{-# OPTIONS --lossy-unification #-}
module Cubical.Homotopy.EilenbergMacLane.GradedCommTensor where
open import Cubical.Homotopy.EilenbergMacLane.GroupStructure
open import Cubical.Homotopy.EilenbergMacLane.Properties
open import Cubical.Homotopy.EilenbergMacLane.Base as EM
open import Cubical.Homotopy.EilenbergMacLane.CupProductTensor
open import Cubical.Homotopy.Loopspace
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Path
open import Cubical.Foundations.Function
open import Cubical.Foundations.Transport
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Pointed.Homogeneous
open import Cubical.Foundations.Isomorphism
open import Cubical.Data.Nat renaming (_+_ to _+ℕ_ ; elim to ℕelim)
open import Cubical.Data.Fin
open import Cubical.Data.Fin.Arithmetic
open import Cubical.Data.Sigma
open import Cubical.Data.Unit
open import Cubical.Data.Empty as ⊥
open import Cubical.Data.Sum
open import Cubical.HITs.EilenbergMacLane1 as EM₁
open import Cubical.HITs.Susp
open import Cubical.HITs.Truncation as TR
open import Cubical.Algebra.CommRing.Base
open import Cubical.Algebra.Group.Properties
open import Cubical.Algebra.AbGroup.Base
open import Cubical.Algebra.AbGroup.TensorProduct
open import Cubical.Algebra.Group.MorphismProperties
open AbGroupStr renaming (_+_ to _+Gr_ ; -_ to -Gr_)
open PlusBis
private
variable
ℓ ℓ' : Level
private
⌣-ℕ-elim2 : {A : ℕ → ℕ → Type ℓ}
→ ((n : ℕ) → A zero n)
→ ((n : ℕ) → A (suc n) zero)
→ ((n m : ℕ) → A (suc n) m × A n (suc m) × A n m → A (suc n) (suc m))
→ (n m : ℕ) → A n m
⌣-ℕ-elim2 l r ind zero m = l m
⌣-ℕ-elim2 l r ind (suc n) zero = r n
⌣-ℕ-elim2 {A = A} l r ind (suc n) (suc m) =
ind n m ((⌣-ℕ-elim2 {A = A} l r ind (suc n) m)
, ⌣-ℕ-elim2 {A = A} l r ind n (suc m)
, ⌣-ℕ-elim2 {A = A} l r ind n m)
cong-subst-lem : (n m : ℕ) (q' : n ≡ m) (q : suc n ≡ suc m)
(A : ℕ → Pointed ℓ)
(F : (x : ℕ) → fst (A x) → fst (Ω (A (suc x))))
(F∙ : (x : ℕ) → F x (pt (A x)) ≡ refl)
→ (x : fst (A n))
→ (s : (n m : ℕ) (q : n ≡ m)
→ snd (A m) ≡ subst (λ n₁ → A n₁ .fst) q (snd (A n)))
→ ((n : ℕ) → s n n refl ≡ sym (transportRefl _))
→ (cong (subst (λ n → A n .fst) q) (F n x))
≡ (sym (s _ _ q) ∙∙ F m (subst (λ n → A n .fst) q' x) ∙∙ s _ _ q)
cong-subst-lem {ℓ} n = J> λ q → lem2 q (isSetℕ _ _ refl q)
where
lem2 : (q : suc n ≡ suc n) (r : refl ≡ q)
→ (A : ℕ → Pointed ℓ)
(F : (x : ℕ) → fst (A x) → fst (Ω (A (suc x)))) →
((x : ℕ) → F x (pt (A x)) ≡ refl) →
(x : fst (A n))
(s : (n₁ m : ℕ) (q₁ : n₁ ≡ m)
→ snd (A m) ≡ subst (λ n₂ → A n₂ .fst) q₁ (snd (A n₁))) →
((n₁ : ℕ) → s n₁ n₁ refl ≡ sym (transportRefl (snd (A n₁)))) →
cong (subst (λ n₁ → A n₁ .fst) q) (F n x) ≡
(sym (s (suc n) (suc n) q) ∙∙ F n (subst (λ n₁ → A n₁ .fst) refl x)
∙∙ s (suc n) (suc n) q)
lem2 = J> λ A F substId k s id
→ rUnit _
∙∙ (λ j → (λ i → transportRefl (snd (A (suc n))) (j ∧ i))
∙∙ (λ i → transportRefl (F n (transportRefl k (~ j)) i) j)
∙∙ λ i → transportRefl (snd (A (suc n))) (j ∧ ~ i))
∙∙ cong (λ p → sym p ∙∙ F n (subst (λ n₁ → A n₁ .fst) refl k) ∙∙ p)
(sym (id (suc n)))
module _ {G' : AbGroup ℓ} where
private
_+G_ = _+Gr_ (snd G')
-G_ = -Gr_ (snd G')
-ₖ^<_·_> : (n m k : ℕ)
→ isEvenT n ⊎ isOddT n
→ isEvenT m ⊎ isOddT m
→ EM G' k → EM G' k
-ₖ^< n · m > zero (inl x) q r = r
-ₖ^< n · m > zero (inr x) (inl x₁) r = r
-ₖ^< n · m > zero (inr x) (inr x₁) r = -ₖ r
-ₖ^< n · m > (suc zero) p q =
EM₁.rec (AbGroup→Group G')
(hLevelEM _ 1)
embase
(eml p q)
(sq p q)
where
eml : (p : isEvenT n ⊎ isOddT n) (q : isEvenT m ⊎ isOddT m)
(g : fst G') → Path (EM G' 1) embase embase
eml (inl x) q g = emloop g
eml (inr x) (inl x₁) g = emloop g
eml (inr x) (inr x₁) g = sym (emloop g)
sq : (p : isEvenT n ⊎ isOddT n) (q : isEvenT m ⊎ isOddT m) (g h : fst G')
→ Square (eml p q g) (eml p q (g +G h)) refl (eml p q h)
sq (inl x) q g h = emcomp g h
sq (inr x) (inl x₁) g h = emcomp g h
sq (inr x) (inr x₁) g h =
sym (emloop-sym _ g)
◁ (flipSquare (flipSquare (emcomp (-G g) (-G h))
▷ emloop-sym _ h)
▷ (cong emloop (+Comm (snd G') (-G g) (-G h)
∙ sym (GroupTheory.invDistr (AbGroup→Group G') g h))
∙ emloop-sym _ (g +G h)))
-ₖ^< n · m > (suc (suc k)) p q =
TR.rec (isOfHLevelTrunc (4 +ℕ k))
λ { north → 0ₖ _
; south → 0ₖ _
; (merid a i) → loopCase p q a i}
where
loopCase : (p : isEvenT n ⊎ isOddT n) (q : isEvenT m ⊎ isOddT m)
(a : EM-raw G' (suc k))
→ Path (EM G' (suc (suc k))) (0ₖ (suc (suc k))) (0ₖ (suc (suc k)))
loopCase (inl x) q a = EM→ΩEM+1 _ (EM-raw→EM G' (suc k) a)
loopCase (inr x) (inl x₁) a = EM→ΩEM+1 _ (EM-raw→EM G' (suc k) a)
loopCase (inr x) (inr x₁) a = sym (EM→ΩEM+1 _ (EM-raw→EM G' (suc k) a))
-ₖ^<_·_>∙ : (n m k : ℕ)
(p : isEvenT n ⊎ isOddT n)
(q : isEvenT m ⊎ isOddT m)
→ -ₖ^< n · m > k p q (0ₖ k) ≡ 0ₖ k
-ₖ^< n · m >∙ zero (inl x) q = refl
-ₖ^< n · m >∙ zero (inr x) (inl x₁) = refl
-ₖ^< n · m >∙ zero (inr x) (inr x₁) =
GroupTheory.inv1g (AbGroup→Group G')
-ₖ^< n · m >∙ (suc zero) p q = refl
-ₖ^< n · m >∙ (suc (suc k)) p q = refl
private
EM→ΩEM+1-PP : (k : _) (a : _)
→ PathP (λ i → EM→ΩEM+1 (suc k) (EM-raw→EM G' (suc k) a) i ≡ ∣ merid a i ∣)
refl
(cong ∣_∣ₕ (merid ptEM-raw))
EM→ΩEM+1-PP k a = flipSquare (EM→ΩEM+1∘EM-raw→EM k a
◁ λ i j → ∣ compPath-filler (merid a) (sym (merid ptEM-raw)) (~ i) j ∣ₕ)
-ₖ^<_·_>-inl-left : (n m k : ℕ)
(p : _)
(q : isEvenT m ⊎ isOddT m)
(x : EM G' k)
→ -ₖ^< n · m > k (inl p) q x ≡ x
-ₖ^< n · m >-inl-left zero p q x = refl
-ₖ^< n · m >-inl-left (suc zero) p q =
EM-raw'-elim _ 1 (λ _ → hLevelEM _ 1 _ _)
λ { embase-raw → refl ; (emloop-raw g i) → refl}
-ₖ^< n · m >-inl-left (suc (suc k)) p q =
TR.elim (λ _ → isOfHLevelPath (4 +ℕ k) (isOfHLevelTrunc (4 +ℕ k)) _ _)
λ { north → refl
