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](etc/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](https://github.com/whonore/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](coq-elpi.opam)). We recommend to look at the [CI setup](.github/workflows) for ocaml versions being tested. Also, we recommend to install `dot-merlin-reader` and `ocaml-lsp-server` (version 1.15).

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)
abs_evars show how to close a term containing holes (evars) with binders
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.
Coercion enable to program coercions in Elpi. It comes bundled with Coq-Elpi.

Quick Reference

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

Vernacular commands

(click to expand)

- `Elpi Command ` creates command named `` containing the preamble [elpi-command](elpi/elpi-command-template.elpi). - `Elpi Tactic ` creates a tactic `` containing the preamble [elpi-tactic](elpi/elpi-tactic-template.elpi). - `Elpi Db

` creates a Db (a program that is accumulated into
  other programs). `` is the initial contents of the Db, including the
  type declaration of its constituting predicates.
  It understands the `#[phase]` attribute, see [synterp-vs-interp](README.md#separation-of-parsing-from-execution-of-vernacular-commands).
- `Elpi Program  ` lower level primitive letting one crate a
  command/tactic with a custom preamble ``.
- `From some.load.path Extra Dependency  as `.
- `Elpi Accumulate [|] [|File |Db ]`
  adds code to the current program (or `` or `` if specified).
  The code can be verbatim, from a file or a Db.
  File names `` must have been previously declared with the above command.
  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.
  It understands the `#[phase]` attribute, see [synterp-vs-interp](README.md#separation-of-parsing-from-execution-of-vernacular-commands)
- `Elpi Typecheck []` typechecks the current program (or `` if
  specified).
  It understands the `#[phase]` attribute, see [synterp-vs-interp](README.md#separation-of-parsing-from-execution-of-vernacular-commands)
- `Elpi Debug ` sets the variable ``, relevant for conditional
  clause compilation (the `:if VARIABLE` clause attribute).
  It understands the `#[phase]` attribute, see [synterp-vs-interp](README.md#separation-of-parsing-from-execution-of-vernacular-commands)
- `Elpi Trace [[ ] *|Off]` enable/disable
  tracing, eventually limiting it to a specific range of execution steps or
  predicate names.
  It understands the `#[phase]` attribute, see [synterp-vs-interp](README.md#separation-of-parsing-from-execution-of-vernacular-commands)
- `Elpi Trace Browser` enable/disable
  tracing for Elpi's [trace browser]().
- `Elpi Bound Steps ` limits the number of steps an Elpi program can
  make.
- `Elpi Print  [ *]` prints the program `` to an
  HTML file named `.html` and a text file called `.txt`
  (or `` if provided) filtering out clauses whose file or clause-name
  matches ``.
  It understands the `#[phase]` attribute, see [synterp-vs-interp](README.md#separation-of-parsing-from-execution-of-vernacular-commands)

where:

- `` is a qualified Coq name, e.g. `derive.eq` or `my_program`.
- `` is like `` but lives in a different namespace. By convention
  `` ends in `.db`, e.g. `derive.eq.db`.
- `` 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!"" "`).
- `` is a string containing the path of an external file, e.g.
  `"this_file.elpi"`.
- `` and `` are numbers, e.g. `17 24`.
- `` is a regexp against which the predicate name is matched,
  e.g. `"derive.*"`.


Separation of parsing from execution of vernacular commands
(click to expand)

Since version 8.18 Coq has separate parsing and execution phases,
respectively called synterp and interp.

Since Coq has an extensible grammar the parsing phase is not entirely
performed by the parser: after parsing one sentence Coq evaluates its
synterp action. The synterp actions of a command like `Import A.` are
the subset of its effect which affect parsing, like enabling a notation.
Later, during the execution phase Coq evaluates the its
interp action, which includes effects like putting lemma names in scope or
enables type class instances etc.

Being able to parse an entire document quickly,
without actually executing any sentence, is important for developing reactive
user interfaces, but requires some extra work when defining new commands,
in particular to separate their synterp actions from their interp ones.
Each command defined with Coq-Elpi is split into two programs,
one running during the parsing phase and the other one during the execution
phase.

##### Declaration of synterp actions

Each `Elpi Command` internally declares two programs with the same name.
One to be run while the Coq document is parsed, the synterp-command,
and the other one while it is executed, the interp command.
`Elpi Accumulate`, by default, adds code to the interp-command.
The `#[phase]` attribute can be used to accumulate code to the synterp-command
or to both commands. `Elpi Typecheck` checks both commands.

Each `Elpi Db` internally declares one db, by default for the interp phase.
The `#[phase]` attribute can be used crate a database for the synterp phase,
or for both phases. Note that databases for the two phases are distinct, no
data is shared among them. In particular the `coq.elpi.accumulate*` API exists
in both phases and only acts on data bases for the current phase.

##### The alignment of phases

All synterp actions, i.e. calls to APIs dealing with modules and sections
like begin/end-module or import/export, have to happen at *both* synterp and
interp time and *in the same order*.

