{-# OPTIONS --without-K #-}
module Container.List where
open import Bag-equivalence
using () renaming (_≈-bag_ to _≈-bagL_; _∈_ to _∈L_; Any to AnyL)
open import Container
open import Equality.Propositional
open import Equivalence using (_⇔_; module _⇔_)
open import Fin
open import Prelude as P hiding (List; []; _∷_; foldr; _++_; id; _∘_)
import Bijection
open Bijection equality-with-J using (_↔_; module _↔_; Σ-≡,≡↔≡)
import Function-universe
open Function-universe equality-with-J
import H-level.Closure
open H-level.Closure equality-with-J
List : Container lzero
List = ℕ ▷ Fin
List⇔List : {A : Set} → ⟦ List ⟧ A ⇔ P.List A
List⇔List {A} = record
{ to = to
; from = λ xs → (length xs , P.lookup xs)
}
where
to : ⟦ List ⟧ A → P.List A
to (zero , f) = P.[]
to (suc n , f) = P._∷_ (f (inj₁ tt)) (to (n , f ∘ inj₂))
List↔List : {A : Set} →
({B : Set} → Extensionality B (λ _ → A)) →
⟦ List ⟧ A ↔ P.List A
List↔List {A} ext = record
{ surjection = record
{ equivalence = List⇔List
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
open _⇔_ List⇔List
to∘from : ∀ xs → to (from xs) ≡ xs
to∘from P.[] = refl
to∘from (P._∷_ x xs) = cong (P._∷_ x) (to∘from xs)
from∘to : ∀ xs → from (to xs) ≡ xs
from∘to (zero , f) = cong (_,_ _) (ext λ ())
from∘to (suc n , f) =
(suc (length (to xs)) , P.lookup (P._∷_ x (to xs))) ≡⟨ lemma₃ (from∘to (n , f ∘ inj₂)) ⟩
(suc n , [ (λ _ → x) , f ∘ inj₂ ]) ≡⟨ lemma₁ ⟩∎
(suc n , f) ∎
where
x = f (inj₁ tt)
xs = (n , f ∘ inj₂)
lemma₁ : ∀ {n f} →
_≡_ {A = ⟦ List ⟧ A}
(suc n , [ (λ _ → f (inj₁ tt)) , f ∘ inj₂ ])
(suc n , f)
lemma₁ = cong (_,_ _) (ext [ (λ { tt → refl }) , (λ _ → refl) ])
lemma₂ : {n : ℕ} {lkup : Fin n → A} →
(≡n : length (to xs) ≡ n) →
subst (λ n → Fin n → A) ≡n (P.lookup (to xs)) ≡ lkup →
_≡_ {A = ⟦ List ⟧ A}
(suc (length (to xs)) , P.lookup (P._∷_ x (to xs)))
(suc n , [ (λ _ → x) , lkup ])
lemma₂ refl refl = sym lemma₁
lemma₃ : {ys : ⟦ List ⟧ A} →
(length (to xs) , P.lookup (to xs)) ≡ ys →
_≡_ {A = ⟦ List ⟧ A}
(suc (length (to xs)) , P.lookup (P._∷_ x (to xs)))
(suc (proj₁ ys) , [ (λ _ → x) , proj₂ ys ])
lemma₃ ≡ys = lemma₂ (proj₁ ≡,≡) (proj₂ ≡,≡)
where ≡,≡ = _↔_.from Σ-≡,≡↔≡ ≡ys
Any↔Any-to : {A : Set} (xs : ⟦ List ⟧ A) (P : A → Set) →
Any P xs ↔ AnyL P (_⇔_.to List⇔List xs)
Any↔Any-to (zero , lkup) P =
