Ott: Tool Support for Semantics
User Guide
version 0.32

Peter Sewell*     Francesco Zappa Nardelli**     Scott Owens *
with Gilles Peskine*, Tom Ridge*,
Susmit Sarkar*, and Rok Strniša*
*University of Cambridge     **INRIA


1  Introduction

Ott is a tool for writing definitions of programming languages and calculi. It takes as input a definition of a language syntax and semantics, in a concise and readable ASCII notation that is close to what one would write in informal mathematics. It generates output:

  1. a LaTeX source file that defines commands to build a typeset version of the definition;
  2. a Coq version of the definition;
  3. a HOL version of the definition;
  4. an Isabelle/HOL version of the definition;
  5. a Lem version of the definition;
  6. an OCaml version of the syntax of the definition.

Additionally, it can be run as a filter, taking a LaTeX/Coq/Isabelle/HOL/Lem/OCaml source file with embedded (symbolic) terms of the defined language, parsing them and replacing them by typeset terms.

This document is a user guide for the tool. The papers

give an overview of the project, including discussion of motivation, design decisions, and related work, and one should look at that together with this manual. The project web page

links to the github source repository, with a BSD-style licence. It also has a range of examples, including untyped and simply typed CBV lambda calculus, ML polymorphism, various first-order systems from Pierce’s TAPL [Pie02], the POPLmark F<: language [ABF+05], a module system by Leroy [Ler96, §4] (extended with a term language and an operational semantics), the LJ Java fragment and LJAM Java module system [SSP07], and a substantial fragment of OCaml.

Our main goal is to support work on large programming language definitions, where the scale makes it hard to keep a definition internally consistent, and hard to keep a tight correspondence between a definition and implementations. We also wish to ease rapid prototyping work with smaller calculi, and to make it easier to exchange definitions and definition fragments between groups. Most simply, the tool can be used to aid completely informal LaTeX mathematics. Here it permits the definition, and terms within proofs and exposition, to be written in a clear, editable, ASCII notation, without LaTeX noise. It generates good-quality typeset output. By parsing (and so sort-checking) this input, it quickly catches a range of simple errors, e.g. inconsistent use of judgement forms or metavariable naming conventions. That same input, extended with some additional data, can be used to generate formal definitions for Coq, HOL, Isabelle, and Lem. It should thereby enable a smooth transition between use of informal and formal mathematics. Further, the tool can automatically generate definitions of functions for free variables, single and multiple substitutions, subgrammar checks (e.g. for value subgrammars), and binding auxiliary functions. Ott supports a ‘fully concrete’ representation, sufficient for many examples but not dealing with general alpha equivalence. An experimental Coq backend generates definitions in locally-nameless style for a subset of the Ott metalanguage. The OCaml backend generates type definitions that may be useful for developing a complete implementation of the language, together with the functions listed above. It does not generate anything for inductively defined relations (the various proof-assistant code extraction facilities can sometimes be used for that). Our focus here is on the problem of writing and editing language definitions, not (directly) on aiding mechanized proof of metatheory. If one is involved in hard proofs about a relatively stable small calculus then it will aid only a small part of the work (and one might choose instead to work just within a single proof assistant), but for larger languages the definition is a more substantial problem — so much so that only a handful of full-scale languages have been given complete definitions. We aim to make this more commonplace, less of a heroic task.

2  Getting started with Ott (the README)

Ott is a tool for writing definitions of programming languages and calculi. It takes as input a definition of a language syntax and semantics, in a concise and readable ASCII notation that is close to what one would write in informal mathematics. With appropriate annotations, it can then generate output:

Additionally, it can be run as a filter, taking a LaTeX/Coq/Isabelle/HOL/Lem/OCaml source file with embedded (symbolic) terms of the defined language, parsing them and replacing them by typeset terms.

Most simply, Ott can be used to aid informal LaTeX mathematics. Here it permits the definition, and terms within proofs and exposition, to be written in a clear, editable, ASCII notation, without LaTeX noise. It generates good-quality typeset output. By parsing (and so sort-checking) this input, it quickly catches a range of simple errors, e.g. inconsistent use of judgement forms or metavariable naming conventions.

That same input can be used to generate formal definitions, for Coq, HOL, Isabelle, and Lem. It should thereby enable a smooth transition between use of informal and formal mathematics. Additionally, the tool can automatically generate definitions of functions for free variables, single and multiple substitutions, subgrammar checks (e.g. for value subgrammars), and binding auxiliary functions. At present only a fully concrete representation of binding, without quotienting by alpha equivalence, is fully supported. An experimental backend generates a locally-nameless representation of terms for a subset of the Ott metalanguage: details can be found at

The distribution includes several examples, in varying levels of completeness: untyped and simply typed lambda-calculus, a calculus with ML polymorphism, the POPLmark Fsub with and without records, an ML module system taken from (Leroy, JFP 1996) and equipped with an operational semantics, and LJ, a lightweight Java fragment. More substantially, Ott has been used for work on iJAM and LJAM, Java Module Systems, by Rok Strnisa, and semantics for OCaml light, by Scott Owens.

As of 2020, Ott remains in continuous use.

2.1  Documentation

2.2  Papers

2.3  People

Ott has been principally devloped by Peter Sewell, Francesco Zappa Nardelli, and Scott Owens, with contributions from many others including Joey Eremondi, Hannes Mehnert, Karl Palmskog, Matthew Parkinson, Thibaut Perami, Gilles Peskine, Alastair Reid, Tom Ridge, Susmit Sarkar, Rok Strnisa, Viktor Vafeiadis.

2.4  To install and build

Ott is available as an opam package and a github repo.

2.4.1  With OPAM (released version)

First, ensure you have opam (the OCaml package manager) installed, version 2.0 or greater (opam 1 versions of ott 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 ott will install the latest Ott version. The Emacs mode will be in $(opam config var prefix)/share/emacs/site-lisp, and documentation in $(opam config var prefix)/doc/ott.

If you want to use Ott with the Coq proof assistant, to install the Ott auxiliary files for Coq, first activate the coq-released OPAM repository:

opam repo add coq-released

and then run opam install coq-ott.

2.4.2  With OPAM (github checkout)

In the checkout directory, run opam pin add ott ..

To rebuild and reinstall after local changes, run opam upgrade --working-dir ott (or opam upgrade -w ott).

2.4.3  Without OPAM

Ott depends on OCaml version 4.00.0 or later. It builds with (at least) OCaml 4.02.3 and 4.10.0.

The command make (make world) builds the ott binary in the bin/ subdirectory.

This will compile Ott using ocamlopt. To force it to compile with ocamlc (which may give significantly slower execution of Ott), do make world.byt.

To build the Ott auxiliary files for Coq, go to the coq/ subdirectory and run make. To install the resulting files in Coq’s user-contrib, run make install.

2.5  To run

Ott runs as a command-line tool. Executing ott shows the usage and options. To run Ott on the test file tests/, generating LaTeX in test10.tex and Coq in test10.v, type:

ott -i tests/ -o test10.tex -o test10.v

Isabelle, HOL, and Lem can be generated with options -o test10.thy, -o test10Script.sml, and -o test10.lem, respectively.

The Makefile has various sample targets, make tests/test10.out, make test7, etc. Typically they generate:

out.texLaTeX source for a definition
out.psthe postscript built from that
out.vCoq source
outScript.smlHOL source
out.thyIsabelle source

from files,, etc., in tests/.

2.6  Editor Plugins

2.6.1  Emacs mode

The file emacs/ott-mode.el defines a very simple Emacs mode for syntax highlighting of Ott source files. It can be used by, for example, adding the following to your .emacs file, replacing PATH by a path to your Ott Emacs directory.

(setq load-path (cons (expand-file-name "PATH") load-path))
(require 'ott-mode)

For installations using OPAM on *nix systems, it is sufficient to use the following code, which will call opam config var prefix at load-time.

(setq opam-share (substring (shell-command-to-string "opam config var share") 0 -1))
(add-to-list 'load-path (concat opam-share "/emacs/site-lisp"))
(require 'ott-mode)

2.6.2  Visual Studio Code

There is a plugin for VSCode, which features syntax highlighting and inline error reporting.

2.7  Mailing lists

2.8  Bug Tracker

2.9  Directory contents

aux/auxiliary code (y2l) used to build the user guide
bin/the Ott binary
built_doc/the user guide, in html, pdf, and ps
coq/auxiliary files for Coq
doc/the user guide sources
emacs/an Ott Emacs mode
examples/some larger example Ott files
tex/auxiliary files for LaTeX
hol/auxiliary files for HOL
menhir/auxiliary files for menhir
ocamlgraph-1.7.tar.gza copy of the ocamlgraph library
regression/regression-test machinery
tests/various small example Ott files
src/the (OCaml) Ott sources
Makefilea Makefile for the examples
LICENCEthe BSD-style licence terms
README.mdthis file (Section 2 of the user guide)
revisionhistory.txtthe revision history

2.10  Examples

The following LaTeX, Coq, HOL, and Isabelle files, except the proof scripts, are all automatically generated from the Ott sources.

  <th rowspan="2">System</th>
  <th rowspan="2">Rules</th>
  <th rowspan="2">Ott sources</th>
  <th rowspan="2">Latex</th>
  <th rowspan="2">Typeset</th>
  <th rowspan="2">Dot</th>
  <th colspan="2">Coq</th>
  <th colspan="2">HOL</th>
  <th colspan="2">Isabelle</th>
  <th> Defn </th>
  <th> Proof </th>
  <th> Defn </th>
  <th> Proof </th>
  <th> Defn </th>
  <th> Proof </th>

Untyped CBV lambda







Simply typed CBV lambda










ML polymorphism







TAPL roughly-full-simple




(Coq (including records))





POPLmark Fsub (*)




(pdf)   (ps)

Leroy JFP96 module system (*)





LJ: Lightweight Java






LJAM: Java Module System






OCaml light









(*) These systems would need explicit alpha conversion in the rules to capture the intended semantics using the fully concrete representation.

2.11  Copyright information

The ocamlgraph library is distributed under the LGPL (from; we include a snapshot for convenience. For its authorship and copyright information see the files therein.

All other files are distributed under the BSD-style licence in LICENCE.

3  A minimal Ott source file: the untyped CBV lambda calculus

Fig. 1 shows an Ott source file for an untyped call-by-value (CBV) lambda calculus. This section explains the basic features that appear there, while in the following sections we show what must be added to generate typeset output, proof assistant definitions, and other things.

% minimal
metavar termvar, x ::=  

t :: ’t_’ ::=              
  | x            ::  :: Var
  | \ x . t      ::  :: Lam
  | t t’         ::  :: App 
  | ( t )        :: S:: Paren
  | { t / x } t’ :: M:: Tsub  

v :: ’v_’ ::=              
  | \ x . t      ::  :: Lam 

  v <:: t

Jop ::::=

 t1 --> t2 :: ::reduce::by

    --------------------------  :: ax_app
    (\x.t12) v2 -->  {v2/x}t12

    t1 --> t1’
    -------------- :: ctx_app_fun
    t1 t --> t1’ t

    t1 --> t1’
    -------------- :: ctx_app_arg
    v t1 --> v t1’
Figure 1: Source:

The figure is colourised, with Ott keywords like this and Ott symbols such as | and ::. Other user-specific input appears like this. Any text between % and the next newline is discarded as a comment.

At the top of the figure, the metavar declaration introduces a sort of metavariables termvar (with synonym x), for term variables. The following grammar introduces two grammar rules, one for terms, with nonterminal root t, and one for values v. This specifies the concrete syntax of object-language terms, the abstract syntax representations for proof-assistant mathematics, and the syntax of symbolic terms to be used in semantic rules.

Each rule has a rule name prefix (e.g. ’t_’) and then a list of productions. Each production, e.g.

  | \ x . t      ::  :: Lam

specifies a syntactic form as a list of elements, here ‘\’, ‘x’, ‘.’, and ‘t’, each of which is either a metavariable (the ‘x’), a nonterminal (the ‘t’), or a terminal (\ . ( ) { } / -->). Within productions all elements must be whitespace-separated, so that the tool can deduce which are terminals. In the symbolic terms in the semantic rules below, however, whitespace is required only where necessary. A few terminals have to be quoted (with '') if they appear in a grammar, e.g. to use | as an object-language token, as they are part of the Ott syntax, but they do not have to be quoted at usage points. (If one accidentally omits inter-token whitespace in the grammar, the output of Ott can be surprising. This is best diagnosed by looking at the colourised ASCII or LaTeX output from Ott.)

Metavariables and nonterminals can be formed from the specified metavariable and nonterminal roots by appending a suffix, e.g. the nonterminal t' in the App and Tsub productions.

Between the ::’s is an optional meta flag M or S. Non-meta productions give rise to clauses of datatype definitions in the Isabelle/Coq/HOL output, whereas meta productions do not. Later, we will see how the user can specify how meta syntax should be translated away when generating proof assistant output. The two flags M and S are identical except that productions with the latter are admitted when parsing example concrete terms; the S tag is thus appropriate for lightweight syntactic sugar, such as productions for parentheses. One can also use another flag, e.g. X, along with the command-line option -tex_suppress_category, to suppress productions in the generated LaTeX.

Each production has a production name (e.g. t_Lam), composed of the rule name prefix (here t_) and the production name kernel that follows the ::’s (here Lam). The production name is used as a constructor name in the generated Isabelle/Coq/HOL.

The tool supports arbitrary context-free grammars, extended with special constructs for list forms (c.f. §12).

Following the grammar in this example is a subrule declaration

    v <:: t

declaring that the v grammar rule (of values) is a subgrammar of the t rule (of terms). The tool checks that there is in fact a subgrammar relationship, i.e. that for each production of the lower rule there exists a production of the higher rule with corresponding elements (up to the subrule relation). The subrule declaration means that, in the semantic rules below, we will be able to use v’s in places where the grammar specifies t’s. In the generated Isabelle/Coq/HOL for this example only one free datatype will be generated, for the t rule, while for the v rule we generate an is_v predicate over the t type. Usages of v nonterminals in the semantic rules will have instances of this predicate automatically inserted.

Finally, we give a collection of definitions of inductive relations. In this example there is just one family of definitions (of operational judgements), introduced by the defns Jop; it contains just one definition of a relation, called reduce. In general there may be many defns blocks, each of which introduces a mutually recursive collection of defns. The relation definition defn ... also includes a grammar production specifying how elements of the relation can be written and typeset, here

  t1 --> t2

As in the main grammar, the tokens of this syntax definition in the header must be space-separated, but usages of the syntax generally need not be. Syntax rules for each family of judgements, and for their union, are implicitly generated. The relation definition is given by a sequence of inference rules, each with a horizontal line separating a number of premises from a conclusion, for example as below.

    t1 --> t1’
    -------------- :: ctx_app_arg
    v t1 --> v t1’

The conclusion must be a symbolic term of the form of the judgement being defined. In simple cases (as here) the premises can be symbolic terms of the form of any of the defined judgements. More generally (see §6) they can be symbolic terms of a user-defined formula grammar, or in-line embedded prover code. Each rule has a name, composed of a definition family prefix (here empty), a definition prefix (here also empty) and a kernel (the ctx_app_arg).

