base.v 56.9 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 *)
43 44 45 46
Section seal.
  Local Set Primitive Projections.
  Record seal {A} (f : A) := { unseal : A; seal_eq : unseal = f }.
End seal.
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.
Robbert Krebbers's avatar
Robbert Krebbers committed
109

110
Inductive TCAnd (P1 P2 : Prop) : Prop := TCAnd_intro : P1  P2  TCAnd P1 P2.
111 112
Existing Class TCAnd.
Existing Instance TCAnd_intro.
113

114 115 116
Inductive TCTrue : Prop := TCTrue_intro : TCTrue.
Existing Class TCTrue.
Existing Instance TCTrue_intro.
117

118 119 120 121 122 123 124
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.

Robbert Krebbers's avatar
Robbert Krebbers committed
125 126 127 128 129 130 131 132
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.

133 134 135 136 137 138 139
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.

140 141 142 143
Inductive TCEq {A} (x : A) : A  Prop := TCEq_refl : TCEq x x.
Existing Class TCEq.
Existing Instance TCEq_refl.

144 145 146 147 148
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.

149 150 151 152 153 154
(** 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.

155
(** Throughout this development we use [stdpp_scope] for all general purpose
156
notations that do not belong to a more specific scope. *)
157 158
Delimit Scope stdpp_scope with stdpp.
Global Open Scope stdpp_scope.
159

160
(** Change [True] and [False] into notations in order to enable overloading.
161 162
We will use this to give [True] and [False] a different interpretation for
embedded logics. *)
163 164
Notation "'True'" := True (format "True") : type_scope.
Notation "'False'" := False (format "False") : type_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
165 166


167
(** * Equality *)
168
(** Introduce some Haskell style like notations. *)
169
Notation "(=)" := eq (only parsing) : stdpp_scope.
170 171
Notation "( x =.)" := (eq x) (only parsing) : stdpp_scope.
Notation "(.= x )" := (λ y, eq y x) (only parsing) : stdpp_scope.
172
Notation "(≠)" := (λ x y, x  y) (only parsing) : stdpp_scope.
173 174
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
175

176 177 178 179
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.
180 181
Notation "X ≠@{ A } Y":= (¬X =@{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.
182

183 184
Hint Extern 0 (_ = _) => reflexivity : core.
Hint Extern 100 (_  _) => discriminate : core.
Robbert Krebbers's avatar
Robbert Krebbers committed
185

186
Instance:  A, PreOrder (=@{A}).
187 188 189
Proof. split; repeat intro; congruence. Qed.

(** ** Setoid equality *)
Ralf Jung's avatar
Ralf Jung committed
190 191 192
(** 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). *)
193
Class Equiv A := equiv: relation A.
194 195 196
(* No Hint Mode set because of Coq bug #5735
Hint Mode Equiv ! : typeclass_instances. *)

197
Infix "≡" := equiv (at level 70, no associativity) : stdpp_scope.
198 199 200
Infix "≡@{ A }" := (@equiv A _)
  (at level 70, only parsing, no associativity) : stdpp_scope.

201
Notation "(≡)" := equiv (only parsing) : stdpp_scope.
202 203
Notation "( X ≡.)" := (equiv X) (only parsing) : stdpp_scope.
Notation "(.≡ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
204 205
Notation "(≢)" := (λ X Y, ¬X  Y) (only parsing) : stdpp_scope.
Notation "X ≢ Y":= (¬X  Y) (at level 70, no associativity) : stdpp_scope.
206 207
Notation "( X ≢.)" := (λ Y, X  Y) (only parsing) : stdpp_scope.
Notation "(.≢ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
208

209 210
Notation "(≡@{ A } )" := (@equiv A _) (only parsing) : stdpp_scope.
Notation "(≢@{ A } )" := (λ X Y, ¬X @{A} Y) (only parsing) : stdpp_scope.
211 212
Notation "X ≢@{ A } Y":= (¬X @{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.
213

214 215 216 217 218
(** 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.
219 220
Hint Mode LeibnizEquiv ! - : typeclass_instances.

