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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
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(** Initial Author : Evgeny Makarov, INRIA, 2007 *)
(** * Axioms for a domain with [zero], [succ], [pred]. *)
From Stdlib.Structures Require Export Equalities Orders.
(** We use the [Equalities] module in order to work with a general decidable equality [eq]. *)
(** The [Orders] module contains module types about orders [lt] and [le] in [Prop]. *)
From Stdlib.Numbers Require Export NumPrelude.
From Stdlib.Structures Require Export GenericMinMax.
(** The [GenericMinMax] module adds specifications and basic lemmas for [min] and [max] operators on ordered types. *)
(** At the end of the day, this file defines the module types [NZDecOrdAxiomsSig] and [NZDecOrdAxiomsSig'] (with notations) : [[ Module Type NZDecOrdAxiomsSig' = Sig Parameter t : Type. Parameter eq : t -> t -> Prop. Parameter eq_equiv : Equivalence eq. Parameter zero : t. Parameter succ : t -> t. Parameter pred : t -> t. Parameter succ_wd : Proper (eq ==> eq) succ. Parameter pred_wd : Proper (eq ==> eq) pred. Parameter pred_succ : forall n : t, eq (pred (succ n)) n. Parameter bi_induction : forall A : t -> Prop, Proper (eq ==> iff) A -> A zero -> (forall n : t, A n <-> A (succ n)) -> forall n : t, A n. Parameter one : t. Parameter two : t. Parameter one_succ : eq one (succ zero). Parameter two_succ : eq two (succ one). Parameter lt : t -> t -> Prop. Parameter le : t -> t -> Prop. Parameter lt_wd : Proper (eq ==> eq ==> iff) lt. Parameter lt_eq_cases : forall n m : t, le n m <-> lt n m \/ eq n m. Parameter lt_irrefl : forall n : t, ~ lt n n. Parameter lt_succ_r : forall n m : t, lt n (succ m) <-> le n m. Parameter add : t -> t -> t. Parameter sub : t -> t -> t. Parameter mul : t -> t -> t. Parameter add_wd : Proper (eq ==> eq ==> eq) add. Parameter sub_wd : Proper (eq ==> eq ==> eq) sub. Parameter mul_wd : Proper (eq ==> eq ==> eq) mul. Parameter add_0_l : forall n : t, eq (add zero n) n. Parameter add_succ_l : forall n m : t, eq (add (succ n) m) (succ (add n m)). Parameter sub_0_r : forall n : t, eq (sub n zero) n. Parameter sub_succ_r : forall n m : t, eq (sub n (succ m)) (pred (sub n m)). Parameter mul_0_l : forall n : t, eq (mul zero n) zero. Parameter mul_succ_l : forall n m : t, eq (mul (succ n) m) (add (mul n m) m). Parameter max : t -> t -> t. Parameter max_l : forall x y : t, le y x -> eq (max x y) x. Parameter max_r : forall x y : t, le x y -> eq (max x y) y. Parameter min : t -> t -> t. Parameter min_l : forall x y : t, le x y -> eq (min x y) x. Parameter min_r : forall x y : t, le y x -> eq (min x y) y. Parameter compare : t -> t -> comparison. Parameter compare_spec : forall x y : t, CompareSpec (eq x y) (lt x y) (lt y x) (compare x y). End ]] *)
(** ** Axiomatization of a domain with [zero], [succ], [pred] and a bi-directional induction principle. *)
(* NB: This was Pierre Letouzey's conclusion in the (now deprecated) NZDomain file. *)
(** We require [P (S n) = n] but not the other way around, since this domain is meant to be either N or Z. In fact it can be a few other things,  S is always injective, P is always surjective  (thanks to [pred_succ]).  I) If S is not surjective, we have an initial point, which is unique. This bottom is below zero: we have N shifted (or not) to the left. P cannot be injective: P init = P (S (P init)). (P init) can be arbitrary.  II) If S is surjective, we have [forall n, S (P n) = n], S and P are bijective and reciprocal.  IIa) if [exists k<>O, 0 == S^k 0], then we have a cyclic structure Z/nZ IIb) otherwise, we have Z *)
(** The [Typ] module type in [Equalities] only has a parameter [t : Type]. *)
Module Type ZeroSuccPred (Import T:Typ).
Parameter Inline(20) zero : t.
Parameter Inline(50) succ : t -> t.
Parameter Inline pred : t -> t.
End ZeroSuccPred.
Module Type ZeroSuccPredNotation (T:Typ)(Import NZ:ZeroSuccPred T).
Notation "0" := zero.
Notation S := succ.
Notation P := pred.
End ZeroSuccPredNotation.
Module Type ZeroSuccPred' (T:Typ) :=
ZeroSuccPred T <+ ZeroSuccPredNotation T.
(** The [Eq'] module type in [Equalities] is a [Type] [t] with a binary predicate [eq] denoted [==]. The negation of [==] is denoted [~=]. *)
Module Type IsNZDomain (Import E:Eq')(Import NZ:ZeroSuccPred' E).
#[global]
Declare Instance succ_wd : Proper (eq ==> eq) S.
#[global]
Declare Instance pred_wd : Proper (eq ==> eq) P.

Axiom bi_induction :
forall A : t -> Prop, Proper (eq==>iff) A ->
A 0 -> (forall n, A n <-> A (S n)) -> forall n, A n.