-- Mathlib/SetTheory/Ordinal/Notation.lean
theorem cmp_compares : ∀ (a b : ONote) [NF a] [NF b], (cmp a b).Compares a b
  | 0, 0, _, _ => rfl
  | oadd _ _ _, 0, _, _ => oadd_pos _ _ _
  | 0, oadd _ _ _, _, _ => oadd_pos _ _ _
  | o₁@(oadd e₁ n₁ a₁), o₂@(oadd e₂ n₂ a₂), h₁, h₂ => by -- TODO: golf
    rw [cmp]
    have IHe := @cmp_compares _ _ h₁.fst h₂.fst
    simp only [Ordering.Compares, gt_iff_lt] at IHe; revert IHe
    cases cmp e₁ e₂
    case lt => intro IHe; exact oadd_lt_oadd_1 h₁ IHe
    case gt => intro IHe; exact oadd_lt_oadd_1 h₂ IHe
    case eq =>
      intro IHe; dsimp at IHe; subst IHe
      unfold _root_.cmp; cases nh : cmpUsing (· < ·) (n₁ : ℕ) n₂ <;>
      rw [cmpUsing, ite_eq_iff, not_lt] at nh
      case lt =>
        rcases nh with nh | nh
        · exact oadd_lt_oadd_2 h₁ nh.left
        · rw [ite_eq_iff] at nh; rcases nh.right with nh | nh <;> cases nh <;> contradiction
      case gt =>
        rcases nh with nh | nh
        · cases nh; contradiction
        · obtain ⟨_, nh⟩ := nh
          rw [ite_eq_iff] at nh; rcases nh with nh | nh
          · exact oadd_lt_oadd_2 h₂ nh.left
          · cases nh; contradiction
      rcases nh with nh | nh
      · cases nh; contradiction
      obtain ⟨nhl, nhr⟩ := nh
      rw [ite_eq_iff] at nhr
      rcases nhr with nhr | nhr
      · cases nhr; contradiction
      obtain rfl := Subtype.eq (nhl.eq_of_not_lt nhr.1)
      have IHa := @cmp_compares _ _ h₁.snd h₂.snd
      revert IHa; cases cmp a₁ a₂ <;> intro IHa <;> dsimp at IHa
      case lt => exact oadd_lt_oadd_3 IHa
      case gt => exact oadd_lt_oadd_3 IHa
      subst IHa; exact rfl

Proof state:
case eq.inr.intro.inr.lt
o₁ e₁ a₁ o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝¹ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹).NF
h✝ : o₁ = e₁.oadd n₂ a₁
h₁ : (namedPattern o₁ (e₁.oadd n₂ a₁) h✝).NF
nhl : ↑n₂ ≤ ↑n₂
nhr : ¬↑n₂ < ↑n₂ ∧ Ordering.eq = Ordering.eq
IHa : a₁ < a₂
⊢ (Ordering.eq.then (Ordering.eq.then Ordering.lt)).Compares (namedPattern o₁ (e₁.oadd n₂ a₁) h✝)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹)

case eq.inr.intro.inr.eq
o₁ e₁ a₁ o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝¹ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹).NF
h✝ : o₁ = e₁.oadd n₂ a₁
h₁ : (namedPattern o₁ (e₁.oadd n₂ a₁) h✝).NF
nhl : ↑n₂ ≤ ↑n₂
nhr : ¬↑n₂ < ↑n₂ ∧ Ordering.eq = Ordering.eq
IHa : a₁ = a₂
⊢ (Ordering.eq.then (Ordering.eq.then Ordering.eq)).Compares (namedPattern o₁ (e₁.oadd n₂ a₁) h✝)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹)

case eq.inr.intro.inr.gt
o₁ e₁ a₁ o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝¹ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹).NF
h✝ : o₁ = e₁.oadd n₂ a₁
h₁ : (namedPattern o₁ (e₁.oadd n₂ a₁) h✝).NF
nhl : ↑n₂ ≤ ↑n₂
nhr : ¬↑n₂ < ↑n₂ ∧ Ordering.eq = Ordering.eq
IHa : a₁ > a₂
⊢ (Ordering.eq.then (Ordering.eq.then Ordering.gt)).Compares (namedPattern o₁ (e₁.oadd n₂ a₁) h✝)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹)

