base.v

(* Copyright (c) 2012-2013, 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.
Require Export Morphisms RelationClasses List Bool Utf8 Program Setoid NArith.

(** * General *)
(** The following coercion allows us to use Booleans as propositions. *)
Coercion Is_true : bool >-> Sortclass.

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

(** Change [True] and [False] into notations in order to enable overloading.
We will use this in the file [assertions] to give [True] and [False] a
different interpretation in [assert_scope] used for assertions of our axiomatic
semantics. *)
Notation "'True'" := True : type_scope.
Notation "'False'" := False : type_scope.

Notation curry := prod_curry.
Notation uncurry := prod_uncurry.

(** 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.

(** 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 (?x = ?x) => reflexivity.

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.

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.

(** Set convenient implicit arguments for [existT] and introduce notations. *)
Arguments existT {_ _} _ _.
Notation "x ↾ p" := (exist _ x p) (at level 20) : C_scope.
Notation "` x" := (proj1_sig x) : C_scope.

(** * Type classes *)
(** ** Provable propositions *)
(** This type class collects provable propositions. It is useful to constraint
type classes by arbitrary propositions. *)
Class PropHolds (P : Prop) := prop_holds: P.

Hint Extern 0 (PropHolds _) => assumption : typeclass_instances.
Instance: Proper (iff ==> iff) PropHolds.
Proof. repeat intro; trivial. Qed.

Ltac solve_propholds :=
  match goal with
  | |- PropHolds (?P) => apply _
  | |- ?P => change (PropHolds P); apply _
  end.

(** ** 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 _ {_}.

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

Instance unit_inhabited: Inhabited unit := populate ().
Instance list_inhabited {A} : Inhabited (list A) := populate [].
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 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 option_inhabited {A} : Inhabited (option A) := populate None.

(** ** 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.

(** ** 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.

Ltac fold_leibniz := repeat
  match goal with
  | H : context [ @equiv ?A _ _ _ ] |- _ =>
    setoid_rewrite (leibniz_equiv (A:=A)) in H
  | |- context [ @equiv ?A _ _ _ ] =>
    setoid_rewrite (leibniz_equiv (A:=A))
  end.
Ltac unfold_leibniz := repeat
  match goal with
  | H : context [ @eq ?A _ _ ] |- _ =>
    setoid_rewrite <-(leibniz_equiv (A:=A)) in H
  | |- context [ @eq ?A _ _ ] =>
    setoid_rewrite <-(leibniz_equiv (A:=A))
  end.

(** 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.

(** ** 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.

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.

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) : 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.

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

Class SubsetEq A := subseteq: A → A → Prop.
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, subseteq 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.

Hint Extern 0 (_ ⊆ _) => reflexivity.

Class Subset A := subset: A → A → Prop.
Instance: Params (@subset) 2.
Infix "⊂" := subset (at level 70) : C_scope.
Notation "(⊂)" := subset (only parsing) : C_scope.
Notation "( X ⊂ )" := (subset X) (only parsing) : C_scope.
Notation "( ⊂ X )" := (λ Y, subset 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.

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, disjoint Y X) (only parsing) : C_scope.

Inductive list_disjoint `{Empty A} `{Union A}
      `{Disjoint A} : list A → Prop :=
  | disjoint_nil :
     list_disjoint []
  | disjoint_cons X Xs :
     X ⊥ ⋃ Xs →
     list_disjoint Xs →
     list_disjoint (X :: Xs).
Lemma list_disjoint_cons_inv `{Empty A} `{Union A} `{Disjoint A} X Xs :
  list_disjoint (X :: Xs) →
  X ⊥ ⋃ Xs ∧ list_disjoint Xs.
Proof. inversion_clear 1; auto. Qed.

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

(* Arguments filter {_ _ _} _ {_} !_ / : simpl nomatch. *)

(** We define variants of the relations [(≡)] and [(⊆)] that are indexed by
an environment. *)
Class EquivEnv A B := equiv_env : A → relation B.
Notation "X ≡@{ E } Y" := (equiv_env E X Y)
  (at level 70, format "X  ≡@{ E }  Y") : C_scope.
Notation "(≡@{ E } )" := (equiv_env E)
  (E at level 1, only parsing) : C_scope.
Instance: Params (@equiv_env) 4.

Class SubsetEqEnv A B := subseteq_env : A → relation B.
Notation "X ⊆@{ E } Y" := (subseteq_env E X Y)
  (at level 70, format "X  ⊆@{ E }  Y") : C_scope.
Notation "(⊆@{ E } )" := (subseteq_env E)
  (E at level 1, only parsing) : C_scope.
Instance: Params (@subseteq_env) 4.

