From eb5587b25f80d277eae2c17aa6b9f44ed73b2c2d Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Mon, 1 Nov 2021 00:07:28 +0000 Subject: [PATCH] added arc0/quest5 and arc1/quest3 --- 0Trinitarianism/Quest4Solutions.agda | 4 +- 0Trinitarianism/Quest5Solutions.agda | 50 ++++++++++++++++++ 1FundamentalGroup/Quest3.agda | 1 + 1FundamentalGroup/Quest3Solutions.agda | 14 ++++- .../0Trinitarianism/Quest5Solutions.agdai | Bin 0 -> 64064 bytes .../2.6.2/agda/1FundamentalGroup/Quest3.agdai | Bin 0 -> 2334 bytes 6 files changed, 66 insertions(+), 3 deletions(-) create mode 100644 0Trinitarianism/Quest5Solutions.agda create mode 100644 1FundamentalGroup/Quest3.agda create mode 100644 _build/2.6.2/agda/0Trinitarianism/Quest5Solutions.agdai create mode 100644 _build/2.6.2/agda/1FundamentalGroup/Quest3.agdai diff --git a/0Trinitarianism/Quest4Solutions.agda b/0Trinitarianism/Quest4Solutions.agda index 7991533..b51fe1b 100644 --- a/0Trinitarianism/Quest4Solutions.agda +++ b/0Trinitarianism/Quest4Solutions.agda @@ -104,10 +104,10 @@ Trans {x = x} {z = z} = J (λ y1 p → y1 ≡ z → x ≡ z) λ q → q _∙_ = Trans -- input \. -_≡⟨_⟩_ : (x : A) → x ≡ y → y ≡ z → x ≡ z +_≡⟨_⟩_ : (x : A) → x ≡ y → y ≡ z → x ≡ z -- input \< and \> _ ≡⟨ x≡y ⟩ y≡z = x≡y ∙ y≡z -_∎ : (x : A) → x ≡ x +_∎ : (x : A) → x ≡ x -- input \qed _ ∎ = refl infixr 30 _∙_ diff --git a/0Trinitarianism/Quest5Solutions.agda b/0Trinitarianism/Quest5Solutions.agda new file mode 100644 index 0000000..682d74d --- /dev/null +++ b/0Trinitarianism/Quest5Solutions.agda @@ -0,0 +1,50 @@ +module 0Trinitarianism.Quest5Solutions where + +open import Cubical.Foundations.Prelude renaming (subst to endPt ; transport to pathToFun) +open import Cubical.HITs.S1 using ( S¹ ; base ; loop ) +open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) +open import Cubical.Foundations.Path +open import 1FundamentalGroup.Quest0Solutions +open import Cubical.Data.Bool + +PathD : {A0 A1 : Type} (A : A0 ≡ A1) (x : A0) (y : A1) → Type +PathD A x y = pathToFun A x ≡ y + +syntax PathD A x y = x ≡ y along A + +outOfS¹P : (B : S¹ → Type) → (b : B base) → PathP (λ i → B (loop i)) b b → (x : S¹) → B x +outOfS¹P B b p base = b +outOfS¹P B b p (loop i) = p i + +outOfS¹D : (B : S¹ → Type) → (b : B base) → b ≡ b along (λ i → B (loop i)) → (x : S¹) → B x +outOfS¹D B b p x = outOfS¹P B b (_≅_.inv (PathPIsoPath (λ i → B (loop i)) b b) p) x + +-- example : (x : S¹) → doubleCover x → Bool +-- example = outOfS¹D (λ x → doubleCover x → Bool) (λ x → true) +-- (pathToFun (λ i → (doubleCover (loop i)) → Bool) (λ x → true) +-- ≡⟨ refl ⟩ +-- pathToFun (λ i → flipPath i → Bool) (λ x → true) +-- ≡⟨ refl ⟩ +-- (λ x → true) +-- ∎) + +example : (x : S¹) → doubleCover x → doubleCover x +example base = Flip +example (loop i) = p i where + + p : PathP (λ i → doubleCover (loop i) → doubleCover (loop i)) Flip Flip + p = {!!} + +example' : (x : S¹) → doubleCover x → doubleCover x +example' = outOfS¹D (λ x → doubleCover x → doubleCover x) Flip + ( + pathToFun (λ i → doubleCover (loop i) → doubleCover (loop i)) Flip + ≡⟨ refl ⟩ + pathToFun (λ i → flipPath i → flipPath i) Flip + ≡⟨ funExt lem ⟩ + Flip + ∎) where + + lem : (x : Bool) → pathToFun (λ i → flipPath i → flipPath i) Flip x ≡ Flip x + lem false = refl + lem true = refl diff --git a/1FundamentalGroup/Quest3.agda b/1FundamentalGroup/Quest3.agda new file mode 100644 index 0000000..15d5b14 --- /dev/null +++ b/1FundamentalGroup/Quest3.agda @@ -0,0 +1 @@ +module 1FundamentalGroup.Quest3 where diff --git a/1FundamentalGroup/Quest3Solutions.agda b/1FundamentalGroup/Quest3Solutions.agda index b999030..849ce06 100644 --- a/1FundamentalGroup/Quest3Solutions.agda +++ b/1FundamentalGroup/Quest3Solutions.agda @@ -1,9 +1,21 @@ module 1FundamentalGroup.Quest3Solutions where open import Cubical.Foundations.Prelude -open import Cubical.HITs.S1 +open import Cubical.HITs.S1 using ( S¹ ; base ; loop ) open import 1FundamentalGroup.Quest1Solutions +open import Cubical.Foundations.Path outOfS¹ : (B : S¹ → Type) → (b : B base) → PathP (λ i → B (loop i)) b b → (x : S¹) → B x outOfS¹ B b p base = b outOfS¹ B b p (loop i) = p i + + + +rewind : (x : S¹) → helix x → base ≡ x +rewind base = loop_times +rewind (loop i) = path i where + + path : PathP (λ i → helix (loop i) → base ≡ loop i) loop_times loop_times + path = {!!} + +thing = PathPIsoPath diff --git a/_build/2.6.2/agda/0Trinitarianism/Quest5Solutions.agdai b/_build/2.6.2/agda/0Trinitarianism/Quest5Solutions.agdai new file mode 100644 index 0000000000000000000000000000000000000000..345b18712d436124b8706b644a712035ed5e5a60 GIT binary patch literal 64064 zcmXVY3p|wh_y2R5o9m4Gh+#xgx=dHM@K8dDiQkr8%9Jj;#}*(*9N&9^L}!ltRAY!t%(e8%p)UfVSDJm)!|bKd8D&gXp26cBx^C$Qs>XImcZ zO!eC~?+6C}>Fd4Kw>`u`Z!reRxUZvd{Ce8^^Q-qmL&K>v{#E^7*rFMWUj6%*yzYGT z`a9fXC#Nraet5yeg1A3*Z~I>PvpV$Tne8JbC2rL}{qoDJ^|mWBcWe5G^yBo+4%#gp zd39)Y6mh)S5R0LGG2B#!>l*MVavN0;jd3JcC+%Cq<0-5uwyFC+7?Hs|G$Y!3FS9_5~UXW^LORw9*RQYliOKG>yr5GZQ_tjAU$e 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