Definition sqrt_le_mono : forall a b, a<=b -> √a <= √b
 := sqrt_le_mono.Definition sqrt_lt_cancel : forall a b, √a < √b -> a < b
 := sqrt_lt_cancel.

(* Specifications: *)
Lemma sqrt_le_square : forall a b, b*b<=a <-> b <= √a.
Lemma sqrt_lt_square : forall a b, a<b*b <-> √a < b.