In order to do so, the synterp-command may need to communicate data to the
corresponding interp-command. There are two ways for doing so.

The first one is to use, as the main entry points, the following ones:
```
pred main-synterp i:list argument, o:any.
pred main-interp i:list argument, i:any.
```
Unlike `main` the former outputs a datum while the latter receives it in input.
During the synterp phase the API `coq.synterp-actions` lists the actions
performed so far. An excerpt from the [coq-builtin-synterp](builtin-doc/coq-builtin-synterp.elpi) file:
```
% Action executed during the parsing phase (aka synterp)
kind synterp-action type.
type begin-module id -> synterp-action.
type end-module modpath -> synterp-action.
```
The synterp-command can output data of that type, but also any other data it
wishes.

The second way to communicate data is implicit, but limited to synterp actions.
Such synterp actions can be recorded into (nested) groups whose structure is
declared using well-bracketed calls to predicates `coq.begin-synterp-group`
and `coq.end-synterp-group` in the synterp phase. In the interp phase, one can
then use predicate `coq.replay-synterp-action-group` to replay all the synterp
actions of the group with the given name at once.

In the case where one wishes to interleave code between the actions of a given
group, it is also possible to match the synterp group structure at interp, via
`coq.begin-synterp-group` and `coq.end-synterp-group`. Individual actions that
are contained in the group then need to be replayed individually.

One can use `coq.replay-next-synterp-actions` to replay all synterp actions
until the next beginning/end of a synterp group. However, this is discouraged
in favour of using groups explicitly, as this is more modular. Code that used
to rely on the now-removed `coq.replay-all-missing-synterp-actions` predicate
can rely on `coq.replay-next-synterp-actions` instead, but this is discouraged
in favour of using groups explicitly)

##### Syntax of the `#[phase]` attribute

- `#[phase="ph"]` where `"ph"` can be `"parsing"`,
  `"execution"` or `"both"`
- `#[synterp]` is a shorthand for `#[phase="parsing"]`
- `#[interp]` is a shorthand for `#[phase="execution]`


Invocation of Elpi code
(click to expand)

- `Elpi  *.` invokes the `main` predicate of the ``
  program passing a possible empty list of arguments. This is how you invoke a
  command.
- `elpi  *.` invokes the `solve` predicate of the ``
  program passing a possible empty list of arguments and the current goal. This
  is how you invoke a tactic.

- `Elpi Export  [As ]` makes it possible to invoke
  command `` (or `` if given) without
  the `Elpi` prefix or invoke tactic `` in the middle of a term just
  writing ` args` instead of `ltac:(elpi  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 `` 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_tactic:(t)` (for `t` of type `tactic_expr`)
- `ltac_attributes:(v)` (for `v` of type `attributes`)
For example:
```coq
Tactic Notation "tac" string(X) ident(Y) int(Z) hyp(T) constr_list(L) simple_intropattern_list(P) :=
  elpi tac ltac_string:(X) ltac_string:(Y) ltac_int:(Z) ltac_term:(T) ltac_term_list:(L) ltac_tactic:(intros P).
```
lets one write `tac "a" b 3 H t1 t2 t3 [|m]` 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:
  ```coq
  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 [] ` runs `` in the current program (or in
  `` if specified).
- `Elpi Query []  ` runs
  `` in the current (synterp) program (or in
`` if specified) and `` in the current program (or ``).
- `elpi query []  *` runs the `` predicate
  (that must have the same signature of the default predicate `solve`).


Supported features of Gallina (core calculus of Coq)
(click to expand)

- [x] functional core (fun, forall, match, application, let-in, sorts)
- [x] evars (unification variables)
- [x] single Inductive and CoInductive types (including parameters, non-uniform
      parameters, indexes)
- [ ] mutual Inductive and CoInductive types
- [x] fixpoints
- [ ] mutual fixpoints
- [ ] cofixpoints
- [x] primitive records
- [x] primitive projections
- [x] primitive integers
- [x] primitive floats
- [ ] primitive arrays
- [x] universe polymorphism
- [x] modules
- [x] module types
- [x] functor application
- [x] 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](builtin-doc/coq-builtin.elpi).

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


Relevant files

coq-builtin documents the HOAS encoding of Coq terms
and the API to access Coq
coq-builtin-synterp documents APIs to interact with Coq at parsing time
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 (execution phase)
elpi-command-template-synterp provides the pre-loaded code for Elpi Command (parsing phase)
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.

LPCIC / coq-elpi

readme

Coq-Elpi

What is Elpi

What is Coq-Elpi

What is the purpose of all that

Installation

Editor Setup

Documentation

Tutorials

Small examples (proofs of concept)

Applications written in Coq-Elpi

Quick Reference

Vernacular commands

Separation of parsing from execution of vernacular commands

Invocation of Elpi code

Supported features of Gallina (core calculus of Coq)

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

Relevant files

Organization of the repository