| import OAI.Probability.SATComputability.MaskDelay |
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| namespace OAI |
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| namespace FixedClauseThreshold.Computability |
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| open DilutedSpinGlass Filter |
| open scoped BigOperators NNReal Topology |
|
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| theorem poissonKillProbability_le_one {n : ℕ} [NeZero n] |
| (U : Finset (DeletionCandidate n)) (k : ℕ) (r : ℝ≥0) : |
| poissonKillProbability U k r ≤ 1 := by |
| have h := FiniteLaw.expect_nonneg (candidateBlock r k U) candidateAlive_nonneg |
| rw [candidateBlock_expect_alive] at h |
| linarith |
|
|
| theorem positive_killing_in_dyadic_bin {n : ℕ} [NeZero n] |
| (U : Finset (DeletionCandidate n)) (hU : U.Nonempty) |
| (hq : 0 < poissonKillProbability U 2 60) : ∃ j, dyadicBin j U := by |
| obtain ⟨j, hj, hj'⟩ := exists_nat_pow_near_of_lt_one hq |
| (poissonKillProbability_le_one U 2 60) |
| (by norm_num : (0 : ℝ) < 2⁻¹) (by norm_num : (2 : ℝ)⁻¹ < 1) |
| refine ⟨j+1, hU, ?_, ?_⟩ |
| · simpa only [dyadicLevel, inv_pow] using hj.le |
| · have he : 2*dyadicLevel (j+1) = ((2 : ℝ)⁻¹)^j := by |
| simp only [dyadicLevel, inv_pow, pow_succ] |
| norm_num |
| ring |
| rwa [he] |
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|
| noncomputable def dyadicLifetimeBound (j : ℕ) : ℝ := |
| (4*(dyadicBlock j : ℝ)/dyadicLevel j)^(1/5 : ℝ) |
|
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| theorem dyadicLifetimeBound_nonneg (j : ℕ) : 0 ≤ dyadicLifetimeBound j := by |
| unfold dyadicLifetimeBound |
| positivity [dyadicLevel_pos j] |
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| theorem dyadic_lifetime_bound {n : ℕ} [NeZero n] (r : ℝ≥0) |
| (hr : (1 : ℝ)/8 ≤ r) (m j : ℕ) (U : Finset (DeletionCandidate n)) |
| (hU : dyadicBin j U) : |
| (FiniteLaw.pi (fun _ : Fin m => candidateBlock (n := n) r 3 Finset.univ)).expect |
| (fun xs => (maskLifetime m U xs)^(1/5 : ℝ)) ≤ dyadicLifetimeBound j := by |
| have h := maskLifetime_fractional_moment r 3 m (dyadicBlock j) U |
| (show 0 < dyadicLevel j/4 by positivity [dyadicLevel_pos j]) |
| (dyadic_block_kills U j r hr hU.2.1) |
| convert h using 1 |
| unfold dyadicLifetimeBound |
| congr 1 |
| field_simp |
|
|
| theorem capped_lifetime_fractional_bound {n : ℕ} [NeZero n] |
| (r : ℝ≥0) (k m M : ℕ) (hm : m ≤ M) (U : Finset (DeletionCandidate n)) : |
| (FiniteLaw.pi (fun _ : Fin m => candidateBlock (n := n) r k Finset.univ)).expect |
| (fun xs => (maskLifetime m U xs)^(1/5 : ℝ)) ≤ (M : ℝ)^(1/5 : ℝ) := by |
| apply (FiniteLaw.expect_mono _ (g := fun _ => (M : ℝ)^(1/5 : ℝ)) (fun xs => Real.rpow_le_rpow |
| (maskLifetime_nonneg U xs) ((maskLifetime_le U xs).trans (by exact_mod_cast hm)) |
| (by norm_num : (0 : ℝ) ≤ 1/5))).trans_eq |
| exact FiniteLaw.expect_const _ _ |
|
|
| theorem dyadicLevel_rpow (j : ℕ) : dyadicLevel j = (2 : ℝ)^(-(j : ℝ)) := by |
| rw [dyadicLevel, Real.rpow_neg (by norm_num : (0 : ℝ) ≤ 2), Real.rpow_natCast] |
|
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| theorem dyadicBlock_growth (j : ℕ) : |
| (dyadicBlock j : ℝ) ≤ 128*((j : ℝ)+122)^2 * (2 : ℝ)^((j : ℝ)/2) := by |
| have he : (((j+1)/2 : ℕ) : ℝ) ≤ (j : ℝ)/2+1 := by |
| have h : 2*((j+1)/2) ≤ j+2 := by omega |
| have h' : (2 : ℝ)*((j+1)/2 : ℕ) ≤ (j : ℝ)+2 := by exact_mod_cast h |
| linarith |
| have hp : (2 : ℝ)^((j+1)/2) ≤ 2*(2 : ℝ)^((j : ℝ)/2) := by |
| rw [← Real.rpow_natCast] |
| apply (Real.rpow_le_rpow_of_exponent_le (by norm_num) he).trans_eq |
| rw [Real.rpow_add (by norm_num : (0 : ℝ) < 2), Real.rpow_one] |
| ring |
| unfold dyadicBlock dyadicMultiplicity dyadicTruncation |
| push_cast |
| nlinarith [mul_le_mul_of_nonneg_left hp (show 0 ≤ 64*((j : ℝ)+122)^2 by positivity)] |
|
|
| noncomputable def dyadicMomentCoefficient (j : ℕ) : ℝ := |
| (dyadicLevel j)^2 * (4*(dyadicBlock j : ℝ)/dyadicLevel j)^(6/5 : ℝ) |
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| theorem dyadicMomentCoefficient_nonneg (j : ℕ) : 0 ≤ dyadicMomentCoefficient j := by |
| unfold dyadicMomentCoefficient |
| positivity [dyadicLevel_pos j] |
|
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| theorem dyadicMomentCoefficient_bound (j : ℕ) : |
| dyadicMomentCoefficient j ≤ |
| (512 : ℝ)^(6/5 : ℝ) * ((j : ℝ)+122)^3 * ((2 : ℝ)^(-1/5 : ℝ))^j := by |
| have hx : 0 < (j : ℝ)+122 := by positivity |
| have hpow : (2 : ℝ)^((j : ℝ)/2) * (2 : ℝ)^(j : ℝ) = |
| (2 : ℝ)^(3*(j : ℝ)/2) := by |
| rw [← Real.rpow_add (by norm_num : (0 : ℝ) < 2)] |
| congr 1 |
| ring |
| have hA : 4*(dyadicBlock j : ℝ)/dyadicLevel j ≤ |
| 512*((j : ℝ)+122)^2 * (2 : ℝ)^(3*(j : ℝ)/2) := by |
| calc |
| _ = 4*(dyadicBlock j : ℝ)*(2 : ℝ)^(j : ℝ) := by |
| simp only [dyadicLevel, div_eq_mul_inv, inv_inv, Real.rpow_natCast] |
| _ ≤ 4*(128*((j : ℝ)+122)^2*(2 : ℝ)^((j : ℝ)/2)) * (2 : ℝ)^(j : ℝ) := by |
| gcongr |
| exact dyadicBlock_growth j |
| _ = _ := by rw [show 4*(128*((j : ℝ)+122)^2*(2 : ℝ)^((j : ℝ)/2)) * |
| (2 : ℝ)^(j : ℝ) = 512*((j : ℝ)+122)^2 * |
| ((2 : ℝ)^((j : ℝ)/2)*(2 : ℝ)^(j : ℝ)) by ring, hpow] |
| have hq : (dyadicLevel j)^2 = (2 : ℝ)^(-2*(j : ℝ)) := by |
| rw [dyadicLevel_rpow, pow_two, ← Real.rpow_add (by norm_num : (0 : ℝ) < 2)] |
| congr 1 |
| ring |
| have hxp : (((j : ℝ)+122)^2)^(6/5 : ℝ) = ((j : ℝ)+122)^(12/5 : ℝ) := by |
| rw [← Real.rpow_natCast, ← Real.rpow_mul hx.le] |
| norm_num |
| have htwo : ((2 : ℝ)^(3*(j : ℝ)/2))^(6/5 : ℝ) = (2 : ℝ)^(9*(j : ℝ)/5) := by |
| rw [← Real.rpow_mul (by norm_num : (0 : ℝ) ≤ 2)] |
| congr 1 |
| ring |
| have he : (2 : ℝ)^(-2*(j : ℝ)) * (2 : ℝ)^(9*(j : ℝ)/5) = |
| ((2 : ℝ)^(-1/5 : ℝ))^j := by |
| rw [← Real.rpow_add (by norm_num : (0 : ℝ) < 2), |
| ← Real.rpow_mul_natCast (by norm_num : (0 : ℝ) ≤ 2)] |
| congr 1 |
| ring |
| calc |
| _ ≤ (dyadicLevel j)^2 * |
| (512*((j : ℝ)+122)^2*(2 : ℝ)^(3*(j : ℝ)/2))^(6/5 : ℝ) := by |
| exact mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow (by positivity [dyadicLevel_pos j]) hA (by norm_num)) (sq_nonneg _) |
| _ = (512 : ℝ)^(6/5 : ℝ) * ((j : ℝ)+122)^(12/5 : ℝ) * |
| ((2 : ℝ)^(-1/5 : ℝ))^j := by |
| rw [Real.mul_rpow (by positivity) (by positivity), |
| Real.mul_rpow (by norm_num : (0 : ℝ) ≤ 512) (sq_nonneg _), hq, hxp, htwo] |
| calc |
| _ = (512 : ℝ)^(6/5 : ℝ) * ((j : ℝ)+122)^(12/5 : ℝ) * |
| ((2 : ℝ)^(-2*(j : ℝ))*(2 : ℝ)^(9*(j : ℝ)/5)) := by ring |
| _ = _ := by rw [he] |
| _ ≤ _ := by |
| gcongr |
| rw [← Real.rpow_natCast] |
| exact Real.rpow_le_rpow_of_exponent_le (by linarith [Nat.cast_nonneg (α := ℝ) j]) (by norm_num) |
|
|
| theorem dyadicMomentCoefficient_summable : Summable dyadicMomentCoefficient := by |
| let ρ : ℝ := 2^(-1/5 : ℝ) |
| have hρ : ‖ρ‖ < 1 := by |
| rw [Real.norm_eq_abs, abs_of_pos (Real.rpow_pos_of_pos (by norm_num) _)] |
| exact Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (by norm_num) |
| have hpoly : Summable (fun j : ℕ => ((j : ℝ)+122)^3*ρ^j) := by |
| have h0 := (summable_geometric_of_norm_lt_one hρ).mul_left (122^3 : ℝ) |
| have h1 := (summable_pow_mul_geometric_of_norm_lt_one 1 hρ).mul_left (3*122^2 : ℝ) |
| have h2 := (summable_pow_mul_geometric_of_norm_lt_one 2 hρ).mul_left (3*122 : ℝ) |
| have h3 := summable_pow_mul_geometric_of_norm_lt_one 3 hρ |
| apply (h3.add (h2.add (h1.add h0))).congr |
| intro j |
| ring |
| apply Summable.of_nonneg_of_le dyadicMomentCoefficient_nonneg dyadicMomentCoefficient_bound |
| simpa only [mul_assoc, ρ] using hpoly.mul_left ((512 : ℝ)^(6/5 : ℝ)) |
|
|
| end FixedClauseThreshold.Computability |
|
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| end OAI |
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