base.v 59.2 KB
Newer Older
1 2
(** This file collects type class interfaces, notations, and general theorems
that are used throughout the whole development. Most importantly it contains
3
abstract interfaces for ordered structures, sets, and various other data
4
structures. *)
5

6
From Coq Require Export Morphisms RelationClasses List Bool Utf8 Setoid.
7
From Coq Require Import Permutation.
8
Set Default Proof Using "Type".
9 10
Export ListNotations.
From Coq.Program Require Export Basics Syntax.
11

Michael Sammler's avatar
Michael Sammler committed
12 13 14 15 16
(** This notation is necessary to prevent [length] from being printed
as [strings.length] if strings.v is imported and later base.v. See
also strings.v and
https://gitlab.mpi-sws.org/iris/stdpp/-/merge_requests/144 and
https://gitlab.mpi-sws.org/iris/stdpp/-/merge_requests/129. *)
17 18
Notation length := Datatypes.length.

Ralf Jung's avatar
Ralf Jung committed
19 20
(** * Enable implicit generalization. *)
(** This option enables implicit generalization in arguments of the form
Robbert Krebbers's avatar
Robbert Krebbers committed
21 22
   [`{...}] (i.e., anonymous arguments).  Unfortunately, it also enables
   implicit generalization in [Instance].  We think that the fact that both
23 24 25 26
   behaviors are coupled together is a [bug in
   Coq](https://github.com/coq/coq/issues/6030). *)
Global Generalizable All Variables.

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
(** * Tweak program *)
(** 1. Since we only use Program to solve logical side-conditions, they should
always be made Opaque, otherwise we end up with performance problems due to
Coq blindly unfolding them.

Note that in most cases we use [Next Obligation. (* ... *) Qed.], for which
this option does not matter. However, sometimes we write things like
[Solve Obligations with naive_solver (* ... *)], and then the obligations
should surely be opaque. *)
Global Unset Transparent Obligations.

(** 2. Do not let Program automatically simplify obligations. The default
obligation tactic is [Tactics.program_simpl], which, among other things,
introduces all variables and gives them fresh names. As such, it becomes
impossible to refer to hypotheses in a robust way. *)
42
Obligation Tactic := idtac.
43 44

(** 3. Hide obligations from the results of the [Search] commands. *)
45
Add Search Blacklist "_obligation_".
Robbert Krebbers's avatar
Robbert Krebbers committed
46

47
(** * Sealing off definitions *)
Ralf Jung's avatar
Ralf Jung committed
48 49 50 51
Section seal.
  Local Set Primitive Projections.
  Record seal {A} (f : A) := { unseal : A; seal_eq : unseal = f }.
End seal.
Ralf Jung's avatar
Ralf Jung committed
52 53
Arguments unseal {_ _} _ : assert.
Arguments seal_eq {_ _} _ : assert.
54

55
(** * Non-backtracking type classes *)
56
(** The type class [TCNoBackTrack P] can be used to establish [P] without ever
57 58 59 60 61 62 63 64 65 66 67
backtracking on the instance of [P] that has been found. Backtracking may
normally happen when [P] contains evars that could be instanciated in different
ways depending on which instance is picked, and type class search somewhere else
depends on this evar.

The proper way of handling this would be by setting Coq's option
`Typeclasses Unique Instances`. However, this option seems to be broken, see Coq
issue #6714.

See https://gitlab.mpi-sws.org/FP/iris-coq/merge_requests/112 for a rationale
of this type class. *)
68 69
Class TCNoBackTrack (P : Prop) := { tc_no_backtrack : P }.
Hint Extern 0 (TCNoBackTrack _) => constructor; apply _ : typeclass_instances.
70

71 72
(* A conditional at the type class level. Note that [TCIf P Q R] is not the same
as [TCOr (TCAnd P Q) R]: the latter will backtrack to [R] if it fails to
Paolo G. Giarrusso's avatar
Paolo G. Giarrusso committed
73
establish [Q], i.e. does not have the behavior of a conditional. Furthermore,
74
note that [TCOr (TCAnd P Q) (TCAnd (TCNot P) R)] would not work; we generally
Robbert Krebbers's avatar
Robbert Krebbers committed
75
would not be able to prove the negation of [P]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
76
Inductive TCIf (P Q R : Prop) : Prop :=
77 78 79 80 81 82 83 84
  | TCIf_true : P  Q  TCIf P Q R
  | TCIf_false : R  TCIf P Q R.
Existing Class TCIf.

Hint Extern 0 (TCIf _ _ _) =>
  first [apply TCIf_true; [apply _|]
        |apply TCIf_false] : typeclass_instances.

85
(** * Typeclass opaque definitions *)
Ralf Jung's avatar
Ralf Jung committed
86
(** The constant [tc_opaque] is used to make definitions opaque for just type
87 88 89 90 91
class search. Note that [simpl] is set up to always unfold [tc_opaque]. *)
Definition tc_opaque {A} (x : A) : A := x.
Typeclasses Opaque tc_opaque.
Arguments tc_opaque {_} _ /.

Ralf Jung's avatar
Ralf Jung committed
92
(** Below we define type class versions of the common logical operators. It is
93 94 95 96 97 98 99 100 101 102 103 104 105 106 107
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. *)
108
Inductive TCOr (P1 P2 : Prop) : Prop :=
109 110 111 112 113
  | 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.
114
Hint Mode TCOr ! ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
115

116
Inductive TCAnd (P1 P2 : Prop) : Prop := TCAnd_intro : P1  P2  TCAnd P1 P2.
117 118
Existing Class TCAnd.
Existing Instance TCAnd_intro.
119
Hint Mode TCAnd ! ! : typeclass_instances.
120

121 122 123
Inductive TCTrue : Prop := TCTrue_intro : TCTrue.
Existing Class TCTrue.
Existing Instance TCTrue_intro.
124

125 126 127 128 129 130
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.
131
Hint Mode TCForall ! ! ! : typeclass_instances.
132

133 134 135
(** The class [TCForall2 P l k] is commonly used to transform an input list [l]
into an output list [k], or the converse. Therefore there are two modes, either
[l] input and [k] output, or [k] input and [l] input. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
136 137 138 139 140 141 142
Inductive TCForall2 {A B} (P : A  B  Prop) : list A  list B  Prop :=
  | TCForall2_nil : TCForall2 P [] []
  | TCForall2_cons x y xs ys :
     P x y  TCForall2 P xs ys  TCForall2 P (x :: xs) (y :: ys).
Existing Class TCForall2.
Existing Instance TCForall2_nil.
Existing Instance TCForall2_cons.
143 144
Hint Mode TCForall2 ! ! ! ! - : typeclass_instances.
Hint Mode TCForall2 ! ! ! - ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
145

146 147 148 149 150 151
Inductive TCElemOf {A} (x : A) : list A  Prop :=
  | TCElemOf_here xs : TCElemOf x (x :: xs)
  | TCElemOf_further y xs : TCElemOf x xs  TCElemOf x (y :: xs).
Existing Class TCElemOf.
Existing Instance TCElemOf_here.
Existing Instance TCElemOf_further.
152
Hint Mode TCElemOf ! ! ! : typeclass_instances.
153

Robbert Krebbers's avatar
Robbert Krebbers committed
154 155 156 157
(** We declare both arguments [x] and [y] of [TCEq x y] as outputs, which means
[TCEq] can also be used to unify evars. This is harmless: since the only
instance of [TCEq] is [TCEq_refl] below, it can never cause loops. See
https://gitlab.mpi-sws.org/iris/iris/merge_requests/391 for a use case. *)
158 159 160
Inductive TCEq {A} (x : A) : A  Prop := TCEq_refl : TCEq x x.
Existing Class TCEq.
Existing Instance TCEq_refl.
Robbert Krebbers's avatar
Robbert Krebbers committed
161
Hint Mode TCEq ! - - : typeclass_instances.
162

