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cdec422 | 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 | import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWWindowedLeaves
namespace OAI
/-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/
noncomputable section
namespace MatrixMultiplication.CWCompleteStatistics
open CWWindowedLeaves CWLeafStatistics CWStrands AllFieldParameters
open scoped BigOperators
attribute [local instance] Classical.propDecidable
def pairEquiv : Raw 2 ≃ (Fin 7 × Fin 7) where
toFun w := (w 0, w 1)
invFun u := ![u.1, u.2]
left_inv w := by funext i; fin_cases i <;> rfl
right_inv u := by cases u; rfl
def orderedEquiv : Raw 4 ≃ ((Fin 7 × Fin 7) × (Fin 7 × Fin 7)) where
toFun w := ((w 0, w 1), (w 2, w 3))
invFun u := ![u.1.1, u.1.2, u.2.1, u.2.2]
left_inv w := by funext i; fin_cases i <;> rfl
right_inv u := by cases u with | mk u v => cases u; cases v; rfl
def twoStatistic (w : Raw 2) : Fin 6 := pairStatistic (pairEquiv w)
def fourStatistic (w : Raw 4) : Fin 6 × Fin 6 :=
(pairStatistic (orderedEquiv w).1, pairStatistic (orderedEquiv w).2)
def complementStatistic : Fin 6 ≃ Fin 6 :=
{ toFun := kappa
invFun := kappa
left_inv := kappa_involution
right_inv := kappa_involution }
def orderedComplementStatistic : (Fin 6 × Fin 6) ≃ (Fin 6 × Fin 6) :=
Equiv.prodCongr complementStatistic complementStatistic
theorem two_weight (w : Raw 2) :
CWStrands.weight w = statisticWeight (twoStatistic w) := by
rw [twoStatistic, statistic_weight]
simp [CWStrands.weight, pairEquiv, Fin.sum_univ_succ]
theorem four_weight (w : Raw 4) :
CWStrands.weight w = statisticWeight (fourStatistic w).1 +
statisticWeight (fourStatistic w).2 := by
simp only [fourStatistic, statistic_weight]
simp [CWStrands.weight, orderedEquiv, Fin.sum_univ_succ, Nat.add_assoc]
theorem two_multiplicity (s : Fin 6) :
Fintype.card {w : Raw 2 // twoStatistic w = s} = AllFieldParameters.multiplicity s := by
let e : {w : Raw 2 // twoStatistic w = s} ≃
{u : Fin 7 × Fin 7 // pairStatistic u = s} :=
{ toFun := fun w => ⟨pairEquiv w.val, w.property⟩
invFun := fun u => ⟨pairEquiv.symm u.val, by simpa [twoStatistic] using u.property⟩
left_inv := by intro w; apply Subtype.ext; simp
right_inv := by intro u; apply Subtype.ext; simp }
rw [Fintype.card_congr e, pair_multiplicity]
theorem four_multiplicity (s : Fin 6 × Fin 6) :
Fintype.card {w : Raw 4 // fourStatistic w = s} =
AllFieldParameters.multiplicity s.1 * AllFieldParameters.multiplicity s.2 := by
let e : {w : Raw 4 // fourStatistic w = s} ≃
{u : (Fin 7 × Fin 7) × (Fin 7 × Fin 7) //
pairStatistic u.1 = s.1 ∧ pairStatistic u.2 = s.2} :=
{ toFun := fun w => ⟨orderedEquiv w.val,
congrArg Prod.fst w.property, congrArg Prod.snd w.property⟩
invFun := fun u => ⟨orderedEquiv.symm u.val, by
apply Prod.ext
· simpa [fourStatistic] using u.property.1
· simpa [fourStatistic] using u.property.2⟩
left_inv := by intro w; apply Subtype.ext; simp
right_inv := by intro u; apply Subtype.ext; simp }
rw [Fintype.card_congr e, ordered_pair_multiplicity]
theorem two_complement (w : Raw 2) :
twoStatistic (complementWord w) = complementStatistic (twoStatistic w) := by
exact complement_statistic (pairEquiv w)
theorem four_complement (w : Raw 4) :
fourStatistic (complementWord w) = orderedComplementStatistic (fourStatistic w) := by
apply Prod.ext
· exact complement_statistic (orderedEquiv w).1
· exact complement_statistic (orderedEquiv w).2
theorem two_fixedWeight_multiplicity (a : ℕ) (s : Fin 6) (hs : statisticWeight s = a) :
Fintype.card {w : Alphabet 2 a // twoStatistic w.val = s} =
AllFieldParameters.multiplicity s := by
rw [ExactStatisticWords.card_fixedWeight_fiber twoStatistic CWStrands.weight
statisticWeight two_weight a s hs, two_multiplicity]
theorem four_fixedWeight_multiplicity (a : ℕ) (s : Fin 6 × Fin 6)
(hs : statisticWeight s.1 + statisticWeight s.2 = a) :
Fintype.card {w : Alphabet 4 a // fourStatistic w.val = s} =
AllFieldParameters.multiplicity s.1 * AllFieldParameters.multiplicity s.2 := by
rw [ExactStatisticWords.card_fixedWeight_fiber fourStatistic CWStrands.weight
(fun s => statisticWeight s.1 + statisticWeight s.2) four_weight a s hs,
four_multiplicity]
end MatrixMultiplication.CWCompleteStatistics
end
end OAI
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