openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexBounds /CW75ReorderedFiniteRates.lean
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedFinite | |
| namespace OAI | |
| /-! Explicit complex square and rectangular matrix multiplication bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CW75ReorderedFiniteRates | |
| open MatrixMultiplication.Foundation CW75LabelHierarchy StageHierarchyResources | |
| open CW75ReorderedFinite ComplexWitness Filter | |
| open scoped BigOperators Topology Classical | |
| theorem tendsto_log_copies (counts : Scalar → ℕ) (hD : 0 < ∑ a, counts a) | |
| (p : FiniteLaw Scalar) | |
| (mass : ∀ a, p.mass a = (counts a : ℝ) / (∑ b, counts b : ℕ)) : | |
| Tendsto (fun t : ℕ => Real.log (copies counts t : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (finiteEntropy p.mass)) := by | |
| have hmass : (fun a => (counts a : ℝ) / (∑ b, counts b : ℕ)) = p.mass := | |
| (funext mass).symm | |
| simpa only [copies, hmass] using tendsto_log_exactWords_card_mul counts hD | |
| theorem tendsto_log_auxiliaryBudget (counts : Scalar → ℕ) (hD : 0 < ∑ a, counts a) | |
| (p : FiniteLaw Scalar) | |
| (mass : ∀ a, p.mass a = (counts a : ℝ) / (∑ b, counts b : ℕ)) : | |
| Tendsto (fun t : ℕ => Real.log (stageAuxiliaryBudget counts curveLabels 5 t : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (finiteEntropy p.mass)) := by | |
| have hmass : (fun a => (counts a : ℝ) / (∑ b, counts b : ℕ)) = p.mass := | |
| (funext mass).symm | |
| simpa only [hmass] using tendsto_log_stageAuxiliaryBudget counts hD curveLabels 5 | |
| curve_completeRecord_injective | |
| theorem tendsto_log_sourceRank (counts : Scalar → ℕ) (hD : 0 < ∑ a, counts a) : | |
| Tendsto (fun t : ℕ => Real.log ((27 ^ blocks counts t : ℕ) : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (3 * Real.log 3)) := by | |
| have hlog27 : Real.log (27 : ℝ) = 3 * Real.log 3 := by | |
| rw [show (27 : ℝ) = 3 ^ 3 by norm_num, Real.log_pow] | |
| norm_num | |
| have hD' : ((∑ a, counts a : ℕ) : ℝ) ≠ 0 := | |
| Nat.cast_ne_zero.mpr (Nat.ne_of_gt hD) | |
| apply tendsto_const_nhds.congr' | |
| filter_upwards [eventually_ne_atTop (0 : ℕ)] with t ht | |
| have htD : (t : ℝ) * (∑ a, counts a : ℕ) ≠ 0 := | |
| mul_ne_zero (Nat.cast_ne_zero.mpr ht) hD' | |
| rw [Nat.cast_pow, Real.log_pow, blocks, Nat.cast_mul] | |
| change 3 * Real.log 3 = | |
| ((t : ℝ) * (∑ a, counts a : ℕ)) * Real.log 27 / | |
| ((t : ℝ) * (∑ a, counts a : ℕ)) | |
| rw [mul_div_cancel_left₀ _ htD, hlog27] | |
| theorem tendsto_log_rankBudget (counts : Scalar → ℕ) (hD : 0 < ∑ a, counts a) | |
| (p : FiniteLaw Scalar) | |
| (mass : ∀ a, p.mass a = (counts a : ℝ) / (∑ b, counts b : ℕ)) : | |
| Tendsto (fun t : ℕ => Real.log (rankBudget counts t : ℝ) / | |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop | |
| (𝓝 (3 * Real.log 3 + finiteEntropy p.mass)) := by | |
| apply ((tendsto_log_sourceRank counts hD).add | |
| (tendsto_log_auxiliaryBudget counts hD p mass)).congr | |
| intro t | |
| have hsource : ((27 ^ blocks counts t : ℕ) : ℝ) ≠ 0 := | |
| Nat.cast_ne_zero.mpr (pow_ne_zero _ (by decide)) | |
| have hauxiliary : (stageAuxiliaryBudget counts curveLabels 5 t : ℝ) ≠ 0 := | |
| Nat.cast_ne_zero.mpr | |
| (Nat.ne_of_gt (stageAuxiliaryBudget_pos counts curveLabels 5 t)) | |
| rw [rankBudget, Nat.cast_mul, Real.log_mul hsource hauxiliary, add_div] | |
| end MatrixMultiplication.CW75ReorderedFiniteRates | |
| end | |
| end OAI | |