openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Polynomial /ComplexConstantPolynomialKernel.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialLocalConstruction | |
| namespace OAI | |
| /-! Polynomial tensor restrictions and exact coefficient extraction. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.Foundation.PolynomialLocalConstruction | |
| open Tensor LocalMaps | |
| variable {X Y Z AX AY AZ OX OY OZ : Type*} | |
| variable [Fintype AX] [Fintype AY] [Fintype AZ] | |
| def constantMatrix (M : X → OX → AX → ℂ) : X → OX → AX → Polynomial ℂ := | |
| fun x u i => Polynomial.C (M x u i) | |
| omit [Fintype AX] in | |
| theorem constantMatrix_degree (M : X → OX → AX → ℂ) (x : X) (u : OX) (i : AX) : | |
| (constantMatrix M x u i).degree ≤ (0 : ℕ) := by | |
| simpa only [constantMatrix, Nat.cast_zero] using | |
| (Polynomial.degree_C_le (a := M x u i)) | |
| theorem fiberTransform_constantMatrix | |
| (MX : X → OX → AX → ℂ) (MY : Y → OY → AY → ℂ) | |
| (MZ : Z → OZ → AZ → ℂ) | |
| (T : Tensor ℂ (X × AX) (Y × AY) (Z × AZ)) : | |
| fiberTransform (constantMatrix MX) (constantMatrix MY) (constantMatrix MZ) | |
| (fun x y z => Polynomial.C (T x y z)) = | |
| fun x y z => Polynomial.C (fiberTransform MX MY MZ T x y z) := by | |
| funext x y z | |
| simp only [fiberTransform, constantMatrix, map_sum, map_mul] | |
| theorem kernel_constantMatrix (auxiliary : Tensor ℂ AX AY AZ) | |
| (MX : X → OX → AX → ℂ) (MY : Y → OY → AY → ℂ) | |
| (MZ : Z → OZ → AZ → ℂ) : | |
| kernel auxiliary (constantMatrix MX) (constantMatrix MY) (constantMatrix MZ) = | |
| fun x y z => Polynomial.C | |
| (fiberTransform MX MY MZ (fun x y z => auxiliary x.2 y.2 z.2) x y z) := | |
| fiberTransform_constantMatrix MX MY MZ _ | |
| theorem kernel_constantMatrix_degree (auxiliary : Tensor ℂ AX AY AZ) | |
| (MX : X → OX → AX → ℂ) (MY : Y → OY → AY → ℂ) | |
| (MZ : Z → OZ → AZ → ℂ) (x : X × OX) (y : Y × OY) (z : Z × OZ) : | |
| (kernel auxiliary (constantMatrix MX) (constantMatrix MY) (constantMatrix MZ) | |
| x y z).degree ≤ (0 : ℕ) := by | |
| rw [kernel_constantMatrix] | |
| exact Polynomial.degree_C_le | |
| @[simp] theorem kernel_constantMatrix_coeff_zero (auxiliary : Tensor ℂ AX AY AZ) | |
| (MX : X → OX → AX → ℂ) (MY : Y → OY → AY → ℂ) | |
| (MZ : Z → OZ → AZ → ℂ) (x : X × OX) (y : Y × OY) (z : Z × OZ) : | |
| (kernel auxiliary (constantMatrix MX) (constantMatrix MY) (constantMatrix MZ) | |
| x y z).coeff 0 = | |
| fiberTransform MX MY MZ (fun x y z => auxiliary x.2 y.2 z.2) x y z := by | |
| rw [kernel_constantMatrix] | |
| exact Polynomial.coeff_C_zero | |
| theorem kernel_constantMatrix_coeff_ne_zero (auxiliary : Tensor ℂ AX AY AZ) | |
| (MX : X → OX → AX → ℂ) (MY : Y → OY → AY → ℂ) | |
| (MZ : Z → OZ → AZ → ℂ) (x : X × OX) (y : Y × OY) (z : Z × OZ) | |
| {j : ℕ} (hj : j ≠ 0) : | |
| (kernel auxiliary (constantMatrix MX) (constantMatrix MY) (constantMatrix MZ) | |
| x y z).coeff j = 0 := by | |
| rw [kernel_constantMatrix] | |
| exact Polynomial.coeff_C_of_ne_zero hj | |
| theorem constantMatrix_product_substitution | |
| (original : Tensor ℂ X Y Z) (auxiliary : Tensor ℂ AX AY AZ) | |
| (MX : X → OX → AX → ℂ) (MY : Y → OY → AY → ℂ) | |
| (MZ : Z → OZ → AZ → ℂ) : | |
| fiberTransform (constantMatrix MX) (constantMatrix MY) (constantMatrix MZ) | |
| (fun x y z => Polynomial.C (Tensor.product original auxiliary x y z)) = | |
| originalScale (fun x y z => Polynomial.C (original x y z)) | |
| (kernel auxiliary (constantMatrix MX) (constantMatrix MY) (constantMatrix MZ)) := by | |
| have h := fiberTransform_originalScale | |
| (constantMatrix MX) (constantMatrix MY) (constantMatrix MZ) | |
| (fun x y z => Polynomial.C (original x y z)) | |
| (fun x y z => Polynomial.C (auxiliary x.2 y.2 z.2)) | |
| have hinput : | |
| (fun x y z => Polynomial.C (Tensor.product original auxiliary x y z)) = | |
| originalScale (fun x y z => Polynomial.C (original x y z)) | |
| (fun x y z => Polynomial.C (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 | |
| section CoordinateSelectors | |
| variable [DecidableEq AX] [DecidableEq AY] [DecidableEq AZ] | |
| def weightedCoordinateMatrix (select : X × OX → AX) (weight : X × OX → ℂ) : | |
| X → OX → AX → ℂ := | |
| fun x u i => if i = select (x, u) then weight (x, u) else 0 | |
| def coordinateMatrix (select : X × OX → AX) : X → OX → AX → ℂ := | |
| weightedCoordinateMatrix select (fun _ => 1) | |
| def weightedCoordinatePolynomialMatrix | |
| (select : X × OX → AX) (weight : X × OX → ℂ) : | |
| X → OX → AX → Polynomial ℂ := | |
| constantMatrix (weightedCoordinateMatrix select weight) | |
| def coordinatePolynomialMatrix (select : X × OX → AX) : | |
| X → OX → AX → Polynomial ℂ := | |
| constantMatrix (coordinateMatrix select) | |
| omit [Fintype AX] in | |
| theorem weightedCoordinatePolynomialMatrix_degree | |
| (select : X × OX → AX) (weight : X × OX → ℂ) (x : X) (u : OX) (i : AX) : | |
| (weightedCoordinatePolynomialMatrix select weight x u i).degree ≤ (0 : ℕ) := | |
| constantMatrix_degree _ x u i | |
| omit [Fintype AX] in | |
| theorem coordinatePolynomialMatrix_degree | |
| (select : X × OX → AX) (x : X) (u : OX) (i : AX) : | |
| (coordinatePolynomialMatrix select x u i).degree ≤ (0 : ℕ) := | |
| constantMatrix_degree _ x u i | |
| theorem fiberTransform_weightedCoordinateMatrix | |
| (fx : X × OX → AX) (fy : Y × OY → AY) (fz : Z × OZ → AZ) | |
| (wx : X × OX → ℂ) (wy : Y × OY → ℂ) (wz : Z × OZ → ℂ) | |
| (T : Tensor ℂ (X × AX) (Y × AY) (Z × AZ)) : | |
| fiberTransform (weightedCoordinateMatrix fx wx) (weightedCoordinateMatrix fy wy) | |
| (weightedCoordinateMatrix fz wz) T = | |
| fun x y z => wx x * wy y * wz z * T (x.1, fx x) (y.1, fy y) (z.1, fz z) := by | |
| funext x y z | |
| simp [fiberTransform, weightedCoordinateMatrix, ite_mul, mul_ite] | |
| theorem fiberTransform_coordinateMatrix | |
| (fx : X × OX → AX) (fy : Y × OY → AY) (fz : Z × OZ → AZ) | |
| (T : Tensor ℂ (X × AX) (Y × AY) (Z × AZ)) : | |
| fiberTransform (coordinateMatrix fx) (coordinateMatrix fy) (coordinateMatrix fz) T = | |
| Tensor.pullback (fun x => (x.1, fx x)) (fun y => (y.1, fy y)) | |
| (fun z => (z.1, fz z)) T := by | |
| funext x y z | |
| have h := fiberTransform_weightedCoordinateMatrix fx fy fz | |
| (fun _ => 1) (fun _ => 1) (fun _ => 1) T | |
| simpa only [coordinateMatrix, one_mul, Tensor.pullback] using | |
| congrFun (congrFun (congrFun h x) y) z | |
| theorem kernel_weightedCoordinatePolynomialMatrix (auxiliary : Tensor ℂ AX AY AZ) | |
| (fx : X × OX → AX) (fy : Y × OY → AY) (fz : Z × OZ → AZ) | |
| (wx : X × OX → ℂ) (wy : Y × OY → ℂ) (wz : Z × OZ → ℂ) : | |
| kernel auxiliary (weightedCoordinatePolynomialMatrix fx wx) | |
| (weightedCoordinatePolynomialMatrix fy wy) (weightedCoordinatePolynomialMatrix fz wz) = | |
| fun x y z => Polynomial.C | |
| (wx x * wy y * wz z * auxiliary (fx x) (fy y) (fz z)) := by | |
| unfold weightedCoordinatePolynomialMatrix | |
