openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticConvergingUpper.lean
| import OAI.Probability.InvariantIsing.Magnetic.RestrictedSpectrumEntropyUpper | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticGroupEntropyLimit | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticGroupInterior | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticTrialApproximation | |
| import OAI.Probability.InvariantIsing.Core.TrialGridApproximation | |
| namespace OAI | |
| /-! Manuscript Proposition up:bound, in the equivalent eventual-epsilon | |
| form against every trial of finite entropy. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter | |
| open scoped BigOperators Topology | |
| namespace InvariantIsing | |
| theorem finiteSpectrum_magneticPressure_converging_upper | |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) | |
| (N : ℕ → ℕ) (hN : ∀ k, 3 ≤ N k) (hNlim : Tendsto N atTop atTop) (m : ℕ) | |
| (μ : (k : ℕ) → Measure (SpecialOrthogonal (N k))) [∀ k, IsProbabilityMeasure (μ k)] | |
| (hμinv : ∀ k, (μ k).IsMulLeftInvariant) | |
| (eig : (k : ℕ) → Fin (N k) → ℝ) | |
| (K : ℝ) (hK : 0 < K) (heig : ∀ k i, |eig k i| ≤ K) | |
| (I : (k : ℕ) → Fin m → Finset (Fin (N k))) | |
| (hdis : ∀ k, Set.PairwiseDisjoint (Set.univ : Set (Fin m)) (I k)) | |
| (hcover : ∀ k, Finset.univ.biUnion (I k) = Finset.univ) | |
| (lam : Fin m → ℝ) (hlam : ∀ k a i, i ∈ I k a → eig k i = lam a) | |
| (ρ : Fin m → ℝ) (hρpos : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) | |
| (hρ : Tendsto (fun k a => ((I k a).card : ℝ) / N k) atTop (𝓝 ρ)) | |
| {A : Type*} [Fintype A] [DecidableEq A] | |
| (group : ∀ k, Fin (N k) → A) (count : ℕ → A → ℕ) | |
| (hcount : ∀ k a, count k a ≤ spinGroupSize (group k) a) | |
| (γs mags : ℕ → A → ℝ) (γ mag : A → ℝ) | |
| (hγs : Tendsto γs atTop (𝓝 γ)) (hmags : Tendsto mags atTop (𝓝 mag)) | |
| (hmag : ∀ a, |mag a|<1) | |
| (hc : ∀ k a, (count k a : ℝ)=spinGroupSize (group k) a*((1+mags k a)/2)) | |
| (hgroup : ∀ k a, (spinGroupSize (group k) a : ℝ)=N k*γs k a) | |
| (p : OverlapPath) (S : ℝ) (hS : magneticEntropyFunctional γ mag p ≤ (S : EReal)) : | |
| ∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop, | |
| (∫ U : SpecialOrthogonal (N k), | |
| restrictedRotatedPressure (spinGroupSlice (group k) (count k)) (eig k) (specialRotation U) (fun _ => 0) ∂μ k) ≤ | |
| S + spectralFunctional (finiteR ρ lam hρpos hρsum) p + ε := by | |
| intro ε hε | |
| have hhalf : 0 < ε / 2 := half_pos hε | |
| have hgrid := lowerGridPath_spectral_tendsto ρ lam hρpos hρsum p | |
| obtain ⟨n, hn⟩ := (hgrid.eventually (gt_mem_nhds | |
| (show spectralFunctional (finiteR ρ lam hρpos hρsum) p < | |
| spectralFunctional (finiteR ρ lam hρpos hρsum) p + ε / 2 by linarith))).exists | |
| obtain ⟨r, hr, hmr, hmkr⟩ := exists_uniform_magnetization_radius mags mag hmags hmag | |
| have hup := finiteSpectrum_restrictedPressure_entropy_upper hhaar hgauss N hN hNlim m n | |
| μ hμinv (fun k => spinGroupSlice (group k) (count k)) | |
| (fun k => spinGroupSlice_nonempty (group k) (count k) (hcount k)) eig K hK heig I hdis hcover lam hlam ρ hρpos hρsum hρ | |
| (uniformCut n) (uniformCut_strict n) (uniformCut_zero n) (uniformCut_last n) | |
| (lowerGridPath p n) | |
| (fun i => (1 - 1 / (n + 1 : ℕ)) * p ((i.val : ℝ) / (n + 1 : ℕ))) | |
| (lowerGridPath_on_cell p n) (1 / (n + 1 : ℕ)) (by positivity) | |
| (ae_of_all _ (lowerGridPath_below_one p n)) S | |
| (magnetic_group_entropy_limit N (fun k => by have := hN k; omega) | |
| group count hcount γs mags γ mag hγs hmags hr hmr hmkr hgroup hc | |
| (lowerGridPath p n) S ((lowerGridPath_magneticEntropy_le γ mag p n).trans hS) | |
| (uniformCut n) (uniformCut_strict n) (uniformCut_zero n) (uniformCut_last n)) (ε / 2) hhalf | |
| filter_upwards [hup] with k hk | |
| linarith | |
| end InvariantIsing | |
| end | |
| end OAI | |