openAI-math-dataset / lean /OAI /Analysis /JiangSu /DimensionFunctions /FiniteSpectralIntegralPositivity.lean
| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.DimensionFunctions.FiniteSpectralIntegralCommuting | |
| import OAI.Analysis.JiangSu.DimensionFunctions.FiniteSpectralIntegralRootBounds | |
| import OAI.Analysis.JiangSu.Comparison.FiniteLscComparison | |
| namespace OAI | |
| noncomputable section | |
| namespace Pureness | |
| open TraceCone MeasureTheory Set OrderedSemigroup | |
| open scoped ENNReal NNReal | |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| theorem finiteSpectralIntegral_cut_lower_bound | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (ε : ℝ≥0) : | |
| d (finitePositiveClass n (Positive.cut ε M)) * (ε : ℝ≥0∞) ≤ finiteSpectralIntegral d n M := by | |
| rw [← ENNReal.ofReal_coe_nnreal,← positive_halfline_indicator_integral] | |
| apply lintegral_mono_ae | |
| filter_upwards [ae_restrict_mem measurableSet_Ioi] with t ht | |
| by_cases htε : t < ε | |
| · rw [Set.indicator_of_mem (show t ∈ Iio (ε : ℝ) from htε)] | |
| have hle : t.toNNReal ≤ ε := by | |
| change (t.toNNReal : ℝ) ≤ ε | |
| rw [Real.coe_toNNReal t (le_of_lt ht)] | |
| exact htε.le | |
| exact finiteCutValue_antitone d hd n M hle | |
| · rw [Set.indicator_of_notMem (show t ∉ Iio (ε : ℝ) from htε)] | |
| exact bot_le | |
| theorem finiteSpectralIntegral_pos | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) | |
| (hpos : 0 < regularizedFiniteDimension d hd (finitePositiveClass n M)) : | |
| 0 < finiteSpectralIntegral d n M := by | |
| rw [regularizedFiniteDimension_eval] at hpos | |
| obtain ⟨ε,hε⟩ := lt_iSup_iff.mp hpos | |
| obtain ⟨hεpos,hcut⟩ := lt_iSup_iff.mp hε | |
| exact (ENNReal.mul_pos hcut.ne' (by exact_mod_cast hεpos.ne')).trans_le | |
| (finiteSpectralIntegral_cut_lower_bound d hd n M ε) | |
| variable [Nontrivial A] | |
| theorem regularizedState_positive_nonzero (hsimple : TopologicallySimple A) | |
| (y : finiteCuntzSubmonoid (A := A)) (hy : y ≠ 0) (f : StateSpace y) | |
| (x : finiteCuntzSubmonoid (A := A)) (hx : x ≠ 0) : | |
| 0 < regularizedFiniteDimension f.toAddHom f.monotone x := by | |
| apply pos_iff_ne_zero.mpr | |
| intro hz | |
| obtain ⟨k,hk⟩ := finiteCuntz_nonzero_orderUnit hsimple x hx finiteUnitClass | |
| have h := regularizedFiniteDimension_monotone f.toAddHom f.monotone hk | |
| rw [map_nsmul,hz,nsmul_zero] at h | |
| exact (not_le_of_gt (finiteState_regularized_unit hsimple y hy f).1) h | |
| omit [Nontrivial A] in | |
| theorem regularizedState_integral_finite (hsimple : TopologicallySimple A) | |
| (y : finiteCuntzSubmonoid (A := A)) (hy : y ≠ 0) (f : StateSpace y) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) : | |
| finiteSpectralIntegral (regularizedFiniteDimension f.toAddHom f.monotone) n M ≠ ⊤ := by | |
| apply finiteSpectralIntegral_ne_top _ (regularizedFiniteDimension_monotone f.toAddHom f.monotone) | |
| exact ne_top_of_le_ne_top (finiteCuntz_state_finite hsimple y hy f _) | |
| (regularizedFiniteDimension_le f.toAddHom f.monotone _) | |
| theorem regularizedState_integral_positive (hsimple : TopologicallySimple A) | |
| (y : finiteCuntzSubmonoid (A := A)) (hy : y ≠ 0) (f : StateSpace y) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (hM : M.1 ≠ 0) : | |
| 0 < finiteSpectralIntegral (regularizedFiniteDimension f.toAddHom f.monotone) n M := by | |
| apply finiteSpectralIntegral_pos _ (regularizedFiniteDimension_monotone f.toAddHom f.monotone) | |
| erw [regularizedFiniteDimension_idempotent] | |
| exact regularizedState_positive_nonzero hsimple y hy f _ | |
| (fun h => hM ((finitePositiveClass_eq_zero_iff n M).mp h)) | |
| omit [Nontrivial A] in | |
| theorem regularizedState_integral_unit (y : finiteCuntzSubmonoid (A := A)) (f : StateSpace y) : | |