(** ** Monadic operations *)
(** We define operational type classes for the monadic operations bind, join 
and fmap. These type classes are defined in a non-standard way by taking the
function as a parameter of the class. For example, we define
<<
  Class FMapD := fmap: ∀ {A B}, (A → B) → M A → M B.
>>
instead of
<<
  Class FMap {A B} (f : A → B) := fmap: M A → M B.
>>
This approach allows us to define [fmap] on lists such that [simpl] unfolds it
in the appropriate way, and so that it can be used for mutual recursion
(the mapped function [f] is not part of the fixpoint) as well. This is a hack,
and should be replaced by something more appropriate in future versions. *)

(** 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.
Instance: Params (@mret) 3.
Arguments mret {_ _ _} _.

Class MBindD (M : Type → Type) {A B} (f : A → M B) := mbind: M A → M B.
Notation MBind M := (∀ {A B} (f : A → M B), MBindD M f)%type.
Instance: Params (@mbind) 5.
Arguments mbind {_ _ _} _ {_} !_ / : simpl nomatch.

Class MJoin (M : Type → Type) := mjoin: ∀ {A}, M (M A) → M A.
Instance: Params (@mjoin) 3.
Arguments mjoin {_ _ _} !_ / : simpl nomatch.

Class FMapD (M : Type → Type) {A B} (f : A → B) := fmap: M A → M B.
Notation FMap M := (∀ {A B} (f : A → B), FMapD M f)%type.
Instance: Params (@fmap) 6.
Arguments fmap {_ _ _} _ {_} !_ / : simpl nomatch.

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, next at level 35, right associativity) : C_scope.
Infix "<$>" := fmap (at level 60, right associativity) : C_scope.

Class MGuard (M : Type → Type) :=
  mguard: ∀ P {dec : Decision P} {A}, M A → M A.
Notation "'guard' P ; o" := (mguard P o)
  (at level 65, only parsing, next at level 35, right associativity) : C_scope.
Arguments mguard _ _ _ !_ _ !_ / : simpl nomatch.

(** ** 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, lookup i m) (only parsing) : C_scope.
Notation "(!! i )" := (lookup i) (only parsing) : C_scope.
Arguments lookup _ _ _ _ !_ !_ / : simpl nomatch.

(** 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) 4.
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) 3.
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 AlterD (K A M : Type) (f : A → A) :=
  alter: K → M → M.
Notation Alter K A M := (∀ (f : A → A), AlterD K A M f)%type.
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 {_} _ {_} !_ / : simpl nomatch, clear implicits.

(** 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.

(** We lift the insert and delete operation to lists of elements. *)
Definition insert_list `{Insert K A M} (l : list (K * A)) (m : M) : M :=
  fold_right (λ p, <[ fst p := snd p ]>) m l.
Instance: Params (@insert_list) 4.
Definition delete_list `{Delete K M} (l : list K) (m : M) : M :=
  fold_right delete m l.
Instance: Params (@delete_list) 3.

Definition insert_consecutive `{Insert nat A M}
    (i : nat) (l : list A) (m : M) : M :=
  fold_right (λ x f i, <[i:=x]>(f (S i))) (λ _, m) l i.
Instance: Params (@insert_consecutive) 3.

(** 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 _ _ _ _ _ !_ /.

(** ** 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 [injective (k ++)] instead of [app_inv_head k]. *)
Class Injective {A B} (R : relation A) S (f : A → B) : Prop :=
  injective: ∀ x y : A, S (f x) (f y) → R x y.
Class Idempotent {A} (R : relation A) (f : A → A → A) : Prop :=
  idempotent: ∀ x, R (f x x) x.
Class Commutative {A B} (R : relation A) (f : B → B → A) : Prop :=
  commutative: ∀ 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 Associative {A} (R : relation A) (f : A → A → A) : Prop :=
  associative: ∀ 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 LeftDistr {A} (R : relation A) (f g : A → A → A) : Prop :=
  left_distr: ∀ x y z, R (f x (g y z)) (g (f x y) (f x z)).
Class RightDistr {A} (R : relation A) (f g : A → A → A) : Prop :=
  right_distr: ∀ y z x, R (f (g y z) x) (g (f y x) (f z x)).
Class AntiSymmetric {A} (R : relation A) : Prop :=
  anti_symmetric: ∀ x y, R x y → R y x → x = y.