Michael Sammler's avatar
Michael Sammler committed
163 164 165
Lemma TCEq_eq {A} (x1 x2 : A) : TCEq x1 x2  x1 = x2.
Proof. split; destruct 1; reflexivity. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
166 167 168 169
Inductive TCDiag {A} (C : A  Prop) : A  A  Prop :=
  | TCDiag_diag x : C x  TCDiag C x x.
Existing Class TCDiag.
Existing Instance TCDiag_diag.
170 171
Hint Mode TCDiag ! ! ! - : typeclass_instances.
Hint Mode TCDiag ! ! - ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
172

173 174 175 176 177 178
(** Given a proposition [P] that is a type class, [tc_to_bool P] will return
[true] iff there is an instance of [P]. It is often useful in Ltac programming,
where one can do [lazymatch tc_to_bool P with true => .. | false => .. end]. *)
Definition tc_to_bool (P : Prop)
  {p : bool} `{TCIf P (TCEq p true) (TCEq p false)} : bool := p.

179
(** Throughout this development we use [stdpp_scope] for all general purpose
180
notations that do not belong to a more specific scope. *)
181 182
Delimit Scope stdpp_scope with stdpp.
Global Open Scope stdpp_scope.
183

184
(** Change [True] and [False] into notations in order to enable overloading.
185 186
We will use this to give [True] and [False] a different interpretation for
embedded logics. *)
187 188
Notation "'True'" := True (format "True") : type_scope.
Notation "'False'" := False (format "False") : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
189 190


191
(** * Equality *)
192
(** Introduce some Haskell style like notations. *)
193
Notation "(=)" := eq (only parsing) : stdpp_scope.
194 195
Notation "( x =.)" := (eq x) (only parsing) : stdpp_scope.
Notation "(.= x )" := (λ y, eq y x) (only parsing) : stdpp_scope.
196
Notation "(≠)" := (λ x y, x  y) (only parsing) : stdpp_scope.
197 198
Notation "( x ≠.)" := (λ y, x  y) (only parsing) : stdpp_scope.
Notation "(.≠ x )" := (λ y, y  x) (only parsing) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
199

200 201 202 203
Infix "=@{ A }" := (@eq A)
  (at level 70, only parsing, no associativity) : stdpp_scope.
Notation "(=@{ A } )" := (@eq A) (only parsing) : stdpp_scope.
Notation "(≠@{ A } )" := (λ X Y, ¬X =@{A} Y) (only parsing) : stdpp_scope.
204 205
Notation "X ≠@{ A } Y":= (¬X =@{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.
206

Tej Chajed's avatar
Tej Chajed committed
207 208
Hint Extern 0 (_ = _) => reflexivity : core.
Hint Extern 100 (_  _) => discriminate : core.
Robbert Krebbers's avatar
Robbert Krebbers committed
209

210
Instance:  A, PreOrder (=@{A}).
211 212 213
Proof. split; repeat intro; congruence. Qed.

(** ** Setoid equality *)
Ralf Jung's avatar
Ralf Jung committed
214 215 216
(** We define an operational type class for setoid equality, i.e., the
"canonical" equivalence for a type. The typeclass is tied to the \equiv
symbol. This is based on (Spitters/van der Weegen, 2011). *)
217
Class Equiv A := equiv: relation A.
218 219 220
(* No Hint Mode set because of Coq bug #5735
Hint Mode Equiv ! : typeclass_instances. *)

221
Infix "≡" := equiv (at level 70, no associativity) : stdpp_scope.
222 223 224
Infix "≡@{ A }" := (@equiv A _)
  (at level 70, only parsing, no associativity) : stdpp_scope.

