base.v 58.8 KB
Newer Older
1
(* Copyright (c) 2012-2019, Coq-std++ developers. *)
2 3 4
(* 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
5
abstract interfaces for ordered structures, sets, and various other data
6
structures. *)
7

8
From Coq Require Export Morphisms RelationClasses List Bool Utf8 Setoid.
9
From Coq Require Import Permutation.
10
Set Default Proof Using "Type".
11 12
Export ListNotations.
From Coq.Program Require Export Basics Syntax.
13

Ralf Jung's avatar
Ralf Jung committed
14 15
(** * Enable implicit generalization. *)
(** This option enables implicit generalization in arguments of the form
16 17 18 19 20 21
   `{...} (i.e., anonymous arguments).  Unfortunately, it also enables
   implicit generalization in `Instance`.  We think that the fact taht both
   behaviors are coupled together is a [bug in
   Coq](https://github.com/coq/coq/issues/6030). *)
Global Generalizable All Variables.

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
(** * 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. *)
37
Obligation Tactic := idtac.
38 39

(** 3. Hide obligations from the results of the [Search] commands. *)
40
Add Search Blacklist "_obligation_".
Robbert Krebbers's avatar
Robbert Krebbers committed
41

42
(** * Sealing off definitions *)
Ralf Jung's avatar
Ralf Jung committed
43 44 45 46
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
47 48
Arguments unseal {_ _} _ : assert.
Arguments seal_eq {_ _} _ : assert.
49

50
(** * Non-backtracking type classes *)
51
(** The type class [TCNoBackTrack P] can be used to establish [P] without ever
52 53 54 55 56 57 58 59 60 61 62
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. *)
63 64
Class TCNoBackTrack (P : Prop) := { tc_no_backtrack : P }.
Hint Extern 0 (TCNoBackTrack _) => constructor; apply _ : typeclass_instances.
65

66 67
(* 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
68
establish [Q], i.e. does not have the behavior of a conditional. Furthermore,
69
note that [TCOr (TCAnd P Q) (TCAnd (TCNot P) R)] would not work; we generally
Robbert Krebbers's avatar
Robbert Krebbers committed
70
would not be able to prove the negation of [P]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
71
Inductive TCIf (P Q R : Prop) : Prop :=
72 73 74 75 76 77 78 79
  | 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.

80
(** * Typeclass opaque definitions *)
Ralf Jung's avatar
Ralf Jung committed
81
(** The constant [tc_opaque] is used to make definitions opaque for just type
82 83 84 85 86
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
87
(** Below we define type class versions of the common logical operators. It is
88 89 90 91 92 93 94 95 96 97 98 99 100 101 102
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. *)
103
Inductive TCOr (P1 P2 : Prop) : Prop :=
104 105 106 107 108
  | 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.
109
Hint Mode TCOr ! ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
110

111
Inductive TCAnd (P1 P2 : Prop) : Prop := TCAnd_intro : P1  P2  TCAnd P1 P2.
112 113
Existing Class TCAnd.
Existing Instance TCAnd_intro.
114
Hint Mode TCAnd ! ! : typeclass_instances.
115

116 117 118
Inductive TCTrue : Prop := TCTrue_intro : TCTrue.
Existing Class TCTrue.
Existing Instance TCTrue_intro.
119

120 121 122 123 124 125
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.
126
Hint Mode TCForall ! ! ! : typeclass_instances.
127

128 129 130
(** 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
131 132 133 134 135 136 137
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.
138 139
Hint Mode TCForall2 ! ! ! ! - : typeclass_instances.
Hint Mode TCForall2 ! ! ! - ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
140

141 142 143 144 145 146
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.
147
Hint Mode TCElemOf ! ! ! : typeclass_instances.
148

149 150 151
(** Similarly to [TCForall2], we declare the modes of [TCEq x y] in both
directions, i.e., either [x] input and [y] output, or [y] input and [x]
output. *)
152 153 154
Inductive TCEq {A} (x : A) : A  Prop := TCEq_refl : TCEq x x.
Existing Class TCEq.
Existing Instance TCEq_refl.
155 156
Hint Mode TCEq ! ! - : typeclass_instances.
Hint Mode TCEq ! - ! : typeclass_instances.
157

Robbert Krebbers's avatar
Robbert Krebbers committed
158 159 160 161
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.
162 163
Hint Mode TCDiag ! ! ! - : typeclass_instances.
Hint Mode TCDiag ! ! - ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
164

165 166 167 168 169 170
(** 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.

