openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Completion /FiniteRealization.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Scripts | |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Execution | |
| import OAI.LinearAlgebra.MatrixMultiplication.Separation.ExecutionApproximation | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexExecutionPairingBudget | |
| import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexFiniteMMRealization | |
| namespace OAI | |
| /-! Readable tensor completion and its finite arithmetic realization. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CompletionFiniteRealization | |
| open MatrixMultiplication.Foundation RecursiveCompletion | |
| open scoped BigOperators Classical | |
| theorem power_coefficient_one {X Y Z : Type*} (T : Tensor ℂ X Y Z) | |
| (unit : ∀ x y z, T x y z ≠ 0 → T x y z = 1) (n : ℕ) | |
| (x : Fin n → X) (y : Fin n → Y) (z : Fin n → Z) | |
| (supported : Tensor.power T n x y z ≠ 0) : Tensor.power T n x y z = 1 := by | |
| apply Finset.prod_eq_one | |
| intro i hi | |
| exact unit _ _ _ ((Finset.prod_ne_zero_iff.mp supported) i hi) | |
| theorem complete_coefficient_one {X Y Z : Type*} (S : FlaggedTensor X Y Z) | |
| (unit : ∀ x y z, S.tensor x y z ≠ 0 → S.tensor x y z = 1) | |
| (center : Color) (m : ℕ) (x : Fin m → X) (y : Fin m → Y) (z : Fin m → Z) | |
| (supported : (complete S center m).tensor x y z ≠ 0) : | |
| (complete S center m).tensor x y z = 1 := by | |
| rw [complete_tensor] at supported ⊢ | |
| split_ifs at supported ⊢ with hweight | |
| · exact power_coefficient_one S.tensor unit m x y z supported | |
| · exact False.elim (supported rfl) | |
| theorem script_coefficient_one {X Y Z : Type*} (S : FlaggedTensor X Y Z) | |
| (unit : ∀ x y z, S.tensor x y z ≠ 0 → S.tensor x y z = 1) (s : Script) : | |
| ∀ x y z, (s.tensor S).tensor x y z ≠ 0 → (s.tensor S).tensor x y z = 1 := by | |
| induction s with | |
| | base => exact unit | |
| | step prior center m ih => exact complete_coefficient_one (prior.tensor S) ih center m | |
| def scriptSourceApproximation {X Y Z : Type*} | |
| [Fintype X] [Fintype Y] [Fintype Z] (S : FlaggedTensor X Y Z) {d D : ℕ} | |
| (source : Tensor.PolynomialApproximation S.tensor 3 d D) (s : Script) (blocks : ℕ) : | |
| Tensor.PolynomialApproximation (Tensor.power (s.tensor S).tensor blocks) | |
| (3 ^ (s.factors * blocks)) (s.order d * blocks) (s.degree D * blocks) := by | |
| simpa only [pow_mul] using (s.approximation S source).power blocks | |
| open CompletionHierarchyWords CompletionExecution ExecutionPairingBudget | |
| variable {A X Y Z : Type} [Fintype A] | |
| variable (H : ReadableHierarchy A X Y Z) (counts : A → ℕ) (t : ℕ) | |
| def blocks : ℕ := t * ∑ a, counts a | |
| def copies : ℕ := Fintype.card (Prefix H counts t H.depth) | |
| def rows : ℕ := (realize H counts t).shape.rows | |
| def inner : ℕ := (realize H counts t).shape.inner | |
| def columns : ℕ := (realize H counts t).shape.columns | |
| def order (d : ℕ) : ℕ := | |
| d * blocks counts t * ((realize H counts t).execution.order + 1) + | |
| (realize H counts t).execution.order | |
| def degree (D : ℕ) : ℕ := | |
| D * blocks counts t * ((realize H counts t).execution.order + 1) + | |
| (realize H counts t).execution.leftDegree + | |
| (realize H counts t).execution.middleDegree + | |
| (realize H counts t).execution.rightDegree | |
