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+********************************
+  Functional Programming HOWTO
+********************************
+
+:Author: \A. M. Kuchling
+:Release: 0.30
+
+(This is a first draft.  Please send comments/error reports/suggestions to
+amk@amk.ca.  This URL is probably not going to be the final location of the
+document, so be careful about linking to it -- you may want to add a
+disclaimer.)
+
+In this document, we'll take a tour of Python's features suitable for
+implementing programs in a functional style.  After an introduction to the
+concepts of functional programming, we'll look at language features such as
+iterators and generators and relevant library modules such as :mod:`itertools`
+and :mod:`functools`.
+
+
+Introduction
+============
+
+This section explains the basic concept of functional programming; if you're
+just interested in learning about Python language features, skip to the next
+section.
+
+Programming languages support decomposing problems in several different ways:
+
+* Most programming languages are **procedural**: programs are lists of
+  instructions that tell the computer what to do with the program's input.  C,
+  Pascal, and even Unix shells are procedural languages.
+
+* In **declarative** languages, you write a specification that describes the
+  problem to be solved, and the language implementation figures out how to
+  perform the computation efficiently.  SQL is the declarative language you're
+  most likely to be familiar with; a SQL query describes the data set you want
+  to retrieve, and the SQL engine decides whether to scan tables or use indexes,
+  which subclauses should be performed first, etc.
+
+* **Object-oriented** programs manipulate collections of objects.  Objects have
+  internal state and support methods that query or modify this internal state in
+  some way. Smalltalk and Java are object-oriented languages.  C++ and Python
+  are languages that support object-oriented programming, but don't force the
+  use of object-oriented features.
+
+* **Functional** programming decomposes a problem into a set of functions.
+  Ideally, functions only take inputs and produce outputs, and don't have any
+  internal state that affects the output produced for a given input.  Well-known
+  functional languages include the ML family (Standard ML, OCaml, and other
+  variants) and Haskell.
+
+The designers of some computer languages have chosen one approach to programming
+that's emphasized.  This often makes it difficult to write programs that use a
+different approach.  Other languages are multi-paradigm languages that support
+several different approaches.  Lisp, C++, and Python are multi-paradigm; you can
+write programs or libraries that are largely procedural, object-oriented, or
+functional in all of these languages.  In a large program, different sections
+might be written using different approaches; the GUI might be object-oriented
+while the processing logic is procedural or functional, for example.
+
+In a functional program, input flows through a set of functions. Each function
+operates on its input and produces some output.  Functional style frowns upon
+functions with side effects that modify internal state or make other changes
+that aren't visible in the function's return value.  Functions that have no side
+effects at all are called **purely functional**.  Avoiding side effects means
+not using data structures that get updated as a program runs; every function's
+output must only depend on its input.
+
+Some languages are very strict about purity and don't even have assignment
+statements such as ``a=3`` or ``c = a + b``, but it's difficult to avoid all
+side effects.  Printing to the screen or writing to a disk file are side
+effects, for example.  For example, in Python a ``print`` statement or a
+``time.sleep(1)`` both return no useful value; they're only called for their
+side effects of sending some text to the screen or pausing execution for a
+second.
+
+Python programs written in functional style usually won't go to the extreme of
+avoiding all I/O or all assignments; instead, they'll provide a
+functional-appearing interface but will use non-functional features internally.
+For example, the implementation of a function will still use assignments to
+local variables, but won't modify global variables or have other side effects.
+
+Functional programming can be considered the opposite of object-oriented
+programming.  Objects are little capsules containing some internal state along
+with a collection of method calls that let you modify this state, and programs
+consist of making the right set of state changes.  Functional programming wants
+to avoid state changes as much as possible and works with data flowing between
+functions.  In Python you might combine the two approaches by writing functions
+that take and return instances representing objects in your application (e-mail
+messages, transactions, etc.).
+
+Functional design may seem like an odd constraint to work under.  Why should you
+avoid objects and side effects?  There are theoretical and practical advantages
+to the functional style:
+
+* Formal provability.
+* Modularity.
+* Composability.
+* Ease of debugging and testing.
+
+Formal provability
+------------------
+
+A theoretical benefit is that it's easier to construct a mathematical proof that
+a functional program is correct.
+
+For a long time researchers have been interested in finding ways to
+mathematically prove programs correct.  This is different from testing a program
+on numerous inputs and concluding that its output is usually correct, or reading
+a program's source code and concluding that the code looks right; the goal is
+instead a rigorous proof that a program produces the right result for all
+possible inputs.
+
+The technique used to prove programs correct is to write down **invariants**,
+properties of the input data and of the program's variables that are always
+true.  For each line of code, you then show that if invariants X and Y are true
+**before** the line is executed, the slightly different invariants X' and Y' are
+true **after** the line is executed.  This continues until you reach the end of
+the program, at which point the invariants should match the desired conditions
+on the program's output.
+
+Functional programming's avoidance of assignments arose because assignments are
+difficult to handle with this technique; assignments can break invariants that
+were true before the assignment without producing any new invariants that can be
+propagated onward.
+
+Unfortunately, proving programs correct is largely impractical and not relevant
+to Python software. Even trivial programs require proofs that are several pages
+long; the proof of correctness for a moderately complicated program would be
+enormous, and few or none of the programs you use daily (the Python interpreter,
+your XML parser, your web browser) could be proven correct.  Even if you wrote
+down or generated a proof, there would then be the question of verifying the
+proof; maybe there's an error in it, and you wrongly believe you've proved the
+program correct.
+
+Modularity
+----------
+
+A more practical benefit of functional programming is that it forces you to
+break apart your problem into small pieces.  Programs are more modular as a
+result.  It's easier to specify and write a small function that does one thing
+than a large function that performs a complicated transformation.  Small
+functions are also easier to read and to check for errors.
+
+
+Ease of debugging and testing 
+-----------------------------
+
+Testing and debugging a functional-style program is easier.
+
+Debugging is simplified because functions are generally small and clearly
+specified.  When a program doesn't work, each function is an interface point
+where you can check that the data are correct.  You can look at the intermediate
+inputs and outputs to quickly isolate the function that's responsible for a bug.
+
+Testing is easier because each function is a potential subject for a unit test.
+Functions don't depend on system state that needs to be replicated before
+running a test; instead you only have to synthesize the right input and then
+check that the output matches expectations.
+
+
+
+Composability
+-------------
+
+As you work on a functional-style program, you'll write a number of functions
+with varying inputs and outputs.  Some of these functions will be unavoidably
+specialized to a particular application, but others will be useful in a wide
+variety of programs.  For example, a function that takes a directory path and
+returns all the XML files in the directory, or a function that takes a filename
+and returns its contents, can be applied to many different situations.
+
+Over time you'll form a personal library of utilities.  Often you'll assemble
+new programs by arranging existing functions in a new configuration and writing
+a few functions specialized for the current task.
+
+
+
+Iterators
+=========
+
+I'll start by looking at a Python language feature that's an important
+foundation for writing functional-style programs: iterators.
+
+An iterator is an object representing a stream of data; this object returns the
+data one element at a time.  A Python iterator must support a method called
+``next()`` that takes no arguments and always returns the next element of the
+stream.  If there are no more elements in the stream, ``next()`` must raise the
+``StopIteration`` exception.  Iterators don't have to be finite, though; it's
+perfectly reasonable to write an iterator that produces an infinite stream of
+data.
+
+The built-in :func:`iter` function takes an arbitrary object and tries to return
+an iterator that will return the object's contents or elements, raising
+:exc:`TypeError` if the object doesn't support iteration.  Several of Python's
+built-in data types support iteration, the most common being lists and
+dictionaries.  An object is called an **iterable** object if you can get an
+iterator for it.
+
+You can experiment with the iteration interface manually::
+
+    >>> L = [1,2,3]
+    >>> it = iter(L)
+    >>> print it
+    <iterator object at 0x8116870>
+    >>> it.next()
+    1
+    >>> it.next()
+    2
+    >>> it.next()
+    3
+    >>> it.next()
+    Traceback (most recent call last):
+      File "<stdin>", line 1, in ?
+    StopIteration
+    >>>      
+
+Python expects iterable objects in several different contexts, the most
+important being the ``for`` statement.  In the statement ``for X in Y``, Y must
+be an iterator or some object for which ``iter()`` can create an iterator.
+These two statements are equivalent::
+
+        for i in iter(obj):
+            print i
+
+        for i in obj:
+            print i
+
+Iterators can be materialized as lists or tuples by using the :func:`list` or
+:func:`tuple` constructor functions::
+
+    >>> L = [1,2,3]
+    >>> iterator = iter(L)
+    >>> t = tuple(iterator)
+    >>> t
+    (1, 2, 3)
+
+Sequence unpacking also supports iterators: if you know an iterator will return
+N elements, you can unpack them into an N-tuple::
+
+    >>> L = [1,2,3]
+    >>> iterator = iter(L)
+    >>> a,b,c = iterator
+    >>> a,b,c
+    (1, 2, 3)
+
+Built-in functions such as :func:`max` and :func:`min` can take a single
+iterator argument and will return the largest or smallest element.  The ``"in"``
+and ``"not in"`` operators also support iterators: ``X in iterator`` is true if
+X is found in the stream returned by the iterator.  You'll run into obvious
+problems if the iterator is infinite; ``max()``, ``min()``, and ``"not in"``
+will never return, and if the element X never appears in the stream, the
+``"in"`` operator won't return either.
+
+Note that you can only go forward in an iterator; there's no way to get the
+previous element, reset the iterator, or make a copy of it.  Iterator objects
+can optionally provide these additional capabilities, but the iterator protocol
+only specifies the ``next()`` method.  Functions may therefore consume all of
+the iterator's output, and if you need to do something different with the same
+stream, you'll have to create a new iterator.
+
+
+
+Data Types That Support Iterators
+---------------------------------
+
+We've already seen how lists and tuples support iterators.  In fact, any Python
+sequence type, such as strings, will automatically support creation of an
+iterator.
+
+Calling :func:`iter` on a dictionary returns an iterator that will loop over the
+dictionary's keys::
+
+    >>> m = {'Jan': 1, 'Feb': 2, 'Mar': 3, 'Apr': 4, 'May': 5, 'Jun': 6,
+    ...      'Jul': 7, 'Aug': 8, 'Sep': 9, 'Oct': 10, 'Nov': 11, 'Dec': 12}
+    >>> for key in m:
+    ...     print key, m[key]
+    Mar 3
+    Feb 2
+    Aug 8
+    Sep 9
+    May 5
+    Jun 6
+    Jul 7
+    Jan 1
+    Apr 4
+    Nov 11
+    Dec 12
+    Oct 10
+
+Note that the order is essentially random, because it's based on the hash
+ordering of the objects in the dictionary.
+
+Applying ``iter()`` to a dictionary always loops over the keys, but dictionaries
+have methods that return other iterators.  If you want to iterate over keys,
+values, or key/value pairs, you can explicitly call the ``iterkeys()``,
+``itervalues()``, or ``iteritems()`` methods to get an appropriate iterator.
+
+The :func:`dict` constructor can accept an iterator that returns a finite stream
+of ``(key, value)`` tuples::
+
+    >>> L = [('Italy', 'Rome'), ('France', 'Paris'), ('US', 'Washington DC')]
+    >>> dict(iter(L))
+    {'Italy': 'Rome', 'US': 'Washington DC', 'France': 'Paris'}
+
+Files also support iteration by calling the ``readline()`` method until there
+are no more lines in the file.  This means you can read each line of a file like
+this::
+
+    for line in file:
+        # do something for each line
+        ...
+
+Sets can take their contents from an iterable and let you iterate over the set's
+elements::
+
+    S = set((2, 3, 5, 7, 11, 13))
+    for i in S:
+        print i
+
+
+
+Generator expressions and list comprehensions
+=============================================
+
+Two common operations on an iterator's output are 1) performing some operation
+for every element, 2) selecting a subset of elements that meet some condition.
+For example, given a list of strings, you might want to strip off trailing
+whitespace from each line or extract all the strings containing a given
+substring.
