openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /PositiveSpectralPartition.lean
| import OAI.Probability.InvariantIsing.Spectral.PositiveSpectralLabels | |
| import OAI.Probability.InvariantIsing.Spectral.SpectralPartitionLaw | |
| import Mathlib.MeasureTheory.Measure.Support | |
| namespace OAI | |
| /-! Positive-mass cells carry the same spectral law as the original partition. -/ | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped Topology Classical BigOperators Function | |
| namespace InvariantIsing | |
| lemma ae_spectralPartitionIndex_pos {ι : Type*} [Fintype ι] | |
| (μ : Measure ℝ) [IsProbabilityMeasure μ] (S : ι → Set ℝ) | |
| (hcover : ∀ x, ∃ i, x∈S i) : | |
| ∀ᵐ x ∂μ, 0 < μ.real (S (spectralPartitionIndex S hcover x)) := by | |
| have hi (i : ι) : ∀ᵐ x ∂μ, x∈S i → 0 < μ.real (S i) := by | |
| by_cases hp : 0 < μ.real (S i) | |
| · exact Filter.Eventually.of_forall (fun _ _ => hp) | |
| · have hz : μ (S i)=0 := (measureReal_eq_zero_iff).mp | |
| (le_antisymm (le_of_not_gt hp) measureReal_nonneg) | |
| filter_upwards [compl_mem_ae_iff.mpr hz] with x hx | |
| exact fun h => (hx h).elim | |
| filter_upwards [ae_all_iff.mpr hi] with x hx | |
| exact hx _ (spectralPartitionIndex_mem S hcover x) | |
| lemma positive_spectral_partition_map {ι : Type*} [Fintype ι] | |
| (μ : Measure ℝ) [IsProbabilityMeasure μ] (S : ι → Set ℝ) | |
| (hcover : ∀ x, ∃ i, x∈S i) (hdis : Pairwise (Disjoint on S)) | |
| (hS : ∀ i, MeasurableSet (S i)) (lam : ι → ℝ) | |
| (fallback : {i // 0 < μ.real (S i)}) : | |
| μ.map (fun x => lam (positiveSpectralLabel (fun i => μ.real (S i)) fallback | |
| (spectralPartitionIndex S hcover x)))= | |
| finiteSpectralMeasure (fun i : {i // 0 < μ.real (S i)} => μ.real (S i)) | |
| (fun i => lam i) := by | |
| have he : (fun x => lam (positiveSpectralLabel (fun i => μ.real (S i)) fallback | |
| (spectralPartitionIndex S hcover x))) =ᵐ[μ] | |
| (fun x => lam (spectralPartitionIndex S hcover x)) := by | |
| filter_upwards [ae_spectralPartitionIndex_pos μ S hcover] with x hx | |
| simp only [positiveSpectralLabel,dite_eq_left hx] | |
| rw [Measure.map_congr he,spectralPartition_pushforward μ S hcover hdis hS lam, | |
| finiteSpectralMeasure_positive _ _ (fun _ => measureReal_nonneg)] | |
| lemma positive_spectral_edge_bounds {ι : Type*} [Fintype ι] | |
| (μ : Measure ℝ) [IsProbabilityMeasure μ] (S : ι → Set ℝ) | |
| (hcover : ∀ x, ∃ i, x∈S i) (lam : ι → ℝ) (δ a b : ℝ) | |
| (ha : a∈μ.support) (hb : b∈μ.support) | |
| (aMin aMax : {i // 0 < μ.real (S i)}) | |
| (hmin : ∀ i : {i // 0 < μ.real (S i)}, lam aMin ≤ lam i) | |
| (hmax : ∀ i : {i // 0 < μ.real (S i)}, lam i ≤ lam aMax) | |
| (hclose : ∀ᵐ x ∂μ, dist (lam (spectralPartitionIndex S hcover x)) x ≤ δ) : | |
| lam aMin-δ ≤ a ∧ b ≤ lam aMax+δ := by | |
| have hl : ∀ᵐ x ∂μ, x∈Ici (lam aMin-δ) := by | |
| filter_upwards [ae_spectralPartitionIndex_pos μ S hcover,hclose] with x hx hc | |
| have hh := hmin ⟨spectralPartitionIndex S hcover x,hx⟩ | |
| rw [Real.dist_eq,abs_le] at hc | |
| change lam aMin-δ ≤ x | |
| linarith | |
| have hu : ∀ᵐ x ∂μ, x∈Iic (lam aMax+δ) := by | |
| filter_upwards [ae_spectralPartitionIndex_pos μ S hcover,hclose] with x hx hc | |
| have hh := hmax ⟨spectralPartitionIndex S hcover x,hx⟩ | |
| rw [Real.dist_eq,abs_le] at hc | |
| change x ≤ lam aMax+δ | |
| linarith | |
| exact ⟨μ.support_subset_of_isClosed isClosed_Ici hl ha, | |
| μ.support_subset_of_isClosed isClosed_Iic hu hb⟩ | |
| end InvariantIsing | |
| end | |
| end OAI | |