Coq-Elpi

Coq plugin embedding Elpi.

What is Elpi

Elpi provides an easy-to-embed implementation of a dialect of λProlog, a programming language well suited to manipulate abstract syntax trees containing binders and unification variables.

What is Coq-Elpi

Coq-Elpi provides a Coq plugin that lets one define new commands and tactics in Elpi. For that purpose it provides an embedding of Coq's terms into λProlog using the Higher-Order Abstract Syntax approach (HOAS). It also exports to Elpi a comprehensive set of Coq's primitives, so that one can print a message, access the environment of theorems and data types, define a new constant, declare implicit arguments, type classes instances, and so on. For convenience it also provides a quotation and anti-quotation for Coq's syntax, so that one can write {{ nat -> lp:X }} in the middle of a λProlog program instead of the equivalent AST prod `_` (global (indt «Coq.Init.Datatypes.nat»)) X.

What is the purpose of all that

In the short term, provide an extension language for Coq well suited to manipulate terms containing binders. One can already use Elpi to implement commands and tactics.

In addition to that Elpi extends λProlog with higher order constraints, a language feature that helps to manipulate terms containing not only binders, but also unification variables (evars, in Coq's slang). As ongoing research we are looking forward to express algorithms like higher order unification and type inference for Coq.

Installation

The simplest way is to use OPAM and type

opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-elpi

Editor Setup

The recommended user interface is VSCoq. We provide an extension for vscode in the market place, just look for Coq Elpi. The extension provides syntax hilighting for both languages even when they are nested via quotations and antiquotations.

Other editors (click to expand)

At the time of writing Proof General does not handle quotations correctly, see ProofGeneral/PG#437. In particular Elpi Accumulate lp:{{ .... }}. is used in tutorials to mix Coq and Elpi code without escaping. Coq-Elpi also accepts Elpi Accumulate " .... ". but strings part of the Elpi code needs to be escaped. Finally, for non-tutorial material, one can always put the code in an external file and use Elpi Accumulate File "filename". instead.

CoqIDE does not handle quotations correctly. The installation process puts coq-elpi.lang in a place where CoqIDE can find it. Then you can select coq-elpi from the menu Edit -> Preferences -> Colors.

If you use Vim, we recommend to add the following lines to ~/.vimrc (in addition to the ones for elpi)

(click to expand)

"coq-elpi
autocmd FileType lprolog syn keyword coqElpiSpecial fun prod sort let match fix axiom indc indt const prop app
autocmd FileType lprolog syn cluster elpiAntiQuotation contains=elpiAntiQuotationVar,elpiAntiQuotationBound,elpiAntiQuotationTerm
autocmd FileType lprolog syn region elpiAntiQuotationTerm start=+lp:"+ end=+"+ contains=elpiQuotation,lprologVariable,coqElpiSpecial,elpiMacro,lprologSpecial
autocmd FileType lprolog syn match elpiAntiQuotationVar "lp:[A-Z_-]\+"ms=s+3
autocmd FileType lprolog syn match elpiAntiQuotationBound "lp:[a-z_-]\+"
autocmd FileType lprolog hi def link elpiAntiQuotationVar Keyword
autocmd FileType lprolog hi def link elpiAntiQuotationBound Normal
autocmd FileType lprolog hi def link coqElpiSpecial Special

Development version (click to expand)

To install the development version one can type

opam pin add coq-elpi https://github.com/LPCIC/coq-elpi.git

One can also clone this repository and type make, but check you have all the dependencies installed first (see coq-elpi.opam).

Documentation

Tutorials

• The Elpi programming language is an Elpi tutorial, there is nothing Coq specific in there even if the tutorial uses Coq to step trough the various examples. If you never heard of λProlog or HOAS based languages (like Twelf or Beluga) then you are strongly encouraged to read this tutorial and have a look at λProlog's home page for additional documentation. Even if you are familiar with λProlog or HOAS it may be worth reading the last sections since they focus on Elpi specific features. Last but not least it covers common pitfalls for people with a background in functional programming and the tracing mechanisms (useful for debugging)
• Using Elpi to extend Coq focuses on the integration of Elpi in Coq, covering the representation of terms and the implementation of commands and tactics. It assumes the reader is familiar with λProlog

Small examples (proofs of concept)

• reification is the typical use case for meta programs: reading the syntax of terms into an inductive representing a sub language on which some decision procedure can be implemented
• data bases shows how Elpi programs can store data and reuse it across multiple runs
• record expansion sketches a program to unpack records in a definition: it replaces and abstraction over a records with abstractions over all of its components
• record to sigma sketches a program that de-sugars a record type to iterated sigma types
• fuzzer sketches a program to alter an inductive type while preserving its well typedness. It makes nothing useful per se, but shows how to map a term and call the type checker deep inside it.
• tactics show how to create simple tactics by using (proof) terms and the elaborator of Coq

Applications written in Coq-Elpi

• Derive shows how to obtain proved equality tests and a few extra gadgets out of inductive type declarations. It comes bundled with Coq-Elpi.
• Hierarchy Builder is a Coq extension to declare hierarchies of algebraic structures.
• Namespace Emulation System implements most of the features of namespaces (on top of Coq's modules).

