Superposition with Datatypes and Codatatypes

International audienceThe absence of a finite axiomatization of the first-order theory of data-types and codatatypes represents a challenge for automatic theorem provers. We propose two approaches to reason by saturation in this theory: one is a conservative theory extension with a finite number of axioms; the other is an extension of the superposition calculus, in conjunction with axioms. Both techniques are refutationally complete with respect to nonstandard models of datatypes and non-branching codatatypes. They take into account the acyclicity of datatype values and the existence and uniqueness of cyclic codatatype values. We implemented them in the first-order prover Vampire and compare them experimentally

Superposition with Datatypes and Codatatypes

The absence of a finite axiomatization of the first-order theory of datatypes and codatatypes represents a challenge for automatic theorem provers. We propose two approaches to reason by saturation in this theory: one is a conservative theory extension with a finite number of axioms; the other is an extension of the superposition calculus, in conjunction with axioms. Both techniques are refutationally complete with respect to nonstandard models of datatypes and nonbranching codatatypes. They take into account the acyclicity of datatype values and the existence and uniqueness of cyclic codatatype values. We implemented them in the first-order prover Vampire and compare them experimentally

A Decision Procedure for (Co)datatypes in SMT Solvers

International audienceWe present a decision procedure that combines reasoning about datatypes and codatatypes. The dual of the acyclicity rule for datatypes is a uniqueness rule that identifies observationally equal codatatype values, including cyclic values. The procedure decides universal problems and is composable via the Nelson–Oppen method. It has been implemented in CVC4, a state-of-the-art SMT solver. An evaluation based on problems generated from theories developed with Isabelle demonstrates the potential of the procedure

Superposition with Structural Induction

International audienceSuperposition-based provers have been successfully used to discharge proof obligations stemming from proof assistants. However, many such obligations require induction to be proved. We present a new extension of typed superposition that can perform structural induction. Several inductive goals can be attempted within a single saturation loop, by leveraging AVATAR [1]. Lemmas obtained by generalization or theory exploration can be introduced during search, used, and proved, all in the same search space. We describe an implementation and present some promising results

Induction with Generalization in Superposition Reasoning

We describe an extension of automating induction in superposition-based reasoning by strengthening inductive properties and generalizing terms over which induction should be applied. We implemented our approach in the first-order theorem prover Vampire and evaluated our work against state-of-the-art reasoners automating induction. We demonstrate the strength of our technique by showing that many interesting mathematical properties of natural numbers and lists can be proved automatically using this extension.European Research Council (ERC)European CommissionAustrian Science Fund (FWF)Vereine, Stiftungen, Preise12313715Engineering and Physical Sciences Research Council (EPSRC)KAW Foundatio