awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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))
neg₂ :
  count U (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ []) =
    count D (List.take (p.firstReturn + 1) (↑p : 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 neg₃ : 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]
-- 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))
neg₂ :
  count U (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ []) =
    count D (List.take (p.firstReturn + 1) (↑p : List DyckStep) ++ [])
neg₃ :
  count U (List.take (p.firstReturn + 1) (↑p : List DyckStep)) =
    count D (List.take (p.firstReturn + 1) (↑p : 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 neg₄ : count D (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]
-- 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
j : ℕ
hj : 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 : decide (count U (List.take (j + 1) (↑p : List DyckStep)) = count D (List.take (j + 1) (↑p : List DyckStep))) = false
Tactic: rw [take_append, show j + 1 - p.toList.length = 0 by lia,
        take_zero, append_nil]
-- 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.firstReturn < (range (↑(p + q) : List DyckStep).length).length
have neg : p.firstReturn < (↑p : List DyckStep).length + (↑q : List DyckStep).length
Tactic: rw [length_range, u, length_append]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p : DyckWord
⊢ p.nest.firstReturn = (↑p : List DyckStep).length + 1
have u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
Tactic: have u : p.nest.toList = U :: p.toList ++ [D] := rfl
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
⊢ p.nest.firstReturn = (↑p : List DyckStep).length + 1
have rw : decide
      (count U
          (List.take
            (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length)
                ((↑p : List DyckStep).length + 1) sorry +
              1)
            (↑p.nest : List DyckStep)) =
        count D
          (List.take
            (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length)
                ((↑p : List DyckStep).length + 1) sorry +
              1)
            (↑p.nest : List DyckStep))) =
    true ∧
  ∀ (j : ℕ) (hji : j < (↑p : List DyckStep).length + 1),
    decide
        (count U
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length) j
                  sorry +
                1)
              (↑p.nest : List DyckStep)) =
          count D
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length) j
                  sorry +
                1)
              (↑p.nest : List DyckStep))) =
      false
Tactic: rw [firstReturn, findIdx_eq]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
rw :
  decide
        (count U
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length)
                  ((↑p : List DyckStep).length + 1) ⋯ +
                1)
              (↑p.nest : List DyckStep)) =
          count D
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length)
                  ((↑p : List DyckStep).length + 1) ⋯ +
                1)
              (↑p.nest : List DyckStep))) =
      true ∧
    ∀ (j : ℕ) (hji : j < (↑p : List DyckStep).length + 1),
      decide
          (count U
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length) j
                    ⋯ +
                  1)
                (↑p.nest : List DyckStep)) =
            count D
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length) j
                    ⋯ +
                  1)
                (↑p.nest : List DyckStep))) =
        false
⊢ p.nest.firstReturn = (↑p : List DyckStep).length + 1
have rw₁ : (↑p : List DyckStep).length + 1 < (range (↑p.nest : List DyckStep).length).length
Tactic: rw [firstReturn, findIdx_eq]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
⊢ decide
        (count U
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length)
                  ((↑p : List DyckStep).length + 1) ⋯ +
                1)
              (↑p.nest : List DyckStep)) =
          count D
            (List.take
              (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length)
                  ((↑p : List DyckStep).length + 1) ⋯ +
                1)
              (↑p.nest : List DyckStep))) =
      true ∧
    ∀ (j : ℕ) (hji : j < (↑p : List DyckStep).length + 1),
      decide
          (count U
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length) j
                    ⋯ +
                  1)
                (↑p.nest : List DyckStep)) =
            count D
              (List.take
