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combinatorics.quiver.arborescence

Arborescences #

A quiver V is an arborescence (or directed rooted tree) when we have a root vertex root : V such that for every b : V there is a unique path from root to b.

Main definitions #

• quiver.arborescence V: a typeclass asserting that V is an arborescence
• arborescence_mk: a convenient way of proving that a quiver is an arborescence
• rooted_connected r: a typeclass asserting that there is at least one path from r to b for every b.
• geodesic_subtree r: given [rooted_conntected r], this is a subquiver of V which contains just enough edges to include a shortest path from r to b for every b.
• geodesic_arborescence : arborescence (geodesic_subtree r): an instance saying that the geodesic subtree is an arborescence. This proves the directed analogue of 'every connected graph has a spanning tree'. This proof avoids the use of Zorn's lemma.
@[class]
structure quiver.arborescence (V : Type u) [quiver V] :
Type (max u v)
• root : V
• unique_path : Π (b : V),

A quiver is an arborescence when there is a unique path from the default vertex to every other vertex.

Instances of this typeclass
Instances of other typeclasses for quiver.arborescence
• quiver.arborescence.has_sizeof_inst
def quiver.root (V : Type u) [quiver V]  :
V

The root of an arborescence.

Equations
@[protected, instance]
def quiver.path.unique {V : Type u} [quiver V] (b : V) :
Equations
noncomputable def quiver.arborescence_mk {V : Type u} [quiver V] (r : V) (height : V → ) (height_lt : ∀ ⦃a b : V⦄, (a b)height a < height b) (unique_arrow : ∀ ⦃a b c : V⦄ (e : a c) (f : b c), a = b e == f) (root_or_arrow : ∀ (b : V), b = r ∃ (a : V), nonempty (a b)) :

To show that [quiver V] is an arborescence with root r : V, it suffices to

• provide a height function V → ℕ such that every arrow goes from a lower vertex to a higher vertex,
• show that every vertex has at most one arrow to it, and
• show that every vertex other than r has an arrow to it.
Equations
• height height_lt unique_arrow root_or_arrow = {root := r, unique_path := λ (b : V),
@[class]
structure quiver.rooted_connected {V : Type u} [quiver V] (r : V) :
Prop
• nonempty_path : ∀ (b : V), nonempty b)

rooted_connected r means that there is a path from r to any other vertex.

Instances of this typeclass
noncomputable def quiver.shortest_path {V : Type u} [quiver V] (r : V) (b : V) :
b

A path from r of minimal length.

Equations
• = _
theorem quiver.shortest_path_spec {V : Type u} [quiver V] (r : V) {a : V} (p : a) :

The length of a path is at least the length of the shortest path

def quiver.geodesic_subtree {V : Type u} [quiver V] (r : V)  :

A subquiver which by construction is an arborescence.

Equations
• = λ (a b : V), {e : a b | ∃ (p : a), = p.cons e}
Instances for ↥quiver.geodesic_subtree
@[protected, instance]
noncomputable def quiver.geodesic_arborescence {V : Type u} [quiver V] (r : V)  :
Equations