221
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@{A})} (x y : A) :
222 223
  x  y  x = y.
Proof. split. apply leibniz_equiv. intros ->; reflexivity. Qed.
224

225 226
Ltac fold_leibniz := repeat
  match goal with
227
  | H : context [ _ @{?A} _ ] |- _ =>
228
    setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
229
  | |- context [ _ @{?A} _ ] =>
230 231 232 233
    setoid_rewrite (leibniz_equiv_iff (A:=A))
  end.
Ltac unfold_leibniz := repeat
  match goal with
234
  | H : context [ _ =@{?A} _ ] |- _ =>
235
    setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
236
  | |- context [ _ =@{?A} _ ] =>
237 238 239 240 241 242 243 244
    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. *)
245
Instance: Params (@equiv) 2 := {}.
246 247 248 249

(** The following instance forces [setoid_replace] to use setoid equality
(for types that have an [Equiv] instance) rather than the standard Leibniz
equality. *)
250
Instance equiv_default_relation `{Equiv A} : DefaultRelation () | 3 := {}.
251 252
Hint Extern 0 (_  _) => reflexivity : core.
Hint Extern 0 (_  _) => symmetry; assumption : core.
253 254 255 256 257


(** * Type classes *)
(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
258
propositions. *)
259
Class Decision (P : Prop) := decide : {P} + {¬P}.
260
Hint Mode Decision ! : typeclass_instances.
261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280
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.
281
Notation EqDecision A := (RelDecision (=@{A})).
282 283 284 285

(** ** Inhabited types *)
(** This type class collects types that are inhabited. *)
Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
286
Hint Mode Inhabited ! : typeclass_instances.
287
Arguments populate {_} _ : assert.
288 289 290 291 292 293

(** ** 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.
294
Hint Mode ProofIrrel ! : typeclass_instances.
295 296 297

(** ** Common properties *)
(** These operational type classes allow us to refer to common mathematical
298 299
properties in a generic way. For example, for injectivity of [(k ++.)] it
allows us to write [inj (k ++.)] instead of [app_inv_head k]. *)
300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329
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}.
330 331 332 333 334

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.
335

336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351
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.
352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404

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.
405
Instance: Params (@strict) 2 := {}.
406 407 408 409 410 411 412 413 414 415
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 *)
416
Notation "(∧)" := and (only parsing) : stdpp_scope.
417 418
Notation "( A ∧.)" := (and A) (only parsing) : stdpp_scope.
Notation "(.∧ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
419

420
Notation "(∨)" := or (only parsing) : stdpp_scope.
421 422
Notation "( A ∨.)" := (or A) (only parsing) : stdpp_scope.
Notation "(.∨ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
423

424
Notation "(↔)" := iff (only parsing) : stdpp_scope.
425 426
Notation "( A ↔.)" := (iff A) (only parsing) : stdpp_scope.
Notation "(.↔ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
427

428 429
Hint Extern 0 (_  _) => reflexivity : core.
Hint Extern 0 (_  _) => symmetry; assumption : core.
430 431 432 433 434 435 436 437 438 439 440

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.
441 442 443 444 445 446
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.
447

448
Instance: Comm () (=@{A}).
449
Proof. red; intuition. Qed.
450
Instance: Comm () (λ x y, y =@{A} x).
451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489
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 *)
490
Notation "(→)" := (λ A B, A  B) (only parsing) : stdpp_scope.
491 492
Notation "( A →.)" := (λ B, A  B) (only parsing) : stdpp_scope.
Notation "(.→ B )" := (λ A, A  B) (only parsing) : stdpp_scope.
493

494
Notation "t $ r" := (t r)
495 496
  (at level 65, right associativity, only parsing) : stdpp_scope.
Notation "($)" := (λ f x, f x) (only parsing) : stdpp_scope.
497
Notation "(.$ x )" := (λ f, f x) (only parsing) : stdpp_scope.
498

499 500
Infix "∘" := compose : stdpp_scope.
Notation "(∘)" := compose (only parsing) : stdpp_scope.
501 502
Notation "( f ∘.)" := (compose f) (only parsing) : stdpp_scope.
Notation "(.∘ f )" := (λ g, compose g f) (only parsing) : stdpp_scope.
503

504 505 506
Instance impl_inhabited {A} `{Inhabited B} : Inhabited (A  B) :=
  populate (λ _, inhabitant).

