base.v 43.6 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, 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.
Robbert Krebbers's avatar
Robbert Krebbers committed
9
Global Set Asymmetric Patterns.
10
Require Export Morphisms RelationClasses List Bool Utf8 Program Setoid.
Robbert Krebbers's avatar
Robbert Krebbers committed
11

12
(** * General *)
13 14 15 16 17
(** 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).
18

19 20
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
21
Arguments id _ _ /.
22
Arguments compose _ _ _ _ _ _ /.
23
Arguments flip _ _ _ _ _ _ /.
24 25
Arguments const _ _ _ _ /.
Typeclasses Transparent id compose flip const.
Robbert Krebbers's avatar
Robbert Krebbers committed
26
Instance: Params (@pair) 2.
27

28 29 30 31
(** 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. *)
32 33
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
34

35 36
Notation curry := prod_curry.
Notation uncurry := prod_uncurry.
37 38 39 40
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.
41

42 43
(** 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
44 45 46
Delimit Scope C_scope with C.
Global Open Scope C_scope.

47
(** Introduce some Haskell style like notations. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
48 49 50 51 52 53 54 55
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.
56
Hint Extern 100 (_  _) => discriminate.
Robbert Krebbers's avatar
Robbert Krebbers committed
57

58 59 60 61
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.

62
Notation "t $ r" := (t r)
63
  (at level 65, right associativity, only parsing) : C_scope.
64 65 66
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
67 68 69 70
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.
71

72 73 74 75 76 77 78 79 80 81 82 83
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.

84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
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").

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 {_ _ _ _ _ _} _ _ !_ !_ /.

100 101
(** Set convenient implicit arguments for [existT] and introduce notations. *)
Arguments existT {_ _} _ _.
102
Arguments proj1_sig {_ _} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
103
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
104
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
105

106 107 108 109
(** * 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
110 111
Class PropHolds (P : Prop) := prop_holds: P.

112 113
Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
114
Proof. repeat intro; trivial. Qed.
115 116 117

Ltac solve_propholds :=
  match goal with
118 119
  | |- PropHolds (?P) => apply _
  | |- ?P => change (PropHolds P); apply _
120 121 122 123 124 125 126
  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
127 128 129
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.

130 131
(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
132
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
133 134 135 136 137 138
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) :=
139
  match iA, iB with populate x, populate y => populate (x,y) end.
140
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
141
  match iA with populate x => populate (inl x) end.
142
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
143
  match iB with populate y => populate (inl y) end.
144 145
Instance option_inhabited {A} : Inhabited (option A) := populate None.

146 147 148 149 150 151
(** ** 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.

152 153 154
(** ** 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
155 156 157
Class Equiv A := equiv: relation A.
Infix "≡" := equiv (at level 70, no associativity) : C_scope.
Notation "(≡)" := equiv (only parsing) : C_scope.
158 159 160 161 162 163
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
164

165 166 167 168 169
Class EquivE E A := equivE: E  relation A.
Instance: Params (@equivE) 4.
Notation "X ≡{ Γ } Y" := (equivE Γ X Y)
  (at level 70, format "X  ≡{ Γ }  Y") : C_scope.
Notation "(≡{ Γ } )" := (equivE Γ) (only parsing, Γ at level 1) : C_scope.
170 171 172 173 174
Notation "X ≡{ Γ1 , Γ2 , .. , Γ3 } Y" :=
  (equivE (pair .. (Γ1, Γ2) .. Γ3) X Y)
  (at level 70, format "'[' X  ≡{ Γ1 , Γ2 , .. , Γ3 }  '/' Y ']'") : C_scope.
Notation "(≡{ Γ1 , Γ2 , .. , Γ3 } )" := (equivE (pair .. (Γ1, Γ2) .. Γ3))
  (only parsing, Γ1 at level 1) : C_scope.
175

176 177 178 179
(** 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. *)
180 181 182 183 184
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.
 
185 186 187
Ltac fold_leibniz := repeat
  match goal with
  | H : context [ @equiv ?A _ _ _ ] |- _ =>
188
    setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
189
  | |- context [ @equiv ?A _ _ _ ] =>
190
    setoid_rewrite (leibniz_equiv_iff (A:=A))
191 192 193 194
  end.
Ltac unfold_leibniz := repeat
  match goal with
  | H : context [ @eq ?A _ _ ] |- _ =>
195
    setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
196
  | |- context [ @eq ?A _ _ ] =>
197
    setoid_rewrite <-(leibniz_equiv_iff (A:=A))
198 199
  end.

200 201
Definition equivL {A} : Equiv A := (=).

202 203 204 205 206 207 208 209
(** 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
210
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
211 212
Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_  _) => symmetry; assumption.
213 214
Hint Extern 0 (_ {_} _) => reflexivity.
Hint Extern 0 (_ {_} _) => symmetry; assumption.
Robbert Krebbers's avatar
Robbert Krebbers committed
215

216
(** ** Operations on collections *)
217
(** We define operational type classes for the traditional operations and
218
relations on collections: the empty collection [∅], the union [(∪)],
219 220
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
221 222 223 224
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

Class Union A := union: A  A  A.
225
Instance: Params (@union) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
226 227 228 229
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.
230 231 232 233 234 235
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
236

237
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
238 239 240
Arguments union_list _ _ _ !_ /.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
241
Class Intersection A := intersection: A  A  A.
242
Instance: Params (@intersection) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
243 244 245 246 247 248
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.
249
Instance: Params (@difference) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
250 251 252 253
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.
254 255 256 257 258 259
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
260

261 262
Class Singleton A B := singleton: A  B.
Instance: Params (@singleton) 3.
263
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
264
Notation "{[ x ; y ; .. ; z ]}" :=
265 266 267 268 269 270
  (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.
271

272
Class SubsetEq A := subseteq: relation A.
273
Instance: Params (@subseteq) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
274 275 276
Infix "⊆" := subseteq (at level 70) : C_scope.
Notation "(⊆)" := subseteq (only parsing) : C_scope.
Notation "( X ⊆ )" := (subseteq X) (only parsing) : C_scope.
277
Notation "( ⊆ X )" := (λ Y, Y  X) (only parsing) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
278 279 280 281
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.
282 283 284 285 286 287 288
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
289

290
Hint Extern 0 (_  _) => reflexivity.
291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321
Hint Extern 0 (_ * _) => reflexivity.
Hint Extern 0 (_ ** _) => reflexivity.

Class SubsetEqE E A := subseteqE: E  relation A.
Instance: Params (@subseteqE) 4.
Notation "X ⊆{ Γ } Y" := (subseteqE Γ X Y)
  (at level 70, format "X  ⊆{ Γ }  Y") : C_scope.
Notation "(⊆{ Γ } )" := (subseteqE Γ) (only parsing, Γ at level 1) : C_scope.
Notation "X ⊈{ Γ } Y" := (¬X {Γ} Y)
  (at level 70, format "X  ⊈{ Γ }  Y") : C_scope.
Notation "(⊈{ Γ } )" := (λ X Y, X {Γ} Y)
  (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" :=
  (subseteqE (pair .. (Γ1, Γ2) .. Γ3) X Y)
  (at level 70, format "'[' X  ⊆{ Γ1 , Γ2 , .. , Γ3 }  '/' Y ']'") : C_scope.
Notation "(⊆{ Γ1 , Γ2 , .. , Γ3 } )" := (subseteqE (pair .. (Γ1, Γ2) .. Γ3))
  (only parsing, Γ1 at level 1) : C_scope.
Notation "X ⊈{ Γ1 , Γ2 , .. , Γ3 } Y" := (¬X {pair .. (Γ1, Γ2) .. Γ3} Y)
  (at level 70, format "X  ⊈{ Γ1 , Γ2 , .. , Γ3 }  Y") : C_scope.
Notation "(⊈{ Γ1 , Γ2 , .. , Γ3 } )" := (λ X Y, X {pair .. (Γ1, Γ2) .. Γ3} Y)
  (only parsing) : C_scope.
Notation "Xs ⊆{ Γ1 , Γ2 , .. , Γ3 }* Ys" :=
  (Forall2 ({pair .. (Γ1, Γ2) .. Γ3}) Xs Ys)
  (at level 70, format "Xs  ⊆{ Γ1 , Γ2 , .. , Γ3 }*  Ys") : C_scope.
Notation "(⊆{ Γ1 , Γ2 , .. , Γ3 }* )" := (Forall2 ({pair .. (Γ1, Γ2) .. Γ3}))
  (only parsing, Γ1 at level 1) : C_scope.
Hint Extern 0 (_ {_} _) => reflexivity.
322

323 324
Definition strict {A} (R : relation A) : relation A := λ X Y, R X Y  ¬R Y X.
Instance: Params (@strict) 2.
325 326 327 328
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.
329 330 331 332
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
333

334 335 336 337 338
(** 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
339
Class ElemOf A B := elem_of: A  B  Prop.
340
Instance: Params (@elem_of) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
341 342 343 344 345 346 347 348 349
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
350 351 352 353
Class Disjoint A := disjoint : A  A  Prop.
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
354
Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380
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 (_  _) => 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.
381 382 383

