| import OAI.LinearAlgebra.MatrixMultiplication.Duality.CompletionLaws |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.UniformInstances |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.TerminationLaw |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.WitnessProgram |
|
|
| 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 |
|
|
| def law : FiniteLaw Support := |
| CompletionTerminationLaw.law source .B .C (laws288 .B) (laws288 .C) state288.rho |
| (by rw [state288_rho]; exact rho288_range.1.le) |
| (by rw [state288_rho]; exact rho288_range.2.le) |
|
|
| def rate (pair : ReaderPair) : ℝ := |
| ConditionalLabels.lawPairingRate law program.labels (1 + readable.depth) |
| (fun n => program.pair n.val) pair |
|
|
| theorem law_uniform : UniformCoordinate law x (fun _ => True) := by |
| have hu := DualUniformMixtures.colorMix_uniform_complement source .B .C |
| (laws288 .B) (laws288 .C) state288.rho |
| (by rw [state288_rho]; exact rho288_range.1.le) |
| (by rw [state288_rho]; exact rho288_range.2.le) |
| source.flagB DualUniform.laws288_uniform.B DualUniform.laws288_uniform.C |
| DualUniform.rho288.symm |
| intro u |
| change ((law.map (fun a : Support => a.val.val.1)).mass u) = _ |
| exact (CompletionTerminationLaw.law_map_mass source .B .C |
| (laws288 .B) (laws288 .C) state288.rho |
| (by rw [state288_rho]; exact rho288_range.1.le) |
| (by rw [state288_rho]; exact rho288_range.2.le) |
| (fun a : Leaf source => a.val.1) u).trans (hu u) |
|
|
| theorem x_entropy : finiteEntropy (law.map x).mass = 288 * Real.log 2 := by |
| rw [uniform_coordinate_entropy law x (fun _ => True) law_uniform] |
| simp only [Fintype.card_subtype_true, word_card, Nat.cast_pow, Nat.cast_ofNat, |
| Real.log_pow] |
|
|
| theorem rate_sum : rate .xy + rate .xz + rate .yz = finiteEntropy law.mass := |
| ConditionalLabels.lawPairingRate_sum law program.labels (1 + readable.depth) |
| (fun n => program.pair n.val) record_injective |
|
|
| end MatrixMultiplication.DualWitness |
|
|
| end |
|
|
| end OAI |
|
|
