| 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 |
|
|
