openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexBounds /CW75ReorderedFinite.lean
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.CW75ReorderedExecution | |
| import OAI.LinearAlgebra.MatrixMultiplication.Separation.ExecutionApproximation | |
| namespace OAI | |
| /-! Explicit complex square and rectangular matrix multiplication bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CW75ReorderedFinite | |
| open MatrixMultiplication.Foundation CW75LabelHierarchy StageHierarchyResources | |
| open ExecutionPairingBudget CW75ReorderedWords CW75ReorderedExecution ComplexWitness | |
| open scoped BigOperators Classical | |
| attribute [local irreducible] CW75LabelHierarchy.retainedWords | |
| def blocks (counts : Scalar → ℕ) (t : ℕ) : ℕ := t * ∑ a, counts a | |
| def copies (counts : Scalar → ℕ) (t : ℕ) : ℕ := | |
| Fintype.card (ExactWords (fun a => t * counts a)) | |
| def rows (counts : Scalar → ℕ) (t : ℕ) : ℕ := (realize counts t).shape.rows | |
| def inner (counts : Scalar → ℕ) (t : ℕ) : ℕ := (realize counts t).shape.inner | |
| def columns (counts : Scalar → ℕ) (t : ℕ) : ℕ := (realize counts t).shape.columns | |
| def rankBudget (counts : Scalar → ℕ) (t : ℕ) : ℕ := | |
| 27 ^ blocks counts t * stageAuxiliaryBudget counts curveLabels 5 t | |
| def order (counts : Scalar → ℕ) (t : ℕ) : ℕ := | |
| 6 * blocks counts t * ((realize counts t).execution.order + 1) + | |
| (realize counts t).execution.order | |
| def degree (counts : Scalar → ℕ) (t : ℕ) : ℕ := | |
| 18 * blocks counts t * ((realize counts t).execution.order + 1) + | |
| (realize counts t).execution.leftDegree + | |
| (realize counts t).execution.middleDegree + | |
| (realize counts t).execution.rightDegree | |
| def sourceApproximation (counts : Scalar → ℕ) (t : ℕ) : | |
| Tensor.PolynomialApproximation (CW75InitialExecution.fullSource (blocks counts t)) | |
| (27 ^ blocks counts t) (6 * blocks counts t) (18 * blocks counts t) := by | |
| simpa only [CW75InitialExecution.fullSource, show (3 : ℕ) ^ 3 = 27 by norm_num, | |
| show (2 : ℕ) * 3 = 6 by norm_num, show (6 : ℕ) * 3 = 18 by norm_num] using | |
| (CoppersmithWinograd.approximation.power 3).power (blocks counts t) | |
| def labelEquiv (counts : Scalar → ℕ) (t : ℕ) : | |
| Fin (copies counts t) ≃ Prefix counts t 5 := | |
| Fintype.equivOfCardEq ((Fintype.card_fin (copies counts t)).trans | |
| (realize_label_card counts t).symm) | |
| def designatedX {counts : Scalar → ℕ} {t : ℕ} (p : Prefix counts t 5) : | |
| Coordinate counts t := longX (terminalWord p).val | |
| def designatedY {counts : Scalar → ℕ} {t : ℕ} (p : Prefix counts t 5) : | |
| Coordinate counts t := longY (terminalWord p).val | |
| def designatedZ {counts : Scalar → ℕ} {t : ℕ} (p : Prefix counts t 5) : | |
| Coordinate counts t := longZ (terminalWord p).val | |
| theorem designated_coefficient_one (counts : Scalar → ℕ) (t : ℕ) | |
| (p : Prefix counts t 5) : | |
| CW75InitialExecution.fullSource (blocks counts t) | |
| (designatedX p) (designatedY p) (designatedZ p) = 1 := by | |
| unfold CW75InitialExecution.fullSource Tensor.power | |
| apply Finset.prod_eq_one | |
| intro i _ | |
| apply Finset.prod_eq_one | |
| intro j _ | |
| have hs := CW75Primitive.original_support _ _ _ | |
| (coefficient_ne_zero ((terminalWord p).val i)) j | |
| simp only [CoppersmithWinograd.tensor, designatedX, designatedY, designatedZ, | |
| longX, longY, longZ] | |
| exact ite_eq_left hs | |
| def leftSelect (counts : Scalar → ℕ) (t : ℕ) | |
| (u : Fin (copies counts t) × (Fin (rows counts t) × Fin (inner counts t))) : | |
| Coordinate counts t × (Prefix counts t 5 × (realize counts t).PX) := | |
| (designatedX (labelEquiv counts t u.1), | |
| (labelEquiv counts t u.1, (realize counts t).shape.left.symm u.2)) | |
| def middleSelect (counts : Scalar → ℕ) (t : ℕ) | |
| (v : Fin (copies counts t) × (Fin (inner counts t) × Fin (columns counts t))) : | |
| Coordinate counts t × (Prefix counts t 5 × (realize counts t).PY) := | |
| (designatedY (labelEquiv counts t v.1), | |
| (labelEquiv counts t v.1, (realize counts t).shape.middle.symm v.2)) | |
| def rightSelect (counts : Scalar → ℕ) (t : ℕ) | |
| (w : Fin (copies counts t) × (Fin (columns counts t) × Fin (rows counts t))) : | |
| Coordinate counts t × (Prefix counts t 5 × (realize counts t).PZ) := | |
| (designatedZ (labelEquiv counts t w.1), | |
| (labelEquiv counts t w.1, (realize counts t).shape.right.symm w.2)) | |
| theorem selected_output (counts : Scalar → ℕ) (t : ℕ) : | |
| Tensor.pullback (leftSelect counts t) (middleSelect counts t) (rightSelect counts t) | |
| (LocalMaps.originalScale (CW75InitialExecution.fullSource (blocks counts t)) | |
| (realize counts t).execution.value) = | |
| Tensor.directSum (fun _ : Fin (copies counts t) => | |
| Tensor.matrixCoefficients (K := ℂ) (Fin (rows counts t)) | |
| (Fin (inner counts t)) (Fin (columns counts t))) := by | |
| funext u v w | |
| simp only [Tensor.pullback, LocalMaps.originalScale, leftSelect, middleSelect, rightSelect] | |
| rw [realize_value] | |
| simp only [MatrixCoordinates.tensor, Tensor.pullback] | |
| by_cases h : u.1 = v.1 ∧ u.1 = w.1 | |
| · have hv : v.1 = u.1 := h.1.symm | |
| have hw : w.1 = u.1 := h.2.symm | |
| have hc := designated_coefficient_one counts t (labelEquiv counts t u.1) | |
| simp only [designatedX, designatedY, designatedZ] at hc | |
| simp only [hv, hw, designatedX, designatedY, designatedZ, hc, | |
| Tensor.directSum, true_and, ite_true, one_mul] | |
| simp only [rows, inner, columns, | |
| (realize counts t).shape.left.apply_symm_apply u.2, | |
| (realize counts t).shape.middle.apply_symm_apply v.2, | |
| (realize counts t).shape.right.apply_symm_apply w.2] | |
| unfold Tensor.matrixCoefficients | |
| split_ifs <;> rfl | |
| · have hn : ¬ (labelEquiv counts t u.1 = labelEquiv counts t v.1 ∧ | |
| labelEquiv counts t u.1 = labelEquiv counts t w.1 ∧ | |
| designatedX (labelEquiv counts t u.1) = | |
| longX (terminalWord (labelEquiv counts t u.1)).val ∧ | |
| designatedY (labelEquiv counts t v.1) = | |
| longY (terminalWord (labelEquiv counts t u.1)).val ∧ | |
| designatedZ (labelEquiv counts t w.1) = | |
| longZ (terminalWord (labelEquiv counts t u.1)).val) := by | |
| intro he | |
| exact h ⟨(labelEquiv counts t).injective he.1, | |
| (labelEquiv counts t).injective he.2.1⟩ | |
