openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /CompactPartitionPressure.lean
| import OAI.Probability.InvariantIsing.Spectral.CompactSpectralPartition | |
| import OAI.Probability.InvariantIsing.Spectral.SpectralPartitionPressure | |
| import OAI.Probability.InvariantIsing.Cavity.CavitySpectralSqueeze | |
| namespace OAI | |
| /-! The finite positive spectral partitions bracket physical mean pressure. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set | |
| open scoped Topology Classical Function | |
| namespace InvariantIsing.CompactSpectralPartition | |
| variable {ν : ProbabilityMeasure ℝ} {a b δ : ℝ} (D : CompactSpectralPartition ν a b δ) | |
| def lowerValue (x : ℝ) : ℝ := | |
| D.value (positiveSpectralLabel D.weight D.lower | |
| (spectralPartitionIndex D.cells (spectralBallCell_cover _) x)) | |
| def upperValue (x : ℝ) : ℝ := | |
| D.value (positiveSpectralLabel D.weight D.upper | |
| (spectralPartitionIndex D.cells (spectralBallCell_cover _) x)) | |
| lemma rounding_bounds (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) | |
| (x : ℝ) (hx : x∈Icc a b) : D.lowerValue x-δ ≤ x ∧ x ≤ D.upperValue x+δ := by | |
| have he := D.endpoint_bounds ha hb | |
| exact positive_spectral_rounding_bounds (ν : Measure ℝ) D.cells (spectralBallCell_cover _) | |
| D.value a b δ D.lower D.upper he.1 he.2 | |
| (positive_spectral_ball_cell_close (ν : Measure ℝ) D.center D.radius a δ | |
| D.radius_le D.residual_zero) x hx | |
| lemma rounded_mem (hab : a ≤ b) (x : ℝ) : | |
| D.lowerValue x∈Icc a b ∧ D.upperValue x∈Icc a b := | |
| ⟨D.value_mem hab _,D.value_mem hab _⟩ | |
| lemma pressure_tendsto | |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) | |
| (hpub : PanchenkoTalagrandFieldPairInput) | |
| (μ : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (μ N)] | |
| [∀ N, (μ N).IsMulRightInvariant] (eig : (N : ℕ) → Fin N → ℝ) | |
| (hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1))) | |
| atTop (𝓝 ν)) : | |
| Tendsto (fun N => ∫ V, rotatedPressure (fun i => D.lowerValue (eig N i)) | |
| (matrixRotation V⁻¹) (fun _ => 0) ∂μ N) atTop | |
| (𝓝 (variationalFunctional (measureR (D.law : Measure ℝ) D.edge)).toReal) ∧ | |
| Tendsto (fun N => ∫ V, rotatedPressure (fun i => D.upperValue (eig N i)) | |
| (matrixRotation V⁻¹) (fun _ => 0) ∂μ N) atTop | |
| (𝓝 (variationalFunctional (measureR (D.law : Measure ℝ) D.edge)).toReal) := by | |
| have hh (fallback : D.Positive) := spectral_partition_pressure_tendsto hhaar hgauss hpub μ eig ν hweak | |
| D.cells (spectralBallCell_cover _) (spectralBallCell_disjoint _) | |
| (spectralBallCell_measurable _ (fun _ => measurableSet_ball)) | |
| (spectralBallCell_null_boundary (ν : Measure ℝ) _ D.null_boundary) | |
| D.value fallback D.upper D.le_upper | |
| exact ⟨hh D.lower,hh D.upper⟩ | |
| lemma mean_pressure_bounds {N : ℕ} (hN : 0 < N) | |
| (μ : Measure (Orthogonal N)) [IsProbabilityMeasure μ] (eig : Fin N → ℝ) | |
| (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) | |
| (hab : a ≤ b) (heig : ∀ i, eig i∈Icc a b) : | |
| (∫ V, rotatedPressure (fun i => D.lowerValue (eig i)) | |
| (matrixRotation V⁻¹) (fun _ => 0) ∂μ)-δ/2 ≤ | |
| ∫ V, rotatedPressure eig (matrixRotation V⁻¹) (fun _ => 0) ∂μ ∧ | |
| (∫ V, rotatedPressure eig (matrixRotation V⁻¹) (fun _ => 0) ∂μ) ≤ | |
| (∫ V, rotatedPressure (fun i => D.upperValue (eig i)) | |
| (matrixRotation V⁻¹) (fun _ => 0) ∂μ)+δ/2 := by | |
| let K := max |a| |b| | |
| have hbound (x : ℝ) (hx : x∈Icc a b) : |x| ≤ K := by | |
| apply abs_le.mpr | |
| constructor | |
| · have hh := neg_abs_le a | |
| have hk : |a| ≤ K := le_max_left _ _ | |
| linarith [hx.1] | |
| · exact hx.2.trans ((le_abs_self b).trans (le_max_right _ _)) | |
| exact cavity_meanPressure_squeeze hN μ eig (fun i => D.lowerValue (eig i)) | |
| (fun i => D.upperValue (eig i)) (fun _ => 0) δ K | |
| (fun i => hbound _ (heig i)) | |
| (fun i => hbound _ (D.rounded_mem hab (eig i)).1) | |
| (fun i => hbound _ (D.rounded_mem hab (eig i)).2) | |
| (fun i => (D.rounding_bounds ha hb _ (heig i)).1) | |
| (fun i => (D.rounding_bounds ha hb _ (heig i)).2) | |
| end InvariantIsing.CompactSpectralPartition | |
| end | |
| end OAI | |