base.v 42.4 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
From Coq Require Export Morphisms RelationClasses List Bool Utf8 Program Setoid.
11
Obligation Tactic := idtac.
Robbert Krebbers's avatar
Robbert Krebbers committed
12

13 14 15 16
(** Throughout this development we use [C_scope] for all general purpose
notations that do not belong to a more specific scope. *)
Delimit Scope C_scope with C.
Global Open Scope C_scope.
17

18
(** Change [True] and [False] into notations in order to enable overloading.
19 20
We will use this to give [True] and [False] a different interpretation for
embedded logics. *)
21 22
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
23 24


25
(** * Equality *)
26
(** Introduce some Haskell style like notations. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
27 28 29 30 31 32 33
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.

34
Hint Extern 0 (_ = _) => reflexivity.
35
Hint Extern 100 (_  _) => discriminate.
Robbert Krebbers's avatar
Robbert Krebbers committed
36

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294
Instance: @PreOrder A (=).
Proof. split; repeat intro; congruence. Qed.

(** ** Setoid equality *)
(** We define an operational type class for setoid equality. This is based on
(Spitters/van der Weegen, 2011). *)
Class Equiv A := equiv: relation A.
Infix "≡" := equiv (at level 70, no associativity) : C_scope.
Notation "(≡)" := equiv (only parsing) : C_scope.
Notation "( X ≡)" := (equiv X) (only parsing) : C_scope.
Notation "(≡ X )" := (λ Y, Y  X) (only parsing) : C_scope.
Notation "(≢)" := (λ X Y, ¬X  Y) (only parsing) : C_scope.
Notation "X ≢ Y":= (¬X  Y) (at level 70, no associativity) : C_scope.
Notation "( X ≢)" := (λ Y, X  Y) (only parsing) : C_scope.
Notation "(≢ X )" := (λ Y, Y  X) (only parsing) : C_scope.

(** The type class [LeibnizEquiv] collects setoid equalities that coincide
with Leibniz equality. We provide the tactic [fold_leibniz] to transform such
setoid equalities into Leibniz equalities, and [unfold_leibniz] for the
reverse. *)
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.
 
Ltac fold_leibniz := repeat
  match goal with
  | H : context [ @equiv ?A _ _ _ ] |- _ =>
    setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
  | |- context [ @equiv ?A _ _ _ ] =>
    setoid_rewrite (leibniz_equiv_iff (A:=A))
  end.
Ltac unfold_leibniz := repeat
  match goal with
  | H : context [ @eq ?A _ _ ] |- _ =>
    setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
  | |- context [ @eq ?A _ _ ] =>
    setoid_rewrite <-(leibniz_equiv_iff (A:=A))
  end.

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

(** A [Params f n] instance forces the setoid rewriting mechanism not to
rewrite in the first [n] arguments of the function [f]. We will declare such
instances for all operational type classes in this development. *)
Instance: Params (@equiv) 2.

(** The following instance forces [setoid_replace] to use setoid equality
(for types that have an [Equiv] instance) rather than the standard Leibniz
equality. *)
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3.
Hint Extern 0 (_  _) => reflexivity.
Hint Extern 0 (_  _) => symmetry; assumption.


(** * Type classes *)
(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
propositions. For example to declare a parameter expressing decidable equality
on a type [A] we write [`{∀ x y : A, Decision (x = y)}] and use it by writing
[decide (x = y)]. *)
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.

(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
Arguments populate {_} _.

(** ** 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.

(** ** 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 [inj (k ++)] instead of [app_inv_head k]. *)
Class Inj {A B} (R : relation A) (S : relation B) (f : A  B) : Prop :=
  inj x y : S (f x) (f y)  R x y.
Class Inj2 {A B C} (R1 : relation A) (R2 : relation B)
    (S : relation C) (f : A  B  C) : Prop :=
  inj2 x1 x2 y1 y2 : S (f x1 x2) (f y1 y2)  R1 x1 y1  R2 x2 y2.
Class Cancel {A B} (S : relation B) (f : A  B) (g : B  A) : Prop :=
  cancel :  x, S (f (g x)) x.
Class Surj {A B} (R : relation B) (f : A  B) :=
  surj y :  x, R (f x) y.
Class IdemP {A} (R : relation A) (f : A  A  A) : Prop :=
  idemp x : R (f x x) x.
Class Comm {A B} (R : relation A) (f : B  B  A) : Prop :=
  comm x y : R (f x y) (f y x).
Class LeftId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
  left_id x : R (f i x) x.
Class RightId {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
  right_id x : R (f x i) x.
Class Assoc {A} (R : relation A) (f : A  A  A) : Prop :=
  assoc x y z : R (f x (f y z)) (f (f x y) z).
Class LeftAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
  left_absorb x : R (f i x) i.
Class RightAbsorb {A} (R : relation A) (i : A) (f : A  A  A) : Prop :=
  right_absorb x : R (f x i) i.
Class AntiSymm {A} (R S : relation A) : Prop :=
  anti_symm x y : S x y  S y x  R x y.
Class Total {A} (R : relation A) := total x y : R x y  R y x.
Class Trichotomy {A} (R : relation A) :=
  trichotomy x y : R x y  x = y  R y x.
Class TrichotomyT {A} (R : relation A) :=
  trichotomyT x y : {R x y} + {x = y} + {R y x}.

