| import OAI.Probability.SATComputability.PoissonBlockBound |
| import OAI.Probability.SATComputability.CandidateSurvival |
|
|
| namespace OAI |
|
|
| namespace FixedClauseThreshold.Computability |
|
|
| open DilutedSpinGlass MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal Classical |
|
|
| noncomputable def dyadicLevel (j : ℕ) : ℝ := ((2 : ℝ)^j)⁻¹ |
| def dyadicTruncation (j : ℕ) : ℕ := j+122 |
| def dyadicMultiplicity (j : ℕ) : ℕ := 2*(j+122)*2^((j+1)/2) |
| def dyadicBlock (j : ℕ) : ℕ := 32*dyadicTruncation j*dyadicMultiplicity j |
|
|
| theorem dyadicLevel_pos (j : ℕ) : 0 < dyadicLevel j := by |
| unfold dyadicLevel |
| positivity |
|
|
| theorem dyadicLevel_le_one (j : ℕ) : dyadicLevel j ≤ 1 := by |
| unfold dyadicLevel |
| exact inv_le_one_of_one_le₀ (one_le_pow₀ (by norm_num)) |
|
|
| theorem dyadic_truncation_error (j : ℕ) : |
| Real.exp 60 / (2 : ℝ)^(dyadicTruncation j) ≤ dyadicLevel j/4 := by |
| have he : Real.exp 60 ≤ (2 : ℝ)^120 := by |
| calc |
| _ = (Real.exp 1)^60 := by rw [← Real.exp_nat_mul]; norm_num |
| _ ≤ (4 : ℝ)^60 := pow_le_pow_left₀ (Real.exp_pos 1).le |
| (Real.exp_one_lt_three.le.trans (by norm_num : (3 : ℝ) ≤ 4)) 60 |
| _ = _ := by norm_num |
| have hden : (2 : ℝ)^(dyadicTruncation j) = (2 : ℝ)^j * (2 : ℝ)^122 := by |
| rw [dyadicTruncation, pow_add] |
| rw [dyadicLevel, hden] |
| apply (div_le_iff₀ (by positivity)).mpr |
| field_simp |
| nlinarith |
|
|
| theorem dyadic_multiplicity_pos (j : ℕ) : 0 < dyadicMultiplicity j := by |
| unfold dyadicMultiplicity |
| positivity |
|
|
| theorem dyadic_replacement_error (j : ℕ) : |
| (dyadicTruncation j : ℝ) / (dyadicMultiplicity j : ℝ)^2 ≤ dyadicLevel j/4 := by |
| have hp : j ≤ (j+1)/2*2 := by omega |
| have hpow : (2 : ℝ)^j ≤ ((2 : ℝ)^((j+1)/2))^2 := by |
| rw [← pow_mul] |
| exact pow_le_pow_right₀ (by norm_num) hp |
| have hL : (1 : ℝ) ≤ j+122 := by exact_mod_cast (show 1 ≤ j+122 by omega) |
| have hmul : (4 : ℝ)*((j:ℝ)+122)*(2:ℝ)^j ≤ |
| (2*((j:ℝ)+122)*(2:ℝ)^((j+1)/2))^2 := by |
| have hprod := mul_le_mul_of_nonneg_left hpow |
| (show (0 : ℝ) ≤ 4*((j:ℝ)+122) by positivity) |
| have hrest := mul_nonneg |
| (show (0 : ℝ) ≤ (j:ℝ)+122-1 by linarith) |
| (show (0 : ℝ) ≤ 4*((j:ℝ)+122)*((2:ℝ)^((j+1)/2))^2 by positivity) |
| nlinarith |
| unfold dyadicTruncation dyadicMultiplicity dyadicLevel |
| push_cast |
| apply (div_le_iff₀ (by positivity)).mpr |
| rw [show (2^j : ℝ)⁻¹ / 4 * (2*((j:ℝ)+122)*2^((j+1)/2))^2 = |
| (2*((j:ℝ)+122)*2^((j+1)/2))^2 / (4*2^j) by field_simp] |
| apply (le_div_iff₀ (by positivity)).mpr |
| nlinarith |
|
|
| theorem dyadic_block_kills {n : ℕ} [NeZero n] |
| (U : Finset (DeletionCandidate n)) (j : ℕ) (rate : ℝ≥0) |
| (hrate : (1 : ℝ)/8 ≤ rate) |
| (hq : dyadicLevel j ≤ poissonKillProbability U 2 60) : |
| dyadicLevel j/4 ≤ poissonKillProbability U 3 (rate*dyadicBlock j) := by |
| apply poisson_two_to_three_block U (dyadic_multiplicity_pos j) |
| (show 0 < dyadicTruncation j by unfold dyadicTruncation; omega) hq |
| (dyadic_truncation_error j) (dyadic_replacement_error j) |
| simp only [dyadicBlock, NNReal.coe_mul, NNReal.coe_natCast, Nat.cast_mul, |
| Nat.cast_ofNat, NNReal.coe_ofNat] |
| have h := mul_le_mul_of_nonneg_right hrate |
| (show (0 : ℝ) ≤ 32*(dyadicTruncation j : ℝ)*dyadicMultiplicity j by positivity) |
| nlinarith |
|
|
| def dyadicBin {n : ℕ} [NeZero n] (j : ℕ) (U : Finset (DeletionCandidate n)) : Prop := |
| U.Nonempty ∧ dyadicLevel j ≤ poissonKillProbability U 2 60 ∧ |
| poissonKillProbability U 2 60 ≤ 2*dyadicLevel j |
|
|
| theorem dyadic_bin_occupation {n : ℕ} [NeZero n] |
| (rate : ℝ≥0) (hrate : (1 : ℝ)/8 ≤ rate) (j M : ℕ) |
| (U : Finset (DeletionCandidate n)) : |
| (∑ t ∈ Finset.range M, (candidateBlock (rate*t) 3 U).expect |
| (fun V => if dyadicBin j V then 1 else 0)) ≤ |
| 4*(dyadicBlock j : ℝ)/dyadicLevel j := by |
| classical |
| have h := candidateBlock_occupation rate 3 (dyadicBlock j) M (dyadicBin j) |
| (show 0 < dyadicLevel j/4 by positivity [dyadicLevel_pos j]) |
| (fun V hV => ⟨hV.1, dyadic_block_kills V j rate hrate hV.2.1⟩) U |
| convert h using 1 |
| ring |
|
|
| end FixedClauseThreshold.Computability |
|
|
| end OAI |
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|
