Definition TotalP (P : R -> float -> Prop) :=
  forall r : R, exists p : float, P r p.Definition UniqueP (P : R -> float -> Prop) :=
  forall (r : R) (p q : float), P r p -> P r q -> p = q :>R.Definition CompatibleP (P : R -> float -> Prop) :=
  forall (r1 r2 : R) (p q : float),
  P r1 p -> r1 = r2 -> p = q :>R -> Fbounded b q -> P r2 q.Definition MinOrMaxP (P : R -> float -> Prop) :=
  forall (r : R) (p : float), P r p -> isMin b radix r p \/ isMax b radix r p.Definition RoundedModeP (P : R -> float -> Prop) :=
  TotalP P /\ CompatibleP P /\ MinOrMaxP P /\ MonotoneP radix P.

Theorem RoundedModeP_inv1 : forall P, RoundedModeP P -> TotalP P.