+
+List comprehensions and generator expressions (short form: "listcomps" and
+"genexps") are a concise notation for such operations, borrowed from the
+functional programming language Haskell (http://www.haskell.org).  You can strip
+all the whitespace from a stream of strings with the following code::
+
+        line_list = ['  line 1\n', 'line 2  \n', ...]
+
+        # Generator expression -- returns iterator
+        stripped_iter = (line.strip() for line in line_list)
+
+        # List comprehension -- returns list
+        stripped_list = [line.strip() for line in line_list]
+
+You can select only certain elements by adding an ``"if"`` condition::
+
+        stripped_list = [line.strip() for line in line_list
+                         if line != ""]
+
+With a list comprehension, you get back a Python list; ``stripped_list`` is a
+list containing the resulting lines, not an iterator.  Generator expressions
+return an iterator that computes the values as necessary, not needing to
+materialize all the values at once.  This means that list comprehensions aren't
+useful if you're working with iterators that return an infinite stream or a very
+large amount of data.  Generator expressions are preferable in these situations.
+
+Generator expressions are surrounded by parentheses ("()") and list
+comprehensions are surrounded by square brackets ("[]").  Generator expressions
+have the form::
+
+    ( expression for expr in sequence1 
+                 if condition1
+                 for expr2 in sequence2
+                 if condition2
+                 for expr3 in sequence3 ...
+                 if condition3
+                 for exprN in sequenceN
+                 if conditionN )
+
+Again, for a list comprehension only the outside brackets are different (square
+brackets instead of parentheses).
+
+The elements of the generated output will be the successive values of
+``expression``.  The ``if`` clauses are all optional; if present, ``expression``
+is only evaluated and added to the result when ``condition`` is true.
+
+Generator expressions always have to be written inside parentheses, but the
+parentheses signalling a function call also count.  If you want to create an
+iterator that will be immediately passed to a function you can write::
+
+        obj_total = sum(obj.count for obj in list_all_objects())
+
+The ``for...in`` clauses contain the sequences to be iterated over.  The
+sequences do not have to be the same length, because they are iterated over from
+left to right, **not** in parallel.  For each element in ``sequence1``,
+``sequence2`` is looped over from the beginning.  ``sequence3`` is then looped
+over for each resulting pair of elements from ``sequence1`` and ``sequence2``.
+
+To put it another way, a list comprehension or generator expression is
+equivalent to the following Python code::
+
+    for expr1 in sequence1:
+        if not (condition1):
+            continue   # Skip this element
+        for expr2 in sequence2:
+            if not (condition2):
+                continue    # Skip this element
+            ...
+            for exprN in sequenceN:
+                 if not (conditionN):
+                     continue   # Skip this element
+
+                 # Output the value of 
+                 # the expression.
+
+This means that when there are multiple ``for...in`` clauses but no ``if``
+clauses, the length of the resulting output will be equal to the product of the
+lengths of all the sequences.  If you have two lists of length 3, the output
+list is 9 elements long::
+
+    seq1 = 'abc'
+    seq2 = (1,2,3)
+    >>> [ (x,y) for x in seq1 for y in seq2]
+    [('a', 1), ('a', 2), ('a', 3), 
+     ('b', 1), ('b', 2), ('b', 3), 
+     ('c', 1), ('c', 2), ('c', 3)]
+
+To avoid introducing an ambiguity into Python's grammar, if ``expression`` is
+creating a tuple, it must be surrounded with parentheses.  The first list
+comprehension below is a syntax error, while the second one is correct::
+
+    # Syntax error
+    [ x,y for x in seq1 for y in seq2]
+    # Correct
+    [ (x,y) for x in seq1 for y in seq2]
+
+
+Generators
+==========
+
+Generators are a special class of functions that simplify the task of writing
+iterators.  Regular functions compute a value and return it, but generators
+return an iterator that returns a stream of values.
+
+You're doubtless familiar with how regular function calls work in Python or C.
+When you call a function, it gets a private namespace where its local variables
+are created.  When the function reaches a ``return`` statement, the local
+variables are destroyed and the value is returned to the caller.  A later call
+to the same function creates a new private namespace and a fresh set of local
+variables. But, what if the local variables weren't thrown away on exiting a
+function?  What if you could later resume the function where it left off?  This
+is what generators provide; they can be thought of as resumable functions.
+
+Here's the simplest example of a generator function::
+
+    def generate_ints(N):
+        for i in range(N):
+            yield i
+
+Any function containing a ``yield`` keyword is a generator function; this is
+detected by Python's bytecode compiler which compiles the function specially as
+a result.
+
+When you call a generator function, it doesn't return a single value; instead it
+returns a generator object that supports the iterator protocol.  On executing
+the ``yield`` expression, the generator outputs the value of ``i``, similar to a
+``return`` statement.  The big difference between ``yield`` and a ``return``
+statement is that on reaching a ``yield`` the generator's state of execution is
+suspended and local variables are preserved.  On the next call to the
+generator's ``.next()`` method, the function will resume executing.
+
+Here's a sample usage of the ``generate_ints()`` generator::
+
+    >>> gen = generate_ints(3)
+    >>> gen
+    <generator object at 0x8117f90>
+    >>> gen.next()
+    0
+    >>> gen.next()
+    1
+    >>> gen.next()
+    2
+    >>> gen.next()
+    Traceback (most recent call last):
+      File "stdin", line 1, in ?
+      File "stdin", line 2, in generate_ints
+    StopIteration
+
+You could equally write ``for i in generate_ints(5)``, or ``a,b,c =
+generate_ints(3)``.
+
+Inside a generator function, the ``return`` statement can only be used without a
+value, and signals the end of the procession of values; after executing a
+``return`` the generator cannot return any further values.  ``return`` with a
+value, such as ``return 5``, is a syntax error inside a generator function.  The
+end of the generator's results can also be indicated by raising
+``StopIteration`` manually, or by just letting the flow of execution fall off
+the bottom of the function.
+
+You could achieve the effect of generators manually by writing your own class
+and storing all the local variables of the generator as instance variables.  For
+example, returning a list of integers could be done by setting ``self.count`` to
+0, and having the ``next()`` method increment ``self.count`` and return it.
+However, for a moderately complicated generator, writing a corresponding class
+can be much messier.
+
+The test suite included with Python's library, ``test_generators.py``, contains
+a number of more interesting examples.  Here's one generator that implements an
+in-order traversal of a tree using generators recursively.
+
+::
+
+    # A recursive generator that generates Tree leaves in in-order.
+    def inorder(t):
+        if t:
+            for x in inorder(t.left):
+                yield x
+
+            yield t.label
+
+            for x in inorder(t.right):
+                yield x
+
+Two other examples in ``test_generators.py`` produce solutions for the N-Queens
+problem (placing N queens on an NxN chess board so that no queen threatens
+another) and the Knight's Tour (finding a route that takes a knight to every
+square of an NxN chessboard without visiting any square twice).
+
+
+
+Passing values into a generator
+-------------------------------
+
+In Python 2.4 and earlier, generators only produced output.  Once a generator's
+code was invoked to create an iterator, there was no way to pass any new
+information into the function when its execution is resumed.  You could hack
+together this ability by making the generator look at a global variable or by
+passing in some mutable object that callers then modify, but these approaches
+are messy.
+
+In Python 2.5 there's a simple way to pass values into a generator.
+:keyword:`yield` became an expression, returning a value that can be assigned to
+a variable or otherwise operated on::
+
+    val = (yield i)
+
+I recommend that you **always** put parentheses around a ``yield`` expression
+when you're doing something with the returned value, as in the above example.
+The parentheses aren't always necessary, but it's easier to always add them
+instead of having to remember when they're needed.
+
+(PEP 342 explains the exact rules, which are that a ``yield``-expression must
+always be parenthesized except when it occurs at the top-level expression on the
+right-hand side of an assignment.  This means you can write ``val = yield i``
+but have to use parentheses when there's an operation, as in ``val = (yield i)
++ 12``.)
+
+Values are sent into a generator by calling its ``send(value)`` method.  This
+method resumes the generator's code and the ``yield`` expression returns the
+specified value.  If the regular ``next()`` method is called, the ``yield``
+returns ``None``.
+
+Here's a simple counter that increments by 1 and allows changing the value of
+the internal counter.
+
+::
+
+    def counter (maximum):
+        i = 0
+        while i < maximum:
+            val = (yield i)
+            # If value provided, change counter
+            if val is not None:
+                i = val
+            else:
+                i += 1
+
+And here's an example of changing the counter:
+
+    >>> it = counter(10)
+    >>> print it.next()
+    0
+    >>> print it.next()
+    1
+    >>> print it.send(8)
+    8
+    >>> print it.next()
+    9
+    >>> print it.next()
+    Traceback (most recent call last):
+      File ``t.py'', line 15, in ?
+        print it.next()
+    StopIteration
+
+Because ``yield`` will often be returning ``None``, you should always check for
+this case.  Don't just use its value in expressions unless you're sure that the
+``send()`` method will be the only method used resume your generator function.
+
+In addition to ``send()``, there are two other new methods on generators:
+
+* ``throw(type, value=None, traceback=None)`` is used to raise an exception
+  inside the generator; the exception is raised by the ``yield`` expression
+  where the generator's execution is paused.
+
+* ``close()`` raises a :exc:`GeneratorExit` exception inside the generator to
+  terminate the iteration.  On receiving this exception, the generator's code
+  must either raise :exc:`GeneratorExit` or :exc:`StopIteration`; catching the
+  exception and doing anything else is illegal and will trigger a
+  :exc:`RuntimeError`.  ``close()`` will also be called by Python's garbage
+  collector when the generator is garbage-collected.
+
+  If you need to run cleanup code when a :exc:`GeneratorExit` occurs, I suggest
+  using a ``try: ... finally:`` suite instead of catching :exc:`GeneratorExit`.
+
+The cumulative effect of these changes is to turn generators from one-way
+producers of information into both producers and consumers.
+
+Generators also become **coroutines**, a more generalized form of subroutines.
+Subroutines are entered at one point and exited at another point (the top of the
+function, and a ``return`` statement), but coroutines can be entered, exited,
+and resumed at many different points (the ``yield`` statements).
+
+
+Built-in functions
+==================
+
+Let's look in more detail at built-in functions often used with iterators.
+
+Two Python's built-in functions, :func:`map` and :func:`filter`, are somewhat
+obsolete; they duplicate the features of list comprehensions but return actual
+lists instead of iterators.
+
+``map(f, iterA, iterB, ...)`` returns a list containing ``f(iterA[0], iterB[0]),
+f(iterA[1], iterB[1]), f(iterA[2], iterB[2]), ...``.
+
+::
+
+    def upper(s):
+        return s.upper()
+    map(upper, ['sentence', 'fragment']) =>
+      ['SENTENCE', 'FRAGMENT']
+
+    [upper(s) for s in ['sentence', 'fragment']] =>
+      ['SENTENCE', 'FRAGMENT']
+
+As shown above, you can achieve the same effect with a list comprehension.  The
+:func:`itertools.imap` function does the same thing but can handle infinite
+iterators; it'll be discussed later, in the section on the :mod:`itertools` module.
+
+``filter(predicate, iter)`` returns a list that contains all the sequence
+elements that meet a certain condition, and is similarly duplicated by list
+comprehensions.  A **predicate** is a function that returns the truth value of
+some condition; for use with :func:`filter`, the predicate must take a single
+value.
+
+::
+
+    def is_even(x):
+        return (x % 2) == 0
+
+    filter(is_even, range(10)) =>
+      [0, 2, 4, 6, 8]
+
+This can also be written as a list comprehension::
+
+    >>> [x for x in range(10) if is_even(x)]
+    [0, 2, 4, 6, 8]
+
+:func:`filter` also has a counterpart in the :mod:`itertools` module,
+:func:`itertools.ifilter`, that returns an iterator and can therefore handle
+infinite sequences just as :func:`itertools.imap` can.
+
+``reduce(func, iter, [initial_value])`` doesn't have a counterpart in the
+:mod:`itertools` module because it cumulatively performs an operation on all the
+iterable's elements and therefore can't be applied to infinite iterables.
