{-# OPTIONS --without-K #-}
open import Equality
module Bijection
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where
open Derived-definitions-and-properties eq
open import Equality.Decision-procedures eq
open import H-level eq
open import Injection eq using (Injective; _↣_)
open import Logical-equivalence using (_⇔_)
open import Prelude as P hiding (id) renaming (_∘_ to _⊚_)
open import Surjection eq as Surjection using (_↠_; module _↠_)
infix 0 _↔_
record _↔_ {f t} (From : Set f) (To : Set t) : Set (f ⊔ t) where
field
surjection : From ↠ To
open _↠_ surjection
field
left-inverse-of : ∀ x → from (to x) ≡ x
injective : Injective to
injective {x = x} {y = y} to-x≡to-y =
x ≡⟨ sym (left-inverse-of x) ⟩
from (to x) ≡⟨ cong from to-x≡to-y ⟩
from (to y) ≡⟨ left-inverse-of y ⟩∎
y ∎
injection : From ↣ To
injection = record
{ to = to
; injective = injective
}
to-from : ∀ {x y} → to x ≡ y → from y ≡ x
to-from {x} {y} to-x≡y =
from y ≡⟨ cong from $ sym to-x≡y ⟩
from (to x) ≡⟨ left-inverse-of x ⟩∎
x ∎
open _↠_ surjection public
id : ∀ {a} {A : Set a} → A ↔ A
id = record
{ surjection = Surjection.id
; left-inverse-of = refl
}
inverse : ∀ {a b} {A : Set a} {B : Set b} → A ↔ B → B ↔ A
inverse A↔B = record
{ surjection = record
{ logical-equivalence = Logical-equivalence.inverse
logical-equivalence
; right-inverse-of = left-inverse-of
}
; left-inverse-of = right-inverse-of
} where open _↔_ A↔B
infixr 9 _∘_
_∘_ : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
B ↔ C → A ↔ B → A ↔ C
f ∘ g = record
{ surjection = Surjection._∘_ (surjection f) (surjection g)
; left-inverse-of = from∘to
}
where
open _↔_
abstract
from∘to : ∀ x → from g (from f (to f (to g x))) ≡ x
from∘to = λ x →
from g (from f (to f (to g x))) ≡⟨ cong (from g) (left-inverse-of f (to g x)) ⟩
from g (to g x) ≡⟨ left-inverse-of g x ⟩∎
x ∎
infix 0 finally-↔
infixr 0 _↔⟨_⟩_
_↔⟨_⟩_ : ∀ {a b c} (A : Set a) {B : Set b} {C : Set c} →
A ↔ B → B ↔ C → A ↔ C
_ ↔⟨ A↔B ⟩ B↔C = B↔C ∘ A↔B
finally-↔ : ∀ {a b} (A : Set a) (B : Set b) → A ↔ B → A ↔ B
finally-↔ _ _ A↔B = A↔B
syntax finally-↔ A B A↔B = A ↔⟨ A↔B ⟩□ B □
Σ-≡,≡↔≡ : ∀ {a b} {A : Set a} {B : A → Set b} {p₁ p₂ : Σ A B} →
(∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) →
subst B p (proj₂ p₁) ≡ proj₂ p₂) ↔
(p₁ ≡ p₂)
Σ-≡,≡↔≡ {A = A} {B} {p₁} {p₂} = record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
from-P = λ {p₁ p₂ : Σ A B} (_ : p₁ ≡ p₂) →
∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) →
