base.v 40.2 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
2
3
4
5
6
7
8
9
(* Copyright (c) 2012-2015, 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. *)
Global Generalizable All Variables.
Global Set Automatic Coercions Import.
Global Set Asymmetric Patterns.
10
From Coq Require Export Morphisms RelationClasses List Bool Utf8 Program Setoid.
11
Obligation Tactic := idtac.
Robbert Krebbers's avatar
Robbert Krebbers committed
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55

(** * General *)
(** Zipping lists. *)
Definition zip_with {A B C} (f : A  B  C) : list A  list B  list C :=
  fix go l1 l2 :=
  match l1, l2 with x1 :: l1, x2 :: l2 => f x1 x2 :: go l1 l2 | _ , _ => [] end.
Notation zip := (zip_with pair).

(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
Arguments id _ _ /.
Arguments compose _ _ _ _ _ _ /.
Arguments flip _ _ _ _ _ _ /.
Arguments const _ _ _ _ /.
Typeclasses Transparent id compose flip const.
Instance: Params (@pair) 2.

(** 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. *)
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.

Notation curry := prod_curry.
Notation uncurry := prod_uncurry.
Definition curry3 {A B C D} (f : A  B  C  D) (p : A * B * C) : D :=
  let '(a,b,c) := p in f a b c.
Definition curry4 {A B C D E} (f : A  B  C  D  E) (p : A * B * C * D) : E :=
  let '(a,b,c,d) := p in f a b c d.

(** Throughout this development we use [C_scope] for all general purpose
notations that do not belong to a more specific scope. *)
Delimit Scope C_scope with C.
Global Open Scope C_scope.

(** Introduce some Haskell style like notations. *)
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.

56
Hint Extern 0 (_ = _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
Hint Extern 100 (_  _) => discriminate.

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.

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

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.

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.

Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_  _) => symmetry; assumption.

Notation "( x ,)" := (pair x) (only parsing) : C_scope.
Notation "(, y )" := (λ x, (x,y)) (only parsing) : C_scope.

Notation "p .1" := (fst p) (at level 10, format "p .1").
Notation "p .2" := (snd p) (at level 10, format "p .2").

Ralf Jung's avatar
Ralf Jung committed
94
95
Definition fun_map {A A' B B'} (f : A' -> A) (g : B -> B')
  (h : A -> B) : A' -> B' := g  h  f.
Robbert Krebbers's avatar
Robbert Krebbers committed
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
Definition prod_map {A A' B B'} (f : A  A') (g : B  B')
  (p : A * B) : A' * B' := (f (p.1), g (p.2)).
Arguments prod_map {_ _ _ _} _ _ !_ /.
Definition prod_zip {A A' A'' B B' B''} (f : A  A'  A'') (g : B  B'  B'')
    (p : A * B) (q : A' * B') : A'' * B'' := (f (p.1) (q.1), g (p.2) (q.2)).
Arguments prod_zip {_ _ _ _ _ _} _ _ !_ !_ /.

(** Set convenient implicit arguments for [existT] and introduce notations. *)
Arguments existT {_ _} _ _.
Arguments proj1_sig {_ _} _.
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : C_scope.

(** * Type classes *)
(** ** Provable propositions *)
(** This type class collects provable propositions. It is useful to constraint
type classes by arbitrary propositions. *)
Class PropHolds (P : Prop) := prop_holds: P.

Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
Proof. repeat intro; trivial. Qed.

Ltac solve_propholds :=
  match goal with
  | |- PropHolds (?P) => apply _
  | |- ?P => change (PropHolds P); apply _
  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)]. *)
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.

(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
135
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
Robbert Krebbers's avatar
Robbert Krebbers committed
136
137
138
Arguments populate {_} _.

Instance unit_inhabited: Inhabited unit := populate ().
Robbert Krebbers's avatar
Robbert Krebbers committed
139
Instance bool_inhabated : Inhabited bool := populate true.
Robbert Krebbers's avatar
Robbert Krebbers committed
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
Instance list_inhabited {A} : Inhabited (list A) := populate [].
Instance prod_inhabited {A B} (iA : Inhabited A)
    (iB : Inhabited B) : Inhabited (A * B) :=
  match iA, iB with populate x, populate y => populate (x,y) end.
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
  match iA with populate x => populate (inl x) end.
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
  match iB with populate y => populate (inl y) end.
Instance option_inhabited {A} : Inhabited (option A) := populate None.

(** ** 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.

(** ** Setoid equality *)
(** We define an operational type class for setoid equality. This is based on
(Spitters/van der Weegen, 2011). *)
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.

(** 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. *)
173
174
175
176
177
Class LeibnizEquiv A `{Equiv A} := leibniz_equiv x y : x  y  x = y.
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@equiv A _)} (x y : A) :
  x  y  x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
 
Robbert Krebbers's avatar
Robbert Krebbers committed
178
179
180
Ltac fold_leibniz := repeat
  match goal with
  | H : context [ @equiv ?A _ _ _ ] |- _ =>
181
    setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
Robbert Krebbers's avatar
Robbert Krebbers committed
182
  | |- context [ @equiv ?A _ _ _ ] =>
183
    setoid_rewrite (leibniz_equiv_iff (A:=A))
Robbert Krebbers's avatar
Robbert Krebbers committed
184
185
186
187
  end.
Ltac unfold_leibniz := repeat
  match goal with
  | H : context [ @eq ?A _ _ ] |- _ =>
188
    setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
Robbert Krebbers's avatar
Robbert Krebbers committed
189
  | |- context [ @eq ?A _ _ ] =>
190
    setoid_rewrite <-(leibniz_equiv_iff (A:=A))
Robbert Krebbers's avatar
Robbert Krebbers committed
191
192
  end.

193
194
Definition equivL {A} : Equiv A := (=).

Robbert Krebbers's avatar
Robbert Krebbers committed
195
196
197
198
199
200
201
202
203
(** 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. *)
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
204
Hint Extern 0 (_  _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
205
206
207
208
209
210
211
212
213
214
Hint Extern 0 (_  _) => symmetry; assumption.

(** ** Operations on collections *)
(** We define operational type classes for the traditional operations and
relations on collections: the empty collection [∅], the union [(∪)],
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

215
216
217
Class Top A := top : A.
Notation "⊤" := top : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
Class Union A := union: A  A  A.
Instance: Params (@union) 2.
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.
Infix "∪*" := (zip_with ()) (at level 50, left associativity) : C_scope.
Notation "(∪*)" := (zip_with ()) (only parsing) : C_scope.
Infix "∪**" := (zip_with (zip_with ()))
  (at level 50, left associativity) : C_scope.
Infix "∪*∪**" := (zip_with (prod_zip () (*)))
  (at level 50, left associativity) : C_scope.

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.

Class Intersection A := intersection: A  A  A.
Instance: Params (@intersection) 2.
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.
Instance: Params (@difference) 2.
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.
Infix "∖*" := (zip_with ()) (at level 40, left associativity) : C_scope.
Notation "(∖*)" := (zip_with ()) (only parsing) : C_scope.
Infix "∖**" := (zip_with (zip_with ()))
  (at level 40, left associativity) : C_scope.
Infix "∖*∖**" := (zip_with (prod_zip () (*)))
  (at level 50, left associativity) : C_scope.

Class Singleton A B := singleton: A  B.
Instance: Params (@singleton) 3.
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
Notation "{[ x ; y ; .. ; z ]}" :=
  (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.

Class SubsetEq A := subseteq: relation A.
Instance: Params (@subseteq) 2.
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, 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.
Infix "⊆*" := (Forall2 ()) (at level 70) : C_scope.
Notation "(⊆*)" := (Forall2 ()) (only parsing) : C_scope.
Infix "⊆**" := (Forall2 (*)) (at level 70) : C_scope.
Infix "⊆1*" := (Forall2 (λ p q, p.1  q.1)) (at level 70) : C_scope.
Infix "⊆2*" := (Forall2 (λ p q, p.2  q.2)) (at level 70) : C_scope.
Infix "⊆1**" := (Forall2 (λ p q, p.1 * q.1)) (at level 70) : C_scope.
Infix "⊆2**" := (Forall2 (λ p q, p.2 * q.2)) (at level 70) : C_scope.

Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_ * _) => reflexivity.
Hint Extern 0 (_ ** _) => reflexivity.

Definition strict {A} (R : relation A) : relation A := λ X Y, R X Y  ¬R Y X.
Instance: Params (@strict) 2.
Infix "⊂" := (strict ()) (at level 70) : C_scope.
Notation "(⊂)" := (strict ()) (only parsing) : C_scope.
Notation "( X ⊂ )" := (strict () X) (only parsing) : C_scope.
Notation "( ⊂ X )" := (λ Y, 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.

(** 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.

Class ElemOf A B := elem_of: A  B  Prop.
Instance: Params (@elem_of) 3.
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.

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.
320
Notation "(.⊥ X )" := (λ Y, Y  X) (only parsing) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
Infix "⊥*" := (Forall2 ()) (at level 70) : C_scope.
Notation "(⊥*)" := (Forall2 ()) (only parsing) : C_scope.
Infix "⊥**" := (Forall2 (*)) (at level 70) : C_scope.
Infix "⊥1*" := (Forall2 (λ p q, p.1  q.1)) (at level 70) : C_scope.
Infix "⊥2*" := (Forall2 (λ p q, p.2  q.2)) (at level 70) : C_scope.
Infix "⊥1**" := (Forall2 (λ p q, p.1 * q.1)) (at level 70) : C_scope.
Infix "⊥2**" := (Forall2 (λ p q, p.2 * q.2)) (at level 70) : C_scope.
Hint Extern 0 (_  _) => symmetry; eassumption.
Hint Extern 0 (_ * _) => symmetry; eassumption.

