{-# OPTIONS --without-K --safe #-}
open import Equality
module Function-universe
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where
open import Bijection eq as Bijection using (_↔_; module _↔_)
open Derived-definitions-and-properties eq
open import Embedding eq as Emb using (Is-embedding; Embedding)
open import Equality.Decidable-UIP eq
open import Equality.Decision-procedures eq
open import Equivalence eq as Eq using (_≃_; module _≃_)
open import H-level eq as H-level
open import H-level.Closure eq
open import Injection eq as Injection using (_↣_; module _↣_; Injective)
open import Logical-equivalence using (_⇔_; module _⇔_)
open import Nat eq hiding (_≟_)
open import Preimage eq using (_⁻¹_)
open import Prelude as P hiding (id) renaming (_∘_ to _⊚_)
open import Surjection eq as Surjection using (_↠_; module _↠_)
data Kind : Set where
implication
logical-equivalence
injection
embedding
surjection
bijection
equivalence : Kind
infix 0 _↝[_]_
_↝[_]_ : ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Kind → Set ℓ₂ → Set _
A ↝[ implication ] B = A → B
A ↝[ logical-equivalence ] B = A ⇔ B
A ↝[ injection ] B = A ↣ B
A ↝[ embedding ] B = Embedding A B
A ↝[ surjection ] B = A ↠ B
A ↝[ bijection ] B = A ↔ B
A ↝[ equivalence ] B = A ≃ B
from-equivalence : ∀ {k a b} {A : Set a} {B : Set b} →
A ≃ B → A ↝[ k ] B
from-equivalence {implication} = _≃_.to
from-equivalence {logical-equivalence} = _≃_.logical-equivalence
from-equivalence {injection} = _≃_.injection
from-equivalence {embedding} = λ f → record
{ to = _≃_.to f
; is-embedding = λ _ _ →
_≃_.is-equivalence
(Eq.inverse (Eq.≃-≡ f))
}
from-equivalence {surjection} = _≃_.surjection
from-equivalence {bijection} = _≃_.bijection
from-equivalence {equivalence} = P.id
from-bijection : ∀ {k a b} {A : Set a} {B : Set b} →
A ↔ B → A ↝[ k ] B
from-bijection {implication} = _↔_.to
from-bijection {logical-equivalence} = _↔_.logical-equivalence
from-bijection {injection} = _↔_.injection
from-bijection {embedding} = from-equivalence ⊚ Eq.↔⇒≃
from-bijection {surjection} = _↔_.surjection
from-bijection {bijection} = P.id
from-bijection {equivalence} = Eq.↔⇒≃
to-implication : ∀ {k a b} {A : Set a} {B : Set b} →
A ↝[ k ] B → A → B
to-implication {implication} = P.id
to-implication {logical-equivalence} = _⇔_.to
to-implication {injection} = _↣_.to
to-implication {embedding} = Embedding.to
to-implication {surjection} = _↠_.to
to-implication {bijection} = _↔_.to
to-implication {equivalence} = _≃_.to
data Symmetric-kind : Set where
logical-equivalence bijection equivalence : Symmetric-kind
⌊_⌋-sym : Symmetric-kind → Kind
⌊ logical-equivalence ⌋-sym = logical-equivalence
⌊ bijection ⌋-sym = bijection
⌊ equivalence ⌋-sym = equivalence
inverse : ∀ {k a b} {A : Set a} {B : Set b} →
A ↝[ ⌊ k ⌋-sym ] B → B ↝[ ⌊ k ⌋-sym ] A
inverse {logical-equivalence} = Logical-equivalence.inverse
inverse {bijection} = Bijection.inverse
inverse {equivalence} = Eq.inverse
data Isomorphism-kind : Set where
bijection equivalence : Isomorphism-kind
⌊_⌋-iso : Isomorphism-kind → Kind
⌊ bijection ⌋-iso = bijection
⌊ equivalence ⌋-iso = 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 {equivalence} = from-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 logical-equivalence = refl _
t∘f bijection injection = refl _
t∘f bijection embedding = refl _
t∘f bijection surjection = refl _
t∘f bijection bijection = refl _
t∘f bijection equivalence = refl _
t∘f equivalence implication = refl _
t∘f equivalence logical-equivalence = refl _
t∘f equivalence injection = refl _
t∘f equivalence embedding = refl _
t∘f equivalence surjection = refl _
t∘f equivalence bijection = refl _
t∘f equivalence 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
_∘_ {logical-equivalence} = Logical-equivalence._∘_
_∘_ {injection} = Injection._∘_
_∘_ {embedding} = Emb._∘_
_∘_ {surjection} = Surjection._∘_
_∘_ {bijection} = Bijection._∘_
_∘_ {equivalence} = Eq._∘_
id : ∀ {k a} {A : Set a} → A ↝[ k ] A
id {implication} = P.id
id {logical-equivalence} = Logical-equivalence.id
id {injection} = Injection.id
id {embedding} = Emb.id
id {surjection} = Surjection.id
id {bijection} = Bijection.id
id {equivalence} = Eq.id
infix -1 finally-↝ finally-↔
infix -1 _□
infixr -2 step-↝ step-↔ _↔⟨⟩_
infix -3 $⟨_⟩_
step-↝ : ∀ {k a b c} (A : Set a) {B : Set b} {C : Set c} →
B ↝[ k ] C → A ↝[ k ] B → A ↝[ k ] C
step-↝ _ = _∘_
syntax step-↝ A B↝C A↝B = A ↝⟨ A↝B ⟩ B↝C
step-↔ : ∀ {k₁ k₂ a b c} (A : Set a) {B : Set b} {C : Set c} →
B ↝[ k₂ ] C → A ↔[ k₁ ] B → A ↝[ k₂ ] C
step-↔ _ B↝C A↔B = step-↝ _ B↝C (from-isomorphism A↔B)
syntax step-↔ A B↝C A↔B = A ↔⟨ 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
syntax finally-↝ A B A↝B = A ↝⟨ A↝B ⟩□ 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 □
$⟨_⟩_ : ∀ {k a b} {A : Set a} {B : Set b} →
A → A ↝[ k ] B → B
$⟨ a ⟩ A↝B = to-implication A↝B a
to-implication-id :
∀ {a} {A : Set a} k →
to-implication {k = k} id ≡ id {A = A}
to-implication-id implication = refl _
to-implication-id logical-equivalence = refl _
to-implication-id injection = refl _
to-implication-id embedding = refl _
to-implication-id surjection = refl _
to-implication-id bijection = refl _
to-implication-id equivalence = refl _
to-implication-∘ :
∀ {a b c} {A : Set a} {B : Set b} {C : Set c} →
(k : Kind) {f : A ↝[ k ] B} {g : B ↝[ k ] C} →
to-implication (g ∘ f) ≡ to-implication g ∘ to-implication f
to-implication-∘ implication = refl _
to-implication-∘ logical-equivalence = refl _
to-implication-∘ injection = refl _
to-implication-∘ embedding = refl _
to-implication-∘ surjection = refl _
to-implication-∘ bijection = refl _
to-implication-∘ equivalence = refl _
to-implication-inverse-id :
∀ {a} {A : Set a} k →
to-implication (inverse {k = k} id) ≡ id {A = A}
to-implication-inverse-id logical-equivalence = refl _
to-implication-inverse-id bijection = refl _
to-implication-inverse-id equivalence = refl _
data Without-extensionality : Set where
implication logical-equivalence : Without-extensionality
⌊_⌋-without : Without-extensionality → Kind
⌊ implication ⌋-without = implication
⌊ logical-equivalence ⌋-without = logical-equivalence
data With-extensionality : Set where
injection embedding surjection bijection equivalence :
With-extensionality
⌊_⌋-with : With-extensionality → Kind
⌊ injection ⌋-with = injection
⌊ embedding ⌋-with = embedding
⌊ surjection ⌋-with = surjection
⌊ bijection ⌋-with = bijection
⌊ equivalence ⌋-with = equivalence
data Extensionality-kind : Kind → Set where
without-extensionality : (k : Without-extensionality) →
Extensionality-kind ⌊ k ⌋-without
with-extensionality : (k : With-extensionality) →
Extensionality-kind ⌊ k ⌋-with
extensionality? : (k : Kind) → Extensionality-kind k
extensionality? implication = without-extensionality implication
extensionality? logical-equivalence = without-extensionality
logical-equivalence
extensionality? injection = with-extensionality injection
extensionality? embedding = with-extensionality embedding
extensionality? surjection = with-extensionality surjection
extensionality? bijection = with-extensionality bijection
extensionality? equivalence = with-extensionality equivalence
Extensionality? : Kind → (a b : Level) → Set (lsuc (a ⊔ b))
Extensionality? k with extensionality? k
... | without-extensionality _ = λ _ _ → ↑ _ ⊤
... | with-extensionality _ = Extensionality
forget-ext? : ∀ k {a b} → Extensionality a b → Extensionality? k a b
forget-ext? k with extensionality? k
... | without-extensionality _ = _
... | with-extensionality _ = id
lower-extensionality? :
∀ k {a} â {b} b̂ →
Extensionality? k (a ⊔ â) (b ⊔ b̂) → Extensionality? k a b
lower-extensionality? k with extensionality? k
... | without-extensionality _ = _
... | with-extensionality _ = lower-extensionality
generalise-ext? :
∀ {a b c d} {A : Set a} {B : Set b} →
A ⇔ B →
(Extensionality c d → A ↔ B) →
∀ {k} → Extensionality? k c d → A ↝[ k ] B
generalise-ext? f⇔ f↔ {k} with extensionality? k
... | without-extensionality implication = λ _ → _⇔_.to f⇔
... | without-extensionality logical-equivalence = λ _ → f⇔
... | with-extensionality _ = λ ext →
from-isomorphism (f↔ ext)
inverse-ext? :
∀ {a b c d} {A : Set a} {B : Set b} →
(∀ {k} → Extensionality? k c d → A ↝[ k ] B) →
(∀ {k} → Extensionality? k c d → B ↝[ k ] A)
inverse-ext? hyp = generalise-ext? (inverse $ hyp _) (inverse ⊚ hyp)
≡⇒↝ : ∀ k {ℓ} {A B : Set ℓ} → A ≡ B → A ↝[ k ] B
≡⇒↝ k = elim (λ {A B} _ → A ↝[ k ] B) (λ _ → id)
abstract
≡⇒↝-refl : ∀ {k a} {A : Set a} →
≡⇒↝ k (refl A) ≡ id
≡⇒↝-refl {k} = elim-refl (λ {A B} _ → A ↝[ k ] B) _
≡⇒↝-sym : ∀ k {ℓ} {A B : Set ℓ} {eq : A ≡ B} →
to-implication (≡⇒↝ ⌊ k ⌋-sym (sym eq)) ≡
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym eq))
≡⇒↝-sym k {A = A} {eq = eq} = elim¹
(λ eq → to-implication (≡⇒↝ ⌊ k ⌋-sym (sym eq)) ≡
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym eq)))
(to-implication (≡⇒↝ ⌊ k ⌋-sym (sym (refl A))) ≡⟨ cong (to-implication ∘ ≡⇒↝ ⌊ k ⌋-sym) sym-refl ⟩
to-implication (≡⇒↝ ⌊ k ⌋-sym (refl A)) ≡⟨ cong (to-implication {k = ⌊ k ⌋-sym}) ≡⇒↝-refl ⟩
to-implication {k = ⌊ k ⌋-sym} id ≡⟨ to-implication-id ⌊ k ⌋-sym ⟩
id ≡⟨ sym $ to-implication-inverse-id k ⟩
to-implication (inverse {k = k} id) ≡⟨ cong (to-implication ∘ inverse {k = k}) $ sym ≡⇒↝-refl ⟩∎
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym (refl A))) ∎)
