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
      refine' mul_lt_omega_opow rr0 this (nat_lt_omega _)
      simpa using (add_lt_add_iff_left (repr a0)).2 e0
    · refine'
        lt_of_lt_of_le Rl
          (opow_le_opow_right omega_pos <|
            mul_le_mul_left' (succ_le_succ_iff.2 (nat_cast_le.2 (le_of_lt k.lt_succ_self))) _)
  calc
    (ω0 ^ (k.succ : Ordinal)) * α' + R'
    _ = (ω0 ^ succ (k : Ordinal)) * α' + ((ω0 ^ (k : Ordinal)) * α' * m + R) :=
        by rw [nat_cast_succ, RR, ← mul_assoc]
    _ = ((ω0 ^ (k : Ordinal)) * α' + R) * α' + ((ω0 ^ (k : Ordinal)) * α' + R) * m := ?_
    _ = (α' + m) ^ succ (k.succ : Ordinal) := by rw [← mul_add, nat_cast_succ, opow_succ, IH.2]
  congr 1
  · have αd : ω ∣ α' :=
      dvd_add (dvd_mul_of_dvd_left (by simpa using opow_dvd_opow ω (one_le_iff_ne_zero.2 e0)) _) d
    rw [mul_add (ω0 ^ (k : Ordinal)), add_assoc, ← mul_assoc, ← opow_succ,
      add_mul_limit _ (isLimit_iff_omega_dvd.2 ⟨ne_of_gt α0, αd⟩), mul_assoc,
      @mul_omega_dvd n (nat_cast_pos.2 n.pos) (nat_lt_omega _) _ αd]
    apply @add_absorp _ (repr a0 * succ ↑k)
    · refine' principal_add_omega_opow _ _ Rl
      rw [opow_mul, opow_succ, Ordinal.mul_lt_mul_iff_left ω00]
      exact No.snd'.repr_lt
    · have := mul_le_mul_left' (one_le_iff_pos.2 <| nat_cast_pos.2 n.pos) (ω0 ^ succ (k : Ordinal))
      rw [opow_mul]
      simpa [-opow_succ]
  · cases m
    · have : R = 0 := by cases k <;> simp [opowAux]
      simp [this]
    · rw [nat_cast_succ, add_mul_succ]
      apply add_absorp Rl
      

Proof state:
case succ.refine'_2.e_a.succ.ba
a0 a' : ONote
N0 : NF a0
Na' : NF a'
d : ω ∣ repr a'
e0 : repr a0 ≠ 0
n : ℕ+
No : NF (oadd a0 n a')
k : ℕ
ω0 : Ordinal.{0} := ω ^ repr a0
α' : Ordinal.{0} := ω0 * ↑↑n + repr a'
α0 : 0 < α'
ω00 : 0 < ω0 ^ ↑k
n✝ : ℕ
h : repr a' + ↑(Nat.succ n✝) < ω ^ repr a0
R' : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) (Nat.succ k) (Nat.succ n✝))
R : Ordinal.{0} := repr (opowAux 0 a0 (oadd a0 n a' * ↑(Nat.succ n✝)) k (Nat.succ n✝))
IH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' + R = (α' + ↑(Nat.succ n✝)) ^ succ ↑k
RR : R' = ω0 ^ ↑k * (α' * ↑(Nat.succ n✝)) + R
Rl : R < ω ^ (repr a0 * succ ↑k)
⊢ ω ^ (repr a0 * succ ↑k) ≤ ω0 ^ ↑k * α'

Next tactic:
rw [opow_mul, opow_succ]