Class DisjointE E A := disjointE : E  A  A  Prop.
Instance: Params (@disjointE) 4.
Notation "X ⊥{ Γ } Y" := (disjointE Γ X Y)
  (at level 70, format "X  ⊥{ Γ }  Y") : C_scope.
Notation "(⊥{ Γ } )" := (disjointE Γ) (only parsing, Γ at level 1) : C_scope.
Notation "Xs ⊥{ Γ }* Ys" := (Forall2 ({Γ}) Xs Ys)
  (at level 70, format "Xs  ⊥{ Γ }*  Ys") : C_scope.
Notation "(⊥{ Γ }* )" := (Forall2 ({Γ}))
  (only parsing, Γ at level 1) : C_scope.
Notation "X ⊥{ Γ1 , Γ2 , .. , Γ3 } Y" := (disjoint (pair .. (Γ1, Γ2) .. Γ3) X Y)
  (at level 70, format "X  ⊥{ Γ1 , Γ2 , .. , Γ3 }  Y") : C_scope.
Notation "Xs ⊥{ Γ1 , Γ2 , .. , Γ3 }* Ys" :=
  (Forall2 (disjoint (pair .. (Γ1, Γ2) .. Γ3)) Xs Ys)
  (at level 70, format "Xs  ⊥{ Γ1 ,  Γ2 , .. , Γ3 }*  Ys") : C_scope.
Hint Extern 0 (_ {_} _) => symmetry; eassumption.

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

Section disjoint_list.
  Context `{Disjoint A, Union A, Empty A}.
  Inductive disjoint_list_default : DisjointList A :=
    | disjoint_nil_2 :  (@nil A)
    | disjoint_cons_2 (X : A) (Xs : list A) : X   Xs   Xs   (X :: Xs).
  Global Existing Instance disjoint_list_default.

  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 disjoint_list.

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

(** ** Monadic operations *)
(** We define operational type classes for the monadic operations bind, join 
and fmap. 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.
Arguments mret {_ _ _} _.
373
Instance: Params (@mret) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
374
375
Class MBind (M : Type  Type) := mbind :  {A B}, (A  M B)  M A  M B.
Arguments mbind {_ _ _ _} _ !_ /.
376
Instance: Params (@mbind) 4.
Robbert Krebbers's avatar
Robbert Krebbers committed
377
378
Class MJoin (M : Type  Type) := mjoin:  {A}, M (M A)  M A.
Arguments mjoin {_ _ _} !_ /.
379
Instance: Params (@mjoin) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
380
381
Class FMap (M : Type  Type) := fmap :  {A B}, (A  B)  M A  M B.
Arguments fmap {_ _ _ _} _ !_ /.
382
Instance: Params (@fmap) 4.
Robbert Krebbers's avatar
Robbert Krebbers committed
383
384
Class OMap (M : Type  Type) := omap:  {A B}, (A  option B)  M A  M B.
Arguments omap {_ _ _ _} _ !_ /.
385
Instance: Params (@omap) 4.
Robbert Krebbers's avatar
Robbert Krebbers committed
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435

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))
  (at level 65, only parsing, right associativity) : C_scope.
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.
Notation "' ( x1 , x2 ) ← y ; z" :=
  (y = (λ x : _, let ' (x1, x2) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "' ( x1 , x2 , x3 ) ← y ; z" :=
  (y = (λ x : _, let ' (x1,x2,x3) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "' ( x1 , x2 , x3  , x4 ) ← y ; z" :=
  (y = (λ x : _, let ' (x1,x2,x3,x4) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "' ( x1 , x2 , x3  , x4 , x5 ) ← y ; z" :=
  (y = (λ x : _, let ' (x1,x2,x3,x4,x5) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "' ( x1 , x2 , x3  , x4 , x5 , x6 ) ← y ; z" :=
  (y = (λ x : _, let ' (x1,x2,x3,x4,x5,x6) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.

Notation "ps .*1" := (fmap (M:=list) fst ps)
  (at level 10, format "ps .*1").
Notation "ps .*2" := (fmap (M:=list) snd ps)
  (at level 10, format "ps .*2").

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

(** ** Operations on maps *)
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
The function look up [m !! k] should yield the element at key [k] in [m]. *)
Class Lookup (K A M : Type) := lookup: K  M  option A.
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, m !! i) (only parsing) : C_scope.
Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.

