#### zipperposition

Automated theorem prover for first-order logic with equality and theories.

Logic toolkit (

`logtk`

), designed primarily

for first-order automated reasoning. It aims

at providing basic types and algorithms (terms, unification, orderings,

indexing, etc.) that can be factored out of several applications.

### Short summary

Zipperposition is intended to be a superposition prover for full first

order logic, plus some extensions

(datatypes, recursive functions, arithmetic, lambda-free higher order).

The accent is on flexibility, modularity and simplicity rather than

performance, to allow quick experimenting on automated theorem proving. It

generates TSTP traces or graphviz files for nice graphical display.

Zipperposition supports several input formats:

TPTP (fof, cnf, tff)

its own native input, extension

`.zf`

(see directory`examples/`

and section below)

Zipperposition is written in the functional and imperative language

OCaml. The name is a bad play on the words "zipper" (a

functional data structure) and "superposition" (the calculus used by the

prover), although the current implementation is written in quite an imperative style.

Superposition-based theorem proving is an active field of research, so

there is a lot of literature about it; for this implementation the main references

for the base calculus are:

the chapter

*paramodulation-based theorem proving*of the*handbook of automated reasoning*,the paper

*E: a brainiac theorem prover*that describes the E prover by S.Schulz,the paper

*Superposition with equivalence reasoning and delayed clause normal form transformation*by H.Ganzinger and J.Stuber

**Disclaimer**: Note that the prover is a research project.

Please don't use it to drive your personal nuclear power plant, nor as a

trusted tool for critical applications.

### License

This project is licensed under the BSD2 license. See the `LICENSE`

file.

### Build

Zipperposition requires OCaml >= 4.03.0, and some libraries that are

available on opam.

#### Via opam

The recommended way to install Zipperposition is through opam.

You need to have GMP (with headers) installed (it's not handled by opam).

Once you have installed GMP and opam, type:

```
$ opam install zipperposition
```

To upgrade to more recent versions:

```
$ opam update
$ opam upgrade
```

If you want to try the development (unstable) version, which has more

dependencies (in particular `dune`

for the build), try:

```
$ opam pin -k git https://github.com/sneeuwballen/zipperposition.git#master
```

NOTE: do *not* install `logtk`

. It now ships with zipperposition itself.

NOTE: if installation fails, you might want to try to `opam update`

and`opam upgrade`

: it might be because some of the dependencies are too old.

#### Manually

If you really need to, you can download a release on the

following github page for releases.

Look in the file `opam`

to see which dependencies you need to install.

They include `menhir`

, `zarith`

, `containers`

,`oclock`

, msat and `sequence`

, but

maybe also other libraries. Consider using opam directly if possible.

```
$ make install
```

Additional sub-libraries can be built if their respective dependencies

are met.

If menhir is installed, the

parsers library `Logtk_parsers`

will automatically be built.

If you have installed qcheck

and alcotest, for instance

via `opam install qcheck alcotest`

, you can enable the property-based testing and

random term generators with

```
$ make test
```

NOTE: in case of build errors, it might be because of outdated dependencies

(see via opam for more details), or stale build files.

Try `rm _build -rf`

to try to build from scratch.

### Documentation

See this page.

There are some examples of how to use the libraries in `src/tools/`

and `src/demo/`

.

### Use

Typical usage:

```
$ zipperposition --help
$ zipperposition problem_file [options]
$ zipperposition --arith examples/ARI114=1.p
$ zipperposition --dot /tmp/foo.dot examples/ind/nat1.zf
```

to run the prover. Help is available with the option `--help`

.

For instance,

```
$ zipperposition examples/pelletier_problems/pb47.p --ord rpo6 --timeout 30
```

To build the library, documentation, and tools, type in a terminal located in

the root directory of the project:

```
$ make
```

If you use `ocamlfind`

(which is strongly recommended),

installation/uninstallation are just:

```
$ make install
$ make uninstall
```

#### Native Syntax

The native syntax, with file extension `.zf`

, resembles a simple fragment of

ML with explicit polymorphism. Many examples

in `examples/`

are written using this syntax.

A vim syntax coloring file is available.

##### Basics

Comments start with `#`

and continue to the end of the line.

Every symbol must be declared, using the builtin type `prop`

for propositions.

A type is declared like this: `val i : type.`

and a parametrized type: `val array: type -> type.`

```
val i : type.
val a : i.
val f : i -> i. # a function
val p : i -> i -> prop. # a binary predicate
```

Then, axioms and the goal:

```
assert forall x y. p x y => p y x.
assert p a (f a).
goal exists (x:i). p (f x) x.
```

We can run the prover on a file containing these declarations.

It will display a proof very quickly:

```
$ ./zipperposition.native example.zf
% done 3 iterations
% SZS status Theorem for 'example.zf'
% SZS output start Refutation
* ⊥/7 by simp simplify with [⊥]/5
* [⊥]/5 by
inf s_sup- with {X2[1] → a[0]}
with [p (f a) a]/4, forall (X2:i). [¬p (f X2) X2]/2
* forall (X2:i). [¬p (f X2) X2]/2 by
esa cnf with ¬ (∃ x/13:i. (p (f x/13) x/13))
* [p (f a) a]/4 by simp simplify with [p (f a) a ∨ ⊥]/3
* [p (f a) a ∨ ⊥]/3 by
inf s_sup- with {X0[0] → f a[1], X1[0] → a[1]}
with [p a (f a)]/1, forall (X0:i) (X1:i). [p X0 X1 ∨ ¬p X1 X0]/0
* ¬ (∃ x/13:i. (p (f x/13) x/13)) by
esa neg_goal negate goal to find a refutation
with ∃ x/13:i. (p (f x/13) x/13)
* ∃ x/13:i. (p (f x/13) x/13) by goal 'example.zf'
* forall (X0:i) (X1:i). [p X0 X1 ∨ ¬p X1 X0]/0 by
esa cnf with ∀ x/9:i y/11:i. ((p x/9 y/11) ⇒ (p y/11 x/9))
* [p a (f a)]/1 by esa cnf with p a (f a)
* p a (f a) by 'example.zf'
* ∀ x/9:i y/11:i. ((p x/9 y/11) ⇒ (p y/11 x/9)) by 'example.zf'
% SZS output end Refutation
```

Each `*`

-prefixed item in the list is an inference step. The top step is

the empty clause: zipperposition works by negating the goal before looking

for proving `false`

. Indeed, proving `a ⇒ b`

is equivalent to deducing`false`

from `a ∧ ¬b`

.

##### Connectives and Quantifiers

The connectives are:

true:

`true`

false:

`false`

conjunction:

`a && b`

disjunction:

`a || b`

negation:

`~ a`

equality:

`a = b`

disequality:

`a != b`

(synonym for`~ (a = b)`

)implication:

`a => b`

equivalence:

`a <=> b`

Implication and equivalence have the same priority as disjunction.

Conjunction binds tighter, meaning that `a && b || c`

is actually parsed as `(a && b) || c`

.

Negation is even stronger: `~ a && b`

means `(~ a) && b`

.

Binders extend as far as possible to their right, and are typed, although

the type constraint can be omitted if it can be inferred:

universal quantification:

`forall x. F`

or in its typed form:`forall (x:ty). F`

existential quantification:

`exists x. F`

Polymorphic symbols can be declare using `pi <var>. type`

,

for instance `val f : pi a b. a -> array a b -> b`

is a polymorphic

function that takes 2 type arguments, then 2 term arguments.

An application of `f`

will look like `f nat (list bool) (Succ Z) empty`

.

Type arguments might be omitted if they can be inferred.

##### Inclusion

It can be convenient to put commonly used axioms in a separate file.

The statement

```
include "foo.zf".
```

will include the corresponding file (whose path is relative to the

current file).

##### Advanced Syntax

There are more advanced concepts that are mostly related to induction:

datatypes: (here, Peano numbers and polymorphic lists)

```
data nat := Zero | Succ nat.
data list a := nil | cons a (list a).
```

simple definitions:

```
def four : nat := Succ (Succ (Succ (Succ Zero))).
```

rewrite rules:

A rewrite rule is similar to an `assert`

statement, except it is much

more efficient. Zipperposition assumes that the set of rewrite rules

in its input is *confluent* and *terminating* (otherwise, no guarantee

applies). Rewriting can be done on terms and on atomic formulas:

```
val set : type -> type.
val member : pi a. a -> set a -> prop.
val union : pi a. set a -> set a -> set a.
rewrite forall a (x:a)(s1:set a)(s2:set a).
member x (union s1 s2) <=> (member x s1 || member x s2).
val subset : pi a. set a -> set a -> prop.
rewrite forall a (s1:set a)(s2:set a).
subset s1 s2 <=> (forall x. member x s1 => member x s2).
val equal_set : pi a. set a -> set a -> prop.
rewrite forall a (s1:set a) s2.
equal_set s1 s2 <=> subset s1 s2 && subset s2 s1.
# now show that union is associative:
goal forall a (s1:set a) s2 s3.
equal_set
(union s1 (union s2 s3))
(union (union s1 s2) s3).
```

there are several variations on literal rewrite rules:

