{-# OPTIONS --without-K #-}
open import Equality
module Function-universe
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where
open Derived-definitions-and-properties eq
open import Equivalence using (_⇔_; module _⇔_)
private
module Bijection where
import Bijection; open Bijection eq public
open Bijection using (_↔_; module _↔_)
import Equality.Decision-procedures as ED; open ED eq
private
module Injection where
import Injection; open Injection eq public
open Injection using (_↣_; module _↣_; Injective)
open import Prelude as P hiding (id) renaming (_∘_ to _⊚_)
private
module Surjection where
import Surjection; open Surjection eq public
open Surjection using (_↠_; module _↠_)
private
module Weak where
import Weak-equivalence; open Weak-equivalence eq public
open Weak using (_≈_; module _≈_)
data Kind : Set where
implication
equivalence
injection
surjection
bijection
weak-equivalence : Kind
infix 0 _↝[_]_
_↝[_]_ : ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Kind → Set ℓ₂ → Set _
A ↝[ implication ] B = A → B
A ↝[ equivalence ] B = A ⇔ B
A ↝[ injection ] B = A ↣ B
A ↝[ surjection ] B = A ↠ B
A ↝[ bijection ] B = A ↔ B
A ↝[ weak-equivalence ] B = A ≈ B
from-bijection : ∀ {k a b} {A : Set a} {B : Set b} →
A ↔ B → A ↝[ k ] B
from-bijection {implication} = _↔_.to
from-bijection {equivalence} = _↔_.equivalence
from-bijection {injection} = _↔_.injection
from-bijection {surjection} = _↔_.surjection
from-bijection {bijection} = P.id
from-bijection {weak-equivalence} = Weak.bijection⇒weak-equivalence
from-weak-equivalence : ∀ {k a b} {A : Set a} {B : Set b} →
A ≈ B → A ↝[ k ] B
from-weak-equivalence {implication} = _≈_.to
from-weak-equivalence {equivalence} = _≈_.equivalence
from-weak-equivalence {injection} = _≈_.injection
from-weak-equivalence {surjection} = _≈_.surjection
from-weak-equivalence {bijection} = _≈_.bijection
from-weak-equivalence {weak-equivalence} = P.id
to-implication : ∀ {k a b} {A : Set a} {B : Set b} →
A ↝[ k ] B → A → B
to-implication {implication} = P.id
to-implication {equivalence} = _⇔_.to
to-implication {injection} = _↣_.to
to-implication {surjection} = _↠_.to
to-implication {bijection} = _↔_.to
to-implication {weak-equivalence} = _≈_.to
data Symmetric-kind : Set where
equivalence bijection weak-equivalence : Symmetric-kind
⌊_⌋-sym : Symmetric-kind → Kind
⌊ equivalence ⌋-sym = equivalence
⌊ bijection ⌋-sym = bijection
⌊ weak-equivalence ⌋-sym = weak-equivalence
inverse : ∀ {k a b} {A : Set a} {B : Set b} →
A ↝[ ⌊ k ⌋-sym ] B → B ↝[ ⌊ k ⌋-sym ] A
inverse {equivalence} = Equivalence.inverse
inverse {bijection} = Bijection.inverse
inverse {weak-equivalence} = Weak.inverse
data Isomorphism-kind : Set where
bijection weak-equivalence : Isomorphism-kind
⌊_⌋-iso : Isomorphism-kind → Kind
