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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 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 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.Labels
import OAI.LinearAlgebra.MatrixMultiplication.Duality.Information
namespace OAI
/-! Dual matrix multiplication exponents and finite rectangular constructions. -/
noncomputable section
namespace MatrixMultiplication.CompletionLabels.TopologicalFlatten.TopologicalSchedule
open MatrixMultiplication.Foundation RecursiveCompletion DualInformation
universe u v
variable {A : Type u} [Fintype A] {Coord : Color → Type v}
[Fintype (Coord .B)] {depth : ℕ}
theorem incident_rate_eq_x_entropy (s : TopologicalSchedule A Coord depth)
(p : FiniteLaw A) (complete : s.toNodeSchedule.Complete depth)
(independent : ∀ n : Fin depth, s.pairs n = .yz →
ConditionalIndependent p (labelRecordOf s.toNodeSchedule.labels n.val)
(s.toNodeSchedule.labels n.val) (s.view .B)) :
ConditionalLabels.lawPairingRate p s.toNodeSchedule.labels depth s.pairs .xy +
ConditionalLabels.lawPairingRate p s.toNodeSchedule.labels depth s.pairs .xz =
finiteEntropy (p.map (s.view .B)).mass := by
apply incident_rate_eq_coordinate_entropy p s.toNodeSchedule.labels depth s.pairs
(s.view .B) (s.completeRecord_injective complete) ?_ independent
intro n hn
have hs : firstSide (s.readerPair n.val) = .B := by
cases hr : s.readerPair n.val <;> simp_all [pairs, firstSide]
let read : LabelRecord s.toNodeSchedule.Label n.val → Coord .B →
s.toNodeSchedule.Label n.val := fun r x =>
s.firstReader n.val r (hs.symm ▸ x)
refine ⟨fun rx => read rx.1 rx.2, ?_⟩
intro a
have transport (c d : Color) (h : c = d) :
h.symm ▸ s.view d a = s.view c a := by
cases h
rfl
have hv : hs.symm ▸ s.view .B a = s.view (firstSide (s.readerPair n.val)) a :=
transport _ _ hs
change s.firstReader n.val (labelRecordOf s.toNodeSchedule.labels n.val a)
(hs.symm ▸ s.view .B a) = s.toNodeSchedule.labels n.val a
rw [hv]
exact s.firstReader_readable n.val n.isLt a
end MatrixMultiplication.CompletionLabels.TopologicalFlatten.TopologicalSchedule
namespace MatrixMultiplication.CompletionLabels.TopologicalFlatten.Program
open MatrixMultiplication.Foundation RecursiveCompletion DualInformation
universe u v w z
variable {A : Type u} [Fintype A] {Coord : Color → Type v}
[Fintype (Coord .B)] {Context : Type z} {Code : Type w} [Fintype Code]
{depth : ℕ}
theorem incident_rate_eq_x_entropy (s : Program A Coord Context Code depth)
(p : FiniteLaw A) (complete : s.Complete) (ctx : Context)
(context_constant : ∀ a, s.context a = ctx)
(independent : ∀ n : Fin depth, s.pair n.val = .yz →
ConditionalIndependent p (labelRecordOf s.labels n.val)
(s.labels n.val) (s.view .B)) :
ConditionalLabels.lawPairingRate p s.labels depth (fun n => s.pair n.val) .xy +
ConditionalLabels.lawPairingRate p s.labels depth (fun n => s.pair n.val) .xz =
finiteEntropy (p.map (s.view .B)).mass := by
have hinj : Function.Injective (labelRecordOf s.labels depth) := by
intro a b h
apply complete a b ((context_constant a).trans (context_constant b).symm)
intro n hn
exact label_eq_of_record_eq s.labels h hn
apply incident_rate_eq_coordinate_entropy p s.labels depth (fun n => s.pair n.val)
(s.view .B) hinj ?_ independent
intro n hn
have hs : .B = firstSide (s.pair n.val) := by
cases hr : s.pair n.val <;> simp_all [firstSide]
refine ⟨fun rx => s.read n.val .B ctx (recordArray n.val rx.1) rx.2, ?_⟩
intro a
change s.read n.val .B ctx (recordArray n.val (labelRecordOf s.labels n.val a))
(s.view .B a) = s.labels n.val a
rw [recordArray_labelRecordOf]
rw [← context_constant a]
exact s.readable n.val n.isLt a .B (Or.inl hs)
end MatrixMultiplication.CompletionLabels.TopologicalFlatten.Program
end
end OAI
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