The symbolic terms in semantic rules are parsed with a scannerless parser, built using parser combinators over character-list inputs. The parser searches for all parses of the input. If none are found, the ASCII and TeX output are annotated no parses, with a copy of the input with *** inserted at the point where the last token was read. This is often at the point of the error (though if, for example, a putative dot form is read but the two element lists cannot be anti-unified, it will be after the point of the error). If multiple parses are found, the TeX output is annotated multiple parses and the different parses are output to the console in detail during the Ott run. If the option picky_multiple_parses is set to true, multiple parses are always reported. If it set to false, a symbolic term is considered ambiguous only if two different parses compile to different strings (for a target). The parser combinators use memoization and continuation-passing to achieve reasonable performance on the small symbolic terms that are typical in semantic rules. Their performance on large (whole-program size) examples is untested. To resolve ambiguity one can add metaproductions for parentheses (as in Fig. 1), or production-name annotations in particular symbolic terms, e.g. the :t_tsub: in the AppAbs rule of the POPLmark example, There is currently no support for precedence or associativity.

This file is included in the distribution as tests/ It can be processed by executing

   bin/ott -i tests/

from the main directory. This simply reads in the file, checking that it is well-formed. Adding options:

   bin/ott -show_sort true -show_defns true -i tests/

it echos a colourised version to the screen, with metavariables in red, nonterminals in yellow, terminals in green, and object variables in white. The colourisation uses vt220 control codes; if they do not work on your screen add -colour false to the middle of the command line. To suppress the echo of the definition, add -show_sort false and -show_defns false.

3.1  Index variables

In addition to the metavar declarations above, the user can declare any number of distinguished index metavariables, e.g. by:

  indexvar index, i, j, n, m ::= {{ isa num }} {{ coq nat }} {{ hol num }} 

Given such a declaration, index, i, j, n and m can be used in suffixes, e.g. in the production

   |  ( t1 , .... , tn )           :: :: Tuple

There is a fixed ad-hoc language of suffixes, including numbers, primes, and index variables (see §23). Index metavariables cannot themselves be suffixed.

4  Generating LaTeX

The example from the previous section can already be used to generate LaTeX, for example by executing

   bin/ott -i tests/ -o out.tex 

to produce a LaTeX file out.tex. One often needs to fine-tune the default typesetting, as illustrated in Figure 2 (the Ott source) and Figure 3 (the resulting LaTeX). (The latter was built using the additional option -tex_show_meta false, to suppress display of the metaproductions.)

% minimal + latex + comments
metavar termvar, x ::=   
  {{ tex \mathit{[[termvar]]} }}

t :: ’t_’ ::=                               {{ com term    }}
  | x            ::  :: Var                   {{ com variable}}
  | \ x . t      ::  :: Lam                   {{ com lambda  }}
  | t t’         ::  :: App                   {{ com app     }}
  | ( t )        :: S:: Paren      
  | { t / x } t’ :: M:: Tsub  

v :: ’v_’ ::=                               {{ com value   }}
  | \ x . t      ::  :: Lam                   {{ com lambda  }}

terminals :: ’terminals_’ ::=
  | \            ::  :: lambda  {{ tex \lambda }}
  | -->          ::  :: red     {{ tex \longrightarrow }}

  v <:: t

Jop ::::=

 t1 --> t2 :: ::reduce::{{ com $[[t1]]$ reduces to $[[t2]]$}} by

    --------------------------  :: ax_app
    (\x.t12) v2 -->  {v2/x}t12

    t1 --> t1’
    -------------- :: ctx_app_fun
    t1 t --> t1’ t

    t1 --> t1’
    -------------- :: ctx_app_arg
    v t1 --> v t1’
Figure 2: Source:

Figure 3: Generated LaTeX: test10.2.tex

The source file has three additions to the previous file. Firstly, the metavar declaration is annotated with a specification of how metavariables should be translated to LaTeX:

  metavar termvar, x ::=   
    {{ tex \mathit{[[termvar]]} }}

Inside the {{ tex }} is some LaTeX code \mathit{$[[termvar]]$} giving the translation of a termvar or x. Here they are typeset in math italic (which in fact is also the default). Within the translation, the metavariable itself can be mentioned inside double square brackets [[ ]].

Secondly, there is a grammar for a distinguished nonterminal root terminals, with a {{ tex }} translation for each, overriding the default typesetting of some terminals. Note that the other terminals (. ( ) { } /) are still given their default typesetting.

  terminals :: ’terminals_’ ::=
    | \            ::  :: lambda  {{ tex \lambda }}
    | -->          ::  :: red     {{ tex \longrightarrow }}

Thirdly, the file has com comments, including the {{ com term }} attached to a grammar rule, the {{ com variable}} attached to a production, and the {{ com [[t1]] reduces to [[t2]]}} attached to a semantic relation. These appear in the LaTeX output as shown in Figure 3.

4.1  Specifying LaTeX for productions

One can also specify tex translations for productions, overriding the default LaTeX typesetting, e.g. as in this example of a type abstraction production.

    |  X <: T . t   :: :: TLam   {{ tex \Lambda [[X]] [[<:]] [[T]]. \, [[t]] }}

These homomorphisms, or homs1, can refer to the metavariables and nonterminals that occur in the production, e.g. the [[X]], [[T]], and [[t]] in the tex hom above, interleaved with arbitrary strings and with typeset elements of the terminals grammar, e.g. the [[<:]].

Homomorphisms are applied recursively down the structure of symbolic terms. For example, an F<: term

  (\X<:T11.t12) [T2]

would be LaTeX-pretty-printed, using the tex clause above, as

( \, \Lambda  \mathit{X} <: \mathit{T_{\mathrm{11}}} . \, \mathit{t_{\mathrm{12}}} \, )
 \, \, [ \, \mathit{T_{\mathrm{2}}} \, ]

which is typeset as below.

Note the X, T11 and t12 of the symbolic term are used to instantiate the formal parameters X, T and t of the homomorphism definition clause. If the t itself had compound term structure, e.g. as below

  (\X<:T. \X'<:T'.x)

the homomorphism would be applied recursively, producing

( \, \Lambda  \mathit{X} <: \mathit{T} . \,  \Lambda  \mathit{X'} <: \mathit{T'} 
. \,  \mathit{x} \,  \, )

typeset as follows.

Where there is no user-supplied homomorphism clause the LaTeX pretty-printing defaults to a sequence of the individual items separated by thin spaces (\,), with reasonable default fonts and making use of the terminals grammar where appropriate.

4.2  Specifying LaTeX for grammar rules

Grammar rules can include a tex hom specifying how all the nonterminal roots should be typeset, e.g.

  type, t, s :: Typ_ ::=  {{ tex \mathsf{[[type]]} }} 
        | unit                  ::   :: unit   
        | type * type’          ::   :: pair   
        | type -> type’         ::   :: fun    

Alternatively, the individual nonterminal roots can have tex homs specifying how they should be typeset:

  G {{ tex \Gamma }} , D {{ tex \Delta }} :: ’G_’ ::= 
        | empty                 ::   :: empty       
        | G , x : T             ::   :: term        

permitting the user to write G', D12 etc. in symbolic terms, to be typeset as etc.

4.3  Using the LaTeX code

The generated LaTeX code can be used in two main ways. By default, Ott generates a stand-alone LaTeX file, with a standard wrapper (including a \documentclass, various macro definitions, and a main body), that gives the complete system definition.

The default header can be overridden by writing embed {{ tex-wrap-pre ... }} and the default footer by writing embed {{ tex-wrap-post ... }} . Alternatively, the program option -tex_wrap false with the -tex_wrap false command-line argument, one can generate a file that can be included in other LaTeX files, that just defines macros to typeset various parts of the system (-tex_wrap false overrides any tex-wrap-pre/tex-wrap-post embeds).

The generated LaTeX output is factored into individual LaTeX commands: for the metavariable declarations, each rule of the syntax definition, the collected syntax (\ottgrammar), each rule of the inductive relation definitions, the collected rules for each relation, the collected rules for each defns block, the union of those (\ottdefns) and the whole (\ottall). This makes it possible to quote individual parts of the definition, possibly out-of-order, in a paper or technical report.

If one needs to include more than one system in a single LaTeX document, the ott prefix can be replaced using the -tex_name_prefix command-line argument.

The generated LaTeX is factored through some common style macros, e.g. to typeset a comment, a production, and a grammar. If necessary these can be redefined in an embed block (see Section 8.1). For example, the file tests/

 {{ tex-preamble
  {\begin{minipage}{\columnwidth}\begin{tabular}{ll}#1\end{tabular}\end{minipage} }
  {$#1$  $#2$ & $#3$}
  { \quad $#1$ \ $#2$ & \quad $#3 #4$  $#5$  $#6$}

defines a more compact style for grammars. Note that the [[TEX_NAME_PREFIX]] is replaced by whatever prefix is in force, so such style files can be reused in different contexts.

A more sophisticated LaTeX package ottlayout.sty, providing fine control of how inference rules and grammars should be typeset, is contained in the tex directory of the distribution. It is described in the manual therein.

5  Generating proof assistant definitions

To generate proof assistant definitions, for Coq, Isabelle, and HOL, the minimal Ott source file of Section 3/Figure 1 must be extended with a modest amount of additional data, as shown in Figure 4. Executing

   bin/ott  -i tests/  -o out.v  -o out.thy  -o outScript.sml 

generates Coq out.v, Isabelle out.thy, and HOL outScript.sml, shown in Figures 5, 6, and 7. The additional data can be combined with the annotations for LaTeX of the previous section, but those are omitted here.

% minimal                    + binding + subst + coq/hol/isa
metavar termvar, x ::=  
{{ isa string}} {{ coq nat}} {{ hol string}} {{ coq-equality }}

t :: ’t_’ ::=                               
  | x            ::  :: Var                  
  | \ x . t      ::  :: Lam     (+ bind x in t +)
  | t t’         ::  :: App                  
  | ( t )        :: S:: Paren   {{ icho [[t]]  }} 
  | { t / x } t’ :: M:: Tsub    {{ icho (tsubst_t [[t]] [[x]] [[t’]])}}

v :: ’v_’ ::=                              
  | \ x . t      ::  :: Lam                

  v <:: t

  single t x :: tsubst 

Jop ::::=

 t1 --> t2 :: ::reduce::by

    --------------------------  :: ax_app
    (\x.t12) v2 -->  {v2/x}t12

    t1 --> t1’
    -------------- :: ctx_app_fun
    t1 t --> t1’ t

    t1 --> t1’
    -------------- :: ctx_app_arg
    v t1 --> v t1’
Figure 4: Source:

We add four things. First, we specify proof assistant types to represent object-language variables — in this example, choosing the string type of Isabelle and HOL, and the nat type for Coq:

  metavar termvar, x ::=  
  {{ isa string}} {{ coq nat}} {{ hol string}} {{ coq-equality }}

For Coq output, one can specify {{ coq-equality proof-script }} to build a decidable equality over the Coq representation type using the proof proof-script. If the script is omitted, as in this example, it defaults to

  decide equality; auto with ott_coq_equality arith.

where the ott_coq_equality database contains the decidable equalities of the representation types defined in the source. It is possible to suppress type generation for specific metavariables or nonterminals, by adding the declaration {{ phantom }}. This is useful in some cases, for instance to avoid duplicate definitions of types already defined in an imported library. Any type homs are taken into account when the metavariable or nonterminal root is output as a type.

Second, we specify what the binding is in the object language, with the (+ bind x in t +) annotation on the Lam production:

  | \ x . t      ::  :: Lam     (+ bind x in t +)

Section 9 describes the full language of binding specifications.

Third, we add a block

    single t x :: tsubst 

to cause Ott to generate Coq/Isabelle/HOL definitions of a substitution function, with name root tsubst, replacing metavariables x by terms t. This is for single substitutions; multiple substitution functions (taking lists of substitutand/substitutee pairs) can also be generated with the keyword multiple. Substitution functions are generated for all rules of the grammar for which they might be required — here, just over t, with a function named tsubst_t.

Finally, we specify translations for the metaproductions:

  | ( t )        :: S:: Paren   {{ icho [[t]]  }} 
  | { t / x } t’ :: M:: Tsub    {{ icho (tsubst_t [[t]] [[x]] [[t’]])}}

These specify that (t) should be translated into just the translation of t, whereas {t/x}t’ should be translated into the proof-assistant application of tsubst_t to the translations of t, x, and t’. The (admittedly terse) icho specifies that these translations should be done uniformly for Isabelle, Coq, HOL, and OCaml output. One can also specify just one of these, writing {{ coq }}, {{ hol }}, {{ isa }}, or {{ ocaml }}, or include several, with different translations for each. There are also abbreviated forms ich, ic, ch, and ih. The body of a proof assistant hom should normally include outer parentheses, as in the Tsub hom above, so that it is parsed correctly by the proof assistant in all contexts.

(* generated by Ott 0.32 from: ../tests/ ../tests/ *)

Require Import Arith.
Require Import Bool.
Require Import List.
Require Import Ott.ott_list_core.

Definition var : Set := nat. (*r term variable *)
Lemma eq_var: forall (x y : var), {x = y} + {x <> y}.
  decide equality; auto with ott_coq_equality arith.
Hint Resolve eq_var : ott_coq_equality.

Inductive term : Set :=  (*r term *)
 | t_var (x:var) (*r variable *)
 | t_lam (x:var) (t:term) (*r lambda *)
 | t_app (t:term) (t':term) (*r app *).

(** subrules *)
Definition is_val_of_term (t5:term) : bool :=
  match t5 with
  | (t_var x) => false
  | (t_lam x t) => (true)
  | (t_app t t') => false

(** library functions *)
Fixpoint list_mem A (eq:forall a b:A,{a=b}+{a<>b}) (x:A) (l:list A) {struct l} : bool :=
  match l with
  | nil => false
  | cons h t => if eq h x then true else list_mem A eq x t
Arguments list_mem [A] _ _ _.

(** substitutions *)
Fixpoint tsubst_term (t5:term) (x5:var) (t_6:term) {struct t_6} : term :=
  match t_6 with
  | (t_var x) => (if eq_var x x5 then t5 else (t_var x))
  | (t_lam x t) => t_lam x (if list_mem eq_var x5 (cons x nil) then t else (tsubst_term t5 x5 t))
  | (t_app t t') => t_app (tsubst_term t5 x5 t) (tsubst_term t5 x5 t')

(** definitions *)

(* defns Jop *)
Inductive reduce : term -> term -> Prop :=    (* defn reduce *)
 | ax_app : forall (x:var) (t1 v2:term),
     Is_true (is_val_of_term v2) ->
     reduce (t_app  (t_lam x t1)  v2)  (tsubst_term  v2   x   t1 ) 
 | ctx_app_fun : forall (t1 t t1':term),
     reduce t1 t1' ->
     reduce (t_app t1 t) (t_app t1' t)
 | ctx_app_arg : forall (v t1 t1':term),
     Is_true (is_val_of_term v) ->
     reduce t1 t1' ->
     reduce (t_app v t1) (t_app v t1').

Figure 5: Generated Coq:test10.v

(* generated by Ott 0.32 from: ../tests/ ../tests/ *)
theory test10
imports Main

type_synonym "var" = "string" \<comment> \<open>term variable\<close>
datatype "term" =  \<comment> \<open>term\<close>
   t_var "var"   \<comment> \<open>variable\<close>
 | t_lam "var" "term"   \<comment> \<open>lambda\<close>
 | t_app "term" "term"   \<comment> \<open>app\<close>

(** subrules *)
is_val_of_term :: "term => bool"
"is_val_of_term (t_var x) = (False)"
| "is_val_of_term (t_lam x t) = ((True))"
| "is_val_of_term (t_app t t') = (False)"

(** substitutions *)
tsubst_term :: "term => var => term => term"
"tsubst_term t5 x5 (t_var x) = ((if x=x5 then t5 else (t_var x)))"
| "tsubst_term t5 x5 (t_lam x t) = (t_lam x (if x5 : set [x] then t else (tsubst_term t5 x5 t)))"
| "tsubst_term t5 x5 (t_app t t') = (t_app (tsubst_term t5 x5 t) (tsubst_term t5 x5 t'))"

(** definitions *)
(* defns Jop *)
inductive reduce :: "term \<Rightarrow> term \<Rightarrow> bool"
(* defn reduce *)

ax_appI: "\<lbrakk>is_val_of_term v2\<rbrakk> \<Longrightarrow>
reduce ((t_app  (t_lam x t1)  v2)) ( (tsubst_term  v2   x   t1 ) )"

| ctx_app_funI: "\<lbrakk>reduce (t1) (t1')\<rbrakk> \<Longrightarrow>
reduce ((t_app t1 t)) ((t_app t1' t))"

| ctx_app_argI: "\<lbrakk>is_val_of_term v ;
reduce (t1) (t1')\<rbrakk> \<Longrightarrow>
reduce ((t_app v t1)) ((t_app v t1'))"


Figure 6: Generated Isabelle:test10.thy

(* generated by Ott 0.32 from: ../tests/ ../tests/ *)
(* to compile: Holmake test10Theory.uo   *)
(* for interactive use:
  app load ["pred_setTheory","finite_mapTheory","stringTheory","containerTheory","ottLib"];

open HolKernel boolLib Parse bossLib ottLib;
infix THEN THENC |-> ## ;
local open arithmeticTheory stringTheory containerTheory pred_setTheory listTheory 
  finite_mapTheory in end;

val _ = new_theory "test10";

val _ = type_abbrev("var", ``:string``); (* term variable *)
val _ = Hol_datatype ` 
term =  (* term *)
   t_var of var (* variable *)
 | t_lam of var => term (* lambda *)
 | t_app of term => term (* app *)

(** subrules *)
val _ = ottDefine "is_val_of_term" `
    ( is_val_of_term (t_var x) = F)
/\  ( is_val_of_term (t_lam x t) = (T))
/\  ( is_val_of_term (t_app t t') = F)

(** substitutions *)
val _ = ottDefine "tsubst_term" `
    ( tsubst_term t5 x5 (t_var x) = (if x=x5 then t5 else (t_var x)))
/\  ( tsubst_term t5 x5 (t_lam x t) = t_lam x (if MEM x5 [x] then t else (tsubst_term t5 x5 t)))
/\  ( tsubst_term t5 x5 (t_app t t') = t_app (tsubst_term t5 x5 t) (tsubst_term t5 x5 t'))
(** definitions *)
(* defns Jop *)

val (Jop_rules, Jop_ind, Jop_cases) = Hol_reln`
(* defn reduce *)

( (* ax_app *) ! (x:var) (t1:term) (v2:term) . (clause_name "ax_app") /\
((is_val_of_term v2))
( ( reduce (t_app  (t_lam x t1)  v2)  (tsubst_term  v2   x   t1 )  )))

/\ ( (* ctx_app_fun *) ! (t1:term) (t:term) (t1':term) . (clause_name "ctx_app_fun") /\
(( ( reduce t1 t1' )))
( ( reduce (t_app t1 t) (t_app t1' t) )))

/\ ( (* ctx_app_arg *) ! (v:term) (t1:term) (t1':term) . (clause_name "ctx_app_arg") /\
((is_val_of_term v) /\
( ( reduce t1 t1' )))
( ( reduce (t_app v t1) (t_app v t1') )))


val _ = export_theory ();

Figure 7: Generated HOL:test10Script.sml

5.1  Proof assistant code for grammar rules

The normal behaviour is to generate a free proof assistant type for each (non-subrule, non-phantom) grammar rule. For example, the Coq compilation for t here generates a free type with three constructors:

  Inductive term : Set := 
   | t_var (x:var)
   | t_lam (x:var) (t:term)
   | t_app (t:term) (t':term).

(note that the metaproductions do not give rise to constructors).

Remark: prior to version 0.20.2, the free type generated for Coq was

  Inductive term : Set := 
   | t_var : var -> term
   | t_lam : var -> term -> term
   | t_app : term -> term -> term.

but we found that trying to preserve the names specified by the user is helpful later, when doing proofs. Whenever a clash is detected, or for list forms, the wildcard _ is used. The old behaviour can be obtained via the top-level option -coq_names_in_rules false.

By default the order of the arguments to those constructors follows the order in which they appear in the production. That can be overridden with an order hom. For example, if for some reason (perhaps compatibility with other Coq code) one wished the arguments to t_Lam to be reversed:

   | t_Lam : t -> termvar -> t

one could add an order hom as below.

  | \ x . t      ::  :: Lam  {{ order [[t]] [[x]] }}

Instead of using the generated free type, one can specify an arbitrary proof assistant representation type, annotating the grammar rule with a coq, isa, hol, or ocaml hom — for example, in the following grammar for substitutions.

  s {{ tex \sigma }} :: ’S_’ ::= {{ com multiple subst }} {{ isa (termvar*t) list }}
       | [ x |-> t ]            ::   :: singleton   {{ isa [ ([[x]],[[t]]) ]  }}
       | s1 , .. , sn           ::   :: list        {{ isa List.concat [[s1 .. sn]] }}

Here the {{ isa (termvar*t) list }} hom specifies that in Isabelle output this type be represented as an Isabelle (termvar*t) list instead of the default free inductive type; all the productions are metaproductions (tagged M); and isa homs for each production specify how they should be translated into that Isabelle type. This feature must be used with care, as any Ott-generated functions, e.g. substitution functions, cannot recurse through such user-defined types.

Grammar rules (whether free or non-free) can also include a coq equality hom, instructing the Coq code generator to derive a decidable equality for the Coq representation type. For example, the ML polymorphism Ott source of includes the following.

  typvar :: TV_ ::= {{ coq-equality decide equality. apply eq_value_name_t. }}
       | ’ ident                ::   :: ident

The Coq/HOL/Isabelle/OCaml type name for a grammar rule, or for a metavariable declaration, is normally taken to be just its primary nonterminal root. Occasionally it is useful to work around a clash between a metavar or nonterminal primary root and a proof assistant symbol, e.g. T in HOL or value in Isabelle. For this, one can add a coq, hol, isa, or ocaml hom to the primary nonterminal root. In the example below, the user can write T, T’ etc. in their Ott source, but the generated HOL type is Typ.

T {{ hol Typ }}, S, U :: ’T_’ ::=                    {{ com type  }}
  | T -> T’                          :: :: Fun          {{ com type of functions }}

The grammar rules within each grammar block of a syntax definition may depend on each other arbitrarily. When generating Isabelle/Coq/HOL/OCaml representation types, however, they are topologically sorted, to simplify the resulting induction principles.

5.2  Proof assistant code for inductive definitions

The semantic relations are defined with the proof-assistant inductive relations packages, Inductive, Hol_reln, and inductive_set or inductive, respectively. Each defns block gives rise to a potentially mutually recursive definition of each defn inside it (it seems clearer not to do a topological sort here). Definition rules are expressed internally with symbolic terms. We give a simplified grammar thereof in Fig. 17, omitting the symbolic terms for list forms. A symbolic term st for a nonterminal root is either an explicit nonterminal or a node, the latter labelled with a production name and containing a list of symterm_elements, which in turn are either symbolic terms, metavariables, or variables. Each definition rule gives rise to an implicational clause, essentially that the premises (Ott symbolic terms of the formula grammar) imply the conclusion (an Ott symbolic term of whichever judgement is being defined). Symbolic terms are compiled in several different ways:

Further, for each nonterminal of a non-free grammar rule, e.g. a usage of v' where v<::t, an additional premise invoking the generated subrule predicate for the non-free rule is added, e.g. is_v v'. For Coq and HOL, explicit quantifiers are introduced for all variables mentioned in the rule. For HOL, rules are tagged with their rule name (using clause_name).

5.3  Representation of binding

At present the generated Isabelle/Coq/HOL uses fully concrete representations of variables in terms, without any notion of alpha equivalence, as one can see in Fig. 6: see the t datatype of terms and the tsubst_t substitution function there. An experimental Coq backend generates definitions in locally-nameless style for a subset of the Ott metalanguage. This is work-in-progress, and it is extensively documented in We intend in future to generate other representations, and in some circumstances homs can be used to implement other representations directly. For a reasonably wide variety of languages, however, one can capture the intended semantics of whole programs in this idiom, subject only to the condition that standard library identifiers are not shadowed within the program, as the operational semantics does not involve reduction under binders — so any substitutions are of terms which (except for standard library identifiers) are closed. This includes the ML polymorphism example of For languages which require a type environment with internal dependencies, however, for example F<:, this is no longer the case. The POPLmark F<: example given in has a type system which disallows all shadowing, a property that is not preserved by reduction. However, a correct translation of F<: is generated by the Coq locally-nameless backend, and can be found in

Further discussion of binding representations is in the Ott ICFP 2007 paper and in a working draft

Binding and Substitition. Susmit Sarkar, Peter Sewell, and Francesco Zappa Nardelli. August 2007.

available from the Ott web page.

5.4  Helper functions for free variable and substitution functions

The generated free variable and substitution functions in the Coq output (e.g., in Figure 5) often rely on a few standard library functions: list_mem, list_assoc, list_minus, list_minus2. In order to avoid dependencies on external libraries for defining those functions, by default Ott generates the definitions for any such functions it uses. It is possible to turn off the generation of definitions for these such functions by writing the following directive early on in the source file:

embed {{ coq-lib list_mem list_minus }}

This instructs Ott to avoid generating definition for list_mem and list_minus, but to continue generating definitions for other functions such as list_assoc and list_minus2.

Note about list_minus2: Instead of using the function list_minus2, earlier versions of Ott generated equivalent code based on list_filter, which was more difficult to reason about. For backwards compatibility, however, we provide the command-line option -coq_use_filter_fn for generating a definition using the older code pattern.

5.5  Correctness of the generated proof assistant code

We have attempted to ensure that the proof assistant definitions generated by Ott are well-formed and what the user would intend. This is not guaranteed, however, for several reasons: (1) There may be name clashes between Ott-generated identifiers and proof assistant built-in identifiers (or, in pathological cases, even among different Ott-generated identifiers). (2) In some cases we depend on automatic proof procedures, e.g. for HOL definitions. These work in our test cases, but it is hard to ensure that they will in all cases. More importantly, (3) the generation process is complex, so it is quite possible that there is either a bug in Ott or a mismatch between the user expectation and what the tool actually does. Ultimately one has to read the generated proof assistant definitions to check that they are as intended — but typically one would do this in any case, many times over, in the process of proving metatheoretic results, so we do not consider it a major issue.

5.6  Using the generated proof assistant code

Note added 2017-11-30: the following is out of date.

Ott builds code for

Coq 8.3
HOL 4 (the current svn version)
Isabelle/HOL (Isabelle 2011)

Given proof assistant files in the top-level directory of the distribution, as produced at the start of this section (Coq out.v, Isabelle out.thy, and HOL outScript.sml), the various proof assistants can be invoked as follows.

5.6.1  Coq

First run


in the coq directory of the distribution, to build the auxiliary files. These include a core file (ott_list_core) of definitions that are used in Ott-generated output. At present these are only required when Coq native lists are used. There are also various lemmas (in ott_list.v) which may be useful; they can be made available with Require Import ott_list.

For batch mode run

  coqc -I coq  out.v

where coq is the path to the coq directory of the distribution.

The experimental locally-nameless backend requires the Metatheory library by Arthur Chargueraud, available from the project web page.

5.6.2  HOL

First run


in the hol directory of the distribution, to build the auxiliary files.

For batch mode run

  Holmake -I hol  outTheory.uo

where hol is the path to the hol directory of the distribution. For interactive mode, run

  hol -I hol

inside an editor window (where the second hol is again the path to the hol directory of the distribution), and in another window view the outScript.sml file. First paste in the app load command from a comment at the top of the file, then paste in the remainder.

5.6.3  Isabelle

For batch mode:

  echo 'ML_command {* (use_thy "Tmp"; OS.Process.exit OS.Process.success) handle e => (OS.Process.exit OS.Process.failure); *}' | /usr/local/Isabelle/bin/isabelle tty 

Interactively, using Proof General:

  isabelle emacs out.thy

6  Judgments and formulae

In a semantic rule, for example

    t1 --> t1’
    -------------- :: ctx_app_arg
    v t1 --> v t1’

the conclusion must be a symbolic term of the form of the judgement being defined, but in general the premises may be symbolic terms of a formula grammar or in-line embedded prover code. By default the formula grammar includes all the defined judgement forms: for the running example Ott will synthesise grammars as below.

The user can also define an explicit formula grammar, to let other forms (not just judgements) appear as rule premises. Below is a fragment of the formula grammar from the LJ example on the Ott web page.

  formula :: formula_ ::=  
   |  judgement                       ::   :: judgement
   |  formula1 .. formulan            ::   :: dots
   |  not formula                     :: M :: not
          {{ tex \neg [[formula]] }} 
          {{ isa \<not> ([[formula]]) }}
   |  ( formula )                     :: M :: brackets
          {{ tex ([[formula]]\!) }}  
          {{ isa [[formula]] }}
   |  formula \/ formula:: M :: or
          {{ tex [[formula]] \vee [[formula]] }}
          {{ isa [[formula]] \<or> [[formula]] }}
   |  formula /\ formula:: M :: and
          {{ tex [[formula]] \wedge [[formula]] }}
          {{ isa [[formula]] \<and> [[formula]] }}
   |  x = x’                          :: M :: xali
          {{ isa [[x]] = [[x’]] }}
   |  X = X’                          :: M :: Xali
          {{ isa [[X]] = [[X’]] }}

This example adds (to the judgement forms) syntax for parenthesised formulae, negation, and, or, and equality testing on two sorts. For each, tex and isa homs specify how they should be typeset and be translated into Isabelle.

If the user defines a formula grammar then (as here) the production name prefix must be formula_ and the name for the judgement production must be judgement.

6.1  Naming of premises for the Coq backend

It is possible to specify the names of premises of inductive predicates; these names are then used by the Coq backend, and are often useful in proofs. For instance, we can call RED the hypothesis in the rule below

    t1 --> t1’  [[:RED]]
    -------------- :: ctx_app_arg
    v t1 --> v t1’

which will then generate the following Coq code:

| ctx_app_arg : forall (v t1 t1’:term)
    (RED: reduce t1 t1’),
    is_val_of_term v ->
    reduce (t_app v t1) (t_app v t1’).

Names of rules cannot contain spaces or other non alpha-numerical characters, and must begin with a letter. The name annotation must at the rightmost place on the hypothesis line, and must be enclosed (without spaces) between the [[: and ]] parentheses.

6.2  In-line embedded prover code in premises

Instead of adding a formula production, one can directly embed prover code as a premise, delimited as below by {{ and }}. Within that, text will be echoed directly to a prover (or given a default LATEX typesetting) except that symbolic terms enclosed within [[ and ]] will be processed as in an embed section.

  {{ type_to_chunk ([[typeof e1]]) = Some [[c]] }}
  ----------------------------------------------------------- :: Assign1
  e1=e2 . k |env   --tau-->    lval(e1) . [__=c e2] . k |env

6.3  User syntax

The tool also synthesises a user_syntax grammar of all the user syntax, for example:

This is used for parsing top-level strings, for example when filtering embedded code (§8).

7  Concrete terms and OCaml generation

In semantic definitions, one typically never uses concrete variables, only metavariables that range over them. In examples, however, one may need either a mix of concrete variables and metavariables, or, for strictly concrete terms, to restrict to just the former (and also to prohibit symbolic nonterminals).

Figure 2 combines the LaTeX and proof assistant annotations of Sections 3 and 4, adding a {{ lex alphanum}} hom to the metavar declaration to specify the lexical form of concrete variables of this sort. At present a lex homomorphism must have body either Alphanum (standing for [A-Z]([A-Z]|[a-z]|[0-9]|'|_)*), alphanum (for ([A-Z]|[a-z])([A-Z]|[a-z]|[0-9]|'|_)*), alphanum0 (for [a-z]([A-Z]|[a-z]|[0-9]|'|_)*), or numeral (for [0-9][0-9]*); more general regular expressions are not supported. An identifier that can be ambiguously lexed as either a concrete or symbolic metavariable, e.g. x in the scope of the above declaration, will be taken to be symbolic. To restrict the parser to strictly concrete terms only, one can add a :concrete: prefix, as shown in Figure 10.

One can also specify how concrete variables should be LaTeX’d or translated into a proof assistant, e.g. with homomorphisms {{ texvar \mathrm{[[termvar]]}} and {{ isavar ’’[[termvar]]’’}} (and similarly coqvar, holvar, and ocamlvar).

Figure 2 also specifies an OCaml representation type for variables, with the metavar hom {{ ocaml int}}. Executing

   bin/ott -i tests/ -o  

produces the OCaml code shown in Figure 9, including OCaml types to represent the abstract syntax, and auxiliary functions for subrules and substitutions. This does not implement the semantic rules. In some cases the various proof assistant code extraction facilities can be used — see Section 21.

% all
metavar termvar, x ::=   {{ com  term variable }} 
{{ isa string}} {{ coq nat}} {{ hol string}} {{ coq-equality }}
{{ ocaml int}} {{ lex alphanum}} {{ tex \mathit{[[termvar]]} }}

t :: ’t_’ ::=                               {{ com term    }}
  | x            ::  :: Var                   {{ com variable}}
  | \ x . t      ::  :: Lam (+ bind x in t +) {{ com lambda  }}
  | t t’         ::  :: App                   {{ com app     }}
  | ( t )        :: S:: Paren                 {{ icho [[t]]  }} 
  | { t / x } t’ :: M:: Tsub  
                        {{ icho (tsubst_t [[t]] [[x]] [[t’]])}}

v :: ’v_’ ::=                               {{ com value   }}
  | \ x . t      ::  :: Lam                   {{ com lambda  }}

terminals :: ’terminals_’ ::=
  | \            ::  :: lambda  {{ tex \lambda }}
  | -->          ::  :: red     {{ tex \longrightarrow }}

  v <:: t

  single t x :: tsubst 

Jop ::::=

 t1 --> t2 :: ::reduce::{{ com $[[t1]]$ reduces to $[[t2]]$}} by

    --------------------------  :: ax_app
    (\x.t12) v2 -->  {v2/x}t12

    t1 --> t1’
    -------------- :: ctx_app_fun
    t1 t --> t1’ t

    t1 --> t1’
    -------------- :: ctx_app_arg
    v t1 --> v t1’
Figure 8: Source:

(* generated by Ott 0.32 from: ../tests/ ../tests/ *)

type var = int (* term variable *)

term =  (* term *)
   T_var of var (* variable *)
 | T_lam of var * term (* lambda *)
 | T_app of term * term (* app *)

(** subrules *)
let is_val_of_term (t5:term) : bool =
  match t5 with
  | (T_var x) -> false
  | (T_lam (x,t)) -> (true)
  | (T_app (t,t')) -> false

(** substitutions *)
let rec tsubst_term (t5:term) (x5:var) (t_6:term) : term =
  match t_6 with
  | (T_var x) -> (if x=x5 then t5 else (T_var x))
  | (T_lam (x,t)) -> T_lam (x,(if List.mem x5 ([x]) then t else (tsubst_term t5 x5 t)))
  | (T_app (t,t')) -> T_app ((tsubst_term t5 x5 t),(tsubst_term t5 x5 t'))

(** definitions *)

Figure 9: Generated OCaml code:

8  Filtering: Using Ott syntax within LaTeX, Coq, Isabelle, HOL, or OCaml

8.1  Filtering embedded code

It is possible to embed arbitrary code in the Ott source using an embed block, which can contain tex, coq, hol, isa, or ocaml homomorphisms, the bodies of which will appear in the respective output. The embed keyword should be on a line by itself). For example, contains the following to define Coq and HOL remove_duplicates functions.

{{ coq
Fixpoint remove_duplicates (l:list typvar_t) : list typvar_t :=
  match l with
  | nil => nil
  | cons h t => if (list_mem eq_typvar_t h t)  then remove_duplicates t
                else cons h (remove_duplicates t)
end. }}

{{ hol
val _ = Define ‘
  (remove_duplicates [] = []) /\
  (remove_duplicates (x::xs) = if (MEM x xs) then remove_duplicates xs 
                               else x::(remove_duplicates xs))
‘; }}

Within the body of an embed homomorphism, any text between [[ and ]] will be parsed as a symbolic term (of the user_syntax grammar) and pretty printed, so one can use user syntax within LaTeX or proof assistant code. An Isabelle example is below, defining an Isabelle function to calculate the order of a type with productions unit, t*t', and t->t'.

{{ isa
order :: "type => nat"
"order [[unit]] = 0"
"order [[t*t’]] = max (order [[t]]) (order [[t’]])"
"order [[t->t’]] = max (1+order [[t]]) (order [[t’]])"


It is often useful to define a proof assistant function, in an embed section, together with a production of the formula grammar with a proof assistant hom that uses that function, thereby introducing syntax that lets the function be used in semantic rules.

Ott also permits embed blocks with tex-preamble, homs, whose contents appear in the generated LATEX preamble. Any definitions of LATEX commands must appear in such a tex-preamble section.

8.2  Filtering files

Similar processing can be carried out on separate files, using the command-line options tex_filter, isa_filter, etc. Each of these takes two arguments, a source filename and a destination filename. In processing the source file, any text between [[ and ]] will be parsed as a symbolic term (of the user_syntax grammar) and pretty printed in the appropriate style. All other text is simply echoed.

Typical usage for LaTeX would be something like this (from the Makefile used to produce this document):

test7.tex: ../src/ott ../tests/ ../tests/test7tt.mng
         cd ../src; make
         ../src/ott                                        \
                -i ../src/              \
                -o test7.tex                               \
                -tex_show_meta false                       \
                -tex_wrap false                            \
                -tex_name_prefix testSeven                 \
                -tex_filter ../tests/test7tt.mng test7tt.tex 

The -tex_wrap false turns off output of the default LaTeX document preamble, so the generated file test7.tex just contains LaTeX definitions. The -tex_name_prefix testSeven sets a prefix for the generated LaTeX commands (so the LaTeX definitions from multiple Ott source files can be included in a single LaTeX document). The -tex_filter argument takes two filenames, a source and a destination. It filters the source file, (roughly) replacing any string found within [[ ]] by the tex pretty-print of its parse. This parsing is done w.r.t. the generated nonterminal user_syntax which is a union of all the user’s grammar.

At present munged strings are not automatically put within $ $, and there is no analogue of the <[ ]> of our previous munger.

The lexing turns any sequence of [ (resp. of ]) of length n+1 for n>2 into a literal sequence of length n.

Figures 10 and 11 show a source file (test7tt.mng) that uses terms of the F<: definition of, and the result of filtering it.

Similar filtering can be performed on Coq, Isabelle, HOL, and OCaml files.

To filter files with respect to a relatively stable system definition, without having to re-process the Ott source files of that system definition each time, there are command-line options

  -writesys <filename>                Output system definition
  -readsys <filename>                 Input system definition

to first write the system definition (generated from some source files) to a file, and then to read one back in (instead of re-reading the Ott source files). The saved system definitions are in an internal format, produced using the OCaml marshaller, and contain OCaml closures. They therefore will not be compatible between different Ott versions. They may also be quite large.

We can TeX-typeset symbolic terms of the language, e.g.
\[ [[ (\X<:Top. \x:X.x) [Top->Top] ]]\]
and concrete terms
\[ [[ :concrete: \Z1<:Top. \x:Z1.x ]]\]
and similarly judgements etc, e.g.
\[ [[G |- t : T ]] \]
Here is an extract of the syntax:


and a single semantic rule:


and a judgement definition:


One can also include a ‘defns’ collection of judgements, or the complete definition.
% \section{Full Definition}
% \testSevenmetavars\\[0pt]
% \testSevengrammar\\[0pt]
% \testSevendefnss
% \testSevenall

Figure 10: F<: Extracts: LaTeX source file to be filtered (test7tt.mng)

Figure 11: F<: Extracts: the filtered output (test7tt.tex)

9  Binding specifications

Our first example involved a production with a single binder:

specified by the source shown in Figure 4:

  | \ x . t      ::  :: Lam     (+ bind x in t +)

in which a single variable binds in a single subterm. Realistic programming languages often have much more complex binding structures, e.g. structured patterns, multiple mutually recursive let definitions, comprehensions, or-patterns, and dependent record patterns.

Ott has a flexible metalanguage for specifying binding structures, expressive enough to cover these. It comprises two forms of annotation on productions. The first, lets one specify that variables bind in nonterminals of the production, as in the Lam production above. Here is a metavariable set expression, e.g. in that lambda production just the singleton metavariable of the production. A variable can bind in multiple nonterminals, as in the example of a simple recursive let below.

More complex examples require one to collect together sets of variables. For example, the grammar below (shown in Ott source and the generated LaTeX) has structured patterns, with a production in which all the binders of the pattern bind in the continuation

t :: E_ ::=
  | x                        ::   :: ident
  | ( t1 , t2 )              ::   :: pair
  | let p = t in t’          ::   :: letrec      (+ bind binders(p) in t’ +)

p :: P_ ::=
  | _                        ::   :: wildcard    (+ binders = {} +)
  | x                        ::   :: ident       (+ binders = x +)
  | ( p1 , p2 )              ::   :: pair        (+ binders = binders(p1) union binders(p2) +)

This is expressed with the second form of annotation: user-defined auxiliary functions such as the binders above. This is an auxiliary function defined over the grammar that identifies a set of variables to be used in the bind annotation on the production. There can be any number of such auxiliary functions; binders is not a distinguished keyword.

The syntax of a precise fragment of the binding metalanguage is given in Fig. 12,

Figure 12: Mini-Ott in Ott: the binding specification metalanguage

where we have used Ott to define part of the Ott metalanguage. A simple type system (not shown) enforces sanity properties, e.g. that each auxiliary function is only applied to nonterminals that it is defined over, and that metavariable set expressions are well-sorted.

Further to that fragment, the tool supports binding for the list forms of §12. Metavariable set expressions can include lists of metavariables and auxiliary functions applied to lists of nonterminals, e.g. as in the record patterns below.

This suffices to express the binding structure of almost all the natural examples we have come across, including definitions of mutually recursive functions with multiple clauses for each, Join calculus definitions [FGL+96], dependent record patterns, and many others.

10  Generating substitution and free variable functions

The tool can generate Isabelle/Coq/HOL/OCaml code for both single and multiple substitution functions. For example, the ML polymorphism Ott source of includes the following.

    single   expr value_name :: subst  
    multiple typexpr typvar  :: tsubst 

This causes the generation of two families of substitution functions, one replacing a single value_name by a expr, the other replacing multiple typvars by typexprs.

Each family contains a function for each datatype for which it is required, so in that example there are functions subst_expr for the first and tsubst_typexpr, tsubst_typscheme and tsubst_G for the second.

The functions for substitutions declared by

    single   this that :: name1 
    multiple this that :: name2 

replaces terms of productions consisting just of a single that by a this. Here this must be a nonterminal root, while that can be either a metavariable root or a nonterminal root (the latter possibility allows substitution for compound identifiers, though it is not clear that this is generally useful enough to be included). Substitution functions are generated for each member of each (mutually recursive) block of grammar rules which either contain such a production or (indirectly) refer to one that does.

At present multiple substitutions are represented by Isabelle/Coq/HOL/OCaml lists, so for the example above we have Isabelle

  tsubst_typexpr :: "(typvar*typexpr) list => typexpr => typexpr"
  tsubst_typscheme :: "(typvar*typexpr) list => typscheme => typscheme"
  tsubst_G :: "(typvar*typexpr) list => G => G"

The generated functions do not substitute bound things, and recursive calls under binders are filtered to remove the bound things.

Similarly, the tool can generate Isabelle/Coq/HOL/OCaml to calculate the free variables of terms. For example, the ML polymorphism Ott source of includes the following.

    typexpr typvar :: ftv

This causes Isabelle functions as below to be generated, calculating the free typvars that occur in singleton productions in the typexpr grammar, within terms of all types.

  ftv_typexpr :: "typexpr => typvar list"
  ftv_typscheme :: "typscheme => typvar list"
  ftv_G :: "G => typvar list"

11  Locally-nameless representation

The Coq backend of Ott includes experimental support for a locally-nameless representation (and co-finite quantification).

The user must specify which metavariables require a locally-nameless representation via the repr-locally-nameless hom, e.g.:

metavar x ::= {{ repr-locally-nameless }} {{ com  term variable  }} 

As usual, metavariables can be bound in productions, using the bindspec language, as in the lam production below:

  t :: ’t_’ ::=                                         {{ com term }}
    | x                   ::   :: Var                     {{ com variable }}         
    | \ x . t             ::   :: Lam  (+ bind x in t +)  {{ com abstraction }}      
    | t t’                ::   :: App                     {{ com application }}      
    | ( t )               :: S :: paren   {{ coq [[t]] }} 
    | { t / x } t’        :: M :: tsub    {{ coq (t_subst_t [[t]][[x  t’]]) }}

This definition gives rise to the datatype term below (here with option -coq_names_in_rules false):

Inductive term : Set := 
| term_var_b : nat -> term 
| term_var_f : var -> term 
| term_lam : term -> term 
| term_app : term -> term -> term.


  1. Productions containing metavariables susceptible to be bound (e.g., term_var) give rise to two distinct constructors, one (term_var_b) for de Bruijn indices to be used when the metavariable is bound, one (term_var_f) for "free" variables. The type var, together with decidable equality and several useful lemmas and functions, is defined in the Metatheory library.

    In the current implementation, metavariables susceptible to be bound in a symbolic term (eg. the x in the term_var production) must be the only element of the production.

  2. Binder metavariables are erased from productions (eg. term_lam), as in de Bruijn representation.

Ott automatically generates the appropriate open functions and lc predicates to test if terms are locally-closed. The other support functions for substitutions and free-variables (subst and fv) are generated once the user declares the relevant substitutions and freevars sections.

Ott automatically compiles the symbolic terms that appear in rule definitions into the appropriate terms in locally-nameless style. For instance, the typing rule for the simply-typed lambda-calculus:

    E,x:S |- t : T
    ------------------ :: lambda
    E |- \x.t : S->T

is compiled into its locally-nameless representation:

Inductive typing : env -> term -> type -> Prop := (* defn typing *) 
| ... 
| typing_lambda : forall (L:vars) (E:env) (t:term) (S T:type), 
   (forall x, x \notin L -> typing (E & x ~ S) (open_term_wrt_term t (term_var_f x)) T) -> 
   typing E (term_lam t) (type_arrow S T).

For that, Ott follows the algorithm below. For each rule,

  1. for each nonterminal that appears in the rule, compute the maximal set of binders under which it appears: for example, in the rule lambda above, the maximal set of binders for the nonterminal t is the singleton x, and it is empty for all the other nonterminals;
  2. for each pair nonterminal / maximal binder set collected in phase 1., go over all the occurrences of the nonterminal in the rule and open them with respect to all the variables in the maximal binding set except those under which this particular occurrence is bound. In the example, this amounts to opening the occurrence of t in the premise with respect to the metavariable x;
  3. quantify using cofinite-quantification each metavariable that has been used to open a nonterminal;
  4. add hypothesis about local-closure to guarantee the invariant that if a derivation holds, then the top-level terms involved are locally-closed.

In some cases the user may want a finer control on which nonterminals are opened and with respect to which metavariables. Consider for instance the CBV beta-reduction rule:

    --------------------------  :: ax_app
    (\x.t1) v2 -->  {v2/x}t1

A naive application of the algorithm described above would open the right hand side occurrence of t1 with respect to a cofinitely-quantified x. Substitution should then be used to replace the occurrences of x with v2, resulting in the awkward term

reduce (term_app (term_lam t1) v2) (subst_term v2 x (open_term_wrt_term t1 (term_var_f x)))

Instead, an idiomatic translation of CBV beta-reduction rule would directly rely on the open function to substitute v2 for the bound occurrences of x in t1, as in:

reduce (term_app (term_lam t1) v2) (open_term_wrt_term t1 v2)

A special syntax for production homomorphisms allow the user to specify this translation:

    | { t / x } t’        :: M :: tsub    {{ coq (t_subst_t [[t]][[x t’]]) }}

In the homomorphism the nonterminal t’ is referred to with [[x t’]] instead of the usual [[t’]]: the prefixed x specifies that occurrences of t’ should not be opened with respect to the metavariable x. If this homomorphism is specified, then the translation of the ax_app rule is exactly idiomatic Coq shown above.

Current limitations: support for single binders only, no auxfn, Coq only.

Disclaimer: to compile rule definitions, Ott applies blindly the algorithm described above. Although in most of the cases, this generates a correct and idiomatic representation of the language, some language constructs might not be faithfully translated. Please, let us know if you find one of these cases.

If Ott is invoked with the -coq_lngen option, then the generated locally-nameless Coq code is compatible with Aydemir’s lngen tool ( sweirich/papers/lngen/).

12  List forms

Ott has direct support for lists, both as dot forms such as and as list comprehensions such as . Figure 13 shows an example semantic rule taken from our OCaml fragment semantics, as both the generated LaTeX and its Ott source, that involves several dot forms. Other types commonly used in semantics, e.g. finite maps or sets, can often be described with this list syntax in conjunction with type and metaproduction homs to specify the proof assistant representation. When using list forms, one usually also wants to add a list-of-formula production to the formula grammar, e.g. (as in

  formula :: formula_ ::=  
   |  judgement                       ::   :: judgement
   |  formula1 .. formulan            ::   :: dots

The proof assistant code generation for such a production (which must be named formula_dots) is special-cased to a list conjunction.

 E |- e1 : t1 ... E |- en : tn
 E |- field_name1 : t->t1 ... E |- field_namen : t->tn
 t = (t1’, ..., tl’) typeconstr_name
 E |- typeconstr_name gives typeconstr_name:kind {field_name1’; ...; field_namem’}
 field_name1...field_namen PERMUTES field_name1’...field_namem’
 length (e1)...(en)>=1
 -------------------------------------------------------------------------- :: record_constr
 E |- {field_name1=e1; ...; field_namen=en} : t
Figure 13: A sample OCaml semantic rule, in LaTeX and Ott source forms

12.1  List dot forms

Example productions for record types, record terms, and record patterns are shown below, in both Ott source and LaTeX, taken from our F<: example.

T, S, U :: ’T_’ ::=                                               {{ com type  }}
  | { l1 : T1 , .. , ln : Tn }      :: :: Rec                       {{ com record }}           

t :: ’t_’ ::=                                                     {{ com  term  }}
  | { l1 = t1 ,  .. , ln = tn }     :: :: Rec                       {{ com record }}
  | let p = t in t’                 :: :: Let (+ bind b(p) in t’ +) {{ com pattern binding}}

p :: ’P_’ ::=                                                     {{ com  pattern }}
  | x : T                           :: :: Var (+ b = x  +)          {{ com variable pattern }}
  | { l1 = p1 , .. , ln = pn }      :: :: Rec (+ b = b(p1 .. pn) +) {{ com  record pattern  }}

Dot forms can be used in symbolic terms in semantic rules:

Individually indexed projections from dot forms can be mentioned, eg the lj below:

Symbolic terms can also include concatenations of two dot forms with a singleton in between:

Multiple dot forms within the same semantic rule can share bounds (e.g. 1..m):

In more detail, productions can have dot tokens interspersed between the elements. Dot tokens consist of two, three or four consecutive dots (.., ..., or ....), indicating lists with minimum lengths 0, 1, and 2 respectively (these length minimums are respected only when parsing concrete lists; they are not present in Isabelle/Coq/HOL output). The tool identifies the maximal sequence of elements on either side of the dots that are identical modulo anti-unification of some index. Optionally, there may also be a single terminal on either side of the dot token, separating instances of the repeated unit. For example, in the production

  | { l1 = t1 ,  .. , ln = tn }  :: :: Rec                     

there is such a terminal (the ‘,’). The tool identifies that l1 = t1 and ln = tn can be anti-unified as (roughly) l_ = t_, taking _ to be the bounds 1 and n. A single production may contain multiple dot forms, but they must not overlap; nested dot forms (including those with multiple changing indices) are not currently supported.

Homomorphisms and binding specifications are generalised to match: an mse can involve a dot form of metavariables; a dot form of nonterminals; or an auxiliary function applied to a dot form of nonterminals (e.g. the b( above). Dot forms on the right of a bind are not currently supported.

LaTeX homomorphisms should not refer to dot forms, as either an error or bad output will be generated. (For LaTeX, there should really be some means to specify a homomorphism for the repeated expression, and also data on how any list separators should be typeset. This would require more special-case treatment, which is not currently supported.)

12.2  List comprehension forms

Lists can also be expressed as explicit list comprehensions, for more concise typesetting. Three different styles are supported, with no bounds, an upper bound, or a lower and upper bound. For example, in a symbolic term, instead of the dot form

  G |- t1:T1  ..  G |- tn:Tn

one can write any of the following

   </ G |- ti:Ti // i           />
   </ G |- ti:Ti // i IN n      />
   </ G |- ti:Ti // i IN 1 .. n />

Similar comprehensions can be used in productions, for example lines 2–4 below. In addition, comprehensions in productions can specify a terminal to be used as a separator in concrete lists, as in lines 5–7 below. (These examples are taken from

  |  l1 = t1 ,  .. , ln = tn            :: :: Rec              {{ com dots }}
  |  </ li = ti // i           />       :: :: Rec_comp_none    {{ com comp }}
  |  </ li = ti // i IN n      />       :: :: Rec_comp_u_none  {{ com compu }} 
  |  </ li = ti // i IN 1 .. n />       :: :: Rec_comp_lu_none {{ com complu }}  
  |  </ li = ti // , // i           />  :: :: Rec_comp_some    {{ com comp with terminal }}
  |  </ li = ti // , // i IN n      />  :: :: Rec_comp_u_some  {{ com compu with terminal }} 
  |  </ li = ti // , // i IN 1 .. n />  :: :: Rec_comp_lu_some {{ com complu with terminal }}  

In Coq, HOL or Isabelle output, list dot forms and the various list comprehension forms are treated almost identically. In LaTeX output, comprension forms are default-typeset with overbars. For example, the rules below

 G|- t:l1:T1,..,ln:Tn
 ----------------------- :: Proj_dotform
 G|- t.lj : Tj

 G|- t:  </ li:Ti // i/> 
 ---------------------------------- :: Proj_comp
 G|- t.lj : Tj

 G|- t:  </ li:Ti // i IN n/> 
 ---------------------------------- :: Proj_comp_u
 G|- t.lj : Tj

 G|- t:  </ li:Ti // i IN 1..n/> 
 ---------------------------------- :: Proj_comp_lu
 G|- t.lj : Tj

are typeset as follows.

Upper bounds of the form n−1 are also permitted, e.g. with

 G|- t:l0:T0,..,ln-1:Tn-1
 ----------------------- :: Proj_dotform_minus
 G|- t.lj : Tj

 G|- t:  </ li:Ti // i IN 0..n-1/> 
 ---------------------------------- :: Proj_comp_lu_minus
 G|- t.lj : Tj

typeset as below. More complex arithmetic expressions are not currently supported.

A list form used in a symbolic term does not have to be in the same style as that in the corresponding production. However, if a metavariable or nonterminal occurs in multiple different list forms in the same inference rule, they must all be in the same style and with the same bounds. Moreover, in a production, a list form in a bindspec or homomorphism must be in the same style and with the same bounds as the corresponding list form in the elements of the production.

The comprehension form without an upper bound, e.g. </ G |- ti:Ti // i />, typeset as is not standard notation, but is often very useful. Many semantic rules involve lists of matched length, e.g. of the and here, but do not need to introduce an identifier for that length; omitting it keeps them concise.

The default visual style for typesetting list comprehensions can be overridden by redefining the LaTeX commands \ottcomp, \ottcompu, and \ottcomplu in an embed section, as in Section 4.3.

In some cases one could make the typeset notation even less noisy, by either omitting the superscript i or omitting both the superscript i and the subscript i’s on t and T. The first is unambiguous if there is at most one index on each element in the comprehension; the second if all the elements are indexed by the same thing (not the case for this example, but common for comprehensions of single elements, e.g. << Ti // i>> for T). It is arguable that that should be automated in future Ott releases, though it would bring the typeset and ASCII versions out of step.

List comprehension forms can also be used in bindspecs and in homomorphisms.

12.3  Proof assistant code for list forms

12.3.1  Types

We have to choose proof assistant representations for productions involving list forms. For example, for a language with records one might write

  metavar label, l ::= {{ hol string }} {{ coq nat }}
  indexvar index, n ::=  {{ hol num }} {{ coq nat }} 
  term, t :: ’t_’ ::=
    | { l1 = t1 ,  .. , ln = tn }      :: :: record

In HOL and Isabelle we represent these simply with contructors whose argument types involve proof-assistant native list types, e.g. the HOL list of pairs of a label and a t:

  val _ = Hol_datatype ` 
  t = E_record of (label#t) list  `;

For Coq we provide two alternatives: one can either use native lists, or lists can be translated away, depending on taste. The choice is determined by the -coq_expand_list_types command-line option. In the former case we generate an appropriate induction principle using nested fixpoints, as the default principle produced by Coq is too weak to be useful. In the latter case we synthesise an additional type for each type of lists-of-tuples that arises in the grammar. In the example, we need a type of lists of pairs of a label and a t:

list_label_t : Set := 
   Nil_list_label_t : list_label_t
 | Cons_list_label_t : label -> t -> list_label_t 
     -> list_label_t

with t : Set := 
   E_record : list_label_t -> t .

These are included in the grammar topological sort, and utility functions, e.g. to make and unmake lists, are synthesised.

12.3.2  Terms (in inductive definition rules)

Supporting list forms in the rules of an inductive definition requires some additional analysis. For example, consider the record typing rule below.

We analyse the symbolic terms in the premises and conclusion to identify lists of nonterminals and metavariables with the same bounds — here t0 .. tn−1, T0 .. Tn−1, and l0 .. ln−1 all have bounds 0..n−1. To make the fact that they have the same length immediate in the generated code, we introduce a single proof assistant variable for each such collection, with appropriate projections and list maps/foralls at the usage points. For example, the HOL for the above is essentially as follows, with an l_t_Typ_list : (label#t#Typ) list.

(* Ty_Rcd *)  !(l_t_Typ_list:(label#t#Typ) list) (G:G) . 
(EVERY (\b.b) 
  (MAP (\(l_,t_,Typ_). (Ty G t_ Typ_)) l_t_Typ_list))
  (E_record (MAP (\(l_,t_,Typ_). (l_,t_)) l_t_Typ_list))
  (T_Rec    (MAP (\(l_,t_,Typ_). (l_,Typ_)) l_t_Typ_list)))

This seems to be a better idiom for later proof development than the alternative of three different list variables coupled with assertions that they have the same length.

With direct support for lists, we need also direct support for symbolic terms involving list projection and concatenation. For example, the rule

gives rise to HOL code as below — note the list-lifted usage of the is_v_of_t predicate, and the list appends (++) in the conclusion.

(* reduce_Rec *)  !(l'_t'_list:(label#t) list) 
       (l_v_list:(label#t) list) (l:label) (t:t) (t':t) . 
((EVERY (\(l_,v_). is_v_of_t v_) l_v_list) /\
(( reduce t t' )))
(( reduce (t_Rec (l_v_list ++ [(l,t)] ++ l'_t'_list)) 
          (t_Rec (l_v_list ++ [(l,t')] ++ l'_t'_list))))

For the Proj typing rule

we need a specific projection (the HOL EL) to pick out the j’th element:

(* Ty_Proj *)  !(l_Typ_list:(label#Typ) list) 
       (j:index) (G:G) (t:t) . 
((( Ty G t (T_Rec (l_Typ_list)) )))
(( Ty 
     (t_Proj t  ((\ (l_,Typ_) . l_) (EL j l_Typ_list)))  
     ((\ (l_,Typ_) . Typ_) (EL  j l_Typ_list))))

For Coq, when translating away lists, we have to introduce yet more list types for these proof assistant variables, in addition to the obvious translation of symbolic terms, and, more substantially, to introduce additional inductive relation definitions to induct over them.

For similar examples in Isabelle, the generated Isabelle for the first three rules of §12.1 is shown below (lightly hand-edited for format). The first involves an Isabelle variable l_t_T_list, and list maps and projections thereof.

Ty_RcdI: "
  [|(formula_formuladots (( (%(l_,t_,T_).( ( G , t_ , T_ ) : Ty)) l_t_T_list)))|] 
  ( G , 
    (t_Rec (( (%(l_,t_,T_).(l_,t_)) l_t_T_list))) , 
    (T_Rec (( (%(l_,t_,T_).(l_,T_)) l_t_T_list))) 
  ) : Ty"

Ty_ProjI: "
  [| ( G , t , (T_Rec (l_T_list)) ) : Ty|] ==> 
  ( G , 
    (t_Proj t (%(l_,T_).l_) (List.nth l_T_list (j - 1))) , 
    (%(l_,T_).T_) (List.nth l_T_list (j - 1)) 
  ) : Ty"

E_Ctx_recordI: "
[| List.list_all (%(l_,v_).is_v v_) l_v_list ;
   ( t , t' ) : E|] 
 ( (t_Rec (l_v_list @ [(l,t)] @ l_'t_'list)) , 
   (t_Rec (l_v_list @ [(l,t')] @ l_'t_'list)) 
 ) : E"

The generated code for substitutions and free variables takes account of such list structure.

