Add lemmas about Qp and fraction

algebra/frac.v

@@ -35,3 +35,6 @@ Global Instance frac_full_exclusive : Exclusive 1%Qp.
 Proof.
   move=> y /Qcle_not_lt [] /=. by rewrite -{1}(Qcplus_0_r 1) -Qcplus_lt_mono_l.
 Qed.
+
+Lemma invalid_plus_q: ∀ (q: Qp), ¬ ✓ (1 + q)%Qp.
+Proof. intros q H. by apply (Qp_ge_1 q). Qed.

prelude/numbers.v

@@ -566,3 +566,10 @@ Proof.
   exists (mk_Qp (q2 - q1 / 2%Z) Hq2'). split; [by rewrite Qp_div_2|].
   apply Qp_eq; simpl. ring.
 Qed.
+
+Lemma Qp_ge_1 (q: Qp): ¬ ((1 + q)%Qp ≤ 1%Qp)%Qc.
+Proof.
+  intros Hle.
+  apply (Qcplus_le_mono_l q 0 1) in Hle.
+  apply Qcle_ngt in Hle. by destruct q.
+Qed.