Class DisjointList A := disjoint_list : list A  Prop.
Instance: Params (@disjoint_list) 2.
384
Notation "⊥ Xs" := (disjoint_list Xs) (at level 20, format "⊥  Xs") : C_scope.
385

386 387 388 389 390 391
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.
392

393
  Lemma disjoint_list_nil  :  @nil A  True.
394 395 396
  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.
397
End disjoint_list.
398 399

Class Filter A B := filter:  (P : A  Prop) `{ x, Decision (P x)}, B  B.
400 401 402

(** ** Monadic operations *)
(** We define operational type classes for the monadic operations bind, join 
403 404 405
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). *)
406 407
Class MRet (M : Type  Type) := mret:  {A}, A  M A.
Arguments mret {_ _ _} _.
408
Instance: Params (@mret) 3.
409 410
Class MBind (M : Type  Type) := mbind :  {A B}, (A  M B)  M A  M B.
Arguments mbind {_ _ _ _} _ !_ /.
411
Instance: Params (@mbind) 4.
412
Class MJoin (M : Type  Type) := mjoin:  {A}, M (M A)  M A.
413
Arguments mjoin {_ _ _} !_ /.
414
Instance: Params (@mjoin) 3.
415 416
Class FMap (M : Type  Type) := fmap :  {A B}, (A  B)  M A  M B.
Arguments fmap {_ _ _ _} _ !_ /.
417
Instance: Params (@fmap) 4.
418 419
Class OMap (M : Type  Type) := omap:  {A B}, (A  option B)  M A  M B.
Arguments omap {_ _ _ _} _ !_ /.
420
Instance: Params (@omap) 4.
421

422 423 424 425 426 427
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))
Robbert Krebbers's avatar
Robbert Krebbers committed
428
  (at level 65, only parsing, right associativity) : C_scope.
429
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.
430
Notation "' ( x1 , x2 ) ← y ; z" :=
431
  (y = (λ x : _, let ' (x1, x2) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
432
  (at level 65, only parsing, right associativity) : C_scope.
433
Notation "' ( x1 , x2 , x3 ) ← y ; z" :=
434
  (y = (λ x : _, let ' (x1,x2,x3) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
435
  (at level 65, only parsing, right associativity) : C_scope.
436
Notation "' ( x1 , x2 , x3  , x4 ) ← y ; z" :=
437
  (y = (λ x : _, let ' (x1,x2,x3,x4) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
438
  (at level 65, only parsing, right associativity) : C_scope.
439 440
Notation "' ( x1 , x2 , x3  , x4 , x5 ) ← y ; z" :=
  (y = (λ x : _, let ' (x1,x2,x3,x4,x5) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
441
  (at level 65, only parsing, right associativity) : C_scope.
442 443
Notation "' ( x1 , x2 , x3  , x4 , x5 , x6 ) ← y ; z" :=
  (y = (λ x : _, let ' (x1,x2,x3,x4,x5,x6) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
444
  (at level 65, only parsing, right associativity) : C_scope.
445

446 447 448 449 450
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").

451
Class MGuard (M : Type  Type) :=
452 453 454
  mguard:  P {dec : Decision P} {A}, (P  M A)  M A.
Arguments mguard _ _ _ !_ _ _ /.
Notation "'guard' P ; o" := (mguard P (λ _, o))
Robbert Krebbers's avatar
Robbert Krebbers committed
455
  (at level 65, only parsing, right associativity) : C_scope.
456
Notation "'guard' P 'as' H ; o" := (mguard P (λ H, o))
Robbert Krebbers's avatar
Robbert Krebbers committed
457
  (at level 65, only parsing, right associativity) : C_scope.
458

459
(** ** Operations on maps *)
460 461
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
462
The function look up [m !! k] should yield the element at key [k] in [m]. *)
463
Class Lookup (K A M : Type) := lookup: K  M  option A.
464 465 466
Instance: Params (@lookup) 4.
Notation "m !! i" := (lookup i m) (at level 20) : C_scope.
Notation "(!!)" := lookup (only parsing) : C_scope.
467
Notation "( m !!)" := (λ i, m !! i) (only parsing) : C_scope.
468
Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
469
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.
470 471 472

(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
473
Class Insert (K A M : Type) := insert: K  A  M  M.
Robbert Krebbers's avatar
Robbert Krebbers committed
474
Instance: Params (@insert) 5.
475 476
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
477
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.
478

