openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialOrbit.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialEntropy | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| namespace MatrixMultiplication.AllFieldInitialOrbit | |
| open AllFieldParameters AllFieldHistory AllFieldInitialEntropy | |
| open scoped BigOperators | |
| theorem source_total (p : PlacedInitial) : | |
| (initialShapeCode p).1.val + (initialShapeCode p).2.1.val + | |
| (initialShapeCode p).2.2.val = 16 := by | |
| unfold initialShapeCode rootShape | |
| rw [encodePhysicalShape_total] | |
| exact (root_shape_spec _ (initialShape_mem _)).2 | |
| theorem decodeShape_injective : Function.Injective decodeShape := by | |
| intro u v h | |
| apply Prod.ext | |
| · exact Fin.ext (congrFun h 0) | |
| · apply Prod.ext | |
| · exact Fin.ext (congrFun h 1) | |
| · exact Fin.ext (congrFun h 2) | |
| theorem decodeShape_mem_shapes (u : JointPopulation.Shape) | |
| (hu : u.1.val + u.2.1.val + u.2.2.val = 16) : | |
| decodeShape u ∈ shapes 16 := by | |
| simp only [shapes, List.mem_flatMap, List.mem_map, List.mem_range] | |
| refine ⟨u.1.val, by omega, u.2.1.val, by omega, ?_⟩ | |
| funext i | |
| fin_cases i | |
| · rfl | |
| · rfl | |
| · change 16 - u.1.val - u.2.1.val = u.2.2.val | |
| omega | |
| theorem initialShapeCode_eq_iff (p : PlacedInitial) (u : JointPopulation.Shape) : | |
| initialShapeCode p = u ↔ | |
| physicalShape p.2 (sortedInitial.get p.1) = decodeShape u := by | |
| have hdecode : decodeShape (initialShapeCode p) = | |
| physicalShape p.2 (sortedInitial.get p.1) := by | |
| funext i | |
| exact initialShapeCode_side p i | |
| rw [← hdecode] | |
| exact decodeShape_injective.eq_iff.symm | |
| theorem weighted_shape_fiber_count : ∀ u ∈ shapes 16, | |
| (∑ p : PlacedInitial, | |
| if physicalShape p.2 (sortedInitial.get p.1) = u | |
| then initialMultiplicity (sortedInitial.get p.1) else 0) = 6 := by | |
| decide +kernel | |
| theorem weighted_fiber_count (u : JointPopulation.Shape) | |
| (hu : u.1.val + u.2.1.val + u.2.2.val = 16) : | |
| (∑ p : PlacedInitial, if initialShapeCode p = u | |
| then initialMultiplicity (sortedInitial.get p.1) else 0) = 6 := by | |
| simp only [initialShapeCode_eq_iff] | |
| exact weighted_shape_fiber_count _ (decodeShape_mem_shapes u hu) | |
| end MatrixMultiplication.AllFieldInitialOrbit | |
| end OAI | |