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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialKernelExecution
namespace OAI
/-! Polynomial tensor restrictions and exact coefficient extraction. -/
noncomputable section
namespace MatrixMultiplication.Foundation.PolynomialKernelExecution.Execution
open Tensor LocalMaps PolynomialLocalConstruction
variable {X Y Z Prefix PX PY PZ AX AY AZ Prefix' PX' PY' PZ' : 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)
variable (ePrefix : Prefix' ≃ Prefix) (eX : PX' ≃ PX) (eY : PY' ≃ PY) (eZ : PZ' ≃ PZ)
def reindexOutput : Execution X Y Z Prefix' PX' PY' PZ' AX AY AZ support where
auxiliary := E.auxiliary
rankBound := E.rankBound
auxiliary_rank := E.auxiliary_rank
order := E.order
leftDegree := E.leftDegree
middleDegree := E.middleDegree
rightDegree := E.rightDegree
leftMap := fun x output input => E.leftMap x (ePrefix output.1, eX output.2) input
middleMap := fun y output input => E.middleMap y (ePrefix output.1, eY output.2) input
rightMap := fun z output input => E.rightMap z (ePrefix output.1, eZ output.2) input
left_degree := fun x output input =>
E.left_degree x (ePrefix output.1, eX output.2) input
middle_degree := fun y output input =>
E.middle_degree y (ePrefix output.1, eY output.2) input
right_degree := fun z output input =>
E.right_degree z (ePrefix output.1, eZ output.2) input
value := fun x y z => E.value
(x.1, (ePrefix x.2.1, eX x.2.2))
(y.1, (ePrefix y.2.1, eY y.2.2))
(z.1, (ePrefix z.2.1, eZ z.2.2))
vanishes := fun x y z hs j hj => E.vanishes
(x.1, (ePrefix x.2.1, eX x.2.2))
(y.1, (ePrefix y.2.1, eY y.2.2))
(z.1, (ePrefix z.2.1, eZ z.2.2)) hs j hj
leading := fun x y z hs => E.leading
(x.1, (ePrefix x.2.1, eX x.2.2))
(y.1, (ePrefix y.2.1, eY y.2.2))
(z.1, (ePrefix z.2.1, eZ z.2.2)) hs
synchronized := by
intro x y z hs hn
have h := E.synchronized
(x.1, (ePrefix x.2.1, eX x.2.2))
(y.1, (ePrefix y.2.1, eY y.2.2))
(z.1, (ePrefix z.2.1, eZ z.2.2)) hs hn
exact ⟨ePrefix.injective h.1, ePrefix.injective h.2⟩
@[simp] theorem reindexOutput_auxiliary :
(E.reindexOutput ePrefix eX eY eZ).auxiliary = E.auxiliary := rfl
@[simp] theorem reindexOutput_rankBound :
(E.reindexOutput ePrefix eX eY eZ).rankBound = E.rankBound := rfl
@[simp] theorem reindexOutput_order :
(E.reindexOutput ePrefix eX eY eZ).order = E.order := rfl
@[simp] theorem reindexOutput_leftDegree :
(E.reindexOutput ePrefix eX eY eZ).leftDegree = E.leftDegree := rfl
@[simp] theorem reindexOutput_middleDegree :
(E.reindexOutput ePrefix eX eY eZ).middleDegree = E.middleDegree := rfl
@[simp] theorem reindexOutput_rightDegree :
(E.reindexOutput ePrefix eX eY eZ).rightDegree = E.rightDegree := rfl
@[simp] theorem reindexOutput_leftMap (x : X) (output : Prefix' × PX') (input : AX) :
(E.reindexOutput ePrefix eX eY eZ).leftMap x output input =
