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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import OAI.RingTheory.DirectFiniteness.BalanceCoarsening
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6
noncomputable section
open scoped Classical
open scoped BigOperators
open Equiv Equiv.Perm
namespace KaplanskyCounterexample.Planar
open Equiv Equiv.Perm
variable {α : Type u_1} {β : Type u_2} {V : Type u_3} {F : Type u_4} {A : Type u_5} {B : Type u_6} [Fintype α] [AddCommGroup A] [AddCommGroup B]
lemma label_sameCycle (p : Perm α) (f : α → β) (hf : ∀ a, f (p a) = f a)
{x y : α} (hxy : p.SameCycle x y) : f x = f y :=
setoid_sameCycle_of_step (Setoid.ker f) p (fun x => (hf x).symm) hxy
lemma label_unplug (p : Perm α) (f : α → β) (hf : ∀ a, f (p a) = f a) (a x : α) :
f (unplug p a x) = f x :=
(label_sameCycle p f hf (unplug_sameCycle p a ((SameCycle.refl _ x).apply_right))).symm
lemma sameCycle_unplug_iff_of_ne (p : Perm α) (a : α) {x y : α}
(hx : x ≠ a) (hy : y ≠ a) : (unplug p a).SameCycle x y ↔ p.SameCycle x y := by
by_cases ha : p a = a
· simp only [unplug, ha, swap_self]
rfl
· exact unplug_sameCycle_iff p a ha hx hy
lemma cycleSet_unplug (p : Perm α) (a x : α) (hx : x ≠ a) :
cycleSet (unplug p a) (cycleClass (unplug p a) x) =
(cycleSet p (cycleClass p x)).erase a := by
classical
ext y
rw [mem_cycleSet, cycleClass_eq, Finset.mem_erase, mem_cycleSet, cycleClass_eq]
constructor
· intro hy
refine ⟨?_, unplug_sameCycle p a hy⟩
intro hya
subst y
exact hx (hy.symm.eq_of_right (unplug_fixed p a))
· rintro ⟨hya, hy⟩
exact (sameCycle_unplug_iff_of_ne p a hya hx).mpr hy
omit [Fintype α] in
lemma sum_erase_of_zero (s : Finset α) (a : α) (f : α → A) (ha : f a = 0) :
∑ x ∈ s.erase a, f x = ∑ x ∈ s, f x := by
classical
apply Finset.sum_subset (Finset.erase_subset _ _)
intro x hx hxa
have : x = a := by simpa [hx] using hxa
simpa [this] using ha
lemma sum_cycle_unplug (p : Perm α) (a x : α) (hx : x ≠ a)
(f : α → A) (ha : f a = 0) :
∑ y ∈ cycleSet (unplug p a) (cycleClass (unplug p a) x), f y =
∑ y ∈ cycleSet p (cycleClass p x), f y := by
rw [cycleSet_unplug p a x hx]
exact sum_erase_of_zero _ a f ha
variable {R : Ribbon α} {Inc : V → F → Prop} {ι : V → F → (A →+ B)}
lemma WeakBalanced.exists_exception_nonzero (D : WeakBalanced R Inc ι) :
∃ e ∈ cycleSet R.white D.exception, D.weight e ≠ 0 := by
classical
by_contra! h
exact D.exceptional_sum (Finset.sum_eq_zero h)
/-- A zero-weight edge can be detached without changing any nonzero balance. -/
def WeakBalanced.detach (D : WeakBalanced R Inc ι) (a : α) (ha : D.weight a = 0) :
WeakBalanced (R.detach a) Inc ι := by
classical
let e := D.exists_exception_nonzero.choose
have he := D.exists_exception_nonzero.choose_spec.1
have hwe := D.exists_exception_nonzero.choose_spec.2
have hea : e ≠ a := by rintro rfl; exact hwe ha
have hec := (mem_cycleSet R.white D.exception e).mp he
refine {
blackLabel := D.blackLabel
whiteLabel := D.whiteLabel
blackLabel_rot := label_unplug R.black D.blackLabel D.blackLabel_rot a
whiteLabel_rot := label_unplug R.white D.whiteLabel D.whiteLabel_rot a
incident := D.incident
weight := D.weight
exception := cycleClass (unplug R.white a) e
black_sum := ?_
white_sum := ?_
exceptional_sum := ?_ }
· dsimp only [Ribbon.detach]
intro c
induction c using Quotient.inductionOn with
| h x =>
change (∑ y ∈ cycleSet (unplug R.black a) (cycleClass (unplug R.black a) x),
ι (D.blackLabel y) (D.whiteLabel y) (D.weight y)) = 0
by_cases hx : x = a
· subst x
rw [cycleSet_singleton_of_fixed _ _ (unplug_fixed R.black a)]
simp [ha]
· rw [sum_cycle_unplug R.black a x hx
(fun y => ι (D.blackLabel y) (D.whiteLabel y) (D.weight y)) (by simp [ha])]
exact D.black_sum _
· dsimp only [Ribbon.detach]
intro c hc
induction c using Quotient.inductionOn with
| h x =>
change (∑ y ∈ cycleSet (unplug R.white a) (cycleClass (unplug R.white a) x),
D.weight y) = 0
by_cases hx : x = a
· subst x
rw [cycleSet_singleton_of_fixed _ _ (unplug_fixed R.white a)]
simpa using ha
· rw [sum_cycle_unplug R.white a x hx _ ha]
apply D.white_sum
intro hxc
apply hc
apply (cycleClass_eq _ _ _).mpr
apply (sameCycle_unplug_iff_of_ne R.white a hx hea).mpr
exact (cycleClass_eq _ _ _).mp (hxc.trans hec.symm)
· dsimp only [Ribbon.detach]
rw [sum_cycle_unplug R.white a e hea _ ha, hec]
exact D.exceptional_sum
end KaplanskyCounterexample.Planar
end
end OAI
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