openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticCountTemplateLower.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticFiniteTemplateLower | |
| import OAI.Probability.InvariantIsing.Fields.PositiveFieldApproximation | |
| namespace OAI | |
| /-! Spectral and field rational populations with different denominators | |
| have a common physical block template. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter | |
| open scoped Topology BigOperators | |
| namespace InvariantIsing | |
| theorem finite_count_template_pressure_lower | |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) | |
| (hpub : PanchenkoTalagrandRestrictedFieldPairInput) | |
| {m n : ℕ} (hm : 2 ≤ m) (hn : 0 < n) | |
| (ρ lam : Fin m → ℝ) (hρ : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a=1) | |
| {K : ℝ} (hK : 0 ≤ K) (hlam : ∀ a, |lam a| ≤ K) | |
| (amax : Fin m) (hmax : ∀ a, lam a ≤ lam amax) | |
| (μ : (M : ℕ) → Measure (Orthogonal M)) [∀ M, IsProbabilityMeasure (μ M)] | |
| [∀ M, (μ M).IsMulRightInvariant] | |
| (spec : Fin m → ℕ) (hsp : ∀ a, 0 < spec a) (hspec : ∑ a, spec a=n) | |
| (hρspec : ∀ a, (spec a : ℝ)=(n : ℝ)*ρ a) | |
| (e : (M : ℕ) → Fin M → Fin m) | |
| (he : Tendsto (fun M a => (spinGroupSize (e M) a : ℝ)/M) atTop (𝓝 ρ)) | |
| {A : Type*} [Fintype A] [DecidableEq A] | |
| (w : A → ℕ) (hd : 0 < ∑ a, w a) | |
| (γ : A → ℝ) (hγ : ∀ a, 0 ≤ γ a) (hγsum : ∑ a, γ a=1) | |
| (hcount : ∀ a, (w a : ℝ)=(∑ a, w a : ℕ)*γ a) | |
| (g : (M : ℕ) → Fin M → A) (b : A → ℝ) | |
| (hg : Tendsto (fun M a => (spinGroupSize (g M) a : ℝ)/M) atTop (𝓝 γ)) | |
| {D : ℝ} (hD : 0 ≤ D) (hb : ∀ a, |b a| ≤ D) | |
| : ∀ ε > 0, ∀ᶠ M in atTop, | |
| (finiteMagneticFunctional (finiteR ρ lam hρ hsum) γ b).toReal-ε ≤ | |
| ∫ V, rotatedPressure (fun i => lam (e M i)) (matrixRotation V⁻¹) | |
| (fun i => b (g M i)) ∂μ M := by | |
| let d := ∑ a, w a | |
| have htot : ∑ a, n*w a=n*d := by rw [← Finset.mul_sum] | |
| let group := countedFieldGroup (fun a => n*w a) htot | |
| have hgroup a : (spinGroupSize group a : ℝ)=(n*d : ℕ)*γ a := by | |
| rw [countedFieldGroup_size,Nat.cast_mul,hcount,Nat.cast_mul] | |
| ring | |
| have hspec' : ∑ a, d*spec a=n*d := by rw [← Finset.mul_sum,hspec,Nat.mul_comm] | |
| have hρspec' a : (d*spec a : ℕ)=(n*d : ℕ)*ρ a := by | |
| rw [Nat.cast_mul,hρspec,Nat.cast_mul] | |
| ring | |
| exact finite_template_magnetic_pressure_lower hhaar hgauss hpub hm (Nat.mul_pos hn hd) | |
| ρ lam hρ hsum hK hlam amax hmax μ (fun a => d*spec a) | |
| (fun a => Nat.mul_pos hd (hsp a)) hspec' hρspec' e he group γ hγ hγsum hgroup | |
| g b hg hD hb | |
| end InvariantIsing | |
| end | |
| end OAI | |