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(************************************************************************)
(* * The Coq Proof Assistant / The Coq 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) *)
(************************************************************************)
(*
Tactic nsatz: proofs of polynomials equalities in an integral domain
(commutative ring without zero divisor).
Examples: see test-suite/success/Nsatz.v
Reification is done using type classes, defined in Ncring_tac.v
*)
Require Import List.
Require Import Setoid.
Require Import BinPos.
Require Import BinList.
Require Import Znumtheory.
Require Export Morphisms Setoid Bool.
Require Export Algebra_syntax.
Require Export Ncring.
Require Export Ncring_initial.
Require Export Ncring_tac.
Require Export Integral_domain.
Require Import DiscrR.
Require Import ZArith.
Require Import Lia.
Require Export NsatzTactic.
(** Make use of [discrR] in [nsatz] *)
Ltac nsatz_internal_discrR ::= discrR.
(* Real numbers *)
Require Import Reals.
Require Import RealField.
Lemma Rsth : Setoid_Theory R (@eq R).
constructor;red;intros;subst;trivial.
Qed.
Instance Rops: (@Ring_ops R 0%R 1%R Rplus Rmult Rminus Ropp (@eq R)).
Defined.
Instance Rri : (Ring (Ro:=Rops)).
constructor;
try (try apply Rsth;
try (unfold respectful, Proper; unfold equality; unfold eq_notation in *;
intros; try rewrite H; try rewrite H0; reflexivity)).
exact Rplus_0_l. exact Rplus_comm. symmetry. apply Rplus_assoc.
exact Rmult_1_l. exact Rmult_1_r. symmetry. apply Rmult_assoc.
exact Rmult_plus_distr_r. intros; apply Rmult_plus_distr_l.
exact Rplus_opp_r.
Defined.
Class can_compute_Z (z : Z) := dummy_can_compute_Z : True.
Hint Extern 0 (can_compute_Z ?v) =>
match isZcst v with true => exact I end : typeclass_instances.
Instance reify_IZR z lvar {_ : can_compute_Z z} : reify (PEc z) lvar (IZR z).
Defined.
Lemma R_one_zero: 1%R <> 0%R.
discrR.
Qed.
Instance Rcri: (Cring (Rr:=Rri)).
red. exact Rmult_comm. Defined.
Instance Rdi : (Integral_domain (Rcr:=Rcri)).
constructor.
exact Rmult_integral. exact R_one_zero. Defined.
(* Rational numbers *)
Require Import QArith.
Instance Qops: (@Ring_ops Q 0%Q 1%Q Qplus Qmult Qminus Qopp Qeq).
Defined.
Instance Qri : (Ring (Ro:=Qops)).
constructor.
try apply Q_Setoid.
apply Qplus_comp.
apply Qmult_comp.
apply Qminus_comp.
apply Qopp_comp.
exact Qplus_0_l. exact Qplus_comm. apply Qplus_assoc.
exact Qmult_1_l. exact Qmult_1_r. apply Qmult_assoc.
apply Qmult_plus_distr_l. intros. apply Qmult_plus_distr_r.
reflexivity. exact Qplus_opp_r.
Defined.
Lemma Q_one_zero: not (Qeq 1%Q 0%Q).
Proof. unfold Qeq. simpl. lia. Qed.
Instance Qcri: (Cring (Rr:=Qri)).
red. exact Qmult_comm. Defined.
Instance Qdi : (Integral_domain (Rcr:=Qcri)).
constructor.
exact Qmult_integral. exact Q_one_zero. Defined.
(* Integers *)
Lemma Z_one_zero: 1%Z <> 0%Z.
Proof. lia. Qed.
Instance Zcri: (Cring (Rr:=Zr)).
red. exact Z.mul_comm. Defined.
Instance Zdi : (Integral_domain (Rcr:=Zcri)).
constructor.
exact Zmult_integral. exact Z_one_zero. Defined.
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