(************************************************************************)
(*         *      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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Require Export Ncring.
Require Export Ncring_polynom.
Set Implicit Arguments.
(* An object to return when an expression is not recognized as a constant *)
Definition NotConstant := false.
(** Z is a ring and a setoid*)

(* Specifications: *)
Lemma Zsth : Equivalence (@eq Z).
Lemma ARgen_phiPOS_Psucc : forall x,
gen_phiPOS1 (Pos.succ x) == 1 + (gen_phiPOS1 x).
Lemma ARgen_phiPOS_add : forall x y,
gen_phiPOS1 (x + y) == (gen_phiPOS1 x) + (gen_phiPOS1 y).
Lemma ARgen_phiPOS_mult :
forall x y, gen_phiPOS1 (x * y) == gen_phiPOS1 x * gen_phiPOS1 y.
(*morphisms are extensionally equal*)
Lemma same_genZ : forall x, [x] == gen_phiZ1 x.
Lemma gen_Zeqb_ok : forall x y,
Zeq_bool x y = true -> [x] == [y].
Lemma gen_phiZ1_add_pos_neg : forall x y,
gen_phiZ1 (Z.pos_sub x y)
== gen_phiPOS1 x + -gen_phiPOS1 y.
Lemma match_compOpp : forall x (B:Type) (be bl bg:B),
match CompOpp x with Eq => be | Lt => bl | Gt => bg end
= match x with Eq => be | Lt => bg | Gt => bl end.
Lemma gen_phiZ_opp : forall x, [- x] == - [x].
Lemma gen_phiZ_mul : forall x y, [x * y] == [x] * [y].
Lemma gen_phiZ_ext : forall x y : Z, x = y -> [x] == [y].