openAI-math-dataset / lean /OAI /NumberTheory /Ostmann /Arithmetic /ArithmeticTwoPrimePatternNormRate.lean
| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticPatternErrorRate | |
| import OAI.NumberTheory.Ostmann.Arithmetic.PrimePatternFrequencyNormSum | |
| namespace OAI | |
| /-! # The full numerical norm bound with the original equality patterns -/ | |
| namespace Ostmann | |
| open Filter | |
| open scoped Classical BigOperators | |
| /-- The actual pattern sum and paired frequency sum have an absolute bulk | |
| constant. The cutoff includes the constructed bottom reserve `2 sqrt(m)`. | |
| Only the genuine comparison errors use the full number of histories. -/ | |
| theorem arithmetic_two_prime_pattern_norm_rate (ψ : SchwartzMap ℝ ℂ) (n k : ℕ) | |
| (hk : 0 < k) (hn : n ≤ k) (B Cfreq Cprior ε : ℝ) | |
| (hCfreq : 0 ≤ Cfreq) (hCprior : 1 ≤ Cprior) (hε : 0 < ε) | |
| (hdepth : 8 * (Cprior + 1) ≤ (k : ℝ) ^ 3) : | |
| ∀ᶠ L : ℝ in atTop, let m := spectatorBulkCount k L | |
| ∃ D : ℝ, 0 ≤ D ∧ | |
| (∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * Cfreq * m) → | |
| (q.divisors.card : ℝ) ≤ D) ∧ | |
| ∀ (S : Finset ℤ) (N V : ℕ) (Δ err : ℝ), | |
| (S.card : ℝ) ≤ Real.exp (Cfreq * m) → | |
| (∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N) → | |
| (N : ℝ) ≤ Real.exp (Cfreq * m) → | |
| (V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) → | |
| 0 ≤ err → err ≤ Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) + | |
| 3 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L)) → | |
| let _ := sampleSetoidFintype (Bool × MovingSampleIndex n) | |
| let K := ((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) * | |
| B ^ (2 ^ n - 1)) ^ 2 * | |
| ((2 : ℝ) ^ (2 ^ n * m) * 4 * 3 ^ (2 ^ n * m)) * 2 ^ (2 ^ n * m) | |
| ∀ R : Setoid (Bool × MovingSampleIndex n) → FrequencyTree (S × S) n → ℂ, | |
| (∀ s t, ‖R s t‖ ≤ | |
| ((4 : ℝ) ^ Fintype.card (Quotient s) * | |
| (Real.exp (Cprior * L)) ^ (4 * n * 2 ^ n - Fintype.card (Quotient s))) * | |
| (err + K * (frequencyLeafWeight (pairedFrequencyLeaf S V) n t * | |
| ((frequencySplitList S n t).map (pairFrequencySupportBound D)).prod))) → | |
| ‖∑ s, ∑ t, R s t‖ ≤ | |
| Real.exp ((2 ^ n : ℕ) * Δ + (Real.log 12 + 1) * (2 ^ n : ℕ) * m + ε * m) + | |
| 4 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by | |
| have hCplus : 1 ≤ Cprior + 1 := by linarith | |
| filter_upwards [arithmetic_prime_pattern_budget_rate ψ n k hk hn B Cfreq (Cprior + 1) ε | |
| hCfreq hCplus hε hdepth, | |
| movingPattern_four_errors_rate n k hk (Cprior + 1) Cfreq hCplus hCfreq, | |
| eventually_ge_atTop (2 : ℝ)] with L hbudget herrors hL | |
| dsimp only | |
| obtain ⟨D, hD, hdiv, hbudget⟩ := hbudget | |
| refine ⟨D, hD, hdiv, ?_⟩ | |
| intro S N V Δ err hcard hS hN hV herr herrbound R hR | |
| have hdivN : ∀ q : ℕ, q ≠ 0 → q ≤ N ^ 2 → (q.divisors.card : ℝ) ≤ D := by | |
| intro q hq hqN | |
| apply hdiv q hq | |
| calc | |
| (q : ℝ) ≤ (N : ℝ) ^ 2 := by exact_mod_cast hqN | |
| _ ≤ (Real.exp (Cfreq * spectatorBulkCount k L)) ^ 2 := | |
| pow_le_pow_left₀ (Nat.cast_nonneg N) hN 2 | |
| _ = Real.exp (2 * Cfreq * spectatorBulkCount k L) := by | |
| rw [← Real.exp_nat_mul] | |
| congr 1 | |
| ring | |
| have hexp : (4 : ℝ) ≤ Real.exp ((Cprior + 1) * L) := by | |
| have hCL := mul_le_mul hCplus hL (by norm_num : (0 : ℝ) ≤ 2) (by linarith) | |
| have hCL' := mul_le_mul_of_nonneg_right hCprior (by linarith : 0 ≤ L) | |
| linarith [Real.add_one_le_exp ((Cprior + 1) * L)] | |
| have hmono : max 4 (Real.exp (Cprior * L)) ≤ max 2 (Real.exp ((Cprior + 1) * L)) := by | |
| apply max_le | |
| · exact hexp.trans (le_max_right _ _) | |
| · exact (Real.exp_le_exp.mpr (by linarith)).trans (le_max_right _ _) | |
| have hpriorcost : | |
| (2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * (max 4 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n) ≤ | |
| (2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * (max 2 (Real.exp ((Cprior + 1) * L))) ^ (4 * n * 2 ^ n) := | |
| mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) hmono _) (by positivity) | |
| have hsum := prime_pattern_frequency_norm_sum_le S n N V D (Real.exp (Cprior * L)) | |
| (((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) * | |
| B ^ (2 ^ n - 1)) ^ 2 * | |
| ((2 : ℝ) ^ (2 ^ n * spectatorBulkCount k L) * 4 * 3 ^ (2 ^ n * spectatorBulkCount k L)) * | |
| 2 ^ (2 ^ n * spectatorBulkCount k L)) err hD (Real.exp_nonneg _) (by positivity) | |
| herr hS hdivN R hR | |
| have hcost := (mul_le_mul_of_nonneg_right hpriorcost (by positivity)).trans (hbudget N V Δ hN hV) | |
| have he := (mul_le_mul_of_nonneg_left herrbound | |
| (show 0 ≤ ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * | |
| (max 4 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) * | |
| (Fintype.card (FrequencyTree (S × S) n) : ℝ) by positivity)).trans | |
| ((mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hpriorcost | |
| (Nat.cast_nonneg _)) (by positivity)).trans (herrors S hcard)) | |
| apply hsum.trans | |
| nlinarith only [hcost, he] | |
| /-- The actual pattern sum and paired frequency sum have an absolute bulk | |
| constant. The cutoff includes the constructed bottom reserve `2 sqrt(m)`. | |
| The fifth error is the literal giant comparison; only comparison errors use the full number of histories. -/ | |
| theorem arithmetic_two_prime_original_pattern_norm_rate (ψ : SchwartzMap ℝ ℂ) (n k : ℕ) | |
| (hk : 0 < k) (hn : n ≤ k) (B Cfreq Cprior ε : ℝ) | |
| (hCfreq : 0 ≤ Cfreq) (hCprior : 1 ≤ Cprior) (hε : 0 < ε) | |
| (hdepth : 8 * (Cprior + 1) ≤ (k : ℝ) ^ 3) : | |
| ∀ᶠ L : ℝ in atTop, let m := spectatorBulkCount k L | |
| ∃ D : ℝ, 0 ≤ D ∧ | |
| (∀ q : ℕ, q ≠ 0 → (q : ℝ) ≤ Real.exp (2 * Cfreq * m) → | |
| (q.divisors.card : ℝ) ≤ D) ∧ | |
| ∀ (S : Finset ℤ) (N V : ℕ) (Δ err : ℝ), | |
| (S.card : ℝ) ≤ Real.exp (Cfreq * m) → | |
| (∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N) → | |
| (N : ℝ) ≤ Real.exp (Cfreq * m) → | |
| (V : ℝ) ≤ Real.exp (Δ + Real.sqrt (4 * m)) → | |
| 0 ≤ err → err ≤ Real.exp (-Real.exp ((125 / 100000 : ℝ) * L)) + | |
| 4 * Real.exp (-Real.exp ((2 / 1000 : ℝ) * L)) → | |
| let _ := sampleSetoidFintype (Bool × MovingSampleIndex n) | |
| let K := ((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) * | |
| B ^ (2 ^ n - 1)) ^ 2 * | |
| ((2 : ℝ) ^ (2 ^ n * m) * 4 * 3 ^ (2 ^ n * m)) * 2 ^ (2 ^ n * m) | |
| ∀ R : Setoid (Bool × MovingSampleIndex n) → FrequencyTree (S × S) n → ℂ, | |
