Equivalent characterization of ordinary $F$-crystals

Let $k$ be a perfect field in characteristic $p$, let $W$ be its ring of Witt vectors. Let $A=W[[t_1,cdots,t_n]]$, let $A_0=A/pA$. Let $H$ be an $F$-crystal over $A_0$.

(An $F$-crystal over $A_0$ is a finite rank free $A$ module equipped with topologically nilpotent integrable connection $(H,nabla)$, such that for any endomorphism $varphicolon Ato A$ lifting the absolute Frobenius on $A_0$, a horizontal homomorphism $F(varphi)colon varphi^*Hto H$ satisfying certain cocycle conditions.)

In "Cristaux Ordinaries et Coordonnees Canoniques", [http://publications.ias.edu/sites/default/files/Number37.pdf], Page 93, Proposition 1.3.2, several equivalent characterization of “ordinary $F$-crystals” are given:

(i) The Newton and Hodge polygon of the $F$-crystal at the closed point $e_0^*H$ coincide, where $e_0colon A_0to k, t_imapsto0$.

(ii)The Hodge filtration and conjugate filration on $H_0:=H/pH$ are opposed, namely $H_0=mathrm{Fil}_0H_0oplus mathrm{Fil}^{1}H_0$, where $$mathrm{Fil}_iH_0=mathrm{Im}(Fcolon H_0^{(p)}to H_0)$$ $$mathrm{Fil}^{i}H_0=mathrm{Ker}(Fcolon H_0to H_0)$$

I am a bit confused about the proof (i) implies (ii):

(a) The goal is to prove the $H_0$ is a sum of $mathrm{Fil}_0H_0$ and $mathrm{Fil}^1H_0$. In the proof, the fibers $s_0^*(mathrm{Fil}_0H_0)$ and $s_0^*(mathrm{Fil}^1H_0)$ at the geometric generic points are shown to have zero intersection, therefore $mathrm{Fil}_0H_0$ has zero intersection with $mathrm{Fil}^1H_0$. But how can we show they span $H_0$?

(b) I tried to think in another way: we always have the morphism of direct sum $mathrm{Fil}_0H_0oplusmathrm{Fil}^1H_0to H_0$. The goal is to show $mathrm{Ker}$ and $mathrm{Coker}$ are zero. These are finitely generated $A_0$ modules, to check they are zero, by Nakayama lemma, it suffices to check on residue field of $A_0$. Does it then follow from the Rank-Nullity formula and locally freeness? I am not sure where I understood this thing wrongly?