File size: 2,913 Bytes
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 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.MatrixBalancing
import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedFinite
namespace OAI
/-! Explicit complex square and rectangular matrix multiplication bounds. -/
noncomputable section
namespace MatrixMultiplication.CW75ReorderedBalanced
open MatrixMultiplication.Foundation CW75LabelHierarchy MatrixBalancing
def witness (counts : Scalar → ℕ) (t : ℕ) : ComplexWitness.FiniteRectangularWitness where
outer := CW75ReorderedFinite.rows counts t * CW75ReorderedFinite.inner counts t
inner := CW75ReorderedFinite.columns counts t ^ 2
multiplicity := CW75ReorderedFinite.copies counts t ^ 2
rank := CW75ReorderedFinite.rankBudget counts t ^ 2
outer_pos := Nat.mul_pos (CW75ReorderedFinite.rows_pos counts t)
(CW75ReorderedFinite.inner_pos counts t)
inner_pos := pow_pos (CW75ReorderedFinite.columns_pos counts t) 2
multiplicity_pos := pow_pos (CW75ReorderedFinite.copies_pos counts t) 2
rank_pos := pow_pos (CW75ReorderedFinite.rankBudget_pos counts t) 2
order := 2 * CW75ReorderedFinite.order counts t
degree := 2 * CW75ReorderedFinite.degree counts t
approximation := finiteBalanced (CW75ReorderedFinite.approximation counts t)
@[simp] theorem witness_outer (counts : Scalar → ℕ) (t : ℕ) :
(witness counts t).outer =
CW75ReorderedFinite.rows counts t * CW75ReorderedFinite.inner counts t := rfl
@[simp] theorem witness_inner (counts : Scalar → ℕ) (t : ℕ) :
(witness counts t).inner = CW75ReorderedFinite.columns counts t ^ 2 := rfl
@[simp] theorem witness_multiplicity (counts : Scalar → ℕ) (t : ℕ) :
(witness counts t).multiplicity = CW75ReorderedFinite.copies counts t ^ 2 := rfl
@[simp] theorem witness_rank (counts : Scalar → ℕ) (t : ℕ) :
(witness counts t).rank = CW75ReorderedFinite.rankBudget counts t ^ 2 := rfl
theorem witness_volume (counts : Scalar → ℕ) (t : ℕ) :
(witness counts t).outer ^ 2 * (witness counts t).inner =
(witness counts t).multiplicity := by
simp only [witness_outer, witness_inner, witness_multiplicity, ← mul_pow,
CW75ReorderedFinite.volume_eq_copies]
theorem witness_dimensions (counts : Scalar → ℕ) (t : ℕ) :
(witness counts t).outer =
ConditionalLabels.stagePairingSize counts ComplexWitness.curveLabels 5
ComplexWitness.curveReaderPair .xz t *
ConditionalLabels.stagePairingSize counts ComplexWitness.curveLabels 5
ComplexWitness.curveReaderPair .xy t ∧
(witness counts t).inner =
ConditionalLabels.stagePairingSize counts ComplexWitness.curveLabels 5
ComplexWitness.curveReaderPair .yz t ^ 2 := by
rcases CW75ReorderedFinite.dimensions_eq_pairingSize counts t with ⟨hr, hi, hc⟩
simp only [witness_outer, witness_inner, hr, hi, hc, and_self]
end MatrixMultiplication.CW75ReorderedBalanced
end
end OAI
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