| def labelEquiv : Fin (copies H counts t) ≃ Prefix H counts t H.depth := | |
| (Fintype.equivFin _).symm | |
| variable (complete : Function.Injective (labelRecordOf H.labels H.depth)) | |
| def designatedX (p : Prefix H counts t H.depth) : Coordinate X counts t := | |
| longX H (terminalWord H complete p).val | |
| def designatedY (p : Prefix H counts t H.depth) : Coordinate Y counts t := | |
| longY H (terminalWord H complete p).val | |
| def designatedZ (p : Prefix H counts t H.depth) : Coordinate Z counts t := | |
| longZ H (terminalWord H complete p).val | |
| theorem leading_shape | |
| (x : Coordinate X counts t × (Prefix H counts t H.depth × (realize H counts t).PX)) | |
| (y : Coordinate Y counts t × (Prefix H counts t H.depth × (realize H counts t).PY)) | |
| (z : Coordinate Z counts t × (Prefix H counts t H.depth × (realize H counts t).PZ)) : | |
| (realize H counts t).execution.value x y z = | |
| if x.2.1 = y.2.1 ∧ x.2.1 = z.2.1 ∧ | |
| x.1 = designatedX H counts t complete x.2.1 ∧ | |
| y.1 = designatedY H counts t complete x.2.1 ∧ | |
| z.1 = designatedZ H counts t complete x.2.1 | |
| then (realize H counts t).shape.tensor x.2.2 y.2.2 z.2.2 else 0 := by | |
| rw [(realize H counts t).normal] | |
| simp only [prefixValue, terminal_eligible_iff H complete, | |
| designatedX, designatedY, designatedZ] | |
| rfl | |
| def leftSelect | |
| (u : Fin (copies H counts t) × (Fin (rows H counts t) × Fin (inner H counts t))) : | |
| Coordinate X counts t × (Prefix H counts t H.depth × (realize H counts t).PX) := | |
| (designatedX H counts t complete (labelEquiv H counts t u.1), | |
| (labelEquiv H counts t u.1, (realize H counts t).shape.left.symm u.2)) | |
| def middleSelect | |
| (v : Fin (copies H counts t) × (Fin (inner H counts t) × Fin (columns H counts t))) : | |
| Coordinate Y counts t × (Prefix H counts t H.depth × (realize H counts t).PY) := | |
| (designatedY H counts t complete (labelEquiv H counts t v.1), | |
| (labelEquiv H counts t v.1, (realize H counts t).shape.middle.symm v.2)) | |
| def rightSelect | |
| (w : Fin (copies H counts t) × (Fin (columns H counts t) × Fin (rows H counts t))) : | |
| Coordinate Z counts t × (Prefix H counts t H.depth × (realize H counts t).PZ) := | |
| (designatedZ H counts t complete (labelEquiv H counts t w.1), | |
| (labelEquiv H counts t w.1, (realize H counts t).shape.right.symm w.2)) | |
| theorem designated_coefficient_one (T : Tensor ℂ X Y Z) | |
| (unit : ∀ a, T (H.x a) (H.y a) (H.z a) = 1) | |
| (p : Prefix H counts t H.depth) : | |
| Tensor.power T (blocks counts t) (designatedX H counts t complete p) | |
| (designatedY H counts t complete p) (designatedZ H counts t complete p) = 1 := by | |
| apply Finset.prod_eq_one | |
| intro i _ | |
| exact unit ((terminalWord H complete p).val i) | |
| theorem selected_output (T : Tensor ℂ X Y Z) | |
| (unit : ∀ a, T (H.x a) (H.y a) (H.z a) = 1) : | |
| Tensor.pullback (leftSelect H counts t complete) (middleSelect H counts t complete) | |
| (rightSelect H counts t complete) | |
| (LocalMaps.originalScale (Tensor.power T (blocks counts t)) | |
| (realize H counts t).execution.value) = | |
| Tensor.directSum (fun _ : Fin (copies H counts t) => | |
| Tensor.matrixMultiplication (rows H counts t) (inner H counts t) (columns H counts t)) := by | |
| funext u v w | |
