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import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexTensorRestrictionComposition
namespace OAI
/-! Polynomial tensor restrictions and exact coefficient extraction. -/
noncomputable section
namespace MatrixMultiplication.Foundation
namespace PolynomialControlledKernel
open Tensor LocalMaps PolynomialLocalConstruction
open scoped BigOperators
section ScalarCoefficients
variable {F : Type*} [Field F]
theorem expanded_mul_coeff_window (P Q : Polynomial F) (d k : ℕ) (a : F)
(hP : ∀ j, j < d → P.coeff j = 0) (hPa : P.coeff d = a)
(j : ℕ) (hj : j ≤ d * (k + 1) + k) :
(Polynomial.expand F (k + 1) P * Q).coeff j =
if d * (k + 1) ≤ j then a * Q.coeff (j - d * (k + 1)) else 0 := by
obtain ⟨P₀, hP₀⟩ := Polynomial.X_pow_dvd_iff.mpr hP
have hconstant : P₀.coeff 0 = a := by
rw [hP₀] at hPa
simpa only [Polynomial.coeff_X_pow_mul', le_refl, ite_true, Nat.sub_self] using hPa
have hfactor : Polynomial.expand F (k + 1) P * Q =
Polynomial.X ^ (d * (k + 1)) *
(Q * Polynomial.expand F (k + 1) P₀) := by
rw [hP₀, map_mul, map_pow, Polynomial.expand_X]
simp only [← pow_mul]
rw [Nat.mul_comm (k + 1) d]
ring
rw [hfactor, Polynomial.coeff_X_pow_mul']
split_ifs with hdj
· have hsmall : j - d * (k + 1) ≤ k := by omega
rw [Tensor.coeff_mul_expand_eq_constant Q P₀ a hconstant k _ hsmall,
Polynomial.coeff_mul_C, mul_comm]
· rfl
theorem expanded_mul_vanishes_of_leading_support (P Q : Polynomial F)
(d k : ℕ) (a : F)
(hP : ∀ j, j < d → P.coeff j = 0) (hPa : P.coeff d = a)
(hQ : a ≠ 0 → ∀ j, j < k → Q.coeff j = 0)
(j : ℕ) (hj : j < d * (k + 1) + k) :
(Polynomial.expand F (k + 1) P * Q).coeff j = 0 := by
rw [expanded_mul_coeff_window P Q d k a hP hPa j (Nat.le_of_lt hj)]
split_ifs with hdj
· by_cases ha : a = 0
· simp [ha]
· rw [hQ ha _ (by omega), mul_zero]
· rfl
theorem expanded_mul_leading_of_leading_support (P Q : Polynomial F)
(d k : ℕ) (a b : F)
(hP : ∀ j, j < d → P.coeff j = 0) (hPa : P.coeff d = a)
(hQb : a ≠ 0 → Q.coeff k = b) :
(Polynomial.expand F (k + 1) P * Q).coeff (d * (k + 1) + k) = a * b := by
rw [expanded_mul_coeff_window P Q d k a hP hPa _ le_rfl]
simp only [Nat.le_add_right, ite_true, Nat.add_sub_cancel_left]
by_cases ha : a = 0
· simp [ha]
· rw [hQb ha]
end ScalarCoefficients
abbrev Output (Prefix New OldCoord NewCoord : Type*) :=
(Prefix × OldCoord) × (New × NewCoord)
section ActualMatrices
variable {X Y Z Prefix New PX PY PZ NX NY NZ AP BP CP AS BS CS : Type*}
variable [Fintype AP] [Fintype BP] [Fintype CP]
variable [Fintype AS] [Fintype BS] [Fintype CS]
def expandMap {O A : Type*} (factor : ℕ) (M : X → O → A → Polynomial ℂ) :
X → O → A → Polynomial ℂ :=
fun x output input => Polynomial.expand ℂ factor (M x output input)
def controlledMap (kStep : ℕ)
(previous : X → (Prefix × PX) → AP → Polynomial ℂ)
(step : Prefix → X → (New × NX) → AS → Polynomial ℂ) :
X → Output Prefix New PX NX → (AP × AS) → Polynomial ℂ :=
fun x output input =>
Polynomial.expand ℂ (kStep + 1) (previous x output.1 input.1) *
step output.1.1 x output.2 input.2
omit [Fintype AP] [Fintype AS] in
theorem controlledMap_degree (kStep dPrevious dStep : ℕ)
(previous : X → (Prefix × PX) → AP → Polynomial ℂ)
(step : Prefix → X → (New × NX) → AS → Polynomial ℂ)
(hp : ∀ x output input, (previous x output input).degree ≤ dPrevious)
