base.v

(** This file collects type class interfaces, notations, and general theorems
that are used throughout the whole development. Most importantly it contains
abstract interfaces for ordered structures, sets, and various other data
structures. *)

(* We want to ensure that [le] and [lt] refer to operations on [nat].
These two functions being defined both in [Coq.Bool] and in [Coq.Peano],
we must export [Coq.Peano] later than any export of [Coq.Bool]. *)
(* We also want to ensure that notations from [Coq.Utf8] take precedence
over the ones of [Coq.Peano] (see Coq PR#12950), so we import [Utf8] last. *)
From Coq Require Export Morphisms RelationClasses List Bool Setoid Peano Utf8.
From Coq Require Import Permutation.
Export ListNotations.
From Coq.Program Require Export Basics Syntax.
From stdpp Require Import options.

(** 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. *)
Notation length := Datatypes.length.

(** * Enable implicit generalization. *)
(** This option enables implicit generalization in arguments of the form
   [`{...}] (i.e., anonymous arguments).  Unfortunately, it also enables
   implicit generalization in [Instance].  We think that the fact that both
   behaviors are coupled together is a [bug in
   Coq](https://github.com/coq/coq/issues/6030). *)
Global Generalizable All Variables.

(** * 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. *)
Obligation Tactic := idtac.

(** 3. Hide obligations from the results of the [Search] commands. *)
Add Search Blacklist "_obligation_".

(** * Sealing off definitions *)
Section seal.
  Local Set Primitive Projections.
  Record seal {A} (f : A) := { unseal : A; seal_eq : unseal = f }.
End seal.
Global Arguments unseal {_ _} _ : assert.
Global Arguments seal_eq {_ _} _ : assert.

(** * Non-backtracking type classes *)
(** The type class [TCNoBackTrack P] can be used to establish [P] without ever
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. *)
Class TCNoBackTrack (P : Prop) := { tc_no_backtrack : P }.
Global Hint Extern 0 (TCNoBackTrack _) => constructor; apply _ : typeclass_instances.

(* 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
establish [Q], i.e. does not have the behavior of a conditional. Furthermore,
note that [TCOr (TCAnd P Q) (TCAnd (TCNot P) R)] would not work; we generally
would not be able to prove the negation of [P]. *)
Inductive TCIf (P Q R : Prop) : Prop :=
  | TCIf_true : P → Q → TCIf P Q R
  | TCIf_false : R → TCIf P Q R.
Existing Class TCIf.

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

(** * Typeclass opaque definitions *)
(** The constant [tc_opaque] is used to make definitions opaque for just type
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.
Global Arguments tc_opaque {_} _ /.

(** Below we define type class versions of the common logical operators. It is
important to note that we duplicate the definitions, and do not declare the
existing logical operators as type classes. That is, we do not say:

  Existing Class or.
  Existing Class and.

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

These classes are useful for two purposes: writing complicated type class
premises in a more concise way, and for efficiency. For example, using the [Or]
class, instead of defining two instances [P → Q1 → R] and [P → Q2 → R] we could
have one instance [P → Or Q1 Q2 → R]. When we declare the instance that way, we
avoid the need to derive [P] twice. *)
Inductive TCOr (P1 P2 : Prop) : Prop :=
  | 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.
Global Hint Mode TCOr ! ! : typeclass_instances.

Inductive TCAnd (P1 P2 : Prop) : Prop := TCAnd_intro : P1 → P2 → TCAnd P1 P2.
Existing Class TCAnd.
Existing Instance TCAnd_intro.
Global Hint Mode TCAnd ! ! : typeclass_instances.

Inductive TCTrue : Prop := TCTrue_intro : TCTrue.
Existing Class TCTrue.
Existing Instance TCTrue_intro.

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.
Global Hint Mode TCForall ! ! ! : typeclass_instances.

(** 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. *)
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.
Global Hint Mode TCForall2 ! ! ! ! - : typeclass_instances.
Global Hint Mode TCForall2 ! ! ! - ! : typeclass_instances.

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.
Global Hint Mode TCElemOf ! ! ! : typeclass_instances.

(** 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. *)
Inductive TCEq {A} (x : A) : A → Prop := TCEq_refl : TCEq x x.
Existing Class TCEq.
Existing Instance TCEq_refl.
Global Hint Mode TCEq ! - - : typeclass_instances.

Lemma TCEq_eq {A} (x1 x2 : A) : TCEq x1 x2 ↔ x1 = x2.
Proof. split; destruct 1; reflexivity. Qed.

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.
Global Hint Mode TCDiag ! ! ! - : typeclass_instances.
Global Hint Mode TCDiag ! ! - ! : typeclass_instances.

(** 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.

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

(** Change [True] and [False] into notations in order to enable overloading.
We will use this to give [True] and [False] a different interpretation for
embedded logics. *)
Notation "'True'" := True (format "True") : type_scope.
Notation "'False'" := False (format "False") : type_scope.

(** Change [forall] into a notation in order to enable overloading. *)
Notation "'forall' x .. y , P" := (forall x, .. (forall y, P) ..)
  (at level 200, x binder, y binder, right associativity,
   only parsing) : type_scope.


(** * Equality *)
(** Introduce some Haskell style like notations. *)
Notation "(=)" := eq (only parsing) : stdpp_scope.
Notation "( x =.)" := (eq x) (only parsing) : stdpp_scope.
Notation "(.= x )" := (λ y, eq y x) (only parsing) : stdpp_scope.
Notation "(≠)" := (λ x y, x ≠ y) (only parsing) : stdpp_scope.
Notation "( x ≠.)" := (λ y, x ≠ y) (only parsing) : stdpp_scope.
Notation "(.≠ x )" := (λ y, y ≠ x) (only parsing) : stdpp_scope.

