Splitting and Updating Hybrid Knowledge Bases (Extended Version) 35 3. G[U] = D[U]and G′[U] = D′[U].

Then:
hE
′, Ei |= π(bU (K)) ∧ hF
′, Fi |= π(eU (K, G′, G)) =⇒ hD′, Di |= π(K) .

Proof Take some φ ∈ π(K). We consider three cases depending on which part of K this formula originates from:

a) If φ belongs to π(bU (K)), then pr(φ) ⊆ U, so we can use Corollary 53 to infer hD′, Di |= φ from hE
′, Ei |= φ.

b) If φ belongs to π(tU (O)), then φ also belongs to π(eU (K, G′, G)) and pr(φ) ⊆
P \ U, so by Corollary 53 we can infer hD′, Di |= φ from hF
′, Fi |= φ.

c) If φ belongs to π(tU (P)), then φ = π(r) for some rule r. If hD′, D**i 6|**= π(B(r)),
then hM′, Mi |= π(r) and we are finished. On the other hand, if hD′, Di |=
π(B(r)), then let B = { L ∈ B(r) | pr(L) ⊆ U }. We have hD′, Di |= π(B) and by Corollary 53 we can conclude that hG′, Gi |= π(B). Consequently, eU (P, G′, G)
contains a rule r
′such that H(r
′) = H(r) and B(r
′) = B(r) \ B. We know that hF
′, Fi |= π(r
′) and since pr(r
′) ⊆ P \ U, we can use Corollary 53 to infer hD′, Di |= π(r
′). Furthermore, hD′, Di |= π(B(r
′)), so hD′, Di |= π(H(r
′)).

Consequently, hD′, Di |= π(r), which is the desired result.
Proposition 59 Let U be a splitting set for a hybrid knowledge base K and D, D′, E, E′, F, F′, G, G′ ∈ M
be such that the following conditions are satisfied:
1. E[U] = D[U]and E′[U] = D′[U];
2. F
[P\U] = D[P\U]and F
′[P\U] = D′[P\U];
3. G[U] = D[U]and G′[U] = D′[U].

Then:
hD′, Di |= π(K) **⇐⇒ h**E
′, Ei |= π(bU (K)) ∧ hF
′, Fi |= π(eU (K, G′, G)) .

Corollary 60 Let U be a splitting set for a hybrid knowledge base K and M, X ∈ M be such that X[U] = M[U]. Then:
M |= π(K) ⇐⇒ M |= π(bU (K)) ∧ M |= π(eU (K, X)) .

Proof Proposition 59 for D = D′ = E = E′ = F = F
′ = M and G = G′ = X implies that hM, Mi |= π(K) ⇐⇒ hM, Mi |= π(bU (K)) ∧ hM, Mi |= π(eU (K**, X, X**)) .

The claim of this corollary now follows from Proposition 51. Proof Follows by Lemmas 57 and 58.