Notation " 1 " := (Q2Qc 1) : Qc_scope.
Definition Qcle (x y : Qc) := (x <= y)%Q.
Definition Qclt (x y : Qc) := (x < y)%Q.
Notation Qcgt := (fun x y : Qc => Qlt y x).
Notation Qcge := (fun x y : Qc => Qle y x).
Infix "<" := Qclt : Qc_scope.
Infix "<=" := Qcle : Qc_scope.
Infix ">" := Qcgt : Qc_scope.
Infix ">=" := Qcge : Qc_scope.
Notation "x <= y <= z" := (x<=y/\y<=z) : Qc_scope.
Notation "x < y < z" := (x<y/\y<z) : Qc_scope.
Definition Qccompare (p q : Qc) := (Qcompare p q).
Notation "p ?= q" := (Qccompare p q) : Qc_scope.

Lemma Qcle_alt : forall p q, (p<=q) <-> (p?=q <> Gt).