openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /DirectionalTail.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.TailProduct | |
| import OAI.Probability.CriticalPercolation.Clusters.InfiniteUniqueness | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.InfiniteEventSubsetHit | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def infiniteVertices (ω : Set E) : Set V := {v | (G.cluster ω v).Infinite} | |
| lemma infiniteVertices_mono {ω τ : Set E} (h : ω⊆τ) : | |
| G.infiniteVertices ω⊆G.infiniteVertices τ := fun v hv => hv.mono (G.cluster_mono h v) | |
| lemma cluster_union_unchanged (ω S : Set E) (o : V) | |
| (hends : ∀ e∈S, ∀ v∈G.ends e, v∉G.cluster ω o) : | |
| G.cluster (ω∪S) o=G.cluster ω o := by | |
| apply Set.Subset.antisymm | |
| · intro z hz | |
| have hr := ((G.openGraph (ω∪S)).reachable_iff_reflTransGen o z).mp hz | |
| clear hz | |
| induction hr with | |
| | refl => exact .refl _ | |
| | @tail x y hr hxy ih => | |
| obtain ⟨⟨e,he,hend⟩,hne⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hxy | |
| rcases he with he|he | |
| · have hadj : (G.openGraph ω).Adj x y := | |
| (SimpleGraph.fromEdgeSet_adj _).mpr ⟨⟨e,he,hend⟩,hne⟩ | |
| exact ih.trans hadj.reachable | |
| · exact False.elim (hends e he x (by simp [hend]) ih) | |
| · exact G.cluster_mono Set.subset_union_left o | |
| lemma infiniteVertices_union_finite_error (ω S : Set E) (hS : S.Finite) : | |
| (G.infiniteVertices (ω∪S)\G.infiniteVertices ω).Finite := by | |
| let K := ⋃ e∈S, ⋃ v∈(G.ends e).toFinset, | |
| {x | x∈G.cluster ω v ∧ (G.cluster ω v).Finite} | |
| have hK : K.Finite := hS.biUnion fun e _ => (G.ends e).toFinset.finite_toSet.biUnion fun v _ => by | |
| by_cases hv : (G.cluster ω v).Finite | |
| · exact hv.subset (fun _ h => h.1) | |
| · have heq : {x | x∈G.cluster ω v ∧ (G.cluster ω v).Finite}=(∅ : Set V) := by | |
| ext x | |
| simp [hv] | |
| rw [heq] | |
| exact Set.finite_empty | |
| apply hK.subset | |
| intro x hx | |
| have hfin : (G.cluster ω x).Finite := Set.not_infinite.mp hx.2 | |
| by_contra hn | |
| have hends : ∀ e∈S, ∀ v∈G.ends e, v∉G.cluster ω x := by | |
| intro e he v hv hvc | |
| apply hn | |
| refine Set.mem_iUnion₂.mpr ⟨e,he,Set.mem_iUnion₂.mpr ⟨v,?_,hvc.symm,?_⟩⟩ | |
| · simpa only [Sym2.mem_toFinset] using hv | |
| · rwa [← G.cluster_eq_of_reachable hvc] | |
| exact hx.1 (by rwa [G.cluster_union_unchanged ω S x hends]) | |
| def directionUnbounded (f : V → ℝ) : Set (Set E) := | |
| {ω | ∀ n : ℕ, ∃ v, (n : ℝ)<f v ∧ v∈G.infiniteVertices ω} | |
| lemma directionUnbounded_mono (f : V → ℝ) : IsUpperSet (G.directionUnbounded f) := by | |
| intro ω τ h hω n | |
| obtain ⟨v,hv,hv'⟩ := hω n | |
| exact ⟨v,hv,G.infiniteVertices_mono h hv'⟩ | |
| lemma directionUnbounded_union_finite_iff (f : V → ℝ) (ω S : Set E) (hS : S.Finite) : | |
| ω∪S∈G.directionUnbounded f ↔ ω∈G.directionUnbounded f := by | |
| constructor | |
| · intro h n | |
| have hK := (G.infiniteVertices_union_finite_error ω S hS).image f | |
| obtain ⟨M,hM⟩ := hK.bddAbove | |
| obtain ⟨N,hN⟩ := exists_nat_gt (max M n) | |
| obtain ⟨v,hv,hinf⟩ := h N | |
| refine ⟨v,lt_trans (lt_of_le_of_lt (le_max_right M n) hN) hv,?_⟩ | |
| by_contra hnot | |
| have hfM := hM (Set.mem_image_of_mem f ⟨hinf,hnot⟩) | |
| have : M<(N : ℝ) := lt_of_le_of_lt (le_max_left M n) hN | |
| linarith | |
| · exact G.directionUnbounded_mono f Set.subset_union_left | |
| lemma directionUnbounded_diff_finite_iff (f : V → ℝ) (ω S : Set E) (hS : S.Finite) : | |
| ω\S∈G.directionUnbounded f ↔ ω∈G.directionUnbounded f := by | |
| constructor | |
| · exact G.directionUnbounded_mono f Set.sdiff_subset | |
| · intro h | |
| have hs : ω⊆(ω\S)∪S := fun e he => by | |
| by_cases hh : e∈S | |
| · exact Or.inr hh | |
| · exact Or.inl ⟨he,hh⟩ | |
| exact (G.directionUnbounded_union_finite_iff f (ω\S) S hS).mp | |
| (G.directionUnbounded_mono f hs h) | |
| lemma directionUnbounded_measurable [Countable V] [Countable E] (f : V → ℝ) : | |
| MeasurableSet (G.directionUnbounded f) := by | |
