base.v

(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects type class interfaces, notations, and general theorems
that are used throughout the whole development. Most importantly it contains
abstract interfaces for ordered structures, collections, and various other data
structures. *)
Global Generalizable All Variables.
Global Set Automatic Coercions Import.
Global Set Asymmetric Patterns.
Global Unset Transparent Obligations.
From Coq Require Export Morphisms RelationClasses List Bool Utf8 Setoid.
Export ListNotations.
From Coq.Program Require Export Basics Syntax.
Obligation Tactic := idtac.

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

(** 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 : type_scope.
Notation "'False'" := False : type_scope.


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

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

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

(** ** Setoid equality *)
(** We define an operational type class for setoid equality. This is based on
(Spitters/van der Weegen, 2011). *)
Class Equiv A := equiv: relation A.
Infix "≡" := equiv (at level 70, no associativity) : C_scope.
Notation "(≡)" := equiv (only parsing) : C_scope.
Notation "( X ≡)" := (equiv X) (only parsing) : C_scope.
Notation "(≡ X )" := (λ Y, Y ≡ X) (only parsing) : C_scope.
Notation "(≢)" := (λ X Y, ¬X ≡ Y) (only parsing) : C_scope.
Notation "X ≢ Y":= (¬X ≡ Y) (at level 70, no associativity) : C_scope.
Notation "( X ≢)" := (λ Y, X ≢ Y) (only parsing) : C_scope.
Notation "(≢ X )" := (λ Y, Y ≢ X) (only parsing) : C_scope.

(** The type class [LeibnizEquiv] collects setoid equalities that coincide
with Leibniz equality. We provide the tactic [fold_leibniz] to transform such
setoid equalities into Leibniz equalities, and [unfold_leibniz] for the
reverse. *)
Class LeibnizEquiv A `{Equiv A} := leibniz_equiv x y : x ≡ y → x = y.
Lemma leibniz_equiv_iff `{LeibnizEquiv A, !Reflexive (@equiv 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 [ @equiv ?A _ _ _ ] |- _ =>
    setoid_rewrite (leibniz_equiv_iff (A:=A)) in H
  | |- context [ @equiv ?A _ _ _ ] =>
    setoid_rewrite (leibniz_equiv_iff (A:=A))
  end.
Ltac unfold_leibniz := repeat
  match goal with
  | H : context [ @eq ?A _ _ ] |- _ =>
    setoid_rewrite <-(leibniz_equiv_iff (A:=A)) in H
  | |- context [ @eq ?A _ _ ] =>
    setoid_rewrite <-(leibniz_equiv_iff (A:=A))
  end.

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

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

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


(** * Type classes *)
(** ** Decidable propositions *)
(** This type class by (Spitters/van der Weegen, 2011) collects decidable
propositions. For example to declare a parameter expressing decidable equality
on a type [A] we write [`{∀ x y : A, Decision (x = y)}] and use it by writing
[decide (x = y)]. *)
Class Decision (P : Prop) := decide : {P} + {¬P}.
Arguments decide _ {_}.
Notation EqDecision A := (∀ x y : A, Decision (x = y)).

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

(** ** 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.

(** ** 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}.

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

Lemma not_symmetry `{R : relation A, !Symmetric R} x y : ¬R x y → ¬R y x.
Proof. intuition. Qed.
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} x y : R x y ↔ R y x.
Proof. intuition. Qed.

Lemma not_inj `{Inj A B R R' f} x y : ¬R x y → ¬R' (f x) (f y).
Proof. intuition. Qed.
Lemma not_inj2_1 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 :
  ¬R x1 x2 → ¬R'' (f x1 y1) (f x2 y2).
Proof. intros HR HR''. destruct (inj2 f x1 y1 x2 y2); auto. Qed.
Lemma not_inj2_2 `{Inj2 A B C R R' R'' f} x1 x2 y1 y2 :
  ¬R' y1 y2 → ¬R'' (f x1 y1) (f x2 y2).
Proof. intros HR' HR''. destruct (inj2 f x1 y1 x2 y2); auto. Qed.

Lemma inj_iff {A B} {R : relation A} {S : relation B} (f : A → B)
  `{!Inj R S f} `{!Proper (R ==> S) f} x y : S (f x) (f y) ↔ R x y.
Proof. firstorder. Qed.
Instance inj2_inj_1 `{Inj2 A B C R1 R2 R3 f} y : Inj R1 R3 (λ x, f x y).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.
Instance inj2_inj_2 `{Inj2 A B C R1 R2 R3 f} x : Inj R2 R3 (f x).
Proof. repeat intro; edestruct (inj2 f); eauto. Qed.

