module Cubical.HITs.Sn.Multiplication where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Path
open import Cubical.Foundations.Function
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Transport
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Pointed.Homogeneous
open import Cubical.Data.Sum
open import Cubical.Data.Bool hiding (elim)
open import Cubical.Data.Nat hiding (elim)
open import Cubical.Data.Sigma
open import Cubical.HITs.S1 renaming (_·_ to _*_)
open import Cubical.HITs.S2
open import Cubical.HITs.Sn
open import Cubical.HITs.Susp renaming (toSusp to σ)
open import Cubical.HITs.Join
open import Cubical.HITs.Pushout
open import Cubical.HITs.SmashProduct
open import Cubical.Homotopy.Loopspace
open Iso
open PlusBis
sphereFun↑ : {n m k : ℕ}
→ (f : S₊ n → S₊ m → S₊ k)
→ S₊ (suc n) → S₊ m → S₊ (suc k)
sphereFun↑ {n = zero} {m = m} f base y = ptSn _
sphereFun↑ {n = zero} {m = m} f (loop i) y = σS (f false y) i
sphereFun↑ {n = suc n} {m = m} f north y = ptSn _
sphereFun↑ {n = suc n} {m = m} f south y = ptSn _
sphereFun↑ {n = suc n} {m = m} f (merid a i) y = σS (f a y) i
_⌣S_ : {n m : ℕ} → S₊ n → S₊ m → S₊ (n + m)
_⌣S_ {n = zero} {m = m} false y = y
_⌣S_ {n = zero} {m = m} true y = ptSn m
_⌣S_ {n = suc n} {m = m} = sphereFun↑ (_⌣S_ {n = n})
IdR⌣S : {n m : ℕ} (y : S₊ m)
→ Path (S₊ (n + m)) (ptSn n ⌣S y) (ptSn (n + m))
IdR⌣S {n = zero} {m = m} y = refl
IdR⌣S {n = suc zero} {m = m} y = refl
IdR⌣S {n = suc (suc n)} {m = m} y = refl
IdL⌣S : {n m : ℕ} (x : S₊ n)
→ Path (S₊ (n + m)) (x ⌣S (ptSn m)) (ptSn (n + m))
IdL⌣S {n = zero} false = refl
IdL⌣S {n = zero} true = refl
IdL⌣S {n = suc zero} base = refl
IdL⌣S {n = suc zero} {zero} (loop i) j = base
IdL⌣S {n = suc zero} {suc m} (loop i) j = rCancel (merid (ptSn (suc m))) j i
IdL⌣S {n = suc (suc n)} {m} north j = north
IdL⌣S {n = suc (suc n)} {m} south j = north
IdL⌣S {n = suc (suc n)} {m} (merid a i) j =
(cong σS (IdL⌣S a)
∙ rCancel (merid (ptSn _))) j i
IdL⌣S≡IdR⌣S : (n m : ℕ)
→ IdL⌣S {n = n} {m = m} (ptSn n) ≡ IdR⌣S (ptSn m)
IdL⌣S≡IdR⌣S zero m = refl
IdL⌣S≡IdR⌣S (suc zero) m = refl
IdL⌣S≡IdR⌣S (suc (suc n)) m = refl
⋀S∙ : (n m : ℕ) → S₊∙ n ⋀∙ S₊∙ m →∙ S₊∙ (n + m)
fst (⋀S∙ n m) (inl x) = ptSn _
fst (⋀S∙ n m) (inr x) = (fst x) ⌣S (snd x)
fst (⋀S∙ n m) (push (inl x) i) = IdL⌣S x (~ i)
fst (⋀S∙ n m) (push (inr x) i) = IdR⌣S x (~ i)
fst (⋀S∙ n m) (push (push a i₁) i) = IdL⌣S≡IdR⌣S n m i₁ (~ i)
snd (⋀S∙ n m) = refl
⋀S : (n m : ℕ) → S₊∙ n ⋀ S₊∙ m → S₊ (n + m)
⋀S n m = fst (⋀S∙ n m)
⋀S-base : (m : ℕ)
→ Iso (S₊∙ zero ⋀ S₊∙ m) (S₊ m)
fun (⋀S-base m) = ⋀S zero m
inv (⋀S-base m) x = inr (false , x)
rightInv (⋀S-base m) x = refl
leftInv (⋀S-base m) =
⋀-fun≡ _ _
(sym (push (inl false)))
(λ { (false , y) → refl
; (true , y) → sym (push (inl false)) ∙ push (inr y)})
(λ { false i j → push (inl false) (i ∨ ~ j)
; true → compPath-filler (sym (push (inl false))) (push (inl true))
▷ cong (sym (push (inl false)) ∙_)
(λ i → push (push tt i) )})
λ x → compPath-filler (sym (push (inl false))) (push (inr x))
⋀S-ind : (n m : ℕ) (x : _)
→ ⋀S (suc n) m x
≡ Iso.inv (IsoSucSphereSusp (n + m))
(suspFun (⋀S n m) (Iso.fun SuspSmashCommIso
(((Iso.fun (IsoSucSphereSusp n) , IsoSucSphereSusp∙' n)
⋀→ idfun∙ (S₊∙ m)) x)))
⋀S-ind zero m = ⋀-fun≡ _ _
(sym (IsoSucSphereSusp∙ m))
(λ x → main m (fst x) (snd x))
(mainₗ m)
mainᵣ
where
F' : (m : ℕ) → Susp ((Bool , true) ⋀ S₊∙ m) → _
F' m = inv (IsoSucSphereSusp (zero + m)) ∘ suspFun (⋀S zero m)
F : (m : ℕ) → Susp∙ Bool ⋀ S₊∙ m → _
F m = F' m ∘ fun SuspSmashCommIso
G : (m : ℕ) → _ → _
G m = _⋀→_ {A = S₊∙ 1} {B = S₊∙ m}
(fun (IsoSucSphereSusp zero) , (λ _ → north))
(idfun∙ (S₊∙ m))
main : (m : ℕ) (x : S¹) (y : S₊ m)
→ x ⌣S y
≡ F m (inr (S¹→SuspBool x , y))
main m base y = sym (IsoSucSphereSusp∙ m)
main zero (loop i) false j =
((cong-∙ (λ x → F zero (inr (x , false)))
(merid false) (sym (merid true)))
∙ sym (rUnit loop)) (~ j) i
main zero (loop i) true j =
F zero (inr (rCancel (merid true) (~ j) i , false))
main (suc m) (loop i) y j =
cong-∙ (λ x → F (suc m) (inr (x , y)))
(merid false) (sym (merid true)) (~ j) i
mainₗ : (m : ℕ) (x : S¹)
→ PathP (λ i → IdL⌣S {m = m} x (~ i)
≡ F m (G m (push (inl x) i)))
