package universo

README.md

Note that a new interactive version of Dedukti is under development on https://github.com/Deducteam/lambdapi.

For interoperability development, the current version of Dedukti is still used.

USER MANUAL FOR DEDUKTI (DEVELOPMENT VERSION)

INSTALL

INSTALL WITH OPAM

opam install dedukti

The current version on opam is too old and we recommend to install Dedukti by cloning this repository.

INSTALL FROM SOURCES

git clone https://github.com/Deducteam/Dedukti.git
cd Dedukti
opam install .

COMPILE SOURCES

git clone https://github.com/Deducteam/Dedukti.git
cd Dedukti
opam install . --deps-only # Install only dependencies with Opam
make

Dependencies to compile Dedukti are listed in dedukti.opam.

QUICK START (ASSUMING INSTALLATION)

The command

dk check examples/append.dk

should output the following.

SUCCESS File 'examples/append.dk' was successfully checked.

EDITOR MODES

See the editors directory. Only the emacs mode is maintained currently.

COMMAND LINE PROGRAMS

The installation provides the following commands:

• dk check is the type-checker for Dedukti,

• dk top is an interactive wrapper around the type-checker,

• dk dep is a dependency generator for Dedukti files,

• dk prune is a program to re-print only the strictly required subset of Dedukti files,

• dk pretty is a source code beautifier for Dedukti,

• dk meta is a tool which uses Dedukti rewrite rules to rewrite a Dedukti file,

• universo is a tool which allows to work with universes using the encoding of The Calculus of Inductive Constructions in Dedukti.

OPTIONS

dk check provides the following options:

• -e Generates an object file .dko;

• -I DIR Adds the directory DIR to the load path;

• -d FLAGS enables debugging for all given flags:

  • q (quiet) disables all warnings,

  • n (notice) notifies about which symbol or rule is currently treated,

  • o (module) notifies about loading of an external module (associated to the command #REQUIRE),

  • c (confluence) notifies about information provided to the confluence checker (when option -cc used),

  • u (rule) provides information about type checking of rules,

  • t (typing) provides information about type-checking of terms,

  • r (reduce) provides information about reduction performed in terms,

  • m (matching) provides information about pattern matching;

• -v Verbose mode (equivalent to -d montru;

• -q Quiet mode (equivalent to -d q;

• --no-color Disables colors in the output;

• --stdin MOD Parses standard input using module name MOD;

• --coc [Experimental] Allows to declare a symbol whose type contains Type in the left-hand side of a product (useful for the Calculus of Construction);

• --type-lhs Forbids rules with untypable left-hand side;

• --snf Normalizes the types in error messages;

• --confluence CMD Sets the external confluence checker command to CMD;

• --beautify Pretty printer. Prints on the standard output;

• --version Prints the version number;

• --help Prints the list of available options.

A SMALL EXAMPLE

Then we can declare constants, giving their name and their type. Dedukti distinguishes two kinds of declarations:

• declaration of a static symbol f of type A is written f : A,

• declaration of a definable symbol f of type A is written def f : A.

Definable symbols can be defined using rewrite rules, static symbols can not be defined.

Nat: Type.
zero: Nat.
succ: Nat -> Nat.
def plus: Nat -> Nat -> Nat.

Let's add rewrite rules to compute additions.

[ n ] plus zero n --> n
[ n ] plus n zero --> n
[ n, m ] plus (succ n) m --> succ (plus n m)
[ n, m ] plus n (succ m) --> succ (plus n m).

When adding rewrite rules, Dedukti checks that they preserve typing. For this, it checks that the left-hand and right-hand sides of the rules have the same type in some context giving types to the free variables (in fact, the criterion used is more general, see below), that the free variables occurring in the right-hand side also occur in the left-hand side and that the left-hand side is a higher-order pattern (see below).

Remark: there is no constraint on the number of rewrite rules associated with a definable symbol. However it is necessary that the rewrite system generated by the rewrite rules together with beta-reduction be confluent and terminating on well-typed terms. Confluence can be checked using the option -cc (see below), termination is not checked (yet?).

Remark: Because static symbols cannot appear at head of rewrite rules, they are injective with respect to conversion and this information can be exploited by Dedukti for type-checking rewrite rules (see below).

ADVANCED FEATURES

SPLITTING A DEVELOPMENT BETWEEN SEVERAL FILES

A development in Dedukti is usually composed of several files corresponding to different modules. Using dk check with the option -e will produce a file my_module.dko that exports the constants and rewrite rules declared in the module my_module. Then you can use these symbols in other files/modules using the prefix notation my_module.identifier.

COMMENTS

In Dedukti comments are delimited by (; and ;).

(; This is a comment ;)

COMMANDS

Supported commands are:

#EVAL t.             (; evaluate t to its strong normal form and display it. ;)
#EVAL[N].            (; same as above, but evaluate in at most N steps. ;)
#EVAL[STRAT].        (; evaluate t with the strategy STRAT. :)
#EVAL[N,STRAT].      (; same as above, but evaluate in at most N steps. :)
#CHECK t1 == t2.     (; display "YES" if t1 and t2 are convertible, "NO" otherwise. ;)
#CHECK t1 : t2.      (; display "YES" if t1 has type t2, "NO" otherwise. ;)
#CHECKNOT t1 == t2.  (; display "YES" if t1 and t2 are not convertible, "NO" otherwise. ;)
#CHECKNOT t1 : t2.   (; display "YES" if t1 does not have type t2, "NO" otherwise. ;)
#ASSERT t1 : t2.     (; fail if t1 does not have type t2. ;)
#ASSERT t1 == t2.    (; fail if t1 is not convertible with t2. ;)
#ASSERTNOT t1 : t2.  (; fail if t1 does have type t2. ;)
#ASSERTNOT t1 == t2. (; fail if t1 is convertible with t2. ;)
#INFER t1.           (; infer the type of t1 and display it. ;)
#PRINT s.            (; print the string s. ;)

The supported evaluation strategies are:

• SNF (strong normal form: a term t is in SNF if no reduction can occur in t),

• WHNF (weak head normal form: a term t is said in WHNF if there is a finite sequence t=t0, t2, ..., tn such that tn is in normal form and for all i, ti reduces to t(i+1) and this reduction does not occur at the head).

Note that the #INFER command accepts the same form of configuration as the #EVAL command. When given, it is used to evaluate the obtained type.

DEFINITIONS

Dedukti supports definitions:

def three : Nat := succ ( succ ( succ ( zero ) ) ).

or, omitting the type,

def three := succ ( succ ( succ ( zero ) ) ).

A definition is syntactic sugar for a declaration followed by a rewrite rule. The definition above is equivalent to:

def three : Nat.
[ ] three --> succ ( succ ( succ ( zero ) ) ).

Using the keyword thm instead of def makes a definition opaque, meaning that the defined symbol do not reduce to the body of the definition. This means that the rewrite rule is not added to the system.

thm three := succ ( succ ( succ ( zero ) ) ).

This can be useful when the body of a definition does not matter (only its existence matters), to avoid adding a useless rewrite rule.

JOKERS

When a variable is not used on the right-hand side of a rewrite rule, it can be replaced by an underscore on the left-hand side. In the following definition:

def mult : Nat -> Nat -> Nat.
[ n ] mult zero n --> zero
[ n, m ] mult (succ n) m --> plus m (mult n m).

the first rule can also be written:

[ ] mult zero _ --> zero.

Similarly underscores can replace unused abstracted variables in lambdas: x => y => z => zero can be written _ => _ => _ => zero. Be mindful that, in a pattern, the expression _ => _ means x => Y where both x and Y are fresh variables occuring nowhere else.

TYPING OF REWRITE RULES

A typical example of the use of dependent types is the type of Vector defined as lists parametrized by their size:

Elt: Type.
Vector: Nat -> Type.
nil: Vector zero.
cons: n:Nat -> Elt -> Vector n -> Vector (succ n).

and a typical operation on vectors is concatenation:

def append: n:Nat -> Vector n -> m:Nat -> Vector m -> Vector (plus n m).
[ n, v ] append zero nil n v --> v
[ n, v1, m, e, v2 ] append (succ n) (cons n e v1) m v2 --> cons (plus n m) e (append n v1 m v2).

These rules verify the typing constraint given above: both left-hand and right-hand sides have the same type.

Also, the second rule is non-left-linear; this is usually an issue because in an untyped setting, non-left-linear rewrite rules usually generate a non-confluent rewrite system when combined with beta-reduction.

However, because we only intend to rewrite well-typed terms, the rule above is computationally equivalent to the following left-linear rule:

[ n, v1, m, e, v2, x ] append x (cons n e v1) m v2 --> cons (plus n m) e (append n v1 m v2).

Dedukti will also accept this rule, even if the left-hand side is not well-typed, because it is able to detect that, because of typing constraints, x can only be instantiated by a term of the form succ n (this comes from the fact that Vector is a static symbol and is hence injective with respect to conversion: from the type-checking constraint Vector x = Vector (succ n), Dedukti deduces x = succ n).

For the same reason, it is not necessary to check that the first argument of append is zero for the first rule:

[ n, v, x ] append x nil n v --> v.

Using underscores, we can write:

[ v ] append _ nil _ v --> v
[ n, v1, m, e, v2 ] append _ (cons n e v1) m v2 --> cons (plus n m) e (append n v1 m v2).

INJECTIVITY

Declaring a symbol as injective may help the type checker. Hence, it is possible to declare a symbol injective. However, no injectivity check is performed by the typechecker but the injectivity will be assumed and used when typechecking rules defined later on.

inj double : Nat -> Nat.
[   ] double zero     --> zero.
[ n ] double (succ n) --> succ (succ (double n)).

Declaring a non-injective symbol as injective may break the injectivity of product, and therefore may break subjection reduction.

TYPE ANNOTATIONS

Variables in the context of a rule may be annotated with their expected type. It is checked that the inferred type for annotated rule variables are convertible with the provided annotation.

[ n : Nat
, v1 : Vector n
, m : Nat
, e : Elt
, v2  : Vector m ]
  append _ (cons n e v1) m v2 --> cons (plus n m) e (append n v1 m v2).

NON-LEFT-LINEAR REWRITE RULES

By default, Dedukti accepts non-left-linear rewrite rules even though they usually generated non confluent rewrite systems when combined with beta-reduction.

eq: Nat -> Nat -> Bool.
[ n ] eq n n --> true.

This behaviour can be changed by invoking dk check with the option --ll (left-linear) to guarantee that non-left-linear rewrite rules are never added to the system.

HIGHER-ORDER REWRITE RULES

In the previous examples, left-hand sides of rewrite rules were first-order terms. In fact, Dedukti supports a larger class of left-hand sides: higher-order patterns.

A higher-order pattern is a beta-normal term whose free variables are applied to (possibly empty) vectors of distinct bound variables.

A classical example of the use of higher-order rules is the encoding the simply types lambda-calculus with beta-reduction:

type: Type.
arrow: type -> type -> type.

term: type -> Type.

def app: a:type -> b:type -> term (arrow a b) -> term a -> term b.
lambda: a:type -> b:type -> (term a -> term b) -> term (arrow a b).

[ f, arg ] app _ _ (lambda _ _ (x => f x)) arg --> f arg.

Remark: type annotations on abstraction must be omitted.

Remark: free variables must be applied to the same number of arguments on the left-hand side and on the right-hand side of the rule.

Remark: with such rewrite rules, matching is done modulo beta in order to preserve confluence. This means that, in the context (o: type)(c:term o), the term App o o (Lam o o (x => x)) c reduces to c.

BRACKET PATTERNS

A different solution to the same problem is to mark with brackets the parts of the left-hand side of the rewrite rules that are constrained by typing.

[ n, v1, m, e, v2 ] append (succ n) (cons {n} e v1) m v2 --> cons (plus n m) e (append n v1 m v2).

The information between brackets will be used when typing the rule but they will not be match against when using the rule (as if they were replaced by unapplied fresh variables).

Remarks:

• In order to make this feature type-safe, Dedukti checks at runtime that the bracket constraint is verified whenever the rule may be used and fails otherwise.

• This feature is not conditional rewriting. When a constraints is not satisfied, Dedukti doesn't just ignore the rule and proceed, it actually raises an error.

• Because they are replaced with unapplied fresh variables, bracket expressions may not contain variables locally bounded previously in the pattern.

• Since they are not used during matching, bracket expressions may not "introduce" variables. All rule variables occurring in bracket expression need to also occur in an other part of the pattern, outside a bracket.

• Bracket expressions and their type may contain variables occurring "before" (to the left of) the pattern.

• The type of a bracket expression may not contain variables occurring for the first time "after" (to the right of) the bracket.

• The bracket expression may contain variable occurring for the first time "after" (to the right of) the bracket on the condition that the inferred types for these variables do not depend on the bracket's fresh variable (no circularity).

CONFLUENCE CHECKING

Dedukti can check the confluence of the rewrite system generated by the rewrite rules and beta-reduction, using an external confluence checker. For this you need to install a confluence checker for higher-order rewrite systems supporting the TPDB format, for instance CSI^HO or ACPH.

To enable confluence checking you need to call dk check with the option -cc followed by the path to the confluence checker:

$ dk check -cc /path/to/csiho.sh examples/append.dk
> File examples/append.dk was successfully checked.

PRIVATE SYMBOLS

A user can declare a symbol as private. A private symbol cannot be used outside the module it is defined. These symbols may freely occur in type annotation, definitions and rewrite rules within the file they are defined, however they are completely inaccessible to outside developments. Note that they may still appear in the normal forms or inferred types of terms that were defined without relying on them.

LICENSE

Dedukti is distributed under the CeCILL-B License.