; south → cong ∣_∣ₕ (merid ptEM-raw)
; (merid a i) → EM→ΩEM+1-PP k a i}
-ₖ^<_·_>-inl-right : (n m k : ℕ)
(p : _)
(q : isEvenT m)
(x : EM G' k)
→ -ₖ^< n · m > k p (inl q) x ≡ x
-ₖ^< n · m >-inl-right zero (inl p) q x = refl
-ₖ^< n · m >-inl-right (suc zero) (inl p) q =
EM-raw'-elim _ 1 (λ _ → hLevelEM _ 1 _ _)
λ { embase-raw → refl ; (emloop-raw g i) → refl}
-ₖ^< n · m >-inl-right (suc (suc k)) (inl p) q =
TR.elim (λ _ → isOfHLevelPath (4 +ℕ k) (isOfHLevelTrunc (4 +ℕ k)) _ _)
λ { north → refl
; south → cong ∣_∣ₕ (merid ptEM-raw)
; (merid a i) → EM→ΩEM+1-PP k a i}
-ₖ^< n · m >-inl-right zero (inr x) q _ = refl
-ₖ^< n · m >-inl-right (suc zero) (inr x) q =
EM-raw'-elim _ 1 (λ _ → hLevelEM _ 1 _ _)
λ { embase-raw → refl ; (emloop-raw g i) → refl}
-ₖ^< n · m >-inl-right (suc (suc k)) (inr x) q =
TR.elim (λ _ → isOfHLevelPath (4 +ℕ k) (isOfHLevelTrunc (4 +ℕ k)) _ _)
λ { north → refl
; south → cong ∣_∣ₕ (merid ptEM-raw)
; (merid a i) → EM→ΩEM+1-PP k a i}
-ₖ^<_·_>-inr : (n m k : ℕ)
(p : _)
(q : isOddT m)
(x : EM G' k)
→ -ₖ^< n · m > k (inr p) (inr q) x ≡ -ₖ x
-ₖ^< n · m >-inr zero p q x = refl
-ₖ^< n · m >-inr (suc zero) p q x = refl
-ₖ^< n · m >-inr (suc (suc k)) p q =
TR.elim (λ _ → isOfHLevelPath (4 +ℕ k) (isOfHLevelTrunc (4 +ℕ k)) _ _)
λ { north → refl ; south → refl ; (merid a i) j → l2 k a j i}
where
l2 : (k : ℕ) (a : EM-raw G' (suc k))
→ cong (-ₖ^< n · m > (suc (suc k)) (inr p) (inr q)) (cong ∣_∣ₕ (merid a))
≡ cong -ₖ_ (cong ∣_∣ₕ (merid a))
l2 zero a = refl
l2 (suc k) a = refl
cong-ₖ^<_·_>₂ : (n m k : ℕ)
(p : isEvenT n ⊎ isOddT n)
(q : isEvenT m ⊎ isOddT m)
→ (x : EM G' (suc k))
→ cong (-ₖ^< n · m > (suc (suc k)) p q) (EM→ΩEM+1 (suc k) x)
≡ EM→ΩEM+1 (suc k) (-ₖ^< n · m > (suc k) p q x)
cong-ₖ^< n · m >₂ k (inl p) q x =
(λ i j → -ₖ^< n · m >-inl-left (suc (suc k)) p q (EM→ΩEM+1 (suc k) x j) i)
∙ cong (EM→ΩEM+1 (suc k)) (sym (-ₖ^< n · m >-inl-left (suc k) p q x))
cong-ₖ^< n · m >₂ k (inr p) (inl q) x =
(λ i j → -ₖ^< n · m >-inl-right (suc (suc k)) (inr p) q
(EM→ΩEM+1 (suc k) x j) i)
∙ cong (EM→ΩEM+1 (suc k)) (sym (-ₖ^< n · m >-inl-right (suc k) (inr p) q x))
cong-ₖ^< n · m >₂ k (inr p) (inr q) x =
(λ i j → -ₖ^< n · m >-inr (suc (suc k)) p q
(EM→ΩEM+1 (suc k) x j) i)
∙∙ cong-₂ k (EM→ΩEM+1 (suc k) x)
∙∙ sym (EM→ΩEM+1-sym (suc k) x)
∙ cong (EM→ΩEM+1 (suc k)) (sym (-ₖ^< n · m >-inr (suc k) p q x))
-ₖ^<_·_>² : (n m k : ℕ)
(p : isEvenT n ⊎ isOddT n)
(q : isEvenT m ⊎ isOddT m)
(x : EM G' k)
→ -ₖ^< n · m > k p q (-ₖ^< n · m > k p q x) ≡ x
-ₖ^< n · m >² k (inl x₁) q x =
-ₖ^< n · m >-inl-left k x₁ q _
∙ -ₖ^< n · m >-inl-left k x₁ q x
-ₖ^< n · m >² k (inr x₁) (inl x₂) x =
-ₖ^< n · m >-inl-right k (inr x₁) x₂ _
∙ -ₖ^< n · m >-inl-right k (inr x₁) x₂ x
-ₖ^< n · m >² k (inr x₁) (inr x₂) x =
-ₖ^< n · m >-inr k x₁ x₂ _
∙ cong -ₖ_ (-ₖ^< n · m >-inr k x₁ x₂ x)
∙ -ₖ² k x
-ₖ^<_·_>²-swap : (n m k : ℕ)
(p : isEvenT n ⊎ isOddT n)
(q : isEvenT m ⊎ isOddT m)
(x : EM G' k)
→ -ₖ^< n · m > k p q (-ₖ^< m · n > k q p x) ≡ x
-ₖ^< n · m >²-swap k (inl p) q x =
-ₖ^< n · m >-inl-left k p q _
∙ -ₖ^< m · n >-inl-right k q p x
-ₖ^< n · m >²-swap k (inr p) (inl q) x =
-ₖ^< n · m >-inl-right k (inr p) q _
∙ -ₖ^< m · n >-inl-left k q (inr p) x
-ₖ^< n · m >²-swap k (inr p) (inr q) x =
-ₖ^< n · m >-inr k p q _
∙ cong -ₖ_ (-ₖ^< m · n >-inr k q p _)
∙ -ₖ² k x
wrap : (n : ℕ) (p : Path (EM G' n) (0ₖ n) (0ₖ n))
→ typ ((Ω^ 2) (EM∙ G' (suc n)))
wrap n p = sym (EM→ΩEM+1-0ₖ n) ∙∙ cong (EM→ΩEM+1 n) p ∙∙ EM→ΩEM+1-0ₖ n
wrapEq : (n : ℕ) (p q : EM→ΩEM+1 {G = G'} n (0ₖ n) ≡ EM→ΩEM+1 n (0ₖ n))
→ p ≡ q → sym (EM→ΩEM+1-0ₖ n) ∙∙ p ∙∙ EM→ΩEM+1-0ₖ n
≡ (sym (EM→ΩEM+1-0ₖ n) ∙∙ q ∙∙ EM→ΩEM+1-0ₖ n)
wrapEq n p q r = cong (sym (EM→ΩEM+1-0ₖ n) ∙∙_∙∙ EM→ΩEM+1-0ₖ n) r
pp-wrap : (n : ℕ) (p : Path (EM G' n) (0ₖ n) (0ₖ n))
→ PathP (λ i → EM→ΩEM+1-0ₖ n i ≡ EM→ΩEM+1-0ₖ n i)
(cong (EM→ΩEM+1 n) p) (wrap n p)
pp-wrap n p =
doubleCompPath-filler
(sym (EM→ΩEM+1-0ₖ n)) (cong (EM→ΩEM+1 n) p) (EM→ΩEM+1-0ₖ n)
cong-cong-ₖ^<_·_>₂ : (n m k : ℕ)
(p : isEvenT n ⊎ isOddT n)
(q : isEvenT m ⊎ isOddT m)
→ (x : Path (EM G' (2 +ℕ k)) (0ₖ (2 +ℕ k)) (0ₖ (2 +ℕ k)))
→ cong (cong (-ₖ^< n · m > (suc (suc (suc k))) p q)) (wrap (suc (suc k)) x)
≡ wrap (suc (suc k)) (cong (-ₖ^< n · m > (suc (suc k)) p q) x)
cong-cong-ₖ^< n · m >₂ k (inl p) q x =
(λ r i j → -ₖ^< n · m >-inl-left (suc (suc (suc k))) p q
(wrap (suc (suc k)) x i j) r)
∙ cong (wrap (suc (suc k)))
λ r i → -ₖ^< n · m >-inl-left (suc (suc k)) p q (x i) (~ r)
cong-cong-ₖ^< n · m >₂ k (inr p) (inl q) x =
(λ r i j → -ₖ^< n · m >-inl-right (suc (suc (suc k))) (inr p) q
(wrap (suc (suc k)) x i j) r)
∙ cong (wrap (suc (suc k)))
λ r i → -ₖ^< n · m >-inl-right (suc (suc k)) (inr p) q (x i) (~ r)
cong-cong-ₖ^< n · m >₂ k (inr p) (inr q) x =
(λ r i j → -ₖ^< n · m >-inr (suc (suc (suc k))) p q
(wrap (suc (suc k)) x i j) r)
∙ rUnit _
∙ (λ r →
((λ i j → cong-₂ (suc k) refl (r ∧ i) j))
∙∙ (λ i j → cong-₂ (suc k) (wrap (suc (suc k)) x i) r j)
∙∙ λ i j → cong-₂ (suc k) refl (r ∧ ~ i) j)
∙ cong₂ (λ x y → x ∙∙ y ∙∙ sym x)
(transportRefl refl)
(sym (sym≡cong-sym (wrap (suc (suc k)) x)))
∙ sym (rUnit _)
∙ cong (wrap (suc (suc k)))
(sym ((λ r i → -ₖ^< n · m >-inr (suc (suc k)) p q (x i) r)
∙ cong-₂ k x))
⌣ₖ-comm-final-lem : (n m k : ℕ)
(p : _) (q : _) (p' : _) (q' : _)
(x : EM G' k)
→ -ₖ^< n · suc m > k p' q x
≡ -ₖ^< suc n · suc m > k p q
(-ₖ -ₖ^< suc n · m > k p q'
(-ₖ^< n · m > k p' q' x))
⌣ₖ-comm-final-lem n m k (inl p) q (inl p') q' x =
⊥.rec (¬evenAndOdd n (p' , p))
⌣ₖ-comm-final-lem n m k (inl p) (inl q) (inr p') (inl q') x =
⊥.rec (¬evenAndOdd m (q' , q))
⌣ₖ-comm-final-lem n m k (inl p) (inl q) (inr p') (inr q') x =
