base.v 24.3 KB
Newer Older
1
2
3
4
5
6
(* Copyright (c) 2012, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects type class interfaces, notations, and general theorems
that are used throughout the whole development. Most importantly it contains
abstract interfaces for ordered structures, collections, and various other data
structures. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
7
8
9
10
Global Generalizable All Variables.
Global Set Automatic Coercions Import.
Require Export Morphisms RelationClasses List Bool Utf8 Program Setoid NArith.

11
12
13
14
15
(** * General *)
(** The following coercion allows us to use Booleans as propositions. *)
Coercion Is_true : bool >-> Sortclass.

(** Ensure that [simpl] unfolds [id] and [compose] when fully applied. *)
16
17
18
Arguments id _ _/.
Arguments compose _ _ _ _ _ _ /.

19
20
21
22
(** Change [True] and [False] into notations in order to enable overloading.
We will use this in the file [assertions] to give [True] and [False] a
different interpretation in [assert_scope] used for assertions of our axiomatic
semantics. *)
23
24
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
25

26
27
(** Throughout this development we use [C_scope] for all general purpose
notations that do not belong to a more specific scope. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
28
29
30
Delimit Scope C_scope with C.
Global Open Scope C_scope.

31
(** Introduce some Haskell style like notations. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
32
33
34
35
36
37
38
39
40
Notation "(=)" := eq (only parsing) : C_scope.
Notation "( x =)" := (eq x) (only parsing) : C_scope.
Notation "(= x )" := (λ y, eq y x) (only parsing) : C_scope.
Notation "(≠)" := (λ x y, x  y) (only parsing) : C_scope.
Notation "( x ≠)" := (λ y, x  y) (only parsing) : C_scope.
Notation "(≠ x )" := (λ y, y  x) (only parsing) : C_scope.

Hint Extern 0 (?x = ?x) => reflexivity.

41
42
43
Notation "(→)" := (λ x y, x  y) (only parsing) : C_scope.
Notation "( T →)" := (λ y, T  y) (only parsing) : C_scope.
Notation "(→ T )" := (λ y, y  T) (only parsing) : C_scope.
44

45
Notation "t $ r" := (t r)
46
  (at level 65, right associativity, only parsing) : C_scope.
47
48
49
Notation "($)" := (λ f x, f x) (only parsing) : C_scope.
Notation "($ x )" := (λ f, f x) (only parsing) : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
50
51
52
53
Infix "∘" := compose : C_scope.
Notation "(∘)" := compose (only parsing) : C_scope.
Notation "( f ∘)" := (compose f) (only parsing) : C_scope.
Notation "(∘ f )" := (λ g, compose g f) (only parsing) : C_scope.
54
55
56

(** Set convenient implicit arguments for [existT] and introduce notations. *)
Arguments existT {_ _} _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
57
58
59
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
Notation "` x" := (proj1_sig x) : C_scope.

60
61
62
63
(** * Type classes *)
(** ** Provable propositions *)
(** This type class collects provable propositions. It is useful to constraint
type classes by arbitrary propositions. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
64
65
Class PropHolds (P : Prop) := prop_holds: P.

66
67
Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
68
Proof. repeat intro; trivial. Qed.
69
70
71

Ltac solve_propholds :=
  match goal with
72
73
  | |- PropHolds (?P) => apply _
  | |- ?P => change (PropHolds P); apply _
74
75
76
77
78
79
80
  end.

(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
propositions. For example to declare a parameter expressing decidable equality
on a type [A] we write [`{∀ x y : A, Decision (x = y)}] and use it by writing
[decide (x = y)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
81
82
83
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.

84
85
86
(** ** Setoid equality *)
(** We define an operational type class for setoid equality. This is based on
(Spitters/van der Weegen, 2011). *)
Robbert Krebbers's avatar
Robbert Krebbers committed
87
88
89
90
91
92
93
94
95
96
Class Equiv A := equiv: relation A.
Infix "≡" := equiv (at level 70, no associativity) : C_scope.
Notation "(≡)" := equiv (only parsing) : C_scope.
Notation "( x ≡)" := (equiv x) (only parsing) : C_scope.
Notation "(≡ x )" := (λ y, y  x) (only parsing) : C_scope.
Notation "(≢)" := (λ x y, ¬x  y) (only parsing) : C_scope.
Notation "x ≢ y":= (¬x  y) (at level 70, no associativity) : C_scope.
Notation "( x ≢)" := (λ y, x  y) (only parsing) : C_scope.
Notation "(≢ x )" := (λ y, y  x) (only parsing) : C_scope.

