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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 | import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexRationalLogCertificate
import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexLogThreeCertificate
namespace OAI
/-! Exact rational intervals, logarithm bounds and arithmetic circuit soundness. -/
namespace MatrixMultiplication.ComplexCertificates
open MatrixMultiplication.Foundation.RationalLogCertificate
theorem log_bounds_of_reduction {x : ℚ} (d : ReducedArgument x) (lo hi : ℚ)
(hl : lo < reducedApprox d - (1 + |(d.exponent : ℚ)|) / 2 ^ 120)
(hu : reducedApprox d + (1 + |(d.exponent : ℚ)|) / 2 ^ 120 < hi) :
(lo : ℝ) < Real.log (x : ℝ) ∧ Real.log (x : ℝ) < (hi : ℝ) := by
have herror := abs_le.mp (reduced_log_error d)
have hl' : (lo : ℝ) < (reducedApprox d : ℝ) -
(1 + |(d.exponent : ℝ)|) / 2 ^ 120 := by exact_mod_cast hl
have hu' : (reducedApprox d : ℝ) +
(1 + |(d.exponent : ℝ)|) / 2 ^ 120 < (hi : ℝ) := by exact_mod_cast hu
constructor <;> linarith
def reduceTwo : ReducedArgument (2 : ℚ) where
exponent := 0
mantissa := 2
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_two_bounds :
(34657 : ℝ) / 50000 < Real.log 2 ∧ Real.log 2 < (17329 : ℝ) / 25000 := by
have h := log_bounds_of_reduction reduceTwo (34657 / 50000) (17329 / 25000)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h
def reduceThree : ReducedArgument (3 : ℚ) where
exponent := 1
mantissa := 3 / 2
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_three_lower : (549 : ℝ) / 500 < Real.log 3 := by
have h := log_bounds_of_reduction reduceThree (549 / 500) (10987 / 10000)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h.1
theorem log_three_upper : Real.log 3 < (10987 : ℝ) / 10000 := by
have h := MatrixMultiplication.Foundation.log_three_upper
norm_num at h ⊢
linarith
def reduceNineteen : ReducedArgument (19 : ℚ) where
exponent := 4
mantissa := 19 / 16
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_nineteen_bounds :
(7361 : ℝ) / 2500 < Real.log 19 ∧ Real.log 19 < (5889 : ℝ) / 2000 := by
have h := log_bounds_of_reduction reduceNineteen (7361 / 2500) (5889 / 2000)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h
def reduceCurveLo : ReducedArgument (246439 / 5000 : ℚ) where
exponent := 5
mantissa := 246439 / 160000
lower := by norm_num
upper := by norm_num
value := by norm_num
def reduceCurveHi : ReducedArgument (49289 / 1000 : ℚ) where
exponent := 5
mantissa := 49289 / 32000
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_curve_lower :
(77953 : ℝ) / 20000 < Real.log ((246439 : ℝ) / 5000) := by
have h := log_bounds_of_reduction reduceCurveLo (77953 / 20000) (97443 / 25000)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h.1
theorem log_curve_upper :
Real.log ((49289 : ℝ) / 1000) < (97443 : ℝ) / 25000 := by
have h := log_bounds_of_reduction reduceCurveHi (77953 / 20000) (97443 / 25000)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h.2
def reduceTen : ReducedArgument (10 : ℚ) where
exponent := 3
mantissa := 5 / 4
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_ten_lower : (921 : ℝ) / 400 < Real.log 10 := by
have h := log_bounds_of_reduction reduceTen (921 / 400) (2303 / 1000)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h.1
def reduceSquareMantissa : ReducedArgument (59 / 20 : ℚ) where
exponent := 1
mantissa := 59 / 40
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_square_mantissa_lower :
(10817 : ℝ) / 10000 < Real.log ((59 : ℝ) / 20) := by
have h := log_bounds_of_reduction reduceSquareMantissa (10817 / 10000) (541 / 500)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h.1
def reduceNineSevenths : ReducedArgument (9 / 7 : ℚ) where
exponent := 0
mantissa := 9 / 7
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_nine_sevenths_bounds :
(1 : ℝ) / 4 < Real.log (9 / 7 : ℝ) ∧ Real.log (9 / 7 : ℝ) < (63 : ℝ) / 250 := by
have h := log_bounds_of_reduction reduceNineSevenths (1 / 4) (63 / 250)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h
def reduceExpLo48 : ReducedArgument (701 / 1000 : ℚ) where
exponent := -1
mantissa := 701 / 500
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem exp_neg_354_lower : (701 : ℝ) / 1000 < Real.exp (-(177 / 500 : ℝ)) := by
have h := log_bounds_of_reduction reduceExpLo48 (-356 / 1000) (-177 / 500)
(by decide +kernel) (by decide +kernel)
have hh : Real.log (701 / 1000 : ℝ) < -(177 / 500 : ℝ) := by
simpa only [Rat.cast_div, Rat.cast_neg, Rat.cast_ofNat, neg_div] using h.2
exact (Real.log_lt_iff_lt_exp (by norm_num)).1 hh
def reduceExpHi48 : ReducedArgument (707 / 1000 : ℚ) where
exponent := -1
mantissa := 707 / 500
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem exp_neg_348_upper : Real.exp (-(87 / 250 : ℝ)) < (707 : ℝ) / 1000 := by
have h := log_bounds_of_reduction reduceExpHi48 (-87 / 250) (-346 / 1000)
(by decide +kernel) (by decide +kernel)
have hh : -(87 / 250 : ℝ) < Real.log (707 / 1000 : ℝ) := by
simpa only [Rat.cast_div, Rat.cast_neg, Rat.cast_ofNat, neg_div] using h.1
exact (Real.lt_log_iff_exp_lt (by norm_num)).1 hh
def reduceExpLo288 : ReducedArgument (169 / 400 : ℚ) where
exponent := -2
mantissa := 169 / 100
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem exp_neg_86_lower : (169 : ℝ) / 400 < Real.exp (-(43 / 50 : ℝ)) := by
have h := log_bounds_of_reduction reduceExpLo288 (-863 / 1000) (-43 / 50)
(by decide +kernel) (by decide +kernel)
have hh : Real.log (169 / 400 : ℝ) < -(43 / 50 : ℝ) := by
simpa only [Rat.cast_div, Rat.cast_neg, Rat.cast_ofNat, neg_div] using h.2
exact (Real.log_lt_iff_lt_exp (by norm_num)).1 hh
def reduceZ48Lo : ReducedArgument (((1 + 701 / 1000) ^ 2 - 1) : ℚ) where
exponent := 0
mantissa := (1 + 701 / 1000) ^ 2 - 1
lower := by norm_num
upper := by norm_num
value := by norm_num
def reduceZ48Hi : ReducedArgument (((1 + 707 / 1000) ^ 2 - 1) : ℚ) where
exponent := 0
mantissa := (1 + 707 / 1000) ^ 2 - 1
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_z48_lower :
(127 : ℝ) / 200 < Real.log ((1 + (701 / 1000 : ℝ)) ^ 2 - 1) := by
have h := log_bounds_of_reduction reduceZ48Lo (127 / 200) (651 / 1000)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h.1
theorem log_z48_upper :
Real.log ((1 + (707 / 1000 : ℝ)) ^ 2 - 1) < (651 : ℝ) / 1000 := by
have h := log_bounds_of_reduction reduceZ48Hi (127 / 200) (651 / 1000)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h.2
def reduceZ288Lo : ReducedArgument (((1 + 169 / 400) ^ 3 - 1) : ℚ) where
exponent := 0
mantissa := (1 + 169 / 400) ^ 3 - 1
lower := by norm_num
upper := by norm_num
value := by norm_num
theorem log_z288_lower :
(157 : ℝ) / 250 < Real.log ((1 + (169 / 400 : ℝ)) ^ 3 - 1) := by
have h := log_bounds_of_reduction reduceZ288Lo (157 / 250) (16 / 25)
(by decide +kernel) (by decide +kernel)
norm_num at h ⊢
exact h.1
end MatrixMultiplication.ComplexCertificates
end OAI
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