case succ l ih =>
  rcases ih with ⟨s, Hs, hls⟩
  have hl : l < lh s := by simp [←hls]
  have : ∃ z, ⟪l, z⟫ ∈ s := Hs.seq.exists hl
  rcases this with ⟨z, hz⟩
  exact ⟨s ⁀' c.succ v l z, Hs.successor hls hz, by simp [Hs.seq, hls]⟩
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚺₁
k : ℕ
p : Formulae k
c : Construction M p
v : Fin k → M
l : M
ih : ∃ s, c.CSeq v s ∧ l + 1 = lh s
⊢ ∃ s, c.CSeq v s ∧ l + 1 + 1 = lh s
no goals