Arguments irreflexivity {_} _ {_} _ _.
Arguments inj {_ _ _ _} _ {_} _ _ _.
Arguments inj2 {_ _ _ _ _ _} _ {_} _ _ _ _ _.
Arguments cancel {_ _ _} _ _ {_} _.
Arguments surj {_ _ _} _ {_} _.
Arguments idemp {_ _} _ {_} _.
Arguments comm {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments assoc {_ _} _ {_} _ _ _.
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
Arguments anti_symm {_ _} _ {_} _ _ _ _.
Arguments total {_} _ {_} _ _.
Arguments trichotomy {_} _ {_} _ _.
Arguments trichotomyT {_} _ {_} _ _.

Lemma not_symmetry `{R : relation A, !Symmetric R} x y : ¬R x y  ¬R y x.
Proof. intuition. Qed.
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} x y : R x y  R y x.
Proof. intuition. Qed.

Lemma not_inj `{Inj A B R R' f} x y : ¬R x y  ¬R' (f x) (f y).
Proof. intuition. Qed.
Lemma not_inj2_1 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 :
  ¬R x1 x2  ¬R'' (f x1 y1) (f x2 y2).
Proof. intros HR HR''. destruct (inj2 f x1 y1 x2 y2); auto. Qed.
Lemma not_inj2_2 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 :
  ¬R' y1 y2  ¬R'' (f x1 y1) (f x2 y2).
Proof. intros HR' HR''. destruct (inj2 f x1 y1 x2 y2); auto. Qed.

Lemma inj_iff {A B} {R : relation A} {S : relation B} (f : A  B)
  `{!Inj R S f} `{!Proper (R ==> S) f} x y : S (f x) (f y)  R x y.
Proof. firstorder. Qed.
Instance inj2_inj_1 `{Inj2 A B C R1 R2 R3 f} y : Inj R1 R3 (λ x, f x y).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.
Instance inj2_inj_2 `{Inj2 A B C R1 R2 R3 f} x : Inj R2 R3 (f x).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.

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

(** The following lemmas are specific versions of the projections of the above
type classes for Leibniz equality. These lemmas allow us to enforce Coq not to
use the setoid rewriting mechanism. *)
Lemma idemp_L {A} f `{!@IdemP A (=) f} x : f x x = x.
Proof. auto. Qed.
Lemma comm_L {A B} f `{!@Comm A B (=) f} x y : f x y = f y x.
Proof. auto. Qed.
Lemma left_id_L {A} i f `{!@LeftId A (=) i f} x : f i x = x.
Proof. auto. Qed.
Lemma right_id_L {A} i f `{!@RightId A (=) i f} x : f x i = x.
Proof. auto. Qed.
Lemma assoc_L {A} f `{!@Assoc A (=) f} x y z : f x (f y z) = f (f x y) z.
Proof. auto. Qed.
Lemma left_absorb_L {A} i f `{!@LeftAbsorb A (=) i f} x : f i x = i.
Proof. auto. Qed.
Lemma right_absorb_L {A} i f `{!@RightAbsorb A (=) i f} x : f x i = i.
Proof. auto. Qed.

(** ** Generic orders *)
(** The classes [PreOrder], [PartialOrder], and [TotalOrder] use an arbitrary
relation [R] instead of [⊆] to support multiple orders on the same type. *)
Definition strict {A} (R : relation A) : relation A := λ X Y, R X Y  ¬R Y X.
Instance: Params (@strict) 2.
Class PartialOrder {A} (R : relation A) : Prop := {
  partial_order_pre :> PreOrder R;
  partial_order_anti_symm :> AntiSymm (=) R
}.
Class TotalOrder {A} (R : relation A) : Prop := {
  total_order_partial :> PartialOrder R;
  total_order_trichotomy :> Trichotomy (strict R)
}.

(** * Logic *)
Notation "(∧)" := and (only parsing) : C_scope.
Notation "( A ∧)" := (and A) (only parsing) : C_scope.
Notation "(∧ B )" := (λ A, A  B) (only parsing) : C_scope.

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

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

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

Lemma or_l P Q : ¬Q  P  Q  P.
Proof. tauto. Qed.
Lemma or_r P Q : ¬P  P  Q  Q.
Proof. tauto. Qed.
Lemma and_wlog_l (P Q : Prop) : (Q  P)  Q  (P  Q).
Proof. tauto. Qed.
Lemma and_wlog_r (P Q : Prop) : P  (P  Q)  (P  Q).
Proof. tauto. Qed.
Lemma impl_transitive (P Q R : Prop) : (P  Q)  (Q  R)  (P  R).
Proof. tauto. Qed.

Instance: Comm () (@eq A).
Proof. red; intuition. Qed.
Instance: Comm () (λ x y, @eq A y x).
Proof. red; intuition. Qed.
Instance: Comm () ().
Proof. red; intuition. Qed.
Instance: Comm () ().
Proof. red; intuition. Qed.
Instance: Assoc () ().
Proof. red; intuition. Qed.
Instance: IdemP () ().
Proof. red; intuition. Qed.
Instance: Comm () ().
Proof. red; intuition. Qed.
Instance: Assoc () ().
Proof. red; intuition. Qed.
Instance: IdemP () ().
Proof. red; intuition. Qed.
Instance: LeftId () True ().
Proof. red; intuition. Qed.
Instance: RightId () True ().
Proof. red; intuition. Qed.
Instance: LeftAbsorb () False ().
Proof. red; intuition. Qed.
Instance: RightAbsorb () False ().
Proof. red; intuition. Qed.
Instance: LeftId () False ().
Proof. red; intuition. Qed.
Instance: RightId () False ().
Proof. red; intuition. Qed.
Instance: LeftAbsorb () True ().
Proof. red; intuition. Qed.
Instance: RightAbsorb () True ().
Proof. red; intuition. Qed.
Instance: LeftId () True impl.
Proof. unfold impl. red; intuition. Qed.
Instance: RightAbsorb () True impl.
Proof. unfold impl. red; intuition. Qed.


(** * Common data types *)
(** ** Functions *)
295 296 297 298
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.

299
Notation "t $ r" := (t r)
300
  (at level 65, right associativity, only parsing) : C_scope.
301 302 303
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
304 305 306 307
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.
308

Robbert Krebbers's avatar
Robbert Krebbers committed
309 310 311
Instance impl_inhabited {A} `{Inhabited B} : Inhabited (A  B) :=
  populate (λ _, inhabitant).

312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
Arguments id _ _ /.
Arguments compose _ _ _ _ _ _ /.
Arguments flip _ _ _ _ _ _ /.
Arguments const _ _ _ _ /.
Typeclasses Transparent id compose flip const.

Definition fun_map {A A' B B'} (f: A'  A) (g: B  B') (h : A  B) : A'  B' :=
  g  h  f.

Instance const_proper `{R1 : relation A, R2 : relation B} (x : B) :
  Reflexive R2  Proper (R1 ==> R2) (λ _, x).
Proof. intros ? y1 y2; reflexivity. Qed.

Instance id_inj {A} : Inj (=) (=) (@id A).
Proof. intros ??; auto. Qed.
Instance compose_inj {A B C} R1 R2 R3 (f : A  B) (g : B  C) :
  Inj R1 R2 f  Inj R2 R3 g  Inj R1 R3 (g  f).
Proof. red; intuition. Qed.

Instance id_surj {A} : Surj (=) (@id A).
Proof. intros y; exists y; reflexivity. Qed.
Instance compose_surj {A B C} R (f : A  B) (g : B  C) :
  Surj (=) f  Surj R g  Surj R (g  f).
Proof.
  intros ?? x. unfold compose. destruct (surj g x) as [y ?].
  destruct (surj f y) as [z ?]. exists z. congruence.
Qed.

Instance id_comm {A B} (x : B) : Comm (=) (λ _ _ : A, x).
Proof. intros ?; reflexivity. Qed.
Instance id_assoc {A} (x : A) : Assoc (=) (λ _ _ : A, x).
Proof. intros ???; reflexivity. Qed.
Instance const1_assoc {A} : Assoc (=) (λ x _ : A, x).
Proof. intros ???; reflexivity. Qed.
Instance const2_assoc {A} : Assoc (=) (λ _ x : A, x).
Proof. intros ???; reflexivity. Qed.
Instance const1_idemp {A} : IdemP (=) (λ x _ : A, x).
Proof. intros ?; reflexivity. Qed.
Instance const2_idemp {A} : IdemP (=) (λ _ x : A, x).
Proof. intros ?; reflexivity. Qed.

(** ** Lists *)
Instance list_inhabited {A} : Inhabited (list A) := populate [].

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).

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

Instance bool_inhabated : Inhabited bool := populate true.
375

376 377 378 379 380
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.
381

382 383 384 385 386 387 388 389
Lemma andb_True b1 b2 : b1 && b2  b1  b2.
Proof. destruct b1, b2; simpl; tauto. Qed.
Lemma orb_True b1 b2 : b1 || b2  b1  b2.
Proof. destruct b1, b2; simpl; tauto. Qed.
Lemma negb_True b : negb b  ¬b.
Proof. destruct b; simpl; tauto. Qed.
Lemma Is_true_false (b : bool) : b = false  ¬b.
Proof. now intros -> ?. Qed.
390

391 392 393 394 395
(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
Instance unit_equivalence : Equivalence (@equiv unit _).
Proof. repeat split. Qed.
Instance unit_inhabited: Inhabited unit := populate ().
396

397
(** ** Products *)
398 399 400 401 402 403
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").

404 405
Instance: Params (@pair) 2.

406 407 408 409 410 411 412 413 414
Notation curry := prod_curry.
Notation uncurry := prod_uncurry.
Definition curry3 {A B C D} (f : A  B  C  D) (p : A * B * C) : D :=
  let '(a,b,c) := p in f a b c.
Definition curry4 {A B C D E} (f : A  B  C  D  E) (p : A * B * C * D) : E :=
  let '(a,b,c,d) := p in f a b c d.

Definition prod_map {A A' B B'} (f: A  A') (g: B  B') (p : A * B) : A' * B' :=
  (f (p.1), g (p.2)).
415
Arguments prod_map {_ _ _ _} _ _ !_ /.
416

417 418 419 420
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 {_ _ _ _ _ _} _ _ !_ !_ /.

421 422 423
Instance prod_inhabited {A B} (iA : Inhabited A)
    (iB : Inhabited B) : Inhabited (A * B) :=
  match iA, iB with populate x, populate y => populate (x,y) end.
424

425 426 427 428 429 430 431 432
Instance pair_inj : Inj2 (=) (=) (=) (@pair A B).
Proof. injection 1; auto. Qed.
Instance prod_map_inj {A A' B B'} (f : A  A') (g : B  B') :
  Inj (=) (=) f  Inj (=) (=) g  Inj (=) (=) (prod_map f g).
Proof.
  intros ?? [??] [??] ?; simpl in *; f_equal;
    [apply (inj f)|apply (inj g)]; congruence.
Qed.
433

434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449
Definition prod_relation {A B} (R1 : relation A) (R2 : relation B) :
  relation (A * B) := λ x y, R1 (x.1) (y.1)  R2 (x.2) (y.2).
Section prod_relation.
  Context `{R1 : relation A, R2 : relation B}.
  Global Instance prod_relation_refl :
    Reflexive R1  Reflexive R2  Reflexive (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance prod_relation_sym :
    Symmetric R1  Symmetric R2  Symmetric (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance prod_relation_trans :
    Transitive R1  Transitive R2  Transitive (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance prod_relation_equiv :
    Equivalence R1  Equivalence R2  Equivalence (prod_relation R1 R2).
  Proof. split; apply _. Qed.
450

451 452
  Global Instance pair_proper' : Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
  Proof. firstorder eauto. Qed.
453 454
  Global Instance pair_inj' : Inj2 R1 R2 (prod_relation R1 R2) pair.
  Proof. inversion_clear 1; eauto. Qed.
455 456 457 458 459
  Global Instance fst_proper' : Proper (prod_relation R1 R2 ==> R1) fst.
  Proof. firstorder eauto. Qed.
  Global Instance snd_proper' : Proper (prod_relation R1 R2 ==> R2) snd.
  Proof. firstorder eauto. Qed.
End prod_relation.
Robbert Krebbers's avatar
Robbert Krebbers committed
460

461 462
Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation () ().
Instance pair_proper `{Equiv A, Equiv B} :
463 464
  Proper (() ==> () ==> ()) (@pair A B) := _.
Instance pair_equiv_inj `{Equiv A, Equiv B} : Inj2 () () () (@pair A B) := _.
465 466 467
Instance fst_proper `{Equiv A, Equiv B} : Proper (() ==> ()) (@fst A B) := _.
Instance snd_proper `{Equiv A, Equiv B} : Proper (() ==> ()) (@snd A B) := _.
Typeclasses Opaque prod_equiv.
468

469 470 471 472 473
Instance prod_leibniz `{LeibnizEquiv A, !Equivalence (() : relation A),
                        LeibnizEquiv B, !Equivalence (() : relation B)}
  : LeibnizEquiv (A * B).
Proof. intros [] [] []; fold_leibniz; simpl; congruence. Qed.