97
98
99
100
101
102
103
104
(** A [Params f n] instance forces the setoid rewriting mechanism not to
rewrite in the first [n] arguments of the function [f]. We will declare such
instances for all operational type classes in this development. *)
Instance: Params (@equiv) 2.

(** The following instance forces [setoid_replace] to use setoid equality
(for types that have an [Equiv] instance) rather than the standard Leibniz
equality. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
105
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
106
107
Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_  _) => symmetry; assumption.
Robbert Krebbers's avatar
Robbert Krebbers committed
108

109
(** ** Operations on collections *)
110
(** We define operational type classes for the traditional operations and
111
relations on collections: the empty collection [∅], the union [(∪)],
112
113
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
114
115
116
117
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

Class Union A := union: A  A  A.
118
Instance: Params (@union) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
119
120
121
122
123
Infix "∪" := union (at level 50, left associativity) : C_scope.
Notation "(∪)" := union (only parsing) : C_scope.
Notation "( x ∪)" := (union x) (only parsing) : C_scope.
Notation "(∪ x )" := (λ y, union y x) (only parsing) : C_scope.

124
125
126
127
128
Definition union_list `{Empty A}
  `{Union A} : list A  A := fold_right () .
Arguments union_list _ _ _ !_ /.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
129
Class Intersection A := intersection: A  A  A.
130
Instance: Params (@intersection) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
131
132
133
134
135
136
Infix "∩" := intersection (at level 40) : C_scope.
Notation "(∩)" := intersection (only parsing) : C_scope.
Notation "( x ∩)" := (intersection x) (only parsing) : C_scope.
Notation "(∩ x )" := (λ y, intersection y x) (only parsing) : C_scope.

Class Difference A := difference: A  A  A.
137
Instance: Params (@difference) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
138
139
140
141
142
Infix "∖" := difference (at level 40) : C_scope.
Notation "(∖)" := difference (only parsing) : C_scope.
Notation "( x ∖)" := (difference x) (only parsing) : C_scope.
Notation "(∖ x )" := (λ y, difference y x) (only parsing) : C_scope.

143
144
145
146
147
148
Class Singleton A B := singleton: A  B.
Instance: Params (@singleton) 3.
Notation "{[ x ]}" := (singleton x) : C_scope.
Notation "{[ x ; y ; .. ; z ]}" :=
  (union .. (union (singleton x) (singleton y)) .. (singleton z)) : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
149
Class SubsetEq A := subseteq: A  A  Prop.
150
Instance: Params (@subseteq) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
151
152
153
154
155
156
157
158
159
Infix "⊆" := subseteq (at level 70) : C_scope.
Notation "(⊆)" := subseteq (only parsing) : C_scope.
Notation "( X ⊆ )" := (subseteq X) (only parsing) : C_scope.
Notation "( ⊆ X )" := (λ Y, subseteq Y X) (only parsing) : C_scope.
Notation "X ⊈ Y" := (¬X  Y) (at level 70) : C_scope.
Notation "(⊈)" := (λ X Y, X  Y) (only parsing) : C_scope.
Notation "( X ⊈ )" := (λ Y, X  Y) (only parsing) : C_scope.
Notation "( ⊈ X )" := (λ Y, Y  X) (only parsing) : C_scope.

160
Hint Extern 0 (_  _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
161
162

Class ElemOf A B := elem_of: A  B  Prop.
163
Instance: Params (@elem_of) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
164
165
166
167
168
169
170
171
172
Infix "∈" := elem_of (at level 70) : C_scope.
Notation "(∈)" := elem_of (only parsing) : C_scope.
Notation "( x ∈)" := (elem_of x) (only parsing) : C_scope.
Notation "(∈ X )" := (λ x, elem_of x X) (only parsing) : C_scope.
Notation "x ∉ X" := (¬x  X) (at level 80) : C_scope.
Notation "(∉)" := (λ x X, x  X) (only parsing) : C_scope.
Notation "( x ∉)" := (λ X, x  X) (only parsing) : C_scope.
Notation "(∉ X )" := (λ x, x  X) (only parsing) : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
173
174
175
176
177
178
179
Class Disjoint A := disjoint : A  A  Prop.
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
Notation "( X ⊥)" := (disjoint X) (only parsing) : C_scope.
Notation "(⊥ X )" := (λ Y, disjoint Y X) (only parsing) : C_scope.

