base.v 41.6 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2014, 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

163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183
(** 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.

184 185 186 187 188 189 190 191
(** 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
192
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
193 194
Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_  _) => symmetry; assumption.
Robbert Krebbers's avatar
Robbert Krebbers committed
195

196
(** ** Operations on collections *)
197
(** We define operational type classes for the traditional operations and
198
relations on collections: the empty collection [∅], the union [(∪)],
199 200
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
201 202 203 204
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

Class Union A := union: A  A  A.
205
Instance: Params (@union) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
206 207 208 209
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.
210 211 212 213 214 215
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
216

217
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
218 219 220
Arguments union_list _ _ _ !_ /.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
221
Class Intersection A := intersection: A  A  A.
222
Instance: Params (@intersection) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
223 224 225 226 227 228
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.
229
Instance: Params (@difference) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
230 231 232 233
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.
234 235 236 237 238 239
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
240

241 242
Class Singleton A B := singleton: A  B.
Instance: Params (@singleton) 3.
243
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
244
Notation "{[ x ; y ; .. ; z ]}" :=
245 246 247 248 249 250
  (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.
251

252
Class SubsetEq A := subseteq: relation A.
253
Instance: Params (@subseteq) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
254 255 256
Infix "⊆" := subseteq (at level 70) : C_scope.
Notation "(⊆)" := subseteq (only parsing) : C_scope.
Notation "( X ⊆ )" := (subseteq X) (only parsing) : C_scope.
257
Notation "( ⊆ X )" := (λ Y, Y  X) (only parsing) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
258 259 260 261
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.
262 263 264 265 266 267 268
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
269

270
Hint Extern 0 (_  _) => reflexivity.
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
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.
302

303 304
Definition strict {A} (R : relation A) : relation A := λ X Y, R X Y  ¬R Y X.
Instance: Params (@strict) 2.
305 306 307 308
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.
309 310 311 312
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
313

314 315 316 317 318
(** 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
319
Class ElemOf A B := elem_of: A  B  Prop.
320
Instance: Params (@elem_of) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
321 322 323 324 325 326 327 328 329
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
330 331 332 333
Class Disjoint A := disjoint : A  A  Prop.
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
334
Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
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
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.
361 362 363

Class DisjointList A := disjoint_list : list A  Prop.
Instance: Params (@disjoint_list) 2.
364
Notation "⊥ Xs" := (disjoint_list Xs) (at level 20, format "⊥  Xs") : C_scope.
365

366 367 368 369 370 371
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.
372

373
  Lemma disjoint_list_nil  :  @nil A  True.
374 375 376
  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.
377
End disjoint_list.
378 379

Class Filter A B := filter:  (P : A  Prop) `{ x, Decision (P x)}, B  B.
380 381 382

(** ** Monadic operations *)
(** We define operational type classes for the monadic operations bind, join 
383 384 385
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). *)
386 387 388
Class MRet (M : Type  Type) := mret:  {A}, A  M A.
Instance: Params (@mret) 3.
Arguments mret {_ _ _} _.
389 390
Class MBind (M : Type  Type) := mbind :  {A B}, (A  M B)  M A  M B.
Arguments mbind {_ _ _ _} _ !_ /.
391 392 393
Instance: Params (@mbind) 5.
Class MJoin (M : Type  Type) := mjoin:  {A}, M (M A)  M A.
Instance: Params (@mjoin) 3.
394
Arguments mjoin {_ _ _} !_ /.
395
Class FMap (M : Type  Type) := fmap :  {A B}, (A  B)  M A  M B.
396
Instance: Params (@fmap) 6.
397 398
Arguments fmap {_ _ _ _} _ !_ /.
Class OMap (M : Type  Type) := omap:  {A B}, (A  option B)  M A  M B.
399
Instance: Params (@omap) 6.
400
Arguments omap {_ _ _ _} _ !_ /.
401

402 403 404 405 406 407
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))
408
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
409
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.
410
Notation "' ( x1 , x2 ) ← y ; z" :=
411 412
  (y = (λ x : _, let ' (x1, x2) := x in z))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
413
Notation "' ( x1 , x2 , x3 ) ← y ; z" :=
414 415
  (y = (λ x : _, let ' (x1,x2,x3) := x in z))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
416
Notation "' ( x1 , x2 , x3  , x4 ) ← y ; z" :=
417 418
  (y = (λ x : _, let ' (x1,x2,x3,x4) := x in z))
  (at level 65, next at level 35, only parsing, right associativity) : C_scope.
419 420 421
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.
422 423 424
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.
425 426

Class MGuard (M : Type  Type) :=
427 428 429 430 431 432
  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.
433

434
(** ** Operations on maps *)
435 436
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
437
The function look up [m !! k] should yield the element at key [k] in [m]. *)
438
Class Lookup (K A M : Type) := lookup: K  M  option A.
439 440 441
Instance: Params (@lookup) 4.
Notation "m !! i" := (lookup i m) (at level 20) : C_scope.
Notation "(!!)" := lookup (only parsing) : C_scope.
442
Notation "( m !!)" := (λ i, m !! i) (only parsing) : C_scope.
443
Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
444
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.
445 446 447

(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
448
Class Insert (K A M : Type) := insert: K  A  M  M.
449 450 451
Instance: Params (@insert) 4.
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
452
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.
453

454 455 456
(** 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. *)
457
Class Delete (K M : Type) := delete: K  M  M.
458 459
Instance: Params (@delete) 3.
Arguments delete _ _ _ !_ !_ / : simpl nomatch.
460 461

(** The function [alter f k m] should update the value at key [k] using the
462
function [f], which is called with the original value. *)
463
Class Alter (K A M : Type) := alter: (A  A)  K  M  M.
464
Instance: Params (@alter) 5.
465
Arguments alter {_ _ _ _} _ !_ !_ / : simpl nomatch.
466 467

(** The function [alter f k m] should update the value at key [k] using the
468 469 470
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]. *)
471 472
Class PartialAlter (K A M : Type) :=
  partial_alter: (option A  option A)  K  M  M.
473
Instance: Params (@partial_alter) 4.
474
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.
475 476 477

(** 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]. *)
478 479 480
Class Dom (M C : Type) := dom: M  C.
Instance: Params (@dom) 3.
Arguments dom {_} _ {_} !_ / : simpl nomatch, clear implicits.
481 482

(** The function [merge f m1 m2] should merge the maps [m1] and [m2] by
483 484 485 486 487
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.
488

489 490 491 492 493
(** 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.
494
Instance: Params (@union_with) 3.
495
Arguments union_with {_ _ _} _ !_ !_ / : simpl nomatch.
496

497 498 499
(** Similarly for intersection and difference. *)
Class IntersectionWith (A M : Type) :=
  intersection_with: (A  A  option A)  M  M  M.
500
Instance: Params (@intersection_with) 3.
501 502
Arguments intersection_with {_ _ _} _ !_ !_ / : simpl nomatch.

503 504
Class DifferenceWith (A M : Type) :=
  difference_with: (A  A  option A)  M  M  M.
505
Instance: Params (@difference_with) 3.
506
Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch.
Robbert Krebbers's avatar
Robbert Krebbers committed
507

508 509 510 511
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 _ _ _ _ _ !_ /.

512 513 514 515 516 517 518 519 520 521 522 523 524
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.

