openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /WitnessRealization.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.FiniteRealization | |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.FiniteRates | |
| import OAI.LinearAlgebra.MatrixMultiplication.Duality.WitnessProgram | |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.HierarchyPrograms | |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Termination | |
| namespace OAI | |
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.DualWitnessRealization | |
| open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels | |
| open CompletionLabels.TopologicalFlatten CompletionHierarchyWords | |
| open scoped BigOperators Classical | |
| theorem program_context (a : DualWitness.Support) : DualWitness.program.context a = PUnit.unit := | |
| rfl | |
| theorem program_joint : | |
| Function.Injective (fun a : DualWitness.Support => | |
| (DualWitness.program.view .B a, DualWitness.program.view .C a, | |
| DualWitness.program.view .A a)) := by | |
| intro a b h | |
| apply Subtype.ext | |
| apply Subtype.ext | |
| exact h | |
| def hierarchy : | |
| ReadableHierarchy DualWitness.Support DualWitness.Word DualWitness.Word DualWitness.Word := | |
| ReadableHierarchy.ofProgram DualWitness.program PUnit.unit program_context program_joint | |
| theorem hierarchy_complete : | |
| Function.Injective (labelRecordOf hierarchy.labels hierarchy.depth) := | |
| ReadableHierarchy.ofProgram_complete DualWitness.program PUnit.unit program_context | |
| program_joint DualWitness.program_complete | |
| @[simp] theorem hierarchy_x (a : DualWitness.Support) : hierarchy.x a = a.val.val.1 := rfl | |
| @[simp] theorem hierarchy_y (a : DualWitness.Support) : hierarchy.y a = a.val.val.2.1 := rfl | |
| @[simp] theorem hierarchy_z (a : DualWitness.Support) : hierarchy.z a = a.val.val.2.2 := rfl | |
| def tensor : Tensor ℂ DualWitness.Word DualWitness.Word DualWitness.Word := | |
| CompletionTermination.tensor DualWitness.source .A | |
| def retainedLeafEquiv : CompletionTermination.Leaf DualWitness.source .A ≃ DualWitness.Support := | |
| CompletionTermination.retainedLeafEquiv DualWitness.source .A .B .C | |
| (by intro c; cases c <;> simp) | |
| def approximation : | |
| Tensor.PolynomialApproximation tensor (2 ^ 288) | |
| (Script.dual.order 1) (Script.dual.degree 3) := | |
| CompletionTermination.approximation DualWitness.source .A DualFiniteRealization.dualSource | |
| theorem base_unit (x y z : Fin 2) (h : BaseTwo.tensor x y z ≠ 0) : | |
| BaseTwo.tensor x y z = 1 := by | |
| unfold BaseTwo.tensor at h ⊢ | |
| split_ifs with hsum | |
| · rfl | |
| · simp [hsum] at h | |
| theorem tensor_unit_coefficients (x y z : DualWitness.Word) (h : tensor x y z ≠ 0) : | |
| tensor x y z = 1 := by | |
| obtain ⟨hs, ha⟩ := (CompletionTermination.support_iff DualWitness.source .A x y z).mp h | |
| change (if ownerFlag DualWitness.source .A (coordinate (x, y, z) .A) then 0 | |
| else DualWitness.source.tensor x y z) = 1 | |
| rw [ite_eq_right ha] | |
| exact CompletionFiniteRealization.script_coefficient_one BaseTwo.flagged base_unit | |
| Script.dual x y z hs | |
| theorem source_supported (x y z : DualWitness.Word) (h : tensor x y z ≠ 0) : | |
| ∃ a : DualWitness.Support, hierarchy.x a = x ∧ hierarchy.y a = y ∧ hierarchy.z a = z := | |
| ⟨retainedLeafEquiv ⟨(x, y, z), h⟩, rfl, rfl, rfl⟩ | |
| theorem source_unit (a : DualWitness.Support) : | |
| tensor (hierarchy.x a) (hierarchy.y a) (hierarchy.z a) = 1 := | |
| tensor_unit_coefficients _ _ _ (retainedLeafEquiv.symm a).property | |
| def witness (counts : DualWitness.Support → ℕ) (t : ℕ) : | |
| ComplexWitness.FiniteRectangularWitness := | |
| DualFiniteRealization.witness hierarchy counts t hierarchy_complete approximation | |
| (pow_pos (by decide : 0 < (2 : ℕ)) 288) source_supported source_unit | |
| theorem witness_rank (counts : DualWitness.Support → ℕ) (t : ℕ) : | |
| (witness counts t).rank = | |
| (2 ^ (288 * CompletionFiniteRealization.blocks counts t) * | |
| StageHierarchyResources.stageAuxiliaryBudget counts hierarchy.labels hierarchy.depth t) ^ 2 := by | |
| rw [witness, DualFiniteRealization.witness_rank_stageBudget, ← pow_mul] | |
| theorem source_log : Real.log ((2 ^ 288 : ℕ) : ℝ) = 288 * Real.log 2 := by | |
| rw [Nat.cast_pow, Real.log_pow] | |
| norm_num | |
| theorem source_log_positive : 0 < Real.log ((2 ^ 288 : ℕ) : ℝ) := by | |
| rw [source_log] | |
| exact mul_pos (by norm_num) (Real.log_pos (by norm_num)) | |
| end MatrixMultiplication.DualWitnessRealization | |
| end | |
| end OAI | |