474
(** ** Sums *)
475 476 477 478
Definition sum_map {A A' B B'} (f: A  A') (g: B  B') (xy : A + B) : A' + B' :=
  match xy with inl x => inl (f x) | inr y => inr (g y) end.
Arguments sum_map {_ _ _ _} _ _ !_ /.

479
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
480
  match iA with populate x => populate (inl x) end.
481
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
482
  match iB with populate y => populate (inl y) end.
483

484 485 486 487
Instance inl_inj : Inj (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Instance inr_inj : Inj (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.
488

489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515
Instance sum_map_inj {A A' B B'} (f : A  A') (g : B  B') :
  Inj (=) (=) f  Inj (=) (=) g  Inj (=) (=) (sum_map f g).
Proof. intros ?? [?|?] [?|?] [=]; f_equal; apply (inj _); auto. Qed.

Inductive sum_relation {A B}
     (R1 : relation A) (R2 : relation B) : relation (A + B) :=
  | inl_related x1 x2 : R1 x1 x2  sum_relation R1 R2 (inl x1) (inl x2)
  | inr_related y1 y2 : R2 y1 y2  sum_relation R1 R2 (inr y1) (inr y2).

Section sum_relation.
  Context `{R1 : relation A, R2 : relation B}.
  Global Instance sum_relation_refl :
    Reflexive R1  Reflexive R2  Reflexive (sum_relation R1 R2).
  Proof. intros ?? [?|?]; constructor; reflexivity. Qed.
  Global Instance sum_relation_sym :
    Symmetric R1  Symmetric R2  Symmetric (sum_relation R1 R2).
  Proof. destruct 3; constructor; eauto. Qed.
  Global Instance sum_relation_trans :
    Transitive R1  Transitive R2  Transitive (sum_relation R1 R2).
  Proof. destruct 3; inversion_clear 1; constructor; eauto. Qed.
  Global Instance sum_relation_equiv :
    Equivalence R1  Equivalence R2  Equivalence (sum_relation R1 R2).
  Proof. split; apply _. Qed.
  Global Instance inl_proper' : Proper (R1 ==> sum_relation R1 R2) inl.
  Proof. constructor; auto. Qed.
  Global Instance inr_proper' : Proper (R2 ==> sum_relation R1 R2) inr.
  Proof. constructor; auto. Qed.
516 517 518 519
  Global Instance inl_inj' : Inj R1 (sum_relation R1 R2) inl.
  Proof. inversion_clear 1; auto. Qed.
  Global Instance inr_inj' : Inj R2 (sum_relation R1 R2) inr.
  Proof. inversion_clear 1; auto. Qed.
520 521 522 523 524
End sum_relation.

Instance sum_equiv `{Equiv A, Equiv B} : Equiv (A + B) := sum_relation () ().
Instance inl_proper `{Equiv A, Equiv B} : Proper (() ==> ()) (@inl A B) := _.
Instance inr_proper `{Equiv A, Equiv B} : Proper (() ==> ()) (@inr A B) := _.
525 526
Instance inl_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inl A B) := _.
Instance inr_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inr A B) := _.
527 528
Typeclasses Opaque sum_equiv.

529 530
(** ** Option *)
Instance option_inhabited {A} : Inhabited (option A) := populate None.
Robbert Krebbers's avatar
Robbert Krebbers committed
531

532 533 534 535 536
(** ** Sigma types *)
Arguments existT {_ _} _ _.
Arguments proj1_sig {_ _} _.
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : C_scope.
537

538 539 540
Lemma proj1_sig_inj {A} (P : A  Prop) x (Px : P x) y (Py : P y) :
  xPx = yPy  x = y.
Proof. injection 1; trivial. Qed.
541

