intros P H p q x H0 H1 H2 H3 H4.
apply Zlt_powerRZ with (e := IZR radix); auto with real zarith.
apply Rle_lt_trans with (FtoRradix (Fabs q)).
replace (powerRZ radix (Fexp q)) with (FtoRradix (Float 1%nat (Fexp q)));
unfold FtoRradix in |- *;
[ apply Fop.RleFexpFabs; auto with arith
| unfold FtoR in |- *; simpl in |- *; ring ].
rewrite (Fabs_correct radix); auto with zarith.
fold FtoRradix in |- *; rewrite H3.
apply Rlt_le_trans with (Fulp p);
[ apply RoundedModeUlp with P; auto | unfold Fulp in |- * ].
apply Rle_powerRZ; auto with real zarith.
apply FcanonicLeastExp with radix b precision; auto with real arith.
apply sym_eq; apply FnormalizeCorrect; auto.
apply FnormalizeCanonic; auto with zarith.
Qed.

Theorem RoundedModeProjectorIdem :
forall P (p : float), RoundedModeP b radix P -> Fbounded b p -> P p p.