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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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.RingTheory.DirectFiniteness.RibbonGeometry
namespace OAI
universe u_1
noncomputable section
open scoped Classical
open scoped BigOperators
open Equiv Equiv.Perm
namespace KaplanskyCounterexample.Planar
open Equiv Equiv.Perm
attribute [local simp] SameCycle.rfl
variable {α : Type u_1} [Fintype α]
lemma sameCycle_map_quotient {p q : Perm α}
(h : ∀ {x y}, p.SameCycle x y → q.SameCycle x y) :
Nat.card (Cycles q) ≤ Nat.card (Cycles p) := by
exact Nat.card_le_card_of_surjective (Quotient.map id (fun _ _ hxy => h hxy))
(Quotient.map_surjective _ Function.surjective_id)
lemma swap_cycles_card_le (p : Perm α) (a b : α) :
Nat.card (Cycles (swap a b * p)) ≤ Nat.card (Cycles p) + 1 := by
let q := swap a b * p
have hrev : swap a b * q = p := by simp [q, ← mul_assoc]
by_cases hab : q.SameCycle a b
· have href : ∀ {x y}, p.SameCycle x y → q.SameCycle x y := by
intro x y h
apply sameCycle_map p q id _ h
intro x
have hx : q.SameCycle x (q x) := SameCycle.rfl.apply_right
have hp : p x = swap a b (q x) := by
rw [← Perm.mul_apply, hrev]
rw [hp]
by_cases hxa : q x = a
· rw [hxa] at hx
simpa [hxa] using hx.trans hab
· by_cases hxb : q x = b
· rw [hxb] at hx
simpa [hxb] using hx.trans hab.symm
· rwa [swap_apply_of_ne_of_ne hxa hxb]
exact (sameCycle_map_quotient href).trans (Nat.le_succ _)
· have h := swap_mul_cycles_card q a b hab
rw [hrev] at h
change Nat.card (Cycles q) ≤ _
omega
lemma cycles_card_le (p : Perm α) : Nat.card (Cycles p) ≤ Fintype.card α := by
simpa using Nat.card_le_card_of_surjective (cycleClass p) Quotient.mk_surjective
lemma cycles_one_card : Nat.card (Cycles (1 : Perm α)) = Fintype.card α := by
have heq : SameCycle.setoid (1 : Perm α) = ⊥ := by
ext x y
exact sameCycle_one
rw [Cycles, heq]
exact (Nat.card_congr Setoid.quotientBotEquiv).trans (Nat.card_eq_fintype_card)
/-- A minimal transposition factorization, with each transposition contained in
one of the original cycles. -/
lemma exists_cycle_factorization (p : Perm α) :
∃ l : List (α × α),
(l.map fun ab => swap ab.1 ab.2).prod = p ∧
l.length + Nat.card (Cycles p) = Fintype.card α ∧
∀ ab ∈ l, ab.1 ≠ ab.2 ∧ p.SameCycle ab.1 ab.2 := by
have hmain : ∀ n : ℕ, ∀ p : Perm α,
Fintype.card α - Nat.card (Cycles p) = n →
∃ l : List (α × α),
(l.map fun ab => swap ab.1 ab.2).prod = p ∧
l.length + Nat.card (Cycles p) = Fintype.card α ∧
∀ ab ∈ l, ab.1 ≠ ab.2 ∧ p.SameCycle ab.1 ab.2 := by
intro n
induction n using Nat.strong_induction_on with
| h n ih =>
intro p hn
by_cases hp : p = 1
· subst p
exact ⟨[], rfl, (Nat.zero_add _).trans (cycles_one_card (α := α)),
fun _ membership => False.elim (List.not_mem_nil membership)⟩
· have hex : ∃ a, p a ≠ a :=
not_forall.mp (fun fixed => hp (Equiv.ext fixed))
obtain ⟨a, ha⟩ := hex
have hc := unplug_cycles_card p a ha
have hle := cycles_card_le (unplug p a)
have hlt : Fintype.card α - Nat.card (Cycles (unplug p a)) < n := by omega
obtain ⟨l, hl, hlen, hmem⟩ := ih _ hlt (unplug p a) rfl
refine ⟨(a, p a) :: l, ?_, ?_, ?_⟩
· simp only [List.map_cons, List.prod_cons, hl]
exact swap_mul_self_mul a (p a) p
· simp only [List.length_cons]
omega
· intro ab hab
rcases List.mem_cons.mp hab with rfl | hab
· exact ⟨Ne.symm ha, SameCycle.rfl.apply_right⟩
· exact ⟨(hmem ab hab).1, unplug_sameCycle p a (hmem ab hab).2⟩
exact hmain _ p rfl
end KaplanskyCounterexample.Planar
end
end OAI
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