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A formalisation of the Calculus of Constructions
License: GNU Lesser General Public License v2.1
This project forked from coq-contribs/coq-in-coq
A formalisation of the Calculus of Constructions
License: GNU Lesser General Public License v2.1
Contribution Rocq/COC ===================== This directory contains: - A formalization in Coq of the metatheory of the Calculus of Constructions and the interface of a standalone proof-checker based on this type system. - The proof-checker, produced by extraction. Author & Date: Bruno Barras INRIA-Rocquencourt October 1997 E-mail : [email protected] WWW : http://pauillac.inria.fr/~barras Installation procedure: ----------------------- To get this contribution compiled, type make or make opt It will compile all the proofs and perform the extraction. Then, it will compile the proof-checker (called coc). As an example, the file newman.coc is checked with coc. Description: ------------ The essential step of the formal verification of a proof-checker such as Coq is the verification of its kernel: a type-checker for the Calculus of Inductive Constructions (CIC) which is its underlying formalism. The present work is a first small-scale attempt on a significative fragment of CIC: the Calculus of Constructions (CC) designed by Huet and Coquand in 1985. It is defined with De Bruijn indices notation. The whole metatheory of this calculus is proved in the following order: - Confluence of beta-reduction - Inversion lemma - Thinning lemma - Subsitution lemma - Type Correctness - Subject Reduction - Strong Normalisation - Decidability of Type Inference and Type Checking From the latter proof, we extract a certified Objective Caml program, which performs type inference (or type-checking) for an arbitrary typing judgement in CC. Integrating this program in a larger system, including a parser and pretty-printer, we obtain a stand-alone proof-checker, called Coc, for the Calculus of Constructions. As an example, the formal proof of Newman's lemma, build with Coq, can be re-verified by Coc with reasonable performance. Upon this kernel, we formalized the interface of a small proof-checker, based on the type-checking functions above, but it seems the ideas can generalize to other type systems, as far as they are based on the proofs-as-terms principle. We suppose that the metatheory of the corresponding type system is proved (up to type decidability). We specify and certify the toplevel loop, the system invariant, and the error messages. Further information on this contribution: ----------------------------------------- A first description of the proofs can be found as an INRIA technical report (in french), number 3026, october 1996. The current updated version was described in a paper (see coqincoq.ps.gz). It also describes the strong normalization proof. Last, the proof-checker was formalized in another paper included in this contribution. See ./proof-checker.ps.gz
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