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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id: NSigNAxioms.v 11040 2008-06-03 00:04:16Z letouzey $ i*)
+
+Require Import ZArith.
+Require Import Nnat.
+Require Import NAxioms.
+Require Import NSig.
+
+(** * The interface [NSig.NType] implies the interface [NAxiomsSig] *)
+
+Module NSig_NAxioms (N:NType) <: NAxiomsSig.
+
+Delimit Scope IntScope with Int.
+Bind Scope IntScope with N.t.
+Open Local Scope IntScope.
+Notation "[ x ]" := (N.to_Z x) : IntScope.
+Infix "=="  := N.eq (at level 70) : IntScope.
+Notation "0" := N.zero : IntScope.
+Infix "+" := N.add : IntScope.
+Infix "-" := N.sub : IntScope.
+Infix "*" := N.mul : IntScope.
+
+Module Export NZOrdAxiomsMod <: NZOrdAxiomsSig.
+Module Export NZAxiomsMod <: NZAxiomsSig.
+
+Definition NZ := N.t.
+Definition NZeq := N.eq.
+Definition NZ0 := N.zero.
+Definition NZsucc := N.succ.
+Definition NZpred := N.pred.
+Definition NZadd := N.add.
+Definition NZsub := N.sub.
+Definition NZmul := N.mul.
+
+Theorem NZeq_equiv : equiv N.t N.eq.
+Proof.
+repeat split; repeat red; intros; auto; congruence.
+Qed.
+
+Add Relation N.t N.eq
+ reflexivity proved by (proj1 NZeq_equiv)
+ symmetry proved by (proj2 (proj2 NZeq_equiv))
+ transitivity proved by (proj1 (proj2 NZeq_equiv))
+ as NZeq_rel.
+
+Add Morphism NZsucc with signature N.eq ==> N.eq as NZsucc_wd.
+Proof.
+unfold N.eq; intros; rewrite 2 N.spec_succ; f_equal; auto.
+Qed.
+
+Add Morphism NZpred with signature N.eq ==> N.eq as NZpred_wd.
+Proof.
+unfold N.eq; intros.
+generalize (N.spec_pos y) (N.spec_pos x) (N.spec_eq_bool x 0).
+destruct N.eq_bool; rewrite N.spec_0; intros.
+rewrite 2 N.spec_pred0; congruence.
+rewrite 2 N.spec_pred; f_equal; auto; try omega.
+Qed.
+
+Add Morphism NZadd with signature N.eq ==> N.eq ==> N.eq as NZadd_wd.
+Proof.
+unfold N.eq; intros; rewrite 2 N.spec_add; f_equal; auto.
+Qed.
+
+Add Morphism NZsub with signature N.eq ==> N.eq ==> N.eq as NZsub_wd.
+Proof.
+unfold N.eq; intros x x' Hx y y' Hy.
+destruct (Z_lt_le_dec [x] [y]).
+rewrite 2 N.spec_sub0; f_equal; congruence.
+rewrite 2 N.spec_sub; f_equal; congruence.
+Qed.
+
+Add Morphism NZmul with signature N.eq ==> N.eq ==> N.eq as NZmul_wd.
+Proof.
+unfold N.eq; intros; rewrite 2 N.spec_mul; f_equal; auto.
+Qed.
+
+Theorem NZpred_succ : forall n, N.pred (N.succ n) == n.
+Proof.
+unfold N.eq; intros.
+rewrite N.spec_pred; rewrite N.spec_succ.
+omega.
+generalize (N.spec_pos n); omega.
+Qed.
+
+Definition N_of_Z z := N.of_N (Zabs_N z).
+
+Section Induction.
+
+Variable A : N.t -> Prop.
+Hypothesis A_wd : predicate_wd N.eq A.
+Hypothesis A0 : A 0.
+Hypothesis AS : forall n, A n <-> A (N.succ n). 
+
+Add Morphism A with signature N.eq ==> iff as A_morph.
+Proof. apply A_wd. Qed.
+
+Let B (z : Z) := A (N_of_Z z).
+
+Lemma B0 : B 0.
+Proof.
+unfold B, N_of_Z; simpl.
+rewrite <- (A_wd 0); auto.
+red; rewrite N.spec_0, N.spec_of_N; auto.
+Qed.
+
+Lemma BS : forall z : Z, (0 <= z)%Z -> B z -> B (z + 1).
+Proof.
+intros z H1 H2.
+unfold B in *. apply -> AS in H2.
+setoid_replace (N_of_Z (z + 1)) with (N.succ (N_of_Z z)); auto.
+unfold N.eq. rewrite N.spec_succ.
+unfold N_of_Z.
+rewrite 2 N.spec_of_N, 2 Z_of_N_abs, 2 Zabs_eq; auto with zarith.
+Qed.
