We present a framework for processing formulas in automatic theorem provers, with generation of detailed proofs. The main components are a generic contextual recursion algorithm and an extensible set of inference rules. Clausification, skolemization, theory-specific simplifications, and expansion of 'let' expressions are instances of this framework. With suitable data structures, proof generation adds only a linear-time overhead, and proofs can be checked in linear time. We implemented the approach in the SMT solver veriT. This allowed us to dramatically simplify the code base while increasing the number of problems for which detailed proofs can be produced, which is important for independent checking and reconstruction in proof assistants.

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The success of superposition-based theorem proving in first-order logic relies in particular on the fact that the superposition calculus can be turned into a decision procedure for various decidable fragments of first-order logic and has been successfully used to identify new decidable classes. In this paper, we extend this story to the hierarchic combination of linear arithmetic and first-order superposition. We show that decidability of reachability in timed automata can be obtained by instantiation of an abstract termination result for SUP(LA), the hierarchic combination of linear arithmetic and first-order superposition

We define a superposition calculus with explicit splitting on the basis of labelled clauses. For the first time we show a superposition calculus with an explicit non-chronological backtracking rule sound and complete. The new backtracking rule advances backtracking with branch condensing known from Spass. An experimental evaluation of an implementation of the new rule shows that it improves considerably on the previous Spass splitting implementation. Finally, we discuss the relationship between labelled first-order splitting and DPLL style splitting with intelligent backtracking and clause learning.