openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexBounds /CW75ReorderedBalanced.lean
| 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 | |