base.v 37.1 KB
Newer Older
1
(* Copyright (c) 2012-2013, Robbert Krebbers. *)
2
3
4
5
6
(* 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
Global Generalizable All Variables.
Global Set Automatic Coercions Import.
9
Require Export Morphisms RelationClasses List Bool Utf8 Program Setoid.
Robbert Krebbers's avatar
Robbert Krebbers committed
10

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

15
16
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
17
Arguments id _ _ /.
18
Arguments compose _ _ _ _ _ _ /.
19
Arguments flip _ _ _ _ _ _ /.
20
Typeclasses Transparent id compose flip.
21

22
23
24
25
(** 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. *)
26
27
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
28

29
30
31
Notation curry := prod_curry.
Notation uncurry := prod_uncurry.

32
33
(** 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
34
35
36
Delimit Scope C_scope with C.
Global Open Scope C_scope.

37
(** Introduce some Haskell style like notations. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
38
39
40
41
42
43
44
45
46
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.

47
48
49
50
Notation "(→)" := (λ A B, A  B) (only parsing) : C_scope.
Notation "( A →)" := (λ B, A  B) (only parsing) : C_scope.
Notation "(→ B )" := (λ A, A  B) (only parsing) : C_scope.

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

61
62
63
64
65
66
67
68
69
70
71
72
Notation "(∧)" := and (only parsing) : C_scope.
Notation "( A ∧)" := (and A) (only parsing) : C_scope.
Notation "(∧ B )" := (λ A, A  B) (only parsing) : C_scope.

Notation "(∨)" := or (only parsing) : C_scope.
Notation "( A ∨)" := (or A) (only parsing) : C_scope.
Notation "(∨ B )" := (λ A, A  B) (only parsing) : C_scope.

Notation "(↔)" := iff (only parsing) : C_scope.
Notation "( A ↔)" := (iff A) (only parsing) : C_scope.
Notation "(↔ B )" := (λ A, A  B) (only parsing) : C_scope.

73
74
(** Set convenient implicit arguments for [existT] and introduce notations. *)
Arguments existT {_ _} _ _.
75
Arguments proj1_sig {_ _} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
76
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
77
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
78

79
80
81
82
(** * 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
83
84
Class PropHolds (P : Prop) := prop_holds: P.

85
86
Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
87
Proof. repeat intro; trivial. Qed.
88
89
90

Ltac solve_propholds :=
  match goal with
91
92
  | |- PropHolds (?P) => apply _
  | |- ?P => change (PropHolds P); apply _
93
94
95
96
97
98
99
  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
100
101
102
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.

103
104
105
106
107
108
109
110
111
(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Prop := populate { _ : A }.
Arguments populate {_} _.

Instance unit_inhabited: Inhabited unit := populate ().
Instance list_inhabited {A} : Inhabited (list A) := populate [].
Instance prod_inhabited {A B} (iA : Inhabited A)
    (iB : Inhabited B) : Inhabited (A * B) :=
112
  match iA, iB with populate x, populate y => populate (x,y) end.
113
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
114
  match iA with populate x => populate (inl x) end.
115
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
116
  match iB with populate y => populate (inl y) end.
117
118
Instance option_inhabited {A} : Inhabited (option A) := populate None.

119
120
121
122
123
124
(** ** Proof irrelevant types *)
(** This type class collects types that are proof irrelevant. That means, all
elements of the type are equal. We use this notion only used for propositions,
but by universe polymorphism we can generalize it. *)
Class ProofIrrel (A : Type) : Prop := proof_irrel (x y : A) : x = y.

125
126
127
(** ** 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
128
129
130
131
132
133
134
135
136
137
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.

138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
(** The type class [LeibnizEquiv] collects setoid equalities that coincide
with Leibniz equality. We provide the tactic [fold_leibniz] to transform such
setoid equalities into Leibniz equalities, and [unfold_leibniz] for the
reverse. *)
Class LeibnizEquiv A `{Equiv A} := leibniz_equiv x y : x  y  x = y.

Ltac fold_leibniz := repeat
  match goal with
  | H : context [ @equiv ?A _ _ _ ] |- _ =>
    setoid_rewrite (leibniz_equiv (A:=A)) in H
  | |- context [ @equiv ?A _ _ _ ] =>
    setoid_rewrite (leibniz_equiv (A:=A))
  end.
Ltac unfold_leibniz := repeat
  match goal with
  | H : context [ @eq ?A _ _ ] |- _ =>
    setoid_rewrite <-(leibniz_equiv (A:=A)) in H
  | |- context [ @eq ?A _ _ ] =>
    setoid_rewrite <-(leibniz_equiv (A:=A))
  end.

159
160
161
162
163
164
165
166
(** 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
167
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
168
169
Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_  _) => symmetry; assumption.
Robbert Krebbers's avatar
Robbert Krebbers committed
170

171
(** ** Operations on collections *)
172
(** We define operational type classes for the traditional operations and
173
relations on collections: the empty collection [∅], the union [(∪)],
174
175
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
176
177
178
179
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

Class Union A := union: A  A  A.
180
Instance: Params (@union) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
181
182
183
184
185
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.

186
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
187
188
189
Arguments union_list _ _ _ !_ /.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
190
Class Intersection A := intersection: A  A  A.
191
Instance: Params (@intersection) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
192
193
194
195
196
197
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.
198
Instance: Params (@difference) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
199
200
201
202
203
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.

204
205
Class Singleton A B := singleton: A  B.
Instance: Params (@singleton) 3.
206
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
207
Notation "{[ x ; y ; .. ; z ]}" :=
208
209
210
211
212
213
  (union .. (union (singleton x) (singleton y)) .. (singleton z))
  (at level 1) : C_scope.
Notation "{[ x , y ]}" := (singleton (x,y))
  (at level 1, y at next level) : C_scope.
Notation "{[ x , y , z ]}" := (singleton (x,y,z))
  (at level 1, y at next level, z at next level) : C_scope.
214

215
Class SubsetEq A := subseteq: relation A.
216
Instance: Params (@subseteq) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
217
218
219
220
221
222
223
224
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.
225
226
Infix "⊆*" := (Forall2 subseteq) (at level 70) : C_scope.
Notation "(⊆*)" := (Forall2 subseteq) (only parsing) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
227

228
229
Hint Extern 0 (_  _) => reflexivity.

230
231
232
233
234
235
Definition strict {A} (R : relation A) : relation A := λ X Y, R X Y  ¬R Y X.
Instance: Params (@strict) 2.
Infix "⊂" := (strict subseteq) (at level 70) : C_scope.
Notation "(⊂)" := (strict subseteq) (only parsing) : C_scope.
Notation "( X ⊂ )" := (strict subseteq X) (only parsing) : C_scope.
Notation "( ⊂ X )" := (λ Y, strict subseteq Y X) (only parsing) : C_scope.
236
237
238
239
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
240

241
242
243
244
245
(** The class [Lexico A] is used for the lexicographic order on [A]. This order
is used to create finite maps, finite sets, etc, and is typically different from
the order [(⊆)]. *)
Class Lexico A := lexico: relation A.

Robbert Krebbers's avatar
Robbert Krebbers committed
246
Class ElemOf A B := elem_of: A  B  Prop.
247
Instance: Params (@elem_of) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
248
249
250
251
252
253
254
255
256
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
257
258
259
260
Class Disjoint A := disjoint : A  A  Prop.
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
Notation "(.⊥ X )" := (λ Y, disjoint Y X) (only parsing) : C_scope.

Class DisjointList A := disjoint_list : list A  Prop.
Instance: Params (@disjoint_list) 2.
Notation "⊥ l" := (disjoint_list l) (at level 20, format "⊥  l") : C_scope.

Section default_disjoint_list.
  Context `{Empty A} `{Union A} `{Disjoint A}.
  Inductive default_disjoint_list : DisjointList A :=
    | disjoint_nil_2 :  []
    | disjoint_cons_2 X Xs : X   Xs   Xs   (X :: Xs).
  Global Existing Instance default_disjoint_list.

  Lemma disjoint_list_nil :  @nil A  True.
  Proof. split; constructor. Qed.
  Lemma disjoint_list_cons X Xs :  (X :: Xs)  X   Xs   Xs.
  Proof. split. inversion_clear 1; auto. intros [??]. constructor; auto. Qed.
End default_disjoint_list.

Class Filter A B := filter:  (P : A  Prop) `{ x, Decision (P x)}, B  B.
282

283
284
285
286
287
(** We define variants of the relations [(≡)] and [(⊆)] that are indexed by
an environment. *)
Class EquivEnv A B := equiv_env : A  relation B.
Notation "X ≡@{ E } Y" := (equiv_env E X Y)
  (at level 70, format "X  ≡@{ E }  Y") : C_scope.
288
Notation "(≡@{ E } )" := (equiv_env E) (E at level 1, only parsing) : C_scope.
289
290
291
Instance: Params (@equiv_env) 4.

Class SubsetEqEnv A B := subseteq_env : A  relation B.
292
293
294
295
296
297
298
299
Instance: Params (@subseteq_env) 4.
Notation "X ⊑@{ E } Y" := (subseteq_env E X Y)
  (at level 70, format "X  ⊑@{ E }  Y") : C_scope.
Notation "(⊑@{ E } )" := (subseteq_env E)
  (E at level 1, only parsing) : C_scope.
Notation "X ⊑@{ E }* Y" := (Forall2 (subseteq_env E) X Y)
  (at level 70, format "X  ⊑@{ E }*  Y") : C_scope.
Notation "(⊑@{ E }*)" := (Forall2 (subseteq_env E))
300
301
302
  (E at level 1, only parsing) : C_scope.
Instance: Params (@subseteq_env) 4.

303
304
305
Hint Extern 0 (_ @{_} _) => reflexivity.
Hint Extern 0 (_ @{_} _) => reflexivity.

306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
(** ** 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. *)

322
(** We use these type classes merely for convenient overloading of notations and
323
324
325
326
327
328
329
330
331
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.
332
Arguments mbind {_ _ _} _ {_} !_ /.
333
334
335

Class MJoin (M : Type  Type) := mjoin:  {A}, M (M A)  M A.
Instance: Params (@mjoin) 3.
336
Arguments mjoin {_ _ _} !_ /.
337
338
339
340

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.
341
Arguments fmap {_ _ _} _ {_} !_ /.
342
343
344
345
346
347
348

Notation "m ≫= f" := (mbind f m) (at level 60, right associativity) : C_scope.
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.

Notation "x ← y ; z" := (y = (λ x : _, z))
349
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
350
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.
351
352

Class MGuard (M : Type  Type) :=
353
354
355
356
357
358
  mguard:  P {dec : Decision P} {A}, (P  M A)  M A.
Arguments mguard _ _ _ !_ _ _ /.
Notation "'guard' P ; o" := (mguard P (λ _, o))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
Notation "'guard' P 'as' H ; o" := (mguard P (λ H, o))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
359

360
(** ** Operations on maps *)
361
362
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
363
The function look up [m !! k] should yield the element at key [k] in [m]. *)
364
Class Lookup (K A M : Type) := lookup: K  M  option A.
365
366
367
368
369
370
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.
371
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.
372
373
374

(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
375
Class Insert (K A M : Type) := insert: K  A  M  M.
376
377
378
Instance: Params (@insert) 4.
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
379
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.
380

381
382
383
(** 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. *)
384
Class Delete (K M : Type) := delete: K  M  M.
385
386
Instance: Params (@delete) 3.
Arguments delete _ _ _ !_ !_ / : simpl nomatch.
387
388

(** The function [alter f k m] should update the value at key [k] using the
389
function [f], which is called with the original value. *)
390
Class AlterD (K A M : Type) (f : A  A) := alter: K  M  M.
391
392
393
Notation Alter K A M := ( (f : A  A), AlterD K A M f)%type.
Instance: Params (@alter) 5.
Arguments alter {_ _ _} _ {_} !_ !_ / : simpl nomatch.
394
395

(** The function [alter f k m] should update the value at key [k] using the
396
397
398
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]. *)
399
400
Class PartialAlter (K A M : Type) :=
  partial_alter: (option A  option A)  K  M  M.
401
Instance: Params (@partial_alter) 4.
402
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.
403
404
405

(** 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]. *)
406
407
408
Class Dom (M C : Type) := dom: M  C.
Instance: Params (@dom) 3.
Arguments dom {_} _ {_} !_ / : simpl nomatch, clear implicits.
409
410

(** The function [merge f m1 m2] should merge the maps [m1] and [m2] by
411
412
413
414
415
constructing a new map whose value at key [k] is [f (m1 !! k) (m2 !! k)].*)
Class Merge (M : Type  Type) :=
  merge:  {A B C}, (option A  option B  option C)  M A  M B  M C.
Instance: Params (@merge) 4.
Arguments merge _ _ _ _ _ _ !_ !_ / : simpl nomatch.
416
417

(** We lift the insert and delete operation to lists of elements. *)
418
Definition insert_list `{Insert K A M} (l : list (K * A)) (m : M) : M :=
419
420
  fold_right (λ p, <[ fst p := snd p ]>) m l.
Instance: Params (@insert_list) 4.
421
Definition delete_list `{Delete K M} (l : list K) (m : M) : M :=
422
  fold_right delete m l.
423
424
425
426
427
428
429
Instance: Params (@delete_list) 3.

(** 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.
430
Instance: Params (@union_with) 3.
431
Arguments union_with {_ _ _} _ !_ !_ / : simpl nomatch.
432

433
434
435
(** Similarly for intersection and difference. *)
Class IntersectionWith (A M : Type) :=
  intersection_with: (A  A  option A)  M  M  M.
436
Instance: Params (@intersection_with) 3.
437
438
Arguments intersection_with {_ _ _} _ !_ !_ / : simpl nomatch.

439
440
Class DifferenceWith (A M : Type) :=
  difference_with: (A  A  option A)  M  M  M.
441
Instance: Params (@difference_with) 3.
442
Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch.
Robbert Krebbers's avatar
Robbert Krebbers committed
443

444
445
446
447
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 _ _ _ _ _ !_ /.

448
449
450
451
(** ** 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]. *)
452
453
454
455
456
Class Injective {A B} (R : relation A) (S : relation B) (f : A  B) : Prop :=
  injective:  x y, S (f x) (f y)  R x y.
Class Injective2 {A B C} (R1 : relation A) (R2 : relation B)
    (S : relation C) (f : A  B  C) : Prop :=
  injective2:  x1 x2  y1 y2, S (f x1 x2) (f y1 y2)  R1 x1 y1  R2 x2 y2.
457
458
459
460
Class Cancel {A B} (S : relation B) (f : A  B) (g : B  A) : Prop :=
  cancel:  x, S (f (g x)) x.
Class Surjective {A B} (R : relation B) (f : A  B) :=
  surjective :  y,  x, R (f x) y.
461
Class Idempotent {A} (R : relation A) (f : A  A  A) : Prop :=
462
  idempotent:  x, R (f x x) x.
463
Class Commutative {A B} (R : relation A) (f : B  B  A) : Prop :=
464
  commutative:  x y, R (f x y) (f y x).
465
Class LeftId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
466
  left_id:  x, R (f i x) x.
467
Class RightId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
468
  right_id:  x, R (f x i) x.
469
Class Associative {A} (R : relation A) (f : A  A  A) : Prop :=
470
  associative:  x y z, R (f x (f y z)) (f (f x y) z).
471
Class LeftAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
472
  left_absorb:  x, R (f i x) i.
473
Class RightAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
474
  right_absorb:  x, R (f x i) i.
475
476
477
478
Class LeftDistr {A} (R : relation A) (f g : A  A  A) : Prop :=
  left_distr:  x y z, R (f x (g y z)) (g (f x y) (f x z)).
Class RightDistr {A} (R : relation A) (f g : A  A  A) : Prop :=
  right_distr:  y z x, R (f (g y z) x) (g (f y x) (f z x)).
479
480
Class AntiSymmetric {A} (R S : relation A) : Prop :=
  anti_symmetric:  x y, S x y  S y x  R x y.
481
482
483
484
485
Class Total {A} (R : relation A) := total x y : R x y  R y x.
Class Trichotomy {A} (R : relation A) :=
  trichotomy :  x y, strict R x y  x = y  strict R y x.
Class TrichotomyT {A} (R : relation A) :=
  trichotomyT :  x y, {strict R x y} + {x = y} + {strict R y x}.
Robbert Krebbers's avatar
Robbert Krebbers committed
486

487
Arguments irreflexivity {_} _ {_} _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
488
Arguments injective {_ _ _ _} _ {_} _ _ _.
489
Arguments injective2 {_ _ _ _ _ _} _ {_} _ _ _ _ _.
490
491
Arguments cancel {_ _ _} _ _ {_} _.
Arguments surjective {_ _ _} _ {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
492
493
494
495
496
Arguments idempotent {_ _} _ {_} _.
Arguments commutative {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments associative {_ _} _ {_} _ _ _.
497
498
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
499
500
Arguments left_distr {_ _} _ _ {_} _ _ _.
Arguments right_distr {_ _} _ _ {_} _ _ _.
501
Arguments anti_symmetric {_ _} _ {_} _ _ _ _.
502
503
504
Arguments total {_} _ {_} _ _.
Arguments trichotomy {_} _ {_} _ _.
Arguments trichotomyT {_} _ {_} _ _.
505

506
507
508
509
(** The following lemmas are specific versions of the projections of the above
type classes for Leibniz equality. These lemmas allow us to enforce Coq not to
use the setoid rewriting mechanism. *)
Lemma idempotent_L {A} (f : A  A  A) `{!Idempotent (=) f} x : f x x = x.
510
Proof. auto. Qed.
511
Lemma commutative_L {A B} (f : B  B  A) `{!Commutative (=) f} x y :
512
  f x y = f y x.
513
Proof. auto. Qed.
514
Lemma left_id_L {A} (i : A) (f : A  A  A) `{!LeftId (=) i f} x : f i x = x.
515
Proof. auto. Qed.
516
Lemma right_id_L {A} (i : A) (f : A  A  A) `{!RightId (=) i f} x : f x i = x.
517
Proof. auto. Qed.
518
Lemma associative_L {A} (f : A  A  A) `{!Associative (=) f} x y z :
519
  f x (f y z) = f (f x y) z.
520
Proof. auto. Qed.
521
Lemma left_absorb_L {A} (i : A) (f : A  A  A) `{!LeftAbsorb (=) i f} x :
522
523
  f i x = i.
Proof. auto. Qed.
524
Lemma right_absorb_L {A} (i : A) (f : A  A  A) `{!RightAbsorb (=) i f} x :
525
526
  f x i = i.
Proof. auto. Qed.
527
Lemma left_distr_L {A} (f g : A  A  A) `{!LeftDistr (=) f g} x y z :
528
529
  f x (g y z) = g (f x y) (f x z).
Proof. auto. Qed.
530
Lemma right_distr_L {A} (f g : A  A  A) `{!RightDistr (=) f g} y z x :
531
532
  f (g y z) x = g (f y x) (f z x).
Proof. auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
533

534
(** ** Axiomatization of ordered structures *)
535
536
537
538
539
540
541
542
543
544
545
(** The classes [PreOrder], [PartialOrder], and [TotalOrder] do not use the
relation [⊆] because we often have multiple orders on the same structure. *)
Class PartialOrder {A} (R : relation A) : Prop := {
  po_preorder :> PreOrder R;
  po_anti_symmetric :> AntiSymmetric (=) R
}.
Class TotalOrder {A} (R : relation A) : Prop := {
  to_po :> PartialOrder R;
  to_trichotomy :> Trichotomy R
}.

