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Require Export NumPrelude.
Module Type NZAxiomsSig.
Parameter Inline NZ : Set.
Parameter Inline NZeq : NZ -> NZ -> Prop.
Parameter Inline NZ0 : NZ.
Parameter Inline NZsucc : NZ -> NZ.
Parameter Inline NZpred : NZ -> NZ.
Parameter Inline NZplus : NZ -> NZ -> NZ.
Parameter Inline NZminus : NZ -> NZ -> NZ.
Parameter Inline NZtimes : NZ -> NZ -> NZ.
Axiom NZE_equiv : equiv NZ NZeq.
Add Relation NZ NZeq
reflexivity proved by (proj1 NZE_equiv)
symmetry proved by (proj2 (proj2 NZE_equiv))
transitivity proved by (proj1 (proj2 NZE_equiv))
as NZE_rel.
Add Morphism NZsucc with signature NZeq ==> NZeq as NZsucc_wd.
Add Morphism NZpred with signature NZeq ==> NZeq as NZpred_wd.
Add Morphism NZplus with signature NZeq ==> NZeq ==> NZeq as NZplus_wd.
Add Morphism NZminus with signature NZeq ==> NZeq ==> NZeq as NZminus_wd.
Add Morphism NZtimes with signature NZeq ==> NZeq ==> NZeq as NZtimes_wd.
Delimit Scope NatIntScope with NatInt.
Open Local Scope NatIntScope.
Notation "x == y" := (NZeq x y) (at level 70) : NatIntScope.
Notation "x ~= y" := (~ NZeq x y) (at level 70) : NatIntScope.
Notation "0" := NZ0 : NatIntScope.
Notation S := NZsucc.
Notation P := NZpred.
Notation "1" := (S 0) : NatIntScope.
Notation "x + y" := (NZplus x y) : NatIntScope.
Notation "x - y" := (NZminus x y) : NatIntScope.
Notation "x * y" := (NZtimes x y) : NatIntScope.
Axiom NZpred_succ : forall n : NZ, P (S n) == n.
Axiom NZinduction :
forall A : NZ -> Prop, predicate_wd NZeq A ->
A 0 -> (forall n : NZ, A n <-> A (S n)) -> forall n : NZ, A n.
Axiom NZplus_0_l : forall n : NZ, 0 + n == n.
Axiom NZplus_succ_l : forall n m : NZ, (S n) + m == S (n + m).
Axiom NZminus_0_r : forall n : NZ, n - 0 == n.
Axiom NZminus_succ_r : forall n m : NZ, n - (S m) == P (n - m).
Axiom NZtimes_0_r : forall n : NZ, n * 0 == 0.
Axiom NZtimes_succ_r : forall n m : NZ, n * (S m) == n * m + n.
End NZAxiomsSig.
Module Type NZOrdAxiomsSig.
Declare Module Export NZAxiomsMod : NZAxiomsSig.
Open Local Scope NatIntScope.
Parameter Inline NZlt : NZ -> NZ -> Prop.
Parameter Inline NZle : NZ -> NZ -> Prop.
Add Morphism NZlt with signature NZeq ==> NZeq ==> iff as NZlt_wd.
Add Morphism NZle with signature NZeq ==> NZeq ==> iff as NZle_wd.
Notation "x < y" := (NZlt x y) : NatIntScope.
Notation "x <= y" := (NZle x y) : NatIntScope.
Notation "x > y" := (NZlt y x) (only parsing) : NatIntScope.
Notation "x >= y" := (NZle y x) (only parsing) : NatIntScope.
Axiom NZle_lt_or_eq : forall n m : NZ, n <= m <-> n < m \/ n == m.
Axiom NZlt_irrefl : forall n : NZ, ~ (n < n).
Axiom NZlt_succ_le : forall n m : NZ, n < S m <-> n <= m.
End NZOrdAxiomsSig.
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