| import OAI.Combinatorics.BalancedRyser.Local.Local |
| import OAI.Combinatorics.BalancedRyser.Probability.ConfigurationStructure |
| import OAI.Combinatorics.BalancedRyser.Probability.TupleExposure |
|
|
| namespace OAI |
|
|
| namespace Balanced |
| open scoped _root_.Balanced |
|
|
| noncomputable section |
| open scoped Classical |
|
|
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
|
|
| theorem Configuration.exists_pencil_reuse_witness (C : Configuration F) |
| (hvalid : C.Valid) (hcompatible : C.Compatible) (p : Point F) |
| (hreuse : 3 ≤ Local.reuseCount C p) : |
| ∃ l : List (Slot F), l.Nodup ∧ |
| (∀ s ∈ l, OnLine p (C.choice s).line) ∧ l.length ≤ 6 ∧ |
| 3 ≤ reuseExposureScore (fun s => generatorSet (C.choice s).tuple) l := by |
| have hcount := ConfigurationStructure.split_multiplicity_eq_slots C hvalid hcompatible p |
| have hreuse' : 3 ≤ |
| 4 * (Finset.univ.filter (fun s : Slot F => OnLine p (C.choice s).line)).card - |
| ((Finset.univ.filter (fun s : Slot F => OnLine p (C.choice s).line)).biUnion |
| (fun s => generatorSet (C.choice s).tuple)).card := by |
| simpa only [Local.reuseCount, Local.generatorUnionAt, hcount] using hreuse |
| obtain ⟨l, hnodup, hsub, hlength, hscore⟩ := |
| C.exists_six_tuple_reuse_order hvalid hcompatible _ hreuse' |
| refine ⟨l, hnodup, ?_, hlength, hscore⟩ |
| intro s hs |
| exact (Finset.mem_filter.mp (hsub (List.mem_toFinset.mpr hs))).2 |
|
|
| theorem Configuration.tuplePoint_ne_pencil_center (C : Configuration F) |
| (hcompatible : C.Compatible) (p : Point F) |
| (hmultiplicity : 2 ≤ C.splitMultiplicity p) (s : Slot F) (i : Fin 4) : |
| tuplePoint C.merged (C.choice s).tuple i ≠ p := by |
| intro heq |
| have hp : p ∈ designated C.merged (C.choice s) := by |
| exact Finset.mem_image.mpr ⟨i, Finset.mem_univ _, heq⟩ |
| exact ConfigurationStructure.center_not_designated C hcompatible hmultiplicity s hp |
|
|
| end |
| end Balanced |
|
|
| end OAI |
|
|