436
437
438
(** The singleton map *)
Class SingletonM K A M := singletonM: K  A  M.
Instance: Params (@singletonM) 5.
439
Notation "{[ k := a ]}" := (singletonM k a) (at level 1) : C_scope.
440

Robbert Krebbers's avatar
Robbert Krebbers committed
441
442
443
(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
Class Insert (K A M : Type) := insert: K  A  M  M.
Robbert Krebbers's avatar
Robbert Krebbers committed
444
Instance: Params (@insert) 5.
Robbert Krebbers's avatar
Robbert Krebbers committed
445
446
447
448
449
450
451
452
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.

(** 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. *)
Class Delete (K M : Type) := delete: K  M  M.
Robbert Krebbers's avatar
Robbert Krebbers committed
453
Instance: Params (@delete) 4.
Robbert Krebbers's avatar
Robbert Krebbers committed
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
Arguments delete _ _ _ !_ !_ / : simpl nomatch.

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

(** The function [alter f k m] should update the value at key [k] using the
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]. *)
Class PartialAlter (K A M : Type) :=
  partial_alter: (option A  option A)  K  M  M.
Instance: Params (@partial_alter) 4.
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.

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

(** 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)].*)
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.

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

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

Class DifferenceWith (A M : Type) :=
  difference_with: (A  A  option A)  M  M  M.
Instance: Params (@difference_with) 3.
Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch.

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 _ _ _ _ _ !_ /.

Class LookupE (E K A M : Type) := lookupE: E  K  M  option A.
Instance: Params (@lookupE) 6.
Notation "m !!{ Γ } i" := (lookupE Γ i m)
  (at level 20, format "m  !!{ Γ }  i") : C_scope.
Notation "(!!{ Γ } )" := (lookupE Γ) (only parsing, Γ at level 1) : C_scope.
Arguments lookupE _ _ _ _ _ _ !_ !_ / : simpl nomatch.

Class InsertE (E K A M : Type) := insertE: E  K  A  M  M.
Robbert Krebbers's avatar
Robbert Krebbers committed
515
Instance: Params (@insertE) 6.
Robbert Krebbers's avatar
Robbert Krebbers committed
516
517
518
519
520
521
522
Notation "<[ k := a ]{ Γ }>" := (insertE Γ k a)
  (at level 5, right associativity, format "<[ k := a ]{ Γ }>") : C_scope.
Arguments insertE _ _ _ _ _ _ !_ _ !_ / : simpl nomatch.

(** ** 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
523
524
525
526
allows us to write [inj (k ++)] instead of [app_inv_head k]. *)
Class Inj {A B} (R : relation A) (S : relation B) (f : A  B) : Prop :=
  inj x y : S (f x) (f y)  R x y.
Class Inj2 {A B C} (R1 : relation A) (R2 : relation B)
Robbert Krebbers's avatar
Robbert Krebbers committed
527
    (S : relation C) (f : A  B  C) : Prop :=
528
  inj2 x1 x2 y1 y2 : S (f x1 x2) (f y1 y2)  R1 x1 y1  R2 x2 y2.
Robbert Krebbers's avatar
Robbert Krebbers committed
529
Class Cancel {A B} (S : relation B) (f : A  B) (g : B  A) : Prop :=
530
531
532
533
534
535
536
  cancel :  x, S (f (g x)) x.
Class Surj {A B} (R : relation B) (f : A  B) :=
  surj y :  x, R (f x) y.
Class IdemP {A} (R : relation A) (f : A  A  A) : Prop :=
  idemp x : R (f x x) x.
Class Comm {A B} (R : relation A) (f : B  B  A) : Prop :=
  comm x y : R (f x y) (f y x).
Robbert Krebbers's avatar
Robbert Krebbers committed
537
Class LeftId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
538
  left_id x : R (f i x) x.
Robbert Krebbers's avatar
Robbert Krebbers committed
539
Class RightId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
540
541
542
  right_id x : R (f x i) x.
Class Assoc {A} (R : relation A) (f : A  A  A) : Prop :=
  assoc x y z : R (f x (f y z)) (f (f x y) z).
Robbert Krebbers's avatar
Robbert Krebbers committed
543
Class LeftAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
544
  left_absorb x : R (f i x) i.
Robbert Krebbers's avatar
Robbert Krebbers committed
545
Class RightAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
546
547
548
  right_absorb x : R (f x i) i.
Class AntiSymm {A} (R S : relation A) : Prop :=
  anti_symm x y : S x y  S y x  R x y.
Robbert Krebbers's avatar
Robbert Krebbers committed
549
550
Class Total {A} (R : relation A) := total x y : R x y  R y x.
Class Trichotomy {A} (R : relation A) :=
551
  trichotomy x y : R x y  x = y  R y x.
Robbert Krebbers's avatar
Robbert Krebbers committed
552
Class TrichotomyT {A} (R : relation A) :=
553
  trichotomyT x y : {R x y} + {x = y} + {R y x}.
Robbert Krebbers's avatar
Robbert Krebbers committed
554
555

