equal x) -> forall p:x = x, P p.
+    forall P:x = x -> Prop, P (eq_refl x) -> forall p:x = x, P p.
   Proof.
     intros.
-    elim eq_proofs_unicity with x (refl_equal x) p.
+    elim eq_proofs_unicity with x (eq_refl x) p.
     trivial.
   Qed.
 
@@ -135,7 +135,7 @@ Require Import EqdepFacts.
 Theorem K_dec_type :
   forall A:Type,
     (forall x y:A, {x = y} + {x <> y}) ->
-    forall (x:A) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p.
+    forall (x:A) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.
 Proof.
   intros A eq_dec x P H p.
   elim p using K_dec; intros.
@@ -146,7 +146,7 @@ Qed.
 Theorem K_dec_set :
   forall A:Set,
     (forall x y:A, {x = y} + {x <> y}) ->
-    forall (x:A) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p.
+    forall (x:A) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.
 Proof fun A => K_dec_type (A:=A).
 
 (** We deduce the [eq_rect_eq] axiom for (decidable) types *)
@@ -212,13 +212,13 @@ Module DecidableEqDep (M:DecidableType).
 
   (** Uniqueness of Reflexive Identity Proofs *)
 
-  Lemma UIP_refl : forall (x:U) (p:x = x), p = refl_equal x.
+  Lemma UIP_refl : forall (x:U) (p:x = x), p = eq_refl x.
   Proof (UIP__UIP_refl U UIP).
 
   (** Streicher's axiom K *)
 
   Lemma Streicher_K :
-    forall (x:U) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p.
+    forall (x:U) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.
   Proof (K_dec_type eq_dec).
 
   (** Injectivity of equality on dependent pairs in [Type] *)
@@ -281,13 +281,13 @@ Module DecidableEqDepSet (M:DecidableSet).
 
   (** Uniqueness of Reflexive Identity Proofs *)
 
-  Lemma UIP_refl : forall (x:U) (p:x = x), p = refl_equal x.
+  Lemma UIP_refl : forall (x:U) (p:x = x), p = eq_refl x.
   Proof N.UIP_refl.
 
   (** Streicher's axiom K *)
 
   Lemma Streicher_K :
-    forall (x:U) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p.
+    forall (x:U) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.
   Proof N.Streicher_K.
 
   (** Proof-irrelevance on subsets of decidable sets *)
@@ -301,7 +301,7 @@ Module DecidableEqDepSet (M:DecidableSet).
 
   Lemma inj_pair2 :
     forall (P:U -> Type) (p:U) (x y:P p),
-      existS P p x = existS P p y -> x = y.
+      existT P p x = existT P p y -> x = y.
   Proof eq_dep_eq__inj_pair2 U N.eq_dep_eq.
 
   (** Injectivity of equality on dependent pairs with second component
diff --git a/theories/Logic/ProofIrrelevanceFacts.v b/theories/Logic/ProofIrrelevanceFacts.v
index 6accc48077..776010c03f 100644
--- a/theories/Logic/ProofIrrelevanceFacts.v
+++ b/theories/Logic/ProofIrrelevanceFacts.v
@@ -25,7 +25,7 @@ Module ProofIrrelevanceTheory (M:ProofIrrelevance).
       forall (U:Type) (p:U) (Q:U -> Type) (x:Q p) (h:p = p),
         x = eq_rect p Q x p h.
     Proof.
-      intros; rewrite M.proof_irrelevance with (p1:=h) (p2:=refl_equal p).
+      intros; rewrite M.proof_irrelevance with (p1:=h) (p2:=eq_refl p).
       reflexivity.
     Qed.
   End Eq_rect_eq.