File size: 3,362 Bytes
cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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
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