openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /InducedAutomorphism.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced | |
| import OAI.Probability.CriticalPercolation.Criticality.ComponentCritical | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma Aut.inducedEdges_mem (phi : G.Aut) (S : Set V) | |
| (hS : ∀ x, phi.vertex x∈S ↔ x∈S) (e : E) : | |
| phi.edge e∈G.inducedEdges S ↔ e∈G.inducedEdges S := by | |
| constructor | |
| · intro h x hx | |
| apply (hS x).mp | |
| apply h (phi.vertex x) | |
| rw [phi.preserves] | |
| exact Sym2.mem_map.mpr ⟨x,hx,rfl⟩ | |
| · intro h y hy | |
| rw [phi.preserves] at hy | |
| obtain ⟨x,hx,rfl⟩ := Sym2.mem_map.mp hy | |
| exact (hS x).mpr (h x hx) | |
| lemma Aut.inducedEdges_image (phi : G.Aut) (S : Set V) | |
| (hS : ∀ x, phi.vertex x∈S ↔ x∈S) : | |
| phi.edge '' G.inducedEdges S=G.inducedEdges S := by | |
| ext e | |
| constructor | |
| · rintro ⟨a,ha,rfl⟩ | |
| exact (Aut.inducedEdges_mem G phi S hS a).mpr ha | |
| · intro he | |
| exact ⟨phi.edge.symm e,(Aut.inducedEdges_mem G phi S hS (phi.edge.symm e)).mp | |
| (by simpa only [Equiv.apply_symm_apply] using he),phi.edge.apply_symm_apply e⟩ | |
| def Aut.restrict (phi : G.Aut) (S : Set V) | |
| (hS : ∀ x, phi.vertex x∈S ↔ x∈S) : (G.induce S).Aut where | |
| vertex := | |
| { toFun := fun x => ⟨phi.vertex x,(hS x).mpr x.property⟩ | |
| invFun := fun x => ⟨phi.vertex.symm x,(hS (phi.vertex.symm x)).mp | |
| (by simpa only [Equiv.apply_symm_apply] using x.property)⟩ | |
| left_inv := fun x => Subtype.ext (phi.vertex.symm_apply_apply x) | |
| right_inv := fun x => Subtype.ext (phi.vertex.apply_symm_apply x) } | |
| edge := | |
| { toFun := fun e => ⟨phi.edge e,(Aut.inducedEdges_mem G phi S hS e).mpr e.property⟩ | |
| invFun := fun e => ⟨phi.edge.symm e,(Aut.inducedEdges_mem G phi S hS (phi.edge.symm e)).mp | |
| (by simpa only [Equiv.apply_symm_apply] using e.property)⟩ | |
| left_inv := fun e => Subtype.ext (phi.edge.symm_apply_apply e) | |
| right_inv := fun e => Subtype.ext (phi.edge.apply_symm_apply e) } | |
| preserves e := by | |
| apply Sym2.map.injective Subtype.val_injective | |
| rw [G.induce_ends_map,Sym2.map_map] | |
| change G.ends (phi.edge e.val)=Sym2.map (fun x : S => phi.vertex x.val) ((G.induce S).ends e) | |
| calc | |
| G.ends (phi.edge e.val)=Sym2.map phi.vertex (G.ends e.val) := phi.preserves e.val | |
| _=Sym2.map phi.vertex (Sym2.map (Subtype.val : S → V) ((G.induce S).ends e)) := | |
| congrArg (Sym2.map phi.vertex) (G.induce_ends_map S e).symm | |
| _=_ := Sym2.map_map _ | |
| lemma Aut.domainComponent_image (phi : G.Aut) (S : Set V) | |
| (hS : ∀ x, phi.vertex x∈S ↔ x∈S) (o : V) : | |
| phi.vertex '' G.domainComponent S o=G.domainComponent S (phi.vertex o) := by | |
| change phi.vertex '' G.cluster (G.inducedEdges S) o=G.cluster (G.inducedEdges S) (phi.vertex o) | |
| rw [Aut.cluster_image,Aut.inducedEdges_image G phi S hS] | |
| lemma domainComponent_eq_of_mem (S : Set V) {o x : V} (hx : x∈G.domainComponent S o) : | |
| G.domainComponent S x=G.domainComponent S o := by | |
| ext y | |
| exact ⟨fun h => hx.trans h,fun h => hx.symm.trans h⟩ | |
| lemma Aut.preserves_domainComponent (phi : G.Aut) (S : Set V) | |
| (hS : ∀ x, phi.vertex x∈S ↔ x∈S) (o x : V) | |
| (hx : x∈G.domainComponent S o) (hpx : phi.vertex x∈G.domainComponent S o) : | |
| ∀ y, phi.vertex y∈G.domainComponent S o ↔ y∈G.domainComponent S o := by | |
| have heq : phi.vertex '' G.domainComponent S o=G.domainComponent S o := calc | |
| phi.vertex '' G.domainComponent S o=phi.vertex '' G.domainComponent S x := | |
| congrArg (Set.image phi.vertex) (G.domainComponent_eq_of_mem S hx).symm | |
| _=G.domainComponent S (phi.vertex x) := Aut.domainComponent_image G phi S hS x | |
| _=G.domainComponent S o := G.domainComponent_eq_of_mem S hpx | |
| intro y | |
| constructor | |
| · intro hy | |
| have him : phi.vertex y∈phi.vertex '' G.domainComponent S o := by rwa [heq] | |
| obtain ⟨z,hz,hzy⟩ := him | |
| exact phi.vertex.injective hzy ▸ hz | |
| · intro hy | |
| rw [←heq] | |
| exact ⟨y,hy,rfl⟩ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |