| 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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|