Note that at present the generated Isabelle code for these functions does not always build without change, in particular if tuples of size 3 or more are required in patterns.

12.3.3  List forms in homomorphisms

Proof assistant homomorphisms in productions can refer to dot-form metavariables and nonterminals. For example, the second production below (taken from test17.9) mentions [[x1 t1 ... xn tn]] in the isa homomorphism. This must exactly match the dot form in the production except that all terminals must be omitted — the metavariables and nonterminals must occur in the same order as in the production, and the bounds must be the same.

    E :: ’E_’ ::= {{ isa ( ident * t ) list  }}
      | < x1 : t1 , .. , xn : tn > :: :: 2  {{ isa  List.rev [[x1 t1 .. xn tn]] }}
    formula :: formula_ ::=          
      | judgement              :: :: judgement
      | formula1 .. formulan   :: :: dots

The generated Isabelle code for symbolic terms mentioning this production will involve a list of pairs. For example, the rules

|- E  :: :: Eok :: Eok_ by

---------------------------- :: 2
|- <x1:t1,..,xn:tn> 

|- t1:K1  .. |- tn:Kn
---------------------------- :: 3
|- <x1:t1,..,xn:tn> 


  Eok :: "E set"
inductive Eok tK

(* defn Eok *)

Eok_2I: " ( List.rev  (x_t_list) ) : Eok"

Eok_3I: "[|
(List.list_all (\<lambda> b . b) (  (( (%(x_,t_,K_). ( t_ , K_ ) : tK) x_t_K_list))  ) )|]
 ( List.rev  (( (%(x_,t_,K_).(x_,t_)) x_t_K_list)) ) : Eok"

Note that in the second the list of pairs is projected out from the x_t_K_list list of triples that is quantified over in the rule.

13  Subrules

Subrule declarations have the form

    nt1 <:: nt2

where nt1 and nt2 are nonterminal roots.

Subrules can be chained, i.e. there can be a pair of subrule declarations nt1 <:: nt2 and nt2 <:: nt3, and they can form a directed acyclic graph, e.g. with nt0 <:: nt1, nt0 <:: nt2, nt1 <:: nt3, and nt2 <:: nt3. However, there cannot be cycles, or nonterminal roots for which there are multiple upper bounds. Subrule declarations should not involve nonterminal roots for which proof-assistant type homs are specified.

We support the case in which the upper rule is also non-free, i.e. it contains productions that mention nonterminals that occur on the left of a subrule declaration. In the example below ( the t rule contains a production Foo v.

  metavar termvar , x ::=
    {{ isa string }} {{ coq nat }} {{ coq-equality }} {{ hol string }} {{ ocaml int }}

    t :: ’t_’ ::=                                         
      | x                   ::   :: Var                   
      | \ x . t             ::   :: Lam  (+ bind x in t +)
      | t t’                ::   :: App                   
      | Foo v               ::   :: Foo

    v :: ’v_’ ::=                        
      | \ x . t             ::   :: Lam   

    v <:: t

    Jb ::::=

      Baz t , v ::  :: Baz ::by 

      --------- :: ax
      Baz t , v

In this case generated Isabelle/Coq/HOL/OCaml will define a single type and both is_v and is_t predicates, and the generated inductive definition clause for ax uses both predicates. The Isabelle clause is below.

  axI: "[|is_t t ; is_v v|] ==>  ( t , v ) : Baz"

14  Context rules

The system supports the definition of single-hole contexts, e.g. for evaluation contexts. For example, suppose one has a term grammar as below:

t :: ’t_’ ::=                               {{ com term    }}
  | x                  ::  :: Var                   {{ com variable}}
  | \ x . t            ::  :: Lam (+ bind x in t +) {{ com lambda  }}
  | t t’               ::  :: App                   {{ com app     }}
  | ( t1 , .... , tn ) ::  :: Tuple                 {{ com tuple }}
  | ( t )              :: S:: Paren                 {{ icho [[t]]  }} 
  | { t / x } t’       :: M:: Tsub  
                        {{ icho (tsubst_t [[t]] [[x]] [[t’]])}}

  | E . t              :: M:: Ctx  
                        {{ icho (appctx_E_t [[E]] [[t]])}}
                        {{ tex [[E]] \cdot [[t]] }}

A context grammar is declared as a normal grammar but with a single occurrence of the terminal __ in each production, e.g. as in the grammar for E below (a rather strange evaluation strategy, admittedly).

E :: ’E_’ ::=                               {{ com evaluation context }}
  | __ t         ::  :: AppL                   {{ com app L    }}
  | v __         ::  :: AppR                   {{ com app R    }}
  | \ x . __     ::  :: Lam                    {{ com reduce under lambda }}
  | ( t1 ( __  t2 ) ) :: :: Nested              {{ com hole nested }}
  | ( v1 , .. , vm , __ , t1 , .. , tn ) :: :: Tuple {{ com tuple }}

A contextrules declaration:

  E _:: t :: t

causes Ott to (a) check that each production of the E grammar is indeed a context for the t grammar, and (b) generates proof assistant functions, e.g. appctx_E_t, to apply a context to a term:

(** context application *)
Definition appctx_E_t (E5:E) (t_6:t) : t :=
  match E5 with
  | (E_AppL t5) => (t_App t_6 t5)
  | (E_AppR v5) => (t_App v5 t_6)
  | (E_Lam x) => (t_Lam x t_6)
  | (E_Nested t1 t2) =>  (t_App t1  (t_App t_6 t2) )
  | (E_Tuple v_list t_list) => (t_Tuple ((app_list_t v_list
     (app_list_t (Cons_list_t t_6 Nil_list_t) (app_list_t t_list Nil_list_t)))))

As the Nested production shows, context productions can involve nested term structure.

Note also that here the E grammar is not free (it mentions the subrule nonterminal v) so an isvalue predicate is_E_of_E is also generated.

In general, context rule declarations have the form

    ntE _:: nt1 :: nt2

where ntE, nt1, and nt2 are nonterminal roots. This declares contexts ntE for the nt1 grammar, with holes in nt2 positions.

Just as for substitutions, the context application function is typically used by adding a metaproduction to the term grammar. Here we add a production E.t to the t grammar with an icho hom that uses appctx_E_t.

t :: ’t_’ ::=                               {{ com term    }}
  | E . t              :: M:: Ctx  
                        {{ icho (appctx_E_t [[E]] [[t]])}}
                        {{ tex [[E]] \cdot [[t]] }}

That can then be used in relations:

    t --> t’
    -------------- :: ctx
    E.t --> E.t’

One would typically also define a terminals production for the hole terminal __, e.g. here we typeset the hole as [·].

terminals :: ’terminals_’ ::=
  | __           ::  :: hole    {{ tex [\cdot] }}

15  Auxiliary Rules

We permit an aux hom on grammar rules. For any rule with such a hom, we transform that rule by appending an _aux to its primary nonterminal root name. We then add a synthesised rule with the original nonterminal root name and a single production, with a shape described by the body of the aux hom, which must be of the form

  {{ aux  foo1 foo2 _ bar1 bar2 bar3 }}

with a single _ and any number of strings fooi and barj before and after. The _ is replaced by the original nonterminal root name.

For example, given a grammar or metavariable l of source locations, one might say

ntr :: 'NTR_'  ::=  {{ aux _ l }}
 | ...

to synthesise grammars ntr_aux and ntr of unannotated and location-annotated terms, the first with all the original productions and the second with a single production

 | ntr l :: :: NTR_aux. 

If the rule has an empty production name wrapper (eg with ’’ in place of ’NTR_’) then the production name is based on the original nonterminal root, capitalised and with _aux appended (eg Ntr_aux), to avoid spurious conflicts.

An additional auxparams hom on rules lets the user add type parameters to the generated OCaml output.

Generation of aux rules is controlled by a command-line option -generate_aux_rules, which one might (eg) set to false for latex output and true for OCaml output.

16  Functions

Ott includes experimental support for writing function definitions. As a simple example, consider the Ott file below:

     n :: ’n_’ ::=
       | 0       :: :: Zero
       | S n     :: :: Succ

      Add ::=   {{ hol-proof ...  }}
      n1 + n2 :: n :: add    {{ com a function of type num -> num -> num }}
      0 + n2 === n2
      S n1 + n2 === n1 + S n2

Here the add function is compiled into the following Coq code:

    Fixpoint add (x1:num) (x2:num) : num:=
      match x1,x2 with
      | n_zero , n2 => n2
      | (n_succ n1) , n2 =>  (add n1 (n_succ n2) ) 

More in detail, the fun n1 + n2 :: n :: add by declaration specifies:

The type of the arguments of the function is defined by the non-terminals appearing in the lhs, the return type by the rhs non-terminal (so numnumnum in the above example). As side-effect, whenever a function of type symb_termnt is defined, a production nt ::= symb_term is added to the definition of the non-terminal nt (in the above example, the production n1 + n2 is added to the grammar of num).

Functions are then defined by case analysis, where the lhs and the rhs are separated by the reserved symbol ===.

The {{ hol-proof }} hom allows the specification of a termination proof, which is required by Hol. Mutually recursive functions can be defined in the same funs block, analogously to mutually recursive rule definitions.

Disclaimer: the different treatment of partial functions by the different provers can result in a function definition being compiled correctly to one prover but not to others.

17  Parsing Priorities

Symbolic terms that can have more than one parse tree are typically considered erroneous; however, certain classes of parse trees are ignored in order to support common idioms that are ambiguous. For example, the production

Γ ::= Γ1, .., Γn

might be used to allow a list of typing contexts to be appended together, but it is highly ambiguous. The following restrictions forbid many unwanted parses that could otherwise occur.

metavar n ::=


e :: e_ ::= 
| n             ::   :: num
| - e           ::   :: neg
| e1 + e2       ::   :: add
| e1 - e2       ::   :: sub
| e1 e2         ::   :: mul
| e1 / e2       ::   :: div
| e1 , .. , e2  ::   :: tup
| ( e )         :: M :: par {{ icho [[e]] }}


e_add left e_add   % #1
e_sub left e_sub
e_add left e_sub   % #3
e_sub left e_add

e_mul left e_mul
e_div left e_div
e_mul left e_div
e_div left e_mul

e_neg <= e_add     % #9
e_neg <= e_sub
e_neg <= e_mul
e_neg <= e_div
e_neg <= e_tup

e_add <= e_div
e_add <= e_mul     % #15
e_add <= e_tup
e_sub <= e_div
e_sub <= e_mul
e_sub <= e_tup

e_mul <= e_tup     % #20
e_div <= e_tup

Figure 14: An Ott source file for basic arithmetic using the typical parsing priorities

18  Combining multiple source files

Ott can be invoked with multiple source files. Input filenames with extensions .tex, .v, .thy, .sml, or ml are simply copied into the relevant output (LATEX, Coq, Isabelle, HOL, or OCaml). By default the source-file and command-line order of blocks is preserved, for grammar, embeds, and inductive definitions.

The prover output can be split into multiple output files: each prover output file specified with -o <filename> will contain the material from the previous input files specified with -i (since the last -o for the same prover).

Alternatively, one can add a -merge true command-line option, in which case the productions of multiple grammars that share the same header are merged into a single grammar, and the rules of multiple inductive definitions that share the same header are merged into a single inductive definition. This rudimentary form of modularity can be very useful, either to split a language definition into separate features, or to define reusable Ott components to define standard formulae, LaTeX pretty printing of terminals, or LaTeX styles. For example, Figure 15 shows the Ott source file for a let feature in isolation, taken from our Ott development of some languages from Pierce’s TAPL [Pie02].

t :: Tm ::=                                        {{ com terms: }}
  | let x = t in t’                   ::   :: Let (+ bind x in t’ +)     {{ com let binding }}

Jop ::::= 

t --> t’ :: :: red :: E_ {{ com Evaluation }} by

----------------------------- :: LetV
let x=v1 in t2 --> [x|->v1]t2 

t1 --> t1’
---------------------------------- :: Let
let x=t1 in t2 --> let x=t1’ in t2

Jtype ::::= 

G |- t : T :: :: typing :: T_ {{ com Typing }} by

G |- t1:T1
G,x:T1 |- t2:T2
------------------------ :: Let
G |- let x=t1 in t2 : T2
Figure 15: An ott source file for the let fragment of TAPL

The original TAPL languages were produced using TinkerType [LP03] to compose features and check for conflicts. In examples/tapl we build a system, similar to the TinkerType sys-fullsimple, from ott source files that correspond roughly to the various TinkerType components, each with syntax and semantic rules for a single feature.

19  Hom blocks

Bindspecs and homomorphisms for productions, and any homomorphisms for definitions, can appear in an Ott source file either attached to the production or definition, as we have shown earlier, or in separate hom blocks. For example, one can write

homs ’t_’
  :: Lam  (+ bind x in t +)  

homs ’t_’
  :: Var     {{ com variable }}         
  :: Lam     {{ com abstraction }}      
  :: App     {{ com application }}      
  :: paren   {{ ich [[t]] }} 
  :: tsub    {{ ich ( tsubst_t [[t]] [[x]] [[t’]] ) }}

homs:: reduce  {{ com [[t1]] reduces to [[t2]] }} 

Each of these begins with a prefix and then has a sequence of production name or definition name kernels, each followed by a sequence of bindspecs and then a sequence of homomorphisms.

The example, in Fig. 16, shows this. It is semantically equivalent to the example of Fig. 8, but the homs have been moved into hom blocks.

metavar termvar , x ::= 
  {{ isa string }} {{ coq nat }} {{ coq-equality }} {{ hol string }} {{ lex alphanum }}
  {{ tex \mathit{[[termvar]]} }} {{ com  term variable  }} 

  t :: ’t_’ ::=    {{ com term }}
    | x                   ::   :: Var   
    | \ x . t             ::   :: Lam  
    | t t’                ::   :: App  
    | ( t )               :: S :: paren 
    | { t / x } t’        :: M :: tsub  

  v :: ’v_’ ::=    {{ com  value }}
    | \ x . t             ::   :: Lam   

  terminals :: ’terminals_’ ::=
    | \                   ::   :: lambda  {{ tex \lambda }}
    | -->                 ::   :: red     {{ tex \longrightarrow }}

homs ’t_’
  :: Lam  (+ bind x in t +)  

homs ’t_’
  :: Var     {{ com variable }}         
  :: Lam     {{ com abstraction }}      
  :: App     {{ com application }}      
  :: paren   {{ ich [[t]] }} 
  :: tsub    {{ ich ( tsubst_t [[t]] [[x]] [[t’]] ) }}

homs:: reduce  {{ com $[[t1]]$ reduces to $[[t2]]$ }} 

  v <:: t

  single t x :: tsubst 

  Jop ::::=

    t1 --> t2 ::  :: reduce ::by 

    --------------------------  :: ax_app
    (\x.t12) v2 -->  {v2/x}t12

    t1 --> t1’
    -------------- :: ctx_app_fun
    t1 t --> t1’ t

    t1 --> t1’
    -------------- :: ctx_app_arg
    v t1 --> v t1’

Figure 16: Hom Sections:

20  Isabelle syntax support

Ott has limited facilities to allow the Isabelle mixfix syntax support and xsymbol to be used. The example shows this in use.