| simp only [Tensor.pullback, LocalMaps.originalScale, leftSelect, middleSelect, rightSelect] | |
| rw [leading_shape H counts t complete] | |
| 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 | |
| simp only [hv, hw, designated_coefficient_one H counts t complete T unit, | |
| Tensor.directSum, true_and, ite_true, one_mul, | |
| Tensor.matrixMultiplication_eq_matrixCoefficients, rows, inner, columns] | |
| rw [(realize H counts t).shape.left.apply_symm_apply u.2, | |
| (realize H counts t).shape.middle.apply_symm_apply v.2, | |
| (realize H counts t).shape.right.apply_symm_apply w.2] | |
| · have hn : ¬ (labelEquiv H counts t u.1 = labelEquiv H counts t v.1 ∧ | |
| labelEquiv H counts t u.1 = labelEquiv H counts t w.1 ∧ | |
| True ∧ | |
| designatedY H counts t complete (labelEquiv H counts t v.1) = | |
| designatedY H counts t complete (labelEquiv H counts t u.1) ∧ | |
| designatedZ H counts t complete (labelEquiv H counts t w.1) = | |
| designatedZ H counts t complete (labelEquiv H counts t u.1)) := by | |
| intro he | |
| exact h ⟨(labelEquiv H counts t).injective he.1, | |
| (labelEquiv H counts t).injective he.2.1⟩ | |
| simp only [Tensor.directSum, ite_eq_right h, ite_eq_right hn, mul_zero] | |
| theorem dimensions_positive : | |
| 0 < rows H counts t ∧ 0 < inner H counts t ∧ 0 < columns H counts t := | |
| (realize H counts t).shape_positive | |
| include complete in | |
| theorem volume_eq_copies : | |
| rows H counts t * inner H counts t * columns H counts t = copies H counts t := | |
| (realize_volume H counts t complete).trans (realize_label_card H counts t complete).symm | |
| include complete in | |
| theorem copies_pos : 0 < copies H counts t := by | |
| rw [← volume_eq_copies H counts t complete] | |
| exact Nat.mul_pos (Nat.mul_pos (dimensions_positive H counts t).1 | |
| (dimensions_positive H counts t).2.1) (dimensions_positive H counts t).2.2 | |
| theorem auxiliaryRank_pos : 0 < (realize H counts t).execution.rankBound := | |
| lt_of_lt_of_le | |
| (Nat.mul_pos (Nat.mul_pos (dimensions_positive H counts t).1 | |
| (dimensions_positive H counts t).2.1) (dimensions_positive H counts t).2.2) | |
| (realize H counts t).pairing_budget | |
| variable [Fintype X] [Fintype Y] [Fintype Z] | |
| def sourceApproximation {T : Tensor ℂ X Y Z} {factors d D : ℕ} | |
| (source : Tensor.PolynomialApproximation T (3 ^ factors) d D) : | |
| Tensor.PolynomialApproximation (Tensor.power T (blocks counts t)) | |
| (3 ^ (factors * blocks counts t)) (d * blocks counts t) (D * blocks counts t) := by | |
| simpa only [pow_mul] using source.power (blocks counts t) | |
| def approximation {T : Tensor ℂ X Y Z} {factors d D : ℕ} | |
| (source : Tensor.PolynomialApproximation T (3 ^ factors) d D) | |
| (supported : ∀ x y z, T x y z ≠ 0 → ∃ a, H.x a = x ∧ H.y a = y ∧ H.z a = z) | |
| (unit : ∀ a, T (H.x a) (H.y a) (H.z a) = 1) : | |
| Tensor.PolynomialApproximation | |
| (Tensor.directSum (fun _ : Fin (copies H counts t) => | |
| Tensor.matrixMultiplication (rows H counts t) (inner H counts t) (columns H counts t))) | |
| (3 ^ (factors * blocks counts t) * (realize H counts t).execution.rankBound) | |
| (order H counts t d) (degree H counts t D) := by | |
| have hs : ∀ x y z, ¬ support H counts t x y z → | |
| Tensor.power T (blocks counts t) x y z = 0 := by | |
| intro x y z outside | |
| by_contra nonzero | |
| apply outside | |
| intro i | |
