l3lab/ntp-mathlib huggingface dataset:
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  

Proof state:
n : ℕ
⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)

Next tactic:
have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by 

Proof state:
n : ℕ
⊢ ↑n + 1 ≠ 0

Next tactic:
norm_cast
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; 

Proof state:
n : ℕ
⊢ ¬n + 1 = 0

Next tactic:
exact n.succ_ne_zero
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  

Proof state:
n : ℕ
this : ↑n + 1 ≠ 0
⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)

Next tactic:
have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  have : (n : ℚ) + 1 + 1 ≠ 0 := by 

Proof state:
n : ℕ
this : ↑n + 1 ≠ 0
⊢ ↑n + 1 + 1 ≠ 0

Next tactic:
norm_cast
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; 

Proof state:
n : ℕ
this : ↑n + 1 ≠ 0
⊢ ¬n + 1 + 1 = 0

Next tactic:
exact (n + 1).succ_ne_zero
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  

Proof state:
n : ℕ
this✝ : ↑n + 1 ≠ 0
this : ↑n + 1 + 1 ≠ 0
⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)

Next tactic:
have h : (n : ℚ) + 2 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  have h : (n : ℚ) + 2 ≠ 0 := by 

Proof state:
n : ℕ
this✝ : ↑n + 1 ≠ 0
this : ↑n + 1 + 1 ≠ 0
⊢ ↑n + 2 ≠ 0

Next tactic:
norm_cast
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  have h : (n : ℚ) + 2 ≠ 0 := by norm_cast; 

Proof state:
n : ℕ
this✝ : ↑n + 1 ≠ 0
this : ↑n + 1 + 1 ≠ 0
⊢ ¬n + 2 = 0

Next tactic:
exact (n + 1).succ_ne_zero
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  have h : (n : ℚ) + 2 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  

Proof state:
n : ℕ
this✝ : ↑n + 1 ≠ 0
this : ↑n + 1 + 1 ≠ 0
h : ↑n + 2 ≠ 0
⊢ gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = ↑(Nat.centralBinom (n + 1)) / (↑n + 2)

Next tactic:
simp only [gosperCatalan, Nat.sub_zero, Nat.centralBinom_zero, Nat.sub_self]
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  have h : (n : ℚ) + 2 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  simp only [gosperCatalan, Nat.sub_zero, Nat.centralBinom_zero, Nat.sub_self]
  

Proof state:
n : ℕ
this✝ : ↑n + 1 ≠ 0
this : ↑n + 1 + 1 ≠ 0
h : ↑n + 2 ≠ 0
⊢ ↑(Nat.centralBinom (n + 1)) * ↑1 * (2 * ↑(n + 1) - ↑(n + 1)) / (2 * ↑(n + 1) * (↑(n + 1) + 1)) -
      ↑1 * ↑(Nat.centralBinom (n + 1)) * (2 * ↑0 - ↑(n + 1)) / (2 * ↑(n + 1) * (↑(n + 1) + 1)) =
    ↑(Nat.centralBinom (n + 1)) / (↑n + 2)

Next tactic:
field_simp
private theorem gosper_catalan_sub_eq_central_binom_div (n : ℕ) : gosperCatalan (n + 1) (n + 1) -
    gosperCatalan (n + 1) 0 = Nat.centralBinom (n + 1) / (n + 2) := by
  have : (n : ℚ) + 1 ≠ 0 := by norm_cast; exact n.succ_ne_zero
  have : (n : ℚ) + 1 + 1 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  have h : (n : ℚ) + 2 ≠ 0 := by norm_cast; exact (n + 1).succ_ne_zero
  simp only [gosperCatalan, Nat.sub_zero, Nat.centralBinom_zero, Nat.sub_self]
  field_simp
  

Proof state:
n : ℕ
this✝ : ↑n + 1 ≠ 0
this : ↑n + 1 + 1 ≠ 0
h : ↑n + 2 ≠ 0
⊢ (↑(Nat.centralBinom (n + 1)) * (2 * (↑n + 1) - (↑n + 1)) - ↑(Nat.centralBinom (n + 1)) * (-1 + -↑n)) * (↑n + 2) =
    ↑(Nat.centralBinom (n + 1)) * (2 * (↑n + 1) * (↑n + 1 + 1))

Next tactic:
ring
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  

Proof state:
n : ℕ
⊢ catalan n = Nat.centralBinom n / (n + 1)

Next tactic:
suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    

Proof state:
n : ℕ
this : ↑(catalan n) = ↑(Nat.centralBinom n) / (↑n + 1)
⊢ catalan n = Nat.centralBinom n / (n + 1)

Next tactic:
have h := Nat.succ_dvd_centralBinom n
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    

Proof state:
n : ℕ
this : ↑(catalan n) = ↑(Nat.centralBinom n) / (↑n + 1)
h : n + 1 ∣ Nat.centralBinom n
⊢ catalan n = Nat.centralBinom n / (n + 1)

Next tactic:
exact mod_cast this
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  

Proof state:
n : ℕ
⊢ ↑(catalan n) = ↑(Nat.centralBinom n) / (↑n + 1)

Next tactic:
induction' n using Nat.case_strong_induction_on with d hd
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · 

Proof state:
case hz
⊢ ↑(catalan 0) = ↑(Nat.centralBinom 0) / (↑0 + 1)

Next tactic:
simp
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · 

Proof state:
case hi
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ ↑(catalan (Nat.succ d)) = ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)

Next tactic:
simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    

Proof state:
case hi
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ ∑ x : Fin (Nat.succ d), ↑(catalan ↑x) * ↑(catalan (d - ↑x)) = ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)

Next tactic:
trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · 

Proof state:
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ ∑ x : Fin (Nat.succ d), ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =
    ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1))

Next tactic:
congr
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      

Proof state:
case e_f
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ (fun x => ↑(catalan ↑x) * ↑(catalan (d - ↑x))) = fun i =>
    ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1))

Next tactic:
ext1 x
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      

Proof state:
case e_f.h
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
x : Fin (Nat.succ d)
⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =
    ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))

Next tactic:
have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by 

Proof state:
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
x : Fin (Nat.succ d)
⊢ ↑x ≤ d

Next tactic:
apply Nat.le_of_lt_succ
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; 

Proof state:
case a
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
x : Fin (Nat.succ d)
⊢ ↑x < Nat.succ d

Next tactic:
apply x.2
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      

Proof state:
case e_f.h
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
x : Fin (Nat.succ d)
m_le_d : ↑x ≤ d
⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =
    ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))

Next tactic:
have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
      

Proof state:
case e_f.h
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
x : Fin (Nat.succ d)
m_le_d : ↑x ≤ d
d_minus_x_le_d : d - ↑x ≤ d
⊢ ↑(catalan ↑x) * ↑(catalan (d - ↑x)) =
    ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))

Next tactic:
rw [hd _ m_le_d, hd _ d_minus_x_le_d]
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
      rw [hd _ m_le_d, hd _ d_minus_x_le_d]
      

Proof state:
case e_f.h
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
x : Fin (Nat.succ d)
m_le_d : ↑x ≤ d
d_minus_x_le_d : d - ↑x ≤ d
⊢ ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑(d - ↑x) + 1)) =
    ↑(Nat.centralBinom ↑x) / (↑↑x + 1) * (↑(Nat.centralBinom (d - ↑x)) / (↑d - ↑↑x + 1))

Next tactic:
norm_cast
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
      rw [hd _ m_le_d, hd _ d_minus_x_le_d]
      norm_cast
    · 

Proof state:
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =
    ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)

Next tactic:
trans (∑ i : Fin d.succ, (gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i))
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
      rw [hd _ m_le_d, hd _ d_minus_x_le_d]
      norm_cast
    · trans (∑ i : Fin d.succ, (gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i))
      · 

Proof state:
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ ∑ i : Fin (Nat.succ d), ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =
    ∑ i : Fin (Nat.succ d), (gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i)

Next tactic:
refine' sum_congr rfl fun i _ => _
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
      rw [hd _ m_le_d, hd _ d_minus_x_le_d]
      norm_cast
    · trans (∑ i : Fin d.succ, (gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i))
      · refine' sum_congr rfl fun i _ => _
        

Proof state:
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
i : Fin (Nat.succ d)
x✝ : i ∈ univ
⊢ ↑(Nat.centralBinom ↑i) / (↑↑i + 1) * (↑(Nat.centralBinom (d - ↑i)) / (↑d - ↑↑i + 1)) =
    gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i

Next tactic:
rw [gosper_trick i.is_le, mul_div]
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
      rw [hd _ m_le_d, hd _ d_minus_x_le_d]
      norm_cast
    · trans (∑ i : Fin d.succ, (gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i))
      · refine' sum_congr rfl fun i _ => _
        rw [gosper_trick i.is_le, mul_div]
      · 