eq
≡⇒↝-trans : ∀ k {ℓ} {A B C : Set ℓ} {A≡B : A ≡ B} {B≡C : B ≡ C} →
to-implication (≡⇒↝ k (trans A≡B B≡C)) ≡
to-implication (≡⇒↝ k B≡C ∘ ≡⇒↝ k A≡B)
≡⇒↝-trans k {B = B} {A≡B = A≡B} = elim¹
(λ B≡C → to-implication (≡⇒↝ k (trans A≡B B≡C)) ≡
to-implication (≡⇒↝ k B≡C ∘ ≡⇒↝ k A≡B))
(to-implication (≡⇒↝ k (trans A≡B (refl B))) ≡⟨ cong (to-implication ∘ ≡⇒↝ k) $ trans-reflʳ _ ⟩
to-implication (≡⇒↝ k A≡B) ≡⟨ sym $ cong (λ f → f ∘ to-implication (≡⇒↝ k A≡B)) $ to-implication-id k ⟩
to-implication {k = k} id ∘ to-implication (≡⇒↝ k A≡B) ≡⟨ sym $ to-implication-∘ k ⟩
to-implication (id ∘ ≡⇒↝ k A≡B) ≡⟨ sym $ cong (λ f → to-implication (f ∘ ≡⇒↝ k A≡B)) ≡⇒↝-refl ⟩∎
to-implication (≡⇒↝ k (refl B) ∘ ≡⇒↝ k A≡B) ∎)
_
≡⇒↝-cong : ∀ {k ℓ p A B} {eq : A ≡ B}
(P : Set ℓ → Set p)
(P-cong : ∀ {A B} → A ↝[ k ] B → P A ↝[ k ] P B) →
P-cong (id {A = A}) ≡ id →
≡⇒↝ _ (cong P eq) ≡ P-cong (≡⇒↝ _ eq)
≡⇒↝-cong {eq = eq} P P-cong P-cong-id = elim¹
(λ eq → ≡⇒↝ _ (cong P eq) ≡ P-cong (≡⇒↝ _ eq))
(≡⇒↝ _ (cong P (refl _)) ≡⟨ cong (≡⇒↝ _) $ cong-refl P ⟩
≡⇒↝ _ (refl _) ≡⟨ elim-refl (λ {A B} _ → A ↝[ _ ] B) _ ⟩
id ≡⟨ sym P-cong-id ⟩
P-cong id ≡⟨ cong P-cong $ sym $
elim-refl (λ {A B} _ → A ↝[ _ ] B) _ ⟩∎
P-cong (≡⇒↝ _ (refl _)) ∎)
eq
subst-in-terms-of-≡⇒↝ :
∀ k {a p} {A : Set a} {x y} (x≡y : x ≡ y) (P : A → Set p) p →
subst P x≡y p ≡ to-implication (≡⇒↝ k (cong P x≡y)) p
subst-in-terms-of-≡⇒↝ k x≡y P p = elim¹
(λ eq → subst P eq p ≡ to-implication (≡⇒↝ k (cong P eq)) p)
(subst P (refl _) p ≡⟨ subst-refl P p ⟩
p ≡⟨ sym $ cong (_$ p) (to-implication-id k) ⟩
to-implication {k = k} id p ≡⟨ sym $ cong (λ f → to-implication {k = k} f p) ≡⇒↝-refl ⟩
to-implication (≡⇒↝ k (refl _)) p ≡⟨ sym $ cong (λ eq → to-implication (≡⇒↝ k eq) p) $ cong-refl P ⟩∎
to-implication (≡⇒↝ k (cong P (refl _))) p ∎)
x≡y
subst-in-terms-of-inverse∘≡⇒↝ :
∀ k {a p} {A : Set a} {x y} (x≡y : x ≡ y) (P : A → Set p) p →
subst P (sym x≡y) p ≡
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym (cong P x≡y))) p
subst-in-terms-of-inverse∘≡⇒↝ k x≡y P p =
subst P (sym x≡y) p ≡⟨ subst-in-terms-of-≡⇒↝ ⌊ k ⌋-sym (sym x≡y) P p ⟩
to-implication (≡⇒↝ ⌊ k ⌋-sym (cong P (sym x≡y))) p ≡⟨ cong (λ eq → to-implication (≡⇒↝ ⌊ k ⌋-sym eq) p) (cong-sym P _) ⟩
to-implication (≡⇒↝ ⌊ k ⌋-sym (sym $ cong P x≡y)) p ≡⟨ cong (_$ p) (≡⇒↝-sym k) ⟩∎
to-implication (inverse (≡⇒↝ ⌊ k ⌋-sym (cong P x≡y))) p ∎
to-implication-≡⇒↝ :
∀ k {ℓ} {A B : Set ℓ} (eq : A ≡ B) →
to-implication (≡⇒↝ k eq) ≡ ≡⇒↝ implication eq
to-implication-≡⇒↝ k =
elim (λ eq → to-implication (≡⇒↝ k eq) ≡ ≡⇒↝ implication eq)
(λ A → to-implication (≡⇒↝ k (refl A)) ≡⟨ cong to-implication (≡⇒↝-refl {k = k}) ⟩
to-implication {k = k} id ≡⟨ to-implication-id k ⟩
id ≡⟨ sym ≡⇒↝-refl ⟩∎
≡⇒↝ implication (refl A) ∎)
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-emb : ∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : Set b₁} {B₂ : Set b₂} →
Embedding A₁ A₂ → Embedding B₁ B₂ →
Embedding (A₁ ⊎ B₁) (A₂ ⊎ B₂)
⊎-cong-emb A₁↣A₂ B₁↣B₂ = record
{ to = to′
; is-embedding = is-embedding′
}
where
open Embedding
to′ = ⊎-map (to A₁↣A₂) (to B₁↣B₂)
is-embedding′ : Is-embedding to′
is-embedding′ (inj₁ x) (inj₁ y) =
_≃_.is-equivalence $
Eq.with-other-function
(inj₁ x ≡ inj₁ y ↔⟨ inverse Bijection.≡↔inj₁≡inj₁ ⟩
x ≡ y ↝⟨ Eq.⟨ _ , is-embedding A₁↣A₂ _ _ ⟩ ⟩
to A₁↣A₂ x ≡ to A₁↣A₂ y ↔⟨ Bijection.≡↔inj₁≡inj₁ ⟩□
inj₁ (to A₁↣A₂ x) ≡ inj₁ (to A₁↣A₂ y) □)
_
(λ eq →
cong inj₁ (cong (to A₁↣A₂) (⊎.cancel-inj₁ eq)) ≡⟨ cong-∘ _ _ _ ⟩
cong (inj₁ ⊚ to A₁↣A₂) (⊎.cancel-inj₁ eq) ≡⟨ cong-∘ _ _ _ ⟩
cong (inj₁ ⊚ to A₁↣A₂ ⊚ [ id , const x ]) eq ≡⟨ sym $ trans-reflʳ _ ⟩
trans (cong (inj₁ ⊚ to A₁↣A₂ ⊚ [ id , const x ]) eq) (refl _) ≡⟨ cong-respects-relevant-equality
{f = inj₁ ⊚ to A₁↣A₂ ⊚ [ id , const x ]}
(if_then true else false)
[ (λ _ _ → refl _) , (λ _ ()) ] ⟩
trans (refl _) (cong (⊎-map (to A₁↣A₂) (to B₁↣B₂)) eq) ≡⟨ trans-reflˡ _ ⟩∎
cong (⊎-map (to A₁↣A₂) (to B₁↣B₂)) eq ∎)
is-embedding′ (inj₂ x) (inj₂ y) =
_≃_.is-equivalence $
Eq.with-other-function
(inj₂ x ≡ inj₂ y ↔⟨ inverse Bijection.≡↔inj₂≡inj₂ ⟩
x ≡ y ↝⟨ Eq.⟨ _ , is-embedding B₁↣B₂ _ _ ⟩ ⟩
to B₁↣B₂ x ≡ to B₁↣B₂ y ↔⟨ Bijection.≡↔inj₂≡inj₂ ⟩□
inj₂ (to B₁↣B₂ x) ≡ inj₂ (to B₁↣B₂ y) □)
_
(λ eq →
cong inj₂ (cong (to B₁↣B₂) (⊎.cancel-inj₂ eq)) ≡⟨ cong-∘ _ _ _ ⟩
cong (inj₂ ⊚ to B₁↣B₂) (⊎.cancel-inj₂ eq) ≡⟨ cong-∘ _ _ _ ⟩
cong (inj₂ ⊚ to B₁↣B₂ ⊚ [ const x , id ]) eq ≡⟨ sym $ trans-reflʳ _ ⟩
trans (cong (inj₂ ⊚ to B₁↣B₂ ⊚ [ const x , id ]) eq) (refl _) ≡⟨ cong-respects-relevant-equality
{f = inj₂ ⊚ to B₁↣B₂ ⊚ [ const x , id ]}
(if_then false else true)
[ (λ _ ()) , (λ _ _ → refl _) ] ⟩
trans (refl _) (cong (⊎-map (to A₁↣A₂) (to B₁↣B₂)) eq) ≡⟨ trans-reflˡ _ ⟩∎
cong (⊎-map (to A₁↣A₂) (to B₁↣B₂)) eq ∎)
is-embedding′ (inj₁ x) (inj₂ y) =
_≃_.is-equivalence $
Eq.with-other-function
(inj₁ x ≡ inj₂ y ↔⟨ inverse $ Bijection.⊥↔uninhabited ⊎.inj₁≢inj₂ ⟩
⊥₀ ↔⟨ Bijection.⊥↔uninhabited ⊎.inj₁≢inj₂ ⟩□
inj₁ (to A₁↣A₂ x) ≡ inj₂ (to B₁↣B₂ y) □)
_
(⊥-elim ⊚ ⊎.inj₁≢inj₂)
is-embedding′ (inj₂ x) (inj₁ y) =
_≃_.is-equivalence $
Eq.with-other-function
(inj₂ x ≡ inj₁ y ↔⟨ inverse $ Bijection.⊥↔uninhabited (⊎.inj₁≢inj₂ ⊚ sym) ⟩
⊥₀ ↔⟨ Bijection.⊥↔uninhabited (⊎.inj₁≢inj₂ ⊚ sym) ⟩□
inj₂ (to B₁↣B₂ x) ≡ inj₁ (to A₁↣A₂ 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
{ logical-equivalence = ⊎-cong-eq (_↠_.logical-equivalence A₁↠A₂)
(_↠_.logical-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_ {logical-equivalence} = ⊎-cong-eq
_⊎-cong_ {injection} = ⊎-cong-inj
_⊎-cong_ {embedding} = ⊎-cong-emb
_⊎-cong_ {surjection} = ⊎-cong-surj
_⊎-cong_ {bijection} = ⊎-cong-bij
_⊎-cong_ {equivalence} = λ A₁≃A₂ B₁≃B₂ →
from-bijection $ ⊎-cong-bij (from-equivalence A₁≃A₂)
(from-equivalence B₁≃B₂)
⊎-comm : ∀ {a b} {A : Set a} {B : Set b} → A ⊎ B ↔ B ⊎ A
⊎-comm = record
{ surjection = record
{ logical-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
{ logical-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
{ logical-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
}
Σ-preserves-embeddings :
∀ {a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
(A₁↣A₂ : Embedding A₁ A₂) →
(∀ x → Embedding (B₁ x) (B₂ (Embedding.to A₁↣A₂ x))) →
Embedding (Σ A₁ B₁) (Σ A₂ B₂)
Σ-preserves-embeddings {B₁ = B₁} {B₂} A₁↣A₂ B₁↣B₂ = record
{ to = Σ-map (to A₁↣A₂) (to (B₁↣B₂ _))
; is-embedding = λ { (x₁ , x₂) (y₁ , y₂) →
_≃_.is-equivalence $
Eq.with-other-function
((x₁ , x₂) ≡ (y₁ , y₂) ↝⟨ inverse $ Eq.↔⇒≃ Bijection.Σ-≡,≡↔≡ ⟩
(∃ λ (eq : x₁ ≡ y₁) → subst B₁ eq x₂ ≡ y₂) ↝⟨ Eq.Σ-preserves (Embedding.equivalence A₁↣A₂) (λ eq →
subst B₁ eq x₂ ≡ y₂ ↝⟨ Embedding.equivalence (B₁↣B₂ y₁) ⟩
to (B₁↣B₂ y₁) (subst B₁ eq x₂) ≡ to (B₁↣B₂ y₁) y₂ ↝⟨ ≡⇒↝ _ (cong (_≡ _) $ lemma₁ eq _ y₂) ⟩□
subst B₂ (cong (to A₁↣A₂) eq) (to (B₁↣B₂ x₁) x₂) ≡
to (B₁↣B₂ y₁) y₂ □) ⟩
(∃ λ (eq : to A₁↣A₂ x₁ ≡ to A₁↣A₂ y₁) →
subst B₂ eq (to (B₁↣B₂ x₁) x₂) ≡ to (B₁↣B₂ y₁) y₂) ↝⟨ Eq.↔⇒≃ Bijection.Σ-≡,≡↔≡ ⟩□
(to A₁↣A₂ x₁ , to (B₁↣B₂ x₁) x₂) ≡
(to A₁↣A₂ y₁ , to (B₁↣B₂ y₁) y₂) □)
_
(elim
(λ { {y = _ , y₂} eq →
uncurry Σ-≡,≡→≡
(Σ-map (cong (to A₁↣A₂))
(_≃_.to (≡⇒↝ _ (cong (_≡ _) $ lemma₁ _ _ y₂)) ⊚
cong (to (B₁↣B₂ _)))
(Σ-≡,≡←≡ eq)) ≡
cong (Σ-map (to A₁↣A₂) (to (B₁↣B₂ _))) eq })
(λ _ →
uncurry Σ-≡,≡→≡
(Σ-map (cong (to A₁↣A₂))
(_≃_.to (≡⇒↝ _ (cong (_≡ _) $ lemma₁ _ _ _)) ⊚
cong (to (B₁↣B₂ _)))
(Σ-≡,≡←≡ (refl _))) ≡⟨ cong (λ eq → uncurry Σ-≡,≡→≡
(Σ-map _
(_≃_.to (≡⇒↝ _ (cong (_≡ _) $ lemma₁ _ _ _)) ⊚
cong (to (B₁↣B₂ _)))
eq))
Σ-≡,≡←≡-refl ⟩
Σ-≡,≡→≡
(cong (to A₁↣A₂) (refl _))
(_≃_.to (≡⇒↝ _ (cong (_≡ to (B₁↣B₂ _) _) $ lemma₁ _ _ _))
(cong (to (B₁↣B₂ _)) (subst-refl B₁ _))) ≡⟨ Σ-≡,≡→≡-cong (cong-refl _) (lemma₂ _ _) ⟩
Σ-≡,≡→≡ (refl _) (subst-refl B₂ _) ≡⟨ Σ-≡,≡→≡-refl-subst-refl ⟩
refl _ ≡⟨ sym $ cong-refl _ ⟩∎
cong (Σ-map (to A₁↣A₂) (to (B₁↣B₂ _))) (refl _) ∎)) }
}
where
open Embedding using (to)
lemma₁ = elim
(λ {x₁ y₁} eq → (x₂ : B₁ x₁) (y₂ : B₁ y₁) →
to (B₁↣B₂ y₁) (subst B₁ eq x₂) ≡
subst B₂ (cong (to A₁↣A₂) eq) (to (B₁↣B₂ x₁) x₂))
(λ z₁ x₂ y₂ →
to (B₁↣B₂ z₁) (subst B₁ (refl z₁) x₂) ≡⟨ cong (to (B₁↣B₂ z₁)) $ subst-refl _ _ ⟩
to (B₁↣B₂ z₁) x₂ ≡⟨ sym $ subst-refl _ _ ⟩
subst B₂ (refl (to A₁↣A₂ z₁)) (to (B₁↣B₂ z₁) x₂) ≡⟨ cong (λ eq → subst B₂ eq _) (sym $ cong-refl _) ⟩∎
subst B₂ (cong (to A₁↣A₂) (refl z₁)) (to (B₁↣B₂ z₁) x₂) ∎)
lemma₂ = λ x y →
let eq₁ = cong (flip (subst B₂) _) (sym (cong-refl _))
eq₂ = cong (to (B₁↣B₂ x)) (subst-refl B₁ y)
in
trans eq₁ (_≃_.to (≡⇒↝ _ (cong (_≡ _) $ lemma₁ (refl x) y y)) eq₂) ≡⟨ cong (λ eq → trans eq₁ (_≃_.to (≡⇒↝ _ (cong (_≡ _) (eq y y))) eq₂)) $
elim-refl (λ {x₁ y₁} eq → (x₂ : B₁ x₁) (y₂ : B₁ y₁) →
to (B₁↣B₂ y₁) (subst B₁ eq x₂) ≡
subst B₂ (cong (to A₁↣A₂) eq) (to (B₁↣B₂ x₁) x₂))
_ ⟩
trans eq₁ (_≃_.to (≡⇒↝ _ $ cong (_≡ _) $
trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁))
eq₂) ≡⟨ cong (trans _) $ sym $ subst-in-terms-of-≡⇒↝ equivalence _ _ _ ⟩
trans eq₁ (subst (_≡ _)
(trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁))
eq₂) ≡⟨ cong (λ eq → trans eq₁ (subst (_≡ _) eq eq₂)) $
sym $ sym-sym (trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁)) ⟩
trans eq₁ (subst (_≡ _)
(sym $ sym $
trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁))
eq₂) ≡⟨ cong (trans _) $ subst-trans _ ⟩
trans eq₁ (trans
(sym $ trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁))
eq₂) ≡⟨ cong (λ eq → trans eq₁ (trans eq eq₂)) $
sym-trans eq₂ (trans (sym $ subst-refl B₂ _) eq₁) ⟩