| (∀ s t, ‖R s t‖ ≤ | |
| ((4 : ℝ) ^ Fintype.card (Quotient s) * | |
| (Real.exp (Cprior * L)) ^ (4 * n * 2 ^ n - Fintype.card (Quotient s))) * | |
| (err + K * (frequencyLeafWeight (pairedFrequencyLeaf S V) n t * | |
| ((frequencySplitList S n t).map (pairFrequencySupportBound D)).prod))) → | |
| ‖∑ s, ∑ t, R s t‖ ≤ | |
| Real.exp ((2 ^ n : ℕ) * Δ + (Real.log 12 + 1) * (2 ^ n : ℕ) * m + ε * m) + | |
| 5 * Real.exp (-Real.exp ((12 / 10000 : ℝ) * L)) := by | |
| have hCplus : 1 ≤ Cprior + 1 := by linarith | |
| filter_upwards [arithmetic_prime_pattern_budget_rate ψ n k hk hn B Cfreq (Cprior + 1) ε | |
| hCfreq hCplus hε hdepth, | |
| movingPattern_five_errors_rate n k hk (Cprior + 1) Cfreq hCplus hCfreq, | |
| eventually_ge_atTop (2 : ℝ)] with L hbudget herrors hL | |
| dsimp only | |
| obtain ⟨D, hD, hdiv, hbudget⟩ := hbudget | |
| refine ⟨D, hD, hdiv, ?_⟩ | |
| intro S N V Δ err hcard hS hN hV herr herrbound R hR | |
| have hdivN : ∀ q : ℕ, q ≠ 0 → q ≤ N ^ 2 → (q.divisors.card : ℝ) ≤ D := by | |
| intro q hq hqN | |
| apply hdiv q hq | |
| calc | |
| (q : ℝ) ≤ (N : ℝ) ^ 2 := by exact_mod_cast hqN | |
| _ ≤ (Real.exp (Cfreq * spectatorBulkCount k L)) ^ 2 := | |
| pow_le_pow_left₀ (Nat.cast_nonneg N) hN 2 | |
| _ = Real.exp (2 * Cfreq * spectatorBulkCount k L) := by | |
| rw [← Real.exp_nat_mul] | |
| congr 1 | |
| ring | |
| have hexp : (4 : ℝ) ≤ Real.exp ((Cprior + 1) * L) := by | |
| have hCL := mul_le_mul hCplus hL (by norm_num : (0 : ℝ) ≤ 2) (by linarith) | |
| have hCL' := mul_le_mul_of_nonneg_right hCprior (by linarith : 0 ≤ L) | |
| linarith [Real.add_one_le_exp ((Cprior + 1) * L)] | |
| have hmono : max 4 (Real.exp (Cprior * L)) ≤ max 2 (Real.exp ((Cprior + 1) * L)) := by | |
| apply max_le | |
| · exact hexp.trans (le_max_right _ _) | |
| · exact (Real.exp_le_exp.mpr (by linarith)).trans (le_max_right _ _) | |
| have hpriorcost : | |
| (2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * (max 4 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n) ≤ | |
| (2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * (max 2 (Real.exp ((Cprior + 1) * L))) ^ (4 * n * 2 ^ n) := | |
| mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) hmono _) (by positivity) | |
| have hsum := prime_pattern_frequency_norm_sum_le S n N V D (Real.exp (Cprior * L)) | |
| (((SchwartzMap.seminorm ℝ 0 0 ψ / Real.sqrt (Real.exp Δ)) ^ (2 ^ n) * | |
| B ^ (2 ^ n - 1)) ^ 2 * | |
| ((2 : ℝ) ^ (2 ^ n * spectatorBulkCount k L) * 4 * 3 ^ (2 ^ n * spectatorBulkCount k L)) * | |
| 2 ^ (2 ^ n * spectatorBulkCount k L)) err hD (Real.exp_nonneg _) (by positivity) | |
| herr hS hdivN R hR | |
| have hcost := (mul_le_mul_of_nonneg_right hpriorcost (by positivity)).trans (hbudget N V Δ hN hV) | |
| have he := (mul_le_mul_of_nonneg_left herrbound | |
| (show 0 ≤ ((2 : ℝ) ^ ((4 * n * 2 ^ n) ^ 2) * | |
| (max 4 (Real.exp (Cprior * L))) ^ (4 * n * 2 ^ n)) * | |
| (Fintype.card (FrequencyTree (S × S) n) : ℝ) by positivity)).trans | |
| ((mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hpriorcost | |
| (Nat.cast_nonneg _)) (by positivity)).trans (herrors S hcard)) | |
| apply hsum.trans | |
| nlinarith only [hcost, he] | |
| end Ostmann | |
| end OAI | |