openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexBounds /SquareWitness.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexWitness | |
| import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexCertificates | |
| namespace OAI | |
| /-! Explicit complex square and rectangular matrix multiplication bounds. -/ | |
| open Filter | |
| namespace MatrixMultiplication.SquareWitness | |
| open MatrixMultiplication.Foundation | |
| def leafCount : ℕ := 2995461856366867388223311649574852466276480577 | |
| theorem leafCount_pos : 0 < leafCount := by decide | |
| theorem exponent_upper {c l : ℝ} | |
| (hc : c < (10987 : ℝ) / 10000) | |
| (hl : (523471 : ℝ) / 5000 < l) : | |
| 216 * c / l < (1186596 : ℝ) / 523471 := by | |
| have hl₀ : 0 < l := by linarith | |
| apply (div_lt_iff₀ hl₀).2 | |
| nlinarith | |
| theorem exponent_lt_target {c l : ℝ} | |
| (hc : c < (10987 : ℝ) / 10000) | |
| (hl : (523471 : ℝ) / 5000 < l) : | |
| 216 * c / l < (2267 : ℝ) / 1000 := by | |
| exact (exponent_upper hc hl).trans | |
| (by norm_num : (1186596 : ℝ) / 523471 < 2267 / 1000) | |
| theorem strict_log_gap | |
| (hthree : Real.log 3 < (10987 : ℝ) / 10000) | |
| (hleaves : (523471 : ℝ) / 5000 < Real.log leafCount) : | |
| Real.log 3 < ((2267 : ℝ) / 1000) / 216 * Real.log leafCount := by | |
| have h := exponent_lt_target hthree hleaves | |
| have hl₀ : 0 < Real.log leafCount := by linarith | |
| have hm := (div_lt_iff₀ hl₀).1 h | |
| nlinarith | |
| theorem omega_lt_of_uniform_square_family | |
| (W : ℕ → FiniteMMRealization) | |
| (hlimit : Tendsto | |
| (fun n => (W n).rates.score ((2267 : ℝ) / 1000)) atTop | |
| (nhds (((2267 : ℝ) / 1000) / 216 * Real.log leafCount))) | |
| (hthree : Real.log 3 < (10987 : ℝ) / 10000) | |
| (hleaves : (523471 : ℝ) / 5000 < Real.log leafCount) : | |
| Arithmetic.omega < (2267 : ℝ) / 1000 := | |
| ComplexWitness.omega_lt_of_score_limit W hlimit (strict_log_gap hthree hleaves) | |
| theorem omega_lt_of_certified_uniform_square_family | |
| (W : ℕ → FiniteMMRealization) | |
| (hlimit : Tendsto | |
| (fun n => (W n).rates.score ((2267 : ℝ) / 1000)) atTop | |
| (nhds (((2267 : ℝ) / 1000) / 216 * Real.log leafCount))) : | |
| Arithmetic.omega < (2267 : ℝ) / 1000 := by | |
| apply omega_lt_of_uniform_square_family W hlimit | |
| ComplexCertificates.log_three_upper | |
| exact ComplexCertificates.square_log_lower | |
| end MatrixMultiplication.SquareWitness | |
| end OAI | |