+``func`` must be a function that takes two elements and returns a single value.
+:func:`reduce` takes the first two elements A and B returned by the iterator and
+calculates ``func(A, B)``.  It then requests the third element, C, calculates
+``func(func(A, B), C)``, combines this result with the fourth element returned,
+and continues until the iterable is exhausted.  If the iterable returns no
+values at all, a :exc:`TypeError` exception is raised.  If the initial value is
+supplied, it's used as a starting point and ``func(initial_value, A)`` is the
+first calculation.
+
+::
+
+    import operator
+    reduce(operator.concat, ['A', 'BB', 'C']) =>
+      'ABBC'
+    reduce(operator.concat, []) =>
+      TypeError: reduce() of empty sequence with no initial value
+    reduce(operator.mul, [1,2,3], 1) =>
+      6
+    reduce(operator.mul, [], 1) =>
+      1
+
+If you use :func:`operator.add` with :func:`reduce`, you'll add up all the
+elements of the iterable.  This case is so common that there's a special
+built-in called :func:`sum` to compute it::
+
+    reduce(operator.add, [1,2,3,4], 0) =>
+      10
+    sum([1,2,3,4]) =>
+      10
+    sum([]) =>
+      0
+
+For many uses of :func:`reduce`, though, it can be clearer to just write the
+obvious :keyword:`for` loop::
+
+    # Instead of:
+    product = reduce(operator.mul, [1,2,3], 1)
+
+    # You can write:
+    product = 1
+    for i in [1,2,3]:
+        product *= i
+
+
+``enumerate(iter)`` counts off the elements in the iterable, returning 2-tuples
+containing the count and each element.
+
+::
+
+    enumerate(['subject', 'verb', 'object']) =>
+      (0, 'subject'), (1, 'verb'), (2, 'object')
+
+:func:`enumerate` is often used when looping through a list and recording the
+indexes at which certain conditions are met::
+
+    f = open('data.txt', 'r')
+    for i, line in enumerate(f):
+        if line.strip() == '':
+            print 'Blank line at line #%i' % i
+
+``sorted(iterable, [cmp=None], [key=None], [reverse=False)`` collects all the
+elements of the iterable into a list, sorts the list, and returns the sorted
+result.  The ``cmp``, ``key``, and ``reverse`` arguments are passed through to
+the constructed list's ``.sort()`` method.
+
+::
+
+    import random
+    # Generate 8 random numbers between [0, 10000)
+    rand_list = random.sample(range(10000), 8)
+    rand_list =>
+      [769, 7953, 9828, 6431, 8442, 9878, 6213, 2207]
+    sorted(rand_list) =>
+      [769, 2207, 6213, 6431, 7953, 8442, 9828, 9878]
+    sorted(rand_list, reverse=True) =>
+      [9878, 9828, 8442, 7953, 6431, 6213, 2207, 769]
+
+(For a more detailed discussion of sorting, see the Sorting mini-HOWTO in the
+Python wiki at http://wiki.python.org/moin/HowTo/Sorting.)
+
+The ``any(iter)`` and ``all(iter)`` built-ins look at the truth values of an
+iterable's contents.  :func:`any` returns True if any element in the iterable is
+a true value, and :func:`all` returns True if all of the elements are true
+values::
+
+    any([0,1,0]) =>
+      True
+    any([0,0,0]) =>
+      False
+    any([1,1,1]) =>
+      True
+    all([0,1,0]) =>
+      False
+    all([0,0,0]) => 
+      False
+    all([1,1,1]) =>
+      True
+
+
+Small functions and the lambda expression
+=========================================
+
+When writing functional-style programs, you'll often need little functions that
+act as predicates or that combine elements in some way.
+
+If there's a Python built-in or a module function that's suitable, you don't
+need to define a new function at all::
+
+        stripped_lines = [line.strip() for line in lines]
+        existing_files = filter(os.path.exists, file_list)
+
+If the function you need doesn't exist, you need to write it.  One way to write
+small functions is to use the ``lambda`` statement.  ``lambda`` takes a number
+of parameters and an expression combining these parameters, and creates a small
+function that returns the value of the expression::
+
+        lowercase = lambda x: x.lower()
+
+        print_assign = lambda name, value: name + '=' + str(value)
+
+        adder = lambda x, y: x+y
+
+An alternative is to just use the ``def`` statement and define a function in the
+usual way::
+
+        def lowercase(x):
+            return x.lower()
+
+        def print_assign(name, value):
+            return name + '=' + str(value)
+
+        def adder(x,y):
+            return x + y
+
+Which alternative is preferable?  That's a style question; my usual course is to
+avoid using ``lambda``.
+
+One reason for my preference is that ``lambda`` is quite limited in the
+functions it can define.  The result has to be computable as a single
+expression, which means you can't have multiway ``if... elif... else``
+comparisons or ``try... except`` statements.  If you try to do too much in a
+``lambda`` statement, you'll end up with an overly complicated expression that's
+hard to read.  Quick, what's the following code doing?
+
+::
+
+    total = reduce(lambda a, b: (0, a[1] + b[1]), items)[1]
+
+You can figure it out, but it takes time to disentangle the expression to figure
+out what's going on.  Using a short nested ``def`` statements makes things a
+little bit better::
+
+    def combine (a, b):
+        return 0, a[1] + b[1]
+
+    total = reduce(combine, items)[1]
+
+But it would be best of all if I had simply used a ``for`` loop::
+
+     total = 0
+     for a, b in items:
+         total += b
+
+Or the :func:`sum` built-in and a generator expression::
+
+     total = sum(b for a,b in items)
+
+Many uses of :func:`reduce` are clearer when written as ``for`` loops.
+
+Fredrik Lundh once suggested the following set of rules for refactoring uses of
+``lambda``:
+
+1) Write a lambda function.
+2) Write a comment explaining what the heck that lambda does.
+3) Study the comment for a while, and think of a name that captures the essence
+   of the comment.
+4) Convert the lambda to a def statement, using that name.
+5) Remove the comment.
+
+I really like these rules, but you're free to disagree that this lambda-free
+style is better.
