openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Separation /ComplexAuxiliarySeparation.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexSeparationAsymptotics | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexSeparatedTensorDecomposition | |
| namespace OAI | |
| /-! Finite type counts, hierarchy separation and tensor execution bounds. -/ | |
| namespace MatrixMultiplication.Foundation | |
| instance auxiliarySeparation_modulus_neZero (K : ℕ) : | |
| NeZero (4 * Separation.gridWidth K) := | |
| ⟨Nat.mul_ne_zero (by decide) | |
| (Nat.ne_of_gt (lt_of_lt_of_le (by decide : 0 < 2) | |
| (Separation.gridWidth_ge_two K)))⟩ | |
| section AuxiliarySeparation | |
| variable {X Y Z S : Type*} [Fintype S] [DecidableEq S] | |
| theorem auxiliary_separation_mem_restrictionOrbit | |
| [Fintype X] [Fintype Y] [Fintype Z] | |
| (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) | |
| (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) : | |
| separatedTensor (U := Fin (Fintype.card S)) lx T ∈ Tensor.restrictionOrbit | |
| (Tensor.product T (groupTensor (Separation.AuxiliaryGroup | |
| (Separation.gridDimension (Fintype.card S)) | |
| (Separation.gridWidth (Fintype.card S))))) := by | |
| exact finite_grid_separation_mem_restrictionOrbit T lx ly | |
| (Separation.grid_population_bound (Fintype.card S)) | |
| (Separation.grid_population_bound (Fintype.card S)) hlabels | |
| theorem auxiliary_separation_rankAtMost | |
| (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ} | |
| (hT : Tensor.RankAtMost T r) | |
| (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) : | |
| Tensor.RankAtMost (separatedTensor (U := Fin (Fintype.card S)) lx T) | |
| (r * Separation.gridGroupOrder (Fintype.card S)) := by | |
| exact finite_grid_separation_rankAtMost T lx ly | |
| (lt_of_lt_of_le (by decide : 0 < 2) | |
| (Separation.gridWidth_ge_two (Fintype.card S))) | |
| (Separation.grid_population_bound (Fintype.card S)) | |
| (Separation.grid_population_bound (Fintype.card S)) hT hlabels | |
| theorem auxiliary_separation_borderRankAtMost | |
| [Fintype X] [Fintype Y] [Fintype Z] | |
| (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ} | |
| (hT : Tensor.BorderRankAtMost T r) | |
| (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) : | |
| Tensor.BorderRankAtMost (separatedTensor (U := Fin (Fintype.card S)) lx T) | |
| (r * Separation.gridGroupOrder (Fintype.card S)) := by | |
| exact finite_grid_separation_borderRankAtMost T lx ly | |
| (lt_of_lt_of_le (by decide : 0 < 2) | |
| (Separation.gridWidth_ge_two (Fintype.card S))) | |
| (Separation.grid_population_bound (Fintype.card S)) | |
| (Separation.grid_population_bound (Fintype.card S)) hT hlabels | |
| theorem auxiliary_separation_directSum_rankAtMost | |
| (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ} | |
| (hT : Tensor.RankAtMost T r) | |
| (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) : | |
| Tensor.RankAtMost (Tensor.dependentDirectSum (fun s => | |
| Tensor.product (labelFiberTensor lx ly T s) | |
| (Tensor.dotPairing (Fin (Fintype.card S))))) | |
| (r * Separation.gridGroupOrder (Fintype.card S)) := by | |
| exact separatedTensor_fiber_directSum_rankAtMost T lx ly hlabels | |
| (auxiliary_separation_rankAtMost T lx ly hT hlabels) | |
| theorem auxiliary_separation_directSum_borderRankAtMost | |
| [Fintype X] [Fintype Y] [Fintype Z] | |
| (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ} | |
| (hT : Tensor.BorderRankAtMost T r) | |
| (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) : | |
| Tensor.BorderRankAtMost (Tensor.dependentDirectSum (fun s => | |
| Tensor.product (labelFiberTensor lx ly T s) | |
| (Tensor.dotPairing (Fin (Fintype.card S))))) | |
| (r * Separation.gridGroupOrder (Fintype.card S)) := by | |
| exact separatedTensor_fiber_directSum_borderRankAtMost T lx ly hlabels | |
| (auxiliary_separation_borderRankAtMost T lx ly hT hlabels) | |
| end AuxiliarySeparation | |
| theorem auxiliary_separation_logarithmic_cost : | |
| Filter.Tendsto | |
| (fun K : ℕ => Real.log (Separation.gridGroupOrder K : ℝ) / Real.log (K : ℝ)) | |
| Filter.atTop (nhds 1) := | |
| Separation.tendsto_log_gridGroupOrder_div_log_population | |
| end MatrixMultiplication.Foundation | |
| end OAI | |