protected theorem IsNormal.isFundamentalSequence {f : Ordinal.{u} → Ordinal.{u}} (hf : IsNormal f)
    {a o} (ha : IsLimit a) {g} (hg : IsFundamentalSequence a o g) :
    IsFundamentalSequence (f a) o fun b hb => f (g b hb) := by
  refine' ⟨_, @fun i j _ _ h => hf.strictMono (hg.2.1 _ _ h), _⟩
  · rcases exists_lsub_cof (f a) with ⟨ι, f', hf', hι⟩
    rw [← hg.cof_eq, ord_le_ord, ← hι]
    suffices (lsub.{u, u} fun i => sInf { b : Ordinal | f' i ≤ f b }) = a by
      rw [← this]
      apply cof_lsub_le
    have H : ∀ i, ∃ b < a, f' i ≤ f b := fun i => by
      have := lt_lsub.{u, u} f' i
      rw [hf', ← IsNormal.blsub_eq.{u, u} hf ha, lt_blsub_iff] at this
      simpa using this
    refine' (lsub_le fun i => _).antisymm (le_of_forall_lt fun b hb => _)
    · rcases H i with ⟨b, hb, hb'⟩
      exact lt_of_le_of_lt (csInf_le' hb') hb
    · have := hf.strictMono hb
      rw [← hf', lt_lsub_iff] at this
      cases' this with i hi
      rcases H i with ⟨b, _, hb⟩
      exact
        ((le_csInf_iff'' ⟨b, by 

Proof state:
α : Type u_1
r : α → α → Prop
f : Ordinal.{u} → Ordinal.{u}
hf : IsNormal f
a o : Ordinal.{u}
ha : IsLimit a
g : (b : Ordinal.{u}) → b < o → Ordinal.{u}
hg : IsFundamentalSequence a o g
ι : Type u
f' : ι → Ordinal.{u}
hf' : lsub f' = f a
hι : #ι = cof (f a)
H : ∀ (i : ι), ∃ b < a, f' i ≤ f b
b✝ : Ordinal.{u}
hb✝ : b✝ < a
i : ι
hi : f b✝ ≤ f' i
b : Ordinal.{u}
left✝ : b < a
hb : f' i ≤ f b
⊢ b ∈ fun c =>
    Quot.lift
      (fun a₁ =>
        Quotient.lift
          ((fun x x_1 =>
              match x with
              | { α := α, r := r, wo := wo } =>
                match x_1 with
                | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s))
            a₁)
          (_ :
            ∀ (a b : WellOrder),
              a ≈ b →
                (match a₁ with
                  | { α := α, r := r, wo := wo } =>
                    match a with
                    | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s)) =
                  match a₁ with
                  | { α := α, r := r, wo := wo } =>
                    match b with
                    | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s))
          (f c))
      (_ :
        ∀ (a b : WellOrder),
          a ≈ b →
            (fun a₁ =>
                  Quotient.lift
                    (fun x =>
                      match a₁ with
                      | { α := α, r := r, wo := wo } =>
                        match x with
                        | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s))
                    (_ :
                      ∀ (a b : WellOrder),
                        a ≈ b →
                          (fun x x_1 =>
                                match x with
                                | { α := α, r := r, wo := wo } =>
                                  match x_1 with
                                  | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s))
                              a₁ a =
                            (fun x x_1 =>
                                match x with
                                | { α := α, r := r, wo := wo } =>
                                  match x_1 with
                                  | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s))
                              a₁ b)
                    (f c))
                a =
              (fun a₁ =>
                  Quotient.lift
                    (fun x =>
                      match a₁ with
                      | { α := α, r := r, wo := wo } =>
                        match x with
                        | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s))
                    (_ :
                      ∀ (a b : WellOrder),
                        a ≈ b →
                          (fun x x_1 =>
                                match x with
                                | { α := α, r := r, wo := wo } =>
                                  match x_1 with
                                  | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s))
                              a₁ a =
                            (fun x x_1 =>
                                match x with
                                | { α := α, r := r, wo := wo } =>
                                  match x_1 with
                                  | { α := α_1, r := s, wo := wo } => Nonempty (r ≼i s))
                              a₁ b)
                    (f c))
                b)
      (f' i)

Next tactic:
exact hb