openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /FaithfulOrbitalModel.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteIndexAction | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitQuotientTransfer | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalFiniteGeneration | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalScales | |
| import OAI.Probability.CriticalPercolation.Modular.ModularLevelGeometry | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def orbitalActionGroup (o : V) (L : ℕ) (Λ : Subgroup G.Aut) : | |
| Subgroup (G.orbitPower o L ≃g G.orbitPower o L) := | |
| ((G.orbitPowerHom o L).comp Λ.subtype).range | |
| def orbitalActionMap (o : V) (L : ℕ) (Λ : Subgroup G.Aut) : | |
| Λ →* G.orbitalActionGroup o L Λ := | |
| ((G.orbitPowerHom o L).comp Λ.subtype).rangeRestrict | |
| lemma orbitalActionMap_surjective (o : V) (L : ℕ) (Λ : Subgroup G.Aut) : | |
| Surjective (G.orbitalActionMap o L Λ) := MonoidHom.rangeRestrict_surjective _ | |
| instance orbitalActionGroup_fg (o : V) (L : ℕ) (Λ : Subgroup G.Aut) [Group.FG Λ] : | |
| Group.FG (G.orbitalActionGroup o L Λ) := | |
| Group.fg_of_surjective (G.orbitalActionMap_surjective o L Λ) | |
| lemma orbitalActionMap_kernel_fixes (o : V) (L : ℕ) (Λ : Subgroup G.Aut) | |
| (φ : (G.orbitalActionMap o L Λ).ker) : φ • o=o := by | |
| have hh := congrArg Subtype.val φ.property | |
| change (G.orbitPowerHom o L φ.val.val) = 1 at hh | |
| let o' : MulAction.orbit G.Aut o := ⟨o,MulAction.mem_orbit_self o⟩ | |
| have ho := congrArg (fun f : G.orbitPower o L ≃g G.orbitPower o L => (f o').val) hh | |
| exact ho | |
| lemma orbitalActionMap_finite_kernel_orbits (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (o : V) (L : ℕ) (Λ : Subgroup G.Aut) : | |
| ∀ v : V, (MulAction.orbit (G.orbitalActionMap o L Λ).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) (G.orbitalActionMap o L Λ) o | |
| apply (Set.finite_singleton o).subset | |
| rintro v ⟨φ,rfl⟩ | |
| exact G.orbitalActionMap_kernel_fixes o L Λ φ | |
| lemma orbitalActionGroup_transitive (o : V) (L : ℕ) (Λ : Subgroup G.Aut) | |
| (horbit : MulAction.orbit Λ o=MulAction.orbit G.Aut o) | |
| (u v : MulAction.orbit G.Aut o) : | |
| ∃ φ : G.orbitalActionGroup o L Λ, φ.val u=v := by | |
| have hu : u.val∈MulAction.orbit Λ o := by rw [horbit]; exact u.property | |
| have hv : v.val∈MulAction.orbit Λ o := by rw [horbit]; exact v.property | |
| obtain ⟨a,ha⟩ := hu | |
| obtain ⟨b,hb⟩ := hv | |
| refine ⟨G.orbitalActionMap o L Λ (b*a⁻¹),?_⟩ | |
| apply Subtype.ext | |
| change (b*a⁻¹) • u.val=v.val | |
| change a • o=u.val at ha | |
| change b • o=v.val at hb | |
| rw [←ha,mul_smul,inv_smul_smul,hb] | |
| theorem sparse_polynomial_faithful_orbital_model [Infinite V] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (o : V) (d : ℕ) | |
| (hpoly : ∀ R : ℕ, ∃ n≥R, ∃ v : V, (G.vertexBall v n).ncard≤n^d) : | |
| ∃ C : ℕ, ∃ Λ : Subgroup G.Aut, Group.FG Λ ∧ | |
| (G.orbitPower o (2*C+1)).Connected ∧ | |
| (MulAction.orbit G.Aut o).Infinite ∧ | |
| (∀ u, ((G.orbitPower o (2*C+1)).neighborSet u).Finite) ∧ | |
| (∀ u v, ∃ φ : G.orbitalActionGroup o (2*C+1) Λ, φ.val u=v) ∧ | |
| (∀ v : V, (MulAction.orbit (G.orbitalActionMap o (2*C+1) Λ).ker v).Finite) ∧ | |
| (∃ S : Set V, S.Finite ∧ ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) ∧ | |
| (∀ R : ℕ, ∃ n≥R, ∀ u : MulAction.orbit G.Aut o, | |
| {v | ∃ p : (G.orbitPower o (2*C+1)).Walk u v, p.length≤n}.ncard≤n^(d+1)) := by | |
| obtain ⟨C,hC,hdense,hscales⟩ := G.orbital_power_polynomial_scales hc hl hq o d hpoly | |
| obtain ⟨Λ,hΛ,horbit,hreps⟩ := G.finitelyGenerated_quasiTransitive_subgroup hc hl hq o | |
| refine ⟨C,Λ,hΛ,hC,G.dense_set_infinite hl hdense,?_,?_,?_,hreps,hscales⟩ | |
| · exact G.powerOn_finite_neighbors hl _ _ | |
| · exact G.orbitalActionGroup_transitive o (2*C+1) Λ horbit | |
| · exact G.orbitalActionMap_finite_kernel_orbits hc hl o (2*C+1) Λ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |