| import OAI.LinearAlgebra.MatrixMultiplication.Duality.HiddenRates |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.UniformInstances |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.WitnessRecurrence |
|
|
| namespace OAI |
|
|
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.DualHiddenRates |
|
|
| open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels CompletionColorLaws |
| open CompletionLaws DualCompletionLaws CompletionPartitions |
| attribute [local instance 10000] Classical.propDecidable Classical.decEq |
| attribute [local instance 11000] instDecidableEqFin |
| |
| private theorem coord_nonempty (s : Script) : Nonempty (Script.Coord (Fin 2) s) := by |
| induction s with |
| | base => exact inferInstanceAs (Nonempty (Fin 2)) |
| | step prior center m ih => |
| let := ih |
| exact inferInstanceAs (Nonempty (Fin m → Script.Coord (Fin 2) prior)) |
| attribute [local instance] coord_nonempty |
|
|
| theorem base_hidden : HiddenInvariant BaseTwo.flagged baseLaws DualWitness.initial := by |
| have hc : Fintype.card {x // BaseTwo.flagged.flagB x} = 1 := |
| @Fintype.card_subtype_eq (Fin 2) 1 |
| (inferInstanceAs (Fintype {x // BaseTwo.flagged.flagB x})) |
| constructor <;> |
| norm_num [baseLaws, DualWitness.initial, activeCount, inactiveCount, |
| BaseTwo.flagged, BaseTwo.flagB, Fintype.card_subtype_compl] |
| all_goals norm_num [hc] |
|
|
| theorem laws2_hidden : HiddenInvariant (Script.tensor BaseTwo.flagged script2) |
| laws2 DualWitness.state2 := |
| minus_hidden BaseTwo.flagged baseLaws DualWitness.initial 2 (by norm_num) |
| (by norm_num [DualWitness.initial]) (by norm_num [DualWitness.initial]) |
| DualUniform.base_activeFraction.symm base_hidden |
|
|
| theorem laws4_hidden : HiddenInvariant (Script.tensor BaseTwo.flagged script4) |
| laws4 DualWitness.state4 := |
| plusA_hidden _ laws2 DualWitness.state2 2 (by norm_num) |
| (by norm_num [DualWitness.state2, DualWitness.initial, DualWitness.minus]) |
| (by norm_num [DualWitness.state2, DualWitness.initial, DualWitness.minus]) |
| DualUniform.rho2.symm laws2_hidden |
|
|
| theorem laws8_hidden : HiddenInvariant (Script.tensor BaseTwo.flagged script8) |
| laws8 DualWitness.state8 := |
| plusC_hidden _ laws4 DualWitness.state4 2 (by norm_num) |
| (by norm_num [DualWitness.state4, DualWitness.state2, DualWitness.initial, |
| DualWitness.minus, DualWitness.plusA]) |
| (by norm_num [DualWitness.state4, DualWitness.state2, DualWitness.initial, |
| DualWitness.minus, DualWitness.plusA]) |
| DualUniform.rho4.symm laws4_hidden |
|
|
| theorem laws24_hidden : HiddenInvariant (Script.tensor BaseTwo.flagged script24) |
| laws24 DualWitness.state24 := |
| minus_hidden _ laws8 DualWitness.state8 3 (by norm_num) |
| (by rw [DualWitness.state8_rho]; exact DualWitness.rho8_range.1) |
| (by rw [DualWitness.state8_rho]; exact DualWitness.rho8_range.2) |
| DualUniform.rho8.symm laws8_hidden |
|
|
| theorem laws48_hidden : HiddenInvariant (Script.tensor BaseTwo.flagged script48) |
| laws48 DualWitness.state48 := |
| plusC_hidden _ laws24 DualWitness.state24 2 (by norm_num) |
| (by rw [DualWitness.state24_rho]; exact DualWitness.rho24_range.1) |
| (by rw [DualWitness.state24_rho]; exact DualWitness.rho24_range.2) |
| DualUniform.rho24.symm laws24_hidden |
|
|
| theorem laws96_hidden : HiddenInvariant (Script.tensor BaseTwo.flagged script96) |
| laws96 DualWitness.state96 := |
| minus_hidden _ laws48 DualWitness.state48 2 (by norm_num) |