225
Notation "(≡)" := equiv (only parsing) : stdpp_scope.
226 227
Notation "( X ≡.)" := (equiv X) (only parsing) : stdpp_scope.
Notation "(.≡ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
228 229
Notation "(≢)" := (λ X Y, ¬X  Y) (only parsing) : stdpp_scope.
Notation "X ≢ Y":= (¬X  Y) (at level 70, no associativity) : stdpp_scope.
230 231
Notation "( X ≢.)" := (λ Y, X  Y) (only parsing) : stdpp_scope.
Notation "(.≢ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
232

233 234
Notation "(≡@{ A } )" := (@equiv A _) (only parsing) : stdpp_scope.
Notation "(≢@{ A } )" := (λ X Y, ¬X @{A} Y) (only parsing) : stdpp_scope.
235 236
Notation "X ≢@{ A } Y":= (¬X @{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.
237

238 239 240 241 242
(** 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.
243 244
Hint Mode LeibnizEquiv ! - : typeclass_instances.

245
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@{A})} (x y : A) :
246 247
  x  y  x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
248

249 250
Ltac fold_leibniz := repeat
  match goal with
251
  | H : context [ _ @{?A} _ ] |- _ =>
252
    setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
253
  | |- context [ _ @{?A} _ ] =>
254 255 256 257
    setoid_rewrite (leibniz_equiv_iff (A:=A))
  end.
Ltac unfold_leibniz := repeat
  match goal with
258
  | H : context [ _ =@{?A} _ ] |- _ =>
259
    setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
260
  | |- context [ _ =@{?A} _ ] =>
261 262 263 264 265 266 267 268
    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. *)
269
Instance: Params (@equiv) 2 := {}.
270 271 272 273

(** The following instance forces [setoid_replace] to use setoid equality
(for types that have an [Equiv] instance) rather than the standard Leibniz
equality. *)
274
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3 := {}.
Tej Chajed's avatar
Tej Chajed committed
275 276
Hint Extern 0 (_  _) => reflexivity : core.
Hint Extern 0 (_  _) => symmetry; assumption : core.
277 278 279 280 281


(** * Type classes *)
(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
282
propositions. *)
283
Class Decision (P : Prop) := decide : {P} + {¬P}.
284
Hint Mode Decision ! : typeclass_instances.
285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304
Arguments decide _ {_} : simpl never, assert.

(** Although [RelDecision R] is just [∀ x y, Decision (R x y)], we make this
an explicit class instead of a notation for two reasons:

- It allows us to control [Hint Mode] more precisely. In particular, if it were
  defined as a notation, the above [Hint Mode] for [Decision] would not prevent
  diverging instance search when looking for [RelDecision (@eq ?A)], which would
  result in it looking for [Decision (@eq ?A x y)], i.e. an instance where the
  head position of [Decision] is not en evar.
- We use it to avoid inefficient computation due to eager evaluation of
  propositions by [vm_compute]. This inefficiency arises for example if
  [(x = y) := (f x = f y)]. Since [decide (x = y)] evaluates to
  [decide (f x = f y)], this would then lead to evaluation of [f x] and [f y].
  Using the [RelDecision], the [f] is hidden under a lambda, which prevents
  unnecessary evaluation. *)
Class RelDecision {A B} (R : A  B  Prop) :=
  decide_rel x y :> Decision (R x y).
Hint Mode RelDecision ! ! ! : typeclass_instances.
Arguments decide_rel {_ _} _ {_} _ _ : simpl never, assert.
305
Notation EqDecision A := (RelDecision (=@{A})).
306 307 308 309

(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
310
Hint Mode Inhabited ! : typeclass_instances.
311
Arguments populate {_} _ : assert.
312 313 314 315 316 317

(** ** 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.
318
Hint Mode ProofIrrel ! : typeclass_instances.
319 320 321

(** ** Common properties *)
(** These operational type classes allow us to refer to common mathematical
322 323
properties in a generic way. For example, for injectivity of [(k ++.)] it
allows us to write [inj (k ++.)] instead of [app_inv_head k]. *)
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
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}.
354 355 356 357 358

Notation Involutive R f := (Cancel R f f).
Lemma involutive {A} {R : relation A} (f : A  A) `{Involutive R f} x :
  R (f (f x)) x.
Proof. auto. Qed.
359

360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375
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.
376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428

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.
429
Instance: Params (@strict) 2 := {}.
430 431 432 433 434 435 436 437 438 439
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 *)
Robbert Krebbers's avatar
Robbert Krebbers committed
440 441
Instance prop_inhabited : Inhabited Prop := populate True.