Non-meta productions can be annotated with isasyn and/or isaprec homomorphisms. For example, contains the production

  | t t’      :: :: App   {{ isasyn [[t]]\<bullet>[[t’]] }}  {{ isaprec 50 }}

The two homs are used to output the Isabelle syntax annotation in the t_App clause of the datatype definition below.

t = 
   t_Var "termvar"  
 | t_Lam "termvar" "t"  ("\<lambda> _ . _" 60)
 | t_App "t" "t"  ("_\<bullet>_" 50)

Definitions can be annotated with isasyn and/or isaprec homomorphisms similarly, e.g. as below.

    t1 --> t2 ::  :: reduce ::{{ isasyn [[t1]] ---> [[t2]] }}  by 

This generates syntax and translations blocks as below.

inductive_set reduce :: "(t*t) set"
and  "reduce'" :: "t => t =>  bool" ("_ ---> _" 50)
where "(t1 ---> t2) ==  ( t1 , t2 ) : reduce"

Symbolic terms in definitions are printed using any production or definition syntax. This (especially with xsymbol turned on) makes the current goal state during Isabelle proof development much more readable.

Further, there is a command line option -isa_syntax true. If this is set then the tool generates Isabelle syntax annotations from the source syntax. For example, the source file production for the t_Lam clause is

    | \ x . t             ::   :: Lam   {{ isaprec 60 }}

and the terminals grammar contains a mapping from \ to \<lambda>:

  terminals :: ’terminals_’ ::=
    | \                   ::   :: lambda  {{ tex \lambda }}  {{ isa \<lambda> }}
    | -->                 ::   :: red     {{ tex \longrightarrow }} {{ isa ---> }}

This is used (just as for LaTeX homs) to generate the ("\<lambda> _ . _" 60) in the datatype definition above.

This functionality is limited in various ways: (1) the full range of Isabelle precedence and associativity specifications are not supported; (2) the automatically generated syntax annotations are somewhat crude, especially w.r.t. spacing and parenthesisation; (3) syntax annotation on meta productions is not propertly supported; and (4) it would be desirable to have more fine-grain control of whether to automatically generate annotations: per-production, per-rule, and per-file.

21  Isabelle code generation example

The Isabelle/Coq/HOL code generation facilities can be sometimes used to generate (variously) OCaml and SML code from the Isabelle/Coq/HOL definitions produced by Ott.

For example, the test10st_codegen.thy file uses Isabelle code generation to produce SML code to calculate the possible reductions of terms in the simply typed lambda calculus.

theory test10st_codegen 
imports test10st_snapshot_out Executable_Set

ML "reset Codegen.quiet_mode"

(* Code generation for the test10st simply typed lambda calculus. *)

 ta :: t
"ta == (t_App (t_Lam ''z'' (t_Var ''z''))) (t_Lam ''y'' (t_Var ''y''))"

code_module Test10st_codegen file "" contains 
reduce_ta  = "(ta,_):reduce"

(* build and demo the resulting code...

Isabelle test10st_codegen.thy
...`use' that... a shell...
use "";
open Test10st_codegen;

...a test term...
val it = t_App (t_Lam (["z"], t_Var ["z"]), t_Lam (["y"], t_Var ["y"]))

...a sample reduction...
DSeq.hd(reducep__1 ta);
val it = t_Lam (["y"], t_Var ["y"]) : Test10st_codegen.t 



22  Reference: Command-line usage

A good place to get started is one of the test make targets in the ott directory, e.g.

test10: tests/
           bin/ott                                                \
                  -i tests/                             \ 
                  -o out.thy -o out.v -o outScript.sml            \
                  -o out.tex                                      \
                  -parse ":t:  (\z.z z) y"                        \                  
           && ($(LATEX) out; $(DVIPS) out -o)

When make test10 is executed, ott:

That LaTeX document is then compiled and converted to postscript.

For convenience, input files can also be listed at the end of the command line:

  ott [options] <file1> ... <filen>

is equivalent to

  ott -i <file1> ... -i <filen> [options]

The %.out Makefile target runs ott with common defaults on the file, so for example executing make tests/test10.out runs ott on tests/, generating all outputs. There are also targets  %.coq.out,  %.hol.out, and  %.isa.out, to generate just LaTeX and the code for one proof assistant, and  %.tex.out, to generate just LaTeX.

The ott command-line options (with default values where applicable) are shown below.

Ott version 0.32   distribution of Wed 9 Mar 16:05:28 GMT 2022

usage: ott <options> <filename1> .. <filenamen> 
  (use "OCAMLRUNPARAM=p  ott ..." to show the ocamlyacc trace)
  (ott <options> <filename1> .. <filenamen>    is equivalent to
   ott -i <filename1> .. -i <filenamen> <options>)

  -i <filename>                      Input file (can be used multiple times)
  -o <filename>                      Output file (can be used multiple times)
  -writesys <filename>               Output system definition
  -readsys <filename>                Input system definition
  -tex_filter <src><dst>             Files to TeX filter
  -coq_filter <src><dst>             Files to Coq filter
  -hol_filter <src><dst>             Files to HOL filter
  -lem_filter <src><dst>             Files to HOL filter
  -isa_filter <src><dst>             Files to Isabelle filter
  -ocaml_filter <src><dst>           Files to OCaml filter
  -merge <false>                     merge grammar and definition rules
  -parse <string>                    Test parse symterm,eg ":nontermroot: term"
  -fast_parse <false>                do not parse :rulename: pseudoterminals
  -signal_parse_errors <false>       return >0 if there are bad defns
  -picky_multiple_parses <false>     Picky about multiple parses
  -quotient_rules <true>             Quotient rules, as per {{ quotient-with ntr }} homs
  -generate_aux_rules <true>         Generate auxiliary rules or constructor arguments from {{ aux ... }} homs
  -aux_style_rules <true>            Auxiliary rules (true) vs constructor arguments (false)
  -output_source_locations <0>       Include source location info in output (0=none, 1=drules, 2=grammar+drules)
  -colour <true>                     Use (vt220) colour for ASCII pretty print
  -show_sort <false>                 Show ASCII pretty print of syntax
  -show_defns <false>                Show ASCII pretty print defns
  -tex_show_meta <true>              Include meta prods and rules in TeX output
  -tex_show_categories <false>       Signal production flags in TeX output
  -tex_suppress_category <[]>        Suppress productions and rules with this category in TeX output
  -tex_suppress_ntr <[]>             Suppress nonterminal root in TeX output
  -tex_colour <true>                 Colour parse errors in TeX output
  -tex_wrap <true>                   Wrap TeX output in document pre/postamble
  -tex_name_prefix <string>          Prefix for tex commands (default "ott")
  -isabelle_primrec <true>           Use "primrec" instead of "fun"
                                       for functions
  -isabelle_inductive <true>         Use "inductive" instead of "inductive_set"
                                       for relations
  -isa_syntax <false>                Use fancy syntax in Isabelle output
  -isa_generate_lemmas <false>       Lemmas for collapsed functions in Isabelle
  -coq_avoid <1>                     coq type-name avoidance
                                       (0=nothing, 1=avoid, 2=secondaryify)
  -coq_expand_list_types <false>     Expand list types in Coq output
  -coq_lngen <false>                 lngen compatibility
  -coq_names_in_rules <true>         Copy user names in rule definitions
  -coq_use_filter_fn <false>         Use list_filter instead of list_minus2 in substitutions
  -ocaml_include_terminals <false>   Include terminals in OCaml output (experimental!)
  -ocaml_pp                          < filename> generate OCaml AST pretty printer files (experimental!) (also included in .mly target)
  -ocaml_pp_ast_module               override default OCaml module name for AST module (experimental!)
  -ocaml_pp_json <false>             Include JSON output in pretty printer (experimental)
  -pp_grammar                        (debug) print term grammar
  -dot <filename>                    (debug) dot graph of syntax dependencies
  -alltt <filename>                  (debug) alltt output of single source file
  -sort <true>                       (debug) do topological sort
  -process_defns <true>              (debug) process inductive reln definitions
  -showraw <false>                   (debug) show raw grammar
  -ugly <false>                      (debug) use ugly ASCII output
  -no_rbcatn <true>                  (debug) remove relevant bind clauses
  -lem_debug                         (debug) print lem debug locations
  -help                              Display this list of options
  --help                             Display this list of options

23  Reference: The language of symbolic terms

A syntax definition conceptually defines two different languages: that of concrete terms of the object language, and that of symbolic terms over the object language. The former includes concrete variables (if nontrivial lex homs have been specified for metavariables). The latter includes the former but also allows symbolic metavariables and nonterminals. Symbolic terms may also include the production-name annotations mentioned in §3. For a syntax definition with list forms (c.f. §12) symbolic terms also include various list constructs. A simplified abstract syntax of symbolic terms is shown in Figure 17, omitting list forms. In this section we give an informal definition of the full concrete syntax of symbolic terms.

Figure 17: Mini-Ott in Ott: symbolic terms

The premises and conclusions of inductive definition rules are symbolic terms. The language of symbolic terms is defined informally below, with interpretation functions [[_]] that map defined entities into grammar clauses.

For a rule rule   =

nontermroot1   ,   ..   ,   nontermrootn ::   ’’   ::=   prod1   ..   prodm 

we have

 |nontermroot suffix(1) 

(1) for each nontermroot in the set {nontermroot1   ,   ..   ,   nontermrootn } and for each nontermroot defined by any rule′ which is declared as a subrule of this rule.

For a production prod   =

|   element1   ..   elementm   ::   ::   prodname

we have

 |[[element1]]   ..   [[elementm]] 
 |:prodname:  [[element1]]   ..   [[elementm]] 

For an element there are various cases.

  1. For a terminal terminal
  2. For a nonterminal nontermroot   suffix
     [[nontermroot   suffix]]::=[[rule]] 
    where rule includes nontermroot among the nonterminal roots it defines. (Note that this does not depend on what suffix was used in the grammar, and similarly for the metavar case below.)
  3. For an index variable indexvarroot
    for each indexvarroot′ defined by the indexvar definition that defines indexvarroot.
  4. For a metavariable metavarroot   suffix
     [[metavarroot   suffix]]::= 
     |metavarroot′ suffix(1) 
    (1) for each metavarroot′ defined by the metavar definition that defines metavarroot. (2) where variable ranges over all the strings defined by the lex regexp of the metavar definition that defines metavarroot, except for any string which can be parsed as a nonterminal, metavariable or terminal of the syntax definition.
  5. A list form element element could be any of the following, either without a separating terminal:
    element1 .. elementn    dots   element1 .. elementn 
    </  element1 .. elementn   //   indexvar   /> 
    </  element1 .. elementn   //   indexvar   IN   indexvar′   /> 
    </  element1 .. elementn   //   indexvar   IN   number   dots   indexvar′   /> 
    </  element1 .. elementn   //   indexvar   IN   number   dots   indexvar-1   /> 
    or with a separating terminal:
    element1 .. elementn    terminal   dots   terminal   element1 .. elementn 
    </  element1 .. elementn   //  terminal //  indexvar   /> 
    </  element1 .. elementn   //  terminal //  indexvar   IN   indexvar′   /> 
    </  element1 .. elementn   //  terminal //   indexvar   IN   number   dots   indexvar′   /> 
    </  element1 .. elementn   //  terminal //   indexvar   IN   number   dots   indexvar-1   /> 
    In any of these cases the interpretation [[element]] is the lists (separated by the terminal if one was specified) of concrete list entries and of list forms. Without a separating terminal, this is:
     [[element]]::=(concrete_list_entry | list_form)*(2), (3)
    concrete_list_entry::=[[element1]]   ..   [[elementn]] 
     |[[element1]]   ..   [[elementn]]   dots′  [[element1]]   ..   [[elementn]](1) 
     |</  [[element1]] .. [[elementn]]   //  indexvar″   /> 
     |</  [[element1]] .. [[elementn]] //  indexvar″   IN   indexvar‴   /> 
     |</  [[element1]] .. [[elementn]] //   indexvar″   IN   number′   dots′   indexvar‴   /> 
     |</  [[element1]] .. [[elementn]] //   indexvar″   IN   number′   dots′   indexvar-1   /> 
    This is subject to constraints: (1) that [[element1]]   ..   [[elementn]] and [[element1]]   ..   [[elementn]] can be anti-unified with exactly one varying index; (2) if the list has only concrete entries (i.e., no list forms), its length must meet the constraint of any dots in the element.

    With a separating terminal, we have:

     [[element]]::=є | (concrete_list_entry | list_form) (terminal (concrete_list_entry | list_form))*

In the above

dots::=.. | ... | .... 
number::=0 | 1  
 |(0 | 1 |2 | 3 |4 | 5 |6 | 7 |8 | 9)+(longest match) 

Further, whitespace (' '|'\010'|'\009'|'\013'|'\012') is allowed before any token except a those in a suffix, and nonterminals, metavariables, index variables, and terminals that end with an alphanumeric character, must not be followed by an alphanumeric character.

The tool also builds a parser for concrete terms, with fake nonterminal roots concrete_ntr for each primary ntr of the syntax definition. One can switch to concrete-term parsing with a :concrete: annotation, as in the example

\[ [[ :concrete: \Z1<:Top. \x:Z1.x ]]\]

shown in Figure 10. Below such an annotation, only concrete terms are permitted, with no further annotation, no symbolic nonterminals or metavariables, no list dot forms or comprehensions, etc.

Parsing of terms is done with a scannerless GLR parser over character-list inputs. The parser searches for all parses of the input. If none are found, the ASCII and TeX output are annotated no parses, with a copy of the input with *** inserted at the point where the last token was read. This is often at the point of the error (though if, for example, a putative dot form is read but the two element lists cannot be anti-unified, it will be after the point of the error). If multiple parses are found, the TeX output is annotated multiple parses and the different parses are output to the console in detail during the Ott run.

The GLR parser achieves reasonable performance on the small symbolic terms that are typical in semantic rules. Its performance on large (whole-program size) examples is untested.

24  Reference: Generation of proof assistant definitions

This section briefly summarises the steps involved in the generation of proof assistant definitions from an Ott source file. For a description of the locally-nameless backend, refer to zappa/projects/ln_ott/.

24.1  Generation of types

24.2  Generation of functions

A small number of library functions (list_mem, list_minus,...) are included in the output if they are required.

Several Coq list functions (map, make, unmake, nth, app) are generated for each synthesized list type.

The definitions of the more interesting functions (subrule predicates, binding auxiliaries, free variable functions, and substitutions) are generated over the free types generated for the maximal elements of the subrule order (generation of these functions for rules with type homs is not supported). The definitions are by pattern-matching and recursion. The patterns are generated by building canonical symbolic terms from the productions of each relevant rule. The recursion is essentially primitive recursion: for Coq we produce Fixpoints or Definitions (the latter is sometimes needed as the former gives an error in the case where there is no recursion); for Isabelle we produce primrecs (or, experimentally, funs); for HOL we use an ottDefine variant of the Define package. In general we have to deal both with the type dependency (the topologically sorted mutually recursive types described above) and with function dependency — for example, for subrule predicates and binding auxiliaries we may have multiple mutually recursive functions over the same type.

For Coq the function generation over productions that involve list types must mirror that, so we generate auxiliary functions that recurse over those list types.