`rewrite forall x. p x`

(short for`p x <=> true`

)`rewrite forall x. ~ p x`

(short for`p x <=> false`

)`rewrite forall x. p x => q x`

(one way rule, will rewrite`p x`

but not`~ p x`

; also called*polarized rewriting*)`rewrite forall x. ~ p x => q x`

(negative polarized rule)

recursive definitions:

one can write recursive functions (assuming they terminate), they

will be desugared to a declaration + a set of rewrite rules:

```
def plus : nat -> nat -> nat where
forall y. plus Zero y = y;
forall x y. plus (Succ x) y = Succ (plus x y).
```

Mutually recursive definitions are separated by `and`

:

```
def even : nat -> prop where
even Zero;
forall x. even (Succ x) = odd x
and odd : nat -> prop where
forall x. odd (Succ x) = even x.
```

Zipperposition is able to do simple inductive proofs using these recursive

functions and datatypes:

```
$ cat doc/plus_assoc.zf
data nat := Zero | Succ nat.
def plus : nat -> nat -> nat where
forall y. plus Zero y = y;
forall x y. plus (Succ x) y = Succ (plus x y).
goal forall (x:nat) y z. plus x (plus y z) = plus (plus x y) z.
$ zipperposition doc/plus_assoc.zf -o none
% done 17 iterations
% SZS status Theorem for 'doc/plus_assoc.zf'
```

conditionals:

tests on boolean formulas are written `if a then b else c`

, where `a:prop`

,`b`

, and `c`

, are terms. `b`

and `c`

must have the same type.

pattern-matching:

shallow pattern matching is written `match <term> with [case]+ end`

where each case is `| <constructor> [var]* -> <term>`

.

AC symbols:

Some symbols can be declared "associative commutative": they satisfy

`forall x y z. f x (f y z) = f (f x y) z`

`forall x y. f x y = f y x`

.

the following statement is a bit more efficient than writing the corresponding

axioms:

```
val[AC] f : foo -> foo -> foo.
```

Axioms in

*Set of Support*:

Some axioms (introduced using `assert [sos] <formula>.`

) will be considered

as part of the so-called "set of support" strategy.

No saturation among SOS axioms is done. They are only used for inferences

(and simplifications) with non-SOS axioms and goals.

Typically this is useful for introducing general lemmas while preventing them

from interacting in ways not related to the current goal.

Named Axioms:

An axiom can be given a name, as in TPTP, to retrieve it easily in proofs.

The syntax is:

```
assert[name "foo"] bar.
```

#### Graphical Display of Proofs

A handy way of displaying the proof is to use graphviz:

```
$ ./zipperposition.native --dot /tmp/example.dot example.zf
$ dot -Txlib /tmp/example.dot
```

One can generate an image from the `.dot`

file:

```
$ dot -Tsvg /tmp/example.dot > some_picture.svg
```

#### Proof Format

It is possible to avoid displaying the proof at all, by using `-o none`

.

A TSTP derivation can be obtained with `-o tstp`

.

#### Library

Zipperposition's library provides several useful

parts for logic-related implementations:

a library packed in a module

`Logtk`

, with terms, formulas, etc.;a library packed in a module

`Logtk_parsers`

, with parsers for input formats;small tools (see directory

`src/tools/`

) to illustrate how to use the library

and provide basic services (type-checking, reduction to CNF, etc.);

### Hacking

Some advices if you want to hack on the code:

`--debug 5`

prints everything the prover does`--debug.foo <n>`

changes the verbosity only for`foo`

(see`--help`

for a list of such flags)`--backtrace`

is very useful to get stack traces when a

wild uncaught exception appears`--stats`

prints some statistics, and you can add your own easily

with`Util.mk_stat`

`--dot <some-file>.dot`

dumps the proof in the given file

in graphviz. This is very useful for reading proofs, e.g.

using`dot -Txlib <some-file>.dot`

.

See Graphical display of proofs for more details.many flags control the behavior of the prover; to dumb heuristics down

a bit you can try:`-cq bfs`

(BFS traversal of the search space, instead of weight-based

clause selection rules)`--ord none`

for disabling term orderings

### StarExec

StarExec is a service for experimental evaluation of logic solvers like Zipperposition.

The easiest way to import Zipperposition as a solver is to pre-compile Zipperposition on the StarExec virtual machine. Download the VM image and open it in VirtualBox.