546
547
(** A pre-order equipped with a smallest element. *)
Class BoundedPreOrder A `{Empty A} `{SubsetEq A} : Prop := {
Robbert Krebbers's avatar
Robbert Krebbers committed
548
549
550
551
  bounded_preorder :>> PreOrder ();
  subseteq_empty x :   x
}.

552
(** We do not include equality in the following interfaces so as to avoid the
553
need for proofs that the relations and operations respect setoid equality.
554
555
Instead, we will define setoid equality in a generic way as
[λ X Y, X ⊆ Y ∧ Y ⊆ X]. *)
556
Class BoundedJoinSemiLattice A `{Empty A} `{SubsetEq A} `{Union A} : Prop := {
557
  bjsl_preorder :>> BoundedPreOrder A;
558
559
  union_subseteq_l x y : x  x  y;
  union_subseteq_r x y : y  x  y;
Robbert Krebbers's avatar
Robbert Krebbers committed
560
561
  union_least x y z : x  z  y  z  x  y  z
}.
562
Class MeetSemiLattice A `{Empty A} `{SubsetEq A} `{Intersection A} : Prop := {
Robbert Krebbers's avatar
Robbert Krebbers committed
563
  msl_preorder :>> BoundedPreOrder A;
564
565
  intersection_subseteq_l x y : x  y  x;
  intersection_subseteq_r x y : x  y  y;
Robbert Krebbers's avatar
Robbert Krebbers committed
566
567
  intersection_greatest x y z : z  x  z  y  z  x  y
}.
568
569
570
571

(** A join distributive lattice with distributivity stated in the order
theoretic way. We will prove that distributivity of join, and distributivity
as an equality can be derived. *)
572
Class LowerBoundedLattice A `{Empty A} `{SubsetEq A}
573
    `{Union A} `{Intersection A} : Prop := {
574
  lbl_bjsl :>> BoundedJoinSemiLattice A;
575
576
  lbl_msl :>> MeetSemiLattice A;
  lbl_distr x y z : (x  y)  (x  z)  x  (y  z)
577
}.
578

579
(** ** Axiomatization of collections *)
580
581
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
582
Instance: Params (@map) 3.
583
Class SimpleCollection A C `{ElemOf A C}
584
    `{Empty C} `{Singleton A C} `{Union C} : Prop := {
585
  not_elem_of_empty (x : A) : x  ;
586
  elem_of_singleton (x y : A) : x  {[ y ]}  x = y;
587
588
589
  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}
590
    `{Union C} `{Intersection C} `{Difference C} : Prop := {
591
  collection_simple :>> SimpleCollection A C;
Robbert Krebbers's avatar
Robbert Krebbers committed
592
  elem_of_intersection X Y (x : A) : x  X  Y  x  X  x  Y;
593
594
595
596
597
598
599
  elem_of_difference X Y (x : A) : x  X  Y  x  X  x  Y
}.
Class CollectionOps A C
    `{ElemOf A C} `{Empty C} `{Singleton A C}
    `{Union C} `{Intersection C} `{Difference C}
    `{IntersectionWith A C} `{Filter A C} : Prop := {
  collection_ops :>> Collection A C;
600
  elem_of_intersection_with (f : A  A  option A) X Y (x : A) :
601
602
603
    x  intersection_with f X Y   x1 x2, x1  X  x2  Y  f x1 x2 = Some x;
  elem_of_filter X P `{ x, Decision (P x)} x :
    x  filter P X  P x  x  X
Robbert Krebbers's avatar
Robbert Krebbers committed
604
605
}.

606
607
608
(** 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
609
Class Elements A C := elements: C  list A.
610
Instance: Params (@elements) 3.
611
612
613
614
615
616
617
618
619
620
621
622
623
624

(** 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).

