From a2157edf1e1b29874a4c90b8a14ed6094425d721 Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Tue, 16 Nov 2021 14:54:06 +0000 Subject: [PATCH] fixed import issues --- 0Trinitarianism/Preambles/P5.agda | 37 ++++++++ 0Trinitarianism/Quest4Solutions.agda | 82 +++++++++--------- 0Trinitarianism/Quest5.agda | 0 0Trinitarianism/Quest5Solutions.agda | 35 ++++++-- 1FundamentalGroup/Preambles/P3.agda | 17 +++- 1FundamentalGroup/Quest3Solutions.agda | 5 -- .../agda/0Trinitarianism/Preambles/P5.agdai | Bin 0 -> 61681 bytes .../0Trinitarianism/Quest4Solutions.agdai | Bin 139967 -> 147577 bytes .../0Trinitarianism/Quest5Solutions.agdai | Bin 69596 -> 82230 bytes .../agda/1FundamentalGroup/Preambles/P2.agdai | Bin 60114 -> 59998 bytes .../agda/1FundamentalGroup/Preambles/P3.agdai | Bin 66309 -> 78026 bytes .../1FundamentalGroup/Quest2Solutions.agdai | Bin 103827 -> 103828 bytes .../1FundamentalGroup/Quest3Solutions.agdai | Bin 94549 -> 90271 bytes 13 files changed, 121 insertions(+), 55 deletions(-) create mode 100644 0Trinitarianism/Preambles/P5.agda create mode 100644 0Trinitarianism/Quest5.agda create mode 100644 _build/2.6.2/agda/0Trinitarianism/Preambles/P5.agdai diff --git a/0Trinitarianism/Preambles/P5.agda b/0Trinitarianism/Preambles/P5.agda new file mode 100644 index 0000000..0f148be --- /dev/null +++ b/0Trinitarianism/Preambles/P5.agda @@ -0,0 +1,37 @@ +module 0Trinitarianism.Preambles.P5 where + +open import Cubical.Foundations.Prelude renaming + (funExt to libFunExt ; + sym to libSym ; + _≡⟨_⟩_ to lib_≡⟨_⟩_ ; + _∎ to lib_∎ ; + _∙_ to lib_∙_ ; + fst to libFst ; + snd to libSnd + ) public +open import Cubical.HITs.S1 using ( S¹ ; base ; loop ) public +open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public +open import Cubical.Foundations.Path public +open import 0Trinitarianism.Quest4Solutions public +open import 1FundamentalGroup.Quest0Solutions public +open import Cubical.Data.Bool public + +pathToFun≡transport : {u : Level} {A B : Type u} (p : A ≡ B) (x : A) + → pathToFun p x ≡ transport p x +pathToFun≡transport {u} {A} = J (λ B p → (x : A) → pathToFun p x ≡ transport p x) + λ x → + pathToFun refl x + ≡⟨ pathToFunReflx x ⟩ + x + ≡⟨ sym (transportRefl x) ⟩ + transport refl x ∎ + +PathPIsoPathD : {u : Level} {A B : Type u} (p : A ≡ B) (x : A) (y : B) → + (PathP (λ i → p i) x y) ≅ (pathToFun p x ≡ y) +PathPIsoPathD {u} {A} {B} p x = + subst (λ b → (y : B) → (PathP (λ i → p i) x y) ≅ (b ≡ y)) + (sym (pathToFun≡transport p x)) + (PathPIsoPath _ x) + + + diff --git a/0Trinitarianism/Quest4Solutions.agda b/0Trinitarianism/Quest4Solutions.agda index b51fe1b..6e1e9bc 100644 --- a/0Trinitarianism/Quest4Solutions.agda +++ b/0Trinitarianism/Quest4Solutions.agda @@ -7,30 +7,30 @@ private u : Level -data Id {A : Type} : (x y : A) → Type where +data Id {A : Type u} : (x