(** The sum of two [log2_up] is less than twice the [log2_up] of the sum. The large inequality is obvious thanks to monotonicity. The strict one requires some more work. This is almost a convexity inequality for points [2a], [2b] and their middle [a+b] : ideally, we would have [2*log(a+b) >= log(2a)+log(2b) = 2+log a+log b]. Here, we cannot do better: consider for instance a=3 b=5, then 2+3<2*3 *)

Lemma add_log2_up_lt : forall a b, 0<a -> 0<b ->
log2_up a + log2_up b < 2 * log2_up (a+b).