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### 7.4 More redundancies: subsumption

We say that a path sequent Π subsumes a path sequent Π′ if Π is a subsequent of Π′. Saturation-based theorem provers for first-order logic often use subsumption as a redundancy criterion: clauses subsumed by other clauses are removed from the search space.

In order to formalize subsumption as a redundancy criterion, we need a new notion of derivation, since subsumed clauses are not redundant per se, but only redundant with respect to other clauses in the search space. So to prove completeness of subsumption, we need to formalize the notion of search space. In this section we give such a formalization and prove that subsumption can be used to remove clauses from the search space. The formalization here ...

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