442
Notation "(∧)" := and (only parsing) : stdpp_scope.
443 444
Notation "( A ∧.)" := (and A) (only parsing) : stdpp_scope.
Notation "(.∧ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
445

446
Notation "(∨)" := or (only parsing) : stdpp_scope.
447 448
Notation "( A ∨.)" := (or A) (only parsing) : stdpp_scope.
Notation "(.∨ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
449

450
Notation "(↔)" := iff (only parsing) : stdpp_scope.
451 452
Notation "( A ↔.)" := (iff A) (only parsing) : stdpp_scope.
Notation "(.↔ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
453

Tej Chajed's avatar
Tej Chajed committed
454 455
Hint Extern 0 (_  _) => reflexivity : core.
Hint Extern 0 (_  _) => symmetry; assumption : core.
456 457 458 459 460 461 462 463 464 465 466

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.
467 468 469 470 471 472
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.
473

474
Instance: Comm () (=@{A}).
475
Proof. red; intuition. Qed.
476
Instance: Comm () (λ x y, y =@{A} x).
477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515
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 *)
516
Notation "(→)" := (λ A B, A  B) (only parsing) : stdpp_scope.
517 518
Notation "( A →.)" := (λ B, A  B) (only parsing) : stdpp_scope.
Notation "(.→ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
519

520
Notation "t $ r" := (t r)
521 522
  (at level 65, right associativity, only parsing) : stdpp_scope.
Notation "($)" := (λ f x, f x) (only parsing) : stdpp_scope.
523
Notation "(.$ x )" := (λ f, f x) (only parsing) : stdpp_scope.
524

525 526
Infix "∘" := compose : stdpp_scope.
Notation "(∘)" := compose (only parsing) : stdpp_scope.
527 528
Notation "( f ∘.)" := (compose f) (only parsing) : stdpp_scope.
Notation "(.∘ f )" := (λ g, compose g f) (only parsing) : stdpp_scope.
529

Robbert Krebbers's avatar
Robbert Krebbers committed
530 531 532
Instance impl_inhabited {A} `{Inhabited B} : Inhabited (A  B) :=
  populate (λ _, inhabitant).

533 534
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
535 536 537 538
Arguments id _ _ / : assert.
Arguments compose _ _ _ _ _ _ / : assert.
Arguments flip _ _ _ _ _ _ / : assert.
Arguments const _ _ _ _ / : assert.
539
Typeclasses Transparent id compose flip const.
540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586

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.
Tej Chajed's avatar
Tej Chajed committed
587 588 589
Hint Unfold Is_true : core.
Hint Immediate Is_true_eq_left : core.
Hint Resolve orb_prop_intro andb_prop_intro : core.
590 591 592 593 594 595
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.
596

597 598 599 600 601
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.
602

603 604 605 606 607 608 609 610
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.
611

612 613
(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
614
Instance unit_equivalence : Equivalence (@{unit}).
615
Proof. repeat split. Qed.
616 617
Instance unit_leibniz : LeibnizEquiv unit.
Proof. intros [] []; reflexivity. Qed.
618
Instance unit_inhabited: Inhabited unit := populate ().
619

Ralf Jung's avatar
Ralf Jung committed
620 621 622 623 624 625 626
(** ** Empty *)
Instance Empty_set_equiv : Equiv Empty_set := λ _ _, True.
Instance Empty_set_equivalence : Equivalence (@{Empty_set}).
Proof. repeat split. Qed.
Instance Empty_set_leibniz : LeibnizEquiv Empty_set.
Proof. intros [] []; reflexivity. Qed.

