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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 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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | 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
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