awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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]
l3 : (↑p : List DyckStep).length - 1 = (↑p : List DyckStep).length - 1 - 1 + 1
j : ℕ := min (1 + i) ((↑p : List DyckStep).length - 1)
ub : j < (↑p : List DyckStep).length
lb : 0 < j
eq : count D (List.take j (↑p : List DyckStep)) < count U (List.take j (↑p : List DyckStep))
⊢ count D [U] + count D (List.drop 1 (List.take j (↑p : List DyckStep))) <
    count U [U] + count U (List.drop 1 (List.take j (↑p : List DyckStep)))

Tactic:
rw [← (p.toList.take j).take_append_drop 1, count_append, count_append, take_take,
      min_eq_left (by lia), l1, head_eq_U] at eq

Premises used: DyckWord.head_eq_U, List.take_append_drop, min_eq_left, List.count_append, List.take_take, OfNat.ofNat, DyckWord.toList, List.take
-- 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]
l3 : (↑p : List DyckStep).length - 1 = (↑p : List DyckStep).length - 1 - 1 + 1
j : ℕ := min (1 + i) ((↑p : List DyckStep).length - 1)
ub : j < (↑p : List DyckStep).length
lb : 0 < j
eq :
  count D [U] + count D (List.drop 1 (List.take j (↑p : List DyckStep))) <
    count U [U] + count U (List.drop 1 (List.take j (↑p : List DyckStep)))
⊢ (if U = D then 1 else 0) + count D (List.drop 1 (List.take j (↑p : List DyckStep))) <
    1 + count U (List.drop 1 (List.take j (↑p : List DyckStep)))

Tactic:
simp only [count_singleton', ite_true] at eq

Premises used: ite_true, List.count_singleton'
-- 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]
l3 : (↑p : List DyckStep).length - 1 = (↑p : List DyckStep).length - 1 - 1 + 1
j : ℕ := min (1 + i) ((↑p : List DyckStep).length - 1)
ub : j < (↑p : List DyckStep).length
lb : 0 < j
eq :
  (if U = D then 1 else 0) + count D (List.drop 1 (List.take j (↑p : List DyckStep))) <
    1 + count U (List.drop 1 (List.take j (↑p : List DyckStep)))
⊢ count D (List.drop 1 (List.take j (↑p : List DyckStep))) ≤ count U (List.drop 1 (List.take j (↑p : List DyckStep)))

Tactic:
lia
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
hn : p.IsNested
⊢ (p.denest hn).nest = p

Tactic:
simpa [DyckWord.ext_iff] using p.cons_tail_dropLast_concat hn.1

Premises used: DyckWord.cons_tail_dropLast_concat, DyckWord.ext_iff, And.left
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
simp_rw : ([U] ++ (↑p : List DyckStep)).tail = (↑p : List DyckStep)
⊢ p.nest.denest ⋯ = p

Tactic:
simp_rw [nest, denest, DyckWord.ext_iff, dropLast_concat]

Premises used: DyckWord.ext_iff, List.dropLast_concat, DyckWord.nest, DyckWord.denest
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
⊢ ([U] ++ (↑p : List DyckStep)).tail = (↑p : List DyckStep)

Tactic:
rfl
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
⊢ semilength 0 = 0

Tactic:
rfl

Premises used: List.countP.go, DyckWord.semilength, rfl
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
⊢ (p + q).semilength = p.semilength + q.semilength

Tactic:
count_append ..

Premises used: DyckWord.semilength, List.count_append
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
⊢ p.nest.semilength = p.semilength + 1

Tactic:
simp [semilength, nest]

Premises used: DyckWord.nest, DyckWord.semilength
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
rw : p.semilength = count U (↑p : List DyckStep)
⊢ p.semilength = count D (↑p : List DyckStep)

Tactic:
rw [← count_U_eq_count_D]

Premises used: DyckWord.count_U_eq_count_D
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
⊢ p.semilength = count U (↑p : List DyckStep)

Tactic:
rfl
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
nth_rw : count D (↑p : List DyckStep) + count U (↑p : List DyckStep) = (↑p : List DyckStep).length
⊢ 2 * p.semilength = (↑p : List DyckStep).length

Tactic:
nth_rw 1 [two_mul, semilength, p.count_U_eq_count_D, semilength]

