openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /CompactPartitionContinuity.lean
| import OAI.Probability.InvariantIsing.Spectral.CompactSpectralPartition | |
| import OAI.Probability.InvariantIsing.Spectral.MeasureVariationalContinuity | |
| namespace OAI | |
| /-! The finite positive-mass partition values converge to the compact spectral value. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| lemma compact_partition_laws_tendsto (ν : ProbabilityMeasure ℝ) (a b : ℝ) | |
| (δ : ℕ → ℝ) (D : (k : ℕ) → CompactSpectralPartition ν a b (δ k)) | |
| (hδ : Tendsto δ atTop (𝓝 0)) : | |
| Tendsto (fun k => (D k).law) atTop (𝓝 ν) := by | |
| have hh := spectral_quantization_weak ν | |
| (fun k => spectralBallQuantizer (D k).center (D k).radius a) | |
| (fun k => measurable_spectralBallQuantizer _ _ _) δ hδ (fun k => (D k).quantizer_close) | |
| apply hh.congr | |
| intro k | |
| apply Subtype.ext | |
| exact (D k).law_eq_map.symm | |
| lemma compact_partition_edges_tendsto (ν : ProbabilityMeasure ℝ) (a b : ℝ) | |
| (δ : ℕ → ℝ) (D : (k : ℕ) → CompactSpectralPartition ν a b (δ k)) | |
| (hab : a ≤ b) (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) | |
| (hδ : Tendsto δ atTop (𝓝 0)) : | |
| Tendsto (fun k => (D k).edge) atTop (𝓝 b) := by | |
| have hlo k : b-δ k ≤ (D k).edge := by linarith [(D k).endpoint_bounds ha hb] | |
| have hhi k : (D k).edge ≤ b := ((D k).value_mem hab (D k).upper).2 | |
| exact tendsto_of_tendsto_of_tendsto_of_le_of_le | |
| (by simpa only [sub_zero] using tendsto_const_nhds.sub hδ) | |
| tendsto_const_nhds hlo hhi | |
| lemma compact_partition_variational_tendsto (ν : ProbabilityMeasure ℝ) (a b : ℝ) | |
| (δ : ℕ → ℝ) (D : (k : ℕ) → CompactSpectralPartition ν a b (δ k)) | |
| (hab : a ≤ b) (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) | |
| (hbound : (ν : Measure ℝ).support ⊆ Icc a b) (hδ : Tendsto δ atTop (𝓝 0)) : | |
| Tendsto (fun k => (variationalFunctional (measureR ((D k).law : Measure ℝ) (D k).edge)).toReal) | |
| atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by | |
| let K := |a|+|b|+1 | |
| have hK : 0 < K := by dsimp [K]; positivity | |
| have haK : -K ≤ a := by dsimp [K]; linarith [neg_abs_le a,abs_nonneg b] | |
| have hbK : b ≤ K := by dsimp [K]; linarith [le_abs_self b,abs_nonneg a] | |
| exact measureVariational_tendsto_weak (fun k => (D k).law) ν | |
| (fun k => (D k).edge) b (compact_partition_laws_tendsto ν a b δ D hδ) | |
| (compact_partition_edges_tendsto ν a b δ D hab ha hb hδ) hK hbK | |
| (fun k => (((D k).value_mem hab (D k).upper).2).trans hbK) | |
| (fun k => ((D k).law_support hab).mono (fun _ hx => ⟨haK.trans hx.1,hx.2⟩)) | |
| (by | |
| filter_upwards [(ν : Measure ℝ).support_mem_ae] with x hx | |
| exact ⟨haK.trans (hbound hx).1,(hbound hx).2⟩) | |
| end InvariantIsing | |
| end | |
| end OAI | |