proposition: Lemma Rabs_triang_inv : forall a b:R, Rabs a - Rabs b <= Rabs (a - b).
proof: Proof. intros; apply (Rplus_le_reg_l (Rabs b) (Rabs a - Rabs b) (Rabs (a - b))); unfold Rminus; rewrite <- (Rplus_assoc (Rabs b) (Rabs a) (- Rabs b)); rewrite (Rplus_comm (Rabs b) (Rabs a)); rewrite (Rplus_assoc (Rabs a) (Rabs b) (- Rabs b)); rewrite (Rplus_opp_r (Rabs b)); rewrite (proj1 (Rplus_ne (Rabs a))); replace (Rabs a) with (Rabs (a + 0)). - rewrite <- (Rplus_opp_r b); rewrite <- (Rplus_assoc a b (- b)); rewrite (Rplus_comm a b); rewrite (Rplus_assoc b a (- b)). exact (Rabs_triang b (a + - b)). - rewrite (proj1 (Rplus_ne a)); trivial. Qed.
imports: Rdefinitions Raxioms RIneq R_Ifp