Proof state:
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ ∑ i : Fin (Nat.succ d), (gosperCatalan (d + 1) (↑i + 1) - gosperCatalan (d + 1) ↑i) =
    ↑(Nat.centralBinom (Nat.succ d)) / (↑(Nat.succ d) + 1)

Next tactic:
rw [← sum_range fun i => gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i,
            sum_range_sub, Nat.succ_eq_add_one]
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
      rw [hd _ m_le_d, hd _ d_minus_x_le_d]
      norm_cast
    · trans (∑ i : Fin d.succ, (gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i))
      · refine' sum_congr rfl fun i _ => _
        rw [gosper_trick i.is_le, mul_div]
      · rw [← sum_range fun i => gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i,
            sum_range_sub, Nat.succ_eq_add_one]
        

Proof state:
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ gosperCatalan (d + 1) (d + 1) - gosperCatalan (d + 1) 0 = ↑(Nat.centralBinom (d + 1)) / (↑(d + 1) + 1)

Next tactic:
rw [gosper_catalan_sub_eq_central_binom_div d]
theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by
  suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
    have h := Nat.succ_dvd_centralBinom n
    exact mod_cast this
  induction' n using Nat.case_strong_induction_on with d hd
  · simp
  · simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
    trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
                             (Nat.centralBinom (d - i) / (d - i + 1)) : ℚ)
    · congr
      ext1 x
      have m_le_d : x.val ≤ d := by apply Nat.le_of_lt_succ; apply x.2
      have d_minus_x_le_d : (d - x.val) ≤ d := tsub_le_self
      rw [hd _ m_le_d, hd _ d_minus_x_le_d]
      norm_cast
    · trans (∑ i : Fin d.succ, (gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i))
      · refine' sum_congr rfl fun i _ => _
        rw [gosper_trick i.is_le, mul_div]
      · rw [← sum_range fun i => gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i,
            sum_range_sub, Nat.succ_eq_add_one]
        rw [gosper_catalan_sub_eq_central_binom_div d]
        

Proof state:
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑(Nat.centralBinom m) / (↑m + 1)
⊢ ↑(Nat.centralBinom (d + 1)) / (↑d + 2) = ↑(Nat.centralBinom (d + 1)) / (↑(d + 1) + 1)

Next tactic:
norm_cast
theorem catalan_two : catalan 2 = 2 := by
  

Proof state:
⊢ catalan 2 = 2

Next tactic:
norm_num [catalan_eq_centralBinom_div, Nat.centralBinom, Nat.choose]
theorem catalan_three : catalan 3 = 5 := by
  

Proof state:
⊢ catalan 3 = 5

Next tactic:
norm_num [catalan_eq_centralBinom_div, Nat.centralBinom, Nat.choose]
/-- Given two finsets, find all trees that can be formed with
  left child in `a` and right child in `b` -/
@[reducible]
def pairwiseNode (a b : Finset (Tree Unit)) : Finset (Tree Unit) :=
  (a ×ˢ b).map ⟨fun x => x.1 △ x.2, fun ⟨x₁, x₂⟩ ⟨y₁, y₂⟩ => fun h => by 

Proof state:
a b : Finset (Tree Unit)
x✝¹ x✝ : Tree Unit × Tree Unit
x₁ x₂ y₁ y₂ : Tree Unit
h : (fun x => node () x.1 x.2) (x₁, x₂) = (fun x => node () x.1 x.2) (y₁, y₂)
⊢ (x₁, x₂) = (y₁, y₂)

Next tactic:
simpa using h
/-- A Finset of all trees with `n` nodes. See `mem_treesOfNodesEq` -/
def treesOfNumNodesEq : ℕ → Finset (Tree Unit)
  | 0 => {nil}
  | n + 1 =>
    (antidiagonal n).attach.biUnion fun ijh =>
      -- Porting note: `unusedHavesSuffices` linter is not happy with this. Commented out.
      -- have := Nat.lt_succ_of_le (fst_le ijh.2)
      -- have := Nat.lt_succ_of_le (snd_le ijh.2)
      pairwiseNode (treesOfNumNodesEq ijh.1.1) (treesOfNumNodesEq ijh.1.2)
  -- Porting note: Add this to satisfy the linter.
  decreasing_by
      

Proof state:
n : ℕ
ijh : { x // x ∈ antidiagonal n }
⊢ (invImage (fun a => sizeOf a) instWellFoundedRelation).1 (↑ijh).1 (Nat.succ n)

Next tactic:
simp_wf