Tactic:
case lt => exact oadd_lt_oadd_3 IHa

Result:
case eq.inr.intro.inr.eq
o₁ e₁ a₁ o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝¹ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹).NF
h✝ : o₁ = e₁.oadd n₂ a₁
h₁ : (namedPattern o₁ (e₁.oadd n₂ a₁) h✝).NF
nhl : ↑n₂ ≤ ↑n₂
nhr : ¬↑n₂ < ↑n₂ ∧ Ordering.eq = Ordering.eq
IHa : a₁ = a₂
⊢ (Ordering.eq.then (Ordering.eq.then Ordering.eq)).Compares (namedPattern o₁ (e₁.oadd n₂ a₁) h✝)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹)

case eq.inr.intro.inr.gt
o₁ e₁ a₁ o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝¹ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹).NF
h✝ : o₁ = e₁.oadd n₂ a₁
h₁ : (namedPattern o₁ (e₁.oadd n₂ a₁) h✝).NF
nhl : ↑n₂ ≤ ↑n₂
nhr : ¬↑n₂ < ↑n₂ ∧ Ordering.eq = Ordering.eq
IHa : a₁ > a₂
⊢ (Ordering.eq.then (Ordering.eq.then Ordering.gt)).Compares (namedPattern o₁ (e₁.oadd n₂ a₁) h✝)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹)

Proof state:
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h✝ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF
nh : ↑n₁ < ↑n₂ ∧ Ordering.lt = Ordering.gt ∨ ↑n₂ ≤ ↑n₁ ∧ (if ↑n₂ < ↑n₁ then Ordering.gt else Ordering.eq) = Ordering.gt
⊢ (Ordering.eq.then (Ordering.gt.then (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝)

Tactic:
rcases nh with nh | nh

Result:
case inl
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h✝ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF
nh : ↑n₁ < ↑n₂ ∧ Ordering.lt = Ordering.gt
⊢ (Ordering.eq.then (Ordering.gt.then (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝)

case inr
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h✝ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF
nh : ↑n₂ ≤ ↑n₁ ∧ (if ↑n₂ < ↑n₁ then Ordering.gt else Ordering.eq) = Ordering.gt
⊢ (Ordering.eq.then (Ordering.gt.then (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝)

Proof state:
case eq.inl
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h✝ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF
nh : ↑n₁ < ↑n₂ ∧ Ordering.lt = Ordering.eq
⊢ (Ordering.eq.then (Ordering.eq.then (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝)

Tactic:
cases nh

Result:
case eq.inl.intro
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h✝ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝).NF
left✝ : ↑n₁ < ↑n₂
right✝ : Ordering.lt = Ordering.eq
⊢ (Ordering.eq.then (Ordering.eq.then (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝)

Proof state:
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
IHe : (e₁.cmp e₂).Compares e₁ e₂
⊢ ((e₁.cmp e₂).then ((_root_.cmp ↑n₁ ↑n₂).then (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹)
    (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)

Tactic:
simp only [Ordering.Compares, gt_iff_lt] at IHe

Result:
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
IHe :
  match e₁.cmp e₂, e₁, e₂ with
  | Ordering.lt, a, b => a < b
  | Ordering.eq, a, b => a = b
  | Ordering.gt, a, b => b < a
⊢ ((e₁.cmp e₂).then ((_root_.cmp ↑n₁ ↑n₂).then (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹)
    (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)

Proof state:
case eq.inr.intro.inr.eq
o₁ e₁ a₁ o₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝¹ : o₂ = e₁.oadd n₂ a₂
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹).NF
h✝ : o₁ = e₁.oadd n₂ a₁
h₁ : (namedPattern o₁ (e₁.oadd n₂ a₁) h✝).NF
nhl : ↑n₂ ≤ ↑n₂
nhr : ¬↑n₂ < ↑n₂ ∧ Ordering.eq = Ordering.eq
IHa : a₁ = a₂
⊢ (Ordering.eq.then (Ordering.eq.then Ordering.eq)).Compares (namedPattern o₁ (e₁.oadd n₂ a₁) h✝)
    (namedPattern o₂ (e₁.oadd n₂ a₂) h✝¹)

Tactic:
subst IHa

Result:
case eq.inr.intro.inr.eq
o₁ e₁ a₁ o₂ : ONote
n₂ : ℕ+
h✝¹ : o₁ = e₁.oadd n₂ a₁
h₁ : (namedPattern o₁ (e₁.oadd n₂ a₁) h✝¹).NF
nhl : ↑n₂ ≤ ↑n₂
nhr : ¬↑n₂ < ↑n₂ ∧ Ordering.eq = Ordering.eq
h✝ : o₂ = e₁.oadd n₂ a₁
h₂ : (namedPattern o₂ (e₁.oadd n₂ a₁) h✝).NF
⊢ (Ordering.eq.then (Ordering.eq.then Ordering.eq)).Compares (namedPattern o₁ (e₁.oadd n₂ a₁) h✝¹)
    (namedPattern o₂ (e₁.oadd n₂ a₁) h✝)