⌊ bijection ⌋-iso = bijection
⌊ weak-equivalence ⌋-iso = weak-equivalence
infix 0 _↔[_]_
_↔[_]_ : ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Isomorphism-kind → Set ℓ₂ → Set _
A ↔[ k ] B = A ↝[ ⌊ k ⌋-iso ] B
from-isomorphism : ∀ {k₁ k₂ a b} {A : Set a} {B : Set b} →
A ↔[ k₁ ] B → A ↝[ k₂ ] B
from-isomorphism {bijection} = from-bijection
from-isomorphism {weak-equivalence} = from-weak-equivalence
to-implication∘from-isomorphism :
∀ {a b} {A : Set a} {B : Set b} k₁ k₂ {A↔B : A ↔[ k₁ ] B} →
to-implication A↔B ≡
to-implication (from-isomorphism {k₂ = k₂} A↔B)
to-implication∘from-isomorphism {A = A} {B} = t∘f
where
t∘f : ∀ k₁ k₂ {A↔B : A ↔[ k₁ ] B} →
to-implication A↔B ≡
to-implication (from-isomorphism {k₂ = k₂} A↔B)
t∘f bijection implication = refl _
t∘f bijection equivalence = refl _
t∘f bijection injection = refl _
t∘f bijection surjection = refl _
t∘f bijection bijection = refl _
t∘f bijection weak-equivalence = refl _
t∘f weak-equivalence implication = refl _
t∘f weak-equivalence equivalence = refl _
t∘f weak-equivalence injection = refl _
t∘f weak-equivalence surjection = refl _
t∘f weak-equivalence bijection = refl _
t∘f weak-equivalence weak-equivalence = refl _
infixr 9 _∘_
_∘_ : ∀ {k a b c} {A : Set a} {B : Set b} {C : Set c} →
B ↝[ k ] C → A ↝[ k ] B → A ↝[ k ] C
_∘_ {implication} = λ f g → f ⊚ g
_∘_ {equivalence} = Equivalence._∘_
_∘_ {injection} = Injection._∘_
_∘_ {surjection} = Surjection._∘_
_∘_ {bijection} = Bijection._∘_
_∘_ {weak-equivalence} = Weak._∘_
id : ∀ {k a} {A : Set a} → A ↝[ k ] A
id {implication} = P.id
id {equivalence} = Equivalence.id
id {injection} = Injection.id
id {surjection} = Surjection.id
id {bijection} = Bijection.id
id {weak-equivalence} = Weak.id
infixr 0 _↝⟨_⟩_ _↔⟨_⟩_ _↔⟨⟩_
infix 0 finally-↝ finally-↔
infix 0 _□
_↝⟨_⟩_ : ∀ {k a b c} (A : Set a) {B : Set b} {C : Set c} →
A ↝[ k ] B → B ↝[ k ] C → A ↝[ k ] C
_ ↝⟨ A↝B ⟩ B↝C = B↝C ∘ A↝B
_↔⟨_⟩_ : ∀ {k₁ k₂ a b c} (A : Set a) {B : Set b} {C : Set c} →
A ↔[ k₁ ] B → B ↝[ k₂ ] C → A ↝[ k₂ ] C
_ ↔⟨ A↔B ⟩ B↝C = _ ↝⟨ from-isomorphism A↔B ⟩ B↝C
_↔⟨⟩_ : ∀ {k a b} (A : Set a) {B : Set b} →
A ↝[ k ] B → A ↝[ k ] B
_ ↔⟨⟩ A↝B = A↝B
_□ : ∀ {k a} (A : Set a) → A ↝[ k ] A
A □ = id
finally-↝ : ∀ {k a b} (A : Set a) (B : Set b) →
A ↝[ k ] B → A ↝[ k ] B
finally-↝ _ _ A↝B = A↝B
finally-↔ : ∀ {k₁ k₂ a b} (A : Set a) (B : Set b) →
A ↔[ k₁ ] B → A ↝[ k₂ ] B
finally-↔ _ _ A↔B = from-isomorphism A↔B
syntax finally-↝ A B A↝B = A ↝⟨ A↝B ⟩□ B □
syntax finally-↔ A B A↔B = A ↔⟨ A↔B ⟩□ B □
≡⇒↝ : ∀ k {ℓ} {A B : Set ℓ} → A ≡ B → A ↝[ k ] B
≡⇒↝ k = elim (λ {A B} _ → A ↝[ k ] B) (λ _ → id)
contractible↔⊤ : ∀ {a} {A : Set a} → Contractible A → A ↔ ⊤
contractible↔⊤ c = record
{ surjection = record
{ equivalence = record
