simp only [if_pos]
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) }
no goals