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f18e7de | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexCertificatesLog
import OAI.LinearAlgebra.MatrixMultiplication.Duality.WitnessArithmetic
namespace OAI
/-! Exact rational intervals, logarithm bounds and arithmetic circuit soundness. -/
namespace MatrixMultiplication.ComplexCertificates
open DualWitness
theorem rho24_bounds : (1597 : ℝ) / 5000 < rho24 ∧ rho24 < (639 : ℝ) / 2000 := by
norm_num [rho24, rho8]
theorem rho48_bounds : (67 : ℝ) / 125 < rho48 ∧ rho48 < (537 : ℝ) / 1000 := by
norm_num [rho48, rho24, rho8]
theorem inactive48_lower : (463 : ℝ) / 1000 < 1 - rho48 := by
norm_num [rho48, rho24, rho8]
theorem inactive288_lower : (9 : ℝ) / 25 < 1 - rho288 := by
norm_num [rho288, rho96, rho48, rho24, rho8]
theorem delta24_bounds :
(87 : ℝ) / 250 < delta24 ∧ delta24 < (177 : ℝ) / 500 := by
obtain ⟨hl, hh⟩ := log_nine_sevenths_bounds
norm_num [delta24, rho24, rho8]
constructor <;> linarith
theorem z48_bounds : (127 : ℝ) / 200 < z48 ∧ z48 < (651 : ℝ) / 1000 := by
obtain ⟨hl, hh⟩ := delta24_bounds
have helo : (701 : ℝ) / 1000 < Real.exp (-delta24) :=
exp_neg_354_lower.trans (Real.exp_lt_exp.mpr (by linarith))
have hehi : Real.exp (-delta24) < (707 : ℝ) / 1000 :=
(Real.exp_lt_exp.mpr (by linarith : -delta24 < -(87 / 250 : ℝ))).trans
exp_neg_348_upper
have harglo : (1 + (701 / 1000 : ℝ)) ^ 2 - 1 <
(1 + Real.exp (-delta24)) ^ 2 - 1 := by gcongr
have harghi : (1 + Real.exp (-delta24)) ^ 2 - 1 <
(1 + (707 / 1000 : ℝ)) ^ 2 - 1 := by gcongr
have hargpos : 0 < (1 + Real.exp (-delta24)) ^ 2 - 1 := by
norm_num at harglo
linarith
exact ⟨log_z48_lower.trans (Real.log_lt_log (by norm_num) harglo),
(Real.log_lt_log hargpos harghi).trans log_z48_upper⟩
theorem delta96_bounds : 0 < delta96 ∧ delta96 < (43 : ℝ) / 50 := by
obtain ⟨hr, hr'⟩ := rho48_bounds
obtain ⟨hz, hz'⟩ := z48_bounds
have hd : 0 < 1 + rho48 := by linarith
constructor
· exact div_pos (by linarith) hd
· apply (div_lt_iff₀ hd).2
linarith
theorem z288_bounds : (157 : ℝ) / 250 < z288 ∧ z288 < 2 := by
obtain ⟨hd, hd'⟩ := delta96_bounds
have helo : (169 : ℝ) / 400 < Real.exp (-delta96) :=
exp_neg_86_lower.trans (Real.exp_lt_exp.mpr (by linarith))
have hehi : Real.exp (-delta96) < 1 := by
simpa using Real.exp_lt_exp.mpr (by linarith : -delta96 < 0)
have harglo : (1 + (169 / 400 : ℝ)) ^ 3 - 1 <
(1 + Real.exp (-delta96)) ^ 3 - 1 := by gcongr
have harghi : (1 + Real.exp (-delta96)) ^ 3 - 1 < 7 := by
have h : (1 + Real.exp (-delta96)) ^ 3 - 1 < (1 + (1 : ℝ)) ^ 3 - 1 := by
gcongr
norm_num at h
exact h
have hargpos : 0 < (1 + Real.exp (-delta96)) ^ 3 - 1 := by
norm_num at harglo
linarith
have hlog7 : Real.log 7 < 2 := by
apply (Real.log_lt_iff_lt_exp (by norm_num)).2
refine lt_of_lt_of_le ?_ (Real.sum_le_exp_of_nonneg (by norm_num : (0 : ℝ) ≤ 2) 6)
norm_num [Finset.sum_range_succ, Nat.factorial]
exact ⟨log_z288_lower.trans (Real.log_lt_log (by norm_num) harglo),
(Real.log_lt_log hargpos harghi).trans hlog7⟩
theorem hiddenRate_lower :
(9 / 64 : ℝ) * (549 / 500) + (463 / 1000 : ℝ) / 48 * (127 / 200) +
(9 / 25 : ℝ) / 288 * (157 / 250) < hiddenRate := by
have h3 := log_three_lower
have h48 := z48_bounds.1
have h288 := z288_bounds.1
have hr48 := inactive48_lower
have hr288 := inactive288_lower
dsimp [hiddenRate]
gcongr
theorem dual_aspect_lower : (1548637 : ℝ) / 3327360 < aspect := by
have h2 := log_two_bounds
have hD := hiddenRate_lower
have hl2 : 0 < Real.log 2 := by linarith [h2.1]
apply (lt_div_iff₀ hl2).2
norm_num at hD
linarith [h2.2]
theorem dual_aspect_lt_one : aspect < 1 := by
have h3 := log_three_upper
have h48 := z48_bounds
have h288 := z288_bounds
have hr48 := DualWitness.rho48_range
have hr288 := DualWitness.rho288_range
have h2 := log_two_bounds
have hD : hiddenRate < (9 / 64 : ℝ) * (10987 / 10000) +
(1 : ℝ) / 48 * (651 / 1000) + (1 : ℝ) / 288 * 2 := by
dsimp [hiddenRate]
gcongr <;> linarith
have hl2 : 0 < Real.log 2 := by linarith [h2.1]
apply (div_lt_iff₀ hl2).2
norm_num at hD
linarith [h2.1]
theorem dual_aspect_milestone : (465 : ℝ) / 1000 < aspect := by
have h := dual_aspect_lower
norm_num at h ⊢
linarith
end MatrixMultiplication.ComplexCertificates
end OAI
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