{ to = const tt
; from = const $ proj₁ c
}
; right-inverse-of = refl
}
; left-inverse-of = proj₂ c
}
private
⊎-cong-eq : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ⇔ A₂ → B₁ ⇔ B₂ → A₁ ⊎ B₁ ⇔ A₂ ⊎ B₂
⊎-cong-eq A₁⇔A₂ B₁⇔B₂ = record
{ to = ⊎-map (to A₁⇔A₂) (to B₁⇔B₂)
; from = ⊎-map (from A₁⇔A₂) (from B₁⇔B₂)
} where open _⇔_
⊎-cong-inj : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↣ A₂ → B₁ ↣ B₂ → A₁ ⊎ B₁ ↣ A₂ ⊎ B₂
⊎-cong-inj A₁↣A₂ B₁↣B₂ = record
{ to = to′
; injective = injective′
}
where
open _↣_
to′ = ⊎-map (to A₁↣A₂) (to B₁↣B₂)
abstract
injective′ : Injective to′
injective′ {x = inj₁ x} {y = inj₁ y} = cong inj₁ ⊚ injective A₁↣A₂ ⊚ ⊎.cancel-inj₁
injective′ {x = inj₂ x} {y = inj₂ y} = cong inj₂ ⊚ injective B₁↣B₂ ⊚ ⊎.cancel-inj₂
injective′ {x = inj₁ x} {y = inj₂ y} = ⊥-elim ⊚ ⊎.inj₁≢inj₂
injective′ {x = inj₂ x} {y = inj₁ y} = ⊥-elim ⊚ ⊎.inj₁≢inj₂ ⊚ sym
⊎-cong-surj : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↠ A₂ → B₁ ↠ B₂ → A₁ ⊎ B₁ ↠ A₂ ⊎ B₂
⊎-cong-surj A₁↠A₂ B₁↠B₂ = record
{ equivalence = ⊎-cong-eq (_↠_.equivalence A₁↠A₂)
(_↠_.equivalence B₁↠B₂)
; right-inverse-of =
[ cong inj₁ ⊚ _↠_.right-inverse-of A₁↠A₂
, cong inj₂ ⊚ _↠_.right-inverse-of B₁↠B₂
]
}
⊎-cong-bij : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↔ A₂ → B₁ ↔ B₂ → A₁ ⊎ B₁ ↔ A₂ ⊎ B₂
⊎-cong-bij A₁↔A₂ B₁↔B₂ = record
{ surjection = ⊎-cong-surj (_↔_.surjection A₁↔A₂)
(_↔_.surjection B₁↔B₂)
; left-inverse-of =
[ cong inj₁ ⊚ _↔_.left-inverse-of A₁↔A₂
, cong inj₂ ⊚ _↔_.left-inverse-of B₁↔B₂
]
}
infixr 1 _⊎-cong_
_⊎-cong_ : ∀ {k a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↝[ k ] A₂ → B₁ ↝[ k ] B₂ → A₁ ⊎ B₁ ↝[ k ] A₂ ⊎ B₂
_⊎-cong_ {implication} = ⊎-map
_⊎-cong_ {equivalence} = ⊎-cong-eq
_⊎-cong_ {injection} = ⊎-cong-inj
_⊎-cong_ {surjection} = ⊎-cong-surj
_⊎-cong_ {bijection} = ⊎-cong-bij
_⊎-cong_ {weak-equivalence} = λ A₁≈A₂ B₁≈B₂ →
from-bijection $ ⊎-cong-bij (from-weak-equivalence A₁≈A₂)
(from-weak-equivalence B₁≈B₂)
⊎-comm : ∀ {a b} {A : Set a} {B : Set b} → A ⊎ B ↔ B ⊎ A
⊎-comm = record
{ surjection = record
{ equivalence = record
{ to = [ inj₂ , inj₁ ]
; from = [ inj₂ , inj₁ ]
}
; right-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}
; left-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}
⊎-assoc : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
A ⊎ (B ⊎ C) ↔ (A ⊎ B) ⊎ C
⊎-assoc = record
{ surjection = record
{ equivalence = record
{ to = [ inj₁ ⊚ inj₁ , [ inj₁ ⊚ inj₂ , inj₂ ] ]
; from = [ [ inj₁ , inj₂ ⊚ inj₁ ] , inj₂ ⊚ inj₂ ]
}
; right-inverse-of =
[ [ refl ⊚ inj₁ ⊚ inj₁ , refl ⊚ inj₁ ⊚ inj₂ ] , refl ⊚ inj₂ ]
}
; left-inverse-of =
[ refl ⊚ inj₁ , [ refl ⊚ inj₂ ⊚ inj₁ , refl ⊚ inj₂ ⊚ inj₂ ] ]
}
⊎-left-identity : ∀ {a ℓ} {A : Set a} → ⊥ {ℓ = ℓ} ⊎ A ↔ A
⊎-left-identity = record
{ surjection = record
