-- Module: Mathlib.Order.RelSeries
Goal:
α : Type u_3
inst : PartialOrder α
inst_1 : WellFoundedLT α
inst_2 : WellFoundedGT α
s : Fin (HAdd.hAdd (α := ℕ) 0 1) → α
h :
  ∀ (i : Fin 0),
    Membership.mem (γ := SetRel α α)
      {x : α × α |
        match (motive := α × α → Prop) x with
        | (a, b) => a < b}
      (s i.castSucc, s i.succ)
⊢ ∃ (t :
    RelSeries (α := α)
      {x : α × α |
        match (motive := α × α → Prop) x with
        | (a, b) => a ⋖ b})
    (i :
    Fin
        ((RelSeries.mk (r :=
                {x : α × α |
                  match (motive := α × α → Prop) x with
                  | (a, b) => a < b})
                0 s h).length
            (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a < b}) +
          1) ↪
      Fin
        (t.length (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) +
          1)),
    t.toFun (r :=
            {x : α × α |
              match (motive := α × α → Prop) x with
              | (a, b) => a ⋖ b}) ∘
          (⇑i : (a : Fin ({ length := 0, toFun := s, step := h }.length + 1)) → Fin (t.length + 1)) =
        (RelSeries.mk (r :=
              {x : α × α |
                match (motive := α × α → Prop) x with
                | (a, b) => a < b})
              0 s h).toFun
          (r :=
          {x : α × α |
            match (motive := α × α → Prop) x with
            | (a, b) => a < b}) ∧
      Eq (α :=
          Fin
            (t.length (r :=
                {x : α × α |
                  match (motive := α × α → Prop) x with
                  | (a, b) => a ⋖ b}) +
              1))
          ((i : (a : Fin ({ length := 0, toFun := s, step := h }.length + 1)) → Fin (t.length + 1)) 0) 0 ∧
        Eq (α :=
          Fin
            (t.length (r :=
                {x : α × α |
                  match (motive := α × α → Prop) x with
                  | (a, b) => a ⋖ b}) +
              1))
          ((i : (a : Fin ({ length := 0, toFun := s, step := h }.length + 1)) → Fin (t.length + 1))
              (Fin.last
                ((RelSeries.mk (r :=
                      {x : α × α |
                        match (motive := α × α → Prop) x with
                        | (a, b) => a < b})
                      0 s h).length
                  (r :=
                  {x : α × α |
                    match (motive := α × α → Prop) x with
                    | (a, b) => a < b}))) :
            Fin
              (t.length (r :=
                  {x : α × α |
                    match (motive := α × α → Prop) x with
                    | (a, b) => a ⋖ b}) +
                1))
          (Fin.last
              (t.length (r :=
                {x : α × α |
                  match (motive := α × α → Prop) x with
                  | (a, b) => a ⋖ b})) :
            Fin
              (t.length (r :=
                  {x : α × α |
                    match (motive := α × α → Prop) x with
                    | (a, b) => a ⋖ b}) +
                1))

Tactic:
⟨⟨0, s, nofun⟩, (Equiv.refl _).toEmbedding, rfl, rfl, rfl⟩

Premises used: Fin.succ, CovBy, setOf, FiniteDimensionalOrder.match_1, Membership.mem, Prod.mk, Equiv.toEmbedding, And.intro, Exists.intro, _private.0.LTSeries.exists_relSeries_covBy.match_1, Fin.castSucc, OfNat.ofNat, Equiv.refl, rfl, RelSeries.mk