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544609e | 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 | import Mathlib
namespace OAI
/-! Finite-index virtual isomorphism graphs. -/
namespace ArithmeticGerms
variable {G H : Type*} [Group G] [Group H]
structure VirtualGraph (G H : Type*) [Group G] [Group H] where
graph : Subgroup (G × H)
fst_injective : Function.Injective ((MonoidHom.fst G H).comp graph.subtype)
snd_injective : Function.Injective ((MonoidHom.snd G H).comp graph.subtype)
fst_finite : (graph.map (MonoidHom.fst G H)).FiniteIndex
snd_finite : (graph.map (MonoidHom.snd G H)).FiniteIndex
attribute [instance] VirtualGraph.fst_finite VirtualGraph.snd_finite
namespace VirtualGraph
variable (C : VirtualGraph G H)
@[ext]
theorem ext {C D : VirtualGraph G H} (h : C.graph = D.graph) : C = D := by
cases C
cases D
cases h
rfl
abbrev domain : Subgroup G := C.graph.map (MonoidHom.fst G H)
abbrev codomain : Subgroup H := C.graph.map (MonoidHom.snd G H)
end VirtualGraph
end ArithmeticGerms
end OAI
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