(sym (IsoSucSphereSusp∙ m))
(main m x (ptSn m))
mainₗ zero =
toPropElim (λ _ → isOfHLevelPathP' 1 (isGroupoidS¹ _ _) _ _)
(flipSquare (cong (cong (F zero)) (rUnit (push (inl north)))))
mainₗ (suc m) x = flipSquare (help x
▷ (cong (cong (F (suc m))) (rUnit (push (inl (S¹→SuspBool x))))))
where
help : (x : S¹)
→ PathP (λ i → north ≡ main (suc m) x (ptSn (suc m)) i)
(sym (IdL⌣S {n = 1} x))
(cong (F (suc m)) (push (inl (S¹→SuspBool x))))
help base = refl
help (loop i) j k =
hcomp (λ r
→ λ {(i = i0) → north
; (i = i1) → F' (suc m)
(merid-fill
{A = Bool , true}
{B = S₊∙ (suc m)} true (~ r) k j)
; (j = i0) → rCancel-filler (merid (ptSn (suc m))) r (~ k) i
; (j = i1) → F (suc m)
(push (inl (compPath-filler
(merid false) (sym (merid true)) r i)) k)
; (k = i0) → north
; (k = i1) → cong-∙∙-filler
(λ x₁ → F (suc m) (inr (x₁ , ptSn (suc m))))
refl (merid false) (sym (merid true)) r (~ j) i})
(F' (suc m) (merid-fill {A = Bool , true} {B = S₊∙ (suc m)} false k i j))
mainᵣ : (x : S₊ m)
→ PathP (λ i → ptSn (suc m) ≡ F m (G m (push (inr x) i)))
(sym (IsoSucSphereSusp∙ m))
(sym (IsoSucSphereSusp∙ m))
mainᵣ x = flipSquare ((λ i j → (IsoSucSphereSusp∙ m) (~ i))
▷ cong (cong (F m)) (rUnit (push (inr x))))
⋀S-ind (suc n) m = ⋀-fun≡ _ _ refl
(λ x → h (fst x) (snd x))
hₗ
λ x → flipSquare (cong (cong (suspFun (⋀S (suc n) m)
∘ fun SuspSmashCommIso))
(rUnit (push (inr x))))
where
h : (x : S₊ (suc (suc n))) (y : S₊ m)
→ (x ⌣S y)
≡ suspFun (⋀S (suc n) m)
(SuspL→Susp⋀ (inr (idfun (Susp (S₊ (suc n))) x , y)))
h north y = refl
h south y = merid (ptSn _)
h (merid a i) y j = compPath-filler
(merid (a ⌣S y)) (sym (merid (ptSn (suc (n + m))))) (~ j) i
hₗ-lem : (x : Susp (S₊ (suc n)))
→ PathP (λ i → north ≡ h x (ptSn m) i)
(sym (IdL⌣S x))
(cong (suspFun (⋀S (suc n) m)
∘ fun SuspSmashCommIso)
(push (inl x)))
hₗ-lem north = refl
hₗ-lem south i j = merid (ptSn (suc (n + m))) (i ∧ j)
hₗ-lem (merid a i) j k = help j k i
where
help : Cube (sym (cong σS
(IdL⌣S {n = suc n} {m = m} a)
∙ rCancel (merid (ptSn (suc n + m)))))
(λ k i → suspFun (⋀S (suc n) m)
(SuspL→Susp⋀ (push (inl (merid a i)) k)))
(λ j i → north)
(λ j i → compPath-filler
(merid (a ⌣S ptSn m))
(sym (merid (ptSn (suc (n + m))))) (~ j) i)
(λ j k → north)
λ j k → merid (ptSn (suc (n + m))) (j ∧ k)
help j k i =
hcomp (λ r
→ λ {(i = i0) → north
; (i = i1) → merid (ptSn (suc (n + m))) (j ∧ k)
; (j = i0) → compPath-filler'
(cong σS
(IdL⌣S {n = suc n} {m = m} a))
(rCancel (merid (ptSn (suc n + m)))) r (~ k) i
; (j = i1) → suspFun (⋀S (suc n) m)
(merid-fill a k i r)
; (k = i0) → north
; (k = i1) → compPath-filler
(merid (IdL⌣S a (~ r)))
(sym (merid (ptSn (suc (n + m))))) (~ j) i})
(hcomp (λ r → λ {(i = i0) → north
; (i = i1) → merid (ptSn (suc (n + m))) ((j ∨ ~ r) ∧ k)
; (j = i0) → rCancel-filler (merid (ptSn _)) r (~ k) i
; (j = i1) → merid (ptSn (suc (n + m))) (i ∧ k)
; (k = i0) → north
; (k = i1) → compPath-filler
(merid (ptSn _))
(sym (merid (ptSn (suc (n + m)))))
(~ j ∧ r) i})
(merid (ptSn (suc (n + m))) (i ∧ k)))
hₗ : (x : Susp (S₊ (suc n)))
→ PathP (λ i → IdL⌣S x (~ i)
≡ inv (IsoSucSphereSusp (suc n + m))
(suspFun (⋀S (suc n) m)
(fun SuspSmashCommIso
(((fun (IsoSucSphereSusp (suc n)) , IsoSucSphereSusp∙' (suc n)) ⋀→
idfun∙ (S₊∙ m))
(push (inl x) i))))) refl (h x (ptSn m))
hₗ x =
flipSquare
((hₗ-lem x
▷ sym (cong (cong (inv (IsoSucSphereSusp (suc n + m))
∘ suspFun (⋀S (suc n) m)
∘ fun SuspSmashCommIso))
(sym (rUnit (push (inl x)))))))
isEquiv-⋀S : (n m : ℕ) → isEquiv (⋀S n m)
isEquiv-⋀S zero m = isoToIsEquiv (⋀S-base m)
isEquiv-⋀S (suc n) m =
subst isEquiv (sym (funExt (⋀S-ind n m)))
(snd (helpEq (isEquiv-⋀S n m)))
where
r = isoToEquiv (IsoSucSphereSusp n)
helpEq : isEquiv (⋀S n m) → (S₊∙ (suc n) ⋀ S₊∙ m) ≃ S₊ (suc n + m)
helpEq iseq =
compEquiv
(compEquiv
(compEquiv
(⋀≃ (r , IsoSucSphereSusp∙' n) (idEquiv (S₊ m) , refl))
(isoToEquiv SuspSmashCommIso))
(isoToEquiv
(congSuspIso (equivToIso (⋀S n m , iseq)))))
(isoToEquiv (invIso (IsoSucSphereSusp (n + m))))
SphereSmashIso : (n m : ℕ) → Iso (S₊∙ n ⋀ S₊∙ m) (S₊ (n + m))
SphereSmashIso n m = equivToIso (⋀S n m , isEquiv-⋀S n m)
join→Sphere : (n m : ℕ)
→ join (S₊ n) (S₊ m) → S₊ (suc (n + m))
join→Sphere n m (inl x) = ptSn _
join→Sphere n m (inr x) = ptSn _
join→Sphere n m (push a b i) = σS (a ⌣S b) i
join→Sphere∙ : (n m : ℕ)
→ join∙ (S₊∙ n) (S₊∙ m) →∙ S₊∙ (suc (n + m))
fst (join→Sphere∙ n m) = join→Sphere n m
snd (join→Sphere∙ n m) = refl
joinSphereIso' : (n m : ℕ)
→ Iso (join (S₊ n) (S₊ m)) (S₊ (suc (n + m)))
joinSphereIso' n m =
compIso (invIso (SmashJoinIso {A = S₊∙ n} {B = S₊∙ m}))
(compIso (congSuspIso (SphereSmashIso n m))
(invIso (IsoSucSphereSusp (n + m))))
sphere→Join : (n m : ℕ)
→ S₊ (suc (n + m)) → join (S₊ n) (S₊ m)
sphere→Join zero zero = Iso.inv (joinSphereIso' 0 0)
sphere→Join zero (suc m) north = inl true
sphere→Join zero (suc m) south = inl true
sphere→Join zero (suc m) (merid a i) =
(push true (ptSn (suc m))
∙ cong (Iso.inv (joinSphereIso' zero (suc m))) (merid a)) i
sphere→Join (suc n) m north = inl (ptSn (suc n))
sphere→Join (suc n) m south = inl (ptSn (suc n))
sphere→Join (suc n) m (merid a i) =
(push (ptSn (suc n)) (ptSn m)
∙ cong (Iso.inv (joinSphereIso' (suc n) m)) (merid a)) i
join→Sphere≡ : (n m : ℕ) (x : _)
→ join→Sphere n m x ≡ joinSphereIso' n m .Iso.fun x
join→Sphere≡ zero zero (inl x) = refl
join→Sphere≡ zero (suc m) (inl x) = refl
join→Sphere≡ (suc n) m (inl x) = refl
join→Sphere≡ zero zero (inr x) = refl
join→Sphere≡ zero (suc m) (inr x) = merid (ptSn (suc m))
join→Sphere≡ (suc n) zero (inr x) = merid (ptSn (suc n + zero))
join→Sphere≡ (suc n) (suc m) (inr x) = merid (ptSn (suc n + suc m))
join→Sphere≡ zero zero (push false false i) j = loop i
join→Sphere≡ zero zero (push false true i) j = base
join→Sphere≡ zero zero (push true b i) j = base
join→Sphere≡ zero (suc m) (push a b i) j =
compPath-filler
(merid (a ⌣S b)) (sym (merid (ptSn (suc m)))) (~ j) i
join→Sphere≡ (suc n) zero (push a b i) j =
compPath-filler
(merid (a ⌣S b)) (sym (merid (ptSn (suc n + zero)))) (~ j) i
join→Sphere≡ (suc n) (suc m) (push a b i) j =
compPath-filler
(merid (a ⌣S b)) (sym (merid (ptSn (suc n + suc m)))) (~ j) i
sphere→Join≡ : (n m : ℕ) (x : _)
→ sphere→Join n m x ≡ joinSphereIso' n m .Iso.inv x
sphere→Join≡ zero zero x = refl
sphere→Join≡ zero (suc m) north = push true (pt (S₊∙ (suc m)))
sphere→Join≡ zero (suc m) south = refl
sphere→Join≡ zero (suc m) (merid a i) j =
compPath-filler' (push true (pt (S₊∙ (suc m))))
(cong (joinSphereIso' zero (suc m) .Iso.inv) (merid a)) (~ j) i
sphere→Join≡ (suc n) m north = push (ptSn (suc n)) (pt (S₊∙ m))
sphere→Join≡ (suc n) m south = refl
sphere→Join≡ (suc n) m (merid a i) j =
compPath-filler' (push (ptSn (suc n)) (pt (S₊∙ m)))
(cong (joinSphereIso' (suc n) m .Iso.inv) (merid a)) (~ j) i
IsoSphereJoin : (n m : ℕ)
→ Iso (join (S₊ n) (S₊ m)) (S₊ (suc (n + m)))
fun (IsoSphereJoin n m) = join→Sphere n m
inv (IsoSphereJoin n m) = sphere→Join n m
rightInv (IsoSphereJoin n m) x =
(λ i → join→Sphere≡ n m (sphere→Join≡ n m x i) i)
∙ joinSphereIso' n m .Iso.rightInv x
leftInv (IsoSphereJoin n m) x =
(λ i → sphere→Join≡ n m (join→Sphere≡ n m x i) i)
∙ joinSphereIso' n m .Iso.leftInv x
joinSphereEquiv∙ : (n m : ℕ) → join∙ (S₊∙ n) (S₊∙ m) ≃∙ S₊∙ (suc (n + m))
fst (joinSphereEquiv∙ n m) = isoToEquiv (IsoSphereJoin n m)
snd (joinSphereEquiv∙ n m) = refl
IsoSphereJoinPres∙ : (n m : ℕ)
→ Iso.fun (IsoSphereJoin n m) (inl (ptSn n)) ≡ ptSn (suc (n + m))
IsoSphereJoinPres∙ n m = refl
IsoSphereJoin⁻Pres∙ : (n m : ℕ)
→ Iso.inv (IsoSphereJoin n m) (ptSn (suc (n + m))) ≡ inl (ptSn n)
IsoSphereJoin⁻Pres∙ zero zero = push true true ⁻¹
IsoSphereJoin⁻Pres∙ zero (suc m) = refl
IsoSphereJoin⁻Pres∙ (suc n) m = refl
⌣S-false : {n : ℕ} (x : S₊ n) → PathP (λ i → S₊ (+-comm n zero i)) (x ⌣S false) x
⌣S-false {n = zero} false = refl
⌣S-false {n = zero} true = refl
⌣S-false {n = suc zero} base = refl
⌣S-false {n = suc zero} (loop i) = refl
⌣S-false {n = suc (suc n)} north i = north
⌣S-false {n = suc (suc n)} south i = merid (ptSn (suc (+-zero n i))) i
⌣S-false {n = suc (suc n)} (merid a i) j =
hcomp (λ k → λ {(i = i0) → north
; (i = i1) → merid (ptSn (suc (+-zero n j))) (j ∨ ~ k)
; (j = i1) → merid a i})
(merid (⌣S-false a j) i)
assoc⌣S : {n m l : ℕ} (x : S₊ n) (y : S₊ m) (z : S₊ l)
→ PathP (λ i → S₊ (+-assoc n m l i))
(x ⌣S (y ⌣S z)) ((x ⌣S y) ⌣S z)
assoc⌣S {n = zero} {m = m} false y z = refl
assoc⌣S {n = zero} {m = m} true y z = sym (IdR⌣S z)
assoc⌣S {n = suc zero} {m = m} base y z = sym (IdR⌣S z)
assoc⌣S {n = suc zero} {m = m} (loop i) y z j = help m y j i
where
help : (m : ℕ) (y : S₊ m)
→ Square (σS (y ⌣S z)) (cong (λ w → sphereFun↑ _⌣S_ w z) (σS y))
(sym (IdR⌣S z)) (sym (IdR⌣S z))
help zero false = refl
help zero true = σS∙
help (suc m) y =
rUnit (σS (y ⌣S z))
∙ cong (σS (y ⌣S z) ∙_)
(cong sym ((sym σS∙)
∙ congS σS (sym (IdR⌣S z))))
∙ sym (cong-∙ (λ k → sphereFun↑ _⌣S_ k z)
(merid y)
(sym (merid (ptSn (suc m)))))
assoc⌣S {n = suc (suc n)} {m = m} north y z k = north
assoc⌣S {n = suc (suc n)} {m = m} south y z k = north
assoc⌣S {n = suc (suc n)} {m = m} {l} (merid a i) y z j =
help m y (assoc⌣S a y z) j i
where
help : (m : ℕ) (y : S₊ m)
→ PathP (λ i₁ → S₊ (suc (+-assoc n m l i₁)))
(a ⌣S (y ⌣S z))
((a ⌣S y) ⌣S z)
→ SquareP (λ j i → S₊ (+-assoc (suc (suc n)) m l j))
(σS (a ⌣S (y ⌣S z)))
(λ i → (merid a i ⌣S y) ⌣S z)
(λ _ → north) (λ _ → north)
help zero false _ =
(λ i j → σ (S₊∙ (suc (+-assoc n zero l i))) (lem i) j)
▷ rUnit _
∙ cong₂ _∙_ (congS σS (λ _ → (a ⌣S false) ⌣S z))
(cong sym (sym σS∙
∙ congS σS
(sym (IdR⌣S z))))
∙ sym (cong-∙ (_⌣S z)
(merid (a ⌣S false))
(sym (merid (ptSn (suc (n + zero))))))
where
lem : PathP (λ i → S₊ (suc (+-assoc n zero l i)))
(a ⌣S z) ((a ⌣S false) ⌣S z)
lem = toPathP ((λ i → subst (S₊ ∘ suc)
(isSetℕ _ _ (+-assoc n zero l)
(λ j → +-zero n (~ j) + l) i)
(a ⌣S z))
∙ fromPathP (λ i → ⌣S-false a (~ i) ⌣S z))
help zero true _ =
(congS σS (IdL⌣S a) ∙ σS∙)
◁ (λ i j → north)
▷ (cong (cong (_⌣S z))
(sym σS∙
∙ congS σS (sym (IdL⌣S a))))
help (suc m) y q =
(λ i j → σS (q i) j)
▷ (rUnit _
∙ cong₂ _∙_ refl
(cong sym (sym σS∙ ∙ cong σS (sym (IdR⌣S z))))
∙ sym (cong-∙ (_⌣S z) (merid (a ⌣S y))
(sym (merid (ptSn (suc (n + suc m)))))))
-S^ : {k : ℕ} (n : ℕ) → S₊ k → S₊ k
-S^ zero x = x
-S^ (suc n) x = invSphere (-S^ n x)
-S^-expl : {k : ℕ} (n m : ℕ)
→ isEvenT n ⊎ isOddT n
→ isEvenT m ⊎ isOddT m
→ S₊ k → S₊ k
-S^-expl {k = zero} n m (inl x₁) q x = x
-S^-expl {k = zero} n m (inr x₁) (inl x₂) x = x
-S^-expl {k = zero} n m (inr x₁) (inr x₂) false = true
-S^-expl {k = zero} n m (inr x₁) (inr x₂) true = false
-S^-expl {k = suc zero} n m p q base = base
-S^-expl {k = suc zero} n m (inl x) q (loop i) = loop i
-S^-expl {k = suc zero} n m (inr x) (inl x₁) (loop i) = loop i
-S^-expl {k = suc zero} n m (inr x) (inr x₁) (loop i) = loop (~ i)
-S^-expl {k = suc (suc k)} n m p q north = north
-S^-expl {k = suc (suc k)} n m p q south = north
-S^-expl {k = suc (suc k)} n m (inl x) q (merid a i) = σS a i
-S^-expl {k = suc (suc k)} n m (inr x) (inl x₁) (merid a i) = σS a i
-S^-expl {k = suc (suc k)} n m (inr x) (inr x₁) (merid a i) = σS a (~ i)
invSphere-S^ : {k : ℕ} (n : ℕ) (x : S₊ k)
→ invSphere (-S^ n x) ≡ -S^ n (invSphere x)
invSphere-S^ zero x = refl
invSphere-S^ (suc n) x = cong invSphere (invSphere-S^ n x)
-S^² : {k : ℕ} (n : ℕ) (x : S₊ k) → -S^ n (-S^ n x) ≡ x
-S^² zero x = refl
-S^² (suc n) x =
cong invSphere (sym (invSphere-S^ n (-S^ n x)))
∙ invSphere² _ (-S^ n (-S^ n x))
∙ -S^² n x
-S^Iso : {k : ℕ} (n : ℕ) → Iso (S₊ k) (S₊ k)
fun (-S^Iso n) = -S^ n
inv (-S^Iso n) = -S^ n
rightInv (-S^Iso n) = -S^² n
leftInv (-S^Iso n) = -S^² n
-S^-comp : {k : ℕ} (n m : ℕ) (x : S₊ k)
→ -S^ n (-S^ m x) ≡ -S^ (n + m) x
-S^-comp zero m x = refl
-S^-comp (suc n) m x = cong invSphere (-S^-comp n m x)
-S^·2 : {k : ℕ} (n : ℕ) (x : S₊ k) → -S^ (n + n) x ≡ x
-S^·2 zero x = refl
-S^·2 (suc n) x =
cong invSphere (λ i → -S^ (+-comm n (suc n) i) x)
∙ invSphere² _ (-S^ (n + n) x)
∙ -S^·2 n x
private
-S^-transp : {k : ℕ} (m : ℕ) (p : k ≡ m) (n : ℕ) (x : S₊ k)
→ subst S₊ p (-S^ n x) ≡ -S^ n (subst S₊ p x)
-S^-transp =
J> λ n x → transportRefl _
∙ sym (cong (-S^ n) (transportRefl x))
sym^ : ∀ {ℓ} {A : Type ℓ} {x : A} (n : ℕ) → x ≡ x → x ≡ x
sym^ zero p = p
sym^ (suc n) p = sym (sym^ n p)
σS-S^ : {k : ℕ} (n : ℕ) (x : S₊ (suc k))
→ σS (-S^ n x) ≡ sym^ n (σS x)
σS-S^ {k = k} zero x = refl
σS-S^ {k = k} (suc n) x =
σ-invSphere k _ ∙ cong sym (σS-S^ n x)
sphereFun↑-subst : {n m : ℕ} (k' k : ℕ) (p : k' ≡ k)
→ (f : S₊ n → S₊ m → S₊ k') (x : S₊ _) (y : S₊ _)
→ sphereFun↑ (λ x y → subst S₊ p (f x y)) x y
≡ subst S₊ (cong suc p) (sphereFun↑ f x y)
sphereFun↑-subst k' = J> λ f x y
→ (λ i → sphereFun↑ (λ x₁ y₁ → transportRefl (f x₁ y₁) i) x y)
∙ sym (transportRefl _)
sphereFun↑-invSphere : {n m k : ℕ}
→ (f : S₊ (suc n) → S₊ (suc m) → S₊ (suc k)) (x : S₊ _) (y : S₊ _)
→ sphereFun↑ (λ x y → invSphere' (f x y)) x y
≡ invSphere' (sphereFun↑ (λ x y → (f x y)) x y)
sphereFun↑-invSphere {n = n} {m = m} {k = k} f north y = refl
sphereFun↑-invSphere {n = n} {m = m} {k = k} f south y = refl
sphereFun↑-invSphere {n = n} {m = m} {k = k} f (merid a i) y j =
lem k (f a y) j i
where
lem : (k : ℕ) (x : S₊ (suc k))
→ (σS (invSphere' x)) ≡ (cong invSphere' (σS x))
lem k x =
sym (cong-∙ invSphere' (merid x) (sym (merid (ptSn _)))
∙∙ cong (cong invSphere' (merid x) ∙_)
(rCancel (merid (ptSn _)))
∙∙ (sym (rUnit _)
∙ sym (σ-invSphere k x)
∙ cong (σ (S₊∙ (suc k)))
(sym (invSphere'≡ x))))
sphereFun↑^ : {n m k : ℕ} (l : ℕ)
→ (f : S₊ (suc n) → S₊ (suc m) → S₊ (suc k)) (x : S₊ _) (y : S₊ _)
→ sphereFun↑ (λ x y → -S^ l (f x y)) x y
≡ -S^ l (sphereFun↑ (λ x y → (f x y)) x y)
sphereFun↑^ zero f x y = refl
sphereFun↑^ (suc l) f x y =
(λ i → sphereFun↑ (λ x₁ y₁ → invSphere'≡ (-S^ l (f x₁ y₁)) (~ i)) x y)
∙ sphereFun↑-invSphere (λ x₁ y₁ → (-S^ l (f x₁ y₁))) x y
∙ invSphere'≡ ((sphereFun↑ (λ x₁ y₁ → -S^ l (f x₁ y₁)) x y))
∙ cong invSphere (sphereFun↑^ l f x y)
S^-even : {k : ℕ} (n : ℕ) (x : S₊ k) → isEvenT n → -S^ n x ≡ x
S^-even zero x p = refl
S^-even (suc (suc n)) x p = invSphere² _ (-S^ n x) ∙ S^-even n x p
private
move-transp-S^ : {k : ℕ} (n : ℕ) (p : k ≡ n) (m : ℕ)
→ (x : S₊ k) (y : S₊ n)
→ subst S₊ p (-S^ m x) ≡ y
→ subst S₊ (sym p) (-S^ m y) ≡ x
move-transp-S^ =
J> λ m x → J> transportRefl _
∙ cong (-S^ m) (transportRefl _)
∙ -S^² m x
master-lem : ∀ {ℓ} {A : Type ℓ} {x : A} (p : x ≡ x) (coh : refl ≡ p)
(q : p ≡ p)
→ Cube (λ j k → coh j (~ k)) (λ j k → coh j (~ k))
(λ i k → q k i) (λ i k → q i (~ k))
(λ j k → coh (~ k) j) λ j k → coh (~ k) j
master-lem = J> λ q → λ i j k → sym≡flipSquare q j (~ k) i
comm⌣S₁ : {m : ℕ} → (x : S¹) (y : S₊ (suc m))
→ (x ⌣S y)
≡ subst S₊ (+-comm (suc m) 1)
(-S^ (suc m) (y ⌣S x))
comm⌣S₁ {m = zero} x y =
(main x y ∙ invSphere'≡ (y ⌣S x))
∙ sym (transportRefl (invSusp (y ⌣S x)))
where
pp-main : (x : S¹) → PathP (λ i → IdL⌣S {m = 1} x i
≡ IdL⌣S {m = 1} x i) (cong (x ⌣S_) loop) (sym (σS x))
pp-main base i j = rCancel (merid base) (~ i) (~ j)
pp-main (loop k) i j =
master-lem _ (sym (rCancel (merid base)))
(λ j k → σS (loop j) k) k i j
pp-help : (x : S¹) → PathP (λ i → IdL⌣S {m = 1} x i
≡ IdL⌣S {m = 1} x i)
(cong (x ⌣S_) loop) (cong invSphere' (σS x))
pp-help x = pp-main x
▷ (rUnit _
∙∙ cong (sym (σS x) ∙_) (sym (rCancel (merid base)))
∙∙ sym (cong-∙ invSphere' (merid x) (sym (merid base))))
main : (x y : S¹) → (x ⌣S y) ≡ invSphere' (y ⌣S x)
main x base = IdL⌣S {m = 1} x
main x (loop i) = flipSquare (pp-help x) i
comm⌣S₁ {m = suc m} x y =
(main-lem x y
∙ sym (transportRefl (invSphere' (sphereFun↑ (λ x₂ x₃ → x₃ ⌣S x₂) y x)))
∙ sym (compSubstℕ {A = S₊} (cong suc (sym (+-comm (suc m) 1)))
(+-comm (suc (suc m)) 1) refl
{x = invSphere' (sphereFun↑ (λ x₂ x₃ → x₃ ⌣S x₂) y x)}))
∙ cong (subst S₊ (+-comm (suc (suc m)) 1))
(cong (subst S₊ (cong suc (sym (+-comm (suc m) 1))))
(sym (S^-lem (suc m) (sphereFun↑ (λ x₂ x₃ → x₃ ⌣S x₂) y x)))
∙ -S^-transp _ (cong suc (sym (+-comm (suc m) 1)))
(suc (suc m) + suc m)
(sphereFun↑ (λ x₂ x₃ → x₃ ⌣S x₂) y x)
∙ sym (-S^-comp (suc (suc m)) (suc m)
(subst S₊ (cong suc (sym (+-comm (suc m) 1)))
(sphereFun↑ (λ x₂ x₃ → x₃ ⌣S x₂) y x))))
∙ cong (subst S₊ (+-comm (suc (suc m)) 1)
∘ -S^ (suc (suc m)))
((sym (-S^-transp _ (cong suc (sym (+-comm (suc m) 1))) (suc m)
(sphereFun↑ (λ x₂ x₃ → x₃ ⌣S x₂) y x))
∙ cong (subst S₊ (cong suc (sym (+-comm (suc m) 1))))
(sym (sphereFun↑^ (suc m)
(λ x₂ x₃ → (x₃ ⌣S x₂)) y x))
∙ sym (sphereFun↑-subst _ _ (sym (+-comm (suc m) 1))
(λ x₂ x₃ → (-S^ (suc m) (x₃ ⌣S x₂))) y x))
∙ cong (λ (s : S₊ (suc m) → S¹ → S₊ (suc m + 1))
→ sphereFun↑ s y x)
(refl
∙ sym (funExt λ x → funExt λ y
→ sym (move-transp-S^ _ (+-comm (suc m) 1)
(suc m) (x ⌣S y) (y ⌣S x)
(sym (comm⌣S₁ y x)))
∙ refl)))
where
S^-lem : {k : ℕ} (m : ℕ) (x : S₊ k)
→ -S^ (suc m + m) x ≡ invSphere' x
S^-lem m x =
sym (invSphere'≡ (-S^ (m + m) x))
∙ cong invSphere' (-S^·2 m x)
⌣S-south : (x : S¹) → x ⌣S south ≡ north
⌣S-south base = refl
⌣S-south (loop i) j =
(cong σS (sym (merid (ptSn (suc m))) )
∙ rCancel (merid (ptSn _))) j i
PathP-main : (x : S¹) (a : S₊ (suc m))
→ PathP (λ i → IdL⌣S x i ≡ ⌣S-south x i) (cong (x ⌣S_) (merid a))
(sym (σS (x ⌣S a)))
PathP-main base a j i = rCancel (merid north) (~ j) (~ i)
PathP-main (loop k) a j i =
hcomp (λ r → λ {(i = i0) → rCancel (merid north) j k
; (i = i1) → compPath-filler'
(cong σS (sym (merid (ptSn (suc m)))))
(rCancel (merid north)) r j k
; (j = i0) → σS (compPath-filler (merid a)
(sym (merid (ptSn (suc m)))) (~ r) i) k
; (j = i1) → σS (σS a k) (~ i)
; (k = i0) → rCancel (merid north) (~ j) (~ i)
; (k = i1) → rCancel (merid north) (~ j) (~ i)})
(master-lem _ (sym (rCancel (merid north)))
(λ i k → σS (loop i ⌣S a) k) k j i)
pp : (x : S¹) (a : S₊ (suc m))
→ PathP (λ i → IdL⌣S x i ≡ ⌣S-south x i) (cong (x ⌣S_) (merid a))
(cong invSphere' (σS (x ⌣S a)))
pp x a = PathP-main x a
▷ (rUnit _
∙∙ cong (sym (σS (x ⌣S a)) ∙_) (sym (rCancel (merid north)))
∙∙ sym (cong-∙ invSphere' (merid (x ⌣S a)) (sym (merid north))))
main-lem : (x : S¹) (y : S₊ (2 + m))
→ (x ⌣S y)
≡ invSphere' (sphereFun↑ (λ x₂ x₃ → x₃ ⌣S x₂) y x)
main-lem x north = IdL⌣S x
main-lem x south = ⌣S-south x
main-lem x (merid a i) j = pp x a j i
comm⌣S-lem : {n m : ℕ}
→ ((x : S₊ (suc n)) (y : S₊ (suc (suc m)))
→ (x ⌣S y) ≡ subst S₊ (+-comm (suc (suc m)) (suc n))
(-S^ (suc (suc m) · (suc n)) (y ⌣S x)))
→ (((x : S₊ (suc m)) (y : S₊ (suc (suc n)))
→ (x ⌣S y) ≡ subst S₊ (+-comm (suc (suc n)) (suc m))
(-S^ ((suc (suc n)) · (suc m)) (y ⌣S x))))
→ (((x : S₊ (suc n)) (y : S₊ (suc m))
→ (y ⌣S x) ≡ subst S₊ (sym (+-comm (suc m) (suc n)))
(-S^ ((suc n) · (suc m)) (x ⌣S y))))
→ (x : S₊ (suc (suc n))) (y : S₊ (suc (suc m)))
→ (x ⌣S y) ≡ subst S₊ (+-comm (suc (suc m)) (suc (suc n)))
(-S^ (suc (suc m) · (suc (suc n))) (y ⌣S x))
comm⌣S-lem {n = n} {m = m} ind1 ind2 ind3 x y =
cong (λ (s : S₊ (suc n) → S₊ (suc (suc m))
→ S₊ ((suc n) + (suc (suc m)))) → sphereFun↑ s x y)
(funExt (λ x → funExt λ y
→ ind1 x y))
∙ (sphereFun↑-subst _ _ (+-comm (suc (suc m)) (suc n))
(λ x y → -S^ (suc (suc m) · suc n) (y ⌣S x)) x y
∙ cong (subst S₊ (cong suc (+-comm (suc (suc m)) (suc n))))
(sphereFun↑^ (suc (suc m) · suc n) (λ x y → y ⌣S x) x y
∙ cong (-S^ (suc (suc m) · suc n))
(cong (λ (s : S₊ (suc n) → S₊ (suc (suc m))
→ S₊ ((suc (suc m)) + (suc n))) → sphereFun↑ s x y)
(funExt (λ x → funExt λ y →
cong (λ (s : S₊ (suc m) → S₊ (suc n) → S₊ ((suc m) + (suc n)))
→ sphereFun↑ s y x)
(funExt λ x
→ funExt λ y
→ ind3 y x)
∙ sphereFun↑-subst _ _ (sym (+-comm (suc m) (suc n)))
(λ x y → -S^ (suc n · suc m) (y ⌣S x)) y x
∙ cong (subst S₊ (cong suc (sym (+-comm (suc m) (suc n)))))
(sphereFun↑^ (suc n · suc m) (λ x y → (y ⌣S x)) y x)))
∙ sphereFun↑-subst _ _ (sym (cong suc (+-comm (suc m) (suc n))))
((λ x₁ x₂ →
(-S^ (suc n · suc m)
(sphereFun↑ (λ x₃ y₁ → y₁ ⌣S x₃) x₂ x₁)))) x y
∙ cong (subst S₊ (sym (cong (suc ∘ suc) (+-comm (suc m) (suc n)))))
((sphereFun↑^ (suc n · suc m)
((λ x₁ x₂ → (sphereFun↑ (λ x₃ y₁ → y₁ ⌣S x₃) x₂ x₁))) x y
∙ cong (-S^ (suc n · suc m)) (lem₃ x y)))))
∙ big-lem (suc n) (suc m)
_ (λ i → suc (suc (+-comm (suc m) (suc n) (~ i))))
_ (λ i → suc (+-comm (suc (suc m)) (suc n) i)) _
(sym (+-comm (suc (suc m)) (suc (suc n))))
(λ i → suc (+-comm (suc (suc n)) (suc m) i))
(sphereFun↑ (λ x₁ y₂ → y₂ ⌣S x₁) y x)
∙ sym (cong (subst S₊ (+-comm (suc (suc m)) (suc (suc n))))
(cong (-S^ (suc (suc m) · suc (suc n)))
((λ i → sphereFun↑ (λ x y → ind2 x y i) y x)
∙ sphereFun↑-subst _ _
(+-comm (suc (suc n)) (suc m))
(λ x y → -S^ (suc (suc n) · suc m) (y ⌣S x)) y x
∙ cong (subst S₊ (cong suc (+-comm (suc (suc n)) (suc m))))
(sphereFun↑^ (suc (suc n) · suc m) (λ x y → y ⌣S x) y x)))))
where
ℕ-p : (n m : ℕ)
→ (suc m · suc n + suc n · m)
≡ (m + m) + ((n · m + n · m) + (suc n))
ℕ-p n m =
cong suc (cong (_+ (m + n · m)) (cong (n +_) (·-comm m (suc n)))
∙ sym (+-assoc n (m + n · m) _)
∙ +-comm n _
∙ cong (_+ n) (+-assoc (m + n · m) m (n · m)
∙ cong (_+ (n · m))
(sym (+-assoc m (n · m) m)
∙ cong (m +_) (+-comm (n · m) m)
∙ +-assoc m m (n · m))
∙ sym (+-assoc (m + m) (n · m) (n · m))))
∙ sym (+-suc (m + m + (n · m + n · m)) n)
∙ sym (+-assoc (m + m) (n · m + n · m) (suc n))
ℕ-p2 : (n m : ℕ) → suc m · n + n · m + 1 ≡ (((n · m) + (n · m)) + (suc n))
ℕ-p2 n m = (λ _ → ((n + m · n) + n · m) + 1)
∙ cong (_+ 1) (sym (+-assoc n (m · n) (n · m))
∙ (λ i → +-comm n ((·-comm m n i) + n · m) i))
∙ sym (+-assoc (n · m + n · m) n 1)
∙ cong (n · m + n · m +_) (+-comm n 1)
big-lem : (n m : ℕ) {x : ℕ} (y : ℕ) (p : x ≡ y) (z : ℕ) (s : y ≡ z)
(d : ℕ) (r : z ≡ d) (t : x ≡ d)
(a : S₊ x)
→ subst S₊ s (-S^ (suc m · n) (subst S₊ p (-S^ (n · m) (invSphere' a))))
≡ subst S₊ (sym r)
(-S^ (suc m · suc n)
(subst S₊ t (-S^ (suc n · m) a)))
big-lem n m =
J> (J> (J> λ t a
→ transportRefl _
∙ cong (-S^ (n + m · n)) (transportRefl _)
∙ sym (transportRefl _
∙ cong (-S^ (suc m · suc n))
((λ i → subst S₊ (isSetℕ _ _ t refl i) (-S^ (m + n · m) a))
∙ transportRefl (-S^ (m + n · m) a) )
∙ -S^-comp (suc m · suc n) (suc n · m) a
∙ ((funExt⁻ (cong -S^ (ℕ-p n m)) a
∙ (sym (-S^-comp (m + m) _ a)
∙ -S^·2 m (-S^ (n · m + n · m + suc n) a))
∙ funExt⁻ (cong -S^ (sym (ℕ-p2 n m))) a)
∙ sym (-S^-comp (suc m · n + n · m) 1 a)
∙ cong (-S^ (suc m · n + n · m))
(sym (invSphere'≡ a)))
∙ sym (-S^-comp (suc m · n) (n · m) (invSphere' a)) )))
lem₁ : (x : S₊ (2 + n))
→ sphereFun↑ (λ x₂ x₃ → sphereFun↑ (λ x₄ y₁ → y₁ ⌣S x₄) x₃ x₂) x north ≡ north
lem₁ north = refl
lem₁ south = refl
lem₁ (merid a i) j = rCancel (merid north) j i
lem₂ : (x : S₊ (2 + n))
→ sphereFun↑ (λ x₂ x₃
→ sphereFun↑ (λ x₄ y₁ → y₁ ⌣S x₄) x₃ x₂) x south
≡ north
lem₂ north = refl
lem₂ south = refl
lem₂ (merid a i) j = rCancel (merid north) j i
lem₃ : (x : S₊ (2 + n)) (y : S₊ (2 + m))
→ (sphereFun↑ (λ x₁ x₂
→ sphereFun↑ (λ x₃ y₁ → y₁ ⌣S x₃) x₂ x₁) x y)
≡ invSphere' (sphereFun↑ (λ x₁ y₂ → y₂ ⌣S x₁) y x)
lem₃ x north = lem₁ x
lem₃ x south = lem₂ x
lem₃ x (merid a i) j = h j i
where
main : (x : _) → PathP (λ i → lem₁ x i ≡ lem₂ x i)
(cong (sphereFun↑ (λ x₂ x₃
→ sphereFun↑ (λ x₄ y₁ → y₁ ⌣S x₄) x₃ x₂) x )
(merid a))
(sym (σS (x ⌣S a)))
main north = cong sym (sym (rCancel (merid north)))
main south = cong sym (sym (rCancel (merid north)))
main (merid z i) j k =
master-lem _
(sym (rCancel (merid north)))
(cong (λ x → σS (x ⌣S a)) (merid z)) i j k
h : PathP (λ i → lem₁ x i ≡ lem₂ x i)
(cong (sphereFun↑
(λ x₂ x₃ → sphereFun↑ (λ x₄ y₁ → y₁ ⌣S x₄) x₃ x₂) x)
(merid a)) (cong invSphere' (σS (x ⌣S a)))
h = main x
▷ ((rUnit _ ∙ cong (sym (σS (x ⌣S a)) ∙_)
(sym (rCancel (merid north))))
∙ sym (cong-∙ invSphere'
(merid (x ⌣S a)) (sym (merid (ptSn _)))))
comm⌣S₀ : (x : Bool) (m : ℕ) (y : S₊ m)
→ PathP (λ i → S₊ (+-zero m (~ i))) (x ⌣S y) (y ⌣S x)
comm⌣S₀ false =
elim+2 (λ { false → refl ; true → refl})
(λ { base → refl ; (loop i) → refl})
ind
where
ind : (n : ℕ) →
((y : S₊ (suc n)) →
PathP (λ i → S₊ (suc (+-zero n (~ i)))) y (y ⌣S false)) →
(y : Susp (S₊ (suc n))) →
PathP (λ i → Susp (S₊ (suc (+-zero n (~ i))))) y (y ⌣S false)
ind n p north i = north
ind n p south i = merid (ptSn (suc (+-zero n (~ i)))) (~ i)
ind n p (merid a j) i =
comp (λ k → Susp (S₊ (suc (+-zero n (~ i ∨ ~ k)))))
(λ r →
λ {(i = i0) → merid a j
; (i = i1) →
σ (S₊∙ (suc (+-zero n (~ r)))) (p a r) j
; (j = i0) → north
; (j = i1) → merid (ptSn (suc (+-zero n (~ i ∨ ~ r)))) (~ i)})
(compPath-filler (merid a) (sym (merid (ptSn _))) i j)
comm⌣S₀ true m y = (λ i → ptSn (+-zero m (~ i))) ▷ (sym (IdL⌣S y))
comm⌣S : {n m : ℕ} (x : S₊ n) (y : S₊ m)
→ (x ⌣S y) ≡ subst S₊ (+-comm m n) (-S^ (m · n) (y ⌣S x))
comm⌣S {n = zero} {m = m} x y =
sym (fromPathP (symP {A = λ i → S₊ (+-zero m i)} ((comm⌣S₀ x m y))))
∙ sym (cong (subst S₊ (+-zero m))
((λ i → -S^ (0≡m·0 m (~ i)) (y ⌣S x))))
comm⌣S {n = suc n} {m = zero} x y =
sym (fromPathP (comm⌣S₀ y (suc n) x))
∙ (λ i → subst S₊ (isSetℕ _ _
(sym (+-comm (suc n) zero))
(+-comm zero (suc n)) i) (y ⌣S x))
comm⌣S {n = suc zero} {m = suc m} x y =
comm⌣S₁ x y
∙ cong (subst S₊ (+-comm (suc m) 1))
λ i → -S^ (·-identityʳ (suc m) (~ i)) (y ⌣S x)
comm⌣S {n = suc (suc n)} {m = suc zero} x y =
sym (substSubst⁻ S₊ (+-comm 1 (suc (suc n))) (x ⌣S y))
∙ cong (subst S₊ (+-comm 1 (suc (suc n))))
