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) [Quiver V] : Setoid V

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

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

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.

def Quiver.WeaklyConnectedComponent.mk {V : Type u_1} [Quiver V] : V → Quiver.WeaklyConnectedComponent V

The weakly connected component corresponding to a vertex.

instance Quiver.WeaklyConnectedComponent.instCoeTCWeaklyConnectedComponent {V : Type u_1} [Quiver V] : CoeTC V (Quiver.WeaklyConnectedComponent V)

instance Quiver.WeaklyConnectedComponent.instInhabitedWeaklyConnectedComponent {V : Type u_1} [Quiver V] [Inhabited V] : Inhabited (Quiver.WeaklyConnectedComponent V)

theorem Quiver.WeaklyConnectedComponent.eq {V : Type u_1} [Quiver V] (a : V) (b : V) : Quiver.WeaklyConnectedComponent.mk a = Quiver.WeaklyConnectedComponent.mk b ↔ Nonempty (Quiver.Path a b)

def Quiver.wideSubquiverSymmetrify {V : Type u_1} [Quiver V] (H : WideSubquiver (Quiver.Symmetrify V)) : WideSubquiver V

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.