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authorGravatar Samuel Mimram <smimram@debian.org>2008-07-25 15:12:53 +0200
committerGravatar Samuel Mimram <smimram@debian.org>2008-07-25 15:12:53 +0200
commita0cfa4f118023d35b767a999d5a2ac4b082857b4 (patch)
treedabcac548e299fee1da464c93b3dba98484f45b1 /theories/Numbers/Natural/Abstract/NAxioms.v
parent2281410e38ef99d025ea77194585a9bc019fdaa9 (diff)
Imported Upstream version 8.2~beta3+dfsgupstream/8.2.beta3+dfsg
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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 *)
+(************************************************************************)
+(* Evgeny Makarov, INRIA, 2007 *)
+(************************************************************************)
+
+(*i $Id: NAxioms.v 11040 2008-06-03 00:04:16Z letouzey $ i*)
+
+Require Export NZAxioms.
+
+Set Implicit Arguments.
+
+Module Type NAxiomsSig.
+Declare Module Export NZOrdAxiomsMod : NZOrdAxiomsSig.
+
+Delimit Scope NatScope with Nat.
+Notation N := NZ.
+Notation Neq := NZeq.
+Notation N0 := NZ0.
+Notation N1 := (NZsucc NZ0).
+Notation S := NZsucc.
+Notation P := NZpred.
+Notation add := NZadd.
+Notation mul := NZmul.
+Notation sub := NZsub.
+Notation lt := NZlt.
+Notation le := NZle.
+Notation min := NZmin.
+Notation max := NZmax.
+Notation "x == y" := (Neq x y) (at level 70) : NatScope.
+Notation "x ~= y" := (~ Neq x y) (at level 70) : NatScope.
+Notation "0" := NZ0 : NatScope.
+Notation "1" := (NZsucc NZ0) : NatScope.
+Notation "x + y" := (NZadd x y) : NatScope.
+Notation "x - y" := (NZsub x y) : NatScope.
+Notation "x * y" := (NZmul x y) : NatScope.
+Notation "x < y" := (NZlt x y) : NatScope.
+Notation "x <= y" := (NZle x y) : NatScope.
+Notation "x > y" := (NZlt y x) (only parsing) : NatScope.
+Notation "x >= y" := (NZle y x) (only parsing) : NatScope.
+
+Open Local Scope NatScope.
+
+Parameter Inline recursion : forall A : Type, A -> (N -> A -> A) -> N -> A.
+Implicit Arguments recursion [A].
+
+Axiom pred_0 : P 0 == 0.
+
+Axiom recursion_wd : forall (A : Type) (Aeq : relation A),
+ forall a a' : A, Aeq a a' ->
+ forall f f' : N -> A -> A, fun2_eq Neq Aeq Aeq f f' ->
+ forall x x' : N, x == x' ->
+ Aeq (recursion a f x) (recursion a' f' x').
+
+Axiom recursion_0 :
+ forall (A : Type) (a : A) (f : N -> A -> A), recursion a f 0 = a.
+
+Axiom recursion_succ :
+ forall (A : Type) (Aeq : relation A) (a : A) (f : N -> A -> A),
+ Aeq a a -> fun2_wd Neq Aeq Aeq f ->
+ forall n : N, Aeq (recursion a f (S n)) (f n (recursion a f n)).
+
+(*Axiom dep_rec :
+ forall A : N -> Type, A 0 -> (forall n : N, A n -> A (S n)) -> forall n : N, A n.*)
+
+End NAxiomsSig.
+