# Documentation

## Weakly connected components #

For a quiver V, define the type WeaklyConnectedComponent V as the quotient of V by the relation which identifies a with b if there is a path from a to b in Symmetrify V. (These zigzags can be seen as a proof-relevant analogue of EqvGen.)

Strongly connected components have not yet been defined.

def Quiver.zigzagSetoid (V : Type u_1) [] :

Two vertices are related in the zigzag setoid if there is a zigzag of arrows from one to the other.

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• = { r := fun (a b : V) => Nonempty (), iseqv := }
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The type of weakly connected components of a directed graph. Two vertices are in the same weakly connected component if there is a zigzag of arrows from one to the other.

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The weakly connected component corresponding to a vertex.

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• Quiver.WeaklyConnectedComponent.mk = Quotient.mk'
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• Quiver.WeaklyConnectedComponent.instCoeTC = { coe := Quiver.WeaklyConnectedComponent.mk }
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• Quiver.WeaklyConnectedComponent.instInhabited = { default := let_fun this := default; }
theorem Quiver.WeaklyConnectedComponent.eq {V : Type u_1} [] (a : V) (b : V) :
def Quiver.wideSubquiverSymmetrify {V : Type u_1} [] (H : ) :

A wide subquiver H of Symmetrify V determines a wide subquiver of V, containing an arrow e if either e or its reversal is in H.

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• = {e : x✝ x | H x✝ x () H x x✝ ()}
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