(** Take-home Exercise: We can use almost the same proof script as above to state how substitution works in the variable case. Try it on your own.  *)

(* Specifications: *)
Lemma subst_eq_var: forall (x : atom) u,
[x ~> u]x = u.
Lemma subst_neq_var : forall (x y : atom) u,
y <> x -> [x ~> u]y = y.
Lemma subst_eq_var: forall (x : atom) u,
[x ~> u]x = u.