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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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
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