| import Mathlib |
| import OAI.Computability.VertexCover.Reduction.FiniteMeanUnionBound |
| import OAI.Computability.VertexCover.Reduction.GridBudget |
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| namespace OAI |
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| namespace VertexCover.LabelCover |
| open MeasureTheory ProbabilityTheory |
|
|
| theorem high_count_lower (Φ : LabelCover) {d : ℕ} {t : ℝ} |
| (A : Finset (Φ.Vertex d)) (hA : (Φ.graph d t).IsIndepSet (A : Set (Φ.Vertex d))) : |
| A.card ≤ (A.filter (fun v => t < Φ.compatibilityNorm (Φ.sumVector v))).card + 1 := by |
| classical |
| have hl : (A.filter (fun v => Φ.compatibilityNorm (Φ.sumVector v) ≤ t)).card ≤ 1 := by |
| apply Finset.card_le_one.mpr |
| intro v hv w hw |
| exact Φ.low_norm_subsingleton (A : Set (Φ.Vertex d)) hA |
| (Finset.mem_filter.mp hv) (Finset.mem_filter.mp hw) |
| have he := Finset.card_filter_add_card_filter_not (s := A) |
| (fun v => t < Φ.compatibilityNorm (Φ.sumVector v)) |
| simp only [not_lt] at he |
| omega |
|
|
| theorem vertex_card_ge_grid (Φ : LabelCover) {d : ℕ} (hd : 0 < d) : |
| 2*d+1 ≤ Fintype.card (Φ.Vertex d) := by |
| classical |
| have hq : 0 < Φ.WeightDimension d := by |
| dsimp [WeightDimension] |
| exact pow_pos (lt_of_lt_of_le Φ.qU_pos (le_max_left _ _)) _ |
| have he : 1 ≤ Φ.WeightDimension d * d := Nat.mul_pos hq hd |
| rw [Φ.vertex_card] |
| have hM : 1 ≤ Φ.M^(d.choose 2) := one_le_pow₀ Φ.M_pos |
| have hp : 2*d+1 ≤ (2*d+1)^(Φ.WeightDimension d*d) := by |
| simpa using Nat.pow_le_pow_right (by omega : 0 < 2*d+1) he |
| exact hp.trans (Nat.le_mul_of_pos_left _ hM) |
|
|
| theorem dense_high_fraction (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m) |
| (A : Finset (Φ.Vertex (Parameters.d m))) |
| (hA : (Φ.graph (Parameters.d m) (Parameters.t m)).IsIndepSet |
| (A : Set (Φ.Vertex (Parameters.d m)))) |
| (hdense : (Fintype.card (Φ.Vertex (Parameters.d m)):ℝ)/(m:ℝ) ≤ A.card) : |
| Parameters.b₀ m ≤ |
| ((A.filter (fun v => (Parameters.t m:ℝ) < Φ.compatibilityNorm (Φ.sumVector v))).card:ℝ) / |
| Fintype.card (Φ.Vertex (Parameters.d m)) := by |
| classical |
| have hmR : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega) |
| have hN : 2*m < Fintype.card (Φ.Vertex (Parameters.d m)) := |
| (Parameters.grid_budget m hm).2.2.trans_le |
| (Φ.vertex_card_ge_grid (by have h := (Parameters.dimensions m hm).1; omega)) |
| have hNR : (2:ℝ)*m < Fintype.card (Φ.Vertex (Parameters.d m)) := by exact_mod_cast hN |
| have hNP : (0:ℝ) < Fintype.card (Φ.Vertex (Parameters.d m)) := by linarith |
| have hc : (A.card:ℝ) ≤ |
| (A.filter (fun v => (Parameters.t m:ℝ) < Φ.compatibilityNorm (Φ.sumVector v))).card + 1 := by |
| exact_mod_cast Φ.high_count_lower A hA |
| unfold Parameters.b₀ |
| apply (le_div_iff₀ hNP).mpr |
| have hd := (div_le_iff₀ hmR).mp hdense |
| have hh : (1/(2*(m:ℝ))) * Fintype.card (Φ.Vertex (Parameters.d m)) = |
| Fintype.card (Φ.Vertex (Parameters.d m))/(2*(m:ℝ)) := by ring |
| rw [hh] |
| apply (div_le_iff₀ (by positivity : (0:ℝ)<2*m)).mpr |
| nlinarith |
|
|
| end VertexCover.LabelCover |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end |
| end |
| end |
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|
| end OAI |
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