| import OAI.NumberTheory.TwoPoint.Circuits.CircuitPolynomialGrowth |
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| namespace OAI |
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| /-! Convert a dyadic accuracy index to the natural logarithm used in the |
| published fixed-depth bounded-independence input. -/ |
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| namespace TwoPointCorrelations |
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| lemma dyadic_index_of_ratio (m : ℕ) (hm : 1 ≤ m) (ε : ℝ) |
| (hε : 0 < ε) (hεhalf : ε ≤ 1 / 2) : |
| ∃ j : ℕ, m ≤ 2 ^ j ∧ (1 / 2 : ℝ) ^ j ≤ ε ∧ |
| (j : ℝ) + 1 ≤ 6 * Real.log ((m : ℝ) / ε) := by |
| let L := Real.log ((m : ℝ) / ε) |
| have hmreal : (1 : ℝ) ≤ m := by exact_mod_cast hm |
| have hratio : (2 : ℝ) ≤ (m : ℝ) / ε := by |
| apply (le_div_iff₀ hε).mpr |
| linarith |
| have hlogtwo : (1 / 2 : ℝ) ≤ Real.log 2 := by |
| have h := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 2) |
| norm_num at h ⊢ |
| exact h |
| have hL : 1 / 2 ≤ L := hlogtwo.trans (Real.log_le_log (by norm_num) hratio) |
| let j := ⌈2 * L⌉₊ |
| have hjlo : 2 * L ≤ (j : ℝ) := Nat.le_ceil _ |
| have hjhi : (j : ℝ) < 2 * L + 1 := Nat.ceil_lt_add_one (by linarith) |
| have hlogbound : L ≤ (j : ℝ) * Real.log 2 := by |
| nlinarith [show (0 : ℝ) ≤ j from Nat.cast_nonneg j] |
| have hx : (m : ℝ) / ε ≤ (2 : ℝ) ^ j := by |
| apply (Real.log_le_log_iff (by positivity) (by positivity)).mp |
| simpa only [Real.log_pow] using hlogbound |
| have hmul : (m : ℝ) ≤ (2 : ℝ) ^ j * ε := (div_le_iff₀ hε).mp hx |
| have hmle : (m : ℝ) ≤ (2 : ℝ) ^ j := by nlinarith [show (0 : ℝ) < 2 ^ j by positivity] |
| have heps : (1 / 2 : ℝ) ^ j ≤ ε := by |
| have hh : (1 : ℝ) / (2 : ℝ) ^ j ≤ ε := |
| (div_le_iff₀ (by positivity)).mpr (by nlinarith) |
| simpa only [div_pow, one_pow] using hh |
| exact ⟨j, by exact_mod_cast hmle, heps, by linarith⟩ |
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| theorem bravermanDegree_logarithmic_budget : |
| ∃ K C : ℕ, 0 < K ∧ 0 < C ∧ ∀ m : ℕ, 1 ≤ m → |
| ∀ ε : ℝ, 0 < ε → ε ≤ 1 / 2 → ∀ t : ℕ, |
| (K : ℝ) * (Real.log ((m : ℝ) / ε)) ^ C ≤ (t : ℝ) → |
| ∃ j : ℕ, m ≤ 2 ^ j ∧ (1 / 2 : ℝ) ^ j ≤ ε ∧ bravermanDegree j ≤ t := by |
| obtain ⟨K, C, hK, hC, hbound⟩ := bravermanDegree_polynomial_positive |
| refine ⟨K * 6 ^ C, C, by positivity, hC, ?_⟩ |
| intro m hm ε hε hεhalf t ht |
| obtain ⟨j, hmj, hjε, hjlog⟩ := dyadic_index_of_ratio m hm ε hε hεhalf |
| refine ⟨j, hmj, hjε, ?_⟩ |
| have hj : (bravermanDegree j : ℝ) ≤ (K : ℝ) * ((j : ℝ) + 1) ^ C := by |
| exact_mod_cast hbound j |
| have hdeg : (bravermanDegree j : ℝ) ≤ (t : ℝ) := by |
| calc |
| _ ≤ _ := hj |
| _ ≤ (K : ℝ) * (6 * Real.log ((m : ℝ) / ε)) ^ C := by gcongr |
| _ = ((K * 6 ^ C : ℕ) : ℝ) * (Real.log ((m : ℝ) / ε)) ^ C := by |
| push_cast |
| rw [mul_pow] |
| ring |
| _ ≤ _ := ht |
| exact_mod_cast hdeg |
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| end TwoPointCorrelations |
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| end OAI |
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