set_option linter.unusedVariables false in
theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω ∣ repr a')
    (e0 : repr a0 ≠ 0) (h : repr a' + m < (ω ^ repr a0)) (n : ℕ+) (k : ℕ) :
    let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m)
    (k ≠ 0 → R < ((ω ^ repr a0) ^ succ (k : Ordinal))) ∧
      ((ω ^ repr a0) ^ (k : Ordinal)) * ((ω ^ repr a0) * (n : ℕ) + repr a') + R =
        ((ω ^ repr a0) * (n : ℕ) + repr a' + m) ^ succ (k : Ordinal) := by
  intro R'
  haveI No : NF (oadd a0 n a') :=
    N0.oadd n (Na'.below_of_lt' <| lt_of_le_of_lt (le_add_right _ _) h)
  induction' k with k IH
  · cases m <;> simp [opowAux]
  -- rename R => R'
  let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m)
  let ω0 := ω ^ repr a0
  let α' := ω0 * n + repr a'
  change (k ≠ 0 → R < (ω0 ^ succ (k : Ordinal))) ∧ (ω0 ^ (k : Ordinal)) * α' + R
    = (α' + m) ^ (succ ↑k : Ordinal) at IH
  have RR : R' = ω0 ^ (k : Ordinal) * (α' * m) + R := by
    by_cases h : m = 0
    · simp only [h, ONote.ofNat, Nat.cast_zero, zero_add, ONote.repr, mul_zero, ONote.opowAux,
        add_zero]
    · simp only [ONote.repr_scale, ONote.repr, ONote.mulNat_eq_mul, ONote.opowAux, ONote.repr_ofNat,
        ONote.repr_mul, ONote.repr_add, Ordinal.opow_mul, ONote.zero_add]
  have α0 : 0 < α' := by simpa [lt_def, repr] using oadd_pos a0 n a'
  have ω00 : 0 < ω0 ^ (k : Ordinal) := opow_pos _ (opow_pos _ omega_pos)
  have Rl : R < ω ^ (repr a0 * succ ↑k) := by
    by_cases k0 : k = 0
    · simp [k0]
      refine' lt_of_lt_of_le _ (opow_le_opow_right omega_pos (one_le_iff_ne_zero.2 e0))
      cases' m with m <;> simp [opowAux, omega_pos]
      rw [← add_one_eq_succ, ← Nat.cast_succ]
      apply nat_lt_omega
    · rw [opow_mul]
      exact IH.1 k0
  refine' ⟨fun _ => _, _⟩
  · rw [RR, ← opow_mul _ _ (succ k.succ)]
    have e0 := Ordinal.pos_iff_ne_zero.2 e0
    have rr0 : 0 < repr a0 + repr a0 := lt_of_lt_of_le e0 (le_add_left _ _)
    apply principal_add_omega_opow
    · simp [opow_mul, opow_add, mul_assoc]
      rw [Ordinal.mul_lt_mul_iff_left ω00, ← Ordinal.opow_add]
      have : _ < ω ^ (repr a0 + repr a0) := (No.below_of_lt ?_).repr_lt
      

Proof state:
case succ.refine'_1.a.refine_2
a0 a' : ONote
N0 : NF a0
Na' : NF a'
m : ℕ
d : ω ∣ repr a'
e0✝ : repr a0 ≠ 0
h : repr a' + ↑m < ω ^ repr a0
n : ℕ+
No : NF (oadd a0 n a')
k : ℕ
R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m)
R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m)
ω0 : Ordinal.{0} := ω ^ repr a0
α' : Ordinal.{0} := ω0 * ↑↑n + repr a'
IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k
RR : R' = ω0 ^ ↑k * (α' * ↑m) + R
α0 : 0 < α'
ω00 : 0 < ω0 ^ ↑k
Rl : R < ω ^ (repr a0 * succ ↑k)
x✝ : Nat.succ k ≠ 0
e0 : 0 < repr a0
rr0 : 0 < repr a0 + repr a0
this : repr (oadd a0 n a') < ω ^ (repr a0 + repr a0)
⊢ (ω ^ repr a0 * ↑↑n + repr a') * ↑m < ω ^ (repr a0 + repr a0)
case succ.refine'_1.a.refine_1
a0 a' : ONote
N0 : NF a0
Na' : NF a'
m : ℕ
d : ω ∣ repr a'
e0✝ : repr a0 ≠ 0
h : repr a' + ↑m < ω ^ repr a0
n : ℕ+
No : NF (oadd a0 n a')
k : ℕ
R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) (Nat.succ k) m)
R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑m) k m)
ω0 : Ordinal.{0} := ω ^ repr a0
α' : Ordinal.{0} := ω0 * ↑↑n + repr a'
IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑m) ^ succ ↑k
RR : R' = ω0 ^ ↑k * (α' * ↑m) + R
α0 : 0 < α'
ω00 : 0 < ω0 ^ ↑k
Rl : R < ω ^ (repr a0 * succ ↑k)
x✝ : Nat.succ k ≠ 0
e0 : 0 < repr a0
rr0 : 0 < repr a0 + repr a0
⊢ repr a0 < repr a0 + repr a0

Next tactic:
refine' mul_lt_omega_opow rr0 this (nat_lt_omega _)