-ₖ^< n · suc m >-inl-right k (inr p') q x
∙ sym (-ₖ^< suc n · suc m >-inl-left k p (inl q) _
∙ cong -ₖ_ (-ₖ^< suc n · m >-inl-left k p (inr q') _
∙ -ₖ^< n · m >-inr k p' q' x)
∙ -ₖ² k x)
⌣ₖ-comm-final-lem n m k (inl p) (inr q) (inr p') (inl q') x =
-ₖ^< n · suc m >-inr k p' q x
∙ cong -ₖ_ (sym (-ₖ^< suc n · m >-inl-left k p (inl q') _
∙ -ₖ^< n · m >-inl-right k (inr p') q' x))
∙ sym (-ₖ^< suc n · suc m >-inl-left k p (inr q) _)
⌣ₖ-comm-final-lem n m k (inl p) (inr q) (inr p') (inr q') x =
⊥.rec (¬evenAndOdd m (q , q'))
⌣ₖ-comm-final-lem n m k (inr p) (inl q) (inl p') (inl q') x =
⊥.rec (¬evenAndOdd (suc m) (q , q'))
⌣ₖ-comm-final-lem n m k (inr p) (inl q) (inl p') (inr q') x =
-ₖ^< n · suc m >-inl-left k p' (inl q) x
∙ sym (-ₖ^< suc n · suc m >-inl-right k (inr p) q _
∙ cong -ₖ_
(-ₖ^< suc n · m >-inr k p q' _
∙ cong -ₖ_ (-ₖ^< n · m >-inl-left k p' (inr q') x))
∙ -ₖ² k x)
⌣ₖ-comm-final-lem n m k (inr p) (inr q) (inl p') (inl q') x =
-ₖ^< n · suc m >-inl-left k p' (inr q) x
∙ sym (-ₖ^< suc n · suc m >-inr k p q _
∙ -ₖ² k _
∙ -ₖ^< suc n · m >-inl-right k (inr p) q' _
∙ -ₖ^< n · m >-inl-left k p' (inl q') x)
⌣ₖ-comm-final-lem n m k (inr p) (inr q) (inl p') (inr q') x =
⊥.rec (¬evenAndOdd m (q , q'))
⌣ₖ-comm-final-lem n m k (inr p) q (inr p') q' x =
⊥.rec (¬evenAndOdd (suc n) (p' , p))
-ₖ^<_·_>-Induced : ∀ {ℓ ℓ'} {G : AbGroup ℓ} {H : AbGroup ℓ'}
(n m k : ℕ) (p : isEvenT n ⊎ isOddT n) (q : isEvenT m ⊎ isOddT m)
(ϕ : AbGroupHom G H) (x : EM G k)
→ inducedFun-EM ϕ k (-ₖ^< n · m > k p q x)
≡ -ₖ^< n · m > k p q (inducedFun-EM ϕ k x)
-ₖ^< n · m >-Induced k (inl x₁) q ϕ x =
cong (inducedFun-EM ϕ k) (-ₖ^< n · m >-inl-left k x₁ q x)
∙ sym (-ₖ^< n · m >-inl-left k x₁ q _)
-ₖ^< n · m >-Induced k (inr x₁) (inl x₂) ϕ x =
cong (inducedFun-EM ϕ k) (-ₖ^< n · m >-inl-right k (inr x₁) x₂ x)
∙ sym (-ₖ^< n · m >-inl-right k (inr x₁) x₂ _)
-ₖ^< n · m >-Induced k (inr x₁) (inr x₂) ϕ x =
cong (inducedFun-EM ϕ k) (-ₖ^< n · m >-inr k x₁ x₂ x)
∙ inducedFun-EM-pres-ₖ ϕ k x
∙ sym (-ₖ^< n · m >-inr k x₁ x₂ _)
⌣ₖ-comm-ind : {G' : AbGroup ℓ} {H' : AbGroup ℓ'} (n m : ℕ)
→ (f g : EM∙ G' (suc n) →∙ (EM∙ H' (suc m)
→∙ EM∙ (G' ⨂ H') (suc n +' suc m) ∙))
→ ((x : EM-raw' G' (suc n)) (y : EM-raw' H' (suc m))
→ f .fst (EM-raw'→EM _ _ x) .fst (EM-raw'→EM _ _ y)
≡ g .fst (EM-raw'→EM _ _ x) .fst (EM-raw'→EM _ _ y))
→ f ≡ g
⌣ₖ-comm-ind {G' = G'} {H' = H'} n m f g ind =
→∙Homogeneous≡ (isHomogeneous→∙ (isHomogeneousEM _))
(funExt (EM-raw'-elim G' (suc n)
(λ _ → isOfHLevelPath' (suc (suc n))
(isOfHLevel↑∙ (suc n) m) _ _)
λ x → →∙Homogeneous≡ (isHomogeneousEM _)
(funExt λ y i → f'≡g' y i .fst x)))
where
f' : EM H' (suc m)
→ EM-raw'∙ G' (suc n) →∙ EM∙ (G' ⨂ H') (suc n +' suc m)
fst (f' y) x = f .fst (EM-raw'→EM _ _ x) .fst y
snd (f' y) = cong (λ x → f .fst x .fst y) (EM-raw'→EM∙ G' (suc n))
∙ funExt⁻ (cong fst (f .snd)) y
g' : EM H' (suc m)
→ EM-raw'∙ G' (suc n) →∙ EM∙ (G' ⨂ H') (suc n +' suc m)
fst (g' y) x = g .fst (EM-raw'→EM _ _ x) .fst y
snd (g' y) = cong (λ x → g .fst x .fst y) (EM-raw'→EM∙ G' (suc n))
∙ funExt⁻ (cong fst (g .snd)) y
f'≡g' : (x : EM H' (suc m)) → f' x ≡ g' x
f'≡g' = EM-raw'-elim H' (suc m)
(λ _ → isOfHLevelPath' (suc (suc m))
(subst (λ x → isOfHLevel (2 +ℕ suc m)
(EM-raw'∙ G' (suc n) →∙ EM∙ (G' ⨂ H') x))
(cong suc (+-comm m (suc n)))
(isOfHLevel↑∙' {G = G'} {H = G' ⨂ H'} (suc m) (suc n))) _ _)
λ y → →∙Homogeneous≡ (isHomogeneousEM _)
(funExt λ x → ind x y)
module _ {G' : AbGroup ℓ} {H' : AbGroup ℓ'} where
cp∙ : (n m : ℕ) → EM G' n → EM∙ H' m →∙ EM∙ (G' ⨂ H') (n +' m)
cp∙ = cup∙
cp∙∙ : (n m : ℕ) → EM∙ G' n →∙ (EM∙ H' m →∙ EM∙ (G' ⨂ H') (n +' m) ∙)
fst (cp∙∙ n m) = cp∙ n m
snd (cp∙∙ n m) = ptd
where
abstract
ptd : cp∙ n m (snd (EM∙ G' n)) ≡ ((λ _ → 0ₖ (n +' m)) , refl)
ptd = →∙Homogeneous≡ (isHomogeneousEM _)
(funExt λ y → 0ₖ-⌣ₖ n m y)
cp'∙ : (n m : ℕ) → EM G' n → EM∙ H' m →∙ EM∙ (H' ⨂ G') (m +' n)
fst (cp'∙ n m x) y = y ⌣ₖ x
snd (cp'∙ n m x) = 0ₖ-⌣ₖ m n x
comm⨂-EM : (n : ℕ) → EM (H' ⨂ G') n → EM (G' ⨂ H') n
comm⨂-EM n = Iso.fun (Iso→EMIso ⨂-comm _)
comm⨂-EM' : (n : ℕ) → EM (H' ⨂ G') n → EM (G' ⨂ H') n
comm⨂-EM' = λ n → Iso.inv (Iso→EMIso ⨂-comm _)
comm⨂-EM≡comm⨂-EM' : (n : ℕ) (x : _) → comm⨂-EM n x ≡ comm⨂-EM' n x
comm⨂-EM≡comm⨂-EM' n =
funExt⁻ (cong (λ F → inducedFun-EM F n) h)
where
h : Path (AbGroupHom (H' ⨂ G') (G' ⨂ H'))
(GroupEquiv→GroupHom ⨂-comm)
(GroupEquiv→GroupHom (invGroupEquiv ⨂-comm))
h = Σ≡Prop (λ _ → isPropIsGroupHom _ _) refl
cong-cong-comm⨂-EM : (n : ℕ) (p : fst (Ω (EM∙ (H' ⨂ G') (suc (suc n)))))
→ cong (cong (comm⨂-EM (suc (suc (suc n))))) (wrap (suc (suc n)) p)
≡ wrap (suc (suc n)) (cong (comm⨂-EM (suc (suc n))) p)
cong-cong-comm⨂-EM n p =
TR.rec (hLevelEM _ (suc (suc (suc n))) _ _ _ _ _ _)
(uncurry (λ x q k i j
→ hcomp (λ r → λ {
(i = i0) → comm⨂-EM (suc (suc (suc n)))
(EM→ΩEM+1-0ₖ (suc (suc n)) (r ∨ k) j)
; (i = i1) → comm⨂-EM (suc (suc (suc n)))
(EM→ΩEM+1-0ₖ (suc (suc n)) (r ∨ k) j)
; (j = i0) → 0ₖ (3 +ℕ n)
; (j = i1) → 0ₖ (3 +ℕ n)
; (k = i0) → comm⨂-EM (suc (suc (suc n)))
(pp-wrap (2 +ℕ n) (q r) r i j)
; (k = i1) → wrap (suc (suc n))
(cong (comm⨂-EM (suc (suc n))) (q r)) i j})
(hcomp (λ r → λ {(i = i0) → comm⨂-EM (suc (suc (suc n)))
∣ rCancel-filler (merid north) r k j ∣ₕ
; (i = i1) → comm⨂-EM (suc (suc (suc n)))
∣ rCancel-filler (merid north) r k j ∣ₕ
; (j = i0) → 0ₖ (3 +ℕ n)
; (j = i1) → ∣ merid north (~ r ∧ ~ k) ∣ₕ
; (k = i0) → comm⨂-EM (suc (suc (suc n)))
∣ compPath-filler (merid (x i))
(sym (merid north)) r j ∣ₕ
; (k = i1) → wrap (suc (suc n)) (cong (comm⨂-EM (suc (suc n)))
(cong ∣_∣ₕ x)) i j})
(hcomp (λ r → λ {
(i = i0) → ∣ rCancel-filler (merid north) (~ r) k j ∣ₕ
; (i = i1) → ∣ rCancel-filler (merid north) (~ r) k j ∣ₕ
; (j = i0) → 0ₖ (3 +ℕ n)
; (j = i1) → ∣ merid north (r ∧ ~ k) ∣ₕ
; (k = i0) → ∣ compPath-filler
(merid (inducedFun-EM-raw
(GroupEquiv→GroupHom ⨂-comm)
(suc (suc n)) (x i)))
(sym (merid north)) (~ r) j ∣ₕ
; (k = i1) → wrap (suc (suc n))
(cong (comm⨂-EM (suc (suc n))) (cong ∣_∣ₕ x)) i j})
(pp-wrap (suc (suc n))
(cong (comm⨂-EM (suc (suc n))) (cong ∣_∣ₕ x)) k i j)))))
(ind-lem n p)
where
ind-lem : (n : ℕ) (p : fst (Ω (EM∙ (H' ⨂ G') (suc (suc n)))))
→ hLevelTrunc (2 +ℕ n)
(Σ[ x ∈ north ≡ north ] cong ∣_∣ₕ x ≡ p)
ind-lem n p =
TR.map (λ {(x , p) → (fst (ind-lem₃ n x))
, (sym (snd (ind-lem₃ n x)) ∙ p)})
(ind-lem₂ n p (ind-lem₁ n p))
where
ind-lem₁ : (n : ℕ) (p : fst (Ω (EM∙ (H' ⨂ G') (suc (suc n)))))
→ Σ[ x ∈ EM (H' ⨂ G') (suc n) ] EM→ΩEM+1 (suc n) x ≡ p
ind-lem₁ n p = _ , Iso.rightInv (Iso-EM-ΩEM+1 (suc n)) p
ind-lem₂ : (n : ℕ) (p : fst (Ω (EM∙ (H' ⨂ G') (suc (suc n)))))
→ Σ[ x ∈ EM (H' ⨂ G') (suc n) ] EM→ΩEM+1 (suc n) x ≡ p
→ hLevelTrunc (2 +ℕ n) (Σ[ x ∈ EM-raw' (H' ⨂ G') (suc n) ]
EM→ΩEM+1 (suc n) (EM-raw'→EM _ _ x) ≡ p)
ind-lem₂ n p =
uncurry (EM-raw'-elim _ _
(λ _ → isOfHLevelΠ (2 +ℕ n)
λ _ → (isOfHLevelTrunc (2 +ℕ n)))
λ x → J (λ p _ → hLevelTrunc (2 +ℕ n)
(Σ[ x ∈ EM-raw' (H' ⨂ G') (suc n) ]
EM→ΩEM+1 (suc n) (EM-raw'→EM _ _ x) ≡ p))
∣ x , refl ∣)
ind-lem₃ : (n : ℕ)
(x : EM-raw' (H' ⨂ G') (suc n))
→ Σ[ r ∈ north ≡ north ] EM→ΩEM+1 (suc n) (EM-raw'→EM _ _ x)
≡ cong ∣_∣ₕ r
ind-lem₃ zero x = _ , refl
ind-lem₃ (suc n) x = _ , refl
cp'∙∙ : (n m : ℕ) (p : isEvenT n ⊎ isOddT n) (q : isEvenT m ⊎ isOddT m)
→ EM∙ G' n →∙ (EM∙ H' m →∙ EM∙ (G' ⨂ H') (n +' m) ∙)
fst (fst (cp'∙∙ n m p q) x) y =
subst (EM (G' ⨂ H')) (+'-comm m n)
(-ₖ^< n · m > (m +' n) p q
(comm⨂-EM (m +' n) (y ⌣ₖ x)))
snd (fst (cp'∙∙ n m p q) x) =
cong (subst (EM (G' ⨂ H')) (+'-comm m n))
(cong (-ₖ^< n · m > (m +' n) p q)
(cong (comm⨂-EM (m +' n)) (0ₖ-⌣ₖ m n x) ∙ Iso→EMIso∙ ⨂-comm _)
∙ -ₖ^< n · m >∙ (m +' n) p q)
∙ substCommSlice (λ _ → EM (G' ⨂ H') (n +' m))
(EM (G' ⨂ H')) (λ n _ → 0ₖ n) (+'-comm m n) (0ₖ _)
snd (cp'∙∙ n m p q) =
→∙Homogeneous≡ (isHomogeneousEM _)
(funExt λ y
→ cong (subst (EM (G' ⨂ H')) (+'-comm m n))
(cong (-ₖ^< n · m > (m +' n) p q)
(cong (comm⨂-EM (m +' n)) (⌣ₖ-0ₖ m n y) ∙ Iso→EMIso∙ ⨂-comm _)
∙ -ₖ^< n · m >∙ (m +' n) p q)
∙ substCommSlice (λ _ → EM (G' ⨂ H') (n +' m))
(EM (G' ⨂ H')) (λ n _ → 0ₖ n) (+'-comm m n) (0ₖ _))
flip-⌣ₖ-comm : (n m n' m' : ℕ) (l r : ℕ) (s : l ≡ r)
(x : EM (H' ⨂ G') l) (y : EM (G' ⨂ H') r)
(p : isEvenT n ⊎ isOddT n) (q : isEvenT m ⊎ isOddT m)
(p' : isEvenT n' ⊎ isOddT n') (q' : isEvenT m' ⊎ isOddT m')
→ y ≡ subst (EM (G' ⨂ H')) s (-ₖ^< n' · m' > l p' q' (comm⨂-EM l x))
→ subst (EM (G' ⨂ H')) s (-ₖ^< n · m > l p q (comm⨂-EM l x))
≡ -ₖ^< n · m > r p q (-ₖ^< n' · m' > r p' q' y)
flip-⌣ₖ-comm n m n' m' l = J> λ x y p q p' q' id
→ transportRefl _
∙ sym (cong (-ₖ^< n · m > l p q) (-ₖ^< n' · m' >² l p' q' (comm⨂-EM l x)))
∙ sym (cong (λ y → -ₖ^< n · m > l p q
(-ₖ^< n' · m' > l p' q' y))
(id ∙ transportRefl _))
⌣ₖ-comm₀ : (m : ℕ) (x : fst G') (y : EM H' (suc m))
→ cp∙ zero (suc m) x .fst y
≡ comm⨂-EM (suc m) (cp'∙ zero (suc m) x .fst y)
⌣ₖ-comm₀ zero x =
EM-raw'-elim _ 1 (λ _ → hLevelEM _ 1 _ _)
λ { embase-raw → refl
; (emloop-raw g i) j → EM→ΩEM+1 _ (x ⊗ g) i}
⌣ₖ-comm₀ (suc m) x =
TR.elim
(λ _ → isOfHLevelPath (4 +ℕ m) (hLevelEM _ (suc (suc m))) _ _)
λ { north → refl
; south → refl
; (merid a i) j → lem a j i}
where
lem : (a : EM-raw H' (suc m))
→ EM→ΩEM+1 (suc m) (cp∙ zero (suc m) x .fst (EM-raw→EM _ _ a))
≡ cong (comm⨂-EM (suc (suc m)))
(EM→ΩEM+1 (suc m) (cp'∙ zero (suc m) x
.fst (EM-raw→EM _ _ a)))
lem a = cong (EM→ΩEM+1 (suc m)) (⌣ₖ-comm₀ m x (EM-raw→EM _ _ a))
∙ EMFun-EM→ΩEM+1 (suc m) (cp'∙ zero (suc m) x .fst (EM-raw→EM _ _ a))
⌣ₖ-comm₁ : (m : ℕ) (p : isEvenT m ⊎ isOddT m)
→ cp∙∙ 1 m ≡ cp'∙∙ 1 m (inr tt) p
⌣ₖ-comm₁ =
elim+2 (λ { (inl x) →
→∙Homogeneous≡
(isHomogeneous→∙ (isHomogeneousEM _))
(funExt (EM-raw'-elim _ 1
(λ _ → isOfHLevelPath' 2 (isOfHLevel→∙ 3 (hLevelEM _ 1)) _ _)
λ x → →∙Homogeneous≡ (isHomogeneousEM _)
(funExt λ y
→ 1-0 x y
∙∙ sym (-ₖ^< 1 · zero >-inl-right 1
(inr tt) tt (comm⨂-EM 1
(_⌣ₖ_ {n = zero} {m = suc zero}
y (EM-raw'→EM G' (suc zero) x))))
∙∙ sym (transportRefl
(-ₖ^< 1 · zero > 1 (inr tt) (inl tt)
(comm⨂-EM 1 (_⌣ₖ_ {n = zero} {m = suc zero} y
(EM-raw'→EM G' (suc zero) x))))))))})
(λ { (inr x)
→ ⌣ₖ-comm-ind 0 0 _ _
λ x y → 1-1 x y
∙ sym (transportRefl
(-ₖ^< 1 · 1 > 2 (inr tt) (inr tt)
(comm⨂-EM 2 (_⌣ₖ_ {n = suc zero} {m = suc zero}
(EM-raw'→EM H' (suc zero) y)
(EM-raw'→EM G' (suc zero) x)))))})
λ { n ind p → ⌣ₖ-comm-ind 0 (suc n) _ _ (1-suc n ind p)}
where
1-0 : (x : EM-raw' G' 1) (y : fst H')
→ cp∙ 1 0 (EM-raw'→EM G' 1 x) .fst y
≡ comm⨂-EM 1 (_⌣ₖ_ {n = zero} y (EM-raw'→EM G' 1 x))
1-0 embase-raw y = refl
1-0 (emloop-raw g i) y = refl
1-1 : (x : EM-raw' G' 1) (y : EM-raw' H' 1)
→ cp∙∙ 1 1 .fst (EM-raw'→EM G' (suc 0) x) .fst (EM-raw'→EM H' (suc 0) y)
≡ -ₖ^< 1 · 1 > 2 (inr tt) (inr tt)
(comm⨂-EM 2
(_⌣ₖ_ {n = suc zero} {m = suc zero}
(EM-raw'→EM H' 1 y) (EM-raw'→EM G' 1 x)))
1-1 x embase-raw = ⌣ₖ-0ₖ (suc zero) (suc zero) (EM-raw'→EM _ 1 x)
1-1 x (emloop-raw h i) = flipSquare PP2 i
where
PP1 : (x : EM-raw' G' 1)
→ PathP (λ i → Path (EM (G' ⨂ H') 2)
(⌣ₖ-0ₖ (suc zero) (suc zero) (EM-raw'→EM _ 1 x) i)
(⌣ₖ-0ₖ (suc zero) (suc zero) (EM-raw'→EM _ 1 x) i))
(cong (cp∙∙ 1 1 .fst (EM-raw'→EM G' (suc 0) x) .fst)
(emloop h))
(EM→ΩEM+1 (suc 0)
(-ₖ (comm⨂-EM 1
(_⌣ₖ_ {n = zero} {m = suc zero} h (EM-raw'→EM G' 1 x)))))
PP1 embase-raw = sym (EM→ΩEM+1-0ₖ 1)
PP1 (emloop-raw g i) j k =
hcomp (λ r → λ {
(i = i0) → ∣ rCancel (merid embase) (~ j ∨ ~ r) k ∣
; (i = i1) → ∣ rCancel (merid embase) (~ j ∨ ~ r) k ∣
; (j = i0) → pp-wrap 1 (λ i →
(_⌣ₖ_ {n = zero} {m = 1} g (emloop h i)))
(~ r) k i
; (j = i1) → pp-wrap 1 (λ i →
(-ₖ (comm⨂-EM 1 (_⌣ₖ_ {n = zero} {m = 1} h (emloop g i)))))
(~ r) i k
; (k = i0) → ⌣ₖ-0ₖ 1 1 (emloop g i) (j ∨ ~ r)
; (k = i1) → ⌣ₖ-0ₖ 1 1 (emloop g i) (j ∨ ~ r)})
(help j i k)
where
help : flipSquare (wrap 1 λ i → (_⌣ₖ_ {n = zero} {m = 1} g (emloop h i)))
≡ wrap 1 λ i →
(-ₖ (comm⨂-EM 1 (_⌣ₖ_ {n = zero} {m = 1} h (emloop g i))))
help =
sym (sym≡flipSquare (wrap 1 λ i →
(_⌣ₖ_ {n = zero} {m = 1} g (emloop h i))))
∙ wrapEq 1 _ _
(cong (cong (EM→ΩEM+1 1))
(sym (cong-₁ (λ i →
comm⨂-EM 1 (_⌣ₖ_ {n = zero} {m = 1} h (emloop g i))))))
PP2 : PathP (λ i → Path (EM (G' ⨂ H') 2)
(⌣ₖ-0ₖ (suc zero) (suc zero) (EM-raw'→EM _ 1 x) i)
(⌣ₖ-0ₖ (suc zero) (suc zero) (EM-raw'→EM _ 1 x) i))
(cong (cp∙∙ 1 1 .fst (EM-raw'→EM G' (suc 0) x) .fst)
(emloop h))
(λ i → -ₖ^< 1 · 1 > 2 (inr tt) (inr tt)
(comm⨂-EM 2 (EM→ΩEM+1 (suc zero)
(_⌣ₖ_ {n = zero} {m = suc zero} h (EM-raw'→EM G' 1 x)) i)))
PP2 = PP1 x
▷ (sym (cong-ₖ^< 1 · 1 >₂ 0 (inr tt) (inr tt)
(comm⨂-EM 1
(_⌣ₖ_ {n = zero} {m = suc zero} h (EM-raw'→EM G' 1 x)))))
∙ cong (cong (-ₖ^< 1 · 1 > 2 (inr tt) (inr tt)))
(EMFun-EM→ΩEM+1 1 (_⌣ₖ_ {n = zero} {m = suc zero}
h (EM-raw'→EM G' 1 x)))
module _ (n : ℕ)
(ind : (p : isEvenT (suc n) ⊎ isOddT (suc n))
→ cp∙∙ 1 (suc n) ≡ cp'∙∙ 1 (suc n) (inr tt) p)
where
south-c : (x : EM-raw' G' 1) (p : _)
→ cp∙∙ 1 (suc (suc n)) .fst (EM-raw'→EM G' (suc 0) x) .fst
(EM-raw'→EM H' (suc (suc n)) south)
≡ cp'∙∙ 1 (suc (suc n)) (inr tt) p .fst (EM-raw'→EM G' (suc 0) x)
.fst (EM-raw'→EM H' (suc (suc n)) south)
south-c embase-raw p = refl
south-c (emloop-raw g i) p j = EM→ΩEM+1-0ₖ (suc (suc n)) j i
1-suc : (p : isEvenT (suc (suc n)) ⊎ isOddT (suc (suc n)))
(x : EM-raw' G' (suc zero))
(y : EM-raw' H' (suc (suc n)))
→ cp∙∙ 1 (suc (suc n)) .fst
(EM-raw'→EM G' (suc 0) x) .fst
(EM-raw'→EM H' (suc (suc n)) y)
≡ cp'∙∙ 1 (suc (suc n)) (inr tt) p .fst
(EM-raw'→EM G' (suc 0) x) .fst
(EM-raw'→EM H' (suc (suc n)) y)
1-suc p x north = ⌣ₖ-0ₖ 1 (suc (suc n)) (EM-raw'→EM G' (suc 0) x)
1-suc p x south = south-c x p
1-suc p x (merid a i) j = main j i
where
good : (x : EM-raw' G' 1)
→ PathP (λ i₁ →
⌣ₖ-0ₖ 1 (suc (suc n)) (EM-raw'→EM G' (suc 0) x) i₁ ≡
south-c x p i₁)
(λ i₁ →
cp∙∙ 1 (suc (suc n)) .fst (EM-raw'→EM G' (suc 0) x) .fst
(EM-raw'→EM H' (suc (suc n)) (merid a i₁)))
(sym
(EM→ΩEM+1 (suc (suc n))
(fst (fst (cp∙∙ 1 (suc n)) (EM-raw'→EM G' 1 x))
(EM-raw→EM H' (suc n) a))))
good embase-raw j k = EM→ΩEM+1-0ₖ (suc (suc n)) (~ j) (~ k)
good (emloop-raw g i) j k =
hcomp (λ r → λ {
(i = i0) → ∣ rCancel (merid north) (~ j ∨ ~ r) (~ k) ∣
; (i = i1) → ∣ rCancel (merid north) (~ j ∨ ~ r) (~ k) ∣
; (j = i0) → l-f (~ r) k i
; (j = i1) → l-f (~ r) i (~ k)
; (k = i0) → ∣ rCancel (merid north) (j ∨ ~ r) i ∣
; (k = i1) → ∣ rCancel (merid north) (j ∨ ~ r) i ∣})
(flipSq-lem j i k)
where
l-f = pp-wrap (suc (suc n))
(EM→ΩEM+1 (suc n)
(_⌣ₖ_ {G' = G'} {H' = H'} {n = 0} {m = suc n}
g (EM-raw→EM H' (suc n) a)))
l = wrap (suc (suc n)) (EM→ΩEM+1 (suc n)
(_⌣ₖ_ {n = 0} {m = suc n} g (EM-raw→EM H' (suc n) a)))
flipSq-lem : flipSquare l ≡ cong sym l
flipSq-lem = sym (sym≡flipSquare l) ∙ sym≡cong-sym l
-ₖ-lem : (n : ℕ) (p : _) (q : _)
→ -ₖ^< 1 · suc (suc n) > (1 +' suc n) (inr tt) p
∘ (-ₖ^< 1 · suc n > (1 +' suc n) (inr tt) q)
≡ -ₖ_ {G = G' ⨂ H'}
-ₖ-lem n (inl x) (inl x₁) = ⊥.rec (¬evenAndOdd _ (x , x₁))
-ₖ-lem n (inl x) (inr x₁) i z =
-ₖ^< 1 · suc (suc n) >-inl-right (1 +' suc n) (inr tt) x
(-ₖ^< 1 · suc n >-inr (1 +' suc n) tt x₁ z i) i
-ₖ-lem n (inr x) (inl x₁) i z =
-ₖ^< 1 · suc (suc n) >-inr (1 +' suc n) tt x
(-ₖ^< 1 · suc n >-inl-right (1 +' suc n) (inr tt) x₁ z i) i
-ₖ-lem n (inr x) (inr x₁) = ⊥.rec (¬evenAndOdd _ (x , x₁))
main : PathP (λ i → ⌣ₖ-0ₖ 1 (suc (suc n)) (EM-raw'→EM G' (suc 0) x) i
≡ south-c x p i)
(λ i → cp∙∙ 1 (suc (suc n)) .fst (EM-raw'→EM G' (suc 0) x) .fst
(EM-raw'→EM H' (suc (suc n)) (merid a i)))
λ i → cp'∙∙ 1 (suc (suc n)) (inr tt) p .fst
(EM-raw'→EM G' (suc 0) x) .fst
(EM-raw'→EM H' (suc (suc n)) (merid a i))
main = good x
▷ ((sym (EM→ΩEM+1-sym (suc (suc n))
(fst (fst (cp∙∙ 1 (suc n)) (EM-raw'→EM G' 1 x))
(EM-raw→EM H' (suc n) a)))
∙ cong (EM→ΩEM+1 (suc (suc n)))
(sym (funExt⁻
(-ₖ-lem n p (evenOrOdd (suc n)))
((fst (fst (cp∙∙ 1 (suc n)) (EM-raw'→EM G' 1 x))
(EM-raw→EM H' (suc n) a))))))
∙ cong (EM→ΩEM+1 (suc (suc n)))
(sym (flip-⌣ₖ-comm 1 (suc (suc n)) 1 (suc n) _ _
(+'-comm (suc n) 1) a⌣x (_⌣ₖ_ {n = 1} {m = suc n}
(EM-raw'→EM G' (suc zero) x) (EM-raw→EM H' (suc n) a))
(inr tt) p (inr tt) (evenOrOdd (suc n))
(funExt⁻
(cong fst (funExt⁻
(cong fst (ind (evenOrOdd (suc n))))
(EM-raw'→EM G' 1 x))) (EM-raw→EM H' (suc n) a))))
∙ sym main-lem
∙ cong (cong (subst (EM (G' ⨂ H')) (+'-comm (suc (suc n)) 1)))
(sym (cong-ₖ^< 1 · (suc (suc n)) >₂ (suc (n +ℕ 0)) (inr tt) p
(comm⨂-EM (suc (suc (n +ℕ 0))) a⌣x))
∙ cong (cong (-ₖ^< 1 · (suc (suc n)) > ((suc (suc n)) +' 1)
(inr tt) p))
(EMFun-EM→ΩEM+1 _ a⌣x)))
where
a⌣x = _⌣ₖ_ {n = suc n} {m = 1}
(EM-raw→EM H' (suc n) a) (EM-raw'→EM G' (suc zero) x)
mid = EM→ΩEM+1 (suc (suc (n +ℕ 0)))
(-ₖ^< 1 · suc (suc n) > (suc (suc (n +ℕ 0))) (inr tt) p
(comm⨂-EM (suc (suc (n +ℕ 0))) a⌣x))
substPres0 : (n₁ m : ℕ) (q : n₁ ≡ m)
→ snd (EM∙ (G' ⨂ H') m)
≡ subst (λ n₂ → EM∙ (G' ⨂ H') n₂ .fst) q (snd (EM∙ (G' ⨂ H') n₁))
substPres0 n = J> sym (transportRefl _)
substPres0-refl : (n : ℕ)
→ sym (substPres0 (suc (suc n +' 1)) (suc (1 +' suc n))
(+'-comm (suc (suc n)) 1))
