Fixpoint Dichotomy_lb (x y:R) (P:R -> bool) (N:nat) {struct N} : R :=
  match N with
    | O => x
    | S n =>
      let down := Dichotomy_lb x y P n in
        let up := Dichotomy_ub x y P n in
          let z := (down + up) / 2 in if P z then down else z
  end

  with Dichotomy_ub (x y:R) (P:R -> bool) (N:nat) {struct N} : R :=
    match N with
      | O => y
      | S n =>
        let down := Dichotomy_lb x y P n in
          let up := Dichotomy_ub x y P n in
            let z := (down + up) / 2 in if P z then z else up
    end.Definition dicho_lb (x y:R) (P:R -> bool) (N:nat) : R := Dichotomy_lb x y P N.Definition dicho_up (x y:R) (P:R -> bool) (N:nat) : R := Dichotomy_ub x y P N.

Lemma dicho_lb_dicho_up : forall (x y:R) (P:R -> bool) (n:nat), x <= y -> dicho_up x y P n - dicho_lb x y P n = (y - x) / 2 ^ n.