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July 13, 2022 09:11
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Direct proof for groupoidal structure of homotopic identity types via path induction in Agda
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| {-# OPTIONS --without-K #-} | |
| import Relation.Binary.PropositionalEquality as Eq | |
| open Eq using (_≡_; refl) | |
| pattern erefl x = refl {x = x} | |
| --- path induction --- | |
| J : ∀ {A : Set} → | |
| ∀ (C : ∀ (x y : A) → x ≡ y → Set) → | |
| (∀ (x : A) → C x x refl) → | |
| (∀ (x y : A) → ∀ (p : x ≡ y) → C x y p) | |
| J {A} C c x x refl = c x | |
| --- identity types form a groupoid --- | |
| _∙_ : ∀ {A : Set} {x y z : A} → | |
| x ≡ y → y ≡ z → x ≡ z | |
| _∙_ {_} {x} {y} {z} p = J (λ a b p → (b ≡ z → a ≡ z)) (λ a p → p) x y p | |
| _⁻¹ : ∀ {A : Set} {x y : A} → | |
| x ≡ y → y ≡ x | |
| _⁻¹ {_} {x} {y} p = J (λ a b _ → b ≡ a) (λ x → erefl x) x y p | |
| infixr 80 _∙_ | |
| infixr 100 _⁻¹ | |
| refl∙refl : ∀ {A : Set} {x : A} → | |
| (erefl x ∙ erefl x) ≡ erefl x | |
| refl∙refl {_} {x} = erefl (erefl x) | |
| ∙-refl : ∀ {A : Set} {x y : A} {p : x ≡ y} → | |
| p ∙ erefl y ≡ p | |
| ∙-refl {_} {x} {y} {p} = J (λ a b q → q ∙ erefl b ≡ q) (λ a → refl∙refl) x y p | |
| refl-∙ : ∀ {A : Set} {x y : A} {p : x ≡ y} → | |
| erefl x ∙ p ≡ p | |
| refl-∙ {_} {x} {y} {p} = J (λ a b q → erefl a ∙ q ≡ q) (λ a → refl∙refl) x y p | |
| ∙-⁻¹ : ∀ {A : Set} {x y : A} {p : x ≡ y} → | |
| p ∙ (p ⁻¹) ≡ erefl x | |
| ∙-⁻¹ {_} {x} {y} {p} = J (λ a b q → q ∙ (q ⁻¹) ≡ erefl a) (λ a → refl∙refl) x y p | |
| ⁻¹-∙ : ∀ {A : Set} {x y : A} {p : x ≡ y} → | |
| (p ⁻¹) ∙ p ≡ erefl y | |
| ⁻¹-∙ {_} {x} {y} {p} = J (λ a b q → (q ⁻¹) ∙ q ≡ erefl b) (λ a → refl∙refl) x y p | |
| ⁻¹⁻¹ : ∀ {A : Set} {x y : A} {p : x ≡ y} → | |
| (p ⁻¹) ⁻¹ ≡ p | |
| ⁻¹⁻¹ {_} {x} {y} {p} = J (λ a b q → (q ⁻¹) ⁻¹ ≡ q) (λ a → erefl (erefl a)) x y p | |
| ∙-assoc : ∀ {A : Set} {w x y z : A} (p : w ≡ x) (q : x ≡ y) (r : y ≡ z) → | |
| (p ∙ q) ∙ r ≡ p ∙ (q ∙ r) | |
| ∙-assoc {A} {w} {x} {y} {z} p q r = (J | |
| (λ w' x' p' → ∀ (y' z' : A) (q' : x' ≡ y') (r' : y' ≡ z') → ((p' ∙ q') ∙ r' ≡ p' ∙ (q' ∙ r'))) | |
| (λ w' y' z' q' r' → erefl (q' ∙ r')) w x p) y z q r |
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