syntax for exponents

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13897 85f007b7-540e-0410-9357-904b9bb8a0f7

4 files changed, 38 insertions, 30 deletions

diff --git a/plugins/setoid_ring/Cring.v b/plugins/setoid_ring/Cring.v
index 5ba016074..4058cd224 100644
--- a/plugins/setoid_ring/Cring.v
+++ b/plugins/setoid_ring/Cring.v
@@ -11,6 +11,7 @@
 Require Import Setoid.
 Require Import BinPos.
 Require Import BinNat.
+Require Export ZArith.
 Require Export Morphisms Setoid Bool.
 Require Import Algebra_syntax.
 Require Import Ring_theory.
@@ -56,8 +57,13 @@ Instance opposite_cring`{R:Type}`{Rr:Cring R} : Opposite R :=
   {opposite x := cring_opp x}.
 Instance equality_cring `{R:Type}`{Rr:Cring R} : Equality :=
   {equality x y := cring_eq x y}.
+Definition ZN(x:Z):=
+  match x with
+    Z0 => N0
+    |Zpos p | Zneg p => Npos p
+end.
 Instance power_cring {R:Type}{Rr:Cring R} : Power:=
-  {power x y := @Ring_theory.pow_N _ cring1 cring_mult x y}.
+  {power x y := @Ring_theory.pow_N _ cring1 cring_mult x (ZN y)}.
 
 Existing Instance cring_setoid.
 Existing Instance cring_plus_comp.
@@ -87,28 +93,24 @@ Instance bracket_cring {C R:Type}`{Cr:Cring C} `{Rr:Cring R}
   {bracket x := cring_morphism_fun x}.
 
 (* Tactics for crings *)
-Axiom eta: forall (A B:Type) (f:A->B), (fun x:A => f x) = f.
-Axiom eta2: forall (A B C:Type) (f:A->B->C), (fun (x:A)(y:B) => f x y) = f.
-
-Ltac etarefl := try apply eta; try apply eta2; try reflexivity.
 
 Lemma cring_syntax1:forall (A:Type)(Ar:Cring A), (@cring_eq _ Ar) = equality.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl ; reflexivity. Qed.
 Lemma cring_syntax2:forall (A:Type)(Ar:Cring A), (@cring_plus _ Ar) = addition.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma cring_syntax3:forall (A:Type)(Ar:Cring A), (@cring_mult _ Ar) = multiplication.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma cring_syntax4:forall (A:Type)(Ar:Cring A), (@cring_sub _ Ar) = subtraction.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma cring_syntax5:forall (A:Type)(Ar:Cring A), (@cring_opp _ Ar) = opposite.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma cring_syntax6:forall (A:Type)(Ar:Cring A), (@cring0 _ Ar) = zero.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma cring_syntax7:forall (A:Type)(Ar:Cring A), (@cring1 _ Ar) = one.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma cring_syntax8:forall (A:Type)(Ar:Cring A)(B:Type)(Br:Cring B)
   (pM:@Cring_morphism A B Ar Br), (@cring_morphism_fun _ _ _ _ pM) = bracket.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 
 Ltac set_cring_notations :=
   repeat (rewrite cring_syntax1);
diff --git a/plugins/setoid_ring/Cring_tac.v b/plugins/setoid_ring/Cring_tac.v
index c83dd4e8d..8df45b56e 100644
--- a/plugins/setoid_ring/Cring_tac.v
+++ b/plugins/setoid_ring/Cring_tac.v
@@ -14,6 +14,7 @@ Require Import Znumtheory.
 Require Import Zdiv_def.
 Require Export Morphisms Setoid Bool.
 Require Import ZArith.
+Open Scope Z_scope.
 Require Import Algebra_syntax.
 Require Import Ring_polynom.
 Require Export Cring.
@@ -212,14 +213,18 @@ Variable R: Type.
 Variable Rr: Cring R.
 Existing Instance Rr.
 
+Goal forall x y z:R, 3%Z * x * (2%Z * y * z) == 6%Z * (x * y) * z.
+cring.
+Qed.
+
 (* reification: 0,7s
 sinon, le reste de la tactique donne le même temps que ring 
 *)
-Goal forall x y z  t u :R, (x + y + z + t + u)^13%N == (u + t + z + y + x) ^13%N.
+Goal forall x y z  t u :R, (x + y + z + t + u)^13 == (u + t + z + y + x) ^13.
 Time cring. (*Finished transaction in 0. secs (0.410938u,0.s)*)
 Qed.
 (*
-Goal forall x y z  t u :R, (x + y + z + t + u)^16%N == (u + t + z + y + x) ^16%N.
+Goal forall x y z  t u :R, (x + y + z + t + u)^16 == (u + t + z + y + x) ^16.
 Time cring.(*Finished transaction in 1. secs (0.968852u,0.001s)*)
 Qed.
 *)
@@ -237,6 +242,10 @@ Time ring.(*Finished transaction in 1. secs (0.914861u,0.s)*)
 Qed.
 *)
 
+Goal forall x y z:R, 6*x^2 + y  == y + 3*2*x^2 + 0 .
+cring.
+Qed.
+
 (*
 Goal forall x y z:R, x + y  == y + x + 0 .
 cring.
@@ -246,7 +255,7 @@ Goal forall x y z:R, x * y * z == x * (y * z).
 cring.
 Qed.
 
-Goal forall x y z:R, 3%Z * x * (2%Z * y * z) == 6%Z * (x * y) * z.
+Goal forall x y z:R, 3 * x * (2%Z * y * z) == 6 * (x * y) * z.
 cring.
 Qed.
 
diff --git a/plugins/setoid_ring/Ring2.v b/plugins/setoid_ring/Ring2.v
index 30a6128f1..a28c1d4ea 100644
--- a/plugins/setoid_ring/Ring2.v
+++ b/plugins/setoid_ring/Ring2.v
@@ -85,28 +85,24 @@ Instance bracket_ring (C R:Type)`{Cr:Ring C} `{Rr:Ring R}
   {bracket x := ring_morphism_fun x}.
 
 (* Tactics for rings *)
-Axiom eta: forall (A B:Type) (f:A->B), (fun x:A => f x) = f.
-Axiom eta2: forall (A B C:Type) (f:A->B->C), (fun (x:A)(y:B) => f x y) = f.
-
-Ltac etarefl := try apply eta; try apply eta2; try reflexivity.
 
 Lemma ring_syntax1:forall (A:Type)(Ar:Ring A), (@ring_eq _ Ar) = equality.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma ring_syntax2:forall (A:Type)(Ar:Ring A), (@ring_plus _ Ar) = addition.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma ring_syntax3:forall (A:Type)(Ar:Ring A), (@ring_mult _ Ar) = multiplication.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma ring_syntax4:forall (A:Type)(Ar:Ring A), (@ring_sub _ Ar) = subtraction.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma ring_syntax5:forall (A:Type)(Ar:Ring A), (@ring_opp _ Ar) = opposite.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma ring_syntax6:forall (A:Type)(Ar:Ring A), (@ring0 _ Ar) = zero.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma ring_syntax7:forall (A:Type)(Ar:Ring A), (@ring1 _ Ar) = one.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 Lemma ring_syntax8:forall (A:Type)(Ar:Ring A)(B:Type)(Br:Ring B)
   (pM:@Ring_morphism A B Ar Br), (@ring_morphism_fun _ _ _ _ pM) = bracket.
-intros. symmetry. simpl; etarefl. Qed.
+intros. symmetry. simpl; reflexivity. Qed.
 
 Ltac set_ring_notations :=
   repeat (rewrite ring_syntax1);
diff --git a/plugins/setoid_ring/Ring2_tac.v b/plugins/setoid_ring/Ring2_tac.v
index 1df1b8e73..15dad95e1 100644
--- a/plugins/setoid_ring/Ring2_tac.v
+++ b/plugins/setoid_ring/Ring2_tac.v
@@ -13,6 +13,7 @@ Require Import BinList.
 Require Import Znumtheory.
 Require Export Morphisms Setoid Bool.
 Require Import ZArith.
+Open Scope Z_scope.
 Require Import Algebra_syntax.
 Require Export Ring2.
 Require Import Ring2_polynom.
@@ -174,11 +175,11 @@ Goal forall x y z:R, x * y * z == x * (y * z).
 ring2.
 Qed.
 
-Goal forall x y z:R, [3%Z]* x *([2%Z]* y * z) == [6%Z] * (x * y) * z.
+Goal forall x y z:R, [3]* x *([2]* y * z) == [6] * (x * y) * z.
 ring2.
 Qed.
 
-Goal forall x y z:R, 3%Z * x * (2%Z * y * z) == 6%Z * (x * y) * z.
+Goal forall x y z:R, 3 * x * (2 * y * z) == 6 * (x * y) * z.
 ring2.
 Qed.