From 4d5a2ded5a8bc01c4b72fddc2ea05001f1637f44 Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Mon, 25 Oct 2021 23:16:53 +0100 Subject: [PATCH] added cheating (cubical) version of funExt --- 0Trinitarianism/Quest4Solutions.agda | 21 +++++++++++++++--- .../0Trinitarianism/Quest4Solutions.agdai | Bin 94718 -> 151976 bytes 2 files changed, 18 insertions(+), 3 deletions(-) diff --git a/0Trinitarianism/Quest4Solutions.agda b/0Trinitarianism/Quest4Solutions.agda index e2557ce..020e198 100644 --- a/0Trinitarianism/Quest4Solutions.agda +++ b/0Trinitarianism/Quest4Solutions.agda @@ -187,7 +187,21 @@ endPt' B p = pathToFun (Cubical.Foundations.Prelude.cong B p ) --------------funExt--------------------- +funExt : (B : A → Type) (f g : (a : A) → B a) → + ((a : A) → f a ≡ g a) → f ≡ g +funExt B f g h = λ i a → h a i +funExtPath : (B : A → Type) (f g : (a : A) → B a) → (f ≡ g) ≡ ((a : A) → f a ≡ g a) +funExtPath {A} B f g = isoToPath (iso fun (funExt B f g) rightInv leftInv) where + + fun : f ≡ g → (a : A) → f a ≡ g a + fun h = λ a i → h i a + + rightInv : section fun (funExt B f g) + rightInv h = refl + + leftInv : retract fun (funExt B f g) + leftInv h = refl --------------Path on Products----------- @@ -218,7 +232,8 @@ snd : A × B → B snd (a , b) = b Path× : {A B : Type} (x y : A × B) → (x ≡ y) ≡ ((fst x ≡ fst y) × (snd x ≡ snd y)) -Path× {A} {B} (a0 , b0) (a1 , b1) = isoToPath (iso fun (inv a0 a1 b0 b1) rightInv leftinv) where +Path× {A} {B} (a0 , b0) (a1 , b1) = + isoToPath (iso fun (inv a0 a1 b0 b1) rightInv leftInv) where fun : {x y : A × B} → x ≡ y → (fst x ≡ fst y) × (snd x ≡ snd y) fun {x} {y} = J (λ y p → (fst x ≡ fst y) × (snd x ≡ snd y)) (refl , refl) @@ -278,5 +293,5 @@ Path× {A} {B} (a0 , b0) (a1 , b1) = isoToPath (iso fun (inv a0 a1 b0 b1) rightI ≡⟨ InvRefl ⟩ refl ∎) - leftinv : {x y : A × B} → retract fun (inv (fst x) (fst y) (snd x) (snd y)) - leftinv {a0 , b0} {a1 , b1} = leftInv + leftinv : (x y : A × B) → retract fun (inv (fst x) (fst y) (snd x) (snd y)) + leftinv 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