intros p q H'1.
elim (LSB_rep_min radix) with (p := p); auto; intros z E.
elim (LSB_rep_min radix) with (p := q); auto; intros z0 E0.
replace (LSB radix (Fminus radix p q)) with
(LSB radix (Fminus radix (Float z (LSB radix p)) (Float z0 (LSB radix q)))).
replace (Zmin (LSB radix p) (LSB radix q)) with
(Fexp (Fminus radix (Float z (LSB radix p)) (Float z0 (LSB radix q))));
[ idtac | simpl in |- *; auto ].
apply Fexp_le_LSB; auto.
apply sym_equal; apply LSB_comp; auto.
repeat rewrite Fminus_correct; auto with arith.
unfold FtoRradix in E; unfold FtoRradix in E0; rewrite E; rewrite E0; auto.
Qed.

Theorem LSBPlus :
forall p q : float,
~ is_Fzero (Fplus radix p q) ->
(Zmin (LSB radix p) (LSB radix q) <= LSB radix (Fplus radix p q))%Z.