trans eq₁ (trans (trans (sym $ trans (sym $ subst-refl B₂ _) eq₁)
(sym eq₂))
eq₂) ≡⟨ cong (trans _) $ trans-[trans-sym]- _ _ ⟩
trans eq₁ (sym $ trans (sym $ subst-refl B₂ _) eq₁) ≡⟨ cong (trans _) $ sym-trans _ _ ⟩
trans eq₁ (trans (sym eq₁) (sym $ sym $ subst-refl B₂ _)) ≡⟨ trans--[trans-sym] _ _ ⟩
sym $ sym $ subst-refl B₂ _ ≡⟨ sym-sym _ ⟩∎
subst-refl B₂ _ ∎
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
{ logical-equivalence = ×-cong-eq (_↠_.logical-equivalence A₁↠A₂)
(_↠_.logical-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_ {logical-equivalence} = ×-cong-eq
_×-cong_ {injection} = ×-cong-inj
_×-cong_ {embedding} = λ A₁↣A₂ B₁↣B₂ →
Σ-preserves-embeddings
A₁↣A₂ (λ _ → B₁↣B₂)
_×-cong_ {surjection} = ×-cong-surj
_×-cong_ {bijection} = ×-cong-bij
_×-cong_ {equivalence} = λ A₁≃A₂ B₁≃B₂ →
from-bijection $ ×-cong-bij (from-equivalence A₁≃A₂)
(from-equivalence B₁≃B₂)
×-comm : ∀ {a b} {A : Set a} {B : Set b} → A × B ↔ B × A
×-comm = record
{ surjection = record
{ logical-equivalence = record
{ to = uncurry λ x y → (y , x)
; from = uncurry λ x y → (y , x)
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
Σ-assoc : ∀ {a b c}
{A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} →
(Σ A λ x → Σ (B x) (C x)) ↔ Σ (Σ A B) (uncurry C)
Σ-assoc = record
{ surjection = record
{ logical-equivalence = record
{ to = λ { (x , (y , z)) → (x , y) , z }
; from = λ { ((x , y) , z) → x , (y , z) }
}
; 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 = Σ-assoc
Σ-left-identity : ∀ {a} {A : ⊤ → Set a} → Σ ⊤ A ↔ A tt
Σ-left-identity = record
{ surjection = record
{ logical-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
{ logical-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 {equivalence} {equivalence} A₁≃A₂ B₁≃B₂ =
Eq.Σ-preserves A₁≃A₂ B₁≃B₂
Σ-cong {equivalence} {bijection} A₁≃A₂ B₁↔B₂ =
Eq.∃-preserves-bijections 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) ↝⟨ ≡⇒↝ _ $ cong (λ f → B₂ (f x)) $
to-implication∘from-isomorphism k₁ 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 = Eq.∃-preserves-functions A₁≃A₂
helper logical-equivalence = Eq.∃-preserves-logical-equivalences A₁≃A₂
helper injection = Eq.∃-preserves-injections A₁≃A₂
helper embedding = Σ-preserves-embeddings
(from-equivalence A₁≃A₂)
helper surjection = Eq.∃-preserves-surjections A₁≃A₂
helper bijection = Eq.∃-preserves-bijections A₁≃A₂
helper equivalence = Eq.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
{ logical-equivalence = ∃-cong-eq (_↠_.logical-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 {logical-equivalence} = ∃-cong-eq
∃-cong {injection} = Σ-cong Bijection.id
∃-cong {embedding} = Σ-preserves-embeddings Emb.id
∃-cong {surjection} = ∃-cong-surj
∃-cong {bijection} = ∃-cong-bij
∃-cong {equivalence} = λ B₁≃B₂ →
from-bijection $ ∃-cong-bij (from-equivalence ⊚ B₁≃B₂)
private
×-cong₂ : ∀ {k a b₁ b₂}
{A : Set a} {B₁ : Set b₁} {B₂ : Set b₂} →
(A → B₁ ↝[ k ] B₂) → A × B₁ ↝[ k ] A × B₂
×-cong₂ = ∃-cong
×-cong₁ : ∀ {k a₁ a₂ b}
{A₁ : Set a₁} {A₂ : Set a₂} {B : Set b} →
(B → A₁ ↝[ k ] A₂) → A₁ × B ↝[ k ] A₂ × B
×-cong₁ {A₁ = A₁} {A₂} {B} A₁↔A₂ =
A₁ × B ↔⟨ ×-comm ⟩
B × A₁ ↝⟨ ∃-cong A₁↔A₂ ⟩
B × A₂ ↔⟨ ×-comm ⟩□
A₂ × B □
drop-⊤-right : ∀ {k a b} {A : Set a} {B : A → Set b} →
((x : A) → B x ↔[ k ] ⊤) → Σ A B ↔ A
drop-⊤-right {A = A} {B} B↔⊤ =
Σ A B ↔⟨ ∃-cong B↔⊤ ⟩
A × ⊤ ↝⟨ ×-right-identity ⟩□
A □
drop-⊤-left-× : ∀ {k a b} {A : Set a} {B : Set b} →
(B → A ↔[ k ] ⊤) → A × B ↔ B
drop-⊤-left-× {A = A} {B} A↔⊤ =
A × B ↝⟨ ×-comm ⟩
B × A ↝⟨ drop-⊤-right A↔⊤ ⟩□
B □
drop-⊤-left-Σ : ∀ {a b} {A : Set a} {B : A → Set b} →
(A↔⊤ : A ↔ ⊤) →
Σ A B ↔ B (_↔_.from A↔⊤ tt)
drop-⊤-left-Σ {A = A} {B} A↔⊤ =
Σ A B ↝⟨ inverse $ Σ-cong (inverse A↔⊤) (λ _ → id) ⟩
Σ ⊤ (B ∘ _↔_.from A↔⊤) ↝⟨ Σ-left-identity ⟩□
B (_↔_.from A↔⊤ tt) □
currying : ∀ {a b c} {A : Set a} {B : A → Set b} {C : Σ A B → Set c} →
((p : Σ A B) → C p) ↔ ((x : A) (y : B x) → C (x , y))
currying = record
{ surjection = record
{ logical-equivalence = record { to = curry; from = uncurry }
; right-inverse-of = refl
}
; left-inverse-of = refl
}
Π⊎↠Π×Π :
∀ {a b c} {A : Set a} {B : Set b} {C : A ⊎ B → Set c} →
((x : A ⊎ B) → C x)
↠
((x : A) → C (inj₁ x)) × ((y : B) → C (inj₂ y))
Π⊎↠Π×Π = record
{ logical-equivalence = record
{ to = λ f → f ⊚ inj₁ , f ⊚ inj₂
; from = uncurry [_,_]
}
; right-inverse-of = refl
}
Π⊎↔Π×Π :
∀ {k a b c} {A : Set a} {B : Set b} {C : A ⊎ B → Set c} →
Extensionality? k (a ⊔ b) c →
((x : A ⊎ B) → C x)
↝[ k ]
((x : A) → C (inj₁ x)) × ((y : B) → C (inj₂ y))
Π⊎↔Π×Π =
generalise-ext? (_↠_.logical-equivalence Π⊎↠Π×Π) λ ext → record
{ surjection = Π⊎↠Π×Π
; left-inverse-of = λ _ → apply-ext ext [ refl ⊚ _ , refl ⊚ _ ]
}
∃-⊎-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
{ logical-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
{ logical-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
{ logical-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 (_⇔_.to contractible⇔↔⊤ (singleton-contractible x)) ⟩
B x × (∃ λ y → y ≡ x) ↔⟨ ∃-comm ⟩
(∃ λ y → B x × y ≡ x) ↔⟨ ∃-cong (λ y → ×-cong₁ (λ y≡x → subst (λ x → B x ↔ B y) y≡x id)) ⟩□
(∃ λ y → B y × y ≡ x) □
∃-introduction :
∀ {a b} {A : Set a} {x : A} (B : (y : A) → x ≡ y → Set b) →
B x (refl x) ↔ ∃ λ y → ∃ λ (x≡y : x ≡ y) → B y x≡y
∃-introduction {x = x} B =
B x (refl x) ↝⟨ ∃-intro (uncurry B) _ ⟩
(∃ λ { (y , x≡y) → B y x≡y × (y , x≡y) ≡ (x , refl x) }) ↝⟨ (∃-cong λ _ → ∃-cong λ _ →
_⇔_.to contractible⇔↔⊤ $
mono₁ 0 (other-singleton-contractible x) _ _) ⟩
(∃ λ { (y , x≡y) → B y x≡y × ⊤ }) ↝⟨ (∃-cong λ _ → ×-right-identity) ⟩
(∃ λ { (y , x≡y) → B y x≡y }) ↝⟨ inverse Σ-assoc ⟩□
(∃ λ y → ∃ λ x≡y → B y x≡y) □
≡×≡↔≡ : ∀ {a b} {A : Set a} {B : Set b} {p₁ p₂ : A × B} →
(proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂) ↔ (p₁ ≡ p₂)
≡×≡↔≡ {B = B} {p₁} {p₂} = record
{ surjection = record
{ logical-equivalence = record
{ to = uncurry (cong₂ _,_)
; from = λ eq → cong proj₁ eq , cong proj₂ eq
}
; right-inverse-of = λ eq →
cong₂ _,_ (cong proj₁ eq) (cong proj₂ eq) ≡⟨ cong₂-cong-cong _ _ _,_ ⟩
cong (λ p → proj₁ p , proj₂ p) eq ≡⟨⟩
cong id eq ≡⟨ sym $ cong-id _ ⟩∎
eq ∎
}
; left-inverse-of = λ { (eq₁ , eq₂) →
cong proj₁ (cong₂ _,_ eq₁ eq₂) , cong proj₂ (cong₂ _,_ eq₁ eq₂) ≡⟨ cong₂ _,_ (cong-proj₁-cong₂-, eq₁ eq₂) (cong-proj₂-cong₂-, eq₁ eq₂) ⟩∎
eq₁ , eq₂ ∎
}
}
ignore-propositional-component :
∀ {a b} {A : Set a} {B : A → Set b} {p q : Σ A B} →
Is-proposition (B (proj₁ q)) →
(proj₁ p ≡ proj₁ q) ↔ (p ≡ q)
ignore-propositional-component {B = B} {p₁ , p₂} {q₁ , q₂} Bq₁-prop =
(p₁ ≡ q₁) ↝⟨ inverse ×-right-identity ⟩
(p₁ ≡ q₁ × ⊤) ↝⟨ ∃-cong (λ _ → inverse $ _⇔_.to contractible⇔↔⊤ (Bq₁-prop _ _)) ⟩
(∃ λ (eq : p₁ ≡ q₁) → subst B eq p₂ ≡ q₂) ↝⟨ Bijection.Σ-≡,≡↔≡ ⟩□
((p₁ , p₂) ≡ (q₁ , q₂)) □
Contractible-commutes-with-× :
∀ {x y} {X : Set x} {Y : Set y} →
Extensionality (x ⊔ y) (x ⊔ y) →
Contractible (X × Y) ≃ (Contractible X × Contractible Y)
Contractible-commutes-with-× {x} {y} ext =
_↔_.to (Eq.⇔↔≃ ext
(Contractible-propositional ext)
(×-closure 1 (Contractible-propositional
(lower-extensionality y y ext))
(Contractible-propositional
(lower-extensionality x x ext))))
(record
{ to = λ cX×Y →
lemma cX×Y ,
lemma (H-level.respects-surjection
(_↔_.surjection ×-comm) 0 cX×Y)
; from = λ { ((x , eq₁) , (y , eq₂)) →
(x , y) ,
λ { (x′ , y′) →
(x , y) ≡⟨ cong₂ _,_ (eq₁ x′) (eq₂ y′) ⟩∎
(x′ , y′) ∎ } }
})
where
lemma : ∀ {x y} {X : Set x} {Y : Set y} →
Contractible (X × Y) → Contractible X
lemma ((x , y) , eq) = x , λ x′ →
x ≡⟨⟩
proj₁ (x , y) ≡⟨ cong proj₁ (eq (x′ , y)) ⟩∎
proj₁ (x′ , y) ∎
≃-to-≡↔≡ :
∀ {a b} →
Extensionality (a ⊔ b) (a ⊔ b) →
{A : Set a} {B : Set b} {p q : A ≃ B} →
(∀ x → _≃_.to p x ≡ _≃_.to q x) ↔ p ≡ q
≃-to-≡↔≡ {a} {b} ext {p = p} {q} =
(∀ x → _≃_.to p x ≡ _≃_.to q x) ↔⟨ Eq.extensionality-isomorphism (lower-extensionality b a ext) ⟩
_≃_.to p ≡ _≃_.to q ↝⟨ ignore-propositional-component (Eq.propositional ext _) ⟩
(_≃_.to p , _≃_.is-equivalence p) ≡ (_≃_.to q , _≃_.is-equivalence q) ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ Eq.≃-as-Σ) ⟩□
p ≡ q □
↔-to-≡↔≡ :
∀ {a b} →
Extensionality (a ⊔ b) (a ⊔ b) →
{A : Set a} {B : Set b} {p q : A ↔ B} →
Is-set A →
(∀ x → _↔_.to p x ≡ _↔_.to q x) ↔ p ≡ q
↔-to-≡↔≡ ext {p = p} {q} A-set =
(∀ x → _↔_.to p x ≡ _↔_.to q x) ↝⟨ id ⟩
(∀ x → _≃_.to (Eq.↔⇒≃ p) x ≡ _≃_.to (Eq.↔⇒≃ q) x) ↝⟨ ≃-to-≡↔≡ ext ⟩
Eq.↔⇒≃ p ≡ Eq.↔⇒≃ q ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ (Eq.↔↔≃ ext A-set)) ⟩□
p ≡ q □
≃-from-≡↔≡ :
∀ {a b} →
Extensionality (a ⊔ b) (a ⊔ b) →
{A : Set a} {B : Set b} {p q : A ≃ B} →
(∀ x → _≃_.from p x ≡ _≃_.from q x) ↔ p ≡ q
≃-from-≡↔≡ ext {p = p} {q} =
(∀ x → _≃_.from p x ≡ _≃_.from q x) ↝⟨ ≃-to-≡↔≡ ext ⟩
inverse p ≡ inverse q ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ (Eq.inverse-isomorphism ext)) ⟩□
p ≡ q □
↔-from-≡↔≡ :
∀ {a b} →
Extensionality (a ⊔ b) (a ⊔ b) →
{A : Set a} {B : Set b} {p q : A ↔ B} →
Is-set A →
(∀ x → _↔_.from p x ≡ _↔_.from q x) ↔ p ≡ q
↔-from-≡↔≡ ext {p = p} {q} A-set =
(∀ x → _↔_.from p x ≡ _↔_.from q x) ↝⟨ id ⟩
(∀ x → _≃_.from (Eq.↔⇒≃ p) x ≡ _≃_.from (Eq.↔⇒≃ q) x) ↝⟨ ≃-from-≡↔≡ ext ⟩
Eq.↔⇒≃ p ≡ Eq.↔⇒≃ q ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ (Eq.↔↔≃ ext A-set)) ⟩□
p ≡ q □
contractible↔≃⊤ :
∀ {a} {A : Set a} →
Extensionality a a →
Contractible A ↔ (A ≃ ⊤)
contractible↔≃⊤ ext = record
{ surjection = record
{ logical-equivalence = record
{ to = Eq.↔⇒≃ ∘ _⇔_.to contractible⇔↔⊤
; from = _⇔_.from contractible⇔↔⊤ ∘ _≃_.bijection
}
; right-inverse-of = λ _ →
Eq.lift-equality ext (refl _)
}
; left-inverse-of = λ _ →
_⇔_.to propositional⇔irrelevant
(Contractible-propositional ext) _ _
}
≃⊥≃¬ :
∀ {a ℓ} {A : Set a} →
Extensionality (a ⊔ ℓ) (a ⊔ ℓ) →
(A ≃ ⊥ {ℓ = ℓ}) ≃ (¬ A)
≃⊥≃¬ {ℓ = ℓ} {A} ext =
_↔_.to (Eq.⇔↔≃ ext (Eq.right-closure ext 0 ⊥-propositional)
(¬-propositional
(lower-extensionality ℓ _ ext))) (record
{ to = λ eq a → ⊥-elim (_≃_.to eq a)
; from = λ ¬a → A ↔⟨ inverse (Bijection.⊥↔uninhabited ¬a) ⟩□
⊥ □
})
×-⊎-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
→-cong-→ : ∀ {a b c d}
{A : Set a} {B : Set b} {C : Set c} {D : Set d} →
(B → A) → (C → D) → (A → C) → (B → D)
→-cong-→ B→A C→D = (C→D ∘_) ∘ (_∘ B→A)
private
→-cong-⇔ : ∀ {a b c d}
{A : Set a} {B : Set b} {C : Set c} {D : Set d} →
A ⇔ B → C ⇔ D → (A → C) ⇔ (B → D)
→-cong-⇔ A⇔B C⇔D = record
{ to = →-cong-→ (from A⇔B) (to C⇔D)
; from = →-cong-→ (to A⇔B) (from C⇔D)
}
where open _⇔_
→-cong-↠ : ∀ {a b c d} → Extensionality b d →
{A : Set a} {B : Set b} {C : Set c} {D : Set d} →
A ↠ B → C ↠ D → (A → C) ↠ (B → D)
→-cong-↠ {a} {b} {c} {d} ext A↠B C↠D = record
{ logical-equivalence = logical-equiv
; right-inverse-of = right-inv
}
where
open _↠_
logical-equiv = →-cong-⇔ (_↠_.logical-equivalence A↠B)
(_↠_.logical-equivalence C↠D)
abstract
right-inv :
∀ f → _⇔_.to logical-equiv (_⇔_.from logical-equiv f) ≡ f
right-inv f = apply-ext ext λ x →
to C↠D (from C↠D (f (to A↠B (from A↠B x)))) ≡⟨ right-inverse-of C↠D _ ⟩
f (to A↠B (from A↠B x)) ≡⟨ cong f $ right-inverse-of A↠B _ ⟩∎
f x ∎
private
→-cong-↔ : ∀ {a b c d}
{A : Set a} {B : Set b} {C : Set c} {D : Set d} →
Extensionality (a ⊔ b) (c ⊔ d) →
A ↔ B → C ↔ D → (A → C) ↔ (B → D)
→-cong-↔ {a} {b} {c} {d} ext A↔B C↔D = record
{ surjection = surj
; left-inverse-of = left-inv
}
where
open _↔_
surj = →-cong-↠ (lower-extensionality a c ext)
(_↔_.surjection A↔B)
(_↔_.surjection C↔D)
abstract
left-inv :
∀ f → _↠_.from surj (_↠_.to surj f) ≡ f
left-inv f = apply-ext (lower-extensionality b d ext) λ x →
from C↔D (to C↔D (f (from A↔B (to A↔B x)))) ≡⟨ left-inverse-of C↔D _ ⟩
f (from A↔B (to A↔B x)) ≡⟨ cong f $ left-inverse-of A↔B _ ⟩∎
f x ∎
→-cong : ∀ {k a b c d} →
Extensionality? ⌊ k ⌋-sym (a ⊔ b) (c ⊔ d) →
{A : Set a} {B : Set b} {C : Set c} {D : Set d} →
A ↝[ ⌊ k ⌋-sym ] B → C ↝[ ⌊ k ⌋-sym ] D →
(A → C) ↝[ ⌊ k ⌋-sym ] (B → D)
→-cong {logical-equivalence} _ A⇔B C⇔D = →-cong-⇔ A⇔B C⇔D
→-cong {bijection} ext A↔B C↔D = →-cong-↔ ext A↔B C↔D
→-cong {equivalence} ext A≃B C≃D = record
{ to = to
; is-equivalence = λ y →
((from y , right-inverse-of y) , irrelevance y)
}
where
A→B≃C→D =
Eq.↔⇒≃ (→-cong-↔ ext (_≃_.bijection A≃B) (_≃_.bijection C≃D))
to = _≃_.to A→B≃C→D
from = _≃_.from A→B≃C→D
abstract
right-inverse-of : ∀ x → to (from x) ≡ x
right-inverse-of = _≃_.right-inverse-of A→B≃C→D
irrelevance : ∀ y (p : to ⁻¹ y) →
(from y , right-inverse-of y) ≡ p
irrelevance = _≃_.irrelevance A→B≃C→D
private
∀-cong-→ :
∀ {a b₁ b₂} {A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x → B₂ x) →
((x : A) → B₁ x) → ((x : A) → B₂ x)
∀-cong-→ B₁→B₂ = B₁→B₂ _ ⊚_
∀-cong-⇔ :
∀ {a b₁ b₂} {A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ⇔ B₂ x) →
((x : A) → B₁ x) ⇔ ((x : A) → B₂ x)
∀-cong-⇔ B₁⇔B₂ = record
{ to = ∀-cong-→ (_⇔_.to ⊚ B₁⇔B₂)
; from = ∀-cong-→ (_⇔_.from ⊚ B₁⇔B₂)
}
∀-cong-surj :
∀ {a b₁ b₂} →
Extensionality a (b₁ ⊔ b₂) →
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↠ B₂ x) →
((x : A) → B₁ x) ↠ ((x : A) → B₂ x)
∀-cong-surj {b₁ = b₁} ext B₁↠B₂ = record
{ logical-equivalence = equiv
; right-inverse-of = right-inverse-of
}
where
equiv = ∀-cong-⇔ (_↠_.logical-equivalence ⊚ B₁↠B₂)
abstract
right-inverse-of : ∀ f → _⇔_.to equiv (_⇔_.from equiv f) ≡ f
right-inverse-of = λ f →
apply-ext (lower-extensionality lzero b₁ ext) λ x →
_↠_.to (B₁↠B₂ x) (_↠_.from (B₁↠B₂ x) (f x)) ≡⟨ _↠_.right-inverse-of (B₁↠B₂ x) (f x) ⟩∎
f x ∎
∀-cong-bij :
∀ {a b₁ b₂} →
Extensionality a (b₁ ⊔ b₂) →
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↔ B₂ x) →
((x : A) → B₁ x) ↔ ((x : A) → B₂ x)
∀-cong-bij {b₂ = b₂} ext B₁↔B₂ = record
{ surjection = surj
; left-inverse-of = left-inverse-of
}
where
surj = ∀-cong-surj ext (_↔_.surjection ⊚ B₁↔B₂)
abstract
left-inverse-of : ∀ f → _↠_.from surj (_↠_.to surj f) ≡ f
left-inverse-of = λ f →
apply-ext (lower-extensionality lzero b₂ ext) λ x →
_↔_.from (B₁↔B₂ x) (_↔_.to (B₁↔B₂ x) (f x)) ≡⟨ _↔_.left-inverse-of (B₁↔B₂ x) (f x) ⟩∎
f x ∎
∀-cong-eq :
∀ {a b₁ b₂} →
Extensionality a (b₁ ⊔ b₂) →
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ≃ B₂ x) →
((x : A) → B₁ x) ≃ ((x : A) → B₂ x)
∀-cong-eq ext = Eq.↔⇒≃ ⊚ ∀-cong-bij ext ⊚ (_≃_.bijection ⊚_)
∀-cong-inj :
∀ {a b₁ b₂} →
Extensionality a (b₁ ⊔ b₂) →
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↣ B₂ x) →
((x : A) → B₁ x) ↣ ((x : A) → B₂ x)
∀-cong-inj {b₁ = b₁} {b₂} ext B₁↣B₂ = record
{ to = to
; injective = injective
}
where
to = ∀-cong-→ (_↣_.to ⊚ B₁↣B₂)
abstract
injective : Injective to
injective {x = f} {y = g} =
(λ x → _↣_.to (B₁↣B₂ x) (f x)) ≡ (λ x → _↣_.to (B₁↣B₂ x) (g x)) ↔⟨ inverse $ Eq.extensionality-isomorphism
(lower-extensionality lzero b₁ ext) ⟩
(∀ x → _↣_.to (B₁↣B₂ x) (f x) ≡ _↣_.to (B₁↣B₂ x) (g x)) ↝⟨ ∀-cong-→ (λ x → _↣_.injective (B₁↣B₂ x)) ⟩
(∀ x → f x ≡ g x) ↔⟨ Eq.extensionality-isomorphism
(lower-extensionality lzero b₂ ext) ⟩□
f ≡ g □
∀-cong-emb :
∀ {a b₁ b₂} →
Extensionality a (b₁ ⊔ b₂) →
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → Embedding (B₁ x) (B₂ x)) →
Embedding ((x : A) → B₁ x) ((x : A) → B₂ x)
∀-cong-emb {b₁ = b₁} {b₂} ext B₁↣B₂ = record
{ to = to
; is-embedding = is-embedding
}
where
to = ∀-cong-→ (Embedding.to ⊚ B₁↣B₂)
ext₂ = lower-extensionality lzero b₁ ext
abstract
is-embedding : Is-embedding to
is-embedding f g = _≃_.is-equivalence $
Eq.with-other-function
(f ≡ g ↝⟨ inverse $ Eq.extensionality-isomorphism
(lower-extensionality lzero b₂ ext) ⟩
(∀ x → f x ≡ g x) ↝⟨ ∀-cong-eq ext (λ x →
Eq.⟨ _ , Embedding.is-embedding (B₁↣B₂ x) (f x) (g x) ⟩) ⟩
(∀ x → Embedding.to (B₁↣B₂ x) (f x) ≡
Embedding.to (B₁↣B₂ x) (g x)) ↝⟨ Eq.extensionality-isomorphism ext₂ ⟩□
(λ x → Embedding.to (B₁↣B₂ x) (f x)) ≡
(λ x → Embedding.to (B₁↣B₂ x) (g x)) □)
_
(λ f≡g →
apply-ext (Eq.good-ext ext₂)
(λ x → cong (Embedding.to (B₁↣B₂ x)) (ext⁻¹ f≡g x)) ≡⟨⟩
apply-ext (Eq.good-ext ext₂)
(λ x → cong (Embedding.to (B₁↣B₂ x)) (cong (_$ x) f≡g)) ≡⟨ cong (apply-ext (Eq.good-ext ext₂)) (apply-ext ext₂ λ _ →
cong-∘ _ _ _) ⟩
apply-ext (Eq.good-ext ext₂)
(λ x → cong (λ h → Embedding.to (B₁↣B₂ x) (h x)) f≡g) ≡⟨ cong (apply-ext (Eq.good-ext ext₂)) (apply-ext ext₂ λ _ → sym $
cong-∘ _ _ _) ⟩
apply-ext (Eq.good-ext ext₂)
(λ x → cong (_$ x)
(cong (λ h x → Embedding.to (B₁↣B₂ x) (h x))
f≡g)) ≡⟨⟩
apply-ext (Eq.good-ext ext₂)
(ext⁻¹ (cong (λ h x → Embedding.to (B₁↣B₂ x) (h x)) f≡g)) ≡⟨ _≃_.right-inverse-of (Eq.extensionality-isomorphism ext₂) _ ⟩∎
cong (λ h x → Embedding.to (B₁↣B₂ x) (h x)) f≡g ∎)
∀-cong :
∀ {k a b₁ b₂} →
Extensionality? k a (b₁ ⊔ b₂) →
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ x → B₁ x ↝[ k ] B₂ x) →
((x : A) → B₁ x) ↝[ k ] ((x : A) → B₂ x)
∀-cong {implication} = λ _ → ∀-cong-→
∀-cong {logical-equivalence} = λ _ → ∀-cong-⇔
∀-cong {injection} = ∀-cong-inj
∀-cong {embedding} = ∀-cong-emb
∀-cong {surjection} = ∀-cong-surj
∀-cong {bijection} = ∀-cong-bij
∀-cong {equivalence} = ∀-cong-eq
implicit-∀-cong :
∀ {k a b₁ b₂} →
Extensionality? k a (b₁ ⊔ b₂) →
{A : Set a} {B₁ : A → Set b₁} {B₂ : A → Set b₂} →
(∀ {x} → B₁ x ↝[ k ] B₂ x) →
({x : A} → B₁ x) ↝[ k ] ({x : A} → B₂ x)
implicit-∀-cong ext {A} {B₁} {B₂} B₁↝B₂ =
({x : A} → B₁ x) ↔⟨ Bijection.implicit-Π↔Π ⟩
((x : A) → B₁ x) ↝⟨ ∀-cong ext (λ _ → B₁↝B₂) ⟩
((x : A) → B₂ x) ↔⟨ inverse Bijection.implicit-Π↔Π ⟩□
({x : A} → B₂ x) □
Π-cong-contra-→ :
∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₂→A₁ : A₂ → A₁) →
(∀ x → B₁ (A₂→A₁ x) → B₂ x) →
((x : A₁) → B₁ x) → ((x : A₂) → B₂ x)
Π-cong-contra-→ {B₁ = B₁} {B₂} A₂→A₁ B₁→B₂ f x =
$⟨ f (A₂→A₁ x) ⟩
B₁ (A₂→A₁ x) ↝⟨ B₁→B₂ x ⟩
B₂ x □
Π-cong-→ :
∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁↠A₂ : A₁ ↠ A₂) →
(∀ x → B₁ x → B₂ (_↠_.to A₁↠A₂ x)) →
((x : A₁) → B₁ x) → ((x : A₂) → B₂ x)
Π-cong-→ {B₁ = B₁} {B₂} A₁↠A₂ B₁→B₂ f x =
$⟨ f (_↠_.from A₁↠A₂ x) ⟩
B₁ (_↠_.from A₁↠A₂ x) ↝⟨ B₁→B₂ (_↠_.from A₁↠A₂ x) ⟩
B₂ (_↠_.to A₁↠A₂ (_↠_.from A₁↠A₂ x)) ↝⟨ subst B₂ (_↠_.right-inverse-of A₁↠A₂ x) ⟩□
B₂ x □
Π-cong-⇔ :
∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁↠A₂ : A₁ ↠ A₂) →
(∀ x → B₁ x ⇔ B₂ (_↠_.to A₁↠A₂ x)) →
((x : A₁) → B₁ x) ⇔ ((x : A₂) → B₂ x)
Π-cong-⇔ {A₁ = A₁} {A₂} {B₁} {B₂} A₁↠A₂ B₁⇔B₂ = record
{ to = Π-cong-→ A₁↠A₂ (_⇔_.to ⊚ B₁⇔B₂)
; from = Π-cong-contra-→ (_↠_.to A₁↠A₂) (_⇔_.from ⊚ B₁⇔B₂)
}
Π-cong-contra-⇔ :
∀ {a₁ a₂ b₁ b₂}
{A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₂↠A₁ : A₂ ↠ A₁) →
(∀ x → B₁ (_↠_.to A₂↠A₁ x) ⇔ B₂ x) →
((x : A₁) → B₁ x) ⇔ ((x : A₂) → B₂ x)
Π-cong-contra-⇔ {A₁ = A₁} {A₂} {B₁} {B₂} A₂↠A₁ B₁⇔B₂ = record
{ to = Π-cong-contra-→ (_↠_.to A₂↠A₁) (_⇔_.to ⊚ B₁⇔B₂)
; from = Π-cong-→ A₂↠A₁ (_⇔_.from ⊚ B₁⇔B₂)
}
Π-cong-↠ :
∀ {a₁ a₂ b₁ b₂} →
Extensionality a₂ b₂ →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁↠A₂ : A₁ ↠ A₂) →
(∀ x → B₁ x ↠ B₂ (_↠_.to A₁↠A₂ x)) →
((x : A₁) → B₁ x) ↠ ((x : A₂) → B₂ x)
