Lem is a tool for lightweight executable mathematics, for writing, managing, and publishing large-scale portable semantic definitions, with export to LaTeX, executable code (currently OCaml) and interactive theorem provers (currently Coq, HOL4, and Isabelle/HOL, though the generated Coq is not necessarily idiomatic). It is also intended as an intermediate language for generating definitions from domain-specific tools, and for porting definitions between interactive theorem proving systems.
The language, originally based on a pure fragment of OCaml, combines features familiar from functional programming languages with logical constructs. From functional programming languages we take pure higher-order functions, general recursion, recursive algebraic datatypes, records, lists, pattern matching, parametric polymorphism, a simple type class mechanism for overloading, and a simple module system. To these we add logical constructs familiar in provers: universal and existential quantification, sets (including set comprehensions), relations, finite maps, inductive relation definitions, and lemma statements. Then there are facilities to let the user tune how Lem definitions are mapped into the various targets (by declaring target representations and controlling notation, renaming, inlining, and type classes), to generate witness types and executable functions from inductive relations, and for assertions.
Lem has been used in several applications, some of which can be found
in the examples directory. As of 2020, Lem remains in continuous
use but is not under active development.
-
Lem: Reusable Engineering of Real-world Semantics. Dominic P. Mulligan, Scott Owens, Kathryn E. Gray, Tom Ridge, and Peter Sewell. In ICFP 2014.
-
Lem: A Lightweight Tool for Heavyweight Semantics. Scott Owens, Peter Böhm, Francesco Zappa Nardelli, and Peter Sewell. In ITP 2011 (Rough Diamond).
Documentation can be found in doc/built-doc, including:
- the manual, in
lem-manual.htmlandlem-manual.pdf; - the Ott grammar and type system for Lem, in
lem.pdf, built from the definition inlanguage/lem.ott; - the Lem library documentation, in
lem-libs.pdf; - the type signatures of the pervasives and pervasives-extra libaries, in
lem-libs-pervasives.txtandlem-libs-pervasives-extra.txt; and - source documentation, in
html-docandlem-doc.pdf, with a dependency diagram of the source modules indep.pdf.
Lem is available as an opam package and a github repo.
First, ensure you have opam (the OCaml package manager) installed,
version 2.0 or greater (opam 1 versions of Lem are no longer
supported). You can use your system's package manager e.g. sudo apt-get install opam (e.g. on Ubuntu 20.04) or follow the
instructions from the opam website.
On older Ubuntu versions you will not be able to use their package
manager's opam 1 version, and will need to install opam 2 following the
instructions on the opam website.
Then opam install lem will install the latest Lem version.
In the checkout directory, run opam pin add lem ..
To rebuild and reinstall after local changes, run opam upgrade --working-dir lem (or opam upgrade -w lem).
The command make (make world) builds the lem binary, and places a symbolic link to it in the top-level directory.
Now set the LEMLIB environment variable to PATH_TO_LEM/library, or alternately pass the -lib PATH_TO_LEM/library flag to lem when you run it.
Running make only generates Lem. It not generate the libraries needed to use Lem's output for certain backends. To generate the libraries for a specific backend, please run
- for OCaml :
make ocaml-libs - for HOL4 :
make hol-libs - for Isabelle:
make isa-libs - for Coq :
make coq-libs
These targets depend on the corresponding tool being installed. If you
just want to generate the input that Lem gives to these tools, please
run make libs.
Lem depends on OCaml. Lem has been tested
against OCaml 4.07.0, 4.12.0 and 5.0.0~beta1. Other versions might or
might not work. Lem requires the OCaml ZArith library for arbitrary
precision arithmetic. Lem has been tested against ZArith version
1.4 and 1.9.1. Other versions might or might not work.
The generated Isabelle theories require the Word_Lib entry of the Archive of
Formal Proofs to be installed and set up (e.g. by
adding the path to Word_Lib to $ISABELLE_HOME_USER/ROOTS). An Isabelle
2022 version of the AFP can be downloaded
here.
Lem has been tested against the following versions of the backend software:
- OCaml:
4.07.0,4.12.0and5.0.0~beta1 - Coq: 8.16.0
- Isabelle 2022
- HOL: HOL4 Kananaskis 14
The examples directory in the repository contains or points to several of the early major examples of Lem usage:
- The operational model for Power/ARM multiprocessor concurrency by Sarkar et al., as described in PLDI11 and extended in POPL12, PLDI12 (ppcmem-model)
- The C/C++11 axiomatic concurrency model by Batty et al., as described in POPL11 and extended in the above two POPL12 and PLDI12 papers (cpp)
- The OCaml_light semantics by Owens, as described in ESOP2008 (ocaml_light)
- The NetSem models for the TCP/IP network protocols and Sockets API by Bishop et al., as described in TACS01, ESOP02, SIGCOMM 2005, POPL 2006, FM08, are available from the github repository https://github.com/PeterSewell/netsem, ported into Lem from the original HOL4.
- The CakeML development by Kumar et al., described in ICFP12, ITP13, and POPL14, is available via its web page: https://cakeml.org.
Lem shares many of the goals of our Ott tool: both emphasize source readability, and multi-prover compatibility. However, Lem is a general-purpose specification language, whereas Ott is a domain-specific language for writing specifications of programming languages (i.e., inductive relations over syntax). Thus, Ott supports rich user-defined syntaxes, whereas Lem supports functional programming idioms. The two can be used together in some cases: Ott can now generate Lem specifications.
- 2018-01-31: moved to github
- 2013-12-13: updated with links to current documentation and examples
- 2013-03-14: Moved page to Kent, removed 0.3 release, and updated for open source development on Bitbucket
- 2011-12-02: Lem 0.3 posted (minor changes from 0.2)
- 2011-08-22: Lem 0.2 posted, manual added
- 2011-05-25: ITP 2011 paper added
- 2011-04-11: Lem 0.1 posted
- 2011-03-24: Web page created
Lem has been developed principally by Dominic Mulligan, Kathryn E. Gray, Scott Owens, Peter Sewell, and Thomas Tuerk; with additional contributions from Basile Clement, Brian Campbell, Christopher Pulte, David Sheets, Fabian Immler Frederic Loulergue, Francesco Zappa Nardelli. Gabriel Kerneis, James Lingard, Jean Pichon-Pharabod, Justus Matthiesen, Kayvan Memarian, Kyndylan Nienhuis, Lars Hupel, Mark Batty, Michael Greenberg Michael Norrish, Ohad Kammar, Peter Boehm, Robert Norton Sami Mäkelä, Shaked Flur Stephen Kell, Thibaut Pérami, Thomas Bauereiss, Thomas Williams, Victor Gomes, and emersion.
Lem is made available under the BSD 3-clause license, with the exception of a few files derived from OCaml, which are under the GNU Library GPL.