Arguments irreflexivity {_} _ {_} _ _.
Arguments injective {_ _ _ _} _ {_} _ _ _.
Arguments idempotent {_ _} _ {_} _.
Arguments commutative {_ _ _} _ {_} _ _.
Arguments left_id {_ _} _ _ {_} _.
Arguments right_id {_ _} _ _ {_} _.
Arguments associative {_ _} _ {_} _ _ _.
Arguments left_absorb {_ _} _ _ {_} _.
Arguments right_absorb {_ _} _ _ {_} _.
Arguments left_distr {_ _} _ _ {_} _ _ _.
Arguments right_distr {_ _} _ _ {_} _ _ _.
Arguments anti_symmetric {_} _ {_} _ _ _ _.

Instance: Commutative (↔) (@eq A).
Proof. red. intuition. Qed.
Instance: Commutative (↔) (λ x y, @eq A y x).
Proof. red. intuition. Qed.
Instance: Commutative (↔) (↔).
Proof. red. intuition. Qed.
Instance: Commutative (↔) (∧).
Proof. red. intuition. Qed.
Instance: Associative (↔) (∧).
Proof. red. intuition. Qed.
Instance: Idempotent (↔) (∧).
Proof. red. intuition. Qed.
Instance: Commutative (↔) (∨).
Proof. red. intuition. Qed.
Instance: Associative (↔) (∨).
Proof. red. intuition. Qed.
Instance: Idempotent (↔) (∨).
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.
Instance: LeftDistr (↔) (∧) (∨).
Proof. red. intuition. Qed.
Instance: RightDistr (↔) (∧) (∨).
Proof. red. intuition. Qed.
Instance: LeftDistr (↔) (∨) (∧).
Proof. red. intuition. Qed.
Instance: RightDistr (↔) (∨) (∧).
Proof. red. intuition. Qed.

(** The following lemmas are more specific versions of the projections of the
above type classes. These lemmas allow us to enforce Coq not to use the setoid
rewriting mechanism. *)
Lemma idempotent_eq {A} (f : A → A → A) `{!Idempotent (=) f} x :
  f x x = x.
Proof. auto. Qed.
Lemma commutative_eq {A B} (f : B → B → A) `{!Commutative (=) f} x y :
  f x y = f y x.
Proof. auto. Qed.
Lemma left_id_eq {A} (i : A) (f : A → A → A) `{!LeftId (=) i f} x :
  f i x = x.
Proof. auto. Qed.
Lemma right_id_eq {A} (i : A) (f : A → A → A) `{!RightId (=) i f} x :
  f x i = x.
Proof. auto. Qed.
Lemma associative_eq {A} (f : A → A → A) `{!Associative (=) f} x y z :
  f x (f y z) = f (f x y) z.
Proof. auto. Qed.
Lemma left_absorb_eq {A} (i : A) (f : A → A → A) `{!LeftAbsorb (=) i f} x :
  f i x = i.
Proof. auto. Qed.
Lemma right_absorb_eq {A} (i : A) (f : A → A → A) `{!RightAbsorb (=) i f} x :
  f x i = i.
Proof. auto. Qed.
Lemma left_distr_eq {A} (f g : A → A → A) `{!LeftDistr (=) f g} x y z :
  f x (g y z) = g (f x y) (f x z).
Proof. auto. Qed.
Lemma right_distr_eq {A} (f g : A → A → A) `{!RightDistr (=) f g} y z x :
  f (g y z) x = g (f y x) (f z x).
Proof. auto. Qed.

(** ** Axiomatization of ordered structures *)
(** A pre-order equipped with a smallest element. *)
Class BoundedPreOrder A `{Empty A} `{SubsetEq A} : Prop := {
  bounded_preorder :>> PreOrder (⊆);
  subseteq_empty x : ∅ ⊆ x
}.
Class PartialOrder A `{SubsetEq A} : Prop := {
  po_preorder :>> PreOrder (⊆);
  po_antisym :> AntiSymmetric (⊆)
}.

(** We do not include equality in the following interfaces so as to avoid the
need for proofs that the relations and operations respect setoid equality.
Instead, we will define setoid equality in a generic way as
[λ X Y, X ⊆ Y ∧ Y ⊆ X]. *)
Class BoundedJoinSemiLattice A `{Empty A} `{SubsetEq A} `{Union A} : Prop := {
  bjsl_preorder :>> BoundedPreOrder A;
  union_subseteq_l x y : x ⊆ x ∪ y;
  union_subseteq_r x y : y ⊆ x ∪ y;
  union_least x y z : x ⊆ z → y ⊆ z → x ∪ y ⊆ z
}.
Class MeetSemiLattice A `{Empty A} `{SubsetEq A} `{Intersection A} : Prop := {
  msl_preorder :>> BoundedPreOrder A;
  intersection_subseteq_l x y : x ∩ y ⊆ x;
  intersection_subseteq_r x y : x ∩ y ⊆ y;
  intersection_greatest x y z : z ⊆ x → z ⊆ y → z ⊆ x ∩ y
}.