                (getElem (valid := fun (as : List ℕ) (i : ℕ) ↦ i < as.length) (range (↑p.nest : List DyckStep).length) j
                    ⋯ +
                  1)
                (↑p.nest : List DyckStep))) =
        false
have simp_rw : count U (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) =
    count D (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) ∧
  ∀ (j : ℕ) (hji : j < (↑p : List DyckStep).length + 1),
    decide
        (count U (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D])) =
          count D (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D]))) =
      false
Tactic: simp_rw [u, decide_eq_true_eq, getElem_range]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
⊢ count U (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) =
      count D (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) ∧
    ∀ (j : ℕ) (hji : j < (↑p : List DyckStep).length + 1),
      decide
          (count U (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D])) =
            count D (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D]))) =
        false
have left : count U (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) =
  count D (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D]))
Tactic: constructor
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
left :
  count U (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) =
    count D (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D]))
⊢ count U (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) =
      count D (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) ∧
    ∀ (j : ℕ) (hji : j < (↑p : List DyckStep).length + 1),
      decide
          (count U (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D])) =
            count D (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D]))) =
        false
have right : ∀ (j : ℕ) (hji : j < (↑p : List DyckStep).length + 1),
  decide
      (count U (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D])) =
        count D (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D]))) =
    false
Tactic: constructor
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
⊢ count U (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) =
    count D (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D]))
have left : count U (U :: (↑p : List DyckStep) ++ [D]) = count D (U :: (↑p : List DyckStep) ++ [D])
Tactic: rw [take_of_length_le (by simp), ← u, p.nest.count_U_eq_count_D]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
left : count U (U :: (↑p : List DyckStep) ++ [D]) = count D (U :: (↑p : List DyckStep) ++ [D])
⊢ count U (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) =
    count D (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D]))
have left₁ : count U (↑p.nest : List DyckStep) = count D (↑p.nest : List DyckStep)
Tactic: rw [take_of_length_le (by simp), ← u, p.nest.count_U_eq_count_D]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
left : count U (U :: (↑p : List DyckStep) ++ [D]) = count D (U :: (↑p : List DyckStep) ++ [D])
left₁ : count U (↑p.nest : List DyckStep) = count D (↑p.nest : List DyckStep)
⊢ count U (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D])) =
    count D (List.take ((↑p : List DyckStep).length + 1 + 1) (U :: (↑p : List DyckStep) ++ [D]))
have left₂ : count D (↑p.nest : List DyckStep) = count D (↑p.nest : List DyckStep)
Tactic: rw [take_of_length_le (by simp), ← u, p.nest.count_U_eq_count_D]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
j : ℕ
hj : j < (↑p : List DyckStep).length + 1
⊢ decide
      (count U (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D])) =
        count D (List.take (j + 1) (U :: (↑p : List DyckStep) ++ [D]))) =
    false
have right : decide (count U (List.take j (↑p : List DyckStep)) + 1 = count D (List.take j (↑p : List DyckStep)) + 0) = false
Tactic: simp_rw [cons_append, take_succ_cons, count_cons, beq_self_eq_true, ite_true,
        beq_iff_eq, reduceCtorEq, ite_false, take_append,
        show j - p.toList.length = 0 by lia, take_zero, append_nil]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
j : ℕ
hj : j < (↑p : List DyckStep).length + 1
⊢ decide (count U (List.take j (↑p : List DyckStep)) + 1 = count D (List.take j (↑p : List DyckStep)) + 0) = false
have this : count D (List.take j (↑p : List DyckStep)) ≤ count U (List.take j (↑p : List DyckStep))
Tactic: have := p.count_D_le_count_U j
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
j : ℕ
hj : j < (↑p : List DyckStep).length + 1
this : count D (List.take j (↑p : List DyckStep)) ≤ count U (List.take j (↑p : List DyckStep))
⊢ decide (count U (List.take j (↑p : List DyckStep)) + 1 = count D (List.take j (↑p : List DyckStep)) + 0) = false
have right : ¬count U (List.take j (↑p : List DyckStep)) + 1 = count D (List.take j (↑p : List DyckStep))
Tactic: simp only [add_zero, decide_eq_false_iff_not, ne_eq]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
u : (↑p.nest : List DyckStep) = U :: (↑p : List DyckStep) ++ [D]
⊢ (↑p : List DyckStep).length + 1 < (range (↑p.nest : List DyckStep).length).length
have simp_rw : (↑p : List DyckStep).length + 1 < (↑p : List DyckStep).length + 1 + (length (α := DyckStep) [] + 1)
Tactic: simp_rw [length_range, u, length_append, length_cons]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h_1 : p ≠ 0
h : ¬p = 0
⊢ p.take (p.firstReturn + 1) ⋯ ≠ 0
have rw : (↑{ toList := List.take (p.firstReturn + 1) (↑p : List DyckStep), count_U_eq_count_D := sorry, count_D_le_count_U := sorry :
        DyckWord } :
    List DyckStep) ≠
  []
Tactic: rw [← toList_ne_nil, take]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h_1 : p ≠ 0
h : ¬p = 0
i : ℕ
lb : 0 < i
ub : i < (↑(p.take (p.firstReturn + 1) ⋯) : List DyckStep).length
⊢ count D (List.take i (↑(p.take (p.firstReturn + 1) ⋯) : List DyckStep)) <
    count U (List.take i (↑(p.take (p.firstReturn + 1) ⋯) : List DyckStep))
have simp : ∀ (ub : i < p.firstReturn + 1 ∧ i < (↑p : List DyckStep).length),
  count D (List.take i (List.take (p.firstReturn + 1) (↑p : List DyckStep))) <
    count U (List.take i (List.take (p.firstReturn + 1) (↑p : List DyckStep)))
Tactic: simp only [take, length_take, lt_min_iff] at ub ⊢
Cleared hypotheses: ub
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p q : DyckWord
h_1 : p ≠ 0
h : ¬p = 0
i : ℕ
lb : 0 < i
ub : i < p.firstReturn + 1 ∧ i < (↑p : List DyckStep).length
⊢ count D (List.take i (List.take (p.firstReturn + 1) (↑p : List DyckStep))) <
    count U (List.take i (List.take (p.firstReturn + 1) (↑p : List DyckStep)))
have ub₁ : i < p.firstReturn + 1
Tactic: replace ub := ub.1
Cleared hypotheses: ub
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h_1 : p ≠ 0
h : ¬p = 0
i : ℕ
lb : 0 < i
ub : i < p.firstReturn + 1
⊢ count D (List.take i (List.take (p.firstReturn + 1) (↑p : List DyckStep))) <
    count U (List.take i (List.take (p.firstReturn + 1) (↑p : List DyckStep)))
have rw : count D (List.take i (↑p : List DyckStep)) < count U (List.take i (↑p : List DyckStep))
Tactic: rw [take_take, min_eq_left ub.le]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h_1 : p ≠ 0
h : ¬p = 0
i : ℕ
lb : 0 < i
ub : i < p.firstReturn + 1
⊢ count D (List.take i (↑p : List DyckStep)) < count U (List.take i (↑p : List DyckStep))
have rw : ∀ (ub : i - 1 + 1 < p.firstReturn + 1),
  count D (List.take (i - 1 + 1) (↑p : List DyckStep)) < count U (List.take (i - 1 + 1) (↑p : List DyckStep))
Tactic: rw [show i = i - 1 + 1 by lia] at ub ⊢
Cleared hypotheses: ub
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p q : DyckWord
h_1 : p ≠ 0
h : ¬p = 0
i : ℕ
lb : 0 < i
ub : i - 1 + 1 < p.firstReturn + 1
⊢ count D (List.take (i - 1 + 1) (↑p : List DyckStep)) < count U (List.take (i - 1 + 1) (↑p : List DyckStep))
have ub₁ : i - 1 < p.firstReturn
Tactic: rw [Nat.add_lt_add_iff_right] at ub
Cleared hypotheses: ub
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : p ≠ 0
⊢ (p + q).insidePart = p.insidePart
have simp_rw : (↑({ toList := List.take (p.firstReturn + 1) (↑(p + q) : List DyckStep), count_U_eq_count_D := sorry,
            count_D_le_count_U := sorry : DyckWord }.denest
        sorry) :
    List DyckStep) =
  (↑({ toList := List.take (p.firstReturn + 1) (↑p : List DyckStep), count_U_eq_count_D := sorry,
            count_D_le_count_U := sorry }.denest
        sorry) :
    List DyckStep)
Tactic: simp_rw [insidePart, firstReturn_add, add_eq_zero', h, false_and, dite_false, ite_false,
    DyckWord.ext_iff, take]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : p ≠ 0
⊢ (↑({ toList := List.take (p.firstReturn + 1) (↑(p + q) : List DyckStep), count_U_eq_count_D := ⋯,
              count_D_le_count_U := ⋯ : DyckWord }.denest
          ⋯) :
      List DyckStep) =
    (↑({ toList := List.take (p.firstReturn + 1) (↑p : List DyckStep), count_U_eq_count_D := ⋯,
              count_D_le_count_U := ⋯ }.denest
          ⋯) :
      List DyckStep)
have e_self : List.take (p.firstReturn + 1) (↑(p + q) : List DyckStep) = List.take (p.firstReturn + 1) (↑p : List DyckStep)
Tactic: congr 3
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p q : DyckWord
h : p ≠ 0
⊢ (p + q).outsidePart = p.outsidePart + q
have simp_rw : List.drop (p.firstReturn + 1) (↑(p + q) : List DyckStep) =
