Splitting and Updating Hybrid Knowledge Bases (Extended Version) 37 Proposition 59 for D = M, D′ = M′, E = X, E′ = X′, F = G = M, F
′ = G′ = M′
implies that hM′, Mi 6|= π(K) ⇐⇒ hX′, Xi 6|= π(bU (K)) ∨ hM′, Mi 6|= π(eU (K, M′, M)) .

The claim of this corollary follows directly from this equivalence.

Corollary 65 Let U be a splitting set for a hybrid knowledge base K and M, M′**, X, Y, Y** ′ be MKNF
interpretations such that M′[U] = X[U] = M[U], Y
[P\U] = M[P\U]and Y
′[P\U] =
M′[P\U]. Then:
hY
′, Y i 6|= π(eU (K, X)) =⇒ hM′, M**i 6|**= π(K) .

Proof Proposition 59 for D = M, D′ = M′, E = E′ = M, F = Y , F
′ = Y
′, G = G′ = X
implies that hM′, Mi 6|= π(K) ⇐⇒ hM, Mi 6|= π(bU (K)) ∨ hY
′, Y i 6|= π(eU (K**, X, X**)) .

The claim of this corollary follows directly from this equivalence.

Proposition 66 Let U be a splitting set for a hybrid knowledge base K, M be an MKNF model of K and X = σ(**M, U**). Then X is an MKNF model of bU (K).

Proof By Proposition 34 we know that M ⊆ X and that X is saturated relative to U. We need to show that X is an MKNF model of bU (K). By Proposition 51, this holds if and only if X is an S5 model of π(bU (K)) and for every X′ ) X it holds that hX′, X**i 6|**= π(bU (K)).

The former follows directly from Corollary 61, so we will focus on the latter condition.

Let's pick some X′ ) X. By Proposition 44 there exists the greatest MKNF interpretation M′that coincides with X′ on U (i.e. M′[U] = X′[U]) and with M on P \ U (i.e.

M′[P\U] = M[P\U]) and which contains X′ ∩ M. Hence, M ⊆ X ∩ M ⊆ X′ ∩ M ⊆ M′. (B4)
Furthermore, we know that X is saturated relative to U, so we can use Proposition 30(2)
to conclude that M[U] = X
[U] ( X
′[U] = M′[U]. (B5)
Consequently, by (B4), (B5) and Proposition 27(3), we obtain M ( M′. This, together with the assumption that M is an MKNF model of K, implies that hM′, M**i 6|**= π(K).

We can now apply Corollary 62 to conclude that hX′, X**i 6|**= π(bU (K)), which is also the desired conclusion.

Proposition 67 Let U be a splitting set for a hybrid knowledge base K, M be an MKNF model of K, X
be the greatest MKNF interpretation that coincides with M on U and Y be the greatest MKNF interpretation that coincides with M on P \ U. Then Y is an MKNF model of eU (K, X).

Proof