2 files changed, 80 insertions, 7 deletions
diff --git a/theories/Numbers/Integer/SpecViaZ/ZSig.v b/theories/Numbers/Integer/SpecViaZ/ZSig.v
index 021f4b1bf..b33ed5f8e 100644
--- a/theories/Numbers/Integer/SpecViaZ/ZSig.v
+++ b/theories/Numbers/Integer/SpecViaZ/ZSig.v
@@ -46,8 +46,9 @@ Module Type ZType.
  Parameter opp : t -> t.
  Parameter mul : t -> t -> t.
  Parameter square : t -> t.
- Parameter power_pos : t -> positive -> t.
- Parameter power : t -> N -> t.
+ Parameter pow_pos : t -> positive -> t.
+ Parameter pow_N : t -> N -> t.
+ Parameter pow : t -> t -> t.
  Parameter sqrt : t -> t.
  Parameter div_eucl : t -> t -> t * t.
  Parameter div : t -> t -> t.
@@ -55,6 +56,8 @@ Module Type ZType.
  Parameter gcd : t -> t -> t.
  Parameter sgn : t -> t.
  Parameter abs : t -> t.
+ Parameter even : t -> bool.
+ Parameter odd : t -> bool.
 
  Parameter spec_compare: forall x y, compare x y = Zcompare [x] [y].
  Parameter spec_eq_bool: forall x y, eq_bool x y = Zeq_bool [x] [y].
@@ -70,8 +73,9 @@ Module Type ZType.
  Parameter spec_opp: forall x, [opp x] = - [x].
  Parameter spec_mul: forall x y, [mul x y] = [x] * [y].
  Parameter spec_square: forall x, [square x] = [x] *  [x].
- Parameter spec_power_pos: forall x n, [power_pos x n] = [x] ^ Zpos n.
- Parameter spec_power: forall x n, [power x n] = [x] ^ Z_of_N n.
+ Parameter spec_pow_pos: forall x n, [pow_pos x n] = [x] ^ Zpos n.
+ Parameter spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z_of_N n.
+ Parameter spec_pow: forall x n, [pow x n] = [x] ^ [n].
  Parameter spec_sqrt: forall x, 0 <= [x] ->
    [sqrt x] ^ 2 <= [x] < ([sqrt x] + 1) ^ 2.
  Parameter spec_div_eucl: forall x y,
@@ -81,6 +85,8 @@ Module Type ZType.
  Parameter spec_gcd: forall a b, [gcd a b] = Zgcd (to_Z a) (to_Z b).
  Parameter spec_sgn : forall x, [sgn x] = Zsgn [x].
  Parameter spec_abs : forall x, [abs x] = Zabs [x].
+ Parameter spec_even : forall x, even x = Zeven_bool [x].
+ Parameter spec_odd : forall x, odd x = Zodd_bool [x].
 
 End ZType.
 
@@ -91,6 +97,7 @@ Module Type ZType_Notation (Import Z:ZType).
  Infix "+" := add.
  Infix "-" := sub.
  Infix "*" := mul.
+ Infix "^" := pow.
  Notation "- x" := (opp x).
  Infix "<=" := le.
  Infix "<" := lt.
diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
index 48cf9e868..90bda6343 100644
--- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
+++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v
@@ -6,7 +6,7 @@
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
 
-Require Import ZArith ZAxioms ZDivFloor ZSig.
+Require Import ZArith Nnat ZAxioms ZDivFloor ZSig.
 
 (** * The interface [ZSig.ZType] implies the interface [ZAxiomsSig]
 
@@ -20,6 +20,7 @@ Hint Rewrite
  spec_0 spec_1 spec_add spec_sub spec_pred spec_succ
  spec_mul spec_opp spec_of_Z spec_div spec_modulo
  spec_compare spec_eq_bool spec_max spec_min spec_abs spec_sgn
+ spec_pow spec_even spec_odd
  : zsimpl.
 
 Ltac zsimpl := autorewrite with zsimpl.
@@ -188,13 +189,15 @@ Qed.
 
 (** Part specific to integers, not natural numbers *)
 
-Program Instance opp_wd : Proper (eq ==> eq) opp.
-
 Theorem succ_pred : forall n, succ (pred n) == n.
 Proof.
 intros. zify. auto with zarith.
 Qed.
 
+(** Opp *)
+
+Program Instance opp_wd : Proper (eq ==> eq) opp.
+
 Theorem opp_0 : - 0 == 0.
 Proof.
 intros. zify. auto with zarith.
@@ -205,6 +208,8 @@ Proof.
 intros. zify. auto with zarith.
 Qed.
 
+(** Abs / Sgn *)
+
 Theorem abs_eq : forall n, 0 <= n -> abs n == n.
 Proof.
 intros n. zify. omega with *.
@@ -230,6 +235,67 @@ Proof.
 intros n. zify. omega with *.
 Qed.
 
+(** Power *)
+
+Program Instance pow_wd : Proper (eq==>eq==>eq) pow.
+
+Local Notation "1" := (succ 0).
+Local Notation "2" := (succ 1).
+
+Lemma pow_0_r : forall a, a^0 == 1.
+Proof.
+ intros. now zify.
+Qed.
+
+Lemma pow_succ_r : forall a b, 0<=b -> a^(succ b) == a * a^b.
+Proof.
+ intros a b. zify. intros Hb.
+ rewrite Zpower_exp; auto with zarith.
+ simpl. unfold Zpower_pos; simpl. ring.
+Qed.
+
+Lemma pow_neg : forall a b, b<0 -> a^b == 0.
+Proof.
+ intros a b. zify. intros Hb.
+ destruct [b]; reflexivity || discriminate.
+Qed.
+
+Lemma pow_pow_N : forall a b, 0<=b -> a^b == pow_N a (Zabs_N (to_Z b)).
+Proof.
+ intros a b. zify. intros Hb.
+ now rewrite spec_pow_N, Z_of_N_abs, Zabs_eq.
+Qed.
+
+Lemma pow_pos_N : forall a p, pow_pos a p == pow_N a (Npos p).
+Proof.
+ intros a b. red. now rewrite spec_pow_N, spec_pow_pos.
+Qed.
+
+(** Even / Odd *)
+
+Definition Even n := exists m, n == 2*m.
+Definition Odd n := exists m, n == 2*m+1.
+
+Lemma even_spec : forall n, even n = true <-> Even n.
+Proof.
+ intros n. unfold Even. zify.
+ rewrite Zeven_bool_iff, Zeven_ex_iff.
+ split; intros (m,Hm).
+ exists (of_Z m). now zify.
+ exists [m]. revert Hm. now zify.
+Qed.
+
+Lemma odd_spec : forall n, odd n = true <-> Odd n.
+Proof.
+ intros n. unfold Odd. zify.
+ rewrite Zodd_bool_iff, Zodd_ex_iff.
+ split; intros (m,Hm).
+ exists (of_Z m). now zify.
+ exists [m]. revert Hm. now zify.
+Qed.
+
+(** Div / Mod *)
+
 Program Instance div_wd : Proper (eq==>eq==>eq) div.
 Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.