openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /SelectedAlphabetPressure.lean
| import OAI.Probability.InvariantIsing.Cavity.CavitySelectedDimensions | |
| import OAI.Probability.InvariantIsing.Spectral.FiniteSpectralMeasure | |
| namespace OAI | |
| /-! The finite-alphabet pressure theorem along diverging dimensions. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter | |
| open scoped Topology BigOperators | |
| namespace InvariantIsing | |
| theorem selected_finite_alphabet_pressure_tendsto | |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) | |
| (hpub : PanchenkoTalagrandFieldPairInput) {m : ℕ} | |
| (μ : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (μ N)] | |
| [∀ N, (μ N).IsMulRightInvariant] (lam : Fin m → ℝ) | |
| (D : ℕ → ℕ) (hD : ∀ k, 0 < D k) (hDlim : Tendsto D atTop atTop) | |
| (g : (k : ℕ) → Fin (D k) → Fin m) | |
| (ρ : Fin m → ℝ) (hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a=1) | |
| (hρlim : Tendsto (fun k a => ((cavitySpectralGroup (g k) a).card : ℝ)/D k) atTop (𝓝 ρ)) : | |
| Tendsto (fun k => ∫ V, rotatedPressure (fun i => lam (g k i)) | |
| (matrixRotation V⁻¹) (fun _ => 0) ∂μ (D k)) atTop | |
| (𝓝 (variationalFunctional (finiteR ρ lam hρ hρsum)).toReal) := by | |
| by_cases hm : 2 ≤ m | |
| · exact cavity_selected_dimensions_pressure_tendsto hhaar hgauss hpub hm μ lam D hD hDlim g ρ hρ hρsum hρlim | |
| have hm0 : 0 < m := by | |
| obtain ⟨a,_⟩ := exists_pos_spectral_weight (fun a => (hρ a).le) hρsum | |
| exact Nat.pos_of_ne_zero (by intro hz; subst m; exact Fin.elim0 a) | |
| have hm1 : m=1 := by omega | |
| subst m | |
| have hlam : lam=fun _ => lam 0 := funext fun a => congrArg lam (Subsingleton.elim a 0) | |
| have hR : finiteR ρ lam hρ hρsum=fun _ => lam 0 := by | |
| funext x | |
| rw [hlam] | |
| exact finiteR_constant ρ hρ hρsum x (lam 0) | |
| rw [hR, variationalFunctional_constant, EReal.toReal_coe] | |
| have he : ∀ᶠ k in atTop, (∫ V, rotatedPressure (fun i => lam (g k i)) | |
| (matrixRotation V⁻¹) (fun _ => 0) ∂μ (D k))=lam 0/2 := by | |
| apply Filter.Eventually.of_forall | |
| intro k | |
| have hpos : 0 < D k := hD k | |
| have heig : (fun i => lam (g k i))=fun _ => lam 0 := | |
| funext fun i => congrArg lam (Subsingleton.elim (g k i) 0) | |
| rw [heig] | |
| simp only [rotatedPressure_constant hpos,integral_const,measureReal_def,measure_univ, | |
| ENNReal.toReal_one,one_smul] | |
| exact tendsto_const_nhds.congr' (he.mono fun _ h => h.symm) | |
| end InvariantIsing | |
| end | |
| end OAI | |