| finiteSpectralIntegral (regularizedFiniteDimension f.toAddHom f.monotone) 1 ⟨1,zero_le_one⟩ = | |
| f.1 finiteUnitClass := by | |
| rw [finiteSpectralIntegral_projection _ _ _ (by simp [IsIdempotentElem])] | |
| exact regularizedFiniteDimension_projection f.toAddHom f.monotone 1 ⟨1,zero_le_one⟩ (by simp [IsIdempotentElem]) | |
| end Pureness | |
| namespace Pureness | |
| open TraceCone MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology | |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| lemma finiteSpectralIntegral_root_cut_lower | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (hM : ‖M.1‖ ≤ 1) | |
| (δ r α : ℝ≥0) (hr : 0 < r) (hα : (α : ℝ) < (δ : ℝ)^(r : ℝ)) : | |
| d (finitePositiveClass n (Positive.cut δ M)) * (α : ℝ≥0∞) ≤ | |
| finiteSpectralIntegral d n (Positive.power r M) := by | |
| have hc : CuntzBelow (Positive.cut δ M) (Positive.cut α (Positive.power r M)) := | |
| CuntzBelow.of_le_smul _ _ _ (one_div_nonneg.mpr (sub_pos.mpr hα).le) | |
| (positive_cut_le_smul_root_cut M hM δ r α hr hα) | |
| have h := hd ((finitePositiveClass_le_iff n _ _).mpr ((cuntzSubequiv_iff_shared _ _).mpr hc)) | |
| exact (mul_le_mul h (le_refl (α : ℝ≥0∞)) bot_le bot_le).trans (finiteSpectralIntegral_cut_lower_bound d hd n _ α) | |
| theorem finiteSpectralIntegral_cut_le_iSup_roots | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (hM : ‖M.1‖ ≤ 1) | |
| (δ : ℝ≥0) (hδ : 0 < δ) : | |
| d (finitePositiveClass n (Positive.cut δ M)) ≤ | |
| ⨆ k, finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M) := by | |
| let R := ⨆ k, finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M) | |
| have hb (α : ℝ≥0) (hα : α < 1) : | |
| d (finitePositiveClass n (Positive.cut δ M))*(α : ℝ≥0∞) ≤ R := by | |
| have he : Tendsto (fun k => (δ : ℝ)^(Positive.rootExponent k : ℝ)) atTop (𝓝 1) := by | |
| have hexp : Tendsto (fun k => (Positive.rootExponent k : ℝ)) atTop (𝓝 0) := by | |
| simpa [Positive.rootExponent] using (tendsto_one_div_add_atTop_nhds_zero_nat : | |
| Tendsto (fun k : ℕ => 1/((k : ℝ)+1)) atTop (𝓝 0)) | |
| simpa using tendsto_const_nhds.rpow hexp (Or.inl (show (δ : ℝ) ≠ 0 by exact_mod_cast hδ.ne')) | |
| obtain ⟨k,hk⟩ := ((tendsto_order.mp he).1 α hα).exists | |
| exact (finiteSpectralIntegral_root_cut_lower d hd n M hM δ _ α | |
| (Positive.rootExponent_pos k) hk).trans (le_iSup (fun j => finiteSpectralIntegral d n (Positive.power (Positive.rootExponent j) M)) k) | |
| let α : ℕ → ℝ≥0 := fun k => (1-(Positive.rootExponent k : ℝ)).toNNReal | |
| have hα (k : ℕ) : α k < 1 := by | |
| dsimp [α] | |
| change ((1-(Positive.rootExponent k : ℝ)).toNNReal : ℝ) < 1 | |
| have hh : (0 : ℝ) ≤ 1-(Positive.rootExponent k : ℝ) := | |
| sub_nonneg.mpr (show (Positive.rootExponent k : ℝ) ≤ 1 from Positive.rootExponent_le_one k) | |
| rw [Real.coe_toNNReal _ hh] | |
| exact sub_lt_self 1 (Positive.rootExponent_pos k) | |
| have ht : Tendsto (fun k => (α k : ℝ≥0∞)) atTop (𝓝 1) := by | |
| have he : Tendsto (fun k => (Positive.rootExponent k : ℝ)) atTop (𝓝 0) := by | |
| simpa [Positive.rootExponent] using (tendsto_one_div_add_atTop_nhds_zero_nat : | |
| Tendsto (fun k : ℕ => 1/((k : ℝ)+1)) atTop (𝓝 0)) | |
| change Tendsto (fun k => ENNReal.ofReal (1-(Positive.rootExponent k : ℝ))) atTop (𝓝 1) | |
| simpa only [Function.comp_def,sub_zero,ENNReal.ofReal_one] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp | |
| ((tendsto_const_nhds (x := (1 : ℝ))).sub he) | |
| have hlim := ENNReal.Tendsto.mul | |
| (tendsto_const_nhds (x := d (finitePositiveClass n (Positive.cut δ M)))) | |
| (Or.inr ENNReal.one_ne_top) ht (Or.inl one_ne_zero) | |
| simpa only [mul_one] using le_of_tendsto hlim (Eventually.of_forall (fun k => hb (α k) (hα k))) | |