180
181
182
183
184
185
186
187
188
189
190
191
Inductive list_disjoint `{Disjoint A} : list A  Prop :=
  | disjoint_nil :
     list_disjoint []
  | disjoint_cons X Xs :
     Forall ( X) Xs 
     list_disjoint Xs 
     list_disjoint (X :: Xs).
Lemma list_disjoint_cons_inv `{Disjoint A} X Xs :
  list_disjoint (X :: Xs) 
  Forall ( X) Xs  list_disjoint Xs.
Proof. inversion_clear 1; auto. Qed.

192
193
194
Instance generic_disjoint `{ElemOf A B} : Disjoint B | 100 :=
  λ X Y,  x, x  X  x  Y.

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
(** ** Monadic operations *)
(** We define operational type classes for the monadic operations bind, join 
and fmap. These type classes are defined in a non-standard way by taking the
function as a parameter of the class. For example, we define
<<
  Class FMapD := fmap: ∀ {A B}, (A → B) → M A → M B.
>>
instead of
<<
  Class FMap {A B} (f : A → B) := fmap: M A → M B.
>>
This approach allows us to define [fmap] on lists such that [simpl] unfolds it
in the appropriate way, and so that it can be used for mutual recursion
(the mapped function [f] is not part of the fixpoint) as well. This is a hack,
and should be replaced by something more appropriate in future versions. *)

(* We use these type classes merely for convenient overloading of notations and
do not formalize any theory on monads (we do not even define a class with the
monad laws). *)
Class MRet (M : Type  Type) := mret:  {A}, A  M A.
Instance: Params (@mret) 3.
Arguments mret {_ _ _} _.

Class MBindD (M : Type  Type) {A B} (f : A  M B) := mbind: M A  M B.
Notation MBind M := ( {A B} (f : A  M B), MBindD M f)%type.
Instance: Params (@mbind) 5.
Arguments mbind {_ _ _} _ {_} !_ / : simpl nomatch.

Class MJoin (M : Type  Type) := mjoin:  {A}, M (M A)  M A.
Instance: Params (@mjoin) 3.
Arguments mjoin {_ _ _} !_ / : simpl nomatch.

Class FMapD (M : Type  Type) {A B} (f : A  B) := fmap: M A  M B.
Notation FMap M := ( {A B} (f : A  B), FMapD M f)%type.
Instance: Params (@fmap) 6.
Arguments fmap {_ _ _} _ {_} !_ / : simpl nomatch.

Notation "m ≫= f" := (mbind f m) (at level 60, right associativity) : C_scope.
Notation "x ← y ; z" := (y = (λ x : _, z))
  (at level 65, only parsing, next at level 35, right associativity) : C_scope.
Infix "<$>" := fmap (at level 65, right associativity, only parsing) : C_scope.

Class MGuard (M : Type  Type) :=
  mguard:  P {dec : Decision P} {A}, M A  M A.
Notation "'guard' P ; o" := (mguard P o)
  (at level 65, only parsing, next at level 35, right associativity) : C_scope.

242
(** ** Operations on maps *)
243
244
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
245
The function lookup [m !! k] should yield the element at key [k] in [m]. *)
246
247
Class Lookup (K M A : Type) :=
  lookup: K  M  option A.
248
249
250
251
252
253
Instance: Params (@lookup) 4.