542 543 544 545 546 547 548 549 550 551 552
Section sig_map.
  Context `{P : A  Prop} `{Q : B  Prop} (f : A  B) (Hf :  x, P x  Q (f x)).
  Definition sig_map (x : sig P) : sig Q := f (`x)  Hf _ (proj2_sig x).
  Global Instance sig_map_inj:
    ( x, ProofIrrel (P x))  Inj (=) (=) f  Inj (=) (=) sig_map.
  Proof.
    intros ?? [x Hx] [y Hy]. injection 1. intros Hxy.
    apply (inj f) in Hxy; subst. rewrite (proof_irrel _ Hy). auto.
  Qed.
End sig_map.
Arguments sig_map _ _ _ _ _ _ !_ /.
553

Robbert Krebbers's avatar
Robbert Krebbers committed
554

555
(** * Operations on collections *)
556
(** We define operational type classes for the traditional operations and
557
relations on collections: the empty collection [∅], the union [(∪)],
558 559
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
560 561 562
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

563 564 565
Class Top A := top : A.
Notation "⊤" := top : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
566
Class Union A := union: A  A  A.
567
Instance: Params (@union) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
568 569 570 571
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.
572 573 574 575 576 577
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
578

579
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
580 581 582
Arguments union_list _ _ _ !_ /.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
583
Class Intersection A := intersection: A  A  A.
584
Instance: Params (@intersection) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
585 586 587 588 589 590
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.
591
Instance: Params (@difference) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
592 593 594 595
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.
596 597 598 599 600 601
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
602

603 604
Class Singleton A B := singleton: A  B.
Instance: Params (@singleton) 3.
605
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
606
Notation "{[ x ; y ; .. ; z ]}" :=
607 608 609 610 611 612
  (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.
613

614
Class SubsetEq A := subseteq: relation A.
615
Instance: Params (@subseteq) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
616 617 618
Infix "⊆" := subseteq (at level 70) : C_scope.
Notation "(⊆)" := subseteq (only parsing) : C_scope.
Notation "( X ⊆ )" := (subseteq X) (only parsing) : C_scope.
619
Notation "( ⊆ X )" := (λ Y, Y  X) (only parsing) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
620 621 622 623
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.
624 625 626 627 628 629 630
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
631

632
Hint Extern 0 (_  _) => reflexivity.
633 634 635 636 637 638 639
Hint Extern 0 (_ * _) => reflexivity.
Hint Extern 0 (_ ** _) => reflexivity.

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.
640 641 642 643
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
644

645 646 647 648 649
Notation "X ⊆ Y ⊆ Z" := (X  Y  Y  Z) (at level 70, Y at next level) : C_scope.
Notation "X ⊆ Y ⊂ Z" := (X  Y  Y  Z) (at level 70, Y at next level) : C_scope.
Notation "X ⊂ Y ⊆ Z" := (X  Y  Y  Z) (at level 70, Y at next level) : C_scope.
Notation "X ⊂ Y ⊂ Z" := (X  Y  Y  Z) (at level 70, Y at next level) : C_scope.

650 651 652 653 654
(** 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
655
Class ElemOf A B := elem_of: A  B  Prop.
656
Instance: Params (@elem_of) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
657 658 659 660 661 662 663 664 665
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
666 667 668 669
Class Disjoint A := disjoint : A  A  Prop.
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
670
Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
671
Notation "(.⊥ X )" := (λ Y, Y  X) (only parsing) : C_scope.
672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696
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.
697 698 699

Class DisjointList A := disjoint_list : list A  Prop.
Instance: Params (@disjoint_list) 2.
700
Notation "⊥ Xs" := (disjoint_list Xs) (at level 20, format "⊥  Xs") : C_scope.
701

702 703 704 705 706 707
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.
708

709
  Lemma disjoint_list_nil  :  @nil A  True.
710 711 712
  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.
713
End disjoint_list.
714 715

Class Filter A B := filter:  (P : A  Prop) `{ x, Decision (P x)}, B  B.
716

717 718

(** * Monadic operations *)
719
(** We define operational type classes for the monadic operations bind, join 
720 721 722
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). *)
723 724
Class MRet (M : Type  Type) := mret:  {A}, A  M A.
Arguments mret {_ _ _} _.
725
Instance: Params (@mret) 3.
726 727
Class MBind (M : Type  Type) := mbind :  {A B}, (A  M B)  M A  M B.
Arguments mbind {_ _ _ _} _ !_ /.
728
Instance: Params (@mbind) 4.
729
Class MJoin (M : Type  Type) := mjoin:  {A}, M (M A)  M A.
730
Arguments mjoin {_ _ _} !_ /.
731
Instance: Params (@mjoin) 3.
732 733
Class FMap (M : Type  Type) := fmap :  {A B}, (A  B)  M A  M B.
Arguments fmap {_ _ _ _} _ !_ /.
734
Instance: Params (@fmap) 4.
735 736
Class OMap (M : Type  Type) := omap:  {A B}, (A  option B)  M A  M B.
Arguments omap {_ _ _ _} _ !_ /.
737
Instance: Params (@omap) 4.
738

739 740 741 742 743 744
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
745
  (at level 65, only parsing, right associativity) : C_scope.
746
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.
747
Notation "' ( x1 , x2 ) ← y ; z" :=
748
  (y = (λ x : _, let ' (x1, x2) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
749
  (at level 65, only parsing, right associativity) : C_scope.
750
Notation "' ( x1 , x2 , x3 ) ← y ; z" :=
751
  (y = (λ x : _, let ' (x1,x2,x3) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
752
  (at level 65, only parsing, right associativity) : C_scope.
753
Notation "' ( x1 , x2 , x3  , x4 ) ← y ; z" :=
754
  (y = (λ x : _, let ' (x1,x2,x3,x4) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
755
  (at level 65, only parsing, right associativity) : C_scope.
756 757
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
758
  (at level 65, only parsing, right associativity) : C_scope.
759 760
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
761
  (at level 65, only parsing, right associativity) : C_scope.
762

763 764 765 766 767
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").

768
Class MGuard (M : Type  Type) :=
769 770 771
  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
772
  (at level 65, only parsing, right associativity) : C_scope.
773
Notation "'guard' P 'as' H ; o" := (mguard P (λ H, o))
Robbert Krebbers's avatar
Robbert Krebbers committed
774
  (at level 65, only parsing, right associativity) : C_scope.
775

776 777

(** * Operations on maps *)
778 779
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
780
The function look up [m !! k] should yield the element at key [k] in [m]. *)
781
Class Lookup (K A M : Type) := lookup: K  M  option A.
782 783 784
Instance: Params (@lookup) 4.
Notation "m !! i" := (lookup i m) (at level 20) : C_scope.
Notation "(!!)" := lookup (only parsing) : C_scope.
785
Notation "( m !!)" := (λ i, m !! i) (only parsing) : C_scope.
786
Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
787
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.
788

789 790 791
(** The singleton map *)
Class SingletonM K A M := singletonM: K  A  M.
Instance: Params (@singletonM) 5.
792
Notation "{[ k := a ]}" := (singletonM k a) (at level 1) : C_scope.
793

794 795
(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
796
Class Insert (K A M : Type) := insert: K  A  M  M.
Robbert Krebbers's avatar
Robbert Krebbers committed
797
Instance: Params (@insert) 5.
798 799
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
800
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.
801

802 803 804
(** 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. *)
805
Class Delete (K M : Type) := delete: K  M  M.
Robbert Krebbers's avatar
Robbert Krebbers committed
806
Instance: Params (@delete) 4.
807
Arguments delete _ _ _ !_ !_ / : simpl nomatch.
808 809

(** The function [alter f k m] should update the value at key [k] using the
810
function [f], which is called with the original value. *)
811
Class Alter (K A M : Type) := alter: (A  A)  K  M  M.
812
Instance: Params (@alter) 5.
813
Arguments alter {_ _ _ _} _ !_ !_ / : simpl nomatch.
814 815

(** The function [alter f k m] should update the value at key [k] using the
816 817 818
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]. *)
819 820
Class PartialAlter (K A M : Type) :=
  partial_alter: (option A  option A)  K  M  M.
821
Instance: Params (@partial_alter) 4.
822
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.
823 824 825

(** 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]. *)
826 827 828
Class Dom (M C : Type) := dom: M  C.
Instance: Params (@dom) 3.
Arguments dom {_} _ {_} !_ / : simpl nomatch, clear implicits.
829 830

(** The function [merge f m1 m2] should merge the maps [m1] and [m2] by
831 832 833 834 835
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.
836

837 838 839 840 841 842 843
(** The function [union_with f m1 m2] is supposed to yield the union of [m1]
and [m2] using the function [f] to combine values of members that are in
both [m1] and [m2]. *)
Class UnionWith (A M : Type) :=
  union_with: (A  A  option A)  M  M  M.
Instance: Params (@union_with) 3.
Arguments union_with {_ _ _} _ !_ !_ / : simpl nomatch.
844

845 846 847 848 849
(** Similarly for intersection and difference. *)
Class IntersectionWith (A M : Type) :=
  intersection_with: (A  A  option A)  M  M  M.
Instance: Params (@intersection_with) 3.
Arguments intersection_with {_ _ _} _ !_ !_ / : simpl nomatch.
850

851 852 853 854
Class DifferenceWith (A M : Type) :=
  difference_with: (A  A  option A)  M  M  M.
Instance: Params (@difference_with) 3.
Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch.
Robbert Krebbers's avatar
Robbert Krebbers committed
855

856 857 858 859 860 861 862 863 864 865
Definition intersection_with_list `{IntersectionWith A M}
  (f : A  A  option A) : M  list M  M := fold_right (intersection_with f).
Arguments intersection_with_list _ _ _ _ _ !_ /.

Class LookupE (E K A M : Type) := lookupE: E  K  M  option A.
Instance: Params (@lookupE) 6.
Notation "m !!{ Γ } i" := (lookupE Γ i m)
  (at level 20, format "m  !!{ Γ }  i") : C_scope.
Notation "(!!{ Γ } )" := (lookupE Γ) (only parsing, Γ at level 1) : C_scope.
Arguments lookupE _ _ _ _ _ _ !_ !_ / : simpl nomatch.
866

867 868 869 870 871 872 873 874
Class InsertE (E K A M : Type) := insertE: E  K  A  M  M.
Instance: Params (@insertE) 6.
Notation "<[ k := a ]{ Γ }>" := (insertE Γ k a)
  (at level 5, right associativity, format "<[ k := a ]{ Γ }>") : C_scope.
Arguments insertE _ _ _ _ _ _ !_ _ !_ / : simpl nomatch.


(** * Ordered structures *)
875 876 877 878 879 880
(** 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 ();
881 882 883
  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
884
}.
885 886
Class MeetSemiLattice A `{SubsetEq A, Intersection A} : Prop := {
  meet_semi_lattice_pre :>> PreOrder ();
887 888 889
  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
890
}.
891 892 893 894
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)
895
}.
896

897
(** ** Axiomatization of collections *)
898 899
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
900 901
Class SimpleCollection A C `{ElemOf A C,
    Empty C, Singleton A C, Union C} : Prop := {
902
  not_elem_of_empty (x : A) : x  ;
903
  elem_of_singleton (x y : A) : x  {[ y ]}  x = y;
904 905
  elem_of_union X Y (x : A) : x  X  Y  x  X  x  Y
}.
906 907
Class Collection A C `{ElemOf A C, Empty C, Singleton A C,
    Union C, Intersection C, Difference C} : Prop := {
908
  collection_simple :>> SimpleCollection A C;
Robbert Krebbers's avatar
Robbert Krebbers committed
909
  elem_of_intersection X Y (x : A) : x  X  Y  x  X  x  Y;
910 911
  elem_of_difference X Y (x : A) : x  X  Y  x  X  x  Y
}.
912 913
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 := {
914
  collection_ops :>> Collection A C;
915
  elem_of_intersection_with (f : A  A  option A) X Y (x : A) :
916
    x  intersection_with f X Y   x1 x2, x1  X  x2  Y  f x1 x2 = Some x;
917
  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
918 919
}.

920 921 922
(** 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
923
Class Elements A C := elements: C  list A.
924
Instance: Params (@elements) 3.
925 926 927 928 929 930 931 932 933 934 935 936 937

(** 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. *)
938 939 940
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
941
  fin_collection :>> Collection A C;
942 943
  elem_of_elements X x : x  elements X  x  X;
  NoDup_elements X : NoDup (elements X)
944 945
}.
Class Size C := size: C  nat.
946
Arguments size {_ _} !_ / : simpl nomatch.
947
Instance: Params (@size) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
948

949 950 951 952 953 954 955 956
(** 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]. *)
957 958 959
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 := {
960 961 962
  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;
963
  elem_of_ret {A} (x y : A) : x  mret y  x = y;
964 965
  elem_of_fmap {A B} (f : A  B) (X : M A) (x : B) :
    x  f <$> X   y, x = f y  y  X;
966
  elem_of_join {A} (X : M (M A)) (x : A) : x  mjoin X   Y, x  Y  Y  X
967 968
}.

969 970 971
(** 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
972
Class Fresh A C := fresh: C  A.
973
Instance: Params (@fresh) 3.
974 975
Class FreshSpec A C `{ElemOf A C,
    Empty C, Singleton A C, Union C, Fresh A C} : Prop := {
976
  fresh_collection_simple :>> SimpleCollection A C;
977
  fresh_proper_alt X Y : ( x, x  X  x  Y)  fresh X = fresh Y;
Robbert Krebbers's avatar
Robbert Krebbers committed
978 979 980
  is_fresh (X : C) : fresh X  X
}.

981
(** * Miscellaneous *)
982
Class Half A := half: A  A.