-- Lean 4 proof step from Mathlib.Combinatorics.SimpleGraph.Acyclic (goal)
Goal:
V : Type u_1
G : SimpleGraph V
hconn : G.Connected
v : V
inst : Fintype (↑(G.neighborSet v) : Type u_1)
hdeg : G.degree v = 1
u : V
adj_vu : G.Adj v u
hu : ∀ (y : V), (fun (w : V) ↦ G.Adj v w) y → y = u
w x : (↑{v}ᶜ : Type u_1)
pwu : G.Walk (↑w : V) u
hpwu : pwu.IsPath
pux : G.Walk u (↑x : V)
hpux : pux.IsPath
w₁ : Walk (induce {v}ᶜ G) w x
h : True
w₂ w₃ :
  ∀ (z : V) (hz : z ∈ (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)).support), Membership.mem (γ := Set V) {v}ᶜ z
w₄ : ∀ (z : V) (hz : z ∈ (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)).support), ¬z = v
w₅ :
  ∀ (z : V) (hz : z ∈ (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)).support) (pwz : G.Walk (↑w : V) z)
    (pzx : G.Walk z (↑x : V)) (p_eq_pwzx : (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)) = pwz.append pzx),
    ¬z = v
w₆ :
  ∀ (z : V) (hz : z ∈ (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)).support) (pwz : G.Walk (↑w : V) z)
    (pzx : G.Walk z (↑x : V)) (p_eq_pwzx : (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)) = pwz.append pzx)
    (this : z = v), False
w₇ :
  ∀ (z : V) [inst : Fintype (↑(G.neighborSet z) : Type u_1)] (hdeg : G.degree z = 1) (adj_vu : G.Adj z u)
    (hu : ∀ (y : V), (fun (w : V) ↦ G.Adj z w) y → y = u) (w x : (↑{z}ᶜ : Type u_1)) (pwu : G.Walk (↑w : V) u)
    (hpwu : pwu.IsPath) (pux : G.Walk u (↑x : V)) (hpux : pux.IsPath)
    (hz : z ∈ (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)).support) (pwz : G.Walk (↑w : V) z)
    (pzx : G.Walk z (↑x : V)) (p_eq_pwzx : (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)) = pwz.append pzx),
    False
⊢ ∃ (x : Walk (induce {v}ᶜ G) w x), True
have w₈ : ∀ (z : V) [inst : Fintype (↑(G.neighborSet z) : Type u_1)] (hdeg : G.degree z = 1) (adj_vu : G.Adj z u)
  (hu : ∀ (y : V), (fun (w : V) ↦ G.Adj z w) y → y = u) (w x : (↑{z}ᶜ : Type u_1)) (pwu : G.Walk (↑w : V) u)
  (hpwu : pwu.IsPath) (pux : G.Walk u (↑x : V)) (hpux : pux.IsPath)
  (hz : z ∈ (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)).support) (pwz : G.Walk (↑w : V) z)
  (pzx : G.Walk z (↑x : V)) (p_eq_pwzx : (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)) = pwz.append pzx),
  List.Duplicate u (↑(pwu.append pux).toPath : G.Walk (↑w : V) (↑x : V)).support
Tactic: classical
  use ((pwu.append pux).toPath.val.induce {v}ᶜ ?_).copy (SetCoe.ext rfl) (SetCoe.ext rfl)
  /- Each path between vertex u and another vertex in G.induce {v}ᶜ
  is contained in G.induce {v}ᶜ. -/
  intro z hz
  rw [Set.mem_compl_iff, Set.mem_singleton_iff]
  obtain ⟨pwz, pzx, p_eq_pwzx⟩ := mem_support_iff_exists_append.mp hz
  /- Prove vertex v is not in the path formed from the concatenated walks
  by showing that vertex u must then be passed twice. -/
  by_contra
  subst_vars
  refine List.nodup_iff_forall_not_duplicate.mp (pwu.append pux).toPath.nodup_support u ?_
  rw [p_eq_pwzx, support_append, List.duplicate_iff_two_le_count, List.count_append]
  have := List.one_le_count_iff.mpr (pwz.getVert_mem_support (pwz.length - 1))
  simp only [hu _ (pwz.adj_penultimate (not_nil_of_ne (by aesop))).symm] at this
  have := List.one_le_count_iff.mpr (pzx.snd_mem_tail_support (not_nil_of_ne (by aesop)))
  rw [hu _ (pzx.adj_snd (not_nil_of_ne (by aesop)))] at this
  lia
Cleared hypotheses: v, inst, hdeg, adj_vu, hu, w, x, pwu, hpwu, pux, hpux