-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
hn : p.IsNested
i : ℕ
h : (↑p : List DyckStep) ≠ []
l1 : List.take 1 (↑p : List DyckStep) = [(↑p : List DyckStep).head h]
nil :
  ∀ (count_U_eq_count_D : count U [] = count D [])
    (count_D_le_count_U : ∀ (i : ℕ), count D (List.take i []) ≤ count U (List.take i []))
    (hn : { toList := [], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U }.IsNested)
    (h :
      (↑{ toList := [], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U :
              DyckWord } :
          List DyckStep) ≠
        [])
    (l1 :
      List.take 1
          (↑{ toList := [], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U :
                DyckWord } :
            List DyckStep) =
        [(↑{ toList := [], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U } :
                List DyckStep).head
            h]),
    (↑{ toList := [], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U : DyckWord } :
            List DyckStep).length -
        1 =
      (↑{ toList := [], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U :
                    DyckWord } :
                List DyckStep).length -
            1 -
          1 +
        1
cons :
  ∀ (s : DyckStep) (count_U_eq_count_D : count U [s] = count D [s])
    (count_D_le_count_U : ∀ (i : ℕ), count D (List.take i [s]) ≤ count U (List.take i [s]))
    (hn :
      { toList := [s], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U }.IsNested)
    (h :
      (↑{ toList := [s], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U :
              DyckWord } :
          List DyckStep) ≠
        [])
    (l1 :
      List.take 1
          (↑{ toList := [s], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U :
                DyckWord } :
            List DyckStep) =
        [(↑{ toList := [s], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U } :
                List DyckStep).head
            h]),
    (↑{ toList := [s], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U : DyckWord } :
            List DyckStep).length -
        1 =
      (↑{ toList := [s], count_U_eq_count_D := count_U_eq_count_D, count_D_le_count_U := count_D_le_count_U :
                    DyckWord } :
                List DyckStep).length -
            1 -
          1 +
        1
cons₁ :
  ∀ (s head : DyckStep) (tail : List DyckStep)
    (count_U_eq_count_D : count U (s :: head :: tail) = count D (s :: head :: tail))
    (count_D_le_count_U :
      ∀ (i : ℕ), count D (List.take i (s :: head :: tail)) ≤ count U (List.take i (s :: head :: tail)))
    (hn :
      { toList := s :: head :: tail, count_U_eq_count_D := count_U_eq_count_D,
          count_D_le_count_U := count_D_le_count_U }.IsNested)
    (h :
      (↑{ toList := s :: head :: tail, count_U_eq_count_D := count_U_eq_count_D,
              count_D_le_count_U := count_D_le_count_U : DyckWord } :
          List DyckStep) ≠
        [])
    (l1 :
      List.take 1
          (↑{ toList := s :: head :: tail, count_U_eq_count_D := count_U_eq_count_D,
                count_D_le_count_U := count_D_le_count_U : DyckWord } :
            List DyckStep) =
        [(↑{ toList := s :: head :: tail, count_U_eq_count_D := count_U_eq_count_D,
                    count_D_le_count_U := count_D_le_count_U } :
                List DyckStep).head
            h]),
    (↑{ toList := s :: head :: tail, count_U_eq_count_D := count_U_eq_count_D,
                count_D_le_count_U := count_D_le_count_U : DyckWord } :
            List DyckStep).length -
        1 =
      (↑{ toList := s :: head :: tail, count_U_eq_count_D := count_U_eq_count_D,
                    count_D_le_count_U := count_D_le_count_U : DyckWord } :
                List DyckStep).length -
            1 -
          1 +
        1
⊢ (↑p : List DyckStep).length - 1 = (↑p : List DyckStep).length - 1 - 1 + 1

Tactic:
rcases p with - | ⟨s, ⟨- | ⟨t, r⟩⟩⟩