File size: 13,561 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 | import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPrefixSeparationPolynomialStage
namespace OAI
/-! Finite type counts, hierarchy separation and tensor execution bounds. -/
noncomputable section
namespace MatrixMultiplication.Foundation.PrefixSeparationOrientations
open Tensor LabelHierarchySeparation PolynomialLocalConstruction
open PrefixSeparationPolynomialStage
variable {Prefix X Y Z : Type*}
def auxiliaryXZ (k : ℕ) :
Tensor ℂ (PoolAuxiliaryGroup k) (PoolAuxiliaryGroup k) (PoolAuxiliaryGroup k) :=
fun a b c => poolAuxiliaryTensor k a c b
def auxiliaryYZ (k : ℕ) :
Tensor ℂ (PoolAuxiliaryGroup k) (PoolAuxiliaryGroup k) (PoolAuxiliaryGroup k) :=
fun a b c => poolAuxiliaryTensor k b c a
theorem auxiliaryXZ_rankAtMost (k : ℕ) :
RankAtMost (auxiliaryXZ k) (poolAuxiliaryCost k) := by
obtain ⟨a, b, c, h⟩ := poolAuxiliaryTensor_rankAtMost k
refine ⟨a, c, b, ?_⟩
funext x y z
change poolAuxiliaryTensor k x z y = _
rw [h]
apply Finset.sum_congr rfl
intro i hi
simp only [rankOne]
ring
theorem auxiliaryYZ_rankAtMost (k : ℕ) :
RankAtMost (auxiliaryYZ k) (poolAuxiliaryCost k) := by
obtain ⟨a, b, c, h⟩ := poolAuxiliaryTensor_rankAtMost k
refine ⟨c, a, b, ?_⟩
funext x y z
change poolAuxiliaryTensor k y z x = _
rw [h]
apply Finset.sum_congr rfl
intro i hi
simp only [rankOne]
ring
def stageLeadingXZ (k : ℕ)
(codeX : Prefix → X → Fin k) (codeZ : Prefix → Z → Fin k)
(keepX : Prefix → X → Prop) (keepY : Prefix → Y → Prop)
(keepZ : Prefix → Z → Prop) (prior : Prefix) :
Tensor ℂ (X × InformedOutput k) (Y × MissingOutput k) (Z × InformedOutput k) :=
fun x y z => stageLeading k codeX codeZ keepX keepZ keepY prior x z y
def stageLeadingYZ (k : ℕ)
(codeY : Prefix → Y → Fin k) (codeZ : Prefix → Z → Fin k)
(keepX : Prefix → X → Prop) (keepY : Prefix → Y → Prop)
(keepZ : Prefix → Z → Prop) (prior : Prefix) :
Tensor ℂ (X × MissingOutput k) (Y × InformedOutput k) (Z × InformedOutput k) :=
fun x y z => stageLeading k codeY codeZ keepY keepZ keepX prior y z x
theorem stageLeadingXZ_synchronized (k : ℕ)
(codeX : Prefix → X → Fin k) (codeZ : Prefix → Z → Fin k)
(keepX : Prefix → X → Prop) (keepY : Prefix → Y → Prop)
(keepZ : Prefix → Z → Prop) (prior : Prefix)
(x : X × InformedOutput k) (y : Y × MissingOutput k) (z : Z × InformedOutput k)
(codes : codeX prior x.1 = codeZ prior z.1)
(nonzero : stageLeadingXZ k codeX codeZ keepX keepY keepZ prior x y z ≠ 0) :
x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ x.2.2 = z.2.2 := by
obtain ⟨hxz, hxy, hij⟩ := stageLeading_synchronized k codeX codeZ
keepX keepZ keepY prior x z y codes nonzero
exact ⟨hxy, hxz, hij⟩
theorem stageLeadingYZ_synchronized (k : ℕ)
(codeY : Prefix → Y → Fin k) (codeZ : Prefix → Z → Fin k)
(keepX : Prefix → X → Prop) (keepY : Prefix → Y → Prop)
(keepZ : Prefix → Z → Prop) (prior : Prefix)
(x : X × MissingOutput k) (y : Y × InformedOutput k) (z : Z × InformedOutput k)
(codes : codeY prior y.1 = codeZ prior z.1)
(nonzero : stageLeadingYZ k codeY codeZ keepX keepY keepZ prior x y z ≠ 0) :
