| import Mathlib |
| import OAI.Probability.Perceptron.Variational.QuantileReference |
| import OAI.Probability.Perceptron.Variational.QuantileLimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal NNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
| |
| instance compactPositivePairLaw_fst_isProbability |
| (ν : ProbabilityMeasure (CompactArray CompactJointOverlap)) : |
| IsProbabilityMeasure ((compactPositivePairLaw ν).map Prod.fst) := |
| inferInstance |
| |
| lemma compactPositivePairLaw_spin_quantile |
| (ν : ProbabilityMeasure (CompactArray CompactJointOverlap)) |
| (hn : ∀ᵐ Q ∂(ν : Measure (CompactArray CompactJointOverlap)), 0≤(Q 0 1).1.val) : |
| (ν : Measure (CompactArray CompactJointOverlap)).map (fun Q => (Q 0 1).1)= |
| (quantileSpinLaw (boundedQuantile ((compactPositivePairLaw ν).map Prod.fst)) |
| (boundedQuantile_measurable _) : Measure CompactOverlap) := by |
| have : IsProbabilityMeasure ((compactPositivePairLaw ν).map Prod.fst) := |
| inferInstance |
| change _=timeLaw.map (timeSpin ∘ boundedQuantile ((compactPositivePairLaw ν).map Prod.fst)) |
| rw [←Measure.map_map timeSpin_continuous.measurable (boundedQuantile_measurable _), |
| timeLaw_eq_volume,boundedQuantile_law] |
| unfold compactPositivePairLaw |
| rw [Measure.map_map (f := fun Q : CompactArray CompactJointOverlap => compactPositivePair (Q 0 1)) |
| measurable_fst (by exact compactPositivePair_measurable.comp (by fun_prop))] |
| rw [Measure.map_map timeSpin_continuous.measurable |
| (show Measurable (Prod.fst ∘ fun Q : CompactArray CompactJointOverlap => compactPositivePair (Q 0 1)) from |
| measurable_fst.comp (compactPositivePair_measurable.comp (by fun_prop)))] |
| apply Measure.map_congr |
| filter_upwards [hn] with Q hQ |
| apply Subtype.ext |
| change (Q 0 1).1.val=max 0 (Q 0 1).1.val |
| exact (max_eq_right hQ).symm |
| |
| theorem compactGG_freshLog_control (ν : ProbabilityMeasure (CompactArray CompactJointOverlap)) |
| (hGG : ∀ (r : ℕ) (i : Fin r) (G : CompactBlock CompactJointOverlap r →ᵇ ℝ) |
| (a : CompactJointOverlap →ᵇ ℝ), compactGGDefect ν r i G a=0) |
| (hgeo : ∀ᵐ Q ∂(ν : Measure (CompactArray CompactJointOverlap)), CompactSpinGeometry Q) |
| (hn : ∀ᵐ Q ∂(ν : Measure (CompactArray CompactJointOverlap)), 0≤(Q 0 1).1.val) |
| (g : Jet3) (η : ProbabilityMeasure (FreshPartitionRange g.f)) |
| (hm : ∀ r, (∫ x, x.val^r ∂(η : Measure (FreshPartitionRange g.f)))= |
| ∫ Q, freshReplicaArrayKernel (expBCF 1 g.f) r Q ∂(ν : Measure _)) |
| (P : Measure BrownianPath) [IsProbabilityMeasure P] (hB : IsBrownianReal brownianEval P) : |
| (∫ x, Real.log x.val ∂(η : Measure (FreshPartitionRange g.f)))= |
| controlValue P g.f (quantileTrial (boundedQuantile ((compactPositivePairLaw ν).map Prod.fst))) := by |
| obtain ⟨ρ,ξ,hG,hge,hp,hξ,hl⟩ := quantile_reference_exists |
| (boundedQuantile ((compactPositivePairLaw ν).map Prod.fst)) (boundedQuantile_monotone _) g P hB |
| rw [freshLog_eq_of_pair_law ν ρ hGG hG hgeo hge |
| ((compactPositivePairLaw_spin_quantile ν hn).trans hp.symm) g.f η ξ hm hξ] |
| exact hl |
| |
| theorem source_contact_freshLog_control (k : ℕ) (g : Jet3) (p d : ℕ → ℕ) (z : Fin k → ℝ) |
| (hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) |
| (t : ℕ → ℝ≥0) (h : ℕ → Fin (k+1) → ℝ) |
| (hh0 : ∀ n l, 0≤h n l) (hh : ∀ n, Monotone (h n)) |
| (u : (n : ℕ) → Fin (n+1) → ℝ) |
| (hu : ∀ n, u n ∈ Icc (fun _ => 1) (fun _ => 2)) |
| (hmin : ∀ n, IsMinOn (fun v => quadraticBoxPenalty (sourcePenaltyWeight (n+1)) v- |
| sourceExpectedPressure n k g.f (fun j => p j.val) (fun j => d j.val) (h n) z (t n) v) |
| (Icc (fun _ : Fin (n+1) => 1) (fun _ => 2)) (u n)) |
| {H T : ℝ} (hH : 0≤H) (hhH : ∀ n, h n 0≤H) (ht : ∀ n, (t n:ℝ)≤T) |
| (hcover : ∀ a b : ℕ, 1≤a+b → ∃ j, p j=a ∧ d j=b) |
| {ν : ProbabilityMeasure (CompactArray CompactJointOverlap)} {s : ℕ → ℕ} (hs : StrictMono s) |
| (hlim : Tendsto (fun n => sourceGibbsArrayLaw (s n) k g.f |
| (fun a => p a.val) (fun a => d a.val) (h (s n)) (u (s n)) z (t (s n))) atTop (𝓝 ν)) |
| (P : Measure BrownianPath) [IsProbabilityMeasure P] (hB : IsBrownianReal brownianEval P) : |
| Tendsto (fun n => sourceFreshLog (s n) k g.f (fun a => p a.val) (fun a => d a.val) |
| (h (s n)) (u (s n)) z (t (s n))) atTop |
| (𝓝 (controlValue P g.f (quantileTrial (boundedQuantile ((compactPositivePairLaw ν).map Prod.fst))))) := by |
| obtain ⟨η,hη,hm,hl⟩ := sourceFreshLog_tendsto k g.f (fun _ a => p a.val) (fun _ a => d a.val) |
| h hh0 hh u z t s hlim |
| have hGG := source_contact_limit_joint_gg k g.f p d z hz hz0 hz1 t h hh0 hh u hu hmin |
| hH hhH ht hs hlim hcover |
| have hgeo := source_contact_limit_spin_geometry k g.f p d z hz hz0 hz1 t h hh0 hh u hu hmin |
| hH hhH ht hcover hs hlim |
| have hn := compact_spin_nonnegative ν (compactRealLaw_gg ν hGG) |
| (sourceGibbsArray_limit_gram k g.f (fun _ a => p a.val) (fun _ a => d a.val) h u z t s hlim).1 |
| rwa [compactGG_freshLog_control ν hGG hgeo hn g η hm P hB] at hl |
| |
| end SphericalPerceptronFreeEnergy |
| end |
| |
| end OAI |
| |
