# A proof theory for description logics by Alexandre Rademaker

By Alexandre Rademaker

Advent -- heritage -- The Sequent Calculus for ALC -- evaluating SC ALC SC with different ALC Deduction platforms -- A usual Deduction for ALC -- in the direction of an evidence conception for ALCQI -- Proofs and factors -- A Prototype Theorem Prover -- end

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The method used in this section is a basis for constructing a proof of SCALCQI completeness. 2. Since the results of Sect. 3 are obtained from a SCALC without cut-rule, we are actually proving the completeness of SCALC without the cut-rule. Given that, the results can also be considered an alternative method of cut-elimination for A. 1007/978-1-4471-4002-3_4, © The Author(s) 2012 37 4 Comparing SCALC with Other ALC Deduction Systems 38 the SCALC presented in Sect. 4, where we followed Gentzen’s original proof for cut elimination.

Moreover, weak rules will be used with the unique purpose of enabling promotion rules applications. 44 4 Comparing SCALC with Other ALC Deduction Systems A more insightful definition of the last item above would be possible if we replace the weak rules in SC[] ALC by the weak ∗ rule below. [Δ ], [Δ, Δ1 ]k , Δ ⇒ Γ, [Γ1 , Γ ]k , [Γ ] [Δ ], Δ, Δ1 ⇒ Γ1 , Γ, [Γ ] weak ∗ Lemma 6 The weak ∗ rule is a derived rule in SC[] ALC . Proof To prove Lemma 6, given a derivation fragment Π with a weak ∗ rule application, we show how to replace it by successive weak-l and weak-r applications.

If SC∗T ALC Δ ⇒ Γ then there is a free-quasi-mix free SC∗T ALC -proof of Δ ⇒ Γ . Theorem 2 is a consequence of the following lemma. Lemma 4 If P is a proof of S (in SC∗T ALC ) which contains only one free-quasi-mix, occurring as the last inference, then S is provable without any free-quasi-mix. Theorem 2 is obtained from Lemma 4 by simple induction over the number of quasi-free-mix occurring in a proof P. We can now concentrate our attention on Lemma 4. First we define three scalars as a measure of the complexity of the proof.