Open the settings of the VM. Set "Network > Adapter 1 > Attached to" to NAT to have internet access from inside the VM. To allow SSH access into the VM open "Network > Adapter 1 > Advanced > Port Forwarding" and create a new rule:

```
Name: ssh
Protocol: TCP
Host Port: 3022
Guest Port: 22
```

Leave the two IP fields empty.

Start the VM. Log in as root using the password "St@rexec".

```
starclone login: root
Password: St@rexec
```

Install the openssh server to get a more convenient access to the machine and to copy the compiled binary later.

```
$ yum install openssh-server
```

Now open a terminal on the host machine while the VM is still running. Tunnel into the VM via SSH:

```
$ ssh -p 3022 root@127.0.0.1
root@127.0.0.1's password: St@rexec
```

Install OPAM:

```
$ wget https://raw.github.com/ocaml/opam/master/shell/opam_installer.sh -O - | sh -s /usr/local/bin
```

So far we have used the superuser root. To download and compile Zipperposition we will use a regular user that we create as follows:

```
$ useradd -m bob
$ passwd bob
New password: bob
BAD PASSWORD: The password is a palindrome
Retype new password: bob
passwd: all authentication tokens updated successfully.
```

Close the SSH connection and reopen it as the new user:

```
$ exit
$ ssh -p 3022 bob@127.0.0.1
bob@127.0.0.1's password: bob
```

Initialize OPAM. Install OCaml 4.05 and the dependencies of Zipperposition (Look in the file `opam`

to see which dependencies you need to install).

```
$ opam init
$ opam switch 4.05.0+flambda
$ eval `opam config env`
$ opam install dune zarith containers sequence msat menhir
```

Clone Zipperposition and compile it:

```
$ git clone https://github.com/sneeuwballen/zipperposition.git --branch dev
$ cd zipperposition
$ make
```

Close the SSH connection and copy the binary from the VM onto your host machine.

```
$ exit
$ scp -P 3022 bob@127.0.0.1:~/zipperposition/zipperposition.native /some/path/on/the/host/machine
bob@127.0.0.1's password: bob
```

As described in the StarExec documentation you need a script whose filename has the prefix `starexec_run_`

to execute your solver. For Zipperposition this script could look like this:

```
#!/bin/sh
./zipperposition.native -o tptp "$1" \
--timeout "$STAREXEC_WALLCLOCK_LIMIT" \
--mem-limit "$STAREXEC_MAX_MEM"
```

Put this script and the file `zipperposition.native`

into a folder called `bin`

. Create a ZIP archive containing that folder. Now Zipperposition is ready to be uploaded to StarExec!

### Docker

(experimental)

to build an image:

`docker build -t zipper .`

to use the image:

`docker run -i zipper < examples/pelletier_problems/pb47.zf`

### Howto (for devs)

#### Make a release

merge

`dev`

into`master`

:`git checkout master; git merge dev`

`make clean all`

(to rerun tests, etc. see if merge was ok)merge

`master`

into`stable`

(branch with only releases):`git checkout stable; git merge master --no-ff`

edit

`*.opam`

files to update the version number (field`version`

).`git commit -a -m "prepare for <version>"`

(to save the changes on the stable branch)`make clean all`

(to check everything builds properly)`git tag <version>`

(e.g.`git tag 1.4`

)`git push origin <version>`

(`origin`

being the name of the github remote)`opam publish prepare zipperposition.1.4 https://github.com/sneeuwballen/zipperposition/archive/1.4.tar.gz`

(using the actual version number).

This might require to`opam install opam-publish`

first, it's a handy opam plugin

for managing releases.if that works properly, then it will create a directory

`zipperposition.<version>`

.

Just run`opam publish submit zipperposition.1.4`

to open a PR against

opam-repository.

If something is wrong with the release, it's possible to change it.

This is a bit brutal, *never* do it for older releases that have been

merged into opam-repo, only for the next release while no one has seen it yet.

`git tag -f <version>; git push origin :<version> ; git push origin <version>`

to change the tagre-run the two

`opam publish`

commands to update the directory and

the PR.

md5=cc320f66f10555c54822da624419e003

sha512=f8d5f7a5ae790bf0388d74261673803cf375f91f92f7b413b70db1ce5841ef55343a208f98727c8551d66f1840ab892f1c0c943a34861d14d79ce469b235a2f2

>= "4.03"

build

>= "0.8" & < "0.9"

>= "1.1"

>= "1.2"

>= "1.0" & < "3.0"

= version

= version