l3lab/ntp-mathlib huggingface dataset:
/-- 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
      simp_wf
      

Proof state:
n : ℕ
ijh : { x // x ∈ antidiagonal n }
⊢ (↑ijh).1 < Nat.succ n

Next tactic:
try exact Nat.lt_succ_of_le (fst_le ijh.2)
/-- 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
      simp_wf
      try 

Proof state:
n : ℕ
ijh : { x // x ∈ antidiagonal n }
⊢ (↑ijh).1 < Nat.succ n

Next tactic:
exact Nat.lt_succ_of_le (fst_le ijh.2)
/-- 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
      simp_wf
      try exact Nat.lt_succ_of_le (fst_le ijh.2)
      

Proof state:

Next tactic:
try exact Nat.lt_succ_of_le (snd_le ijh.2)
/-- 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
      simp_wf
      try exact Nat.lt_succ_of_le (fst_le ijh.2)
      try 

Proof state:

Next tactic:
exact Nat.lt_succ_of_le (snd_le ijh.2)
/-- 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).2 (Nat.succ n)

Next tactic:
simp_wf
/-- 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
      simp_wf
      

Proof state:
n : ℕ
ijh : { x // x ∈ antidiagonal n }
⊢ (↑ijh).2 < Nat.succ n

Next tactic:
try exact Nat.lt_succ_of_le (fst_le ijh.2)
/-- 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
      simp_wf
      try 

Proof state:
n : ℕ
ijh : { x // x ∈ antidiagonal n }
⊢ (↑ijh).2 < Nat.succ n

Next tactic:
exact Nat.lt_succ_of_le (fst_le ijh.2)
/-- 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
      simp_wf
      try exact Nat.lt_succ_of_le (fst_le ijh.2)
      

Proof state:
n : ℕ
ijh : { x // x ∈ antidiagonal n }
⊢ (↑ijh).2 < Nat.succ n

Next tactic:
try exact Nat.lt_succ_of_le (snd_le ijh.2)
/-- 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
      simp_wf
      try exact Nat.lt_succ_of_le (fst_le ijh.2)
      try 

Proof state:
n : ℕ
ijh : { x // x ∈ antidiagonal n }
⊢ (↑ijh).2 < Nat.succ n

Next tactic:
exact Nat.lt_succ_of_le (snd_le ijh.2)
@[simp]
theorem treesOfNumNodesEq_zero : treesOfNumNodesEq 0 = {nil} := by 

Proof state:
⊢ treesOfNumNodesEq 0 = {nil}

Next tactic:
rw [treesOfNumNodesEq]
theorem treesOfNumNodesEq_succ (n : ℕ) :
    treesOfNumNodesEq (n + 1) =
      (antidiagonal n).biUnion fun ij =>
        pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2) := by
  

Proof state:
n : ℕ
⊢ treesOfNumNodesEq (n + 1) =
    Finset.biUnion (antidiagonal n) fun ij => pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2)

Next tactic:
rw [treesOfNumNodesEq]
theorem treesOfNumNodesEq_succ (n : ℕ) :
    treesOfNumNodesEq (n + 1) =
      (antidiagonal n).biUnion fun ij =>
        pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2) := by
  rw [treesOfNumNodesEq]
  

Proof state:
n : ℕ
⊢ (Finset.biUnion (attach (antidiagonal n)) fun ijh =>
      pairwiseNode (treesOfNumNodesEq (↑ijh).1) (treesOfNumNodesEq (↑ijh).2)) =
    Finset.biUnion (antidiagonal n) fun ij => pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2)

Next tactic:
ext
theorem treesOfNumNodesEq_succ (n : ℕ) :
    treesOfNumNodesEq (n + 1) =
      (antidiagonal n).biUnion fun ij =>
        pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2) := by
  rw [treesOfNumNodesEq]
  ext
  

Proof state:
case a
n : ℕ
a✝ : Tree Unit
⊢ (a✝ ∈
      Finset.biUnion (attach (antidiagonal n)) fun ijh =>
        pairwiseNode (treesOfNumNodesEq (↑ijh).1) (treesOfNumNodesEq (↑ijh).2)) ↔
    a✝ ∈ Finset.biUnion (antidiagonal n) fun ij => pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2)

Next tactic:
simp
@[simp]
theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
    x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by
  

Proof state:
x : Tree Unit
n : ℕ
⊢ x ∈ treesOfNumNodesEq n ↔ numNodes x = n

Next tactic:
induction x using Tree.unitRecOn generalizing n
@[simp]
theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
    x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by
  induction x using Tree.unitRecOn generalizing n <;> 

