Coq is built around a well-delimited kernel that performs type checking for definitions in a variant of the Calculus of Inductive Constructions ( CIC ). Although the metatheory of CIC is very stable and reliable, the correctness of its implementation in Coq is less clear. Indeed, implementing an efficient type checker for CIC is a rather complex task, and many parts of the code rely on implicit invariants which can easily be broken by further evolution of the code. Therefore, on average, one critical bug has been found every year in Coq . This article presents the first implementation of a type checker for the kernel of Coq (without the module system, template polymorphism and η-conversion), which is proven sound and complete in Coq with respect to its formal specification. Note that because of Gödel’s second incompleteness theorem, there is no hope to prove completely the soundness of the specification of Coq inside Coq (in particular strong normalization), but it is possible to prove the correctness and completeness of the implementation assuming soundness of the specification, thus moving from a trusted code base (TCB) to a trusted theory base (TTB) paradigm. Our work is based on the MetaCoq project which provides meta-programming facilities to work with terms and declarations at the level of the kernel. We verify a relatively efficient type checker based on the specification of the typing relation of the Polymorphic, Cumulative Calculus of Inductive Constructions ( PCUIC ) at the basis of Coq . It is worth mentioning that during the verification process, we have found a source of incompleteness in Coq ’s official type checker, which has then been fixed in Coq 8.14 thanks to our work. In addition to the kernel implementation, another essential feature of Coq is the so-called extraction mechanism: the production of executable code in functional languages from Coq definitions. We present a verified version of this subtle type and proof erasure step, therefore enabling the verified extraction of a safe type checker for Coq in the future.

Abstract We implement extraction of Coq programs to functional languages based on MetaCoq’s certified erasure. We extend the MetaCoq erasure output language with typing information and use it as an intermediate representation, which we call ${\lambda^T_\square}$ . We complement the extraction functionality with a full pipeline that includes several standard transformations (e.g. eta-expansion and inlining) implemented in a proof-generating manner along with a verified optimisation pass removing unused arguments. We prove the pass correct wrt. a conventional call-by-value operational semantics of functional languages. From the optimised ${\lambda^T_\square}$ representation, we obtain code in two functional smart contract languages, Liquidity and CameLIGO, the functional language Elm, and a subset of the multi-paradigm language for systems programming Rust. Rust is currently gaining popularity as a language for smart contracts, and we demonstrate how our extraction can be used to extract smart contract code for the Concordium network. The development is done in the context of the ConCert framework that enables smart contract verification. We contribute with two verified real-world smart contracts (boardroom voting and escrow), which we use, among other examples, to exemplify the applicability of the pipeline. In addition, we develop a verified web application and extract it to fully functional Elm code. In total, this gives us a way to write dependently typed programs in Coq, verify, and then extract them to several target languages while retaining a small trusted computing base of only MetaCoq and the pretty-printers into these languages.

We implement extraction of Coq programs to functional languages based on MetaCoq's certified erasure. As part of this, we implement an optimisation pass removing unused arguments. We prove the pass correct wrt. a conventional call-by-value operational semantics of functional languages. We apply this to two functional smart contract languages, Liquidity and Midlang, and to the functional language Elm.

We present a new way of embedding functional languages into the Coq proof assistant by using meta-programming. This allows us to develop the meta-theory of the language using the deep embedding and provides a convenient way for reasoning about concrete programs using the shallow embedding. We connect the deep and the shallow embeddings by a soundness theorem. As an instance of our approach, we develop an embedding of a core smart contract language into Coq and verify several important properties of a crowdfunding contract based on a previous formalisation of smart contract execution in blockchains.