openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticBlockTrialLower.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticEntropyMaximum | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticGroupConsistency | |
| namespace OAI | |
| /-! The original constrained block trial attains the magnetic | |
| variational lower bound once its physical overlap is self-consistent. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter | |
| open scoped Topology BigOperators | |
| namespace InvariantIsing | |
| theorem magnetic_block_trial_eventually_lower {A ι : Type*} [Fintype A] [DecidableEq A] [Fintype ι] | |
| (ρ eig : ι → ℝ) (hρ : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a=1) | |
| (N : ℕ → ℕ) (hN : ∀ r, 0 < N r) (hNlim : Tendsto N atTop atTop) | |
| (group : ∀ r, Fin (N r) → A) (k : ℕ → A → ℕ) | |
| (hk : ∀ r a, k r a ≤ spinGroupSize (group r) a) (γ mag : A → ℝ) | |
| (hγ : ∀ a, 0 ≤ γ a) (hγsum : ∑ a, γ a=1) | |
| (hcount : ∀ r a, (spinGroupSize (group r) a : ℝ) = N r * γ a) | |
| {s H : ℝ} (hs : s<1) (hmag : ∀ a, |mag a| ≤ s) | |
| (hc : ∀ r a, (k r a : ℝ) = spinGroupSize (group r) a*((1+mag a)/2)) | |
| (h : ℕ → FieldStep) (hH : ∀ r, (h r).height (Fin.last (h r).depth) ≤ H) | |
| (p : OverlapPath) (q : ℕ → OverlapPath) | |
| (hself : Tendsto (fun r => ∫ t, |restrictedBlockOverlapPath (hN r) | |
| (spinGroupSlice (group r) (k r)) (spinGroupSlice_nonempty (group r) (k r) (hk r)) | |
| (h r) t-p t| ∂pathMeasure) atTop (𝓝 0)) | |
| (hq : Tendsto (fun r => ∫ t, |q r t-p t| ∂pathMeasure) atTop (𝓝 0)) : | |
| ∀ ε>0, ∀ᶠ r in atTop, | |
| (magneticVariationalFunctional (finiteR ρ eig hρ hsum) γ mag).toReal-ε ≤ | |
| constrainedBlockValue (spinGroupSlice (group r) (k r)) (h r)+ | |
| fieldPairing (q r) (h r)/2+spectralFunctional (finiteR ρ eig hρ hsum) (q r) := by | |
| have hgroup := magnetic_group_l1_of_block_consistency N hN hNlim group k hk γ mag | |
| hγ hγsum hcount hs hmag hc h hH p hself | |
| have hlower := magnetic_entropy_trial_eventually_lower ρ eig hρ hsum γ mag hγ hγsum | |
| (fun a => (hmag a).trans_lt hs) p q h hH hgroup hq | |
| have hgap := magneticBlockGap_tendsto N hN hNlim group k hk (fun _ => mag) hs | |
| (fun _ => hmag) hc h hH | |
| have he r : (N r : ℝ)⁻¹*(∑ j, constrainedFieldValue (h r) (mag (group r j))) = | |
| magneticGroupValue γ mag (h r) := | |
| magneticGroupValue_eq_block (hN r) (group r) γ mag (hcount r) (h r) | |
| simp_rw [he] at hgap | |
| intro ε hε | |
| filter_upwards [hlower (ε/2) (half_pos hε), | |
| hgap.eventually (Iio_mem_nhds (half_pos hε))] with r hr hgapr | |
| linarith | |
| end InvariantIsing | |
| end | |
| end OAI | |