File size: 3,082 Bytes
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 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 | import OAI.Combinatorics.BalancedRyser.Construction.ValidChoice
import OAI.Combinatorics.BalancedRyser.Incidence.AffineIncidence
import OAI.Combinatorics.BalancedRyser.Probability.ConfigurationStructure
namespace OAI
namespace Balanced
open scoped _root_.Balanced
namespace Configuration
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
private theorem mem_merged_of_on_constituent (M : MergedPairs F)
(d : Direction F) (s : Bool) {x : Point F}
(hx : OnLine x (constituent M d s)) : x ∈ mergedPoints M d := by
simp only [mergedPoints, linePoints, Finset.mem_union, Finset.mem_filter,
Finset.mem_univ, true_and]
cases s
· exact Or.inl hx
· exact Or.inr hx
theorem points_nonempty (C : Configuration F) : C.points.Nonempty := by
let s : Slot F := (none, false)
refine ⟨tuplePoint C.merged (C.choice s).tuple 0, ?_⟩
apply SplitGeometry.selected_designated_mem_points C s
exact Finset.mem_image.mpr ⟨0, Finset.mem_univ _, rfl⟩
theorem pair_common_block (C : Configuration F) (hV : C.Valid) (hC : C.Compatible)
{x y : Point F} (hx : x ∈ C.points) (hy : y ∈ C.points) :
∃ b ∈ C.blockLabels, x ∈ C.blockPoints b ∧ y ∈ C.blockPoints b := by
classical
by_cases hs : joiningLine x y ∈ C.splitLines
· obtain ⟨s, _, hline⟩ := Finset.mem_image.mp hs
have hxd : x ∈ designated C.merged (C.choice s) := by
rw [← ConfigurationStructure.split_inter_points C hV hC s]
refine Finset.mem_inter.mpr ⟨?_, hx⟩
simpa only [linePoints, Finset.mem_filter, Finset.mem_univ, true_and, hline]
using onLine_joiningLine_left x y
have hyd : y ∈ designated C.merged (C.choice s) := by
rw [← ConfigurationStructure.split_inter_points C hV hC s]
refine Finset.mem_inter.mpr ⟨?_, hy⟩
simpa only [linePoints, Finset.mem_filter, Finset.mem_univ, true_and, hline]
using onLine_joiningLine_right x y
exact SplitGeometry.selected_pair_common_block C hV s hxd hyd
· by_cases hm : joiningLine x y ∈ constituentLines C.merged
· obtain ⟨ds, _, hline⟩ := Finset.mem_image.mp hm
refine ⟨Sum.inr (Sum.inl ds.1), ?_, ?_, ?_⟩
· simp [blockLabels, IsBlockLabel]
· apply Finset.mem_inter.mpr
refine ⟨mem_merged_of_on_constituent C.merged ds.1 ds.2 ?_, hx⟩
rw [hline]
exact onLine_joiningLine_left x y
· apply Finset.mem_inter.mpr
refine ⟨mem_merged_of_on_constituent C.merged ds.1 ds.2 ?_, hy⟩
rw [hline]
exact onLine_joiningLine_right x y
· refine ⟨Sum.inl (joiningLine x y), ?_, ?_, ?_⟩
· simp [blockLabels, IsBlockLabel, hs, hm]
· apply Finset.mem_inter.mpr
refine ⟨?_, hx⟩
simpa only [linePoints, Finset.mem_filter, Finset.mem_univ, true_and]
using onLine_joiningLine_left x y
· apply Finset.mem_inter.mpr
refine ⟨?_, hy⟩
simpa only [linePoints, Finset.mem_filter, Finset.mem_univ, true_and]
using onLine_joiningLine_right x y
end Configuration
end Balanced
end OAI
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