## Logic – Finite Modal Property via Selection-Question for Bisimulation

To let $$tau$$ be a modal similarity type containing only diamonds, and be $$phi$$ be a $$tau$$-Formula. If $$phi$$ is satisfactory, then it is satisfactory on a finite model.

I just prove it for the basic modal language, that is, we only have one diamond $$diamond$$

The proof is:

Fix a modal formula $$phi$$ With $$deg ( phi) = k$$, We limit our modal similarity type $$tau$$ and our collection of sentence letters to the modal operators and sentence letters that actually occur in Germany $$phi$$, To let $$M_1, w_1$$ be so $$M_1, w_1 vdash phi$$
, By Theorem 2.15, a tree-like model exists $$M_2$$ with root $$w_2$$ so that $$M_2, w_2 vdash phi$$
, To let $$M_3: = (M_2 upharpoonright k)$$, With Lemma 2.33 we have $$M_2, w_2$$ is $$k$$-Imilar to $$M_3, w_2$$and by Theorem 2.31 it follows $$M_3, w_2 vdash phi$$,

The construction of $$M_4$$ starts here.

By induction on $$n le k$$ We define endless sets of states $$S_0, …, S_k$$ and a (last) model $$M_4$$ with domain $$S_0 cup cdots cup S_k$$; the points in each $$S_n$$ will have height $$n$$, Define $$S_0$$ to be the singleton $${w_2 }$$, Next, accept this $$S_0, …, S_n$$ have already been defined. Fix an item $$v$$ from $$S_n$$, By Theorem 2.29, there are only infinitely many non-equivalent modal formulas whose degree is at most $$k$$, say $$1, …, m$$, For each of these formulas is this form $$diamond phi$$ and stops in $$M_3$$ at the $$v$$, Choose a state $$u$$ from $$M_3$$ so that $$Rvu$$ and $$M_3, u vdash phi$$, Add all these to us $$S_n + 1$$and repeat this selection process for each status in $$S_n$$, $$S_ {n + 1}$$ is defined as the set of all points selected in this way.

Finally define $$M_4$$ as follows. Your domain is $$S_0 cup cdots S_k$$; like everyone else $$S_i$$ is finite, $$M_4$$ is finally. The relationships and assessments are obtained by restricting the relationships and evaluations of M3 to the domain of M4. We can prove that $$M_4, w_2$$ is $$k$$-Imilar to $$M_3, w_2$$,

The last sentence is actually an exercise that is not easy enough to solve within a few hours. Could someone please tell me how I can prove it (use some relative phrases?) Or give a hint, please?

Thanks for any help.