Towards Closed World Reasoning in Dynamic Open Worlds 29 C′ ⊇ C[ψ]**. The claim follows similarly as in the previous case:**

$\langle I,M,N\rangle\models\phi\Longleftrightarrow(\exists J\in N)\left(\langle J,M,N\rangle\not\equiv\psi\right)$  $$\Longleftrightarrow(\exists J\in N)\left(\left\langle J[\mathbb{P}^{\prime},\mathcal{C}^{\prime}],M[\mathbb{P}^{\prime},\mathcal{C}^{\prime}],N[\mathbb{P}^{\prime},\mathcal{C}^{\prime}]\right\rangle\not\equiv\psi\right)$$ $$\Longleftrightarrow\left(\exists J\in N[\mathbb{P}^{\prime},\mathcal{C}^{\prime}]\right)\left(\left\langle J,M[\mathbb{P}^{\prime},\mathcal{C}^{\prime}],N[\mathbb{P}^{\prime},\mathcal{C}^{\prime}]\right\rangle\not\equiv\psi\right)$$ $$\Longleftrightarrow\left\langle I[\mathbb{P}^{\prime},\mathcal{C}^{\prime}],M[\mathbb{P}^{\prime},\mathcal{C}^{\prime}],N[\mathbb{P}^{\prime},\mathcal{C}^{\prime}]\right\rangle\models\phi\.\qed$$

## Appendix C Properties Of Hybrid Knowledge Bases

Lemma C.1 Let Z **be a set of first-order theories. Then**
mod [Z
=\mod(Z)
where mod(Z) = { mod(S) **| S ∈ Z }**.

Proof The following sequence of equivalences proves the claim:

I ∈ mod [Z Lemma B.5 ⇐=======⇒ ∀φ ∈[Z (I |= φ) ⇐=======⇒ (∀S ∈ Z)(∀φ ∈ S)(I |= φ) Lemma B.5 ⇐=======⇒ (∀S ∈ Z)(I ∈ mod(S)) ⇐=======⇒ I ∈\mod(Z)
Definition C.2 (**Hybrid Immediate Consequence Operator** )
The immediate consequence operator associated with the definite P**-ground hybrid**
knowledge base K = hO,Pi is a mapping TK : M → M defined for any M ∈ M as TK(M) = mod(O ∪ { H∗(r) | r **∈ P ∧** M |= B(r) })
Lemma C.3 Let K = hO,Pi be definite P**-ground hybrid knowledge base. Then for every** M ∈
M **it holds that**
TK(M) = mod(O) ∩ mod({ H∗(r) | r **∈ P ∧** M |= B(r) })
Proof Let S = { H∗(r) | r ∈ P ∧ M |= B(r) }**. We need to show that**
mod(O ∪ S) = mod(O) ∩ mod(S) .

This follows from Lemma C.1.