Lightweight proof by reflection using a posteriori simulation of effectful computation

Guillaume Claret 1, 2 Lourdes del Carmen Gonzalez Huesca 1, 2 Yann Régis-Gianas 2 Beta Ziliani 3
2 PI.R2 - Design, study and implementation of languages for proofs and programs
PPS - Preuves, Programmes et Systèmes, Inria Paris-Rocquencourt, UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique : UMR7126
Abstract : Proof-by-reflection is a well-established technique that em- ploys decision procedures to reduce the size of proof-terms. Currently, decision procedures can be written either in Type Theory--in a purely functional way that also ensures termination-- or in an effectful program- ming language, where they are used as oracles for the certified checker. The first option offers strong correctness guarantees, while the second one permits more efficient implementations. We propose a novel technique for proof-by-reflection that marries, in Type Theory, an effectful language with (partial) proofs of correctness. The key to our approach is to use simulable monads, where a monad is simulable if, for all terminating reduction sequences in its equivalent effectful computational model, there exists a witness from which the same reduction may be simulated a posteriori by the monad. We encode several examples using simulable monads and demonstrate the advantages of the technique over previous approaches.
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Guillaume Claret, Lourdes del Carmen Gonzalez Huesca, Yann Régis-Gianas, Beta Ziliani. Lightweight proof by reflection using a posteriori simulation of effectful computation. Interactive Theorem Proving, Jul 2013, Rennes, France. ⟨hal-00870110⟩

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