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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 plus := NZplus.
Notation times := NZtimes.
Notation minus := NZminus.
Notation lt := NZlt.
Notation le := NZle.
Notation "x == y" := (NZeq x y) (at level 70) : NatScope.
Notation "x ~= y" := (~ NZeq x y) (at level 70) : NatScope.
Notation "0" := NZ0 : NatScope.
Notation "1" := (NZsucc NZ0) : NatScope.
Notation "x + y" := (NZplus x y) : NatScope.
Notation "x - y" := (NZminus x y) : NatScope.
Notation "x * y" := (NZtimes 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 : Set, A -> (N -> A -> A) -> N -> A.
Implicit Arguments recursion [A].
Axiom pred_0 : P 0 == 0.
Axiom recursion_wd : forall (A : Set) (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 : Set) (a : A) (f : N -> A -> A), recursion a f 0 = a.
Axiom recursion_succ :
forall (A : Set) (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)).
End NAxiomsSig.
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