≡ refl
substPres0-refl n = transportRefl _
lem = cong-subst-lem _ _
(+'-comm (suc n) 1) (+'-comm (suc (suc n)) 1) (EM∙ (G' ⨂ H'))
EM→ΩEM+1 (λ x → EM→ΩEM+1-0ₖ _)
((-ₖ^< 1 · suc (suc n) > (suc (suc (n +ℕ 0))) (inr tt) p
(comm⨂-EM (suc (suc (n +ℕ 0))) a⌣x)))
substPres0
λ n → transportRefl (sym (transportRefl _))
main-lem : cong (subst (EM (G' ⨂ H')) (+'-comm (suc (suc n)) 1)) mid
≡ EM→ΩEM+1 (suc (suc n))
(subst (EM (G' ⨂ H')) (+'-comm (suc n) 1)
(-ₖ^< 1 · suc (suc n) > (suc (suc (n +ℕ 0))) (inr tt) p
(comm⨂-EM (suc (suc (n +ℕ 0))) a⌣x)))
main-lem = lem
∙ cong (λ x → sym x
∙∙ EM→ΩEM+1 (suc (suc n))
(subst (EM (G' ⨂ H')) (+'-comm (suc n) 1)
(-ₖ^< 1 · suc (suc n) > (suc (suc (n +ℕ 0))) (inr tt) p
(comm⨂-EM (suc (suc (n +ℕ 0))) a⌣x)))
∙∙ x) (substPres0-refl n)
∙ sym (rUnit _)
module _ {G' : AbGroup ℓ} {H' : AbGroup ℓ'} where
cp∙∙₂ = cp∙∙ {G' = G'} {H' = H'}
cp'∙∙₂ = cp'∙∙ {G' = G'} {H' = H'}
⌣ₖ-comm₂ : (n m : ℕ)
(p : isEvenT (suc n) ⊎ isOddT (suc n))
(q : isEvenT (suc m) ⊎ isOddT (suc m))
→ cp∙∙₂ (suc n) (suc m) ≡ cp'∙∙₂ (suc n) (suc m) p q
⌣ₖ-comm₂ =
⌣-ℕ-elim2
(λ {n (inr tt) q → ⌣ₖ-comm₁ (suc n) q})
(λ {n p (inr tt) →
→∙Homogeneous≡ (isHomogeneous→∙ (isHomogeneousEM _))
(funExt λ x → →∙Homogeneous≡ (isHomogeneousEM _)
(funExt λ y
→ (((sym (Iso.leftInv (Iso→EMIso (⨂-comm {A = G'} {B = H'})
(suc (suc n) +' 1)) (cp∙∙₂ (suc (suc n)) 1 .fst x .fst y))
∙ sym (comm⨂-EM≡comm⨂-EM' {G' = G'} {H' = H'} (suc (suc n) +' 1) _)
∙ cong (comm⨂-EM _) (sym (-ₖ^< suc (suc n) · 1 >²-swap _ p (inr tt) _))
∙ -ₖ^< suc (suc n) · 1 >-Induced (suc (suc n) +' 1) p (inr tt)
(GroupEquiv→GroupHom ⨂-comm) _)
∙ sym (transportRefl _)
∙ sym (compSubstℕ {A = EM (G' ⨂ H')} (+'-comm (suc (suc n)) 1)
(+'-comm 1 (suc (suc n))) refl))
∙ cong (subst (EM (G' ⨂ H')) (+'-comm 1 (suc (suc n))))
(substCommSlice (EM (H' ⨂ G')) (EM (G' ⨂ H'))
(λ k x → -ₖ^< suc (suc n) · 1 > k p (inr tt)
(comm⨂-EM k x))
(+'-comm (suc (suc n)) 1)
(-ₖ^< 1 · suc (suc n) > _ (inr tt) p (comm⨂-EM _ (x ⌣ₖ y)))))
∙ λ i → subst (EM (G' ⨂ H')) (+'-comm 1 (suc (suc n)))
(-ₖ^< suc (suc n) · 1 > (1 +' suc (suc n)) p (inr tt)
(comm⨂-EM (1 +' suc (suc n)) (⌣ₖ-comm₁ {G' = H'} {H' = G'}
(suc (suc n)) p (~ i) .fst y .fst x)))))})
λ n m ind p q → ⌣ₖ-comm-ind _ _ _ _
λ x y → main n m (ind .fst) (ind .snd .fst) (ind .snd .snd) p q x y
∙ λ i → subst (EM (G' ⨂ H'))
(isSetℕ _ _
(cong (3 +ℕ_) (+-comm m (suc n) ∙ sym (+-suc n m)))
(+'-comm (suc (suc m)) (suc (suc n))) i)
(-ₖ^< (2 +ℕ n) · (2 +ℕ m) > ((2 +ℕ m) +' (2 +ℕ n)) p q
(comm⨂-EM ((2 +ℕ m) +' (2 +ℕ n))
(_⌣ₖ_ {n = suc (suc m)} {m = suc (suc n)}
(EM-raw'→EM H' (suc (suc m)) y)
(EM-raw'→EM G' (suc (suc n)) x))))
where
module _ (n m : ℕ)
(indₗ : ((p₁ : isEvenT (suc (suc n)) ⊎ isOddT (suc (suc n)))
(q₁ : isEvenT (suc m) ⊎ isOddT (suc m))
→ cp∙∙ (suc (suc n)) (suc m) ≡ cp'∙∙ (suc (suc n)) (suc m) p₁ q₁))
(indᵣ : ((p₁ : isEvenT (suc n) ⊎ isOddT (suc n))
(q₁ : isEvenT (suc (suc m)) ⊎ isOddT (suc (suc m)))
→ cp∙∙ (suc n) (suc (suc m)) ≡ cp'∙∙ (suc n) (suc (suc m)) p₁ q₁))
(indₘ : (p₁ : isEvenT (suc n) ⊎ isOddT (suc n))
(q₁ : isEvenT (suc m) ⊎ isOddT (suc m)) →
cp∙∙ (suc n) (suc m) ≡ cp'∙∙ (suc n) (suc m) p₁ q₁)
where
ℕpath = cong (2 +ℕ_) (+-comm m (suc n) ∙ sym (+-suc n m))
Fᵣ : (p : _) (q : _)
→ EM (H' ⨂ G') (suc (suc m) +' suc n)
→ EM (G' ⨂ H') (2 +ℕ (n +ℕ suc m))
Fᵣ p q x =
subst (EM (G' ⨂ H')) (cong (2 +ℕ_) (+-comm (suc m) n))
(-ₖ^< suc n · suc (suc m) > (suc (suc m) +' (suc n)) p q
(comm⨂-EM (suc (suc m) +' suc n) x))
Fₗ : (p : _) (q : _)
→ EM (G' ⨂ H') (suc (suc n) +' suc m)
→ EM (H' ⨂ G') (suc m +' suc (suc n))
Fₗ p q x =
Iso.inv (Iso→EMIso ⨂-comm _)
(-ₖ^< suc (suc n) · suc m > _ p q
(subst (EM (G' ⨂ H'))
(cong (2 +ℕ_) (sym (+-comm m (suc n)))) x))
improveᵣ : (p : _) (q : _) (x : EM G' (suc n))
(y : EM H' (suc (suc m)))
→ _⌣ₖ_ {n = suc n} {m = (suc (suc m))} x y
≡ Fᵣ p q (_⌣ₖ_ {n = suc (suc m)} {m = suc n} y x)
improveᵣ p q x y =
(λ i → indᵣ p q i .fst x .fst y)
∙ λ i → subst (EM (G' ⨂ H'))
(isSetℕ _ _ (+'-comm (suc (suc m)) (suc n))
(cong (2 +ℕ_) (+-comm (suc m) n)) i)
(-ₖ^< suc n · suc (suc m) > (suc (suc m) +' (suc n)) p q
(comm⨂-EM (suc (suc m) +' suc n)
(_⌣ₖ_ {n = suc (suc m)} {m = suc n} y x)))
improveₗ : (p : _) (q : _) (x : EM G' (suc (suc n))) (y : EM H' (suc m))
→ _⌣ₖ_ {n = suc m} {m = suc (suc n)} y x
≡ Fₗ p q (_⌣ₖ_ {n = suc (suc n)} {m = (suc m)} x y)
improveₗ p q x y =
(sym (Iso.leftInv (Iso→EMIso ⨂-comm (suc m +' (suc (suc n)))) (y ⌣ₖ x))
∙ cong (Iso.inv (Iso→EMIso ⨂-comm (suc m +' (suc (suc n)))))
(sym (-ₖ^< suc (suc n) · suc m >² _ p q _)
∙ cong (-ₖ^< suc (suc n) · suc m > _ p q)
(sym (transport⁻Transport
(λ i → EM (G' ⨂ H') (2 +ℕ (+-comm m (suc n) i))) _)
∙ cong (subst (EM (G' ⨂ H'))
(cong (2 +ℕ_) (sym (+-comm m (suc n)))))
λ i → subst (EM (G' ⨂ H'))
(isSetℕ _ _
(cong (2 +ℕ_) (+-comm m (suc n)))
(+'-comm (suc m) (suc (suc n))) i)
(-ₖ^< suc (suc n) · suc m > (suc m +' suc (suc n)) p q
(comm⨂-EM (suc m +' suc (suc n)) (y ⌣ₖ x)))))
∙ cong (Fₗ p q) (λ i → indₗ p q (~ i) .fst x .fst y))
st : (p : isEvenT (suc (suc n)) ⊎ isOddT (suc (suc n)))
(q : isEvenT (suc (suc m)) ⊎ isOddT (suc (suc m)))
→ EM (H' ⨂ G') ((2 +ℕ m) +' (2 +ℕ n))
→ EM (G' ⨂ H') ((2 +ℕ n) +' (2 +ℕ m))
st p q x = subst (EM (G' ⨂ H')) (cong suc ℕpath)
(-ₖ^< (2 +ℕ n) · (2 +ℕ m) > ((2 +ℕ m) +' (2 +ℕ n)) p q
(comm⨂-EM ((2 +ℕ m) +' (2 +ℕ n)) x ))
main : (p : _) (q : _)
→ (x : EM-raw' G' (suc (suc n))) (y : EM-raw' H' (suc (suc m)))
→ cp∙∙ (suc (suc n)) (suc (suc m)) .fst
(EM-raw'→EM G' (suc (suc n)) x) .fst
(EM-raw'→EM H' (suc (suc m)) y)
≡ st p q (_⌣ₖ_ {n = suc (suc m)} {m = suc (suc n)}
(EM-raw'→EM H' (suc (suc m)) y)
(EM-raw'→EM G' (suc (suc n)) x))
main p q north north = refl
main p q south north = refl
main p q (merid a i) north k =
(cong (EM→ΩEM+1 (suc (suc (n +ℕ suc m))))
(improveᵣ (evenOrOdd (suc n)) q (EM-raw→EM _ _ a) ∣ north ∣)
∙ EM→ΩEM+1-0ₖ _) k i
main p q north south = refl
main p q south south = refl
main p q (merid a i) south k =
(cong (EM→ΩEM+1 (suc (suc (n +ℕ suc m))))
(improveᵣ (evenOrOdd (suc n)) q (EM-raw→EM _ _ a) ∣ south ∣)
∙ EM→ΩEM+1-0ₖ _) k i
main p q north (merid a i) k =
st p q ((cong (EM→ΩEM+1 (suc (suc (m +ℕ suc n))))
(improveₗ p (evenOrOdd (suc m)) ∣ north ∣ (EM-raw→EM _ _ a))
∙ EM→ΩEM+1-0ₖ _) (~ k) i)
main p q south (merid a i) k =
st p q ((cong (EM→ΩEM+1 (suc (suc (m +ℕ suc n))))
(improveₗ p (evenOrOdd (suc m)) ∣ south ∣ (EM-raw→EM _ _ a))
∙ EM→ΩEM+1-0ₖ _) (~ k) i)
main p q (merid b i) (merid a j) k =
hcomp (λ r → λ {
(i = i0) → st p q
(compPath-filler'
(cong (EM→ΩEM+1 (suc (suc (m +ℕ suc n))))
(improveₗ p (evenOrOdd (suc m)) ∣ north ∣ (EM-raw→EM _ _ a)))
(EM→ΩEM+1-0ₖ _) r (~ k) j)
; (i = i1) → st p q
(compPath-filler'
(cong (EM→ΩEM+1 (suc (suc (m +ℕ suc n))))
(improveₗ p (evenOrOdd (suc m)) ∣ south ∣ (EM-raw→EM _ _ a)))
(EM→ΩEM+1-0ₖ _) r (~ k) j)
; (j = i0) →
compPath-filler'
(cong (EM→ΩEM+1 (suc (suc (n +ℕ suc m))))
(improveᵣ (evenOrOdd (suc n)) q (EM-raw→EM _ _ b) ∣ north ∣))
(EM→ΩEM+1-0ₖ _) r k i
; (j = i1) →
compPath-filler'
(cong (EM→ΩEM+1 (suc (suc (n +ℕ suc m))))
(improveᵣ (evenOrOdd (suc n)) q (EM-raw→EM _ _ b) ∣ south ∣))
(EM→ΩEM+1-0ₖ _) r k i
; (k = i0) → EM→ΩEM+1 (suc (suc (n +ℕ suc m)))
(improveᵣ (evenOrOdd (suc n)) q (EM-raw→EM _ _ b)
∣ merid a j ∣ (~ r)) i
; (k = i1) → st p q
(EM→ΩEM+1 _
(improveₗ p (evenOrOdd (suc m)) ∣ (merid b i) ∣
(EM-raw→EM _ _ a) (~ r)) j)})
(hcomp (λ r → λ {
(i = i0) → st p q ((EM→ΩEM+1-0ₖ _) (~ k ∨ ~ r) j)
; (i = i1) → st p q ((EM→ΩEM+1-0ₖ _) (~ k ∨ ~ r) j)
; (j = i0) → EM→ΩEM+1-0ₖ (suc (suc (n +ℕ suc m))) (k ∨ ~ r) i
; (j = i1) → EM→ΩEM+1-0ₖ (suc (suc (n +ℕ suc m))) (k ∨ ~ r) i
; (k = i0) → pp-wrap _ (cong (Fᵣ (evenOrOdd (suc n)) q)
(EM→ΩEM+1 (suc m +' suc n)
(_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b)))) (~ r) j i
; (k = i1) → st p q (pp-wrap (suc (suc (m +ℕ suc n)))
(cong (Fₗ p (evenOrOdd (suc m)))
(EM→ΩEM+1 (suc n +' suc m)
(_⌣ₖ_ {n = suc n} {m = suc m}
(EM-raw→EM G' (suc n) b)
(EM-raw→EM H' (suc m) a)))) (~ r) i j)})
(final k i j))
where
cong-transpLem : (x : _)
→ cong (subst (EM (G' ⨂ H'))
(cong (2 +ℕ_) (sym (+-comm m (suc n)))))
(EM→ΩEM+1 (suc n +' suc m) x)
≡ EM→ΩEM+1 (suc (m +ℕ suc n))
(subst (EM (G' ⨂ H'))
(cong suc (sym (+-comm m (suc n)))) x)
cong-transpLem x j i =
transp (λ i → EM (G' ⨂ H')
(2 +ℕ (+-comm m (suc n) (~ i ∧ ~ j)))) j
(EM→ΩEM+1 (suc (+-comm m (suc n) (~ j)))
(transp (λ i → EM (G' ⨂ H')
(suc (+-comm m (suc n) (~ i ∨ ~ j)))) (~ j) x) i)
expr₁ =
cong (Fₗ p (evenOrOdd (suc m)))
(EM→ΩEM+1 (suc n +' suc m)
(_⌣ₖ_ {n = suc n} {m = suc m}
(EM-raw→EM G' (suc n) b)
(EM-raw→EM H' (suc m) a)))
↑expr₁ = transport (λ i → fst (Ω (EM∙ (H' ⨂ G') (ℕpath i)))) expr₁
expr₂ : EM (H' ⨂ G') (suc (m +ℕ suc n))
expr₂ =
subst (EM (H' ⨂ G')) (cong suc (sym (+-suc m n)))
(-ₖ^< suc (suc n) · suc m > (suc m +' suc n) p (evenOrOdd (suc m))
(-ₖ^< suc n · suc m > (suc m +' suc n)
(evenOrOdd (suc n)) (evenOrOdd (suc m))
(_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a) (EM-raw→EM G' (suc n) b))))
expr₃ =
Iso.inv (Iso→EMIso ⨂-comm _)
(-ₖ^< suc (suc n) · suc m > (suc (m +ℕ suc n)) p (evenOrOdd (suc m))
(subst (EM (G' ⨂ H')) (cong suc (sym (+-comm m (suc n))))
(_⌣ₖ_ {n = suc n} {m = suc m}
(EM-raw→EM G' (suc n) b)
(EM-raw→EM H' (suc m) a))))
expr₃≡expr₂ : expr₃ ≡ expr₂
expr₃≡expr₂ =
cong (Iso.inv (Iso→EMIso ⨂-comm _))
(cong (-ₖ^< suc (suc n) · suc m >
(suc (m +ℕ suc n)) p (evenOrOdd (suc m)))
(cong (subst (EM (G' ⨂ H'))
(cong suc (sym (+-comm m (suc n)))))
(λ i → indₘ (evenOrOdd (suc n))
(evenOrOdd (suc m)) i .fst
(EM-raw→EM G' (suc n) b) .fst
(EM-raw→EM H' (suc m) a)
)
∙ compSubstℕ (+'-comm (suc m) (suc n))
(sym (cong suc (+-comm m (suc n))))
(cong suc (sym (+-suc m n)))
∙ cong (subst (EM (G' ⨂ H')) (cong suc (sym (+-suc m n))))
(sym (-ₖ^< suc n · suc m >-Induced (suc m +' suc n)
(evenOrOdd (suc n)) (evenOrOdd (suc m))
(GroupEquiv→GroupHom ⨂-comm)
(_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b))))))
∙ sym (substCommSlice (EM (G' ⨂ H')) (EM (H' ⨂ G'))
(λ k x → Iso.inv (Iso→EMIso ⨂-comm k)
(-ₖ^< suc (suc n) · suc m > k p
(evenOrOdd (suc m)) x))
(cong suc (sym (+-suc m n))) _)
∙ cong (subst (EM (H' ⨂ G')) (cong suc (sym (+-suc m n))))
(cong (Iso.inv (Iso→EMIso ⨂-comm _))
(sym (-ₖ^< suc (suc n) · suc m >-Induced (suc (suc (m +ℕ n)))
p (evenOrOdd (suc m)) (GroupEquiv→GroupHom ⨂-comm) _))
∙ Iso.leftInv (Iso→EMIso ⨂-comm (suc (suc (m +ℕ n)))) _)
cong-Fₗ-expr₁ : expr₁ ≡ EM→ΩEM+1 (suc (m +ℕ suc n)) expr₂
cong-Fₗ-expr₁ = cong (cong (Iso.inv (Iso→EMIso ⨂-comm _) ∘
(-ₖ^< suc (suc n) · suc m > _ p (evenOrOdd (suc m)))))
(cong-transpLem (_⌣ₖ_ {n = suc n} {m = suc m}
(EM-raw→EM G' (suc n) b)
(EM-raw→EM H' (suc m) a)))
∙ cong (cong (Iso.inv (Iso→EMIso ⨂-comm _)))
(cong-ₖ^< suc (suc n) · suc m >₂ _
p (evenOrOdd (suc m)) _)
∙ sym (EMFun-EM→ΩEM+1 _ _)
∙ cong (EM→ΩEM+1 (suc (m +ℕ suc n))) expr₃≡expr₂
lem₁ :
cong (-ₖ^< 2 +ℕ n · 2 +ℕ m > (suc (suc (n +ℕ suc m))) p q)
(λ i₁ → comm⨂-EM (suc (suc (n +ℕ suc m))) (↑expr₁ (~ i₁)))
≡ transport (λ i → fst (Ω (EM∙ (G' ⨂ H') (ℕpath i))))
(cong (-ₖ^< 2 +ℕ n · 2 +ℕ m > (suc (suc (m +ℕ suc n))) p q)
(cong (comm⨂-EM (suc (suc (m +ℕ suc n))))
(EM→ΩEM+1 (suc (m +ℕ suc n))
(-ₖ expr₂))))
lem₁ = (λ i → transp
(λ j → fst (Ω (EM∙ (G' ⨂ H') (ℕpath (~ i ∨ j))))) (~ i)