Premises used: DyckWord.count_U_eq_count_D, two_mul, DyckWord.semilength
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
⊢ countP (fun (x : DyckStep) ↦ x == D) (↑p : List DyckStep) +
      countP (fun (a : DyckStep) ↦ decide ¬(a == D) = true) (↑p : List DyckStep) =
    (↑p : List DyckStep).length

Tactic:
(p.toList.length_eq_countP_add_countP (· == D)).symm

Premises used: List.length_eq_countP_add_countP, BEq.beq, DyckStep.D, Eq.symm
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : count U = countP fun (a : DyckStep) ↦ decide ¬(a == D) = true
⊢ count D (↑p : List DyckStep) + count U (↑p : List DyckStep) = (↑p : List DyckStep).length

Tactic:
convert! (p.toList.length_eq_countP_add_countP (· == D)).symm

Premises used: List.length_eq_countP_add_countP, BEq.beq, DyckStep.D, Eq.symm
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : ∀ (x : DyckStep), (x == U) = decide ¬(x == D) = true
⊢ Eq (α := List DyckStep → ℕ) (countP fun (x : DyckStep) ↦ x == U) (countP fun (a : DyckStep) ↦ decide ¬(a == D) = true)

Tactic:
congr!
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
s : DyckStep
h : (U == U) = decide ¬(U == D) = true
h₁ : (D == U) = decide ¬(D == D) = true
⊢ (s == U) = decide ¬(s == D) = true

Tactic:
cases s
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
⊢ (U == U) = decide ¬(U == D) = true

Tactic:
tauto
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
⊢ (D == U) = decide ¬(D == D) = true

Tactic:
tauto
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
⊢ firstReturn 0 = 0

Tactic:
rfl

Premises used: DyckWord.firstReturn, List.findIdx.go, rfl
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
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)
rw₁ : 0 < (range (↑p : List DyckStep).length).length
⊢ 0 < p.firstReturn

Tactic:
rw [← not_le, Nat.le_zero, firstReturn, findIdx_eq, getElem_range]

Premises used: not_le, Nat.le_zero, List.getElem_range, List.findIdx_eq, DyckWord.firstReturn
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
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 ⋯ +
                    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 ⋯ +
                    1)
                  (U :: (↑p : List DyckStep).dropLast.tail ++ [D]))) =
          false)
⊢ ¬(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)

Tactic:
rw! [← p.cons_tail_dropLast_concat h]

Premises used: DyckWord.cons_tail_dropLast_concat
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
⊢ ¬(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 ⋯ +
                    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 ⋯ +
                    1)
                  (U :: (↑p : List DyckStep).dropLast.tail ++ [D]))) =
          false)

Tactic:
simp
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
rw : (↑p : List DyckStep) ≠ []
⊢ 0 < (range (↑p : List DyckStep).length).length

Tactic:
rw [length_range, length_pos_iff]

Premises used: List.length_range, List.length_pos_iff
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
⊢ (↑p : List DyckStep) ≠ []

Tactic:
toList_ne_nil.mpr h

Premises used: Iff.mpr, DyckWord.toList_ne_nil
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
⊢ (↑p : List DyckStep) ≠ []

Tactic:
exact toList_ne_nil.mpr h

Premises used: Iff.mpr, DyckWord.toList_ne_nil
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
⊢ 0 < (↑p : List DyckStep).length

Tactic:
have lp := length_pos_of_ne_nil (toList_ne_nil.mpr h)
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
rw : p.firstReturn < (range (↑p : List DyckStep).length).length
⊢ p.firstReturn < (↑p : List DyckStep).length

Tactic:
rw [← length_range (n := p.toList.length)]

Premises used: List.length_range
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
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
⊢ p.firstReturn < (range (↑p : List DyckStep).length).length

Tactic:
apply findIdx_lt_length_of_exists

Premises used: List.findIdx_lt_length_of_exists
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
h₁ :
  ∃ x < (↑p : List DyckStep).length,
    count U (List.take (x + 1) (↑p : List DyckStep)) = count D (List.take (x + 1) (↑p : List DyckStep))
⊢ ∃ 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:
simp only [mem_range, decide_eq_true_eq]