627
(** ** Products *)
628 629
Notation "( x ,.)" := (pair x) (only parsing) : stdpp_scope.
Notation "(., y )" := (λ x, (x,y)) (only parsing) : stdpp_scope.
630

631 632
Notation "p .1" := (fst p) (at level 2, left associativity, format "p .1").
Notation "p .2" := (snd p) (at level 2, left associativity, format "p .2").
633

634 635 636
Instance: Params (@pair) 2 := {}.
Instance: Params (@fst) 2 := {}.
Instance: Params (@snd) 2 := {}.
637

638 639 640 641 642 643 644
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.

Robbert Krebbers's avatar
Robbert Krebbers committed
645 646 647 648 649
Definition uncurry3 {A B C D} (f : A * B * C  D) (a : A) (b : B) (c : C) : D :=
  f (a, b, c).
Definition uncurry4 {A B C D E} (f : A * B * C * D  E)
  (a : A) (b : B) (c : C) (d : D) : E := f (a, b, c, d).

650 651
Definition prod_map {A A' B B'} (f: A  A') (g: B  B') (p : A * B) : A' * B' :=
  (f (p.1), g (p.2)).
652
Arguments prod_map {_ _ _ _} _ _ !_ / : assert.
653

654 655
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)).
656
Arguments prod_zip {_ _ _ _ _ _} _ _ !_ !_ / : assert.
657

658 659 660
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.
661

662 663 664 665 666 667 668 669
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.
670

671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686
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.
687

688 689
  Global Instance pair_proper' : Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
  Proof. firstorder eauto. Qed.
690 691
  Global Instance pair_inj' : Inj2 R1 R2 (prod_relation R1 R2) pair.
  Proof. inversion_clear 1; eauto. Qed.
692 693 694 695 696
  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
697

698 699
Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation () ().
Instance pair_proper `{Equiv A, Equiv B} :
700 701
  Proper (() ==> () ==> ()) (@pair A B) := _.
Instance pair_equiv_inj `{Equiv A, Equiv B} : Inj2 () () () (@pair A B) := _.
702 703 704
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.
705

Robbert Krebbers's avatar
Robbert Krebbers committed
706 707
Instance prod_leibniz `{LeibnizEquiv A, LeibnizEquiv B} : LeibnizEquiv (A * B).
Proof. intros [??] [??] [??]; f_equal; apply leibniz_equiv; auto. Qed.
708

709
(** ** Sums *)
710 711
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.
712
Arguments sum_map {_ _ _ _} _ _ !_ / : assert.
713

714
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
715
  match iA with populate x => populate (inl x) end.
716
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
717
  match iB with populate y => populate (inl y) end.
718

719 720 721 722
Instance inl_inj : Inj (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Instance inr_inj : Inj (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.
723

724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750
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.
751 752 753 754
  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.
755 756 757 758 759
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) := _.
760 761
Instance inl_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inl A B) := _.
Instance inr_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inr A B) := _.
762 763
Typeclasses Opaque sum_equiv.

764 765
(** ** Option *)
Instance option_inhabited {A} : Inhabited (option A) := populate None.
Robbert Krebbers's avatar
Robbert Krebbers committed
766

767
(** ** Sigma types *)
768 769 770
Arguments existT {_ _} _ _ : assert.
Arguments projT1 {_ _} _ : assert.
Arguments projT2 {_ _} _ : assert.
771

772 773 774
Arguments exist {_} _ _ _ : assert.
Arguments proj1_sig {_ _} _ : assert.
Arguments proj2_sig {_ _} _ : assert.
775 776
Notation "x ↾ p" := (exist _ x p) (at level 20) : stdpp_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : stdpp_scope.
777

778 779 780
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.
781

782 783 784 785 786 787 788 789 790 791
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.
792
Arguments sig_map _ _ _ _ _ _ !_ / : assert.
793

