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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.Computability.PerfectCompleteness.Reduction.SourceGame
import OAI.Computability.UniqueGames.Reduction.FinalParametersLemmas
namespace OAI
section
namespace PerfectCompleteness.SourceAmplification
open SourceClause UniqueGamesTheorem.Foundations Games
open UniqueGamesTheorem.Integration
theorem logTwo_le_four {x : ℝ} (hx : 0 < x) (hbound : x ≤ 16) :
Repetition.logTwo x ≤ 4 := by
calc
Repetition.logTwo x ≤ Repetition.logTwo 16 := by
unfold Repetition.logTwo
exact div_le_div_of_nonneg_right (Real.log_le_log hx hbound)
(Real.log_pos (by norm_num : (1 : ℝ) < 2)).le
_ = 4 := SourceParameters.logTwo_sixteen
theorem game_repetition_rate_le_sixteen {Q₁ Q₂ A₁ A₂ : Type*}
[Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
[Nonempty A₁] [Nonempty A₂] [DecidableEq Q₁] [DecidableEq Q₂]
(G : Game Q₁ Q₂ A₁ A₂) {gap : ℚ} (hgap : 0 < gap) (hgap1 : gap ≤ 1)
(hcards : Fintype.card A₁ * Fintype.card A₂ ≤ 16)
(hvalue : G.value ≤ 1 - (gap : ℝ)) (k : Nat) :
(G.repetition k).value ≤ (SourceParameters.rate gap : ℝ) ^ k := by
have hg : (0 : ℝ) < gap := by exact_mod_cast hgap
have ha : (0 : ℝ) < Fintype.card A₁ := by exact_mod_cast Fintype.card_pos (α := A₁)
have hb : (0 : ℝ) < Fintype.card A₂ := by exact_mod_cast Fintype.card_pos (α := A₂)
have hbound : (Fintype.card A₁ : ℝ) * (Fintype.card A₂ : ℝ) ≤ 16 := by
exact_mod_cast hcards
have halphabet := logTwo_le_four (mul_pos ha hb) hbound
have h := Repetition.holenstein_repetition_value G k hvalue
(by linarith) (by norm_num : (1 : ℝ) ≤ 4) halphabet
simp only [sub_sub_cancel] at h
exact h.trans (SourceParameters.four_bit_rate_comparison hgap hgap1 k)
def ClauseGap {n m : Nat} (clauses : Fin m → NormalizedClause n) (γ : ℚ) : Prop :=
∀ assignment : Fin n → Bool,
γ * (m : ℚ) ≤ (SourceGame.violatedCount clauses assignment : ℚ)
theorem source_value_le {n m : Nat} [NeZero m]
(clauses : Fin m → NormalizedClause n) {γ : ℚ} (gap : ClauseGap clauses γ) :
(SourceGame.game clauses).value ≤ 1 - ((γ / 3 : ℚ) : ℝ) := by
apply ((SourceGame.game clauses).value_le_iff _).mpr
intro strategy
have hm : (0 : ℝ) < m := by exact_mod_cast Nat.pos_of_neZero m
have hcount : (γ : ℝ) * (m : ℝ) ≤
(SourceGame.violatedCount clauses strategy.2 : ℝ) := by
exact_mod_cast gap strategy.2
have hfrac := (le_div_iff₀ hm).mpr hcount
have h := SourceGame.success_le_one_sub_third_violations clauses strategy
push_cast
linarith
theorem source_repetition_rate {n m : Nat} [NeZero m]
(clauses : Fin m → NormalizedClause n) {γ : ℚ}
(hγ : 0 < γ) (hγ1 : γ ≤ 1) (gap : ClauseGap clauses γ) (k : Nat) :
((SourceGame.game clauses).repetition k).value ≤
(SourceParameters.rate (γ / 3) : ℝ) ^ k := by
have hthird : (0 : ℚ) < γ / 3 := div_pos hγ (by norm_num)
have hthird1 : γ / 3 ≤ (1 : ℚ) := by linarith
have hv : (SourceGame.questionGame clauses).value ≤ 1 - ((γ / 3 : ℚ) : ℝ) := by
simpa only [SourceGame.questionGame, OccurrenceGame.toGame_value] using
source_value_le clauses gap
have h := game_repetition_rate_le_sixteen (SourceGame.questionGame clauses)
hthird hthird1 (by decide) hv k
simpa only [SourceGame.questionGame, OccurrenceGame.toGame_repetition_value] using h
theorem exists_repetition_length {γ σ : ℚ}
(hγ : 0 < γ) (hγ1 : γ ≤ 1) (hσ : 0 < σ) :
∃ k : Nat, 0 < k ∧ SourceParameters.rate (γ / 3) ^ k < σ := by
have hthird : (0 : ℚ) < γ / 3 := div_pos hγ (by norm_num)
have hthird1 : γ / 3 ≤ (1 : ℚ) := by linarith
have hr := SourceParameters.rate_bounds hthird hthird1
exact UniqueGamesTheorem.Soundness.RepetitionUpper.exists_power_lt
(SourceParameters.rate (γ / 3)) σ hr.1.le hr.2 hσ
def repetitionLength (γ σ : ℚ) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (hσ : 0 < σ) : Nat :=
Nat.find (exists_repetition_length hγ hγ1 hσ)
theorem repetitionLength_spec (γ σ : ℚ) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (hσ : 0 < σ) :
0 < repetitionLength γ σ hγ hγ1 hσ ∧
SourceParameters.rate (γ / 3) ^ repetitionLength γ σ hγ hγ1 hσ < σ :=
Nat.find_spec (exists_repetition_length hγ hγ1 hσ)
theorem repeated_source_value_lt {n m : Nat} [NeZero m]
(clauses : Fin m → NormalizedClause n) (γ σ : ℚ)
(hγ : 0 < γ) (hγ1 : γ ≤ 1) (hσ : 0 < σ) (gap : ClauseGap clauses γ) :
((SourceGame.game clauses).repetition (repetitionLength γ σ hγ hγ1 hσ)).value < (σ : ℝ) := by
have hsmall : (SourceParameters.rate (γ / 3) : ℝ) ^
repetitionLength γ σ hγ hγ1 hσ < (σ : ℝ) := by
exact_mod_cast (repetitionLength_spec γ σ hγ hγ1 hσ).2
exact (source_repetition_rate clauses hγ hγ1 gap _).trans_lt hsmall
theorem repeated_completeness_success {n m : Nat} [NeZero m]
(clauses : Fin m → NormalizedClause n) (assignment : Fin n → Bool)
(satisfying : ∀ c, (clauses c).clause.eval assignment = true) (k : Nat) :
((SourceGame.game clauses).repetition k).success
(Game.repeatStrategy (SourceGame.completenessStrategy clauses assignment satisfying) k) = 1 := by
rw [← (SourceGame.game clauses).toGame_repetition_success (SourceGame.presentation clauses)]
change ((SourceGame.questionGame clauses).repetition k).success
(Game.repeatStrategy (SourceGame.completenessStrategy clauses assignment satisfying) k) = 1
rw [Game.success_repeatStrategy, SourceGame.questionGame_success,
SourceGame.completeness_success, one_pow]
theorem repeated_completeness_value {n m : Nat} [NeZero m]
(clauses : Fin m → NormalizedClause n) (assignment : Fin n → Bool)
(satisfying : ∀ c, (clauses c).clause.eval assignment = true) (k : Nat) :
((SourceGame.game clauses).repetition k).value = 1 := by
apply le_antisymm ((SourceGame.game clauses).repetition k).value_le_one
rw [← repeated_completeness_success clauses assignment satisfying k]
exact ((SourceGame.game clauses).repetition k).success_le_value _
end PerfectCompleteness.SourceAmplification
end
end OAI
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