openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Separation /ExecutionApproximation.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialKernelExecution | |
| namespace OAI | |
| /-! Finite type counts, hierarchy separation and tensor execution bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.ExecutionApproximation | |
| open MatrixMultiplication.Foundation | |
| open Tensor LocalMaps PolynomialLocalConstruction PolynomialKernelExecution | |
| variable {X Y Z Prefix PX PY PZ AX AY AZ : Type*} | |
| variable [Fintype AX] [Fintype AY] [Fintype AZ] | |
| variable {support : X → Y → Z → Prop} | |
| variable (E : Execution X Y Z Prefix PX PY PZ AX AY AZ support) | |
| theorem polynomial_substitution (original : Tensor ℂ X Y Z) : | |
| fiberTransform E.leftMap E.middleMap E.rightMap | |
| (fun x y z => Polynomial.C (Tensor.product original E.auxiliary x y z)) = | |
| originalScale (fun x y z => Polynomial.C (original x y z)) | |
| (kernel E.auxiliary E.leftMap E.middleMap E.rightMap) := by | |
| have h := fiberTransform_originalScale E.leftMap E.middleMap E.rightMap | |
| (fun x y z => Polynomial.C (original x y z)) | |
| (fun x y z => Polynomial.C (E.auxiliary x.2 y.2 z.2)) | |
| have hinput : | |
| (fun x y z => Polynomial.C (Tensor.product original E.auxiliary x y z)) = | |
| originalScale (fun x y z => Polynomial.C (original x y z)) | |
| (fun x y z => Polynomial.C (E.auxiliary x.2 y.2 z.2)) := by | |
| funext x y z | |
| simp only [Tensor.product, originalScale, map_mul] | |
| rw [hinput] | |
| simpa only [kernel] using h | |
| variable [Fintype X] [Fintype Y] [Fintype Z] | |
| def degeneration (original : Tensor ℂ X Y Z) | |
| (supported : ∀ x y z, ¬ support x y z → original x y z = 0) : | |
| PolynomialRestrictionDegeneration (Tensor.product original E.auxiliary) | |
| (originalScale original E.value) E.order E.leftDegree E.middleDegree E.rightDegree := by | |
| classical | |
| refine { | |
| leftMap := fiberMatrix E.leftMap | |
| middleMap := fiberMatrix E.middleMap | |
| rightMap := fiberMatrix E.rightMap | |
| left_degree := ?_ | |
| middle_degree := ?_ | |
| right_degree := ?_ | |
| vanishes := ?_ | |
| leading := ?_ | |
| } | |
| · intro x i | |
| by_cases h : x.1 = i.1 | |
| · simpa only [fiberMatrix, ite_eq_left h] using E.left_degree x.1 x.2 i.2 | |
| · simp only [fiberMatrix, ite_eq_right h, Polynomial.degree_zero] | |
| exact bot_le | |
| · intro y j | |
| by_cases h : y.1 = j.1 | |
| · simpa only [fiberMatrix, ite_eq_left h] using E.middle_degree y.1 y.2 j.2 | |
| · simp only [fiberMatrix, ite_eq_right h, Polynomial.degree_zero] | |
| exact bot_le | |
| · intro z k | |
| by_cases h : z.1 = k.1 | |
| · simpa only [fiberMatrix, ite_eq_left h] using E.right_degree z.1 z.2 k.2 | |
| · simp only [fiberMatrix, ite_eq_right h, Polynomial.degree_zero] | |
| exact bot_le | |
| · intro x y z j hj | |
| rw [← fiberTransform_eq_restrict, polynomial_substitution E original] | |
| simp only [originalScale, Polynomial.coeff_C_mul] | |
| by_cases hs : support x.1 y.1 z.1 | |
| · rw [E.vanishes x y z hs j hj, mul_zero] | |
| · rw [supported x.1 y.1 z.1 hs, zero_mul] | |
| · intro x y z | |
| rw [← fiberTransform_eq_restrict, polynomial_substitution E original] | |
| simp only [originalScale, Polynomial.coeff_C_mul] | |
| by_cases hs : support x.1 y.1 z.1 | |
| · rw [E.leading x y z hs] | |
| · rw [supported x.1 y.1 z.1 hs] | |
| simp only [zero_mul] | |
| def approximation {original : Tensor ℂ X Y Z} {r d D : ℕ} | |
| (source : PolynomialApproximation original r d D) | |
| (supported : ∀ x y z, ¬ support x y z → original x y z = 0) : | |
| PolynomialApproximation (originalScale original E.value) | |
| (r * E.rankBound) (d * (E.order + 1) + E.order) | |
| (D * (E.order + 1) + E.leftDegree + E.middleDegree + E.rightDegree) := by | |
| have combined := source.productExact E.auxiliary E.rankBound E.auxiliary_rank | |
| exact (degeneration E original supported).compose combined | |
| end MatrixMultiplication.ExecutionApproximation | |
| end | |
| end OAI | |