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
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
|
(************************************************************************)
(* 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 *)
(************************************************************************)
(** Basic specifications : sets that may contain logical information *)
Set Implicit Arguments.
Require Import Notations.
Require Import Datatypes.
Require Import Logic.
(** Subsets and Sigma-types *)
(** [(sig A P)], or more suggestively [{x:A | P x}], denotes the subset
of elements of the type [A] which satisfy the predicate [P].
Similarly [(sig2 A P Q)], or [{x:A | P x & Q x}], denotes the subset
of elements of the type [A] which satisfy both [P] and [Q]. *)
Inductive sig (A:Type) (P:A -> Prop) : Type :=
exist : forall x:A, P x -> sig P.
Inductive sig2 (A:Type) (P Q:A -> Prop) : Type :=
exist2 : forall x:A, P x -> Q x -> sig2 P Q.
(** [(sigT A P)], or more suggestively [{x:A & (P x)}] is a Sigma-type.
Similarly for [(sigT2 A P Q)], also written [{x:A & (P x) & (Q x)}]. *)
Inductive sigT (A:Type) (P:A -> Type) : Type :=
existT : forall x:A, P x -> sigT P.
Inductive sigT2 (A:Type) (P Q:A -> Type) : Type :=
existT2 : forall x:A, P x -> Q x -> sigT2 P Q.
(* Notations *)
Arguments sig (A P)%type.
Arguments sig2 (A P Q)%type.
Arguments sigT (A P)%type.
Arguments sigT2 (A P Q)%type.
Notation "{ x | P }" := (sig (fun x => P)) : type_scope.
Notation "{ x | P & Q }" := (sig2 (fun x => P) (fun x => Q)) : type_scope.
Notation "{ x : A | P }" := (sig (A:=A) (fun x => P)) : type_scope.
Notation "{ x : A | P & Q }" := (sig2 (A:=A) (fun x => P) (fun x => Q)) :
type_scope.
Notation "{ x : A & P }" := (sigT (A:=A) (fun x => P)) : type_scope.
Notation "{ x : A & P & Q }" := (sigT2 (A:=A) (fun x => P) (fun x => Q)) :
type_scope.
Add Printing Let sig.
Add Printing Let sig2.
Add Printing Let sigT.
Add Printing Let sigT2.
(** Projections of [sig]
An element [y] of a subset [{x:A | (P x)}] is the pair of an [a]
of type [A] and of a proof [h] that [a] satisfies [P]. Then
[(proj1_sig y)] is the witness [a] and [(proj2_sig y)] is the
proof of [(P a)] *)
(* Set Universe Polymorphism. *)
Section Subset_projections.
Variable A : Type.
Variable P : A -> Prop.
Definition proj1_sig (e:sig P) := match e with
| exist _ a b => a
end.
Definition proj2_sig (e:sig P) :=
match e return P (proj1_sig e) with
| exist _ a b => b
end.
End Subset_projections.
(** [sig2] of a predicate can be projected to a [sig].
This allows [proj1_sig] and [proj2_sig] to be usable with [sig2].
The [let] statements occur in the body of the [exist] so that
[proj1_sig] of a coerced [X : sig2 P Q] will unify with [let (a,
_, _) := X in a] *)
Definition sig_of_sig2 (A : Type) (P Q : A -> Prop) (X : sig2 P Q) : sig P
:= exist P
(let (a, _, _) := X in a)
(let (x, p, _) as s return (P (let (a, _, _) := s in a)) := X in p).
(** Projections of [sig2]
An element [y] of a subset [{x:A | (P x) & (Q x)}] is the triple
of an [a] of type [A], a of a proof [h] that [a] satisfies [P],
and a proof [h'] that [a] satisfies [Q]. Then
[(proj1_sig (sig_of_sig2 y))] is the witness [a],
[(proj2_sig (sig_of_sig2 y))] is the proof of [(P a)], and
[(proj3_sig y)] is the proof of [(Q a)]. *)
Section Subset_projections2.
Variable A : Type.
Variables P Q : A -> Prop.
Definition proj3_sig (e : sig2 P Q) :=
let (a, b, c) return Q (proj1_sig (sig_of_sig2 e)) := e in c.
End Subset_projections2.
(** Projections of [sigT]
An element [x] of a sigma-type [{y:A & P y}] is a dependent pair
made of an [a] of type [A] and an [h] of type [P a]. Then,
[(projT1 x)] is the first projection and [(projT2 x)] is the
second projection, the type of which depends on the [projT1]. *)
Section Projections.
Variable A : Type.
Variable P : A -> Type.
Definition projT1 (x:sigT P) : A := match x with
| existT _ a _ => a
end.
Definition projT2 (x:sigT P) : P (projT1 x) :=
match x return P (projT1 x) with
| existT _ _ h => h
end.
End Projections.
(** [sigT2] of a predicate can be projected to a [sigT].
This allows [projT1] and [projT2] to be usable with [sigT2].
The [let] statements occur in the body of the [existT] so that
[projT1] of a coerced [X : sigT2 P Q] will unify with [let (a,
_, _) := X in a] *)
Definition sigT_of_sigT2 (A : Type) (P Q : A -> Type) (X : sigT2 P Q) : sigT P
:= existT P
(let (a, _, _) := X in a)
(let (x, p, _) as s return (P (let (a, _, _) := s in a)) := X in p).