  (↑({ toList := List.drop (p.firstReturn + 1) (↑p : List DyckStep), count_U_eq_count_D := sorry, count_D_le_count_U := sorry :
          DyckWord } +
        q) :
    List DyckStep)
Tactic: simp_rw [outsidePart, firstReturn_add, add_eq_zero', h, false_and, dite_false, ite_false,
    DyckWord.ext_iff, drop]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ p.nest.insidePart = p
have simp_rw : (p.nest.take ((↑p : List DyckStep).length + 1 + 1) sorry).denest sorry = p
Tactic: simp_rw [insidePart, nest_ne_zero, dite_false, firstReturn_nest]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ (p.nest.take ((↑p : List DyckStep).length + 1 + 1) ⋯).denest ⋯ = p
have h : p.nest.take ((↑p : List DyckStep).length + 1 + 1) sorry = p.nest
Tactic: convert! p.denest_nest
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ p.nest.take ((↑p : List DyckStep).length + 1 + 1) ⋯ = p.nest
have h : (↑(p.nest.take ((↑p : List DyckStep).length + 1 + 1) sorry) : List DyckStep) = (↑p.nest : List DyckStep)
Tactic: rw [DyckWord.ext_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ (↑(p.nest.take ((↑p : List DyckStep).length + 1 + 1) ⋯) : List DyckStep) = (↑p.nest : List DyckStep)
have h : (↑p.nest : List DyckStep).length ≤ (↑p : List DyckStep).length + 1 + 1
Tactic: apply take_of_length_le
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
⊢ (↑p.nest : List DyckStep).length ≤ (↑p : List DyckStep).length + 1 + 1
have h : 1 + (↑p : List DyckStep).length + 1 ≤ (↑p : List DyckStep).length + 1 + 1
Tactic: simp_rw [nest, length_append, length_singleton]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
⊢ p.insidePart.nest + p.outsidePart = p
have simp_rw : (↑(p.take (p.firstReturn + 1) sorry + p.drop (p.firstReturn + 1) sorry) : List DyckStep) = (↑p : List DyckStep)
Tactic: simp_rw [insidePart, outsidePart, h, dite_false, nest_denest, DyckWord.ext_iff]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
⊢ p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength
have rw : p.insidePart.semilength + p.outsidePart.semilength + 1 = (p.insidePart.nest + p.outsidePart).semilength
Tactic: rw [← congrArg semilength (nest_insidePart_add_outsidePart h), semilength_add, semilength_nest,
    add_right_comm]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
rw : p.insidePart.semilength + p.outsidePart.semilength + 1 = (p.insidePart.nest + p.outsidePart).semilength
⊢ p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength
have rw₁ : p.insidePart.semilength + p.outsidePart.semilength + 1 = p.insidePart.nest.semilength + p.outsidePart.semilength
Tactic: rw [← congrArg semilength (nest_insidePart_add_outsidePart h), semilength_add, semilength_nest,
    add_right_comm]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
rw : p.insidePart.semilength + p.outsidePart.semilength + 1 = (p.insidePart.nest + p.outsidePart).semilength
rw₁ : p.insidePart.semilength + p.outsidePart.semilength + 1 = p.insidePart.nest.semilength + p.outsidePart.semilength
⊢ p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength
have rw₂ : p.insidePart.semilength + p.outsidePart.semilength + 1 = p.insidePart.semilength + 1 + p.outsidePart.semilength
Tactic: rw [← congrArg semilength (nest_insidePart_add_outsidePart h), semilength_add, semilength_nest,
    add_right_comm]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (goal)
Goal:
p : DyckWord
h : p ≠ 0
rw : p.insidePart.semilength + p.outsidePart.semilength + 1 = (p.insidePart.nest + p.outsidePart).semilength
rw₁ : p.insidePart.semilength + p.outsidePart.semilength + 1 = p.insidePart.nest.semilength + p.outsidePart.semilength
rw₂ : p.insidePart.semilength + p.outsidePart.semilength + 1 = p.insidePart.semilength + 1 + p.outsidePart.semilength
⊢ p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength
have rw₃ : p.insidePart.semilength + 1 + p.outsidePart.semilength = p.insidePart.semilength + 1 + p.outsidePart.semilength
Tactic: rw [← congrArg semilength (nest_insidePart_add_outsidePart h), semilength_add, semilength_nest,
    add_right_comm]
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p : DyckWord
h : p ≠ 0
⊢ p.insidePart.semilength < p.semilength
have this : p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength
Tactic: have := semilength_insidePart_add_semilength_outsidePart_add_one h
-- Lean 4 proof step from Mathlib.Combinatorics.Enumerative.DyckWord (hypothesis)
Goal:
p : DyckWord
h : p ≠ 0
⊢ p.outsidePart.semilength < p.semilength
have this : p.insidePart.semilength + p.outsidePart.semilength + 1 = p.semilength
Tactic: have := semilength_insidePart_add_semilength_outsidePart_add_one h