| (by rw [DualWitness.state48_rho]; exact DualWitness.rho48_range.1) |
| (by rw [DualWitness.state48_rho]; exact DualWitness.rho48_range.2) |
| DualUniform.rho48.symm laws48_hidden |
|
|
| theorem laws288_hidden : HiddenInvariant (Script.tensor BaseTwo.flagged Script.dual) |
| laws288 DualWitness.state288 := |
| plusA_hidden _ laws96 DualWitness.state96 3 (by norm_num) |
| (by rw [DualWitness.state96_rho]; exact DualWitness.rho96_range.1) |
| (by rw [DualWitness.state96_rho]; exact DualWitness.rho96_range.2) |
| DualUniform.rho96.symm laws96_hidden |
|
|
| theorem laws288_hidden_B : |
| finiteEntropy (laws288 .B).mass - |
| finiteEntropy ((laws288 .B).map (fun a => a.val.val.1)).mass = |
| DualWitness.state288.hB := by |
| rw [laws288_hidden.B, |
| DualUniform.UniformLaws.entropy_B _ _ DualUniform.laws288_uniform] |
| ring |
|
|
| theorem laws288_hidden_A : |
| finiteEntropy (laws288 .A).mass - |
| finiteEntropy ((laws288 .A).map (fun a => a.val.val.1)).mass = |
| DualWitness.state288.hA := by |
| rw [laws288_hidden.A, |
| DualUniform.UniformLaws.entropy_A _ _ DualUniform.laws288_uniform] |
| ring |
|
|
| theorem laws288_hidden_C : |
| finiteEntropy (laws288 .C).mass - |
| finiteEntropy ((laws288 .C).map (fun a => a.val.val.1)).mass = |
| DualWitness.state288.hC := by |
| rw [laws288_hidden.C, |
| DualUniform.UniformLaws.entropy_C _ _ DualUniform.laws288_uniform] |
| ring |
|
|
| theorem laws288_normalizedHidden : |
| (DualWitness.state288.rho * (finiteEntropy (laws288 .B).mass - |
| finiteEntropy ((laws288 .B).map (fun a => a.val.val.1)).mass) + |
| (1 - DualWitness.state288.rho) * (finiteEntropy (laws288 .C).mass - |
| finiteEntropy ((laws288 .C).map (fun a => a.val.val.1)).mass)) / 288 = |
| DualWitness.hiddenRate := by |
| rw [laws288_hidden_B, laws288_hidden_C] |
| have h := DualWitness.state288_normalizedHidden |
| rw [DualWitness.normalizedHidden, DualWitness.state288_max, |
| DualWitness.state_factors] at h |
| exact h |
|
|
| theorem laws288_weightedHidden : |
| DualWitness.state288.rho * (finiteEntropy (laws288 .B).mass - |
| finiteEntropy ((laws288 .B).map (fun a => a.val.val.1)).mass) + |
| (1 - DualWitness.state288.rho) * (finiteEntropy (laws288 .C).mass - |
| finiteEntropy ((laws288 .C).map (fun a => a.val.val.1)).mass) = |
| 288 * DualWitness.hiddenRate := by |
| linarith only [laws288_normalizedHidden] |
|
|
| theorem laws288_coordinateEntropy : |
| CompletionEntropyAlgebra.binaryEntropy DualWitness.state288.rho + |
| DualWitness.state288.rho * |
| Real.log (activeCount (Script.tensor BaseTwo.flagged Script.dual)) + |
| (1 - DualWitness.state288.rho) * |
| Real.log (inactiveCount (Script.tensor BaseTwo.flagged Script.dual)) = |
| Real.log (Fintype.card (Script.Coord (Fin 2) Script.dual)) := by |
| have hc := counts_pos (Script.tensor BaseTwo.flagged Script.dual) |
| (by |
| rw [DualUniform.rho288, DualWitness.state288_rho] |
| exact DualWitness.rho288_range.1) |
| (by |
| rw [DualUniform.rho288, DualWitness.state288_rho] |
| exact DualWitness.rho288_range.2) |
| have h := CompletionEntropyAlgebra.binaryEntropy_cardinal_mixture |
| (activeCount (Script.tensor BaseTwo.flagged Script.dual)) |
| (inactiveCount (Script.tensor BaseTwo.flagged Script.dual)) hc.1 hc.2 |
| dsimp only at h |
| rw [← activeFraction_ratio, DualUniform.rho288, ← Nat.cast_add, partition_count] at h |
| exact h |
|
|
| end MatrixMultiplication.DualHiddenRates |
|
|
| end |
|
|
| end OAI |
|
|