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.
Notation "X ≠@{ A } Y":= (¬X =@{ A } Y)
  (at level 70, only parsing, no associativity) : stdpp_scope.

Global Hint Extern 0 (_ = _) => reflexivity : core.
Global Hint Extern 100 (_ ≠ _) => discriminate : core.

Instance: ∀ A, PreOrder (=@{A}).
Proof. split; repeat intro; congruence. Qed.

(** ** Setoid equality *)
(** 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). *)
Class Equiv A := equiv: relation A.
(* No Hint Mode set because of Coq bug #5735
Global Hint Mode Equiv ! : typeclass_instances. *)

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

Notation "(≡)" := equiv (only parsing) : stdpp_scope.
Notation "( X ≡.)" := (equiv X) (only parsing) : stdpp_scope.
Notation "(.≡ X )" := (λ Y, Y ≡ X) (only parsing) : stdpp_scope.
Notation "(≢)" := (λ X Y, ¬X ≡ Y) (only parsing) : stdpp_scope.
Notation "X ≢ Y":= (¬X ≡ Y) (at level 70, no associativity) : stdpp_scope.
Notation "( X ≢.)" := (λ Y, X ≢ Y) (only parsing) : stdpp_scope.
Notation "(.≢ X )" := (λ Y, Y ≢ X) (only parsing) : stdpp_scope.

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

(** The type class [LeibnizEquiv] collects setoid equalities that coincide
with Leibniz equality. We provide the tactic [fold_leibniz] to transform such
setoid equalities into Leibniz equalities, and [unfold_leibniz] for the
reverse. *)
Class LeibnizEquiv A `{Equiv A} := leibniz_equiv x y : x ≡ y → x = y.
Global Hint Mode LeibnizEquiv ! - : typeclass_instances.

Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (≡@{A})} (x y : A) :
  x ≡ y ↔ x = y.
Proof. split; [apply leibniz_equiv|]. intros ->; reflexivity. Qed.

Ltac fold_leibniz := repeat
  match goal with
  | H : context [ _ ≡@{?A} _ ] |- _ =>
    setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
  | |- context [ _ ≡@{?A} _ ] =>
    setoid_rewrite (leibniz_equiv_iff (A:=A))
  end.
Ltac unfold_leibniz := repeat
  match goal with
  | H : context [ _ =@{?A} _ ] |- _ =>
    setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
  | |- context [ _ =@{?A} _ ] =>
    setoid_rewrite <-(leibniz_equiv_iff (A:=A))
  end.

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

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

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


(** * Type classes *)
(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
propositions. *)
Class Decision (P : Prop) := decide : {P} + {¬P}.
Global Hint Mode Decision ! : typeclass_instances.
Global 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).
Global Hint Mode RelDecision ! ! ! : typeclass_instances.
Global Arguments decide_rel {_ _} _ {_} _ _ : simpl never, assert.
Notation EqDecision A := (RelDecision (=@{A})).

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

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

(** ** Common properties *)
(** These operational type classes allow us to refer to common mathematical
properties in a generic way. For example, for injectivity of [(k ++.)] it
allows us to write [inj (k ++.)] instead of [app_inv_head k]. *)
Class Inj {A B} (R : relation A) (S : relation B) (f : A → B) : Prop :=
  inj x y : S (f x) (f y) → R x y.
Class Inj2 {A B C} (R1 : relation A) (R2 : relation B)
    (S : relation C) (f : A → B → C) : Prop :=
  inj2 x1 x2 y1 y2 : S (f x1 x2) (f y1 y2) → R1 x1 y1 ∧ R2 x2 y2.
Class Cancel {A B} (S : relation B) (f : A → B) (g : B → A) : Prop :=
  cancel : ∀ x, S (f (g x)) x.
Class Surj {A B} (R : relation B) (f : A → B) :=
  surj y : ∃ x, R (f x) y.
Class IdemP {A} (R : relation A) (f : A → A → A) : Prop :=
  idemp x : R (f x x) x.
Class Comm {A B} (R : relation A) (f : B → B → A) : Prop :=
  comm x y : R (f x y) (f y x).
Class LeftId {A} (R : relation A) (i : A) (f : A → A → A) : Prop :=
  left_id x : R (f i x) x.
Class RightId {A} (R : relation A) (i : A) (f : A → A → A) : Prop :=
  right_id x : R (f x i) x.
Class Assoc {A} (R : relation A) (f : A → A → A) : Prop :=
  assoc x y z : R (f x (f y z)) (f (f x y) z).
Class LeftAbsorb {A} (R : relation A) (i : A) (f : A → A → A) : Prop :=
  left_absorb x : R (f i x) i.
Class RightAbsorb {A} (R : relation A) (i : A) (f : A → A → A) : Prop :=
  right_absorb x : R (f x i) i.
Class AntiSymm {A} (R S : relation A) : Prop :=
  anti_symm x y : S x y → S y x → R x y.
Class Total {A} (R : relation A) := total x y : R x y ∨ R y x.
Class Trichotomy {A} (R : relation A) :=
  trichotomy x y : R x y ∨ x = y ∨ R y x.
Class TrichotomyT {A} (R : relation A) :=
  trichotomyT x y : {R x y} + {x = y} + {R y x}.

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.

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

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.
Global 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.
Global Instance inj2_inj_2 `{Inj2 A B C R1 R2 R3 f} x : Inj R2 R3 (f x).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.

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

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

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

(** * Logic *)
Global Instance prop_inhabited : Inhabited Prop := populate True.

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

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

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

Global Hint Extern 0 (_ ↔ _) => reflexivity : core.
Global Hint Extern 0 (_ ↔ _) => symmetry; assumption : core.

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.
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.

Global Instance eq_comm {A} : Comm (↔) (=@{A}).
Proof. red; intuition. Qed.
Global Instance flip_eq_comm {A} : Comm (↔) (λ x y, y =@{A} x).
Proof. red; intuition. Qed.
Global Instance iff_comm : Comm (↔) (↔).
Proof. red; intuition. Qed.
Global Instance and_comm : Comm (↔) (∧).
Proof. red; intuition. Qed.
Global Instance and_assoc : Assoc (↔) (∧).
Proof. red; intuition. Qed.
Global Instance and_idemp : IdemP (↔) (∧).
Proof. red; intuition. Qed.
Global Instance or_comm : Comm (↔) (∨).
Proof. red; intuition. Qed.
Global Instance or_assoc : Assoc (↔) (∨).
Proof. red; intuition. Qed.
Global Instance or_idemp : IdemP (↔) (∨).
Proof. red; intuition. Qed.
Global Instance True_and : LeftId (↔) True (∧).
Proof. red; intuition. Qed.
Global Instance and_True : RightId (↔) True (∧).
Proof. red; intuition. Qed.
Global Instance False_and : LeftAbsorb (↔) False (∧).
Proof. red; intuition. Qed.
Global Instance and_False : RightAbsorb (↔) False (∧).
Proof. red; intuition. Qed.
Global Instance False_or : LeftId (↔) False (∨).
Proof. red; intuition. Qed.
Global Instance or_False : RightId (↔) False (∨).
Proof. red; intuition. Qed.
Global Instance True_or : LeftAbsorb (↔) True (∨).
Proof. red; intuition. Qed.
Global Instance or_True : RightAbsorb (↔) True (∨).
Proof. red; intuition. Qed.
Global Instance True_impl : LeftId (↔) True impl.
Proof. unfold impl. red; intuition. Qed.
Global Instance impl_True : RightAbsorb (↔) True impl.
Proof. unfold impl. red; intuition. Qed.


(** * Common data types *)
(** ** Functions *)
Notation "(→)" := (λ A B, A → B) (only parsing) : stdpp_scope.
Notation "( A →.)" := (λ B, A → B) (only parsing) : stdpp_scope.
Notation "(.→ B )" := (λ A, A → B) (only parsing) : stdpp_scope.

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

Infix "∘" := compose : stdpp_scope.
Notation "(∘)" := compose (only parsing) : stdpp_scope.
Notation "( f ∘.)" := (compose f) (only parsing) : stdpp_scope.
Notation "(.∘ f )" := (λ g, compose g f) (only parsing) : stdpp_scope.

Global Instance impl_inhabited {A} `{Inhabited B} : Inhabited (A → B) :=
  populate (λ _, inhabitant).

(** Ensure that [simpl] unfolds [id], [compose], and [flip] when fully
applied. *)
Global Arguments id _ _ / : assert.
Global Arguments compose _ _ _ _ _ _ / : assert.
Global Arguments flip _ _ _ _ _ _ / : assert.
Global Arguments const _ _ _ _ / : assert.
Typeclasses Transparent id compose flip const.

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

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

Global Instance id_inj {A} : Inj (=) (=) (@id A).
Proof. intros ??; auto. Qed.
Global 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.

Global Instance id_surj {A} : Surj (=) (@id A).
Proof. intros y; exists y; reflexivity. Qed.
Global 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.

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

(** ** Lists *)
Global 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.
Global Hint Unfold Is_true : core.
Global Hint Immediate Is_true_eq_left : core.
Global Hint Resolve orb_prop_intro andb_prop_intro : core.
Notation "(&&)" := andb (only parsing).
Notation "(||)" := orb (only parsing).
Infix "&&*" := (zip_with (&&)) (at level 40).
Infix "||*" := (zip_with (||)) (at level 50).

Global Instance bool_inhabated : Inhabited bool := populate true.

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.

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.

(** ** Unit *)
Global Instance unit_equiv : Equiv unit := λ _ _, True.
Global Instance unit_equivalence : Equivalence (≡@{unit}).
Proof. repeat split. Qed.
Global Instance unit_leibniz : LeibnizEquiv unit.
Proof. intros [] []; reflexivity. Qed.
Global Instance unit_inhabited: Inhabited unit := populate ().

(** ** Empty *)
Global Instance Empty_set_equiv : Equiv Empty_set := λ _ _, True.
Global Instance Empty_set_equivalence : Equivalence (≡@{Empty_set}).
Proof. repeat split. Qed.
Global Instance Empty_set_leibniz : LeibnizEquiv Empty_set.
Proof. intros [] []; reflexivity. Qed.

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

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").

Instance: Params (@pair) 2 := {}.
Instance: Params (@fst) 2 := {}.
Instance: Params (@snd) 2 := {}.

Notation curry := prod_curry.
Notation uncurry := prod_uncurry.
Definition curry3 {A B C D} (f : A → B → C → D) (p : A * B * C) : D :=
  let '(a,b,c) := p in f a b c.
Definition curry4 {A B C D E} (f : A → B → C → D → E) (p : A * B * C * D) : E :=
  let '(a,b,c,d) := p in f a b c d.

Definition 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).