| have heq : G.directionUnbounded f=⋂ n : ℕ, ⋃ v∈{v | (n : ℝ)<f v}, G.infiniteEvent v := by | |
| ext ω | |
| simp only [directionUnbounded,infiniteVertices,infiniteEvent,Set.mem_ofPred_eq, | |
| Set.mem_iInter,Set.mem_iUnion,exists_prop] | |
| rw [heq] | |
| exact MeasurableSet.iInter fun n => MeasurableSet.biUnion (Set.to_countable _) fun v _ => | |
| G.measurableSet_infiniteEvent v | |
| lemma directionUnbounded_zero_one [Countable V] [Countable E] | |
| (q : E → unitInterval) (f : V → ℝ) : | |
| BernoulliProduct.law q (G.directionUnbounded f)=0 ∨ | |
| BernoulliProduct.law q (G.directionUnbounded f)=1 := by | |
| apply BernoulliProduct.zero_one_of_finite_erasure_invariant q _ (G.directionUnbounded_measurable f) | |
| intro S hS ω | |
| have heq : BernoulliProduct.forceOn S ∅ ω=ω\S := by | |
| ext e | |
| change (if e∈S then False else e∈ω) ↔ e∈ω ∧ e∉S | |
| split_ifs <;> tauto | |
| rw [heq] | |
| exact G.directionUnbounded_diff_finite_iff f ω S hS | |
| lemma directionUnbounded_positive [Countable V] [Countable E] | |
| (p : unitInterval) (f : V → ℝ) (hf : ∀ n : ℕ, ∃ v, (n : ℝ)<f v) | |
| (theta : ℝ) (htheta : 0<theta) | |
| (hroot : ∀ v, theta≤(G.law p).real (G.infiniteEvent v)) : | |
| 0<G.law p (G.directionUnbounded f) := by | |
| let A := fun n : ℕ => ⋃ v∈{v | (n : ℝ)<f v}, G.infiniteEvent v | |
| have hA : ∀ n, MeasurableSet (A n) := fun n => | |
| MeasurableSet.biUnion (Set.to_countable _) fun v _ => G.measurableSet_infiniteEvent v | |
| have hanti : Antitone A := by | |
| intro n m hnm ω hω | |
| obtain ⟨v,hv,hh⟩ := Set.mem_iUnion₂.mp hω | |
| exact Set.mem_iUnion₂.mpr ⟨v,(show (n : ℝ)<f v from lt_of_le_of_lt (Nat.cast_le.mpr hnm) hv),hh⟩ | |
| have heq : (⋂ n, A n)=G.directionUnbounded f := by | |
| ext ω | |
| simp only [A,directionUnbounded,infiniteVertices,infiniteEvent,Set.mem_ofPred_eq, | |
| Set.mem_iInter,Set.mem_iUnion,exists_prop] | |
| have ht := tendsto_measure_iInter_atTop (μ := G.law p) | |
| (fun n => (hA n).nullMeasurableSet) hanti ⟨0,measure_ne_top _ _⟩ | |
| rw [heq] at ht | |
| have hreal := (ENNReal.tendsto_toReal (measure_ne_top _ _)).comp ht | |
| have hbound : ∀ n, theta≤(G.law p).real (A n) := by | |
| intro n | |
| obtain ⟨v,hv⟩ := hf n | |
| have hs : G.infiniteEvent v⊆A n := fun ω hω => Set.mem_iUnion₂.mpr ⟨v,hv,hω⟩ | |
| exact (hroot v).trans (ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono hs)) | |
| have hp : 0<(G.law p).real (G.directionUnbounded f) := | |
| htheta.trans_le (ge_of_tendsto hreal (Eventually.of_forall hbound)) | |
| exact ENNReal.toReal_pos_iff.mp hp |>.1 | |
| theorem directionUnbounded_one [Countable V] [Countable E] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (p : unitInterval) (hpositive : 0<G.law p G.percolates) | |
| (f : V → ℝ) (hf : ∀ n : ℕ, ∃ v, (n : ℝ)<f v) : | |
| G.law p (G.directionUnbounded f)=1 := by | |
| obtain ⟨theta,htheta,hroot⟩ := G.uniform_rooted_probability hc hl hq p hpositive | |
| have hp := G.directionUnbounded_positive p f hf theta htheta hroot | |
| have hz := G.directionUnbounded_zero_one (fun _ => p) f | |
| rw [BernoulliProduct.law_const] at hz | |
| exact hz.resolve_left (ne_of_gt hp) | |
| theorem unique_cluster_direction_unbounded_ae [Countable V] [Countable E] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (hg : G.SubexponentialGrowth) (p : unitInterval) | |
| (hp : (p : ℝ)∈Set.Ioo 0 1) (hpositive : 0<G.law p G.percolates) | |
| (f : V → ℝ) (hf : ∀ n : ℕ, ∃ v, (n : ℝ)<f v) : | |
| ∀ᵐ ω ∂G.law p, ∀ v, (G.cluster ω v).Infinite → | |
| ∀ n : ℕ, ∃ w∈G.cluster ω v, (n : ℝ)<f w := by | |
| have hu : ∀ᵐ ω ∂G.law p, ω∈G.directionUnbounded f := by | |
| rw [ae_iff] | |
| change G.law p (G.directionUnbounded f)ᶜ=0 | |
| rw [measure_compl (G.directionUnbounded_measurable f) (measure_ne_top _ _), | |
| G.directionUnbounded_one hc hl hq p hpositive f hf,measure_univ,tsub_self] | |
| filter_upwards [hu,G.atMostOneInfinite_ae hc hl hq hg p hp] with ω hω huniq | |
| intro v hv n | |
| obtain ⟨w,hw,hwinf⟩ := hω n | |
| exact ⟨w,huniq v w hv hwinf,hw⟩ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |