openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /LevelContactMeasurability.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall | |
| import OAI.Probability.CriticalPercolation.Gluing.LevelContactGeometry | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.TailProduct | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical BigOperators ENNReal | |
| namespace CriticalPercolation | |
| def finiteSetChoice {E : Type*} (S : Set E) : Finset E := | |
| if h : S.Finite then h.toFinset else ∅ | |
| lemma finiteSetChoice_coe {E : Type*} {S : Set E} (hS : S.Finite) : | |
| (finiteSetChoice S : Set E)=S := by simp [finiteSetChoice,hS] | |
| lemma finiteSetChoice_eq_iff {E : Type*} (S : Set E) (U : Finset E) : | |
| finiteSetChoice S=U ↔ S=(U : Set E) ∨ (¬S.Finite ∧ U=∅) := by | |
| by_cases hS : S.Finite | |
| · have hh : finiteSetChoice S=U ↔ S=(U : Set E) := by | |
| calc | |
| finiteSetChoice S=U ↔ (finiteSetChoice S : Set E)=(U : Set E) := Finset.coe_inj.symm | |
| _ ↔ S=(U : Set E) := by rw [finiteSetChoice_coe hS] | |
| simpa [hS] using hh | |
| · have hne : S≠(U : Set E) := fun h => hS (h ▸ U.finite_toSet) | |
| simp [finiteSetChoice,hS,hne,eq_comm] | |
| lemma measurableSet_set_eq_const {Ω E : Type*} {m : MeasurableSpace Ω} [Countable E] | |
| (S : Ω → Set E) (hS : @Measurable Ω (Set E) m inferInstance S) (U : Set E) : | |
| MeasurableSet[m] {ω | S ω=U} := by | |
| let : MeasurableSpace Ω := m | |
| have he : {ω | S ω=U}={ω | ∀ e, e∈S ω ↔ e∈U} := by | |
| ext ω | |
| exact Set.ext_iff | |
| rw [he] | |
| exact (Measurable.forall (fun e => ((measurable_pi_apply e).comp hS).iff measurable_const)).setOf | |
| lemma measurableSet_finiteSetChoice {Ω E : Type*} [MeasurableSpace Ω] [Countable E] | |
| (S : Ω → Set E) (hS : Measurable S) (U : Finset E) : | |
| MeasurableSet {ω | finiteSetChoice (S ω)=U} := by | |
| have hmfinite : MeasurableSet {ω | (S ω).Finite} := by | |
| have he : {ω | (S ω).Finite}=⋃ F : Finset E, {ω | S ω=(F : Set E)} := by | |
| ext ω | |
| simp only [Set.mem_ofPred_eq,Set.mem_iUnion] | |
| exact ⟨fun h => ⟨h.toFinset,h.coe_toFinset.symm⟩, | |
| fun ⟨F,hF⟩ => hF ▸ F.finite_toSet⟩ | |
| rw [he] | |
| exact MeasurableSet.iUnion (fun F => measurableSet_set_eq_const S hS _) | |
| simp_rw [finiteSetChoice_eq_iff] | |
| exact (measurableSet_set_eq_const S hS _).union | |
| (hmfinite.compl.inter (MeasurableSet.const _)) | |
| namespace BernoulliProduct | |
| variable {E : Type*} | |
| lemma measurable_inter_block (X : Set E) : | |
| @Measurable (Set E) (Set E) (blockSigma X) inferInstance (fun ω => ω∩X) := by | |
| have hh := measurable_forceOn_block Xᶜ (∅ : Set E) | |
| rw [compl_compl] at hh | |
| have he : forceOn Xᶜ (∅ : Set E)=(fun ω : Set E => ω∩X) := by | |
| unfold forceOn | |
| funext configuration | |
| apply Set.ext | |
| intro edge | |
| rw [Set.mem_inter_iff] | |
| simp only [Membership.mem, Set.Mem] | |
| split_ifs with hedge | |
| · exact ⟨False.elim, fun hmem => hedge hmem.2⟩ | |
| · exact ⟨fun hmem => ⟨hmem, not_not.mp hedge⟩, And.left⟩ | |
| rw [he] at hh | |
| exact hh | |
| end BernoulliProduct | |
| namespace BondGraph | |
| open BernoulliProduct | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma incident_ncard_le (S : Set V) (hS : S.Finite) | |
| (D : ℕ) (hD : ∀ v∈S, (G.incidentEdges {v}).ncard ≤ D) : | |
| (G.incidentEdges S).ncard ≤ S.ncard*D := by | |
| have he : G.incidentEdges S=⋃ v∈hS.toFinset, G.incidentEdges {v} := by | |
| ext e | |
| simp [incidentEdges] | |
| rw [he] | |
| calc | |
| _ ≤ ∑ v∈hS.toFinset, (G.incidentEdges {v}).ncard := | |
