{-# OPTIONS --cubical-compatible --safe #-}
open import Equality
module Equality.Decidable-UIP
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where
open Derived-definitions-and-properties eq
open import Logical-equivalence using (module _⇔_)
open import H-level eq
open import Prelude
Constant : ∀ {a b} {A : Type a} {B : Type b} → (A → B) → Type (a ⊔ b)
Constant f = ∀ x y → f x ≡ f y
_Left-inverse-of_ : ∀ {a b} {A : Type a} {B : Type b} →
(B → A) → (A → B) → Type a
g Left-inverse-of f = ∀ x → g (f x) ≡ x
abstract
proposition : ∀ {a} {A : Type a} →
(f : ∃ λ (f : A → A) → Constant f) →
(∃ λ g → g Left-inverse-of (proj₁ f)) →
Is-proposition A
proposition (f , constant) (g , left-inverse) x y =
x ≡⟨ sym (left-inverse x) ⟩
g (f x) ≡⟨ cong g (constant x y) ⟩
g (f y) ≡⟨ left-inverse y ⟩∎
y ∎
left-inverse :
∀ {a} {A : Type a} (f : (x y : A) → x ≡ y → x ≡ y) →
∀ {x y} → ∃ λ g → g Left-inverse-of f x y
left-inverse {A = A} f {x} {y} =
(λ x≡y →
x ≡⟨ x≡y ⟩
y ≡⟨ sym (f y y (refl y)) ⟩∎
y ∎) ,
elim (λ {x y} x≡y → trans (f x y x≡y) (sym (f y y (refl y))) ≡ x≡y)
(λ _ → trans-symʳ _)
constant⇒set :
∀ {a} {A : Type a} →
((x y : A) → ∃ λ (f : x ≡ y → x ≡ y) → Constant f) →
Is-set A
constant⇒set constant {x} {y} =
proposition (constant x y)
(left-inverse (λ x y → proj₁ $ constant x y))
decidable⇒constant : ∀ {a} {A : Type a} → Dec A →
∃ λ (f : A → A) → Constant f
decidable⇒constant (yes x) = (const x , λ _ _ → refl x)
decidable⇒constant (no ¬x) = (id , λ _ → ⊥-elim ∘ ¬x)
decidable⇒set : ∀ {a} {A : Type a} → Decidable-equality A → Is-set A
decidable⇒set dec =
constant⇒set (λ x y → decidable⇒constant (dec x y))
propositional-domain⇒constant :
∀ {a b} {A : Type a} {B : Type b} →
Is-proposition A → (f : A → B) → Constant f
propositional-domain⇒constant A-prop f = λ x y →
cong f (A-prop x y)
propositional-identity⇒set :
∀ {a b} {A : Type a}
(B : A → A → Type b) →
(∀ x y → Is-proposition (B x y)) →
(∀ x → B x x) →
(∀ x y → B x y → x ≡ y) →
Is-set A
propositional-identity⇒set B B-prop B-refl f =
constant⇒set λ x y →
(λ eq → f x y (subst (B x) eq (B-refl x))) ,
(λ _ _ → propositional-domain⇒constant (B-prop x y) (f x y) _ _)
cong-preserves-Constant :
∀ {a b} {A : Type a} {B : Type b} {f : A → B} {x y : A} →
Constant f → Constant (cong {x = x} {y = y} f)
cong-preserves-Constant {f = f} {x = x} {y = y} c p q =
cong f p ≡⟨ lemma p ⟩
trans (sym (c x x)) (c x y) ≡⟨ sym (lemma q) ⟩∎
cong f q ∎
where
lemma : ∀ p → cong {x = x} {y = y} f p ≡ trans (sym (c x x)) (c x y)
lemma = elim
(λ {x y} p → cong {x = x} {y = y} f p ≡ trans (sym (c x x)) (c x y))
(λ x →
cong f (refl x) ≡⟨ cong-refl _ ⟩
refl (f x) ≡⟨ sym $ trans-symˡ _ ⟩∎
trans (sym (c x x)) (c x x) ∎)
cong-constant :
∀ {a b} {A : Type a} {B : Type b} {f : A → B} {x} {x≡x : x ≡ x} →
Constant f →
cong f x≡x ≡ refl (f x)
cong-constant {f = f} {x} {x≡x} c =
cong f x≡x ≡⟨ cong-preserves-Constant c _ _ ⟩
cong f (refl x) ≡⟨ cong-refl _ ⟩∎
refl (f x) ∎
fixpoint-lemma :
∀ {a} {A : Type a} →
(f : A → A) →
Constant f →
Is-proposition (∃ λ x → f x ≡ x)
fixpoint-lemma f constant (x , fx≡x) (y , fy≡y) =
let x≡y = x ≡⟨ sym fx≡x ⟩
f x ≡⟨ constant x y ⟩
f y ≡⟨ fy≡y ⟩∎
y ∎
x≡x = x ≡⟨ sym fx≡x ⟩
f x ≡⟨ subst (λ z → f z ≡ z) (sym x≡y) fy≡y ⟩∎
x ∎
lemma =
subst (λ z → f z ≡ z) x≡x fx≡x ≡⟨ subst-in-terms-of-trans-and-cong ⟩
trans (sym (cong f x≡x)) (trans fx≡x (cong id x≡x)) ≡⟨ cong₂ (λ p q → trans (sym p) (trans _ q))
(cong-constant constant) (sym $ cong-id _) ⟩
trans (sym (refl (f x))) (trans fx≡x x≡x) ≡⟨ cong (λ p → trans p (trans fx≡x x≡x)) sym-refl ⟩
trans (refl (f x)) (trans fx≡x x≡x) ≡⟨ trans-reflˡ _ ⟩
trans fx≡x x≡x ≡⟨ sym $ trans-assoc _ _ _ ⟩
trans (trans fx≡x (sym fx≡x))
(subst (λ z → f z ≡ z) (sym x≡y) fy≡y) ≡⟨ cong (λ p → trans p (subst (λ z → f z ≡ z) (sym x≡y) fy≡y)) $
trans-symʳ _ ⟩
trans (refl (f x))
(subst (λ z → f z ≡ z) (sym x≡y) fy≡y) ≡⟨ trans-reflˡ _ ⟩∎
subst (λ z → f z ≡ z) (sym x≡y) fy≡y ∎
in
x , fx≡x ≡⟨ Σ-≡,≡→≡ x≡x lemma ⟩
x , subst (λ z → f z ≡ z) (sym x≡y) fy≡y ≡⟨ sym $ Σ-≡,≡→≡ (sym x≡y) (refl _) ⟩∎
y , fy≡y ∎