Π-cong-↠ ext {B₂ = B₂} A₁↠A₂ B₁↠B₂ = record
{ logical-equivalence = equiv
; right-inverse-of = to∘from
}
where
equiv = Π-cong-⇔ A₁↠A₂ (_↠_.logical-equivalence ⊚ B₁↠B₂)
abstract
to∘from : ∀ f → _⇔_.to equiv (_⇔_.from equiv f) ≡ f
to∘from f = apply-ext ext λ x →
subst B₂ (_↠_.right-inverse-of A₁↠A₂ x)
(_↠_.to (B₁↠B₂ (_↠_.from A₁↠A₂ x))
(_↠_.from (B₁↠B₂ (_↠_.from A₁↠A₂ x))
(f (_↠_.to A₁↠A₂ (_↠_.from A₁↠A₂ x))))) ≡⟨ cong (subst B₂ (_↠_.right-inverse-of A₁↠A₂ x)) $ _↠_.right-inverse-of (B₁↠B₂ _) _ ⟩
subst B₂ (_↠_.right-inverse-of A₁↠A₂ x)
(f (_↠_.to A₁↠A₂ (_↠_.from A₁↠A₂ x))) ≡⟨ dependent-cong f _ ⟩∎
f x ∎
Π-cong-contra-↣ :
∀ {a₁ a₂ b₁ b₂} →
Extensionality a₁ b₁ →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₂↠A₁ : A₂ ↠ A₁) →
(∀ x → B₁ (_↠_.to A₂↠A₁ x) ↣ B₂ x) →
((x : A₁) → B₁ x) ↣ ((x : A₂) → B₂ x)
Π-cong-contra-↣ ext A₂↠A₁ B₁↣B₂ = record
{ to = to
; injective = injective
}
where
to = Π-cong-contra-→ (_↠_.to A₂↠A₁) (_↣_.to ⊚ B₁↣B₂)
abstract
injective : Injective to
injective {x = f} {y = g} to-f≡to-g = apply-ext ext λ x →
let x′ = _↠_.to A₂↠A₁ (_↠_.from A₂↠A₁ x) in
$⟨ to-f≡to-g ⟩
(λ x → _↣_.to (B₁↣B₂ x) (f (_↠_.to A₂↠A₁ x))) ≡
(λ x → _↣_.to (B₁↣B₂ x) (g (_↠_.to A₂↠A₁ x))) ↝⟨ cong (_$ _) ⟩
_↣_.to (B₁↣B₂ (_↠_.from A₂↠A₁ x)) (f x′) ≡
_↣_.to (B₁↣B₂ (_↠_.from A₂↠A₁ x)) (g x′) ↝⟨ _↣_.injective (B₁↣B₂ _) ⟩
f x′ ≡ g x′ ↝⟨ subst (λ x → f x ≡ g x) $ _↠_.right-inverse-of A₂↠A₁ x ⟩□
f x ≡ g x □
private
Π-cong-contra-↠ :
∀ {a₁ a₂ b₁ b₂} →
Extensionality a₂ b₂ →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₂≃A₁ : A₂ ≃ A₁) →
(∀ x → B₁ (_≃_.to A₂≃A₁ x) ↠ B₂ x) →
((x : A₁) → B₁ x) ↠ ((x : A₂) → B₂ x)
Π-cong-contra-↠ ext {B₁ = B₁} A₂≃A₁ B₁↠B₂ = record
{ logical-equivalence = equiv
; right-inverse-of = to∘from
}
where
equiv = Π-cong-contra-⇔ (_≃_.surjection A₂≃A₁)
(_↠_.logical-equivalence ⊚ B₁↠B₂)
abstract
to∘from : ∀ f → _⇔_.to equiv (_⇔_.from equiv f) ≡ f
to∘from f = apply-ext ext λ x →
_↠_.to (B₁↠B₂ x)
(subst B₁ (_≃_.right-inverse-of A₂≃A₁ (_≃_.to A₂≃A₁ x))
(_↠_.from (B₁↠B₂ (_≃_.from A₂≃A₁ (_≃_.to A₂≃A₁ x)))
(f (_≃_.from A₂≃A₁ (_≃_.to A₂≃A₁ x))))) ≡⟨ cong (λ eq → _↠_.to (B₁↠B₂ x) (subst B₁ eq _)) $ sym $
_≃_.left-right-lemma A₂≃A₁ _ ⟩
_↠_.to (B₁↠B₂ x)
(subst B₁ (cong (_≃_.to A₂≃A₁) $ _≃_.left-inverse-of A₂≃A₁ x)
(_↠_.from (B₁↠B₂ (_≃_.from A₂≃A₁ (_≃_.to A₂≃A₁ x)))
(f (_≃_.from A₂≃A₁ (_≃_.to A₂≃A₁ x))))) ≡⟨ cong (_↠_.to (B₁↠B₂ x)) $ sym $ subst-∘ _ _ _ ⟩
_↠_.to (B₁↠B₂ x)
(subst (B₁ ∘ _≃_.to A₂≃A₁) (_≃_.left-inverse-of A₂≃A₁ x)
(_↠_.from (B₁↠B₂ (_≃_.from A₂≃A₁ (_≃_.to A₂≃A₁ x)))
(f (_≃_.from A₂≃A₁ (_≃_.to A₂≃A₁ x))))) ≡⟨ cong (_↠_.to (B₁↠B₂ x)) $
dependent-cong (λ x → _↠_.from (B₁↠B₂ x) (f x)) _ ⟩
_↠_.to (B₁↠B₂ x) (_↠_.from (B₁↠B₂ x) (f x)) ≡⟨ _↠_.right-inverse-of (B₁↠B₂ x) _ ⟩∎
f x ∎
Π-cong-↔ :
∀ {a₁ a₂ b₁ b₂} →
Extensionality (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁≃A₂ : A₁ ≃ A₂) →
(∀ x → B₁ x ↔ B₂ (_≃_.to A₁≃A₂ x)) →
((x : A₁) → B₁ x) ↔ ((x : A₂) → B₂ x)
Π-cong-↔ {a₁} {a₂} {b₁} {b₂} ext {B₂ = B₂} A₁≃A₂ B₁↔B₂ = record
{ surjection = surj
; left-inverse-of = from∘to
}
where
surj = Π-cong-↠ (lower-extensionality a₁ b₁ ext)
(_≃_.surjection A₁≃A₂) (_↔_.surjection ⊚ B₁↔B₂)
abstract
from∘to : ∀ f → _↠_.from surj (_↠_.to surj f) ≡ f
from∘to =
_↠_.right-inverse-of $
Π-cong-contra-↠ (lower-extensionality a₂ b₂ ext)
{B₁ = B₂}
A₁≃A₂
(_↔_.surjection ⊚ inverse ⊚ B₁↔B₂)
Π-cong-contra-↔ :
∀ {a₁ a₂ b₁ b₂} →
Extensionality (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₂≃A₁ : A₂ ≃ A₁) →
(∀ x → B₁ (_≃_.to A₂≃A₁ x) ↔ B₂ x) →
((x : A₁) → B₁ x) ↔ ((x : A₂) → B₂ x)
Π-cong-contra-↔ {a₁} {a₂} {b₁} {b₂} ext {B₂ = B₂} A₂≃A₁ B₁↔B₂ = record
{ surjection = surj
; left-inverse-of = from∘to
}
where
surj = Π-cong-contra-↠ (lower-extensionality a₁ b₁ ext)
A₂≃A₁ (_↔_.surjection ⊚ B₁↔B₂)
abstract
from∘to : ∀ f → _↠_.from surj (_↠_.to surj f) ≡ f
from∘to =
_↠_.right-inverse-of $
Π-cong-↠ (lower-extensionality a₂ b₂ ext)
(_≃_.surjection A₂≃A₁)
(_↔_.surjection ⊚ inverse ⊚ B₁↔B₂)
Π-cong-≃ :
∀ {a₁ a₂ b₁ b₂} →
Extensionality (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁≃A₂ : A₁ ≃ A₂) →
(∀ x → B₁ x ≃ B₂ (_≃_.to A₁≃A₂ x)) →
((x : A₁) → B₁ x) ≃ ((x : A₂) → B₂ x)
Π-cong-≃ ext A₁≃A₂ =
from-isomorphism ⊚ Π-cong-↔ ext A₁≃A₂ ⊚ (from-isomorphism ⊚_)
Π-cong-contra-≃ :
∀ {a₁ a₂ b₁ b₂} →
Extensionality (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₂≃A₁ : A₂ ≃ A₁) →
(∀ x → B₁ (_≃_.to A₂≃A₁ x) ≃ B₂ x) →
((x : A₁) → B₁ x) ≃ ((x : A₂) → B₂ x)
Π-cong-contra-≃ ext A₂≃A₁ =
from-isomorphism ⊚ Π-cong-contra-↔ ext A₂≃A₁ ⊚ (from-isomorphism ⊚_)
Π-cong-↣ :
∀ {a₁ a₂ b₁ b₂} →
Extensionality a₁ b₁ →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁≃A₂ : A₁ ≃ A₂) →
(∀ x → B₁ x ↣ B₂ (_≃_.to A₁≃A₂ x)) →
((x : A₁) → B₁ x) ↣ ((x : A₂) → B₂ x)
Π-cong-↣ ext {A₁} {A₂} {B₁} {B₂} A₁≃A₂ =
(∀ x → B₁ x ↣ B₂ (_≃_.to A₁≃A₂ x)) ↝⟨ Π-cong-contra-→ (_≃_.from A₁≃A₂) (λ _ → id) ⟩
(∀ x → B₁ (_≃_.from A₁≃A₂ x) ↣ B₂ (_≃_.to A₁≃A₂ (_≃_.from A₁≃A₂ x))) ↝⟨ (∀-cong _ λ _ →
subst ((B₁ _ ↣_) ⊚ B₂) (_≃_.right-inverse-of A₁≃A₂ _)) ⟩
(∀ x → B₁ (_≃_.from A₁≃A₂ x) ↣ B₂ x) ↝⟨ Π-cong-contra-↣ ext (_≃_.surjection $ inverse A₁≃A₂) ⟩□
((x : A₁) → B₁ x) ↣ ((x : A₂) → B₂ x) □
Π-cong-contra-Emb :
∀ {a₁ a₂ b₁ b₂} →
Extensionality (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₂≃A₁ : A₂ ≃ A₁) →
(∀ x → Embedding (B₁ (_≃_.to A₂≃A₁ x)) (B₂ x)) →
Embedding ((x : A₁) → B₁ x) ((x : A₂) → B₂ x)
Π-cong-contra-Emb {a₁} {a₂} {b₁} {b₂} ext A₂≃A₁ B₁↣B₂ = record
{ to = to
; is-embedding = is-embedding
}
where
to = Π-cong-contra-→ (_≃_.to A₂≃A₁) (Embedding.to ⊚ B₁↣B₂)
abstract
ext₁₁ : Extensionality a₁ b₁
ext₁₁ = lower-extensionality a₂ b₂ ext
ext₂₁ : Extensionality a₂ b₁
ext₂₁ = lower-extensionality a₁ b₂ ext
ext₂₂ : Extensionality a₂ b₂
ext₂₂ = lower-extensionality a₁ b₁ ext
is-embedding : Is-embedding to
is-embedding f g =
_≃_.is-equivalence $
Eq.with-other-function
(f ≡ g ↝⟨ inverse $ Eq.extensionality-isomorphism ext₁₁ ⟩
(∀ x → f x ≡ g x) ↝⟨ (inverse $ Π-cong-≃ ext A₂≃A₁ λ x →
inverse $ Embedding.equivalence (B₁↣B₂ x)) ⟩
(∀ x → Embedding.to (B₁↣B₂ x) (f (_≃_.to A₂≃A₁ x)) ≡
Embedding.to (B₁↣B₂ x) (g (_≃_.to A₂≃A₁ x))) ↝⟨ Eq.extensionality-isomorphism ext₂₂ ⟩
(λ {x} → Embedding.to (B₁↣B₂ x)) ⊚ f ⊚ _≃_.to A₂≃A₁ ≡
(λ {x} → Embedding.to (B₁↣B₂ x)) ⊚ g ⊚ _≃_.to A₂≃A₁ ↔⟨⟩
to f ≡ to g □)
_
(λ f≡g →
apply-ext (Eq.good-ext ext₂₂)
(cong (Embedding.to (B₁↣B₂ _)) ⊚
ext⁻¹ f≡g ⊚ _≃_.to A₂≃A₁) ≡⟨ sym $ Eq.cong-post-∘-good-ext ext₂₁ ext₂₂ _ ⟩
cong (Embedding.to (B₁↣B₂ _) ⊚_)
(apply-ext (Eq.good-ext ext₂₁)
(ext⁻¹ f≡g ⊚ _≃_.to A₂≃A₁)) ≡⟨ cong (cong (Embedding.to (B₁↣B₂ _) ⊚_)) $ sym $
Eq.cong-pre-∘-good-ext ext₂₁ ext₁₁ _ ⟩
cong (Embedding.to (B₁↣B₂ _) ⊚_)
(cong (_⊚ _≃_.to A₂≃A₁)
(apply-ext (Eq.good-ext ext₁₁) (ext⁻¹ f≡g))) ≡⟨ cong-∘ _ _ _ ⟩
cong to (apply-ext (Eq.good-ext ext₁₁) (ext⁻¹ f≡g)) ≡⟨ cong (cong to) $
_≃_.right-inverse-of (Eq.extensionality-isomorphism ext₁₁) _ ⟩∎
cong to f≡g ∎)
Π-cong-Emb :
∀ {a₁ a₂ b₁ b₂} →
Extensionality (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
∀ {A₁ : Set a₁} {A₂ : Set a₂}
{B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₁≃A₂ : A₁ ≃ A₂) →
(∀ x → Embedding (B₁ x) (B₂ (_≃_.to A₁≃A₂ x))) →
Embedding ((x : A₁) → B₁ x) ((x : A₂) → B₂ x)
Π-cong-Emb ext {A₁} {A₂} {B₁} {B₂} A₁≃A₂ =
(∀ x → Embedding (B₁ x) (B₂ (_≃_.to A₁≃A₂ x))) ↝⟨ Π-cong-contra-→ (_≃_.from A₁≃A₂) (λ _ → id) ⟩
(∀ x → Embedding (B₁ (_≃_.from A₁≃A₂ x))
(B₂ (_≃_.to A₁≃A₂ (_≃_.from A₁≃A₂ x)))) ↝⟨ (∀-cong _ λ _ → subst (Embedding (B₁ _) ⊚ B₂) (_≃_.right-inverse-of A₁≃A₂ _)) ⟩
(∀ x → Embedding (B₁ (_≃_.from A₁≃A₂ x)) (B₂ x)) ↝⟨ Π-cong-contra-Emb ext (inverse A₁≃A₂) ⟩□
Embedding ((x : A₁) → B₁ x) ((x : A₂) → B₂ x) □
Π-cong :
∀ {k₁ k₂ a₁ a₂ b₁ b₂} →
Extensionality? 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)) →
((x : A₁) → B₁ x) ↝[ k₂ ] ((x : A₂) → B₂ x)
Π-cong {k₁} {k₂} {a₁} {a₂} {b₁} {b₂}
ext {A₁} {A₂} {B₁} {B₂} A₁↔A₂ B₁↝B₂ =
helper k₂ ext (B₁↝B₂′ k₁ A₁↔A₂ B₁↝B₂)
where
B₁↝B₂′ :
∀ k₁ (A₁↔A₂ : A₁ ↔[ k₁ ] A₂) →
(∀ x → B₁ x ↝[ k₂ ] B₂ (to-implication A₁↔A₂ x)) →
∀ k x →
B₁ x ↝[ k₂ ] B₂ (to-implication {k = k} (from-isomorphism A₁↔A₂) x)
B₁↝B₂′ bijection _ B₁↝B₂ equivalence = B₁↝B₂
B₁↝B₂′ bijection _ B₁↝B₂ surjection = B₁↝B₂
B₁↝B₂′ equivalence _ B₁↝B₂ equivalence = B₁↝B₂
B₁↝B₂′ equivalence _ B₁↝B₂ surjection = B₁↝B₂
B₁↝B₂′ k₁ A₁↔A₂ B₁↝B₂ k = λ x →
B₁ x ↝⟨ B₁↝B₂ x ⟩
B₂ (to-implication A₁↔A₂ x) ↝⟨ ≡⇒↝ _ $ cong (λ f → B₂ (f x)) $
to-implication∘from-isomorphism k₁ k ⟩□
B₂ (to-implication {k = k} (from-isomorphism A₁↔A₂) x) □
A₁↝A₂ : ∀ {k} → A₁ ↝[ k ] A₂
A₁↝A₂ = from-isomorphism A₁↔A₂
l₁ = lower-extensionality a₁ b₁
l₂ = lower-extensionality a₂ b₂
helper :
∀ k₂ →
Extensionality? k₂ (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
(∀ k x → B₁ x ↝[ k₂ ]
B₂ (to-implication {k = k} (from-isomorphism A₁↔A₂) x)) →
((x : A₁) → B₁ x) ↝[ k₂ ] ((x : A₂) → B₂ x)
helper implication _ = Π-cong-→ A₁↝A₂ ⊚ (_$ surjection)
helper logical-equivalence _ = Π-cong-⇔ A₁↝A₂ ⊚ (_$ surjection)
helper injection ext = Π-cong-↣ (l₂ ext) A₁↝A₂ ⊚ (_$ equivalence)
helper embedding ext = Π-cong-Emb ext A₁↝A₂ ⊚ (_$ equivalence)
helper surjection ext = Π-cong-↠ (l₁ ext) A₁↝A₂ ⊚ (_$ surjection)
helper bijection ext = Π-cong-↔ ext A₁↝A₂ ⊚ (_$ equivalence)
helper equivalence ext = Π-cong-≃ ext A₁↝A₂ ⊚ (_$ equivalence)
Π-cong-contra :
∀ {k₁ k₂ a₁ a₂ b₁ b₂} →
Extensionality? k₂ (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