(** A join distributive lattice with distributivity stated in the order
theoretic way. We will prove that distributivity of join, and distributivity
as an equality can be derived. *)
Class LowerBoundedLattice A `{Empty A} `{SubsetEq A}
    `{Union A} `{Intersection A} : Prop := {
  lbl_bjsl :>> BoundedJoinSemiLattice A;
  lbl_msl :>> MeetSemiLattice A;
  lbl_distr x y z : (x ∪ y) ∩ (x ∪ z) ⊆ x ∪ (y ∩ z)
}.

(** ** Axiomatization of collections *)
(** The class [SimpleCollection A C] axiomatizes a collection of type [C] with
elements of type [A]. *)
Instance: Params (@map) 3.
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
}.
Class Collection A C `{ElemOf A C} `{Empty C} `{Singleton A C}
    `{Union C} `{Intersection C} `{Difference C} : Prop := {
  collection_simple :>> SimpleCollection A C;
  elem_of_intersection X Y (x : A) : x ∈ X ∩ Y ↔ x ∈ X ∧ x ∈ Y;
  elem_of_difference X Y (x : A) : x ∈ X ∖ Y ↔ x ∈ X ∧ x ∉ Y
}.
Class CollectionOps A C
    `{ElemOf A C} `{Empty C} `{Singleton A C}
    `{Union C} `{Intersection C} `{Difference C}
    `{IntersectionWith A C} `{Filter A C} : Prop := {
  collection_ops :>> Collection A C;
  elem_of_intersection_with (f : A → A → option A) X Y (x : A) :
    x ∈ intersection_with f X Y ↔ ∃ x1 x2, x1 ∈ X ∧ x2 ∈ Y ∧ f x1 x2 = Some x;
  elem_of_filter X P `{∀ x, Decision (P x)} x :
    x ∈ filter P X ↔ P x ∧ x ∈ X
}.

(** We axiomative a finite collection as a collection 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.
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.

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

(** Decidability of equality of the carrier set is admissible, but we add it
anyway so as to avoid cycles in type class search. *)
Class FinCollection A C `{ElemOf A C} `{Empty C} `{Singleton A C}
    `{Union C} `{Intersection C} `{Difference C}
    `{Elements A C} `{∀ x y : A, Decision (x = y)} : Prop := {
  fin_collection :>> Collection A C;
  elements_spec X x : x ∈ X ↔ x ∈ elements X;
  elements_nodup X : NoDup (elements X)
}.
Class Size C := size: C → nat.
Arguments size {_ _} !_ / : simpl nomatch.
Instance: Params (@size) 2.

(** The class [Collection M] axiomatizes a type constructor [M] that can be
used to construct a collection [M A] with elements of type [A]. The advantage
of this class, compared to [Collection], 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 removed). More interesting implementations typically need
decidability of equality, or a total order on the elements, which do not fit
in a type constructor of type [Type → Type]. *)
Class CollectionMonad 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 := {
  collection_monad_simple A :> SimpleCollection 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 ∈ 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, x ∈ Y ∧ Y ∈ X
}.

(** The function [fresh X] yields an element that is not contained in [X]. We
will later prove that [fresh] is [Proper] with respect to the induced setoid
equality on collections. *)
Class Fresh A C := fresh: C → A.
Instance: Params (@fresh) 3.
Class FreshSpec A C `{ElemOf A C}
    `{Empty C} `{Singleton A C} `{Union C} `{Fresh A C} : Prop := {
  fresh_collection_simple :>> SimpleCollection A C;
  fresh_proper_alt X Y : (∀ x, x ∈ X ↔ x ∈ Y) → fresh X = fresh Y;
  is_fresh (X : C) : fresh X ∉ X
}.

(** * Miscellaneous *)
Class Half A := half: A → A.
Notation "x .½" := (half x) (at level 20, format "x .½") : 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.
Lemma not_symmetry `{R : relation A} `{!Symmetric R} (x y : A) :
  ¬R x y → ¬R y x.
Proof. intuition. Qed.
Lemma symmetry_iff `(R : relation A) `{!Symmetric R} (x y : A) :
  R x y ↔ R y x.
Proof. intuition. Qed.

