File size: 2,318 Bytes
e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.RingTheory.DirectFiniteness.BalanceObstruction
namespace OAI
universe u_1
noncomputable section
open scoped Classical
open scoped BigOperators
open Equiv Equiv.Perm
namespace KaplanskyCounterexample.Planar
open Equiv Equiv.Perm
variable {α : Type u_1} [Fintype α]
lemma sameCycle_after_swap {p : Perm α} {a b x y : α} (hab : p.SameCycle a b)
(hxy : (swap a b * p).SameCycle x y) : p.SameCycle x y := by
apply setoid_sameCycle_of_step (SameCycle.setoid p) (swap a b * p) _ hxy
intro z
exact (SameCycle.apply_right SameCycle.rfl).trans
(setoid_swap_related (SameCycle.setoid p) hab (p z))
lemma sign_eq_cycles (p : Perm α) :
sign p = (-1 : ℤˣ) ^ (Fintype.card α - Nat.card (Cycles p)) := by
classical
obtain ⟨l, hl, hlen, hmem⟩ := exists_cycle_factorization p
have hs := sign_prod_list_swap (l := l.map fun ab => swap ab.1 ab.2) (by
intro g hg
obtain ⟨ab, hab, rfl⟩ := List.mem_map.mp hg
exact ⟨ab.1, ab.2, (hmem ab hab).1, rfl⟩)
rw [hl, List.length_map] at hs
have hn : Fintype.card α - Nat.card (Cycles p) = l.length := by omega
rwa [hn]
lemma not_sameCycle_swap_of_sameCycle (p : Perm α) (a b : α)
(hne : a ≠ b) (hab : p.SameCycle a b) : ¬ (swap a b * p).SameCycle a b := by
classical
intro hq
let q := swap a b * p
have hrev : swap a b * q = p := by simp [q, ← mul_assoc]
have heq : SameCycle.setoid p = SameCycle.setoid q := by
ext x y
constructor
· intro h
apply sameCycle_after_swap hq
rwa [hrev]
· exact sameCycle_after_swap hab
have hc : Nat.card (Cycles p) = Nat.card (Cycles q) := by rw [Cycles, heq]
have hs : sign p = sign q := by rw [sign_eq_cycles, sign_eq_cycles, hc]
have hsign : (-1 : ℤˣ) = 1 := by
have := hs
change sign p = sign (swap a b * p) at this
rw [Perm.sign_mul, sign_swap hne] at this
exact mul_right_cancel (b := sign p) (by simpa using this.symm)
norm_num at hsign
lemma swap_mul_cycles_card_split (p : Perm α) (a b : α)
(hne : a ≠ b) (hab : p.SameCycle a b) :
Nat.card (Cycles (swap a b * p)) = Nat.card (Cycles p) + 1 := by
have h := swap_mul_cycles_card (swap a b * p) a b
(not_sameCycle_swap_of_sameCycle p a b hne hab)
simpa [← mul_assoc] using h.symm
end KaplanskyCounterexample.Planar
end
end OAI
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