Tutorials

This page is under-construction. For now, it contains a copy of the files within the examples/ folder of the pySMT repository.

If you are interested in helping us create better tutorials, please let us know at info@pysmt.org .

First example

# Checking satisfiability of a formula.
#
# This example shows:
#  1. How to build a formula
#  2. How to perform substitution
#  3. Printing
#  4. Satisfiability checking
from pysmt.shortcuts import Symbol, And, Not, is_sat

varA = Symbol("A") # Default type is Boolean
varB = Symbol("B")
f = And([varA, Not(varB)])
g = f.substitute({varB:varA})

res = is_sat(f)
assert res # SAT
print("f := %s is SAT? %s" % (f, res))

res = is_sat(g)
print("g := %s is SAT? %s" % (g, res))
assert not res # UNSAT

Hello World word puzzle

# This is the tutorial example of pySMT.
#
# This example shows how to:
# 1. Deal with Theory atoms
# 2. Specify a solver in the shortcuts (get_model, is_sat etc.)
# 3. Obtain an print a model
#
#
# The goal of the puzzle is to assign a value from 1 to 10 to each letter s.t.
#    H+E+L+L+O = W+O+R+L+D = 25
#
from pysmt.shortcuts import Symbol, And, GE, LT, Plus, Equals, Int, get_model
from pysmt.typing import INT

hello = [Symbol(s, INT) for s in "hello"]
world = [Symbol(s, INT) for s in "world"]
letters = set(hello+world)
domains = And([And(GE(l, Int(1)),
                   LT(l, Int(10))) for l in letters])

sum_hello = Plus(hello) # n-ary operators can take lists
sum_world = Plus(world) # as arguments
problem = And(Equals(sum_hello, sum_world),
              Equals(sum_hello, Int(25)))
formula = And(domains, problem)

print("Serialization of the formula:")
print(formula)

model = get_model(formula)
if model:
    print(model)
else:
    print("No solution found")

Hello World word puzzle using infix-notation

# This is a different take on the puzzle.py example
#
# This examples shows how to:
# 1. Enable and use infix notation
# 2. Use a solver context
#
from pysmt.shortcuts import Symbol, And, Plus, Int
from pysmt.shortcuts import Solver
from pysmt.typing import INT

# Infix-Notation is automatically enabled whenever you import pysmt.shortcuts.
#
# To enable it without using shortcuts, do:
#
#   from pysmt.environment import get_env
#   get_env().enable_infix_notation = True
#
# Similarly, you can disable infix_notation to prevent its accidental use.
#

hello = [Symbol(s, INT) for s in "hello"]
world = [Symbol(s, INT) for s in "world"]
letters = set(hello+world)

# Infix notation for Theory atoms does the overloading of python
# operator. For boolean connectors, we use e.g., x.And(y) This
# increases readability without running into problems of operator
# precedence.
#
# Note how you can mix prefix and infix notation.
domains = And([(Int(1) <= l).And(Int(10) >= l) for l in letters])

sum_hello = Plus(hello) # n-ary operators can take lists
sum_world = Plus(world) # as arguments
problem = (sum_hello.Equals(sum_world)).And(sum_hello.Equals(Int(25)))
formula = domains.And(problem)

print("Serialization of the formula:")
print(formula)

# A context (with-statment) lets python take care of creating and
# destroying the solver.
with Solver() as solver:
    solver.add_assertion(formula)
    if solver.solve():
        for l in letters:
            print("%s = %s" %(l, solver.get_value(l)))
    else:
        print("No solution found")

Combine multiple solvers

# This example requires Z3 and MathSAT to be installed (but you can
#  replace MathSAT with any other solver for QF_LRA)
#
# This examples shows how to:
# 1. Define Real valued constants using floats and fractions
# 2. Perform quantifier elimination
# 3. Pass results from one solver to another
#
from pysmt.shortcuts import Symbol, Or, ForAll, GE, LT, Real, Plus
from pysmt.shortcuts import qelim, is_sat
from pysmt.typing import REAL

x, y, z = [Symbol(s, REAL) for s in "xyz"]

f = ForAll([x], Or(LT(x, Real(5.0)),
                   GE(Plus(x, y, z), Real((17,2))))) # (17,2) ~> 17/2
print("f := %s" % f)
#f := (forall x . ((x < 5.0) | (17/2 <= (x + y + z))))

qf_f = qelim(f, solver_name="z3")
print("Quantifier-Free equivalent: %s" % qf_f)
#Quantifier-Free equivalent: (7/2 <= (z + y))

res = is_sat(qf_f, solver_name="msat")
print("SAT check using MathSAT: %s" % res)
#SAT check using MathSAT: True

Model-Checking an infinite state system (BMC+K-Induction) in ~150 lines

# This example shows a more advance use of pySMT.
#
# It provides a simple implementation of Bounded Model Checking [1]
# and K-Induction [2] and applies it on a simple infinite-state
# transition system.
#
# [1] ...
#
# [2] ...
#
from six.moves import xrange

from pysmt.shortcuts import Symbol, Not, Equals, And, Times, Int, Plus, LE
from pysmt.shortcuts import is_sat, is_unsat
from pysmt.typing import INT


def next_var(v):
    """Returns the 'next' of the given variable"""
    return Symbol("next(%s)" % v.symbol_name(), v.symbol_type())

def at_time(v, t):
    """Builds an SMT variable representing v at time t"""
    return Symbol("%s@%d" % (v.symbol_name(), t), v.symbol_type())


class TransitionSystem(object):
    """Trivial representation of a Transition System."""

    def __init__(self, variables, init, trans):
        self.variables = variables
        self.init = init
        self.trans = trans

# EOC TransitionSystem



def get_subs(system, i):
    """Builds a map from x to x@i and from x' to x@(i+1), for all x in system."""
    subs_i = {}
    for v in system.variables:
        subs_i[v] = at_time(v, i)
        subs_i[next_var(v)] = at_time(v, i+1)
    return subs_i


def get_unrolling(system, k):
    """Unrolling of the transition relation from 0 to k:

    E.g. T(0,1) & T(1,2) & ... & T(k-1,k)
    """
    res = []
    for i in xrange(k):
        subs_i = get_subs(system, i)
        res.append(system.trans.substitute(subs_i))
    return And(res)


def get_simple_path(system, k):
    """Simple path constraint for k-induction:
    each time encodes a different state
    """
    res = []
    for i in xrange(k):
        subs_i = get_subs(system, i)
        for j in xrange(i+1, k):
            subs_j = get_subs(system, j)
            for v in system.variables:
                v_i = v.substitute(subs_i)
                v_j = v.substitute(subs_j)
                res.append(Not(Equals(v_i, v_j)))
    return And(res)


def get_k_hypothesis(system, prop, k):
    """Hypothesis for k-induction: each state up to k fulfills the property"""
    res = []
    for i in xrange(k):
        subs_i = get_subs(system, i)
        res.append(prop.substitute(subs_i))
    return And(res)


def get_bmc(system, prop, k):
    """Returns the BMC encoding at step k"""
    init_0 = system.init.substitute(get_subs(system, 0))
    prop_k = prop.substitute(get_subs(system, k))
    return And(get_unrolling(system, k), init_0, Not(prop_k))

def get_k_induction(system, prop, k):
    """Returns the K-Induction encoding at step K"""
    subs_k = get_subs(system, k)
    prop_k = prop.substitute(subs_k)
    return And(get_unrolling(system, k),
               get_k_hypothesis(system, prop, k),
               get_simple_path(system, k),
               Not(prop_k))

def check_property(system, prop):
    """Interleaves BMC and K-Ind to verify the property."""
    print("Checking property %s..." % prop)
    for b in xrange(100):
        f = get_bmc(system, prop, b)
        print("   [BMC]    Checking bound %d..." % (b+1))
        if is_sat(f):
            print("--> Bug found at step %d" % (b+1))
            return

        f = get_k_induction(system, prop, b)
        print("   [K-IND]  Checking bound %d..." % (b+1))
        if is_unsat(f):
            print("--> The system is safe!")
            return


def main():
    # Example Transition System (SMV-like syntax)
    #
    # VAR x: integer;
    #     y: integer;
    #
    # INIT: x = 1 & y = 2;
    #
    # TRANS: next(x) = x + 1;
    # TRANS: next(y) = y + 2;

    x, y = [Symbol(s, INT) for s in "xy"]
    nx, ny = [next_var(Symbol(s, INT)) for s in "xy"]

    example = TransitionSystem(variables = [x, y],
               init = And(Equals(x, Int(1)),
                          Equals(y, Int(2))),
               trans = And(Equals(nx, Plus(x, Int(1))),
                           Equals(ny, Plus(y, Int(2)))))

    # A true invariant property: y = x * 2
    true_prop = Equals(y, Times(x, Int(2)))

    # A false invariant property: x <= 10
    false_prop = LE(x, Int(10))

    for prop in [true_prop, false_prop]:
        check_property(example, prop)
        print("")

if __name__ == "__main__":
    main()

How to access functionalities of solvers not currently wrapped by pySMT

# This example requires MathSAT to be installed.
#
# This example shows how to use a solver converter to access
# functionalities of the solver that are not wrapped by pySMT.
#
# Our goal is to call the method msat_all_sat from the MathSAT API.
#
import mathsat
from pysmt.shortcuts import Or, Symbol, Solver, And

def callback(model, converter, result):
    """Callback for msat_all_sat.

    This function is called by the MathSAT API everytime a new model
    is found. If the function returns 1, the search continues,
    otherwise it stops.
    """
    # Elements in model are msat_term .
    # Converter.back() provides the pySMT representation of a solver term.
    py_model = [converter.back(v) for v in model]
    result.append(And(py_model))
    return 1 # go on

x, y = Symbol("x"), Symbol("y")
f = Or(x, y)

msat = Solver(name="msat")
converter = msat.converter # .converter is a property implemented by all solvers
msat.add_assertion(f) # This is still at pySMT level

result = []
# Directly invoke the mathsat API !!!
# The second term is a list of "important variables"
mathsat.msat_all_sat(msat.msat_env(),
      [converter.convert(x)],      # Convert the pySMT term into a MathSAT term
      lambda model : callback(model, converter, result))

print("'exists y . %s' is equivalent to '%s'" %(f, Or(result)))
#exists y . (x | y) is equivalent to ((! x) | x)

How to use any SMT-LIB complaint SMT solver

# This example shows how to define a generic SMT-LIB solver, and use
# it within pySMT. The example looks for mathsat in /tmp, you can
# create a symlink there.
#
# Using this process you can experiment with any SMT-LIB 2 compliant
# solver even if it does not have python bindings, or has not been
# integrated with pySMT.
#
# Note: When using the SMT-LIB wrapper, you can only use logics
# supported by pySMT. If the version of pySMT in use does not support
# Arrays, then you cannot represent arrays.
#

# To define a Generic Solver you need to provide:
#
# - A name to associate to the solver
# - The path to the script + Optional arguments
# - The list of logics supported by the solver
#
# It is usually convenient to wrap the solver in a simple shell script.
# See examples for Z3, Mathsat and Yices in pysmt/test/smtlib/bin/*.template
#
from pysmt.logics import QF_UFLRA, QF_UFIDL, QF_LRA, QF_IDL, QF_LIA
from pysmt.shortcuts import get_env, GT, Solver, Symbol
from pysmt.typing import REAL, INT
from pysmt.exceptions import NoSolverAvailableError

name = "mathsat" # Note: The API version is called 'msat'
path = ["/tmp/mathsat"] # Path to the solver
logics = [QF_UFLRA, QF_UFIDL] # Some of the supported logics

env = get_env()

# Add the solver to the environment
env.factory.add_generic_solver(name, path, logics)

r, s = Symbol("r", REAL), Symbol("s", REAL)
p, q = Symbol("p", INT), Symbol("q", INT)

f_lra = GT(r, s)
f_idl = GT(p, q)

# PySMT takes care of recognizing that QF_LRA can be solved by a QF_UFLRA solver.
with Solver(name=name, logic=QF_LRA) as s:
    res = s.solve()
    assert res, "Was expecting '%s' to be SAT" % f_lra


with Solver(name=name, logic=QF_IDL) as s:
    s.add_assertion(f_idl)
    res = s.solve()
    assert res, "Was expecting '%s' to be SAT" % f_idl

try:
    with Solver(name=name, logic=QF_LIA) as s:
        pass
except NoSolverAvailableError:
    # If we ask for a logic that is not contained by any of the
    # supported logics an exception is thrown
    print("%s does not support QF_LIA" % name)

How to combine two different solvers to solve an Exists Forall problem

# EF-SMT solver implementation
#
# This example shows:
#  1. How to combine 2 different solvers
#  2. How to extract information from a model
#
from pysmt.shortcuts import Solver, get_model
from pysmt.shortcuts import Symbol, Bool, Real, Implies, And, Not, Equals
from pysmt.shortcuts import GT, LT, LE, Minus, Times
from pysmt.logics import AUTO, QF_LRA
from pysmt.typing import REAL
from pysmt.exceptions import SolverReturnedUnknownResultError

def efsmt(y, phi, logic=AUTO, maxloops=None,
          esolver_name=None, fsolver_name=None,
          verbose=False):
    """Solves exists x. forall y. phi(x, y)"""

    y = set(y)
    x = phi.get_free_variables() - y

    with Solver(logic=logic, name=esolver_name) as esolver:

        esolver.add_assertion(Bool(True))
        loops = 0
        while maxloops is None or loops <= maxloops:
            loops += 1

            eres = esolver.solve()
            if not eres:
                return False
            else:
                tau = {v: esolver.get_value(v) for v in x}
                sub_phi = phi.substitute(tau).simplify()
                if verbose: print("%d: Tau = %s" % (loops, tau))

                fmodel = get_model(Not(sub_phi),
                                   logic=logic, solver_name=fsolver_name)
                if fmodel is None:
                    return tau
                else:
                    sigma = {v: fmodel[v] for v in y}
                    sub_phi = phi.substitute(sigma).simplify()
                    if verbose: print("%d: Sigma = %s" % (loops, sigma))
                    esolver.add_assertion(sub_phi)

        raise SolverReturnedUnknownResultError


def run_test(y, f):
    print("Testing " + str(f))
    try:
        res = efsmt(y, f, logic=QF_LRA, maxloops=20, verbose=True)
        if res == False:
            print("unsat")
        else:
            print("sat : %s" % str(res))
    except SolverReturnedUnknownResultError:
        print("unknown")
    print("\n\n")


def main():
    x,y = [Symbol(n, REAL) for n in "xy"]
    f_sat = Implies(And(GT(y, Real(0)), LT(y, Real(10))),
                    LT(Minus(y, Times(x, Real(2))), Real(7)))

    f_incomplete = And(GT(x, Real(0)), LE(x, Real(10)),
                       Implies(And(GT(y, Real(0)), LE(y, Real(10)),
                                   Not(Equals(x, y))),
                               GT(y, x)))

    run_test([y], f_sat)
    run_test([y], f_incomplete)


if __name__ == "__main__":
    main()

How to detect the logic of a formula and perform model enumeration

# Perform ALL-SMT on Theory atoms
#
# This example shows:
#  - How to get the logic of a formula
#  - How to enumerate models
#  - How to extract a partial model
#  - How to use the special operator EqualsOrIff
#
from pysmt.shortcuts import Solver, Not, And, Symbol, Or
from pysmt.shortcuts import LE, GE, Int, Plus, Equals, EqualsOrIff
from pysmt.typing import INT
from pysmt.oracles import get_logic

def all_smt(formula, keys):
    target_logic = get_logic(formula)
    print("Target Logic: %s" % target_logic)
    with Solver(logic=target_logic) as solver:
        solver.add_assertion(formula)
        while solver.solve():
            partial_model = [EqualsOrIff(k, solver.get_value(k)) for k in keys]
            print(partial_model)
            solver.add_assertion(Not(And(partial_model)))


A0 = Symbol("A0", INT)
A1 = Symbol("A1", INT)
A2 = Symbol("A2", INT)

f = And(GE(A0, Int(0)), LE(A0, Int(5)),
        GE(A1, Int(0)), LE(A1, Int(5)),
        GE(A2, Int(0)), LE(A2, Int(5)),
        Equals(Plus(A0, A1, A2), Int(8)))

all_smt(f, [A0, A1, A2])

# By using the operator EqualsOrIff, we can mix theory and bool variables
x = Symbol("x")
y = Symbol("y")
f = And(f, Or(x,y))

all_smt(f, [A0, A1, A2, x])

Shows how to use multi-processing to perform parallel and asynchronous solving

# This example requires Z3
#
# This example shows how to parallelize solving using pySMT, by using
# the multiprocessing library.
#
# All shortcuts (is_sat, is_unsat, get_model, etc.) can be easily
# used. We sho two techniques: map and apply_async.
#
# Map applies a given function to a list of elements, and returns a
# list of results. We show an example using is_sat.
#
# Apply_async provides a way to perform operations in an asynchronous
# way. This allows us to start multiple solving processes in parallel,
# and collect the results whenever they become available.
#
# More information can be found in the Python Standard Library
# documentation for the multiprocessing module.
#
#
# NOTE: When running this example, a warning will appear saying:
#   "Contextualizing formula during is_sat"
#
# Each process will inherit a different formula manager, but the
# formulas that we are solving have been all created in the same
# formula manager. pySMT takes care of moving the formula across
# formula managers. This step is called contextualization, and occurs
# only when using multiprocessing.
# To disable the warning see the python module warnings.
#
from multiprocessing import Pool, TimeoutError
from time import sleep

from pysmt.test.examples import get_example_formulae
from pysmt.shortcuts import is_sat, is_valid, is_unsat
from pysmt.shortcuts import And

# Ignore this for now
def check_validity_and_test(args):
    """Checks expression and compare the outcome against a known value."""

    expr, expected = args # IMPORTANT: Unpack args !!!
    local_res = is_valid(expr)
    return local_res == expected


# Create the Pool with 4 workers.
pool = Pool(4)

# We use the examples formula from the test suite.
# This generator iterates over all the expressions
f_gen = (f.expr for f in get_example_formulae())

# Call the functino is_sat on each expression
res = pool.map(is_sat, f_gen)

# The result is a list of True/False, in the same order as the input.
print(res)

sleep(1) # Have some time to look at the result

# Notice that all shortcuts (is_sat, qelim, etc) require only one
# mandatory argument. To pass multiple arguments, we need to pack them
# together.

# Lets create a list of (formula, result), where result is the
# expected result of the is_valid query.
f_gen = (f.expr for f in get_example_formulae())
res_gen = (f.is_valid for f in get_example_formulae())
args_gen = zip(f_gen, res_gen)

# We now define a function that check the formula against the expected
# value of is_valid: See check_validity_and_test(...) above.
# Due to the way multiprocessing works, we need to define the function
# _before_ we create the Pool.

# As before, we call the map on the pool
res = pool.map(check_validity_and_test, args_gen)

# We can also apply a map-reduce strategy, by making sure that all
# results are as expected.
if all(res):
    print("Everything is ok!")
else:
    print("Ooops, something is wrong!")
    print(res)

# A different option is to run solvers in an asynchronous way.  This
# does not provide us with any guarantee on the execution order, but
# it is particular convenient, if we want to perform solving together
# with another long operation (e.g., I/O, network etc.)  or if we want
# to run multiple solvers.

# Create a formula
big_f = And(f.expr for f in get_example_formulae() \
            if not f.logic.theory.bit_vectors and \
               not f.logic.theory.arrays and \
                   f.logic.theory.linear)

# Create keyword arguments for the function call.
# This is the simplest way to pass multiple arguments to apply_async.
kwargs = {"formula": big_f, "solver_name": "z3"}
future_res_sat = pool.apply_async(is_sat, kwds=kwargs)
future_res_unsat = pool.apply_async(is_unsat, kwds=kwargs)

# In the background, the solving is taking place... We can do other
# stuff in the meanwhile.
print("This is non-blocking...")

# Get the result with a deadline.
# See multiprocessing.pool.AsyncResult for more options
sat_res = future_res_sat.get(10)  # Get result after 10 seconds or kill
try:
    unsat_res = future_res_unsat.get(0) # No wait
except TimeoutError:
    print("UNSAT result was not ready!")
    unsat_res = None
print(sat_res, unsat_res)

Demonstrates how to perform SMT-LIB parsing, dumping and extension

# The last part of the example requires a QF_LIA solver to be installed.
#
#
# This example shows how to interact with files in the SMT-LIB
# format. In particular:
#
# 1. How to read a file in SMT-LIB format
# 2. How to write a file in SMT-LIB format
# 3. Formulas and SMT-LIB script
# 4. How to access annotations from SMT-LIB files
# 5. How to extend the parser with custom commands
#
from six.moves import cStringIO # Py2-Py3 Compatibility

from pysmt.smtlib.parser import SmtLibParser


# To make the example self contained, we store the example SMT-LIB
# script in a string.
DEMO_SMTLIB=\
"""
(set-logic QF_LIA)
(declare-fun p () Int)
(declare-fun q () Int)
(declare-fun x () Bool)
(declare-fun y () Bool)
(define-fun .def_1 () Bool (! (and x y) :cost 1))
(assert (=> x (> p q)))
(check-sat)
(push)
(assert (=> y (> q p)))
(check-sat)
(assert .def_1)
(check-sat)
(pop)
(check-sat)
"""

# We read the SMT-LIB Script by creating a Parser.
# From here we can get the SMT-LIB script.
parser = SmtLibParser()

# The method SmtLibParser.get_script takes a buffer in input. We use
# cStringIO to simulate an open file.
# See SmtLibParser.get_script_fname() if to pass the path of a file.
script = parser.get_script(cStringIO(DEMO_SMTLIB))

# The SmtLibScript provides an iterable representation of the commands
# that are present in the SMT-LIB file.
#
# Printing a summary of the issued commands
for cmd in script:
    print(cmd.name)
print("*"*50)

# SmtLibScript provides some utilities to perform common operations: e.g,
#
# - Checking if a command is present
assert script.contains_command("check-sat")
# - Counting the occurrences of a command
assert script.count_command_occurrences("assert") == 3
# - Obtain all commands of a particular type
decls = script.filter_by_command_name("declare-fun")
for d in decls:
    print(d)
print("*"*50)

# Most SMT-LIB scripts define a single SAT call. In these cases, the
# result can be obtained by conjoining multiple assertions.  The
# method to do that is SmtLibScript.get_strict_formula() that, raises
# an exception if there are push/pop calls.  To obtain the formula at
# the end of the execution of the Script (accounting for push/pop) we
# use get_last_formula
#
f = script.get_last_formula()
print(f)

# Finally, we serialize the script back into SMT-Lib format.  This can
# be dumped into a file (see SmtLibScript.to_file).  The flag daggify,
# specifies whether the printing is done as a DAG or as a tree.

buf_out = cStringIO()
script.serialize(buf_out, daggify=True)
print(buf_out.getvalue())

print("*"*50)
# Expressions can be annotated in order to provide additional
# information. The semantic of annotations is solver/problem
# dependent. For example, VMT uses annotations to identify two
# expressions as 1) the Transition Relation and 2) Initial Condition
#
# Here we pretend that we make up a ficticious Weighted SMT format
# and label .def1 with cost 1
#
# The class pysmt.smtlib.annotations.Annotations deals with the
# handling of annotations.
#
ann = script.annotations
print(ann.all_annotated_formulae("cost"))

print("*"*50)

# Annotations are part of the SMT-LIB standard, and are the
# recommended way to perform inter-operable operations. However, in
# many cases, we are interested in prototyping some algorithm/idea and
# need to write the input files by hand. In those cases, using an
# extended version of SMT-LIB usually provides a more readable input.
# We provide now an example on how to define a symbolic transition
# system as an extension of SMT-LIB.
# (A more complete version of this example can be found in :
#    pysmt.tests.smtlib.test_parser_extensibility.py)
#
EXT_SMTLIB="""\
(declare-fun A () Bool)
(declare-fun B () Bool)
(init (and A B))
(trans (=> A (next A)))
(exit)
"""

# We define two new commands (init, trans) and a new
# operator (next). In order to parse this file, we need to create a
# sub-class of the SmtLibParser, and add handlers for he new commands
# and operators.
from pysmt.smtlib.script import SmtLibCommand

class TSSmtLibParser(SmtLibParser):
    def __init__(self, env=None, interactive=False):
        SmtLibParser.__init__(self, env, interactive)

        # Add new commands
        #
        # The mapping function takes care of consuming the command
        # name from the input stream, e.g., '(init' . Therefore,
        # _cmd_init will receive the rest of the stream, in our
        # example, '(and A B)) ...'
        self.commands["init"] = self._cmd_init
        self.commands["trans"] = self._cmd_trans

        # Remove unused commands
        #
        # If some commands are not compatible with the extension, they
        # can be removed from the parser. If found, they will cause
        # the raising of the exception UnknownSmtLibCommandError
        del self.commands["check-sat"]
        del self.commands["get-value"]
        # ...

        # Add 'next' function
        #
        # New operators can be added similarly as done for commands.
        # e.g., 'next'. The helper function _operator_adapter,
        # simplifies the writing of such extensions.  In this example,
        # we will rewrite the content of the next without the need of
        # introducing a new pySMT operator. If you are interested in a
        # simple way of handling new operators in pySMT see
        # pysmt.test.test_dwf.
        self.interpreted["next"] = self._operator_adapter(self._next_var)

    def _cmd_init(self, current, tokens):
        # This cmd performs the parsing of:
        #   <expr> )
        # and returns a new SmtLibCommand
        expr = self.get_expression(tokens)
        self.consume_closing(tokens, current)
        return SmtLibCommand(name="init", args=(expr,))

    def _cmd_trans(self, current, tokens):
        # This performs the same parsing as _cmd_init, but returns a
        # different object. The parser is not restricted to return
        # SmtLibCommand, but using them makes handling them
        # afterwards easier.
        expr = self.get_expression(tokens)
        self.consume_closing(tokens, current)
        return SmtLibCommand(name="trans", args=(expr,))

    def _next_var(self, symbol):
        # The function is called with the arguments obtained from
        # parsing the rest of the SMT-LIB file. In this case, 'next'
        # is a unary function, thus we have only 1 argument. 'symbol'
        # is an FNode. We require that 'symbol' is _really_ a symbol:
        if symbol.is_symbol():
            name = symbol.symbol_name()
            ty = symbol.symbol_type()
            # The return type MUST be an FNode, because this is part
            # of an expression.
            return self.env.formula_manager.Symbol("next_" + name, ty)
        else:
            raise ValueError("'next' operator can be applied only to symbols")

    def get_ts(self, script):
        # New Top-Level command that takes a script stream in input.
        # We return a pair (Init, Trans) that defines the symbolic
        # transition system.
        init = self.env.formula_manager.TRUE()
        trans = self.env.formula_manager.TRUE()

        for cmd in self.get_command_generator(script):
            if cmd.name=="init":
                init = cmd.args[0]
            elif cmd.name=="trans":
                trans = cmd.args[0]
            else:
                # Ignore other commands
                pass

        return (init, trans)

# Time to try out the parser !!!
#
# First we check that the standard SMT-Lib parser cannot handle the new format.
from pysmt.exceptions import UnknownSmtLibCommandError

try:
    parser.get_script(cStringIO(EXT_SMTLIB))
except UnknownSmtLibCommandError as ex:
    print("Unsupported command: %s" % ex)

# The new parser can parse our example, and returns the (init, trans) pair
ts_parser = TSSmtLibParser()
init, trans = ts_parser.get_ts(cStringIO(EXT_SMTLIB))
print("INIT: %s" % init.serialize())
print("TRANS: %s" % trans.serialize())

Shows the use of UNSAT Core as debugging tools

#
# This example requires MathSAT or Z3
#
# In this example, we will encode a more complex puzzle and see:
#
# - Use of UNSAT cores as a debugging tool
# - Conjunctive partitioning
# - Symbol handling delegation to auxiliary functions
#

# This puzzle is known as Einstein Puzzle
#
# There are five houses in five different colours in a row. In each
# house lives a person with a different nationality. The five owners
# drink a certain type of beverage, smoke a certain brand of cigar and
# keep a certain pet.
#
# No owners have the same pet, smoke the same brand of cigar, or drink
# the same beverage.
#
# The Brit lives in the red house.
# The Swede keeps dogs as pets.
# The Dane drinks tea.
# The green house is on the immediate left of the white house.
# The green house owner drinks coffee.
# The owner who smokes Pall Mall rears birds.
# The owner of the yellow house smokes Dunhill.
# The owner living in the center house drinks milk.
# The Norwegian lives in the first house.
# The owner who smokes Blends lives next to the one who keeps cats.
# The owner who keeps the horse lives next to the one who smokes Dunhill.
# The owner who smokes Bluemasters drinks beer.
# The German smokes Prince.
# The Norwegian lives next to the blue house.
# The owner who smokes Blends lives next to the one who drinks water.
#
# The question is: who owns the fish?

from pysmt.shortcuts import Symbol, ExactlyOne, Or, And, FALSE, Iff
from pysmt.shortcuts import get_model, get_unsat_core, is_sat, is_unsat

#
# Lets start by expliciting all values for all dimensions

Color = "white", "yellow", "blue", "red", "green"
Nat = "german", "swedish", "british", "norwegian", "danish"
Pet =  "birds", "cats", "horses", "fish", "dogs"
Drink = "beer", "water", "tea", "milk", "coffee"
Smoke = "blends", "pall_mall", "prince", "bluemasters", "dunhill"
Houses = range(0,5)
#
# We number the houses from 0 to 4, and create the macros to assert
# properties of the i-th house:
#
#  e.g., color(1, "green")  to indicate that the house 1 is Green
#
# This is not strictly necessary, but it is a way of making programs
# more readable.
#
def color(number, name):
    assert name in Color
    if number in Houses:
        return Symbol("%d_color_%s" % (number, name))
    return FALSE()

def nat(number, name):
    assert name in Nat
    if number in Houses:
        return Symbol("%d_nat_%s" % (number, name))
    return FALSE()

def pet(number, name):
    assert name in Pet
    if number in Houses:
        return Symbol("%d_pet_%s" % (number, name))
    return FALSE()

def drink(number, name):
    assert name in Drink
    if number in Houses:
        return Symbol("%d_drink_%s" % (number, name))
    return FALSE()

def smoke(number, name):
    assert name in Smoke
    if number in Houses:
        return Symbol("%d_smoke_%s" % (number, name))
    return FALSE()

#
# We can encode the facts
#

facts = And(
    # The Brit lives in the red house.
    And( Iff(nat(i, "british"), color(i, "red")) for i in Houses ),

    # The Swede keeps dogs as pets.
    And( Iff(nat(i, "swedish"), pet(i, "dogs")) for i in Houses ),

    # The Dane drinks tea.
    And( Iff(nat(i, "danish"), drink(i, "tea")) for i in Houses ) ,

    # The green house is on the immediate left of the white house.
    And( Iff(color(i, "green"), color(i+1, "white")) for i in Houses) ,

    # The green house owner drinks coffee.
    And( Iff(color(i, "green"), drink(i, "coffee")) for i in Houses ) ,

    # The owner who smokes Pall Mall rears birds.
    And( Iff(smoke(i, "pall_mall"), pet(i, "birds")) for i in Houses ) ,

    # The owner of the yellow house smokes Dunhill.
    And( Iff(color(i, "yellow"), smoke(i, "dunhill")) for i in Houses ) ,

