lean/OAI/Computability/VertexCover/Reduction/ReductionSound.lean

import Mathlib
import OAI.Computability.VertexCover.Analysis.PowerValueRepetition

namespace OAI

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namespace VertexCover.ClauseProjection
open UniqueGames.Foundations
open LabelCover
noncomputable section

def baseGap : ℝ := (PCP.PCPIteration.finalClauseGap : ℝ) / 3

theorem baseGap_pos : 0 < baseGap := by
  exact div_pos (Rat.cast_pos.mpr PCP.PCPIteration.finalClauseGap_positive) (by norm_num)

theorem baseGap_le_third : baseGap ≤ 1/3 := by
  exact div_le_div_of_nonneg_right
    (by exact_mod_cast PCP.TableGapReduction.finalClauseGap_le_one) (by norm_num)

def repetitionAlphabetBudget : ℝ := max 1 (Repetition.logTwo 14)

theorem repetitionAlphabetBudget_pos : 0 < repetitionAlphabetBudget :=
  lt_of_lt_of_le zero_lt_one (le_max_left _ _)

def repetitionBase : ℝ := 1 - baseGap ^ 3 / 6000

theorem repetitionBase_mem : 0 < repetitionBase ∧ repetitionBase < 1 := by
  have hp := baseGap_pos
  have hu := baseGap_le_third
  have hc : baseGap ^ 3 ≤ 1 := by
    calc
      _ ≤ (1 : ℝ)^3 := pow_le_pow_left₀ hp.le (by linarith) 3
      _ = 1 := by norm_num
  constructor
  · unfold repetitionBase; linarith
  · unfold repetitionBase; have : 0 < baseGap^3 / 6000 := by positivity
    linarith

theorem exists_repetition_length (σ : ℝ) (hσ : 0 < σ) :
    ∃ n : ℕ, repetitionBase ^ ((n : ℝ) / repetitionAlphabetBudget) < σ := by
  have hr := Real.rpow_lt_one repetitionBase_mem.1.le repetitionBase_mem.2
    (one_div_pos.mpr repetitionAlphabetBudget_pos)
  obtain ⟨n, hn⟩ := exists_pow_lt_of_lt_one hσ hr
  refine ⟨n, ?_⟩
  have he : (n : ℝ) / repetitionAlphabetBudget =
      (1 / repetitionAlphabetBudget) * (n : ℝ) := by ring
  rw [he, Real.rpow_mul repetitionBase_mem.1.le, Real.rpow_natCast]
  exact hn

def repetitionLength (σ : ℝ) (hσ : 0 < σ) : ℕ :=
  (exists_repetition_length σ hσ).choose

def reduction (σ : ℝ) (hσ : 0 < σ) (F : Target.Formula) : LabelCover :=
  (baseReduction F).parallelPower (repetitionLength σ hσ)

theorem reduction_alphabets (σ : ℝ) (hσ : 0 < σ) (F : Target.Formula) :
    (reduction σ hσ F).qU = 7 ^ repetitionLength σ hσ ∧
    (reduction σ hσ F).qV = 2 ^ repetitionLength σ hσ := ⟨rfl, rfl⟩

theorem reduction_complete (σ : ℝ) (hσ : 0 < σ) (F : Target.Formula)
    (hF : F.Satisfiable) :
    ∃ A : (reduction σ hσ F).Labeling, ∀ e, (reduction σ hσ F).Satisfies A e := by
  obtain ⟨A, hA⟩ := baseReduction_complete F hF
  exact ⟨_, LabelCover.repeat_perfect _ _ A hA⟩

theorem reduction_sound (σ : ℝ) (hσ : 0 < σ) (F : Target.Formula)
    (hF : ¬F.Satisfiable) : (reduction σ hσ F).value < σ := by
  let Φ := baseReduction F
  let n := repetitionLength σ hσ
  let : Nonempty (Fin Φ.qU) := ⟨⟨0, Φ.qU_pos⟩⟩
  let : Nonempty (Fin Φ.qV) := ⟨⟨0, Φ.qV_pos⟩⟩
  have hv : Φ.asGame.value ≤ 1 - baseGap := by
    change Φ.asGameValue ≤ 1 - baseGap
    rw [LabelCover.asGameValue_eq Φ (baseReduction_coherent F)]
    exact baseReduction_sound F hF
  have hb := Repetition.holenstein_repetition_value Φ.asGame n hv
    (by have := baseGap_pos; linarith : 1 - baseGap < 1)
    (le_max_left 1 (Repetition.logTwo 14))
    (show Repetition.logTwo ((Fintype.card (Fin Φ.qU) : ℝ) *
        (Fintype.card (Fin Φ.qV) : ℝ)) ≤ repetitionAlphabetBudget from by
      change Repetition.logTwo ((7 : ℝ) * 2) ≤ repetitionAlphabetBudget
      norm_num only [show (7 : ℝ) * 2 = 14 by norm_num]
      exact le_max_right _ _)
  have he : 1 - (1 - (1 - baseGap)) ^ 3 / 6000 = repetitionBase := by
    unfold repetitionBase; ring
  rw [he] at hb
  exact lt_of_le_of_lt ((LabelCover.power_value_le_repetition Φ
    (baseReduction_coherent F) n).trans hb)
    (exists_repetition_length σ hσ).choose_spec

end
end VertexCover.ClauseProjection


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end OAI