| Finset.set_ncard_biUnion_le hS.toFinset _ | |
| _ ≤ ∑ _v∈hS.toFinset, D := Finset.sum_le_sum (fun v hv => hD v (hS.mem_toFinset.mp hv)) | |
| _ = _ := by simp [Set.ncard_eq_toFinset_card S hS] | |
| lemma finite_contactDrop (hlocal : G.LocallyFinite) (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) | |
| (hS : (G.levelContacts π j o ω).Finite) : (G.contactDrop π j o ω).Finite := | |
| (G.finite_incident hlocal hS).subset (G.contactDrop_subset_incident π j o ω) | |
| lemma contactDrop_ncard_le (hlocal : G.LocallyFinite) (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E) | |
| (hS : (G.levelContacts π j o ω).Finite) (D : ℕ) | |
| (hD : ∀ v∈G.levelContacts π j o ω, (G.incidentEdges {v}).ncard ≤ D) : | |
| (G.contactDrop π j o ω).ncard ≤ (G.levelContacts π j o ω).ncard*D := | |
| (Set.ncard_le_ncard (G.contactDrop_subset_incident π j o ω) (G.finite_incident hlocal hS)).trans | |
| (G.incident_ncard_le _ hS D hD) | |
| variable [Countable V] [Countable E] | |
| lemma measurable_exteriorCluster (π : V → ℤ) (j : ℤ) (o : V) : | |
| @Measurable (Set E) (Set V) (blockSigma (G.inducedEdges {v | j ≤ π v})) inferInstance | |
| (G.exteriorCluster π j o) := | |
| (G.measurable_cluster o).comp (measurable_inter_block _) | |
| lemma measurable_levelContacts (π : V → ℤ) (j : ℤ) (o : V) : | |
| @Measurable (Set E) (Set V) (blockSigma (G.inducedEdges {v | j ≤ π v})) inferInstance | |
| (G.levelContacts π j o) := by | |
| refine @Measurable.of_eval (Set E) V (fun _ => Prop) | |
| (blockSigma (G.inducedEdges {v | j ≤ π v})) (fun _ => inferInstance) _ ?_ | |
| intro v | |
| exact @Measurable.and (Set E) (blockSigma (G.inducedEdges {v | j ≤ π v})) _ _ | |
| ((measurable_pi_apply v).comp (G.measurable_exteriorCluster π j o)) measurable_const | |
| lemma measurable_contactDrop (π : V → ℤ) (j : ℤ) (o : V) : | |
| @Measurable (Set E) (Set E) (blockSigma (G.inducedEdges {v | j ≤ π v})) inferInstance | |
| (G.contactDrop π j o) := by | |
| refine @Measurable.of_eval (Set E) E (fun _ => Prop) | |
| (blockSigma (G.inducedEdges {v | j ≤ π v})) (fun _ => inferInstance) _ ?_ | |
| intro e | |
| exact @Measurable.exists (Set E) V (blockSigma (G.inducedEdges {v | j ≤ π v})) _ _ (fun v => | |
| @Measurable.and (Set E) (blockSigma (G.inducedEdges {v | j ≤ π v})) _ _ | |
| ((measurable_pi_apply v).comp (G.measurable_levelContacts π j o)) measurable_const) | |
| def fewContacts (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) : Set (Set E) := | |
| {ω | ∃ U : Finset V, G.levelContacts π j o ω=(U : Set V) ∧ U.Nonempty ∧ U.card<k} | |
| lemma measurableSet_fewContacts (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) : | |
| MeasurableSet[blockSigma (G.inducedEdges {v | j ≤ π v})] (G.fewContacts π j o k) := by | |
| have he : G.fewContacts π j o k=⋃ U : Finset V, | |
| ({ω | G.levelContacts π j o ω=(U : Set V)}∩{_ω | U.Nonempty ∧ U.card<k}) := by | |
| ext ω | |
| simp [fewContacts] | |
| rw [he] | |
| exact MeasurableSet.iUnion (fun U => | |
| (measurableSet_set_eq_const _ (G.measurable_levelContacts π j o) (U : Set V)).inter | |
| (MeasurableSet.const _)) | |
| omit [Countable V] [Countable E] in | |
| lemma fewContacts_finite (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) (ω : Set E) | |
| (hω : ω∈G.fewContacts π j o k) : | |
| (G.levelContacts π j o ω).Finite ∧ (G.levelContacts π j o ω).Nonempty ∧ | |
| (G.levelContacts π j o ω).ncard<k := by | |
| obtain ⟨U,hU,hne,hcard⟩ := hω | |
| rw [hU] | |
| exact ⟨U.finite_toSet,hne,by simpa using hcard⟩ | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end | |
| end OAI | |