-- Module: Mathlib.SetTheory.Ordinal.Notation
Goal:
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e : b.fundamentalSequence = Sum.inl none
oadd :
  ∀ (e : a.fundamentalSequence = Sum.inl none),
    match (motive := Option ONote ⊕ (ℕ → ONote) → Prop)
      match (motive := Option ONote ⊕ (ℕ → ONote) → ℕ → Option ONote ⊕ (ℕ → ONote)) Sum.inl none, m.natPred with
      | Sum.inl none, 0 => Sum.inl (some zero)
      | Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
      | Sum.inl (some a'), 0 => Sum.inr fun (i : ℕ) ↦ a'.oadd i.succPNat zero
      | Sum.inl (some a'), m.succ => Sum.inr fun (i : ℕ) ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
      | Sum.inr f, 0 => Sum.inr fun (i : ℕ) ↦ (f i).oadd 1 zero
      | Sum.inr f, m.succ => Sum.inr fun (i : ℕ) ↦ a.oadd m.succPNat ((f i).oadd 1 zero) with
    | Sum.inl none => a.oadd m b = 0
    | Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
    | Sum.inr f =>
      IsSuccLimit (a.oadd m b).repr ∧
        (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
          ∀ a_1 < (a.oadd m b).repr, ∃ (i : ℕ), a_1 < (f i).repr
oadd₁ :
  ∀ (a' : ONote) (e : a.fundamentalSequence = Sum.inl (some a')),
    match (motive := Option ONote ⊕ (ℕ → ONote) → Prop)
      match (motive := Option ONote ⊕ (ℕ → ONote) → ℕ → Option ONote ⊕ (ℕ → ONote)) Sum.inl (some a'), m.natPred with
      | Sum.inl none, 0 => Sum.inl (some zero)
      | Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
      | Sum.inl (some a'), 0 => Sum.inr fun (i : ℕ) ↦ a'.oadd i.succPNat zero
      | Sum.inl (some a'), m.succ => Sum.inr fun (i : ℕ) ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
      | Sum.inr f, 0 => Sum.inr fun (i : ℕ) ↦ (f i).oadd 1 zero
      | Sum.inr f, m.succ => Sum.inr fun (i : ℕ) ↦ a.oadd m.succPNat ((f i).oadd 1 zero) with
    | Sum.inl none => a.oadd m b = 0
    | Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
    | Sum.inr f =>
      IsSuccLimit (a.oadd m b).repr ∧
        (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
          ∀ a_1 < (a.oadd m b).repr, ∃ (i : ℕ), a_1 < (f i).repr
oadd₂ :
  ∀ (f : ℕ → ONote) (e : a.fundamentalSequence = Sum.inr f),
    match (motive := Option ONote ⊕ (ℕ → ONote) → Prop)
      match (motive := Option ONote ⊕ (ℕ → ONote) → ℕ → Option ONote ⊕ (ℕ → ONote)) Sum.inr f, m.natPred with
      | Sum.inl none, 0 => Sum.inl (some zero)
      | Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
      | Sum.inl (some a'), 0 => Sum.inr fun (i : ℕ) ↦ a'.oadd i.succPNat zero
      | Sum.inl (some a'), m.succ => Sum.inr fun (i : ℕ) ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
      | Sum.inr f, 0 => Sum.inr fun (i : ℕ) ↦ (f i).oadd 1 zero
      | Sum.inr f, m.succ => Sum.inr fun (i : ℕ) ↦ a.oadd m.succPNat ((f i).oadd 1 zero) with
    | Sum.inl none => a.oadd m b = 0
    | Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
    | Sum.inr f =>
      IsSuccLimit (a.oadd m b).repr ∧
        (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
          ∀ a_1 < (a.oadd m b).repr, ∃ (i : ℕ), a_1 < (f i).repr
⊢ match (motive := Option ONote ⊕ (ℕ → ONote) → Prop)
    match (motive := Option ONote ⊕ (ℕ → ONote) → ℕ → Option ONote ⊕ (ℕ → ONote)) a.fundamentalSequence, m.natPred with
    | Sum.inl none, 0 => Sum.inl (some zero)
    | Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
    | Sum.inl (some a'), 0 => Sum.inr fun (i : ℕ) ↦ a'.oadd i.succPNat zero
    | Sum.inl (some a'), m.succ => Sum.inr fun (i : ℕ) ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
    | Sum.inr f, 0 => Sum.inr fun (i : ℕ) ↦ (f i).oadd 1 zero
    | Sum.inr f, m.succ => Sum.inr fun (i : ℕ) ↦ a.oadd m.succPNat ((f i).oadd 1 zero) with
  | Sum.inl none => a.oadd m b = 0
  | Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
  | Sum.inr f =>
    IsSuccLimit (a.oadd m b).repr ∧
      (∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
        ∀ a_1 < (a.oadd m b).repr, ∃ (i : ℕ), a_1 < (f i).repr

Tactic:
rcases e : a.fundamentalSequence with (⟨_ | a'⟩ | f)

Premises used: ONote.fundamentalSequence