For Isabelle the primrec package does not support definitions involving several mutually recursive functions over the same type, so for these we generate single functions calculating tuples of results, define the intended functions as projections of these, and generate lemmas (and simple proof scripts) characterising them in terms of the intended definitions. Further, it does not support pattern matching involving nested constructors. We therefore generate auxiliary functions for productions with embedded list types. Isabelle tuples are treated as iterated pairs, so we do the same for productions with tuples of size 3 or more. Isabelle also requires a function definition for each recursive type. In the case where there are multiple uses of the same type (e.g. several uses of t list in different productions) all the functions we wish to generate need identical auxiliaries. As yet, the tool does not generate the identical copies required.

If the option -isabelle_primrec is set to false, then Ott uses the fun package instead of the primrec package. Since at the time of writing Isabelle 2008 is not capable of proving automatically termination of all the funs that Ott generates, this feature should be considered experimental.

For HOL the standard Define package tries an automatic termination proof. For productions that involve list types our generated functions involve various list functions which prevent those proofs working in all cases. We therefore use an ottDefine variant (due to Scott Owens), with slightly stronger support for proving termination of definitions involving list operators.

24.2.1  Subrule predicates

We generate subrule predicates to carve out the subsets of each free proof assistant type (from the maximal elements of the subrule order) that represent the rules of the grammar. The non-free rules are the least subset of the rules that either (1) occur on the left of a subrule (<::) declaration, or (2) have a (non-meta) production that mentions a non-free rule. Note that these can include rules that are maximal elements of the subrule order, e.g. if an expression grammar included a production involving packaged values. The subrule predicate for a type is defined by pattern matching over constructors of the maximal type above it — for each non-meta production of the maximal type it calculates a disjunction over all the productions of the lower type that are subproductions of it, invoking other subrule predicates as appropriate.

24.2.2  Binding auxiliaries

The binding auxiliary functions calculate the intuitive semantics of auxiliary functions defined in bindspecs of the Ott source file. Currently these are represented as proof assistant lists of metavariables or nonterminals (arguably set types should be used instead, at least in Isabelle).

24.2.3  Free variables

The free variable functions simply walk over the structure of the free proof assistant types, using any bind specifications (and binding auxiliaries) as appropriate. For these, and for substitutions, we simplify the generated functions by using the dependency analysis of the syntax to exclude recursive calls where there is no dependency.

24.2.4  Substitutions

The generated substitution functions also walk over the structure of the free proof assistant types. For each production, for each occurrence of a nonterminal nt within it, we first calculate the things (of whatever type is in question) binding in that nt, i.e. those that should be removed from the domain of any substitution pushed down into it. There are two cases: (1) the mse' from any bind mse' in nt; (2) nt itself if it occurs in the mse'' of any bind mse'' in nt'', i.e.  nt itself if it is directly used to bind elsewhere. List forms within bindspecs are dealt with analogously.

The substitution function clause for a production is then of one of two forms: either (1) the production comprises a single element, of the nonterminal or metavariable that we are substituting for, and this is within the rule of the nonterminal that it is being replaced by, or (2) all other cases. For (1) the element is compared with the domain of the substitution, and replaced by the corresponding value from the range if it is found. For (2) the substitution functions are mapped over the subelements, having first removed any bound things from the domain of the substitution.

This is all done similarly, but with differences in detail, for single and for multiple substitutions.

24.3  Generation of relations

The semantic relations are defined with the proof-assistant inductive relations packages (coq: Inductive, isa: inductive, hol: Hol_reln). They use the mutual recursion structure that is given by the user, with each defns block giving rise to a potentially mutually recursive definition of each defn inside it. (It is debatable whether it would be preferable to do an automatic dependency analysis and topological sort, as for the syntax.) Each definition rule gives rise to an implicational clause, essentially that the premises (Ott formulas) imply the conclusion (an Ott symbolic term of whichever judgement is being defined). In addition:

25  Reference: Summary of homomorphisms

Homomorphisms can appear in various positions in an Ott source file. The table below summarises their meanings. A * indicates that arguments are meaningful for that usage (e.g. [[e1]] in a production mentioning a nonterminal or metavariable e1).

a metavar or indexvar declaration, after one of the defined metavar/indexvar roots, or
a rule, after one of the defined nonterminal roots
tex*LaTeX typesetting for symbolic variables with that root
isa/coq/hol/ocaml Isabelle/Coq/HOL/OCaml root overriding string (1)
repr-locally-nameless use a locally-nameless representation (Coq backend only)
a metavar or indexvar declaration, after the ::=
isa Isabelle representation type
coq Coq representation type
hol HOL representation type
ocaml OCaml representation type
tex*LaTeX typesetting for symbolic variables
com comment to appear in LaTeX syntax definition
coq-equality Coq proof script to decide equality over the representation type
coq-universe Coq universe (e.g. Type) for the representation type
repr-locally-nameless (Coq only) use a locally-nameless representation
phantom suppress the representation type definition in theorem prover output
lex regular expression for lexing concrete variables
texvar*LaTeX typesetting for concrete variables
isavar*Isabelle output for concrete variables
holvar*HOL output for concrete variables
ocamlvar*OCaml output for concrete variables
a rule, after the ::=
isa Isabelle representation type, if a non-free type is required
coq Coq representation type, if a non-free type is required
hol HOL representation type, if a non-free type is required
ocaml OCaml representation type, if a non-free type is required
tex*LaTeX typesetting for symbolic variables
com*comment to appear in LaTeX syntax definition
coq-equality Coq proof script to decide equality over the representation type
coq-universe Coq universe (e.g. Type) for the representation type
phantom suppress the representation type definition in theorem prover output
aux(*)construct an auxiliary grammar rule with a single production
icho*shorthand for identical coq, isa, hol, and ocaml homs
ichlo*shorthand for identical coq, isa, hol, lem, and ocaml homs
ich*shorthand for identical coq, isa and hol homs
ic*shorthand for identical coq and isa homs
ch*shorthand for identical coq and hol homs
ih*shorthand for identical isa and hol homs
a production
isa*Isabelle output, for a non-free (meta) production
coq*Coq output, for a non-free (meta) production
hol*HOL output, for a non-free (meta) production
ocaml*OCaml output, for a non-free (meta) production
tex*LaTeX typesetting for symbolic terms
texlong typeset as long production
com*comment to appear in LaTeX syntax definition
order*specify order of arguments to prover or Ocaml constructor
isasyn*Isabelle mixfix syntax output
isaprec Isabelle mixfix syntax precedence string
ich*shorthand for identical coq, isa and holhoms
ic*shorthand for identical coq and isa homs
ch*shorthand for identical coq and hol homs
ih*shorthand for identical isa and hol homs
a production of the terminals grammar
isa Isabelle output, for terminals in default generated Isabelle mixfix declarations
tex LaTeX default typesetting for terms
com*comment to appear in LaTeX syntax definition
a defn, before the by
tex*LaTeX typesetting for symbolic terms
com*comment to appear in LaTeX syntax definition
isasyn*Isabelle mixfix syntax output
isaprec Isabelle mixfix syntax precedence string
lemwcf Lem witness, check, and functions spec for indrelns
a homs section clause (for a production or a definition)
as in the above production and defn forms
a group of defns, after the ::=
coq-universe Coq universe (e.g. Type) for the representation type
an embed section
isa embedded Isabelle output
coq embedded Coq output
hol embedded HOL output
ocaml embedded OCaml output
tex embedded LaTeX output
tex-preamble embedded LaTeX output, appearing in the LaTeX preamble
coq-lib do not generate definitions for the listed helper functions
isa-auxfn-proof Isabelle proof script
isa-subrule-proof Isabelle proof script
in a subrule, substitution or function definition
isa-proof Isabelle proof script

(1) This is occasionally useful to work around a clash between a metavar or nonterminal primary root and a proof assistant symbol, e.g. value in Isabelle or T in HOL.

26  Reference: The Ott source grammar

This is automatically generated (by mly-y2l) from the ocamlyacc grammar for Ott.

The lexing of Ott source files is context-dependent; this does not show that.

Not everything in the grammar is fully supported — in particular, option element forms, non-dotted element list forms, the three names distinctness forms of bindspecs, and context rules.

item items
RULES rules
DEFNCLASS defnclass
FUNDEFNCLASS fundefnclass
SUBRULES subrules
CONTEXTRULES contextrules
SUBSTITUTIONS substitutions
FREEVARS freevars
EMBED embed
PARSING parsing_annotations
HOMS hom_section
COQSECTIONBEGIN coq_section_begin
COQSECTIONEND coq_section_end
COQVARIABLE coq_variable
METAVAR ne_ident_hom_desc_list CCE metavarrep
INDEXVAR ne_ident_hom_desc_list CCE metavarrep
{ homomorphism }
{ hom_spec_el }
ident_and_blanks_list dots ident_and_blanks_list
COMP_LEFT ident_and_blanks_list COMP_MIDDLE comp_bound COMP_RIGHT blanks_list
{ embedmorphism }
{ embed_spec_el }
BIND mse IN ident
ident EQ mse
{ bindspec }
ident dots ident
ident LPAREN ident RPAREN
ident LPAREN ident dots ident RPAREN
ident COMP_LEFT ident COMP_MIDDLE comp_bound COMP_RIGHT
mse UNION mse
COMP_LEFT element_list COMP_MIDDLE comp_bound COMP_RIGHT
COMP_LEFT element_list COMP_MIDDLE ident COMP_MIDDLE comp_bound COMP_RIGHT
ident [ COMP_IN ident [ dots ident ] ]
{ element }
{ rule }
ne_ident_hom_desc_list COLONCOLON STRING CCE homomorphism_list prodlist
flavour element_list COLONCOLON ident_list COLONCOLON STRING bindspecall homomorphism_list
{ prod }
ident_desc homomorphism_list [ COMMA ne_ident_hom_desc_list ]
{ ( ident | BLANKS ) }
{ ( ident | blanks ) }
STRING COLONCOLON STRING CCE homomorphism_list defn_list
{ defn }
DEFN element_list COLONCOLON ident_list COLONCOLON STRING COLONCOLON STRING homomorphism_list BY semiraw_rule_list
STRING CCE homomorphism_list fundefn_list
{ fundefn }
FUNDEFN element_list COLONCOLON ident COLONCOLON STRING homomorphism_list BY BLANKLINE anylinelist
{ ( line | COMMENTLINE ) }
[ ( line | COMMENTLINE ) linelist ]
linelist lineline linelist
neblanklinelist semiraw_rule semiraw_rule_list
neblanklinelist defncom semiraw_rule_list
{ subrule }
ident_desc LTCOLONCOLON ident_desc homomorphism_list
{ contextrule }
ident_desc UNDERSCORECOLONCOLON ident_desc COLONCOLON ident_desc
{ substitution }
substitution_kind ident_desc ident_desc COLONCOLON ident_desc homomorphism_list
{ freevar }
ident_desc ident_desc COLONCOLON ident_desc homomorphism_list
{ parsing_annotation }
ident_desc parsing_annotation_type ident_desc
COLONCOLON STRING bindspecall homomorphism_list
{ hom_section_item }
STRING hom_section_item_list

27  Reference: Examples

The project web page

gives a variety of examples. Some of these, and additional small examples, are included in the distribution in the tests directory. Typically they can be built using the Makefile in the ott directory, e.g. typing make test10 or (more generally) make tests/test10.out) there.

test10.ottuntyped CBV lambda
test10st.ottsimply typed CBV lambda
test8.ottML polymorphism example
test7a.ottPOPLmark Fsub example (without records)
test7b.ottPOPLmark Fsub example (with records)
leroy-jfp96.ottLeroy module system
lj.ottLJ: Lightweight Java
test7t.mngwhole-document tex mng source
 (make test7afilter to build)
test7tt.mngfragment tex mng source
test11.ottsubrule example
test12.otttopological sort example
test13.ottsmall bindspec fragment
test10st_snapshot_out.thysnapshot of generated Isabelle from
test10st_metatheory_autoed.thyIsabelle proof script for type preservation and progress
test10st_codegen.thyIsabelle code generation script for reduction
test10_isasyn.ottIsabelle mixfix syntax example
test10st_metatheoryScript.smlHOL proof script for type preservation and progress
test17.10.ottlist comprehension examples

The examples/tapl directory contains several examples taken from the book ‘Types and Programming Languages’ by Benjamin Pierce. The make targets, listed below, combine Ott source files following roughly the TinkerType component structure used in TAPL.

sys-boolbooleans (p34)
sys-aritharithmetic expressions (p41)
sys-untypeduntyped lambda-calculus with booleans
sys-puresimplesimply-typed lambda-calculus
sys-tybooltyped booleans

Other examples can be found on the locally-nameless backend web page.


We thank the Ott users for their feedback, especially Matthew Parkinson; the other members of the POPLmark team, especially Benjamin Pierce, Stephanie Weirich, and Steve Zdancewic, for discussions; and Keith Wansbrough, Matthew Fairbairn, and Tom Wilkie, for their work on various Ott predecessors.

We acknowledge the support of EPSRC grants GR/T11715, EP/C510712, and EP/F036345, a Royal Society University Research Fellowship (Sewell), an EPSRC Leadership Fellowship (Sewell), EPSRC Programme Grant, EP/K008528/1 REMS Rigorous Engineering of Mainstrem Systems, and ANR grant ANR-06-SETI-010-02 (Zappa Nardelli).


Brian E. Aydemir, Aaron Bohannon, Matthew Fairbairn, J. Nathan Foster, Benjamin C. Pierce, Peter Sewell, Dimitrios Vytiniotis, Geoffrey Washburn, Stephanie Weirich, and Steve Zdancewic. Mechanized metatheory for the masses: The POPLmark Challenge. In Proc. TPHOLs, LNCS 3603, 2005.
Cédric Fournet, Georges Gonthier, Jean-Jacques Lévy, Luc Maranget, and Didier Rémy. A calculus of mobile agents. In Proc. CONCUR ’96, LNCS 1119, 1996.
Xavier Leroy. A syntactic theory of type generativity and sharing. Journal of Functional Programming, 6(5):667–698, 1996.
Michael Y. Levin and Benjamin C. Pierce. Tinkertype: A language for playing with formal systems. Journal of Functional Programming, 13(2), March 2003.
Benjamin C. Pierce. Types and Programming Languages. MIT Press, 2002.
Rok Strniša, Peter Sewell, and Matthew Parkinson. The Java Module System: core design and semantic definition. In Proceedings of OOPSLA 2007, the 22nd ACM SIGPLAN International Conference on Object-Oriented Programming, Systems, Languages and Applications (Montreál), October 2007. 15pp.
Peter Sewell, Francesco Zappa Nardelli, Scott Owens, Gilles Peskine, Thomas Ridge, Susmit Sarkar, and Rok Strniša. Ott: Effective tool support for the working semanticist. In Proceedings of ICFP 2007: the 12th ACM SIGPLAN International Conference on Functional Programming (Freiburg), October 2007. 12pp.
Peter Sewell, Francesco Zappa Nardelli, Scott Owens, Gilles Peskine, Thomas Ridge, Susmit Sarkar, and Rok Strniša. Ott: Effective tool support for the working semanticist. Journal of Functional Programming, 20(1):70–122, January 2010. Invited submission from ICFP 2007.

Strictly, clauses of primitive recursive function definitions from symbolic terms to strings, here of LaTeX code.

This document was translated from LATEX by HEVEA.