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.

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

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

Equations

• Quiver.zigzagSetoid V = { r := fun a b => Nonempty (Quiver.Path a b), iseqv := (_ : Equivalence fun a b => Nonempty (Quiver.Path a b)) }

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

Equations

• Quiver.WeaklyConnectedComponent V = Quotient (Quiver.zigzagSetoid V)

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

The weakly connected component corresponding to a vertex.

Equations

• Quiver.WeaklyConnectedComponent.mk = Quotient.mk'

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

Equations

• Quiver.WeaklyConnectedComponent.instCoeTCWeaklyConnectedComponent = { coe := Quiver.WeaklyConnectedComponent.mk }

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

Equations

• Quiver.WeaklyConnectedComponent.instInhabitedWeaklyConnectedComponent = { default := let_fun this := default; Quiver.WeaklyConnectedComponent.mk this }

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

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

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

Equations

• Quiver.wideSubquiverSymmetrify H x x = { e | H x x (Sum.inl e) ∨ H x x (Sum.inr e) }