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
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
Require Import OrderedRing.
Require Import RingMicromega.
Require Import ZArith.
Require Import InitialRing.
Require Import Setoid.
Import OrderedRingSyntax.
Set Implicit Arguments.
Section InitialMorphism.
Variable R : Type.
Variables rO rI : R.
Variables rplus rtimes rminus: R -> R -> R.
Variable ropp : R -> R.
Variables req rle rlt : R -> R -> Prop.
Variable sor : SOR rO rI rplus rtimes rminus ropp req rle rlt.
Notation "0" := rO.
Notation "1" := rI.
Notation "x + y" := (rplus x y).
Notation "x * y " := (rtimes x y).
Notation "x - y " := (rminus x y).
Notation "- x" := (ropp x).
Notation "x == y" := (req x y).
Notation "x ~= y" := (~ req x y).
Notation "x <= y" := (rle x y).
Notation "x < y" := (rlt x y).
Lemma req_refl : forall x, req x x.
Proof.
destruct sor.(SORsetoid) as (Equivalence_Reflexive,_,_).
apply Equivalence_Reflexive.
Qed.
Lemma req_sym : forall x y, req x y -> req y x.
Proof.
destruct sor.(SORsetoid) as (_,Equivalence_Symmetric,_).
apply Equivalence_Symmetric.
Qed.
Lemma req_trans : forall x y z, req x y -> req y z -> req x z.
Proof.
destruct sor.(SORsetoid) as (_,_,Equivalence_Transitive).
apply Equivalence_Transitive.
Qed.
Add Relation R req
reflexivity proved by sor.(SORsetoid).(@Equivalence_Reflexive _ _)
symmetry proved by sor.(SORsetoid).(@Equivalence_Symmetric _ _)
transitivity proved by sor.(SORsetoid).(@Equivalence_Transitive _ _)
as sor_setoid.
Add Morphism rplus with signature req ==> req ==> req as rplus_morph.
Proof.
exact sor.(SORplus_wd).
Qed.
Add Morphism rtimes with signature req ==> req ==> req as rtimes_morph.
Proof.
exact sor.(SORtimes_wd).
Qed.
Add Morphism ropp with signature req ==> req as ropp_morph.
Proof.
exact sor.(SORopp_wd).
Qed.
Add Morphism rle with signature req ==> req ==> iff as rle_morph.
Proof.
exact sor.(SORle_wd).
Qed.
Add Morphism rlt with signature req ==> req ==> iff as rlt_morph.
Proof.
exact sor.(SORlt_wd).
Qed.
Add Morphism rminus with signature req ==> req ==> req as rminus_morph.
Proof.
exact (rminus_morph sor).
Qed.
Ltac le_less := rewrite (Rle_lt_eq sor); left; try assumption.
Ltac le_equal := rewrite (Rle_lt_eq sor); right; try reflexivity; try assumption.
Definition gen_order_phi_Z : Z -> R := gen_phiZ 0 1 rplus rtimes ropp.
Declare Equivalent Keys gen_order_phi_Z gen_phiZ.
Notation phi_pos := (gen_phiPOS 1 rplus rtimes).
Notation phi_pos1 := (gen_phiPOS1 1 rplus rtimes).
Notation "[ x ]" := (gen_order_phi_Z x).
Lemma ring_ops_wd : ring_eq_ext rplus rtimes ropp req.
Proof.
constructor.
exact rplus_morph.
exact rtimes_morph.
exact ropp_morph.
Qed.
Lemma Zring_morph :
ring_morph 0 1 rplus rtimes rminus ropp req
0%Z 1%Z Z.add Z.mul Z.sub Z.opp
Zeq_bool gen_order_phi_Z.
Proof.
exact (gen_phiZ_morph sor.(SORsetoid) ring_ops_wd sor.(SORrt)).
Qed.
Lemma phi_pos1_pos : forall x : positive, 0 < phi_pos1 x.
Proof.
induction x as [x IH | x IH |]; simpl;
try apply (Rplus_pos_pos sor); try apply (Rtimes_pos_pos sor); try apply (Rplus_pos_pos sor);
try apply (Rlt_0_1 sor); assumption.
Qed.
Lemma phi_pos1_succ : forall x : positive, phi_pos1 (Pos.succ x) == 1 + phi_pos1 x.
Proof.
exact (ARgen_phiPOS_Psucc sor.(SORsetoid) ring_ops_wd
(Rth_ARth sor.(SORsetoid) ring_ops_wd sor.(SORrt))).
Qed.
Lemma clt_pos_morph : forall x y : positive, (x < y)%positive -> phi_pos1 x < phi_pos1 y.
Proof.
intros x y H. pattern y; apply Pos.lt_ind with x.
rewrite phi_pos1_succ; apply (Rlt_succ_r sor).
clear y H; intros y _ H. rewrite phi_pos1_succ. now apply (Rlt_lt_succ sor).
assumption.
Qed.
Lemma clt_morph : forall x y : Z, (x < y)%Z -> [x] < [y].
Proof.
intros x y H.
do 2 rewrite (same_genZ sor.(SORsetoid) ring_ops_wd sor.(SORrt));
destruct x; destruct y; simpl in *; try discriminate.
apply phi_pos1_pos.
now apply clt_pos_morph.
apply <- (Ropp_neg_pos sor); apply phi_pos1_pos.
apply (Rlt_trans sor) with 0. apply <- (Ropp_neg_pos sor); apply phi_pos1_pos.
apply phi_pos1_pos.
apply -> (Ropp_lt_mono sor); apply clt_pos_morph.
red. now rewrite Pos.compare_antisym.
Qed.
Lemma Zcleb_morph : forall x y : Z, Z.leb x y = true -> [x] <= [y].
Proof.
unfold Z.leb; intros x y H.
case_eq (x ?= y)%Z; intro H1; rewrite H1 in H.
le_equal. apply Zring_morph.(morph_eq). unfold Zeq_bool; now rewrite H1.
le_less. now apply clt_morph.
discriminate.
Qed.
Lemma Zcneqb_morph : forall x y : Z, Zeq_bool x y = false -> [x] ~= [y].
Proof.
intros x y H. unfold Zeq_bool in H.
case_eq (Z.compare x y); intro H1; rewrite H1 in *; (discriminate || clear H).
apply (Rlt_neq sor). now apply clt_morph.
fold (x > y)%Z in H1. rewrite Z.gt_lt_iff in H1.
apply (Rneq_symm sor). apply (Rlt_neq sor). now apply clt_morph.
Qed.
End InitialMorphism.
|
