For each of the following simply-typed applied lambda calculus expressions, give a typing derivation or show that the expression is ill-typed by showing that there can't exist a typing derivation for it. (Hint: the final result of typechecking each expression can be checked with the provided "readTypecheck" function in LambdaTypecheck.hs.)

a) if (if true then false else true) then false else true b) (x: num). x + x * 3 c) (a: bool). if a then 1 else a d) ((f: num -> bool) (a: num). if f a then a + 1 else a + 2) ((x: num). false)