awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
i : ℕ
apply :
  count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count D (List.take i [U]) ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U (List.take i [U])
⊢ count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count D (List.take i [U]) +
      count D (List.take (i - [U].length) (↑p : List DyckStep)) ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U (List.take i [U]) +
      count U (List.take (i - [U].length) (↑p : List DyckStep))

Tactic:
apply add_le_add _ (p.count_D_le_count_U _)

Premises used: add_le_add, DyckWord.count_D_le_count_U
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
i : ℕ
inl :
  ∀ (hi : i = 0),
    count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count D (List.take i [U]) ≤
      count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U (List.take i [U])
inr :
  ∀ (hi : i > 0),
    count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count D (List.take i [U]) ≤
      count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U (List.take i [U])
⊢ count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count D (List.take i [U]) ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U (List.take i [U])

Tactic:
rcases i.eq_zero_or_pos with hi | hi

Premises used: Nat.eq_zero_or_pos
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
i : ℕ
hi : i = 0
⊢ count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count D (List.take i [U]) ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U (List.take i [U])

Tactic:
simp [hi]
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
i : ℕ
hi : i > 0
rwa rwa₁ rwa₂ rwa₃ : 1 ≤ i
⊢ [U].length ≤ i

Tactic:
rwa [length_singleton]

Premises used: List.length_singleton
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
i : ℕ
hi : i > 0
inr :
  (count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + if U = D then 1 else 0) ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U [U]
⊢ count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count D (List.take i [U]) ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U (List.take i [U])

Tactic:
rw [take_of_length_le (show [U].length ≤ i by rwa [length_singleton]), count_singleton']

Premises used: List.take_of_length_le, List.count_singleton'
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
i : ℕ
hi : i > 0
inr :
  0 + count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U [U]
⊢ count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + 0 ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U [U]

Tactic:
rw [add_comm]

Premises used: add_comm
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
i : ℕ
hi : i > 0
⊢ (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]).length ≤ count U [U]

Tactic:
simp
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
i : ℕ
hi : i > 0
⊢ 0 + count D (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) ≤
    count U (List.take (i - ([U] ++ (↑p : List DyckStep)).length) [D]) + count U [U]

Tactic:
exact add_le_add zero_le (count_le_length.trans (by simp))

Premises used: zero_le, add_le_add, List.count_le_length, LE.le.trans
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
⊢ p.nest ≠ 0

Tactic:
simp [← toList_ne_nil, nest]

Premises used: DyckWord.toList_ne_nil, DyckWord.nest
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
lb : 0 < i
ub : i < (↑p.nest : List DyckStep).length
⊢ i < 1 + (↑p : List DyckStep).length + 1

Tactic:
simp_rw [nest, length_append, length_singleton] at ub ⊢

Premises used: List.length_singleton, List.length_append, DyckWord.nest
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
lb : 0 < i
ub : i < 1 + (↑p : List DyckStep).length + 1
⊢ i ≤ (↑p : List DyckStep).length + 1

Tactic:
lia
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
lb : 0 < i
ub : i < 1 + (↑p : List DyckStep).length + 1
⊢ 1 ≤ i

Tactic:
lia
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
lb : 0 < i
ub : i < 1 + (↑p : List DyckStep).length + 1
⊢ U ≠ D

Tactic:
simp
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
lb : 0 < i
ub : i < 1 + (↑p : List DyckStep).length + 1
rw : count D (List.take (i - 1) (↑p : List DyckStep)) ≤ count U (List.take (i - 1) (↑p : List DyckStep))
⊢ count D (List.take i ([U] ++ (↑p : List DyckStep) ++ [D])) <
    count U (List.take i ([U] ++ (↑p : List DyckStep) ++ [D]))

Tactic:
rw [take_append_of_le_length (by rw [singleton_append, length_cons]; lia),
    take_append, take_of_length_le (by rw [length_singleton]; lia),
    length_singleton, singleton_append, count_cons_of_ne (by simp), count_cons_self,
    Nat.lt_add_one_iff]

Premises used: List.take_append_of_le_length, List.count_cons_of_ne, List.take_append, List.length_singleton, Nat.lt_add_one_iff, List.take_of_length_le, List.singleton_append, List.count_cons_self
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
lb : 0 < i
ub : i < 1 + (↑p : List DyckStep).length + 1
⊢ count D (List.take (i - 1) (↑p : List DyckStep)) ≤ count U (List.take (i - 1) (↑p : List DyckStep))

Tactic:
p.count_D_le_count_U _

Premises used: DyckWord.count_D_le_count_U
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p : DyckWord
i : ℕ
lb : 0 < i
ub : i < 1 + (↑p : List DyckStep).length + 1
⊢ count D (List.take (i - 1) (↑p : List DyckStep)) ≤ count U (List.take (i - 1) (↑p : List DyckStep))

Tactic:
exact p.count_D_le_count_U _

Premises used: DyckWord.count_D_le_count_U
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
hn : p.IsNested
⊢ count U (↑p : List DyckStep) = count D (↑p : List DyckStep)

Tactic:
have := p.count_U_eq_count_D
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
hn : p.IsNested
this : count U (↑p : List DyckStep) = count D (↑p : List DyckStep)
⊢ (count U (↑p : List DyckStep).dropLast.tail + if (U == U) = true then 1 else 0) + count U [D] =
    count D (U :: (↑p : List DyckStep).dropLast.tail ++ [D])

Tactic:
rw [← cons_tail_dropLast_concat hn.1, count_append, count_cons] at this

Premises used: DyckWord.cons_tail_dropLast_concat, List.count_cons, List.count_append, And.left
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
hn : p.IsNested
this :
  (count U (↑p : List DyckStep).dropLast.tail + if (U == U) = true then 1 else 0) + count U [D] =
    count D (U :: (↑p : List DyckStep).dropLast.tail ++ [D])
⊢ count U (↑p : List DyckStep).dropLast.tail = count D (↑p : List DyckStep).dropLast.tail

Tactic:
simpa using this
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
h : p ≠ 0
hn : p.IsNested
i : ℕ
⊢ (↑p : List DyckStep) ≠ []

Tactic:
replace h := toList_ne_nil.mpr hn.1
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
hn : p.IsNested
i : ℕ
h : (↑p : List DyckStep) ≠ []
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) ≠
        []),
    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]
cons :
  ∀ (head : DyckStep) (tail : List DyckStep) (count_U_eq_count_D : count U (head :: tail) = count D (head :: tail))
    (count_D_le_count_U : ∀ (i : ℕ), count D (List.take i (head :: tail)) ≤ count U (List.take i (head :: tail)))
    (h :
      (↑{ toList := 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) ≠
        []),
    List.take 1
        (↑{ toList := 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 := 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]
⊢ List.take 1 (↑p : List DyckStep) = [(↑p : List DyckStep).head h]

Tactic:
rcases p with - | -
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
q : DyckWord
i : ℕ
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) ≠
    []
⊢ 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]

Tactic:
tauto
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
q : DyckWord
i : ℕ
head : DyckStep
tail : List DyckStep
count_U_eq_count_D : count U (head :: tail) = count D (head :: tail)
count_D_le_count_U : ∀ (i : ℕ), count D (List.take i (head :: tail)) ≤ count U (List.take i (head :: tail))
h :
  (↑{ toList := 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) ≠
    []
⊢ List.take 1
      (↑{ toList := 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 := 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]

Tactic:
tauto
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
p q : DyckWord
hn : p.IsNested
i : ℕ
h : (↑p : List DyckStep) ≠ []
⊢ List.take 1 (↑p : List DyckStep) = [(↑p : List DyckStep).head h]

Tactic:
have l1 : p.toList.take 1 = [p.toList.head h] := by rcases p with - | - <;> tauto
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
q : DyckWord
i : ℕ
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

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

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

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

Tactic:
simp at bal
-- Module: Mathlib.Combinatorics.Enumerative.DyckWord
Goal:
q : DyckWord
i : ℕ
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