| theorem finiteSpectralIntegral_iSup_roots | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (hlsc : ∀ n, LowerSemicontinuous (fun M : Positive (FinMatrix A n) => d (finitePositiveClass n M))) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (hM : ‖M.1‖ ≤ 1) : | |
| (⨆ k, finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M)) = | |
| d (finitePositiveClass n M) := by | |
| apply le_antisymm (iSup_le (finiteSpectralIntegral_root_le d hd n M hM)) | |
| have he : d (finitePositiveClass n M) = finiteRegularizedValue d n M := by | |
| rw [← regularizedFiniteDimension_eval d hd n M, regularizedFiniteDimension_eq_of_lowerSemicontinuous d hd hlsc] | |
| rw [he] | |
| exact iSup_le fun δ => iSup_le fun hδ => finiteSpectralIntegral_cut_le_iSup_roots d hd n M hM δ hδ | |
| theorem finiteSpectralIntegral_roots_tendsto_contraction | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (hlsc : ∀ n, LowerSemicontinuous (fun M : Positive (FinMatrix A n) => d (finitePositiveClass n M))) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (hM : ‖M.1‖ ≤ 1) : | |
| Tendsto (fun k => finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M)) atTop | |
| (𝓝 (d (finitePositiveClass n M))) := by | |
| rw [← finiteSpectralIntegral_iSup_roots d hd hlsc n M hM] | |
| exact tendsto_atTop_iSup (finiteSpectralIntegral_roots_monotone d hd n M hM) | |
| theorem finiteSpectralIntegral_roots_tendsto | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (hlsc : ∀ n, LowerSemicontinuous (fun M : Positive (FinMatrix A n) => d (finitePositiveClass n M))) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) : | |
| Tendsto (fun k => finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M)) atTop | |
| (𝓝 (d (finitePositiveClass n M))) := by | |
| let s : ℝ≥0 := max ‖M.1‖₊ 1 | |
| have hs : 0 < s := lt_of_lt_of_le zero_lt_one (le_max_right _ _) | |
| have he : Positive.smul s (Positive.normalize M) = M := by | |
| apply Subtype.ext | |
| change (s : ℝ) • ((s⁻¹ : ℝ≥0) : ℝ) • M.1 = M.1 | |
| rw [NNReal.coe_inv, smul_smul, mul_inv_cancel₀ (show (s : ℝ) ≠ 0 by exact_mod_cast hs.ne'), one_smul] | |
| have hc : finitePositiveClass n (Positive.normalize M) = finitePositiveClass n M := | |
| finitePositiveClass_smul n M s⁻¹ (inv_pos.mpr hs) | |
| have hlim := finiteSpectralIntegral_roots_tendsto_contraction d hd hlsc n (Positive.normalize M) | |
| (Positive.normalize_norm_le_one M) | |
| rw [hc] at hlim | |
| have hscalar := TraceCone.ExtendedTrace.root_scalars_tendsto_one s hs | |
| have h := ENNReal.Tendsto.mul hscalar (Or.inl one_ne_zero) hlim (Or.inr ENNReal.one_ne_top) | |
| simp only [one_mul] at h | |
| convert h using 1 | |
| funext k | |
| nth_rw 1 [← he] | |
| rw [Positive.power_smul _ (Positive.rootExponent_pos k),finiteSpectralIntegral_smul d hd] | |
| theorem finiteSpectralIntegral_root_rank | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (hlsc : ∀ n, LowerSemicontinuous (fun M : Positive (FinMatrix A n) => d (finitePositiveClass n M))) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) : | |
| liminf (fun k => finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M)) atTop = | |
| d (finitePositiveClass n M) := | |
| (finiteSpectralIntegral_roots_tendsto d hd hlsc n M).liminf_eq | |
| theorem regularizedFiniteDimension_integral_roots_tendsto | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) : | |
| Tendsto (fun k => finiteSpectralIntegral (regularizedFiniteDimension d hd) n | |
| (Positive.power (Positive.rootExponent k) M)) atTop | |
| (𝓝 (regularizedFiniteDimension d hd (finitePositiveClass n M))) := by | |
| apply finiteSpectralIntegral_roots_tendsto _ (regularizedFiniteDimension_monotone d hd) | |
| intro m | |
| simpa only [regularizedFiniteDimension_eval] using finiteRegularizedValue_lowerSemicontinuous d hd m | |
| end Pureness | |
| end | |
| end OAI | |