File size: 3,102 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 | import OAI.LinearAlgebra.MatrixMultiplication.Numerical.CurveLawsCertificates
import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalContinuity
import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalRateBridge
namespace OAI
/-! Finite entropy, rate estimates and ordered asymptotic limits. -/
noncomputable section
namespace MatrixMultiplication.CurveLaws
open MatrixMultiplication.Foundation ComplexWitness ConditionalLabels Filter
open scoped BigOperators Topology
theorem exists_rational_rate_bounds :
∃ q : RationalLaw CurveLeaf,
(31519 : ℝ) / 10000 <
lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .xy +
lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .xz ∧
lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .xy +
lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .xz <
(15761 : ℝ) / 5000 ∧
(27961 : ℝ) / 25000 <
lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .yz ∧
lawPairingRate q.toFiniteLaw curveLabels 5 curveReaderPair .yz <
(6991 : ℝ) / 6250 := by
obtain ⟨q, hq⟩ := exists_rational_law_sequence (leafLaw (2 / 3 : ℝ))
have hi := (tendsto_lawPairingRate hq curveLabels 5 curveReaderPair .xy).add
(tendsto_lawPairingRate hq curveLabels 5 curveReaderPair .xz)
have hh := tendsto_lawPairingRate hq curveLabels 5 curveReaderPair .yz
rw [incident_rate_two_thirds] at hi
rw [hidden_rate_two_thirds] at hh
have hilo := hi.eventually_const_lt ComplexCertificates.curveH_bounds.1
have hihi := hi.eventually_lt_const ComplexCertificates.curveH_bounds.2
have hhlo := hh.eventually_const_lt ComplexCertificates.curveG_bounds.1
have hhhi := hh.eventually_lt_const ComplexCertificates.curveG_bounds.2
obtain ⟨n, hn⟩ := (hilo.and (hihi.and (hhlo.and hhhi))).exists
exact ⟨q n, hn⟩
theorem exists_count_rate_bounds :
∃ counts : CurveLeaf → ℕ, ∃ hD : 0 < ∑ a, counts a,
(31519 : ℝ) / 10000 <
lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .xy +
lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .xz ∧
lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .xy +
lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .xz <
(15761 : ℝ) / 5000 ∧
(27961 : ℝ) / 25000 <
lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .yz ∧
lawPairingRate (normalizedCountLaw counts hD) curveLabels 5 curveReaderPair .yz <
(6991 : ℝ) / 6250 := by
obtain ⟨q, hq⟩ := exists_rational_rate_bounds
obtain ⟨counts, hD, hmass⟩ := q.exists_type_representation
have law_eq (p r : FiniteLaw CurveLeaf) (h : p.mass = r.mass) : p = r := by
cases p
cases r
cases h
rfl
have heq : normalizedCountLaw counts hD = q.toFiniteLaw :=
law_eq _ _ (funext hmass)
refine ⟨counts, hD, ?_⟩
simpa only [heq] using hq
end MatrixMultiplication.CurveLaws
end
end OAI
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