Notation "m !! i" := (lookup i m) (at level 20) : C_scope.
Notation "(!!)" := lookup (only parsing) : C_scope.
Notation "( m !!)" := (λ i, lookup i m) (only parsing) : C_scope.
Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
254
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.
255
256
257

(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
258
259
Class Insert (K M A : Type) :=
  insert: K  A  M  M.
260
261
262
Instance: Params (@insert) 4.
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
263
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.
264

265
266
267
(** The function delete [delete k m] should delete the value at key [k] in
[m]. If the key [k] is not a member of [m], the original map should be
returned. *)
268
269
270
271
Class Delete (K M : Type) :=
  delete: K  M  M.
Instance: Params (@delete) 3.
Arguments delete _ _ _ !_ !_ / : simpl nomatch.
272
273

(** The function [alter f k m] should update the value at key [k] using the
274
function [f], which is called with the original value. *)
275
276
277
278
279
Class AlterD (K M A : Type) (f : A  A) :=
  alter: K  M  M.
Notation Alter K M A := ( (f : A  A), AlterD K M A f)%type.
Instance: Params (@alter) 5.
Arguments alter {_ _ _} _ {_} !_ !_ / : simpl nomatch.
280
281

(** The function [alter f k m] should update the value at key [k] using the
282
283
284
function [f], which is called with the original value at key [k] or [None]
if [k] is not a member of [m]. The value at [k] should be deleted if [f] 
yields [None]. *)
285
286
Class PartialAlter (K M A : Type) :=
  partial_alter: (option A  option A)  K  M  M.
287
Instance: Params (@partial_alter) 4.
288
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.
289
290
291

(** The function [dom C m] should yield the domain of [m]. That is a finite
collection of type [C] that contains the keys that are a member of [m]. *)
292
293
294
295
Class Dom (K M : Type) :=
  dom:  C `{Empty C} `{Union C} `{Singleton K C}, M  C.
Instance: Params (@dom) 7.
Arguments dom _ _ _ _ _ _ _ !_ / : simpl nomatch.
296
297
298
299

(** The function [merge f m1 m2] should merge the maps [m1] and [m2] by
constructing a new map whose value at key [k] is [f (m1 !! k) (m2 !! k)]
provided that [k] is a member of either [m1] or [m2].*)
300
Class Merge (M : Type  Type) :=
301
302
  merge:  {A}, (option A  option A  option A)  M A  M A  M A.
Instance: Params (@merge) 3.
303
Arguments merge _ _ _ _ !_ !_ / : simpl nomatch.
304
305

(** We lift the insert and delete operation to lists of elements. *)
306
Definition insert_list `{Insert K M A} (l : list (K * A)) (m : M) : M :=
307
308
  fold_right (λ p, <[ fst p := snd p ]>) m l.
Instance: Params (@insert_list) 4.
309
Definition delete_list `{Delete K M} (l : list K) (m : M) : M :=
310
  fold_right delete m l.
311
312
313
314
315
316
Instance: Params (@delete_list) 3.

Definition insert_consecutive `{Insert nat M A}
    (i : nat) (l : list A) (m : M) : M :=
  fold_right (λ x f i, <[i:=x]>(f (S i))) (λ _, m) l i.
Instance: Params (@insert_consecutive) 3.
317
318
319
320

(** The function [union_with f m1 m2] should yield the union of [m1] and [m2]
using the function [f] to combine values of members that are in both [m1] and
[m2]. *)
321
Class UnionWith (M : Type  Type) :=
322
  union_with:  {A}, (A  A  A)  M A  M A  M A.
323
324
325
Instance: Params (@union_with) 3.

(** Similarly for the intersection and difference. *)
326
Class IntersectionWith (M : Type  Type) :=
327
  intersection_with:  {A}, (A  A  A)  M A  M A  M A.
328
Instance: Params (@intersection_with) 3.
329
Class DifferenceWith (M : Type  Type) :=
330
  difference_with:  {A}, (A  A  option A)  M A  M A  M A.
331
Instance: Params (@difference_with) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
332

333
334
335
336
(** ** Common properties *)
(** These operational type classes allow us to refer to common mathematical
properties in a generic way. For example, for injectivity of [(k ++)] it
allows us to write [injective (k ++)] instead of [app_inv_head k]. *)
337
338
339
340
341
342
343
344
345
346
347
348
Class Injective {A B} R S (f : A  B) :=
  injective:  x y : A, S (f x) (f y)  R x y.
Class Idempotent {A} R (f : A  A  A) :=
  idempotent:  x, R (f x x) x.
Class Commutative {A B} R (f : B  B  A) :=
  commutative:  x y, R (f x y) (f y x).
Class LeftId {A} R (i : A) (f : A  A  A) :=
  left_id:  x, R (f i x) x.
Class RightId {A} R (i : A) (f : A  A  A) :=
  right_id:  x, R (f x i) x.
Class Associative {A} R (f : A  A  A) :=
  associative:  x y z, R (f x (f y z)) (f (f x y) z).
349
350
351
352
Class LeftAbsorb {A} R (i : A) (f : A  A  A) :=
  left_absorb:  x, R (f i x) i.
Class RightAbsorb {A} R (i : A) (f : A  A  A) :=
  right_absorb:  x, R (f x i) i.
Robbert Krebbers's avatar
Robbert Krebbers committed
353
354
355
356
357
358
359