y : A) → Type u where rfl : {x : A} → Id x x -idSym : (A : Type) (x y : A) → Id x y → Id y x +idSym : (A : Type u) (x y : A) → Id x y → Id y x idSym A x .x rfl = rfl -Sym : {A : Type} {x y : A} → Id x y → Id y x +Sym : {A : Type u} {x y : A} → Id x y → Id y x Sym rfl = rfl -_*_ : {A : Type} {x y z : A} → Id x y → Id y z → Id x z +_*_ : {A : Type u} {x y z : A} → Id x y → Id y z → Id x z rfl * q = q -_*0_ : {A : Type} {x y z : A} → Id x y → Id y z → Id x z +_*0_ : {A : Type u} {x y z : A} → Id x y → Id y z → Id x z p *0 rfl = p -_*1_ : {A : Type} {x y z : A} → Id x y → Id y z → Id x z +_*1_ : {A : Type u} {x y z : A} → Id x y → Id y z → Id x z rfl *1 rfl = rfl ------------Cong------------------------- private variable - A B : Type + A B : Type u w x y z : A ------------Groupoid Laws---------------- @@ -54,14 +54,14 @@ Assoc rfl q r = rfl -------------Mapping Out---------------- -outOfId : (M : (y : A) → Id x y → Type) → M x rfl +outOfId : (M : (y : A) → Id x y → Type u) → M x rfl → {y : A} (p : Id x y) → M y p outOfId M h rfl = h ------------Path vs Id-------------------- Path→Id : x ≡ y → Id x y -Path→Id {A} {x} = J (λ y p → Id x y) rfl +Path→Id {u} {A} {x} = J (λ y p → Id x y) rfl Id→Path : Id x y → x ≡ y Id→Path rfl = refl @@ -83,21 +83,21 @@ Path→IdRefl : Path→Id (refl {x = x}) ≡ rfl Path→IdRefl {x = x} = JRefl (λ y p → Id x y) rfl Path≡Id : (x ≡ y) ≡ (Id x y) -Path≡Id {A} {x} {y} = isoToPath (iso Path→Id Id→Path rightInv leftInv) where +Path≡Id {u} {A} {x} {y} = isoToPath (iso Path→Id Id→Path rightInv leftInv) where - rightInv : section (Path→Id {A} {x} {y}) Id→Path + rightInv : section (Path→Id {u} {A} {x} {y}) Id→Path rightInv rfl = Path→IdRefl - leftInv : retract (Path→Id {A} {x} {y}) Id→Path + leftInv : retract (Path→Id {u} {A} {x} {y}) Id→Path leftInv = J (λ y p → Id→Path (Path→Id p) ≡ p) (cong (λ p → Id→Path p) Path→IdRefl) -----------Basics about Path Space----------------- sym : {x y : A} → x ≡ y → y ≡ x -sym {A} {x} = J (λ y1 p → y1 ≡ x) refl +sym {u} {A} {x} = J (λ y1 p → y1 ≡ x) refl -symRefl : {x : A} → sym {A} {x} {x} (refl) ≡ refl -symRefl {A} {x} = JRefl (λ y1 p → y1 ≡ x) refl +symRefl : {x : A} → sym {u} {A} {x} {x} (refl) ≡ refl +symRefl {u} {A} {x} = JRefl (λ y1 p → y1 ≡ x) refl Trans : {x y z : A} → x ≡ y → y ≡ z → x ≡ z Trans {x = x} {z = z} = J (λ y1 p → y1 ≡ z → x ≡ z) λ q → q @@ -114,20 +114,20 @@ infixr 30 _∙_ infix 3 _∎ infixr 2 _≡⟨_⟩_ -TransRefl : {x y : A} → Trans {A} {x} {x} {y} refl ≡ λ q → q +TransRefl : {x y : A} → Trans {u} {A} {x} {x} {y} refl ≡ λ q → q TransRefl {x = x} {y = y} = JRefl ((λ y1 p → y1 ≡ y → x ≡ y)) λ q → q refl∙refl : {x : A} → refl {_} {A} {x} ∙ refl ≡ refl refl∙refl = cong (λ f → f refl) TransRefl ∙refl : {x y : A} (p : x ≡ y) → Trans p refl ≡ p -∙refl {A} {x} {y} = J (λ y p → Trans p refl ≡ p) refl∙refl +∙refl {u} {A} {x} {y} = J (λ y p → Trans p refl ≡ p) refl∙refl -refl∙ : {A : Type} {x y : A} (p : x ≡ y) → refl ∙ p ≡ p +refl∙ : {A : Type u} {x y : A} (p : x ≡ y) → refl ∙ p ≡ p refl∙ = J (λ y p → refl ∙ p ≡ p) refl∙refl -∙sym : {A : Type} {x y : A} (p : x ≡ y) → p ∙ sym p ≡ refl +∙sym : {A : Type u} {x y : A} (p : x ≡ y) → p ∙ sym p ≡ refl ∙sym = J (λ y p → p ∙ sym p ≡ refl) ( refl ∙ sym refl @@ -138,7 +138,7 @@ refl∙ = J (λ y p → refl ∙ p ≡ p) refl∙refl ) -sym∙ : {A : Type} {x y : A} (p : x ≡ y) → (sym p) ∙ p ≡ refl +sym∙ : {A : Type u} {x y : A} (p : x ≡ y) → (sym p) ∙ p ≡ refl sym∙ = J (λ y p → (sym p) ∙ p ≡ refl) ( (sym refl) ∙ refl @@ -148,9 +148,9 @@ sym∙ = J (λ y p → (sym p) ∙ p ≡ refl) refl ∎ ) -assoc : {A : Type} {w x : A} (p : w ≡ x) {y z : A} (q : x ≡ y) (r : y ≡ z) +assoc : {A : Type u} {w x : A} (p : w ≡ x) {y z : A} (q : x ≡ y) (r : y ≡ z) → (p ∙ q) ∙ r ≡ p ∙ (q ∙ r) -assoc {A} = J +assoc {u} {A} = J -- casing on p (λ x p → {y z : A} (q : x ≡ y) (r : y ≡ z) → (p ∙ q) ∙ r ≡ p ∙ (q ∙ r)) -- reduce to showing when p = refl @@ -166,50 +166,50 @@ id : A → A id x = x pathToFun : A ≡ B → A → B -pathToFun {A} = J (λ B p → (A → B)) id +pathToFun {u} {A} = J (λ B p → (A → B)) id pathToFunRefl : pathToFun (refl {x = A}) ≡ id -pathToFunRefl {A} = JRefl (λ B p → (A → B)) id +pathToFunRefl {u} {A} = JRefl (λ B p → (A → B)) id -pathToFunReflx : pathToFun (refl {x = A}) x ≡ x -pathToFunReflx {x = x} = cong (λ f → f x) pathToFunRefl +pathToFunReflx : (x : A) → pathToFun (refl {x = A}) x ≡ x +pathToFunReflx x = cong (λ f → f x) pathToFunRefl -endPt : (B : A → Type) (p : x ≡ y) → B x → B y +endPt : (B : A → Type u) (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 : (B : A → Type u) → 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 : A → Type u) (p : x ≡ y) → B x → B y endPt' B p = pathToFun (cong B p ) --------------funExt--------------------- -funExt : (B : A → Type) (f g : (a : A) → B a) → +funExt : {B : A → Type u} {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 +funExt 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 +funExtPath : (B : A → Type u) (f g : (a : A) → B a) → (f ≡ g) ≡ ((a : A) → f a ≡ g a) +funExtPath {u} {A} B f g = isoToPath (iso fun funExt 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 : section fun funExt rightInv h = refl - leftInv : retract fun (funExt B f g) + leftInv : retract fun funExt leftInv h = refl --------------Path on Products----------- -data _×_ (A B : Type) : Type where +data _×_ (A B : Type u) : Type u 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 +Id× : (a0 a1 : A) (b0 b1 : B) + → (Id {u} {A × B} ( a0 , b0 ) ( a1 , b1 )) ≡ (Id a0 a1 × Id b0 b1) +Id× {u} {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 @@ -229,8 +229,8 @@ 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) = +Path× : {A B : Type u} (x y : A × B) → (x ≡ y) ≡ ((fst x ≡ fst y) × (snd x ≡ snd y)) +Path× {u} {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) diff --git a/0Trinitarianism/Quest5.agda b/0Trinitarianism/Quest5.agda new file mode 100644 index 0000000..e69de29 diff --git a/0Trinitarianism/Quest5Solutions.agda b/0Trinitarianism/Quest5Solutions.agda index d309e33..33290ea 100644 --- a/0Trinitarianism/Quest5Solutions.agda +++ b/0Trinitarianism/Quest5Solutions.agda @@ -1,11 +1,6 @@ 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 renaming (PathPIsoPath to PathPIsoPathD) -open import 1FundamentalGroup.Quest0Solutions -open import Cubical.Data.Bool +open import 0Trinitarianism.Preambles.P5 PathD : {A0 A1 : Type} (A : A0 ≡ A1) (x : A0) (y : A1) → Type PathD A x y = pathToFun A x ≡ y @@ -16,7 +11,7 @@ outOfS¹P : (B : S¹ → Type) (b : B base) → PathP (λ i → B (loop i)) b b 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) → PathD (λ i → B (loop i)) b b → (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 @@ -42,3 +37,29 @@ example' = outOfS¹D (λ x → doubleCover x → doubleCover x) Flip (funExt lem 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 + +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 (sym refl) a))) + ( + pathToFun refl f + ≡⟨ pathToFunReflx f ⟩ + f + ≡⟨ funExt (λ a → + f a + ≡⟨ cong f (sym (pathToFunReflx a)) ⟩ + f (pathToFun refl a) + ≡⟨ cong (λ p → f (pathToFun p a)) (sym symRefl) ⟩ + f (pathToFun (sym refl) a) + ≡⟨ sym (pathToFunReflx (f (pathToFun (sym refl) a))) ⟩ + pathToFun refl (f (pathToFun (sym refl) a)) ∎ + ) + ⟩ + (λ a → pathToFun refl (f (pathToFun (sym refl) a))) ∎ + ) + B + ) + A diff --git a/1FundamentalGroup/Preambles/P3.agda b/1FundamentalGroup/Preambles/P3.agda index de086cd..02a167e 100644 --- a/1FundamentalGroup/Preambles/P3.agda +++ b/1FundamentalGroup/Preambles/P3.agda @@ -4,17 +4,30 @@ 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.Isomorphism renaming (Iso to _≅_) 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 +PathD : {A0 A1 : Type} (A : A0 ≡ A1) (x : A0) (y : A1) → Type +PathD A x y = pathToFun A x ≡ y + 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) + +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) → PathD (λ i → B (loop i)) b b → (x : S¹) → B x +outOfS¹D B b p x = outOfS¹P B b (_≅_.inv (PathPIsoPath (λ i → B (loop i)) b b) p) x + +outOfS¹DBase : (B : S¹ → Type) (b : B base) + (p : PathD (λ i → B (loop i)) b b) → outOfS¹D B b p base ≡ b +outOfS¹DBase B b p = refl diff --git a/1FundamentalGroup/Quest3Solutions.agda b/1FundamentalGroup/Quest3Solutions.agda index 78ec3b8..363ea07 100644 --- a/1FundamentalGroup/Quest3Solutions.agda +++ b/1FundamentalGroup/Quest3Solutions.agda @@ -30,11 +30,6 @@ 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 = 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