File size: 3,584 Bytes
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 | import Mathlib
import OAI.Probability.Perceptron.Interpolation.ArrayGeometry
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set TopologicalSpace Matrix
open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
lemma sourceGibbsArray_closed_full (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ)
(h : Fin (k+1) → ℝ) (u : Fin (n+1) → ℝ) (z : Fin k → ℝ) (t : ℝ≥0)
{F : Set (CompactArray CompactJointOverlap)} (hF : IsClosed F)
(hfull : ∀ x : ℕ→NormalizedSpin (n+1)×IndexedLeaf k, sourceJointArray x ∈ F) :
sourceGibbsArrayLaw n k f p d h u z t F=1 := by
have hp : (fun a : (SourceBaseData n k × (ℕ→ℝ)) ×
(ℕ → NormalizedSpin (n+1) × IndexedLeaf k) => sourceJointArray a.2) ⁻¹' F=univ := by
ext a
simp only [mem_preimage,mem_univ,iff_true]
exact hfull a.2
apply ENNReal.coe_injective
rw [ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure]
change (Measure.map (fun a : (SourceBaseData n k × (ℕ→ℝ)) ×
(ℕ → NormalizedSpin (n+1) × IndexedLeaf k) => sourceJointArray a.2) _) F=1
rw [Measure.map_apply (show Measurable (fun a : (SourceBaseData n k × (ℕ→ℝ)) ×
(ℕ → NormalizedSpin (n+1) × IndexedLeaf k) => sourceJointArray a.2) from
(sourceJointArray_measurable (n+1) k).comp measurable_snd) hF.measurableSet,hp]
exact measure_univ
lemma compact_spin_gram_closed (r : ℕ) :
IsClosed {Q : CompactArray CompactJointOverlap | Matrix.PosSemidef (fun i j : Fin r => (Q i j).1.val)} :=
(isClosed_matrix_posSemidef r).preimage (by fun_prop)
lemma compact_tree_gram_closed (r : ℕ) :
IsClosed {Q : CompactArray CompactJointOverlap | Matrix.PosSemidef (fun i j : Fin r => (Q i j).2.val)} :=
(isClosed_matrix_posSemidef r).preimage (by fun_prop)
lemma sourceGibbsArray_limit_gram (k : ℕ) (f : ℝ →ᵇ ℝ)
(p d : (n : ℕ) → Fin (n+1) → ℕ) (h : ℕ → Fin (k+1) → ℝ)
(u : (n : ℕ) → Fin (n+1) → ℝ) (z : Fin k → ℝ) (t : ℕ → ℝ≥0)
(s : ℕ→ℕ) {ν : ProbabilityMeasure (CompactArray CompactJointOverlap)}
(hlim : Tendsto (fun n => sourceGibbsArrayLaw (s n) k f (p (s n)) (d (s n))
(h (s n)) (u (s n)) z (t (s n))) atTop (𝓝 ν)) :
(∀ᵐ Q : CompactArray CompactJointOverlap ∂(ν : Measure _), ∀ r, Matrix.PosSemidef (fun i j : Fin r => (Q i j).1.val)) ∧
(∀ᵐ Q : CompactArray CompactJointOverlap ∂(ν : Measure _), ∀ r, Matrix.PosSemidef (fun i j : Fin r => (Q i j).2.val)) := by
constructor
· rw [ae_all_iff]
intro r
apply (mem_ae_iff_prob_eq_one (compact_spin_gram_closed r).measurableSet).mpr
have hv := weak_limit_closed_full hlim (compact_spin_gram_closed r)
(fun n => sourceGibbsArray_closed_full (s n) k f (p (s n)) (d (s n)) (h (s n)) (u (s n)) z (t (s n))
(compact_spin_gram_closed r) (fun x => sourceSpinArray_posSemidef x r))
simp only [← ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure,hv,ENNReal.coe_one]
· rw [ae_all_iff]
intro r
apply (mem_ae_iff_prob_eq_one (compact_tree_gram_closed r).measurableSet).mpr
have hv := weak_limit_closed_full hlim (compact_tree_gram_closed r)
(fun n => sourceGibbsArray_closed_full (s n) k f (p (s n)) (d (s n)) (h (s n)) (u (s n)) z (t (s n))
(compact_tree_gram_closed r) (fun x => sourceTreeArray_posSemidef x r))
simp only [← ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure,hv,ENNReal.coe_one]
end SphericalPerceptronFreeEnergy
end
end OAI
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