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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 | import OAI.RingTheory.DirectFiniteness.BalanceDetachment
namespace OAI
universe u_1 u_2 u_3 u_4 u_5
noncomputable section
open scoped Classical
open scoped BigOperators
open Equiv Equiv.Perm
namespace KaplanskyCounterexample.Planar
open Equiv Equiv.Perm
variable {α : Type u_1} {V : Type u_2} {F : Type u_3} {A : Type u_4} {B : Type u_5} [Fintype α] [AddCommGroup A] [AddCommGroup B]
/-- The inclusion of the cycles of an invariant subset. -/
def cycleSubtypeMap (p : Perm α) (P : α → Prop) (hP : ∀ x, P (p x) ↔ P x) :
Cycles (p.subtypePerm hP) → Cycles p :=
Quotient.map Subtype.val (fun _ _ h => (sameCycle_subtypePerm (h := hP)).mp h)
omit [Fintype α] in
lemma cycleSubtypeMap_injective (p : Perm α) (P : α → Prop) (hP : ∀ x, P (p x) ↔ P x) :
Function.Injective (cycleSubtypeMap p P hP) := by
intro c d
induction c using Quotient.inductionOn with
| h x =>
induction d using Quotient.inductionOn with
| h y =>
intro h
exact Quotient.sound ((sameCycle_subtypePerm (h := hP)).mpr (Quotient.exact h))
lemma sum_cycle_subtype (p : Perm α) (P : α → Prop) (hP : ∀ x, P (p x) ↔ P x)
(c : Cycles (p.subtypePerm hP)) (f : α → A) :
∑ a ∈ cycleSet (p.subtypePerm hP) c, f a.val =
∑ a ∈ cycleSet p (cycleSubtypeMap p P hP c), f a := by
classical
induction c using Quotient.inductionOn with
| h x =>
apply Finset.sum_bij (fun y _ => y.val)
· intro y hy
apply (mem_cycleSet _ _ _).mpr
exact Quotient.sound ((sameCycle_subtypePerm (h := hP)).mp
(Quotient.exact ((mem_cycleSet _ _ _).mp hy)))
· intro y _ z _ h
exact Subtype.ext h
· intro y hy
have hcycle : p.SameCycle y x.val := (cycleClass_eq _ _ _).mp ((mem_cycleSet _ _ _).mp hy)
have hPy : P y := (perm_invariant_cycle p P hP hcycle).mpr x.property
refine ⟨⟨y, hPy⟩, ?_, rfl⟩
apply (mem_cycleSet _ _ _).mpr
exact Quotient.sound ((sameCycle_subtypePerm (h := hP)).mpr hcycle)
· intro _ _
rfl
variable {R : Ribbon α} {Inc : V → F → Prop} {ι : V → F → (A →+ B)}
/-- Discard components not containing the exception. -/
def WeakBalanced.restrict (D : WeakBalanced R Inc ι) (P : α → Prop)
(hb : ∀ x, P (R.black x) ↔ P x) (hw : ∀ x, P (R.white x) ↔ P x)
(e : α) (he : cycleClass R.white e = D.exception) (hPe : P e) :
WeakBalanced (R.restrict P hb hw) Inc ι where
blackLabel a := D.blackLabel a.val
whiteLabel a := D.whiteLabel a.val
blackLabel_rot a := D.blackLabel_rot a.val
whiteLabel_rot a := D.whiteLabel_rot a.val
incident a := D.incident a.val
weight a := D.weight a.val
exception := cycleClass (R.white.subtypePerm hw) ⟨e, hPe⟩
black_sum c := by
dsimp only [Ribbon.restrict] at c ⊢
rw [sum_cycle_subtype R.black P hb c
(fun x => ι (D.blackLabel x) (D.whiteLabel x) (D.weight x))]
exact D.black_sum _
white_sum c hc := by
dsimp only [Ribbon.restrict] at c hc ⊢
rw [sum_cycle_subtype]
apply D.white_sum
intro h
apply hc
apply cycleSubtypeMap_injective R.white P hw
exact h.trans he.symm
exceptional_sum := by
change (∑ a ∈ cycleSet (R.white.subtypePerm hw)
(cycleClass (R.white.subtypePerm hw) ⟨e, hPe⟩), D.weight a.val) ≠ 0
rw [sum_cycle_subtype]
change (∑ a ∈ cycleSet R.white (cycleClass R.white e), D.weight a) ≠ 0
rw [he]
exact D.exceptional_sum
end KaplanskyCounterexample.Planar
end
end OAI
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