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

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

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

(** * Logic *)
Notation "(∧)" := and (only parsing) : C_scope.
Notation "( A ∧)" := (and A) (only parsing) : C_scope.
Notation "(∧ B )" := (λ A, A ∧ B) (only parsing) : C_scope.

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

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

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

Lemma or_l P Q : ¬Q → P ∨ Q ↔ P.
Proof. tauto. Qed.
Lemma or_r P Q : ¬P → P ∨ Q ↔ Q.
Proof. tauto. Qed.
Lemma and_wlog_l (P Q : Prop) : (Q → P) → Q → (P ∧ Q).
Proof. tauto. Qed.
Lemma and_wlog_r (P Q : Prop) : P → (P → Q) → (P ∧ Q).
Proof. tauto. Qed.
Lemma impl_transitive (P Q R : Prop) : (P → Q) → (Q → R) → (P → R).
Proof. tauto. Qed.
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.

Instance: Comm (↔) (@eq A).
Proof. red; intuition. Qed.
Instance: Comm (↔) (λ x y, @eq A y x).
Proof. red; intuition. Qed.
Instance: Comm (↔) (↔).
Proof. red; intuition. Qed.
Instance: Comm (↔) (∧).
Proof. red; intuition. Qed.
Instance: Assoc (↔) (∧).
Proof. red; intuition. Qed.
Instance: IdemP (↔) (∧).
Proof. red; intuition. Qed.
Instance: Comm (↔) (∨).
Proof. red; intuition. Qed.
Instance: Assoc (↔) (∨).
Proof. red; intuition. Qed.
Instance: IdemP (↔) (∨).
Proof. red; intuition. Qed.
Instance: LeftId (↔) True (∧).
Proof. red; intuition. Qed.
Instance: RightId (↔) True (∧).
Proof. red; intuition. Qed.
Instance: LeftAbsorb (↔) False (∧).
Proof. red; intuition. Qed.
Instance: RightAbsorb (↔) False (∧).
Proof. red; intuition. Qed.
Instance: LeftId (↔) False (∨).
Proof. red; intuition. Qed.
Instance: RightId (↔) False (∨).
Proof. red; intuition. Qed.
Instance: LeftAbsorb (↔) True (∨).
Proof. red; intuition. Qed.
Instance: RightAbsorb (↔) True (∨).
Proof. red; intuition. Qed.
Instance: LeftId (↔) True impl.
Proof. unfold impl. red; intuition. Qed.
Instance: RightAbsorb (↔) True impl.
Proof. unfold impl. red; intuition. Qed.


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

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

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

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

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

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

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

Instance id_inj {A} : Inj (=) (=) (@id A).
Proof. intros ??; auto. Qed.
Instance compose_inj {A B C} R1 R2 R3 (f : A → B) (g : B → C) :
  Inj R1 R2 f → Inj R2 R3 g → Inj R1 R3 (g ∘ f).
Proof. red; intuition. Qed.

Instance id_surj {A} : Surj (=) (@id A).
Proof. intros y; exists y; reflexivity. Qed.
Instance compose_surj {A B C} R (f : A → B) (g : B → C) :
  Surj (=) f → Surj R g → Surj R (g ∘ f).
Proof.
  intros ?? x. unfold compose. destruct (surj g x) as [y ?].
  destruct (surj f y) as [z ?]. exists z. congruence.
Qed.

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

(** ** Lists *)
Instance list_inhabited {A} : Inhabited (list A) := populate [].

Definition zip_with {A B C} (f : A → B → C) : list A → list B → list C :=
  fix go l1 l2 :=
  match l1, l2 with x1 :: l1, x2 :: l2 => f x1 x2 :: go l1 l2 | _ , _ => [] end.
Notation zip := (zip_with pair).

(** ** Booleans *)
(** The following coercion allows us to use Booleans as propositions. *)
Coercion Is_true : bool >-> Sortclass.
Hint Unfold Is_true.
Hint Immediate Is_true_eq_left.
Hint Resolve orb_prop_intro andb_prop_intro.
Notation "(&&)" := andb (only parsing).
Notation "(||)" := orb (only parsing).
Infix "&&*" := (zip_with (&&)) (at level 40).
Infix "||*" := (zip_with (||)) (at level 50).

