(* Boolean equalities for syntax *)
Variable variable_eq_dec : forall (x1 x2 : variable), {x1=x2}+{x1<>x2}.
Definition variable_beq (x1 x2 : variable) : bool :=
if variable_eq_dec x1 x2 then true else false.

Lemma variable_beq_eq : forall x1 x2, variable_beq x1 x2 = true <-> x1 = x2.
Variable function_eq_dec : forall (x1 x2 : function), {x1=x2}+{x1<>x2}.