Quick Reference

In order to load Coq-Elpi use From elpi Require Import elpi.

Vernacular commands

(click to expand)

• Elpi Command <qname> creates command named <qname> containing the preamble elpi-command.

• Elpi Tactic <qname> creates a tactic <qname> containing the preamble elpi-tactic.

• Elpi Db <dbname> <code> creates a Db (a program that is accumulated into other programs). <code> is the initial contents of the Db, including the type declaration of its constituting predicates.

• Elpi Program <qname> <code> lower level primitive letting one crate a command/tactic with a custom preamble <code>.

• Elpi Accumulate [<qname>] [<code>|File <filename>|Db <dbname>] adds code to the current program (or <qname> if specified). The code can be verbatim, from a file or a Db.

• Elpi Typecheck [<qname>] typechecks the current program (or <qname> if specified).

• Elpi Debug <string> sets the variable <string>, relevant for conditional clause compilation (the :if VARIABLE clause attribute).

• Elpi Trace [[<start> <stop>] <predicate-filter>*|Off] enable/disable tracing, eventually limiting it to a specific range of execution steps or predicate names.

• Elpi Bound Steps <number> limits the number of steps an Elpi program can make.

• Elpi Print <qname> [<string> <filter>*] prints the program <qname> to an HTML file named <qname>.html (or <string> if provided filtering out clauses whose file/clause name matches <filter>.

where:

• <qname> is a qualified Coq name, e.g. derive.eq or my_program.
• <dbname> is like <qname> but lives in a different namespace. By convention <dbname> ends in .db, e.g. derive.eq.db.
• <code> is verbatim Elpi code, either lp:{{ ... }} or " ... " (in the latter case, strings delimiters need to be escaped following Coq rules, e.g. lp:{{ coq.say "hello!" }} becomes " coq.say ""hello!"" ").
• <filename> is a string containing the path of an external file, e.g. "this_file.elpi".
• <start> and <stop> are numbers, e.g. 17 24.
• <predicate-filter> is a regexp against which the predicate name is matched, e.g. "derive.*".

Invocation of Elpi code

(click to expand)

• Elpi <qname> <argument>*. invokes the main predicate of the <qname> program passing a possible empty list of arguments. This is how you invoke a command.

• elpi <qname> <argument>*. invokes the solve predicate of the <qname> program passing a possible empty list of arguments and the current goal. This is how you invoke a tactic.

• Elpi Export <qname> makes it possible to invoke command <qname> without the Elpi prefix.

where <argument> can be:

• a number, e.g. 3, represented in Elpi as (int 3)
• a string, e.g. "foo" or bar.baz, represented in Elpi as (str "foo") and (str "bar.baz"). Coq keywords and symbols are recognized as strings, eg => requires no quotes. Quotes are necessary if the string contains a space or a character that is not accepted for qualified identifiers or if the string is Definition, Axiom, Record or Context.
• a term, e.g. (3) or (f x), represented in Elpi as (trm ...). Note that terms always require parentheses, that is 3 is a number while (3) is a Coq term and depending on the context could be a natural number (i.e. S (S (S O))) or a Z or ... See also the section Terms as arguments down below.

Commands also accept the following arguments (the syntax is as close as possible to the Coq one: [...] means optional, * means 0 or more). See the argument data type in coq-builtin.elpi for their HOAS encoding. See also the section Terms as arguments down below.

• Definition name binder* [: term] := term
• Axiom name : term
• Record name binder* [: sort] := [name] { name : term ; * }
• Inductive name binder* [| binder] [: term] := | name binder : term *
• Context binder*

Testing/debugging:

• Elpi Query [<qname>] <code> runs <code> in the current program (or in <qname> if specified).
• elpi query [<qname>] <string> <argument>* runs the <string> predicate (that must have the same signature of the default predicate solve).

Terms as arguments

Terms are passed to Elpi code in raw format. Notations are unfolded, implicit arguments are expanded (holes _ are added) and lexical analysis is performed (global names and bound names are identified, holes are applied to bound names in scope). Type checking/inference is not performed: the coq.typecheck API can be used to fill in implicit arguments.

Relevant files

• coq-builtin documents the HOAS encoding of Coq terms and the API to access Coq
• elpi-buitin documents Elpi's standard library, you may look here for list processing code
• coq-lib provides some utilities to manipulate Coq terms; it is an addendum to coq-builtin
• elpi-command-template provides the pre-loaded code for Elpi Command
• elpi-tactic-template provides the pre-loaded code for Elpi Tactic

Organization of the repository

The code of the Coq plugin is at the root of the repository in the src, elpi and theories directories.

The apps directory contains client applications written in Coq-Elpi.