Instance bool_inhabated : Inhabited bool := populate true.

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

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

Notation "p .1" := (fst p) (at level 10, format "p .1").
Notation "p .2" := (snd p) (at level 10, format "p .2").

Instance: Params (@pair) 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 prod_map {A A' B B'} (f: A → A') (g: B → B') (p : A * B) : A' * B' :=
  (f (p.1), g (p.2)).
Arguments prod_map {_ _ _ _} _ _ !_ /.

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)).
Arguments prod_zip {_ _ _ _ _ _} _ _ !_ !_ /.

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.

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.

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.

Instance prod_equiv `{Equiv A,Equiv B} : Equiv (A * B) := prod_relation (≡) (≡).
Instance pair_proper `{Equiv A, Equiv B} :
  Proper ((≡) ==> (≡) ==> (≡)) (@pair A B) := _.
Instance pair_equiv_inj `{Equiv A, Equiv B} : Inj2 (≡) (≡) (≡) (@pair A B) := _.
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.

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.
Arguments sum_map {_ _ _ _} _ _ !_ /.

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

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

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.

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) := _.
Instance inl_equiv_inj `{Equiv A, Equiv B} : Inj (≡) (≡) (@inl A B) := _.
Instance inr_equiv_inj `{Equiv A, Equiv B} : Inj (≡) (≡) (@inr A B) := _.
Typeclasses Opaque sum_equiv.

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

(** ** Sigma types *)
Arguments existT {_ _} _ _.
Arguments proj1_sig {_ _} _.
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
Notation "` x" := (proj1_sig x) (at level 10, format "` x") : C_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.
Arguments sig_map _ _ _ _ _ _ !_ /.


(** * Operations on collections *)
(** We define operational type classes for the traditional operations and
relations on collections: the empty collection [∅], the union [(∪)],
intersection [(∩)], and difference [(∖)], the singleton [{[_]}], the subset
[(⊆)] and element of [(∈)] relation, and disjointess [(⊥)]. *)
Class Empty A := empty: A.
Notation "∅" := empty : C_scope.

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

Class Top A := top : A.
Notation "⊤" := top : C_scope.

Class Union A := union: A → A → A.
Instance: Params (@union) 2.
Infix "∪" := union (at level 50, left associativity) : C_scope.
Notation "(∪)" := union (only parsing) : C_scope.
Notation "( x ∪)" := (union x) (only parsing) : C_scope.
Notation "(∪ x )" := (λ y, union y x) (only parsing) : C_scope.
Infix "∪*" := (zip_with (∪)) (at level 50, left associativity) : C_scope.
Notation "(∪*)" := (zip_with (∪)) (only parsing) : C_scope.
Infix "∪**" := (zip_with (zip_with (∪)))
  (at level 50, left associativity) : C_scope.
Infix "∪*∪**" := (zip_with (prod_zip (∪) (∪*)))
  (at level 50, left associativity) : C_scope.

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

Class Intersection A := intersection: A → A → A.
Instance: Params (@intersection) 2.
Infix "∩" := intersection (at level 40) : C_scope.
Notation "(∩)" := intersection (only parsing) : C_scope.
Notation "( x ∩)" := (intersection x) (only parsing) : C_scope.
Notation "(∩ x )" := (λ y, intersection y x) (only parsing) : C_scope.

Class Difference A := difference: A → A → A.
Instance: Params (@difference) 2.
Infix "∖" := difference (at level 40, left associativity) : C_scope.
Notation "(∖)" := difference (only parsing) : C_scope.
Notation "( x ∖)" := (difference x) (only parsing) : C_scope.
Notation "(∖ x )" := (λ y, difference y x) (only parsing) : C_scope.
Infix "∖*" := (zip_with (∖)) (at level 40, left associativity) : C_scope.
Notation "(∖*)" := (zip_with (∖)) (only parsing) : C_scope.
Infix "∖**" := (zip_with (zip_with (∖)))
  (at level 40, left associativity) : C_scope.
Infix "∖*∖**" := (zip_with (prod_zip (∖) (∖*)))
  (at level 50, left associativity) : C_scope.