{A₁ : Set a₁} {A₂ : Set a₂} {B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂} →
(A₂↔A₁ : A₂ ↔[ k₁ ] A₁) →
(∀ x → B₁ (to-implication A₂↔A₁ x) ↝[ k₂ ] B₂ x) →
((x : A₁) → B₁ x) ↝[ k₂ ] ((x : A₂) → B₂ x)
Π-cong-contra {k₁} {k₂} {a₁} {a₂} {b₁} {b₂}
ext {A₁} {A₂} {B₁} {B₂} A₂↔A₁ B₁↝B₂ =
helper k₂ ext (B₁↝B₂′ k₁ A₂↔A₁ B₁↝B₂)
where
B₁↝B₂′ :
∀ k₁ (A₂↔A₁ : A₂ ↔[ k₁ ] A₁) →
(∀ x → B₁ (to-implication A₂↔A₁ x) ↝[ k₂ ] B₂ x) →
∀ k x →
B₁ (to-implication {k = k} (from-isomorphism A₂↔A₁) x) ↝[ k₂ ] B₂ x
B₁↝B₂′ bijection _ B₁↝B₂ equivalence = B₁↝B₂
B₁↝B₂′ bijection _ B₁↝B₂ implication = B₁↝B₂
B₁↝B₂′ bijection _ B₁↝B₂ surjection = B₁↝B₂
B₁↝B₂′ equivalence _ B₁↝B₂ equivalence = B₁↝B₂
B₁↝B₂′ equivalence _ B₁↝B₂ implication = B₁↝B₂
B₁↝B₂′ equivalence _ B₁↝B₂ surjection = B₁↝B₂
B₁↝B₂′ k₁ A₂↔A₁ B₁↝B₂ k = λ x →
B₁ (to-implication {k = k} (from-isomorphism A₂↔A₁) x) ↝⟨ ≡⇒↝ _ $ cong (λ f → B₁ (f x)) $ sym $ to-implication∘from-isomorphism k₁ k ⟩
B₁ (to-implication A₂↔A₁ x) ↝⟨ B₁↝B₂ x ⟩□
B₂ x □
A₂↝A₁ : ∀ {k} → A₂ ↝[ k ] A₁
A₂↝A₁ = from-isomorphism A₂↔A₁
l₁ = lower-extensionality a₁ b₁
l₂ = lower-extensionality a₂ b₂
helper :
∀ k₂ →
Extensionality? k₂ (a₁ ⊔ a₂) (b₁ ⊔ b₂) →
(∀ k x → B₁ (to-implication {k = k} (from-isomorphism A₂↔A₁) x)
↝[ k₂ ]
B₂ x) →
((x : A₁) → B₁ x) ↝[ k₂ ] ((x : A₂) → B₂ x)
helper implication _ = Π-cong-contra-→ A₂↝A₁ ⊚ (_$ implication)
helper logical-equivalence _ = Π-cong-contra-⇔ A₂↝A₁ ⊚ (_$ surjection)
helper injection ext = Π-cong-contra-↣ (l₂ ext) A₂↝A₁ ⊚ (_$ surjection)
helper embedding ext = Π-cong-contra-Emb ext A₂↝A₁ ⊚ (_$ equivalence)
helper surjection ext = Π-cong-contra-↠ (l₁ ext) A₂↝A₁ ⊚ (_$ equivalence)
helper bijection ext = Π-cong-contra-↔ ext A₂↝A₁ ⊚ (_$ equivalence)
helper equivalence ext = Π-cong-contra-≃ ext A₂↝A₁ ⊚ (_$ equivalence)
Π-left-identity : ∀ {a} {A : ⊤ → Set a} → ((x : ⊤) → A x) ↔ A tt
Π-left-identity = record
{ surjection = record
{ logical-equivalence = record
{ to = λ f → f tt
; from = λ x _ → x
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
drop-⊤-left-Π :
∀ {k a b} {A : Set a} {B : A → Set b} →
Extensionality? k a b →
(A↔⊤ : A ↔ ⊤) →
((x : A) → B x) ↝[ k ] B (_↔_.from A↔⊤ tt)
drop-⊤-left-Π {A = A} {B} ext A↔⊤ =
((x : A) → B x) ↝⟨ Π-cong-contra ext (inverse A↔⊤) (λ _ → id) ⟩
((x : ⊤) → B (_↔_.from A↔⊤ x)) ↔⟨ Π-left-identity ⟩□
B (_↔_.from A↔⊤ tt) □
→-right-zero : ∀ {a} {A : Set a} → (A → ⊤) ↔ ⊤
→-right-zero = record
{ surjection = record
{ logical-equivalence = record
{ to = λ _ → tt
; from = λ _ _ → tt
}
; right-inverse-of = λ _ → refl tt
}
; left-inverse-of = λ _ → refl (λ _ → tt)
}
Π⊥↔⊤ : ∀ {k ℓ a} {A : ⊥ {ℓ = ℓ} → Set a} →
Extensionality? k ℓ a →
((x : ⊥) → A x) ↝[ k ] ⊤
Π⊥↔⊤ = generalise-ext? Π⊥⇔⊤ λ ext → record
{ surjection = record
{ logical-equivalence = Π⊥⇔⊤
; right-inverse-of = λ _ → refl _
}
; left-inverse-of = λ _ → apply-ext ext (λ x → ⊥-elim x)
}
where
Π⊥⇔⊤ = record
{ to = _
; from = λ _ x → ⊥-elim x
}
¬⊥↔⊤ : ∀ {k ℓ} →
Extensionality? k ℓ lzero →
¬ ⊥ {ℓ = ℓ} ↝[ k ] ⊤
¬⊥↔⊤ = Π⊥↔⊤
→→↠→ :
∀ {a b} {A : Set a} {B : Set b} →
(A → A → B) ↠ (A → B)
→→↠→ = record
{ logical-equivalence = record
{ to = λ f x → f x x
; from = λ f x _ → f x
}
; right-inverse-of = refl
}
→→⊥↔→⊥ :
∀ {a ℓ} {A : Set a} →
Extensionality a (a ⊔ ℓ) →
(A → A → ⊥ {ℓ = ℓ}) ↔ (A → ⊥ {ℓ = ℓ})
→→⊥↔→⊥ ext = record
{ surjection = →→↠→
; left-inverse-of = λ f → apply-ext ext λ x → ⊥-elim (f x x)
}
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
{ logical-equivalence = record { to = flip; from = flip }
; right-inverse-of = refl
}
; left-inverse-of = refl
}
ΠΣ-comm :
∀ {a b c} {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} →
(∀ x → ∃ λ (y : B x) → C x y)
↔
(∃ λ (f : ∀ x → B x) → ∀ x → C x (f x))
ΠΣ-comm = record
{ surjection = record
{ logical-equivalence = record
{ to = λ f → proj₁ ⊚ f , proj₂ ⊚ f
; from = λ { (f , g) x → f x , g x }
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
implicit-ΠΣ-comm :
∀ {a b c} {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} →
(∀ {x} → ∃ λ (y : B x) → C x y)
↔
(∃ λ (f : ∀ {x} → B x) → ∀ {x} → C x f)
implicit-ΠΣ-comm {A = A} {B} {C} =
(∀ {x} → ∃ λ (y : B x) → C x y) ↝⟨ Bijection.implicit-Π↔Π ⟩
(∀ x → ∃ λ (y : B x) → C x y) ↝⟨ ΠΣ-comm ⟩
(∃ λ (f : ∀ x → B x) → ∀ x → C x (f x)) ↝⟨ inverse $ Σ-cong Bijection.implicit-Π↔Π (λ _ → Bijection.implicit-Π↔Π) ⟩□
(∃ λ (f : ∀ {x} → B x) → ∀ {x} → C x f) □
implicit-extensionality-isomorphism :
∀ {k a b} →
Extensionality a b →
{A : Set a} {B : A → Set b} {f g : {x : A} → B x} →
(∀ x → f {x} ≡ g {x}) ↔[ k ] ((λ {x} → f {x}) ≡ g)
implicit-extensionality-isomorphism ext {f = f} {g} =
(∀ x → f {x} ≡ g {x}) ↔⟨ Eq.extensionality-isomorphism ext ⟩
((λ x → f {x}) ≡ (λ x → g {x})) ↔⟨ inverse $ Eq.≃-≡ (Eq.↔⇒≃ (inverse Bijection.implicit-Π↔Π)) ⟩□
((λ {x} → f {x}) ≡ g) □
private
to-implicit-extensionality-isomorphism :
∀ {a b}
(ext : Extensionality a b) {A : Set a} {B : A → Set b}
{f g : {x : A} → B x} (f≡g : ∀ x → f {x} ≡ g {x}) →
_↔_.to (implicit-extensionality-isomorphism ext) f≡g
≡
implicit-extensionality (Eq.good-ext ext) f≡g
to-implicit-extensionality-isomorphism _ _ = refl _
yoneda :
∀ {a b X} →
Extensionality (lsuc a) (lsuc a ⊔ b) →
(F : SET a → SET b) →
(map : ∀ {A B} → (Type A → Type B) → Type (F A) → Type (F B)) →
(∀ {A} {x : Type (F A)} → map id x ≡ x) →
(∀ {A B C f g x} →
(map {A = B} {B = C} f ∘ map {A = A} g) x ≡ map (f ∘ g) x) →
Type (F X)
↔
∃ λ (γ : ∀ Y → (Type X → Type Y) → Type (F Y)) →
∀ Y₁ Y₂ f g → map f (γ Y₁ g) ≡ γ Y₂ (f ∘ g)
yoneda {a} {X = X} ext F map map-id map-∘ = record
{ surjection = record
{ logical-equivalence = record
{ to = λ x → (λ _ f → map f x) , λ _ _ f g →
map f (map g x) ≡⟨ map-∘ ⟩∎
map (f ∘ g) x ∎
; from = λ { (γ , _) → γ X id }
}
; right-inverse-of = λ { (γ , h) → Σ-≡,≡→≡
((λ _ f → map f (γ X id)) ≡⟨ (apply-ext (lower-extensionality lzero (lsuc a) ext) λ Y →
apply-ext (lower-extensionality _ (lsuc a) ext) λ f →
h X Y f id) ⟩∎
(λ Y f → γ Y f) ∎)
(_⇔_.to propositional⇔irrelevant
(Π-closure ext 1 λ _ →
Π-closure (lower-extensionality lzero (lsuc a) ext) 1 λ Y₂ →
Π-closure (lower-extensionality _ (lsuc a) ext) 1 λ _ →
Π-closure (lower-extensionality _ (lsuc a) ext) 1 λ _ →
proj₂ (F Y₂) _ _)
_ _) }
}
; left-inverse-of = λ x →
map id x ≡⟨ map-id ⟩∎
x ∎
}
Π≡↔≡-↠-≡ : ∀ k {a} {A : Set a} (x y : A) →
(∀ z → (z ≡ x) ↔[ k ] (z ≡ y)) ↠ (x ≡ y)
Π≡↔≡-↠-≡ k x y = record
{ logical-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
{ logical-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} → Extensionality a a →
{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≡↔≡-↠-≡ equivalence x y
open _↠_ surj
abstract
from∘to : ∀ f → from (to f) ≡ f
from∘to f =
apply-ext ext λ z → Eq.lift-equality ext $ apply-ext 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 ∎
∀-intro : ∀ {k a b} →
Extensionality? k a (a ⊔ b) →
{A : Set a} {x : A} (B : (y : A) → x ≡ y → Set b) →
B x (refl x) ↝[ k ] (∀ y (x≡y : x ≡ y) → B y x≡y)
∀-intro ext B =
generalise-ext? (∀-intro-⇔ B) (λ ext → ∀-intro-↔ ext B) ext
where
∀-intro-⇔ : ∀ {a b} {A : Set a} {x : A}
(B : (y : A) → x ≡ y → Set b) →
B x (refl x) ⇔ (∀ y (x≡y : x ≡ y) → B y x≡y)
∀-intro-⇔ {x = x} B = record
{ to = λ b y x≡y →
subst (uncurry B)
(proj₂ (other-singleton-contractible x) (y , x≡y))
b
; from = λ f → f x (refl x)
}
∀-intro-↔ : ∀ {a b} →
Extensionality a (a ⊔ b) →
{A : Set a} {x : A} (B : (y : A) → x ≡ y → Set b) →
B x (refl x) ↔ (∀ y (x≡y : x ≡ y) → B y x≡y)
∀-intro-↔ {a} ext {x = x} B = record
{ surjection = record
{ logical-equivalence = ∀-intro-⇔ B
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
abstract
from∘to :
∀ b → _⇔_.from (∀-intro-⇔ B) (_⇔_.to (∀-intro-⇔ B) b) ≡ b
from∘to b =
subst (uncurry B)
(proj₂ (other-singleton-contractible x) (x , refl x)) b ≡⟨ cong (λ eq → subst (uncurry B) eq b) $
other-singleton-contractible-refl x ⟩
subst (uncurry B) (refl (x , refl x)) b ≡⟨ subst-refl (uncurry B) _ ⟩∎
b ∎
to∘from :
∀ b → _⇔_.to (∀-intro-⇔ B) (_⇔_.from (∀-intro-⇔ B) b) ≡ b
to∘from f =
apply-ext ext λ y →
apply-ext (lower-extensionality lzero a ext) λ x≡y →
elim¹ (λ {y} x≡y →
subst (uncurry B)
(proj₂ (other-singleton-contractible x)
(y , x≡y))
(f x (refl x)) ≡
f y x≡y)
(subst (uncurry B)
(proj₂ (other-singleton-contractible x)
(x , refl x))
(f x (refl x)) ≡⟨ from∘to (f x (refl x)) ⟩∎
f x (refl x) ∎)
x≡y
private
∀-intro′ : ∀ {k a b} →
Extensionality? k a (a ⊔ b) →
{A : Set a} {x : A} (B : (y : A) → x ≡ y → Set b) →
B x (refl x) ↝[ k ] (∀ y (x≡y : x ≡ y) → B y x≡y)
∀-intro′ {k} {a} ext {x = x} B =
B x (refl x) ↔⟨ inverse Π-left-identity ⟩
(⊤ → B x (refl x)) ↝⟨ Π-cong-contra (lower-extensionality? k lzero a ext)
(_⇔_.to contractible⇔↔⊤ c) (λ _ → id) ⟩
((∃ λ y → x ≡ y) → B x (refl x)) ↔⟨ currying ⟩
(∀ y (x≡y : x ≡ y) → B x (refl x)) ↝⟨ (∀-cong ext λ y →
∀-cong (lower-extensionality? k lzero a ext) λ x≡y → from-isomorphism $
Eq.subst-as-equivalence (uncurry B) (proj₂ c (y , x≡y))) ⟩□
(∀ y (x≡y : x ≡ y) → B y x≡y) □
where
c : Contractible (∃ λ y → x ≡ y)
c = other-singleton-contractible x
→-intro :
∀ {a p} {A : Set a} {P : A → Set p} →
Extensionality a (a ⊔ p) →
(∀ x → Is-proposition (P x)) →
(∀ x → P x) ↔ (A → ∀ x → P x)
→-intro {a = a} ext P-prop = record
{ surjection = record
{ logical-equivalence = record
{ to = λ f _ x → f x
; from = λ f x → f x x
}
; right-inverse-of = λ _ →
_⇔_.to propositional⇔irrelevant
(Π-closure ext 1 λ _ →
