The next theorem states that the JFP semantics is a special case of the social semantics. JFP(P1, · · · ,Pn**) denotes the set of the joint fixpoints of the collection** of COLP programs P1, **· · ·** ,Pn.

Theorem 3 Let P1, · · · ,Pn (n ≥ 1) be COLP programs and C = {Q1, · · · , Qn} **be a collection**
of SOLP programs such that Qi = σ n(Pi**). Then:**

$${\mathcal{S O S}}(Q_{1},\cdots,Q_{n})=\left\{\bigcup_{1\leq i\leq n}(F)_{Q_{i}}\mid F\in J F P({\mathcal{P}}_{1},\cdots,{\mathcal{P}}_{n})\right\}.$$

Proof (⊇**). First, we show that**

First, we show that  $$\forall F\in J F P(\mathcal{P}_{1},\cdots,\mathcal{P}_{n}),\quad\bigcup_{1\leq i\leq n}(F)_{Q_{i}}\in\mathcal{S O S}(Q_{1},\cdots,Q_{n}).$$
By contradiction, let us assume that ∃M¯ =S1≤i≤n
(F)Qi| M¯ 6∈ SOS(Q1, **· · ·** , Qn).

Thus, there exists an integer j, 1 ≤ j ≤ n**, such that either:**
(1) F 6∈ AFP(Qj**), or**
(2) F ∈ AFP(Qj) ∧ STC (M¯ ) 6= M¯ .

In case (1), by virtue of Lemma 6 (page 24), F 6∈ FP(Pˆj**) which contradicts the**
hypothesis that F ∈ JFP(P1, **· · ·** ,Pn).

In case (2), it holds that either:
(a) ∃h, Qj| Qj ∈ C ∧ hQj ∈ M¯ ∧ hQj 6∈ STC (M¯ **), or** (b) ∃h, Qj| Qj ∈ C ∧ hQj 6∈ M¯ ∧ hQj ∈ STC (M¯ ).

If condition (a) occurs, then, for each rule r ∈ Qj, s.t. head(r ) = h or head(r ) =
okay(h), it results that body(r ) is false w.r.t. M¯ , because hQj 6∈ STC (M¯ **(recall**
Definition 9, page 12).

Now, since h ∈ F and F ∈ AFP(Qj**), according to Definition 22 (page 24), it**
holds that for each rule r ∈ Qj, s.t. head(r ) = h or head(r ) = okay(h**), the SC**
[n − 1, n − 1]h (introduced by the transformation σ) is false for Qj **w.r.t.** M¯ .

Thus, according to Definition 8 (page 11), there exists k (1 ≤ k 6= j ≤ n)
s.t. hQk 6∈ M¯ . Now, we have obtained that hQj ∈ M¯ and hQk 6∈ M¯ **. Since** M¯ = S1≤i≤n
(F)Qi and F ∈ JFP(P1, · · · ,Pn**), by virtue of Lemma 7 (page 24), we have**
reached a contradiction. This concludes the proof in case (2), when condition (a) holds.

Consider, now, that condition (b**) is true in case (2). According to Definition 9**
(page 12), there exists r ∈ Qj s.t. head(r ) = h or head(r ) = okay(h) and **body**(r )
is true w.r.t. M¯ **. As a consequence and according to Definition 22 (page 24), [**n −
1, n − 1]h is true for Qj w.r.t. M¯ **. Now, according to Definition 8 (page 11), for**
each k (1 ≤ k 6= j ≤ n), hQk ∈ M¯ and hQj 6∈ M¯ **. Since** M¯ =S1≤i≤n
(F)Qi and F ∈ JFP(P1, · · · ,Pn**), by virtue of Lemma 7 (page 24), we have reached a**
contradiction. This concludes the proof in case (2).