{ equivalence = record
{ to = λ { (inj₁ ()); (inj₂ x) → x }
; from = inj₂
}
; right-inverse-of = refl
}
; left-inverse-of = λ { (inj₁ ()); (inj₂ x) → refl (inj₂ x) }
}
⊎-right-identity : ∀ {a ℓ} {A : Set a} → A ⊎ ⊥ {ℓ = ℓ} ↔ A
⊎-right-identity {A = A} =
A ⊎ ⊥ ↔⟨ ⊎-comm ⟩
⊥ ⊎ A ↔⟨ ⊎-left-identity ⟩□
A □
⊎-idempotent : ∀ {a} {A : Set a} → A ⊎ A ⇔ A
⊎-idempotent = record
{ to = [ id , id ]
; from = inj₁
}
private
×-cong-eq : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ⇔ A₂ → B₁ ⇔ B₂ → A₁ × B₁ ⇔ A₂ × B₂
×-cong-eq A₁⇔A₂ B₁⇔B₂ = record
{ to = Σ-map (to A₁⇔A₂) (to B₁⇔B₂)
; from = Σ-map (from A₁⇔A₂) (from B₁⇔B₂)
} where open _⇔_
×-cong-inj : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↣ A₂ → B₁ ↣ B₂ → A₁ × B₁ ↣ A₂ × B₂
×-cong-inj {A₁ = A₁} {A₂} {B₁} {B₂} A₁↣A₂ B₁↣B₂ = record
{ to = to′
; injective = injective′
}
where
open _↣_
to′ : A₁ × B₁ → A₂ × B₂
to′ = Σ-map (to A₁↣A₂) (to B₁↣B₂)
abstract
injective′ : Injective to′
injective′ to′-x≡to′-y =
cong₂ _,_ (injective A₁↣A₂ (cong proj₁ to′-x≡to′-y))
(injective B₁↣B₂ (cong proj₂ to′-x≡to′-y))
×-cong-surj : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↠ A₂ → B₁ ↠ B₂ → A₁ × B₁ ↠ A₂ × B₂
×-cong-surj A₁↠A₂ B₁↠B₂ = record
{ equivalence = ×-cong-eq (_↠_.equivalence A₁↠A₂)
(_↠_.equivalence B₁↠B₂)
; right-inverse-of = uncurry λ x y →
cong₂ _,_ (_↠_.right-inverse-of A₁↠A₂ x)
(_↠_.right-inverse-of B₁↠B₂ y)
}
×-cong-bij : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↔ A₂ → B₁ ↔ B₂ → A₁ × B₁ ↔ A₂ × B₂
×-cong-bij A₁↔A₂ B₁↔B₂ = record
{ surjection = ×-cong-surj (_↔_.surjection A₁↔A₂)
(_↔_.surjection B₁↔B₂)
; left-inverse-of = uncurry λ x y →
cong₂ _,_ (_↔_.left-inverse-of A₁↔A₂ x)
(_↔_.left-inverse-of B₁↔B₂ y)
}
infixr 2 _×-cong_
_×-cong_ : ∀ {k a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
A₁ ↝[ k ] A₂ → B₁ ↝[ k ] B₂ → A₁ × B₁ ↝[ k ] A₂ × B₂
_×-cong_ {implication} = λ f g → Σ-map f g
_×-cong_ {equivalence} = ×-cong-eq
_×-cong_ {injection} = ×-cong-inj
_×-cong_ {surjection} = ×-cong-surj
_×-cong_ {bijection} = ×-cong-bij
_×-cong_ {weak-equivalence} = λ A₁≈A₂ B₁≈B₂ →
from-bijection $ ×-cong-bij (from-weak-equivalence A₁≈A₂)
(from-weak-equivalence B₁≈B₂)
×-comm : ∀ {a b} {A : Set a} {B : Set b} → A × B ↔ B × A
×-comm = record
{ surjection = record
{ equivalence = record
{ to = uncurry λ x y → (y , x)
; from = uncurry λ x y → (y , x)
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
×-assoc : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
A × (B × C) ↔ (A × B) × C
×-assoc = record
{ surjection = record
{ equivalence = record
{ to = uncurry λ x → uncurry λ y z → ((x , y) , z)
; from = uncurry (flip λ z → uncurry λ x y → (x , (y , z)))
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
Σ-left-identity : ∀ {a} {A : ⊤ → Set a} → Σ ⊤ A ↔ A tt
Σ-left-identity = record
{ surjection = record
{ equivalence = record
{ to = proj₂
; from = λ x → (tt , x)
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
×-left-identity : ∀ {a} {A : Set a} → ⊤ × A ↔ A
×-left-identity = Σ-left-identity
×-right-identity : ∀ {a} {A : Set a} → A × ⊤ ↔ A
×-right-identity {A = A} =
A × ⊤ ↔⟨ ×-comm ⟩
⊤ × A ↔⟨ ×-left-identity ⟩□
A □
Σ-left-zero : ∀ {ℓ₁ a ℓ₂} {A : ⊥ {ℓ = ℓ₁} → Set a} →
Σ ⊥ A ↔ ⊥ {ℓ = ℓ₂}
Σ-left-zero = record
{ surjection = record
{ equivalence = record
{ to = λ { (() , _) }
; from = λ ()
}
; right-inverse-of = λ ()
}
; left-inverse-of = λ { (() , _) }
}
×-left-zero : ∀ {a ℓ₁ ℓ₂} {A : Set a} → ⊥ {ℓ = ℓ₁} × A ↔ ⊥ {ℓ = ℓ₂}
×-left-zero = Σ-left-zero
×-right-zero : ∀ {a ℓ₁ ℓ₂} {A : Set a} → A × ⊥ {ℓ = ℓ₁} ↔ ⊥ {ℓ = ℓ₂}
×-right-zero {A = A} =
A × ⊥ ↔⟨ ×-comm ⟩
⊥ × A ↔⟨ ×-left-zero ⟩□
⊥ □
Σ-cong : ∀ {k₁ k₂ a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁↔A₂ : A₁ ↔[ k₁ ] A₂) →
(∀ x → B₁ x ↝[ k₂ ] B₂ (to-implication A₁↔A₂ x)) →
Σ A₁ B₁ ↝[ k₂ ] Σ A₂ B₂
Σ-cong {weak-equivalence} {weak-equivalence} A₁≈A₂ B₁≈B₂ =
Weak.Σ-preserves A₁≈A₂ B₁≈B₂
Σ-cong {k₁} {k₂} {A₁ = A₁} {A₂} {B₁} {B₂} A₁↔A₂ B₁↝B₂ = helper k₂ B₁↝B₂′
where
A₁≈A₂ : A₁ ≈ A₂
A₁≈A₂ = from-isomorphism A₁↔A₂
B₁↝B₂′ : ∀ x → B₁ x ↝[ k₂ ] B₂ (_≈_.to A₁≈A₂ x)
B₁↝B₂′ x =
B₁ x ↝⟨ B₁↝B₂ x ⟩
B₂ (to-implication A₁↔A₂ x) ↔⟨ ≡⇒↝ bijection $ cong (λ f → B₂ (f x)) $
to-implication∘from-isomorphism k₁ weak-equivalence ⟩
B₂ (_≈_.to (from-isomorphism A₁↔A₂) x) □
helper : ∀ k₂ → (∀ x → B₁ x ↝[ k₂ ] B₂ (_≈_.to A₁≈A₂ x)) →
Σ A₁ B₁ ↝[ k₂ ] Σ A₂ B₂
helper implication = Weak.∃-preserves-functions A₁≈A₂
helper equivalence = Weak.∃-preserves-equivalences A₁≈A₂
helper injection = Weak.∃-preserves-injections A₁≈A₂
helper surjection = Weak.∃-preserves-surjections A₁≈A₂
helper bijection = Weak.∃-preserves-bijections A₁≈A₂
helper weak-equivalence = Weak.Σ-preserves A₁≈A₂
private
∃-cong-impl : ∀ {a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x → B₂ x) → ∃ B₁ → ∃ B₂
∃-cong-impl B₁→B₂ = Σ-map id (λ {x} → B₁→B₂ x)
∃-cong-eq : ∀ {a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ⇔ B₂ x) → ∃ B₁ ⇔ ∃ B₂
∃-cong-eq B₁⇔B₂ = record
{ to = ∃-cong-impl (to ⊚ B₁⇔B₂)
; from = ∃-cong-impl (from ⊚ B₁⇔B₂)
} where open _⇔_
∃-cong-surj : ∀ {a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↠ B₂ x) → ∃ B₁ ↠ ∃ B₂
∃-cong-surj B₁↠B₂ = record
{ equivalence = ∃-cong-eq (_↠_.equivalence ⊚ B₁↠B₂)
; right-inverse-of = uncurry λ x y →
cong (_,_ x) (_↠_.right-inverse-of (B₁↠B₂ x) y)
}
∃-cong-bij : ∀ {a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↔ B₂ x) → ∃ B₁ ↔ ∃ B₂
∃-cong-bij B₁↔B₂ = record
{ surjection = ∃-cong-surj (_↔_.surjection ⊚ B₁↔B₂)
; left-inverse-of = uncurry λ x y →
cong (_,_ x) (_↔_.left-inverse-of (B₁↔B₂ x) y)
}
∃-cong : ∀ {k a b₁ b₂}
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↝[ k ] B₂ x) → ∃ B₁ ↝[ k ] ∃ B₂
∃-cong {implication} = ∃-cong-impl
∃-cong {equivalence} = ∃-cong-eq
∃-cong {injection} = Σ-cong Bijection.id
∃-cong {surjection} = ∃-cong-surj
∃-cong {bijection} = ∃-cong-bij
∃-cong {weak-equivalence} = λ B₁≈B₂ →
from-bijection $ ∃-cong-bij (from-weak-equivalence ⊚ B₁≈B₂)
∃-⊎-distrib-left :
∀ {a b c} {A : Set a} {B : A → Set b} {C : A → Set c} →
(∃ λ x → B x ⊎ C x) ↔ ∃ B ⊎ ∃ C
∃-⊎-distrib-left = record
{ surjection = record
{ equivalence = record
{ to = uncurry λ x → [ inj₁ ⊚ _,_ x , inj₂ ⊚ _,_ x ]
; from = [ Σ-map id inj₁ , Σ-map id inj₂ ]
}
; right-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}
; left-inverse-of =
uncurry λ x → [ refl ⊚ _,_ x ⊚ inj₁ , refl ⊚ _,_ x ⊚ inj₂ ]
}
∃-⊎-distrib-right :
∀ {a b c} {A : Set a} {B : Set b} {C : A ⊎ B → Set c} →
Σ (A ⊎ B) C ↔ Σ A (C ⊚ inj₁) ⊎ Σ B (C ⊚ inj₂)
∃-⊎-distrib-right {A = A} {B} {C} = record
{ surjection = record
{ equivalence = record
{ to = to
; from = from
}
; right-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}
; left-inverse-of = from∘to
}
where
to : Σ (A ⊎ B) C → Σ A (C ⊚ inj₁) ⊎ Σ B (C ⊚ inj₂)
to (inj₁ x , y) = inj₁ (x , y)
to (inj₂ x , y) = inj₂ (x , y)
from = [ Σ-map inj₁ id , Σ-map inj₂ id ]
from∘to : ∀ p → from (to p) ≡ p
from∘to (inj₁ x , y) = refl _
from∘to (inj₂ x , y) = refl _
∃-comm : ∀ {a b c} {A : Set a} {B : Set b} {C : A → B → Set c} →
(∃ λ x → ∃ λ y → C x y) ↔ (∃ λ y → ∃ λ x → C x y)
∃-comm = record
{ surjection = record
{ equivalence = record
{ to = uncurry λ x → uncurry λ y z → (y , (x , z))
; from = uncurry λ x → uncurry λ y z → (y , (x , z))
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
∃-intro : ∀ {a b} {A : Set a} (B : A → Set b) (x : A) →
B x ↔ ∃ λ y → B y × y ≡ x
∃-intro B x =
B x ↔⟨ inverse ×-right-identity ⟩
B x × ⊤ ↔⟨ id ×-cong inverse (contractible↔⊤ (singleton-contractible x)) ⟩
B x × (∃ λ y → y ≡ x) ↔⟨ ∃-comm ⟩
(∃ λ y → B x × y ≡ x) ↔⟨ ∃-cong (λ _ → ×-comm) ⟩
(∃ λ y → y ≡ x × B x) ↔⟨ ∃-cong (λ y → ∃-cong (λ y≡x → subst (λ x → B x ↔ B y) y≡x id)) ⟩
(∃ λ y → y ≡ x × B y) ↔⟨ ∃-cong (λ _ → ×-comm) ⟩□
(∃ λ y → B y × y ≡ x) □
×-⊎-distrib-left : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
A × (B ⊎ C) ↔ (A × B) ⊎ (A × C)
×-⊎-distrib-left = ∃-⊎-distrib-left
×-⊎-distrib-right : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
(A ⊎ B) × C ↔ (A × C) ⊎ (B × C)
×-⊎-distrib-right = ∃-⊎-distrib-right
Π-left-identity : ∀ {a} {A : ⊤ → Set a} → ((x : ⊤) → A x) ↔ A tt
Π-left-identity = record
{ surjection = record
{ equivalence = record
{ to = λ f → f tt
; from = λ x _ → x
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
→-right-zero : ∀ {a} {A : Set a} → (A → ⊤) ↔ ⊤
→-right-zero = record
{ surjection = record
{ equivalence = record
{ to = λ _ → tt
; from = λ _ _ → tt
}
; right-inverse-of = λ _ → refl tt
}
; left-inverse-of = λ _ → refl (λ _ → tt)
}
Π-comm : ∀ {a b c} {A : Set a} {B : Set b} {C : A → B → Set c} →
(∀ x y → C x y) ↔ (∀ y x → C x y)
Π-comm = record
{ surjection = record
{ equivalence = record { to = flip; from = flip }
; right-inverse-of = refl
}
; left-inverse-of = refl
}
Π≡↔≡-↠-≡ : ∀ k {a} {A : Set a} (x y : A) →
(∀ z → (z ≡ x) ↔[ k ] (z ≡ y)) ↠ (x ≡ y)
Π≡↔≡-↠-≡ k x y = record
{ equivalence = record { to = to; from = from }
; right-inverse-of = to∘from
}
where
to : (∀ z → (z ≡ x) ↔[ k ] (z ≡ y)) → x ≡ y
to f = to-implication (f x) (refl x)
from′ : x ≡ y → ∀ z → (z ≡ x) ↔ (z ≡ y)
from′ x≡y z = record
{ surjection = record
{ equivalence = record
{ to = λ z≡x → trans z≡x x≡y
; from = λ z≡y → trans z≡y (sym x≡y)
}
; right-inverse-of = λ z≡y → trans-[trans-sym] z≡y x≡y
}
; left-inverse-of = λ z≡x → trans-[trans]-sym z≡x x≡y
}
from : x ≡ y → ∀ z → (z ≡ x) ↔[ k ] (z ≡ y)
from x≡y z = from-bijection (from′ x≡y z)
abstract
to∘from : ∀ x≡y → to (from x≡y) ≡ x≡y
to∘from x≡y =
to (from x≡y) ≡⟨ sym $ cong (λ f → f (refl x)) $ to-implication∘from-isomorphism bijection ⌊ k ⌋-iso ⟩
trans (refl x) x≡y ≡⟨ trans-reflˡ _ ⟩∎
x≡y ∎
Π≡≈≡-↔-≡ : ∀ {a} →
({A : Set a} {B : A → Set a} → Extensionality A B) →
{A : Set a} (x y : A) →
(∀ z → (z ≡ x) ≈ (z ≡ y)) ↔ (x ≡ y)
Π≡≈≡-↔-≡ ext x y = record
{ surjection = surj
; left-inverse-of = from∘to
}
where
surj = Π≡↔≡-↠-≡ weak-equivalence x y
open _↠_ surj
abstract
from∘to : ∀ f → from (to f) ≡ f
from∘to f = ext λ z → Weak.lift-equality ext λ z≡x →
trans z≡x (_≈_.to (f x) (refl x)) ≡⟨ elim (λ {u v} u≡v →
(f : ∀ z → (z ≡ v) ≈ (z ≡ y)) →
trans u≡v (_≈_.to (f v) (refl v)) ≡
_≈_.to (f u) u≡v)
(λ _ _ → trans-reflˡ _)
z≡x f ⟩∎
_≈_.to (f z) z≡x ∎
if-lemma : ∀ {a b p} {A : Set a} {B : Set b} (P : Bool → Set p) →
A ↔ P true → B ↔ P false →
∀ b → T b × A ⊎ T (not b) × B ↔ P b
if-lemma {A = A} {B} P A↔ B↔ true =
⊤ × A ⊎ ⊥ × B ↔⟨ ×-left-identity ⊎-cong ×-left-zero ⟩
A ⊎ ⊥₀ ↔⟨ ⊎-right-identity ⟩
A ↔⟨ A↔ ⟩
P true □
if-lemma {A = A} {B} P A↔ B↔ false =
⊥ × A ⊎ ⊤ × B ↔⟨ ×-left-zero ⊎-cong ×-left-identity ⟩
⊥₀ ⊎ B ↔⟨ ⊎-left-identity ⟩
B ↔⟨ B↔ ⟩
P false □
if-encoding : ∀ {ℓ} {A B : Set ℓ} →
∀ b → if b then A else B ↔ T b × A ⊎ T (not b) × B
if-encoding {A = A} {B} =
inverse ⊚ if-lemma (λ b → if b then A else B) id id
ℕ↔ℕ⊎⊤ : ℕ ↔ ℕ ⊎ ⊤
ℕ↔ℕ⊎⊤ = record
{ surjection = record
{ equivalence = record
{ to = ℕ-rec (inj₂ tt) (λ n _ → inj₁ n)
; from = [ suc , const zero ]
}
; right-inverse-of = [ refl ⊚ inj₁ , refl ⊚ inj₂ ]
}
; left-inverse-of = ℕ-rec (refl 0) (λ n _ → refl (suc n))
}
¬-⊎-left-cancellative :
∀ k → ¬ ((A B C : Set) → A ⊎ B ↝[ k ] A ⊎ C → B ↝[ k ] C)
¬-⊎-left-cancellative k cancel =
¬B→C $ to-implication $ cancel A B C (from-bijection A⊎B↔A⊎C)
where
A = ℕ
B = ⊤
C = ⊥
A⊎B↔A⊎C : A ⊎ B ↔ A ⊎ C
A⊎B↔A⊎C =
ℕ ⊎ ⊤ ↔⟨ inverse ℕ↔ℕ⊎⊤ ⟩
ℕ ↔⟨ inverse ⊎-right-identity ⟩
ℕ ⊎ ⊥ □
¬B→C : ¬ (B → C)
¬B→C B→C = B→C tt
Well-behaved : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
(A ⊎ B → A ⊎ C) → Set _
Well-behaved f =
∀ {b a a′} → f (inj₂ b) ≡ inj₁ a → f (inj₁ a) ≢ inj₁ a′
⊎-left-cancellative :
∀ {a b c} {A : Set a} {B : Set b} {C : Set c}
(f : A ⊎ B ↔ A ⊎ C) →
Well-behaved (_↔_.to f) →
Well-behaved (_↔_.from f) →
B ↔ C
⊎-left-cancellative inv to-hyp from-hyp = record
{ surjection = record
{ equivalence = record
{ to = g (to inv) to-hyp
; from = g (from inv) from-hyp
}
; right-inverse-of = g∘g (inverse inv) from-hyp to-hyp
}
; left-inverse-of = g∘g inv to-hyp from-hyp
}
where
open _↔_
g : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c}
(f : A ⊎ B → A ⊎ C) → Well-behaved f → (B → C)
g f hyp b with inspect (f (inj₂ b))
g f hyp b | inj₂ c with-≡ eq₁ = c
g f hyp b | inj₁ a with-≡ eq₁ with inspect (f (inj₁ a))
g f hyp b | inj₁ a with-≡ eq₁ | inj₂ c with-≡ eq₂ = c
g f hyp b | inj₁ a with-≡ eq₁ | inj₁ a′ with-≡ eq₂ =
⊥-elim $ hyp eq₁ eq₂
abstract
g∘g : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c}
(f : A ⊎ B ↔ A ⊎ C) →
(to-hyp : Well-behaved (to f)) →
(from-hyp : Well-behaved (from f)) →
∀ b → g (from f) from-hyp (g (to f) to-hyp b) ≡ b
g∘g f to-hyp from-hyp b = g∘g′
where
g∘g′ : g (from f) from-hyp (g (to f) to-hyp b) ≡ b
g∘g′ with inspect (to f (inj₂ b))
g∘g′ | inj₂ c with-≡ eq₁ with inspect (from f (inj₂ c))
g∘g′ | inj₂ c with-≡ eq₁ | inj₂ b′ with-≡ eq₂ = ⊎.cancel-inj₂ (
inj₂ b′ ≡⟨ sym eq₂ ⟩
from f (inj₂ c) ≡⟨ to-from f eq₁ ⟩∎
inj₂ b ∎)
g∘g′ | inj₂ c with-≡ eq₁ | inj₁ a with-≡ eq₂ = ⊥-elim $ ⊎.inj₁≢inj₂ (
inj₁ a ≡⟨ sym eq₂ ⟩
from f (inj₂ c) ≡⟨ to-from f eq₁ ⟩∎
inj₂ b ∎)
g∘g′ | inj₁ a with-≡ eq₁ with inspect (to f (inj₁ a))
g∘g′ | inj₁ a with-≡ eq₁ | inj₁ a′ with-≡ eq₂ = ⊥-elim $ to-hyp eq₁ eq₂
g∘g′ | inj₁ a with-≡ eq₁ | inj₂ c with-≡ eq₂ with inspect (from f (inj₂ c))
g∘g′ | inj₁ a with-≡ eq₁ | inj₂ c with-≡ eq₂ | inj₂ b′ with-≡ eq₃ = ⊥-elim $ ⊎.inj₁≢inj₂ (
inj₁ a ≡⟨ sym $ to-from f eq₂ ⟩
from f (inj₂ c) ≡⟨ eq₃ ⟩∎
inj₂ b′ ∎)
g∘g′ | inj₁ a with-≡ eq₁ | inj₂ c with-≡ eq₂ | inj₁ a′ with-≡ eq₃ with inspect (from f (inj₁ a′))
g∘g′ | inj₁ a with-≡ eq₁ | inj₂ c with-≡ eq₂ | inj₁ a′ with-≡ eq₃ | inj₁ a″ with-≡ eq₄ = ⊥-elim $ from-hyp eq₃ eq₄
g∘g′ | inj₁ a with-≡ eq₁ | inj₂ c with-≡ eq₂ | inj₁ a′ with-≡ eq₃ | inj₂ b′ with-≡ eq₄ = ⊎.cancel-inj₂ (
let lemma =
inj₁ a′ ≡⟨ sym eq₃ ⟩
from f (inj₂ c) ≡⟨ to-from f eq₂ ⟩∎
inj₁ a ∎
in
inj₂ b′ ≡⟨ sym eq₄ ⟩
from f (inj₁ a′) ≡⟨ cong (from f ⊚ inj₁) $ ⊎.cancel-inj₁ lemma ⟩
from f (inj₁ a) ≡⟨ to-from f eq₁ ⟩∎
inj₂ b ∎)