simp only [Nat.succ_eq_add_one, Set.mem_setOf_eq, Walk.length_cons, add_left_inj,
  Set.mem_iUnion, Set.mem_image, exists_prop]
case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ Walk.cons huw pwv ∈ {p | Walk.length p = Nat.succ n} ↔
    Walk.cons huw pwv ∈ ⋃ (w : V) (h : G.Adj u w), Walk.cons h '' {p' | Walk.length p' = n}
case h.cons
V : Type u
V' : Type v
V'' : Type w
G : Digraph V
G' : Digraph V'
G'' : Digraph V''
u v : V
n : ℕ
w : V
huw : G.Adj u w
pwv : Walk G w v
⊢ Walk.length pwv = n ↔ ∃ i h x, Walk.length x = n ∧ Walk.cons (_ : G.Adj u i) x = Walk.cons huw pwv