| rw [kernel_constantMatrix, fiberTransform_weightedCoordinateMatrix] | |
| theorem kernel_coordinatePolynomialMatrix (auxiliary : Tensor ℂ AX AY AZ) | |
| (fx : X × OX → AX) (fy : Y × OY → AY) (fz : Z × OZ → AZ) : | |
| kernel auxiliary (coordinatePolynomialMatrix fx) | |
| (coordinatePolynomialMatrix fy) (coordinatePolynomialMatrix fz) = | |
| fun x y z => Polynomial.C (auxiliary (fx x) (fy y) (fz z)) := by | |
| simpa only [coordinatePolynomialMatrix, coordinateMatrix, | |
| weightedCoordinatePolynomialMatrix, one_mul] using | |
| kernel_weightedCoordinatePolynomialMatrix auxiliary fx fy fz | |
| (fun _ => 1) (fun _ => 1) (fun _ => 1) | |
| @[simp] theorem kernel_weightedCoordinatePolynomialMatrix_coeff_zero | |
| (auxiliary : Tensor ℂ AX AY AZ) | |
| (fx : X × OX → AX) (fy : Y × OY → AY) (fz : Z × OZ → AZ) | |
| (wx : X × OX → ℂ) (wy : Y × OY → ℂ) (wz : Z × OZ → ℂ) | |
| (x : X × OX) (y : Y × OY) (z : Z × OZ) : | |
| (kernel auxiliary (weightedCoordinatePolynomialMatrix fx wx) | |
| (weightedCoordinatePolynomialMatrix fy wy) (weightedCoordinatePolynomialMatrix fz wz) | |
| x y z).coeff 0 = wx x * wy y * wz z * auxiliary (fx x) (fy y) (fz z) := by | |
| rw [kernel_weightedCoordinatePolynomialMatrix] | |
| exact Polynomial.coeff_C_zero | |
| @[simp] theorem kernel_coordinatePolynomialMatrix_coeff_zero | |
| (auxiliary : Tensor ℂ AX AY AZ) | |
| (fx : X × OX → AX) (fy : Y × OY → AY) (fz : Z × OZ → AZ) | |
| (x : X × OX) (y : Y × OY) (z : Z × OZ) : | |
| (kernel auxiliary (coordinatePolynomialMatrix fx) | |
| (coordinatePolynomialMatrix fy) (coordinatePolynomialMatrix fz) x y z).coeff 0 = | |
| auxiliary (fx x) (fy y) (fz z) := by | |
| rw [kernel_coordinatePolynomialMatrix] | |
| exact Polynomial.coeff_C_zero | |
| theorem kernel_coordinatePolynomialMatrix_coeff_ne_zero | |
| (auxiliary : Tensor ℂ AX AY AZ) | |
| (fx : X × OX → AX) (fy : Y × OY → AY) (fz : Z × OZ → AZ) | |
| (x : X × OX) (y : Y × OY) (z : Z × OZ) {j : ℕ} (hj : j ≠ 0) : | |
| (kernel auxiliary (coordinatePolynomialMatrix fx) | |
| (coordinatePolynomialMatrix fy) (coordinatePolynomialMatrix fz) x y z).coeff j = 0 := by | |
| rw [kernel_coordinatePolynomialMatrix] | |
| exact Polynomial.coeff_C_of_ne_zero hj | |
| end CoordinateSelectors | |
| section ExactDegeneration | |
| variable [Fintype X] [Fintype Y] [Fintype Z] | |
| def constantFiberDegeneration (original : Tensor ℂ X Y Z) | |
| (auxiliary : Tensor ℂ AX AY AZ) | |
| (MX : X → OX → AX → ℂ) (MY : Y → OY → AY → ℂ) | |
| (MZ : Z → OZ → AZ → ℂ) : | |
| PolynomialRestrictionDegeneration (Tensor.product original auxiliary) | |
| (fiberTransform MX MY MZ (Tensor.product original auxiliary)) 0 0 0 0 := by | |
| classical | |
| refine { | |
| leftMap := fiberMatrix (constantMatrix MX) | |
| middleMap := fiberMatrix (constantMatrix MY) | |
| rightMap := fiberMatrix (constantMatrix MZ) | |
| 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 constantMatrix_degree MX 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 constantMatrix_degree MY 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 constantMatrix_degree MZ 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 | |
| exact (Nat.not_lt_zero j hj).elim | |
| · intro x y z | |
| rw [← fiberTransform_eq_restrict, fiberTransform_constantMatrix] | |
| exact Polynomial.coeff_C_zero | |
| end ExactDegeneration | |
| end MatrixMultiplication.Foundation.PolynomialLocalConstruction | |
| end | |
| end OAI | |