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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 | import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialControlledKernel
namespace OAI
/-! Finite coefficient tensors and their algebraic transformations. -/
noncomputable section
namespace MatrixMultiplication.Foundation.InitialTargetKernel
open Tensor LocalMaps PolynomialLocalConstruction
variable {X Y Z RawLabel Label : Type*}
def Chosen (window : X → Prop) (read : X → RawLabel) (embed : Label → RawLabel)
(label : Label) (x : X) : Prop :=
window x ∧ read x = embed label
def Eligible (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
(lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
(embed : Label → RawLabel) (label : Label) : X → Y → Z → Prop :=
fun x y z => Chosen PX lx embed label x ∧ Chosen PY ly embed label y ∧
Chosen PZ lz embed label z
def auxiliary : Tensor ℂ Unit Unit Unit := fun _ _ _ => 1
theorem auxiliary_rank : RankAtMost auxiliary 1 := by
have haux : auxiliary = rankOne (fun _ : Unit => (1 : ℂ))
(fun _ : Unit => (1 : ℂ)) (fun _ : Unit => (1 : ℂ)) := by
funext x y z
simp only [auxiliary, rankOne, one_mul]
rw [haux]
exact rankOne_rankAtMost _ _ _
def sideMap (window : X → Prop) (read : X → RawLabel) (embed : Label → RawLabel) :
X → (Label × (Fin 1 × Fin 1)) → Unit → Polynomial ℂ := by
classical
exact fun x output _ => Polynomial.C (if Chosen window read embed output.1 x then 1 else 0)
theorem sideMap_degree (window : X → Prop) (read : X → RawLabel)
(embed : Label → RawLabel) (x : X) (output : Label × (Fin 1 × Fin 1)) (input : Unit) :
(sideMap window read embed x output input).degree ≤ (0 : ℕ) := by
classical
exact Polynomial.degree_C_le
def rawLeading (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
(lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
(embed : Label → RawLabel) :
Tensor ℂ (X × (Label × (Fin 1 × Fin 1)))
(Y × (Label × (Fin 1 × Fin 1))) (Z × (Label × (Fin 1 × Fin 1))) := by
classical
exact fun x y z => if Chosen PX lx embed x.2.1 x.1 ∧
Chosen PY ly embed y.2.1 y.1 ∧ Chosen PZ lz embed z.2.1 z.1 then 1 else 0
def previousLeading (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
(lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
(embed : Label → RawLabel) :
Tensor ℂ (X × (Label × (Fin 1 × Fin 1)))
(Y × (Label × (Fin 1 × Fin 1))) (Z × (Label × (Fin 1 × Fin 1))) := by
classical
exact fun x y z => if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1 then 1 else 0
theorem kernel_eq_C_rawLeading (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
(lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
(embed : Label → RawLabel)
(x : X × (Label × (Fin 1 × Fin 1)))
(y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1))) :
kernel auxiliary (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed)
x y z = Polynomial.C (rawLeading PX PY PZ lx ly lz embed x y z) := by
classical
by_cases hx : Chosen PX lx embed x.2.1 x.1 <;>
by_cases hy : Chosen PY ly embed y.2.1 y.1 <;>
by_cases hz : Chosen PZ lz embed z.2.1 z.1 <;>
simp [kernel, fiberTransform, sideMap, auxiliary, rawLeading, hx, hy, hz]
theorem rawLeading_eq_previousLeading
(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)
(x : X × (Label × (Fin 1 × Fin 1)))
(y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1)))
(hs : support x.1 y.1 z.1) :
rawLeading PX PY PZ lx ly lz embed x y z =
previousLeading PX PY PZ lx ly lz embed x y z := by
classical
have hcondition :
(Chosen PX lx embed x.2.1 x.1 ∧ Chosen PY ly embed y.2.1 y.1 ∧
Chosen PZ lz embed z.2.1 z.1) ↔
(x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1) := by
constructor
· rintro ⟨hx, hy, hz⟩
have hsync := compatible x.1 y.1 z.1 hs hx.1 hy.1 hz.1
have hxy : x.2.1 = y.2.1 := injective (hx.2.symm.trans (hsync.1.trans hy.2))
have hxz : x.2.1 = z.2.1 := injective (hx.2.symm.trans (hsync.2.trans hz.2))
refine ⟨hxy, hxz, hx, ?_, ?_⟩
· simpa only [hxy] using hy
· simpa only [hxz] using hz
· rintro ⟨hxy, hxz, hx, hy, hz⟩
refine ⟨hx, ?_, ?_⟩
· simpa only [hxy] using hy
· simpa only [hxz] using hz
simp only [rawLeading, previousLeading, hcondition]
theorem kernel_leading (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)
(x : X × (Label × (Fin 1 × Fin 1)))
(y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1)))
(hs : support x.1 y.1 z.1) :
(kernel auxiliary (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed)
x y z).coeff 0 = previousLeading PX PY PZ lx ly lz embed x y z := by
rw [kernel_eq_C_rawLeading, Polynomial.coeff_C_zero]
exact rawLeading_eq_previousLeading PX PY PZ lx ly lz embed injective support
compatible x y z hs
theorem kernel_low (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
(lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
(embed : Label → RawLabel)
(x : X × (Label × (Fin 1 × Fin 1)))
(y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1)))
(j : ℕ) (hj : j < 0) :
(kernel auxiliary (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed)
x y z).coeff j = 0 :=
(Nat.not_lt_zero j hj).elim
theorem previousLeading_ne_zero_iff (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
(lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
(embed : Label → RawLabel)
(x : X × (Label × (Fin 1 × Fin 1)))
(y : Y × (Label × (Fin 1 × Fin 1))) (z : Z × (Label × (Fin 1 × Fin 1))) :
previousLeading PX PY PZ lx ly lz embed x y z ≠ 0 ↔
x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1 := by
classical
by_cases h : x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧
Eligible PX PY PZ lx ly lz embed x.2.1 x.1 y.1 z.1 <;>
simp [previousLeading, h]
theorem kernel_substitution (PX : X → Prop) (PY : Y → Prop) (PZ : Z → Prop)
(lx : X → RawLabel) (ly : Y → RawLabel) (lz : Z → RawLabel)
(embed : Label → RawLabel) (original : Tensor ℂ X Y Z) :
fiberTransform (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed)
(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 (sideMap PX lx embed) (sideMap PY ly embed) (sideMap PZ lz embed)) := by
have h := fiberTransform_originalScale (sideMap PX lx embed)
(sideMap PY ly embed) (sideMap PZ lz embed)
(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
end MatrixMultiplication.Foundation.InitialTargetKernel
end
end OAI
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