Proving f (f bool) = bool

Question

How can I in coq, prove that a function f that accepts a bool true|false and returns a bool true|false (shown below), when applied twi

Answer 1

A tad shorter proof:

Require Import Sumbool.

Goal forall (f : bool -> bool) (b:bool), f (f (f b)) = f b.
Proof.
  destruct b;                             (* case analysis on [b] *)
    destruct (sumbool_of_bool (f true));  (* case analysis on [f true] *)
    destruct (sumbool_of_bool (f false)); (* case analysis on [f false] *)
    congruence.                           (* equational reasoning *)
Qed.