# Library Coq.Numbers.Natural.Abstract.NAxioms

Require Export Bool NZAxioms NZParity NZPow NZSqrt NZLog NZDiv NZGcd NZBits.
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.
Module Type NAxiomsSig := NAxiomsMiniSig <+ OrderFunctions
<+ NZParity.NZParity <+ NZPow.NZPow <+ NZSqrt.NZSqrt <+ NZLog.NZLog2
<+ NZGcd.NZGcd <+ NZDiv.NZDiv <+ NZBits.NZBits <+ NZSquare.
Module Type NAxiomsSig' := NAxiomsMiniSig' <+ OrderFunctions'
<+ NZParity.NZParity <+ NZPow.NZPow' <+ NZSqrt.NZSqrt' <+ NZLog.NZLog2
<+ NZGcd.NZGcd' <+ NZDiv.NZDiv' <+ NZBits.NZBits' <+ NZSquare.
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.