| simp only [Tensor.directSum, ite_eq_right h, ite_eq_right hn, mul_zero] | |
| def approximation (counts : Scalar → ℕ) (t : ℕ) : | |
| Tensor.PolynomialApproximation | |
| (Tensor.directSum (fun _ : Fin (copies counts t) => | |
| Tensor.matrixCoefficients (K := ℂ) (Fin (rows counts t)) | |
| (Fin (inner counts t)) (Fin (columns counts t)))) | |
| (rankBudget counts t) (order counts t) (degree counts t) := by | |
| have supported : ∀ x y z : Coordinate counts t, | |
| ¬ CW75InitialExecution.support x y z → | |
| CW75InitialExecution.fullSource (blocks counts t) x y z = 0 := by | |
| intro x y z hs | |
| exact not_ne_iff.mp hs | |
| have full := ExecutionApproximation.approximation (realize counts t).execution | |
| (sourceApproximation counts t) supported | |
| have selected := full.pullback (leftSelect counts t) (middleSelect counts t) | |
| (rightSelect counts t) | |
| rw [selected_output] at selected | |
| simpa only [rankBudget, order, degree, realize_rankBound] using selected | |
| theorem dimensions_positive (counts : Scalar → ℕ) (t : ℕ) : | |
| 0 < rows counts t ∧ 0 < inner counts t ∧ 0 < columns counts t := | |
| (realize counts t).shape_positive | |
| theorem volume_eq_copies (counts : Scalar → ℕ) (t : ℕ) : | |
| rows counts t * inner counts t * columns counts t = copies counts t := | |
| realize_volume counts t | |
| theorem rows_pos (counts : Scalar → ℕ) (t : ℕ) : 0 < rows counts t := | |
| (dimensions_positive counts t).1 | |
| theorem inner_pos (counts : Scalar → ℕ) (t : ℕ) : 0 < inner counts t := | |
| (dimensions_positive counts t).2.1 | |
| theorem columns_pos (counts : Scalar → ℕ) (t : ℕ) : 0 < columns counts t := | |
| (dimensions_positive counts t).2.2 | |
| theorem copies_pos (counts : Scalar → ℕ) (t : ℕ) : 0 < copies counts t := by | |
| rw [← volume_eq_copies] | |
| exact Nat.mul_pos (Nat.mul_pos (rows_pos counts t) (inner_pos counts t)) | |
| (columns_pos counts t) | |
| theorem rankBudget_pos (counts : Scalar → ℕ) (t : ℕ) : 0 < rankBudget counts t := by | |
| apply Nat.mul_pos (pow_pos (by decide) _) | |
| rw [← realize_rankBound] | |
| exact lt_of_lt_of_le (by | |
| change 0 < rows counts t * inner counts t * columns counts t | |
| rw [volume_eq_copies] | |
| exact copies_pos counts t) (realize_pairing_budget counts t) | |
| theorem dimensions_eq_pairingSize (counts : Scalar → ℕ) (t : ℕ) : | |
| rows counts t = | |
| ConditionalLabels.stagePairingSize counts curveLabels 5 curveReaderPair .xz t ∧ | |
| inner counts t = | |
| ConditionalLabels.stagePairingSize counts curveLabels 5 curveReaderPair .xy t ∧ | |
| columns counts t = | |
| ConditionalLabels.stagePairingSize counts curveLabels 5 curveReaderPair .yz t := | |
| realize_dimensions counts t | |
| theorem rank_power (counts : Scalar → ℕ) (t n : ℕ) : | |
| Tensor.RankAtMost | |
| (Tensor.power (Tensor.directSum (fun _ : Fin (copies counts t) => | |
| Tensor.matrixCoefficients (K := ℂ) (Fin (rows counts t)) | |
| (Fin (inner counts t)) (Fin (columns counts t)))) n) | |
| ((degree counts t * n + 1) * rankBudget counts t ^ n) := | |
| (approximation counts t).rank_power n | |
| end MatrixMultiplication.CW75ReorderedFinite | |
| end | |
| end OAI | |