| import Mathlib.Algebra.BigOperators.Group.Finset.Basic |
| import Mathlib.Tactic |
| import OAI.Combinatorics.BalancedRyser.Construction.Construction |
| import OAI.Combinatorics.BalancedRyser.Incidence.AffineIncidence |
|
|
| namespace OAI |
|
|
| namespace Balanced.Configuration |
|
|
| open scoped BigOperators |
| noncomputable section |
|
|
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
| variable (C : Configuration F) |
|
|
| theorem linePoints_inter_card_le_one (l m : Line F) (hne : l ≠ m) : |
| (linePoints l ∩ linePoints m).card ≤ 1 := by |
| apply Finset.card_le_one.mpr |
| intro x hx y hy |
| rcases Finset.mem_inter.mp hx with ⟨hxl, hxm⟩ |
| rcases Finset.mem_inter.mp hy with ⟨hyl, hym⟩ |
| exact common_point_eq_of_lines_ne l m hne x y |
| (Finset.mem_filter.mp hxl).2 (Finset.mem_filter.mp hxm).2 |
| (Finset.mem_filter.mp hyl).2 (Finset.mem_filter.mp hym).2 |
|
|
| theorem valid_split_not_constituent (hvalid : C.Valid) |
| (l : Line F) (hl : l ∈ C.splitLines) : l ∉ constituentLines C.merged := by |
| obtain ⟨s, _, rfl⟩ := Finset.mem_image.mp hl |
| exact (hvalid.2 s).2.1 |
|
|
| theorem constituent_not_splitLinesAt (hvalid : C.Valid) |
| (p : Point F) (d : Direction F) (bit : Bool) : |
| constituent C.merged d bit ∉ C.splitLinesAt p := by |
| intro h |
| apply C.valid_split_not_constituent hvalid _ (Finset.mem_filter.mp h).1 |
| exact Finset.mem_image.mpr ⟨(d, bit), Finset.mem_univ _, rfl⟩ |
|
|
| theorem designatedAt_subset_splitLinesAt_union (hvalid : C.Valid) (p : Point F) : |
| C.designatedAt p ⊆ (C.splitLinesAt p).biUnion linePoints := by |
| intro x hx |
| obtain ⟨s, hs, hxs⟩ := Finset.mem_biUnion.mp hx |
| obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hxs |
| rcases hvalid.2 s with ⟨_, _, _, _, _, hpoints, _, _, _⟩ |
| refine Finset.mem_biUnion.mpr ⟨(C.choice s).line, ?_, ?_⟩ |
| · exact Finset.mem_filter.mpr ⟨Finset.mem_image.mpr |
| ⟨s, Finset.mem_univ _, rfl⟩, (Finset.mem_filter.mp hs).2⟩ |
| · exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, hpoints i⟩ |
|
|
| theorem designatedAt_inter_line_card_le (hvalid : C.Valid) |
| (p : Point F) (l : Line F) (hline : l ∉ C.splitLinesAt p) : |
| (C.designatedAt p ∩ linePoints l).card ≤ C.splitMultiplicity p := by |
| have hsub : C.designatedAt p ∩ linePoints l ⊆ |
| (C.splitLinesAt p).biUnion (fun m => linePoints m ∩ linePoints l) := by |
| intro x hx |
| obtain ⟨hxd, hxl⟩ := Finset.mem_inter.mp hx |
| obtain ⟨m, hm, hxm⟩ := Finset.mem_biUnion.mp |
| (C.designatedAt_subset_splitLinesAt_union hvalid p hxd) |
| exact Finset.mem_biUnion.mpr ⟨m, hm, Finset.mem_inter.mpr ⟨hxm, hxl⟩⟩ |
| calc |
| (C.designatedAt p ∩ linePoints l).card ≤ |
| ((C.splitLinesAt p).biUnion (fun m => linePoints m ∩ linePoints l)).card := |
| Finset.card_le_card hsub |
| _ ≤ ∑ m ∈ C.splitLinesAt p, (linePoints m ∩ linePoints l).card := |