Arguments irreflexivity {_} _ {_} _ _.
556
557
Arguments inj {_ _ _ _} _ {_} _ _ _.
Arguments inj2 {_ _ _ _ _ _} _ {_} _ _ _ _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
558
Arguments cancel {_ _ _} _ _ {_} _.
559
560
561
Arguments surj {_ _ _} _ {_} _.
Arguments idemp {_ _} _ {_} _.
Arguments comm {_ _ _} _ {_} _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
562
563
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
564
Arguments assoc {_ _} _ {_} _ _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
565
566
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
567
Arguments anti_symm {_ _} _ {_} _ _ _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
568
569
570
571
Arguments total {_} _ {_} _ _.
Arguments trichotomy {_} _ {_} _ _.
Arguments trichotomyT {_} _ {_} _ _.

572
Instance id_inj {A} : Inj (=) (=) (@id A).
Robbert Krebbers's avatar
Robbert Krebbers committed
573
574
575
576
577
Proof. intros ??; auto. Qed.

(** 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. *)
578
Lemma idemp_L {A} (f : A  A  A) `{!IdemP (=) f} x : f x x = x.
Robbert Krebbers's avatar
Robbert Krebbers committed
579
Proof. auto. Qed.
580
Lemma comm_L {A B} (f : B  B  A) `{!Comm (=) f} x y :
Robbert Krebbers's avatar
Robbert Krebbers committed
581
582
583
584
585
586
  f x y = f y x.
Proof. auto. Qed.
Lemma left_id_L {A} (i : A) (f : A  A  A) `{!LeftId (=) i f} x : f i x = x.
Proof. auto. Qed.
Lemma right_id_L {A} (i : A) (f : A  A  A) `{!RightId (=) i f} x : f x i = x.
Proof. auto. Qed.
587
Lemma assoc_L {A} (f : A  A  A) `{!Assoc (=) f} x y z :
Robbert Krebbers's avatar
Robbert Krebbers committed
588
589
590
591
592
593
594
595
596
597
598
599
600
601
  f x (f y z) = f (f x y) z.
Proof. auto. Qed.
Lemma left_absorb_L {A} (i : A) (f : A  A  A) `{!LeftAbsorb (=) i f} x :
  f i x = i.
Proof. auto. Qed.
Lemma right_absorb_L {A} (i : A) (f : A  A  A) `{!RightAbsorb (=) i f} x :
  f x i = i.
Proof. auto. Qed.

(** ** Axiomatization of ordered structures *)
(** The classes [PreOrder], [PartialOrder], and [TotalOrder] use an arbitrary
relation [R] instead of [⊆] to support multiple orders on the same type. *)
Class PartialOrder {A} (R : relation A) : Prop := {
  partial_order_pre :> PreOrder R;
602
  partial_order_anti_symm :> AntiSymm (=) R
Robbert Krebbers's avatar
Robbert Krebbers committed
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
}.
Class TotalOrder {A} (R : relation A) : Prop := {
  total_order_partial :> PartialOrder R;
  total_order_trichotomy :> Trichotomy (strict R)
}.

(** We do not use a setoid equality in the following interfaces to avoid the
need for proofs that the relations and operations are proper. Instead, we
define setoid equality generically [λ X Y, X ⊆ Y ∧ Y ⊆ X]. *)
Class EmptySpec A `{Empty A, SubsetEq A} : Prop := subseteq_empty X :   X.
Class JoinSemiLattice A `{SubsetEq A, Union A} : Prop := {
  join_semi_lattice_pre :>> PreOrder ();
  union_subseteq_l X Y : X  X  Y;
  union_subseteq_r X Y : Y  X  Y;
  union_least X Y Z : X  Z  Y  Z  X  Y  Z
}.
Class MeetSemiLattice A `{SubsetEq A, Intersection A} : Prop := {
  meet_semi_lattice_pre :>> PreOrder ();
  intersection_subseteq_l X Y : X  Y  X;
  intersection_subseteq_r X Y : X  Y  Y;
  intersection_greatest X Y Z : Z  X  Z  Y  Z  X  Y
}.
Class Lattice A `{SubsetEq A, Union A, Intersection A} : Prop := {
  lattice_join :>> JoinSemiLattice A;
  lattice_meet :>> MeetSemiLattice A;
  lattice_distr X Y Z : (X  Y)  (X  Z)  X  (Y  Z)
}.

