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).
(* we show that equivf is an equivalence *)
Canonical equivf_equiv := EquivRel equivf equivf_refl equivf_sym equivf_trans.
Definition type := {eq_quot equivf}.
Definition type_of of phant R := type.
Notation "{ 'fraction' T }" := (type_of (Phant T)).
(* we recover some structure for the quotient *)
Canonical frac_quotType := [quotType of type].
Canonical frac_eqType := [eqType of type].
Canonical frac_choiceType := [choiceType of type].
Canonical frac_eqQuotType := [eqQuotType equivf of type].
Canonical frac_of_quotType := [quotType of {fraction R}].
Canonical frac_of_eqType := [eqType of {fraction R}].
Canonical frac_of_choiceType := [choiceType of {fraction R}].
Canonical frac_of_eqQuotType := [eqQuotType equivf of {fraction R}].
Definition tofrac := lift_embed {fraction R} (fun x : R => Ratio x 1).
Canonical tofrac_pi_morph := PiEmbed tofrac.
Notation "x %:F"  := (@tofrac x).
Implicit Types a b c : type.
Definition addf x y : dom := Ratio (\n_x * \d_y + \n_y * \d_x) (\d_x * \d_y).
Definition add := lift_op2 {fraction R} addf.
Canonical pi_add_morph := PiMorph2 pi_add.
Definition oppf x : dom := Ratio (- \n_x) \d_x.
Definition opp := lift_op1 {fraction R} oppf.
Canonical pi_opp_morph := PiMorph1 pi_opp.
Definition mulf x y : dom := Ratio (\n_x * \n_y) (\d_x * \d_y).
Definition mul := lift_op2 {fraction R} mulf.
Canonical pi_mul_morph := PiMorph2 pi_mul.
Definition invf x : dom := Ratio \d_x \n_x.
Definition inv := lift_op1 {fraction R} invf.
Canonical pi_inv_morph := PiMorph1 pi_inv.
(* fracions form an abelian group *)
Definition frac_zmodMixin :=  ZmodMixin addA addC add0_l addN_l.
Canonical frac_zmodType := Eval hnf in ZmodType type frac_zmodMixin.
(* fractions form a ring with explicit unit *)
Definition RatFieldUnitMixin := FieldUnitMixin mulV_l inv0.
Canonical frac_unitRingType := Eval hnf in UnitRingType type RatFieldUnitMixin.
Canonical frac_comUnitRingType := [comUnitRingType of type].
Canonical tofrac_additive := Additive tofrac_is_additive.
Canonical tofrac_rmorphism := AddRMorphism tofrac_is_multiplicative.

Lemma tofrac_eq (p q : R): (p%:F == q%:F) = (p == q).