| exact supported _ _ _ ((Finset.prod_ne_zero_iff.mp nonzero) i (Finset.mem_univ i)) | |
| have full := ExecutionApproximation.approximation (realize H counts t).execution | |
| (sourceApproximation counts t source) hs | |
| have selected := full.pullback (leftSelect H counts t complete) | |
| (middleSelect H counts t complete) (rightSelect H counts t complete) | |
| rw [selected_output H counts t complete T unit] at selected | |
| exact selected | |
| def realization {T : Tensor ℂ X Y Z} {factors d D : ℕ} | |
| (source : Tensor.PolynomialApproximation T (3 ^ factors) d D) | |
| (supported : ∀ x y z, T x y z ≠ 0 → ∃ a, H.x a = x ∧ H.y a = y ∧ H.z a = z) | |
| (unit : ∀ a, T (H.x a) (H.y a) (H.z a) = 1) | |
| (factors_pos : 0 < factors) (blocks_pos : 0 < blocks counts t) : FiniteMMRealization where | |
| occurrences := factors * blocks counts t | |
| multiplicity := copies H counts t | |
| auxiliaryRank := (realize H counts t).execution.rankBound | |
| rows := rows H counts t | |
| inner := inner H counts t | |
| columns := columns H counts t | |
| occurrences_pos := Nat.mul_pos factors_pos blocks_pos | |
| multiplicity_pos := copies_pos H counts t complete | |
| auxiliaryRank_pos := auxiliaryRank_pos H counts t | |
| rows_pos := (dimensions_positive H counts t).1 | |
| inner_pos := (dimensions_positive H counts t).2.1 | |
| columns_pos := (dimensions_positive H counts t).2.2 | |
| order := order H counts t d | |
| degree := degree H counts t D | |
| approximation := approximation H counts t complete source supported unit | |
| section Projections | |
| variable {T : Tensor ℂ X Y Z} {factors d D : ℕ} | |
| variable (source : Tensor.PolynomialApproximation T (3 ^ factors) d D) | |
| variable (supported : ∀ x y z, T x y z ≠ 0 → | |
| ∃ a, H.x a = x ∧ H.y a = y ∧ H.z a = z) | |
| variable (unit : ∀ a, T (H.x a) (H.y a) (H.z a) = 1) | |
| variable (factors_pos : 0 < factors) (blocks_pos : 0 < blocks counts t) | |
| @[simp] theorem realization_occurrences : | |
| (realization H counts t complete source supported unit factors_pos blocks_pos).occurrences = | |
| factors * blocks counts t := rfl | |
| @[simp] theorem realization_multiplicity : | |
| (realization H counts t complete source supported unit factors_pos blocks_pos).multiplicity = | |
| copies H counts t := rfl | |
| @[simp] theorem realization_auxiliaryRank : | |
| (realization H counts t complete source supported unit factors_pos blocks_pos).auxiliaryRank = | |
| StageHierarchyResources.stageAuxiliaryBudget counts H.labels H.depth t := | |
| realize_rankBound H counts t | |
| @[simp] theorem realization_rows : | |
| (realization H counts t complete source supported unit factors_pos blocks_pos).rows = | |
| rows H counts t := rfl | |
| @[simp] theorem realization_inner : | |
| (realization H counts t complete source supported unit factors_pos blocks_pos).inner = | |
| inner H counts t := rfl | |
| @[simp] theorem realization_columns : | |
| (realization H counts t complete source supported unit factors_pos blocks_pos).columns = | |
| columns H counts t := rfl | |
| @[simp] theorem realization_volume : | |
| (realization H counts t complete source supported unit factors_pos blocks_pos).rates.volume = | |
| copies H counts t := | |
| volume_eq_copies H counts t complete | |
| end Projections | |
| end MatrixMultiplication.CompletionFiniteRealization | |
| end | |
| end OAI | |