Arguments injective {_ _ _ _} _ {_} _ _ _.
Arguments idempotent {_ _} _ {_} _.
Arguments commutative {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments associative {_ _} _ {_} _ _ _.
360
361
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
362

363
364
365
(** The following lemmas are more specific versions of the projections of the
above type classes. These lemmas allow us to enforce Coq not to use the setoid
rewriting mechanism. *)
366
367
Lemma idempotent_eq {A} (f : A  A  A) `{!Idempotent (=) f} x :
  f x x = x.
368
Proof. auto. Qed.
369
370
Lemma commutative_eq {A B} (f : B  B  A) `{!Commutative (=) f} x y :
  f x y = f y x.
371
Proof. auto. Qed.
372
373
Lemma left_id_eq {A} (i : A) (f : A  A  A) `{!LeftId (=) i f} x :
  f i x = x.
374
Proof. auto. Qed.
375
376
Lemma right_id_eq {A} (i : A) (f : A  A  A) `{!RightId (=) i f} x :
  f x i = x.
377
Proof. auto. Qed.
378
379
Lemma associative_eq {A} (f : A  A  A) `{!Associative (=) f} x y z :
  f x (f y z) = f (f x y) z.
380
Proof. auto. Qed.
381
382
383
384
385
386
Lemma left_absorb_eq {A} (i : A) (f : A  A  A) `{!LeftAbsorb (=) i f} x :
  f i x = i.
Proof. auto. Qed.
Lemma right_absorb_eq {A} (i : A) (f : A  A  A) `{!RightAbsorb (=) i f} x :
  f x i = i.
Proof. auto. Qed.
387

388
389
(** ** Axiomatization of ordered structures *)
(** A pre-order equiped with a smallest element. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
390
391
392
393
394
Class BoundedPreOrder A `{Empty A} `{SubsetEq A} := {
  bounded_preorder :>> PreOrder ();
  subseteq_empty x :   x
}.

395
396
397
398
(** We do not include equality in the following interfaces so as to avoid the
need for proofs that the  relations and operations respect setoid equality.
Instead, we will define setoid equality in a generic way as
[λ X Y, X ⊆ Y ∧ Y ⊆ X]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
399
Class BoundedJoinSemiLattice A `{Empty A} `{SubsetEq A} `{Union A} := {
400
  bjsl_preorder :>> BoundedPreOrder A;
Robbert Krebbers's avatar
Robbert Krebbers committed
401
402
403
404
405
406
407
408
409
410
  subseteq_union_l x y : x  x  y;
  subseteq_union_r x y : y  x  y;
  union_least x y z : x  z  y  z  x  y  z
}.
Class MeetSemiLattice A `{Empty A} `{SubsetEq A} `{Intersection A} := {
  msl_preorder :>> BoundedPreOrder A;
  subseteq_intersection_l x y : x  y  x;
  subseteq_intersection_r x y : x  y  y;
  intersection_greatest x y z : z  x  z  y  z  x  y
}.
411
412
413
414
415
Class LowerBoundedLattice A `{Empty A} `{SubsetEq A}
    `{Union A} `{Intersection A} := {
  lbl_bjsl :>> BoundedJoinSemiLattice A;
  lbl_msl :>> MeetSemiLattice A
}.
416
(** ** Axiomatization of collections *)
417
418
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
419
Instance: Params (@map) 3.
420
421
Class SimpleCollection A C `{ElemOf A C}
  `{Empty C} `{Singleton A C} `{Union C} := {
422
  not_elem_of_empty (x : A) : x  ;
423
  elem_of_singleton (x y : A) : x  {[ y ]}  x = y;
424
425
426
427
428
  elem_of_union X Y (x : A) : x  X  Y  x  X  x  Y
}.
Class Collection A C `{ElemOf A C} `{Empty C} `{Singleton A C} `{Union C}
    `{Intersection C} `{Difference C} := {
  collection_simple :>> SimpleCollection A C;
Robbert Krebbers's avatar
Robbert Krebbers committed
429
  elem_of_intersection X Y (x : A) : x  X  Y  x  X  x  Y;
430
  elem_of_difference X Y (x : A) : x  X  Y  x  X  x  Y
Robbert Krebbers's avatar
Robbert Krebbers committed
431
432
}.

