openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Separation /AuxiliarySeparation.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexAuxiliarySeparation | |
| import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexStageHierarchyResources | |
| namespace OAI | |
| /-! Finite type counts, hierarchy separation and tensor execution bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication | |
| open MatrixMultiplication.Foundation Filter | |
| open scoped Topology BigOperators | |
| namespace AuxiliarySeparation | |
| variable {X Y Z Label : Type*} [Fintype Label] [DecidableEq Label] | |
| theorem directSum_rankAtMost (T : Tensor ℂ X Y Z) | |
| (readX : X → Label) (readY : Y → Label) {R : ℕ} | |
| (hT : Tensor.RankAtMost T R) | |
| (readable : ∀ x y z, T x y z ≠ 0 → readX x = readY y) : | |
| Tensor.RankAtMost (Tensor.dependentDirectSum (fun label => | |
| Tensor.product (labelFiberTensor readX readY T label) | |
| (Tensor.dotPairing (Fin (Fintype.card Label))))) | |
| (R * Separation.gridGroupOrder (Fintype.card Label)) := | |
| auxiliary_separation_directSum_rankAtMost T readX readY hT readable | |
| theorem directSum_borderRankAtMost [Fintype X] [Fintype Y] [Fintype Z] | |
| (T : Tensor ℂ X Y Z) (readX : X → Label) (readY : Y → Label) {R : ℕ} | |
| (hT : Tensor.BorderRankAtMost T R) | |
| (readable : ∀ x y z, T x y z ≠ 0 → readX x = readY y) : | |
| Tensor.BorderRankAtMost (Tensor.dependentDirectSum (fun label => | |
| Tensor.product (labelFiberTensor readX readY T label) | |
| (Tensor.dotPairing (Fin (Fintype.card Label))))) | |
| (R * Separation.gridGroupOrder (Fintype.card Label)) := | |
| auxiliary_separation_directSum_borderRankAtMost T readX readY hT readable | |
| theorem logarithmic_cost : | |
| Tendsto (fun K : ℕ => Real.log (Separation.gridGroupOrder K : ℝ) / | |
| Real.log (K : ℝ)) atTop (𝓝 1) := | |
| auxiliary_separation_logarithmic_cost | |
| theorem cost_minus_population_tendsto_zero {population : ℕ → ℕ} {rate : ℝ} | |
| (positive : ∀ n, 0 < population n) (rate_nonneg : 0 ≤ rate) | |
| (population_rate : Tendsto | |
| (fun n => Real.log (population n : ℝ) / (n : ℝ)) atTop (𝓝 rate)) : | |
| Tendsto (fun n => | |
| (Real.log (LabelHierarchySeparation.poolAuxiliaryCost (population n) : ℝ) - | |
| Real.log (population n : ℝ)) / (n : ℝ)) atTop (𝓝 0) := by | |
| have h := (LabelHierarchySeparation.tendsto_log_poolAuxiliaryCost_div_nat | |
| positive rate_nonneg population_rate).sub population_rate | |
| simpa only [sub_self, sub_div] using h | |
| end AuxiliarySeparation | |
| end MatrixMultiplication | |
| end | |
| end OAI | |