    # The owner living in the center house drinks milk.
    And( drink(2, "milk") ) ,

    # The Norwegian lives in the first house.
    And( nat(0, "norwegian") ) ,

    # The owner who smokes Blends lives next to the one who keeps cats.
    And( Iff(smoke(i, "blends"), Or(pet(i-1, "cats"), pet(i+1, "cats"))) for i in Houses ) ,

    # The owner who keeps the horse lives next to the one who smokes Dunhill.
    And( Iff(pet(i, "horses"), Or(smoke(i-1, "dunhill"), smoke(i+1, "dunhill"))) for i in Houses ) ,

    # The owner who smokes Bluemasters drinks beer.
    And( Iff(smoke(i, "bluemasters"), drink(i, "beer")) for i in Houses ) ,

    # The German smokes Prince.
    And( Iff(nat(i, "german"), smoke(i, "prince")) for i in Houses ) ,

    # The Norwegian lives next to the blue house.
    # Careful with this!!!
    And( Iff(nat(i, "norwegian"), Or(color(i-1, "blue"), color(i+1, "blue"))) for i in Houses ) ,

    # The owner who smokes Blends lives next to the one who drinks water.
    And( Iff(smoke(i, "blends"), Or(drink(i-1, "water"), drink(i+1, "water"))) for i in Houses )
    )

domain = And(
    And(ExactlyOne(color(i, c) for i in Houses) for c in Color),
    And(ExactlyOne(nat(i, c) for i in Houses) for c in Nat),
    And(ExactlyOne(pet(i, c) for i in Houses) for c in Pet),
    And(ExactlyOne(drink(i, c) for i in Houses) for c in Drink),
    And(ExactlyOne(smoke(i, c) for i in Houses) for c in Smoke),
    #
    And(ExactlyOne(color(i, c) for c in Color) for i in Houses),
    And(ExactlyOne(nat(i, c) for c in Nat) for i in Houses),
    And(ExactlyOne(pet(i, c) for c in Pet) for i in Houses),
    And(ExactlyOne(drink(i, c) for c in Drink) for i in Houses),
    And(ExactlyOne(smoke(i, c) for c in Smoke) for i in Houses),
    )

problem = And(domain, facts)

model = get_model(problem)

if model is None:
    print("UNSAT")
    # We first check whether the constraints on the domain and problem
    # are satisfiable in isolation.
    assert is_sat(facts)
    assert is_sat(domain)
    assert is_unsat(problem)

    # In isolation they are both fine, rules from both are probably
    # interacting.
    #
    # The problem is given by a nesting of And().
    # conjunctive_partition can be used to obtain a "flat"
    # structure, i.e., a list of conjuncts.
    #
    from pysmt.rewritings import conjunctive_partition
    conj = conjunctive_partition(problem)
    ucore = get_unsat_core(conj)
    print("UNSAT-Core size '%d'" % len(ucore))
    for f in ucore:
        print(f.serialize())

    # The exact version of the UNSAT-Core depends on the solver in
    # use.  Nevertheless, this represents a starting point for your
    # debugging.  A possible way to approach the result is to look for
    # clauses of size 1 (i.e., unit clauses). In the facts list there
    # are only 2 facts:
    #   2_drink_milk
    #   0_nat_norwegian
    #
    # The clause ("1_color_blue" <-> "0_nat_norwegian")
    # Implies that "1_color_blue"
    # But (("3_color_blue" | "1_color_blue") <-> "2_nat_norwegian")
    # Requires "2_nat_norwegian"
    # The ExactlyOne constraint forbids that both 0 and 2 are nowegian
    # thus, we have a better idea of where the problem might be.
    #
    # Please go back to the comment '# Careful with this!!!' in the
    # facts list, and change the Iff with an Implies.
    #
    # Done?
    #
    # Good, you should be getting a model, now!
else:
    for h in Houses:
        # Extract the relevants bits to get some pretty-printing
        c = [x for x in Color if model[color(h, x)].is_true()][0]
        n = [x for x in Nat if model[nat(h, x)].is_true()][0]
        p = [x for x in Pet if model[pet(h, x)].is_true()][0]
        d = [x for x in Drink if model[drink(h, x)].is_true()][0]
        s = [x for x in Smoke if model[smoke(h, x)].is_true()][0]
        print(h, c, n, p, d, s)
        if p == "fish":
            sol = "The '%s' owns the fish!" % n
    print(sol)