| import Mathlib |
| import OAI.Computability.VertexCover.Reduction.DecodingUpperThreeHalves |
| import OAI.Computability.VertexCover.Analysis.SignAtom |
| import OAI.Computability.VertexCover.Analysis.GridCentralCard |
|
|
| namespace OAI |
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| namespace VertexCover.FiniteAntiConcentration |
| noncomputable section |
| open AntiConcentration |
|
|
| def gridTailDensity (d n t : ℕ) : ℝ := |
| ((Finset.univ.filter (fun x : Fin n → Fin (2*d+1) => |
| (t : ℝ) < ∑ i, grid d (x i))).card : ℝ) / ((2*d+1 : ℕ)^n : ℝ) |
| |
| theorem gridTailDensity_lower (d k t : ℕ) (hk : 0 < k) (ht : t < d) : |
| 1/2 - Real.sqrt 2 * ((t : ℝ)+1) / Real.sqrt (2*(k : ℝ)) < |
| gridTailDensity d (2*k) t := by |
| classical |
| let C : ℝ := ((Finset.univ.filter (fun x : Fin (2*k) → Fin (2*d+1) => |
| -(t : ℝ) ≤ ∑ i, grid d (x i) ∧ (∑ i, grid d (x i)) ≤ t)).card : ℝ) |
| let Q : ℝ := ((2*d+1 : ℕ)^(2*k) : ℝ) |
| have hQ : 0 < Q := by dsimp [Q]; positivity |
| have hP : 0 < (2:ℝ)^(2*k) := by positivity |
| have hsqrt : 0 < Real.sqrt (2*(k:ℝ)) := by positivity |
| have hc : C / Q ≤ (2*(t:ℝ)+1) * centralProbability k := by |
| have h := grid_central_card d (2*k) t ht |
| have hatom := sign_atom_le k (2*k/2) |
| have hfirst : C/Q ≤ (2*(t:ℝ)+1) * (((2*k).choose (2*k/2) : ℝ)/(2:ℝ)^(2*k)) := by |
| rw [mul_div, div_le_div_iff₀ hQ hP] |
| dsimp [C, Q] at * |
| nlinarith [h] |
| exact hfirst.trans (mul_le_mul_of_nonneg_left hatom (by positivity)) |
| have hstrict : C / Q < (2*(t:ℝ)+2) * (Real.sqrt 2 / Real.sqrt (2*(k:ℝ))) := by |
| have hstep := mul_lt_mul_of_pos_left (centralProbability_lt k hk) |
| (by positivity : (0:ℝ) < 2*t+1) |
| have hnonneg : 0 ≤ Real.sqrt 2 / Real.sqrt (2*(k:ℝ)) := by positivity |
| exact lt_of_le_of_lt hc (hstep.trans_le (mul_le_mul_of_nonneg_right (by linarith) hnonneg)) |
| have htail : 2 * gridTailDensity d (2*k) t + C / Q = 1 := by |
| have h := grid_tail_count d (2*k) t |
| unfold gridTailDensity |
| apply (div_eq_one_iff_eq (ne_of_gt hQ)).mpr at h |
| dsimp [C, Q] at * |
| convert h using 1 ; ring |
| have he : (2*(t:ℝ)+2) * (Real.sqrt 2 / Real.sqrt (2*(k:ℝ))) = |
| 2 * (Real.sqrt 2 * ((t:ℝ)+1) / Real.sqrt (2*(k:ℝ))) := by ring |
| rw [he] at hstrict |
| linarith |
| |
| end |
| end VertexCover.FiniteAntiConcentration |
| |
| namespace VertexCover.Parameters |
| noncomputable section |
| |
| theorem t_lt_d (m : ℕ) (hm : 4 ≤ m) : t m < d m := by |
| have hm1 : 1 ≤ m := by omega |
| have hpow : m^4 ≤ m^10 := Nat.pow_le_pow_right hm1 (by decide) |
| have hp : 1 ≤ m^4 := one_le_pow₀ hm1 |
| unfold t d |
| norm_num only [Nat.reducePow] |
| omega |
| |
| theorem tail_error_lt (m : ℕ) (hm : 4 ≤ m) : |
| Real.sqrt 2 * (a m + 1/Real.sqrt (d m)) < 1/(m:ℝ) := by |
| have hmpos : (0:ℝ) < m := by exact_mod_cast (show 0 < m by omega) |
| have ha : 0 < a m := by unfold a; positivity |
| have hinv : (1:ℝ)/Real.sqrt (d m) < a m := by |
| exact one_div_lt_one_div_of_lt (by positivity) (sqrt_d_gt m hm) |
| have hsq : Real.sqrt (2:ℝ) < 3/2 := by |
| have hs := Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2) |
| have hp := Real.sqrt_nonneg (2:ℝ) |
| nlinarith |
| have hmul : Real.sqrt 2 * (a m + 1/Real.sqrt (d m)) < 3*a m := by |
| have h1 := mul_lt_mul_of_pos_left hinv (Real.sqrt_pos.mpr (by norm_num : (0:ℝ)<2)) |
| have h2 := mul_lt_mul_of_pos_right hsq (by linarith : 0 < 2*a m) |
| nlinarith |
| have he : 3*a m < 1/(m:ℝ) := by |
| unfold a |
| rw [mul_one_div] |
| apply (div_lt_div_iff₀ (by positivity : (0:ℝ)<4*m) hmpos).mpr |
| nlinarith |
| exact hmul.trans he |
| |
| theorem midpoint_grid_tail (m : ℕ) (hm : 4 ≤ m) : |
| 1/2 - Real.sqrt 2 * (a m + 1/Real.sqrt (d m)) < |
| FiniteAntiConcentration.gridTailDensity (d m) (d m) (t m) ∧ |
| 1/2 - 1/(m:ℝ) < |
| 1/2 - Real.sqrt 2 * (a m + 1/Real.sqrt (d m)) := by |
| obtain ⟨k, hk⟩ := (dimensions m hm).2.2.2 |
| have hd : d m = 2*k := by omega |
| have hkpos : 0 < k := by have := (dimensions m hm).1; omega |
| have ht := FiniteAntiConcentration.gridTailDensity_lower (d m) k (t m) hkpos (t_lt_d m hm) |
| have hs : Real.sqrt (2*(k:ℝ)) = Real.sqrt (d m) := by rw [hd]; push_cast; rfl |
| rw [← hd, hs] at ht |
| have hnonzero : Real.sqrt (d m) ≠ 0 := ne_of_gt (lt_trans (by positivity) (sqrt_d_gt m hm)) |
| have he : Real.sqrt 2 * ((t m : ℝ)+1) / Real.sqrt (d m) = |
| Real.sqrt 2 * (a m + 1/Real.sqrt (d m)) := by |
| rw [t_eq m (by omega)] |
| field_simp |
| exact ⟨by simpa only [he] using ht, by linarith [tail_error_lt m hm]⟩ |
| |
| end |
| end VertexCover.Parameters |
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| end |
| end |
| end |
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| end |
| end |
| |
| end OAI |
| |