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

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)).
Global Arguments prod_zip {_ _ _ _ _ _} _ _ !_ !_ / : assert.

Global 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.

Global Instance pair_inj {A B} : Inj2 (=) (=) (=) (@pair A B).
Proof. injection 1; auto. Qed.
Global 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.

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.

  Global Instance pair_proper' : Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
  Proof. firstorder eauto. Qed.
  Global Instance pair_inj' : Inj2 R1 R2 (prod_relation R1 R2) pair.
  Proof. inversion_clear 1; eauto. Qed.
  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.

Global Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation (≡) (≡).
Global Instance pair_proper `{Equiv A, Equiv B} :
  Proper ((≡) ==> (≡) ==> (≡)) (@pair A B) := _.
Global Instance pair_equiv_inj `{Equiv A, Equiv B} : Inj2 (≡) (≡) (≡) (@pair A B) := _.
Global Instance fst_proper `{Equiv A, Equiv B} : Proper ((≡) ==> (≡)) (@fst A B) := _.
Global Instance snd_proper `{Equiv A, Equiv B} : Proper ((≡) ==> (≡)) (@snd A B) := _.
Typeclasses Opaque prod_equiv.

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

(** ** Sums *)
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.
Global Arguments sum_map {_ _ _ _} _ _ !_ / : assert.

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

Global Instance inl_inj {A B} : Inj (=) (=) (@inl A B).
Proof. injection 1; auto. Qed.
Global Instance inr_inj {A B} : Inj (=) (=) (@inr A B).
Proof. injection 1; auto. Qed.

Global 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.
  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.
End sum_relation.

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

(** ** Option *)
Global Instance option_inhabited {A} : Inhabited (option A) := populate None.

(** ** Sigma types *)
Global Arguments existT {_ _} _ _ : assert.
Global Arguments projT1 {_ _} _ : assert.
Global Arguments projT2 {_ _} _ : assert.

Global Arguments exist {_} _ _ _ : assert.
Global Arguments proj1_sig {_ _} _ : assert.
Global Arguments proj2_sig {_ _} _ : assert.
Notation "x ↾ p" := (exist _ x p) (at level 20) : stdpp_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : stdpp_scope.

Lemma proj1_sig_inj {A} (P : A → Prop) x (Px : P x) y (Py : P y) :
  x↾Px = y↾Py → x = y.
Proof. injection 1; trivial. Qed.

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.
Global Arguments sig_map _ _ _ _ _ _ !_ / : assert.

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.

(** * Operations on sets *)
(** We define operational type classes for the traditional operations and
relations on sets: the empty set [∅], the union [(∪)],
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(##)]. *)
Class Empty A := empty: A.
Global Hint Mode Empty ! : typeclass_instances.
Notation "∅" := empty (format "∅") : stdpp_scope.

Global Instance empty_inhabited `(Empty A) : Inhabited A := populate ∅.

Class Union A := union: A → A → A.
Global Hint Mode Union ! : typeclass_instances.
Instance: Params (@union) 2 := {}.
Infix "∪" := union (at level 50, left associativity) : stdpp_scope.
Notation "(∪)" := union (only parsing) : stdpp_scope.
Notation "( x ∪.)" := (union x) (only parsing) : stdpp_scope.
Notation "(.∪ x )" := (λ y, union y x) (only parsing) : stdpp_scope.
Infix "∪*" := (zip_with (∪)) (at level 50, left associativity) : stdpp_scope.
Notation "(∪*)" := (zip_with (∪)) (only parsing) : stdpp_scope.

Definition union_list `{Empty A} `{Union A} : list A → A := fold_right (∪) ∅.
Global Arguments union_list _ _ _ !_ / : assert.
Notation "⋃ l" := (union_list l) (at level 20, format "⋃  l") : stdpp_scope.

Class DisjUnion A := disj_union: A → A → A.
Global 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.
Notation "( x ⊎.)" := (disj_union x) (only parsing) : stdpp_scope.
Notation "(.⊎ x )" := (λ y, disj_union y x) (only parsing) : stdpp_scope.

Class Intersection A := intersection: A → A → A.
Global Hint Mode Intersection ! : typeclass_instances.
Instance: Params (@intersection) 2 := {}.
Infix "∩" := intersection (at level 40) : stdpp_scope.
Notation "(∩)" := intersection (only parsing) : stdpp_scope.
Notation "( x ∩.)" := (intersection x) (only parsing) : stdpp_scope.
Notation "(.∩ x )" := (λ y, intersection y x) (only parsing) : stdpp_scope.

Class Difference A := difference: A → A → A.
Global Hint Mode Difference ! : typeclass_instances.
Instance: Params (@difference) 2 := {}.
Infix "∖" := difference (at level 40, left associativity) : stdpp_scope.
Notation "(∖)" := difference (only parsing) : stdpp_scope.
Notation "( x ∖.)" := (difference x) (only parsing) : stdpp_scope.
Notation "(.∖ x )" := (λ y, difference y x) (only parsing) : stdpp_scope.
Infix "∖*" := (zip_with (∖)) (at level 40, left associativity) : stdpp_scope.
Notation "(∖*)" := (zip_with (∖)) (only parsing) : stdpp_scope.

Class Singleton A B := singleton: A → B.
Global Hint Mode Singleton - ! : typeclass_instances.
Instance: Params (@singleton) 3 := {}.
Notation "{[ x ]}" := (singleton x) (at level 1) : stdpp_scope.
Notation "{[ x ; y ; .. ; z ]}" :=
  (union .. (union (singleton x) (singleton y)) .. (singleton z))
  (at level 1) : stdpp_scope.
Notation "{[ x , y ]}" := (singleton (x,y))
  (at level 1, y at next level) : stdpp_scope.
Notation "{[ x , y , z ]}" := (singleton (x,y,z))
  (at level 1, y at next level, z at next level) : stdpp_scope.

Class SubsetEq A := subseteq: relation A.
Global Hint Mode SubsetEq ! : typeclass_instances.
Instance: Params (@subseteq) 2 := {}.
Infix "⊆" := subseteq (at level 70) : stdpp_scope.
Notation "(⊆)" := subseteq (only parsing) : stdpp_scope.
Notation "( X ⊆.)" := (subseteq X) (only parsing) : stdpp_scope.
Notation "(.⊆ X )" := (λ Y, Y ⊆ X) (only parsing) : stdpp_scope.
Notation "X ⊈ Y" := (¬X ⊆ Y) (at level 70) : stdpp_scope.
Notation "(⊈)" := (λ X Y, X ⊈ Y) (only parsing) : stdpp_scope.
Notation "( X ⊈.)" := (λ Y, X ⊈ Y) (only parsing) : stdpp_scope.
Notation "(.⊈ X )" := (λ Y, Y ⊈ X) (only parsing) : stdpp_scope.

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

Infix "⊆*" := (Forall2 (⊆)) (at level 70) : stdpp_scope.
Notation "(⊆*)" := (Forall2 (⊆)) (only parsing) : stdpp_scope.

Global Hint Extern 0 (_ ⊆ _) => reflexivity : core.
Global Hint Extern 0 (_ ⊆* _) => reflexivity : core.

Infix "⊂" := (strict (⊆)) (at level 70) : stdpp_scope.
Notation "(⊂)" := (strict (⊆)) (only parsing) : stdpp_scope.
Notation "( X ⊂.)" := (strict (⊆) X) (only parsing) : stdpp_scope.
Notation "(.⊂ X )" := (λ Y, Y ⊂ X) (only parsing) : stdpp_scope.
Notation "X ⊄ Y" := (¬X ⊂ Y) (at level 70) : stdpp_scope.
Notation "(⊄)" := (λ X Y, X ⊄ Y) (only parsing) : stdpp_scope.
Notation "( X ⊄.)" := (λ Y, X ⊄ Y) (only parsing) : stdpp_scope.
Notation "(.⊄ X )" := (λ Y, Y ⊄ X) (only parsing) : stdpp_scope.

Infix "⊂@{ A }" := (strict (⊆@{A})) (at level 70, only parsing) : stdpp_scope.
Notation "(⊂@{ A } )" := (strict (⊆@{A})) (only parsing) : 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.
Notation "X ⊂ Y ⊂ Z" := (X ⊂ Y ∧ Y ⊂ Z) (at level 70, Y at next level) : stdpp_scope.

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.

(** 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.
Global Hint Mode Lexico ! : typeclass_instances.

Class ElemOf A B := elem_of: A → B → Prop.
Global Hint Mode ElemOf - ! : typeclass_instances.
Instance: Params (@elem_of) 3 := {}.
Infix "∈" := elem_of (at level 70) : stdpp_scope.
Notation "(∈)" := elem_of (only parsing) : stdpp_scope.
Notation "( x ∈.)" := (elem_of x) (only parsing) : stdpp_scope.
Notation "(.∈ X )" := (λ x, elem_of x X) (only parsing) : stdpp_scope.
Notation "x ∉ X" := (¬x ∈ X) (at level 80) : stdpp_scope.
Notation "(∉)" := (λ x X, x ∉ X) (only parsing) : stdpp_scope.
Notation "( x ∉.)" := (λ X, x ∉ X) (only parsing) : stdpp_scope.
Notation "(.∉ X )" := (λ x, x ∉ X) (only parsing) : stdpp_scope.

Infix "∈@{ B }" := (@elem_of _ B _) (at level 70, only parsing) : stdpp_scope.
Notation "(∈@{ B } )" := (@elem_of _ B _) (only parsing) : stdpp_scope.

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.

Class Disjoint A := disjoint : A → A → Prop.
Global Hint Mode Disjoint ! : typeclass_instances.
Instance: Params (@disjoint) 2 := {}.
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.

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

Infix "##*" := (Forall2 (##)) (at level 70) : stdpp_scope.
Notation "(##*)" := (Forall2 (##)) (only parsing) : stdpp_scope.

Global Hint Extern 0 (_ ## _) => symmetry; eassumption : core.
Global Hint Extern 0 (_ ##* _) => symmetry; eassumption : core.

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

Class UpClose A B := up_close : A → B.
Global Hint Mode UpClose - ! : typeclass_instances.
Notation "↑ x" := (up_close x) (at level 20, format "↑ x").

(** * Monadic operations *)
(** We define operational type classes for the monadic operations bind, join
and fmap. We use these type classes merely for convenient overloading of
notations and do not formalize any theory on monads (we do not even define a
class with the monad laws). *)
Class MRet (M : Type → Type) := mret: ∀ {A}, A → M A.
Global Arguments mret {_ _ _} _ : assert.
Instance: Params (@mret) 3 := {}.
Class MBind (M : Type → Type) := mbind : ∀ {A B}, (A → M B) → M A → M B.
Global Arguments mbind {_ _ _ _} _ !_ / : assert.
Instance: Params (@mbind) 4 := {}.
Class MJoin (M : Type → Type) := mjoin: ∀ {A}, M (M A) → M A.
Global Arguments mjoin {_ _ _} !_ / : assert.
Instance: Params (@mjoin) 3 := {}.
Class FMap (M : Type → Type) := fmap : ∀ {A B}, (A → B) → M A → M B.
Global Arguments fmap {_ _ _ _} _ !_ / : assert.
Instance: Params (@fmap) 4 := {}.
Class OMap (M : Type → Type) := omap: ∀ {A B}, (A → option B) → M A → M B.
Global Arguments omap {_ _ _ _} _ !_ / : assert.
Instance: Params (@omap) 4 := {}.

