openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /InformationSchedule.lean
| 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 | |