Require Import Arith.

Inductive exp : Set :=
| Constant : nat -> exp
| Plus : exp -> exp -> exp
| Times : exp -> exp -> exp.

Fixpoint eval (e : exp) : nat :=
  match e with
  | Constant n => n
  | Plus e1 e2 => eval e1 + eval e2
  | Times e1 e2 => eval e1 * eval e2
  end.

Fixpoint commuter (e: exp) : exp :=
  match e with
  | Constant _ => e
  | Plus e1 e2 => Plus (commuter e2) (commuter e1)
  | Times e1 e2 => Times (commuter e2) (commuter e1)
  end.

Theorem eval_commuter : forall e, eval(commuter e) = eval e.
Proof.
  induction e; simpl.
  reflexivity.
  rewrite IHe1, IHe2.
  ring.
  rewrite IHe1, IHe2.
  ring.
  Qed.