openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /Terminal.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.FiniteFamily | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.TerminalRelabel | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldTerminal | |
| open MatrixMultiplication.Foundation AllFieldFiniteFamily CommonDimensions | |
| open scoped BigOperators Classical | |
| variable {F : Type*} {H : Type} [Field F] [Fintype H] | |
| {X Y Z : H → Type} [∀ h, Fintype (X h)] [∀ h, Fintype (Y h)] [∀ h, Fintype (Z h)] | |
| (T : ∀ h, Tensor F (X h) (Y h) (Z h)) (A B C : H → Type) | |
| [∀ h, Fintype (A h)] [∀ h, Fintype (B h)] [∀ h, Fintype (C h)] | |
| (maps : ∀ h, LocalMap (T h) (Tensor.matrixCoefficients (A h) (B h) (C h))) | |
| def terminalProductMap : | |
| LocalMap (familyProduct T) (Tensor.matrixCoefficients (∀ h, A h) (∀ h, B h) (∀ h, C h)) := | |
| LocalMap.ofExists (TerminalProducts.matrix_restriction_of_history_restrictions | |
| T A B C (fun h => (maps h).x) (fun h => (maps h).y) (fun h => (maps h).z) | |
| (fun h => (maps h).coefficient)) | |
| variable {I : Type} [Fintype I] {K N : ℕ} | |
| def terminalExecution | |
| (E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T))) : | |
| Execution (cwSource F K N) | |
| (AllFieldSource.matrixTarget F (Fintype.card (∀ h, A h)) | |
| (Fintype.card (∀ h, B h)) (Fintype.card (∀ h, C h)) (Fintype.card I)) := | |
| (E.restrict _ ((terminalProductMap T A B C maps).parallelCopies I)).restrict _ | |
| (LocalMap.ofExists (TerminalRelabel.directSum_matrix_restrict_fin (K := F) | |
| I (∀ h, A h) (∀ h, B h) (∀ h, C h))) | |
| @[simp] theorem terminalExecution_copies | |
| (E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T))) : | |
| Fintype.card (terminalExecution T A B C maps E).Copies = Fintype.card E.Copies := rfl | |
| def finish | |
| (E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T))) | |
| (hA : 0 < Fintype.card (∀ h, A h)) (hB : 0 < Fintype.card (∀ h, B h)) | |
| (hC : 0 < Fintype.card (∀ h, C h)) (hI : 0 < Fintype.card I) : AllFieldWitness F := | |
| (terminalExecution T A B C maps E).toWitness hA hB hC hI | |
| @[simp] theorem finish_multiplicity | |
| (E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T))) | |
| (hA : 0 < Fintype.card (∀ h, A h)) (hB : 0 < Fintype.card (∀ h, B h)) | |
| (hC : 0 < Fintype.card (∀ h, C h)) (hI : 0 < Fintype.card I) : | |
| (finish T A B C maps E hA hB hC hI).multiplicity = Fintype.card I := rfl | |
| theorem finish_volume | |
| (E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T))) | |
| (hA : 0 < Fintype.card (∀ h, A h)) (hB : 0 < Fintype.card (∀ h, B h)) | |
| (hC : 0 < Fintype.card (∀ h, C h)) (hI : 0 < Fintype.card I) : | |
| (finish T A B C maps E hA hB hC hI).volume = | |
| ∏ h, Fintype.card (A h) * Fintype.card (B h) * Fintype.card (C h) := by | |
| change Fintype.card (∀ h, A h) * Fintype.card (∀ h, B h) * | |
| Fintype.card (∀ h, C h) = _ | |
| simp only [Fintype.card_pi, Finset.prod_mul_distrib] | |
| @[simp] theorem finish_rankBound | |
| (E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T))) | |
| (hA : 0 < Fintype.card (∀ h, A h)) (hB : 0 < Fintype.card (∀ h, B h)) | |
| (hC : 0 < Fintype.card (∀ h, C h)) (hI : 0 < Fintype.card I) : | |
| (finish T A B C maps E hA hB hC hI).rankBound = | |
| AllFieldSource.rankBudget E.Copies K N := rfl | |
| end MatrixMultiplication.AllFieldTerminal | |
| end | |
| end OAI | |