Hint Resolve denq_neq0 denq_gt0 denq_ge0 : core.
Definition subq (x y : rat) : rat := (addq x (oppq y)).
Definition divq (x y : rat) : rat := (mulq x (invq y)).
Notation "0" := zeroq : rat_scope.
Notation "1" := oneq : rat_scope.
Infix "+" := addq : rat_scope.
Notation "- x" := (oppq x) : rat_scope.
Infix "*" := mulq : rat_scope.
Notation "x ^-1" := (invq x) : rat_scope.
Infix "-" := subq : rat_scope.
Infix "/" := divq : rat_scope.

Lemma numqN x : numq (- x) = - numq x.