| 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 |
|
|