Proof state:
case base
n : ℕ
⊢ nil ∈ treesOfNumNodesEq n ↔ numNodes nil = n

Next tactic:
cases n
@[simp]
theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
    x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by
  induction x using Tree.unitRecOn generalizing n <;> 

Proof state:
case ind
x✝ y✝ : Tree Unit
a✝¹ : ∀ {n : ℕ}, x✝ ∈ treesOfNumNodesEq n ↔ numNodes x✝ = n
a✝ : ∀ {n : ℕ}, y✝ ∈ treesOfNumNodesEq n ↔ numNodes y✝ = n
n : ℕ
⊢ node () x✝ y✝ ∈ treesOfNumNodesEq n ↔ numNodes (node () x✝ y✝) = n

Next tactic:
cases n
@[simp]
theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
    x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by
  induction x using Tree.unitRecOn generalizing n <;> cases n <;>
    

Proof state:
case base.zero
⊢ nil ∈ treesOfNumNodesEq Nat.zero ↔ numNodes nil = Nat.zero

Next tactic:
simp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]
@[simp]
theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
    x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by
  induction x using Tree.unitRecOn generalizing n <;> cases n <;>
    

Proof state:
case base.succ
n✝ : ℕ
⊢ nil ∈ treesOfNumNodesEq (Nat.succ n✝) ↔ numNodes nil = Nat.succ n✝

Next tactic:
simp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]
@[simp]
theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
    x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by
  induction x using Tree.unitRecOn generalizing n <;> cases n <;>
    

Proof state:
case ind.zero
x✝ y✝ : Tree Unit
a✝¹ : ∀ {n : ℕ}, x✝ ∈ treesOfNumNodesEq n ↔ numNodes x✝ = n
a✝ : ∀ {n : ℕ}, y✝ ∈ treesOfNumNodesEq n ↔ numNodes y✝ = n
⊢ node () x✝ y✝ ∈ treesOfNumNodesEq Nat.zero ↔ numNodes (node () x✝ y✝) = Nat.zero

Next tactic:
simp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]
@[simp]
theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
    x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by
  induction x using Tree.unitRecOn generalizing n <;> cases n <;>
    

Proof state:
case ind.succ
x✝ y✝ : Tree Unit
a✝¹ : ∀ {n : ℕ}, x✝ ∈ treesOfNumNodesEq n ↔ numNodes x✝ = n
a✝ : ∀ {n : ℕ}, y✝ ∈ treesOfNumNodesEq n ↔ numNodes y✝ = n
n✝ : ℕ
⊢ node () x✝ y✝ ∈ treesOfNumNodesEq (Nat.succ n✝) ↔ numNodes (node () x✝ y✝) = Nat.succ n✝

Next tactic:
simp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]
@[simp]
theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
    x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by
  induction x using Tree.unitRecOn generalizing n <;> cases n <;>
    simp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]
  

Proof state:
case base.succ
n✝ : ℕ
⊢ ¬0 = n✝ + 1

Next tactic:
exact (Nat.succ_ne_zero _).symm
@[simp, norm_cast]
theorem coe_treesOfNumNodesEq (n : ℕ) :
    ↑(treesOfNumNodesEq n) = { x : Tree Unit | x.numNodes = n } :=
  Set.ext (by 

Proof state:
n : ℕ
⊢ ∀ (x : Tree Unit), x ∈ ↑(treesOfNumNodesEq n) ↔ x ∈ {x | numNodes x = n}

Next tactic:
simp
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  

Proof state:
n : ℕ
⊢ card (treesOfNumNodesEq n) = catalan n

Next tactic:
induction' n using Nat.case_strong_induction_on with n ih
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · 

Proof state:
case hz
⊢ card (treesOfNumNodesEq 0) = catalan 0

Next tactic:
simp
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  

Proof state:
case hi
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
⊢ card (treesOfNumNodesEq (Nat.succ n)) = catalan (Nat.succ n)

Next tactic:
rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · 

Proof state:
case hi
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
⊢ ∑ u in antidiagonal n, card (pairwiseNode (treesOfNumNodesEq u.1) (treesOfNumNodesEq u.2)) =
    ∑ ij in antidiagonal n, catalan ij.1 * catalan ij.2

Next tactic:
apply sum_congr rfl
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    

Proof state:
case hi
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
⊢ ∀ x ∈ antidiagonal n, card (pairwiseNode (treesOfNumNodesEq x.1) (treesOfNumNodesEq x.2)) = catalan x.1 * catalan x.2

Next tactic:
rintro ⟨i, j⟩ H
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    

Proof state:
case hi.mk
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
H : (i, j) ∈ antidiagonal n
⊢ card (pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2)) = catalan (i, j).1 * catalan (i, j).2

Next tactic:
rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · 

Proof state:
case hi
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
⊢ ∀ x ∈ antidiagonal n,
    ∀ y ∈ antidiagonal n,
      x ≠ y →
        Disjoint (pairwiseNode (treesOfNumNodesEq x.1) (treesOfNumNodesEq x.2))
          (pairwiseNode (treesOfNumNodesEq y.1) (treesOfNumNodesEq y.2))

Next tactic:
simp_rw [disjoint_left]
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    

Proof state:
case hi
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
⊢ ∀ x ∈ antidiagonal n,
    ∀ y ∈ antidiagonal n,
      x ≠ y →
        ∀ ⦃a : Tree Unit⦄,
          a ∈ pairwiseNode (treesOfNumNodesEq x.1) (treesOfNumNodesEq x.2) →
            a ∉ pairwiseNode (treesOfNumNodesEq y.1) (treesOfNumNodesEq y.2)

Next tactic:
rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    

Proof state:
case hi.mk.mk
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
⊢ (i, j) ≠ (i', j') →
    ∀ ⦃a : Tree Unit⦄,
      a ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2) →
        a ∉ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)

Next tactic:
intros h a
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    

Proof state:
case hi.mk.mk
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h : (i, j) ≠ (i', j')
a : Tree Unit
⊢ a ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2) →
    a ∉ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)

Next tactic:
cases' a with a l r
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    cases' a with a l r
    · 

Proof state:
case hi.mk.mk.nil
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h : (i, j) ≠ (i', j')
⊢ nil ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2) →
    nil ∉ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)

Next tactic:
intro h
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    cases' a with a l r
    · intro h; 

Proof state:
case hi.mk.mk.nil
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h✝ : (i, j) ≠ (i', j')
h : nil ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2)
⊢ nil ∉ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)

Next tactic:
simp at h
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    cases' a with a l r
    · intro h; simp at h
    · 

Proof state:
case hi.mk.mk.node
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h : (i, j) ≠ (i', j')
a : Unit
l r : Tree Unit
⊢ node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2) →
    node a l r ∉ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)

Next tactic:
intro h1 h2
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    cases' a with a l r
    · intro h; simp at h
    · intro h1 h2
      

Proof state:
case hi.mk.mk.node
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h : (i, j) ≠ (i', j')
a : Unit
l r : Tree Unit
h1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2)
h2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)
⊢ False

Next tactic:
apply h
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    cases' a with a l r
    · intro h; simp at h
    · intro h1 h2
      apply h
      

Proof state:
case hi.mk.mk.node
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h : (i, j) ≠ (i', j')
a : Unit
l r : Tree Unit
h1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2)
h2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)
⊢ (i, j) = (i', j')

Next tactic:
trans (numNodes l, numNodes r)
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    cases' a with a l r
    · intro h; simp at h
    · intro h1 h2
      apply h
      trans (numNodes l, numNodes r)
      · 

Proof state:
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h : (i, j) ≠ (i', j')
a : Unit
l r : Tree Unit
h1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2)
h2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)
⊢ (i, j) = (numNodes l, numNodes r)

Next tactic:
simp at h1
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    cases' a with a l r
    · intro h; simp at h
    · intro h1 h2
      apply h
      trans (numNodes l, numNodes r)
      · simp at h1; 

Proof state:
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h : (i, j) ≠ (i', j')
a : Unit
l r : Tree Unit
h2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)
h1 : numNodes l = i ∧ numNodes r = j
⊢ (i, j) = (numNodes l, numNodes r)

Next tactic:
simp [h1]
theorem treesOfNumNodesEq_card_eq_catalan (n : ℕ) : (treesOfNumNodesEq n).card = catalan n := by
  induction' n using Nat.case_strong_induction_on with n ih
  · simp
  rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']
  · apply sum_congr rfl
    rintro ⟨i, j⟩ H
    rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]
  · simp_rw [disjoint_left]
    rintro ⟨i, j⟩ _ ⟨i', j'⟩ _
    -- Porting note: was clear * -; tidy
    intros h a
    cases' a with a l r
    · intro h; simp at h
    · intro h1 h2
      apply h
      trans (numNodes l, numNodes r)
      · simp at h1; simp [h1]
      · 

Proof state:
n : ℕ
ih : ∀ m ≤ n, card (treesOfNumNodesEq m) = catalan m
i j : ℕ
a✝¹ : (i, j) ∈ antidiagonal n
i' j' : ℕ
a✝ : (i', j') ∈ antidiagonal n
h : (i, j) ≠ (i', j')
a : Unit
l r : Tree Unit
h1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).1) (treesOfNumNodesEq (i, j).2)
h2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').1) (treesOfNumNodesEq (i', j').2)
⊢ (numNodes l, numNodes r) = (i', j')

Next tactic:
simp at h2