File size: 6,842 Bytes
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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | 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
|