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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 47 48 49 50 51 52 53 54 55 56 57 58 | 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
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