openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Modular /ModularLevelGeometry.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ComponentAction | |
| import OAI.Probability.CriticalPercolation.Modular.Unimodularity | |
| import OAI.Probability.CriticalPercolation.Modular.ModularCharacter | |
| import OAI.Probability.CriticalPercolation.Modular.ActionUnimodularRoot | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| @[instance_reducible] def autEdgeAction : MulAction G.Aut E where | |
| smul φ e := φ.edge e | |
| one_smul _ := rfl | |
| mul_smul _ _ _ := rfl | |
| local instance : MulAction G.Aut E := G.autEdgeAction | |
| lemma aut_preservesEnds : G.PreservesEnds (Λ := G.Aut) := fun φ => φ.preserves | |
| lemma stabilizer_le_levelComponentGroup (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (o : V) (Q : Set ℝˣ) (v : V) | |
| (x : G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v) : | |
| MulAction.stabilizer G.Aut x.val ≤ componentKernel (G.modularUnitCharacter hc hl o) | |
| (G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v) := by | |
| intro φ hφ | |
| have hker := G.unit_stabilizer_le_kernel hc hl o x.val hφ | |
| change φ.vertex x.val = x.val at hφ | |
| refine ⟨hker,?_⟩ | |
| have hS : ∀ y, φ.vertex y ∈ G.modularCoordinate hc hl o ⁻¹' Q ↔ | |
| y ∈ G.modularCoordinate hc hl o ⁻¹' Q := by | |
| intro y | |
| change G.modularCoordinate hc hl o (φ • y) ∈ Q ↔ _ | |
| rw [G.modularCoordinate_equivariant hc hl o φ y,hker,one_mul] | |
| rfl | |
| exact Aut.preserves_domainComponent G φ _ hS v x.val x.property | |
| (by rw [hφ]; exact x.property) | |
| lemma levelComponent_suborbit_image (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (o : V) (Q : Set ℝˣ) (v : V) : | |
| let C := G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v | |
| let L := componentKernel (G.modularUnitCharacter hc hl o) C | |
| let _ := restrictedVertexAction C L (componentKernel_le _ _) | |
| ∀ x y : C, Subtype.val '' ActionOrbitTransport.stabilizerOrbit L C x y = | |
| G.stabilizerOrbit x.val y.val := by | |
| intro C L _ x y | |
| ext z | |
| constructor | |
| · rintro ⟨w,hw,rfl⟩ | |
| obtain ⟨φ,hx,hy⟩ := (ActionOrbitTransport.mem_stabilizerOrbit L C).mp hw | |
| exact G.mem_stabilizerOrbit.mpr ⟨φ.val,congrArg Subtype.val hx,congrArg Subtype.val hy⟩ | |
| · intro hz | |
| obtain ⟨φ,hx,hy⟩ := G.mem_stabilizerOrbit.mp hz | |
| have hm : φ ∈ L := G.stabilizer_le_levelComponentGroup hc hl o Q v x hx | |
| let ψ : L := ⟨φ,hm⟩ | |
| refine ⟨ψ • y,?_,hy⟩ | |
| exact (ActionOrbitTransport.mem_stabilizerOrbit L C).mpr | |
| ⟨ψ,Subtype.ext hx,rfl⟩ | |
| lemma levelComponent_suborbit_finite (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (o : V) (Q : Set ℝˣ) (v : V) : | |
| let C := G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v | |
| let L := componentKernel (G.modularUnitCharacter hc hl o) C | |
| let _ := restrictedVertexAction C L (componentKernel_le _ _) | |
| ∀ x y : C, (ActionOrbitTransport.stabilizerOrbit L C x y).Finite := by | |
| intro C L _ x y | |
| apply Set.Finite.of_finite_image (f := Subtype.val) ?_ Subtype.val_injective.injOn | |
| rw [G.levelComponent_suborbit_image hc hl o Q v] | |
| exact G.stabilizerOrbit_finite hc hl x.val y.val | |
| lemma levelComponent_suborbitCard (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (o : V) (Q : Set ℝˣ) (v : V) : | |
| let C := G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v | |
| let L := componentKernel (G.modularUnitCharacter hc hl o) C | |
| let _ := restrictedVertexAction C L (componentKernel_le _ _) | |
| ∀ x y : C, ActionOrbitTransport.suborbitCard L C x y = G.suborbitCard x.val y.val := by | |
| intro C L _ x y | |
| unfold ActionOrbitTransport.suborbitCard | |
| rw [←Set.ncard_image_of_injective _ Subtype.val_injective, | |
| G.levelComponent_suborbit_image hc hl o Q v] | |
| rfl | |
| def levelComponent_unimodularRoot (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (hq : G.QuasiTransitive) (o : V) | |
| (Q : Set ℝˣ) (hQ : Q.Finite) (v : V) (hv : G.modularCoordinate hc hl o v ∈ Q) : | |
| (G.induce (G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v)).UnimodularRoot := Classical.choice (by | |
| let C := G.domainComponent (G.modularCoordinate hc hl o ⁻¹' Q) v | |
| let L := componentKernel (G.modularUnitCharacter hc hl o) C | |
| let := restrictedVertexAction C L (componentKernel_le _ _) | |
| let := G.restrictedEdgeAction G.aut_preservesEnds C L (componentKernel_le _ _) | |
| have ha := G.restricted_preservesEnds G.aut_preservesEnds C L (componentKernel_le _ _) | |
| obtain ⟨R,hR,hcover⟩ := hq | |
| have hcover' : ∀ x, ∃ u∈R, x∈MulAction.orbit G.Aut u := by | |
| intro x | |
| obtain ⟨u,hu,φ,hφ⟩ := hcover x | |
| exact ⟨u,hu,φ,hφ⟩ | |
| obtain ⟨S,hS,hScover⟩ := G.componentKernel_finite_orbits G.aut_preservesEnds | |
| R hR hcover' (G.modularUnitCharacter hc hl o) (G.modularCoordinate hc hl o) | |
| (G.modularCoordinate_equivariant hc hl o) Q hQ v hv | |
| let w : C → ℝ := fun x => G.modularRatio o x.val | |
| have hw (φ : L) (x : C) : w (φ • x) = w x := by | |
| have hchar : G.modularUnitCharacter hc hl o φ.val = 1 := φ.property.1 | |
| have hh := congrArg (fun u : ℝˣ => (u : ℝ)) | |
| (G.modularCoordinate_equivariant hc hl o φ.val x.val) | |
| change G.modularRatio o (φ.val.vertex x.val) = _ at hh | |
| change G.modularRatio o (φ.val.vertex x.val) = _ | |
| simpa only [hchar,one_mul,modularCoordinate_val] using hh | |
| have hb (x y : C) : w x * (ActionOrbitTransport.suborbitCard L C x y : ℝ) = | |
| w y * (ActionOrbitTransport.suborbitCard L C y x : ℝ) := by | |
| rw [G.levelComponent_suborbitCard hc hl o Q v, | |
| G.levelComponent_suborbitCard hc hl o Q v] | |
| exact G.modularRatio_card_balance hc hl o x.val y.val | |
| exact ⟨unimodularRoot_of_action (G.induce C) ha S hS hScover | |
| ⟨v,G.domainComponent_self _ _⟩ w (fun x => G.modularRatio_pos hc hl o x.val) | |
| hw (G.levelComponent_suborbit_finite hc hl o Q v) hb⟩) | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |