```------------------------------------------------------------------------
-- The Agda standard library
--
-- Some code related to indexed containers that uses heterogeneous
-- equality
------------------------------------------------------------------------

-- The notation and presentation here is perhaps close to those used
-- by Hancock and Hyvernat in "Programming interfaces and basic
-- topology" (2006).

{-# OPTIONS --with-K --safe #-}

module Data.Container.Indexed.WithK where

open import Axiom.Extensionality.Heterogeneous using (Extensionality)
open import Data.Container.Indexed hiding (module PlainMorphism)
open import Data.Product as Prod hiding (map)
open import Function renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
open import Level
open import Relation.Unary using (Pred; _⊆_)
open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
open import Relation.Binary.HeterogeneousEquality as H using (_≅_; refl)
open import Relation.Binary.Indexed.Heterogeneous

------------------------------------------------------------------------

-- Equality, parametrised on an underlying relation.

Eq : ∀ {i o c r ℓ} {I : Set i} {O : Set o} (C : Container I O c r)
(X Y : Pred I ℓ) → IREL X Y ℓ → IREL (⟦ C ⟧ X) (⟦ C ⟧ Y) _
Eq C _ _ _≈_ {o₁} {o₂} (c , k) (c′ , k′) =
o₁ ≡ o₂ × c ≅ c′ × (∀ r r′ → r ≅ r′ → k r ≈ k′ r′)

private

-- Note that, if propositional equality were extensional, then Eq _≅_
-- and _≅_ would coincide.

Eq⇒≅ : ∀ {i o c r ℓ} {I : Set i} {O : Set o}
{C : Container I O c r} {X : Pred I ℓ} {o₁ o₂ : O}
{xs : ⟦ C ⟧ X o₁} {ys : ⟦ C ⟧ X o₂} → Extensionality r ℓ →
Eq C X X (λ x₁ x₂ → x₁ ≅ x₂) xs ys → xs ≅ ys
Eq⇒≅ {xs = c , k} {.c , k′} ext (refl , refl , k≈k′) =
H.cong (_,_ c) (ext (λ _ → refl) (λ r → k≈k′ r r refl))

setoid : ∀ {i o c r s} {I : Set i} {O : Set o} →
Container I O c r → IndexedSetoid I s _ → IndexedSetoid O _ _
setoid C X = record
{ Carrier       = ⟦ C ⟧ X.Carrier
; _≈_           = _≈_
; isEquivalence = record
{ refl  = refl , refl , λ { r .r refl → X.refl }
; sym   = sym
; trans = λ { {_} {i = xs} {ys} {zs} → trans {_} {i = xs} {ys} {zs}  }
}
}
where
module X = IndexedSetoid X

_≈_ : IRel (⟦ C ⟧ X.Carrier) _
_≈_ = Eq C X.Carrier X.Carrier X._≈_

sym : Symmetric (⟦ C ⟧ X.Carrier) _≈_
sym {_} {._} {_ , _} {._ , _} (refl , refl , k) =
refl , refl , λ { r .r refl → X.sym (k r r refl) }

trans : Transitive (⟦ C ⟧ X.Carrier) _≈_
trans {._} {_} {._} {_ , _} {._ , _} {._ , _}
(refl , refl , k) (refl , refl , k′) =
refl , refl , λ { r .r refl → X.trans (k r r refl) (k′ r r refl) }

------------------------------------------------------------------------
-- Functoriality

module Map where

identity : ∀ {i o c r s} {I : Set i} {O : Set o} (C : Container I O c r)
(X : IndexedSetoid I s _) → let module X = IndexedSetoid X in
∀ {o} {xs : ⟦ C ⟧ X.Carrier o} → Eq C X.Carrier X.Carrier
X._≈_ xs (map C {X.Carrier} ⟨id⟩ xs)
identity C X = IndexedSetoid.refl (setoid C X)

composition : ∀ {i o c r s ℓ₁ ℓ₂} {I : Set i} {O : Set o}
(C : Container I O c r) {X : Pred I ℓ₁} {Y : Pred I ℓ₂}
(Z : IndexedSetoid I s _) → let module Z = IndexedSetoid Z in
{f : Y ⊆ Z.Carrier} {g : X ⊆ Y} {o : O} {xs : ⟦ C ⟧ X o} →
Eq C Z.Carrier Z.Carrier Z._≈_
(map C {Y} f (map C {X} g xs))
(map C {X} (f ⟨∘⟩ g) xs)
composition C Z = IndexedSetoid.refl (setoid C Z)

------------------------------------------------------------------------
-- Plain morphisms

module PlainMorphism {i o c r} {I : Set i} {O : Set o} where

open Data.Container.Indexed.PlainMorphism

-- Naturality.

