From 81dbe9004cfbb95a4051a43ea7dc6b81d089618f Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Sat, 13 Nov 2021 00:42:58 +0000 Subject: [PATCH] finished rewind --- 1FundamentalGroup/Quest3Solutions.agda | 122 +++++++++++++++++- .../1FundamentalGroup/Quest3Solutions.agdai | Bin 39214 -> 89830 bytes 2 files changed, 120 insertions(+), 2 deletions(-) diff --git a/1FundamentalGroup/Quest3Solutions.agda b/1FundamentalGroup/Quest3Solutions.agda index 8f4be97..423f14a 100644 --- a/1FundamentalGroup/Quest3Solutions.agda +++ b/1FundamentalGroup/Quest3Solutions.agda @@ -2,14 +2,132 @@ module 1FundamentalGroup.Quest3Solutions where open import Cubical.HITs.S1 using ( S¹ ; base ; loop ) open import 1FundamentalGroup.Quest1Solutions -open import Cubical.Foundations.Prelude renaming (transport to pathToFun) +open import Cubical.Foundations.Prelude + renaming (transport to pathToFun ; transportRefl to pathToFunRefl) +open import Cubical.Foundations.GroupoidLaws + renaming (lCancel to sym∙ ; rCancel to ∙sym ; lUnit to Refl∙ ; rUnit to ∙Refl) open import Cubical.Foundations.Path open import 0Trinitarianism.Quest5Solutions +open import Cubical.Data.Int using (ℤ) +open import Cubical.Data.Nat + +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 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 + + + +{- + +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 +loopSucℤtimes (pos n) = refl +loopSucℤtimes (negsuc zero) = sym∙ loop +loopSucℤtimes (negsuc (suc n)) = + (loop (negsuc n) times ∙ sym loop) ∙ loop + ≡⟨ sym (assoc _ _ _) ⟩ + loop (negsuc n) times ∙ (sym loop ∙ loop) + ≡⟨ cong (λ p → loop (negsuc n) times ∙ p) (sym∙ _) ⟩ + loop (negsuc n) times ∙ refl + ≡⟨ sym (∙Refl _) ⟩ + loop (negsuc n) times ∎ + +sucℤPredℤ : (n : ℤ) → sucℤ (predℤ n) ≡ n +sucℤPredℤ (pos zero) = refl +sucℤPredℤ (pos (suc n)) = refl +sucℤPredℤ (negsuc n) = refl + +pathToFunPathFibration : {A : Type} {x y z : A} (q : x ≡ y) (p : y ≡ z) → + pathToFun (λ i → x ≡ p i) q ≡ q ∙ p +pathToFunPathFibration {A} {x} {y} q = J (λ z p → pathToFun (λ i → x ≡ p i) q ≡ q ∙ p) + ( + pathToFun refl q + ≡⟨ pathToFunRefl q ⟩ + q + ≡⟨ ∙Refl q ⟩ + q ∙ refl ∎ + ) rewind : (x : S¹) → helix x → base ≡ x rewind = outOfS¹D (λ x → helix x → base ≡ x) loop_times ( pathToFun (λ i → sucℤPath i → base ≡ loop i) loop_times - ≡⟨ {!!} ⟩ + ≡⟨ pathToFun→ sucℤPath (λ i → base ≡ loop i) loop_times ⟩ -- how pathToFun interacts with → + (λ n → pathToFun (λ i → base ≡ loop i) (loop_times (pathToFun (sym sucℤPath) n))) + ≡⟨ refl ⟩ -- sucℤPath is just taking successor, and so its inverse is definitionally taking predecessor + (λ n → pathToFun (λ i → base ≡ loop i) (loop_times (predℤ n))) + ≡⟨ funExt (λ n → pathToFunPathFibration _ _) ⟩ -- how pathToFun interacts with the "path fibration" + (λ n → (loop (predℤ n) times) ∙ loop) + ≡⟨ funExt (λ n → + loop predℤ n times ∙ loop + ≡⟨ loopSucℤtimes (predℤ n) ⟩ + loop (sucℤ (predℤ n)) times + ≡⟨ cong loop_times (sucℤPredℤ n) ⟩ + loop n times ∎) ⟩ loop_times ∎ ) + + diff --git a/_build/2.6.2/agda/1FundamentalGroup/Quest3Solutions.agdai b/_build/2.6.2/agda/1FundamentalGroup/Quest3Solutions.agdai index bcd1697b8e8dd40b3fb85a96e293e17b7a30dd37..87c2ed446f15b72580f436b1ac907338fbcb988d 100644 GIT binary patch literal 89830 zcmXV22|SeR_kWk!Fc?c2`ztD12NhXgk|mYFz3oM*8|}$bgf|pQOr`Bs43Y7_E$U7i zWoePruc)~#RO-r2Q&O&_M);p++&iDoHJ|r==Dg=Q=X<{AJm-1doJXDfU*Oo2r+x3n zzjzY6b`1vq@wMLZE%))zSByonYBi+edTw~W{V?!j@KNF!$8Bd~8?w*bojLxmm?hKY z_VXUe=eA^OW{F=uKj_}@d+(Ua^%@6SX4juzUwx<5E?czs-zv`Ox%QcV{4B^wtJXE! zX^ly7O`IiWIJVPhOuG@^onud95sZ#)3?IX@Nf&9Oy$&zjmF60{jK+yY{0_$ZUL1ps z!?nf7Fjl!c{Cppdf96FpxAbYul{ArxGuW{Ur#XAx)mSSdMwb7q(-dhn}Zk9+8&Ss5`7mfNjtgkdZ>)r|>$=bq_ 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