On the strength of dependent products in the type theory of Martin-L\"of

One may formulate the dependent product types of Martin-L\"of type theory either in terms of abstraction and application operators like those for the lambda-calculus; or in terms of introduction and elimination rules like those for the other constructors of type theory. It is known that the latter rules are at least as strong as the former: we show that they are in fact strictly stronger. We also show, in the presence of the identity types, that the elimination rule for dependent products--which is a "higher-order" inference rule in the sense of Schroeder-Heister--can be reformulated in a first-order manner. Finally, we consider the principle of function extensionality in type theory, which asserts that two elements of a dependent product type which are pointwise propositionally equal, are themselves propositionally equal. We demonstrate that the usual formulation of this principle fails to verify a number of very natural propositional equalities; and suggest an alternative formulation which rectifies this deficiency.Comment: 18 pages; v2: final journal versio

Cut-Simulation and Impredicativity

We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -- in our case a sequent calculus for classical type theory -- is like adding cut. The phenomenon equally applies to prominent axioms like Boolean- and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.Comment: 21 page

Encoding TLA+ set theory into many-sorted first-order logic

We present an encoding of Zermelo-Fraenkel set theory into many-sorted first-order logic, the input language of state-of-the-art SMT solvers. This translation is the main component of a back-end prover based on SMT solvers in the TLA+ Proof System

New results on rewrite-based satisfiability procedures

Program analysis and verification require decision procedures to reason on theories of data structures. Many problems can be reduced to the satisfiability of sets of ground literals in theory T. If a sound and complete inference system for first-order logic is guaranteed to terminate on T-satisfiability problems, any theorem-proving strategy with that system and a fair search plan is a T-satisfiability procedure. We prove termination of a rewrite-based first-order engine on the theories of records, integer offsets, integer offsets modulo and lists. We give a modularity theorem stating sufficient conditions for termination on a combinations of theories, given termination on each. The above theories, as well as others, satisfy these conditions. We introduce several sets of benchmarks on these theories and their combinations, including both parametric synthetic benchmarks to test scalability, and real-world problems to test performances on huge sets of literals. We compare the rewrite-based theorem prover E with the validity checkers CVC and CVC Lite. Contrary to the folklore that a general-purpose prover cannot compete with reasoners with built-in theories, the experiments are overall favorable to the theorem prover, showing that not only the rewriting approach is elegant and conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page

Reverse Mathematics and parameter-free Transfer

Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi