base.v 46 KB
Newer Older
1
(* Copyright (c) 2012-2017, Coq-std++ developers. *)
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
Global Unset Transparent Obligations.
11
From Coq Require Export Morphisms RelationClasses List Bool Utf8 Setoid.
12
Set Default Proof Using "Type".
13 14
Export ListNotations.
From Coq.Program Require Export Basics Syntax.
15 16 17

(* Tweak program: don't let it automatically simplify obligations and hide
them from the results of the [Search] commands. *)
18
Obligation Tactic := idtac.
19
Add Search Blacklist "_obligation_".
Robbert Krebbers's avatar
Robbert Krebbers committed
20

21 22 23
(** Sealing off definitions *)
Set Primitive Projections.
Record seal {A} (f : A) := { unseal : A; seal_eq : unseal = f }.
24 25
Arguments unseal {_ _} _ : assert.
Arguments seal_eq {_ _} _ : assert.
26 27
Unset Primitive Projections.

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
(* Below we define type class versions of the common logical operators. It is
important to note that we duplicate the definitions, and do not declare the
existing logical operators as type classes. That is, we do not say:

  Existing Class or.
  Existing Class and.

If we could define the existing logical operators as classes, there is no way
of disambiguating whether a premise of a lemma should be solved by type class
resolution or not.

These classes are useful for two purposes: writing complicated type class
premises in a more concise way, and for efficiency. For example, using the [Or]
class, instead of defining two instances [P → Q1 → R] and [P → Q2 → R] we could
have one instance [P → Or Q1 Q2 → R]. When we declare the instance that way, we
avoid the need to derive [P] twice. *)
44
Inductive TCOr (P1 P2 : Prop) : Prop :=
45 46 47 48 49
  | TCOr_l : P1  TCOr P1 P2
  | TCOr_r : P2  TCOr P1 P2.
Existing Class TCOr.
Existing Instance TCOr_l | 9.
Existing Instance TCOr_r | 10.
Robbert Krebbers's avatar
Robbert Krebbers committed
50

51
Inductive TCAnd (P1 P2 : Prop) : Prop := TCAnd_intro : P1  P2  TCAnd P1 P2.
52 53
Existing Class TCAnd.
Existing Instance TCAnd_intro.
54

55 56 57
Inductive TCTrue : Prop := TCTrue_intro : TCTrue.
Existing Class TCTrue.
Existing Instance TCTrue_intro.
58

59 60 61 62 63 64 65
Inductive TCForall {A} (P : A  Prop) : list A  Prop :=
  | TCForall_nil : TCForall P []
  | TCForall_cons x xs : P x  TCForall P xs  TCForall P (x :: xs).
Existing Class TCForall.
Existing Instance TCForall_nil.
Existing Instance TCForall_cons.

66 67 68 69
(** 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.
70

71
(** Change [True] and [False] into notations in order to enable overloading.
72 73
We will use this to give [True] and [False] a different interpretation for
embedded logics. *)
74 75
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
76 77


78
(** * Equality *)
79
(** Introduce some Haskell style like notations. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
80 81 82 83 84 85 86
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.

87
Hint Extern 0 (_ = _) => reflexivity.
88
Hint Extern 100 (_  _) => discriminate.
Robbert Krebbers's avatar
Robbert Krebbers committed
89

90 91 92 93 94 95 96
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.
97 98 99
(* No Hint Mode set because of Coq bug #5735
Hint Mode Equiv ! : typeclass_instances. *)

100 101 102 103 104 105 106 107 108 109 110 111 112 113
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.
114 115
Hint Mode LeibnizEquiv ! - : typeclass_instances.

116 117 118
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@equiv A _)} (x y : A) :
  x  y  x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
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
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}.
157
Hint Mode Decision ! : typeclass_instances.
158
Arguments decide _ {_} : assert.
159
Notation EqDecision A := ( x y : A, Decision (x = y)).
160 161 162 163

(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
164
Hint Mode Inhabited ! : typeclass_instances.
165
Arguments populate {_} _ : assert.
166 167 168 169 170 171

(** ** 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.
172
Hint Mode ProofIrrel ! : typeclass_instances.
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

(** ** 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}.

209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224
Arguments irreflexivity {_} _ {_} _ _ : assert.
Arguments inj {_ _ _ _} _ {_} _ _ _ : assert.
Arguments inj2 {_ _ _ _ _ _} _ {_} _ _ _ _ _: assert.
Arguments cancel {_ _ _} _ _ {_} _ : assert.
Arguments surj {_ _ _} _ {_} _ : assert.
Arguments idemp {_ _} _ {_} _ : assert.
Arguments comm {_ _ _} _ {_} _ _ : assert.
Arguments left_id {_ _} _ _ {_} _ : assert.
Arguments right_id {_ _} _ _ {_} _ : assert.
Arguments assoc {_ _} _ {_} _ _ _ : assert.
Arguments left_absorb {_ _} _ _ {_} _ : assert.
Arguments right_absorb {_ _} _ _ {_} _ : assert.
Arguments anti_symm {_ _} _ {_} _ _ _ _ : assert.
Arguments total {_} _ {_} _ _ : assert.
Arguments trichotomy {_} _ {_} _ _ : assert.
Arguments trichotomyT {_} _ {_} _ _ : assert.
225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313

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.
314 315 316 317 318 319
Lemma forall_proper {A} (P Q : A  Prop) :
  ( x, P x  Q x)  ( x, P x)  ( x, Q x).
Proof. firstorder. Qed.
Lemma exist_proper {A} (P Q : A  Prop) :
  ( x, P x  Q x)  ( x, P x)  ( x, Q x).
Proof. firstorder. Qed.
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

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 *)
363 364 365 366
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.

367
Notation "t $ r" := (t r)
368
  (at level 65, right associativity, only parsing) : C_scope.
369 370 371
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
372 373 374 375
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.
376

Robbert Krebbers's avatar
Robbert Krebbers committed
377 378 379
Instance impl_inhabited {A} `{Inhabited B} : Inhabited (A  B) :=
  populate (λ _, inhabitant).

380 381
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
382 383 384 385
Arguments id _ _ / : assert.
Arguments compose _ _ _ _ _ _ / : assert.
Arguments flip _ _ _ _ _ _ / : assert.
Arguments const _ _ _ _ / : assert.
386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442
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.
443

444 445 446 447 448
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.
449

450 451 452 453 454 455 456 457
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.
458

459 460 461 462
(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
Instance unit_equivalence : Equivalence (@equiv unit _).
Proof. repeat split. Qed.
463 464
Instance unit_leibniz : LeibnizEquiv unit.
Proof. intros [] []; reflexivity. Qed.
465
Instance unit_inhabited: Inhabited unit := populate ().
466

467
(** ** Products *)
468 469 470 471 472 473
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").

474
Instance: Params (@pair) 2.
475 476
Instance: Params (@fst) 2.
Instance: Params (@snd) 2.
477

478 479 480 481 482 483 484 485 486
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)).
487
Arguments prod_map {_ _ _ _} _ _ !_ / : assert.
488

489 490
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)).
491
Arguments prod_zip {_ _ _ _ _ _} _ _ !_ !_ / : assert.
492

493 494 495
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.
496

497 498 499 500 501 502 503 504
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.
505

506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521
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.
522

523 524
  Global Instance pair_proper' : Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
  Proof. firstorder eauto. Qed.
525 526
  Global Instance pair_inj' : Inj2 R1 R2 (prod_relation R1 R2) pair.
  Proof. inversion_clear 1; eauto. Qed.
527 528 529 530 531
  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
532

533 534
Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation () ().
Instance pair_proper `{Equiv A, Equiv B} :
535 536
  Proper (() ==> () ==> ()) (@pair A B) := _.
Instance pair_equiv_inj `{Equiv A, Equiv B} : Inj2 () () () (@pair A B) := _.
537 538 539
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.
540

Robbert Krebbers's avatar
Robbert Krebbers committed
541 542
Instance prod_leibniz `{LeibnizEquiv A, LeibnizEquiv B} : LeibnizEquiv (A * B).
Proof. intros [??] [??] [??]; f_equal; apply leibniz_equiv; auto. Qed.
543

544
(** ** Sums *)
545 546
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.
547
Arguments sum_map {_ _ _ _} _ _ !_ / : assert.
548

549
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
550
  match iA with populate x => populate (inl x) end.
551
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
552
  match iB with populate y => populate (inl y) end.
553

554 555 556 557
Instance inl_inj : Inj (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Instance inr_inj : Inj (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.
558

559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585
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.
586 587 588 589
  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.
590 591 592 593 594
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) := _.
595 596
Instance inl_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inl A B) := _.
Instance inr_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inr A B) := _.
597 598
Typeclasses Opaque sum_equiv.

599 600
(** ** Option *)
Instance option_inhabited {A} : Inhabited (option A) := populate None.
Robbert Krebbers's avatar
Robbert Krebbers committed
601

602
(** ** Sigma types *)
603 604 605
Arguments existT {_ _} _ _ : assert.
Arguments projT1 {_ _} _ : assert.
Arguments projT2 {_ _} _ : assert.
606

607 608 609
Arguments exist {_} _ _ _ : assert.
Arguments proj1_sig {_ _} _ : assert.
Arguments proj2_sig {_ _} _ : assert.
610 611
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : C_scope.
612

613 614 615
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.
616

617 618 619 620 621 622 623 624 625 626
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.
627
Arguments sig_map _ _ _ _ _ _ !_ / : assert.
628

Robbert Krebbers's avatar
Robbert Krebbers committed
629

630
(** * Operations on collections *)
631
(** We define operational type classes for the traditional operations and
632
relations on collections: the empty collection [∅], the union [(∪)],
633 634
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
635
Class Empty A := empty: A.
636
Hint Mode Empty ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
637 638
Notation "∅" := empty : C_scope.

639 640
Instance empty_inhabited `(Empty A) : Inhabited A := populate .

641
Class Top A := top : A.
642
Hint Mode Top ! : typeclass_instances.
643 644
Notation "⊤" := top : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
645
Class Union A := union: A  A  A.
646
Hint Mode Union ! : typeclass_instances.
647
Instance: Params (@union) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
648 649 650 651
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.
652 653 654 655 656 657
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
658

659
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
660
Arguments union_list _ _ _ !_ / : assert.
661 662
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : C_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
663
Class Intersection A := intersection: A  A  A.
664
Hint Mode Intersection ! : typeclass_instances.
665
Instance: Params (@intersection) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
666 667 668 669 670 671
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.
672
Hint Mode Difference ! : typeclass_instances.
673
Instance: Params (@difference) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
674
Infix "∖" := difference (at level 40, left associativity) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
675 676 677
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.
678 679 680 681 682 683
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
684

685
Class Singleton A B := singleton: A  B.
686
Hint Mode Singleton - ! : typeclass_instances.
687
Instance: Params (@singleton) 3.
688
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
689
Notation "{[ x ; y ; .. ; z ]}" :=
690 691 692 693 694 695
  (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.
696

697
Class SubsetEq A := subseteq: relation A.
698
Hint Mode SubsetEq ! : typeclass_instances.
699
Instance: Params (@subseteq) 2.
Robbert Krebbers's avatar
Robbert Krebbers committed
700 701
Infix "⊆" := subseteq (at level 70) : C_scope.
Notation "(⊆)" := subseteq (only parsing) : C_scope.
702 703
Notation "( X ⊆)" := (subseteq X) (only parsing) : C_scope.
Notation "(⊆ X )" := (λ Y, Y  X) (only parsing) : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
704 705
Notation "X ⊈ Y" := (¬X  Y) (at level 70) : C_scope.
Notation "(⊈)" := (λ X Y, X  Y) (only parsing) : C_scope.
706 707
Notation "( X ⊈)" := (λ Y, X  Y) (only parsing) : C_scope.
Notation "(⊈ X )" := (λ Y, Y  X) (only parsing) : C_scope.
708 709 710 711 712 713 714
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
715

716
Hint Extern 0 (_  _) => reflexivity.
717 718 719 720 721
Hint Extern 0 (_ * _) => reflexivity.
Hint Extern 0 (_ ** _) => reflexivity.

Infix "⊂" := (strict ()) (at level 70) : C_scope.
Notation "(⊂)" := (strict ()) (only parsing) : C_scope.
722 723 724
Notation "( X ⊂)" := (strict () X) (only parsing) : C_scope.
Notation "(⊂ X )" := (λ Y, Y  X) (only parsing) : C_scope.
Notation "X ⊄ Y" := (¬X  Y) (at level 70) : C_scope.
725
Notation "(⊄)" := (λ X Y, X  Y) (only parsing) : C_scope.
726 727
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
728

729 730 731 732 733
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.

734 735 736 737
(** 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.
738
Hint Mode Lexico ! : typeclass_instances.
739

Robbert Krebbers's avatar
Robbert Krebbers committed
740
Class ElemOf A B := elem_of: A  B  Prop.
741
Hint Mode ElemOf - ! : typeclass_instances.
742
Instance: Params (@elem_of) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
743 744 745 746 747 748 749 750 751
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
752
Class Disjoint A := disjoint : A  A  Prop.
753
Hint Mode Disjoint ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
754 755 756
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
757
Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
758
Notation "(.⊥ X )" := (λ Y, Y  X) (only parsing) : C_scope.
759 760 761 762 763 764 765 766 767 768 769
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.
770
Hint Mode DisjointE - ! : typeclass_instances.
771 772 773 774 775 776 777 778 779 780 781 782 783 784
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.
785 786

Class DisjointList A := disjoint_list : list A  Prop.
787
Hint Mode DisjointList ! : typeclass_instances.
788
Instance: Params (@disjoint_list) 2.
789
Notation "⊥ Xs" := (disjoint_list Xs) (at level 20, format "⊥  Xs") : C_scope.
790

791 792
Section disjoint_list.
  Context `{Disjoint A, Union A, Empty A}.
793 794
  Implicit Types X : A.

795 796 797 798
  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.
799

800
  Lemma disjoint_list_nil  :  @nil A  True.
801 802 803
  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.
804
End disjoint_list.
805 806

Class Filter A B := filter:  (P : A  Prop) `{ x, Decision (P x)}, B  B.
807
Hint Mode Filter - ! : typeclass_instances.
808

809
Class UpClose A B := up_close : A  B.
810
Hint Mode UpClose - ! : typeclass_instances.
811
Notation "↑ x" := (up_close x) (at level 20, format "↑ x").
812 813

(** * Monadic operations *)
814
(** We define operational type classes for the monadic operations bind, join 
815 816 817
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). *)
818
Class MRet (M : Type  Type) := mret:  {A}, A  M A.
819
Arguments mret {_ _ _} _ : assert.
820
Instance: Params (@mret) 3.
821
Class MBind (M : Type  Type) := mbind :  {A B}, (A  M B)  M A  M B.
822
Arguments mbind {_ _ _ _} _ !_ / : assert.
823
Instance: Params (@mbind) 4.
824
Class MJoin (M : Type  Type) := mjoin:  {A}, M (M A)  M A.
825
Arguments mjoin {_ _ _} !_ / : assert.
826
Instance: Params (@mjoin) 3.
827
Class FMap (M : Type  Type) := fmap :  {A B}, (A  B)  M A  M B.
828
Arguments fmap {_ _ _ _} _ !_ / : assert.
829
Instance: Params (@fmap) 4.
830
Class OMap (M : Type  Type) := omap:  {A B}, (A  option B)  M A  M B.
831
Arguments omap {_ _ _ _} _ !_ / : assert.
832
Instance: Params (@omap) 4.
833

834 835 836 837 838 839
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
840
  (at level 65, only parsing, right associativity) : C_scope.
841
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.
842
Notation "' ( x1 , x2 ) ← y ; z" :=
843
  (y = (λ x : _, let ' (x1, x2) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
844
  (at level 65, only parsing, right associativity) : C_scope.
845
Notation "' ( x1 , x2 , x3 ) ← y ; z" :=
846
  (y = (λ x : _, let ' (x1,x2,x3) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
847
  (at level 65, only parsing, right associativity) : C_scope.
848
Notation "' ( x1 , x2 , x3  , x4 ) ← y ; z" :=
849
  (y = (λ x : _, let ' (x1,x2,x3,x4) := x in z))
Robbert Krebbers's avatar
Robbert Krebbers committed
850
  (at level 65, only parsing, right associativity) : C_scope.
851 852
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
853
  (at level 65, only parsing, right associativity) : C_scope.
854 855
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
856
  (at level 65, only parsing, right associativity) : C_scope.
857

858 859 860 861 862
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").

863
Class MGuard (M : Type  Type) :=
864
  mguard:  P {dec : Decision P} {A}, (P  M A)  M A.
865
Arguments mguard _ _ _ !_ _ _ / : assert.
866
Notation "'guard' P ; o" := (mguard P (λ _, o))
Robbert Krebbers's avatar
Robbert Krebbers committed
867
  (at level 65, only parsing, right associativity) : C_scope.