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Metatheory à la carte
 In POPL ’13
, 2013
"... Formalizing metatheory, or proofs about programming languages, in a proof assistant has many wellknown benefits. However, the considerable effort involved in mechanizing proofs has prevented it from becoming standard practice. This cost can be amortized by reusing as much of an existing formalizat ..."
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Formalizing metatheory, or proofs about programming languages, in a proof assistant has many wellknown benefits. However, the considerable effort involved in mechanizing proofs has prevented it from becoming standard practice. This cost can be amortized by reusing as much of an existing formalization as possible when building a new language or extending an existing one. Unfortunately reuse of components is typically adhoc, with the language designer cutting and pasting existing definitions and proofs, and expending considerable effort to patch up the results. This paper presents a more structured approach to the reuse of formalizations of programming language semantics through the composition of modular definitions and proofs. The key contribution is the development of an approach to induction for extensible Church encodings which uses a novel reinterpretation of the universal property of folds. These encodings provide the foundation for a framework, formalized in Coq, which uses type classes to automate the composition of proofs from modular components. Several interesting language features, including binders and general recursion, illustrate the capabilities of our framework. We reuse these features to build fully mechanized definitions and proofs for a number of languages, including a version of miniML. Bounded induction enables proofs of properties for noninductive semantic functions, and mediating type classes enable proof adaptation for more featurerich languages. 1.
Generic Datatypes a ̀ la Carte
"... Formal reasoning in proof assistants, also known as mechanization, has high development costs. Building modular reusable components is a key issue in reducing these costs. A stumbling block for reuse is that inductive definitions and proofs are closed to extension. This is a manifestation of the ..."
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Formal reasoning in proof assistants, also known as mechanization, has high development costs. Building modular reusable components is a key issue in reducing these costs. A stumbling block for reuse is that inductive definitions and proofs are closed to extension. This is a manifestation of the expression problem that has been addressed by the MetaTheory a ̀ la Carte (MTC) framework in the context of programming language metatheory. However, MTC’s use of extensible Churchencodings is unsatisfactory. This paper takes a better approach to the problem with datatypegeneric programming (DGP). It applies wellknown DGP techniques to represent modular datatypes, to build functions from functor algebras with folds and to compose proofs from proof algebras by means of induction. Moreover, for certain functionality and proofs our approach can achieve more reuse than MTC: instead of composing modular components we provide a single generic definition once and for all.