433
434
435
(** We axiomative a finite collection as a collection whose elements can be
enumerated as a list. These elements, given by the [elements] function, may be
in any order and should not contain duplicates. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
436
Class Elements A C := elements: C  list A.
437
Instance: Params (@elements) 3.
438

439
440
441
442
443
444
445
446
447
448
(** We redefine the standard library's [In] and [NoDup] using type classes. *)
Inductive elem_of_list {A} : ElemOf A (list A) :=
  | elem_of_list_here (x : A) l : x  x :: l
  | elem_of_list_further (x y : A) l : x  l  x  y :: l.
Existing Instance elem_of_list.

Inductive NoDup {A} : list A  Prop :=
  | NoDup_nil_2 : NoDup []
  | NoDup_cons_2 x l : x  l  NoDup l  NoDup (x :: l).

449
450
451
(** Decidability of equality of the carrier set is admissible, but we add it
anyway so as to avoid cycles in type class search. *)
Class FinCollection A C `{ElemOf A C} `{Empty C} `{Union C}
452
    `{Intersection C} `{Difference C} `{Singleton A C}
453
    `{Elements A C} `{ x y : A, Decision (x = y)} := {
Robbert Krebbers's avatar
Robbert Krebbers committed
454
  fin_collection :>> Collection A C;
455
  elements_spec X x : x  X  x  elements X;
Robbert Krebbers's avatar
Robbert Krebbers committed
456
  elements_nodup X : NoDup (elements X)
457
458
}.
Class Size C := size: C  nat.
459
Arguments size {_ _} !_ / : simpl nomatch.
460
Instance: Params (@size) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
461

462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
(** The class [Collection M] axiomatizes a type constructor [M] that can be
used to construct a collection [M A] with elements of type [A]. The advantage
of this class, compared to [Collection], is that it also axiomatizes the
the monadic operations. The disadvantage, is that not many inhabits are
possible (we will only provide an inhabitant using unordered lists without
duplicates removed). More interesting implementations typically need
decidability of equality, or a total order on the elements, which do not fit
in a type constructor of type [Type → Type]. *)
Class CollectionMonad M `{ A, ElemOf A (M A)}
    `{ A, Empty (M A)} `{ A, Singleton A (M A)} `{ A, Union (M A)}
    `{!MBind M} `{!MRet M} `{!FMap M} `{!MJoin M} := {
  collection_monad_simple A :> SimpleCollection A (M A);
  elem_of_bind {A B} (f : A  M B) (x : B) (X : M A) :
    x  X = f   y, x  f y  y  X;
  elem_of_ret {A} (x y : A) :
    x  mret y  x = y;
  elem_of_fmap {A B} (f : A  B) (x : B) (X : M A) :
    x  f <$> X   y, x = f y  y  X;
  elem_of_join {A} (x : A) (X : M (M A)) :
    x  mjoin X   Y, x  Y  Y  X
}.

484
485
486
(** The function [fresh X] yields an element that is not contained in [X]. We
will later prove that [fresh] is [Proper] with respect to the induced setoid
equality on collections. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
487
Class Fresh A C := fresh: C  A.
488
Instance: Params (@fresh) 3.
489
490
491
Class FreshSpec A C `{ElemOf A C}
    `{Empty C} `{Singleton A C} `{Union C} `{Fresh A C} := {
  fresh_collection_simple :>> SimpleCollection A C;
492
  fresh_proper_alt X Y : ( x, x  X  x  Y)  fresh X = fresh Y;
Robbert Krebbers's avatar
Robbert Krebbers committed
493
494
495
  is_fresh (X : C) : fresh X  X
}.

