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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 36 37 38 39 40 41 42 43 44 45 46 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.OrderedLimits
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
namespace MatrixMultiplication.AllFieldConditional
open AllFieldNativeCapacity
structure NumericalConditions : Prop where
denominator_gap : nativeB < nativeHigh
stageA_positive : ∀ i : Fin 3, 0 < nativeLA i
stageB_positive : ∀ i : Fin 3, 0 < nativeLB i
stageC_positive : ∀ i : Fin 3,
0 < nativeCstar - (nativeLA i + nativeLB i)
stageC_feasible : ∀ i : Fin 3,
nativeCstar - (nativeLA i + nativeLB i) < nativeHigh
volume_positive : 0 < AllFieldTerminalRates.S
strict_rate_bound :
3 * (8 * Real.log 7 -
(AllFieldInitialEntropy.nativeH0 + nativeCstar)) / AllFieldTerminalRates.S <
(2371054887 : ℝ) / 10 ^ 9
theorem omega_lt_2371054887_div_1000000000_of_numerical_conditions
(F : Type*) [Field F] (h : NumericalConditions) :
Arithmetic.omega F < (2371054887 : ℝ) / 10 ^ 9 := by
have hc (i : Fin 3) : 0 < nativeLC nativeLambda i := by
rw [nativeLC_balanced h.denominator_gap i]
exact h.stageC_positive i
have hbound := AllFieldOrderedLimits.omega_le_native_ratio F
h.denominator_gap h.stageC_feasible
h.stageA_positive h.stageB_positive hc h.volume_positive
exact lt_of_le_of_lt hbound h.strict_rate_bound
end MatrixMultiplication.AllFieldConditional
end OAI
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