x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ y.2.2 = z.2.2 := by
obtain ⟨hyz, hyx, hij⟩ := stageLeading_synchronized k codeY codeZ
keepY keepZ keepX prior y z x codes nonzero
exact ⟨hyx.symm, hyx.symm.trans hyz, hij⟩
theorem stageLeadingXZ_retains_pairing (k : ℕ)
(codeX : Prefix → X → Fin k) (codeZ : Prefix → Z → Fin k)
(keepX : Prefix → X → Prop) (keepY : Prefix → Y → Prop)
(keepZ : Prefix → Z → Prop) (prior : Prefix) (x : X) (y : Y) (z : Z)
(hx : keepX prior x) (hy : keepY prior y) (hz : keepZ prior z)
(codes : codeX prior x = codeZ prior z) (i : Fin k) :
stageLeadingXZ k codeX codeZ keepX keepY keepZ prior
(x, (codeX prior x, i)) (y, (codeX prior x, PUnit.unit))
(z, (codeX prior x, i)) = 1 :=
stageLeading_retains_pairing k codeX codeZ keepX keepZ keepY prior x z y hx hz hy codes i
theorem stageLeadingYZ_retains_pairing (k : ℕ)
(codeY : Prefix → Y → Fin k) (codeZ : Prefix → Z → Fin k)
(keepX : Prefix → X → Prop) (keepY : Prefix → Y → Prop)
(keepZ : Prefix → Z → Prop) (prior : Prefix) (x : X) (y : Y) (z : Z)
(hx : keepX prior x) (hy : keepY prior y) (hz : keepZ prior z)
(codes : codeY prior y = codeZ prior z) (i : Fin k) :
stageLeadingYZ k codeY codeZ keepX keepY keepZ prior
(x, (codeY prior y, PUnit.unit)) (y, (codeY prior y, i))
(z, (codeY prior y, i)) = 1 :=
stageLeading_retains_pairing k codeY codeZ keepY keepZ keepX prior y z x hy hz hx codes i
theorem exists_prefix_stage_XZ (k : ℕ)
(codeX : Prefix → X → Fin k) (codeZ : Prefix → Z → Fin k)
(keepX : Prefix → X → Prop) (keepY : Prefix → Y → Prop)
(keepZ : Prefix → Z → Prop)
(eligible : Prefix → X → Y → Z → Prop)
(agree : ∀ p x y z, eligible p x y z → codeX p x = codeZ p z) :
∃ (MX : Prefix → X → InformedOutput k → PoolAuxiliaryGroup k → Polynomial ℂ)
(MY : Prefix → Y → MissingOutput k → PoolAuxiliaryGroup k → Polynomial ℂ)
(MZ : Prefix → Z → InformedOutput k → PoolAuxiliaryGroup k → Polynomial ℂ),
(∀ p x u i, (MX p x u i).degree ≤ (0 : ℕ)) ∧
(∀ p y v j, (MY p y v j).degree ≤ (0 : ℕ)) ∧
(∀ p z w l, (MZ p z w l).degree ≤ (0 : ℕ)) ∧
∀ p x y z, eligible p x.1 y.1 z.1 →
kernel (auxiliaryXZ k) (MX p) (MY p) (MZ p) x y z =
Polynomial.C (stageLeadingXZ k codeX codeZ keepX keepY keepZ p x y z) := by
classical
obtain ⟨left, right, tag, coefficients⟩ := exists_auxiliary_coordinates k
let fx : Prefix → X × InformedOutput k → PoolAuxiliaryGroup k :=
fun p x => left (codeX p x.1) x.2.2
let fy : Prefix → Y × MissingOutput k → PoolAuxiliaryGroup k := fun _ y => tag y.2.1
let fz : Prefix → Z × InformedOutput k → PoolAuxiliaryGroup k :=
fun p z => right (codeZ p z.1) z.2.2
let MX := fun p => weightedCoordinatePolynomialMatrix (fx p)
(informedWeight (keepX p) (codeX p))
let MY := fun p => weightedCoordinatePolynomialMatrix (fy p) (missingWeight (keepY p))
let MZ := fun p => weightedCoordinatePolynomialMatrix (fz p)
(informedWeight (keepZ p) (codeZ p))
refine ⟨MX, MY, MZ, ?_, ?_, ?_, ?_⟩
· intro p x u i
exact weightedCoordinatePolynomialMatrix_degree _ _ x u i
· intro p y v j
exact weightedCoordinatePolynomialMatrix_degree _ _ y v j
