inductive formula :=
eqf  : nat → nat → formula,
impf : formula → formula → formula

namespace formula
  definition denote : formula → Prop,
  denote (eqf n1 n2)  := n1 = n2,
  denote (impf f1 f2) := denote f1 → denote f2

  theorem denote_eqf (n1 n2 : nat) : denote (eqf n1 n2) = (n1 = n2) :=
  rfl

  theorem denote_impf (f1 f2 : formula) : denote (impf f1 f2) = (denote f1 → denote f2) :=
  rfl
end formula