(** 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} `{Singleton A C}
625
626
    `{Union C} `{Intersection C} `{Difference C}
    `{Elements A C} `{ x y : A, Decision (x = y)} : Prop := {
Robbert Krebbers's avatar
Robbert Krebbers committed
627
  fin_collection :>> Collection A C;
628
  elements_spec X x : x  X  x  elements X;
Robbert Krebbers's avatar
Robbert Krebbers committed
629
  elements_nodup X : NoDup (elements X)
630
631
}.
Class Size C := size: C  nat.
632
Arguments size {_ _} !_ / : simpl nomatch.
633
Instance: Params (@size) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
634

635
636
637
638
639
640
641
642
643
644
(** 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)}
645
    `{!MBind M} `{!MRet M} `{!FMap M} `{!MJoin M} : Prop := {
646
647
648
  collection_monad_simple A :> SimpleCollection A (M A);
  elem_of_bind {A B} (f : A  M B) (X : M A) (x : B) :
    x  X = f   y, x  f y  y  X;
649
  elem_of_ret {A} (x y : A) : x  mret y  x = y;
650
651
  elem_of_fmap {A B} (f : A  B) (X : M A) (x : B) :
    x  f <$> X   y, x = f y  y  X;
652
  elem_of_join {A} (X : M (M A)) (x : A) : x  mjoin X   Y, x  Y  Y  X
653
654
}.

655
656
657
(** 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
658
Class Fresh A C := fresh: C  A.
659
Instance: Params (@fresh) 3.
660
Class FreshSpec A C `{ElemOf A C}
661
    `{Empty C} `{Singleton A C} `{Union C} `{Fresh A C} : Prop := {
662
  fresh_collection_simple :>> SimpleCollection A C;
663
  fresh_proper_alt X Y : ( x, x  X  x  Y)  fresh X = fresh Y;
Robbert Krebbers's avatar
Robbert Krebbers committed
664
665
666
  is_fresh (X : C) : fresh X  X
}.

667
(** * Miscellaneous *)
668
669
670
Class Half A := half: A  A.
Notation "x .½" := (half x) (at level 20, format "x .½") : C_scope.

671
672
Lemma proj1_sig_inj {A} (P : A  Prop) x (Px : P x) y (Py : P y) :
  xPx = yPy  x = y.
673
Proof. injection 1; trivial. Qed.
674
Lemma not_symmetry `{R : relation A} `{!Symmetric R} x y : ¬R x y  ¬R y x.
675
Proof. intuition. Qed.
676
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} x y : R x y  R y x.
677
678
Proof. intuition. Qed.

679
680
681
(** ** 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
682
683
684
685
686
687
688
689
690
691
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.

692
(** ** Products *)
693
694
695
696
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)).
697
698
699
700
701
Arguments fst_map {_ _ _} _ !_ /.
Arguments snd_map {_ _ _} _ !_ /.

Instance:  {A A' B} (f : A  A'),
  Injective (=) (=) f  Injective (=) (=) (@fst_map A A' B f).
702
Proof. intros ????? [??] [??]; injection 1; firstorder congruence. Qed.
703
704
Instance:  {A B B'} (f : B  B'),
  Injective (=) (=) f  Injective (=) (=) (@snd_map A B B' f).
705
Proof. intros ????? [??] [??]; injection 1; firstorder congruence. Qed.
706

707
708
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
709
710
711

Section prod_relation.
  Context `{R1 : relation A} `{R2 : relation B}.
712
713
  Global Instance:
    Reflexive R1  Reflexive R2  Reflexive (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
714
  Proof. firstorder eauto. Qed.
715
716
  Global Instance:
    Symmetric R1  Symmetric R2  Symmetric (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
717
  Proof. firstorder eauto. Qed.
718
719
  Global Instance:
    Transitive R1  Transitive R2  Transitive (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
720
  Proof. firstorder eauto. Qed.
721
722
  Global Instance:
    Equivalence R1  Equivalence R2  Equivalence (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
723
724
725
726
727
728
729
730
731
  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.

