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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 | 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
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