Π-closure (lower-extensionality a a ext) 1 λ _ →
P-prop _)
_ _
}
; left-inverse-of = refl
}
from≡↔≡to : ∀ {a b k} →
{A : Set a} {B : Set b}
(A≃B : A ≃ B) {x : B} {y : A} →
(_≃_.from A≃B x ≡ y) ↔[ k ] (x ≡ _≃_.to A≃B y)
from≡↔≡to A≃B {x} {y} =
(_≃_.from A≃B x ≡ y) ↔⟨ inverse $ Eq.≃-≡ A≃B ⟩
(_≃_.to A≃B (_≃_.from A≃B x) ≡ _≃_.to A≃B y) ↝⟨ ≡⇒↝ _ $ cong (λ z → z ≡ _≃_.to A≃B y) $ _≃_.right-inverse-of A≃B x ⟩□
(x ≡ _≃_.to A≃B y) □
∘from≡↔≡∘to : ∀ {a b c k} →
Extensionality (a ⊔ b) c →
{A : Set a} {B : Set b} {C : Set c}
(A≃B : A ≃ B) {f : A → C} {g : B → C} →
(f ∘ _≃_.from A≃B ≡ g) ↔[ k ] (f ≡ g ∘ _≃_.to A≃B)
∘from≡↔≡∘to ext A≃B = from≡↔≡to (→-cong ext (inverse A≃B) Eq.id)
to∘≡↔≡from∘ : ∀ {a b c k} →
Extensionality a (b ⊔ c) →
{A : Set a} {B : A → Set b} {C : A → Set c}
(B≃C : ∀ {x} → B x ≃ C x)
{f : (x : A) → B x} {g : (x : A) → C x} →
(_≃_.to B≃C ⊚ f ≡ g) ↔[ k ] (f ≡ _≃_.from B≃C ⊚ g)
to∘≡↔≡from∘ ext B≃C =
from≡↔≡to (∀-cong ext (λ _ → inverse B≃C))
private
↑-cong-→ :
∀ {a b c} {B : Set b} {C : Set c} →
(B → C) → ↑ a B → ↑ a C
↑-cong-→ B→C = lift ⊚ B→C ⊚ lower
↑-cong-⇔ :
∀ {a b c} {B : Set b} {C : Set c} →
B ⇔ C → ↑ a B ⇔ ↑ a C
↑-cong-⇔ B⇔C = record
{ to = ↑-cong-→ to
; from = ↑-cong-→ from
} where open _⇔_ B⇔C
↑-cong-↣ :
∀ {a b c} {B : Set b} {C : Set c} →
B ↣ C → ↑ a B ↣ ↑ a C
↑-cong-↣ {a} B↣C = record
{ to = to′
; injective = injective′
}
where
open _↣_ B↣C
to′ = ↑-cong-→ {a = a} to
abstract
injective′ : Injective to′
injective′ = cong lift ⊚ injective ⊚ cong lower
↑-cong-Embedding :
∀ {a b c} {B : Set b} {C : Set c} →
Embedding B C → Embedding (↑ a B) (↑ a C)
↑-cong-Embedding {a} {B = B} B↣C = record
{ to = ↑-cong-→ to
; is-embedding = λ x y →
_≃_.is-equivalence $
Eq.with-other-function
(x ≡ y ↔⟨⟩
lift (lower x) ≡ lift (lower y) ↔⟨ inverse lift-lemma ⟩
lower x ≡ lower y ↝⟨ Eq.⟨ _ , is-embedding _ _ ⟩ ⟩
to (lower x) ≡ to (lower y) ↔⟨ lift-lemma ⟩□
lift (to (lower x)) ≡ lift (to (lower y)) □)
_
(λ p → cong lift (cong to (cong lower p)) ≡⟨ cong-∘ _ _ _ ⟩
cong (lift ⊚ to) (cong lower p) ≡⟨ cong-∘ _ _ _ ⟩∎
cong (lift ⊚ to ⊚ lower) p ∎)
}
where
open Embedding B↣C
lift-lemma : ∀ {ℓ a} {A : Set a} {x y : A} →
(x ≡ y) ↔ (lift {ℓ = ℓ} x ≡ lift y)
lift-lemma {ℓ} = record
{ surjection = record
{ logical-equivalence = record
{ to = cong lift
; from = cong lower
}
; right-inverse-of = λ eq →
cong lift (cong lower eq) ≡⟨ cong-∘ _ _ _ ⟩
cong (lift ⊚ lower) eq ≡⟨ sym (cong-id _) ⟩∎
eq ∎
}
; left-inverse-of = λ eq →
cong lower (cong lift eq) ≡⟨ cong-∘ _ _ _ ⟩
cong (lower {ℓ = ℓ} ⊚ lift) eq ≡⟨ sym (cong-id _) ⟩∎
eq ∎
}
↑-cong-↠ :
∀ {a b c} {B : Set b} {C : Set c} →
B ↠ C → ↑ a B ↠ ↑ a C
↑-cong-↠ {a} B↠C = record
{ logical-equivalence = logical-equivalence′
; right-inverse-of = right-inverse-of′
}
where
open _↠_ B↠C renaming (logical-equivalence to logical-equiv)
logical-equivalence′ = ↑-cong-⇔ {a = a} logical-equiv
abstract
right-inverse-of′ : ∀ x →
_⇔_.to logical-equivalence′
(_⇔_.from logical-equivalence′ x) ≡
x
right-inverse-of′ = cong lift ⊚ right-inverse-of ⊚ lower
↑-cong-↔ :
∀ {a b c} {B : Set b} {C : Set c} →
B ↔ C → ↑ a B ↔ ↑ a C
↑-cong-↔ {a} B↔C = record
{ surjection = surjection′
; left-inverse-of = left-inverse-of′
}
where
open _↔_ B↔C renaming (surjection to surj)
surjection′ = ↑-cong-↠ {a = a} surj
abstract
left-inverse-of′ :
∀ x → _↠_.from surjection′ (_↠_.to surjection′ x) ≡ x
left-inverse-of′ = cong lift ⊚ left-inverse-of ⊚ lower
↑-cong : ∀ {k a b c} {B : Set b} {C : Set c} →
B ↝[ k ] C → ↑ a B ↝[ k ] ↑ a C
↑-cong {implication} = ↑-cong-→
↑-cong {logical-equivalence} = ↑-cong-⇔
↑-cong {injection} = ↑-cong-↣
↑-cong {embedding} = ↑-cong-Embedding
↑-cong {surjection} = ↑-cong-↠
↑-cong {bijection} = ↑-cong-↔
↑-cong {equivalence} =
from-bijection ∘ ↑-cong-↔ ∘ from-equivalence
tt≡tt↔⊤ : tt ≡ tt ↔ ⊤
tt≡tt↔⊤ = _⇔_.to contractible⇔↔⊤ $
propositional⇒inhabited⇒contractible
(mono (zero≤ 2) ⊤-contractible _ _) (refl _)
⊥↔⊥ : ∀ {ℓ₁ ℓ₂} → ⊥ {ℓ = ℓ₁} ↔ ⊥ {ℓ = ℓ₂}
⊥↔⊥ = Bijection.⊥↔uninhabited ⊥-elim
¬↔→⊥ : ∀ {a ℓ} {A : Set a} →
Extensionality a ℓ →
¬ A ↔ (A → ⊥ {ℓ = ℓ})
¬↔→⊥ {A = A} ext =
(A → ⊥₀) ↝⟨ →-cong ext id ⊥↔⊥ ⟩□
(A → ⊥) □
¬[⇔¬] : ∀ {a} {A : Set a} → ¬ (A ⇔ ¬ A)
¬[⇔¬] {A = A} =
A ⇔ ¬ A ↝⟨ (λ eq → (λ a → _⇔_.to eq a a) , eq) ⟩
¬ A × (A ⇔ ¬ A) ↝⟨ (λ { (¬a , eq) → ¬a , _⇔_.from eq ¬a }) ⟩
¬ A × A ↝⟨ uncurry _$_ ⟩□
⊥ □
H-level-cong :
∀ {k a b} {A : Set a} {B : Set b} →
Extensionality (a ⊔ b) (a ⊔ b) →
∀ n → A ↔[ k ] B → H-level n A ↔[ k ] H-level n B
H-level-cong {a = a} {b} ext n A↔B′ =
from-equivalence $
_↔_.to (Eq.⇔↔≃ ext (H-level-propositional
(lower-extensionality b b ext) n)
(H-level-propositional
(lower-extensionality a a ext) n)) (record
{ to = respects-surjection (_↔_.surjection A↔B) n
; from = respects-surjection (_↔_.surjection (inverse A↔B)) n
})
where
A↔B = from-isomorphism A↔B′
propositional≃irrelevant :
∀ {a} {A : Set a} →
Extensionality a a →
Is-proposition A ≃ Proof-irrelevant A
propositional≃irrelevant ext =
_↔_.to (Eq.⇔↔≃ ext
(H-level-propositional ext 1)
(Proof-irrelevant-propositional ext))
propositional⇔irrelevant
set≃UIP :
∀ {a} {A : Set a} →
Extensionality a a →
Is-set A ≃ Uniqueness-of-identity-proofs A
set≃UIP ext =
_↔_.to (Eq.⇔↔≃ ext
(H-level-propositional ext 2)
(UIP-propositional ext))
set⇔UIP
→≃→↠≃ :
∀ {n ℓ} {A B : Set ℓ} →
Extensionality ℓ ℓ →
(hA : H-level n A) (hB : H-level n B) →
(∃ λ (f : (C : Set ℓ) → H-level n C → (A → C) ≃ (B → C)) →
((C : Set ℓ) (hC : H-level n C) (g : A → C) →
g ∘ _≃_.to (f A hA) id ≡ _≃_.to (f C hC) g) ×
((C : Set ℓ) (hC : H-level n C) (g : B → C) →
g ∘ _≃_.from (f B hB) id ≡ _≃_.from (f C hC) g))
↠
(A ≃ B)
→≃→↠≃ {A = A} {B} ext hA hB = record
{ logical-equivalence = record
{ from = λ A≃B → (λ _ _ → →-cong ext A≃B id)
, (λ _ _ g → refl (g ∘ _≃_.from A≃B))
, (λ _ _ g → refl (g ∘ _≃_.to A≃B))
; to = λ { (A→≃B→ , ∘to≡ , ∘from≡) → Eq.↔⇒≃ (record
{ surjection = record
{ logical-equivalence = record
{ to = _≃_.from (A→≃B→ B hB) id
; from = _≃_.to (A→≃B→ A hA) id
}
; right-inverse-of = λ x →
_≃_.from (A→≃B→ B hB) id (_≃_.to (A→≃B→ A hA) id x) ≡⟨⟩
(_≃_.from (A→≃B→ B hB) id ∘ _≃_.to (A→≃B→ A hA) id) x ≡⟨ cong (_$ x) $ ∘to≡ _ _ _ ⟩
(_≃_.to (A→≃B→ B hB) (_≃_.from (A→≃B→ B hB) id)) x ≡⟨ cong (_$ x) $ _≃_.right-inverse-of (A→≃B→ B hB) _ ⟩∎
x ∎
}
; left-inverse-of = λ x →
_≃_.to (A→≃B→ A hA) id (_≃_.from (A→≃B→ B hB) id x) ≡⟨⟩
(_≃_.to (A→≃B→ A hA) id ∘ _≃_.from (A→≃B→ B hB) id) x ≡⟨ cong (_$ x) $ ∘from≡ _ _ _ ⟩
(_≃_.from (A→≃B→ A hA) (_≃_.to (A→≃B→ A hA) id)) x ≡⟨ cong (_$ x) $ _≃_.left-inverse-of (A→≃B→ A hA) _ ⟩∎
x ∎
}) }
}
; right-inverse-of = λ A≃B → _↔_.to (≃-to-≡↔≡ ext) λ x →
refl (_≃_.to A≃B x)
}
→≃→↔≃ :
∀ {ℓ} {A B : Set ℓ} →
Extensionality (lsuc ℓ) ℓ →
(hA : Is-set A) (hB : Is-set B) →
(∃ λ (f : (C : Set ℓ) → Is-set C → (A → C) ≃ (B → C)) →
((C : Set ℓ) (hC : Is-set C) (g : A → C) →
g ∘ _≃_.to (f A hA) id ≡ _≃_.to (f C hC) g) ×
((C : Set ℓ) (hC : Is-set C) (g : B → C) →
g ∘ _≃_.from (f B hB) id ≡ _≃_.from (f C hC) g))
↔
(A ≃ B)
→≃→↔≃ {A = A} {B} ext hA hB = record
{ surjection = →≃→↠≃ ext′ hA hB
; left-inverse-of = λ { (A→≃B→ , ∘to≡ , _) →
Σ-≡,≡→≡
(apply-ext ext λ C →
apply-ext ext′ λ hC →
_↔_.to (≃-to-≡↔≡ ext′) λ f →
f ∘ _≃_.to (A→≃B→ A hA) id ≡⟨ ∘to≡ _ _ _ ⟩∎
_≃_.to (A→≃B→ C hC) f ∎)
(_⇔_.to propositional⇔irrelevant
(×-closure 1
(Π-closure ext 1 λ _ →
Π-closure ext′ 1 λ hC →
Π-closure ext′ 1 λ _ →
(Π-closure ext′ 2 λ _ → hC) _ _)
(Π-closure ext 1 λ _ →
Π-closure ext′ 1 λ hC →
Π-closure ext′ 1 λ _ →
(Π-closure ext′ 2 λ _ → hC) _ _))
_ _) }
}
where
ext′ = lower-extensionality _ lzero ext
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
{ logical-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))
}
ℕ↔ℕ⊎ℕ : ℕ ↔ ℕ ⊎ ℕ
ℕ↔ℕ⊎ℕ = record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
step : ℕ ⊎ ℕ → ℕ ⊎ ℕ
step = [ inj₂ , inj₁ ∘ suc ]
to : ℕ → ℕ ⊎ ℕ
to zero = inj₁ zero
to (suc n) = step (to n)
double : ℕ → ℕ
double zero = zero
double (suc n) = suc (suc (double n))
from : ℕ ⊎ ℕ → ℕ
from = [ double , suc ∘ double ]
from∘to : ∀ n → from (to n) ≡ n
from∘to zero = zero ∎
from∘to (suc n) with to n | from∘to n
... | inj₁ m | eq =
suc (double m) ≡⟨ cong suc eq ⟩∎
suc n ∎
... | inj₂ m | eq =
suc (suc (double m)) ≡⟨ cong suc eq ⟩∎
suc n ∎
to∘double : ∀ n → to (double n) ≡ inj₁ n
to∘double zero = inj₁ zero ∎
to∘double (suc n) =
to (double (suc n)) ≡⟨⟩
to (suc (suc (double n))) ≡⟨⟩
step (step (to (double n))) ≡⟨ cong (step ∘ step) (to∘double n) ⟩
step (step (inj₁ n)) ≡⟨⟩
inj₁ (suc n) ∎
to∘from : ∀ x → to (from x) ≡ x
to∘from =
[ to∘double
, (λ n →
to (from (inj₂ n)) ≡⟨⟩
to (suc (double n)) ≡⟨⟩
step (to (double n)) ≡⟨ cong step (to∘double n) ⟩
step (inj₁ n) ≡⟨⟩
inj₂ n ∎)
]
ℕ↔ℕ² : ℕ ↔ ℕ × ℕ
ℕ↔ℕ² = record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
step : ℕ × ℕ → ℕ × ℕ
step (m , zero) = (zero , suc m)
step (m , suc n) = (suc m , n)
to : ℕ → ℕ × ℕ
to zero = (zero , zero)
to (suc n) = step (to n)
from′ : (m n sum : ℕ) → n + m ≡ sum → ℕ
from′ zero zero _ _ = zero
from′ zero (suc n) zero eq = ⊥-elim (0≢+ (sym eq))
from′ zero (suc n) (suc sum) eq = suc (from′ n zero sum (cancel-suc
(suc n ≡⟨ cong suc (sym +-right-identity) ⟩
suc (n + 0) ≡⟨ eq ⟩∎
suc sum ∎)))
from′ (suc m) n sum eq = suc (from′ m (suc n) sum
(suc n + m ≡⟨ suc+≡+suc n ⟩
n + suc m ≡⟨ eq ⟩∎
sum ∎))
from : ℕ × ℕ → ℕ
from (m , n) = from′ m n _ (refl _)
from′-irr : ∀ m {n sum₁ eq₁ sum₂ eq₂} →
