. intro l hl ivs' ivs'_nodup ivs'_lt
  if hx : x[l]'(Nat.lt_trans hl hk) > x[k] then
    simp only [dif_pos hx, Array.nodup_push, Array.mem_push, ivs'_nodup, true_and]
    constructor
    . intro h
      specialize ivs'_lt _ h; dsimp at ivs'_lt
      refine ivs'_lt.elim ?_ ?_
      . exact Nat.lt_irrefl k
      . intro h; exact Nat.lt_irrefl l h.2
    . intro iv hiv
      cases hiv with
      | inl hmem =>
        refine Or.imp_right ?_ <| ivs'_lt iv hmem
        exact And.imp_right Nat.le.step
      | inr heq =>
        cases heq; dsimp
        exact Or.inr ⟨rfl, l.lt_succ_self⟩
  else
    simp only [dif_neg hx]
    apply And.intro ivs'_nodup
    intro iv hiv; specialize ivs'_lt iv hiv
    refine Or.imp_right ?_ ivs'_lt
    exact And.imp_right Nat.le.step
case refine_2.refl.refine_2
n : Nat
x : Permutation n
k : Nat
hk : k < n
ivs : Array (Inversion x)
ivs_nodup : Array.Nodup ivs
ivs_lt : ∀ (iv : Inversion x), iv ∈ ivs → ↑iv.snd < k
⊢ ∀ (i : Nat) (hi : i < k) (a : Array (Inversion x)),
    Array.Nodup a →
      (∀ (iv : Inversion x), iv ∈ a → ↑iv.snd < k ∨ ↑iv.snd = k ∧ ↑iv.fst < i) →
        Array.Nodup
            (if h : x[i] > x[k] then
              Array.push a
                { fst := { val := i, isLt := (_ : i < n) }, snd := { val := k, isLt := hk },
                  lt := (_ : { val := i, isLt := (_ : i < n) } < { val := k, isLt := hk }),
                  inverted := (_ : x[↑{ val := i, isLt := (_ : i < n) }] > x[↑{ val := k, isLt := hk }]) }
            else a) ∧
          ∀ (iv : Inversion x),
            (iv ∈
                if h : x[i] > x[k] then
                  Array.push a
                    { fst := { val := i, isLt := (_ : i < n) }, snd := { val := k, isLt := hk },
                      lt := (_ : { val := i, isLt := (_ : i < n) } < { val := k, isLt := hk }),
                      inverted := (_ : x[↑{ val := i, isLt := (_ : i < n) }] > x[↑{ val := k, isLt := hk }]) }
                else a) →
              ↑iv.snd < k ∨ ↑iv.snd = k ∧ ↑iv.fst < i + 1
no goals