openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /FullOrbitalStructureTransfer.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphTopology.GraphNeighborhoodCompact | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalAction | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitQuotientTransfer | |
| import OAI.Probability.CriticalPercolation.Modular.ModularLevelGeometry | |
| import OAI.Probability.CriticalPercolation.Nilpotent.StructuredVirtuallyNilpotent | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory Topology | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation | |
| namespace StructurePullback | |
| variable {A B : Type*} [Group A] [Group B] | |
| lemma nilpotent_comap_of_injective (f : A →* B) (hf : Injective f) | |
| (N : Subgroup B) [Group.IsNilpotent N] : Group.IsNilpotent (N.comap f) := by | |
| let F : N.comap f →* N := | |
| {toFun := fun g => ⟨f g.val,g.property⟩ | |
| map_one' := Subtype.ext (map_one f) | |
| map_mul' := fun g h => Subtype.ext (map_mul f g.val h.val)} | |
| have hF : Injective F := by | |
| intro g h he | |
| apply Subtype.ext | |
| exact hf (congrArg Subtype.val he) | |
| apply Subgroup.isNilpotent_of_ker_le_center F | |
| rw [(MonoidHom.ker_eq_bot_iff F).mpr hF] | |
| exact bot_le | |
| lemma finiteIndex_comap (f : A →* B) (N : Subgroup B) (hN : N.index≠0) : | |
| (N.comap f).index≠0 := by | |
| rw [Subgroup.index_comap] | |
| intro h | |
| exact hN (Subgroup.index_eq_zero_of_relIndex_eq_zero h) | |
| end StructurePullback | |
| namespace BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma finite_stabilizer_images_from_one_vertex | |
| {A B : Type*} [Group A] [Group B] [MulAction A V] [MulAction A E] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (ha : G.PreservesEnds (Λ := A)) (ρ : A →* B) (o : V) | |
| (ho : ((MulAction.stabilizer A o).map ρ : Set B).Finite) : | |
| ∀ v : V, ((MulAction.stabilizer A v).map ρ : Set B).Finite := by | |
| let a : V → A := fun w => if h : ∃ g : A, g • o=w then h.choose else 1 | |
| have ha' : ∀ {w : V}, (∃ g : A, g • o=w) → a w • o=w := by | |
| intro w hw | |
| simp only [a, dite_eq_left hw] | |
| exact hw.choose_spec | |
| intro v | |
| obtain ⟨n,hn⟩ := G.reachable_mem_vertexBall (hc v o) | |
| apply ((G.vertexBall_finite hl v n).biUnion | |
| (fun w _ => ho.image (fun b => ρ (a w)*b))).subset | |
| rintro b ⟨g,hg,rfl⟩ | |
| have hgo : g • o∈G.vertexBall v n := by | |
| have hh := (Aut.vertexBall_mem G (G.smulAut ha g) v o n).mpr hn | |
| change g • o∈G.vertexBall (g • v) n at hh | |
| exact hg ▸ hh | |
| refine Set.mem_iUnion₂.mpr ⟨g • o,hgo,ρ ((a (g • o))⁻¹*g),?_,?_⟩ | |
| · refine ⟨(a (g • o))⁻¹*g,?_,rfl⟩ | |
| change ((a (g • o))⁻¹*g) • o=o | |
| rw [mul_smul] | |
| calc | |
| (a (g • o))⁻¹ • (g • o) = (a (g • o))⁻¹ • (a (g • o) • o) := | |
| congrArg (fun x => (a (g • o))⁻¹ • x) (ha' ⟨g,rfl⟩).symm | |
| _ = o := inv_smul_smul _ _ | |
| · simp only [map_mul, map_inv, mul_inv_cancel_left] | |
| def fullOrbitalQuotientHom (o : V) (L : ℕ) (Λ : Subgroup G.Aut) | |
| (H : Subgroup (G.orbitPower o L ≃g G.orbitPower o L)) [H.Normal] : | |
| Λ →* (G.orbitPower o L ≃g G.orbitPower o L) ⧸ H := | |
| (QuotientGroup.mk' H).comp ((G.orbitPowerHom o L).comp Λ.subtype) | |
| lemma fullOrbital_finite_kernel_orbits (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (o : V) (L : ℕ) (Λ : Subgroup G.Aut) | |
| (H : Subgroup (G.orbitPower o L ≃g G.orbitPower o L)) [H.Normal] | |
| (hH : ((fun g : G.orbitPower o L ≃g G.orbitPower o L => | |
| g (⟨o,MulAction.mem_orbit_self o⟩ : MulAction.orbit G.Aut o)) '' (H : Set _)).Finite) : | |
| ∀ v : V, (MulAction.orbit (G.fullOrbitalQuotientHom o L Λ H).rangeRestrict.ker v).Finite := by | |
| let _ : MulAction G.Aut E := G.autEdgeAction | |
