From ff2594dd5e8a7a55b6baf7759b7395d80ab2c54c Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Sat, 13 Nov 2021 23:18:44 +0000 Subject: [PATCH] added quest5 solutions --- 0Trinitarianism/Quest5Solutions.agda | 8 +- 1FundamentalGroup/Preambles/P3.agda | 20 +++++ 1FundamentalGroup/Quest3Solutions.agda | 83 +++++++++++++----- .../0Trinitarianism/Quest4Solutions.agdai | Bin 139857 -> 139967 bytes .../0Trinitarianism/Quest5Solutions.agdai | Bin 67287 -> 69596 bytes .../agda/1FundamentalGroup/Preambles/P3.agdai | Bin 0 -> 66309 bytes .../1FundamentalGroup/Quest3Solutions.agdai | Bin 89830 -> 94549 bytes 7 files changed, 88 insertions(+), 23 deletions(-) create mode 100644 1FundamentalGroup/Preambles/P3.agda create mode 100644 _build/2.6.2/agda/1FundamentalGroup/Preambles/P3.agdai diff --git a/0Trinitarianism/Quest5Solutions.agda b/0Trinitarianism/Quest5Solutions.agda index 6af97f0..d309e33 100644 --- a/0Trinitarianism/Quest5Solutions.agda +++ b/0Trinitarianism/Quest5Solutions.agda @@ -12,11 +12,11 @@ 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 : 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 : 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 (PathPIsoPathD (λ i → B (loop i)) b b) p) x example : (x : S¹) → doubleCover x → doubleCover x @@ -38,3 +38,7 @@ example' = outOfS¹D (λ x → doubleCover x → doubleCover x) Flip (funExt lem lem : (x : Bool) → pathToFun (λ i → flipPath i → flipPath i) Flip x ≡ Flip x lem false = refl lem true = refl + +outOfS¹DBase : (B : S¹ → Type) (b : B base) + (p : b ≡ b along (λ i → B (loop i)))→ outOfS¹D B b p base ≡ b +outOfS¹DBase B b p = refl diff --git a/1FundamentalGroup/Preambles/P3.agda b/1FundamentalGroup/Preambles/P3.agda new file mode 100644 index 0000000..de086cd --- /dev/null +++ b/1FundamentalGroup/Preambles/P3.agda @@ -0,0 +1,20 @@ +module 1FundamentalGroup.Preambles.P3 where + +open import Cubical.Foundations.Prelude public + renaming (transport to pathToFun ; + transportRefl to pathToFunRefl ; + subst to endPt) public +open import Cubical.Foundations.Isomorphism public +open import Cubical.Foundations.GroupoidLaws + renaming (lCancel to sym∙ ; rCancel to ∙sym ; lUnit to Refl∙ ; rUnit to ∙Refl) public +open import Cubical.Foundations.Path public +open import Cubical.Data.Int using (ℤ ; isSetℤ) public +open import Cubical.Data.Nat public +open import Cubical.HITs.S1 using ( S¹ ; base ; loop ) public +open import 0Trinitarianism.Quest5Solutions public +open import 1FundamentalGroup.Quest1Solutions public + +open ℤ public + +endPtRefl : {A : Type} {x : A} (B : A → Type) → endPt B (refl {x = x}) ≡ λ b → b +endPtRefl {x = x} B = funExt (λ b → substRefl {B = B} b) diff --git a/1FundamentalGroup/Quest3Solutions.agda b/1FundamentalGroup/Quest3Solutions.agda index 21d7532..78ec3b8 100644 --- a/1FundamentalGroup/Quest3Solutions.agda +++ b/1FundamentalGroup/Quest3Solutions.agda @@ -1,25 +1,6 @@ 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 ; 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 - - - -open ℤ +open import 1FundamentalGroup.Preambles.P3 loopSucℤtimes : (n : ℤ) → loop n times ∙ loop ≡ loop sucℤ n times loopSucℤtimes (pos n) = refl @@ -49,11 +30,16 @@ pathToFunPathFibration {A} {x} {y} q = J (λ z p → pathToFun (λ i → x ≡ p q ∙ refl ∎ ) +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 = 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 → + ≡⟨ refl ⟩ -- 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))) @@ -68,4 +54,59 @@ rewind = outOfS¹D (λ x → helix x → base ≡ x) loop_times loop_times ∎ ) +windingNumberRewindBase : (n : ℤ) → windingNumber base (rewind base n) ≡ n +windingNumberRewindBase (pos zero) = refl +windingNumberRewindBase (pos (suc n)) = + windingNumber base (rewind base (pos (suc n))) + ≡⟨ refl ⟩ + windingNumber base (loop (pos n) times ∙ loop) + ≡⟨ refl ⟩ + endPt helix (loop (pos n) times ∙ loop) (pos zero) + ≡⟨ refl ⟩ + sucℤ (endPt helix (loop (pos n) times) (pos zero)) + ≡⟨ cong sucℤ (windingNumberRewindBase (pos n)) ⟩ + sucℤ (pos n) + ≡⟨ refl ⟩ + pos (suc n) ∎ +windingNumberRewindBase (negsuc zero) = refl +windingNumberRewindBase (negsuc (suc n)) = cong predℤ (windingNumberRewindBase (negsuc n)) +----------------------------------- + -- windingNumber base (rewind base n) + -- ≡⟨ refl ⟩ + -- windingNumber base (loop n times) + -- ≡⟨ refl ⟩ + -- endPt helix (loop n times) (pos zero) + -- ≡⟨ {!!} ⟩ + -- n ∎ +------------------------------------- + +windingNumberRewind : (x : S¹) (n : helix x) → windingNumber x (rewind x n) ≡ n +windingNumberRewind = -- must case on x / use recursor / outOfS¹ since that is def of rewind + outOfS¹D (λ x → (n : helix x) → windingNumber x (rewind x n) ≡ n) + windingNumberRewindBase ( + pathToFun + (λ i → (n : helix (loop i)) → windingNumber (loop i) (rewind (loop i) n) ≡ n) + windingNumberRewindBase + ≡⟨ funExt (λ x → isSetℤ _ _ _ _ ) ⟩ -- they are just functions so use funExt. + -- two equalities in ℤ so must be equal by isSetℤ + windingNumberRewindBase ∎) + +-- must case on p / use recursor / J since that is def of windingNumber / endPt +-- actually endPt and J are both just transport in cubical agda. +rewindWindingNumber : (x : S¹) (p : base ≡ x) → rewind x (windingNumber x p) ≡ p +rewindWindingNumber x = J (λ x p → rewind x (windingNumber x p) ≡ p) + (rewind base (windingNumber base refl) + ≡⟨ refl ⟩ + loop_times (endPt helix (refl {x = base}) (pos zero)) -- reduce both definitions + ≡⟨ cong loop_times (cong (λ g → g (pos zero)) (endPtRefl {x = base} helix)) ⟩ + loop (pos zero) times + ≡⟨ refl ⟩ + refl ∎) + +pathFibration≡helix : (x : S¹) → (base ≡ x) ≡ helix x +pathFibration≡helix x = + isoToPath (iso (windingNumber x) (rewind x) (windingNumberRewind x) (rewindWindingNumber x)) + +loopSpaceS¹≡ℤ : loopSpace S¹ base ≡ ℤ +loopSpaceS¹≡ℤ = pathFibration≡helix base diff --git a/_build/2.6.2/agda/0Trinitarianism/Quest4Solutions.agdai b/_build/2.6.2/agda/0Trinitarianism/Quest4Solutions.agdai index 8b0208a953e9a689eb7dbb0e0c4d317fa3103ac7..e6de3ba6bacaff02120d3885c9f8ba209f36b2ea 100644 GIT binary patch literal 139967 zcmY(r2~-o=);4^qQW*$CB4Zd61O!Kdpdw>|ir69qwUyf{1W}2t2;#_GponM`P+RHi zK!8^9w_EWxqP7abA>bS}Z)@X#HbD`k(FiD!z_%;h_xsmptwrLgs$Ki+;n~AEmA9Q; zHft^a>8pHW{@~!Qn>!KwGc$c{Cfv=TzaoM`G_jSQ=?3=4CvOdgq1-KBo`<|`X}Ng+ z^@odZzigO%+ruq9LM96D)2trrX0v#Hu*dX;bqB1ESJwG?r*u#3t_(UQ8+t#SxxV*= 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