Definition t := bool.Definition eq       := @Logic.eq       bool.Definition eq_refl  := @Logic.eq_refl  bool.Definition eq_sym   := @Logic.eq_sym   bool.Definition eq_trans := @Logic.eq_trans bool.Definition eq_dec   := bool_dec.Definition lt (b1 b2 : bool) : Prop := b1 = false /\ b2 = true.

Theorem lt_not_eq : forall x y : bool, lt x y -> ~ eq x y.