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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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import Mathlib
import OAI.Probability.Perceptron.Variational.TripleIndex
import OAI.Probability.Perceptron.Interpolation.ArrayGeometry
namespace OAI
noncomputable section
namespace SphericalPerceptronFreeEnergy
open MeasureTheory ProbabilityTheory Filter Set TopologicalSpace Matrix
open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction
lemma compactGG_events (μ : ProbabilityMeasure (CompactArray CompactJointOverlap))
(n : ℕ) (hn : 0<n) (i : Fin n)
(hGG : ∀ (f : CompactBlock CompactJointOverlap n →ᵇ ℝ) (g : CompactJointOverlap →ᵇ ℝ),
compactGGDefect μ n i f g=0)
(s : Set (CompactBlock CompactJointOverlap n)) (hs : MeasurableSet s)
(t : Set CompactJointOverlap) (ht : MeasurableSet t) :
(μ : Measure _).real (compactBlock n ⁻¹' s ∩ {Q | Q i n∈t}) =
(μ : Measure _).real (compactBlock n ⁻¹' s)*(μ : Measure _).real {Q : CompactArray CompactJointOverlap | Q 0 1∈t}/n+
(∑ j ∈ Finset.univ.erase i, (μ : Measure _).real (compactBlock n ⁻¹' s ∩ {Q | Q i j∈t}))/n := by
classical
have he := congrArg (fun ρ : Measure (CompactBlock CompactJointOverlap n × CompactJointOverlap) =>
ρ.real (s×ˢt)) (compactGG_joint_law μ hn i hGG)
have hb := (compactBlock_continuous (K:=CompactJointOverlap) n).measurable
have hm (i j : ℕ) : Measurable (fun Q : CompactArray CompactJointOverlap => Q i j) := by fun_prop
rw [map_measureReal_apply (hb.prodMk (hm _ _)) (hs.prod ht),
measureReal_nnreal_smul_apply,measureReal_add_apply] at he
have hsum : (∑ j ∈ Finset.univ.erase i,
(μ : Measure _).map (fun Q => (compactBlock n Q,Q i j))).real (s×ˢt) =
∑ j ∈ Finset.univ.erase i, (μ : Measure _).real (compactBlock n ⁻¹' s ∩ {Q | Q i j∈t}) := by
simp only [measureReal_def,Measure.finsetSum_apply]
rw [ENNReal.toReal_sum (by intro _ _; finiteness)]
apply Finset.sum_congr rfl
intro j hj
rw [Measure.map_apply (hb.prodMk (hm _ _)) (hs.prod ht)]
rfl
rw [hsum] at he
have hprod : (((μ : Measure _).map (compactBlock n)).prod
((μ : Measure _).map (fun Q : CompactArray CompactJointOverlap => Q 0 1))).real (s×ˢt) =
(μ : Measure _).real (compactBlock n ⁻¹' s)*(μ : Measure _).real {Q : CompactArray CompactJointOverlap | Q 0 1∈t} := by
simp only [measureReal_def,Measure.prod_prod,ENNReal.toReal_mul,
Measure.map_apply hb hs,Measure.map_apply (hm 0 1) ht]
rfl
rw [hprod] at he
simp only [NNReal.coe_inv,NNReal.coe_natCast] at he
change (μ : Measure _).real (compactBlock n ⁻¹' s ∩ {Q | Q i n∈t}) = (n:ℝ)⁻¹*(_+_) at he
rw [he]
ring
def compactPairReal (x : CompactJointOverlap) : ℝ×ℝ := (x.1.val,x.2.val)
def compactArrayReal (Q : CompactArray CompactJointOverlap) : PairedOverlapArray :=
fun i j => compactPairReal (Q i j)
def compactBlockReal {n : ℕ} (Q : CompactBlock CompactJointOverlap n) : PairedOverlapBlock n :=
fun i j => compactPairReal (Q i j)
lemma compactPairReal_measurable : Measurable compactPairReal := by unfold compactPairReal; fun_prop
lemma compactArrayReal_measurable : Measurable compactArrayReal := by
unfold compactArrayReal; exact Measurable.of_eval fun row => Measurable.of_eval fun column =>
compactPairReal_measurable.comp ((measurable_pi_apply column).comp (measurable_pi_apply row))
lemma compactBlockReal_measurable (n : ℕ) : Measurable (compactBlockReal (n:=n)) := by
unfold compactBlockReal; exact Measurable.of_eval fun row => Measurable.of_eval fun column =>
compactPairReal_measurable.comp ((measurable_pi_apply column).comp (measurable_pi_apply row))
def compactRealLaw (μ : ProbabilityMeasure (CompactArray CompactJointOverlap)) :
Measure PairedOverlapArray := (μ : Measure _).map compactArrayReal
instance compactRealLaw_probability (μ : ProbabilityMeasure (CompactArray CompactJointOverlap)) :
IsProbabilityMeasure (compactRealLaw μ) :=
(Measure.isProbabilityMeasure_map_iff compactArrayReal_measurable.aemeasurable).2 inferInstance
lemma compactRealLaw_gg (μ : ProbabilityMeasure (CompactArray CompactJointOverlap))
(hGG : ∀ (n : ℕ) (i : Fin n) (f : CompactBlock CompactJointOverlap n →ᵇ ℝ)
(g : CompactJointOverlap →ᵇ ℝ), compactGGDefect μ n i f g=0) :
JointGhirlandaGuerra (compactRealLaw μ) := by
intro n hn i s hs t ht
have hb : Measurable (pairedBlock n) := by unfold pairedBlock; fun_prop
have hedge (a b : ℕ) : Measurable (fun Q : PairedOverlapArray => Q a b) := by fun_prop
have hmap (u : Set PairedOverlapArray) (hu : MeasurableSet u) :
(compactRealLaw μ).real u=(μ : Measure _).real (compactArrayReal ⁻¹' u) :=
map_measureReal_apply compactArrayReal_measurable hu
rw [hmap (pairedBlock n ⁻¹' s ∩ {Q | Q i n∈t}) ((hs.preimage hb).inter (ht.preimage (hedge i n))),
hmap _ (hs.preimage hb), hmap {Q | Q 0 1∈t} (ht.preimage (hedge 0 1))]
have hmj (j : Fin n) := hmap (pairedBlock n ⁻¹' s ∩ {Q | Q i j∈t})
((hs.preimage hb).inter (ht.preimage (hedge i j)))
simp_rw [hmj]
exact compactGG_events μ n (by omega) i (hGG n i)
(compactBlockReal ⁻¹' s) (hs.preimage (compactBlockReal_measurable n))
(compactPairReal ⁻¹' t) (ht.preimage compactPairReal_measurable)
lemma compactRealLaw_exchangeable (μ : ProbabilityMeasure (CompactArray CompactJointOverlap))
(hEx : ∀ e : Equiv.Perm ℕ, MeasurePreserving (compactRelabel (K:=CompactJointOverlap) e) (μ : Measure _) (μ : Measure _)) :
PairedSwapInvariant (compactRealLaw μ) := by
intro i j
have hp : Measurable (pairedRelabel (Equiv.swap i j)) := by unfold pairedRelabel; fun_prop
refine ⟨hp,?_⟩
unfold compactRealLaw
rw [Measure.map_map hp compactArrayReal_measurable]
change Measure.map (compactArrayReal ∘ compactRelabel (Equiv.swap i j)) _ = _
rw [← Measure.map_map compactArrayReal_measurable (hEx _).measurable,(hEx _).map_eq]
lemma compactRealLaw_gram (μ : ProbabilityMeasure (CompactArray CompactJointOverlap))
(hR : ∀ᵐ Q : CompactArray CompactJointOverlap ∂(μ : Measure _), ∀ n, Matrix.PosSemidef (fun i j : Fin n => (Q i j).1.val))
(hT : ∀ᵐ Q : CompactArray CompactJointOverlap ∂(μ : Measure _), ∀ n, Matrix.PosSemidef (fun i j : Fin n => (Q i j).2.val)) :
(∀ᵐ Q ∂compactRealLaw μ, ∀ n, Matrix.PosSemidef (overlapBlock id n (scalarArray Prod.fst Q))) ∧
(∀ᵐ Q ∂compactRealLaw μ, ∀ n, Matrix.PosSemidef (overlapBlock id n (scalarArray Prod.snd Q))) := by
constructor
· exact ae_map_iff compactArrayReal_measurable.aemeasurable (measurableSet_gram_arrays.preimage
(scalarArray_measurable measurable_fst)) |>.mpr hR
· exact ae_map_iff compactArrayReal_measurable.aemeasurable (measurableSet_gram_arrays.preimage
(scalarArray_measurable measurable_snd)) |>.mpr hT
lemma compactRealLaw_diagonals (μ : ProbabilityMeasure (CompactArray CompactJointOverlap)) :
(∀ᵐ Q ∂compactRealLaw μ, ∀ i, (Q i i).1≤1) ∧
(∀ᵐ Q ∂compactRealLaw μ, ∀ i, (Q i i).2≤1) := by
constructor
· apply ae_map_iff compactArrayReal_measurable.aemeasurable (by measurability) |>.mpr
exact Filter.Eventually.of_forall (fun Q i => (Q i i).1.property.2)
· apply ae_map_iff compactArrayReal_measurable.aemeasurable (by measurability) |>.mpr
exact Filter.Eventually.of_forall (fun Q i => (Q i i).2.property.2)
lemma compact_gg_no_crossing (μ : ProbabilityMeasure (CompactArray CompactJointOverlap))
(hEx : ∀ e : Equiv.Perm ℕ, MeasurePreserving (compactRelabel (K:=CompactJointOverlap) e) (μ : Measure _) (μ : Measure _))
(hGG : ∀ (n : ℕ) (i : Fin n) (f : CompactBlock CompactJointOverlap n →ᵇ ℝ)
(g : CompactJointOverlap →ᵇ ℝ), compactGGDefect μ n i f g=0)
(hR : ∀ᵐ Q : CompactArray CompactJointOverlap ∂(μ : Measure _), ∀ n, Matrix.PosSemidef (fun i j : Fin n => (Q i j).1.val))
(hT : ∀ᵐ Q : CompactArray CompactJointOverlap ∂(μ : Measure _), ∀ n, Matrix.PosSemidef (fun i j : Fin n => (Q i j).2.val)) :
∀ᵐ Q : CompactArray CompactJointOverlap ∂(μ : Measure _), ¬ ((Q 0 1).1.val < (Q 0 2).1.val ∧ (Q 0 2).2.val < (Q 0 1).2.val) := by
obtain ⟨hgr,hgt⟩ := compactRealLaw_gram μ hR hT
obtain ⟨hdr,hdt⟩ := compactRealLaw_diagonals μ
have hh := joint_gg_no_crossing (compactRealLaw μ) (compactRealLaw_exchangeable μ hEx)
(compactRealLaw_gg μ hGG) hgr hgt hdr hdt
exact (ae_map_iff compactArrayReal_measurable.aemeasurable (by measurability)).mp hh
end SphericalPerceptronFreeEnergy
end
end OAI
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