Natural : ∀ {ℓ} {C₁ C₂ : Container I O c r} →
((X : Pred I ℓ) → ⟦ C₁ ⟧ X ⊆ ⟦ C₂ ⟧ X) → Set _
Natural {C₁ = C₁} {C₂} m =
∀ {X} Y → let module Y = IndexedSetoid Y in (f : X ⊆ Y.Carrier) →
∀ {o} (xs : ⟦ C₁ ⟧ X o) →
Eq C₂ Y.Carrier Y.Carrier Y._≈_
(m Y.Carrier \$ map C₁ {X} f xs) (map C₂ {X} f \$ m X xs)

-- Natural transformations.

NT : ∀ {ℓ} (C₁ C₂ : Container I O c r) → Set _
NT {ℓ} C₁ C₂ = ∃ λ (m : (X : Pred I ℓ) → ⟦ C₁ ⟧ X ⊆ ⟦ C₂ ⟧ X) →
Natural m

-- Container morphisms are natural.

natural : ∀ {ℓ} (C₁ C₂ : Container I O c r) (m : C₁ ⇒ C₂) → Natural {ℓ} ⟪ m ⟫
natural _ _ m {X} Y f _ = refl , refl , λ { r .r refl → lemma (coherent m) }
where
module Y = IndexedSetoid Y

lemma : ∀ {i j} (eq : i ≡ j) {x} →
P.subst Y.Carrier eq (f x) Y.≈ f (P.subst X eq x)
lemma refl = Y.refl

-- In fact, all natural functions of the right type are container
-- morphisms.

complete : ∀ {C₁ C₂ : Container I O c r} (nt : NT C₁ C₂) →
∃ λ m → (X : IndexedSetoid I _ _) →
let module X = IndexedSetoid X in
∀ {o} (xs : ⟦ C₁ ⟧ X.Carrier o) →
Eq C₂ X.Carrier X.Carrier X._≈_
(proj₁ nt X.Carrier xs) (⟪ m ⟫ X.Carrier {o} xs)
complete {C₁} {C₂} (nt , nat) = m , (λ X xs → nat X
(λ { (r , eq) → P.subst (IndexedSetoid.Carrier X) eq (proj₂ xs r) })
(proj₁ xs , (λ r → r , refl)))
where

m : C₁ ⇒ C₂
m = record
{ command  = λ      c₁       → proj₁        (lemma c₁)
; response = λ {_} {c₁}  r₂  → proj₁ (proj₂ (lemma c₁) r₂)
; coherent = λ {_} {c₁} {r₂} → proj₂ (proj₂ (lemma c₁) r₂)
}
where
lemma : ∀ {o} (c₁ : Command C₁ o) → Σ[ c₂ ∈ Command C₂ o ]
((r₂ : Response C₂ c₂) → Σ[ r₁ ∈ Response C₁ c₁ ]
next C₁ c₁ r₁ ≡ next C₂ c₂ r₂)
lemma c₁ = nt (λ i → Σ[ r₁ ∈ Response C₁ c₁ ] next C₁ c₁ r₁ ≡ i)
(c₁ , λ r₁ → r₁ , refl)

-- Composition commutes with ⟪_⟫.

∘-correct : {C₁ C₂ C₃ : Container I O c r}
(f : C₂ ⇒ C₃) (g : C₁ ⇒ C₂) (X : IndexedSetoid I (c ⊔ r) _) →
let module X = IndexedSetoid X in
∀ {o} {xs : ⟦ C₁ ⟧ X.Carrier o} →
Eq C₃ X.Carrier X.Carrier X._≈_
(⟪ f ∘ g ⟫ X.Carrier xs)
(⟪ f ⟫ X.Carrier (⟪ g ⟫ X.Carrier xs))
∘-correct f g X = refl , refl , λ { r .r refl → lemma (coherent g)
(coherent f) }
where
module X = IndexedSetoid X

lemma : ∀ {i j k} (eq₁ : i ≡ j) (eq₂ : j ≡ k) {x} →
P.subst X.Carrier (P.trans eq₁ eq₂) x
X.≈
P.subst X.Carrier eq₂ (P.subst X.Carrier eq₁ x)
lemma refl refl = X.refl

------------------------------------------------------------------------
-- All and any

-- Membership.

infix 4 _∈_

_∈_ : ∀ {i o c r ℓ} {I : Set i} {O : Set o}
{C : Container I O c r} {X : Pred I (i ⊔ ℓ)} → IREL X (⟦ C ⟧ X) _
_∈_ {C = C} {X} x xs = ◇ C {X = X} ((x ≅_) ⟨∘⟩ proj₂) (-, xs)
```