| import Mathlib |
| import OAI.Computability.VertexCover.Analysis.PowerValueRepetition |
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| namespace OAI |
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| namespace VertexCover.ClauseProjection |
| open UniqueGames.Foundations |
| open LabelCover |
| noncomputable section |
|
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| def baseGap : ℝ := (PCP.PCPIteration.finalClauseGap : ℝ) / 3 |
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| theorem baseGap_pos : 0 < baseGap := by |
| exact div_pos (Rat.cast_pos.mpr PCP.PCPIteration.finalClauseGap_positive) (by norm_num) |
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| 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) |
|
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| def repetitionAlphabetBudget : ℝ := max 1 (Repetition.logTwo 14) |
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| theorem repetitionAlphabetBudget_pos : 0 < repetitionAlphabetBudget := |
| lt_of_lt_of_le zero_lt_one (le_max_left _ _) |
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| def repetitionBase : ℝ := 1 - baseGap ^ 3 / 6000 |
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| 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 |
|
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| 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 |
| 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 |
| 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 |
| exact le_max_right _ _) |
| have he : 1 - (1 - (1 - baseGap)) ^ 3 / 6000 = repetitionBase := by |
| unfold repetitionBase; ring |
| rw 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 |
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| end |
| end VertexCover.ClauseProjection |
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| end |
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| end |
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| end OAI |
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