(∃ λ (p : Fin zero) → P (lkup p)) ↔⟨ ∃-Fin-zero _ ⟩
⊥ □
Any↔Any-to (suc n , lkup) P =
(∃ λ (p : Fin (suc n)) → P (lkup p)) ↔⟨ ∃-Fin-suc _ ⟩
P (lkup (inj₁ tt)) ⊎ Any {C = List} P (n , lkup ∘ inj₂) ↔⟨ id ⊎-cong Any↔Any-to (n , lkup ∘ inj₂) P ⟩
P (lkup (inj₁ tt)) ⊎ AnyL P (_⇔_.to List⇔List (n , lkup ∘ inj₂)) □
Any-from↔Any : {A : Set} (xs : P.List A) (P : A → Set) →
Any P (_⇔_.from List⇔List xs) ↔ AnyL P xs
Any-from↔Any P.[] P =
(∃ λ (p : Fin zero) → P (P.lookup P.[] p)) ↔⟨ ∃-Fin-zero _ ⟩
⊥ □
Any-from↔Any (P._∷_ x xs) P =
(∃ λ (p : Fin (suc (P.length xs))) → P (P.lookup (P._∷_ x xs) p)) ↔⟨ ∃-Fin-suc _ ⟩
P x ⊎ Any {C = List} P (_⇔_.from List⇔List xs) ↔⟨ id ⊎-cong Any-from↔Any xs P ⟩
P x ⊎ AnyL P xs □
≈-⇔-to-≈-to :
{A : Set} {xs ys : ⟦ List ⟧ A} →
xs ≈-bag ys ⇔ _⇔_.to List⇔List xs ≈-bagL _⇔_.to List⇔List ys
≈-⇔-to-≈-to {xs = xs} {ys} = record
{ to = λ xs≈ys z →
z ∈L (_⇔_.to List⇔List xs) ↔⟨ inverse $ Any↔Any-to xs _ ⟩
z ∈ xs ↔⟨ xs≈ys z ⟩
z ∈ ys ↔⟨ Any↔Any-to ys _ ⟩
z ∈L (_⇔_.to List⇔List ys) □
; from = λ xs≈ys z →
z ∈ xs ↔⟨ Any↔Any-to xs _ ⟩
z ∈L (_⇔_.to List⇔List xs) ↔⟨ xs≈ys z ⟩
z ∈L (_⇔_.to List⇔List ys) ↔⟨ inverse $ Any↔Any-to ys _ ⟩
z ∈ ys □
}
≈-⇔-from-≈-from :
{A : Set} {xs ys : P.List A} →
xs ≈-bagL ys ⇔ _⇔_.from List⇔List xs ≈-bag _⇔_.from List⇔List ys
≈-⇔-from-≈-from {xs = xs} {ys} = record
{ to = λ xs≈ys z →
z ∈ (_⇔_.from List⇔List xs) ↔⟨ Any-from↔Any xs _ ⟩
z ∈L xs ↔⟨ xs≈ys z ⟩
z ∈L ys ↔⟨ inverse $ Any-from↔Any ys _ ⟩
z ∈ (_⇔_.from List⇔List ys) □
; from = λ xs≈ys z →
z ∈L xs ↔⟨ inverse $ Any-from↔Any xs _ ⟩
z ∈ (_⇔_.from List⇔List xs) ↔⟨ xs≈ys z ⟩
z ∈ (_⇔_.from List⇔List ys) ↔⟨ Any-from↔Any ys _ ⟩
z ∈L ys □
}
[] : {A : Set} → ⟦ List ⟧ A
[] = (zero , λ ())
infixr 5 _∷_
_∷_ : {A : Set} → A → ⟦ List ⟧ A → ⟦ List ⟧ A
x ∷ (n , lkup) = (suc n , [ (λ _ → x) , lkup ])
[]≈ : {A : Set} {lkup : _ → A} →
_≈-bag_ {C₂ = List} [] (zero , lkup)
[]≈ _ = record
{ surjection = record
{ equivalence = record
{ to = λ { (() , _) }
; from = λ { (() , _) }
}
; right-inverse-of = λ { (() , _) }
}
; left-inverse-of = λ { (() , _) }
}
∷≈ : ∀ {A : Set} {n} {lkup : _ → A} →
_≈-bag_ {C₂ = List}
(lkup (inj₁ tt) ∷ (n , lkup ∘ inj₂))
(suc n , lkup)
∷≈ _ = record
{ surjection = record
{ equivalence = record
{ to = λ { (inj₁ tt , eq) → (inj₁ tt , eq)
; (inj₂ s , eq) → (inj₂ s , eq)
}
; from = λ { (inj₁ tt , eq) → (inj₁ tt , eq)
; (inj₂ s , eq) → (inj₂ s , eq)
}
}
; right-inverse-of = λ { (inj₁ tt , eq) → refl
; (inj₂ s , eq) → refl
}
}
; left-inverse-of = λ { (inj₁ tt , eq) → refl
; (inj₂ s , eq) → refl
}
}
Any-[] : {A : Set} (P : A → Set) →
Any P [] ↔ ⊥₀
Any-[] _ = record
{ surjection = record
{ equivalence = record
{ to = λ { (() , _) }
; from = λ ()
}
; right-inverse-of = λ ()
}
; left-inverse-of = λ { (() , _) }
}
Any-∷ : ∀ {A : Set} (P : A → Set) {x xs} →
Any P (x ∷ xs) ↔ P x ⊎ Any P xs
Any-∷ _ = record
{ surjection = record
{ equivalence = record
{ to = λ { (inj₁ tt , eq) → inj₁ eq
; (inj₂ s , eq) → inj₂ (s , eq)
}
; from = λ { (inj₁ eq) → (inj₁ tt , eq)
; (inj₂ (s , eq)) → (inj₂ s , eq)
}
}
; right-inverse-of = λ { (inj₁ eq) → refl
; (inj₂ (s , eq)) → refl
}
}
; left-inverse-of = λ { (inj₁ tt , eq) → refl
; (inj₂ s , eq) → refl
}
}
fold : {A B : Set} → B → (A → ⟦ List ⟧ A → B → B) → ⟦ List ⟧ A → B
fold nl cns (zero , lkup) = nl
fold nl cns (suc n , lkup) =
cns (lkup (inj₁ tt)) (n , lkup ∘ inj₂)
(fold nl cns (n , lkup ∘ inj₂))
fold-lemma : ∀ {A B : Set} {nl : B} {cns : A → ⟦ List ⟧ A → B → B}
(P : ⟦ List ⟧ A → B → Set) →
(∀ xs ys → xs ≈-bag ys → ∀ b → P xs b → P ys b) →
P [] nl →
(∀ x xs b → P xs b → P (x ∷ xs) (cns x xs b)) →
∀ xs → P xs (fold nl cns xs)
fold-lemma Q resp nl cns (zero , lkup) = resp _ _ []≈ _ nl
fold-lemma Q resp nl cns (suc n , lkup) = resp _ _ ∷≈ _ $
cns _ _ _ $ fold-lemma Q resp nl cns (n , lkup ∘ inj₂)
infixr 5 _++_
_++_ : {A : Set} → ⟦ List ⟧ A → ⟦ List ⟧ A → ⟦ List ⟧ A
xs ++ ys = fold ys (λ z _ zs → z ∷ zs) xs
Any-++ : ∀ {A : Set} (P : A → Set) xs ys →
Any P (xs ++ ys) ↔ Any P xs ⊎ Any P ys
Any-++ P xs ys = fold-lemma
(λ xs xs++ys → Any P xs++ys ↔ Any P xs ⊎ Any P ys)
(λ us vs us≈vs us++ys hyp →
Any P us++ys ↔⟨ hyp ⟩
Any P us ⊎ Any P ys ↔⟨ _⇔_.to (∼⇔∼″ us vs) us≈vs P ⊎-cong id ⟩
Any P vs ⊎ Any P ys □)
(Any P ys ↔⟨ inverse ⊎-left-identity ⟩
⊥ ⊎ Any P ys ↔⟨ inverse (Any-[] P) ⊎-cong id ⟩
Any P [] ⊎ Any P ys □)
(λ x xs xs++ys ih →
Any P (x ∷ xs++ys) ↔⟨ Any-∷ P ⟩
P x ⊎ Any P xs++ys ↔⟨ id ⊎-cong ih ⟩
P x ⊎ Any P xs ⊎ Any P ys ↔⟨ ⊎-assoc ⟩
(P x ⊎ Any P xs) ⊎ Any P ys ↔⟨ inverse (Any-∷ P) ⊎-cong id ⟩
Any P (x ∷ xs) ⊎ Any P ys □)
xs