Notation "m ≫= f" := (mbind f m) (at level 60, right associativity) : stdpp_scope.
Notation "( m ≫=.)" := (λ f, mbind f m) (only parsing) : stdpp_scope.
Notation "(.≫= f )" := (mbind f) (only parsing) : stdpp_scope.
Notation "(≫=)" := (λ m f, mbind f m) (only parsing) : stdpp_scope.

Notation "x ← y ; z" := (y ≫= (λ x : _, z))
  (at level 20, y at level 100, z at level 200, only parsing) : stdpp_scope.

Notation "' x ← y ; z" := (y ≫= (λ x : _, z))
  (at level 20, x pattern, y at level 100, z at level 200, only parsing) : stdpp_scope.

Infix "<$>" := fmap (at level 61, left associativity) : stdpp_scope.

Notation "x ;; z" := (x ≫= λ _, z)
  (at level 100, z at level 200, only parsing, right associativity): stdpp_scope.

Notation "ps .*1" := (fmap (M:=list) fst ps)
  (at level 2, left associativity, format "ps .*1").
Notation "ps .*2" := (fmap (M:=list) snd ps)
  (at level 2, left associativity, format "ps .*2").

Class MGuard (M : Type → Type) :=
  mguard: ∀ P {dec : Decision P} {A}, (P → M A) → M A.
Global Arguments mguard _ _ _ !_ _ _ / : assert.
Notation "'guard' P ; z" := (mguard P (λ _, z))
  (at level 20, z at level 200, only parsing, right associativity) : stdpp_scope.
Notation "'guard' P 'as' H ; z" := (mguard P (λ H, z))
  (at level 20, z at level 200, only parsing, right associativity) : stdpp_scope.

(** * Operations on maps *)
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
The function look up [m !! k] should yield the element at key [k] in [m]. *)
Class Lookup (K A M : Type) := lookup: K → M → option A.
Global Hint Mode Lookup - - ! : typeclass_instances.
Instance: Params (@lookup) 4 := {}.
Notation "m !! i" := (lookup i m) (at level 20) : stdpp_scope.
Notation "(!!)" := lookup (only parsing) : stdpp_scope.
Notation "( m !!.)" := (λ i, m !! i) (only parsing) : stdpp_scope.
Notation "(.!! i )" := (lookup i) (only parsing) : stdpp_scope.
Global Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch, assert.

(** The function [lookup_total] should be the total over-approximation
of the partial [lookup] function. *)
Class LookupTotal (K A M : Type) := lookup_total : K → M → A.
Global Hint Mode LookupTotal - - ! : typeclass_instances.
Instance: Params (@lookup_total) 4 := {}.
Notation "m !!! i" := (lookup_total i m) (at level 20) : stdpp_scope.
Notation "(!!!)" := lookup_total (only parsing) : stdpp_scope.
Notation "( m !!!.)" := (λ i, m !!! i) (only parsing) : stdpp_scope.
Notation "(.!!! i )" := (lookup_total i) (only parsing) : stdpp_scope.
Global Arguments lookup_total _ _ _ _ !_ !_ / : simpl nomatch, assert.

(** The singleton map *)
Class SingletonM K A M := singletonM: K → A → M.
Global Hint Mode SingletonM - - ! : typeclass_instances.
Instance: Params (@singletonM) 5 := {}.
Notation "{[ k := a ]}" := (singletonM k a) (at level 1) : stdpp_scope.

(** The function insert [<[k:=a]>m] should update the element at key [k] with
value [a] in [m]. *)
Class Insert (K A M : Type) := insert: K → A → M → M.
Global Hint Mode Insert - - ! : typeclass_instances.
Instance: Params (@insert) 5 := {}.
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : stdpp_scope.
Global Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch, assert.

(** The function delete [delete k m] should delete the value at key [k] in
[m]. If the key [k] is not a member of [m], the original map should be
returned. *)
Class Delete (K M : Type) := delete: K → M → M.
Global Hint Mode Delete - ! : typeclass_instances.
Instance: Params (@delete) 4 := {}.
Global Arguments delete _ _ _ !_ !_ / : simpl nomatch, assert.

(** The function [alter f k m] should update the value at key [k] using the
function [f], which is called with the original value. *)
Class Alter (K A M : Type) := alter: (A → A) → K → M → M.
Global Hint Mode Alter - - ! : typeclass_instances.
Instance: Params (@alter) 5 := {}.
Global Arguments alter {_ _ _ _} _ !_ !_ / : simpl nomatch, assert.

(** The function [partial_alter f k m] should update the value at key [k] using the
function [f], which is called with the original value at key [k] or [None]
if [k] is not a member of [m]. The value at [k] should be deleted if [f]
yields [None]. *)
Class PartialAlter (K A M : Type) :=
  partial_alter: (option A → option A) → K → M → M.
Global Hint Mode PartialAlter - - ! : typeclass_instances.
Instance: Params (@partial_alter) 4 := {}.
Global Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch, assert.

(** The function [dom C m] should yield the domain of [m]. That is a finite
set of type [C] that contains the keys that are a member of [m]. *)
Class Dom (M C : Type) := dom: M → C.
Global Hint Mode Dom ! ! : typeclass_instances.
Instance: Params (@dom) 3 := {}.
Global Arguments dom : clear implicits.
Global Arguments dom {_} _ {_} !_ / : simpl nomatch, assert.

(** The function [merge f m1 m2] should merge the maps [m1] and [m2] by
constructing a new map whose value at key [k] is [f (m1 !! k) (m2 !! k)].*)
Class Merge (M : Type → Type) :=
  merge: ∀ {A B C}, (option A → option B → option C) → M A → M B → M C.
Global Hint Mode Merge ! : typeclass_instances.
Instance: Params (@merge) 4 := {}.
Global Arguments merge _ _ _ _ _ _ !_ !_ / : simpl nomatch, assert.

(** The function [union_with f m1 m2] is supposed to yield the union of [m1]
and [m2] using the function [f] to combine values of members that are in
both [m1] and [m2]. *)
Class UnionWith (A M : Type) :=
  union_with: (A → A → option A) → M → M → M.
Global Hint Mode UnionWith - ! : typeclass_instances.
Instance: Params (@union_with) 3 := {}.
Global Arguments union_with {_ _ _} _ !_ !_ / : simpl nomatch, assert.

(** Similarly for intersection and difference. *)
Class IntersectionWith (A M : Type) :=
  intersection_with: (A → A → option A) → M → M → M.
Global Hint Mode IntersectionWith - ! : typeclass_instances.
Instance: Params (@intersection_with) 3 := {}.
Global Arguments intersection_with {_ _ _} _ !_ !_ / : simpl nomatch, assert.

Class DifferenceWith (A M : Type) :=
  difference_with: (A → A → option A) → M → M → M.
Global Hint Mode DifferenceWith - ! : typeclass_instances.
Instance: Params (@difference_with) 3 := {}.
Global Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch, assert.

Definition intersection_with_list `{IntersectionWith A M}
  (f : A → A → option A) : M → list M → M := fold_right (intersection_with f).
Global Arguments intersection_with_list _ _ _ _ _ !_ / : assert.

(** * Notations for lattices. *)
(** SqSubsetEq registers the "canonical" partial order for a type, and is used
for the \sqsubseteq symbol. *)
Class SqSubsetEq A := sqsubseteq: relation A.
Global Hint Mode SqSubsetEq ! : typeclass_instances.
Instance: Params (@sqsubseteq) 2 := {}.
Infix "⊑" := sqsubseteq (at level 70) : stdpp_scope.
Notation "(⊑)" := sqsubseteq (only parsing) : stdpp_scope.
Notation "( x ⊑.)" := (sqsubseteq x) (only parsing) : stdpp_scope.
Notation "(.⊑ y )" := (λ x, sqsubseteq x y) (only parsing) : stdpp_scope.

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

Global Instance sqsubseteq_rewrite `{SqSubsetEq A} : RewriteRelation (⊑@{A}) := {}.

Global Hint Extern 0 (_ ⊑ _) => reflexivity : core.

Class Meet A := meet: A → A → A.
Global Hint Mode Meet ! : typeclass_instances.
Instance: Params (@meet) 2 := {}.
Infix "⊓" := meet (at level 40) : stdpp_scope.
Notation "(⊓)" := meet (only parsing) : stdpp_scope.
Notation "( x ⊓.)" := (meet x) (only parsing) : stdpp_scope.
Notation "(.⊓ y )" := (λ x, meet x y) (only parsing) : stdpp_scope.

Class Join A := join: A → A → A.
Global Hint Mode Join ! : typeclass_instances.
Instance: Params (@join) 2 := {}.
Infix "⊔" := join (at level 50) : stdpp_scope.
Notation "(⊔)" := join (only parsing) : stdpp_scope.
Notation "( x ⊔.)" := (join x) (only parsing) : stdpp_scope.
Notation "(.⊔ y )" := (λ x, join x y) (only parsing) : stdpp_scope.

Class Top A := top : A.
Global Hint Mode Top ! : typeclass_instances.
Notation "⊤" := top (format "⊤") : stdpp_scope.

Class Bottom A := bottom : A.
Global Hint Mode Bottom ! : typeclass_instances.
Notation "⊥" := bottom (format "⊥") : stdpp_scope.


(** * Axiomatization of sets *)
(** The classes [SemiSet A C], [Set_ A C], and [TopSet A C] axiomatize sets of
type [C] with elements of type [A]. The first class, [SemiSet] does not include
intersection and difference. It is useful for the case of lists, where decidable
equality is needed to implement intersection and difference, but not union.

Note that we cannot use the name [Set] since that is a reserved keyword. Hence
we use [Set_]. *)
Class SemiSet A C `{ElemOf A C,
    Empty C, Singleton A C, Union C} : Prop := {
  not_elem_of_empty (x : A) : x ∉@{C} ∅; (* We prove
  [elem_of_empty : x ∈@{C} ∅ ↔ False] in [sets.v], which is more convenient for
  rewriting. *)
  elem_of_singleton (x y : A) : x ∈@{C} {[ y ]} ↔ x = y;
  elem_of_union (X Y : C) (x : A) : x ∈ X ∪ Y ↔ x ∈ X ∨ x ∈ Y
}.
Class Set_ A C `{ElemOf A C, Empty C, Singleton A C,
    Union C, Intersection C, Difference C} : Prop := {
  set_semi_set :> SemiSet A C;
  elem_of_intersection (X Y : C) (x : A) : x ∈ X ∩ Y ↔ x ∈ X ∧ x ∈ Y;
  elem_of_difference (X Y : C) (x : A) : x ∈ X ∖ Y ↔ x ∈ X ∧ x ∉ Y
}.
Class TopSet A C `{ElemOf A C, Empty C, Top C, Singleton A C,
    Union C, Intersection C, Difference C} : Prop := {
  top_set_set :> Set_ A C;
  elem_of_top' (x : A) : x ∈@{C} ⊤; (* We prove [elem_of_top : x ∈@{C} ⊤ ↔ True]
  in [sets.v], which is more convenient for rewriting. *)
}.

(** We axiomative a finite set as a set whose elements can be
enumerated as a list. These elements, given by the [elements] function, may be
in any order and should not contain duplicates. *)
Class Elements A C := elements: C → list A.
Global Hint Mode Elements - ! : typeclass_instances.
Instance: Params (@elements) 3 := {}.

(** We redefine the standard library's [In] and [NoDup] using type classes. *)
Inductive elem_of_list {A} : ElemOf A (list A) :=
  | elem_of_list_here (x : A) l : x ∈ x :: l
  | elem_of_list_further (x y : A) l : x ∈ l → x ∈ y :: l.
Existing Instance elem_of_list.

Lemma elem_of_list_In {A} (l : list A) x : x ∈ l ↔ In x l.
Proof.
  split.
  - induction 1; simpl; auto.
  - induction l; destruct 1; subst; constructor; auto.
Qed.

Inductive NoDup {A} : list A → Prop :=
  | NoDup_nil_2 : NoDup []
  | NoDup_cons_2 x l : x ∉ l → NoDup l → NoDup (x :: l).

Lemma NoDup_ListNoDup {A} (l : list A) : NoDup l ↔ List.NoDup l.
Proof.
  split.
  - induction 1; constructor; rewrite <-?elem_of_list_In; auto.
  - induction 1; constructor; rewrite ?elem_of_list_In; auto.
Qed.

(** Decidability of equality of the carrier set is admissible, but we add it
anyway so as to avoid cycles in type class search. *)
Class FinSet A C `{ElemOf A C, Empty C, Singleton A C, Union C,
    Intersection C, Difference C, Elements A C, EqDecision A} : Prop := {
  fin_set_set :> Set_ A C;
  elem_of_elements (X : C) x : x ∈ elements X ↔ x ∈ X;
  NoDup_elements (X : C) : NoDup (elements X)
}.
Class Size C := size: C → nat.
Global Hint Mode Size ! : typeclass_instances.
Global Arguments size {_ _} !_ / : simpl nomatch, assert.
Instance: Params (@size) 2 := {}.

(** The class [MonadSet M] axiomatizes a type constructor [M] that can be
used to construct a set [M A] with elements of type [A]. The advantage
of this class, compared to [Set_], is that it also axiomatizes the
the monadic operations. The disadvantage, is that not many inhabits are
possible (we will only provide an inhabitant using unordered lists without
duplicates). More interesting implementations typically need
decidable equality, or a total order on the elements, which do not fit
in a type constructor of type [Type → Type]. *)
Class MonadSet M `{∀ A, ElemOf A (M A),
    ∀ A, Empty (M A), ∀ A, Singleton A (M A), ∀ A, Union (M A),
    !MBind M, !MRet M, !FMap M, !MJoin M} : Prop := {
  monad_set_semi_set A :> SemiSet A (M A);
  elem_of_bind {A B} (f : A → M B) (X : M A) (x : B) :
    x ∈ X ≫= f ↔ ∃ y, x ∈ f y ∧ y ∈ X;
  elem_of_ret {A} (x y : A) : x ∈@{M A} mret y ↔ x = y;
  elem_of_fmap {A B} (f : A → B) (X : M A) (x : B) :
    x ∈ f <$> X ↔ ∃ y, x = f y ∧ y ∈ X;
  elem_of_join {A} (X : M (M A)) (x : A) :
    x ∈ mjoin X ↔ ∃ Y : M A, x ∈ Y ∧ Y ∈ X
}.

(** The [Infinite A] class axiomatizes types [A] with infinitely many elements.
It contains a function [fresh : list A → A] that given a list [xs] gives an
element [fresh xs ∉ xs].

We do not directly make [fresh] a field of the [Infinite] class, but use a
separate operational type class [Fresh] for it. That way we can overload [fresh]
to pick fresh elements from other data structure like sets. See the file
[fin_sets], where we define [fresh : C → A] for any finite set implementation
[FinSet C A].

Note: we require [fresh] to respect permutations, which is needed to define the
aforementioned [fresh] function on finite sets that respects set equality.

Instead of instantiating [Infinite] directly, consider using [max_infinite] or
[inj_infinite] from the [infinite] module. *)
Class Fresh A C := fresh: C → A.
Global Hint Mode Fresh - ! : typeclass_instances.
Instance: Params (@fresh) 3 := {}.
Global Arguments fresh : simpl never.

Class Infinite A := {
  infinite_fresh :> Fresh A (list A);
  infinite_is_fresh (xs : list A) : fresh xs ∉ xs;
  infinite_fresh_Permutation :> Proper (@Permutation A ==> (=)) fresh;
}.
Global Arguments infinite_fresh : simpl never.

(** * Miscellaneous *)
Class Half A := half: A → A.
Global Hint Mode Half ! : typeclass_instances.
Notation "½" := half (format "½") : stdpp_scope.
Notation "½*" := (fmap (M:=list) half) : stdpp_scope.