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 quotations and anti-quotations for Coq's syntax, so that one can write {{ nat -> lp:X }} in the middle of a λProlog program instead of the equivalent AST.

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. As ongoing research we are looking forward to express algorithms like higher order unification and type inference, and to provide an alternative elaborator 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 declared with From some.load.path Extra Dependency "filename" as f. and use Elpi Accumulate File f..

CoqIDE does handle quotations. 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.

For Vim users, Coqtail provides syntax highlighting and handles quotations.

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).
• HOAS of Coq terms focuses on how Coq terms are represented in Elpi, how to inspect them and call Coq APIs under a context of binders, and finally how holes ("evars" in Coq slang) are represented. It assumes the reader is familiar with Elpi.
• Writing commands in Elpi focuses on how to write commands, in particular how to store a state across calls via so called DBs and how to handled command arguments. It assumes the reader is familiar with Elpi and the HOAS of Coq terms.
• Writing tactics in Elpi describes how goals and tactics are represented, how to handle tactic arguments and finally how to define tactic notations. It assumes the reader is familiar with Elpi and the HOAS of Coq terms.
• Coq-Elpi in 20 minutes video recording of a talk given at the Coq Users and Developers Workshop 2020.

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 an 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
• generalize show how to abstract subterms out (one way to skin the cat, there are many)
• record import gives short names to record projections applied to the given record instance.
• reduction surgery implements a tactic fine tuning cbv with a list of allowed unfoldings taken from a module.

Applications written in Coq-Elpi

• Derive shows how to obtain proved equality tests and a few extra gadgets out of inductive type declarations. See the README for the list of derivations. It comes bundled with Coq-Elpi.
• Locker lets one hide the computational contents of definitions via modules or opaque locks. It comes bundled with Coq-Elpi.
• Hierarchy Builder is a Coq extension to declare hierarchies of algebraic structures.
• Algebra Tactics is a port of the ring and field tactics to the Mathematical Components library.
• Trakt is a generic goal preprocessing tool for proof automation tactics in Coq.
• Namespace Emulation System implements most of the features of namespaces (on top of Coq's modules).
• Dx uses elpi to generate an intermediate representation of Coq terms, to be later tranformed into C.

Quick Reference

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

Vernacular commands

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• 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> From <loadpath>|Db <dbname>] adds code to the current program (or <qname> if specified). The code can be verbatim, from a file or a Db. It understands the #[skip="rex"] and #[only="rex"] which make the command a no op if the Coq version is matched (or not) by the given regular expression. File names are relative to the directory mapped to <loadpath>; if more than one such directory exists, the <filename> must exists only once.

• 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 and a text file called <qname>.txt (or <string> if provided) filtering out clauses whose file or 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 or invoke tactic <qname> in the middle of a term just writing <qname> args instead of ltac:(elpi <qname> args). Note that in the case of tactics, all arguments are considered to be terms. Moreover, remember that one can use Tactic Notation to give the tactic a better syntax and a shorter name when used in the middle of a proof script.

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, Structure, Inductive, CoInductive, Variant 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, and the syntax for Ltac variables 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 | Structure ] name binder* [: sort] := [name] { name : term ; * }
• [ Inductive | CoInductive | Variant ] name binder* [| binder] [: term] := | name binder : term *
• Context binder*

Ltac Variables

Tactics also accept Ltac variables as follows:

• ltac_string:(v) (for v of type string or ident)
• ltac_int:(v) (for v of type int or integer)
• ltac_term:(v) (for v of type constr or open_constr or uconstr or hyp)
• ltac_(string|int|term)_list:(v) (for v of type list of ...)
• ltac_attributes:(v) (for v of type attributes) For example:

Tactic Notation "tac" string(X) ident(Y) int(Z) hyp(T) constr_list(L) :=
  elpi tac ltac_string:(X) ltac_string:(Y) ltac_int:(Z) ltac_term:(T) ltac_term_list:(L).

lets one write tac "a" b 3 H t1 t2 t3 in any Ltac context. Arguments are first interpreted by Ltac according to the types declared in the tactic notation and then injected in the corresponding Elpi argument. For example H must be an existing hypothesis, since it is typed with the hyp Ltac type, but in Elpi it will appear as a term, eg trm c0. Similarly t1, t2 and t3 are checked to be well typed and to contain no unresolved implicit arguments, since this is what the constr Ltac type means If they were typed as open_constr or uconstr, the last or both checks would be respectively skipped. In any case they are passed to the Elpi code as trm .... Both "a" and b are passed to Elpi as str .... Finally, ltac_term:(T) and (T) are not synonyms: but the former must be used when defining tactic notations, the latter when invoking elpi tactics directly.

Attributes

Attributes are supported in both commands and tactics. Examples:

• #[ att ] Elpi cmd
• #[ att ] cmd for a command cmd exported via Elpi Export cmd
• #[ att ] elpi tac
• Tactic Notation ... attributes(A) ... := ltac_attributes:(A) elpi tac. Due to a parsing conflict in Coq grammar, at the time of writing this code:

  Tactic Notation "#[" attributes(A) "]" "tac" :=
    ltac_attributes:(A) elpi tac.

  has the following limitation:
  • #[ att ] tac. does not parse
  • (#[ att ] tac). works
  • idtac; #[ att ] tac. works

Terms as arguments

Since version 1.15, terms passed to Elpi commands code via (term) or via a declaration (like Record, Inductive ...) are in elaborated format by default. This means that all Coq notational facilities are available, like deep pattern matching, or tactics in terms. One can use the attribute #[arguments(raw)] to declare a command which instead takes arguments in raw format. In that case, 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), but deep pattern matching or tactics in terms are not supported, and in particular type checking/inference is not performed. Once can use the coq.typecheck or coq.elaborate-skeleton APIs to fill in implicit arguments and insert coercions on raw terms.

Terms passed to Elpi tactics via tactic notations can be forced to be elaborated beforehand by declaring the parameters to be of type constr or open_constr. Arguments of type uconstr are passed raw.

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).

Supported features of Gallina (core calculus of Coq)

(click to expand)

• functional core (fun, forall, match, application, let-in, sorts)
• evars (unification variables)
• single Inductive and CoInductive types (including parameters, non-uniform parameters, indexes)
• mutual Inductive and CoInductive types
• fixpoints
• mutual fixpoints
• cofixpoints
• primitive records
• primitive projections
• primitive integers
• primitive floats
• primitive arrays
• universe polymorphism
• modules
• module types
• functor application
• functor definition

Supported features of Gallina's extensions (extra logical features, APIs)

(click to expand)

Checked boxes are available, unchecked boxes are planned, missing items are not planned. This is a high level list, for the details see coq-builtin.

• i/o: messages, warnings, errors, Coq version
• logical environment: read, write, locate
  • dependencies between objects
• type classes database: read, write
  • take over resolution
• canonical structures database: read, write
  • take over resolution
• coercions database: read, write
• sections: open, close
• scope management: import, export
• hints: mode, opaque, resolve, strategy
• arguments: implicit, name, scope, simpl
• abbreviations: read, write, locate
• typing and elaboration
• unification
• reduction: lazy, cbv, vm, native
  • flags for lazy and cbv
• ltac1: bridge to call ltac1 code, mono and multi-goal tactics
• option system: get, set, add
• pretty printer: boxes, printing width
• attributes: read

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.