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$$\mathrm{LF}^+$$ in Coq for fast-and-loose reasoning. (English) Zbl 1427.68344
Summary: We develop the metatheory and the implementation, in Coq, of $$\mathrm{LF}^+$$, and discuss several applications. $$\mathrm{LF}^+$$ generalises research work, carried out by the authors over more than a decade, on Logical Frameworks conservatively extending LF and featuring lock-type constructors $$\mathcal L^{\mathcal P(N:\sigma)}\llbracket\cdot\rrbracket$$. Lock-types capture monadically the concept of inhabitability up-to. They were originally introduced for factoring-out, postponing, or delegating to external tools the verification of time-consuming judgments, which are morally proof-irrelevant, thus allowing for integrating different sources of epistemic evidence in a unique Logical Framework. Besides introducing $$\mathrm{LF}^+$$ and its “shallow” implementation in Coq, the main novelty of the paper is to show that lock-types are also a very flexible tool for expressing in Type Theory several diverse cognitive attitudes and mental strategies used in ordinary reasoning, which essentially amount to reasoning up-to, as in e.g. Typical Ambiguity provisos or co-inductive Coq proofs. In particular we address the encoding of the emerging paradigm of fast-and-loose reasoning, which trades off efficiency for correctness. This paradigm, implicitly used normally in naïve Set Theory, is producing considerable impact also in computer architecture and distributed systems, when branch prediction and optimistic concurrency control are implemented
##### MSC:
 68V15 Theorem proving (automated and interactive theorem provers, deduction, resolution, etc.) 03B38 Type theory 03B70 Logic in computer science 68V20 Formalization of mathematics in connection with theorem provers
##### Software:
LLFp; Coq; GitHub; MMT
Full Text:
##### References:
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