Library stdpp.base

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.

From Coq Require Export Morphisms RelationClasses List Bool Utf8 Setoid.
From Coq Require Import Permutation.
Set Default Proof Using "Type".
Export ListNotations.
From Coq.Program Require Export Basics Syntax.

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 taht 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.
Arguments unseal {_ _} _ : assert.
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 }.
Hint Extern 0 (TCNoBackTrack _) ⇒ constructor; apply _ : typeclass_instances.

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.

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.
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.

Inductive TCAnd (P1 P2 : Prop) : Prop := TCAnd_intro : P1 → P2 → TCAnd P1 P2.
Existing Class TCAnd.
Existing Instance TCAnd_intro.

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.

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.

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.

Inductive TCEq {A} (x : A) : A → Prop := TCEq_refl : TCEq x x.
Existing Class TCEq.
Existing Instance TCEq_refl.

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.

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.

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.

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.

Hint Extern 0 (_ = _) ⇒ reflexivity : core.
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.

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.
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.

Instance equiv_default_relation `{Equiv A} : DefaultRelation (≡) | 3 := {}.
Hint Extern 0 (_ ≡ _) ⇒ reflexivity : core.
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 }.
Hint Mode Decision ! : typeclass_instances.
Arguments decide _ {_} : simpl never, assert.

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

It allows us to control Hint Mode more precisely. In particular, if it were defined as a notation, the above Hint Mode for Decision would not prevent diverging instance search when looking for RelDecision (@eq ?A), which would result in it looking for Decision (@eq ?A x y), i.e. an instance where the head position of Decision is not en evar.
We use it to avoid inefficient computation due to eager evaluation of propositions by vm_compute. This inefficiency arises for example if (x = y) := (f x = f y). Since decide (x = y) evaluates to decide (f x = f y), this would then lead to evaluation of f x and f y. Using the RelDecision, the f is hidden under a lambda, which prevents unnecessary evaluation.

Class RelDecision {A B} (R : A → B → Prop) :=
decide_rel x y :> Decision (R x y).
Hint Mode RelDecision ! ! ! : typeclass_instances.
Arguments decide_rel {_ _} _ {_} _ _ : simpl never, assert.
Notation EqDecision A := (RelDecision (=@{A})).

Inhabited types

This type class collects types that are inhabited.

Class Inhabited (A : Type) : Type := populate { inhabitant : A }.
Hint Mode Inhabited ! : typeclass_instances.
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.
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.

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

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. ∃ (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

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.

Hint Extern 0 (_ ↔ _) ⇒ reflexivity : core.
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.

Instance: Comm (↔) (=@{A}).
Proof. red; intuition. Qed.
Instance: Comm (↔) (λ x y, y =@{A} 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) : 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.

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 _ _ / : assert.
Arguments compose _ _ _ _ _ _ / : assert.
Arguments flip _ _ _ _ _ _ / : assert.
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.

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; ∃ 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 ?]. ∃ 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 : core.
Hint Immediate Is_true_eq_left : core.
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).

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 (≡@{unit }).
Proof. repeat split. Qed.
Instance unit_leibniz : LeibnizEquiv unit.
Proof. intros [] []; reflexivity. Qed.
Instance unit_inhabited: Inhabited unit := populate ().

Empty

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

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

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 {_ _ _ _} _ _ !_ / : assert.

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 {_ _} _ _ : assert.
Arguments projT1 {_ _} _ : assert.
Arguments projT2 {_ _} _ : assert.

Arguments exist {_} _ _ _ : assert.
Arguments proj1_sig {_ _} _ : assert.
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.
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.
Hint Mode Empty ! : typeclass_instances.
Notation "∅" := empty (format "∅") : stdpp_scope.

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

Class Union A := union: A → A → A.
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.
Infix "∪**" := (zip_with (zip_with (∪)))
  (at level 50, left associativity) : stdpp_scope.
Infix "∪*∪**" := (zip_with (prod_zip (∪) (∪*)))
  (at level 50, left associativity) : stdpp_scope.

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

Class DisjUnion A := disj_union: A → A → A.
Hint Mode DisjUnion ! : typeclass_instances.
Instance: Params (@disj_union) 2 := {}.
Infix "⊎" := disj_union (at level 50, left associativity) : stdpp_scope.
Notation "(⊎)" := disj_union (only parsing) : stdpp_scope.
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.
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.
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.
Infix "∖**" := (zip_with (zip_with (∖)))
  (at level 40, left associativity) : stdpp_scope.
Infix "∖*∖**" := (zip_with (prod_zip (∖) (∖*)))
  (at level 50, left associativity) : stdpp_scope.

