End Double_round_sqrt_beta_ge_4_FTZ.
End Double_round_sqrt_beta_ge_4.
End Double_round_sqrt.
Section Double_round_div.

Lemma double_round_eq_mid_beta_even :
forall (fexp1 fexp2 : Z -> Z),
Valid_exp fexp1 -> Valid_exp fexp2 ->
forall (choice1 choice2 : Z -> bool),
(exists n, (beta = 2 * n :> Z)%Z) ->
forall x,
0 < x ->
(fexp2 (ln_beta x) <= fexp1 (ln_beta x) - 1)%Z ->
(fexp1 (ln_beta x) <= ln_beta x)%Z ->
x = midp fexp1 x ->
double_round_eq fexp1 fexp2 choice1 choice2 x.