| Finset.card_biUnion_le |
| _ ≤ ∑ _m ∈ C.splitLinesAt p, 1 := by |
| apply Finset.sum_le_sum |
| intro m hm |
| exact linePoints_inter_card_le_one m l (by intro heq; exact hline (heq ▸ hm)) |
| _ = C.splitMultiplicity p := by simp [splitMultiplicity] |
|
|
| theorem designatedAt_inter_merged_card_le (hvalid : C.Valid) |
| (p : Point F) (d : Direction F) : |
| (C.designatedAt p ∩ mergedPoints C.merged d).card ≤ 2 * C.splitMultiplicity p := by |
| rw [mergedPoints, Finset.inter_union_distrib_left] |
| calc |
| _ ≤ (C.designatedAt p ∩ linePoints (constituent C.merged d false)).card + |
| (C.designatedAt p ∩ linePoints (constituent C.merged d true)).card := |
| Finset.card_union_le _ _ |
| _ ≤ C.splitMultiplicity p + C.splitMultiplicity p := |
| Nat.add_le_add |
| (C.designatedAt_inter_line_card_le hvalid p _ |
| (C.constituent_not_splitLinesAt hvalid p d false)) |
| (C.designatedAt_inter_line_card_le hvalid p _ |
| (C.constituent_not_splitLinesAt hvalid p d true)) |
| _ = 2 * C.splitMultiplicity p := by omega |
|
|
| theorem designatedAt_inter_block_card_le (hvalid : C.Valid) |
| (p : Point F) (hb : 1 ≤ C.splitMultiplicity p) |
| (label : BlockLabel F) (hlabel : C.IsBlockLabel label) : |
| (C.designatedAt p ∩ C.blockPoints label).card ≤ 2 * C.splitMultiplicity p := by |
| classical |
| rcases label with l | d | ⟨s, bit⟩ |
| · have hline : l ∉ C.splitLinesAt p := by |
| intro hl |
| exact hlabel (Finset.mem_union_left _ (Finset.mem_filter.mp hl).1) |
| have hsub : C.designatedAt p ∩ C.blockPoints (Sum.inl l) ⊆ |
| C.designatedAt p ∩ linePoints l := by |
| intro x hx |
| exact Finset.mem_inter.mpr ⟨(Finset.mem_inter.mp hx).1, |
| (Finset.mem_inter.mp (Finset.mem_inter.mp hx).2).1⟩ |
| exact (Finset.card_le_card hsub).trans |
| ((C.designatedAt_inter_line_card_le hvalid p l hline).trans (by omega)) |
| · have hsub : C.designatedAt p ∩ C.blockPoints (Sum.inr (Sum.inl d)) ⊆ |
| C.designatedAt p ∩ mergedPoints C.merged d := by |
| intro x hx |
| exact Finset.mem_inter.mpr ⟨(Finset.mem_inter.mp hx).1, |
| (Finset.mem_inter.mp (Finset.mem_inter.mp hx).2).1⟩ |
| exact (Finset.card_le_card hsub).trans (C.designatedAt_inter_merged_card_le hvalid p d) |
| · have hsmall : (C.blockPoints (Sum.inr (Sum.inr (s, bit)))).card ≤ 2 := by |
| cases bit <;> exact Finset.card_le_two |
| exact (Finset.card_le_card Finset.inter_subset_right).trans (hsmall.trans (by omega)) |
| |
| theorem two_split_block_capacity (hvalid : C.Valid) |
| (p : Point F) (hb : C.splitMultiplicity p = 2) |
| (label : BlockLabel F) (hlabel : C.IsBlockLabel label) : |
| (C.designatedAt p ∩ C.blockPoints label).card ≤ 4 := by |
| simpa [hb] using C.designatedAt_inter_block_card_le hvalid p (by omega) label hlabel |
|
|
| end |
| end Balanced.Configuration |
|
|
| end OAI |
|
|
