Module: Mathlib.SetTheory.Ordinal.Notation

Goal state before:
pp: case oadd.inl.none
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b.FundamentalSequenceProp b.fundamentalSequence
e : b.fundamentalSequence = Sum.inl none
⊢ match
    match 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 ↦ a'.oadd i.succPNat zero
    | Sum.inl (some a'), m.succ => Sum.inr fun i ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
    | Sum.inr f, 0 => Sum.inr fun i ↦ (f i).oadd 1 zero
    | Sum.inr f, m.succ => Sum.inr fun i ↦ 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, a_1 < (f i).repr

pp: case oadd.inl.some
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b.FundamentalSequenceProp b.fundamentalSequence
b' : ONote
e : b.fundamentalSequence = Sum.inl (some b')
⊢ (a.oadd m b).repr = succ (a.oadd m b').repr ∧ ((a.oadd m b).NF → (a.oadd m b').NF)

pp: case oadd.inr
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b.FundamentalSequenceProp b.fundamentalSequence
f : ℕ → ONote
e : b.fundamentalSequence = Sum.inr f
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a.oadd m (f i) < a.oadd m (f (i + 1)) ∧ a.oadd m (f i) < a.oadd m b ∧ ((a.oadd m b).NF → (a.oadd m (f i)).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (a.oadd m (f i)).repr

Tactic:
<;>

Goal state after:
case oadd.inl.none
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b.FundamentalSequenceProp b.fundamentalSequence
e : b.fundamentalSequence = Sum.inl none
⊢ match
    match 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 ↦ a'.oadd i.succPNat zero
    | Sum.inl (some a'), m.succ => Sum.inr fun i ↦ a.oadd m.succPNat (a'.oadd i.succPNat zero)
    | Sum.inr f, 0 => Sum.inr fun i ↦ (f i).oadd 1 zero
    | Sum.inr f, m.succ => Sum.inr fun i ↦ 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, a_1 < (f i).repr

case oadd.inl.some
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b.FundamentalSequenceProp b.fundamentalSequence
b' : ONote
e : b.fundamentalSequence = Sum.inl (some b')
⊢ (a.oadd m b).repr = succ (a.oadd m b').repr ∧ ((a.oadd m b).NF → (a.oadd m b').NF)

case oadd.inr
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b.FundamentalSequenceProp b.fundamentalSequence
f : ℕ → ONote
e : b.fundamentalSequence = Sum.inr f
⊢ IsSuccLimit (a.oadd m b).repr ∧
    (∀ (i : ℕ),
        a.oadd m (f i) < a.oadd m (f (i + 1)) ∧ a.oadd m (f i) < a.oadd m b ∧ ((a.oadd m b).NF → (a.oadd m (f i)).NF)) ∧
      ∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (a.oadd m (f i)).repr