+
+Lemma B_holds : forall z : Z, (0 <= z)%Z -> B z.
+Proof.
+exact (natlike_ind B B0 BS).
+Qed.
+
+Theorem NZinduction : forall n, A n.
+Proof.
+intro n. setoid_replace n with (N_of_Z (N.to_Z n)).
+apply B_holds. apply N.spec_pos.
+red; unfold N_of_Z.
+rewrite N.spec_of_N, Z_of_N_abs, Zabs_eq; auto.
+apply N.spec_pos.
+Qed.
+
+End Induction.
+
+Theorem NZadd_0_l : forall n, 0 + n == n.
+Proof.
+intros; red; rewrite N.spec_add, N.spec_0; auto with zarith.
+Qed.
+
+Theorem NZadd_succ_l : forall n m, (N.succ n) + m == N.succ (n + m).
+Proof.
+intros; red; rewrite N.spec_add, 2 N.spec_succ, N.spec_add; auto with zarith.
+Qed.
+
+Theorem NZsub_0_r : forall n, n - 0 == n.
+Proof.
+intros; red; rewrite N.spec_sub; rewrite N.spec_0; auto with zarith.
+apply N.spec_pos.
+Qed.
+
+Theorem NZsub_succ_r : forall n m, n - (N.succ m) == N.pred (n - m).
+Proof.
+intros; red.
+destruct (Z_lt_le_dec [n] [N.succ m]) as [H|H].
+rewrite N.spec_sub0; auto.
+rewrite N.spec_succ in H.
+rewrite N.spec_pred0; auto.
+destruct (Z_eq_dec [n] [m]).
+rewrite N.spec_sub; auto with zarith.
+rewrite N.spec_sub0; auto with zarith.
+
+rewrite N.spec_sub, N.spec_succ in *; auto.
+rewrite N.spec_pred, N.spec_sub; auto with zarith.
+rewrite N.spec_sub; auto with zarith.
+Qed.
+
+Theorem NZmul_0_l : forall n, 0 * n == 0.
+Proof.
+intros; red.
+rewrite N.spec_mul, N.spec_0; auto with zarith.
+Qed.
+
+Theorem NZmul_succ_l : forall n m, (N.succ n) * m == n * m + m.
+Proof.
+intros; red.
+rewrite N.spec_add, 2 N.spec_mul, N.spec_succ; ring.
+Qed.
+
+End NZAxiomsMod.
+
+Definition NZlt := N.lt.
+Definition NZle := N.le.
+Definition NZmin := N.min.
+Definition NZmax := N.max.
+
+Infix "<=" := N.le : IntScope.
+Infix "<" := N.lt : IntScope.
+
+Lemma spec_compare_alt : forall x y, N.compare x y = ([x] ?= [y])%Z.
+Proof.
+ intros; generalize (N.spec_compare x y).
+ destruct (N.compare x y); auto.
+ intros H; rewrite H; symmetry; apply Zcompare_refl.
+Qed.
+
+Lemma spec_lt : forall x y, (x<y) <-> ([x]<[y])%Z.
+Proof.
+ intros; unfold N.lt, Zlt; rewrite spec_compare_alt; intuition.
+Qed.
+
+Lemma spec_le : forall x y, (x<=y) <-> ([x]<=[y])%Z.
+Proof.
+ intros; unfold N.le, Zle; rewrite spec_compare_alt; intuition.
+Qed.
+
+Lemma spec_min : forall x y, [N.min x y] = Zmin [x] [y].
+Proof.
+ intros; unfold N.min, Zmin.
+ rewrite spec_compare_alt; destruct Zcompare; auto.
+Qed.
+
+Lemma spec_max : forall x y, [N.max x y] = Zmax [x] [y].
+Proof.
+ intros; unfold N.max, Zmax.
+ rewrite spec_compare_alt; destruct Zcompare; auto.
+Qed.
+
+Add Morphism N.compare with signature N.eq ==> N.eq ==> (@eq comparison) as compare_wd.
+Proof. 
+intros x x' Hx y y' Hy.
+rewrite 2 spec_compare_alt; rewrite Hx, Hy; intuition.
+Qed.
+
+Add Morphism N.lt with signature N.eq ==> N.eq ==> iff as NZlt_wd.
+Proof.
+intros x x' Hx y y' Hy; unfold N.lt; rewrite Hx, Hy; intuition.
+Qed.
+
+Add Morphism N.le with signature N.eq ==> N.eq ==> iff as NZle_wd.
+Proof.
+intros x x' Hx y y' Hy; unfold N.le; rewrite Hx, Hy; intuition.
+Qed.
+
+Add Morphism N.min with signature N.eq ==> N.eq ==> N.eq as NZmin_wd.
+Proof.
+intros; red; rewrite 2 spec_min; congruence.
+Qed.
+
+Add Morphism N.max with signature N.eq ==> N.eq ==> N.eq as NZmax_wd.
+Proof.
+intros; red; rewrite 2 spec_max; congruence.
+Qed.
+
+Theorem NZlt_eq_cases : forall n m, n <= m <-> n < m \/ n == m.
+Proof.
+intros.
+unfold N.eq; rewrite spec_lt, spec_le; omega.
+Qed.
+
+Theorem NZlt_irrefl : forall n, ~ n < n.
+Proof.
+intros; rewrite spec_lt; auto with zarith.
+Qed.
+
+Theorem NZlt_succ_r : forall n m, n < (N.succ m) <-> n <= m.
+Proof.
+intros; rewrite spec_lt, spec_le, N.spec_succ; omega.
+Qed.
+
+Theorem NZmin_l : forall n m, n <= m -> N.min n m == n.
+Proof.
+intros n m; unfold N.eq; rewrite spec_le, spec_min.
+generalize (Zmin_spec [n] [m]); omega.
+Qed.
+
+Theorem NZmin_r : forall n m, m <= n -> N.min n m == m.
+Proof.
+intros n m; unfold N.eq; rewrite spec_le, spec_min.
+generalize (Zmin_spec [n] [m]); omega.
+Qed.
+
+Theorem NZmax_l : forall n m, m <= n -> N.max n m == n.
+Proof.
+intros n m; unfold N.eq; rewrite spec_le, spec_max.
+generalize (Zmax_spec [n] [m]); omega.
+Qed.
+
+Theorem NZmax_r : forall n m, n <= m -> N.max n m == m.
+Proof.
+intros n m; unfold N.eq; rewrite spec_le, spec_max.
+generalize (Zmax_spec [n] [m]); omega.
+Qed.
+
+End NZOrdAxiomsMod.
+
+Theorem pred_0 : N.pred 0 == 0.
+Proof.
+red; rewrite N.spec_pred0; rewrite N.spec_0; auto.
+Qed.
+
+Definition recursion (A : Type) (a : A) (f : N.t -> A -> A) (n : N.t) :=
+  Nrect (fun _ => A) a (fun n a => f (N.of_N n) a) (N.to_N n).
+Implicit Arguments recursion [A].
+
+Theorem recursion_wd :
+forall (A : Type) (Aeq : relation A),
+  forall a a' : A, Aeq a a' ->
+    forall f f' : N.t -> A -> A, fun2_eq N.eq Aeq Aeq f f' ->
+      forall x x' : N.t, x == x' ->
+        Aeq (recursion a f x) (recursion a' f' x').
+Proof.
+unfold fun2_wd, N.eq, fun2_eq.
+intros A Aeq a a' Eaa' f f' Eff' x x' Exx'.
+unfold recursion.
+unfold N.to_N.
+rewrite <- Exx'; clear x' Exx'.
+replace (Zabs_N [x]) with (N_of_nat (Zabs_nat [x])).
+induction (Zabs_nat [x]).
+simpl; auto.
+rewrite N_of_S, 2 Nrect_step; auto.
+destruct [x]; simpl; auto.
+change (nat_of_P p) with (nat_of_N (Npos p)); apply N_of_nat_of_N.
+change (nat_of_P p) with (nat_of_N (Npos p)); apply N_of_nat_of_N.
+Qed.
+
+Theorem recursion_0 :
+  forall (A : Type) (a : A) (f : N.t -> A -> A), recursion a f 0 = a.
+Proof.
+intros A a f; unfold recursion, N.to_N; rewrite N.spec_0; simpl; auto.
+Qed.
+
+Theorem recursion_succ :
+  forall (A : Type) (Aeq : relation A) (a : A) (f : N.t -> A -> A),
+    Aeq a a -> fun2_wd N.eq Aeq Aeq f ->
+      forall n, Aeq (recursion a f (N.succ n)) (f n (recursion a f n)).
+Proof.
+unfold N.eq, recursion, fun2_wd; intros A Aeq a f EAaa f_wd n.
+replace (N.to_N (N.succ n)) with (Nsucc (N.to_N n)).
+rewrite Nrect_step.
+apply f_wd; auto.
+unfold N.to_N.
+rewrite N.spec_of_N, Z_of_N_abs, Zabs_eq; auto.
+ apply N.spec_pos.
+
+fold (recursion a f n).
+apply recursion_wd; auto.
+red; auto.
+red; auto.
+unfold N.to_N.
+
+rewrite N.spec_succ.
+change ([n]+1)%Z with (Zsucc [n]).
+apply Z_of_N_eq_rev.
+rewrite Z_of_N_succ.
+rewrite 2 Z_of_N_abs.
+rewrite 2 Zabs_eq; auto.
+generalize (N.spec_pos n); auto with zarith.
+apply N.spec_pos; auto.
+Qed.
+
+End NSig_NAxioms.