+
+
+The itertools module
+====================
+
+The :mod:`itertools` module contains a number of commonly-used iterators as well
+as functions for combining several iterators.  This section will introduce the
+module's contents by showing small examples.
+
+The module's functions fall into a few broad classes:
+
+* Functions that create a new iterator based on an existing iterator.
+* Functions for treating an iterator's elements as function arguments.
+* Functions for selecting portions of an iterator's output.
+* A function for grouping an iterator's output.
+
+Creating new iterators
+----------------------
+
+``itertools.count(n)`` returns an infinite stream of integers, increasing by 1
+each time.  You can optionally supply the starting number, which defaults to 0::
+
+        itertools.count() =>
+          0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
+        itertools.count(10) =>
+          10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...
+
+``itertools.cycle(iter)`` saves a copy of the contents of a provided iterable
+and returns a new iterator that returns its elements from first to last.  The
+new iterator will repeat these elements infinitely.
+
+::
+
+        itertools.cycle([1,2,3,4,5]) =>
+          1, 2, 3, 4, 5, 1, 2, 3, 4, 5, ...
+
+``itertools.repeat(elem, [n])`` returns the provided element ``n`` times, or
+returns the element endlessly if ``n`` is not provided.
+
+::
+
+    itertools.repeat('abc') =>
+      abc, abc, abc, abc, abc, abc, abc, abc, abc, abc, ...
+    itertools.repeat('abc', 5) =>
+      abc, abc, abc, abc, abc
+
+``itertools.chain(iterA, iterB, ...)`` takes an arbitrary number of iterables as
+input, and returns all the elements of the first iterator, then all the elements
+of the second, and so on, until all of the iterables have been exhausted.
+
+::
+
+    itertools.chain(['a', 'b', 'c'], (1, 2, 3)) =>
+      a, b, c, 1, 2, 3
+
+``itertools.izip(iterA, iterB, ...)`` takes one element from each iterable and
+returns them in a tuple::
+
+    itertools.izip(['a', 'b', 'c'], (1, 2, 3)) =>
+      ('a', 1), ('b', 2), ('c', 3)
+
+It's similiar to the built-in :func:`zip` function, but doesn't construct an
+in-memory list and exhaust all the input iterators before returning; instead
+tuples are constructed and returned only if they're requested.  (The technical
+term for this behaviour is `lazy evaluation
+<http://en.wikipedia.org/wiki/Lazy_evaluation>`__.)
+
+This iterator is intended to be used with iterables that are all of the same
+length.  If the iterables are of different lengths, the resulting stream will be
+the same length as the shortest iterable.
+
+::
+
+    itertools.izip(['a', 'b'], (1, 2, 3)) =>
+      ('a', 1), ('b', 2)
+
+You should avoid doing this, though, because an element may be taken from the
+longer iterators and discarded.  This means you can't go on to use the iterators
+further because you risk skipping a discarded element.
+
+``itertools.islice(iter, [start], stop, [step])`` returns a stream that's a
+slice of the iterator.  With a single ``stop`` argument, it will return the
+first ``stop`` elements.  If you supply a starting index, you'll get
+``stop-start`` elements, and if you supply a value for ``step``, elements will
+be skipped accordingly.  Unlike Python's string and list slicing, you can't use
+negative values for ``start``, ``stop``, or ``step``.
+
+::
+
+    itertools.islice(range(10), 8) =>
+      0, 1, 2, 3, 4, 5, 6, 7
+    itertools.islice(range(10), 2, 8) =>
+      2, 3, 4, 5, 6, 7
+    itertools.islice(range(10), 2, 8, 2) =>
+      2, 4, 6
+
+``itertools.tee(iter, [n])`` replicates an iterator; it returns ``n``
+independent iterators that will all return the contents of the source iterator.
+If you don't supply a value for ``n``, the default is 2.  Replicating iterators
+requires saving some of the contents of the source iterator, so this can consume
+significant memory if the iterator is large and one of the new iterators is
+consumed more than the others.
+
+::
+
+        itertools.tee( itertools.count() ) =>
+           iterA, iterB
+
+        where iterA ->
+           0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
+
+        and   iterB ->
+           0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
+
+
+Calling functions on elements
+-----------------------------
+
+Two functions are used for calling other functions on the contents of an
+iterable.
+
+``itertools.imap(f, iterA, iterB, ...)`` returns a stream containing
+``f(iterA[0], iterB[0]), f(iterA[1], iterB[1]), f(iterA[2], iterB[2]), ...``::
+
+    itertools.imap(operator.add, [5, 6, 5], [1, 2, 3]) =>
+      6, 8, 8
+
+The ``operator`` module contains a set of functions corresponding to Python's
+operators.  Some examples are ``operator.add(a, b)`` (adds two values),
+``operator.ne(a, b)`` (same as ``a!=b``), and ``operator.attrgetter('id')``
+(returns a callable that fetches the ``"id"`` attribute).
+
+``itertools.starmap(func, iter)`` assumes that the iterable will return a stream
+of tuples, and calls ``f()`` using these tuples as the arguments::
+
+    itertools.starmap(os.path.join, 
+                      [('/usr', 'bin', 'java'), ('/bin', 'python'),
+                       ('/usr', 'bin', 'perl'),('/usr', 'bin', 'ruby')])
+    =>
+      /usr/bin/java, /bin/python, /usr/bin/perl, /usr/bin/ruby
+
+
+Selecting elements
+------------------
+
+Another group of functions chooses a subset of an iterator's elements based on a
+predicate.
+
+``itertools.ifilter(predicate, iter)`` returns all the elements for which the
+predicate returns true::
+
+    def is_even(x):
+        return (x % 2) == 0
+
+    itertools.ifilter(is_even, itertools.count()) =>
+      0, 2, 4, 6, 8, 10, 12, 14, ...
+
+``itertools.ifilterfalse(predicate, iter)`` is the opposite, returning all
+elements for which the predicate returns false::
+
+    itertools.ifilterfalse(is_even, itertools.count()) =>
+      1, 3, 5, 7, 9, 11, 13, 15, ...
+
+``itertools.takewhile(predicate, iter)`` returns elements for as long as the
+predicate returns true.  Once the predicate returns false, the iterator will
+signal the end of its results.
+
+::
+
+    def less_than_10(x):
+        return (x < 10)
+
+    itertools.takewhile(less_than_10, itertools.count()) =>
+      0, 1, 2, 3, 4, 5, 6, 7, 8, 9
+
+    itertools.takewhile(is_even, itertools.count()) =>
+      0
+
+``itertools.dropwhile(predicate, iter)`` discards elements while the predicate
+returns true, and then returns the rest of the iterable's results.
+
+::
+
+    itertools.dropwhile(less_than_10, itertools.count()) =>
+      10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...
+
+    itertools.dropwhile(is_even, itertools.count()) =>
+      1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
+
+
+Grouping elements
+-----------------
+
+The last function I'll discuss, ``itertools.groupby(iter, key_func=None)``, is
+the most complicated.  ``key_func(elem)`` is a function that can compute a key
+value for each element returned by the iterable.  If you don't supply a key
+function, the key is simply each element itself.
+
+``groupby()`` collects all the consecutive elements from the underlying iterable
+that have the same key value, and returns a stream of 2-tuples containing a key
+value and an iterator for the elements with that key.
+
+::
+
+    city_list = [('Decatur', 'AL'), ('Huntsville', 'AL'), ('Selma', 'AL'), 
+                 ('Anchorage', 'AK'), ('Nome', 'AK'),
+                 ('Flagstaff', 'AZ'), ('Phoenix', 'AZ'), ('Tucson', 'AZ'), 
+                 ...
+                ]
+
+    def get_state ((city, state)):
+        return state
+
+    itertools.groupby(city_list, get_state) =>
+      ('AL', iterator-1),
+      ('AK', iterator-2),
+      ('AZ', iterator-3), ...
+
+    where
+    iterator-1 =>
+      ('Decatur', 'AL'), ('Huntsville', 'AL'), ('Selma', 'AL')
+    iterator-2 => 
+      ('Anchorage', 'AK'), ('Nome', 'AK')
+    iterator-3 =>
+      ('Flagstaff', 'AZ'), ('Phoenix', 'AZ'), ('Tucson', 'AZ')
+
+``groupby()`` assumes that the underlying iterable's contents will already be
+sorted based on the key.  Note that the returned iterators also use the
+underlying iterable, so you have to consume the results of iterator-1 before
+requesting iterator-2 and its corresponding key.
+
+
+The functools module
+====================
+
+The :mod:`functools` module in Python 2.5 contains some higher-order functions.
+A **higher-order function** takes one or more functions as input and returns a
+new function.  The most useful tool in this module is the
+:func:`functools.partial` function.
+
+For programs written in a functional style, you'll sometimes want to construct
+variants of existing functions that have some of the parameters filled in.
+Consider a Python function ``f(a, b, c)``; you may wish to create a new function
+``g(b, c)`` that's equivalent to ``f(1, b, c)``; you're filling in a value for
+one of ``f()``'s parameters.  This is called "partial function application".
+
+The constructor for ``partial`` takes the arguments ``(function, arg1, arg2,
+... kwarg1=value1, kwarg2=value2)``.  The resulting object is callable, so you
+can just call it to invoke ``function`` with the filled-in arguments.
+
+Here's a small but realistic example::
+
+    import functools
+
+    def log (message, subsystem):
+        "Write the contents of 'message' to the specified subsystem."
+        print '%s: %s' % (subsystem, message)
+        ...
+
+    server_log = functools.partial(log, subsystem='server')
+    server_log('Unable to open socket')
+
+
+The operator module
+-------------------
+
+The :mod:`operator` module was mentioned earlier.  It contains a set of
+functions corresponding to Python's operators.  These functions are often useful
+in functional-style code because they save you from writing trivial functions
+that perform a single operation.
+
+Some of the functions in this module are:
+
+* Math operations: ``add()``, ``sub()``, ``mul()``, ``div()``, ``floordiv()``,
+  ``abs()``, ...
+* Logical operations: ``not_()``, ``truth()``.
+* Bitwise operations: ``and_()``, ``or_()``, ``invert()``.
+* Comparisons: ``eq()``, ``ne()``, ``lt()``, ``le()``, ``gt()``, and ``ge()``.
+* Object identity: ``is_()``, ``is_not()``.
+
+Consult the operator module's documentation for a complete list.
+
+
+
+The functional module
+---------------------
+
+Collin Winter's `functional module <http://oakwinter.com/code/functional/>`__
+provides a number of more advanced tools for functional programming. It also
+reimplements several Python built-ins, trying to make them more intuitive to
+those used to functional programming in other languages.
+
+This section contains an introduction to some of the most important functions in
+``functional``; full documentation can be found at `the project's website
+<http://oakwinter.com/code/functional/documentation/>`__.
+
+``compose(outer, inner, unpack=False)``
+
+The ``compose()`` function implements function composition.  In other words, it
+returns a wrapper around the ``outer`` and ``inner`` callables, such that the
+return value from ``inner`` is fed directly to ``outer``.  That is,
+
+::
+
+        >>> def add(a, b):
+        ...     return a + b
+        ...
+        >>> def double(a):
+        ...     return 2 * a
+        ...
+        >>> compose(double, add)(5, 6)
+        22
+
+is equivalent to
+
+::
+
+        >>> double(add(5, 6))
+        22
+                    
+The ``unpack`` keyword is provided to work around the fact that Python functions
+are not always `fully curried <http://en.wikipedia.org/wiki/Currying>`__.  By
+default, it is expected that the ``inner`` function will return a single object
+and that the ``outer`` function will take a single argument. Setting the
+``unpack`` argument causes ``compose`` to expect a tuple from ``inner`` which
+will be expanded before being passed to ``outer``. Put simply,
+
+::
+
+        compose(f, g)(5, 6)
+                    
+is equivalent to::
+
+        f(g(5, 6))
+                    
+while
+
+::
+
+        compose(f, g, unpack=True)(5, 6)
+                    
+is equivalent to::
+
+        f(*g(5, 6))
+
+Even though ``compose()`` only accepts two functions, it's trivial to build up a
+version that will compose any number of functions. We'll use ``reduce()``,
+``compose()`` and ``partial()`` (the last of which is provided by both
+``functional`` and ``functools``).
