eapply fun_le_bounded_func_op_simpl ; eauto.
}
move: F_spec => /(_ _ H) []i [] ineq fi.
move : spec_F1 => /(_ _ tree1) [] i1 [] ineq1 fi1.
move : spec_F2 => /(_ _ tree2) [] i2 [] ineq2 fi2.
exists (S(i1 + (i2 * N1) + (i * N1 * N2))); split.
{ apply lt_n_S.
replace (N * N1 * N2) with
(N1 + N1 * (N2 * N -1)).
- eapply (NPeano.Nat.lt_le_trans).
+ instantiate (1:= (N1 + i2 * N1 + i * N1 * N2)).
apply /ltP.
rewrite ltn_add2r;
rewrite ltn_add2r;
apply /ltP; auto.
+ apply /leP.
rewrite -addnA.
rewrite leq_add2l.
apply /leP.
replace (i * N1 * N2) with
((i * N2) * N1).
*
rewrite -mulnDl.
rewrite mulnC.
apply /leP.
rewrite leq_pmul2l; try (apply /ltP; omega).
apply /leP.
eapply lt_n_Sm_le.
rewrite - addn1.
rewrite subnK.
2:
rewrite muln_gt0;
apply /andP; split;
try (apply /ltP; omega).
eapply (NPeano.Nat.lt_le_trans).
-- instantiate (1:= N2 + i * N2).
apply /ltP.
rewrite ltn_add2r.
apply /leP; auto.
-- replace (N2 + i * N2) with
(N2 * (1 + i)).
apply /leP.
rewrite leq_pmul2l.
rewrite add1n.
apply /ltP; auto.
apply /ltP; omega.
rewrite mulnDr.
rewrite mulnC.
f_equal.
compute; auto.
rewrite mulnC; auto.
* do 2 rewrite - mulnA.
f_equal. rewrite mulnC; auto.
- replace (N1 + N1 * (N2 * N - 1))
with (N1 * 1 + N1 * (N2 * N - 1)).
+ rewrite -mulnDr.
rewrite addnC.
rewrite subnK.
2:
rewrite muln_gt0;
apply /andP; split;
try (apply /ltP; omega).
rewrite -mulnA.
rewrite mulnA.
rewrite mulnC; auto.
+ f_equal.
rewrite mulnC.
compute; auto. }
-- simpl; f_equal.
++ rewrite - fi1.
f_equal.
rewrite -addn1.
rewrite -addnBA. 2: ssromega.
replace (i1 + i2 * N1 + i * N1 * N2 + (1 - 1)) with
(i1 + i2 * N1 + i * N1 * N2) by ssromega.
replace (i1 + i2 * N1 + i * N1 * N2) with
(i1 + (i2  + i *  N2) * N1).
2:
rewrite mulnDl addnA; f_equal;
do 2 rewrite -mulnA; f_equal; rewrite mulnC; auto.
rewrite NPeano.Nat.mod_add.
apply NPeano.Nat.mod_small; auto.
destruct N1; omega.
++ rewrite - fi.
f_equal.
rewrite -addn1.
rewrite -addnBA. 2: ssromega.
replace (i1 + i2 * N1 + i * N1 * N2 + (1 - 1)) with
(i1 + i2 * N1 + i * N1 * N2) by ssromega.
assert (i1 + i2 * N1 + i * N1 * N2 =
((N1 * N2) * i) + (i1 + i2 * N1)).
{ rewrite addnC. f_equal.
rewrite - mulnA mulnC; auto. }
eapply NPeano.Nat.div_unique in H0; auto.
eapply (NPeano.Nat.lt_le_trans).
** instantiate (1:= N1 + i2 * N1).
apply /ltP; rewrite ltn_add2r.
apply /ltP; auto.
** replace (N1 + i2 * N1) with ( (1 + i2) * N1).
rewrite add1n.
rewrite mulnC.
apply /leP; rewrite leq_pmul2l.
apply /ltP; auto.
destruct N1; ssromega.
rewrite mulnDl; f_equal.
ssromega.
++ rewrite - fi2.
f_equal.
rewrite -addn1.
rewrite -addnBA. 2: ssromega.
replace (i1 + i2 * N1 + i * N1 * N2 + (1 - 1)) with
(i1 + i2 * N1 + i * N1 * N2) by ssromega.
assert (i1 + i2 * N1 + i * N1 * N2 =
(N1 * (i2 + i * N2)) + i1).
{ rewrite -addnA.
replace (i * N1 * N2) with
(i  * N2 * N1).
rewrite - mulnDl.
rewrite mulnC addnC; auto.
do 2 rewrite -mulnA; f_equal.
rewrite mulnC. auto. }
eapply NPeano.Nat.div_unique in H0; auto.
rewrite - H0.
rewrite NPeano.Nat.mod_add.
apply NPeano.Nat.mod_small; auto.
destruct N2; omega.
+ exists (S( N1 * N2)).
exists (fun n => if n == 0
then PTree.Leaf
else
PTree.Node
(F1 ( (n-1) mod N1))
(None )
(F2 (((n-1) / N1 ) mod N2))).
intros x spec.
destruct x.
* exists 0%nat; split; auto.
omega.
* move: spec .
rewrite /sub_map /= => [] [] FUN_lq [] tree1 tree2.
move : spec_F1 => /(_ _ tree1) [] i1 [] ineq1 fi1.
move : spec_F2 => /(_ _ tree2) [] i2 [] ineq2 fi2.
exists (S(i1 + (i2 * N1))); split.
{ apply lt_n_S.
replace (N1 * N2) with
(N1 + N1 * (N2 -1)).
- eapply (NPeano.Nat.lt_le_trans).
+ instantiate (1:= (N1 + i2 * N1)).
apply /ltP.
rewrite ltn_add2r;
apply /ltP; auto.
+ apply /leP.
rewrite leq_add2l.
apply /leP.
rewrite mulnC.
apply /leP.
rewrite leq_pmul2l; try (apply /ltP; omega).
apply /leP.
eapply lt_n_Sm_le.
rewrite - addn1.
rewrite subnK; auto.
destruct N2; ssromega.
- replace (N1 + N1 * (N2 - 1))
with (N1 * 1 + N1 * (N2 - 1)).
+ rewrite -mulnDr.
rewrite addnC.
rewrite subnK.
2: ssromega.
rewrite mulnC; auto.
+ f_equal.
rewrite mulnC.
compute; auto. }
-- simpl; f_equal.
++ rewrite - fi1.
f_equal.
rewrite -addn1.
rewrite -addnBA. 2: ssromega.
replace (i1 + i2 * N1 + (1 - 1)) with
(i1 + i2 * N1) by ssromega.
rewrite NPeano.Nat.mod_add.
apply NPeano.Nat.mod_small; auto.
destruct N1; omega.
++ destruct o; auto; inversion FUN_lq.
++ rewrite - fi2.
f_equal.
rewrite -addn1.
rewrite -addnBA. 2: ssromega.
replace (i1 + i2 * N1 + (1 - 1)) with
(i1 + i2 * N1 ) by ssromega.
assert (i1 + i2 * N1 =
(N1 * (i2) + i1)).
{ rewrite mulnC addnC; auto. }
eapply NPeano.Nat.div_unique in H0; auto.
rewrite - H0.
apply NPeano.Nat.mod_small; auto.
Qed.

Lemma fun_leq_trans: forall {A B C} f1 f2 f3, @fun_leq A B f1 f2 -> @fun_leq B C f2 f3 ->
@fun_leq A C f1 f3.