cases hik
n : Nat
x : Permutation n
i j : Fin n
k : Nat
hk : k < n
hjk : ¬↑j = k
hik : ↑i = k
⊢ x[↑(if ↑j = k then i else if ↑i = k then j else { val := k, isLt := hk })] =
    if ↑j = k then x[↑i] else if ↑i = k then x[↑j] else x[k]
case refl
n : Nat
x : Permutation n
i j : Fin n
hk : ↑i < n
hjk : ¬↑j = ↑i
⊢ x[↑(if ↑j = ↑i then i else if ↑i = ↑i then j else { val := ↑i, isLt := hk })] =
    if ↑j = ↑i then x[↑i] else if ↑i = ↑i then x[↑j] else x[↑i]