subst B p (proj₂ p₁) ≡ proj₂ p₂
from : {p₁ p₂ : Σ A B} →
p₁ ≡ p₂ →
∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) →
subst B p (proj₂ p₁) ≡ proj₂ p₂
from = Σ-≡,≡←≡
to : {p₁ p₂ : Σ A B} →
(∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) →
subst B p (proj₂ p₁) ≡ proj₂ p₂) →
p₁ ≡ p₂
to = uncurry Σ-≡,≡→≡
abstract
to∘from : ∀ eq → to (from {p₁ = p₁} {p₂ = p₂} eq) ≡ eq
to∘from = elim (λ p≡q → to (from p≡q) ≡ p≡q) λ x →
let lem = subst-refl B (proj₂ x) in
to (from (refl x)) ≡⟨ cong to (elim-refl from-P _) ⟩
to (refl (proj₁ x) , lem) ≡⟨ Σ-≡,≡→≡-reflˡ _ ⟩
cong (_,_ (proj₁ x)) (trans (sym lem) lem) ≡⟨ cong (cong (_,_ (proj₁ x))) $ trans-symˡ lem ⟩
cong (_,_ (proj₁ x)) (refl (proj₂ x)) ≡⟨ cong-refl (_,_ (proj₁ x)) {x = proj₂ x} ⟩∎
refl x ∎
from∘to : ∀ p → from (to {p₁ = p₁} {p₂ = p₂} p) ≡ p
from∘to p = elim
(λ {x₁ x₂} x₁≡x₂ →
∀ {y₁ y₂} (y₁′≡y₂ : subst B x₁≡x₂ y₁ ≡ y₂) →
from (to (x₁≡x₂ , y₁′≡y₂)) ≡ (x₁≡x₂ , y₁′≡y₂))
(λ x {y₁} y₁′≡y₂ → elim
(λ {y₁ y₂} (y₁≡y₂ : y₁ ≡ y₂) →
(y₁′≡y₂ : subst B (refl x) y₁ ≡ y₂) →
y₁≡y₂ ≡ trans (sym $ subst-refl B y₁) y₁′≡y₂ →
from (to (refl x , y₁′≡y₂)) ≡ (refl x , y₁′≡y₂))
(λ y y′≡y eq →
let lem = subst-refl B y in
from (to (refl x , y′≡y)) ≡⟨ cong from $ Σ-≡,≡→≡-reflˡ _ ⟩
from (cong (_,_ x) (trans (sym lem) y′≡y)) ≡⟨ cong (from ⊚ cong (_,_ x)) $ sym eq ⟩
from (cong (_,_ x) (refl y)) ≡⟨ cong from $ cong-refl (_,_ x) {x = y} ⟩
from (refl (x , y)) ≡⟨ elim-refl from-P _ ⟩
(refl x , lem) ≡⟨ cong (_,_ (refl x)) (
lem ≡⟨ sym $ trans-reflʳ _ ⟩
trans lem (refl _) ≡⟨ cong (trans lem) eq ⟩
trans lem (trans (sym lem) y′≡y) ≡⟨ sym $ trans-assoc _ _ _ ⟩
trans (trans lem (sym lem)) y′≡y ≡⟨ cong (λ p → trans p y′≡y) $ trans-symʳ lem ⟩
trans (refl _) y′≡y ≡⟨ trans-reflˡ _ ⟩∎
y′≡y ∎) ⟩∎
(refl x , y′≡y) ∎)
(trans (sym $ subst-refl B y₁) y₁′≡y₂)
y₁′≡y₂
(refl _))
(proj₁ p) (proj₂ p)
≡↔inj₁≡inj₁ : ∀ {a b} {A : Set a} {B : Set b} {x y : A} →
(x ≡ y) ↔ _≡_ {A = A ⊎ B} (inj₁ x) (inj₁ y)
≡↔inj₁≡inj₁ {A = A} {B} {x} {y} = record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
to : x ≡ y → _≡_ {A = A ⊎ B} (inj₁ x) (inj₁ y)
to = cong inj₁
from = ⊎.cancel-inj₁
abstract
to∘from : ∀ ix≡iy → to (from ix≡iy) ≡ ix≡iy
to∘from ix≡iy =
cong inj₁ (⊎.cancel-inj₁ ix≡iy) ≡⟨ cong-∘ inj₁ [ P.id , const x ] ix≡iy ⟩
cong f ix≡iy ≡⟨ cong-roughly-id f p ix≡iy _ _ f≡id ⟩
trans (refl _) (trans ix≡iy $ sym (refl _)) ≡⟨ trans-reflˡ _ ⟩
trans ix≡iy (sym $ refl _) ≡⟨ cong (trans ix≡iy) sym-refl ⟩
trans ix≡iy (refl _) ≡⟨ trans-reflʳ _ ⟩∎
