openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexBounds /CW75ReorderedRectangular.lean
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedBalancedRates | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.CurveRationalLaw | |
| namespace OAI | |
| /-! Explicit complex square and rectangular matrix multiplication bounds. -/ | |
| namespace MatrixMultiplication.CW75ReorderedRectangular | |
| open MatrixMultiplication.Foundation ConditionalLabels ComplexWitness | |
| open CW75ReorderedBalanced CW75ReorderedBalancedRates | |
| open scoped BigOperators | |
| theorem rectangularOmega_lt_target : | |
| Arithmetic.rectangularOmega ℂ ((709 : ℝ) / 1000) < (523 : ℝ) / 250 := by | |
| obtain ⟨counts, hD, h_lower, h_upper, g_lower, _g_upper⟩ := | |
| CurveLaws.exists_count_rate_bounds | |
| let p := normalizedCountLaw counts hD | |
| have mass : ∀ a, p.mass a = (counts a : ℝ) / (∑ b, counts b : ℕ) := fun _ => rfl | |
| exact RectangularWitness.omega_lt_target_of_nearby_balanced_family | |
| (witness counts) (fun t => (t : ℝ) * (∑ a, counts a : ℕ)) | |
| (lawPairingRate p curveLabels 5 curveReaderPair .yz) | |
| (lawPairingRate p curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate p curveLabels 5 curveReaderPair .xz) | |
| (finiteEntropy p.mass) g_lower h_lower h_upper | |
| (scale_eventually_pos counts hD) | |
| (tendsto_log_witness_outer counts hD p mass) | |
| (tendsto_log_witness_inner counts hD p mass) | |
| (tendsto_log_witness_multiplicity counts hD p mass) | |
| (tendsto_log_witness_rank counts hD p mass) | |
| end MatrixMultiplication.CW75ReorderedRectangular | |
| end OAI | |