openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Arrays /CoordinateReplicaAncestry.lean
| import OAI.Probability.InvariantIsing.Fields.LabeledReplicaAncestry | |
| namespace OAI | |
| /-! Sampled ancestry with the forest represented by its independent coordinates. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron | |
| namespace InvariantIsing | |
| theorem coordinate_sampled_ancestry_ae {A S : Type} [MeasurableSpace A] [MeasurableSpace S] | |
| (n : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) (μ : ℕ → ProbabilityMeasure A) | |
| (c : ℕ → S × A → ℝ) (u : ℕ → S × A → S) | |
| (hc : ∀ i, Measurable (c i)) (hu : ∀ i, Measurable (u i)) | |
| (hcne : ∀ i p, c i p ≠ 0) (s : S) | |
| (ν : LabeledTree n × (ForestVertex n → A) → Measure (LabeledLeaf n)) | |
| (hν : Measurable ν) [∀ p, IsProbabilityMeasure (ν p)] | |
| (habs : ∀ p, ν p ≪ labeledLeafLaw n p.1) : | |
| let P := (labeledCascadeLaw n b : Measure (LabeledTree n)).prod | |
| (Measure.infinitePi fun a : ForestVertex n => (μ (forestVertexDepth n a) : Measure A)) | |
| ∀ᵐ q ∂P ⊗ₘ probabilityReplicaKernel ν hν, | |
| noiseLeafCommonDepth n | |
| (noiseLeafKeep n c u s (labeledNoiseLeaf A n | |
| (q.1.1,markForestOfCoords A n q.1.2) (q.2 0))) | |
| (noiseLeafKeep n c u s (labeledNoiseLeaf A n | |
| (q.1.1,markForestOfCoords A n q.1.2) (q.2 1))) = | |
| labeledCommonDepth n (q.2 0) (q.2 1) := by | |
| intro P | |
| have hcoord : MeasurePreserving (fun p : LabeledTree n × (ForestVertex n → A) => | |
| (p.1,markForestOfCoords A n p.2)) P | |
| ((labeledCascadeLaw n b : Measure (LabeledTree n)).prod | |
| (markForestLaw A n μ : Measure (MarkForest A n))) := | |
| (MeasurePreserving.id _).prod | |
| ⟨measurable_markForestOfCoords A n,markForestOfCoords_law A n μ⟩ | |
| have hroot := hcoord.quasiMeasurePreserving.tendsto_ae.eventually | |
| (labeled_ancestry_preserved_ae n b (markForestLaw A n μ : Measure (MarkForest A n)) c u hc hu hcne s) | |
| have hm : Measurable (fun p : (LabeledTree n × (ForestVertex n → A)) × LabeledLeaf n => | |
| noiseLeafKeep n c u s (labeledNoiseLeaf A n | |
| (p.1.1,markForestOfCoords A n p.1.2) p.2)) := by | |
| apply measurable_from_prod_countable_left | |
| intro α | |
| exact (measurable_noiseLeafKeep n hc hu).comp | |
| (measurable_const.prodMk ((measurable_labeledNoiseLeaf A n α).comp hcoord.measurable)) | |
| have hp (i : ℕ) : Measurable (fun q : (LabeledTree n × (ForestVertex n → A)) × | |
| (ℕ → LabeledLeaf n) => (q.1,q.2 i)) := | |
| measurable_fst.prodMk ((measurable_pi_apply i).comp measurable_snd) | |
| apply Measure.ae_compProd_of_ae_ae | |
| (measurableSet_eq_fun ((measurable_noiseLeafCommonDepth n).comp | |
| ((hm.comp (hp 0)).prodMk (hm.comp (hp 1)))) | |
| ((measurable_of_countable (fun p : LabeledLeaf n × LabeledLeaf n => | |
| labeledCommonDepth n p.1 p.2)).comp | |
| (((measurable_pi_apply 0).comp measurable_snd).prodMk | |
| ((measurable_pi_apply 1).comp measurable_snd)))) | |
| have ht := (measurePreserving_fst (μ := (labeledCascadeLaw n b : Measure (LabeledTree n))) | |
| (ν := Measure.infinitePi fun a : ForestVertex n => | |
| (μ (forestVertexDepth n a) : Measure A))).quasiMeasurePreserving.tendsto_ae.eventually | |
| (labeledCascade_total_ae n b hb) | |
| filter_upwards [hroot,ht] with p hpath htotal | |
| have hpos := (habs p).ae_le (labeledLeafLaw_positive_weight n p.1 htotal) | |
| have h0 := (measurePreserving_eval_infinitePi (fun _ : ℕ => ν p) 0).quasiMeasurePreserving.tendsto_ae.eventually hpos | |
| have h1 := (measurePreserving_eval_infinitePi (fun _ : ℕ => ν p) 1).quasiMeasurePreserving.tendsto_ae.eventually hpos | |
| filter_upwards [h0,h1] with σ hσ0 hσ1 | |
| exact hpath (σ 0) (σ 1) hσ0 hσ1 | |
| end InvariantIsing | |
| end | |
| end OAI | |