ix≡iy ∎
where
f : A ⊎ B → A ⊎ B
f = inj₁ ⊚ [ P.id , const x ]
p : A ⊎ B → Bool
p = [ const true , const false ]
f≡id : ∀ z → T (p z) → f z ≡ z
f≡id (inj₁ x) = const (refl (inj₁ x))
f≡id (inj₂ y) = ⊥-elim
from∘to : ∀ x≡y → from (to x≡y) ≡ x≡y
from∘to x≡y =
cong [ P.id , const x ] (cong inj₁ x≡y) ≡⟨ cong-∘ [ P.id , const x ] inj₁ _ ⟩
cong P.id x≡y ≡⟨ sym $ cong-id _ ⟩∎
x≡y ∎
≡↔inj₂≡inj₂ : ∀ {a b} {A : Set a} {B : Set b} {x y : B} →
(x ≡ y) ↔ _≡_ {A = A ⊎ B} (inj₂ x) (inj₂ y)
≡↔inj₂≡inj₂ {A = A} {B} {x} {y} = record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
to : x ≡ y → _≡_ {A = A ⊎ B} (inj₂ x) (inj₂ y)
to = cong inj₂
from = ⊎.cancel-inj₂
abstract
to∘from : ∀ ix≡iy → to (from ix≡iy) ≡ ix≡iy
to∘from ix≡iy =
cong inj₂ (⊎.cancel-inj₂ ix≡iy) ≡⟨ cong-∘ inj₂ [ const x , P.id ] ix≡iy ⟩
cong f ix≡iy ≡⟨ cong-roughly-id f p ix≡iy _ _ f≡id ⟩
trans (refl _) (trans ix≡iy $ sym (refl _)) ≡⟨ trans-reflˡ _ ⟩
trans ix≡iy (sym $ refl _) ≡⟨ cong (trans ix≡iy) sym-refl ⟩
trans ix≡iy (refl _) ≡⟨ trans-reflʳ _ ⟩∎
ix≡iy ∎
where
f : A ⊎ B → A ⊎ B
f = inj₂ ⊚ [ const x , P.id ]
p : A ⊎ B → Bool
p = [ const false , const true ]
f≡id : ∀ z → T (p z) → f z ≡ z
f≡id (inj₁ x) = ⊥-elim
f≡id (inj₂ y) = const (refl (inj₂ y))
from∘to : ∀ x≡y → from (to x≡y) ≡ x≡y
from∘to x≡y =
cong [ const x , P.id ] (cong inj₂ x≡y) ≡⟨ cong-∘ [ const x , P.id ] inj₂ _ ⟩
cong P.id x≡y ≡⟨ sym $ cong-id _ ⟩∎
x≡y ∎
decidable-equality-respects :
∀ {a b} {A : Set a} {B : Set b} →
A ↔ B → Decidable-equality A → Decidable-equality B
decidable-equality-respects A↔B _≟A_ x y =
⊎-map (_↔_.injective (inverse A↔B))
(λ from-x≢from-y → from-x≢from-y ⊚ cong from)
(from x ≟A from y)
where open _↔_ A↔B
contractible-isomorphic :
∀ {a b} {A : Set a} {B : Set b} →
Contractible A → Contractible B → A ↔ B
contractible-isomorphic {A} {B} cA cB = record
{ surjection = record
{ logical-equivalence = record
{ to = const (proj₁ cB)
; from = const (proj₁ cA)
}
; right-inverse-of = proj₂ cB
}
; left-inverse-of = proj₂ cA
}
implicit-Π↔Π : ∀ {a b} {A : Set a} {B : A → Set b} →
({x : A} → B x) ↔ ((x : A) → B x)
implicit-Π↔Π = record
{ surjection = record
{ logical-equivalence = record
{ to = λ f x → f {x}
; from = λ f {x} → f x
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
↑↔ : ∀ {a b} {A : Set a} → ↑ b A ↔ A
↑↔ {b = b} {A} = record
{ surjection = record
{ logical-equivalence = record
{ to = lower
; from = lift
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}