(cong (-ₖ^< 2 +ℕ n · 2 +ℕ m > (ℕpath (~ i)) p q)
(cong (comm⨂-EM (ℕpath (~ i)))
(transp (λ j → fst (Ω (EM∙ (H' ⨂ G') (ℕpath (~ i ∧ j))))) i
(sym expr₁)))))
∙ cong (transport (λ i → fst (Ω (EM∙ (G' ⨂ H') (ℕpath i))))
∘ (cong (-ₖ^< 2 +ℕ n · 2 +ℕ m > (suc (suc (m +ℕ suc n))) p q)
∘ (cong (comm⨂-EM (suc (suc (m +ℕ suc n)))))))
(cong sym cong-Fₗ-expr₁
∙ sym (EM→ΩEM+1-sym (suc (m +ℕ suc n)) expr₂))
substLem : {n : ℕ} (m : ℕ) (p : n ≡ m) (x : typ (Ω (EM∙ (G' ⨂ H') _)))
→ subst (λ n → typ (Ω (EM∙ (G' ⨂ H') n))) (cong (suc ∘ suc) p) x
≡ cong (subst (EM (G' ⨂ H')) (cong (suc ∘ suc) p)) x
substLem =
J> λ x → transportRefl x ∙ λ j i → transportRefl (x i) (~ j)
final : flipSquare
(wrap (suc (suc (n +ℕ suc m)))
(cong (Fᵣ (evenOrOdd (suc n)) q)
(EM→ΩEM+1 (suc m +' suc n)
(_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b)))))
≡ cong (cong (st p q))
(wrap (suc m +' suc (suc n)) expr₁)
final =
sym (sym≡flipSquare _)
∙ cong (sym ∘ wrap (suc (suc (n +ℕ suc m))))
((λ _ → (cong (Fᵣ (evenOrOdd (suc n)) q)
(EM→ΩEM+1 (suc m +' suc n)
(_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b)))))
∙ (sym (substLem _ (+-comm (suc m) n) (cong
(-ₖ^< suc n · suc (suc m) > (suc (suc m) +' (suc n))
(evenOrOdd (suc n)) q
∘ (comm⨂-EM (suc (suc m) +' suc n)))
(EM→ΩEM+1 (suc m +' suc n)
(_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b)))))
∙ cong (subst (λ n → typ (Ω (EM∙ (G' ⨂ H') n)))
(cong (suc ∘ suc) (+-comm (suc m) n)))
(cong (cong (-ₖ^< suc n · suc (suc m) >
(suc (suc m) +' suc n)
(evenOrOdd (suc n)) q))
(sym (EMFun-EM→ΩEM+1 _
(_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b))))
∙ cong-ₖ^< suc n · suc (suc m) >₂ _ (evenOrOdd (suc n)) q
(comm⨂-EM _ (_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b)))
∙ cong (EM→ΩEM+1 (suc (suc (m +ℕ n))))
(sym (-ₖ^< suc n · suc (suc m) >-Induced
(suc m +' suc n)
(evenOrOdd (suc n)) q
(GroupEquiv→GroupHom ⨂-comm)
((_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b))))
∙ cong (comm⨂-EM (suc m +' suc n))
(⌣ₖ-comm-final-lem (suc n) (suc m)
(suc m +' suc n) p q
(evenOrOdd (suc n)) (evenOrOdd (suc m))
(_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b)))
∙ -ₖ^< suc (suc n) · suc (suc m) >-Induced
(suc (suc (m +ℕ n))) p q
(GroupEquiv→GroupHom ⨂-comm) _)))
∙ sym (compSubstℕ {A = λ n → typ (Ω (EM∙ (G' ⨂ H') n))}
(cong (suc ∘ suc) (sym (+-suc m n))) ℕpath
(cong (suc ∘ suc) (+-comm (suc m) n)))
∙ cong (transport (λ i₁ → fst (Ω (EM∙ (G' ⨂ H') (ℕpath i₁)))))
((substCommSlice (EM (H' ⨂ G'))
(λ n → typ (Ω (EM∙ (G' ⨂ H') (suc n))))
(λ k x → EM→ΩEM+1 k (-ₖ^< 2 +ℕ n · 2 +ℕ m >
k p q (comm⨂-EM k (-ₖ x))))
(cong suc (sym (+-suc m n)))
((-ₖ^< suc (suc n) · suc m > (suc m +' suc n) p
(evenOrOdd (suc m))
(-ₖ^< suc n · suc m > (suc m +' suc n)
(evenOrOdd (suc n)) (evenOrOdd (suc m))
((_⌣ₖ_ {n = suc m} {m = suc n}
(EM-raw→EM H' (suc m) a)
(EM-raw→EM G' (suc n) b)))))))
∙ sym (cong-ₖ^< 2 +ℕ n · 2 +ℕ m >₂ (m +ℕ suc n) p q _)
∙ cong (cong (-ₖ^< 2 +ℕ n · 2 +ℕ m >
(suc (suc (m +ℕ suc n))) p q))
(EMFun-EM→ΩEM+1 _ _))
∙ sym lem₁)
∙ cong sym (sym (cong-cong-ₖ^< (2 +ℕ n) · (2 +ℕ m) >₂
(n +ℕ suc m) p q
(cong (comm⨂-EM (suc (suc (n +ℕ suc m))))
(sym ↑expr₁))))
∙∙ cong (cong (cong (-ₖ^< (2 +ℕ n) · (2 +ℕ m) >
(suc (ℕpath i1)) p q)))
(sym (cong-cong-comm⨂-EM _ ↑expr₁))
∙∙ λ k i j
→ transp (λ j
→ EM (G' ⨂ H') (suc (ℕpath (j ∨ ~ k))))
(~ k)
(-ₖ^< (2 +ℕ n) · (2 +ℕ m) >
(suc (ℕpath (~ k))) p q
(comm⨂-EM (suc (ℕpath (~ k)))
(wrap (ℕpath (~ k))
(transp (λ i → fst (Ω (EM∙ (H' ⨂ G')
(ℕpath (~ k ∧ i))))) k
expr₁) i j)))
module _ {G : AbGroup ℓ} where
-ₖ^[_·_] : (n m : ℕ) {k : ℕ} (x : EM G k) → EM G k
-ₖ^[ n · m ] {k = k} = -ₖ^< n · m > k (evenOrOdd n) (evenOrOdd m)
-ₖ^[_·_]-even : (n m : ℕ) → isEvenT n ⊎ isEvenT m
→ {k : ℕ} (x : EM G k) → -ₖ^[ n · m ] x ≡ x
-ₖ^[_·_]-even n m (inl p) {k = k} x =
(λ i → -ₖ^< n · m > k
(isPropEvenOrOdd n (evenOrOdd n) (inl p) i)
(evenOrOdd m) x)
∙ -ₖ^< n · m >-inl-left k p (evenOrOdd m) x
-ₖ^[_·_]-even n m (inr p) {k = k} x =
(λ i → -ₖ^< n · m > k
(evenOrOdd n)
(isPropEvenOrOdd m (evenOrOdd m) (inl p) i) x)
∙ -ₖ^< n · m >-inl-right k (evenOrOdd n) p x
-ₖ^[_·_]-odd : (n m : ℕ) → isOddT n × isOddT m
→ {k : ℕ} (x : EM G k) → -ₖ^[ n · m ] x ≡ -ₖ x
-ₖ^[_·_]-odd n m (p , q) {k = k} x =
cong₂ (λ p q → -ₖ^< n · m > k p q x)
(isPropEvenOrOdd n _ _) (isPropEvenOrOdd m _ _)
∙ -ₖ^< n · m >-inr k p q x
module _ {G' : AbGroup ℓ} {H' : AbGroup ℓ'} where
⌣ₖ-comm : (n m : ℕ) (x : EM G' n) (y : EM H' m)
→ x ⌣ₖ y ≡ subst (EM (G' ⨂ H')) (+'-comm m n)
(-ₖ^[ n · m ] (comm⨂-EM (m +' n) (y ⌣ₖ x)))
⌣ₖ-comm zero zero x y =
sym (transportRefl
(comm⨂-EM zero (_⌣ₖ_ {n = zero} {m = zero} y x)))
⌣ₖ-comm zero (suc m) x y =
⌣ₖ-comm₀ m x y
∙ sym ((λ i → subst (EM (G' ⨂ H'))
(isSetℕ _ _ (+'-comm (suc m) zero) refl i)
((-ₖ^< zero · suc m > (suc m) (inl tt) (evenOrOdd (suc m))
(comm⨂-EM (suc m) (_⌣ₖ_ {m = zero} y x)))))
∙ transportRefl _
∙ -ₖ^< zero · suc m >-inl-left (suc m) tt
(evenOrOdd (suc m))
(comm⨂-EM (suc m) (_⌣ₖ_ {m = zero} y x)))
⌣ₖ-comm (suc n) zero x y =
sym (Iso.leftInv (Iso→EMIso ⨂-comm _)
(_⌣ₖ_ {n = suc n} {m = zero} x y))
∙ sym (comm⨂-EM≡comm⨂-EM' (suc n) _)
∙ cong (comm⨂-EM (suc n)) (sym (⌣ₖ-comm₀ n y x))
∙ sym (-ₖ^< suc n · zero >-inl-right (suc n) (evenOrOdd (suc n)) tt
(comm⨂-EM (suc n) (_⌣ₖ_ {n = zero} {m = suc n} y x)))
∙ sym (transportRefl _)
∙ λ i → subst (EM (G' ⨂ H')) (isSetℕ _ _ refl (+'-comm zero (suc n)) i)
(-ₖ^< suc n · zero > (suc n) (evenOrOdd (suc n)) (inl tt)
(comm⨂-EM (suc n) (_⌣ₖ_ {n = zero} {m = suc n} y x)))
⌣ₖ-comm (suc n) (suc m) x y i =
⌣ₖ-comm₂ n m (evenOrOdd (suc n)) (evenOrOdd (suc m)) i .fst x .fst y