Author: | Enrico Tassi |
---|

This little tutorial does not talk about Coq, but rather focuses on
Elpi as a programming language. It assumes no previous knowledge of
Prolog, λProlog or Elpi. Coq is used as an environment for stepping trough
the tutorial one paragraph at a time. The text between `lp:{{`

and `}}`

is
Elpi code, while the rest are Coq directives to drive the Elpi interpreter.

Contents

Elpi is a dialect of λProlog enriched with constraints. We start by introducing the first order fragment of λProlog, i.e. the terms will not contain binders. Later we cover terms with binders and constraints.

Our first program is called `tutorial`

.
We begin by declaring the signature of our terms.
Here we declare that `person`

is a type, and that
`mallory`

, `bob`

and `alice`

are terms of that type.

```
Elpi Program tutorial lp:{{
kind person type.
type mallory, bob, alice person.
}}.
```

An Elpi program is made of rules that declare when predicates hold and that are accumulated one after the other. Rules are also called clauses in Prolog's slang, so we may use both terms interchangeably.

The next code snippet accumulates on top
of the current `tutorial`

program a predicate declaration for `age`

and three rules representing our knowledge about our terms.

```
Elpi Accumulate lp:{{
pred age o:person, o:int.
age mallory 23.
age bob 23.
age alice 20.
}}.
```

The predicate `age`

has two arguments, the former is a person while
the latter is an integer. The label `o:`

(standing for output)
is a mode declaration, which we will explain later (ignore it for now).

Note

int is the built-in data type of integers

Integers come with usual arithmetic operators, see the calc built-in.

In order to run our program we have to write a query, i.e. a predicate expression containing variables such as:

```
age alice A
```

The execution of the program is expected to assign a value to `A`

, which
represents the age of `alice`

.

Important

Syntactic conventions:

- variables are identifiers starting with a capital letter, eg
`A`

,`B`

,`FooBar`

,`Foo_bar`

,`X1`

- constants (for individuals or predicates) are identifiers
starting with a lowercase letter, eg
`foo`

,`bar`

,`this_that`

,`camelCase`

,`dash-allowed`

,`qmark_too?`

,`arrows->and.dots.as<-well`

A query can be composed of many predicate expressions separated by `,`

that stands for conjunction: we want to get an answer to all the
predicate expressions.

coq.say is a built-in predicate provided by Coq-Elpi which
prints its arguments.
You can look at the output buffer of Coq to see the value for `A`

or hover
or toggle the little bubble after `}}.`

if you are reading the tutorial with a
web browser.

Note

string is a built-in data type

Strings are delimited by double quotes and `\` is the escape symbol.

The predicate `age`

represents a *relation* (in contrast to a function)
and it computes both ways: we can ask Elpi which person `P`

is 23 years old.

Operationally the query `age P 23`

is *unified* with each
and every rule present in the program starting from the first one.

Unification compares two terms structurally and eventually assigns variables. For example for the first rule of the program we obtain the following unification problem:

age P 23 = age mallory 23

This problem can be simplified into smaller unification problems following the structure of the terms:

age = age P = mallory 23 = 23

The first and last are trivial, while the second one can be satisfied by
assigning `mallory`

to `P`

. All equations are solved,
hence unification succeeds.

See also the Wikipedia page on Unification.

Since the first part of the query is succesful the rest of
the query is run: the value of `P`

is printed as well as
the `"is 23 years old"`

string.

Note

`=`

is a regular predicate

The query `age P 23`

can be also written as follows:

A = 23, age P A, Msg = "is 23 years old", coq.say P Msg

Let's try a query harder to solve!

This time the unification problem for the first rule in the program is:

age P 20 = age mallory 23

that is simplified to:

age = age P = mallory 20 = 23

The second equation can be solved by assigning `mallory`

to `P`

,
but the third one has no solution, so unification fails.

When failure occurs all assignements are undone (i.e. `P`

is unset again)
and the next rule in the program is tried. This operation is called
*backtracking*.