Premises used: List.mem_range, decide_eq_true_eq
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
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))
⊢ ∃ x < (↑p : List DyckStep).length,
    count U (List.take (x + 1) (↑p : List DyckStep)) = count D (List.take (x + 1) (↑p : List DyckStep))

Tactic:
use p.toList.length - 1

Premises used: DyckWord.toList, List.length
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
⊢ (↑p : List DyckStep).length - 1 < (↑p : List DyckStep).length

Tactic:
lia
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
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)
rw₂ rw₃ : count D (↑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))

Tactic:
rw [Nat.sub_add_cancel lp, take_of_length_le (le_refl _),
    p.count_U_eq_count_D]

Premises used: le_refl, DyckWord.count_U_eq_count_D, List.take_of_length_le, Nat.sub_add_cancel
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
lp : 0 < (↑p : List DyckStep).length
⊢ (↑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:
exact ⟨by lia, by rw [Nat.sub_add_cancel lp, take_of_length_le (le_refl _),
    p.count_U_eq_count_D]⟩
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
⊢ 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))
                ⋯ +
              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))
                ⋯ +
              1)
            (↑p : List DyckStep))) =
    true

Tactic:
have := findIdx_getElem
    (w := (length_range (n := p.toList.length)).symm ▸ firstReturn_lt_length h)
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
h : p ≠ 0
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))
                ⋯ +
              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))
                ⋯ +
              1)
            (↑p : List DyckStep))) =
    true
⊢ count U (List.take (p.firstReturn + 1) (↑p : List DyckStep)) =
    count D (List.take (p.firstReturn + 1) (↑p : List DyckStep))

Tactic:
simpa using this
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
hi : i < p.firstReturn
⊢ 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

Tactic:
have ne := not_of_lt_findIdx hi
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
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 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

Premises used: List.getElem_range, ne_eq, decide_eq_false_iff_not
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
hi : i < p.firstReturn
ne : count U (List.take (i + 1) (↑p : List DyckStep)) ≠ count D (List.take (i + 1) (↑p : List DyckStep))
⊢ count D (List.take (i + 1) (↑p : List DyckStep)) < count U (List.take (i + 1) (↑p : List DyckStep))

Tactic:
lt_of_le_of_ne (p.count_D_le_count_U (i + 1)) ne.symm

Premises used: DyckWord.count_D_le_count_U, Ne.symm, HAdd.hAdd, OfNat.ofNat, lt_of_le_of_ne
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
hi : i < p.firstReturn
ne : count U (List.take (i + 1) (↑p : List DyckStep)) ≠ count D (List.take (i + 1) (↑p : List DyckStep))
⊢ count D (List.take (i + 1) (↑p : List DyckStep)) < count U (List.take (i + 1) (↑p : List DyckStep))

Tactic:
exact lt_of_le_of_ne (p.count_D_le_count_U (i + 1)) ne.symm

Premises used: DyckWord.count_D_le_count_U, Ne.symm, HAdd.hAdd, OfNat.ofNat, lt_of_le_of_ne
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
pos : ∀ (h : p = 0), (p + q).firstReturn = q.firstReturn
neg : ∀ (h : ¬p = 0), (p + q).firstReturn = p.firstReturn
⊢ (p + q).firstReturn = if p = 0 then q.firstReturn else p.firstReturn

Tactic:
split_ifs with h
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p = 0
⊢ (p + q).firstReturn = q.firstReturn

Tactic:
simp [h]
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : ¬p = 0
⊢ (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)

Tactic:
have u : (p + q).toList = p.toList ++ q.toList := rfl
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
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
neg₁ : p.firstReturn < (range (↑(p + q) : List DyckStep).length).length
⊢ (p + q).firstReturn = p.firstReturn

Tactic:
rw [firstReturn, findIdx_eq]

Premises used: List.findIdx_eq, DyckWord.firstReturn
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
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
⊢ 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

Tactic:
simp_rw [u, decide_eq_true_eq, getElem_range]

Premises used: List.getElem_range, decide_eq_true_eq
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : ¬p = 0
u : (↑(p + q) : List DyckStep) = (↑p : List DyckStep) ++ (↑q : List DyckStep)
⊢ p.firstReturn < (↑p : List DyckStep).length

Tactic:
have v := firstReturn_lt_length h
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
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)))
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
⊢ 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:
constructor