Robert L. Constable

The semantics of reflected proof

The authors lay the foundations for reasoning about proofs whose steps include both invocations of programs to build subproofs (tactics) and references to representations of proofs themselves (reflected proofs). The main result is the definition of a single type of proof which can mention itself, using a novel technique which finds a fixed point of a mapping between metalanguage and object language. This single type contrasts with hierarchies of types used in other approaches to accomplish the same classification. It is shown that these proofs are valid, and that every proof can be reduced to a proof involving only primitive inference rules. The extension of the results to proofs from which programs (such as tactics) can be derive and to proofs that can refer to a library of definitions and previously proven theorems is shown. It is believed that the mechanism of reflection is fundamental in building proof development systems, and its power is illustrated with applications to automating reasoning and describing modes of computation.<<ETX>>

On Classes of Program Schemata

We define the following classes of program schemata:

Innovations in computational type theory using Nuprl

{\text{P}} =

Computability concepts for programming language semantics

class of schemes using a finite number of simple variables;

The Nuprl Open Logical Environment

{\text{P}}_{\text{A}} =

MetaPRL – A Modular Logical Environment

class of schemes using simple and subscripted variables (arrays);

Type two computational complexity

{\text{P}}_{{\text{Ae}}} =

The Type Theory of PL/CV3

class of schemes in

Writing programs that construct proofs

{\text{P}}_{\text{A}} =

Protocol switching: exploiting meta-properties

, with the addition of an equality test on subscript values;