aboutsummaryrefslogtreecommitdiffhomepage
path: root/theories/Numbers/Rational/BigQ/BigQ.v
blob: 850afe53459964517807c334465d8cff899c0cca (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
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
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(** * BigQ: an efficient implementation of rational numbers *)

(** Initial authors: Benjamin Gregoire, Laurent Thery, INRIA, 2007 *)

Require Export BigZ.
Require Import Field Qfield QSig QMake Orders GenericMinMax.

(** We choose for BigQ an implemention with
    multiple representation of 0: 0, 1/0, 2/0 etc.
    See [QMake.v] *)

(** First, we provide translations functions between [BigN] and [BigZ] *)

Module BigN_BigZ <: NType_ZType BigN.BigN BigZ.
 Definition Z_of_N := BigZ.Pos.
 Lemma spec_Z_of_N : forall n, BigZ.to_Z (Z_of_N n) = BigN.to_Z n.
 Proof.
 reflexivity.
 Qed.
 Definition Zabs_N := BigZ.to_N.
 Lemma spec_Zabs_N : forall z, BigN.to_Z (Zabs_N z) = Z.abs (BigZ.to_Z z).
 Proof.
 unfold Zabs_N; intros.
 rewrite BigZ.spec_to_Z, Z.mul_comm; apply Z.sgn_abs.
 Qed.
End BigN_BigZ.

(** This allows building [BigQ] out of [BigN] and [BigQ] via [QMake] *)

Delimit Scope bigQ_scope with bigQ.

Module BigQ <: QType <: OrderedTypeFull <: TotalOrder.
 Include QMake.Make BigN BigZ BigN_BigZ
  <+ !QProperties <+ HasEqBool2Dec
  <+ !MinMaxLogicalProperties <+ !MinMaxDecProperties.
 Ltac order := Private_Tac.order.
End BigQ.

(** Notations about [BigQ] *)

Local Open Scope bigQ_scope.

Notation bigQ := BigQ.t.
Bind Scope bigQ_scope with bigQ BigQ.t BigQ.t_.
(** As in QArith, we use [#] to denote fractions *)
Notation "p # q" := (BigQ.Qq p q) (at level 55, no associativity) : bigQ_scope.
Local Notation "0" := BigQ.zero : bigQ_scope.
Local Notation "1" := BigQ.one : bigQ_scope.
Infix "+" := BigQ.add : bigQ_scope.
Infix "-" := BigQ.sub : bigQ_scope.
Notation "- x" := (BigQ.opp x) : bigQ_scope.
Infix "*" := BigQ.mul : bigQ_scope.
Infix "/" := BigQ.div : bigQ_scope.
Infix "^" := BigQ.power : bigQ_scope.
Infix "?=" := BigQ.compare : bigQ_scope.
Infix "==" := BigQ.eq : bigQ_scope.
Notation "x != y" := (~x==y) (at level 70, no associativity) : bigQ_scope.
Infix "<" := BigQ.lt : bigQ_scope.
Infix "<=" := BigQ.le : bigQ_scope.
Notation "x > y" := (BigQ.lt y x) (only parsing) : bigQ_scope.
Notation "x >= y" := (BigQ.le y x) (only parsing) : bigQ_scope.
Notation "x < y < z" := (x<y /\ y<z) : bigQ_scope.
Notation "x < y <= z" := (x<y /\ y<=z) : bigQ_scope.
Notation "x <= y < z" := (x<=y /\ y<z) : bigQ_scope.
Notation "x <= y <= z" := (x<=y /\ y<=z) : bigQ_scope.
Notation "[ q ]" := (BigQ.to_Q q) : bigQ_scope.

(** [BigQ] is a field *)

Lemma BigQfieldth :
 field_theory 0 1 BigQ.add BigQ.mul BigQ.sub BigQ.opp
  BigQ.div BigQ.inv BigQ.eq.
Proof.
constructor.
constructor.
exact BigQ.add_0_l. exact BigQ.add_comm. exact BigQ.add_assoc.
exact BigQ.mul_1_l. exact BigQ.mul_comm. exact BigQ.mul_assoc.
exact BigQ.mul_add_distr_r. exact BigQ.sub_add_opp.
exact BigQ.add_opp_diag_r. exact BigQ.neq_1_0.
exact BigQ.div_mul_inv. exact BigQ.mul_inv_diag_l.
Qed.

Declare Equivalent Keys pow_N pow_pos.

Lemma BigQpowerth :
 power_theory 1 BigQ.mul BigQ.eq Z.of_N BigQ.power.
Proof.
constructor. intros. BigQ.qify.
replace ([r] ^ Z.of_N n)%Q with (pow_N 1 Qmult [r] n)%Q by (now destruct n).
destruct n. reflexivity.
induction p; simpl; auto; rewrite ?BigQ.spec_mul, ?IHp; reflexivity.
Qed.

Ltac isBigQcst t :=
 match t with
 | BigQ.Qz ?t => isBigZcst t
 | BigQ.Qq ?n ?d => match isBigZcst n with
             | true => isBigNcst d
             | false => constr:(false)
             end
 | BigQ.zero => constr:(true)
 | BigQ.one => constr:(true)
 | BigQ.minus_one => constr:(true)
 | _ => constr:(false)
 end.

Ltac BigQcst t :=
 match isBigQcst t with
 | true => constr:(t)
 | false => constr:(NotConstant)
 end.

Add Field BigQfield : BigQfieldth
 (decidable BigQ.eqb_correct,
  completeness BigQ.eqb_complete,
  constants [BigQcst],
  power_tac BigQpowerth [Qpow_tac]).

Section TestField.

Let ex1 : forall x y z, (x+y)*z ==  (x*z)+(y*z).
  intros.
  ring.
Qed.

Let ex8 : forall x, x ^ 2 == x*x.
  intro.
  ring.
Qed.

Let ex10 : forall x y, y!=0 -> (x/y)*y == x.
intros.
field.
auto.
Qed.

End TestField.

(** [BigQ] can also benefit from an "order" tactic *)

Ltac bigQ_order := BigQ.order.

Section TestOrder.
Let test : forall x y : bigQ, x<=y -> y<=x -> x==y.
Proof. bigQ_order. Qed.
End TestOrder.

(** We can also reason by switching to QArith thanks to tactic
    BigQ.qify. *)

Section TestQify.
Let test : forall x : bigQ, 0+x == 1*x.
Proof. intro x. BigQ.qify. ring. Qed.
End TestQify.