Class Singleton A B := singleton: A → B.
Instance: Params (@singleton) 3.
Notation "{[ x ]}" := (singleton x) (at level 1) : C_scope.
Notation "{[ x ; y ; .. ; z ]}" :=
  (union .. (union (singleton x) (singleton y)) .. (singleton z))
  (at level 1) : C_scope.
Notation "{[ x , y ]}" := (singleton (x,y))
  (at level 1, y at next level) : C_scope.
Notation "{[ x , y , z ]}" := (singleton (x,y,z))
  (at level 1, y at next level, z at next level) : C_scope.

Class SubsetEq A := subseteq: relation A.
Instance: Params (@subseteq) 2.
Infix "⊆" := subseteq (at level 70) : C_scope.
Notation "(⊆)" := subseteq (only parsing) : C_scope.
Notation "( X ⊆ )" := (subseteq X) (only parsing) : C_scope.
Notation "( ⊆ X )" := (λ Y, Y ⊆ X) (only parsing) : C_scope.
Notation "X ⊈ Y" := (¬X ⊆ Y) (at level 70) : C_scope.
Notation "(⊈)" := (λ X Y, X ⊈ Y) (only parsing) : C_scope.
Notation "( X ⊈ )" := (λ Y, X ⊈ Y) (only parsing) : C_scope.
Notation "( ⊈ X )" := (λ Y, Y ⊈ X) (only parsing) : C_scope.
Infix "⊆*" := (Forall2 (⊆)) (at level 70) : C_scope.
Notation "(⊆*)" := (Forall2 (⊆)) (only parsing) : C_scope.
Infix "⊆**" := (Forall2 (⊆*)) (at level 70) : C_scope.
Infix "⊆1*" := (Forall2 (λ p q, p.1 ⊆ q.1)) (at level 70) : C_scope.
Infix "⊆2*" := (Forall2 (λ p q, p.2 ⊆ q.2)) (at level 70) : C_scope.
Infix "⊆1**" := (Forall2 (λ p q, p.1 ⊆* q.1)) (at level 70) : C_scope.
Infix "⊆2**" := (Forall2 (λ p q, p.2 ⊆* q.2)) (at level 70) : C_scope.

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

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

Notation "X ⊆ Y ⊆ Z" := (X ⊆ Y ∧ Y ⊆ Z) (at level 70, Y at next level) : C_scope.
Notation "X ⊆ Y ⊂ Z" := (X ⊆ Y ∧ Y ⊂ Z) (at level 70, Y at next level) : C_scope.
Notation "X ⊂ Y ⊆ Z" := (X ⊂ Y ∧ Y ⊆ Z) (at level 70, Y at next level) : C_scope.
Notation "X ⊂ Y ⊂ Z" := (X ⊂ Y ∧ Y ⊂ Z) (at level 70, Y at next level) : C_scope.

(** 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.

Class ElemOf A B := elem_of: A → B → Prop.
Instance: Params (@elem_of) 3.
Infix "∈" := elem_of (at level 70) : C_scope.
Notation "(∈)" := elem_of (only parsing) : C_scope.
Notation "( x ∈)" := (elem_of x) (only parsing) : C_scope.
Notation "(∈ X )" := (λ x, elem_of x X) (only parsing) : C_scope.
Notation "x ∉ X" := (¬x ∈ X) (at level 80) : C_scope.
Notation "(∉)" := (λ x X, x ∉ X) (only parsing) : C_scope.
Notation "( x ∉)" := (λ X, x ∉ X) (only parsing) : C_scope.
Notation "(∉ X )" := (λ x, x ∉ X) (only parsing) : C_scope.

Class Disjoint A := disjoint : A → A → Prop.
Instance: Params (@disjoint) 2.
Infix "⊥" := disjoint (at level 70) : C_scope.
Notation "(⊥)" := disjoint (only parsing) : C_scope.
Notation "( X ⊥.)" := (disjoint X) (only parsing) : C_scope.
Notation "(.⊥ X )" := (λ Y, Y ⊥ X) (only parsing) : C_scope.
Infix "⊥*" := (Forall2 (⊥)) (at level 70) : C_scope.
Notation "(⊥*)" := (Forall2 (⊥)) (only parsing) : C_scope.
Infix "⊥**" := (Forall2 (⊥*)) (at level 70) : C_scope.
Infix "⊥1*" := (Forall2 (λ p q, p.1 ⊥ q.1)) (at level 70) : C_scope.
Infix "⊥2*" := (Forall2 (λ p q, p.2 ⊥ q.2)) (at level 70) : C_scope.
Infix "⊥1**" := (Forall2 (λ p q, p.1 ⊥* q.1)) (at level 70) : C_scope.
Infix "⊥2**" := (Forall2 (λ p q, p.2 ⊥* q.2)) (at level 70) : C_scope.
Hint Extern 0 (_ ⊥ _) => symmetry; eassumption.
Hint Extern 0 (_ ⊥* _) => symmetry; eassumption.

