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Prove identity law for coercions.

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Frederik Hanghøj Iversen 2018-03-20 12:12:09 +01:00
parent 30725d71b6
commit 2188e690a0
1 changed files with 2 additions and 11 deletions
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13
src/Cat/Categories/Sets.agda
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@ -28,21 +28,12 @@ sym≃ = Equivalence.symmetry
infixl 10 _⊙_ infixl 10 _⊙_
module _ {ℓ : Level} {A : Set ℓ} {a : A} where module _ {ℓ : Level} {A : Set ℓ} {a : A} where
-- "This follows from the propositional equiality for `J`" - Vezzosi.
id-coe : coe refl a ≡ a id-coe : coe refl a ≡ a
id-coe = begin id-coe = begin
coe refl a ≡⟨⟩ coe refl a ≡⟨⟩
-- pathJprop : pathJ _ refl ≡ d pathJ (λ y x → A) _ A refl ≡⟨ pathJprop {x = a} (λ y x → A) _ ⟩
pathJ (λ y x → A) scary A refl ≡⟨ pathJprop {x = scary} (λ y x → A) scary ⟩ _ ≡⟨ pathJprop {x = a} (λ y x → A) _ ⟩
scary ≡⟨ {!!} ⟩
-- pathJ (λ y x → A) scary A refl ≡⟨ pathJprop {A = A} (λ y x → A) scary ⟩
-- scary ≡⟨ {!!} ⟩
-- pathJ (λ y _ → B _ → B y) (λ x → x) _ p ≡⟨ {!!} ⟩
a ∎ a ∎
where
scary : A
scary = (primComp (λ j → A) i0 (λ j → p[ a ] i0 (~ j)) a)
module _ {ℓ : Level} {A B : Set ℓ} {a : A} where module _ {ℓ : Level} {A B : Set ℓ} {a : A} where
inv-coe : (p : A ≡ B) → coe (sym p) (coe p a) ≡ a inv-coe : (p : A ≡ B) → coe (sym p) (coe p a) ≡ a