507 508
(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
509 510 511 512
Arguments id _ _ / : assert.
Arguments compose _ _ _ _ _ _ / : assert.
Arguments flip _ _ _ _ _ _ / : assert.
Arguments const _ _ _ _ / : assert.
513
Typeclasses Transparent id compose flip const.
514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560

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.
561 562 563
Hint Unfold Is_true : core.
Hint Immediate Is_true_eq_left : core.
Hint Resolve orb_prop_intro andb_prop_intro : core.
564 565 566 567 568 569
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.
570

571 572 573 574 575
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.
576

577 578 579 580 581 582 583 584
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.
585

586 587
(** ** Unit *)
Instance unit_equiv : Equiv unit := λ _ _, True.
588
Instance unit_equivalence : Equivalence (@{unit}).
589
Proof. repeat split. Qed.
590 591
Instance unit_leibniz : LeibnizEquiv unit.
Proof. intros [] []; reflexivity. Qed.
592
Instance unit_inhabited: Inhabited unit := populate ().
593

594 595 596 597 598 599 600
(** ** 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.

601
(** ** Products *)
602 603
Notation "( x ,.)" := (pair x) (only parsing) : stdpp_scope.
Notation "(., y )" := (λ x, (x,y)) (only parsing) : stdpp_scope.
604

605 606
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").
607

608 609 610
Instance: Params (@pair) 2 := {}.
Instance: Params (@fst) 2 := {}.
Instance: Params (@snd) 2 := {}.
611

612 613 614 615 616 617 618
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.

619 620 621 622 623
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).

624 625
Definition prod_map {A A' B B'} (f: A  A') (g: B  B') (p : A * B) : A' * B' :=
  (f (p.1), g (p.2)).
626
Arguments prod_map {_ _ _ _} _ _ !_ / : assert.
627

628 629
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)).
630
Arguments prod_zip {_ _ _ _ _ _} _ _ !_ !_ / : assert.
631

632 633 634
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.
635

636 637 638 639 640 641 642 643
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.
644

645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660
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.
661

662 663
  Global Instance pair_proper' : Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
  Proof. firstorder eauto. Qed.
664 665
  Global Instance pair_inj' : Inj2 R1 R2 (prod_relation R1 R2) pair.
  Proof. inversion_clear 1; eauto. Qed.
666 667 668 669 670
  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
671

672 673
Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation () ().
Instance pair_proper `{Equiv A, Equiv B} :
674 675
  Proper (() ==> () ==> ()) (@pair A B) := _.
Instance pair_equiv_inj `{Equiv A, Equiv B} : Inj2 () () () (@pair A B) := _.
676 677 678
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.
679

680 681
Instance prod_leibniz `{LeibnizEquiv A, LeibnizEquiv B} : LeibnizEquiv (A * B).
Proof. intros [??] [??] [??]; f_equal; apply leibniz_equiv; auto. Qed.
682

683
(** ** Sums *)
684 685
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.
686
Arguments sum_map {_ _ _ _} _ _ !_ / : assert.
687

688
Instance sum_inhabited_l {A B} (iA : Inhabited A) : Inhabited (A + B) :=
689
  match iA with populate x => populate (inl x) end.
690
Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
691
  match iB with populate y => populate (inl y) end.
692

693 694 695 696
Instance inl_inj : Inj (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Instance inr_inj : Inj (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.
697

698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724
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.
725 726 727 728
  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.
729 730 731 732 733
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) := _.
734 735
Instance inl_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inl A B) := _.
Instance inr_equiv_inj `{Equiv A, Equiv B} : Inj () () (@inr A B) := _.
736 737
Typeclasses Opaque sum_equiv.

