openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /MeasureVariationalContinuity.lean
| import OAI.Probability.InvariantIsing.Spectral.MeasureSpectralContinuity | |
| namespace OAI | |
| /-! Uniform transform convergence and continuity of the variational value | |
| for the compact spectral laws in manuscript Lemma `gen:R-continuity`. -/ | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology Classical | |
| namespace InvariantIsing | |
| theorem measureR_tendsto_weak_nonneg | |
| (μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ) | |
| (es : ℕ → ℝ) (e : ℝ) (hweak : Tendsto μs atTop (𝓝 μ)) | |
| (hedge : Tendsto es atTop (𝓝 e)) | |
| {K : ℝ} (hK : 0 < K) (he : e ≤ K) (hes : ∀ k, es k ≤ K) | |
| (hs : ∀ k, ∀ᵐ y ∂(μs k : Measure ℝ), y ∈ Icc (-K) (es k)) | |
| (hμ : ∀ᵐ y ∂(μ : Measure ℝ), y ∈ Icc (-K) e) | |
| (xs : ℕ → ℝ) {x : ℝ} (hxs : Tendsto xs atTop (𝓝 x)) (hx : 0 ≤ x) : | |
| Tendsto (fun k => measureR (μs k : Measure ℝ) (es k) (xs k)) atTop | |
| (𝓝 (measureR (μ : Measure ℝ) e x)) := by | |
| rcases hx.eq_or_lt with hx | hx | |
| · subst x | |
| exact measureR_tendsto_weak_zero μs μ es e hweak hK he hes hs hμ xs hxs | |
| · exact measureR_tendsto_weak_pos μs μ es e hweak hedge | |
| (fun k => (hs k).mono fun _ h => h.2) (hμ.mono fun _ h => h.2) xs hxs hx | |
| theorem measureR_uniform_weak | |
| (μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ) | |
| (es : ℕ → ℝ) (e : ℝ) (hweak : Tendsto μs atTop (𝓝 μ)) | |
| (hedge : Tendsto es atTop (𝓝 e)) | |
| {K : ℝ} (hK : 0 < K) (he : e ≤ K) (hes : ∀ k, es k ≤ K) | |
| (hs : ∀ k, ∀ᵐ y ∂(μs k : Measure ℝ), y ∈ Icc (-K) (es k)) | |
| (hμ : ∀ᵐ y ∂(μ : Measure ℝ), y ∈ Icc (-K) e) : | |
| ∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop, ∀ x ∈ Icc (0 : ℝ) 1, | |
| |measureR (μs k : Measure ℝ) (es k) x - measureR (μ : Measure ℝ) e x| < ε := by | |
| intro ε hε | |
| by_contra hbad | |
| have hbad' : ∃ᶠ k in atTop, ∃ x ∈ Icc (0 : ℝ) 1, | |
| ε ≤ |measureR (μs k : Measure ℝ) (es k) x - measureR (μ : Measure ℝ) e x| := by | |
| simpa only [Filter.not_eventually, not_forall, not_imp, not_lt, exists_prop] using hbad | |
| obtain ⟨φ, hφ, hφbad⟩ := extraction_of_frequently_atTop hbad' | |
| choose xs hxs herr using hφbad | |
| obtain ⟨x, hx, ψ, hψ, hlim⟩ := isCompact_Icc.tendsto_subseq hxs | |
| have hR := measureR_tendsto_weak_nonneg (fun k => μs (φ (ψ k))) μ | |
| (fun k => es (φ (ψ k))) e (hweak.comp (hφ.comp hψ).tendsto_atTop) | |
| (hedge.comp (hφ.comp hψ).tendsto_atTop) hK he (fun k => hes (φ (ψ k))) | |
| (fun k => hs (φ (ψ k))) hμ (fun k => xs (ψ k)) hlim hx.1 | |
| have hfixed := (continuous_measureR (μ : Measure ℝ) hK he hμ).continuousAt.tendsto.comp hlim | |
| have hzero : Tendsto (fun k => | |
| |measureR (μs (φ (ψ k)) : Measure ℝ) (es (φ (ψ k))) (xs (ψ k)) - | |
| measureR (μ : Measure ℝ) e (xs (ψ k))|) atTop (𝓝 0) := by | |
| simpa only [Function.comp_apply, sub_self, abs_zero] using (hR.sub hfixed).abs | |
| obtain ⟨k, hk⟩ := (hzero.eventually (Iio_mem_nhds hε)).exists | |
| exact (not_le_of_gt hk) (herr (ψ k)) | |
| theorem measureVariational_ne_top_bot (μ : Measure ℝ) [IsProbabilityMeasure μ] | |
