openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Numerical /ComplexCertificates.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexCertificatesLog | |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real | |
| namespace OAI | |
| /-! Exact rational intervals, logarithm bounds and arithmetic circuit soundness. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.ComplexCertificates | |
| def curveU : ℝ := (2 : ℝ) ^ ((2 : ℝ) / 3) | |
| def curveV : ℝ := (19 : ℝ) ^ ((2 : ℝ) / 3) | |
| def curveZ : ℝ := 8 + 12 * curveU + 6 * curveU ^ 2 + curveV | |
| def curveG : ℝ := | |
| (12 * curveU * (1 + curveU) * Real.log 2 + curveV * Real.log 19) / curveZ | |
| def curveH : ℝ := Real.log curveZ - (2 / 3 : ℝ) * curveG | |
| theorem curveU_cube : curveU ^ 3 = 4 := by | |
| rw [curveU, ← Real.rpow_mul_natCast (by norm_num : (0 : ℝ) ≤ 2)] | |
| norm_num | |
| theorem curveV_cube : curveV ^ 3 = 361 := by | |
| rw [curveV, ← Real.rpow_mul_natCast (by norm_num : (0 : ℝ) ≤ 19)] | |
| norm_num | |
| theorem curveU_bounds : | |
| (158739 : ℝ) / 100000 < curveU ∧ curveU < (79371 : ℝ) / 50000 := by | |
| have h0 : 0 ≤ curveU := Real.rpow_nonneg (by norm_num) _ | |
| constructor | |
| · apply (pow_lt_pow_iff_left₀ (by norm_num) h0 (by norm_num : (3 : ℕ) ≠ 0)).1 | |
| rw [curveU_cube] | |
| norm_num | |
| · apply (pow_lt_pow_iff_left₀ h0 (by norm_num) (by norm_num : (3 : ℕ) ≠ 0)).1 | |
| rw [curveU_cube] | |
| norm_num | |
| theorem curveV_bounds : | |
| (71203 : ℝ) / 10000 < curveV ∧ curveV < (14241 : ℝ) / 2000 := by | |
| have h0 : 0 ≤ curveV := Real.rpow_nonneg (by norm_num) _ | |
| constructor | |
| · apply (pow_lt_pow_iff_left₀ (by norm_num) h0 (by norm_num : (3 : ℕ) ≠ 0)).1 | |
| rw [curveV_cube] | |
| norm_num | |
| · apply (pow_lt_pow_iff_left₀ h0 (by norm_num) (by norm_num : (3 : ℕ) ≠ 0)).1 | |
| rw [curveV_cube] | |
| norm_num | |
| theorem curveZ_bounds : | |
| (246439 : ℝ) / 5000 < curveZ ∧ curveZ < (49289 : ℝ) / 1000 := by | |
| obtain ⟨hu, hu'⟩ := curveU_bounds | |
| obtain ⟨hv, hv'⟩ := curveV_bounds | |
| have hu0 : 0 ≤ curveU := by linarith | |
| constructor | |
| · have h : 8 + 12 * ((158739 : ℝ) / 100000) + | |
| 6 * ((158739 : ℝ) / 100000) ^ 2 + (71203 : ℝ) / 10000 ≤ curveZ := by | |
| dsimp [curveZ] | |
| gcongr | |
| norm_num at h ⊢ | |
| linarith | |
| · have h : curveZ ≤ 8 + 12 * ((79371 : ℝ) / 50000) + | |
| 6 * ((79371 : ℝ) / 50000) ^ 2 + (14241 : ℝ) / 2000 := by | |
| dsimp [curveZ] | |
| gcongr | |
| norm_num at h ⊢ | |
| linarith | |
| theorem curveG_bounds : | |
| (27961 : ℝ) / 25000 < curveG ∧ curveG < (6991 : ℝ) / 6250 := by | |
| obtain ⟨hu, hu'⟩ := curveU_bounds | |
| obtain ⟨hv, hv'⟩ := curveV_bounds | |
| obtain ⟨hz, hz'⟩ := curveZ_bounds | |
| obtain ⟨hl2, hh2⟩ := log_two_bounds | |
| obtain ⟨hl19, hh19⟩ := log_nineteen_bounds | |
| have hu0 : 0 ≤ curveU := by linarith | |
| have hv0 : 0 ≤ curveV := by linarith | |
| have hz0 : 0 < curveZ := by linarith | |
| have hl20 : 0 ≤ Real.log 2 := by linarith | |
| have hl190 : 0 ≤ Real.log 19 := by linarith | |
| have hnlo : | |
| 12 * ((158739 : ℝ) / 100000) * (1 + (158739 : ℝ) / 100000) * | |
| ((34657 : ℝ) / 50000) + ((71203 : ℝ) / 10000) * ((7361 : ℝ) / 2500) ≤ | |
| 12 * curveU * (1 + curveU) * Real.log 2 + curveV * Real.log 19 := by | |
| gcongr | |
| have hnhi : | |
| 12 * curveU * (1 + curveU) * Real.log 2 + curveV * Real.log 19 ≤ | |
| 12 * ((79371 : ℝ) / 50000) * (1 + (79371 : ℝ) / 50000) * | |
| ((17329 : ℝ) / 25000) + ((14241 : ℝ) / 2000) * ((5889 : ℝ) / 2000) := by | |
| gcongr | |
| constructor | |
| · apply (lt_div_iff₀ hz0).2 | |
| norm_num at hnlo | |
| nlinarith | |
| · apply (div_lt_iff₀ hz0).2 | |
| norm_num at hnhi | |
| nlinarith | |
| theorem curveH_bounds : | |
| (31519 : ℝ) / 10000 < curveH ∧ curveH < (15761 : ℝ) / 5000 := by | |
| obtain ⟨hg, hg'⟩ := curveG_bounds | |
| obtain ⟨hz, hz'⟩ := curveZ_bounds | |
| have hz0 : 0 < curveZ := by linarith | |
| have hlo : (77953 : ℝ) / 20000 < Real.log curveZ := | |
| log_curve_lower.trans (Real.log_lt_log (by norm_num) hz) | |
| have hhi : Real.log curveZ < (97443 : ℝ) / 25000 := | |
| (Real.log_lt_log hz0 hz').trans log_curve_upper | |
| dsimp [curveH] | |
| constructor <;> linarith | |
| def squareLeafCount : ℕ := 2995461856366867388223311649574852466276480577 | |
| theorem square_log_lower : | |
| (523471 : ℝ) / 5000 < Real.log (squareLeafCount : ℝ) := by | |
| have hc : (59 / 20 : ℝ) * 10 ^ 45 < (squareLeafCount : ℝ) := by | |
| norm_num [squareLeafCount] | |
| have hlo := Real.log_lt_log (by positivity : (0 : ℝ) < (59 / 20 : ℝ) * 10 ^ 45) hc | |
| rw [Real.log_mul (by norm_num) (by norm_num), Real.log_pow] at hlo | |
| have h1 := log_square_mantissa_lower | |
| have h2 := log_ten_lower | |
| norm_num at hlo | |
| linarith | |
| end MatrixMultiplication.ComplexCertificates | |
| end | |
| end OAI | |