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