(** ** Pointwise relations *)
(** These instances are in Coq trunk since revision 15455, but are not in Coq
8.4 yet. *)
Instance pointwise_reflexive {A} `{R : relation B} :
  Reflexive R → Reflexive (pointwise_relation A R) | 9.
Proof. firstorder. Qed.
Instance pointwise_symmetric {A} `{R : relation B} :
  Symmetric R → Symmetric (pointwise_relation A R) | 9.
Proof. firstorder. Qed.
Instance pointwise_transitive {A} `{R : relation B} :
  Transitive R → Transitive (pointwise_relation A R) | 9.
Proof. firstorder. Qed.

(** ** Products *)
Definition fst_map {A A' B} (f : A → A') (p : A * B) : A' * B :=
  (f (fst p), snd p).
Definition snd_map {A B B'} (f : B → B') (p : A * B) : A * B' :=
  (fst p, f (snd p)).
Arguments fst_map {_ _ _} _ !_ /.
Arguments snd_map {_ _ _} _ !_ /.

Instance: ∀ {A A' B} (f : A → A'),
  Injective (=) (=) f → Injective (=) (=) (@fst_map A A' B f).
Proof.
  intros ????? [??] [??]; simpl; intro; f_equal.
  * apply (injective f). congruence.
  * congruence.
Qed.
Instance: ∀ {A B B'} (f : B → B'),
  Injective (=) (=) f → Injective (=) (=) (@snd_map A B B' f).
Proof.
  intros ????? [??] [??]; simpl; intro; f_equal.
  * congruence.
  * apply (injective f). congruence.
Qed.

Definition prod_relation {A B} (R1 : relation A) (R2 : relation B) :
  relation (A * B) := λ x y, R1 (fst x) (fst y) ∧ R2 (snd x) (snd y).

Section prod_relation.
  Context `{R1 : relation A} `{R2 : relation B}.
  Global Instance:
    Reflexive R1 → Reflexive R2 → Reflexive (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance:
    Symmetric R1 → Symmetric R2 → Symmetric (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance:
    Transitive R1 → Transitive R2 → Transitive (prod_relation R1 R2).
  Proof. firstorder eauto. Qed.
  Global Instance:
    Equivalence R1 → Equivalence R2 → Equivalence (prod_relation R1 R2).
  Proof. split; apply _. Qed.
  Global Instance: Proper (R1 ==> R2 ==> prod_relation R1 R2) pair.
  Proof. firstorder eauto. Qed.
  Global Instance: Proper (prod_relation R1 R2 ==> R1) fst.
  Proof. firstorder eauto. Qed.
  Global Instance: Proper (prod_relation R1 R2 ==> R2) snd.
  Proof. firstorder eauto. Qed.
End prod_relation.

(** ** Other *)
Definition proj_relation {A B} (R : relation A)
  (f : B → A) : relation B := λ x y, R (f x) (f y).
Definition proj_relation_equivalence {A B} (R : relation A) (f : B → A) :
  Equivalence R → Equivalence (proj_relation R f).
Proof. unfold proj_relation. firstorder auto. Qed.

Instance: ∀ A B (x : B), Commutative (=) (λ _ _ : A, x).
Proof. red. trivial. Qed.
Instance: ∀ A (x : A), Associative (=) (λ _ _ : A, x).
Proof. red. trivial. Qed.
Instance: ∀ A, Associative (=) (λ x _ : A, x).
Proof. red. trivial. Qed.
Instance: ∀ A, Associative (=) (λ _ x : A, x).
Proof. red. trivial. Qed.
Instance: ∀ A, Idempotent (=) (λ x _ : A, x).
Proof. red. trivial. Qed.
Instance: ∀ A, Idempotent (=) (λ _ x : A, x).
Proof. red. trivial. Qed.

Instance left_id_propholds {A} (R : relation A) i f :
  LeftId R i f → ∀ x, PropHolds (R (f i x) x).
Proof. red. trivial. Qed.
Instance right_id_propholds {A} (R : relation A) i f :
  RightId R i f → ∀ x, PropHolds (R (f x i) x).
Proof. red. trivial. Qed.
Instance left_absorb_propholds {A} (R : relation A) i f :
  LeftAbsorb R i f → ∀ x, PropHolds (R (f i x) i).
Proof. red. trivial. Qed.
Instance right_absorb_propholds {A} (R : relation A) i f :
  RightAbsorb R i f → ∀ x, PropHolds (R (f x i) i).
Proof. red. trivial. Qed.
Instance idem_propholds {A} (R : relation A) f :
  Idempotent R f → ∀ x, PropHolds (R (f x x) x).
Proof. red. trivial. Qed.