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By Alessandrini L., Bassanelli G.

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By construction, we also know that A ⊆ Q as well as Q ⊆ δ. However, since δ is minimal with respect to set inclusion by assumption, we can conclude Q = δ. ❑ Having established this correspondence between the standard model and realization trees, it is not difficult to prove that any cumulated clause valid in the standard model is DR-derivable, which (as the subsequent corollary shows) gives us the completeness of DR. 20 Let K be a set of cumulated clauses and let k be a cumulated clause. Then, − → K (K) |= k =⇒ K ⊢ k.

If for δ1 , δ2 ∈ ∆ we have (δ1 , δ2 )IR R for some R ∈ MR , we call δ2 the R-neighbor of δ1 . g. [Wi97]). 3 2 In DL terminology, it is usual to denote concept descriptions as well as role names by capital letters. We decided to abide by this convention, but – in order to avoid possible confusion (with other capital letters denoting sets) – we use typewriter font (A, B, C) for FLE concept descriptions and calligraphic letters (A, B, C) for sets of concept descriptions. Furthermore, we use the symbols and for “role quantification” to clearly distinguish them from the “ordinary” quantifiers ∃ and ∀ occurring in some proofs and definitions.

Bn } cumulated premise extension (PE*) A ∪ C ⊸ {B1 , . . , Bn } A ⊸ {B1 , . . , Bn , C} C ⊸ {D1 , . . , Dk } pure substitution (SUB*) A ⊸ {B1 , . . , Bn , D1 , . . , Dk } A ⊸ {B1 , . . , Bn } restriction (RES) A ∪ C ⊸ {B1 ∪ C, . . , Bn ∪ C} A ⊸ {B1 , . . , Bn } C ⊸ {D1 , . . , Dm } distribution (D) A ∪ C ⊸ {Bi ∪ Dj | 1 ≤ i ≤ n, 1 ≤ j ≤ m} cumulated contradiction (CONT*) E E A [ R]A ∪ [ R]B ⊸ {[ R](A ∪ B)} cumulated A ⊥ ⊸ {A} -propagation (AP*) Proof: The validity of these rules is proven by providing the DR proof trees that realize them.

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A balanced proper modification of P 3 by Alessandrini L., Bassanelli G.

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