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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
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.
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