794 795 796 797
Definition proj1_ex {P : Prop} {Q : P  Prop} (p :  x, Q x) : P :=
  let '(ex_intro _ x _) := p in x.
Definition proj2_ex {P : Prop} {Q : P  Prop} (p :  x, Q x) : Q (proj1_ex p) :=
  let '(ex_intro _ x H) := p in H.
Robbert Krebbers's avatar
Robbert Krebbers committed
798

799
(** * Operations on sets *)
800
(** We define operational type classes for the traditional operations and
801
relations on sets: the empty set [∅], the union [(∪)],
802
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
803
[(⊆)] and element of [(∈)] relation, and disjointess [(##)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
804
Class Empty A := empty: A.
805
Hint Mode Empty ! : typeclass_instances.
806
Notation "∅" := empty (format "∅") : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
807

808 809
Instance empty_inhabited `(Empty A) : Inhabited A := populate .

Robbert Krebbers's avatar
Robbert Krebbers committed
810
Class Union A := union: A  A  A.
811
Hint Mode Union ! : typeclass_instances.
812
Instance: Params (@union) 2 := {}.
813 814
Infix "∪" := union (at level 50, left associativity) : stdpp_scope.
Notation "(∪)" := union (only parsing) : stdpp_scope.
815 816
Notation "( x ∪.)" := (union x) (only parsing) : stdpp_scope.
Notation "(.∪ x )" := (λ y, union y x) (only parsing) : stdpp_scope.
817 818
Infix "∪*" := (zip_with ()) (at level 50, left associativity) : stdpp_scope.
Notation "(∪*)" := (zip_with ()) (only parsing) : stdpp_scope.
819
Infix "∪**" := (zip_with (zip_with ()))
820
  (at level 50, left associativity) : stdpp_scope.
821
Infix "∪*∪**" := (zip_with (prod_zip () (*)))
822
  (at level 50, left associativity) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
823

824
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
825
Arguments union_list _ _ _ !_ / : assert.
826
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : stdpp_scope.
827

828 829 830 831 832
Class DisjUnion A := disj_union: A  A  A.
Hint Mode DisjUnion ! : typeclass_instances.
Instance: Params (@disj_union) 2 := {}.
Infix "⊎" := disj_union (at level 50, left associativity) : stdpp_scope.
Notation "(⊎)" := disj_union (only parsing) : stdpp_scope.
833 834
Notation "( x ⊎.)" := (disj_union x) (only parsing) : stdpp_scope.
Notation "(.⊎ x )" := (λ y, disj_union y x) (only parsing) : stdpp_scope.
835

Robbert Krebbers's avatar
Robbert Krebbers committed
836
Class Intersection A := intersection: A  A  A.
837
Hint Mode Intersection ! : typeclass_instances.
838
Instance: Params (@intersection) 2 := {}.
839 840
Infix "∩" := intersection (at level 40) : stdpp_scope.
Notation "(∩)" := intersection (only parsing) : stdpp_scope.
841 842
Notation "( x ∩.)" := (intersection x) (only parsing) : stdpp_scope.
Notation "(.∩ x )" := (λ y, intersection y x) (only parsing) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
843 844

Class Difference A := difference: A  A  A.
845
Hint Mode Difference ! : typeclass_instances.
846
Instance: Params (@difference) 2 := {}.
847 848
Infix "∖" := difference (at level 40, left associativity) : stdpp_scope.
Notation "(∖)" := difference (only parsing) : stdpp_scope.
849 850
Notation "( x ∖.)" := (difference x) (only parsing) : stdpp_scope.
Notation "(.∖ x )" := (λ y, difference y x) (only parsing) : stdpp_scope.
851 852
Infix "∖*" := (zip_with ()) (at level 40, left associativity) : stdpp_scope.
Notation "(∖*)" := (zip_with ()) (only parsing) : stdpp_scope.
853
Infix "∖**" := (zip_with (zip_with ()))
854
  (at level 40, left associativity) : stdpp_scope.