awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ p.semilength = count D (↑p : List DyckStep)
have rw : p.semilength = count U (↑p : List DyckStep)
Tactic: rw [← count_U_eq_count_D]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ 2 * p.semilength = (↑p : List DyckStep).length
have nth_rw : count D (↑p : List DyckStep) + count U (↑p : List DyckStep) = (↑p : List DyckStep).length
Tactic: nth_rw 1 [two_mul, semilength, p.count_U_eq_count_D, semilength]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ count D (↑p : List DyckStep) + count U (↑p : List DyckStep) = (↑p : List DyckStep).length
have h : count U = countP fun (a : DyckStep) ↦ decide ¬(a == D) = true
Tactic: convert! (p.toList.length_eq_countP_add_countP (· == D)).symm
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ Eq (α := List DyckStep → ℕ) (countP fun (x : DyckStep) ↦ x == U) (countP fun (a : DyckStep) ↦ decide ¬(a == D) = true)
have h : ∀ (x : DyckStep), (x == U) = decide ¬(x == D) = true
Tactic: congr!
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
s : DyckStep
⊢ (s == U) = decide ¬(s == D) = true
have h : (U == U) = decide ¬(U == D) = true
Tactic: cases s
Cleared hypotheses: s
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
s : DyckStep
h : (U == U) = decide ¬(U == D) = true
⊢ (s == U) = decide ¬(s == D) = true
have h₁ : (D == U) = decide ¬(D == D) = true
Tactic: cases s
Cleared hypotheses: s
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
⊢ 0 < p.firstReturn
have rw : ¬(decide
        (count U (List.take (HAdd.hAdd (α := ℕ) 0 1) (↑p : List DyckStep)) =
          count D (List.take (HAdd.hAdd (α := ℕ) 0 1) (↑p : List DyckStep))) =
      true ∧
    ∀ (j : ℕ) (hji : j < 0),
      decide
          (count U
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) j sorry +
                  1)
                (↑p : List DyckStep)) =
            count D
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) j sorry +
                  1)
                (↑p : List DyckStep))) =
        false)
Tactic: rw [← not_le, Nat.le_zero, firstReturn, findIdx_eq, getElem_range]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
rw :
  ¬(decide
          (count U (List.take (HAdd.hAdd (α := ℕ) 0 1) (↑p : List DyckStep)) =
            count D (List.take (HAdd.hAdd (α := ℕ) 0 1) (↑p : List DyckStep))) =
        true ∧
      ∀ (j : ℕ) (hji : j < 0),
        decide
            (count U
                (List.take
                  (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) j
                      ⋯ +
                    1)
                  (↑p : List DyckStep)) =
              count D
                (List.take
                  (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) j
                      ⋯ +
                    1)
                  (↑p : List DyckStep))) =
          false)
⊢ 0 < p.firstReturn
have rw₁ : 0 < (range (↑p : List DyckStep).length).length
Tactic: rw [← not_le, Nat.le_zero, firstReturn, findIdx_eq, getElem_range]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
⊢ ¬(decide
          (count U (List.take (HAdd.hAdd (α := ℕ) 0 1) (↑p : List DyckStep)) =
            count D (List.take (HAdd.hAdd (α := ℕ) 0 1) (↑p : List DyckStep))) =
        true ∧
      ∀ (j : ℕ) (hji : j < 0),
        decide
            (count U
                (List.take
                  (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) j
                      ⋯ +
                    1)
                  (↑p : List DyckStep)) =
              count D
                (List.take
                  (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) j
                      ⋯ +
                    1)
                  (↑p : List DyckStep))) =
          false)
have rw! : ¬(decide
        (count U (List.take (HAdd.hAdd (α := ℕ) 0 1) (U :: (↑p : List DyckStep).dropLast.tail ++ [D])) =
          count D (List.take (HAdd.hAdd (α := ℕ) 0 1) (U :: (↑p : List DyckStep).dropLast.tail ++ [D]))) =
      true ∧
    ∀ (j : ℕ) (hji : j < 0),
      decide
          (count U
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length)
                    (range (U :: (↑p : List DyckStep).dropLast.tail ++ [D]).length) j sorry +
                  1)
                (U :: (↑p : List DyckStep).dropLast.tail ++ [D])) =
            count D
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length)
                    (range (U :: (↑p : List DyckStep).dropLast.tail ++ [D]).length) j sorry +
                  1)
                (U :: (↑p : List DyckStep).dropLast.tail ++ [D]))) =
        false)
Tactic: rw! [← p.cons_tail_dropLast_concat h]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
⊢ 0 < (range (↑p : List DyckStep).length).length
have rw : (↑p : List DyckStep) ≠ []
Tactic: rw [length_range, length_pos_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p : DyckWord
h : p ≠ 0
⊢ p.firstReturn < (↑p : List DyckStep).length
have lp : 0 < (↑p : List DyckStep).length
Tactic: have lp := length_pos_of_ne_nil (toList_ne_nil.mpr h)
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
⊢ p.firstReturn < (↑p : List DyckStep).length
have rw : p.firstReturn < (range (↑p : List DyckStep).length).length
Tactic: rw [← length_range (n := p.toList.length)]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
⊢ p.firstReturn < (range (↑p : List DyckStep).length).length
have h₁ : ∃ x ∈ range (↑p : List DyckStep).length,
  decide (count U (List.take (x + 1) (↑p : List DyckStep)) = count D (List.take (x + 1) (↑p : List DyckStep))) = true
Tactic: apply findIdx_lt_length_of_exists
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
⊢ ∃ x ∈ range (↑p : List DyckStep).length,
    decide (count U (List.take (x + 1) (↑p : List DyckStep)) = count D (List.take (x + 1) (↑p : List DyckStep))) = true
have h₁ : ∃ x < (↑p : List DyckStep).length,
  count U (List.take (x + 1) (↑p : List DyckStep)) = count D (List.take (x + 1) (↑p : List DyckStep))
Tactic: simp only [mem_range, decide_eq_true_eq]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
⊢ ∃ x < (↑p : List DyckStep).length,
    count U (List.take (x + 1) (↑p : List DyckStep)) = count D (List.take (x + 1) (↑p : List DyckStep))
have h₁ : (↑p : List DyckStep).length - 1 < (↑p : List DyckStep).length ∧
  count U (List.take ((↑p : List DyckStep).length - 1 + 1) (↑p : List DyckStep)) =
    count D (List.take ((↑p : List DyckStep).length - 1 + 1) (↑p : List DyckStep))
Tactic: use p.toList.length - 1
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
⊢ count U (List.take ((↑p : List DyckStep).length - 1 + 1) (↑p : List DyckStep)) =
    count D (List.take ((↑p : List DyckStep).length - 1 + 1) (↑p : List DyckStep))
have rw : count U (List.take (↑p : List DyckStep).length (↑p : List DyckStep)) =
  count D (List.take (↑p : List DyckStep).length (↑p : List DyckStep))
Tactic: rw [Nat.sub_add_cancel lp, take_of_length_le (le_refl _),
    p.count_U_eq_count_D]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
rw :
  count U (List.take (↑p : List DyckStep).length (↑p : List DyckStep)) =
    count D (List.take (↑p : List DyckStep).length (↑p : List DyckStep))
⊢ count U (List.take ((↑p : List DyckStep).length - 1 + 1) (↑p : List DyckStep)) =
    count D (List.take ((↑p : List DyckStep).length - 1 + 1) (↑p : List DyckStep))
have rw₁ : count U (↑p : List DyckStep) = count D (↑p : List DyckStep)
Tactic: rw [Nat.sub_add_cancel lp, take_of_length_le (le_refl _),
    p.count_U_eq_count_D]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
rw :
  count U (List.take (↑p : List DyckStep).length (↑p : List DyckStep)) =
    count D (List.take (↑p : List DyckStep).length (↑p : List DyckStep))
rw₁ : count U (↑p : List DyckStep) = count D (↑p : List DyckStep)
⊢ count U (List.take ((↑p : List DyckStep).length - 1 + 1) (↑p : List DyckStep)) =
    count D (List.take ((↑p : List DyckStep).length - 1 + 1) (↑p : List DyckStep))
have rw₂ : count D (↑p : List DyckStep) = count D (↑p : List DyckStep)
Tactic: rw [Nat.sub_add_cancel lp, take_of_length_le (le_refl _),
    p.count_U_eq_count_D]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p : DyckWord
h : p ≠ 0
⊢ count U (List.take (p.firstReturn + 1) (↑p : List DyckStep)) =
    count D (List.take (p.firstReturn + 1) (↑p : List DyckStep))
have this : decide
    (count U
        (List.take
          (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length)
              (findIdx
                (fun (i : ℕ) ↦
                  decide
                    (count U (List.take (i + 1) (↑p : List DyckStep)) =
                      count D (List.take (i + 1) (↑p : List DyckStep))))
                (range (↑p : List DyckStep).length))
              sorry +
            1)
          (↑p : List DyckStep)) =
      count D
        (List.take
          (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length)
              (findIdx
                (fun (i : ℕ) ↦
                  decide
                    (count U (List.take (i + 1) (↑p : List DyckStep)) =
                      count D (List.take (i + 1) (↑p : List DyckStep))))
                (range (↑p : List DyckStep).length))
              sorry +
            1)
          (↑p : List DyckStep))) =
  true
Tactic: have := findIdx_getElem
    (w := (length_range (n := p.toList.length)).symm ▸ firstReturn_lt_length h)
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p : DyckWord
i : ℕ
hi : i < p.firstReturn
⊢ count D (List.take (i + 1) (↑p : List DyckStep)) < count U (List.take (i + 1) (↑p : List DyckStep))
have ne : decide
    (count U
        (List.take
