openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphTopology /GraphAutomorphismTopology.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Topology | |
| open scoped Classical | |
| namespace CriticalPercolation.GraphTopology | |
| variable {X : Type*} [TopologicalSpace X] [DiscreteTopology X] | |
| lemma continuous_variable_eval : Continuous (fun p : (X → X) × X => p.1 p.2) := | |
| continuous_prod_of_discrete_right.mpr fun x => continuous_apply x | |
| variable (Γ : SimpleGraph X) | |
| def coordinates (g : Γ ≃g Γ) : (X → X) × (X → X) := (g,g.symm) | |
| @[instance_reducible] def autTopology : TopologicalSpace (Γ ≃g Γ) := | |
| TopologicalSpace.induced (coordinates Γ) inferInstance | |
| local instance : TopologicalSpace (Γ ≃g Γ) := autTopology Γ | |
| omit [DiscreteTopology X] in | |
| lemma coordinates_embedding : IsEmbedding (coordinates Γ) where | |
| eq_induced := rfl | |
| injective := by | |
| intro g h he | |
| apply RelIso.ext | |
| intro x | |
| exact congrFun (congrArg Prod.fst he) x | |
| omit [DiscreteTopology X] in | |
| lemma continuous_eval (x : X) : Continuous (fun g : Γ ≃g Γ => g x) := | |
| (continuous_apply x).comp (continuous_fst.comp (coordinates_embedding Γ).continuous) | |
| omit [DiscreteTopology X] in | |
| lemma continuous_inv_eval (x : X) : Continuous (fun g : Γ ≃g Γ => g.symm x) := | |
| (continuous_apply x).comp (continuous_snd.comp (coordinates_embedding Γ).continuous) | |
| omit [TopologicalSpace X] [DiscreteTopology X] in | |
| lemma coordinates_range : Set.range (coordinates Γ) = | |
| {p : (X → X) × (X → X) | | |
| (∀ x, p.2 (p.1 x)=x) ∧ (∀ x, p.1 (p.2 x)=x) ∧ | |
| (∀ x y, Γ.Adj (p.1 x) (p.1 y) ↔ Γ.Adj x y)} := by | |
| ext p | |
| constructor | |
| · rintro ⟨g,rfl⟩ | |
| exact ⟨g.symm_apply_apply,g.apply_symm_apply,fun _ _ => g.map_adj_iff⟩ | |
| · rintro ⟨hleft,hright,hadj⟩ | |
| refine ⟨{toEquiv := ⟨p.1,p.2,hleft,hright⟩,map_rel_iff' := fun {x y} => hadj x y},?_⟩ | |
| exact Prod.eta p | |
| lemma coordinates_closed : IsClosed (Set.range (coordinates Γ)) := by | |
| rw [coordinates_range] | |
| simp only [Set.ofPred_and,Set.ofPred_forall] | |
| refine (isClosed_iInter fun x => ?_).inter ((isClosed_iInter fun x => ?_).inter | |
| (isClosed_iInter fun x => isClosed_iInter fun y => ?_)) | |
| · change IsClosed {p : (X → X) × (X → X) | p.2 (p.1 x)=x} | |
| exact isClosed_eq ((continuous_variable_eval (X := X)).comp (continuous_snd.prodMk | |
| ((continuous_apply x).comp continuous_fst))) continuous_const | |
| · change IsClosed {p : (X → X) × (X → X) | p.1 (p.2 x)=x} | |
| exact isClosed_eq ((continuous_variable_eval (X := X)).comp (continuous_fst.prodMk | |
| ((continuous_apply x).comp continuous_snd))) continuous_const | |
| · change IsClosed {p : (X → X) × (X → X) | Γ.Adj (p.1 x) (p.1 y) ↔ Γ.Adj x y} | |
| exact (isClosed_discrete {p : X × X | Γ.Adj p.1 p.2 ↔ Γ.Adj x y}).preimage | |
| (show Continuous (fun p : (X → X) × (X → X) => (p.1 x,p.1 y)) from | |
| ((continuous_apply x).comp continuous_fst).prodMk | |
| ((continuous_apply y).comp continuous_fst)) | |
| lemma coordinates_closedEmbedding : IsClosedEmbedding (coordinates Γ) := | |
| ⟨coordinates_embedding Γ,coordinates_closed Γ⟩ | |
| instance aut_t2 : T2Space (Γ ≃g Γ) := (coordinates_embedding Γ).t2Space | |
| instance aut_topologicalGroup : IsTopologicalGroup (Γ ≃g Γ) where | |
| continuous_mul := by | |
| apply (coordinates_embedding Γ).continuous_iff.mpr | |
| apply Continuous.prodMk | |
| · apply continuous_pi | |
| intro x | |
| exact continuous_variable_eval.comp | |
| (((continuous_fst.comp (coordinates_embedding Γ).continuous).comp continuous_fst).prodMk | |
| ((continuous_eval Γ x).comp continuous_snd)) | |
| · apply continuous_pi | |
| intro x | |
| exact continuous_variable_eval.comp | |
| (((continuous_snd.comp (coordinates_embedding Γ).continuous).comp continuous_snd).prodMk | |
| ((continuous_inv_eval Γ x).comp continuous_fst)) | |
| continuous_inv := by | |
| apply (coordinates_embedding Γ).continuous_iff.mpr | |
| exact (continuous_snd.comp (coordinates_embedding Γ).continuous).prodMk | |
| (continuous_fst.comp (coordinates_embedding Γ).continuous) | |
| lemma compact_coordinate_bounds (F : X → Set X) (hF : ∀ x, (F x).Finite) : | |
| IsCompact {g : Γ ≃g Γ | (∀ x, g x∈F x) ∧ (∀ x, g.symm x∈F x)} := by | |
| have hK : IsCompact {f : X → X | ∀ x, f x∈F x} := | |
| isCompact_pi_infinite fun x => (hF x).isCompact | |
| exact (coordinates_closedEmbedding Γ).isCompact_preimage (hK.prod hK) | |
| lemma isClosed_eval_mem (x : X) (S : Set X) : | |
| IsClosed {g : Γ ≃g Γ | g x∈S} := | |
| (isClosed_discrete S).preimage (continuous_eval Γ x) | |
| lemma isOpen_eval_mem (x : X) (S : Set X) : | |
| IsOpen {g : Γ ≃g Γ | g x∈S} := | |
| (isOpen_discrete S).preimage (continuous_eval Γ x) | |
| end CriticalPercolation.GraphTopology | |
| end | |
| end | |
| end OAI | |