| import OAI.RingTheory.DirectFiniteness.CyclicLabels |
|
|
| namespace OAI |
|
|
| universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 |
|
|
| 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] |
|
|
| structure GenericStep (Q : Type u_5) (R : Type u_6) (E : Type u_7) [Zero E] where |
| label : R × Q |
| scalar : Q |
| component : E |
| nonzero : component ≠ 0 |
|
|
| structure ExceptionalStep (Q : Type u_8) (R : Type u_9) [Zero Q] where |
| label : R |
| scalar : Q |
| nonzero : scalar ≠ 0 |
|
|
| abbrev PolygonStep (Q : Type u_10) (R : Type u_11) (E : Type u_12) [Zero Q] [Zero E] := GenericStep Q R E ⊕ ExceptionalStep Q R |
|
|
| namespace PolygonStep |
|
|
| def label : PolygonStep Q R E → (R × Q) ⊕ R |
| | .inl g => .inl g.label |
| | .inr e => .inr e.label |
|
|
| def bundle : PolygonStep Q R E → R |
| | .inl g => g.label.1 |
| | .inr e => e.label |
|
|
| def wcoord : PolygonStep Q R E → Q × Q |
| | .inl g => (g.scalar,g.scalar*g.label.2) |
| | .inr e => (0,e.scalar) |
|
|
| def ecoord : PolygonStep Q R E → E |
| | .inl g => g.component |
| | .inr _ => 0 |
|
|
| def vector (F : DirectionFrame Q A R E) : PolygonStep Q R E → A |
| | .inl g => F.u g.label (g.scalar,g.component) |
| | .inr e => F.exc e.label e.scalar |
|
|
| omit [Module Q E] in |
| lemma ecoord_nonzero {s : PolygonStep Q R E} {j : R × Q} (hs : s.label = .inl j) : |
| s.ecoord ≠ 0 := by |
| cases s with |
| | inl g => exact g.nonzero |
| | inr e => cases hs |
|
|
| lemma vector_ne_zero (F : DirectionFrame Q A R E) (s : PolygonStep Q R E) : |
| s.vector F ≠ 0 := by |
| cases s with |
| | inl g => |
| intro h |
| have hz : (g.scalar,g.component) = 0 := F.u_injective g.label (h.trans (F.u _).map_zero.symm) |
| exact g.nonzero (congrArg Prod.snd hz) |
| | inr e => |
| intro h |
| exact e.nonzero (F.exc_injective e.label (h.trans (F.exc _).map_zero.symm)) |
|
|
| def blockLabel (s : PolygonStep Q R E) (b : Fin 2) : R ⊕ (R × Q) := |
| if b = 0 then .inl s.bundle else |
| match s with |
| | .inl g => .inr g.label |
| | .inr e => .inl e.label |
|
|
| def blockVector (F : DirectionFrame Q A R E) (s : PolygonStep Q R E) (b : Fin 2) : A := |
| if b = 0 then F.w s.bundle s.wcoord else |
| match s with |
| | .inl g => F.e g.label g.component |
| | .inr _ => 0 |
|
|
| lemma blockVector_mem (F : DirectionFrame Q A R E) (s : PolygonStep Q R E) (b : Fin 2) : |
| s.blockVector F b ∈ Sum.elim (fun r => (F.w r).range) (fun j => (F.e j).range) |
| (s.blockLabel b) := by |
| fin_cases b |
| · cases s with |
| | inl g => exact ⟨(g.scalar,g.scalar*g.label.2),rfl⟩ |
| | inr e => exact ⟨(0,e.scalar),rfl⟩ |
| · cases s with |
| | inl g => exact ⟨g.component,rfl⟩ |
| | inr e => exact ⟨0,(F.w e.label).map_zero⟩ |
|
|
| lemma sum_blocks (F : DirectionFrame Q A R E) (s : PolygonStep Q R E) : |
| ∑ b : Fin 2, s.blockVector F b = s.vector F := by |
| cases s <;> simp [blockVector,vector,bundle,wcoord,DirectionFrame.u_apply, |
| Fin.sum_univ_succ] |
| |
| lemma filter_blocks_E (F : DirectionFrame Q A R E) (s : PolygonStep Q R E) (j : R × Q) : |
| ∑ b : Fin 2, (if s.blockLabel b = .inr j then s.blockVector F b else 0) = |
| if s.label = .inl j then F.e j s.ecoord else 0 := by |
| rw [Fin.sum_univ_two] |
| cases s with |
| | inl g => |
| dsimp [blockLabel,blockVector,label,bundle,wcoord,ecoord] |
| by_cases h : g.label = j <;> simp_all |
| | inr e => simp [blockLabel,label,bundle] |
| |
| lemma filter_blocks_W (F : DirectionFrame Q A R E) (s : PolygonStep Q R E) (r : R) : |
| ∑ b : Fin 2, (if s.blockLabel b = .inl r then s.blockVector F b else 0) = |
| if s.bundle = r then F.w r s.wcoord else 0 := by |
| rw [Fin.sum_univ_two] |
| cases s with |
| | inl g => |
| dsimp [blockLabel,blockVector,bundle,wcoord] |
| by_cases h : g.label.1 = r <;> simp_all |
| | inr e => |
| dsimp [blockLabel,blockVector,bundle,wcoord] |
| by_cases h : e.label = r <;> simp_all |
| |
| end PolygonStep |
| |
| namespace DirectionFrame |
| variable (F : DirectionFrame Q A R E) |
|
|
| lemma polygon_component_sums {k : ℕ} (s : Fin k → PolygonStep Q R E) (hk : k ≤ 5) |
| (hz : ∑ i, (s i).vector F = 0) : |
| (∀ j : R × Q, ∑ i ∈ Finset.univ.filter (fun i => (s i).label = .inl j), (s i).ecoord = 0) ∧ |
| (∀ r : R, ∑ i ∈ Finset.univ.filter (fun i => (s i).bundle = r), (s i).wcoord = 0) := by |
| let lab (p : Fin k × Fin 2) := (s p.1).blockLabel p.2 |
| let v (p : Fin k × Fin 2) := (s p.1).blockVector F p.2 |
| have hcard : (Finset.univ : Finset (Fin k × Fin 2)).card ≤ 10 := by simp; omega |
| have hvz : ∑ p : Fin k × Fin 2, v p = 0 := by |
| rw [Fintype.sum_prod_type] |
| simp_rw [v,PolygonStep.sum_blocks] |
| exact hz |
| have hsep := F.direct.grouped_sum Finset.univ hcard lab v |
| (fun p _ => PolygonStep.blockVector_mem F _ _) hvz |
| constructor |
| · intro j |
| have hh := hsep (.inr j) |
| rw [Finset.sum_filter,Fintype.sum_prod_type] at hh |
| simp_rw [lab,v,PolygonStep.filter_blocks_E] at hh |
| apply F.einj j |
| rw [map_zero,map_sum] |
| simpa only [Finset.sum_filter] using hh |
| · intro r |
| have hh := hsep (.inl r) |
| rw [Finset.sum_filter,Fintype.sum_prod_type] at hh |
| simp_rw [lab,v,PolygonStep.filter_blocks_W] at hh |
| apply F.winj r |
| rw [map_zero,map_sum] |
| simpa only [Finset.sum_filter] using hh |
|
|
| /-- Every occurring generic label occurs precisely twice in a candidate with |
| at most five steps. No exceptional-vector cancellation has been assumed. -/ |
| lemma generic_occurs_twice {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) |
| (j : R × Q) (hj : ∃ i, (s i).label = .inl j) : |
| (Finset.univ.filter (fun i => (s i).label = .inl j)).card = 2 := by |
| exact cyclic_label_count_eq_two (fun i => (s i).label) (fun i => (s i).ecoord) |
| hk hadj (.inl j) hj (fun _ hi => PolygonStep.ecoord_nonzero hi) |
| ((F.polygon_component_sums s hk hz).1 j) |
|
|
| end DirectionFrame |
| end KaplanskyCounterexample.FiniteGeometry |
| end |
|
|
| end OAI |
|
|
