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Solving for Set Variables in HigherOrder Theorem Proving
 Proceedings of the 18th International Conference on Automated Deduction
, 2002
"... In higherorder logic, we must consider literals with exible (set variable) heads. Set variables may be instantiated with logical formulas of arbitrary complexity. An alternative to guessing the logical structures of instantiations for set variables is to solve for sets satisfying constraints. U ..."
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In higherorder logic, we must consider literals with exible (set variable) heads. Set variables may be instantiated with logical formulas of arbitrary complexity. An alternative to guessing the logical structures of instantiations for set variables is to solve for sets satisfying constraints. Using the KnasterTarski Fixed Point Theorem [ 15 ] , constraints whose solutions require recursive de nitions can be solved as xed points of monotone set functions. In this paper, we consider an approach to higherorder theorem proving which intertwines conventional theorem proving in the form of mating search with generating and solving set constraints.
Sequent Style Proof Terms for HOL
"... Abstract. In this work we present proof terms for a Gentzen sequent style presentation of HOL. Existing implementations of proof terms for HOL are natural deduction style systems. Sequent style proof terms have many advantages over natural deduction style proof terms. For example, we can translate p ..."
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Abstract. In this work we present proof terms for a Gentzen sequent style presentation of HOL. Existing implementations of proof terms for HOL are natural deduction style systems. Sequent style proof terms have many advantages over natural deduction style proof terms. For example, we can translate proof terms directly into tactics, which we can execute at the tactic level of HOL implementations. We describe several applications of our work, such as an implementation of theory interpretation, and an approach to optimising proof terms by rewriting. 1
Use and Meaning of Open Terms in Interactive Formal Problem Solving
"... By formal problem solving, we mean problem solving where the problem and the possible solutions are stated in a precise way, and so is the notion of correctness, stating whether a solution solves a problem. Formal problem solving occurs in many places in computer science, like a program editor (wher ..."
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By formal problem solving, we mean problem solving where the problem and the possible solutions are stated in a precise way, and so is the notion of correctness, stating whether a solution solves a problem. Formal problem solving occurs in many places in computer science, like a program editor (where a program has to meet a specification), a structure editor (where syntactically correct expressions have to be built), verification (of protocols), testing, theorem proving and data bases (where information is extracted using queries). An important aspect of computer systems for problem solving is that they are interactive. The machine will in general not be able to solve our problems by itself; solving problems requires humanmachine interaction. Ideally, one would let the computer construct a part of the solution, returning a `solutionwithholes', and then the user provides information on how to proceed to refine the holes. Such a hole in a `solutionwithholes' is also known as an ope...
Automatic Deduction for Theories of Algebraic Data Types
, 2011
"... In this thesis we present formal logical systems, concerned with reasoning about algebraic data types. The first formal system is based on the quantifierfree calculus (outermost universally quantified). This calculus is comprised of state change rules, and computations are performed by successive a ..."
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In this thesis we present formal logical systems, concerned with reasoning about algebraic data types. The first formal system is based on the quantifierfree calculus (outermost universally quantified). This calculus is comprised of state change rules, and computations are performed by successive applications of these rules. Thereby, our calculus gives rise to an abstract decision procedure. This decision procedure determines if a given formula involving algebraic type members is valid. It is shown that this calculus is sound and complete. We also examine how this system performs practically and give experimental results. Our main contribution, as compared to previous work on this subject, is a new and more efficient decision procedure for checking satisfiability of the universal fragment within the theory of algebraic data types. The second formal system, called Term Builder, is the deductive system based on higher order type theory, which subsumes second order and higher order logics. The main purpose of this calculus is to formulate and prove theorems about algebraic or other arbitrary userdefined types. Term Builder supports proof objects and is