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