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f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexInitialTargetKernel
import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialKernelExecution
namespace OAI
/-! Finite type counts, hierarchy separation and tensor execution bounds. -/
noncomputable section
namespace MatrixMultiplication.Foundation.InitialTargetExecution
open Tensor LocalMaps PolynomialLocalConstruction PolynomialKernelExecution
variable {X Y Z RawLabel Label : Type*}
def scalarOutput : Label × Unit → Label × (Fin 1 × Fin 1) :=
fun output => (output.1, (0, 0))
def initial (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
(lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
(embed : Label → RawLabel) (injective : Function.Injective embed)
(support : X → Y → Z → Prop)
(compatible : ∀ x y z, support x y z → PX x → PY y → PZ z →
lx x = ly y ∧ lx x = lz z) :
Execution X Y Z Label Unit Unit Unit Unit Unit Unit support where
auxiliary := InitialTargetKernel.auxiliary
rankBound := 1
auxiliary_rank := InitialTargetKernel.auxiliary_rank
order := 0
leftDegree := 0
middleDegree := 0
rightDegree := 0
leftMap := fun x output input =>
InitialTargetKernel.sideMap PX lx embed x (scalarOutput output) input
middleMap := fun y output input =>
InitialTargetKernel.sideMap PY ly embed y (scalarOutput output) input
rightMap := fun z output input =>
InitialTargetKernel.sideMap PZ lz embed z (scalarOutput output) input
left_degree := fun x output input =>
InitialTargetKernel.sideMap_degree PX lx embed x (scalarOutput output) input
middle_degree := fun y output input =>
InitialTargetKernel.sideMap_degree PY ly embed y (scalarOutput output) input
right_degree := fun z output input =>
InitialTargetKernel.sideMap_degree PZ lz embed z (scalarOutput output) input
value := fun x y z => InitialTargetKernel.previousLeading PX PY PZ lx ly lz embed
(x.1, scalarOutput x.2) (y.1, scalarOutput y.2) (z.1, scalarOutput z.2)
vanishes := fun x y z _ j hj =>
InitialTargetKernel.kernel_low PX PY PZ lx ly lz embed
(x.1, scalarOutput x.2) (y.1, scalarOutput y.2) (z.1, scalarOutput z.2) j hj
leading := fun x y z hs =>
InitialTargetKernel.kernel_leading PX PY PZ lx ly lz embed injective support compatible
(x.1, scalarOutput x.2) (y.1, scalarOutput y.2) (z.1, scalarOutput z.2) hs
synchronized := by
intro x y z _ hn
have h := (InitialTargetKernel.previousLeading_ne_zero_iff PX PY PZ lx ly lz embed
(x.1, scalarOutput x.2) (y.1, scalarOutput y.2) (z.1, scalarOutput z.2)).mp hn
exact ⟨h.1, h.2.1⟩
variable (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
variable (lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
variable (embed : Label → RawLabel) (injective : Function.Injective embed)
variable (support : X → Y → Z → Prop)
variable (compatible : ∀ x y z, support x y z → PX x → PY y → PZ z →
lx x = ly y ∧ lx x = lz z)
@[simp] theorem initial_rankBound :
(initial PX PY PZ lx ly lz embed injective support compatible).rankBound = 1 := rfl
@[simp] theorem initial_order :
(initial PX PY PZ lx ly lz embed injective support compatible).order = 0 := rfl
theorem initial_value_ne_zero_iff (x : X × (Label × Unit))
(y : Y × (Label × Unit)) (z : Z × (Label × Unit)) :
(initial PX PY PZ lx ly lz embed injective support compatible).value x y z ≠ 0 ↔
x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
InitialTargetKernel.Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1 :=
InitialTargetKernel.previousLeading_ne_zero_iff PX PY PZ lx ly lz embed
(x.1, scalarOutput x.2) (y.1, scalarOutput y.2) (z.1, scalarOutput z.2)
theorem initial_eligible (x : X × (Label × Unit))
(y : Y × (Label × Unit)) (z : Z × (Label × Unit))
(hn : (initial PX PY PZ lx ly lz embed injective support compatible).value x y z ≠ 0) :
InitialTargetKernel.Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1 :=
((initial_value_ne_zero_iff PX PY PZ lx ly lz embed injective support compatible
x y z).mp hn).2.2
theorem initial_substitution (original : Tensor ℂ X Y Z) :
let E := initial PX PY PZ lx ly lz embed injective support compatible
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
let E := initial PX PY PZ lx ly lz embed injective support compatible
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] at h
simpa only [E, kernel] using h
end MatrixMultiplication.Foundation.InitialTargetExecution
end
end OAI
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