| import OAI.Algebra.DepthFive.BinomialRatios |
|
|
| namespace OAI |
|
|
| namespace Problem335 |
|
|
| /-- A uniform Bernoulli bound for the rounding correction in the descending |
| homogeneous-space dimension ratio. -/ |
| theorem inverse_one_sub_pow_le_two (N : ℝ) (I : ℕ) |
| (hN : 1 ≤ N) (hI : 2 * (I : ℝ) ≤ N) : |
| ((1 - 1 / N) ^ I)⁻¹ ≤ 2 := by |
| have hNpos : 0 < N := by linarith |
| have hinv : 1 / N ≤ 1 := (div_le_one hNpos).2 hN |
| have hber := one_add_mul_le_pow (a := -(1 / N)) (by linarith : -2 ≤ -(1 / N)) I |
| have hfrac : (I : ℝ) / N ≤ 1 / 2 := (div_le_iff₀ hNpos).2 (by linarith) |
| have hhalf : (1 / 2 : ℝ) ≤ (1 - 1 / N) ^ I := by |
| simp only [mul_neg, mul_one_div, ← sub_eq_add_neg] at hber |
| nlinarith |
| have hpow : 0 < (1 - 1 / N) ^ I := lt_of_lt_of_le (by norm_num) hhalf |
| rw [inv_eq_one_div] |
| apply (div_le_iff₀ hpow).2 |
| linarith |
|
|
| /-- Real-exponent version of the preceding natural-power estimate. -/ |
| theorem one_sub_rpow_neg_nat_le_two (N : ℝ) (I : ℕ) |
| (hN : 1 ≤ N) (hI : 2 * (I : ℝ) ≤ N) : |
| (1 - 1 / N) ^ (-(I : ℝ)) ≤ 2 := by |
| have hbase : 0 ≤ 1 - 1 / N := by |
| have hNpos : 0 < N := by linarith |
| have := (div_le_one hNpos).2 hN |
| linarith |
| rw [Real.rpow_neg hbase, Real.rpow_natCast] |
| exact inverse_one_sub_pow_le_two N I hN hI |
|
|
| /-- Rounding loss and the balanced multiplication cost combine into one |
| absolute-discrepancy exponent. -/ |
| theorem corrected_ratio_product_le (α β N ρ : ℝ) (I J : ℕ) |
| (hα : 0 < α) (hβ : α ^ ρ ≤ β) |
| (hN : 1 ≤ N) (hI : 2 * (I : ℝ) ≤ N) : |
| (α / (1 - 1 / N)) ^ I * β ^ (-(J : ℝ)) ≤ |
| 2 * α ^ ((I : ℝ) - ρ * J) := by |
| have hβpos : 0 < β := lt_of_lt_of_le (Real.rpow_pos_of_pos hα _) hβ |
| have hdown : (α / (1 - 1 / N)) ^ I ≤ 2 * α ^ I := by |
| rw [div_pow, div_eq_mul_inv] |
| have := mul_le_mul_of_nonneg_left (inverse_one_sub_pow_le_two N I hN hI) |
| (pow_nonneg hα.le I) |
| simpa only [mul_comm] using this |
| have hup := Real.rpow_le_rpow_of_nonpos (Real.rpow_pos_of_pos hα ρ) hβ |
| (neg_nonpos.mpr (Nat.cast_nonneg J)) |
| rw [← Real.rpow_mul hα.le] at hup |
| calc |
| _ ≤ (2 * α ^ I) * α ^ (ρ * -(J : ℝ)) := |
| mul_le_mul hdown hup (Real.rpow_nonneg hβpos.le _) (by positivity) |
| _ = 2 * α ^ ((I : ℝ) - ρ * J) := by |
| rw [mul_assoc, ← Real.rpow_natCast, ← Real.rpow_add hα] |
| congr 2 |
| ring |
|
|
| /-- The exact intermediate-dimension bound after the harmless rounding factor |
| has been absorbed. The hypotheses are purely numerical and field-independent. -/ |
| theorem intermediate_dimension_ratio_le_balanced {v u a b I J : ℕ} |
| (hv : 1 ≤ v) (hu : 1 ≤ u) (ha : 0 < a) (hb : 0 < b) (hI : I ≤ a) |
| (hbudget : 2 * I ≤ a + v) (ρ : ℝ) |
| (hbalance : ((a : ℝ) / ((a : ℝ) + v)) ^ ρ ≤ |
| (b : ℝ) / ((b : ℝ) + u)) : |
| ((homogeneousDim v (a - I) : ℝ) * homogeneousDim u (b + J)) / |
| ((homogeneousDim v a : ℝ) * homogeneousDim u b) ≤ |
| 2 * ((a : ℝ) / ((a : ℝ) + v)) ^ ((I : ℝ) - ρ * J) := by |
| have ha' : (0 : ℝ) < a := by exact_mod_cast ha |
| have hb' : (0 : ℝ) < b := by exact_mod_cast hb |
| have hv' : (1 : ℝ) ≤ v := by exact_mod_cast hv |
| have hu' : (1 : ℝ) ≤ u := by exact_mod_cast hu |
| have hN : (1 : ℝ) < (a : ℝ) + v := by linarith |
| have hdown : (a : ℝ) / ((a : ℝ) + v - 1) = |
| ((a : ℝ) / ((a : ℝ) + v)) / (1 - 1 / ((a : ℝ) + v)) := by |
| field_simp |
| have hup : (((b : ℝ) + u) / b) ^ J = |
| ((b : ℝ) / ((b : ℝ) + u)) ^ (-(J : ℝ)) := by |
| rw [Real.rpow_neg (by positivity), Real.rpow_natCast, ← inv_pow, inv_div] |
| calc |
| _ ≤ ((a : ℝ) / ((a : ℝ) + v - 1)) ^ I * |
| (((b : ℝ) + u) / b) ^ J := intermediate_dimension_ratio_le hv hu hb hI |
| _ = ((((a : ℝ) / ((a : ℝ) + v)) / (1 - 1 / ((a : ℝ) + v))) ^ I) * |
| ((b : ℝ) / ((b : ℝ) + u)) ^ (-(J : ℝ)) := by rw [hdown, hup] |
| _ ≤ _ := corrected_ratio_product_le _ _ _ ρ I J (by positivity) hbalance hN.le |
| (by exact_mod_cast hbudget) |
|
|
| /-- Unnormalized form of the balanced intermediate-dimension estimate. -/ |
| theorem intermediate_dimension_le_balanced {v u a b I J : ℕ} |
| (hv : 1 ≤ v) (hu : 1 ≤ u) (ha : 0 < a) (hb : 0 < b) (hI : I ≤ a) |
| (hbudget : 2 * I ≤ a + v) (ρ : ℝ) |
| (hbalance : ((a : ℝ) / ((a : ℝ) + v)) ^ ρ ≤ |
| (b : ℝ) / ((b : ℝ) + u)) : |
| (homogeneousDim v (a - I) : ℝ) * homogeneousDim u (b + J) ≤ |
| 2 * ((homogeneousDim v a : ℝ) * homogeneousDim u b) * |
| ((a : ℝ) / ((a : ℝ) + v)) ^ ((I : ℝ) - ρ * J) := by |
| have hd : (0 : ℝ) < (homogeneousDim v a : ℝ) * homogeneousDim u b := by |
| exact mul_pos (by exact_mod_cast homogeneousDim_pos hv a) |
| (by exact_mod_cast homogeneousDim_pos hu b) |
| have h := (div_le_iff₀ hd).mp |
| (intermediate_dimension_ratio_le_balanced (J := J) hv hu ha hb hI hbudget ρ hbalance) |
| simpa only [mul_assoc, mul_left_comm, mul_comm] using h |
|
|
| end Problem335 |
|
|
| end OAI |
|
|