171
(** Throughout this development we use [stdpp_scope] for all general purpose
172
notations that do not belong to a more specific scope. *)
173 174
Delimit Scope stdpp_scope with stdpp.
Global Open Scope stdpp_scope.
175

176
(** Change [True] and [False] into notations in order to enable overloading.
177 178
We will use this to give [True] and [False] a different interpretation for
embedded logics. *)
179 180
Notation "'True'" := True (format "True") : type_scope.
Notation "'False'" := False (format "False") : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
181 182


183
(** * Equality *)
184
(** Introduce some Haskell style like notations. *)
185
Notation "(=)" := eq (only parsing) : stdpp_scope.
186 187
Notation "( x =.)" := (eq x) (only parsing) : stdpp_scope.
Notation "(.= x )" := (λ y, eq y x) (only parsing) : stdpp_scope.
188
Notation "(≠)" := (λ x y, x  y) (only parsing) : stdpp_scope.
189 190
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
191

192 193 194 195
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.
196 197
Notation "X ≠@{ A } Y":= (¬X =@{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.
198

Tej Chajed's avatar
Tej Chajed committed
199 200
Hint Extern 0 (_ = _) => reflexivity : core.
Hint Extern 100 (_  _) => discriminate : core.
Robbert Krebbers's avatar
Robbert Krebbers committed
201

202
Instance:  A, PreOrder (=@{A}).
203 204 205
Proof. split; repeat intro; congruence. Qed.

(** ** Setoid equality *)
Ralf Jung's avatar
Ralf Jung committed
206 207 208
(** 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). *)
209
Class Equiv A := equiv: relation A.
210 211 212
(* No Hint Mode set because of Coq bug #5735
Hint Mode Equiv ! : typeclass_instances. *)

213
Infix "≡" := equiv (at level 70, no associativity) : stdpp_scope.
214 215 216
Infix "≡@{ A }" := (@equiv A _)
  (at level 70, only parsing, no associativity) : stdpp_scope.

217
Notation "(≡)" := equiv (only parsing) : stdpp_scope.
218 219
Notation "( X ≡.)" := (equiv X) (only parsing) : stdpp_scope.
Notation "(.≡ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
220 221
Notation "(≢)" := (λ X Y, ¬X  Y) (only parsing) : stdpp_scope.
Notation "X ≢ Y":= (¬X  Y) (at level 70, no associativity) : stdpp_scope.
222 223
Notation "( X ≢.)" := (λ Y, X  Y) (only parsing) : stdpp_scope.
Notation "(.≢ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
224

225 226
Notation "(≡@{ A } )" := (@equiv A _) (only parsing) : stdpp_scope.
Notation "(≢@{ A } )" := (λ X Y, ¬X @{A} Y) (only parsing) : stdpp_scope.
227 228
Notation "X ≢@{ A } Y":= (¬X @{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.
229

230 231 232 233 234
(** 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.
235 236
Hint Mode LeibnizEquiv ! - : typeclass_instances.

237
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@{A})} (x y : A) :
238 239
  x  y  x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
240

241 242
Ltac fold_leibniz := repeat
  match goal with
243
  | H : context [ _ @{?A} _ ] |- _ =>
244
    setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
245
  | |- context [ _ @{?A} _ ] =>
246 247 248 249
    setoid_rewrite (leibniz_equiv_iff (A:=A))
  end.
Ltac unfold_leibniz := repeat
  match goal with
250
  | H : context [ _ =@{?A} _ ] |- _ =>
251
    setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
252
  | |- context [ _ =@{?A} _ ] =>
253 254 255 256 257 258 259 260
    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. *)
261
Instance: Params (@equiv) 2 := {}.
262 263 264 265

(** The following instance forces [setoid_replace] to use setoid equality
(for types that have an [Equiv] instance) rather than the standard Leibniz
equality. *)
266
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3 := {}.
Tej Chajed's avatar
Tej Chajed committed
267 268
Hint Extern 0 (_  _) => reflexivity : core.
Hint Extern 0 (_  _) => symmetry; assumption : core.
269 270 271 272 273


(** * Type classes *)
(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
274
propositions. *)
275
Class Decision (P : Prop) := decide : {P} + {¬P}.
276
Hint Mode Decision ! : typeclass_instances.
277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296
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.
297
Notation EqDecision A := (RelDecision (=@{A})).
298 299 300 301

(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
302
Hint Mode Inhabited ! : typeclass_instances.
303
Arguments populate {_} _ : assert.
304 305 306 307 308 309

(** ** 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.
310
Hint Mode ProofIrrel ! : typeclass_instances.
311 312 313

(** ** Common properties *)
(** These operational type classes allow us to refer to common mathematical
314 315
properties in a generic way. For example, for injectivity of [(k ++.)] it
allows us to write [inj (k ++.)] instead of [app_inv_head k]. *)
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
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}.
346 347 348 349 350

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.
351

352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367
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.
368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420

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.
421
Instance: Params (@strict) 2 := {}.
422 423 424 425 426 427 428 429 430 431
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
432 433
Instance prop_inhabited : Inhabited Prop := populate True.

434
Notation "(∧)" := and (only parsing) : stdpp_scope.
435 436
Notation "( A ∧.)" := (and A) (only parsing) : stdpp_scope.
Notation "(.∧ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
437

438
Notation "(∨)" := or (only parsing) : stdpp_scope.
439 440
Notation "( A ∨.)" := (or A) (only parsing) : stdpp_scope.
Notation "(.∨ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
441

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

Tej Chajed's avatar
Tej Chajed committed
446 447
Hint Extern 0 (_  _) => reflexivity : core.
Hint Extern 0 (_  _) => symmetry; assumption : core.
448 449 450 451 452 453 454 455 456 457 458

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.
459 460 461 462 463 464
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.
465

466
Instance: Comm () (=@{A}).
467
Proof. red; intuition. Qed.
468
Instance: Comm () (λ x y, y =@{A} x).
469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507
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 *)
508
Notation "(→)" := (λ A B, A  B) (only parsing) : stdpp_scope.
509 510
Notation "( A →.)" := (λ B, A  B) (only parsing) : stdpp_scope.
Notation "(.→ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
511

512
Notation "t $ r" := (t r)
513 514
  (at level 65, right associativity, only parsing) : stdpp_scope.
Notation "($)" := (λ f x, f x) (only parsing) : stdpp_scope.
515
Notation "(.$ x )" := (λ f, f x) (only parsing) : stdpp_scope.
516

517 518
Infix "∘" := compose : stdpp_scope.
Notation "(∘)" := compose (only parsing) : stdpp_scope.
519 520
Notation "( f ∘.)" := (compose f) (only parsing) : stdpp_scope.
Notation "(.∘ f )" := (λ g, compose g f) (only parsing) : stdpp_scope.
521

Robbert Krebbers's avatar
Robbert Krebbers committed
522 523 524
Instance impl_inhabited {A} `{Inhabited B} : Inhabited (A  B) :=
  populate (λ _, inhabitant).

525 526
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
527 528 529 530
Arguments id _ _ / : assert.
Arguments compose _ _ _ _ _ _ / : assert.
Arguments flip _ _ _ _ _ _ / : assert.
Arguments const _ _ _ _ / : assert.
531
Typeclasses Transparent id compose flip const.
532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578

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
579 580 581
Hint Unfold Is_true : core.
Hint Immediate Is_true_eq_left : core.
Hint Resolve orb_prop_intro andb_prop_intro : core.
582 583 584 585 586 587
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.
588

589 590 591 592 593
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.
594

595 596 597 598 599 600 601 602
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.
603

604 605
(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
606
Instance unit_equivalence : Equivalence (@{unit}).
607
Proof. repeat split. Qed.
608 609
Instance unit_leibniz : LeibnizEquiv unit.
Proof. intros [] []; reflexivity. Qed.
610
Instance unit_inhabited: Inhabited unit := populate ().
611

Ralf Jung's avatar
Ralf Jung committed
612 613 614 615 616 617 618
(** ** 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.

619
(** ** Products *)
620 621
Notation "( x ,.)" := (pair x) (only parsing) : stdpp_scope.
Notation "(., y )" := (λ x, (x,y)) (only parsing) : stdpp_scope.
622

623 624
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").
625

626 627 628
Instance: Params (@pair) 2 := {}.
Instance: Params (@fst) 2 := {}.
Instance: Params (@snd) 2 := {}.
629

630 631 632 633 634 635 636
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
637 638 639 640 641
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).

642 643
Definition prod_map {A A' B B'} (f: A  A') (g: B  B') (p : A * B) : A' * B' :=
  (f (p.1), g (p.2)).
644
Arguments prod_map {_ _ _ _} _ _ !_ / : assert.
645

646 647
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)).
648
Arguments prod_zip {_ _ _ _ _ _} _ _ !_ !_ / : assert.
649

650 651 652
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.
653

654 655 656 657 658 659 660 661
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.
662

663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678
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.
679

680 681
  Global Instance pair_proper' : Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
  Proof. firstorder eauto. Qed.
682 683
  Global Instance pair_inj' : Inj2 R1 R2 (prod_relation R1 R2) pair.
  Proof. inversion_clear 1; eauto. Qed.
684 685 686 687 688
  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
689

690 691
Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation () ().
Instance pair_proper `{Equiv A, Equiv B} :
692 693
  Proper (() ==> () ==> ()) (@pair A B) := _.
Instance pair_equiv_inj `{Equiv A, Equiv B} : Inj2 () () () (@pair A B) := _.
694 695 696
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.
697

Robbert Krebbers's avatar
Robbert Krebbers committed
698 699
Instance prod_leibniz `{LeibnizEquiv A, LeibnizEquiv B} : LeibnizEquiv (A * B).
Proof. intros [??] [??] [??]; f_equal; apply leibniz_equiv; auto. Qed.
700

701
(** ** Sums *)
702 703
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.
704
Arguments sum_map {_ _ _ _} _ _ !_ / : assert.
705

706
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
707
  match iA with populate x => populate (inl x) end.
708
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
709
  match iB with populate y => populate (inl y) end.
710

711 712 713 714
Instance inl_inj : Inj (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Instance inr_inj : Inj (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.
715

716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742
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.
743 744 745 746
  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.
747 748 749 750 751
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) := _.
752 753
Instance inl_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inl A B) := _.
Instance inr_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inr A B) := _.
754 755
Typeclasses Opaque sum_equiv.

756 757
(** ** Option *)
Instance option_inhabited {A} : Inhabited (option A) := populate None.
Robbert Krebbers's avatar
Robbert Krebbers committed
758

759
(** ** Sigma types *)
760 761 762
Arguments existT {_ _} _ _ : assert.
Arguments projT1 {_ _} _ : assert.
Arguments projT2 {_ _} _ : assert.
763

764 765 766
Arguments exist {_} _ _ _ : assert.
Arguments proj1_sig {_ _} _ : assert.
Arguments proj2_sig {_ _} _ : assert.
767 768
Notation "x ↾ p" := (exist _ x p) (at level 20) : stdpp_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : stdpp_scope.
769

770 771 772
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.
773

774 775 776 777 778 779 780 781 782 783
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.
784
Arguments sig_map _ _ _ _ _ _ !_ / : assert.
785

786 787 788 789
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
790

791
(** * Operations on sets *)
792
(** We define operational type classes for the traditional operations and
793
relations on sets: the empty set [∅], the union [(∪)],
794
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
795
[(⊆)] and element of [(∈)] relation, and disjointess [(##)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
796
Class Empty A := empty: A.
797
Hint Mode Empty ! : typeclass_instances.
798
Notation "∅" := empty (format "∅") : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
799

800 801
Instance empty_inhabited `(Empty A) : Inhabited A := populate .

Robbert Krebbers's avatar
Robbert Krebbers committed
802
Class Union A := union: A  A  A.
803
Hint Mode Union ! : typeclass_instances.
804
Instance: Params (@union) 2 := {}.
805 806
Infix "∪" := union (at level 50, left associativity) : stdpp_scope.
Notation "(∪)" := union (only parsing) : stdpp_scope.
807 808
Notation "( x ∪.)" := (union x) (only parsing) : stdpp_scope.
Notation "(.∪ x )" := (λ y, union y x) (only parsing) : stdpp_scope.
809 810
Infix "∪*" := (zip_with ()) (at level 50, left associativity) : stdpp_scope.
Notation "(∪*)" := (zip_with ()) (only parsing) : stdpp_scope.
811
Infix "∪**" := (zip_with (zip_with ()))
812
  (at level 50, left associativity) : stdpp_scope.
813
Infix "∪*∪**" := (zip_with (prod_zip () (*)))
814
  (at level 50, left associativity) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
815

816
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
817
Arguments union_list _ _ _ !_ / : assert.
818
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : stdpp_scope.
819

820 821 822 823 824
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.
825 826
Notation "( x ⊎.)" := (disj_union x) (only parsing) : stdpp_scope.
Notation "(.⊎ x )" := (λ y, disj_union y x) (only parsing) : stdpp_scope.
827

Robbert Krebbers's avatar
Robbert Krebbers committed
828
Class Intersection A := intersection: A  A  A.
829
Hint Mode Intersection ! : typeclass_instances.
830
Instance: Params (@intersection) 2 := {}.
831 832
Infix "∩" := intersection (at level 40) : stdpp_scope.
Notation "(∩)" := intersection (only parsing) : stdpp_scope.
833 834
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
835 836

Class Difference A := difference: A  A  A.
837
Hint Mode Difference ! : typeclass_instances.
838
Instance: Params (@difference) 2 := {}.
839 840
Infix "∖" := difference (at level 40, left associativity) : stdpp_scope.
Notation "(∖)" := difference (only parsing) : stdpp_scope.
841 842
Notation "( x ∖.)" := (difference x) (only parsing) : stdpp_scope.
Notation "(.∖ x )" := (λ y, difference y x) (only parsing) : stdpp_scope.
843 844
Infix "∖*" := (zip_with ()) (at level 40, left associativity) : stdpp_scope.
Notation "(∖*)" := (zip_with ()) (only parsing) : stdpp_scope.
845
Infix "∖**" := (zip_with (zip_with ()))
846
  (at level 40, left associativity) : stdpp_scope.
847
Infix "∖*∖**" := (zip_with (prod_zip () (*)))
848
  (at level 50, left associativity) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
849

850
Class Singleton A B := singleton: A  B.