(** ** Axiomatization of collections *)
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
Class SimpleCollection A C `{ElemOf A C,
    Empty C, Singleton A C, Union C} : Prop := {
  not_elem_of_empty (x : A) : x  ;
  elem_of_singleton (x y : A) : x  {[ y ]}  x = y;
  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} : Prop := {
  collection_simple :>> SimpleCollection A C;
  elem_of_intersection X Y (x : A) : x  X  Y  x  X  x  Y;
  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;
  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;
  elem_of_filter X P `{ x, Decision (P x)} x : x  filter P X  P x  x  X
}.

(** 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. *)
Class Elements A C := elements: C  list A.
Instance: Params (@elements) 3.

(** 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,
    Union C, Intersection C, Difference C,
    Elements A C,  x y : A, Decision (x = y)} : Prop := {
  fin_collection :>> Collection A C;
  elem_of_elements X x : x  elements X  x  X;
  NoDup_elements X : NoDup (elements X)
}.
Class Size C := size: C  nat.
Arguments size {_ _} !_ / : simpl nomatch.
Instance: Params (@size) 2.

(** 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} : Prop := {
  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;
  elem_of_ret {A} (x y : A) : x  mret y  x = y;
  elem_of_fmap {A B} (f : A  B) (X : M A) (x : B) :
    x  f <$> X   y, x = f y  y  X;
  elem_of_join {A} (X : M (M A)) (x : A) : x  mjoin X   Y, x  Y  Y  X
}.

(** 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. *)
Class Fresh A C := fresh: C  A.
Instance: Params (@fresh) 3.
Class FreshSpec A C `{ElemOf A C,
    Empty C, Singleton A C, Union C, Fresh A C} : Prop := {
  fresh_collection_simple :>> SimpleCollection A C;
  fresh_proper_alt X Y : ( x, x  X  x  Y)  fresh X = fresh Y;
  is_fresh (X : C) : fresh X  X
}.

(** * Booleans *)
(** The following coercion allows us to use Booleans as propositions. *)
Coercion Is_true : bool >-> Sortclass.
Hint Unfold Is_true.
Hint Immediate Is_true_eq_left.
Hint Resolve orb_prop_intro andb_prop_intro.
Notation "(&&)" := andb (only parsing).
Notation "(||)" := orb (only parsing).
Infix "&&*" := (zip_with (&&)) (at level 40).
Infix "||*" := (zip_with (||)) (at level 50).

Definition bool_le (β1 β2 : bool) : Prop := negb β1 || β2.
Infix "=.>" := bool_le (at level 70).
Infix "=.>*" := (Forall2 bool_le) (at level 70).
Instance: PartialOrder bool_le.
Proof. repeat split; repeat intros [|]; compute; tauto. Qed.

732
733
734
735
736
737
738
739
740
Lemma andb_True b1 b2 : b1 && b2  b1  b2.
Proof. destruct b1, b2; simpl; tauto. Qed.
Lemma orb_True b1 b2 : b1 || b2  b1  b2.
Proof. destruct b1, b2; simpl; tauto. Qed.
Lemma negb_True b : negb b  ¬b.
Proof. destruct b; simpl; tauto. Qed.
Lemma Is_true_false (b : bool) : b = false  ¬b.
Proof. now intros -> ?. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
(** * Miscellaneous *)
Class Half A := half: A  A.
Notation "½" := half : C_scope.
Notation "½*" := (fmap (M:=list) half) : C_scope.

Lemma proj1_sig_inj {A} (P : A  Prop) x (Px : P x) y (Py : P y) :
  xPx = yPy  x = y.
Proof. injection 1; trivial. Qed.
Lemma not_symmetry `{R : relation A, !Symmetric R} x y : ¬R x y  ¬R y x.
Proof. intuition. Qed.
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} x y : R x y  R y x.
Proof. intuition. Qed.

(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
Instance unit_equivalence : Equivalence (@equiv unit _).
Proof. repeat split. Qed.

(** ** Products *)
760
761
Instance prod_map_inj {A A' B B'} (f : A  A') (g : B  B') :
  Inj (=) (=) f  Inj (=) (=) g  Inj (=) (=) (prod_map f g).
Robbert Krebbers's avatar
Robbert Krebbers committed
762
763
Proof.
  intros ?? [??] [??] ?; simpl in *; f_equal;
764
    [apply (inj f)|apply (inj g)]; congruence.
Robbert Krebbers's avatar
Robbert Krebbers committed
765
766
767
768
769
770
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
797
798
799
800
801
802
803
804
805
806
807
808
Qed.

