openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /DeterminantRegularization.lean
| import OAI.Analysis.IrrationalRotation.Spectral.DeterminantNormBound | |
| namespace OAI | |
| noncomputable section | |
| open Filter | |
| open scoped ENNReal Topology CStarAlgebra | |
| namespace Invariant233 | |
| variable {A : Type*} [CStarAlgebra A] | |
| theorem regularizedDeterminant_monotone (τ : TracialState A) (a : A) : | |
| Monotone (fun ε : {r : ℝ // 0 < r} => ENNReal.ofReal | |
| (Real.exp ((τ (CFC.log (star a * a + ε.val • (1 : A)))).re / 2))) := by | |
| let : PartialOrder A := CStarAlgebra.spectralOrder A | |
| let : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A | |
| intro ε δ hεδ | |
| apply ENNReal.ofReal_le_ofReal | |
| apply Real.exp_le_exp.mpr | |
| apply div_le_div_of_nonneg_right _ (by norm_num) | |
| apply τ.real_mono | |
| apply CFC.log_le_log | |
| · apply add_le_add le_rfl | |
| exact smul_le_smul_of_nonneg_right hεδ zero_le_one | |
| · apply (StarOrderedRing.isStrictlyPositive_iff_spectrum_pos | |
| (R := ℝ) _ (by simp [IsSelfAdjoint])).mpr | |
| exact regularizedNormSquare_spectrum_pos a ε.val ε.property | |
| theorem tendsto_positive_atBot_of_tendsto_zero {ι : Type*} {l : Filter ι} | |
| (ε : ι → ℝ) (hε : ∀ i, 0 < ε i) (hlim : Tendsto ε l (𝓝 0)) : | |
| Tendsto (fun i => (⟨ε i, hε i⟩ : {r : ℝ // 0 < r})) l atBot := by | |
| apply tendsto_atBot.2 | |
| intro δ | |
| filter_upwards [hlim.eventually (gt_mem_nhds δ.property)] with i hi | |
| show ε i ≤ δ.val | |
| exact hi.le | |
| theorem tendsto_monotone_positive_iInf (f : {r : ℝ // 0 < r} → ℝ≥0∞) | |
| (hf : Monotone f) {ι : Type*} {l : Filter ι} | |
| (ε : ι → ℝ) (hε : ∀ i, 0 < ε i) (hlim : Tendsto ε l (𝓝 0)) : | |
| Tendsto (fun i => f ⟨ε i, hε i⟩) l (𝓝 (⨅ δ, f δ)) := by | |
| have h : Tendsto f atBot (𝓝 (⨅ δ, f δ)) := tendsto_atBot_iInf hf | |
| exact h.comp (tendsto_positive_atBot_of_tendsto_zero ε hε hlim) | |
| theorem tendsto_regularizedDeterminant | |
| (τ : TracialState A) (a : A) {ι : Type*} {l : Filter ι} | |
| (ε : ι → ℝ) (hε : ∀ i, 0 < ε i) (hlim : Tendsto ε l (𝓝 0)) : | |
| Tendsto (fun i => ENNReal.ofReal | |
| (Real.exp ((τ (CFC.log (star a * a + ε i • (1 : A)))).re / 2))) | |
| l (𝓝 (fugledeKadisonDeterminant τ a)) := by | |
| unfold fugledeKadisonDeterminant | |
| exact tendsto_monotone_positive_iInf _ (regularizedDeterminant_monotone τ a) ε hε hlim | |
| theorem tendsto_regularizedDeterminant_reciprocal (τ : TracialState A) (a : A) : | |
| Tendsto (fun n : ℕ => ENNReal.ofReal | |
| (Real.exp ((τ (CFC.log (star a * a + (1 / ((n : ℝ) + 1)) • (1 : A)))).re / 2))) | |
| atTop (𝓝 (fugledeKadisonDeterminant τ a)) := by | |
| exact tendsto_regularizedDeterminant τ a _ (fun n => by positivity) | |
| (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) | |
| end Invariant233 | |
| end | |
| end OAI | |