Added templates and some code done
This commit is contained in:
@@ -0,0 +1,12 @@
|
||||
module 1FundamentalGroup.Preambles.P0 where
|
||||
|
||||
open import Cubical.Data.Empty using (⊥) public
|
||||
open import Cubical.Data.Unit renaming ( Unit to ⊤ ) public
|
||||
open import Cubical.Data.Bool renaming ( elim to Bool-elim ) public
|
||||
open import Cubical.Foundations.Prelude
|
||||
renaming ( subst to endPt
|
||||
; transport to pathToFun
|
||||
) public
|
||||
open import Cubical.Foundations.Isomorphism renaming ( Iso to _≅_ ) public
|
||||
open import Cubical.Foundations.Path public
|
||||
open import Cubical.HITs.S1 renaming ( elim to S¹-elim ) public
|
||||
@@ -0,0 +1,12 @@
|
||||
module 1FundamentalGroup.Preambles.P1 where
|
||||
|
||||
open import Cubical.HITs.S1 using (S¹ ; base ; loop) public
|
||||
open import Cubical.Data.Nat using (ℕ ; suc ; zero) public
|
||||
open import Cubical.Data.Int using (ℤ ; pos ; negsuc ; -_) public
|
||||
open import Cubical.Data.Empty public
|
||||
open import Cubical.Foundations.Prelude
|
||||
renaming ( subst to endPt
|
||||
; transport to pathToFun
|
||||
) public
|
||||
open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public
|
||||
open import 1FundamentalGroup.Quest0Solutions using ( Refl ; Refl≢loop ) public
|
||||
@@ -0,0 +1,19 @@
|
||||
module 1FundamentalGroup.Preambles.P2 where
|
||||
|
||||
open import Cubical.Data.Nat public
|
||||
open import Cubical.Data.Int using (ℤ ; pos ; negsuc ; -_) public
|
||||
open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public
|
||||
open import Cubical.Data.Empty using (⊥) public
|
||||
open import Cubical.Data.Unit renaming (Unit to ⊤) public
|
||||
open import Cubical.Foundations.Prelude
|
||||
renaming ( subst to endPt
|
||||
; transport to pathToFun
|
||||
) public
|
||||
open import Cubical.HITs.S1 using (S¹ ; base ; loop) public
|
||||
open import 1FundamentalGroup.Quest1Solutions public
|
||||
|
||||
refl∙refl : {A : Type} {a : A} → refl ∙ refl ≡ refl {x = a}
|
||||
refl∙refl {a = a} = sym (λ i j → compPath-filler (refl {x = a}) refl i j)
|
||||
|
||||
symRefl : {A : Type} {a : A} → sym refl ≡ refl {x = a}
|
||||
symRefl = refl
|
||||
@@ -0,0 +1,33 @@
|
||||
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 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 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
|
||||
@@ -0,0 +1,64 @@
|
||||
-- ignore
|
||||
module 1FundamentalGroup.Quest0 where
|
||||
open import 1FundamentalGroup.Preambles.P0
|
||||
|
||||
Refl : base ≡ base
|
||||
Refl = {!!}
|
||||
|
||||
Flip : Bool → Bool
|
||||
Flip x = {!!}
|
||||
|
||||
flipIso : Bool ≅ Bool
|
||||
flipIso = {!!}
|
||||
|
||||
flipPath : Bool ≡ Bool
|
||||
flipPath = {!!}
|
||||
|
||||
doubleCover : S¹ → Type
|
||||
doubleCover x = {!!}
|
||||
|
||||
endPtOfTrue : base ≡ base → doubleCover base
|
||||
endPtOfTrue p = {!!}
|
||||
|
||||
Refl≢loop : Refl ≡ loop → ⊥
|
||||
Refl≢loop p = {!!}
|
||||
|
||||
------------------- Side Quest - Empty -------------------------
|
||||
|
||||
{-
|
||||
-- This is a comment box,
|
||||
-- remove the {- and -} to do the side quests
|
||||
|
||||
toEmpty : (A : Type) → Type
|
||||
toEmpty A = {!!}
|
||||
|
||||
pathEmpty : (A : Type) → Type₁
|
||||
pathEmpty A = {!!}
|
||||
|
||||
isoEmpty : (A : Type) → Type
|
||||
isoEmpty A = {!!}
|
||||
|
||||
outOf⊥ : (A : Type) → ⊥ → A
|
||||
outOf⊥ A ()
|
||||
|
||||
toEmpty→isoEmpty : (A : Type) → toEmpty A → isoEmpty A
|
||||
toEmpty→isoEmpty A = {!!}
|
||||
|
||||
isoEmpty→pathEmpty : (A : Type) → isoEmpty A → pathEmpty A
|
||||
isoEmpty→pathEmpty A = {!!}
|
||||
|
||||
pathEmpty→toEmpty : (A : Type) → pathEmpty A → toEmpty A
|
||||
pathEmpty→toEmpty A = {!!}
|
||||
-}
|
||||
|
||||
------------------- Side Quests - true≢false --------------------
|
||||
|
||||
{-
|
||||
-- This is a comment box,
|
||||
-- remove the {- and -} to do the side quests
|
||||
|
||||
true≢false' : true ≡ false → ⊥
|
||||
true≢false' = {!!}
|
||||
|
||||
-}
|
||||
|
||||
@@ -0,0 +1,35 @@
|
||||
-- ignore
|
||||
module 1FundamentalGroup.Quest1 where
|
||||
open import 1FundamentalGroup.Preambles.P1
|
||||
|
||||
|
||||
loopSpace : (A : Type) (a : A) → Type
|
||||
loopSpace A a = a ≡ a
|
||||
|
||||
loop_times : ℤ → loopSpace S¹ base
|
||||
loop n times = {!!}
|
||||
|
||||
{-
|
||||
The definition of sucℤ goes here.
|
||||
-}
|
||||
|
||||
{-
|
||||
The definition of predℤ goes here.
|
||||
-}
|
||||
|
||||
{-
|
||||
The definition of sucℤIso goes here.
|
||||
-}
|
||||
|
||||
{-
|
||||
The definition of sucℤPath goes here.
|
||||
-}
|
||||
|
||||
helix : S¹ → Type
|
||||
helix = {!!}
|
||||
|
||||
windingNumberBase : base ≡ base → ℤ
|
||||
windingNumberBase = {!!}
|
||||
|
||||
windingNumber : (x : S¹) → base ≡ x → helix x
|
||||
windingNumber = {!!}
|
||||
@@ -0,0 +1,23 @@
|
||||
module 1FundamentalGroup.Quest1SideQuests.Sn where
|
||||
|
||||
open import Cubical.Data.Nat
|
||||
open import Cubical.Data.Empty
|
||||
open import Cubical.Data.Unit renaming (Unit to ⊤)
|
||||
open import Cubical.Data.Bool
|
||||
open import Cubical.Foundations.Prelude
|
||||
|
||||
|
||||
data susp (X : Type) : Type where
|
||||
north : {!!}
|
||||
south : {!!}
|
||||
merid : {!!}
|
||||
|
||||
Sphere : ℕ → Type
|
||||
Sphere = {!!}
|
||||
|
||||
Disk : (n : ℕ) → Type
|
||||
Disk zero = {!!}
|
||||
Disk (suc n) = {!!}
|
||||
|
||||
SphereToDisk : {n : ℕ} → Sphere n → Disk n
|
||||
SphereToDisk {n} s = {!!}
|
||||
@@ -0,0 +1,27 @@
|
||||
-- ignore
|
||||
module 1FundamentalGroup.Quest2 where
|
||||
open import 1FundamentalGroup.Preambles.P2
|
||||
|
||||
isSet→LoopSpace≡⊤ : {A : Type} (x : A) → isSet A → (x ≡ x) ≡ ⊤
|
||||
isSet→LoopSpace≡⊤ = {!!}
|
||||
|
||||
data _⊔_ (A B : Type) : Type where
|
||||
|
||||
inl : A → A ⊔ B
|
||||
inr : B → A ⊔ B
|
||||
|
||||
{-
|
||||
|
||||
Your definition of ℤ≡ℕ⊔ℕ goes here.
|
||||
|
||||
Your definition of ⊔NoConfusion goes here.
|
||||
|
||||
Your definition of Path≡⊔NoConfusion goes here.
|
||||
|
||||
Your definition of isSet⊔NoConfusion goes here.
|
||||
|
||||
Your definition of isSet⊔ goes here.
|
||||
|
||||
Your definition of isSetℤ goes here.
|
||||
|
||||
-}
|
||||
@@ -0,0 +1,3 @@
|
||||
module 1FundamentalGroup.Quest3 where
|
||||
|
||||
open import 1FundamentalGroup.Preambles.P3
|
||||
Reference in New Issue
Block a user