| {K e : ℝ} (hK : 0 < K) (he : e ≤ K) (hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) : | |
| variationalFunctional (measureR μ e) ≠ ⊤ ∧ variationalFunctional (measureR μ e) ≠ ⊥ := by | |
| have hR := continuous_measureR μ hK he hμ | |
| have hb x : |measureR μ e x - 0| ≤ K := by | |
| have h := measureR_bounds_all μ hμ x | |
| rw [sub_zero, abs_le] | |
| exact ⟨h.1, h.2.trans he⟩ | |
| have hspec p : |spectralFunctional (measureR μ e) p| ≤ K / 2 := by | |
| simpa only [spectralFunctional_constant, zero_div, sub_zero] using | |
| abs_spectralFunctional_sub_le (measureR μ e) (fun _ => 0) hR continuous_const | |
| K (fun x _ => hb x) p | |
| have hupper : variationalFunctional (measureR μ e) ≤ (K / 2 : ℝ) := by | |
| let p := constantPath 0 (by constructor <;> norm_num) | |
| have hi := iInf_le (fun p : OverlapPath => entropyFunctional p + | |
| (spectralFunctional (measureR μ e) p : EReal)) p | |
| simp only [p, entropyFunctional_zero, zero_add] at hi | |
| exact hi.trans (by exact_mod_cast (abs_le.mp (hspec p)).2) | |
| have hlower : ((-(K / 2) : ℝ) : EReal) ≤ variationalFunctional (measureR μ e) := by | |
| apply le_iInf | |
| intro p | |
| have hl : ((-(K / 2) : ℝ) : EReal) ≤ (spectralFunctional (measureR μ e) p : EReal) := by | |
| exact_mod_cast (abs_le.mp (hspec p)).1 | |
| have hnonneg := add_le_add (entropyFunctional_nonneg p) | |
| (le_rfl : (spectralFunctional (measureR μ e) p : EReal) ≤ _) | |
| exact hl.trans (by simpa only [zero_add] using hnonneg) | |
| exact ⟨ne_top_of_le_ne_top (EReal.coe_ne_top _) hupper, | |
| ne_bot_of_le_ne_bot (EReal.coe_ne_bot _) hlower⟩ | |
| theorem measureVariational_tendsto_weak | |
| (μs : ℕ → ProbabilityMeasure ℝ) (μ : ProbabilityMeasure ℝ) | |
| (es : ℕ → ℝ) (e : ℝ) (hweak : Tendsto μs atTop (𝓝 μ)) | |
| (hedge : Tendsto es atTop (𝓝 e)) | |
| {K : ℝ} (hK : 0 < K) (he : e ≤ K) (hes : ∀ k, es k ≤ K) | |
| (hs : ∀ k, ∀ᵐ y ∂(μs k : Measure ℝ), y ∈ Icc (-K) (es k)) | |
| (hμ : ∀ᵐ y ∂(μ : Measure ℝ), y ∈ Icc (-K) e) : | |
| Tendsto (fun k => (variationalFunctional (measureR (μs k : Measure ℝ) (es k))).toReal) | |
| atTop (𝓝 (variationalFunctional (measureR (μ : Measure ℝ) e)).toReal) := by | |
| have hf := measureVariational_ne_top_bot (μ : Measure ℝ) hK he hμ | |
| have hR := continuous_measureR (μ : Measure ℝ) hK he hμ | |
| apply Metric.tendsto_nhds.mpr | |
| intro ε hε | |
| filter_upwards [measureR_uniform_weak μs μ es e hweak hedge hK he hes hs hμ ε hε] with k hk | |
| have hfk := measureVariational_ne_top_bot (μs k : Measure ℝ) hK (hes k) (hs k) | |
| have hRk := continuous_measureR (μs k : Measure ℝ) hK (hes k) (hs k) | |
| have hl := variationalFunctional_le_add _ _ hRk hR ε (fun x hx => (hk x hx).le) | |
| have hr := variationalFunctional_le_add _ _ hR hRk ε | |
| (fun x hx => by simpa only [abs_sub_comm] using (hk x hx).le) | |
| rw [← EReal.coe_toReal hfk.1 hfk.2, ← EReal.coe_toReal hf.1 hf.2, | |
| ← EReal.coe_add, EReal.coe_le_coe_iff] at hl hr | |
| rw [Real.dist_eq, abs_lt] | |
| constructor <;> linarith | |
| end InvariantIsing | |
| end | |
| end OAI | |