Tactic:
tauto
-- 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]
⊢ (↑p : List DyckStep).length - 1 = (↑p : List DyckStep).length - 1 - 1 + 1

Tactic:
have l3 : p.toList.length - 1 = p.toList.length - 1 - 1 + 1 := by
      rcases p with - | ⟨s, ⟨- | ⟨t, r⟩⟩⟩
      · tauto
      · rename_i bal _
        cases s <;> simp at bal
      · tauto
-- 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
rw :
  count D (List.drop 1 (List.take (min (1 + i) ((↑p : List DyckStep).length - 1)) (↑p : List DyckStep))) ≤
    count U (List.drop 1 (List.take (min (1 + i) ((↑p : List DyckStep).length - 1)) (↑p : List DyckStep)))
⊢ count D (List.take i (↑p : List DyckStep).dropLast.tail) ≤ count U (List.take i (↑p : List DyckStep).dropLast.tail)

Tactic:
rw [← drop_one, take_drop, dropLast_eq_take, take_take]

Premises used: List.take_drop, List.dropLast_eq_take, List.drop_one, List.take_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
⊢ min (1 + i) ((↑p : List DyckStep).length - 1) < (↑p : List DyckStep).length

Tactic:
have ub : min (1 + i) (p.toList.length - 1) < p.toList.length :=
      (min_le_right _ p.toList.length.pred).trans_lt (Nat.pred_lt ((length_pos_iff.mpr h).ne'))
-- 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
ub : min (1 + i) ((↑p : List DyckStep).length - 1) < (↑p : List DyckStep).length
⊢ 0 < min (1 + i) ((↑p : List DyckStep).length - 1)

Tactic:
omega
-- 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
ub : min (1 + i) ((↑p : List DyckStep).length - 1) < (↑p : List DyckStep).length
⊢ 0 < min (1 + i) ((↑p : List DyckStep).length - 1)

Tactic:
have lb : 0 < min (1 + i) (p.toList.length - 1) := by omega
-- 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
ub : min (1 + i) ((↑p : List DyckStep).length - 1) < (↑p : List DyckStep).length
lb : 0 < min (1 + i) ((↑p : List DyckStep).length - 1)
⊢ count D (List.take (min (1 + i) ((↑p : List DyckStep).length - 1)) (↑p : List DyckStep)) <
    count U (List.take (min (1 + i) ((↑p : List DyckStep).length - 1)) (↑p : List DyckStep))

Tactic:
have eq := hn.2 lb ub
-- 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
ub : min (1 + i) ((↑p : List DyckStep).length - 1) < (↑p : List DyckStep).length
lb : 0 < min (1 + i) ((↑p : List DyckStep).length - 1)
eq :
  count D (List.take (min (1 + i) ((↑p : List DyckStep).length - 1)) (↑p : List DyckStep)) <
    count U (List.take (min (1 + i) ((↑p : List DyckStep).length - 1)) (↑p : List DyckStep))
set :
  let j := 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 (List.drop 1 (List.take j (↑p : List DyckStep))) ≤ count U (List.drop 1 (List.take j (↑p : List DyckStep)))
⊢ count D (List.drop 1 (List.take (min (1 + i) ((↑p : List DyckStep).length - 1)) (↑p : List DyckStep))) ≤
    count U (List.drop 1 (List.take (min (1 + i) ((↑p : List DyckStep).length - 1)) (↑p : List DyckStep)))

Tactic:
set j := min (1 + i) (p.toList.length - 1)

Premises used: HSub.hSub, HAdd.hAdd, OfNat.ofNat, DyckWord.toList, List.length, Min.min
-- 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 (min 1 j) (↑p : List DyckStep)) + count D (List.drop 1 (List.take j (↑p : List DyckStep))) <
    count U (List.take (min 1 j) (↑p : List DyckStep)) + count U (List.drop 1 (List.take j (↑p : List DyckStep)))
⊢ 1 ≤ j

Tactic:
lia