base.v 26.4 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
(** * General *)
(** The following coercion allows us to use Booleans as propositions. *)
Coercion Is_true : bool >-> Sortclass.

Robbert Krebbers's avatar
Robbert Krebbers committed
15
16
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
17
18
Arguments id _ _/.
Arguments compose _ _ _ _ _ _ /.
Robbert Krebbers's avatar
Robbert Krebbers committed
19
Arguments flip _ _ _ _ _ _/.
20

21
22
23
24
(** 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. *)
25
26
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
27

Robbert Krebbers's avatar
Robbert Krebbers committed
28
29
30
Notation curry := prod_curry.
Notation uncurry := prod_uncurry.

31
32
(** 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
33
34
35
Delimit Scope C_scope with C.
Global Open Scope C_scope.

36
(** Introduce some Haskell style like notations. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
37
38
39
40
41
42
43
44
45
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.

46
47
48
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.
49

50
Notation "t $ r" := (t r)
51
  (at level 65, right associativity, only parsing) : C_scope.
52
53
54
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
55
56
57
58
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.
59
60
61

(** Set convenient implicit arguments for [existT] and introduce notations. *)
Arguments existT {_ _} _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
62
63
64
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
Notation "` x" := (proj1_sig x) : C_scope.

65
66
67
68
(** * 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
69
70
Class PropHolds (P : Prop) := prop_holds: P.

71
72
Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
73
Proof. repeat intro; trivial. Qed.
74
75
76

Ltac solve_propholds :=
  match goal with
77
78
  | |- PropHolds (?P) => apply _
  | |- ?P => change (PropHolds P); apply _
79
80
81
82
83
84
85
  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
86
87
88
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.

89
90
91
(** ** 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
92
93
94
95
96
97
98
99
100
101
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.

102
103
104
105
106
107
108
109
(** 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
110
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
111
112
Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_  _) => symmetry; assumption.
Robbert Krebbers's avatar
Robbert Krebbers committed
113

114
(** ** Operations on collections *)
115
(** We define operational type classes for the traditional operations and
116
relations on collections: the empty collection [∅], the union [(∪)],
117
118
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
119
120
121
122
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

Class Union A := union: A  A  A.
123
Instance: Params (@union) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
124
125
126
127
128
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.

129
130
131
132
133
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
134
Class Intersection A := intersection: A  A  A.
135
Instance: Params (@intersection) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
136
137
138
139
140
141
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.
142
Instance: Params (@difference) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
143
144
145
146
147
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.

148
149
150
151
152
153
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
154
Class SubsetEq A := subseteq: A  A  Prop.
155
Instance: Params (@subseteq) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
156
157
158
159
160
161
162
163
164
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.

165
Hint Extern 0 (_  _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
166

Robbert Krebbers's avatar
Robbert Krebbers committed
167
168
169
170
171
172
173
174
175
176
177
Class Subset A := subset: A  A  Prop.
Instance: Params (@subset) 2.
Infix "⊂" := subset (at level 70) : C_scope.
Notation "(⊂)" := subset (only parsing) : C_scope.
Notation "( X ⊂ )" := (subset X) (only parsing) : C_scope.
Notation "( ⊂ X )" := (λ Y, subset 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.

Robbert Krebbers's avatar
Robbert Krebbers committed
178
Class ElemOf A B := elem_of: A  B  Prop.
179
Instance: Params (@elem_of) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
180
181
182
183
184
185
186
187
188
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
189
190
191
192
193
194
195
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.

196
197
198
199
200
201
202
203
204
205
206
207
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.

208
209
210
Instance generic_disjoint `{ElemOf A B} : Disjoint B | 100 :=
  λ X Y,  x, x  X  x  Y.

Robbert Krebbers's avatar
Robbert Krebbers committed
211
212
213
214
Class Filter A B :=
  filter:  (P : A  Prop) `{ x, Decision (P x)}, B  B.
(* Arguments filter {_ _ _} _ {_} !_ / : simpl nomatch. *)

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
(** ** 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.
Robbert Krebbers's avatar
Robbert Krebbers committed
253
254
255
256
Notation "( m ≫=)" := (λ f, mbind f m) (only parsing) : C_scope.
Notation "(≫= f )" := (mbind f) (only parsing) : C_scope.
Notation "(≫=)" := (λ m f, mbind f m) (only parsing) : C_scope.

257
258
Notation "x ← y ; z" := (y = (λ x : _, z))
  (at level 65, only parsing, next at level 35, right associativity) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
259
Infix "<$>" := fmap (at level 65, right associativity) : C_scope.
260
261
262
263
264
265

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.

266
(** ** Operations on maps *)
267
268
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
269
The function lookup [m !! k] should yield the element at key [k] in [m]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
270
Class Lookup (K A M : Type) :=
271
  lookup: K  M  option A.
272
273
274
275
276
277
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.
278
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.
279
280
281

(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
282
Class Insert (K A M : Type) :=
283
  insert: K  A  M  M.
284
285
286
Instance: Params (@insert) 4.
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
287
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.
288

289
290
291
(** 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. *)
292
293
294
295
Class Delete (K M : Type) :=
  delete: K  M  M.
Instance: Params (@delete) 3.
Arguments delete _ _ _ !_ !_ / : simpl nomatch.
296
297

(** The function [alter f k m] should update the value at key [k] using the
298
function [f], which is called with the original value. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
299
Class AlterD (K A M : Type) (f : A  A) :=
300
  alter: K  M  M.
Robbert Krebbers's avatar
Robbert Krebbers committed
301
Notation Alter K A M := ( (f : A  A), AlterD K A M f)%type.
302
303
Instance: Params (@alter) 5.
Arguments alter {_ _ _} _ {_} !_ !_ / : simpl nomatch.
304
305

(** The function [alter f k m] should update the value at key [k] using the
306
307
308
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]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
309
Class PartialAlter (K A M : Type) :=
310
  partial_alter: (option A  option A)  K  M  M.
311
Instance: Params (@partial_alter) 4.
312
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.
313
314
315

(** 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]. *)
316
317
318
319
Class Dom (K M : Type) :=
  dom:  C `{Empty C} `{Union C} `{Singleton K C}, M  C.
Instance: Params (@dom) 7.
Arguments dom _ _ _ _ _ _ _ !_ / : simpl nomatch.
320
321
322
323

(** 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].*)
Robbert Krebbers's avatar
Robbert Krebbers committed
324
325
Class Merge (A M : Type) :=
  merge: (option A  option A  option A)  M  M  M.
326
Instance: Params (@merge) 3.
327
Arguments merge _ _ _ _ !_ !_ / : simpl nomatch.
328
329

(** We lift the insert and delete operation to lists of elements. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
330
Definition insert_list `{Insert K A M} (l : list (K * A)) (m : M) : M :=
331
332
  fold_right (λ p, <[ fst p := snd p ]>) m l.
Instance: Params (@insert_list) 4.
333
Definition delete_list `{Delete K M} (l : list K) (m : M) : M :=
334
  fold_right delete m l.
335
336
Instance: Params (@delete_list) 3.

Robbert Krebbers's avatar
Robbert Krebbers committed
337
Definition insert_consecutive `{Insert nat A M}
338
339
340
    (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.
341

Robbert Krebbers's avatar
Robbert Krebbers committed
342
343
344
345
346
(** The function [union_with f m1 m2] is supposed to yield the union of [m1]
and [m2] using the function [f] to combine values of members that are in
both [m1] and [m2]. *)
Class UnionWith (A M : Type) :=
  union_with: (A  A  option A)  M  M  M.
347
348
Instance: Params (@union_with) 3.

Robbert Krebbers's avatar
Robbert Krebbers committed
349
350
351
(** Similarly for intersection and difference. *)
Class IntersectionWith (A M : Type) :=
  intersection_with: (A  A  option A)  M  M  M.
352
Instance: Params (@intersection_with) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
353
354
Class DifferenceWith (A M : Type) :=
  difference_with: (A  A  option A)  M  M  M.
355
Instance: Params (@difference_with) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
356

Robbert Krebbers's avatar
Robbert Krebbers committed
357
358
359
360
Definition intersection_with_list `{IntersectionWith A M}
  (f : A  A  option A) : M  list M  M := fold_right (intersection_with f).
Arguments intersection_with_list _ _ _ _ _ !_ /.

361
362
363
364
(** ** 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]. *)
365
366
367
368
369
370
371
372
373
374
375
376
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).
377
378
379
380
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
381
382
Class AntiSymmetric {A} (R : A  A  Prop) :=
  anti_symmetric:  x y, R x y  R y x  x = y.
Robbert Krebbers's avatar
Robbert Krebbers committed
383
384
385
386
387
388
389

Arguments injective {_ _ _ _} _ {_} _ _ _.
Arguments idempotent {_ _} _ {_} _.
Arguments commutative {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments associative {_ _} _ {_} _ _ _.
390
391
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
392
Arguments anti_symmetric {_} _ {_} _ _ _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
393

394
395
396
(** 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. *)
397
398
Lemma idempotent_eq {A} (f : A  A  A) `{!Idempotent (=) f} x :
  f x x = x.
399
Proof. auto. Qed.
400
401
Lemma commutative_eq {A B} (f : B  B  A) `{!Commutative (=) f} x y :
  f x y = f y x.
402
Proof. auto. Qed.
403
404
Lemma left_id_eq {A} (i : A) (f : A  A  A) `{!LeftId (=) i f} x :
  f i x = x.
405
Proof. auto. Qed.
406
407
Lemma right_id_eq {A} (i : A) (f : A  A  A) `{!RightId (=) i f} x :
  f x i = x.
408
Proof. auto. Qed.
409
410
Lemma associative_eq {A} (f : A  A  A) `{!Associative (=) f} x y z :
  f x (f y z) = f (f x y) z.
411
Proof. auto. Qed.
412
413
414
415
416
417
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.
418

419
420
(** ** Axiomatization of ordered structures *)
(** A pre-order equiped with a smallest element. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
421
422
423
424
Class BoundedPreOrder A `{Empty A} `{SubsetEq A} := {
  bounded_preorder :>> PreOrder ();
  subseteq_empty x :   x
}.
Robbert Krebbers's avatar
Robbert Krebbers committed
425
426
427
428
Class PartialOrder A `{SubsetEq A} := {
  po_preorder :>> PreOrder ();
  po_antisym :> AntiSymmetric ()
}.
Robbert Krebbers's avatar
Robbert Krebbers committed
429

430
431
432
433
(** 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
434
Class BoundedJoinSemiLattice A `{Empty A} `{SubsetEq A} `{Union A} := {
435
  bjsl_preorder :>> BoundedPreOrder A;
Robbert Krebbers's avatar
Robbert Krebbers committed
436
437
438
439
440
441
442
443
444
445
  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
}.
446
447
448
449
450
Class LowerBoundedLattice A `{Empty A} `{SubsetEq A}
    `{Union A} `{Intersection A} := {
  lbl_bjsl :>> BoundedJoinSemiLattice A;
  lbl_msl :>> MeetSemiLattice A
}.
451
(** ** Axiomatization of collections *)
452
453
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
454
Instance: Params (@map) 3.
455
456
Class SimpleCollection A C `{ElemOf A C}
  `{Empty C} `{Singleton A C} `{Union C} := {
457
  not_elem_of_empty (x : A) : x  ;
458
  elem_of_singleton (x y : A) : x  {[ y ]}  x = y;
459
460
  elem_of_union X Y (x : A) : x  X  Y  x  X  x  Y
}.
Robbert Krebbers's avatar
Robbert Krebbers committed
461
462
Class Collection A C `{ElemOf A C} `{Empty C} `{Singleton A C}
    `{Union C} `{Intersection C} `{Difference C} `{IntersectionWith A C} := {
463
  collection_simple :>> SimpleCollection A C;
Robbert Krebbers's avatar
Robbert Krebbers committed
464
  elem_of_intersection X Y (x : A) : x  X  Y  x  X  x  Y;
Robbert Krebbers's avatar
Robbert Krebbers committed
465
466
467
  elem_of_difference X Y (x : A) : x  X  Y  x  X  x  Y;
  elem_of_intersection_with (f : A  A  option A) X Y (x : A) :
    x  intersection_with f X Y   x1 x2, x1  X  x2  Y  f x1 x2 = Some x
Robbert Krebbers's avatar
Robbert Krebbers committed
468
469
}.

470
471
472
(** 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
473
Class Elements A C := elements: C  list A.
474
Instance: Params (@elements) 3.
475

476
477
478
479
480
481
482
483
484
485
(** 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).

486
487
(** Decidability of equality of the carrier set is admissible, but we add it
anyway so as to avoid cycles in type class search. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
488
489
490
Class FinCollection A C `{ElemOf A C} `{Empty C} `{Singleton A C}
    `{Union C} `{Intersection C} `{Difference C} `{IntersectionWith A C}
    `{Filter A C} `{Elements A C} `{ x y : A, Decision (x = y)} := {
Robbert Krebbers's avatar
Robbert Krebbers committed
491
  fin_collection :>> Collection A C;
Robbert Krebbers's avatar
Robbert Krebbers committed
492
493
  elem_of_filter X P `{ x, Decision (P x)} x :
    x  filter P X  P x  x  X;
494
  elements_spec X x : x  X  x  elements X;
Robbert Krebbers's avatar
Robbert Krebbers committed
495
  elements_nodup X : NoDup (elements X)
496
497
}.
Class Size C := size: C  nat.
498
Arguments size {_ _} !_ / : simpl nomatch.
499
Instance: Params (@size) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
500