+
+::
+
+        from functional import compose, partial
+        
+        multi_compose = partial(reduce, compose)
+        
+    
+We can also use ``map()``, ``compose()`` and ``partial()`` to craft a version of
+``"".join(...)`` that converts its arguments to string::
+
+        from functional import compose, partial
+        
+        join = compose("".join, partial(map, str))
+
+
+``flip(func)``
+                    
+``flip()`` wraps the callable in ``func`` and causes it to receive its
+non-keyword arguments in reverse order.
+
+::
+
+        >>> def triple(a, b, c):
+        ...     return (a, b, c)
+        ...
+        >>> triple(5, 6, 7)
+        (5, 6, 7)
+        >>>
+        >>> flipped_triple = flip(triple)
+        >>> flipped_triple(5, 6, 7)
+        (7, 6, 5)
+
+``foldl(func, start, iterable)``
+                    
+``foldl()`` takes a binary function, a starting value (usually some kind of
+'zero'), and an iterable.  The function is applied to the starting value and the
+first element of the list, then the result of that and the second element of the
+list, then the result of that and the third element of the list, and so on.
+
+This means that a call such as::
+
+        foldl(f, 0, [1, 2, 3])
+
+is equivalent to::
+
+        f(f(f(0, 1), 2), 3)
+
+    
+``foldl()`` is roughly equivalent to the following recursive function::
+
+        def foldl(func, start, seq):
+            if len(seq) == 0:
+                return start
+
+            return foldl(func, func(start, seq[0]), seq[1:])
+
+Speaking of equivalence, the above ``foldl`` call can be expressed in terms of
+the built-in ``reduce`` like so::
+
+        reduce(f, [1, 2, 3], 0)
+
+
+We can use ``foldl()``, ``operator.concat()`` and ``partial()`` to write a
+cleaner, more aesthetically-pleasing version of Python's ``"".join(...)``
+idiom::
+
+        from functional import foldl, partial
+        from operator import concat
+        
+        join = partial(foldl, concat, "")
+
+
+Revision History and Acknowledgements
+=====================================
+
+The author would like to thank the following people for offering suggestions,
+corrections and assistance with various drafts of this article: Ian Bicking,
+Nick Coghlan, Nick Efford, Raymond Hettinger, Jim Jewett, Mike Krell, Leandro
+Lameiro, Jussi Salmela, Collin Winter, Blake Winton.
+
+Version 0.1: posted June 30 2006.
+
+Version 0.11: posted July 1 2006.  Typo fixes.
+
+Version 0.2: posted July 10 2006.  Merged genexp and listcomp sections into one.
+Typo fixes.
+
+Version 0.21: Added more references suggested on the tutor mailing list.
+
+Version 0.30: Adds a section on the ``functional`` module written by Collin
+Winter; adds short section on the operator module; a few other edits.
+
+
+References
+==========
+
+General
+-------
+
+**Structure and Interpretation of Computer Programs**, by Harold Abelson and
+Gerald Jay Sussman with Julie Sussman.  Full text at
+http://mitpress.mit.edu/sicp/.  In this classic textbook of computer science,
+chapters 2 and 3 discuss the use of sequences and streams to organize the data
+flow inside a program.  The book uses Scheme for its examples, but many of the
+design approaches described in these chapters are applicable to functional-style
+Python code.
+
+http://www.defmacro.org/ramblings/fp.html: A general introduction to functional
+programming that uses Java examples and has a lengthy historical introduction.
+
+http://en.wikipedia.org/wiki/Functional_programming: General Wikipedia entry
+describing functional programming.
+
+http://en.wikipedia.org/wiki/Coroutine: Entry for coroutines.
+
+http://en.wikipedia.org/wiki/Currying: Entry for the concept of currying.
+
+Python-specific
+---------------
+
+http://gnosis.cx/TPiP/: The first chapter of David Mertz's book
+:title-reference:`Text Processing in Python` discusses functional programming
+for text processing, in the section titled "Utilizing Higher-Order Functions in
+Text Processing".
+
+Mertz also wrote a 3-part series of articles on functional programming
+for IBM's DeveloperWorks site; see 
+`part 1 <http://www-128.ibm.com/developerworks/library/l-prog.html>`__,
+`part 2 <http://www-128.ibm.com/developerworks/library/l-prog2.html>`__, and
+`part 3 <http://www-128.ibm.com/developerworks/linux/library/l-prog3.html>`__,
+
+
+Python documentation
+--------------------
+
+Documentation for the :mod:`itertools` module.
+
+Documentation for the :mod:`operator` module.
+
+:pep:`289`: "Generator Expressions"
+
+:pep:`342`: "Coroutines via Enhanced Generators" describes the new generator
+features in Python 2.5.
+
+.. comment
+
+    Topics to place
+    -----------------------------
+
+    XXX os.walk()
+
+    XXX Need a large example.
+
+    But will an example add much?  I'll post a first draft and see
+    what the comments say.
+
+.. comment
+
+    Original outline:
+    Introduction
+            Idea of FP
+                    Programs built out of functions
+                    Functions are strictly input-output, no internal state
+            Opposed to OO programming, where objects have state
+
+            Why FP?
+                    Formal provability
+                            Assignment is difficult to reason about
+                            Not very relevant to Python
+                    Modularity
+                            Small functions that do one thing
+                    Debuggability:
+                            Easy to test due to lack of state
+                            Easy to verify output from intermediate steps
+                    Composability
+                            You assemble a toolbox of functions that can be mixed
+
+    Tackling a problem
+            Need a significant example
+
+    Iterators
+    Generators
+    The itertools module
+    List comprehensions
+    Small functions and the lambda statement
+    Built-in functions
+            map
+            filter
+            reduce
+
+.. comment
+
+    Handy little function for printing part of an iterator -- used
+    while writing this document.
+
+    import itertools
+    def print_iter(it):
+         slice = itertools.islice(it, 10)
+         for elem in slice[:-1]:
+             sys.stdout.write(str(elem))
+             sys.stdout.write(', ')
+        print elem[-1]
+
+