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The Semantics of Reflected Proof
 IN PROC. OF FIFTH SYMP. ON LOGIC IN COMP. SCI
, 1990
"... We begin to 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, ..."
Abstract

Cited by 88 (11 self)
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We begin to 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 new 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. We show that these proofs are valid, and that every proof can be reduced to a proof involving only primitive inference rules. We also show how to extend the results to proofs from which programs (such as tactics) can be derived, and to proofs that can refer to a library of definitions and previously proven theorems. We believe that the mechanism of reflection is fundamental in building proof development systems, and we illustrate its power with applications to automating reasoning and describing modes of computation.
Reflection and PropositionsasTypes
"... Reection is the ability of a deductive system to internalize aspects of its own structure and thereby reason to some extent about itself. In this paper we present a theoretical framework for exploring reection in type theories that use the \PropositionsasTypes" principle, such as MartinLof s ..."
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Reection is the ability of a deductive system to internalize aspects of its own structure and thereby reason to some extent about itself. In this paper we present a theoretical framework for exploring reection in type theories that use the \PropositionsasTypes" principle, such as MartinLof style theories. One of the main results is that it is unnecessary to build a complete Godel style \reection" layer on top of the logical theory. This makes it possible to use our framework for an ecient implementation of reection in theorem provers for such type theories. We are doing this for the NuPRL and MetaPRL systems.