openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexBounds /CW75ReorderedBalancedRates.lean
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedBalanced | |
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedFiniteRates | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalRateBridge | |
| import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.Witness | |
| namespace OAI | |
| /-! Explicit complex square and rectangular matrix multiplication bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CW75ReorderedBalancedRates | |
| open MatrixMultiplication.Foundation Filter CW75LabelHierarchy | |
| open ConditionalLabels ComplexWitness CW75ReorderedBalanced | |
| open scoped BigOperators Topology Classical | |
| variable (counts : Scalar → ℕ) (hD : 0 < ∑ a, counts a) | |
| (p : FiniteLaw Scalar) | |
| (mass : ∀ a, p.mass a = (counts a : ℝ) / (∑ b, counts b : ℕ)) | |
| include hD mass | |
| theorem tendsto_log_rows : | |
| Tendsto (fun t => Real.log (CW75ReorderedFinite.rows counts t : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (lawPairingRate p curveLabels 5 curveReaderPair .xz)) := by | |
| have h := tendsto_log_stagePairingSize_law counts hD p mass | |
| curveLabels 5 curveReaderPair .xz | |
| apply h.congr | |
| intro t | |
| rw [(CW75ReorderedFinite.dimensions_eq_pairingSize counts t).1] | |
| theorem tendsto_log_inner : | |
| Tendsto (fun t => Real.log (CW75ReorderedFinite.inner counts t : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (lawPairingRate p curveLabels 5 curveReaderPair .xy)) := by | |
| have h := tendsto_log_stagePairingSize_law counts hD p mass | |
| curveLabels 5 curveReaderPair .xy | |
| apply h.congr | |
| intro t | |
| rw [(CW75ReorderedFinite.dimensions_eq_pairingSize counts t).2.1] | |
| theorem tendsto_log_columns : | |
| Tendsto (fun t => Real.log (CW75ReorderedFinite.columns counts t : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (lawPairingRate p curveLabels 5 curveReaderPair .yz)) := by | |
| have h := tendsto_log_stagePairingSize_law counts hD p mass | |
| curveLabels 5 curveReaderPair .yz | |
| apply h.congr | |
| intro t | |
| rw [(CW75ReorderedFinite.dimensions_eq_pairingSize counts t).2.2] | |
| theorem tendsto_log_witness_outer : | |
| Tendsto (fun t => Real.log ((witness counts t).outer : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (lawPairingRate p curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate p curveLabels 5 curveReaderPair .xz)) := by | |
| have h := (tendsto_log_rows counts hD p mass).add | |
| (tendsto_log_inner counts hD p mass) | |
| rw [add_comm (lawPairingRate p curveLabels 5 curveReaderPair .xz)] at h | |
| apply h.congr | |
| intro t | |
| rw [witness_outer, Nat.cast_mul, Real.log_mul | |
| (Nat.cast_ne_zero.mpr (Nat.ne_of_gt (CW75ReorderedFinite.rows_pos counts t))) | |
| (Nat.cast_ne_zero.mpr (Nat.ne_of_gt (CW75ReorderedFinite.inner_pos counts t))), add_div] | |
| theorem tendsto_log_witness_inner : | |
| Tendsto (fun t => Real.log ((witness counts t).inner : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (2 * lawPairingRate p curveLabels 5 curveReaderPair .yz)) := by | |
| simpa only [witness_inner, Nat.cast_pow, Real.log_pow, Nat.cast_ofNat, | |
| mul_div_assoc] using (tendsto_log_columns counts hD p mass).const_mul 2 | |
| theorem tendsto_log_witness_multiplicity : | |
| Tendsto (fun t => Real.log ((witness counts t).multiplicity : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (2 * finiteEntropy p.mass)) := by | |
| simpa only [witness_multiplicity, Nat.cast_pow, Real.log_pow, Nat.cast_ofNat, | |
| mul_div_assoc] using | |
| (CW75ReorderedFiniteRates.tendsto_log_copies counts hD p mass).const_mul 2 | |
| theorem tendsto_log_witness_rank : | |
| Tendsto (fun t => Real.log ((witness counts t).rank : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (6 * Real.log 3 + 2 * finiteEntropy p.mass)) := by | |
| have h := (CW75ReorderedFiniteRates.tendsto_log_rankBudget counts hD p mass).const_mul 2 | |
| have hcost : 2 * (3 * Real.log 3 + finiteEntropy p.mass) = | |
| 6 * Real.log 3 + 2 * finiteEntropy p.mass := by ring | |
| simpa only [witness_rank, Nat.cast_pow, Real.log_pow, Nat.cast_ofNat, | |
| mul_div_assoc, hcost] using h | |
| omit mass | |
| theorem scale_eventually_pos : | |
| ∀ᶠ t : ℕ in atTop, 0 < (t : ℝ) * (∑ a, counts a : ℕ) := by | |
| filter_upwards [eventually_ge_atTop 1] with t ht | |
| exact mul_pos (Nat.cast_pos.mpr (by omega)) (Nat.cast_pos.mpr hD) | |
| include mass | |
| theorem omega_lt_of_fixed_count_law (k target : ℝ) | |
| (k_nonneg : 0 ≤ k) | |
| (incident_positive : 0 < lawPairingRate p curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate p curveLabels 5 curveReaderPair .xz) | |
| (aspect_gap : k * (lawPairingRate p curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate p curveLabels 5 curveReaderPair .xz) < | |
| 2 * lawPairingRate p curveLabels 5 curveReaderPair .yz) | |
| (cost_gap : 6 * Real.log 3 < target * | |
| (lawPairingRate p curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate p curveLabels 5 curveReaderPair .xz)) : | |
| Arithmetic.rectangularOmega ℂ k < target := | |
| RectangularWitness.omega_lt_of_balanced_family (witness counts) | |
| (fun t => (t : ℝ) * (∑ a, counts a : ℕ)) k target | |
| (lawPairingRate p curveLabels 5 curveReaderPair .yz) | |
| (lawPairingRate p curveLabels 5 curveReaderPair .xy + | |
| lawPairingRate p curveLabels 5 curveReaderPair .xz) | |
| (finiteEntropy p.mass) k_nonneg incident_positive aspect_gap cost_gap | |
| (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.CW75ReorderedBalancedRates | |
| end | |
| end OAI | |