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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | 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
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