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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.Duality.Witness
import OAI.LinearAlgebra.MatrixMultiplication.Duality.HiddenInstances
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.DualWitness
attribute [local instance 10000] Classical.propDecidable Classical.decEq
attribute [local instance 11000] instDecidableEqFin
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels
open CompletionLaws CompletionColorLaws DualCompletionLaws CompletionPartitions
theorem law_entropy : finiteEntropy law.mass =
288 * Real.log 2 + 288 * hiddenRate := by
have hhidden := DualHiddenRates.laws288_weightedHidden
rw [DualUniform.UniformLaws.entropy_B _ _ DualUniform.laws288_uniform,
DualUniform.UniformLaws.entropy_C _ _ DualUniform.laws288_uniform] at hhidden
have hcoord := DualHiddenRates.laws288_coordinateEntropy
change CompletionEntropyAlgebra.binaryEntropy state288.rho +
state288.rho * Real.log (activeCount source) +
(1 - state288.rho) * Real.log (inactiveCount source) =
Real.log (Fintype.card Word) at hcoord
rw [word_card, Nat.cast_pow, Nat.cast_ofNat, Real.log_pow] at hcoord
unfold CompletionEntropyAlgebra.binaryEntropy at hcoord
dsimp only [source] at hcoord
norm_num only [Nat.cast_ofNat] at hcoord
change finiteEntropy (CompletionTerminationLaw.law
(Script.tensor BaseTwo.flagged Script.dual) .B .C (laws288 .B) (laws288 .C)
state288.rho _ _).mass = _
rw [CompletionTerminationLaw.law_entropy _ .B .C (by intro h; cases h)]
linarith only [hhidden, hcoord]
theorem total_hidden_entropy : finiteEntropy law.mass - finiteEntropy (law.map x).mass =
288 * hiddenRate := by rw [law_entropy, x_entropy]; ring
end MatrixMultiplication.DualWitness
end
end OAI
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