proposition: Lemma Qle_plus_plus : forall a b c d : Q, Qle a b -> Qle c d -> Qle (Qplus a c) (Qplus b d).
proof: Proof. intros a b c d Hab Hcd. apply Qle_trans with (Qplus b c); unfold Qlt in |- *; [ rewrite (Qplus_sym a); rewrite (Qplus_sym b) | idtac ]; [ case (Qle_lt_eq_dec _ _ Hab) | case (Qle_lt_eq_dec _ _ Hcd) ]; intro H_; [ apply Qlt_le_weak; apply Qle_lt_reg; [ unfold Qle in |- *; apply Qlt_irreflexive | assumption ] | rewrite H_; unfold Qle in |- *; apply Qlt_irreflexive | apply Qlt_le_weak; apply Qle_lt_reg; [ unfold Qle in |- *; apply Qlt_irreflexive | assumption ] | rewrite H_; unfold Qle in |- *; apply Qlt_irreflexive ]. Qed.
imports: Coq(FunInd) QArithSternBrocot(Field_Theory_Q)