738 739
(** ** Option *)
Instance option_inhabited {A} : Inhabited (option A) := populate None.
Robbert Krebbers's avatar
Robbert Krebbers committed
740

741
(** ** Sigma types *)
742 743 744
Arguments existT {_ _} _ _ : assert.
Arguments projT1 {_ _} _ : assert.
Arguments projT2 {_ _} _ : assert.
745

746 747 748
Arguments exist {_} _ _ _ : assert.
Arguments proj1_sig {_ _} _ : assert.
Arguments proj2_sig {_ _} _ : assert.
749 750
Notation "x ↾ p" := (exist _ x p) (at level 20) : stdpp_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : stdpp_scope.
751

752 753 754
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.
755

756 757 758 759 760 761 762 763 764 765
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.
766
Arguments sig_map _ _ _ _ _ _ !_ / : assert.
767

768 769 770 771
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
772

773
(** * Operations on sets *)
774
(** We define operational type classes for the traditional operations and
775
relations on sets: the empty set [∅], the union [(∪)],
776
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
777
[(⊆)] and element of [(∈)] relation, and disjointess [(##)]. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
778
Class Empty A := empty: A.
779
Hint Mode Empty ! : typeclass_instances.
780
Notation "∅" := empty (format "∅") : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
781

782 783
Instance empty_inhabited `(Empty A) : Inhabited A := populate .

Robbert Krebbers's avatar
Robbert Krebbers committed
784
Class Union A := union: A  A  A.
785
Hint Mode Union ! : typeclass_instances.
786
Instance: Params (@union) 2 := {}.
787 788
Infix "∪" := union (at level 50, left associativity) : stdpp_scope.
Notation "(∪)" := union (only parsing) : stdpp_scope.
789 790
Notation "( x ∪.)" := (union x) (only parsing) : stdpp_scope.
Notation "(.∪ x )" := (λ y, union y x) (only parsing) : stdpp_scope.
791 792
Infix "∪*" := (zip_with ()) (at level 50, left associativity) : stdpp_scope.
Notation "(∪*)" := (zip_with ()) (only parsing) : stdpp_scope.
793
Infix "∪**" := (zip_with (zip_with ()))
794
  (at level 50, left associativity) : stdpp_scope.
795
Infix "∪*∪**" := (zip_with (prod_zip () (*)))
796
  (at level 50, left associativity) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
797

798
Definition union_list `{Empty A} `{Union A} : list A  A := fold_right () .
799
Arguments union_list _ _ _ !_ / : assert.
800
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : stdpp_scope.
801

802 803 804 805 806
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.
807 808
Notation "( x ⊎.)" := (disj_union x) (only parsing) : stdpp_scope.
Notation "(.⊎ x )" := (λ y, disj_union y x) (only parsing) : stdpp_scope.
809

Robbert Krebbers's avatar
Robbert Krebbers committed
810
Class Intersection A := intersection: A  A  A.
811
Hint Mode Intersection ! : typeclass_instances.
812
Instance: Params (@intersection) 2 := {}.
813 814
Infix "∩" := intersection (at level 40) : stdpp_scope.
Notation "(∩)" := intersection (only parsing) : stdpp_scope.
815 816
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
817 818

Class Difference A := difference: A  A  A.
819
Hint Mode Difference ! : typeclass_instances.
820
Instance: Params (@difference) 2 := {}.
821 822
Infix "∖" := difference (at level 40, left associativity) : stdpp_scope.
Notation "(∖)" := difference (only parsing) : stdpp_scope.
823 824
Notation "( x ∖.)" := (difference x) (only parsing) : stdpp_scope.
Notation "(.∖ x )" := (λ y, difference y x) (only parsing) : stdpp_scope.
825 826
Infix "∖*" := (zip_with ()) (at level 40, left associativity) : stdpp_scope.
Notation "(∖*)" := (zip_with ()) (only parsing) : stdpp_scope.
827
Infix "∖**" := (zip_with (zip_with ()))
828
  (at level 40, left associativity) : stdpp_scope.
829
Infix "∖*∖**" := (zip_with (prod_zip () (*)))
830
  (at level 50, left associativity) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
831

832
Class Singleton A B := singleton: A  B.
833
Hint Mode Singleton - ! : typeclass_instances.
834
Instance: Params (@singleton) 3 := {}.
835
Notation "{[ x ]}" := (singleton x) (at level 1) : stdpp_scope.
836
Notation "{[ x ; y ; .. ; z ]}" :=
837
  (union .. (union (singleton x) (singleton y)) .. (singleton z))
838
  (at level 1) : stdpp_scope.
839
Notation "{[ x , y ]}" := (singleton (x,y))
840
  (at level 1, y at next level) : stdpp_scope.
841
Notation "{[ x , y , z ]}" := (singleton (x,y,z))
842
  (at level 1, y at next level, z at next level) : stdpp_scope.
843

844
Class SubsetEq A := subseteq: relation A.
845
Hint Mode SubsetEq ! : typeclass_instances.
846
Instance: Params (@subseteq) 2 := {}.
847 848
Infix "⊆" := subseteq (at level 70) : stdpp_scope.
Notation "(⊆)" := subseteq (only parsing) : stdpp_scope.
849 850
Notation "( X ⊆.)" := (subseteq X) (only parsing) : stdpp_scope.
Notation "(.⊆ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
851 852
Notation "X ⊈ Y" := (¬X  Y) (at level 70) : stdpp_scope.
Notation "(⊈)" := (λ X Y, X  Y) (only parsing) : stdpp_scope.
853 854
Notation "( X ⊈.)" := (λ Y, X  Y) (only parsing) : stdpp_scope.
Notation "(.⊈ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
855 856 857 858

Infix "⊆@{ A }" := (@subseteq A _) (at level 70, only parsing) : stdpp_scope.
Notation "(⊆@{ A } )" := (@subseteq A _) (only parsing) : stdpp_scope.

859 860 861 862 863 864 865
Infix "⊆*" := (Forall2 ()) (at level 70) : stdpp_scope.
Notation "(⊆*)" := (Forall2 ()) (only parsing) : stdpp_scope.
Infix "⊆**" := (Forall2 (*)) (at level 70) : stdpp_scope.
Infix "⊆1*" := (Forall2 (λ p q, p.1  q.1)) (at level 70) : stdpp_scope.
Infix "⊆2*" := (Forall2 (λ p q, p.2  q.2)) (at level 70) : stdpp_scope.
Infix "⊆1**" := (Forall2 (λ p q, p.1 * q.1)) (at level 70) : stdpp_scope.
Infix "⊆2**" := (Forall2 (λ p q, p.2 * q.2)) (at level 70) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
866

867 868 869
Hint Extern 0 (_  _) => reflexivity : core.
Hint Extern 0 (_ * _) => reflexivity : core.
Hint Extern 0 (_ ** _) => reflexivity : core.
870

871 872
Infix "⊂" := (strict ()) (at level 70) : stdpp_scope.
Notation "(⊂)" := (strict ()) (only parsing) : stdpp_scope.
873 874
Notation "( X ⊂.)" := (strict () X) (only parsing) : stdpp_scope.
Notation "(.⊂ X )" := (λ Y, Y  X) (only parsing) : stdpp_scope.
875 876
Notation "X ⊄ Y" := (¬X  Y) (at level 70) : stdpp_scope.
Notation "(⊄)" := (λ X Y, X  Y) (only parsing) : stdpp_scope.
877 878
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
879

880 881 882
Infix "⊂@{ A }" := (strict (@{A})) (at level 70, only parsing) : stdpp_scope.
Notation "(⊂@{ A } )" := (strict (@{A})) (only parsing) : stdpp_scope.