Definition prod_relation {A B} (R1 : relation A) (R2 : relation B) :
  relation (A * B) := λ x y, R1 (x.1) (y.1)  R2 (x.2) (y.2).
Section prod_relation.
  Context `{R1 : relation A, R2 : relation B}.
  Global Instance:
    Reflexive R1  Reflexive R2  Reflexive (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance:
    Symmetric R1  Symmetric R2  Symmetric (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance:
    Transitive R1  Transitive R2  Transitive (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance:
    Equivalence R1  Equivalence R2  Equivalence (prod_relation R1 R2).
  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.

Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation () ().
Instance pair_proper `{Equiv A, Equiv B} :
  Proper (() ==> () ==> ()) (@pair A B) | 0 := _.
Instance fst_proper `{Equiv A, Equiv B} :
  Proper (() ==> ()) (@fst A B) | 0 := _.
Instance snd_proper `{Equiv A, Equiv B} :
  Proper (() ==> ()) (@snd A B) | 0 := _.
Typeclasses Opaque prod_equiv.

(** ** Other *)
Lemma or_l P Q : ¬Q  P  Q  P.
Proof. tauto. Qed.
Lemma or_r P Q : ¬P  P  Q  Q.
Proof. tauto. Qed.
Lemma and_wlog_l (P Q : Prop) : (Q  P)  Q  (P  Q).
Proof. tauto. Qed.
Lemma and_wlog_r (P Q : Prop) : P  (P  Q)  (P  Q).
Proof. tauto. Qed.
809
Instance:  A B (x : B), Comm (=) (λ _ _ : A, x).
Robbert Krebbers's avatar
Robbert Krebbers committed
810
Proof. red. trivial. Qed.
811
Instance:  A (x : A), Assoc (=) (λ _ _ : A, x).
Robbert Krebbers's avatar
Robbert Krebbers committed
812
Proof. red. trivial. Qed.
813
Instance:  A, Assoc (=) (λ x _ : A, x).
Robbert Krebbers's avatar
Robbert Krebbers committed
814
Proof. red. trivial. Qed.
815
Instance:  A, Assoc (=) (λ _ x : A, x).
Robbert Krebbers's avatar
Robbert Krebbers committed
816
Proof. red. trivial. Qed.
817
Instance:  A, IdemP (=) (λ x _ : A, x).
Robbert Krebbers's avatar
Robbert Krebbers committed
818
Proof. red. trivial. Qed.
819
Instance:  A, IdemP (=) (λ _ x : A, x).
Robbert Krebbers's avatar
Robbert Krebbers committed
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
Proof. red. trivial. Qed.

Instance left_id_propholds {A} (R : relation A) i f :
  LeftId R i f   x, PropHolds (R (f i x) x).
Proof. red. trivial. Qed.
Instance right_id_propholds {A} (R : relation A) i f :
  RightId R i f   x, PropHolds (R (f x i) x).
Proof. red. trivial. Qed.
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.
Instance idem_propholds {A} (R : relation A) f :
835
  IdemP R f   x, PropHolds (R (f x x) x).
Robbert Krebbers's avatar
Robbert Krebbers committed
836
837
838
839
840
841
842
Proof. red. trivial. Qed.

Instance:  `{R1 : relation A, R2 : relation B} (x : B),
  Reflexive R2  Proper (R1 ==> R2) (λ _, x).
Proof. intros A R1 B R2 x ? y1 y2; reflexivity. Qed.
Instance: @PreOrder A (=).
Proof. split; repeat intro; congruence. Qed.
843
844
Lemma inj_iff {A B} {R : relation A} {S : relation B} (f : A  B)
  `{!Inj R S f} `{!Proper (R ==> S) f} x y : S (f x) (f y)  R x y.
Robbert Krebbers's avatar
Robbert Krebbers committed
845
Proof. firstorder. Qed.
846
Instance: Inj (=) (=) (@inl A B).
Robbert Krebbers's avatar
Robbert Krebbers committed
847
Proof. injection 1; auto. Qed.
848
Instance: Inj (=) (=) (@inr A B).
Robbert Krebbers's avatar
Robbert Krebbers committed
849
Proof. injection 1; auto. Qed.
850
Instance: Inj2 (=) (=) (=) (@pair A B).
Robbert Krebbers's avatar
Robbert Krebbers committed
851
Proof. injection 1; auto. Qed.
852
853
854
855
Instance:  `{Inj2 A B C R1 R2 R3 f} y, Inj R1 R3 (λ x, f x y).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.
Instance:  `{Inj2 A B C R1 R2 R3 f} x, Inj R2 R3 (f x).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
856

857
858
Lemma cancel_inj `{Cancel A B R1 f g}
  `{!Equivalence R1} `{!Proper (R2 ==> R1) f} : Inj R1 R2 g.
Robbert Krebbers's avatar
Robbert Krebbers committed
859
860
861
Proof.
  intros x y E. rewrite <-(cancel f g x), <-(cancel f g y), E. reflexivity.
Qed.
862
Lemma cancel_surj `{Cancel A B R1 f g} : Surj R1 f.
Robbert Krebbers's avatar
Robbert Krebbers committed
863
864
865
866
Proof. intros y. exists (g y). auto. Qed.