732
(** ** Other *)
733
734
735
736
737
738
739
740
741
Definition proj_eq {A B} (f : B  A) : relation B := λ x y, f x = f y.
Global Instance proj_eq_equivalence `(f : B  A) : Equivalence (proj_eq f).
Proof. unfold proj_eq. repeat split; red; intuition congruence. Qed.
Notation "x ~{ f } y" := (proj_eq f x y)
  (at level 70, format "x  ~{ f }  y") : C_scope.
Notation "(~{ f } )" := (proj_eq f) (f at level 10, only parsing) : C_scope.

Hint Extern 0 (_ ~{_} _) => reflexivity.
Hint Extern 0 (_ ~{_} _) => symmetry; assumption.
Robbert Krebbers's avatar
Robbert Krebbers committed
742
743

Instance:  A B (x : B), Commutative (=) (λ _ _ : A, x).
744
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
745
Instance:  A (x : A), Associative (=) (λ _ _ : A, x).
746
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
747
Instance:  A, Associative (=) (λ x _ : A, x).
748
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
749
Instance:  A, Associative (=) (λ _ x : A, x).
750
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
751
Instance:  A, Idempotent (=) (λ x _ : A, x).
752
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
753
Instance:  A, Idempotent (=) (λ _ x : A, x).
754
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
755

756
757
Instance left_id_propholds {A} (R : relation A) i f :
  LeftId R i f   x, PropHolds (R (f i x) x).
758
Proof. red. trivial. Qed.
759
760
Instance right_id_propholds {A} (R : relation A) i f :
  RightId R i f   x, PropHolds (R (f x i) x).
761
Proof. red. trivial. Qed.
762
763
764
765
766
767
Instance left_absorb_propholds {A} (R : relation A) i f :
  LeftAbsorb R i f   x, PropHolds (R (f i x) i).
Proof. red. trivial. Qed.
Instance right_absorb_propholds {A} (R : relation A) i f :
  RightAbsorb R i f   x, PropHolds (R (f x i) i).
Proof. red. trivial. Qed.
768
769
Instance idem_propholds {A} (R : relation A) f :
  Idempotent R f   x, PropHolds (R (f x x) x).
770
Proof. red. trivial. Qed.
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796

Lemma injective_iff {A B} {R : relation A} {S : relation B} (f : A  B)
  `{!Injective R S f} `{!Proper (R ==> S) f} x y : S (f x) (f y)  R x y.
Proof. firstorder. Qed.
Instance: Injective (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Instance: Injective (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.
Instance: Injective2 (=) (=) (=) (@pair A B).
Proof. injection 1; auto. Qed.
Instance:  `{Injective2 A B C R1 R2 R3 f} y, Injective R1 R3 (λ x, f x y).
Proof. repeat intro; edestruct (injective2 f); eauto. Qed.
Instance:  `{Injective2 A B C R1 R2 R3 f} x, Injective R2 R3 (f x).
Proof. repeat intro; edestruct (injective2 f); eauto. Qed.

Lemma cancel_injective `{Cancel A B R1 f g}
  `{!Equivalence R1} `{!Proper (R2 ==> R1) f} : Injective R1 R2 g.
Proof.
  intros x y E. rewrite <-(cancel f g x), <-(cancel f g y), E. reflexivity.
Qed.
Lemma cancel_surjective `{Cancel A B R1 f g} : Surjective R1 f.
Proof. intros y. exists (g y). auto. Qed.

Lemma impl_transitive (P Q R : Prop) : (P  Q)  (Q  R)  (P  R).
Proof. tauto. Qed.
Instance: Commutative () (@eq A).
797
Proof. red; intuition. Qed.
798
Instance: Commutative () (λ x y, @eq A y x).
799
Proof. red; intuition. Qed.
800
Instance: Commutative () ().
801
Proof. red; intuition. Qed.
802
Instance: Commutative () ().
803
Proof. red; intuition. Qed.
804
Instance: Associative () ().
805
Proof. red; intuition. Qed.
806
Instance: Idempotent () ().
807
Proof. red; intuition. Qed.
808
Instance: Commutative () ().
809
Proof. red; intuition. Qed.
810
Instance: Associative () ().
811
Proof. red; intuition. Qed.
812
Instance: Idempotent