The Rooster and the Syntactic Bracket

We propose an extension of pure type systems with an algebraic presentation of inductive and co-inductive type families with proper indices. This type theory supports coercions toward from smaller sorts to bigger sorts via explicit type construction, as well as impredicative sorts. Type families in impredicative sorts are constructed with a bracketing operation. The necessary restrictions of pattern-matching from impredicative sorts to types are confined to the bracketing construct. This type theory gives an alternative presentation to the calculus of inductive constructions on which the Coq proof assistant is an implementation.Comment: To appear in the proceedings of the 19th International Conference on Types for Proofs and Program

Beating the Productivity Checker Using Embedded Languages

Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures that programs are productive, i.e. that every finite prefix of an infinite value can be computed in finite time. However, many productive programs are not guarded, and it can be nontrivial to put them in guarded form. This paper gives a method for turning a productive program into a guarded program. The method amounts to defining a problem-specific language as a data type, writing the program in the problem-specific language, and writing a guarded interpreter for this language.Comment: In Proceedings PAR 2010, arXiv:1012.455

A proof repository for formal verification of software

We present a proof repository that provides a uniform theorem proving interface to virtually any first-order theorem prover. Instead of taking the greatest common divisor of features supported by the first-order theorem provers, the design allows us to support any extension of the logic that can be expressed in first-order logic. If a theorem prover has native support for such a logic, this is exploited. If the prover has no such support, the repository automatically uses the first-order encoding of the extension. A built-in proof assistant is provided that allows the user to manually guide the proving process when all provers fail to prove a theorem. To prove sub-theorems, the proof assistant is able to use the repository’s full capabilities. The repository also maintains a database of proven theorems. When a requested theorem has been proved before, the result from the database is re-used instead of reconstructing the proof all over again. To test the repository, we constructed a tool for static verification of a basic programming language. This language is also described in this paper

A proof repository for formal verification of software

We present a proof repository that provides a uniform theorem proving interface to virtually any first-order theorem prover. Instead of taking the greatest common divisor of features supported by the first-order theorem provers, the design allows us to support any extension of the logic that can be expressed in first-order logic. If a theorem prover has native support for such a logic, this is exploited. If the prover has no such support, the repository automatically uses the first-order encoding of the extension. A built-in proof assistant is provided that allows the user to manually guide the proving process when all provers fail to prove a theorem. To prove sub-theorems, the proof assistant is able to use the repository’s full capabilities. The repository also maintains a database of proven theorems. When a requested theorem has been proved before, the result from the database is re-used instead of reconstructing the proof all over again. To test the repository, we constructed a tool for static verification of a basic programming language. This language is also described in this paper

A relaxation of Coq's guard condition

National audienceIn a convenient language to handle dependent algebraic data types, this article describes how commutative cuts are used to tackle dependency in pattern matching. It defines a new termination criteria by structural guard condition that allows commutative cuts. Everything exposed scales directly to the Coq proof assistant and describes its implementation. The proof of strong normalisation does not and is still work to do