(hs : ∀ prior x output input, (step prior x output input).degree ≤ dStep)
(x : X) (output : Output Prefix New PX NX) (input : AP × AS) :
(controlledMap kStep previous step x output input).degree ≤
dPrevious * (kStep + 1) + dStep := by
apply Polynomial.degree_le_of_natDegree_le
apply Polynomial.natDegree_mul_le_of_le
· rw [Polynomial.natDegree_expand]
exact Nat.mul_le_mul_right (kStep + 1)
(Polynomial.natDegree_le_of_degree_le (hp x output.1 input.1))
· exact Polynomial.natDegree_le_of_degree_le
(hs output.1.1 x output.2 input.2)
theorem kernel_expandMap (factor : ℕ) (auxiliary : Tensor ℂ AP BP CP)
(MX : X → (Prefix × PX) → AP → Polynomial ℂ)
(MY : Y → (Prefix × PY) → BP → Polynomial ℂ)
(MZ : Z → (Prefix × PZ) → CP → Polynomial ℂ)
(x : X × (Prefix × PX)) (y : Y × (Prefix × PY)) (z : Z × (Prefix × PZ)) :
kernel auxiliary (expandMap factor MX) (expandMap factor MY) (expandMap factor MZ)
x y z = Polynomial.expand ℂ factor (kernel auxiliary MX MY MZ x y z) := by
simp only [kernel, fiberTransform, expandMap, map_sum, map_mul, Polynomial.expand_C]
def mixedStageKernel (auxiliary : Tensor ℂ AS BS CS)
(SX : Prefix → X → (New × NX) → AS → Polynomial ℂ)
(SY : Prefix → Y → (New × NY) → BS → Polynomial ℂ)
(SZ : Prefix → Z → (New × NZ) → CS → Polynomial ℂ) :
Tensor (Polynomial ℂ) (X × Output Prefix New PX NX)
(Y × Output Prefix New PY NY) (Z × Output Prefix New PZ NZ) :=
fun x y z => kernel auxiliary (SX x.2.1.1) (SY y.2.1.1) (SZ z.2.1.1)
(x.1, x.2.2) (y.1, y.2.2) (z.1, z.2.2)
def controlledKernel (kStep : ℕ)
(previousAuxiliary : Tensor ℂ AP BP CP) (stepAuxiliary : Tensor ℂ AS BS CS)
(MX : X → (Prefix × PX) → AP → Polynomial ℂ)
(MY : Y → (Prefix × PY) → BP → Polynomial ℂ)
(MZ : Z → (Prefix × PZ) → CP → Polynomial ℂ)
(SX : Prefix → X → (New × NX) → AS → Polynomial ℂ)
(SY : Prefix → Y → (New × NY) → BS → Polynomial ℂ)
(SZ : Prefix → Z → (New × NZ) → CS → Polynomial ℂ) :
Tensor (Polynomial ℂ) (X × Output Prefix New PX NX)
(Y × Output Prefix New PY NY) (Z × Output Prefix New PZ NZ) :=
kernel (Tensor.product previousAuxiliary stepAuxiliary)
(controlledMap kStep MX SX) (controlledMap kStep MY SY) (controlledMap kStep MZ SZ)
theorem controlledKernel_factor (kStep : ℕ)
(previousAuxiliary : Tensor ℂ AP BP CP) (stepAuxiliary : Tensor ℂ AS BS CS)
(MX : X → (Prefix × PX) → AP → Polynomial ℂ)
(MY : Y → (Prefix × PY) → BP → Polynomial ℂ)
(MZ : Z → (Prefix × PZ) → CP → Polynomial ℂ)
(SX : Prefix → X → (New × NX) → AS → Polynomial ℂ)
(SY : Prefix → Y → (New × NY) → BS → Polynomial ℂ)
(SZ : Prefix → Z → (New × NZ) → CS → Polynomial ℂ)
(x : X × Output Prefix New PX NX) (y : Y × Output Prefix New PY NY)
(z : Z × Output Prefix New PZ NZ) :
controlledKernel kStep previousAuxiliary stepAuxiliary MX MY MZ SX SY SZ x y z =
Polynomial.expand ℂ (kStep + 1)
(kernel previousAuxiliary MX MY MZ (x.1, x.2.1) (y.1, y.2.1) (z.1, z.2.1)) *
mixedStageKernel stepAuxiliary SX SY SZ x y z := by
have h := congrFun (congrFun (congrFun
(Tensor.restrict_product (expandMap (kStep + 1) MX x.1)
(expandMap (kStep + 1) MY y.1) (expandMap (kStep + 1) MZ z.1)
(SX x.2.1.1 x.1) (SY y.2.1.1 y.1) (SZ z.2.1.1 z.1)
(fun a b c => Polynomial.C (previousAuxiliary a b c))
(fun a b c => Polynomial.C (stepAuxiliary a b c))) x.2) y.2) z.2
have hproduct :
controlledKernel kStep previousAuxiliary stepAuxiliary MX MY MZ SX SY SZ x y z =
kernel previousAuxiliary (expandMap (kStep + 1) MX)
(expandMap (kStep + 1) MY) (expandMap (kStep + 1) MZ)
(x.1, x.2.1) (y.1, y.2.1) (z.1, z.2.1) *
mixedStageKernel stepAuxiliary SX SY SZ x y z := by
simpa only [controlledKernel, kernel, fiberTransform, Tensor.restrict, Tensor.product,
controlledMap, expandMap, mixedStageKernel, map_mul] using h
rw [hproduct, kernel_expandMap]
theorem controlledKernel_substitution (kStep : ℕ)
(previousAuxiliary : Tensor ℂ AP BP CP) (stepAuxiliary : Tensor ℂ AS BS CS)
(MX : X → (Prefix × PX) → AP → Polynomial ℂ)
(MY : Y → (Prefix × PY) → BP → Polynomial ℂ)
(MZ : Z → (Prefix × PZ) → CP → Polynomial ℂ)
(SX : Prefix → X → (New × NX) → AS → Polynomial ℂ)
(SY : Prefix → Y → (New × NY) → BS → Polynomial ℂ)
(SZ : Prefix → Z → (New × NZ) → CS → Polynomial ℂ)
(original : Tensor ℂ X Y Z) :
fiberTransform (controlledMap kStep MX SX) (controlledMap kStep MY SY)
(controlledMap kStep MZ SZ)
(fun x y z => Polynomial.C
(Tensor.product original (Tensor.product previousAuxiliary stepAuxiliary) x y z)) =
originalScale (fun x y z => Polynomial.C (original x y z))
(controlledKernel kStep previousAuxiliary stepAuxiliary MX MY MZ SX SY SZ) := by
have h := fiberTransform_originalScale (controlledMap kStep MX SX)
(controlledMap kStep MY SY) (controlledMap kStep MZ SZ)
(fun x y z => Polynomial.C (original x y z))
(fun x y z => Polynomial.C (Tensor.product previousAuxiliary stepAuxiliary x.2 y.2 z.2))
have hinput :
(fun x y z => Polynomial.C
(Tensor.product original (Tensor.product previousAuxiliary stepAuxiliary) x y z)) =
originalScale (fun x y z => Polynomial.C (original x y z))
(fun x y z => Polynomial.C
(Tensor.product previousAuxiliary stepAuxiliary x.2 y.2 z.2)) := by
funext x y z
simp only [Tensor.product, originalScale, map_mul]
rw [hinput]
simpa only [controlledKernel, kernel] using h
def controlledLeading
(previous : Tensor ℂ (X × (Prefix × PX)) (Y × (Prefix × PY)) (Z × (Prefix × PZ)))
(step : Prefix → Tensor ℂ (X × (New × NX)) (Y × (New × NY)) (Z × (New × NZ))) :
Tensor ℂ (X × Output Prefix New PX NX) (Y × Output Prefix New PY NY)
(Z × Output Prefix New PZ NZ) :=
fun x y z => previous (x.1, x.2.1) (y.1, y.2.1) (z.1, z.2.1) *
step x.2.1.1 (x.1, x.2.2) (y.1, y.2.2) (z.1, z.2.2)
theorem controlledKernel_coefficients (dPrevious kStep : ℕ)
(previousAuxiliary : Tensor ℂ AP BP CP) (stepAuxiliary : Tensor ℂ AS BS CS)
(MX : X → (Prefix × PX) → AP → Polynomial ℂ)
(MY : Y → (Prefix × PY) → BP → Polynomial ℂ)
(MZ : Z → (Prefix × PZ) → CP → Polynomial ℂ)
(SX : Prefix → X → (New × NX) → AS → Polynomial ℂ)
(SY : Prefix → Y → (New × NY) → BS → Polynomial ℂ)
(SZ : Prefix → Z → (New × NZ) → CS → Polynomial ℂ)
(previous : Tensor ℂ (X × (Prefix × PX)) (Y × (Prefix × PY)) (Z × (Prefix × PZ)))
(step : Prefix → Tensor ℂ (X × (New × NX)) (Y × (New × NY)) (Z × (New × NZ)))
(support : X → Y → Z → Prop) (eligible : Prefix → X → Y → Z → Prop)
(hprevious_low : ∀ x y z, support x.1 y.1 z.1 → ∀ j, j < dPrevious →
(kernel previousAuxiliary MX MY MZ x y z).coeff j = 0)
(hprevious_leading : ∀ x y z, support x.1 y.1 z.1 →
(kernel previousAuxiliary MX MY MZ x y z).coeff dPrevious = previous x y z)
(hprevious_support : ∀ x y z, support x.1 y.1 z.1 → previous x y z ≠ 0 →
