# Library Coq.Numbers.Natural.Abstract.NAxioms

From NZ, we obtain natural numbers just by stating that pred 0 == 0

Module Type NAxiom (Import NZ : NZDomainSig').
Axiom pred_0 : P 0 == 0.
End NAxiom.

Module Type NAxiomsMiniSig := NZOrdAxiomsSig <+ NAxiom.
Module Type NAxiomsMiniSig' := NZOrdAxiomsSig' <+ NAxiom.

Let's now add some more functions and their specification
Division Function : we reuse NZDiv.DivMod and NZDiv.NZDivCommon, and add to that a N-specific constraint.

Module Type NDivSpecific (Import N : NAxiomsMiniSig')(Import DM : DivMod' N).
Axiom mod_upper_bound : forall a b, b ~= 0 -> a mod b < b.
End NDivSpecific.

For all other functions, the NZ axiomatizations are enough.
We now group everything together.
It could also be interesting to have a constructive recursor function.

Module Type NAxiomsRec (Import NZ : NZDomainSig').

Parameter Inline recursion : forall {A : Type}, A -> (t -> A -> A) -> t -> A.

Declare Instance recursion_wd {A : Type} (Aeq : relation A) :
Proper (Aeq ==> (eq==>Aeq==>Aeq) ==> eq ==> Aeq) recursion.

Axiom recursion_0 :
forall {A} (a : A) (f : t -> A -> A), recursion a f 0 = a.

Axiom recursion_succ :
forall {A} (Aeq : relation A) (a : A) (f : t -> A -> A),
Aeq a a -> Proper (eq==>Aeq==>Aeq) f ->
forall n, Aeq (recursion a f (S n)) (f n (recursion a f n)).

End NAxiomsRec.

Module Type NAxiomsRecSig := NAxiomsMiniSig <+ NAxiomsRec.
Module Type NAxiomsRecSig' := NAxiomsMiniSig' <+ NAxiomsRec.

Module Type NAxiomsFullSig := NAxiomsSig <+ NAxiomsRec.
Module Type NAxiomsFullSig' := NAxiomsSig' <+ NAxiomsRec.