Section FracDomain.
Variable R : ringType.
(* ratios are pairs of R, such that the second member is nonzero *)
Inductive ratio := mkRatio { frac :> R * R; _ : frac.2 != 0 }.
Definition ratio_of of phant R := ratio.
Local Notation "{ 'ratio' T }" := (ratio_of (Phant T)).
Canonical ratio_subType := Eval hnf in [subType for frac].
Canonical ratio_of_subType := Eval hnf in [subType of {ratio R}].
Definition ratio_EqMixin := [eqMixin of ratio by <:].
Canonical ratio_eqType := EqType ratio ratio_EqMixin.
Canonical ratio_of_eqType := Eval hnf in [eqType of {ratio R}].
Definition ratio_ChoiceMixin := [choiceMixin of ratio by <:].
Canonical ratio_choiceType := ChoiceType ratio ratio_ChoiceMixin.
Canonical ratio_of_choiceType := Eval hnf in [choiceType of {ratio R}].
Definition ratio0 := (@mkRatio (0, 1) (oner_neq0 _)).
Definition Ratio x y : {ratio R} := insubd ratio0 (x, y).

Lemma numer_Ratio x y : y != 0 -> (Ratio x y).1 = x.