from′ m n sum₁ eq₁ ≡ from′ m n sum₂ eq₂
from′-irr m {n} {sum₁} {eq₁} {sum₂} {eq₂} =
from′ m n sum₁ eq₁ ≡⟨ cong (uncurry (from′ m n)) (Σ-≡,≡→≡ lemma (_⇔_.to set⇔UIP ℕ-set _ _)) ⟩∎
from′ m n sum₂ eq₂ ∎
where
lemma =
sum₁ ≡⟨ sym eq₁ ⟩
n + m ≡⟨ eq₂ ⟩∎
sum₂ ∎
from∘step : ∀ p → from (step p) ≡ suc (from p)
from∘step (m , zero) = from (zero , suc m) ≡⟨ cong suc (from′-irr m) ⟩∎
suc (from (m , zero)) ∎
from∘step (m , suc n) = from (suc m , n) ≡⟨ cong suc (from′-irr m) ⟩∎
suc (from (m , suc n)) ∎
from∘to : ∀ n → from (to n) ≡ n
from∘to zero = refl _
from∘to (suc n) =
from (to (suc n)) ≡⟨⟩
from (step (to n)) ≡⟨ from∘step (to n) ⟩
suc (from (to n)) ≡⟨ cong suc (from∘to n) ⟩∎
suc n ∎
to∘from′ : ∀ m n sum eq → to (from′ m n sum eq) ≡ (m , n)
to∘from′ zero zero _ _ = refl _
to∘from′ zero (suc n) zero eq = ⊥-elim (0≢+ (sym eq))
to∘from′ zero (suc n) (suc sum) eq =
step (to (from′ n zero _ _)) ≡⟨ cong step (to∘from′ n zero sum _) ⟩
step (n , zero) ≡⟨⟩
(zero , suc n) ∎
to∘from′ (suc m) n sum eq =
step (to (from′ m (suc n) _ _)) ≡⟨ cong step (to∘from′ m (suc n) sum _) ⟩
step (m , suc n) ≡⟨⟩
(suc m , n) ∎
to∘from : ∀ p → to (from p) ≡ p
to∘from _ = to∘from′ _ _ _ _
¬-⊎-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′
private
module ⊎-left-cancellative
{a b c} {A : Set a} {B : Set b} {C : Set c}
(f : A ⊎ B → A ⊎ C)
(hyp : Well-behaved f)
where
mutual
g : B → C
g b = g′ (inspect (f (inj₂ b)))
g′ : ∀ {b} → Other-singleton (f (inj₂ b)) → C
g′ (inj₂ c , _) = c
g′ (inj₁ a , eq) = g″ eq (inspect (f (inj₁ a)))
g″ : ∀ {a b} →
f (inj₂ b) ≡ inj₁ a → Other-singleton (f (inj₁ a)) → C
g″ _ (inj₂ c , _) = c
g″ eq₁ (inj₁ _ , eq₂) = ⊥-elim $ hyp eq₁ eq₂
⊎-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 {A = A} = λ inv to-hyp from-hyp → record
{ surjection = record
{ logical-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 _↔_
open ⊎-left-cancellative
abstract
g∘g : ∀ {b c} {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 , eq₁ with inspect (from f (inj₂ c))
g∘g′ | inj₂ c , eq₁ | inj₂ b′ , eq₂ = ⊎.cancel-inj₂ (
inj₂ b′ ≡⟨ sym eq₂ ⟩
from f (inj₂ c) ≡⟨ to-from f eq₁ ⟩∎
inj₂ b ∎)
g∘g′ | inj₂ c , eq₁ | inj₁ a , eq₂ = ⊥-elim $ ⊎.inj₁≢inj₂ (
inj₁ a ≡⟨ sym eq₂ ⟩
from f (inj₂ c) ≡⟨ to-from f eq₁ ⟩∎
inj₂ b ∎)
g∘g′ | inj₁ a , eq₁ with inspect (to f (inj₁ a))
g∘g′ | inj₁ a , eq₁ | inj₁ a′ , eq₂ = ⊥-elim $ to-hyp eq₁ eq₂
g∘g′ | inj₁ a , eq₁ | inj₂ c , eq₂ with inspect (from f (inj₂ c))
g∘g′ | inj₁ a , eq₁ | inj₂ c , eq₂ | inj₂ b′ , eq₃ = ⊥-elim $ ⊎.inj₁≢inj₂ (
inj₁ a ≡⟨ sym $ to-from f eq₂ ⟩
from f (inj₂ c) ≡⟨ eq₃ ⟩∎
inj₂ b′ ∎)
g∘g′ | inj₁ a , eq₁ | inj₂ c , eq₂ | inj₁ a′ , eq₃ with inspect (from f (inj₁ a′))
g∘g′ | inj₁ a , eq₁ | inj₂ c , eq₂ | inj₁ a′ , eq₃ | inj₁ a″ , eq₄ = ⊥-elim $ from-hyp eq₃ eq₄
g∘g′ | inj₁ a , eq₁ | inj₂ c , eq₂ | inj₁ a′ , eq₃ | inj₂ b′ , 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 ∎)
⊎-left-cancellative-⊤ :
∀ {a b} {A : Set a} {B : Set b} →
(⊤ ⊎ A) ↔ (⊤ ⊎ B) → A ↔ B
⊎-left-cancellative-⊤ = λ ⊤⊎A↔⊤⊎B →
⊎-left-cancellative ⊤⊎A↔⊤⊎B
(wb ⊤⊎A↔⊤⊎B)
(wb $ inverse ⊤⊎A↔⊤⊎B)
where
open _↔_
abstract
wb : ∀ {a b} {A : Set a} {B : Set b}
(⊤⊎A↔⊤⊎B : (⊤ ⊎ A) ↔ (⊤ ⊎ B)) →
Well-behaved (_↔_.to ⊤⊎A↔⊤⊎B)
wb ⊤⊎A↔⊤⊎B {b = b} eq₁ eq₂ = ⊎.inj₁≢inj₂ (
inj₁ tt ≡⟨ sym $ to-from ⊤⊎A↔⊤⊎B eq₂ ⟩
from ⊤⊎A↔⊤⊎B (inj₁ tt) ≡⟨ to-from ⊤⊎A↔⊤⊎B eq₁ ⟩∎
inj₂ b ∎)
[⊤⊎↔⊤⊎]↔[⊤⊎×↔] :
∀ {a b} {A : Set a} {B : Set b} →
Extensionality (a ⊔ b) (a ⊔ b) →
Decidable-equality B →
((⊤ ⊎ A) ↔ (⊤ ⊎ B)) ↔ (⊤ ⊎ B) × (A ↔ B)
[⊤⊎↔⊤⊎]↔[⊤⊎×↔] {A = A} {B} ext _≟B_ = record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
_≟_ : Decidable-equality (⊤ ⊎ B)
_≟_ = ⊎.Dec._≟_ ⊤._≟_ _≟B_
if⌊_⌋then_else_ : ∀ {a p} {A : Set a} {P : Set p} → Dec P → A → A → A
if⌊ yes _ ⌋then t else e = t
if⌊ no _ ⌋then t else e = e
if-not : ∀ {a p} {A : Set a} {P : Set p} (d : Dec P) (t e : A) →
¬ P → if⌊ d ⌋then t else e ≡ e
if-not (yes p) t e ¬p = ⊥-elim (¬p p)
if-not (no _) t e ¬p = refl _
to : (⊤ ⊎ A) ↔ (⊤ ⊎ B) → (⊤ ⊎ B) × (A ↔ B)
to ⊤⊎A↔⊤⊎B = _↔_.to ⊤⊎A↔⊤⊎B (inj₁ tt) , ⊎-left-cancellative-⊤ ⊤⊎A↔⊤⊎B
from : (⊤ ⊎ B) × (A ↔ B) → (⊤ ⊎ A) ↔ (⊤ ⊎ B)
from (⊤⊎B , A↔B) = record
{ surjection = record
{ logical-equivalence = record
{ to = t ⊤⊎B
; from = f ⊤⊎B
}
; right-inverse-of = t∘f ⊤⊎B
}
; left-inverse-of = f∘t ⊤⊎B
}
where
t : ⊤ ⊎ B → ⊤ ⊎ A → ⊤ ⊎ B
t ⊤⊎B (inj₁ tt) = ⊤⊎B
t ⊤⊎B (inj₂ a) = if⌊ b ≟ ⊤⊎B ⌋then inj₁ tt else b
where
b = inj₂ (_↔_.to A↔B a)
f : ⊤ ⊎ B → ⊤ ⊎ B → ⊤ ⊎ A
f ⊤⊎B (inj₁ tt) = [ const (inj₁ tt) , inj₂ ∘ _↔_.from A↔B ] ⊤⊎B
f ⊤⊎B (inj₂ b) =
if⌊ ⊤⊎B ≟ inj₂ b ⌋then inj₁ tt else inj₂ (_↔_.from A↔B b)
abstract
t∘f : ∀ ⊤⊎B x → t ⊤⊎B (f ⊤⊎B x) ≡ x
t∘f (inj₁ tt) (inj₁ tt) = refl _
t∘f (inj₁ tt) (inj₂ b′) = inj₂ (_↔_.to A↔B (_↔_.from A↔B b′)) ≡⟨ cong inj₂ $ _↔_.right-inverse-of A↔B _ ⟩∎
inj₂ b′ ∎
t∘f (inj₂ b) (inj₁ tt) with _↔_.to A↔B (_↔_.from A↔B b) ≟B b
t∘f (inj₂ b) (inj₁ tt) | yes _ = refl _
t∘f (inj₂ b) (inj₁ tt) | no b≢b = ⊥-elim $ b≢b (
_↔_.to A↔B (_↔_.from A↔B b) ≡⟨ _↔_.right-inverse-of A↔B _ ⟩∎
b ∎)
t∘f (inj₂ b) (inj₂ b′) with b ≟B b′
t∘f (inj₂ b) (inj₂ b′) | yes b≡b′ = inj₂ b ≡⟨ cong inj₂ b≡b′ ⟩∎
inj₂ b′ ∎
t∘f (inj₂ b) (inj₂ b′) | no b≢b′ =
t (inj₂ b) (inj₂ (_↔_.from A↔B b′)) ≡⟨⟩
if⌊ inj₂ (_↔_.to A↔B (_↔_.from A↔B b′)) ≟ inj₂ b ⌋then inj₁ tt
else inj₂ (_↔_.to A↔B (_↔_.from A↔B b′)) ≡⟨ cong (λ b′ → if⌊ inj₂ b′ ≟ inj₂ b ⌋then inj₁ tt else inj₂ b′) $
_↔_.right-inverse-of A↔B _ ⟩
if⌊ inj₂ b′ ≟ inj₂ b ⌋then inj₁ tt else inj₂ b′ ≡⟨ if-not (inj₂ b′ ≟ inj₂ b) (inj₁ tt) _ (b≢b′ ∘ sym ∘ ⊎.cancel-inj₂) ⟩∎
inj₂ b′ ∎
f∘t : ∀ ⊤⊎B x → f ⊤⊎B (t ⊤⊎B x) ≡ x
f∘t (inj₁ tt) (inj₁ tt) = refl _
f∘t (inj₁ tt) (inj₂ a) = inj₂ (_↔_.from A↔B (_↔_.to A↔B a)) ≡⟨ cong inj₂ $ _↔_.left-inverse-of A↔B _ ⟩∎
inj₂ a ∎
f∘t (inj₂ b) (inj₁ tt) with b ≟B b
f∘t (inj₂ b) (inj₁ tt) | yes _ = refl _
f∘t (inj₂ b) (inj₁ tt) | no b≢b = ⊥-elim $ b≢b (refl _)
f∘t (inj₂ b) (inj₂ a) with _↔_.to A↔B a ≟B b
f∘t (inj₂ b) (inj₂ a) | yes to-a≡b = inj₂ (_↔_.from A↔B b) ≡⟨ cong inj₂ $ _↔_.to-from A↔B to-a≡b ⟩∎
inj₂ a ∎
f∘t (inj₂ b) (inj₂ a) | no to-a≢b with b ≟B _↔_.to A↔B a
f∘t (inj₂ b) (inj₂ a) | no to-a≢b | yes b≡to-a = ⊥-elim $ to-a≢b
(_↔_.to A↔B a ≡⟨ sym b≡to-a ⟩∎
b ∎)
f∘t (inj₂ b) (inj₂ a) | no to-a≢b | no b≢to-a =
inj₂ (_↔_.from A↔B (_↔_.to A↔B a)) ≡⟨ cong inj₂ $ _↔_.left-inverse-of A↔B _ ⟩∎
inj₂ a ∎
to∘from : ∀ x → to (from x) ≡ x
to∘from (⊤⊎B , A↔B) =
cong (⊤⊎B ,_) (_↔_.to (↔-to-≡↔≡ ext A-set) (lemma ⊤⊎B))
where
A-set : Is-set A
A-set = $⟨ _≟B_ ⟩
Decidable-equality B ↝⟨ decidable⇒set ⟩
Is-set B ↝⟨ H-level.respects-surjection
(_↔_.surjection $ inverse A↔B) 2 ⟩□
Is-set A □
lemma :
∀ ⊤⊎B a →
_↔_.to (⊎-left-cancellative-⊤ (from (⊤⊎B , A↔B))) a ≡ _↔_.to A↔B a
lemma (inj₁ tt) a = refl _
lemma (inj₂ b) a with inspect (_↔_.to (from (inj₂ b , A↔B))
(inj₂ a))
lemma (inj₂ b) a | (inj₁ tt , eq) with _↔_.to A↔B a ≟B b
lemma (inj₂ b) a | (inj₁ tt , eq) | yes to-a≡b = sym to-a≡b
lemma (inj₂ b) a | (inj₁ tt , eq) | no _ = ⊥-elim $ ⊎.inj₁≢inj₂ $ sym eq
lemma (inj₂ b) a | (inj₂ _ , eq) with _↔_.to A↔B a ≟B b
lemma (inj₂ b) a | (inj₂ _ , eq) | yes _ = ⊥-elim $ ⊎.inj₁≢inj₂ eq
lemma (inj₂ b) a | (inj₂ _ , eq) | no _ = ⊎.cancel-inj₂ $ sym eq
from∘to : ∀ x → from (to x) ≡ x
from∘to ⊤⊎A↔⊤⊎B = _↔_.to (↔-to-≡↔≡ ext ⊤⊎A-set) lemma₁
where
open ⊎-left-cancellative
⊤⊎A-set : Is-set (⊤ ⊎ A)
⊤⊎A-set = $⟨ _≟B_ ⟩
Decidable-equality B ↝⟨ decidable⇒set ⟩
Is-set B ↝⟨ ⊎-closure 0 (mono (zero≤ 2) ⊤-contractible) ⟩
Is-set (⊤ ⊎ B) ↝⟨ H-level.respects-surjection
(_↔_.surjection $ inverse ⊤⊎A↔⊤⊎B) 2 ⟩□
Is-set (⊤ ⊎ A) □
mutual
lemma₁ : ∀ ⊤⊎A →
_↔_.to (from (to ⊤⊎A↔⊤⊎B)) ⊤⊎A ≡ _↔_.to ⊤⊎A↔⊤⊎B ⊤⊎A
lemma₁ (inj₁ tt) = refl _
lemma₁ (inj₂ a) = lemma₂ (inspect _) (inspect _)
lemma₂ :
∀ {a} {wb : Well-behaved (_↔_.to ⊤⊎A↔⊤⊎B)}
(x : Other-singleton (_↔_.to ⊤⊎A↔⊤⊎B (inj₂ a)))
(y : Other-singleton (_↔_.to ⊤⊎A↔⊤⊎B (inj₁ tt))) →
let b = g′ (_↔_.to ⊤⊎A↔⊤⊎B) wb x in
if⌊ inj₂ b ≟ proj₁ y ⌋then inj₁ tt else inj₂ b ≡ proj₁ x
lemma₂ {a} (inj₁ tt , eq₁) (inj₁ tt , eq₂) = ⊥-elim $ ⊎.inj₁≢inj₂ (
inj₁ tt ≡⟨ sym $ _↔_.left-inverse-of ⊤⊎A↔⊤⊎B _ ⟩
_↔_.from ⊤⊎A↔⊤⊎B (_↔_.to ⊤⊎A↔⊤⊎B (inj₁ tt)) ≡⟨ cong (_↔_.from ⊤⊎A↔⊤⊎B) eq₂ ⟩
_↔_.from ⊤⊎A↔⊤⊎B (inj₁ tt) ≡⟨ cong (_↔_.from ⊤⊎A↔⊤⊎B) $ sym eq₁ ⟩
_↔_.from ⊤⊎A↔⊤⊎B (_↔_.to ⊤⊎A↔⊤⊎B (inj₂ a)) ≡⟨ _↔_.left-inverse-of ⊤⊎A↔⊤⊎B _ ⟩∎
inj₂ a ∎)
lemma₂ (inj₁ tt , eq₁) (inj₂ b′ , eq₂) = lemma₃ eq₁ (inspect _) eq₂ (inj₂ _ ≟ inj₂ b′)
lemma₂ (inj₂ b , eq₁) (inj₁ tt , eq₂) = refl _
lemma₂ (inj₂ b , eq₁) (inj₂ b′ , eq₂) with b ≟B b′
lemma₂ (inj₂ b , eq₁) (inj₂ b′ , eq₂) | no _ = refl _
lemma₂ {a} (inj₂ b , eq₁) (inj₂ b′ , eq₂) | yes b≡b′ =
⊥-elim $ ⊎.inj₁≢inj₂ (
inj₁ tt ≡⟨ sym $ _↔_.left-inverse-of ⊤⊎A↔⊤⊎B _ ⟩
_↔_.from ⊤⊎A↔⊤⊎B (_↔_.to ⊤⊎A↔⊤⊎B (inj₁ tt)) ≡⟨ cong (_↔_.from ⊤⊎A↔⊤⊎B) eq₂ ⟩
_↔_.from ⊤⊎A↔⊤⊎B (inj₂ b′) ≡⟨ cong (_↔_.from ⊤⊎A↔⊤⊎B ∘ inj₂) $ sym b≡b′ ⟩
_↔_.from ⊤⊎A↔⊤⊎B (inj₂ b) ≡⟨ cong (_↔_.from ⊤⊎A↔⊤⊎B) $ sym eq₁ ⟩
_↔_.from ⊤⊎A↔⊤⊎B (_↔_.to ⊤⊎A↔⊤⊎B (inj₂ a)) ≡⟨ _↔_.left-inverse-of ⊤⊎A↔⊤⊎B _ ⟩∎
inj₂ a ∎)
lemma₃ :
∀ {a b′} {wb : Well-behaved (_↔_.to ⊤⊎A↔⊤⊎B)}
(eq : _↔_.to ⊤⊎A↔⊤⊎B (inj₂ a) ≡ inj₁ tt)
(x : Other-singleton (_↔_.to ⊤⊎A↔⊤⊎B (inj₁ tt))) →
proj₁ x ≡ inj₂ b′ →
let b = g″ (_↔_.to ⊤⊎A↔⊤⊎B) wb eq x in
(d : Dec (inj₂ {A = ⊤} b ≡ inj₂ b′)) →
if⌊ d ⌋then inj₁ tt else inj₂ b ≡ inj₁ tt
lemma₃ eq₁ (inj₁ _ , eq₂) eq₃ _ = ⊥-elim $ ⊎.inj₁≢inj₂ eq₃
lemma₃ eq₁ (inj₂ b″ , eq₂) eq₃ (yes b″≡b′) = refl _
lemma₃ eq₁ (inj₂ b″ , eq₂) eq₃ (no b″≢b′) = ⊥-elim $ b″≢b′ eq₃