(** Projections of [sigT2]
An element [x] of a sigma-type [{y:A & P y & Q y}] is a dependent
pair made of an [a] of type [A], an [h] of type [P a], and an [h']
of type [Q a]. Then, [(projT1 (sigT_of_sigT2 x))] is the first
projection, [(projT2 (sigT_of_sigT2 x))] is the second projection,
and [(projT3 x)] is the third projection, the types of which
depends on the [projT1]. *)
Section Projections2.
Variable A : Type.
Variables P Q : A -> Type.
Definition projT3 (e : sigT2 P Q) :=
let (a, b, c) return Q (projT1 (sigT_of_sigT2 e)) := e in c.
End Projections2.
(** [sigT] of a predicate is equivalent to [sig] *)
Definition sig_of_sigT (A : Type) (P : A -> Prop) (X : sigT P) : sig P
:= exist P (projT1 X) (projT2 X).
Definition sigT_of_sig (A : Type) (P : A -> Prop) (X : sig P) : sigT P
:= existT P (proj1_sig X) (proj2_sig X).
(** [sigT2] of a predicate is equivalent to [sig2] *)
Definition sig2_of_sigT2 (A : Type) (P Q : A -> Prop) (X : sigT2 P Q) : sig2 P Q
:= exist2 P Q (projT1 (sigT_of_sigT2 X)) (projT2 (sigT_of_sigT2 X)) (projT3 X).
Definition sigT2_of_sig2 (A : Type) (P Q : A -> Prop) (X : sig2 P Q) : sigT2 P Q
:= existT2 P Q (proj1_sig (sig_of_sig2 X)) (proj2_sig (sig_of_sig2 X)) (proj3_sig X).
(** [sumbool] is a boolean type equipped with the justification of
their value *)
Inductive sumbool (A B:Prop) : Set :=
| left : A -> {A} + {B}
| right : B -> {A} + {B}
where "{ A } + { B }" := (sumbool A B) : type_scope.
Add Printing If sumbool.
Arguments left {A B} _, [A] B _.
Arguments right {A B} _ , A [B] _.
(** [sumor] is an option type equipped with the justification of why
it may not be a regular value *)
Inductive sumor (A:Type) (B:Prop) : Type :=
| inleft : A -> A + {B}
| inright : B -> A + {B}
where "A + { B }" := (sumor A B) : type_scope.
Add Printing If sumor.
Arguments inleft {A B} _ , [A] B _.
Arguments inright {A B} _ , A [B] _.
(* Unset Universe Polymorphism. *)
(** Various forms of the axiom of choice for specifications *)
Section Choice_lemmas.
Variables S S' : Set.
Variable R : S -> S' -> Prop.
Variable R' : S -> S' -> Set.
Variables R1 R2 : S -> Prop.
Lemma Choice :
(forall x:S, {y:S' | R x y}) -> {f:S -> S' | forall z:S, R z (f z)}.
Proof.
intro H.
exists (fun z => proj1_sig (H z)).
intro z; destruct (H z); assumption.
Defined.
Lemma Choice2 :
(forall x:S, {y:S' & R' x y}) -> {f:S -> S' & forall z:S, R' z (f z)}.
Proof.
intro H.
exists (fun z => projT1 (H z)).
intro z; destruct (H z); assumption.
Defined.
Lemma bool_choice :
(forall x:S, {R1 x} + {R2 x}) ->
{f:S -> bool | forall x:S, f x = true /\ R1 x \/ f x = false /\ R2 x}.
Proof.
intro H.
exists (fun z:S => if H z then true else false).
intro z; destruct (H z); auto.
Defined.
End Choice_lemmas.
Section Dependent_choice_lemmas.
Variables X : Set.
Variable R : X -> X -> Prop.
Lemma dependent_choice :
(forall x:X, {y | R x y}) ->
forall x0, {f : nat -> X | f O = x0 /\ forall n, R (f n) (f (S n))}.
Proof.
intros H x0.
set (f:=fix f n := match n with O => x0 | S n' => proj1_sig (H (f n')) end).
exists f.
split. reflexivity.
induction n; simpl; apply proj2_sig.
Defined.
End Dependent_choice_lemmas.
(** A result of type [(Exc A)] is either a normal value of type [A] or
an [error] :
[Inductive Exc [A:Type] : Type := value : A->(Exc A) | error : (Exc A)].
It is implemented using the option type. *)
Section Exc.
Variable A : Type.
Definition Exc := option A.
Definition value := @Some A.
Definition error := @None A.
End Exc.
Arguments error {A}.
Definition except := False_rec. (* for compatibility with previous versions *)
Arguments except [P] _.
Theorem absurd_set : forall (A:Prop) (C:Set), A -> ~ A -> C.
Proof.
intros A C h1 h2.
apply False_rec.
apply (h2 h1).
Defined.
Hint Resolve left right inleft inright: core v62.
Hint Resolve exist exist2 existT existT2: core.
(* Compatibility *)
Notation sigS := sigT (compat "8.2").
Notation existS := existT (compat "8.2").
Notation sigS_rect := sigT_rect (compat "8.2").
Notation sigS_rec := sigT_rec (compat "8.2").
Notation sigS_ind := sigT_ind (compat "8.2").
Notation projS1 := projT1 (compat "8.2").
Notation projS2 := projT2 (compat "8.2").
Notation sigS2 := sigT2 (compat "8.2").
Notation existS2 := existT2 (compat "8.2").
Notation sigS2_rect := sigT2_rect (compat "8.2").
Notation sigS2_rec := sigT2_rec (compat "8.2").
Notation sigS2_ind := sigT2_ind (compat "8.2").
|
