openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /ComplexBounds /SquareBound.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.BaseConditionalReaders | |
| import OAI.LinearAlgebra.MatrixMultiplication.Arithmetic.Compatibility | |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.HierarchyPrograms | |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.SquareRates | |
| import OAI.LinearAlgebra.MatrixMultiplication.ComplexBounds.SquareCounts | |
| namespace OAI | |
| /-! Explicit complex square and rectangular matrix multiplication bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.SquareBound | |
| open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels | |
| open CompletionLabels.TopologicalFlatten CompletionHierarchyWords Filter | |
| open scoped BigOperators Topology Classical | |
| abbrev Coordinate := Script.Coord (Fin 3) Script.square | |
| abbrev Carrier := RetainedLeaf SquareCounts.completed .B .C | |
| def conditional : ReadableTensor SquareCounts.completed := | |
| ReadableTensor.script BaseThree.flagged BaseConditionalReaders.baseThreeReadable Script.square | |
| def program : | |
| Program Carrier (CompletionLabels.Coordinate Coordinate Coordinate Coordinate) | |
| PUnit (Color ⊕ conditional.Code) (1 + conditional.depth) := | |
| terminationProgram SquareCounts.completed .B .C (by decide) | |
| conditional.program conditional.context_eq conditional.view_eq | |
| theorem program_context (a : Carrier) : program.context a = PUnit.unit := rfl | |
| theorem program_view (side : Color) (a : Carrier) : | |
| program.view side a = coordinate a.val.val side := rfl | |
| theorem program_joint : | |
| Function.Injective (fun a : Carrier => (program.view .B a, program.view .C a, | |
| program.view .A a)) := by | |
| intro a b h | |
| apply Subtype.ext | |
| apply Subtype.ext | |
| exact h | |
| def hierarchy : ReadableHierarchy Carrier Coordinate Coordinate Coordinate := | |
| ReadableHierarchy.ofProgram program PUnit.unit program_context program_joint | |
| theorem hierarchy_complete : | |
| Function.Injective (labelRecordOf hierarchy.labels hierarchy.depth) := by | |
| apply ReadableHierarchy.ofProgram_complete program PUnit.unit program_context program_joint | |
| exact terminationProgram_complete SquareCounts.completed .B .C (by decide) | |
| conditional.program conditional.context_eq conditional.view_eq conditional.complete | |
| @[simp] theorem hierarchy_x (a : Carrier) : hierarchy.x a = a.val.val.1 := rfl | |
| @[simp] theorem hierarchy_y (a : Carrier) : hierarchy.y a = a.val.val.2.1 := rfl | |
| @[simp] theorem hierarchy_z (a : Carrier) : hierarchy.z a = a.val.val.2.2 := rfl | |
| theorem source_supported (x y z : Coordinate) (h : SquareCounts.tensor x y z ≠ 0) : | |
| ∃ a : Carrier, hierarchy.x a = x ∧ hierarchy.y a = y ∧ hierarchy.z a = z := by | |
| exact ⟨SquareCounts.retainedLeafEquiv ⟨(x, y, z), h⟩, rfl, rfl, rfl⟩ | |
| theorem source_unit (a : Carrier) : | |
| SquareCounts.tensor (hierarchy.x a) (hierarchy.y a) (hierarchy.z a) = 1 := by | |
| exact SquareCounts.tensor_unit_coefficients _ _ _ | |
| (SquareCounts.retainedLeafEquiv.symm a).property | |
| def counts (_ : Carrier) : ℕ := 1 | |
| theorem card_carrier : Fintype.card Carrier = SquareCounts.leafCount := | |
| (Fintype.card_congr SquareCounts.retainedLeafEquiv.symm).trans SquareCounts.card_leaf | |
| theorem total_counts : (∑ a : Carrier, counts a) = SquareCounts.leafCount := by | |
| exact (Finset.card_eq_sum_ones Finset.univ).symm.trans card_carrier | |
| theorem total_counts_pos : 0 < ∑ a : Carrier, counts a := by | |
| rw [total_counts] | |
| exact SquareCounts.leafCount_pos | |
| theorem count_entropy : | |
| finiteEntropy (CompletionSquareRates.countMass counts) = Real.log SquareCounts.leafCount := by | |
| have hm : CompletionSquareRates.countMass counts = SquareCounts.retainedUniformLaw.mass := by | |
| funext a | |
| rw [SquareCounts.retainedUniformLaw_mass] | |
| rw [CompletionSquareRates.countMass, total_counts] | |
| simp only [counts, Nat.cast_one, one_div] | |
| rw [hm, SquareCounts.retainedUniformLaw_entropy] | |
| def family (n : ℕ) : FiniteMMRealization := | |
| CompletionSquareRates.positiveRealization hierarchy counts total_counts_pos | |
| hierarchy_complete SquareCounts.approximation source_supported source_unit (by decide) n | |
| theorem score_limit (β : ℝ) : | |
| Tendsto (fun n => (family n).rates.score β) atTop | |
| (𝓝 (β / 216 * Real.log SquareWitness.leafCount)) := by | |
| have h := CompletionSquareRates.score_limit hierarchy counts total_counts_pos | |
| hierarchy_complete SquareCounts.approximation source_supported source_unit (by decide) β | |
| rw [count_entropy] at h | |
| convert h using 1 <;> norm_num [family, SquareCounts.leafCount, SquareWitness.leafCount] | |
| theorem omega_lt_2267_div_1000 : | |
| MatrixMultiplication.Foundation.Arithmetic.omega < (2267 : ℝ) / 1000 := | |
| SquareWitness.omega_lt_of_certified_uniform_square_family family | |
| (score_limit ((2267 : ℝ) / 1000)) | |
| theorem complex_omega_lt_2267_div_1000 : | |
| MatrixMultiplication.Arithmetic.omega ℂ < (2267 : ℝ) / 1000 := by | |
| rw [MatrixMultiplication.Arithmetic.complex_omega_eq] | |
| exact omega_lt_2267_div_1000 | |
| end MatrixMultiplication.SquareBound | |
| end | |
| end OAI | |