501
502
503
504
505
506
507
508
509
510
511
512
(** 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);
Robbert Krebbers's avatar
Robbert Krebbers committed
513
  elem_of_bind {A B} (f : A  M B) (X : M A) (x : B) :
514
515
516
    x  X = f   y, x  f y  y  X;
  elem_of_ret {A} (x y : A) :
    x  mret y  x = y;
Robbert Krebbers's avatar
Robbert Krebbers committed
517
  elem_of_fmap {A B} (f : A  B) (X : M A) (x : B) :
518
    x  f <$> X   y, x = f y  y  X;
Robbert Krebbers's avatar
Robbert Krebbers committed
519
  elem_of_join {A} (X : M (M A)) (x : A) :
520
521
522
    x  mjoin X   Y, x  Y  Y  X
}.

523
524
525
(** 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
526
Class Fresh A C := fresh: C  A.
527
Instance: Params (@fresh) 3.
528
529
530
Class FreshSpec A C `{ElemOf A C}
    `{Empty C} `{Singleton A C} `{Union C} `{Fresh A C} := {
  fresh_collection_simple :>> SimpleCollection A C;
531
  fresh_proper_alt X Y : ( x, x  X  x  Y)  fresh X = fresh Y;
Robbert Krebbers's avatar
Robbert Krebbers committed
532
533
534
  is_fresh (X : C) : fresh X  X
}.

535
536
537
(** * Miscellaneous *)
Lemma proj1_sig_inj {A} (P : A  Prop) x (Px : P x) y (Py : P y) :
  xPx = yPy  x = y.
538
Proof. injection 1; trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
539

540
541
542
543
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} (x y : A) :
  R x y  R y x.
Proof. intuition. Qed.

544
545
546
(** ** 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
547
548
549
550
551
552
553
554
555
556
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.

557
(** ** Products *)
558
559
560
561
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)).
Robbert Krebbers's avatar
Robbert Krebbers committed
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
Arguments fst_map {_ _ _} _ !_ /.
Arguments snd_map {_ _ _} _ !_ /.

Instance:  {A A' B} (f : A  A'),
  Injective (=) (=) f  Injective (=) (=) (@fst_map A A' B f).
Proof.
  intros ????? [??] [??]; simpl; intro; f_equal.
  * apply (injective f). congruence.
  * congruence.
Qed.
Instance:  {A B B'} (f : B  B'),
  Injective (=) (=) f  Injective (=) (=) (@snd_map A B B' f).
Proof.
  intros ????? [??] [??]; simpl; intro; f_equal.
  * congruence.
  * apply (injective f). congruence.
Qed.

580
581
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
582
583
584

Section prod_relation.
  Context `{R1 : relation A} `{R2 : relation B}.
585
586
  Global Instance:
    Reflexive R1  Reflexive R2  Reflexive (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
587
  Proof. firstorder eauto. Qed.
588
589
  Global Instance:
    Symmetric R1  Symmetric R2  Symmetric (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
590
  Proof. firstorder eauto. Qed.
591
592
  Global Instance:
    Transitive R1  Transitive R2  Transitive (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
593
  Proof. firstorder eauto. Qed.
594
595
  Global Instance:
    Equivalence R1  Equivalence R2  Equivalence (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
596
597
598
599
600
601
602
603
604
  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.

605
(** ** Other *)
606
607
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
608
609
Definition lift_relation_equivalence {A B} (R : relation A) (f : B  A) :
  Equivalence R  Equivalence (lift_relation R f).
610
Proof. unfold lift_relation. firstorder auto. Qed.
611
612
Hint Extern 0 (Equivalence (lift_relation _ _)) =>
  eapply @lift_relation_equivalence : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
613
614

Instance:  A B (x : B), Commutative (=) (λ _ _ : A, x).
615
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
616
Instance:  A (x : A), Associative (=) (λ _ _ : A, x).
617
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
618
Instance:  A, Associative (=) (λ x _ : A, x).
619
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
620
Instance:  A, Associative (=) (λ _ x : A, x).
621
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
622
Instance:  A, Idempotent (=) (λ x _ : A, x).
623
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
624
Instance:  A, Idempotent (=) (λ _ x : A, x).
625
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
626

627
628
Instance left_id_propholds {A} (R : relation A) i f :
  LeftId R i f   x, PropHolds (R (f i x) x).
629
Proof. red. trivial. Qed.
630
631
Instance right_id_propholds {A} (R : relation A) i f :
  RightId R i f   x, PropHolds (R (f x i) x).
632
Proof. red. trivial. Qed.
633
634
Instance idem_propholds {A} (R : relation A) f :
  Idempotent R f   x, PropHolds (R (f x x) x).
635
Proof. red. trivial. Qed.