Require Import String.

Inductive term : Set :=
| Var : string -> term
| Abs : string -> term -> term
| App: term -> term -> term.

Fixpoint subst (rep : term) (x: string) (e:term) : term :=
  match e with
  | Var y => if string_dec y x then rep else Var y
  | Abs y e1 => Abs y (if string_dec y x then e1 else subst rep x e1)
  | App e1 e2 => App (subst rep x e1) (subst rep x e2)
  end.

Parameter s : string.
Parameter z : string.
Definition zero := Abs s (Abs z (Var(z))).
Definition one :=  Abs s (Abs z (App (Var(s)) (Var(z)))).

Theorem mytheorem: forall x y t,
       x <> y
    -> y <> t ->
    (subst (Var(t)) y (Abs x (App (Var(x)) (Var(y)))))
                      = (Abs x (App (Var(x)) (Var(t)))).   
Proof.
  intros;simpl.
  destruct (string_dec x y).
  congruence.
  destruct (string_dec y y).
  congruence.
  congruence.
Qed.