Fact fracq_eq0 x : (fracq x == zeroq) = (x.1 == 0) || (x.2 == 0).
Fact fracqMM x n d : x != 0 -> fracq (x * n, x * d) = fracq (n, d).
Definition addq_subdef (x y : int * int) := (x.1 * y.2 + y.1 * x.2, x.2 * y.2).
Definition addq (x y : rat) := nosimpl fracq (addq_subdef (valq x) (valq y)).
Definition oppq_subdef (x : int * int) := (- x.1, x.2).
Definition oppq (x : rat) := nosimpl fracq (oppq_subdef (valq x)).
Fact addq_subdefC : commutative addq_subdef.
Fact addq_frac x y : x.2 != 0 -> y.2 != 0 ->
(addq (fracq x) (fracq y)) = fracq (addq_subdef x y).
Fact ratzD : {morph ratz : x y / x + y >-> addq x y}.
Fact oppq_frac x : oppq (fracq x) = fracq (oppq_subdef x).
Fact ratzN : {morph ratz : x / - x >-> oppq x}.
Fact add0q : left_id zeroq addq.
Fact addNq : left_inverse (fracq (0, 1)) oppq addq.
Definition rat_ZmodMixin := ZmodMixin addqA addqC add0q addNq.
Canonical rat_ZmodType := ZmodType rat rat_ZmodMixin.
Definition mulq_subdef (x y : int * int) := nosimpl (x.1 * y.1, x.2 * y.2).
Definition mulq (x y : rat) := nosimpl fracq (mulq_subdef (valq x) (valq y)).
Fact mulq_subdefC : commutative mulq_subdef.
Fact ratzM : {morph ratz : x y / x * y >-> mulq x y}.
Fact mulqC : commutative mulq.
Fact mul1q : left_id oneq mulq.
Fact mulq_addl : left_distributive mulq addq.
Fact nonzero1q : oneq != zeroq. Proof. by []. Qed.
Definition rat_comRingMixin :=
ComRingMixin mulqA mulqC mul1q mulq_addl nonzero1q.
Canonical rat_Ring := Eval hnf in RingType rat rat_comRingMixin.
Canonical rat_comRing := Eval hnf in ComRingType rat mulqC.
Fact mulVq x : x != 0 -> mulq (invq x) x = 1.
Fact invq0 : invq 0 = 0. Proof. by apply/eqP. Qed.
Definition RatFieldUnitMixin := FieldUnitMixin mulVq invq0.
Canonical rat_unitRing :=
Eval hnf in UnitRingType rat RatFieldUnitMixin.
Canonical rat_comUnitRing := Eval hnf in [comUnitRingType of rat].
Fact rat_field_axiom : GRing.Field.mixin_of rat_unitRing. Proof. exact. Qed.
Definition RatFieldIdomainMixin := (FieldIdomainMixin rat_field_axiom).
Canonical rat_iDomain :=
Eval hnf in IdomainType rat (FieldIdomainMixin rat_field_axiom).
Canonical rat_fieldType := FieldType rat rat_field_axiom.
Canonical rat_countZmodType := [countZmodType of rat].
Canonical rat_countRingType := [countRingType of rat].
Canonical rat_countComRingType := [countComRingType of rat].
Canonical rat_countUnitRingType := [countUnitRingType of rat].
Canonical rat_countComUnitRingType := [countComUnitRingType of rat].
Canonical rat_countIdomainType := [countIdomainType of rat].
Canonical rat_countFieldType := [countFieldType of rat].

Lemma numq_eq0 x : (numq x == 0) = (x == 0).
Notation "n %:Q" := ((n : int)%:~R : rat)
(at level 2, left associativity, format "n %:Q")  : ring_scope.