((λ i → subst S₊ (isSetℕ _ _
(sym (+-comm 1 (suc (suc n)))) (+-comm (suc (suc n)) 1) i)
(x ⌣S y))
∙ (sym (sym
(-S^-transp _ (+-comm (suc (suc n)) 1) (1 · suc (suc n))
(-S^ (suc (suc n)) (x ⌣S y)))
∙ cong (subst S₊ (+-comm (suc (suc n)) 1))
(cong (-S^ (1 · suc (suc n)))
(λ i → -S^ (·-identityˡ (suc (suc n)) (~ i)) (x ⌣S y))
∙ -S^² (1 · suc (suc n)) (x ⌣S y)))))
∙ sym (cong (subst S₊ (+-comm 1 (suc (suc n)))
∘ -S^ (1 · suc (suc n))) (comm⌣S₁ y x))
comm⌣S {n = suc (suc n)} {m = suc (suc m)} x y =
comm⌣S-lem comm⌣S comm⌣S
(λ x y → (sym (cong (subst S₊ (sym (+-comm (suc m) (suc n))))
(sym (-S^-transp _ (+-comm (suc m) (suc n))
(suc n · suc m) (-S^ (suc m · suc n) (y ⌣S x)))
∙ cong (subst S₊ (+-comm (suc m) (suc n)))
(cong (-S^ (suc n · suc m))
(λ i → -S^ (·-comm (suc m) (suc n) i) (y ⌣S x))
∙ -S^² (suc n · suc m) (y ⌣S x) ))
∙ subst⁻Subst S₊ (+-comm (suc m) (suc n)) (y ⌣S x) ))
∙ sym (cong (subst S₊ (sym (+-comm (suc m) (suc n)))
∘ -S^ (suc n · suc m))
(comm⌣S x y))) x y
diag⌣ : {n : ℕ} (x : S₊ (suc n)) → x ⌣S x ≡ ptSn _
diag⌣ {n = zero} base = refl
diag⌣ {n = zero} (loop i) j = help j i
where
help : cong₂ _⌣S_ loop loop ≡ refl
help = cong₂Funct _⌣S_ loop loop
∙ sym (rUnit _)
∙ rCancel (merid base)
diag⌣ {n = suc n} north = refl
diag⌣ {n = suc n} south = refl
diag⌣ {n = suc n} (merid a i) j = help j i
where
help : cong₂ _⌣S_ (merid a) (merid a) ≡ refl
help = cong₂Funct _⌣S_ (merid a) (merid a)
∙ sym (rUnit _)
∙ flipSquare (cong IdL⌣S (merid a))
⌣₁,₁-distr : (a b : S¹) → (b * a) ⌣S b ≡ a ⌣S b
⌣₁,₁-distr a base = refl
⌣₁,₁-distr a (loop i) j = lem j i
where
lem : cong₂ (λ (b1 b2 : S¹) → (b1 * a) ⌣S b2) loop loop
≡ (λ i → a ⌣S loop i)
lem = (cong₂Funct (λ (b1 b2 : S¹) → (b1 * a) ⌣S b2) loop loop
∙ cong₂ _∙_ (PathP→compPathR
(flipSquare (cong (IdL⌣S {n = 1} {1} ∘ (_* a)) loop))
∙ cong₂ _∙_ refl (sym (lUnit _))
∙ rCancel _)
refl
∙ sym (lUnit _))
⌣₁,₁-distr' : (a b : S¹) → (a * b) ⌣S b ≡ a ⌣S b
⌣₁,₁-distr' a b = cong (_⌣S b) (commS¹ a b) ∙ ⌣₁,₁-distr a b
⌣Sinvₗ : {n m : ℕ} (x : S₊ (suc n)) (y : S₊ (suc m))
→ (invSphere x) ⌣S y ≡ invSphere (x ⌣S y)
⌣Sinvₗ {n = zero} {m} base y = merid (ptSn (suc m))
⌣Sinvₗ {n = zero} {m} (loop i) y j = lem j i
where
lem : Square (σS y ⁻¹)
(λ i → invSphere (loop i ⌣S y))
(merid (ptSn (suc m))) (merid (ptSn (suc m)))
lem = sym (cong-∙ invSphere' (merid y) (sym (merid (ptSn (suc m))))
∙ cong₂ _∙_ refl (rCancel _)
∙ sym (rUnit _))
◁ (λ j i → invSphere'≡ (loop i ⌣S y) j)
⌣Sinvₗ {n = suc n} {m} north y = merid (ptSn (suc (n + suc m)))
⌣Sinvₗ {n = suc n} {m} south y = merid (ptSn (suc (n + suc m)))
⌣Sinvₗ {n = suc n} {m} (merid a i) y j = lem j i
where
p = ptSn (suc (n + suc m))
lem : Square (σS (a ⌣S y) ⁻¹)
(λ i → invSphere (merid a i ⌣S y))
(merid p) (merid p)
lem = (sym (cong-∙ invSphere' (merid (a ⌣S y)) (sym (merid p))
∙ cong₂ _∙_ refl (rCancel _)
∙ sym (rUnit _)))
◁ λ j i → invSphere'≡ (merid a i ⌣S y) j
⌣Sinvᵣ : {n m : ℕ} (x : S₊ (suc n)) (y : S₊ (suc m))
→ x ⌣S (invSphere y) ≡ invSphere (x ⌣S y)
⌣Sinvᵣ {n = n} {m} x y =
comm⌣S x (invSphere y)
∙ cong (subst S₊ (+-comm (suc m) (suc n)))
(cong (-S^ (suc m · suc n)) (⌣Sinvₗ y x)
∙ sym (invSphere-S^ (suc m · suc n) (y ⌣S x)))
∙ -S^-transp _ (+-comm (suc m) (suc n)) 1
(-S^ (suc m · suc n) (y ⌣S x))
∙ cong invSphere
(cong (subst S₊ (+-comm (suc m) (suc n)))
(cong (-S^ (suc m · suc n)) (comm⌣S y x)
∙ sym (-S^-transp _ (+-comm (suc n) (suc m))
(suc m · suc n)
(-S^ (suc n · suc m) (x ⌣S y)))
∙ cong (subst S₊ (+-comm (suc n) (suc m)))
((cong (-S^ (suc m · suc n))
(λ i → -S^ (·-comm (suc n) (suc m) i) (x ⌣S y)))
∙ -S^² (suc m · suc n) (x ⌣S y))
∙ cong (λ p → subst S₊ p (x ⌣S y))
(isSetℕ _ _ (+-comm (suc n) (suc m))
(+-comm (suc m) (suc n) ⁻¹)))
∙ subst⁻Subst S₊ (+-comm (suc m) (suc n) ⁻¹) (x ⌣S y))
SuspS¹→S²-S¹×S¹→S² : (a b : S¹)
→ SuspS¹→S² (a ⌣S b) ≡ S¹×S¹→S² a b
SuspS¹→S²-S¹×S¹→S² base b = refl
SuspS¹→S²-S¹×S¹→S² (loop i) b j = main b j i
where
lem : (b : _) → cong SuspS¹→S² (merid b) ≡ (λ j → S¹×S¹→S² (loop j) b)
lem base = refl
lem (loop i) = refl
main : (b : _) → cong SuspS¹→S² (σS b) ≡ (λ j → S¹×S¹→S² (loop j) b)
main b = cong-∙ SuspS¹→S² (merid b) (merid base ⁻¹)
∙ sym (rUnit _)
∙ lem b