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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | 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
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