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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
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