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f0f05de | 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 | 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
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