diff --git a/1FundamentalGroup/Quest3Solutions.agda b/1FundamentalGroup/Quest3Solutions.agda index 423f14a..21d7532 100644 --- a/1FundamentalGroup/Quest3Solutions.agda +++ b/1FundamentalGroup/Quest3Solutions.agda @@ -19,68 +19,6 @@ pathToFun→ A B f = -{- - -private - variable - ℓ : Level - -transport→ : {A B : I → Type ℓ} (f : A i0 → B i0) (x : A i0) → - transport (λ i → A i → B i) f (transport (λ i → A i) x) ≡ transport (λ i → B i) (f x) -transport→ {A = A} {B = B} f x = - J - (λ A1 A → - transport (λ i → A i → B i) f (transport (λ i → A i) x) ≡ transport (λ i → B i) (f x)) - (J - (λ B1 B → - transport (λ i → A i0 → B i) f (transport (λ i → A i0) x) ≡ transport (λ i → B i) (f x)) - ( - transport (λ i → A i0 → B i0) f (transport (λ i → A i0) x) - ≡⟨ cong (transport (λ i → A i0 → B i0) f) (transportRefl x) ⟩ - transport (λ i → A i0 → B i0) f x - ≡⟨ cong (λ g → g x) (transportRefl f) ⟩ - f x - ≡⟨ sym (transportRefl (f x)) ⟩ - transport (λ i → B i0) (f x) ∎ - ) - λ i → B i - ) - λ i → A i - -- J (λ A1 A → transport (λ i → A i → B i) f ≡ λ a1 → transport B (f (transport (sym A) a1))) - -- (funExt (λ a1 → refl)) A - -transport→' : {A0 A1 B0 B1 : Type} {A : A0 ≡ A1} {B : B0 ≡ B1} (f : A0 → B0) → - transport (λ i → A i → B i) f ≡ λ a1 → transport B (f (transport (sym A) a1)) -transport→' {A = A} {B = B} f = - J (λ A1 A → transport (λ i → A i → B i) f ≡ λ a1 → transport B (f (transport (sym A) a1))) - refl A - -pathToFun→ : {A0 A1 B0 B1 : Type} {A : A0 ≡ A1} {B : B0 ≡ B1} (f : A0 → B0) → - pathToFun (λ i → A i → B i) f ≡ λ a1 → pathToFun B (f (pathToFun (sym A) a1)) -pathToFun→ {A = A} {B = B} f = - J (λ A1 A → pathToFun (λ i → A i → B i) f ≡ λ a1 → pathToFun B (f (pathToFun (sym A) a1))) - refl A - -pathToFun→' : {A0 A1 B0 B1 : Type} {A : A0 ≡ A1} {B : B0 ≡ B1} (f : A0 → B0) → - pathToFun (λ i → A i → B i) f ≡ λ a1 → pathToFun B (f (pathToFun (sym A) a1)) -pathToFun→' {A0} {A1} {B0} {B1} {A} {B} f = - J (λ A1 A → pathToFun (λ i → A i → B i) f ≡ λ a1 → pathToFun B (f (pathToFun (sym A) a1))) - ( - J (λ B1 B → pathToFun (λ i → A0 → B i) f ≡ λ a → pathToFun B (f (pathToFun refl a))) - ( - funExt λ a → - pathToFun refl f a - ≡⟨ cong (λ g → g a) (pathToFunRefl f) ⟩ - f a - ≡⟨ sym (pathToFunRefl (f a)) ⟩ - pathToFun refl (f a) - ≡⟨ cong (λ x → pathToFun refl (f x)) (sym (pathToFunRefl a)) ⟩ - pathToFun refl (f (pathToFun refl a)) ∎ - ) - B - ) - A -} - open ℤ loopSucℤtimes : (n : ℤ) → loop n times ∙ loop ≡ loop sucℤ n times