883 884 885 886
Notation "X ⊆ Y ⊆ Z" := (X  Y  Y  Z) (at level 70, Y at next level) : stdpp_scope.
Notation "X ⊆ Y ⊂ Z" := (X  Y  Y  Z) (at level 70, Y at next level) : stdpp_scope.
Notation "X ⊂ Y ⊆ Z" := (X  Y  Y  Z) (at level 70, Y at next level) : stdpp_scope.
Notation "X ⊂ Y ⊂ Z" := (X  Y  Y  Z) (at level 70, Y at next level) : stdpp_scope.
887

888 889 890 891 892 893 894
Definition option_to_set `{Singleton A C, Empty C} (mx : option A) : C :=
  match mx with None =>  | Some x => {[ x ]} end.
Fixpoint list_to_set `{Singleton A C, Empty C, Union C} (l : list A) : C :=
  match l with [] =>  | x :: l => {[ x ]}  list_to_set l end.
Fixpoint list_to_set_disj `{Singleton A C, Empty C, DisjUnion C} (l : list A) : C :=
  match l with [] =>  | x :: l => {[ x ]}  list_to_set_disj l end.

895 896 897 898
(** 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.
899
Hint Mode Lexico ! : typeclass_instances.
900

Robbert Krebbers's avatar
Robbert Krebbers committed
901
Class ElemOf A B := elem_of: A  B  Prop.
902
Hint Mode ElemOf - ! : typeclass_instances.
903
Instance: Params (@elem_of) 3 := {}.
904 905
Infix "∈" := elem_of (at level 70) : stdpp_scope.
Notation "(∈)" := elem_of (only parsing) : stdpp_scope.
906 907
Notation "( x ∈.)" := (elem_of x) (only parsing) : stdpp_scope.
Notation "(.∈ X )" := (λ x, elem_of x X) (only parsing) : stdpp_scope.
908 909
Notation "x ∉ X" := (¬x  X) (at level 80) : stdpp_scope.
Notation "(∉)" := (λ x X, x  X) (only parsing) : stdpp_scope.
910 911
Notation "( x ∉.)" := (λ X, x  X) (only parsing) : stdpp_scope.
Notation "(.∉ X )" := (λ x, x  X) (only parsing) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
912

Robbert Krebbers's avatar
Robbert Krebbers committed
913 914 915
Infix "∈@{ B }" := (@elem_of _ B _) (at level 70, only parsing) : stdpp_scope.
Notation "(∈@{ B } )" := (@elem_of _ B _) (only parsing) : stdpp_scope.

916 917 918
Notation "x ∉@{ B } X" := (¬x @{B} X) (at level 80, only parsing) : stdpp_scope.
Notation "(∉@{ B } )" := (λ x X, x @{B} X) (only parsing) : stdpp_scope.

919
Class Disjoint A := disjoint : A  A  Prop.
920
 Hint Mode Disjoint ! : typeclass_instances.
921
Instance: Params (@disjoint) 2 := {}.
922 923 924 925
Infix "##" := disjoint (at level 70) : stdpp_scope.
Notation "(##)" := disjoint (only parsing) : stdpp_scope.
Notation "( X ##.)" := (disjoint X) (only parsing) : stdpp_scope.
Notation "(.## X )" := (λ Y, Y ## X) (only parsing) : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
926 927 928 929

Infix "##@{ A }" := (@disjoint A _) (at level 70, only parsing) : stdpp_scope.
Notation "(##@{ A } )" := (@disjoint A _) (only parsing) : stdpp_scope.

930 931 932 933 934 935 936
Infix "##*" := (Forall2 (##)) (at level 70) : stdpp_scope.
Notation "(##*)" := (Forall2 (##)) (only parsing) : stdpp_scope.
Infix "##**" := (Forall2 (##*)) (at level 70) : stdpp_scope.
Infix "##1*" := (Forall2 (λ p q, p.1 ## q.1)) (at level 70) : stdpp_scope.
Infix "##2*" := (Forall2 (λ p q, p.2 ## q.2)) (at level 70) : stdpp_scope.
Infix "##1**" := (Forall2 (λ p q, p.1 ##* q.1)) (at level 70) : stdpp_scope.
Infix "##2**" := (Forall2 (λ p q, p.2 ##* q.2)) (at level 70) : stdpp_scope.
937 938
Hint Extern 0 (_ ## _) => symmetry; eassumption : core.
Hint Extern 0 (_ ##* _) => symmetry; eassumption : core.
939 940

Class DisjointE E A := disjointE : E  A  A  Prop.
941
Hint Mode DisjointE - ! : typeclass_instances.
942
Instance: Params (@disjointE) 4 := {}.
943
Notation "X ##{ Γ } Y" := (disjointE Γ X Y)
944 945
  (at level 70, format "X  ##{ Γ }  Y") : stdpp_scope.
Notation "(##{ Γ } )" := (disjointE Γ) (only parsing, Γ at level 1) : stdpp_scope.
946
Notation "Xs ##{ Γ }* Ys" := (Forall2 (##{Γ}) Xs Ys)
947
  (at level 70, format "Xs  ##{ Γ }*  Ys") : stdpp_scope.
948
Notation "(##{ Γ }* )" := (Forall2 (##{Γ}))
949
  (only parsing, Γ at level 1) : stdpp_scope.
950
Notation "X ##{ Γ1 , Γ2 , .. , Γ3 } Y" := (disjoint (pair .. (Γ1, Γ2) .. Γ3) X Y)
951
  (at level 70, format "X  ##{ Γ1 , Γ2 , .. , Γ3 }  Y") : stdpp_scope.
952
Notation "Xs ##{ Γ1 , Γ2 , .. , Γ3 }* Ys" :=
953
  (Forall2 (disjoint (pair .. (Γ1, Γ2) .. Γ3)) Xs Ys)
954
  (at level 70, format "Xs  ##{ Γ1 ,  Γ2 , .. , Γ3 }*  Ys") : stdpp_scope.
955
Hint Extern 0 (_ ##{_} _) => symmetry; eassumption : core.
956 957

Class DisjointList A := disjoint_list : list A  Prop.
958
Hint Mode DisjointList ! : typeclass_instances.
959
Instance: Params (@disjoint_list) 2 := {}.
960
Notation "## Xs" := (disjoint_list Xs) (at level 20, format "##  Xs") : stdpp_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
961 962
Notation "##@{ A } Xs" :=
  (@disjoint_list A _ Xs) (at level 20, only parsing) : stdpp_scope.
963

964 965
Section disjoint_list.
  Context `{Disjoint A, Union A, Empty A}.
966 967
  Implicit Types X : A.

968
  Inductive disjoint_list_default : DisjointList A :=
Robbert Krebbers's avatar
Robbert Krebbers committed
969
    | disjoint_nil_2 : ##@{A} []
970
    | disjoint_cons_2 (X : A) (Xs : list A) : X ##  Xs  ## Xs  ## (X :: Xs).
971
  Global Existing Instance disjoint_list_default.
972

Robbert Krebbers's avatar
Robbert Krebbers committed
973
  Lemma disjoint_list_nil  : ##@{A} []  True.
974
  Proof. split; constructor. Qed.
975
  Lemma disjoint_list_cons X Xs : ## (X :: Xs)  X ##  Xs  ## Xs.
976
  Proof. split. inversion_clear 1; auto. intros [??]. constructor; auto. Qed.
977
End disjoint_list.
978 979

Class Filter A B := filter:  (P : A  Prop) `{ x, Decision (P x)}, B  B.
980
Hint Mode Filter - ! : typeclass_instances.
981