496
497
498
(** * Miscellaneous *)
Lemma proj1_sig_inj {A} (P : A  Prop) x (Px : P x) y (Py : P y) :
  xPx = yPy  x = y.
499
Proof. injection 1; trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
500

501
502
503
504
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} (x y : A) :
  R x y  R y x.
Proof. intuition. Qed.

505
506
507
(** ** Pointwise relations *)
(** These instances are in Coq trunk since revision 15455, but are not in Coq
8.4 yet. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
508
509
510
511
512
513
514
515
516
517
Instance pointwise_reflexive {A} `{R : relation B} :
  Reflexive R  Reflexive (pointwise_relation A R) | 9.
Proof. firstorder. Qed.
Instance pointwise_symmetric {A} `{R : relation B} :
  Symmetric R  Symmetric (pointwise_relation A R) | 9.
Proof. firstorder. Qed.
Instance pointwise_transitive {A} `{R : relation B} :
  Transitive R  Transitive (pointwise_relation A R) | 9.
Proof. firstorder. Qed.

518
(** ** Products *)
519
520
521
522
523
524
Definition fst_map {A A' B} (f : A  A') (p : A * B) : A' * B :=
  (f (fst p), snd p).
Definition snd_map {A B B'} (f : B  B') (p : A * B) : A * B' :=
  (fst p, f (snd p)).
Definition prod_relation {A B} (R1 : relation A) (R2 : relation B) :
  relation (A * B) := λ x y, R1 (fst x) (fst y)  R2 (snd x) (snd y).
Robbert Krebbers's avatar
Robbert Krebbers committed
525
526
527

Section prod_relation.
  Context `{R1 : relation A} `{R2 : relation B}.
528
529
  Global Instance:
    Reflexive R1  Reflexive R2  Reflexive (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
530
  Proof. firstorder eauto. Qed.
531
532
  Global Instance:
    Symmetric R1  Symmetric R2  Symmetric (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
533
  Proof. firstorder eauto. Qed.
534
535
  Global Instance:
    Transitive R1  Transitive R2  Transitive (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
536
  Proof. firstorder eauto. Qed.
537
538
  Global Instance:
    Equivalence R1  Equivalence R2  Equivalence (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
539
540
541
542
543
544
545
546
547
  Proof. split; apply _. Qed.
  Global Instance: Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
  Proof. firstorder eauto. Qed.
  Global Instance: Proper (prod_relation R1 R2 ==> R1) fst.
  Proof. firstorder eauto. Qed.
  Global Instance: Proper (prod_relation R1 R2 ==> R2) snd.
  Proof. firstorder eauto. Qed.
End prod_relation.

548
(** ** Other *)
549
550
Definition lift_relation {A B} (R : relation A)
  (f : B  A) : relation B := λ x y, R (f x) (f y).
Robbert Krebbers's avatar
Robbert Krebbers committed
551
552
Definition lift_relation_equivalence {A B} (R : relation A) (f : B  A) :
  Equivalence R  Equivalence (lift_relation R f).
553
Proof. unfold lift_relation. firstorder auto. Qed.
554
555
Hint Extern 0 (Equivalence (lift_relation _ _)) =>
  eapply @lift_relation_equivalence : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
556
557

Instance:  A B (x : B), Commutative (=) (λ _ _ : A, x).
558
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
559
Instance:  A (x : A), Associative (=) (λ _ _ : A, x).
560
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
561
Instance:  A, Associative (=) (λ x _ : A, x).
562
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
563
Instance:  A, Associative (=) (λ _ x : A, x).
564
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
565
Instance:  A, Idempotent (=) (λ x _ : A, x).
566
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
567
Instance:  A, Idempotent (=) (λ _ x : A, x).
568
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
569

570
571
Instance left_id_propholds {A} (R : relation A) i f :
  LeftId R i f   x, PropHolds (R (f i x) x).
572
Proof. red. trivial. Qed.
573
574
Instance right_id_propholds {A} (R : relation A) i f :
  RightId R i f   x, PropHolds (R (f x i) x).
575
Proof. red. trivial. Qed.
576
577
Instance idem_propholds {A} (R : relation A) f :
  Idempotent R f   x, PropHolds (R (f x x) x).
578
Proof. red. trivial. Qed.