lean/OAI/Computability/VertexCover/Analysis/MidpointGridTail.lean

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 OAI