| apply G.finite_kernel_orbits_from_one_vertex hc hl | |
| (G.subgroup_preservesEnds G.aut_preservesEnds Λ) _ o | |
| apply (hH.image Subtype.val).subset | |
| rintro v ⟨g,rfl⟩ | |
| have he := congrArg Subtype.val g.property | |
| have hg : G.orbitPowerHom o L g.val.val∈H := (QuotientGroup.eq_one_iff _).mp he | |
| exact ⟨⟨g • o,⟨g.val.val,rfl⟩⟩,⟨G.orbitPowerHom o L g.val.val,hg,rfl⟩,rfl⟩ | |
| lemma fullOrbital_finite_stabilizer_images (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (o : V) (L : ℕ) (Λ : Subgroup G.Aut) | |
| (H : Subgroup (G.orbitPower o L ≃g G.orbitPower o L)) [H.Normal] | |
| (hstab : (((QuotientGroup.mk' H) : (G.orbitPower o L ≃g G.orbitPower o L) → _) | |
| '' {g | g (⟨o,MulAction.mem_orbit_self o⟩ : MulAction.orbit G.Aut o)= | |
| ⟨o,MulAction.mem_orbit_self o⟩}).Finite) : | |
| ∀ v : V, ((MulAction.stabilizer Λ v).map | |
| (G.fullOrbitalQuotientHom o L Λ H).rangeRestrict : | |
| Set (G.fullOrbitalQuotientHom o L Λ H).range).Finite := by | |
| let _ : MulAction G.Aut E := G.autEdgeAction | |
| apply G.finite_stabilizer_images_from_one_vertex hc hl | |
| (G.subgroup_preservesEnds G.aut_preservesEnds Λ) _ o | |
| apply (hstab.preimage Subtype.val_injective.injOn).subset | |
| rintro g ⟨a,ha,rfl⟩ | |
| refine ⟨G.orbitPowerHom o L a.val,?_,rfl⟩ | |
| apply Subtype.ext | |
| exact ha | |
| theorem critical_null_of_compact_open_orbital_quotient | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hg : G.SubexponentialGrowth) | |
| (o : V) (L : ℕ) (hconn : (G.orbitPower o L).Connected) | |
| (Λ : Subgroup G.Aut) [Group.FG Λ] | |
| (S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) | |
| (H : Subgroup (G.orbitPower o L ≃g G.orbitPower o L)) [H.Normal] | |
| (N : Subgroup ((G.orbitPower o L ≃g G.orbitPower o L) ⧸ H)) [Group.IsNilpotent N] | |
| (hN : N.index≠0) | |
| (hH : let _ : TopologicalSpace (MulAction.orbit G.Aut o) := ⊥ | |
| let _ : TopologicalSpace (G.orbitPower o L ≃g G.orbitPower o L) := | |
| GraphTopology.autTopology (G.orbitPower o L) | |
| IsCompact (H : Set (G.orbitPower o L ≃g G.orbitPower o L)) ∧ | |
| IsOpen (H : Set (G.orbitPower o L ≃g G.orbitPower o L))) | |
| (hpc : G.criticalProbability<1) : G.law G.criticalProbability G.percolates=0 := by | |
| let _ : TopologicalSpace (MulAction.orbit G.Aut o) := ⊥ | |
| let _ : DiscreteTopology (MulAction.orbit G.Aut o) := ⟨rfl⟩ | |
| let _ : TopologicalSpace (G.orbitPower o L ≃g G.orbitPower o L) := | |
| GraphTopology.autTopology (G.orbitPower o L) | |
| let _ : DiscreteTopology ((G.orbitPower o L ≃g G.orbitPower o L) ⧸ H) := | |
| QuotientGroup.discreteTopology hH.2 | |
| let o' : MulAction.orbit G.Aut o := ⟨o,MulAction.mem_orbit_self o⟩ | |
| have hstab : (((QuotientGroup.mk' H) : (G.orbitPower o L ≃g G.orbitPower o L) → _) | |
| '' {g | g o'=o'}).Finite := by | |
| apply isCompact_iff_finite.mp | |
| apply (GraphTopology.compact_stabilizer (G.orbitPower o L) hconn | |
| (G.powerOn_finite_neighbors hl _ _) o').image | |
| exact QuotientGroup.continuous_mk | |
| let f := G.fullOrbitalQuotientHom o L Λ H | |
| let M := N.comap f.range.subtype | |
| let _ : Group.IsNilpotent M := StructurePullback.nilpotent_comap_of_injective | |
| f.range.subtype Subtype.val_injective N | |
| let _ : MulAction G.Aut E := G.autEdgeAction | |
| exact G.critical_null_of_virtually_nilpotent_quotient hc hl hg | |
| (G.subgroup_preservesEnds G.aut_preservesEnds Λ) S hS hcover | |
| f.rangeRestrict (MonoidHom.rangeRestrict_surjective f) | |
| (G.fullOrbital_finite_kernel_orbits hc hl o L Λ H | |
| (GraphTopology.compact_finite_vertex_images _ _ hH.1 o')) | |
| (G.fullOrbital_finite_stabilizer_images hc hl o L Λ H hstab) M | |
| (StructurePullback.finiteIndex_comap f.range.subtype N hN) hpc | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end OAI | |