base.v 42 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.
9
Require Export Morphisms RelationClasses List Bool Utf8 Program Setoid.
Robbert Krebbers's avatar
Robbert Krebbers committed
10

11
(** * General *)
12 13 14 15 16
(** 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).
17

18 19
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
20
Arguments id _ _ /.
21
Arguments compose _ _ _ _ _ _ /.
22
Arguments flip _ _ _ _ _ _ /.
23 24
Arguments const _ _ _ _ /.
Typeclasses Transparent id compose flip const.
25

26 27 28 29
(** 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. *)
30 31
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
32

33 34
Notation curry := prod_curry.
Notation uncurry := prod_uncurry.
35 36 37 38
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.
39

40 41
(** 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
42 43 44
Delimit Scope C_scope with C.
Global Open Scope C_scope.

45
(** Introduce some Haskell style like notations. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
46 47 48 49 50 51 52 53
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.
54
Hint Extern 100 (_  _) => discriminate.
Robbert Krebbers's avatar
Robbert Krebbers committed
55

56 57 58 59
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.

60
Notation "t $ r" := (t r)
61
  (at level 65, right associativity, only parsing) : C_scope.
62 63 64
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
65 66 67 68
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.
69

70 71 72 73 74 75 76 77 78 79 80 81
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.

82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97
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 {_ _ _ _ _ _} _ _ !_ !_ /.

98 99
(** Set convenient implicit arguments for [existT] and introduce notations. *)
Arguments existT {_ _} _ _.
100
Arguments proj1_sig {_ _} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
101
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
102
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
103

104 105 106 107
(** * 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
108 109
Class PropHolds (P : Prop) := prop_holds: P.

110 111
Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
112
Proof. repeat intro; trivial. Qed.
113 114 115

Ltac solve_propholds :=
  match goal with
116 117
  | |- PropHolds (?P) => apply _
  | |- ?P => change (PropHolds P); apply _
118 119 120 121 122 123 124
  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
125 126 127
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.

128 129 130 131 132 133 134 135 136
(** ** 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) :=
137
  match iA, iB with populate x, populate y => populate (x,y) end.
138
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
139
  match iA with populate x => populate (inl x) end.
140
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
141
  match iB with populate y => populate (inl y) end.
142 143
Instance option_inhabited {A} : Inhabited (option A) := populate None.

144 145 146 147 148 149
(** ** 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.

150 151 152
(** ** 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
153 154 155
Class Equiv A := equiv: relation A.
Infix "≡" := equiv (at level 70, no associativity) : C_scope.
Notation "(≡)" := equiv (only parsing) : C_scope.
156 157 158 159 160 161
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
162

163 164 165 166 167 168
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.

169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
(** 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.

190 191 192 193 194 195 196 197
(** 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
198
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
199 200
Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_  _) => symmetry; assumption.
201 202
Hint Extern 0 (_ {_} _) => reflexivity.
Hint Extern 0 (_ {_} _) => symmetry; assumption.
Robbert Krebbers's avatar
Robbert Krebbers committed
203

204
(** ** Operations on collections *)
205
(** We define operational type classes for the traditional operations and
206
relations on collections: the empty collection [∅], the union [(∪)],
207 208
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
209 210 211 212
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

Class Union A := union: A  A  A.
213
Instance: Params (@union) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
214 215 216 217
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.
218 219 220 221 222 223
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
224

225
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
226 227 228
Arguments union_list _ _ _ !_ /.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
229
Class Intersection A := intersection: A  A  A.
230
Instance: Params (@intersection) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
231 232 233 234 235 236
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.
237
Instance: Params (@difference) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
238 239 240 241
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.
242 243 244 245 246 247
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
248

249 250
Class Singleton A B := singleton: A  B.
Instance: Params (@singleton) 3.
251
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
252
Notation "{[ x ; y ; .. ; z ]}" :=
253 254 255 256 257 258
  (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.
259

260
Class SubsetEq A := subseteq: relation A.
261
Instance: Params (@subseteq) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
262 263 264
Infix "⊆" := subseteq (at level 70) : C_scope.
Notation "(⊆)" := subseteq (only parsing) : C_scope.
Notation "( X ⊆ )" := (subseteq X) (only parsing) : C_scope.
265
Notation "( ⊆ X )" := (λ Y, Y  X) (only parsing) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
266 267 268 269
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.
270 271 272 273 274 275 276
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
277

278
Hint Extern 0 (_  _) => reflexivity.
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
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.
310

311 312
Definition strict {A} (R : relation A) : relation A := λ X Y, R X Y  ¬R Y X.
Instance: Params (@strict) 2.
313 314 315 316
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.
317 318 319 320
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
321

322 323 324 325 326
(** 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
327
Class ElemOf A B := elem_of: A  B  Prop.
328
Instance: Params (@elem_of) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
329 330 331 332 333 334 335 336 337
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
338 339 340 341
Class Disjoint A := disjoint : A  A  Prop.
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
342
Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
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
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.
369 370 371

Class DisjointList A := disjoint_list : list A  Prop.
Instance: Params (@disjoint_list) 2.
372
Notation "⊥ Xs" := (disjoint_list Xs) (at level 20, format "⊥  Xs") : C_scope.
373

374 375 376 377 378 379
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.
380

381
  Lemma disjoint_list_nil  :  @nil A  True.
382 383 384
  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.
385
End disjoint_list.
386 387

Class Filter A B := filter:  (P : A  Prop) `{ x, Decision (P x)}, B  B.
388 389 390

(** ** Monadic operations *)
(** We define operational type classes for the monadic operations bind, join 
391 392 393
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). *)
394 395 396
Class MRet (M : Type  Type) := mret:  {A}, A  M A.
Instance: Params (@mret) 3.
Arguments mret {_ _ _} _.
397 398
Class MBind (M : Type  Type) := mbind :  {A B}, (A  M B)  M A  M B.
Arguments mbind {_ _ _ _} _ !_ /.
399 400 401
Instance: Params (@mbind) 5.
Class MJoin (M : Type  Type) := mjoin:  {A}, M (M A)  M A.
Instance: Params (@mjoin) 3.
402
Arguments mjoin {_ _ _} !_ /.
403
Class FMap (M : Type  Type) := fmap :  {A B}, (A  B)  M A  M B.
404
Instance: Params (@fmap) 6.
405 406
Arguments fmap {_ _ _ _} _ !_ /.
Class OMap (M : Type  Type) := omap:  {A B}, (A  option B)  M A  M B.
407
Instance: Params (@omap) 6.
408
Arguments omap {_ _ _ _} _ !_ /.
409

410 411 412 413 414 415
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))
416
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
417
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.
418
Notation "' ( x1 , x2 ) ← y ; z" :=
419 420
  (y = (λ x : _, let ' (x1, x2) := x in z))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
421
Notation "' ( x1 , x2 , x3 ) ← y ; z" :=
422 423
  (y = (λ x : _, let ' (x1,x2,x3) := x in z))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
424
Notation "' ( x1 , x2 , x3  , x4 ) ← y ; z" :=
425 426
  (y = (λ x : _, let ' (x1,x2,x3,x4) := x in z))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
427 428 429
Notation "' ( x1 , x2 , x3  , x4 , x5 ) ← y ; z" :=
  (y = (λ x : _, let ' (x1,x2,x3,x4,x5) := x in z))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
430 431 432
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, next at level 35, only parsing, right associativity) : C_scope.
433 434

Class MGuard (M : Type  Type) :=
435 436 437 438 439 440
  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.
441

442
(** ** Operations on maps *)
443 444
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
445
The function look up [m !! k] should yield the element at key [k] in [m]. *)
446
Class Lookup (K A M : Type) := lookup: K  M  option A.
447 448 449
Instance: Params (@lookup) 4.
Notation "m !! i" := (lookup i m) (at level 20) : C_scope.
Notation "(!!)" := lookup (only parsing) : C_scope.
450
Notation "( m !!)" := (λ i, m !! i) (only parsing) : C_scope.
451
Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
452
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.
453 454 455

(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
456
Class Insert (K A M : Type) := insert: K  A  M  M.
457 458 459
Instance: Params (@insert) 4.
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
460
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.
461

462 463 464
(** 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. *)
465
Class Delete (K M : Type) := delete: K  M  M.
466 467
Instance: Params (@delete) 3.
Arguments delete _ _ _ !_ !_ / : simpl nomatch.
468 469

(** The function [alter f k m] should update the value at key [k] using the
470
function [f], which is called with the original value. *)
471
Class Alter (K A M : Type) := alter: (A  A)  K  M  M.
472
Instance: Params (@alter) 5.
473
Arguments alter {_ _ _ _} _ !_ !_ / : simpl nomatch.
474 475

(** The function [alter f k m] should update the value at key [k] using the
476 477 478
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]. *)
479 480
Class PartialAlter (K A M : Type) :=
  partial_alter: (option A  option A)  K  M  M.
481
Instance: Params (@partial_alter) 4.
482
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.
483 484 485

(** 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]. *)
486 487 488
Class Dom (M C : Type) := dom: M  C.
Instance: Params (@dom) 3.
Arguments dom {_} _ {_} !_ / : simpl nomatch, clear implicits.
489 490

(** The function [merge f m1 m2] should merge the maps [m1] and [m2] by
491 492 493 494 495
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.
496

497 498 499 500 501
(** 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.
502
Instance: Params (@union_with) 3.
503
Arguments union_with {_ _ _} _ !_ !_ / : simpl nomatch.
504

505 506 507
(** Similarly for intersection and difference. *)
Class IntersectionWith (A M : Type) :=
  intersection_with: (A  A  option A)  M  M  M.
508
Instance: Params (@intersection_with) 3.
509 510
Arguments intersection_with {_ _ _} _ !_ !_ / : simpl nomatch.

511 512
Class DifferenceWith (A M : Type) :=
  difference_with: (A  A  option A)  M  M  M.
513
Instance: Params (@difference_with) 3.
514
Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch.
Robbert Krebbers's avatar
Robbert Krebbers committed
515

516 517 518 519
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 _ _ _ _ _ !_ /.

520 521 522 523 524 525 526 527 528 529 530 531 532
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.
Instance: Params (@insert) 6.
Notation "<[ k := a ]{ Γ }>" := (insertE Γ k a)
  (at level 5, right associativity, format "<[ k := a ]{ Γ }>") : C_scope.
Arguments insertE _ _ _ _ _ _ !_ _ !_ / : simpl nomatch.

533 534 535 536
(** ** 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]. *)
537 538 539 540 541
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.
542 543 544 545
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.
546
Class Idempotent {A} (R : relation A) (f : A  A  A) : Prop :=
547
  idempotent:  x, R (f x x) x.
548
Class Commutative {A B} (R : relation A) (f : B  B  A) : Prop :=
549
  commutative:  x y, R (f x y) (f y x).
550
Class LeftId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
551
  left_id:  x, R (f i x) x.
552
Class RightId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
553
  right_id:  x, R (f x i) x.
554
Class Associative {A} (R : relation A) (f : A  A  A) : Prop :=
555
  associative:  x y z, R (f x (f y z)) (f (f x y) z).
556
Class LeftAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
557
  left_absorb:  x, R (f i x) i.
558
Class RightAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
559
  right_absorb:  x, R (f x i) i.
560 561 562 563
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)).
564 565
Class AntiSymmetric {A} (R S : relation A) : Prop :=
  anti_symmetric:  x y, S x y  S y x  R x y.
566 567
Class Total {A} (R : relation A) := total x y : R x y  R y x.
Class Trichotomy {A} (R : relation A) :=
568
  trichotomy :  x y, R x y  x = y  R y x.
569
Class TrichotomyT {A} (R : relation A) :=
570
  trichotomyT :  x y, {R x y} + {x = y} + {R y x}.
Robbert Krebbers's avatar
Robbert Krebbers committed
571

572
Arguments irreflexivity {_} _ {_} _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
573
Arguments injective {_ _ _ _} _ {_} _ _ _.
574
Arguments injective2 {_ _ _ _ _ _} _ {_} _ _ _ _ _.
575 576
Arguments cancel {_ _ _} _ _ {_} _.
Arguments surjective {_ _ _} _ {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
577 578 579 580 581
Arguments idempotent {_ _} _ {_} _.
Arguments commutative {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments associative {_ _} _ {_} _ _ _.
582 583
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
584 585
Arguments left_distr {_ _} _ _ {_} _ _ _.
Arguments right_distr {_ _} _ _ {_} _ _ _.
586
Arguments anti_symmetric {_ _} _ {_} _ _ _ _.
587 588 589
Arguments total {_} _ {_} _ _.
Arguments trichotomy {_} _ {_} _ _.
Arguments trichotomyT {_} _ {_} _ _.
590

591 592 593
Instance id_injective {A} : Injective (=) (=) (@id A).
Proof. intros ??; auto. Qed.

594 595 596 597
(** 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.
598
Proof. auto. Qed.
599
Lemma commutative_L {A B} (f : B  B  A) `{!Commutative (=) f} x y :
600
  f x y = f y x.
601
Proof. auto. Qed.
602
Lemma left_id_L {A} (i : A) (f : A  A  A) `{!LeftId (=) i f} x : f i x = x.
603
Proof. auto. Qed.
604
Lemma right_id_L {A} (i : A) (f : A  A  A) `{!RightId (=) i f} x : f x i = x.
605
Proof. auto. Qed.
606
Lemma associative_L {A} (f : A  A  A) `{!Associative (=) f} x y z :
607
  f x (f y z) = f (f x y) z.
608
Proof. auto. Qed.
609
Lemma left_absorb_L {A} (i : A) (f : A  A  A) `{!LeftAbsorb (=) i f} x :
610 611
  f i x = i.
Proof. auto. Qed.
612
Lemma right_absorb_L {A} (i : A) (f : A  A  A) `{!RightAbsorb (=) i f} x :
613 614
  f x i = i.
Proof. auto. Qed.
615
Lemma left_distr_L {A} (f g : A  A  A) `{!LeftDistr (=) f g} x y z :
616 617
  f x (g y z) = g (f x y) (f x z).
Proof. auto. Qed.
618
Lemma right_distr_L {A} (f g : A  A  A) `{!RightDistr (=) f g} y z x :
619 620
  f (g y z) x = g (f y x) (f z x).
Proof. auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
621

622
(** ** Axiomatization of ordered structures *)
623 624
(** The classes [PreOrder], [PartialOrder], and [TotalOrder] use an arbitrary
relation [R] instead of [⊆] to support multiple orders on the same type. *)
625
Class PartialOrder {A} (R : relation A) : Prop := {
626 627
  partial_order_pre :> PreOrder R;
  partial_order_anti_symmetric :> AntiSymmetric (=) R
628 629
}.
Class TotalOrder {A} (R : relation A) : Prop := {
630 631
  total_order_partial :> PartialOrder R;
  total_order_trichotomy :> Trichotomy (strict R)
632 633
}.

634 635 636 637 638 639
(** 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 ();
640 641 642
  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
643
}.
644 645
Class MeetSemiLattice A `{SubsetEq A, Intersection A} : Prop := {
  meet_semi_lattice_pre :>> PreOrder ();
646 647 648
  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
649
}.
650 651 652 653
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)
654
}.
655

656
(** ** Axiomatization of collections *)
657 658
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
659
Instance: Params (@map) 3.
660 661
Class SimpleCollection A C `{ElemOf A C,
    Empty C, Singleton A C, Union C} : Prop := {
662
  not_elem_of_empty (x : A) : x  ;
663
  elem_of_singleton (x y : A) : x  {[ y ]}  x = y;
664 665
  elem_of_union X Y (x : A) : x  X  Y  x  X  x  Y
}.
666 667
Class Collection A C `{ElemOf A C, Empty C, Singleton A C,
    Union C, Intersection C, Difference C} : Prop := {
668
  collection_simple :>> SimpleCollection A C;
Robbert Krebbers's avatar
Robbert Krebbers committed
669
  elem_of_intersection X Y (x : A) : x  X  Y  x  X  x  Y;
670 671
  elem_of_difference X Y (x : A) : x  X  Y  x  X  x  Y
}.
672 673
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 := {
674
  collection_ops :>> Collection A C;
675
  elem_of_intersection_with (f : A  A  option A) X Y (x : A) :
676
    x  intersection_with f X Y   x1 x2, x1  X  x2  Y  f x1 x2 = Some x;
677
  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
678 679
}.

680 681 682
(** 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
683
Class Elements A C := elements: C  list A.
684
Instance: Params (@elements) 3.
685 686 687 688 689 690 691 692 693 694 695 696 697

(** 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. *)
698 699 700
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
701
  fin_collection :>> Collection A C;
702 703
  elem_of_elements X x : x  elements X  x  X;
  NoDup_elements X : NoDup (elements X)
704 705
}.
Class Size C := size: C  nat.
706
Arguments size {_ _} !_ / : simpl nomatch.
707
Instance: Params (@size) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
708

709 710 711 712 713 714 715 716
(** 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]. *)
717 718 719
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 := {
720 721 722
  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;
723
  elem_of_ret {A} (x y : A) : x  mret y  x = y;
724 725
  elem_of_fmap {A B} (f : A  B) (X : M A) (x : B) :
    x  f <$> X   y, x = f y  y  X;
726
  elem_of_join {A} (X : M (M A)) (x : A) : x  mjoin X   Y, x  Y  Y  X
727 728
}.

729 730 731
(** 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
732
Class Fresh A C := fresh: C  A.
733
Instance: Params (@fresh) 3.
734 735
Class FreshSpec A C `{ElemOf A C,
    Empty C, Singleton A C, Union C, Fresh A C} : Prop := {
736
  fresh_collection_simple :>> SimpleCollection A C;
737
  fresh_proper_alt X Y : ( x, x  X  x  Y)  fresh X = fresh Y;
Robbert Krebbers's avatar
Robbert Krebbers committed
738 739 740
  is_fresh (X : C) : fresh X  X
}.

741 742 743
(** * Booleans *)
(** The following coercion allows us to use Booleans as propositions. *)
Coercion Is_true : bool >-> Sortclass.
744
Hint Unfold Is_true.
745
Hint Immediate Is_true_eq_left.
746
Hint Resolve orb_prop_intro andb_prop_intro.
747 748 749 750 751 752 753 754 755 756 757
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 [|];