          (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) i sorry + 1)
          (↑p : List DyckStep)) =
      count D
        (List.take
          (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) i sorry + 1)
          (↑p : List DyckStep))) =
  false
Tactic: have ne := not_of_lt_findIdx hi
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p : DyckWord
i : ℕ
hi : i < p.firstReturn
ne :
  decide
      (count U
          (List.take
            (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) i ⋯ + 1)
            (↑p : List DyckStep)) =
        count D
          (List.take
            (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p : List DyckStep).length) i ⋯ + 1)
            (↑p : List DyckStep))) =
    false
⊢ count D (List.take (i + 1) (↑p : List DyckStep)) < count U (List.take (i + 1) (↑p : List DyckStep))
have ne₁ : count U (List.take (i + 1) (↑p : List DyckStep)) ≠ count D (List.take (i + 1) (↑p : List DyckStep))
Tactic: rw [decide_eq_false_iff_not, ← ne_eq, getElem_range] at ne
Cleared hypotheses: ne
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
⊢ (p + q).firstReturn = if p = 0 then q.firstReturn else p.firstReturn
have pos : ∀ (h : p = 0), (p + q).firstReturn = q.firstReturn
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
pos : ∀ (h : p = 0), (p + q).firstReturn = q.firstReturn
⊢ (p + q).firstReturn = if p = 0 then q.firstReturn else p.firstReturn
have neg : ∀ (h : ¬p = 0), (p + q).firstReturn = p.firstReturn
Tactic: split_ifs with h
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p q : DyckWord
h : ¬p = 0
⊢ (p + q).firstReturn = p.firstReturn
have u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
Tactic: have u : (p + q).toList = p.toList ++ q.toList := rfl
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
⊢ (p + q).firstReturn = p.firstReturn
have neg : decide
      (count U
          (List.take
            (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                p.firstReturn sorry +
              1)
            (↑(p + q) : List DyckStep)) =
        count D
          (List.take
            (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                p.firstReturn sorry +
              1)
            (↑(p + q) : List DyckStep))) =
    true ∧
  ∀ (j : ℕ) (hji : j < p.firstReturn),
    decide
        (count U
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length) j
                  sorry +
                1)
              (↑(p + q) : List DyckStep)) =
          count D
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length) j
                  sorry +
                1)
              (↑(p + q) : List DyckStep))) =
      false
Tactic: rw [firstReturn, findIdx_eq]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
neg :
  decide
        (count U
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                  p.firstReturn ⋯ +
                1)
              (↑(p + q) : List DyckStep)) =
          count D
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                  p.firstReturn ⋯ +
                1)
              (↑(p + q) : List DyckStep))) =
      true ∧
    ∀ (j : ℕ) (hji : j < p.firstReturn),
      decide
          (count U
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                    j ⋯ +
                  1)
                (↑(p + q) : List DyckStep)) =
            count D
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                    j ⋯ +
                  1)
                (↑(p + q) : List DyckStep))) =
        false
⊢ (p + q).firstReturn = p.firstReturn
have neg₁ : p.firstReturn < (range (↑(p + q) : List DyckStep).length).length
Tactic: rw [firstReturn, findIdx_eq]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
⊢ decide
        (count U
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                  p.firstReturn ⋯ +
                1)
              (↑(p + q) : List DyckStep)) =
          count D
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                  p.firstReturn ⋯ +
                1)
              (↑(p + q) : List DyckStep))) =
      true ∧
    ∀ (j : ℕ) (hji : j < p.firstReturn),
      decide
          (count U
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                    j ⋯ +
                  1)
                (↑(p + q) : List DyckStep)) =
            count D
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑(p + q) : List DyckStep).length)
                    j ⋯ +
                  1)
                (↑(p + q) : List DyckStep))) =
        false
have neg : count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
    count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) ∧
  ∀ (j : ℕ) (hji : j < p.firstReturn),
    decide
        (count U (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