525 526 527 528
(** ** 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]. *)
529 530 531 532 533
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.
534 535 536 537
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.
538
Class Idempotent {A} (R : relation A) (f : A  A  A) : Prop :=
539
  idempotent:  x, R (f x x) x.
540
Class Commutative {A B} (R : relation A) (f : B  B  A) : Prop :=
541
  commutative:  x y, R (f x y) (f y x).
542
Class LeftId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
543
  left_id:  x, R (f i x) x.
544
Class RightId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
545
  right_id:  x, R (f x i) x.
546
Class Associative {A} (R : relation A) (f : A  A  A) : Prop :=
547
  associative:  x y z, R (f x (f y z)) (f (f x y) z).
548
Class LeftAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
549
  left_absorb:  x, R (f i x) i.
550
Class RightAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
551
  right_absorb:  x, R (f x i) i.
552 553 554 555
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)).
556 557
Class AntiSymmetric {A} (R S : relation A) : Prop :=
  anti_symmetric:  x y, S x y  S y x  R x y.
558 559
Class Total {A} (R : relation A) := total x y : R x y  R y x.
Class Trichotomy {A} (R : relation A) :=
560
  trichotomy :  x y, R x y  x = y  R y x.
561
Class TrichotomyT {A} (R : relation A) :=
562
  trichotomyT :  x y, {R x y} + {x = y} + {R y x}.
Robbert Krebbers's avatar
Robbert Krebbers committed
563

564
Arguments irreflexivity {_} _ {_} _ _.
Robbert Krebbers's avatar
Robbert Krebbers committed
565
Arguments injective {_ _ _ _} _ {_} _ _ _.
566
Arguments injective2 {_ _ _ _ _ _} _ {_} _ _ _ _ _.
567 568
Arguments cancel {_ _ _} _ _ {_} _.
Arguments surjective {_ _ _} _ {_} _.
Robbert Krebbers's avatar
Robbert Krebbers committed
569 570 571 572 573
Arguments idempotent {_ _} _ {_} _.
Arguments commutative {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments associative {_ _} _ {_} _ _ _.
574 575
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
576 577
Arguments left_distr {_ _} _ _ {_} _ _ _.
Arguments right_distr {_ _} _ _ {_} _ _ _.
578
Arguments anti_symmetric {_ _} _ {_} _ _ _ _.
579 580 581
Arguments total {_} _ {_} _ _.
Arguments trichotomy {_} _ {_} _ _.
Arguments trichotomyT {_} _ {_} _ _.
582

583 584 585
Instance id_injective {A} : Injective (=) (=) (@id A).
Proof. intros ??; auto. Qed.

586 587 588 589
(** 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.
590
Proof. auto. Qed.
591
Lemma commutative_L {A B} (f : B  B  A) `{!Commutative (=) f} x y :
592
  f x y = f y x.
593
Proof. auto. Qed.
594
Lemma left_id_L {A} (i : A) (f : A  A  A) `{!LeftId (=) i f} x : f i x = x.
595
Proof. auto. Qed.
596
Lemma right_id_L {A} (i : A) (f : A  A  A) `{!RightId (=) i f} x : f x i = x.
597
Proof. auto. Qed.
598
Lemma associative_L {A} (f : A  A  A) `{!Associative (=) f} x y z :
599
  f x (f y z) = f (f x y) z.
600
Proof. auto. Qed.
601
Lemma left_absorb_L {A} (i : A) (f : A  A  A) `{!LeftAbsorb (=) i f} x :
602 603
  f i x = i.
Proof. auto. Qed.
604
Lemma right_absorb_L {A} (i : A) (f : A  A  A) `{!RightAbsorb (=) i f} x :
605 606
  f x i = i.
Proof. auto. Qed.
607
Lemma left_distr_L {A} (f g : A  A  A) `{!LeftDistr (=) f g} x y z :
608 609
  f x (g y z) = g (f x y) (f x z).
Proof. auto. Qed.
610
Lemma right_distr_L {A} (f g : A  A  A) `{!RightDistr (=) f g} y z x :
611 612
  f (g y z) x = g (f y x) (f z x).
Proof. auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
613

614
(** ** Axiomatization of ordered structures *)
615 616 617 618 619 620 621 622 623 624 625
(** The classes [PreOrder], [PartialOrder], and [TotalOrder] do not use the
relation [⊆] because we often have multiple orders on the same structure. *)
Class PartialOrder {A} (R : relation A) : Prop := {
  po_preorder :> PreOrder R;
  po_anti_symmetric :> AntiSymmetric (=) R
}.
Class TotalOrder {A} (R : relation A) : Prop := {
  to_po :> PartialOrder R;
  to_trichotomy :> Trichotomy R
}.

626
(** We do not include equality in the following interfaces so as to avoid the
627
need for proofs that the relations and operations respect setoid equality.
628 629
Instead, we will define setoid equality in a generic way as
[λ X Y, X ⊆ Y ∧ Y ⊆ X]. *)
630 631 632 633 634
Class BoundedPreOrder A `{Empty A, SubsetEq A} : Prop := {
  bounded_preorder :>> PreOrder ();
  subseteq_empty X :   X
}.
Class BoundedJoinSemiLattice A `{Empty A, SubsetEq A, Union A} : Prop := {
635
  bjsl_preorder :>> BoundedPreOrder A;
636 637 638
  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
639
}.
640
Class MeetSemiLattice A `{Empty A, SubsetEq A, Intersection A} : Prop := {
Robbert Krebbers's avatar
Robbert Krebbers committed
641
  msl_preorder :>> BoundedPreOrder A;
642 643 644
  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
645
}.
646 647 648 649

(** A join distributive lattice with distributivity stated in the order
theoretic way. We will prove that distributivity of join, and distributivity
as an equality can be derived. *)
650 651
Class LowerBoundedLattice A
    `{Empty A, SubsetEq A, Union A, Intersection A} : Prop := {
652
  lbl_bjsl :>> BoundedJoinSemiLattice A;
653
  lbl_msl :>> MeetSemiLattice A;
654
  lbl_distr X Y Z : (X  Y)  (X  Z)  X  (Y  Z)
655
}.
656

657
(** ** Axiomatization of collections *)
658 659
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
660
Instance: Params (@map) 3.
661 662
Class SimpleCollection A C `{ElemOf A C,
    Empty C, Singleton A C, Union C} : Prop := {
663
  not_elem_of_empty (x : A) : x  ;
664
  elem_of_singleton (x y : A) : x  {[ y ]}  x = y;
665 666
  elem_of_union X Y (x : A) : x  X  Y  x  X  x  Y
}.
667 668
Class Collection A C `{ElemOf A C, Empty C, Singleton A C,
    Union C, Intersection C, Difference C} : Prop := {
669
  collection_simple :>> SimpleCollection A C;
Robbert Krebbers's avatar
Robbert Krebbers committed
670
  elem_of_intersection X Y (x : A) : x  X  Y  x  X  x  Y;
671 672
  elem_of_difference X Y (x : A) : x  X  Y  x  X  x  Y
}.
673 674 675
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 := {
676
  collection_ops :>> Collection A C;
677
  elem_of_intersection_with (f : A  A  option A) X Y (x : A) :
678 679 680
    x  intersection_with f X Y   x1 x2, x1  X  x2  Y  f x1 x2 = Some x;
  elem_of_filter X P `{ x, Decision (P x)} x :
    x  filter P X  P x  x  X
Robbert Krebbers's avatar
Robbert Krebbers committed
681 682
}.

683 684 685
(** 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
686
Class Elements A C := elements: C  list A.
687
Instance: Params (@elements) 3.
688 689 690 691 692 693 694 695 696 697 698 699 700

(** 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. *)
701 702 703
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
704
  fin_collection :>> Collection A C;
705 706
  elem_of_elements X x : x  elements X  x  X;
  NoDup_elements X : NoDup (elements X)
707 708
}.
Class Size C := size: C  nat.
709
Arguments size {_ _} !_ / : simpl nomatch.
710
Instance: Params (@size) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
711

712 713 714 715 716 717 718 719
(** 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]. *)
720 721 722
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 := {
723 724 725
  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;
726
  elem_of_ret {A} (x y : A) : x  mret y  x = y;
727 728
  elem_of_fmap {A B} (f : A  B) (X : M A) (x : B) :
    x  f <$> X   y, x = f y  y  X;
729
  elem_of_join {A} (X : M (M A)) (x : A) : x  mjoin X   Y, x  Y  Y  X
730 731
}.

732 733 734
(** 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
735
Class Fresh A C := fresh: C  A.
736
Instance: Params (@fresh) 3.
737 738
Class FreshSpec A C `{ElemOf A C,
    Empty C, Singleton A C, Union C, Fresh A C} : Prop := {
739
  fresh_collection_simple :>> SimpleCollection A C;
740
  fresh_proper_alt X Y : ( x, x  X  x  Y)  fresh X = fresh Y;
Robbert Krebbers's avatar
Robbert Krebbers committed
741 742 743
  is_fresh (X : C) : fresh X  X
}.

744 745 746 747 748 749 750 751 752 753 754 755 756 757
(** * Booleans *)
(** The following coercion allows us to use Booleans as propositions. *)
Coercion Is_true : bool >-> Sortclass.
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.

758
(** * Miscellaneous *)
759
Class Half A := half: A  A.
760 761
Notation "½" := half : C_scope.
Notation "½*" := (fmap (M:=list) half) : C_scope.
762

763 764
Lemma proj1_sig_inj {A} (P : A  Prop) x (Px : P x) y (Py : P y) :
  xPx = yPy  x = y.
765
Proof. injection 1; trivial. Qed.
766
Lemma not_symmetry `{R : relation A, !Symmetric R} x y : ¬R x y  ¬R y x.
767
Proof. intuition. Qed.
768
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} x y : R x y  R y x.
769 770
Proof. intuition. Qed.

771 772 773
(** ** Pointwise relations *)
(** These instances are in Coq trunk since revision 15455, but are not in Coq
8.4 yet. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
774 775 776 777 778 779 780 781 782 783
Instance pointwise_reflexive {A} `{R : relation B} :
  Reflexive R  Reflexive (pointwise_relation A R) | 9.
Proof. firstorder. Qed.
Instance pointwise_symmetric {A} `{R : relation B} :
  Symmetric R  Symmetric (pointwise_relation A R) | 9.
Proof. firstorder. Qed.
Instance pointwise_transitive {A} `{R : relation B} :
  Transitive R  Transitive (pointwise_relation A R) | 9.
Proof. firstorder. Qed.

784
(** ** Products *)
785 786 787 788 789 790 791
Instance prod_map_injective {A A' B B'} (f : A  A') (g : B  B') :
  Injective (=) (=) f  Injective (=) (=) g 
  Injective (=) (=) (prod_map f g).
Proof.
  intros ?? [??] [??] ?; simpl in *; f_equal;
    [apply (injective f)|apply (injective g)]; congruence.
Qed.
792

793
Definition prod_relation {A B} (R1 : relation A) (R2 : relation B) :
794
  relation (A * B) := λ x y, R1 (x.1) (y.1)  R2 (x.2) (y.2).
Robbert Krebbers's avatar
Robbert Krebbers committed
795
Section prod_relation.
796
  Context `{R1 : relation A, R2 : relation B}.
797 798
  Global Instance:
    Reflexive R1  Reflexive R2  Reflexive (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
799
  Proof. firstorder eauto. Qed.
800 801
  Global Instance:
    Symmetric R1  Symmetric R2  Symmetric (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
802
  Proof. firstorder eauto. Qed.
803 804
  Global Instance:
    Transitive R1  Transitive R2  Transitive (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
805
  Proof. firstorder eauto. Qed.
806 807
  Global Instance:
    Equivalence R1  Equivalence R2  Equivalence (prod_relation R1 R2).
Robbert Krebbers's avatar
Robbert Krebbers committed
808 809 810 811 812 813 814 815 816
  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.

817
(** ** Other *)
818 819 820 821
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
822
Instance:  A B (x : B), Commutative (=) (λ _ _ : A, x).
823
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
824
Instance:  A (x : A), Associative (=) (λ _ _ : A, x).
825
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
826
Instance:  A, Associative (=) (λ x _ : A, x).
827
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
828
Instance:  A, Associative (=) (λ _ x : A, x).
829
Proof. red. trivial. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
830
Instance:  A, Idempotent (=) (λ x _ : A, x).
831
Proof. red. trivial. Qed.