Class DisjointE E A := disjointE : E → A → A → Prop.
Instance: Params (@disjointE) 4.
Notation "X ⊥{ Γ } Y" := (disjointE Γ X Y)
  (at level 70, format "X  ⊥{ Γ }  Y") : C_scope.
Notation "(⊥{ Γ } )" := (disjointE Γ) (only parsing, Γ at level 1) : C_scope.
Notation "Xs ⊥{ Γ }* Ys" := (Forall2 (⊥{Γ}) Xs Ys)
  (at level 70, format "Xs  ⊥{ Γ }*  Ys") : C_scope.
Notation "(⊥{ Γ }* )" := (Forall2 (⊥{Γ}))
  (only parsing, Γ at level 1) : C_scope.
Notation "X ⊥{ Γ1 , Γ2 , .. , Γ3 } Y" := (disjoint (pair .. (Γ1, Γ2) .. Γ3) X Y)
  (at level 70, format "X  ⊥{ Γ1 , Γ2 , .. , Γ3 }  Y") : C_scope.
Notation "Xs ⊥{ Γ1 , Γ2 , .. , Γ3 }* Ys" :=
  (Forall2 (disjoint (pair .. (Γ1, Γ2) .. Γ3)) Xs Ys)
  (at level 70, format "Xs  ⊥{ Γ1 ,  Γ2 , .. , Γ3 }*  Ys") : C_scope.
Hint Extern 0 (_ ⊥{_} _) => symmetry; eassumption.

Class DisjointList A := disjoint_list : list A → Prop.
Instance: Params (@disjoint_list) 2.
Notation "⊥ Xs" := (disjoint_list Xs) (at level 20, format "⊥  Xs") : C_scope.

Section disjoint_list.
  Context `{Disjoint A, Union A, Empty A}.
  Inductive disjoint_list_default : DisjointList A :=
    | disjoint_nil_2 : ⊥ (@nil A)
    | disjoint_cons_2 (X : A) (Xs : list A) : X ⊥ ⋃ Xs → ⊥ Xs → ⊥ (X :: Xs).
  Global Existing Instance disjoint_list_default.

  Lemma disjoint_list_nil  : ⊥ @nil A ↔ True.
  Proof. split; constructor. Qed.
  Lemma disjoint_list_cons X Xs : ⊥ (X :: Xs) ↔ X ⊥ ⋃ Xs ∧ ⊥ Xs.
  Proof. split. inversion_clear 1; auto. intros [??]. constructor; auto. Qed.
End disjoint_list.

Class Filter A B := filter: ∀ (P : A → Prop) `{∀ x, Decision (P x)}, B → B.


(** * 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.
Arguments mret {_ _ _} _.
Instance: Params (@mret) 3.
Class MBind (M : Type → Type) := mbind : ∀ {A B}, (A → M B) → M A → M B.
Arguments mbind {_ _ _ _} _ !_ /.
Instance: Params (@mbind) 4.
Class MJoin (M : Type → Type) := mjoin: ∀ {A}, M (M A) → M A.
Arguments mjoin {_ _ _} !_ /.
Instance: Params (@mjoin) 3.
Class FMap (M : Type → Type) := fmap : ∀ {A B}, (A → B) → M A → M B.
Arguments fmap {_ _ _ _} _ !_ /.
Instance: Params (@fmap) 4.
Class OMap (M : Type → Type) := omap: ∀ {A B}, (A → option B) → M A → M B.
Arguments omap {_ _ _ _} _ !_ /.
Instance: Params (@omap) 4.

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

Notation "x ← y ; z" := (y ≫= (λ x : _, z))
  (at level 65, only parsing, right associativity) : C_scope.
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.
Notation "' ( x1 , x2 ) ← y ; z" :=
  (y ≫= (λ x : _, let ' (x1, x2) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "' ( x1 , x2 , x3 ) ← y ; z" :=
  (y ≫= (λ x : _, let ' (x1,x2,x3) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "' ( x1 , x2 , x3  , x4 ) ← y ; z" :=
  (y ≫= (λ x : _, let ' (x1,x2,x3,x4) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "' ( x1 , x2 , x3  , x4 , x5 ) ← y ; z" :=
  (y ≫= (λ x : _, let ' (x1,x2,x3,x4,x5) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "' ( x1 , x2 , x3  , x4 , x5 , x6 ) ← y ; z" :=
  (y ≫= (λ x : _, let ' (x1,x2,x3,x4,x5,x6) := x in z))
  (at level 65, only parsing, right associativity) : C_scope.

Notation "ps .*1" := (fmap (M:=list) fst ps)
  (at level 10, format "ps .*1").
Notation "ps .*2" := (fmap (M:=list) snd ps)
  (at level 10, format "ps .*2").

Class MGuard (M : Type → Type) :=
  mguard: ∀ P {dec : Decision P} {A}, (P → M A) → M A.
Arguments mguard _ _ _ !_ _ _ /.
Notation "'guard' P ; o" := (mguard P (λ _, o))
  (at level 65, only parsing, right associativity) : C_scope.
Notation "'guard' P 'as' H ; o" := (mguard P (λ H, o))
  (at level 65, only parsing, right associativity) : C_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.
Instance: Params (@lookup) 4.
Notation "m !! i" := (lookup i m) (at level 20) : C_scope.
Notation "(!!)" := lookup (only parsing) : C_scope.
Notation "( m !!)" := (λ i, m !! i) (only parsing) : C_scope.
Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.

(** The singleton map *)
Class SingletonM K A M := singletonM: K → A → M.
Instance: Params (@singletonM) 5.
Notation "{[ k := a ]}" := (singletonM k a) (at level 1) : C_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.
Instance: Params (@insert) 5.
Notation "<[ k := a ]>" := (insert k a)
  (at level 5, right associativity, format "<[ k := a ]>") : C_scope.
Arguments insert _ _ _ _ !_ _ !_ / : simpl nomatch.

(** 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.
Instance: Params (@delete) 4.
Arguments delete _ _ _ !_ !_ / : simpl nomatch.

(** 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.
Instance: Params (@alter) 5.
Arguments alter {_ _ _ _} _ !_ !_ / : simpl nomatch.

(** The function [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.
Instance: Params (@partial_alter) 4.
Arguments partial_alter _ _ _ _ _ !_ !_ / : simpl nomatch.

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

(** 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.
Instance: Params (@merge) 4.
Arguments merge _ _ _ _ _ _ !_ !_ / : simpl nomatch.

(** 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.
Instance: Params (@union_with) 3.
Arguments union_with {_ _ _} _ !_ !_ / : simpl nomatch.

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

Class DifferenceWith (A M : Type) :=
  difference_with: (A → A → option A) → M → M → M.
Instance: Params (@difference_with) 3.
Arguments difference_with {_ _ _} _ !_ !_ / : simpl nomatch.

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

Class LookupE (E K A M : Type) := lookupE: E → K → M → option A.
Instance: Params (@lookupE) 6.
Notation "m !!{ Γ } i" := (lookupE Γ i m)
  (at level 20, format "m  !!{ Γ }  i") : C_scope.
Notation "(!!{ Γ } )" := (lookupE Γ) (only parsing, Γ at level 1) : C_scope.
Arguments lookupE _ _ _ _ _ _ !_ !_ / : simpl nomatch.

Class InsertE (E K A M : Type) := insertE: E → K → A → M → M.
Instance: Params (@insertE) 6.
Notation "<[ k := a ]{ Γ }>" := (insertE Γ k a)
  (at level 5, right associativity, format "<[ k := a ]{ Γ }>") : C_scope.
Arguments insertE _ _ _ _ _ _ !_ _ !_ / : simpl nomatch.


(** * Axiomatization of collections *)
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
Class SimpleCollection A C `{ElemOf A C,
    Empty C, Singleton A C, Union C} : Prop := {
  not_elem_of_empty (x : A) : x ∉ ∅;
  elem_of_singleton (x y : A) : x ∈ {[ y ]} ↔ x = y;
  elem_of_union X Y (x : A) : x ∈ X ∪ Y ↔ x ∈ X ∨ x ∈ Y
}.