openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Numerical /ComplexCertificatesLog.lean
| 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 | |