Lemma impl_transitive (P Q R : Prop) : (P  Q)  (Q  R)  (P  R).
Proof. tauto. Qed.
867
Instance: Comm () (@eq A).
Robbert Krebbers's avatar
Robbert Krebbers committed
868
Proof. red; intuition. Qed.
869
Instance: Comm () (λ x y, @eq A y x).
Robbert Krebbers's avatar
Robbert Krebbers committed
870
Proof. red; intuition. Qed.
871
Instance: Comm () ().
Robbert Krebbers's avatar
Robbert Krebbers committed
872
Proof. red; intuition. Qed.
873
Instance: Comm () ().
Robbert Krebbers's avatar
Robbert Krebbers committed
874
Proof. red; intuition. Qed.
875
Instance: Assoc () ().
Robbert Krebbers's avatar
Robbert Krebbers committed
876
Proof. red; intuition. Qed.
877
Instance: IdemP () ().
Robbert Krebbers's avatar
Robbert Krebbers committed
878
Proof. red; intuition. Qed.
879
Instance: Comm () ().
Robbert Krebbers's avatar
Robbert Krebbers committed
880
Proof. red; intuition. Qed.
881
Instance: Assoc () ().
Robbert Krebbers's avatar
Robbert Krebbers committed
882
Proof. red; intuition. Qed.
883
Instance: IdemP () ().
Robbert Krebbers's avatar
Robbert Krebbers committed
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
Proof. red; intuition. Qed.
Instance: LeftId () True ().
Proof. red; intuition. Qed.
Instance: RightId () True ().
Proof. red; intuition. Qed.
Instance: LeftAbsorb () False ().
Proof. red; intuition. Qed.
Instance: RightAbsorb () False ().
Proof. red; intuition. Qed.
Instance: LeftId () False ().
Proof. red; intuition. Qed.
Instance: RightId () False ().
Proof. red; intuition. Qed.
Instance: LeftAbsorb () True ().
Proof. red; intuition. Qed.
Instance: RightAbsorb () True ().
Proof. red; intuition. Qed.
Instance: LeftId () True impl.
Proof. unfold impl. red; intuition. Qed.
Instance: RightAbsorb () True impl.
Proof. unfold impl. red; intuition. Qed.
905
Lemma not_inj `{Inj A B R R' f} x y : ¬R x y  ¬R' (f x) (f y).
Robbert Krebbers's avatar
Robbert Krebbers committed
906
Proof. intuition. Qed.
907
908
909
910
911
912
Lemma not_inj2_1 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 :
  ¬R x1 x2  ¬R'' (f x1 y1) (f x2 y2).
Proof. intros HR HR''. destruct (inj2 f x1 y1 x2 y2); auto. Qed.
Lemma not_inj2_2 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 :
  ¬R' y1 y2  ¬R'' (f x1 y1) (f x2 y2).
Proof. intros HR' HR''. destruct (inj2 f x1 y1 x2 y2); auto. Qed.
913
914
Instance inj_compose {A B C} R1 R2 R3 (f : A  B) (g : B  C) :
  Inj R1 R2 f  Inj R2 R3 g  Inj R1 R3 (g  f).
Robbert Krebbers's avatar
Robbert Krebbers committed
915
Proof. red; intuition. Qed.
916
917
Instance surj_compose {A B C} R (f : A  B) (g : B  C) :
  Surj (=) f  Surj R g  Surj R (g  f).
Robbert Krebbers's avatar
Robbert Krebbers committed
918
Proof.
919
920
  intros ?? x. unfold compose. destruct (surj g x) as [y ?].
  destruct (surj f y) as [z ?]. exists z. congruence.
Robbert Krebbers's avatar
Robbert Krebbers committed
921
922
923
924
925
Qed.

Section sig_map.
  Context `{P : A  Prop} `{Q : B  Prop} (f : A  B) (Hf :  x, P x  Q (f x)).
  Definition sig_map (x : sig P) : sig Q := f (`x)  Hf _ (proj2_sig x).
926
927
  Global Instance sig_map_inj:
    ( x, ProofIrrel (P x))  Inj (=) (=) f  Inj (=) (=) sig_map.
Robbert Krebbers's avatar
Robbert Krebbers committed
928
929
  Proof.
    intros ?? [x Hx] [y Hy]. injection 1. intros Hxy.
930
    apply (inj f) in Hxy; subst. rewrite (proof_irrel _ Hy). auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
931
932
933
  Qed.
End sig_map.
Arguments sig_map _ _ _ _ _ _ !_ /.