          count D (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))) =
      false
Tactic: simp_rw [u, decide_eq_true_eq, getElem_range]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
⊢ count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
      count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) ∧
    ∀ (j : ℕ) (hji : j < p.firstReturn),
      decide
          (count U (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
            count D (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))) =
        false
have v : p.firstReturn < (↑p : List DyckStep).length
Tactic: have v := firstReturn_lt_length h
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
v : p.firstReturn < (↑p : List DyckStep).length
⊢ count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
      count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) ∧
    ∀ (j : ℕ) (hji : j < p.firstReturn),
      decide
          (count U (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
            count D (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))) =
        false
have neg : count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
  count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))
Tactic: constructor
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
v : p.firstReturn < (↑p : List DyckStep).length
neg :
  count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
    count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))
⊢ count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
      count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) ∧
    ∀ (j : ℕ) (hji : j < p.firstReturn),
      decide
          (count U (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
            count D (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))) =
        false
have neg₁ : ∀ (j : ℕ) (hji : j < p.firstReturn),
  decide
      (count U (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
        count D (List.take (j + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))) =
    false
Tactic: constructor
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
v : p.firstReturn < (↑p : List DyckStep).length
⊢ count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
    count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))
have neg : count U
    (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++
      List.take (p.firstReturn + 1 - (↑p : List DyckStep).length) (↑q : List DyckStep)) =
  count D
    (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++
      List.take (p.firstReturn + 1 - (↑p : List DyckStep).length) (↑q : List DyckStep))
Tactic: rw [take_append, show p.firstReturn + 1 - p.toList.length = 0 by lia,
        take_zero, append_nil, count_take_firstReturn_add_one h]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
v : p.firstReturn < (↑p : List DyckStep).length
neg :
  count U
      (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++
        List.take (p.firstReturn + 1 - (↑p : List DyckStep).length) (↑q : List DyckStep)) =
    count D
      (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++
        List.take (p.firstReturn + 1 - (↑p : List DyckStep).length) (↑q : List DyckStep))
⊢ count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
    count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))
have neg₁ : count U (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ List.take 0 (↑q : List DyckStep)) =
  count D (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ List.take 0 (↑q : List DyckStep))
Tactic: rw [take_append, show p.firstReturn + 1 - p.toList.length = 0 by lia,
        take_zero, append_nil, count_take_firstReturn_add_one h]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
v : p.firstReturn < (↑p : List DyckStep).length
neg :
  count U
      (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++
        List.take (p.firstReturn + 1 - (↑p : List DyckStep).length) (↑q : List DyckStep)) =
    count D
      (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++
        List.take (p.firstReturn + 1 - (↑p : List DyckStep).length) (↑q : List DyckStep))
neg₁ :
  count U (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ List.take 0 (↑q : List DyckStep)) =
    count D (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ List.take 0 (↑q : List DyckStep))
⊢ count U (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep))) =
    count D (List.take (p.firstReturn + 1) ((↑p : List DyckStep) ++ (↑q : List DyckStep)))
have neg₂ : count U (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ []) =
  count D (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ [])
Tactic: rw [take_append, show p.firstReturn + 1 - p.toList.length = 0 by lia,
        take_zero, append_nil, count_take_firstReturn_add_one h]