The unification problem for the next rule is:

age P 20 = age bob 23

This one also fails. The unification problem for the last rule is:

age P 20 = age alice 20

This one works, and the assigment `P = alice`

is kept as the result
of the first part of the query. Then `P`

is printed and the program
ends.

An even harder query is the following one where we ask for two distinct individuals to have the same age.

This example shows that backtracking is global. The first solution for
`age P A`

and `age Q A`

picks `P`

and `Q`

to
be the same individual `mallory`

,
but then `not(P = Q)`

fails and forces the last choice that was made to be
reconsidered, so `Q`

becomes `bob`

.

Look at the output of the following code to better understand how backtracking works.

Note

`not`

is a black hole

The `not(P)`

predicate tries to solve the query `P`

: it fails if
`P`

succeeds, and succeeds if `P`

fails. In any case no trace is left
of the computation for `P`

. E.g. `not(X = 1, 2 < 1)`

suceeds, but
the assignment for `X`

is undone. See also the section
about the foundations of λProlog.

The rules we have seen so far are *facts*: they always hold.
In general rules can only be applied if some *condition* holds. Conditions are
also called premises, we may use the two terms interchangeably.

Here we add to our program a rule that defines what `older P Q`

means
in terms of the `age`

of `P`

and `Q`

.

Note

`:-`

separates the *head* of a rule from the *premises*

```
Elpi Accumulate lp:{{
pred older o:person, o:person.
older P Q :- age P N, age Q M, N > M.
}}.
```

The rule reads: `P`

is older than `Q`

if
`N`

is the age of `P`

*and* `M`

is the age of `Q`

*and* `N`

is greater than `M`

.

Let's run a query using older:

The query `older bob X`

is unified with the head of
the program rule `older P Q`

assigning `P = bob`

and `X = Q`

. Then three new queries are run:

age bob N age Q M N > M

The former assigns `N = 23`

, the second one first
sets `Q = mallory`

and `M = 23`

. This makes the last
query to fail, since `23 > 23`

is false. Hence the
second query is run again and again until `Q`

is
set to `alice`

and `M`

to `20`

.

Variables in the query are said to be existentially quantified because Elpi will try to find one possible value for them.

Conversely, the variables used in rules are universally quantified in the front of the rule. This means that the same program rule can be used multiple times, and each time the variables are fresh.

In the following example the variable `P`

in `older P Q :- ...`

once takes `bob`

and another time takes `mallory`

.

So far the syntax of terms is based on constants
(eg `age`

or `mallory`

) and variables (eg `X`

).

λProlog adds another term constructor:
λ-abstraction (written `x\ ...`

).

Note

the variable name before the `\` can be a capital

Given that it is explicitly bound Elpi needs not to guess if it is a global symbol or a rule variable (that required the convention of using capitals for variables in the first place).

Functions built using λ-abstraction can be applied to arguments and honor the usual β-reduction rule (the argument is substituted for the bound variable).

In the following example `F 23`

reads, once
the β-reduction is performed, `age alice 23`

.

Let's now write the "hello world" of λProlog: an
interpreter and type checker for the simply
typed λ-calculus. We call this program `stlc`

.

We start by declaring that `term`

is a type and
that `app`

and `fun`

are constructors of that type.

```
Elpi Program stlc lp:{{
kind term type.
type app term -> term -> term.
type fun (term -> term) -> term.
}}.
```

The constructor `app`

takes two terms
while `fun`

only one (of functional type).

Note that

- there is no constructor for variables, we will use the notion of bound variable of λProlog in order to represent variables
`fun`

takes a function as subterm, i.e. something we can build using the λ-abstraction`x\ ...`

As a consequence, the identity function λx.x is written like this:

fun (x\ x)

while the first projection λx.λy.x is written:

fun (x\ fun (y\ x))

Another consequence of this approach is that there is no such thing as a free variable in our representation of the λ-calculus. Variables are only available under the λ-abstraction of the programming language, that gives them a well defined scope and substitution operation (β-reduction).

This approach is called HOAS.

We can now implement weak head reduction, that is we stop reducing
when the term is a `fun`

or a global constant (potentially applied).
If the term is `app (fun F) A`

then we compute the reduct `F A`

.
Note that `F`

is a λProlog function, so passing an argument to it
implements the substitution of the actual argument for the bound variable.

We first give a type and a mode for our predicate `whd`

. It reads
"whd takes a term in input and gives a term in output". We will
explain what input means precisely later, for now just think about it
as a comment.

```
Elpi Accumulate lp:{{
pred whd i:term, o:term.
% when the head "Hd" of an "app" (lication) is a
% "fun" we substitute and continue
whd (app Hd Arg) Reduct :- whd Hd (fun F), !,
whd (F Arg) Reduct.
% otherwise a term X is already in normal form.
whd X Reduct :- Reduct = X.
}}.
```

Recall that, due to backtracking, all rules are potentially used. Here whenever the first premise of the first rule succeeds we want the second rule to be skipped, since we found a redex.

The premises of a rule are run in order and the `!`

operator discards all
other rules following the current one. Said otherwise it commits to
the currently chosen rule for the current query (but leaves
all rules available for subsequent queries, they are not erased from the
program). So, as soon as `whd Hd (fun F)`

succeeds we discard the second
rule.

Another little test using global constants:

Elpi Accumulate lp:{{ type foo, bar term. }}.

A last test with a lambda term that has no weak head normal form:

Elpi Bound Steps 1000. (* Let's be cautious *)Elpi Bound Steps 0.

`pi x\ `

and `=>`

We have seen how to implement subtitution using the binders of λProlog. More often than not we need to move under binders rather than remove them by substituting some term in place of the bound variable.

In order to move under a binder and inspect the body of a function λProlog
provides the `pi`

quantifier and the `=>`

connective.

A good showcase for these features is to implement a type checker for the simply typed lambda calculus. See also the Wikipedia page on the simply typed lambda calculus.

We start by defining the data type of simple types.
We then declare a new predicate `of`

(for "type of") and finally
we provide two rules, one for each term constructor.

```
Elpi Accumulate lp:{{
kind ty type. % the data type of types
type arr ty -> ty -> ty. % our type constructor
pred of i:term, o:ty. % the type checking algorithm
% for the app node we ensure the head is a function from
% A to B, and that the argument is of type A
of (app Hd Arg) B :-
of Hd (arr A B), of Arg A.
% for lambda, instead of using a context (a list) of bound
% variables we use pi and => , explained below
of (fun F) (arr A B) :-
pi x\ of x A => of (F x) B.
}}.
```

The `pi name\ code`

syntax is reserved, as well as
`rule => code`

.

Operationally `pi x\ code`

introduces a fresh
constant `c`

for `x`

and then runs `code`

.
Operationally `rule => code`

adds `rule`

to
the program and runs `code`

. Such extra rule is
said to be hypothetical.
Both the constant for `x`

and `rule`

are
removed once `code`

terminates.

Important

hypothetical rules are added at the *top* of the program

Hypothetical rules hence take precedence over static rules, since they are tried first.

Note that in this last example the hypothetical rule is going to be
`of c A`

for a fixed `A`

and a fresh constant `c`

.
The variable `A`

is fixed but not assigned yet, meaning
that `c`

has a type, and only one, but we may not know it yet.

Now let's assign a type to λx.λy.x:

Let's run this example step by step:

The rule for `fun`

is used:

`arrow A1 B1`

is assigned to`Ty`

by unification- the
`pi x\`

quantifier creates a fresh constant`c1`

to play the role of`x`

- the
`=>`

connective adds the rule`of c1 A1`

the program - the new query
`of (fun y\ c1) B1`

is run.

Again, the rule for `fun`

is used (since its variables are
universally quantified, we use `A2`

, `B2`

... this time):

`arrow A2 B2`

is assigned to`B1`

by unification- the
`pi x\`

quantifier creates a fresh constant`c2`

to play the role of`x`

- the
`=>`

connective adds the rule`of c2 A2`

the program - the new query
`of c1 B2`

is run.

The (hypotetical) rule `of c1 A1`

is used:

- unification assigns
`A1`

to`B2`

The value of `Ty`

is hence `arr A1 (arr A2 A1)`

, a good type
for λx.λy.x (the first argument and the output have the same type `A1`

).

What about the term λx.(x x) ?

The `;`

infix operator stands for disjunction. Since we see the message
`of`

failed: the term `fun (x\ app x x)`

is not well typed.

First, the rule for elpi:`fun`

is selected:

`arrow A1 B1`

is assigned to`Ty`

by unification- the
`pi x\`

quantifier creates a fresh constant`c1`

to play the role of`x`

- the
`=>`

connective adds the rule`of c1 A1`

the program - the new query
`of (app c1 c1) B1`

is run.

Then it's the turn of typing the application:

the query

`of c1 (arr A2 B2)`

assignes to`A1`

the value`arr A2 B2`

. This means that the hypothetical rule is now`of c1 (arr A2 B2)`

.the query

`of c1 A2`

fails because the unificationof c1 A2 = of c1 (arr A2 B2)

has no solution, in particular the sub problem

`A2 = (arr A2 B2)`

fails the so called occur check.

This section tries to link, informally, λProlog with logic, assuming the reader has some familiarity with first order intuitionistic logic and proof theory. The reader which is not familiar with that can probably skip this section, although section functional-style contains some explanations about the scope of variables which are based on the logical foundations of the language.

The semantics of a λProlog program is given by interpreting it in terms of logical formulas and proof search in intuitionistic logic.

A rule

p A B :- q A C, r C B.

has to be understood as a formula

\begin{equation*}
\forall A~B~C, (\mathrm{q}~A~C \wedge \mathrm{r}~C~B) \rightarrow \mathrm{p}~A~B
\end{equation*}

A query is a goal that is proved by backchaining
rules. For example `p 3 X`

is solved by unifying it with the conclusion of
the formula above (that sets `A`

to `3`

) and
generating two new goals, `q 3 C`

and
`r C B`

. Note that `C`

is an argument to both
`q`

and `r`

and acts as a link: if solving `q`

fixes `C`

then the query for `r`

sees that.
Similarly for `B`

, that is identified with `X`

,
and is hence a link from the solution of `r`

to
the solution of `p`

.

A rule like:

of (fun F) (arr A B) :- pi x\ of x A => of (F x) B.

reads, as a logical formula:

\begin{equation*}
\forall F~A~B, (\forall x, \mathrm{of}~x~A \rightarrow \mathrm{of}~(F~x)~B) \rightarrow \mathrm{of}~(\mathrm{fun}~F)~(\mathrm{arr}~A~B)
\end{equation*}

where \(F\) stands for a function. Alternatively, using the inference rule notation typically used for type systems:

\begin{equation*}
\frac{\Gamma, \mathrm{of}~x~A \vdash \mathrm{of}~(F~x)~B \quad x~\mathrm{fresh}}{\Gamma \vdash \mathrm{of}~(\mathrm{fun}~F)~(\mathrm{arr}~A~B)}
\end{equation*}

Hence, `x`

and `of x A`

are available only
temporarily to prove `of (F x) B`

and this is
also why `A`

cannot change during this sub proof (`A`

is
quantified once and forall outside).

Each program execution is a proof (tree) of the query and is made of the program rules seen as proof rules or axioms.

As we hinted before negation is a black hole, indeed the usual definition of \(\neg A\) as \(A \to \bot\) is the one of a function with no output (see also the the Wikipedia page on the Curry-Howard correspondence).

Elpi extends λProlog with *syntactic constraints*
and rules to manipulate the store of constraints.

Syntactic constraints are goals suspended on a variable which are resumed as soon as that variable gets instantiated. While suspended they are kept in a store which can be manipulated by dedicated rules.

A companion facility is the declaration of *modes*.
The argument of a predicate can be marked as input
to avoid instantiating the goal when it is unified
with the head of a rule (an input argument
is matched, rather than unified).

A simple example: Peano's addition:

Elpi Program peano lp:{{ kind nat type. type z nat. type s nat -> nat. pred add o:nat, o:nat, o:nat. add (s X) Y (s Z) :- add X Y Z. add z X X. }}.

Unfortunately the relation does not work well
when the first argument is a variable. Depending on the
order of the rules for `add`

Elpi can either diverge or pick
`z`

as a value for `X`

(that may not be what one wants)

Elpi Bound Steps 100.Elpi Bound Steps 0.

Indeed the first rule for add can be applied forever.
If one exchanges the two rules in the program, then Elpi
terminates picking `z`

for `X`

.

We can use the mode directive in order to
*match* arguments marked as `i:`

against the patterns
in the head of rules, rather than unifying them.

Elpi Program peano2 lp:{{ kind nat type. type z nat. type s nat -> nat. pred sum i:nat, i:nat, o:nat. sum (s X) Y (s Z) :- sum X Y Z. sum z X X. sum _ _ _ :- coq.error "nothing matched but for this catch all clause!". }}.

The query fails because no rule first argument matches `X`

.

Instead of failing we can suspend goals and turn them into syntactic constraints

Elpi Program peano3 lp:{{ kind nat type. type z nat. type s nat -> nat. pred sum i:nat, i:nat, o:nat. sum (s X) Y (s Z) :- sum X Y Z. sum z X X. sum X Y Z :- % the head of the rule always unifies with the query, so % we double check X is a variable (we could also be % here because the other rules failed) var X, % then we declare the constraint and schedule its resumption % on the assignment of X declare_constraint (sum X Y Z) [X]. }}.

Syntactic constraints are resumed when the variable they are suspended on is assigned:

Here a couple more examples. Keep in mind that:

- resumption can cause failure
- recall that
`;`

stands for disjunction

In this example the computation suspends, then makes progess, then suspends again...

Sometimes the set of syntactic constraints becomes unsatisfiable and we would like to be able to fail early.

Elpi Accumulate lp:{{ pred even i:nat. pred odd i:nat. even z. even (s X) :- odd X. odd (s X) :- even X. odd X :- var X, declare_constraint (odd X) [X]. even X :- var X, declare_constraint (even X) [X]. }}.

A constraint (handling) rule can see the store of syntactic constraints as a whole, remove constraints and/or create new goals:

Elpi Accumulate lp:{{ constraint even odd { % if two distinct, conflicting, constraints about the same X % are part of the constraint store rule (even X) (odd X) <=> % generate the following goal (coq.say X "can't be even and odd at the same time", fail). } }}.

Note

`fail`

is a predicate with no solution

See also the Wikipedia page on Constraint Handling Rules for an introduction to the sub language to manipulate constraints.

Elpi is a relational language, not a functional one. Still some features typical of functional programming are available, with some caveats.

Chaining "relations" can be painful, especially when they look like functions. Here we use std.append and std.rev to build a palindrome:

Elpi Program function lp:{{ pred make-palindrome i:list A, o:list A. make-palindrome L Result :- std.rev L TMP, std.append L TMP Result. }}.

Note

variables (capital letters) can be used in types in order to describe ML-like polymorphism.

Note

`list A`

is a built-in data type

The empty list is written `[]`

, while the cons constructor
is written `[Hd | Tail]`

. Iterated cons can be written
`[ E1, E2 | Tail ]`

and `| Tail`

can be omitted if the list
is nil terminated.

The `make-palindrome`

predicate has to use a temporary variable
just to pass the output of a function as the input to another function.

Spilling is a syntactic elaboration which does that for you.
Expressions between `{`

and `}`

are
said to be spilled out and placed just before the predicate
that contains them.

Elpi Accumulate lp:{{ pred make-palindrome2 i:list A, o:list A. make-palindrome2 L Result :- std.append L {std.rev L} Result. }}.

The two versions of `make-palindrome`

are equivalent.
Actually the latter is elaborated into the former.

Functions about built-in data types are available via the
calc predicate or its infix version `is`

. Example:

The calc predicate works nicely with spilling:

The language let's one use λ-abstraction also to write anonymous rules but one has to be wary of where variables are bound (allocated really).

In our example we use the higher order predicate std.map:

pred std.map i:list A, i:(A -> B -> prop), o:list B.

Note

`prop`

is the type of predicates

The actual type of the `std.map`

symbol is:

type std.map list A -> (A -> B -> prop) -> list B -> prop.

The `pred`

directive complements a type declaration for predicates
(the trailing `-> prop`

is implicit) with a mode declaration for
each argument.

The type of the second argument of `std.map`

is the one of a predicate relating `A`

with `B`

.

Let's try to call `std.map`

passing an anonymous rule (as we
would do in a functional language by passing an anonymous function):

Elpi Accumulate lp:{{ pred bad i:list int, o:list int. bad L Result :- std.map L (x\ r\ TMP is x + 1, r = TMP) Result. pred good i:list int, o:list int. good L Result :- std.map L good.aux Result. good.aux X R :- TMP is X + 1, R = TMP. pred good2 i:list int, o:list int. good2 L Result :- std.map L (x\ r\ sigma TMP\ TMP is x + 1, r = TMP) Result. }}.

The problem with `bad`

is that `TMP`

is fresh each time the rule
is used, but not every time the anonymous rule passed to map
is used. Technically `TMP`

is quantified (allocated) where `L`

and `Result`

are.

There are two ways to quantify `TMP`

correctly, that is inside the
anonymous predicate. One is to actually name the predicate. Another one is
to use the `sigma x\ `

quantifier to allocate `TMP`

at every call.
We recommend to name the auxiliary predicate.

Tip

predicates whose name ends in `.aux`

don't trigger a missing type
declaration warning

One last way to skin the cat is to use `=>`

as follows. It gives us
the occasion to clarify further the scope of variables.

Elpi Accumulate lp:{{ pred good3 i:list int, o:list int. good3 L Result :- pi aux\ (pi TMP X R\ aux X R :- TMP is X + 1, R = TMP) => std.map L aux Result. }}.

In this case the auxiliary predicate `aux`

is only visible inside `good3`

.
What is interesting to remark is that the quantifications are explicit
in the hypothetical rule, and they indicate clearly that each and every
time `aux`

is used `TMP`

, `X`

and `R`

are fresh.

The `pi x\ `

quantifier is dual to `sigma x\ `

: since here it
occurs negatively it has the same meaning. That is, the hypothetical rule
could be written `pi X R\ aux X R :- sigma TMP\ TMP is X + 1, R = TMP`

.

Tip

`pi x\ `

and `sigma x\ `

can quantify on a bunch of variables
at once

That is, `pi x y\ ...`

is equivalent to `pi x\ pi y\ ...`

and
`sigma x y\ ...`

is equivalent to `sigma x\ sigma y\ ...`

.

Tip

`=>`

can load more than one clause at once

It is sufficient to put a list on the left hand side, eg `[ rule1, rule2 ] => code`

.
Moreover one can synthesize a rule before loading it, eg:

Rules = [ one-more-rule | ExtraRules ], Rules => code

The last remark worth making is that bound variables are intimately related to universal quantification, while unification variables are related to existential quantification. It goes without saying that the following two formulas are not equivalent and while the former is trivial the latter is in general false:

\begin{align*}
\forall x, \exists Y, Y = x\\
\exists Y, \forall x, Y = x
\end{align*}

Let's run these two corresponding queries:

Another way to put it: `x`

is in the scope of `Y`

only in the first
formula since it is quantified before it. Hence `x`

can be assigned to
`Y`

in that case, but not in the second query, where it is quantified
after.

More in general, λProlog tracks the bound variables that are in scope of each unification variable. There are only two ways to put a bound variable in the scope:

quantify the unification variable under the bound one (first formula)

pass the bound variable to the unification variable explicitly: in this case the unification variable needs to have a functional type. Indeed \(\exists Y, \forall x, (Y x) = x\) has a solution:

`Y`

can be the identity function.

If we look again at the rule for type checking λ-abstraction:

of (fun F) (arr A B) :- pi x\ of x A => of (F x) B.

we can see that the only unification variable that sees the fresh
`x`

is `F`

, because we pass `x`

to `F`

explicitly
(recall all unification variables such as `F`

, `A`

, `B`

are
quantified upfront, before the `pi x\ `

).
Indeed when we write:

\begin{equation*}
\frac{\Gamma, x : A \vdash f : B}{\Gamma \vdash \lambda x.f : A \rightarrow B}
\end{equation*}

on paper, the variable denoted by `x`

being bound there can only occur in
\(f\), not in \(\Gamma\) or \(B\) for example (although a
*different* variable could be named the same, hence the usual freshness side
conditions which are not really necessary using HOAS).

Remark that in the premise the variable \(x\) is still bound, this time
not by a λ-abstraction but by the context \(\Gamma, x : A\).
In λProlog the context is the set of hypothetical rules and `pi\ `

-quantified variables and is implicitly handled by the runtime of the
programming language.

A slogan to keep in mind is that:

Important

There is no such thing as a free variable!

Indeed the variable bound by the λ-abstraction (of our data) is replaced by a fresh variable bound by the context (of our program). This is called binder mobility. See also the paper Mechanized metatheory revisited by Dale Miller which is an excellent introduction to these concepts.

A common λProlog idiom is to have a debug rule
lying around. The `:if`

attribute can be used to
make the rule conditionally interpreted (only if the
given debug variable is set).

Elpi Debug "DEBUG_MYPRED". Elpi Program debug lp:{{ pred mypred i:int. :if "DEBUG_MYPRED" mypred X :- coq.say "calling mypred on " X, fail. mypred 0 :- coq.say "ok". mypred M :- N is M - 1, mypred N. }}.

As a slightly more sophisticated debugging feature, the Elpi interpreter provides tracing facilities.

Elpi Trace.

The trace can be limited to a range of steps. Look at the
numbers `run HERE {{{`.

Elpi Trace 6 8.

The trace can be limited to a (list of) predicates as follows:

Elpi Trace "of".

One can combine the range of steps with the predicate:

Elpi Trace 6 8 "of".

To switch traces off:

`Elpi Trace Off.`

Given that programs are not written in a single place, but rather obtained by accumulating code, Elpi is able to print a (full) program to an html file as follows. The obtained file provides a facility to filter rules by their predicate.

`Elpi Print stlc "stlc.html".`

Look at the generated page
and type `of`

in the filter.

Finally, one can bound the number of backchaining steps performed by the interpreter:

Elpi Query lp:{{ 0 = 0, 1 = 1 }}. Elpi Bound Steps 1.Elpi Bound Steps 0. (* Go back to no bound *)

Well, no programming language is perfect.

Backtracking can lead to weird execution traces. The std.do! predicate should be used to write non-backtracking code.

pred not-a-backtracking-one. not-a-backtracking-one :- condition, !, std.do! [ step, (generate, test), step, ].

In the example above once `condition`

holds we start a sequence of
steps which we will not reconsider. Locally, backtracking is still
available, e.g. between `generate`

and `test`

.
See also the std.spy-do! predicate which prints each and every step,
and the std.spy one which can be used to spy on a single one.

Unification variables sit in between variables in imperative programming and functional programming. In imperative programming a variable can hold a value, and that value can change over time via assignment. In functional languages variables always hold a value, and that value never changes. In logic programming a unification variable can either be unset (no value) or set to a value that never changes. Backtracking goes back in time, it is not visible to the program.

As a result of this, code like

pred bad-example. bad-example :- X is 1 + 2, X is 4 + 5.

fails, because `X`

cannot be at the same time 3 and 9. Initially
`X`

is unset, then it is set to 3, and finally the programmer is
asserting that 3 (the value hold by `X`

) is equal to 9, not changing
the value carried by `X`

.
Unification, and hence the `=`

pradicate, plays two roles.
When `X`

is unset, `X = v`

sets the variable.
When `X`

is set to `u`

, `X = v`

checks if the value
of `X`

is equal to `u`

: it is equivalent to `u = v`

.

The λProlog website contains useful links to λProlog related material.

Papers and other documentation about Elpi can be found at the Elpi home on github.

Three more tutorials specific to Elpi as an extension language for Coq can be found in the examples folder. You can continue by reading the one about the HOAS for Coq terms.