· intro p z w l
exact weightedCoordinatePolynomialMatrix_degree _ _ z w l
· intro p x y z h
have codes := agree p x.1 y.1 z.1 h
change kernel (auxiliaryXZ k)
(weightedCoordinatePolynomialMatrix (fx p) (informedWeight (keepX p) (codeX p)))
(weightedCoordinatePolynomialMatrix (fy p) (missingWeight (keepY p)))
(weightedCoordinatePolynomialMatrix (fz p) (informedWeight (keepZ p) (codeZ p))) x y z = _
rw [kernel_weightedCoordinatePolynomialMatrix]
dsimp only [fx, fy, fz, auxiliaryXZ]
rw [← codes, coefficients]
congr 1
unfold informedWeight missingWeight stageLeadingXZ stageLeading
split_ifs <;> simp_all
theorem exists_prefix_stage_YZ (k : ℕ)
(codeY : Prefix → Y → Fin k) (codeZ : Prefix → Z → Fin k)
(keepX : Prefix → X → Prop) (keepY : Prefix → Y → Prop)
(keepZ : Prefix → Z → Prop)
(eligible : Prefix → X → Y → Z → Prop)
(agree : ∀ p x y z, eligible p x y z → codeY p y = codeZ p z) :
∃ (MX : Prefix → X → MissingOutput k → PoolAuxiliaryGroup k → Polynomial ℂ)
(MY : Prefix → Y → InformedOutput k → PoolAuxiliaryGroup k → Polynomial ℂ)
(MZ : Prefix → Z → InformedOutput k → PoolAuxiliaryGroup k → Polynomial ℂ),
(∀ p x u i, (MX p x u i).degree ≤ (0 : ℕ)) ∧
(∀ p y v j, (MY p y v j).degree ≤ (0 : ℕ)) ∧
(∀ p z w l, (MZ p z w l).degree ≤ (0 : ℕ)) ∧
∀ p x y z, eligible p x.1 y.1 z.1 →
kernel (auxiliaryYZ k) (MX p) (MY p) (MZ p) x y z =
Polynomial.C (stageLeadingYZ k codeY codeZ keepX keepY keepZ p x y z) := by
classical
obtain ⟨left, right, tag, coefficients⟩ := exists_auxiliary_coordinates k
let fx : Prefix → X × MissingOutput k → PoolAuxiliaryGroup k := fun _ x => tag x.2.1
let fy : Prefix → Y × InformedOutput k → PoolAuxiliaryGroup k :=
fun p y => left (codeY p y.1) y.2.2
let fz : Prefix → Z × InformedOutput k → PoolAuxiliaryGroup k :=
fun p z => right (codeZ p z.1) z.2.2
let MX := fun p => weightedCoordinatePolynomialMatrix (fx p) (missingWeight (keepX p))
let MY := fun p => weightedCoordinatePolynomialMatrix (fy p)
(informedWeight (keepY p) (codeY p))
let MZ := fun p => weightedCoordinatePolynomialMatrix (fz p)
(informedWeight (keepZ p) (codeZ p))
refine ⟨MX, MY, MZ, ?_, ?_, ?_, ?_⟩
· intro p x u i
exact weightedCoordinatePolynomialMatrix_degree _ _ x u i
· intro p y v j
exact weightedCoordinatePolynomialMatrix_degree _ _ y v j
· intro p z w l
exact weightedCoordinatePolynomialMatrix_degree _ _ z w l
· intro p x y z h
have codes := agree p x.1 y.1 z.1 h
change kernel (auxiliaryYZ k)
(weightedCoordinatePolynomialMatrix (fx p) (missingWeight (keepX p)))
(weightedCoordinatePolynomialMatrix (fy p) (informedWeight (keepY p) (codeY p)))
(weightedCoordinatePolynomialMatrix (fz p) (informedWeight (keepZ p) (codeZ p))) x y z = _
rw [kernel_weightedCoordinatePolynomialMatrix]
dsimp only [fx, fy, fz, auxiliaryYZ]
rw [← codes, coefficients]
congr 1
unfold informedWeight missingWeight stageLeadingYZ stageLeading
split_ifs <;> simp_all
section RawPools
variable {RawLabel : Type*} [DecidableEq RawLabel]
def rawPoolLeadingXZ (pool : Prefix → Finset RawLabel) (k : ℕ) (positive : 0 < k)
(counts : ∀ p, (pool p).card = k)
(readX : Prefix → X → RawLabel) (readZ : Prefix → Z → RawLabel) (prior : Prefix) :
Tensor ℂ (X × InformedOutput k) (Y × MissingOutput k) (Z × InformedOutput k) :=
fun x y z => rawPoolLeading pool k positive counts readX readZ prior x z y
def rawPoolLeadingYZ (pool : Prefix → Finset RawLabel) (k : ℕ) (positive : 0 < k)
(counts : ∀ p, (pool p).card = k)
(readY : Prefix → Y → RawLabel) (readZ : Prefix → Z → RawLabel) (prior : Prefix) :
Tensor ℂ (X × MissingOutput k) (Y × InformedOutput k) (Z × InformedOutput k) :=
fun x y z => rawPoolLeading pool k positive counts readY readZ prior y z x
theorem exists_raw_pool_stage_XZ (pool : Prefix → Finset RawLabel) (k : ℕ)
(positive : 0 < k) (counts : ∀ p, (pool p).card = k)
(readX : Prefix → X → RawLabel) (readZ : Prefix → Z → RawLabel)
(eligible : Prefix → X → Y → Z → Prop)
(agree : ∀ p x y z, eligible p x y z → readX p x = readZ p z) :
∃ (MX : Prefix → X → InformedOutput k → PoolAuxiliaryGroup k → Polynomial ℂ)
(MY : Prefix → Y → MissingOutput k → PoolAuxiliaryGroup k → Polynomial ℂ)
(MZ : Prefix → Z → InformedOutput k → PoolAuxiliaryGroup k → Polynomial ℂ),
(∀ p x u i, (MX p x u i).degree ≤ (0 : ℕ)) ∧
(∀ p y v j, (MY p y v j).degree ≤ (0 : ℕ)) ∧
(∀ p z w l, (MZ p z w l).degree ≤ (0 : ℕ)) ∧
∀ p x y z, eligible p x.1 y.1 z.1 →
kernel (auxiliaryXZ k) (MX p) (MY p) (MZ p) x y z =
Polynomial.C (rawPoolLeadingXZ pool k positive counts readX readZ p x y z) := by
apply exists_prefix_stage_XZ k
(fun p x => maskedPoolCode pool k positive counts p (readX p x))
(fun p z => maskedPoolCode pool k positive counts p (readZ p z))
(fun p x => readX p x ∈ pool p) (fun _ _ => True)
(fun p z => readZ p z ∈ pool p) eligible
intro p x y z h
exact congrArg (maskedPoolCode pool k positive counts p) (agree p x y z h)
theorem exists_raw_pool_stage_YZ (pool : Prefix → Finset RawLabel) (k : ℕ)
(positive : 0 < k) (counts : ∀ p, (pool p).card = k)
(readY : Prefix → Y → RawLabel) (readZ : Prefix → Z → RawLabel)
(eligible : Prefix → X → Y → Z → Prop)
(agree : ∀ p x y z, eligible p x y z → readY p y = readZ p z) :
∃ (MX : Prefix → X → MissingOutput k → PoolAuxiliaryGroup k → Polynomial ℂ)
(MY : Prefix → Y → InformedOutput k → PoolAuxiliaryGroup k → Polynomial ℂ)
(MZ : Prefix → Z → InformedOutput k → PoolAuxiliaryGroup k → Polynomial ℂ),
(∀ p x u i, (MX p x u i).degree ≤ (0 : ℕ)) ∧
(∀ p y v j, (MY p y v j).degree ≤ (0 : ℕ)) ∧
(∀ p z w l, (MZ p z w l).degree ≤ (0 : ℕ)) ∧
∀ p x y z, eligible p x.1 y.1 z.1 →
kernel (auxiliaryYZ k) (MX p) (MY p) (MZ p) x y z =
Polynomial.C (rawPoolLeadingYZ pool k positive counts readY readZ p x y z) := by
apply exists_prefix_stage_YZ k
(fun p y => maskedPoolCode pool k positive counts p (readY p y))
(fun p z => maskedPoolCode pool k positive counts p (readZ p z))
(fun _ _ => True) (fun p y => readY p y ∈ pool p)
(fun p z => readZ p z ∈ pool p) eligible
intro p x y z h
exact congrArg (maskedPoolCode pool k positive counts p) (agree p x y z h)
end RawPools
end MatrixMultiplication.Foundation.PrefixSeparationOrientations
end
end OAI
|