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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 | 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
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