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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 | import OAI.RingTheory.DirectFiniteness.CandidateRelations
namespace OAI
universe u_1 u_2 u_3 u_4 u_5
noncomputable section
open scoped Classical BigOperators
namespace KaplanskyCounterexample.FiniteGeometry
variable {Q : Type u_1} {A : Type u_2} {R : Type u_3} {E : Type u_4} [Field Q] [AddCommGroup A] [Module Q A]
[AddCommGroup E] [Module Q E]
namespace DirectionFrame
variable (F : DirectionFrame Q A R E)
def transverse (α : Q) : (Q × Q) →ₗ[Q] Q where
toFun x := x.2 - x.1 * α
map_add' x y := by dsimp; ring
map_smul' c x := by dsimp; ring
lemma exceptional_scalar_sum {k : ℕ} (s : Fin k → PolygonStep Q R E) (hk : k ≤ 5)
(hz : ∑ i, (s i).vector F = 0) (r : R) (α : Q)
(hslope : ∀ i (g : GenericStep Q R E), s i = .inl g → g.label.1 = r → g.label.2 = α) :
∑ i ∈ Finset.univ.filter (fun i => (s i).label = .inr r),
Sum.elim (fun _ => 0) (fun exceptional => exceptional.scalar) (s i) = 0 := by
have h := congrArg (transverse α) ((F.polygon_component_sums s hk hz).2 r)
rw [map_sum,map_zero,Finset.sum_filter] at h
rw [Finset.sum_filter]
convert h using 1
apply Finset.sum_congr rfl
intro i _
cases hi : s i with
| inl g =>
simp only [PolygonStep.label,reduceCtorEq,ite_false,
PolygonStep.bundle,PolygonStep.wcoord]
by_cases hr : g.label.1 = r
· simp [hr,transverse,hslope i g hi hr]
· simp [hr]
| inr e =>
simp only [PolygonStep.label,Sum.inr.injEq,PolygonStep.bundle,PolygonStep.wcoord]
by_cases hr : e.label = r <;> simp [hr,transverse]
lemma exceptional_occurs_twice_single_slope {k : ℕ} (s : Fin k → PolygonStep Q R E)
(hk : k ≤ 5) (hz : ∑ i, (s i).vector F = 0)
(hadj : ∀ i, (s i).label ≠ (s (finRotate k i)).label) (r : R)
(hr : ∃ i, (s i).label = .inr r) (α : Q)
(hslope : ∀ i (g : GenericStep Q R E), s i = .inl g → g.label.1 = r → g.label.2 = α) :
(Finset.univ.filter (fun i => (s i).label = .inr r)).card = 2 := by
apply cyclic_label_count_eq_two (fun i => (s i).label)
(fun i => Sum.elim (fun _ => 0) (fun exceptional => exceptional.scalar) (s i))
hk hadj (.inr r) hr
· intro i hi
cases h : s i with
| inl g => simp [h,PolygonStep.label] at hi
| inr e => exact e.nonzero
· exact F.exceptional_scalar_sum s hk hz r α hslope
lemma all_labels_twice_length {k : ℕ} {L : Type u_5} [DecidableEq L] (lab : Fin k → L)
(hk3 : 3 ≤ k) (hk5 : k ≤ 5)
(hcount : ∀ l ∈ Finset.univ.image lab,
(Finset.univ.filter (fun i => lab i = l)).card = 2) :
k = 4 ∧ (Finset.univ.image lab).card = 2 := by
have h := sum_label_counts lab
have he : (∑ l ∈ Finset.univ.image lab,
(Finset.univ.filter (fun i => lab i = l)).card) =
(Finset.univ.image lab).card * 2 := by
calc
_ = ∑ _l ∈ Finset.univ.image lab, 2 := Finset.sum_congr rfl hcount
_ = _ := by simp
rw [he,Fintype.card_fin] at h
omega
/-- With at most one generic label every occurring label occurs twice, so the
candidate is a four-step rectangle. -/
lemma one_generic_rectangle {k : ℕ} (s : Fin k → PolygonStep Q R E)
(hk3 : 3 ≤ k) (hk5 : k ≤ 5) (hz : ∑ i, (s i).vector F = 0)
(hadj : ∀ i, (s i).label ≠ (s (finRotate k i)).label)
(j : R × Q) (honly : ∀ i (g : GenericStep Q R E), s i = .inl g → g.label = j) :
k = 4 ∧ (Finset.univ.image (fun i => (s i).label)).card = 2 := by
apply all_labels_twice_length _ hk3 hk5
intro l hl
obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hl
cases l with
| inl j' => exact F.generic_occurs_twice s hk5 hz hadj j' ⟨i,hi⟩
| inr r =>
apply F.exceptional_occurs_twice_single_slope s hk5 hz hadj r ⟨i,hi⟩ j.2
intro i g hg _
exact congrArg Prod.snd (honly i g hg)
/-- A lone exceptional step requires two different generic slopes in its own
bundle. This is the cancellation needed for the pentagon count. -/
lemma singleton_exceptional_two_slopes {k : ℕ} (s : Fin k → PolygonStep Q R E)
(hk : k ≤ 5) (hz : ∑ i, (s i).vector F = 0) (r : R)
(hc : (Finset.univ.filter (fun i => (s i).label = .inr r)).card = 1) :
∃ (i i' : Fin k) (g g' : GenericStep Q R E), s i = .inl g ∧ s i' = .inl g' ∧
g.label.1 = r ∧ g'.label.1 = r ∧ g.label.2 ≠ g'.label.2 := by
by_contra hh
push Not at hh
have hslope : ∃ α : Q, ∀ i (g : GenericStep Q R E),
s i = .inl g → g.label.1 = r → g.label.2 = α := by
by_cases hex : ∃ (i : Fin k) (g : GenericStep Q R E), s i = .inl g ∧ g.label.1 = r
· obtain ⟨i,g,hi,hr⟩ := hex
refine ⟨g.label.2,?_⟩
intro i' g' hi' hr'
exact (hh i i' g g' hi hi' hr hr').symm
· refine ⟨0,?_⟩
intro i g hi hr
exact False.elim (hex ⟨i,g,hi,hr⟩)
obtain ⟨α,hα⟩ := hslope
have hn := count_ne_one_of_sum_eq_zero (fun i => (s i).label)
(fun i => Sum.elim (fun _ => 0) (fun exceptional => exceptional.scalar) (s i)) (.inr r)
(by
intro i hi
cases h : s i with
| inl g => simp [h,PolygonStep.label] at hi
| inr e => exact e.nonzero)
(F.exceptional_scalar_sum s hk hz r α hα)
exact hn hc
end DirectionFrame
end KaplanskyCounterexample.FiniteGeometry
end
end OAI
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