x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ eligible x.2.1 x.1 y.1 z.1)
(hstep_low : ∀ prior x y z, eligible prior x.1 y.1 z.1 →
∀ j, j < kStep →
(kernel stepAuxiliary (SX prior) (SY prior) (SZ prior) x y z).coeff j = 0)
(hstep_leading : ∀ prior x y z, eligible prior x.1 y.1 z.1 →
(kernel stepAuxiliary (SX prior) (SY prior) (SZ prior) x y z).coeff kStep =
step prior x y z) :
(∀ x y z, support x.1 y.1 z.1 → ∀ j, j < dPrevious * (kStep + 1) + kStep →
(controlledKernel kStep previousAuxiliary stepAuxiliary MX MY MZ SX SY SZ x y z).coeff j = 0) ∧
(∀ x y z, support x.1 y.1 z.1 →
(controlledKernel kStep previousAuxiliary stepAuxiliary MX MY MZ SX SY SZ x y z).coeff
(dPrevious * (kStep + 1) + kStep) = controlledLeading previous step x y z) := by
have hselected : ∀ (x : X × Output Prefix New PX NX)
(y : Y × Output Prefix New PY NY) (z : Z × Output Prefix New PZ NZ),
support x.1 y.1 z.1 →
previous (x.1, x.2.1) (y.1, y.2.1) (z.1, z.2.1) ≠ 0 →
(∀ j, j < kStep → (mixedStageKernel stepAuxiliary SX SY SZ x y z).coeff j = 0) ∧
(mixedStageKernel stepAuxiliary SX SY SZ x y z).coeff kStep =
step x.2.1.1 (x.1, x.2.2) (y.1, y.2.2) (z.1, z.2.2) := by
intro x y z hs hn
obtain ⟨hxy, hxz, he⟩ :=
hprevious_support (x.1, x.2.1) (y.1, y.2.1) (z.1, z.2.1) hs hn
have hy : y.2.1.1 = x.2.1.1 := hxy.symm
have hz : z.2.1.1 = x.2.1.1 := hxz.symm
constructor
· intro j hj
simpa only [mixedStageKernel, hy, hz] using
hstep_low x.2.1.1 (x.1, x.2.2) (y.1, y.2.2) (z.1, z.2.2) he j hj
· simpa only [mixedStageKernel, hy, hz] using
hstep_leading x.2.1.1 (x.1, x.2.2) (y.1, y.2.2) (z.1, z.2.2) he
constructor
· intro x y z hs j hj
rw [controlledKernel_factor]
exact expanded_mul_vanishes_of_leading_support _ _ dPrevious kStep
(previous (x.1, x.2.1) (y.1, y.2.1) (z.1, z.2.1))
(hprevious_low _ _ _ hs) (hprevious_leading _ _ _ hs)
(fun hn => (hselected x y z hs hn).1) j hj
· intro x y z hs
rw [controlledKernel_factor]
exact expanded_mul_leading_of_leading_support _ _ dPrevious kStep
(previous (x.1, x.2.1) (y.1, y.2.1) (z.1, z.2.1))
(step x.2.1.1 (x.1, x.2.2) (y.1, y.2.2) (z.1, z.2.2))
(hprevious_low _ _ _ hs) (hprevious_leading _ _ _ hs)
(fun hn => (hselected x y z hs hn).2)
theorem controlledKernel_rank [Fintype X] [Fintype Y] [Fintype Z]
(kStep : ℕ) (previousAuxiliary : Tensor ℂ AP BP CP) (stepAuxiliary : Tensor ℂ AS BS CS)
(MX : X → (Prefix × PX) → AP → Polynomial ℂ)
(MY : Y → (Prefix × PY) → BP → Polynomial ℂ)
(MZ : Z → (Prefix × PZ) → CP → Polynomial ℂ)
(SX : Prefix → X → (New × NX) → AS → Polynomial ℂ)
(SY : Prefix → Y → (New × NY) → BS → Polynomial ℂ)
(SZ : Prefix → Z → (New × NZ) → CS → Polynomial ℂ)
{rPrevious rStep : ℕ} (hp : RankAtMost previousAuxiliary rPrevious)
(hs : RankAtMost stepAuxiliary rStep) :
RankAtMost (controlledKernel kStep previousAuxiliary stepAuxiliary MX MY MZ SX SY SZ)
(rPrevious * rStep) := by
classical
have hsource := ((hp.product hs).map Polynomial.C).pullback
(fun x : X × (AP × AS) => x.2) (fun y : Y × (BP × BS) => y.2)
(fun z : Z × (CP × CS) => z.2)
unfold controlledKernel kernel
rw [fiberTransform_eq_restrict]
exact hsource.restrict _ _ _
end ActualMatrices
end PolynomialControlledKernel
end MatrixMultiplication.Foundation
end
end OAI
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