Class Singleton A B := singleton: A → B.
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.
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.
Infix "⊆**" := (Forall2 (⊆*)) (at level 70) : stdpp_scope.
Infix "⊆1*" := (Forall2 (λ p q, p .1 ⊆ q .1)) (at level 70) : stdpp_scope.
Infix "⊆2*" := (Forall2 (λ p q, p .2 ⊆ q .2)) (at level 70) : stdpp_scope.
Infix "⊆1**" := (Forall2 (λ p q, p .1 ⊆* q .1)) (at level 70) : stdpp_scope.
Infix "⊆2**" := (Forall2 (λ p q, p .2 ⊆* q .2)) (at level 70) : stdpp_scope.

Hint Extern 0 (_ ⊆ _) ⇒ reflexivity : core.
Hint Extern 0 (_ ⊆* _) ⇒ reflexivity : core.
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.
Hint Mode Lexico ! : typeclass_instances.

Class ElemOf A B := elem_of: A → B → Prop.
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.
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.
Infix "##**" := (Forall2 (##*)) (at level 70) : stdpp_scope.
Infix "##1*" := (Forall2 (λ p q, p .1 ## q .1)) (at level 70) : stdpp_scope.
Infix "##2*" := (Forall2 (λ p q, p .2 ## q .2)) (at level 70) : stdpp_scope.
Infix "##1**" := (Forall2 (λ p q, p .1 ##* q .1)) (at level 70) : stdpp_scope.
Infix "##2**" := (Forall2 (λ p q, p .2 ##* q .2)) (at level 70) : stdpp_scope.
Hint Extern 0 (_ ## _) ⇒ symmetry; eassumption : core.
Hint Extern 0 (_ ##* _) ⇒ symmetry; eassumption : core.

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

Class DisjointList A := disjoint_list : list A → Prop.
Hint Mode DisjointList ! : typeclass_instances.
Instance: Params (@disjoint_list) 2 := {}.
Notation "## Xs" := (disjoint_list Xs) (at level 20, format "## Xs") : stdpp_scope.
Notation "##@{ A } Xs" :=
  (@disjoint_list A _ Xs) (at level 20, only parsing) : stdpp_scope.

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

  Inductive disjoint_list_default : DisjointList A :=
    | disjoint_nil_2 : ##@{A } []
    | disjoint_cons_2 (X : A) (Xs : list A) : X ## ⋃ Xs → ## Xs → ## (X :: Xs ).
  Global Existing Instance disjoint_list_default.

  Lemma disjoint_list_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.
Hint Mode Filter - ! : typeclass_instances.

Class UpClose A B := up_close : A → B.
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.
Arguments mret {_ _ _} _ : assert.
Instance: Params (@mret) 3 := {}.
Class MBind (M : Type → Type) := mbind : ∀ {A B}, (A → M B ) → M A → M B.
Arguments mbind {_ _ _ _} _ !_ / : assert.
Instance: Params (@mbind) 4 := {}.
Class MJoin (M : Type → Type) := mjoin: ∀ {A}, M (M A) → M A.
Arguments mjoin {_ _ _} !_ / : assert.
Instance: Params (@mjoin) 3 := {}.
Class FMap (M : Type → Type) := fmap : ∀ {A B}, (A → B ) → M A → M B.
Arguments fmap {_ _ _ _} _ !_ / : assert.
Instance: Params (@fmap) 4 := {}.
Class OMap (M : Type → Type) := omap: ∀ {A B}, (A → option B ) → M A → M B.
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 "' x1 .. xn ← y ; z" := (y ≫= (λ x1, .. (λ xn, z) .. ))
  (at level 20, x1 binder, xn binder, y at level 100, z at level 200,
   only parsing, right associativity) : 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.
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.
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.
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.
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.
Arguments lookup_total _ _ _ _ !_ !_ / : simpl nomatch, assert.

The singleton map

Class SingletonM K A M := singletonM: K → A → M.
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.
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.
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.
Hint Mode Delete - ! : typeclass_instances.
Instance: Params (@delete) 4 := {}.
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.
Hint Mode Alter - - ! : typeclass_instances.
Instance: Params (@alter) 5 := {}.
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.
Hint Mode PartialAlter - - ! : typeclass_instances.
Instance: Params (@partial_alter) 4 := {}.
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.
Hint Mode Dom ! ! : typeclass_instances.
Instance: Params (@dom) 3 := {}.
Arguments dom : clear implicits.
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.
Hint Mode Merge ! : typeclass_instances.
Instance: Params (@merge) 4 := {}.
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.
Hint Mode UnionWith - ! : typeclass_instances.
Instance: Params (@union_with) 3 := {}.
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.
Hint Mode IntersectionWith - ! : typeclass_instances.
Instance: Params (@intersection_with) 3 := {}.
Arguments intersection_with {_ _ _} _ !_ !_ / : simpl nomatch, assert.

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

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

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

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

Hint Extern 0 (_ ⊑ _) ⇒ reflexivity : core.

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

Class Bottom A := bottom : A.
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 } ∅;
  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 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.
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.
Hint Mode Size ! : typeclass_instances.
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.
Hint Mode Fresh - ! : typeclass_instances.
Instance: Params (@fresh) 3 := {}.
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;
}.
Arguments infinite_fresh : simpl never.

Miscellaneous

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