openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /DescentCritical.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Classification.StructuredCoordinates | |
| import OAI.Probability.CriticalPercolation.Classification.RestrictedQuotient | |
| import OAI.Probability.CriticalPercolation.Classification.AlgebraicDescent | |
| import OAI.Probability.CriticalPercolation.Criticality.IntegerBoxes | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.BondGraph | |
| universe groupUniverse vertexUniverse edgeUniverse actionUniverse | |
| theorem critical_null_of_algebraic_descent {N : Type groupUniverse} [Group N] | |
| (hN : AlgebraicDescent N) : | |
| ∀ {V : Type vertexUniverse} {E : Type edgeUniverse} {Λ : Type actionUniverse} | |
| [Group Λ] [MulAction Λ V] [MulAction Λ E] | |
| (G : BondGraph V E), | |
| G.fullGraph.Connected → G.LocallyFinite → G.SubexponentialGrowth → | |
| G.PreservesEnds (Λ := Λ) → | |
| ∀ (S : Set V), S.Finite → (∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) → | |
| ∀ (ρ : Λ →* N), Surjective ρ → | |
| (∀ v : V, (MulAction.orbit ρ.ker v).Finite) → | |
| (∀ v : V, ((MulAction.stabilizer Λ v).map ρ : Set N).Finite) → | |
| G.criticalProbability<1 → G.law G.criticalProbability G.percolates=0 := by | |
| induction hN with | |
| | @finite N _ _ => | |
| intro V E Λ _ _ _ G _ _ _ _ S hS hcover ρ _ hker _ _ | |
| have hV := EquivariantCoordinates.finite_vertices_of_finite_quotient ρ hker S hS hcover | |
| have hempty : G.percolates=∅ := by | |
| apply Set.eq_empty_iff_forall_notMem.mpr | |
| rintro ω ⟨v,hv⟩ | |
| exact hv (hV.subset (Set.subset_univ _)) | |
| rw [hempty,measure_empty] | |
| | @cyclic N _ φ hφ hf => | |
| intro V E Λ _ _ _ G _ hl _ ha S hS hcover ρ hρ hker hstab hpc | |
| have heq := G.finite_by_integer_critical_one hl ha S hS hcover ρ hρ φ hφ hf hker hstab | |
| exact False.elim ((ne_of_lt hpc) heq) | |
| | @plane N _ φ hφ lower ih => | |
| intro V E Λ _ _ _ G hc hl hg ha S hS hcover ρ hρ hker hstab hpc | |
| let _ := hc.nonempty | |
| let _ := G.countable_vertices hc hl | |
| let _ := G.countable_bonds hc hl | |
| obtain ⟨z,hzonto,hz⟩ := exists_integer_pair_coordinates ρ hρ φ hφ hstab | |
| apply G.critical_null_of_action_coordinates hc hl ha S hS hcover hg hpc | |
| (fun g => (φ (ρ g)).toAdd) z hzonto hz | |
| by_contra hbad | |
| obtain ⟨k,o,ho,hcrit,hpos⟩ := G.bounded_ray_critical_component | |
| (fun v => ((z v).1 : ℝ)) (fun v => ((z v).2 : ℝ)) (by simpa only [Int.cast_abs] using hbad) | |
| let zM : V → Multiplicative (ℤ×ℤ) := fun v => Multiplicative.ofAdd (z v) | |
| let Q := integerPairBox k | |
| let C := G.domainComponent (zM ⁻¹' Q) o | |
| let L := componentKernel (φ.comp ρ) C | |
| let H := G.induce C | |
| let _ := restrictedVertexAction C L (componentKernel_le (φ.comp ρ) C) | |
| let _ := G.restrictedEdgeAction ha C L (componentKernel_le (φ.comp ρ) C) | |
| have hzm : ∀ g v, zM (g • v)=(φ.comp ρ) g*zM v := | |
| fun g v => congrArg Multiplicative.ofAdd (hz g v) | |
| obtain ⟨R,hR,hRcover⟩ := G.componentKernel_finite_orbits ha S hS hcover | |
| (φ.comp ρ) zM hzm Q (integerPairBox_finite k) o ho | |
| let f := ρ.comp L.subtype | |
| have hfker : ∀ v : C, (MulAction.orbit f.rangeRestrict.ker v).Finite := | |
| rangeRestrict_kernel_orbits f | |
| (restricted_kernel_orbits ρ hker C L (componentKernel_le (φ.comp ρ) C)) | |
| have hfstab : ∀ v : C, | |
| ((MulAction.stabilizer L v).map f.rangeRestrict : Set f.range).Finite := | |
| rangeRestrict_stabilizer_images f | |
| (restricted_stabilizer_images ρ hstab C L (componentKernel_le (φ.comp ρ) C)) | |
| have hle : f.range≤φ.ker := component_quotient_le_kernel ρ φ C | |
| have hnull := ih f.range hle H | |
| (G.domainComponent_connected (zM ⁻¹' Q) o) (G.induce_locallyFinite hl C) | |
| (G.induce_subexponential hl hg C) | |
| (G.restricted_preservesEnds ha C L (componentKernel_le (φ.comp ρ) C)) | |
| R hR hRcover f.rangeRestrict f.rangeRestrict_surjective hfker hfstab | |
| (by change H.criticalProbability=G.criticalProbability at hcrit | |
| rw [hcrit]; exact hpc) | |
| exact (ne_of_gt hpos) hnull | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |