From eab9a55318c96e0d690ba63af7a825febfed9825 Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Sun, 24 Oct 2021 22:42:28 +0100 Subject: [PATCH] =?UTF-8?q?Path=20type=20on=20=C3=97=20type=20is=20a=20pai?= =?UTF-8?q?n?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 0Trinitarianism/Quest4Solutions.agda | 155 ++++++++++++++---- .../0Trinitarianism/Quest4Solutions.agdai | Bin 86737 -> 94718 bytes 2 files changed, 121 insertions(+), 34 deletions(-) diff --git a/0Trinitarianism/Quest4Solutions.agda b/0Trinitarianism/Quest4Solutions.agda index 470809f..e2557ce 100644 --- a/0Trinitarianism/Quest4Solutions.agda +++ b/0Trinitarianism/Quest4Solutions.agda @@ -12,7 +12,7 @@ private u : Level -data Id {A : Type u} : (x y : A) → Type u where +data Id {A : Type} : (x y : A) → Type where rfl : {x : A} → Id x x @@ -31,26 +31,6 @@ p *0 rfl = p _*1_ : {A : Type} {x y z : A} → Id x y → Id y z → Id x z rfl *1 rfl = rfl -data _×_ (A B : Type) : Type where - - _,_ : A → B → A × B - --- id× : {A B : Type} (a0 a1 : A) (b0 b1 : B) → --- (Id a0 a1 × Id b0 b1) ≅ Id {A × B} ( a0 , b0 ) ( a1 , b1 ) --- id× {A} {B} a0 a1 b0 b1 = iso fun inv rightInv leftInv where - --- fun : Id a0 a1 × Id b0 b1 → Id {A × B} ( a0 , b0 ) ( a1 , b1 ) --- fun (rfl , rfl) = rfl - --- inv : Id {A × B} ( a0 , b0 ) ( a1 , b1 ) → Id a0 a1 × Id b0 b1 --- inv rfl = rfl , rfl - --- rightInv : section fun inv --- rightInv rfl = refl - --- leftInv : retract fun inv --- leftInv (rfl , rfl) = refl - ------------Cong------------------------- private @@ -79,8 +59,6 @@ Assoc rfl q r = rfl -------------Mapping Out---------------- -thing = JRefl - outOfId : (M : (y : A) → Id x y → Type) → M x rfl → {y : A} (p : Id x y) → M y p outOfId M h rfl = h @@ -118,17 +96,6 @@ Path≡Id {A} {x} {y} = isoToPath (iso Path→Id Id→Path rightInv leftInv) whe leftInv : retract (Path→Id {A} {x} {y}) Id→Path leftInv = J (λ y p → Id→Path (Path→Id p) ≡ p) (cong (λ p → Id→Path p) Path→IdRefl) - - -- leftInv : retract (Path→Id {A} {x} {y}) Id→Path - -- leftInv = J (λ y p → Id→Path (Path→Id p) ≡ p) - -- ( - -- Id→Path (Path→Id refl) - -- ≡⟨ cong (λ p → Id→Path p) Path→IdRefl ⟩ - -- Id→Path rfl - -- ≡⟨ refl ⟩ - -- refl ∎ - -- ) - -----------Basics about Path Space----------------- sym : {x y : A} → x ≡ y → y ≡ x @@ -193,3 +160,123 @@ assoc {A} = J (q ∙ r) ≡⟨ sym (refl∙ (q ∙ r)) ⟩ (refl ∙ (q ∙ r)) ∎ + +------------------Transport--------------------- + +id : A → A +id x = x + +pathToFun : A ≡ B → A → B +pathToFun {A} = J (λ B p → (A → B)) id + +pathToFunRefl : pathToFun (refl {x = A}) ≡ id +pathToFunRefl {A} = JRefl (λ B p → (A → B)) id + +pathToFunReflx : pathToFun (refl {x = A}) x ≡ x +pathToFunReflx {x = x} = cong (λ f → f x) pathToFunRefl + +endPt : (B : A → Type) (p : x ≡ y) → B x → B y +endPt {x = x} B = J (λ y p → B x → B y) id + +endPtRefl : (B : A → Type) → endPt B (refl {x = x}) ≡ id +endPtRefl {x = x} B = JRefl ((λ y p → B x → B y)) id + +endPt' : (B : A → Type) (p : x ≡ y) → B x → B y +endPt' B p = pathToFun (Cubical.Foundations.Prelude.cong B p ) + + +--------------funExt--------------------- + + + +--------------Path on Products----------- + +data _×_ (A B : Type) : Type where + + _,_ : A → B → A × B + +Id× : {A B : Type} (a0 a1 : A) (b0 b1 : B) + → (Id {A × B} ( a0 , b0 ) ( a1 , b1 )) ≡ (Id a0 a1 × Id b0 b1) +Id× {A} {B} a0 a1 b0 b1 = isoToPath (iso fun inv rightInv leftInv) where + + fun : Id {A = A × B} ( a0 , b0 ) ( a1 , b1 ) → Id a0 a1 × Id b0 b1 + fun rfl = rfl , rfl + + inv : Id a0 a1 × Id b0 b1 → Id {A = A × B} ( a0 , b0 ) ( a1 , b1 ) + inv (rfl , rfl) = rfl + + rightInv : section fun inv + rightInv (rfl , rfl) = refl + + leftInv : retract fun inv + leftInv rfl = refl + +fst : A × B → A +fst (a , b) = a + +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 + + 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) + + funRefl : (x : A × B) → fun (refl {x = x}) ≡ ( refl , refl ) + funRefl x = JRefl {_} {A × B} {x} + ((λ y p → (fst x ≡ fst y) × (snd x ≡ snd y))) ( refl , refl ) + + inv : (a0 a1 : A) (b0 b1 : B) → (a0 ≡ a1) × (b0 ≡ b1) + → _≡_ {A = A × B} ( a0 , b0 ) ( a1 , b1 ) + inv a0 a1 b0 b1 (p , q) = + J -- first induct on p + (λ y p → _≡_ {A = A × B} ( a0 , b0 ) ( y , b1 )) + ( + J -- then induct on q + (λ y q → _≡_ {A = A × B} ( a0 , b0 ) ( a0 , y )) + refl + q + ) p + + invRefl : (a : A) (b : B) → inv a a b b (refl , refl) ≡ refl + invRefl a b = + inv a a b b (refl , refl) + ≡⟨ JRefl (λ y p → (a , b) ≡ (y , b)) (J (λ y q → (a , b) ≡ (a , y)) refl refl) ⟩ + J (λ y q → _≡_ {A = A × B} ( a , b ) ( a , y )) refl refl + ≡⟨ JRefl ((λ y q → _≡_ {A = A × B} ( a , b ) ( a , y ))) refl ⟩ + refl ∎ + + Inv : (x y : A × B) → (fst x ≡ fst y) × (snd x ≡ snd y) → x ≡ y + Inv (a0 , b0) (a1 , b1) = inv a0 a1 b0 b1 + + InvRefl : {x : A × B} → Inv x x (refl , refl) ≡ refl {x = x} + InvRefl {a , b} = invRefl a b + + rightInv : section fun (inv a0 a1 b0 b1) + rightInv (p , q) = + J -- first induct on p + (λ y p → fun (inv a0 y b0 b1 (p , q)) ≡ (p , q)) + ( + J -- then induct on q + (λ y q → fun (inv a0 a0 b0 y (refl , q)) ≡ (refl , q)) + ( + fun (inv a0 a0 b0 b0 (refl , refl)) + ≡⟨ cong fun (invRefl a0 b0) ⟩ + fun (refl {x = (a0 , b0)}) + ≡⟨ funRefl (a0 , b0) ⟩ + (refl , refl) ∎) + q) + p + + leftInv : {x y : A × B} → retract fun (Inv x y) + leftInv {x} = J (λ y p → Inv x y (fun p) ≡ p) + ( + Inv x x (fun {x = x} refl) + ≡⟨ cong (Inv x x) (funRefl x) ⟩ + Inv x x (refl , refl) + ≡⟨ InvRefl ⟩ + refl ∎) + + leftinv : {x y : A × B} → retract fun (inv (fst x) (fst y) (snd x) (snd y)) + leftinv {a0 , b0} {a1 , b1} = leftInv diff --git a/_build/2.6.2/agda/0Trinitarianism/Quest4Solutions.agdai b/_build/2.6.2/agda/0Trinitarianism/Quest4Solutions.agdai index 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