479 480 481
(** 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. *)
482
Class Delete (K M : Type) := delete: K  M  M.
Robbert Krebbers's avatar
Robbert Krebbers committed
483
Instance: Params (@delete) 4.
484
Arguments delete _ _ _ !_ !_ / : simpl nomatch.
485 486

(** The function [alter f k m] should update the value at key [k] using the
487
function [f], which is called with the original value. *)
488
Class Alter (K A M : Type) := alter: (A  A)  K  M  M.
489
Instance: Params (@alter) 5.
490
Arguments alter {_ _ _ _} _ !_ !_ / : simpl nomatch.
491 492

(** The function [alter f k m] should update the value at key [k] using the
493 494 495
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]. *)
496 497
Class PartialAlter (K A M : Type) :=
  partial_alter: (option A  option A)  K  M  M.
498
Instance: Params (@partial_alter) 4.
499
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.
500 501 502

(** 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]. *)
503 504 505
Class Dom (M C : Type) := dom: M  C.
Instance: Params (@dom) 3.
Arguments dom {_} _ {_} !_ / : simpl nomatch, clear implicits.
506 507

(** The function [merge f m1 m2] should merge the maps [m1] and [m2] by
508 509 510 511 512
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.
513

514 515 516 517 518
(** 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.
519
Instance: Params (@union_with) 3.
520
Arguments union_with {_ _ _} _ !_ !_ / : simpl nomatch.
521

522 523 524
(** Similarly for intersection and difference. *)
Class IntersectionWith (A M : Type) :=
  intersection_with: (A  A  option A)  M  M  M.
525
Instance: Params (@intersection_with) 3.
526 527
Arguments intersection_with {_ _ _} _ !_ !_ / : simpl nomatch.

528 529
Class DifferenceWith (A M : Type) :=
  difference_with: (A  A  option A)  M  M  M.
530
Instance: Params (@difference_with) 3.
531
Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch.
Robbert Krebbers's avatar
Robbert Krebbers committed
532

533 534 535 536
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 _ _ _ _ _ !_ /.

537 538 539 540 541 542 543 544
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
545
Instance: Params (@insertE) 6.
546 547 548 549
Notation "<[ k := a ]{ Γ }>" := (insertE Γ k a)
  (at level 5, right associativity, format "<[ k := a ]{ Γ }>") : C_scope.
Arguments insertE _ _ _ _ _ _ !_ _ !_ / : simpl nomatch.

550 551 552 553
(** ** 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]. *)
554 555 556 557 558
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.
559 560 561 562
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.
563
Class Idempotent {A} (R : relation A) (f : A  A  A) : Prop :=
564
  idempotent:  x, R (f x x) x.
565
Class Commutative {A B} (R : relation A) (f : B  B  A) : Prop :=
566
  commutative:  x y, R (f x y) (f y x).
567
Class LeftId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
568
  left_id:  x, R (f i x) x.
569
Class RightId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
570
  right_id:  x, R (f x i) x.
571
Class Associative {A} (R : relation A) (f : A  A  A) : Prop :=
572
  associative:  x y z, R (f x (f y z)) (f (f x y) z).
573
Class LeftAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
574
  left_absorb:  x, R (f i x) i.
575
Class RightAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
576
  right_absorb:  x, R (f x i) i.
577 578 579 580
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)).
581 582
Class AntiSymmetric {A} (R S : relation A) : Prop :=
  anti_symmetric:  x y, S x y  S y x  R x y.
583 584
Class Total {A} (R : relation A) := total x y : R x y  R y x.
Class Trichotomy {A} (R : relation A) :=
585
  trichotomy :  x y, R x y  x = y  R y x.
586
Class TrichotomyT {A} (R : relation A) :=
587
  trichotomyT :  x y, {R x y} + {x = y} + {R y x}.
Robbert Krebbers's avatar
Robbert Krebbers committed
588

589
Arguments irreflexivity {_} _ {_} _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
590
Arguments injective {_ _ _ _} _ {_} _ _ _.
591
Arguments injective2 {_ _ _ _ _ _} _ {_} _ _ _ _ _.
592 593
Arguments cancel {_ _ _} _ _ {_} _.
Arguments surjective {_ _ _} _ {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
594 595 596 597 598
Arguments idempotent {_ _} _ {_} _.
Arguments commutative {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments associative {_ _} _ {_} _ _ _.
599 600
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
601 602
Arguments left_distr {_ _} _ _ {_} _ _ _.
Arguments right_distr {_ _} _ _ {_} _ _ _.
603
Arguments anti_symmetric {_ _} _ {_} _ _ _ _.
604 605 606
Arguments total {_} _ {_} _ _.
Arguments trichotomy {_} _ {_} _ _.
Arguments trichotomyT {_} _ {_} _ _.
607

608 609 610
Instance id_injective {A} : Injective (=) (=) (@id A).
Proof. intros ??; auto. Qed.

611 612 613 614
(** 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.
615
Proof. auto. Qed.
616
Lemma commutative_L {A B} (f : B  B  A) `{!Commutative (=) f} x y :
617
  f x y = f y x.
618
Proof. auto. Qed.
619
Lemma left_id_L {A} (i : A) (f : A  A  A) `{!LeftId (=) i f} x : f i x = x.
620
Proof. auto. Qed.
621
Lemma right_id_L {A} (i : A) (f : A  A  A) `{!RightId (=) i f} x : f x i = x.
622
Proof. auto. Qed.
623
Lemma associative_L {A} (f : A  A  A) `{!Associative (=) f} x y z :
624
  f x (f y z) = f (f x y) z.
625
Proof. auto. Qed.
626
Lemma left_absorb_L {A} (i : A) (f : A  A  A) `{!LeftAbsorb (=) i f} x :
627 628
  f i x = i.
Proof. auto. Qed.
629
Lemma right_absorb_L {A} (i : A) (f : A  A  A) `{!RightAbsorb (=) i f} x :
630 631
  f x i = i.
Proof. auto. Qed.
632
Lemma left_distr_L {A} (f g : A  A  A) `{!LeftDistr (=) f g} x y z :
633 634
  f x (g y z) = g (f x y) (f x z).
Proof. auto. Qed.
635
Lemma right_distr_L {A} (f g : A  A  A) `{!RightDistr (=) f g} y z x :
636 637
  f (g y z) x = g (f y x) (f z x).
Proof. auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
638

639
(** ** Axiomatization of ordered structures *)
640 641
(** The classes [PreOrder], [PartialOrder], and [TotalOrder] use an arbitrary
relation [R] instead of [⊆] to support multiple orders on the same type. *)
642
Class PartialOrder {A} (R : relation A) : Prop := {
643 644
  partial_order_pre :> PreOrder R;
  partial_order_anti_symmetric :> AntiSymmetric (=) R
645 646
}.
Class TotalOrder {A} (R : relation A) : Prop := {
647 648
  total_order_partial :> PartialOrder R;
  total_order_trichotomy :> Trichotomy (strict R)
649 650
}.

651 652 653 654 655 656
(** 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 ();
657 658 659
  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
Robbert Krebbers's avatar
Robbert Krebbers committed
660
}.
661 662
Class MeetSemiLattice A `{SubsetEq A, Intersection A} : Prop := {
  meet_semi_lattice_pre :>> PreOrder ();
663 664 665
  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
Robbert Krebbers's avatar
Robbert Krebbers committed
666
}.
667 668 669 670
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)
671
}.
672

673
(** ** Axiomatization of collections *)
674 675
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
676
Instance: Params (@map) 3.
677 678
Class SimpleCollection A C `{ElemOf A C,
    Empty C, Singleton A C, Union C} : Prop := {
679
  not_elem_of_empty (x : A) : x  ;
680
  elem_of_singleton (x y : A) : x  {[ y ]}  x = y;
681 682
  elem_of_union X Y (x : A) : x  X  Y  x  X  x  Y
}.
683 684
Class Collection A C `{ElemOf A C, Empty C, Singleton A C,
    Union C, Intersection C, Difference C} : Prop := {
685
  collection_simple :>> SimpleCollection A C;
Robbert Krebbers's avatar
Robbert Krebbers committed
686
  elem_of_intersection X Y (x : A) : x  X  Y  x  X  x  Y;
687 688
  elem_of_difference X Y (x : A) : x  X  Y  x  X  x  Y
}.
689 690
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 := {
691
  collection_ops :>> Collection A C;
692
  elem_of_intersection_with (f : A  A  option A) X Y (x : A) :
693
    x  intersection_with f X Y   x1 x2, x1  X  x2  Y  f x1 x2 = Some x;
694
  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
695 696
}.

697 698 699
(** 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
700
Class Elements A C := elements: C  list A.
701
Instance: Params (@elements) 3.
702 703 704 705 706 707 708 709 710 711 712 713 714

(** 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. *)
715 716 717
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 := {
Robbert Krebbers's avatar
Robbert Krebbers committed
718
  fin_collection :>> Collection A C;
719 720
  elem_of_elements X x : x  elements X  x  X;
  NoDup_elements X : NoDup (elements X)
721 722
}.
Class Size C := size: C  nat.
723
Arguments size {_ _} !_ / : simpl nomatch.
724
Instance: Params (@size) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
725