(* norm + add *)

(* Specifications: *)
Lemma normr_real x : `|x| \is real. Proof. by rewrite ger0_real. Qed.
Hint Resolve normr_real : core.
Lemma ler_norm_sum I r (G : I -> R) (P : pred I):
`|\sum_(i <- r | P i) G i| <= \sum_(i <- r | P i) `|G i|.
Lemma ler_norm_sub x y : `|x - y| <= `|x| + `|y|.
Lemma ler_sub_dist x y : `|x| - `|y| <= `|x - y|.
Lemma ler_dist_norm_add x y : `| `|x| - `|y| | <= `| x + y |.
Lemma real_ler_normlP x y :
x \is real -> reflect ((-x <= y) * (x <= y)) (`|x| <= y).