openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Duality /ExponentConsumer.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Arithmetic.Exponent | |
| import OAI.LinearAlgebra.MatrixMultiplication.Numerical.ComplexCertificatesDual | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexWitness | |
| import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.ASI | |
| import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.RateLimit | |
| namespace OAI | |
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.DualExponentBound | |
| open Filter | |
| open scoped Topology | |
| def fixedAspect : ℝ := 4651 / 10000 | |
| theorem target_lt_fixedAspect : (93 : ℝ) / 200 < fixedAspect := by | |
| norm_num [fixedAspect] | |
| theorem fixedAspect_lt_certificate : fixedAspect < (1548637 : ℝ) / 3327360 := by | |
| norm_num [fixedAspect] | |
| theorem fixedAspect_lt_dual_aspect : fixedAspect < DualWitness.aspect := | |
| fixedAspect_lt_certificate.trans ComplexCertificates.dual_aspect_lower | |
| theorem fixedAspect_mem_unit : fixedAspect ∈ Set.Icc (0 : ℝ) 1 := by | |
| norm_num [fixedAspect] | |
| structure ActualFamily (k budget : ℝ) where | |
| witness : ℕ → ComplexWitness.FiniteRectangularWitness | |
| scale : ℕ → ℝ | |
| outerRate : ℝ | |
| innerRate : ℝ | |
| multiplicityRate : ℝ | |
| rankRate : ℝ | |
| scale_positive : ∀ᶠ n in atTop, 0 < scale n | |
| outer_positive : 0 < outerRate | |
| outer_limit : Tendsto | |
| (fun n => Real.log ((witness n).outer : ℝ) / scale n) atTop (𝓝 outerRate) | |
| inner_limit : Tendsto | |
| (fun n => Real.log ((witness n).inner : ℝ) / scale n) atTop (𝓝 innerRate) | |
| multiplicity_limit : Tendsto | |
| (fun n => Real.log ((witness n).multiplicity : ℝ) / scale n) atTop (𝓝 multiplicityRate) | |
| rank_limit : Tendsto | |
| (fun n => Real.log ((witness n).rank : ℝ) / scale n) atTop (𝓝 rankRate) | |
| aspect_gap : k * outerRate < innerRate | |
| cost_bound : rankRate - multiplicityRate ≤ budget * outerRate | |
| namespace ActualFamily | |
| theorem omega_le {k budget : ℝ} (P : ActualFamily k budget) (hk : 0 ≤ k) : | |
| Arithmetic.rectangularOmega ℂ k ≤ budget := by | |
| have hbound := RectangularRateLimit.exponent_le_of_finite_asi | |
| (fun n => (P.witness n).outer_pos) (fun n => (P.witness n).inner_pos) | |
| P.scale_positive P.outer_limit P.inner_limit P.multiplicity_limit P.rank_limit | |
| P.outer_positive P.aspect_gap | |
| (fun n ha hab => Arithmetic.equalRectangular_log_inequality_of_polynomialApproximation | |
| ha (P.witness n).multiplicity_pos hk hab (P.witness n).approximation) | |
| exact hbound.trans ((div_le_iff₀ P.outer_positive).2 P.cost_bound) | |
| end ActualFamily | |
| theorem omega_eq_two_of_arbitrarily_close_families {k : ℝ} (hk : 0 ≤ k) | |
| (families : ∀ ε : ℝ, 0 < ε → Nonempty (ActualFamily k (2 + ε))) : | |
| Arithmetic.rectangularOmega ℂ k = 2 := by | |
| apply le_antisymm _ (Arithmetic.rectangularOmega_two_le k) | |
| by_contra h | |
| have hω : 2 < Arithmetic.rectangularOmega ℂ k := lt_of_not_ge h | |
| have hε : 0 < (Arithmetic.rectangularOmega ℂ k - 2) / 2 := by linarith | |
| obtain ⟨P⟩ := families ((Arithmetic.rectangularOmega ℂ k - 2) / 2) hε | |
| have hbound := P.omega_le hk | |
| linarith | |
| theorem alpha_gt_target_of_arbitrarily_close_families | |
| (families : ∀ ε : ℝ, 0 < ε → Nonempty (ActualFamily fixedAspect (2 + ε))) : | |
| (93 : ℝ) / 200 < Arithmetic.complexAlpha := by | |
| have hω := omega_eq_two_of_arbitrarily_close_families fixedAspect_mem_unit.1 families | |
| exact target_lt_fixedAspect.trans_le (Arithmetic.le_complexAlpha fixedAspect_mem_unit hω) | |
| end MatrixMultiplication.DualExponentBound | |
| end | |
| end OAI | |