openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /MeasureMagneticContinuity.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticSpectralContinuity | |
| import OAI.Probability.InvariantIsing.Spectral.MeasureVariationalContinuity | |
| namespace OAI | |
| /-! Compact spectral weak convergence preserves the finite-field magnetic value. -/ | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology Classical | |
| namespace InvariantIsing | |
| theorem measureMagnetic_tendsto_weak {A : Type*} [Fintype A] | |
| (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) | |
| (γ b : A → ℝ) (hγ : ∀ a, 0 ≤ γ a) (hγsum : ∑ a, γ a=1) : | |
| Tendsto (fun k => (finiteMagneticFunctional (measureR (μs k : Measure ℝ) (es k)) γ b).toReal) | |
| atTop (𝓝 (finiteMagneticFunctional (measureR (μ : Measure ℝ) e) γ b).toReal) := by | |
| have hf0 := measureVariational_ne_top_bot (μ : Measure ℝ) hK he hμ | |
| have hf := finiteMagneticFunctional_ne_top_bot _ γ b hγ hγsum hf0.1 hf0.2 | |
| 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 hfk0 := measureVariational_ne_top_bot (μs k : Measure ℝ) hK (hes k) (hs k) | |
| have hfk := finiteMagneticFunctional_ne_top_bot _ γ b hγ hγsum hfk0.1 hfk0.2 | |
| have hRk := continuous_measureR (μs k : Measure ℝ) hK (hes k) (hs k) | |
| have hl := finiteMagneticFunctional_spectral_le_add γ b _ _ hRk hR ε (fun x hx => (hk x hx).le) | |
| have hr := finiteMagneticFunctional_spectral_le_add γ b _ _ 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 | |