blob: 21f0f30140f027a99fccf5e673f3e25680e61604 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
|
(************************************************************************)
(* * 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.
#[global]
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.
|
