File size: 2,258 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 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexWitness
import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexCertificates
namespace OAI
/-! Explicit complex square and rectangular matrix multiplication bounds. -/
open Filter
namespace MatrixMultiplication.SquareWitness
open MatrixMultiplication.Foundation
def leafCount : ℕ := 2995461856366867388223311649574852466276480577
theorem leafCount_pos : 0 < leafCount := by decide
theorem exponent_upper {c l : ℝ}
(hc : c < (10987 : ℝ) / 10000)
(hl : (523471 : ℝ) / 5000 < l) :
216 * c / l < (1186596 : ℝ) / 523471 := by
have hl₀ : 0 < l := by linarith
apply (div_lt_iff₀ hl₀).2
nlinarith
theorem exponent_lt_target {c l : ℝ}
(hc : c < (10987 : ℝ) / 10000)
(hl : (523471 : ℝ) / 5000 < l) :
216 * c / l < (2267 : ℝ) / 1000 := by
exact (exponent_upper hc hl).trans
(by norm_num : (1186596 : ℝ) / 523471 < 2267 / 1000)
theorem strict_log_gap
(hthree : Real.log 3 < (10987 : ℝ) / 10000)
(hleaves : (523471 : ℝ) / 5000 < Real.log leafCount) :
Real.log 3 < ((2267 : ℝ) / 1000) / 216 * Real.log leafCount := by
have h := exponent_lt_target hthree hleaves
have hl₀ : 0 < Real.log leafCount := by linarith
have hm := (div_lt_iff₀ hl₀).1 h
nlinarith
theorem omega_lt_of_uniform_square_family
(W : ℕ → FiniteMMRealization)
(hlimit : Tendsto
(fun n => (W n).rates.score ((2267 : ℝ) / 1000)) atTop
(nhds (((2267 : ℝ) / 1000) / 216 * Real.log leafCount)))
(hthree : Real.log 3 < (10987 : ℝ) / 10000)
(hleaves : (523471 : ℝ) / 5000 < Real.log leafCount) :
Arithmetic.omega < (2267 : ℝ) / 1000 :=
ComplexWitness.omega_lt_of_score_limit W hlimit (strict_log_gap hthree hleaves)
theorem omega_lt_of_certified_uniform_square_family
(W : ℕ → FiniteMMRealization)
(hlimit : Tendsto
(fun n => (W n).rates.score ((2267 : ℝ) / 1000)) atTop
(nhds (((2267 : ℝ) / 1000) / 216 * Real.log leafCount))) :
Arithmetic.omega < (2267 : ℝ) / 1000 := by
apply omega_lt_of_uniform_square_family W hlimit
ComplexCertificates.log_three_upper
exact ComplexCertificates.square_log_lower
end MatrixMultiplication.SquareWitness
end OAI
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