E.leftMap x (ePrefix output.1, eX output.2) input := rfl
@[simp] theorem reindexOutput_middleMap (y : Y) (output : Prefix' × PY') (input : AY) :
(E.reindexOutput ePrefix eX eY eZ).middleMap y output input =
E.middleMap y (ePrefix output.1, eY output.2) input := rfl
@[simp] theorem reindexOutput_rightMap (z : Z) (output : Prefix' × PZ') (input : AZ) :
(E.reindexOutput ePrefix eX eY eZ).rightMap z output input =
E.rightMap z (ePrefix output.1, eZ output.2) input := rfl
@[simp] theorem reindexOutput_value
(x : X × (Prefix' × PX')) (y : Y × (Prefix' × PY')) (z : Z × (Prefix' × PZ')) :
(E.reindexOutput ePrefix eX eY eZ).value x y z = E.value
(x.1, (ePrefix x.2.1, eX x.2.2))
(y.1, (ePrefix y.2.1, eY y.2.2))
(z.1, (ePrefix z.2.1, eZ z.2.2)) := rfl
@[simp] theorem reindexOutput_kernel
(x : X × (Prefix' × PX')) (y : Y × (Prefix' × PY')) (z : Z × (Prefix' × PZ')) :
kernel (E.reindexOutput ePrefix eX eY eZ).auxiliary
(E.reindexOutput ePrefix eX eY eZ).leftMap
(E.reindexOutput ePrefix eX eY eZ).middleMap
(E.reindexOutput ePrefix eX eY eZ).rightMap x y z =
kernel E.auxiliary E.leftMap E.middleMap E.rightMap
(x.1, (ePrefix x.2.1, eX x.2.2))
(y.1, (ePrefix y.2.1, eY y.2.2))
(z.1, (ePrefix z.2.1, eZ z.2.2)) := rfl
theorem reindexOutput_fiberTransform
(P : Tensor (Polynomial ℂ) (X × AX) (Y × AY) (Z × AZ))
(x : X × (Prefix' × PX')) (y : Y × (Prefix' × PY')) (z : Z × (Prefix' × PZ')) :
fiberTransform (E.reindexOutput ePrefix eX eY eZ).leftMap
(E.reindexOutput ePrefix eX eY eZ).middleMap
(E.reindexOutput ePrefix eX eY eZ).rightMap P x y z =
fiberTransform E.leftMap E.middleMap E.rightMap P
(x.1, (ePrefix x.2.1, eX x.2.2))
(y.1, (ePrefix y.2.1, eY y.2.2))
(z.1, (ePrefix z.2.1, eZ z.2.2)) := rfl
theorem reindexOutput_value_preimage
(x : X × (Prefix × PX)) (y : Y × (Prefix × PY)) (z : Z × (Prefix × PZ)) :
(E.reindexOutput ePrefix eX eY eZ).value
(x.1, (ePrefix.symm x.2.1, eX.symm x.2.2))
(y.1, (ePrefix.symm y.2.1, eY.symm y.2.2))
(z.1, (ePrefix.symm z.2.1, eZ.symm z.2.2)) = E.value x y z := by
simp only [reindexOutput_value, Equiv.apply_symm_apply]
theorem reindexOutput_kernel_preimage
(x : X × (Prefix × PX)) (y : Y × (Prefix × PY)) (z : Z × (Prefix × PZ)) :
kernel (E.reindexOutput ePrefix eX eY eZ).auxiliary
(E.reindexOutput ePrefix eX eY eZ).leftMap
(E.reindexOutput ePrefix eX eY eZ).middleMap
(E.reindexOutput ePrefix eX eY eZ).rightMap
(x.1, (ePrefix.symm x.2.1, eX.symm x.2.2))
(y.1, (ePrefix.symm y.2.1, eY.symm y.2.2))
(z.1, (ePrefix.symm z.2.1, eZ.symm z.2.2)) =
kernel E.auxiliary E.leftMap E.middleMap E.rightMap x y z := by
simp only [reindexOutput_kernel, Equiv.apply_symm_apply]
end MatrixMultiplication.Foundation.PolynomialKernelExecution.Execution
end
end OAI
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