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 <div class="section" id="functional-programming-howto">
<h1>Functional Programming HOWTO<a class="headerlink" href="#functional-programming-howto" title="Permalink to this headline">¶</a></h1>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Author:</th><td class="field-body">A. M. Kuchling</td>
</tr>
<tr class="field-even field"><th class="field-name">Release:</th><td class="field-body">0.32</td>
</tr>
</tbody>
</table>
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
<a class="reference internal" href="../glossary.html#term-iterator">iterator</a>s and <a class="reference internal" href="../glossary.html#term-generator">generator</a>s and relevant library modules such as
<a class="reference internal" href="../library/itertools.html#module-itertools" title="itertools: Functions creating iterators for efficient looping."><code class="xref py py-mod docutils literal notranslate">itertools</code></a> and <a class="reference internal" href="../library/functools.html#module-functools" title="functools: Higher-order functions and operations on callable objects."><code class="xref py py-mod docutils literal notranslate">functools</code></a>.
<div class="section" id="introduction">
<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
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 on <a class="reference internal" href="#functional-howto-iterators">Iterators</a>.
Programming languages support decomposing problems in several different ways:
<ul class="simple">
<li>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.</li>
<li>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.</li>
<li>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.</li>
<li>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.</li>
</ul>
The designers of some computer languages choose to emphasize one
particular approach to programming. 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 discourages
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 <code class="docutils literal notranslate">a=3</code> or <code class="docutils literal notranslate">c = a + b</code>, 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 call to the <a class="reference internal" href="../library/functions.html#print" title="print"><code class="xref py py-func docutils literal notranslate">print()</code></a> or
<a class="reference internal" href="../library/time.html#time.sleep" title="time.sleep"><code class="xref py py-func docutils literal notranslate">time.sleep()</code></a> function 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:
<ul class="simple">
<li>Formal provability.</li>
<li>Modularity.</li>
<li>Composability.</li>
<li>Ease of debugging and testing.</li>
</ul>
<div class="section" id="formal-provability">
<h3>Formal provability<a class="headerlink" href="#formal-provability" title="Permalink to this headline">¶</a></h3>
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.
</div>
<div class="section" id="modularity">
<h3>Modularity<a class="headerlink" href="#modularity" title="Permalink to this headline">¶</a></h3>
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.
</div>
<div class="section" id="ease-of-debugging-and-testing">
<h3>Ease of debugging and testing<a class="headerlink" href="#ease-of-debugging-and-testing" title="Permalink to this headline">¶</a></h3>
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.
</div>
<div class="section" id="composability">
<h3>Composability<a class="headerlink" href="#composability" title="Permalink to this headline">¶</a></h3>
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.
</div>
</div>
<div class="section" id="iterators">
<h2>Iterators<a class="headerlink" href="#iterators" title="Permalink to this headline">¶</a></h2>
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
<a class="reference internal" href="../library/stdtypes.html#iterator.__next__" title="iterator.__next__"><code class="xref py py-meth docutils literal notranslate">__next__()</code></a> that takes no arguments and always returns the next
element of the stream. If there are no more elements in the stream,
<a class="reference internal" href="../library/stdtypes.html#iterator.__next__" title="iterator.__next__"><code class="xref py py-meth docutils literal notranslate">__next__()</code></a> must raise the <a class="reference internal" href="../library/exceptions.html#StopIteration" title="StopIteration"><code class="xref py py-exc docutils literal notranslate">StopIteration</code></a> 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 <a class="reference internal" href="../library/functions.html#iter" title="iter"><code class="xref py py-func docutils literal notranslate">iter()</code></a> function takes an arbitrary object and tries to return
an iterator that will return the object’s contents or elements, raising
<a class="reference internal" href="../library/exceptions.html#TypeError" title="TypeError"><code class="xref py py-exc docutils literal notranslate">TypeError</code></a> 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 <a class="reference internal" href="../glossary.html#term-iterable">iterable</a> if you can get an iterator
for it.
You can experiment with the iteration interface manually:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; L = [1, 2, 3]
&gt;&gt;&gt; it = iter(L)
&gt;&gt;&gt; it 
&lt;...iterator object at ...&gt;
&gt;&gt;&gt; it.__next__() # same as next(it)
1
&gt;&gt;&gt; next(it)
2
&gt;&gt;&gt; next(it)
3
&gt;&gt;&gt; next(it)
Traceback (most recent call last):
 File &quot;&lt;stdin&gt;&quot;, line 1, in &lt;module&gt;
StopIteration
&gt;&gt;&gt;
</pre></div>
</div>
Python expects iterable objects in several different contexts, the most
important being the <a class="reference internal" href="../reference/compound_stmts.html#for"><code class="xref std std-keyword docutils literal notranslate">for</code></a> statement. In the statement <code class="docutils literal notranslate">for X in Y</code>,
Y must be an iterator or some object for which <a class="reference internal" href="../library/functions.html#iter" title="iter"><code class="xref py py-func docutils literal notranslate">iter()</code></a> can create an
iterator. These two statements are equivalent:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>for i in iter(obj):
 print(i)

for i in obj:
 print(i)
</pre></div>
</div>
Iterators can be materialized as lists or tuples by using the <a class="reference internal" href="../library/stdtypes.html#list" title="list"><code class="xref py py-func docutils literal notranslate">list()</code></a> or
<a class="reference internal" href="../library/stdtypes.html#tuple" title="tuple"><code class="xref py py-func docutils literal notranslate">tuple()</code></a> constructor functions:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; L = [1, 2, 3]
&gt;&gt;&gt; iterator = iter(L)
&gt;&gt;&gt; t = tuple(iterator)
&gt;&gt;&gt; t
(1, 2, 3)
</pre></div>
</div>
Sequence unpacking also supports iterators: if you know an iterator will return
N elements, you can unpack them into an N-tuple:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; L = [1, 2, 3]
&gt;&gt;&gt; iterator = iter(L)
&gt;&gt;&gt; a, b, c = iterator
&gt;&gt;&gt; a, b, c
(1, 2, 3)
</pre></div>
</div>
Built-in functions such as <a class="reference internal" href="../library/functions.html#max" title="max"><code class="xref py py-func docutils literal notranslate">max()</code></a> and <a class="reference internal" href="../library/functions.html#min" title="min"><code class="xref py py-func docutils literal notranslate">min()</code></a> can take a single
iterator argument and will return the largest or smallest element. The <code class="docutils literal notranslate">&quot;in&quot;</code>
and <code class="docutils literal notranslate">&quot;not in&quot;</code> operators also support iterators: <code class="docutils literal notranslate">X in iterator</code> is true if
X is found in the stream returned by the iterator. You’ll run into obvious
problems if the iterator is infinite; <a class="reference internal" href="../library/functions.html#max" title="max"><code class="xref py py-func docutils literal notranslate">max()</code></a>, <a class="reference internal" href="../library/functions.html#min" title="min"><code class="xref py py-func docutils literal notranslate">min()</code></a>
will never return, and if the element X never appears in the stream, the
<code class="docutils literal notranslate">&quot;in&quot;</code> and <code class="docutils literal notranslate">&quot;not in&quot;</code> operators 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 <a class="reference internal" href="../library/stdtypes.html#iterator.__next__" title="iterator.__next__"><code class="xref py py-meth docutils literal notranslate">__next__()</code></a> 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.
<div class="section" id="data-types-that-support-iterators">
<h3>Data Types That Support Iterators<a class="headerlink" href="#data-types-that-support-iterators" title="Permalink to this headline">¶</a></h3>
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 <a class="reference internal" href="../library/functions.html#iter" title="iter"><code class="xref py py-func docutils literal notranslate">iter()</code></a> on a dictionary returns an iterator that will loop over the
dictionary’s keys:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; m = {&#39;Jan&#39;: 1, &#39;Feb&#39;: 2, &#39;Mar&#39;: 3, &#39;Apr&#39;: 4, &#39;May&#39;: 5, &#39;Jun&#39;: 6,
... &#39;Jul&#39;: 7, &#39;Aug&#39;: 8, &#39;Sep&#39;: 9, &#39;Oct&#39;: 10, &#39;Nov&#39;: 11, &#39;Dec&#39;: 12}
&gt;&gt;&gt; for key in m: 
... print(key, m[key])
Mar 3
Feb 2
Aug 8
Sep 9
Apr 4
Jun 6
Jul 7
Jan 1
May 5
Nov 11
Dec 12
Oct 10
</pre></div>
</div>
Note that the order is essentially random, because it’s based on the hash
ordering of the objects in the dictionary.
Applying <a class="reference internal" href="../library/functions.html#iter" title="iter"><code class="xref py py-func docutils literal notranslate">iter()</code></a> to a dictionary always loops over the keys, but
dictionaries have methods that return other iterators. If you want to iterate
over values or key/value pairs, you can explicitly call the
<a class="reference internal" href="../library/stdtypes.html#dict.values" title="dict.values"><code class="xref py py-meth docutils literal notranslate">values()</code></a> or <a class="reference internal" href="../library/stdtypes.html#dict.items" title="dict.items"><code class="xref py py-meth docutils literal notranslate">items()</code></a> methods to get an appropriate
iterator.
The <a class="reference internal" href="../library/stdtypes.html#dict" title="dict"><code class="xref py py-func docutils literal notranslate">dict()</code></a> constructor can accept an iterator that returns a finite stream
of <code class="docutils literal notranslate">(key, value)</code> tuples:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; L = [(&#39;Italy&#39;, &#39;Rome&#39;), (&#39;France&#39;, &#39;Paris&#39;), (&#39;US&#39;, &#39;Washington DC&#39;)]
&gt;&gt;&gt; dict(iter(L)) 
{&#39;Italy&#39;: &#39;Rome&#39;, &#39;US&#39;: &#39;Washington DC&#39;, &#39;France&#39;: &#39;Paris&#39;}
</pre></div>
</div>
Files also support iteration by calling the <a class="reference internal" href="../library/io.html#io.TextIOBase.readline" title="io.TextIOBase.readline"><code class="xref py py-meth docutils literal notranslate">readline()</code></a>
method until there are no more lines in the file. This means you can read each
line of a file like this:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>for line in file:
 # do something for each line
 ...
</pre></div>
</div>
Sets can take their contents from an iterable and let you iterate over the set’s
elements:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>S = {2, 3, 5, 7, 11, 13}
for i in S:
 print(i)
</pre></div>
</div>
</div>
</div>
<div class="section" id="generator-expressions-and-list-comprehensions">
<h2>Generator expressions and list comprehensions<a class="headerlink" href="#generator-expressions-and-list-comprehensions" title="Permalink to this headline">¶</a></h2>
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 (<a class="reference external" href="https://www.haskell.org/">https://www.haskell.org/</a>). You can strip
all the whitespace from a stream of strings with the following code:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>line_list = [&#39; line 1\n&#39;, &#39;line 2 \n&#39;, ...]

# 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]
</pre></div>
</div>
You can select only certain elements by adding an <code class="docutils literal notranslate">&quot;if&quot;</code> condition:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>stripped_list = [line.strip() for line in line_list
 if line != &quot;&quot;]
</pre></div>
</div>
With a list comprehension, you get back a Python list; <code class="docutils literal notranslate">stripped_list</code> 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:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>( 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 )
</pre></div>
</div>
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
<code class="docutils literal notranslate">expression</code>. The <code class="docutils literal notranslate">if</code> clauses are all optional; if present, <code class="docutils literal notranslate">expression</code>
is only evaluated and added to the result when <code class="docutils literal notranslate">condition</code> 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:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>obj_total = sum(obj.count for obj in list_all_objects())
</pre></div>
</div>
The <code class="docutils literal notranslate">for...in</code> 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 <code class="docutils literal notranslate">sequence1</code>,
<code class="docutils literal notranslate">sequence2</code> is looped over from the beginning. <code class="docutils literal notranslate">sequence3</code> is then looped
over for each resulting pair of elements from <code class="docutils literal notranslate">sequence1</code> and <code class="docutils literal notranslate">sequence2</code>.
To put it another way, a list comprehension or generator expression is
equivalent to the following Python code:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>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.
</pre></div>
</div>
This means that when there are multiple <code class="docutils literal notranslate">for...in</code> clauses but no <code class="docutils literal notranslate">if</code>
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:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; seq1 = &#39;abc&#39;
&gt;&gt;&gt; seq2 = (1, 2, 3)
&gt;&gt;&gt; [(x, y) for x in seq1 for y in seq2] 
[(&#39;a&#39;, 1), (&#39;a&#39;, 2), (&#39;a&#39;, 3),
 (&#39;b&#39;, 1), (&#39;b&#39;, 2), (&#39;b&#39;, 3),
 (&#39;c&#39;, 1), (&#39;c&#39;, 2), (&#39;c&#39;, 3)]
</pre></div>
</div>
To avoid introducing an ambiguity into Python’s grammar, if <code class="docutils literal notranslate">expression</code> 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:
<div class="highlight-python3 notranslate"><div class="highlight"><pre># Syntax error
[x, y for x in seq1 for y in seq2]
# Correct
[(x, y) for x in seq1 for y in seq2]
</pre></div>
</div>
</div>
<div class="section" id="generators">
<h2>Generators<a class="headerlink" href="#generators" title="Permalink to this headline">¶</a></h2>
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 <code class="docutils literal notranslate">return</code> 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:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; def generate_ints(N):
... for i in range(N):
... yield i
</pre></div>
</div>
Any function containing a <a class="reference internal" href="../reference/simple_stmts.html#yield"><code class="xref std std-keyword docutils literal notranslate">yield</code></a> keyword is a generator function;
this is detected by Python’s <a class="reference internal" href="../glossary.html#term-bytecode">bytecode</a> 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 <code class="docutils literal notranslate">yield</code> expression, the generator outputs the value of <code class="docutils literal notranslate">i</code>, similar to a
<code class="docutils literal notranslate">return</code> statement. The big difference between <code class="docutils literal notranslate">yield</code> and a <code class="docutils literal notranslate">return</code>
statement is that on reaching a <code class="docutils literal notranslate">yield</code> the generator’s state of execution is
suspended and local variables are preserved. On the next call to the
generator’s <a class="reference internal" href="../reference/expressions.html#generator.__next__" title="generator.__next__"><code class="xref py py-meth docutils literal notranslate">__next__()</code></a> method, the function will resume
executing.
Here’s a sample usage of the <code class="docutils literal notranslate">generate_ints()</code> generator:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; gen = generate_ints(3)
&gt;&gt;&gt; gen 
&lt;generator object generate_ints at ...&gt;
&gt;&gt;&gt; next(gen)
0
&gt;&gt;&gt; next(gen)
1
&gt;&gt;&gt; next(gen)
2
&gt;&gt;&gt; next(gen)
Traceback (most recent call last):
 File &quot;stdin&quot;, line 1, in &lt;module&gt;
 File &quot;stdin&quot;, line 2, in generate_ints
StopIteration
</pre></div>
</div>
You could equally write <code class="docutils literal notranslate">for i in generate_ints(5)</code>, or <code class="docutils literal notranslate">a, b, c =
generate_ints(3)</code>.
Inside a generator function, <code class="docutils literal notranslate">return value</code> causes <code class="docutils literal notranslate">StopIteration(value)</code>
to be raised from the <a class="reference internal" href="../reference/expressions.html#generator.__next__" title="generator.__next__"><code class="xref py py-meth docutils literal notranslate">__next__()</code></a> method. Once this happens, or
the bottom of the function is reached, the procession of values ends and the
generator cannot yield any further values.
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 <code class="docutils literal notranslate">self.count</code> to
0, and having the <a class="reference internal" href="../library/stdtypes.html#iterator.__next__" title="iterator.__next__"><code class="xref py py-meth docutils literal notranslate">__next__()</code></a> method increment <code class="docutils literal notranslate">self.count</code> 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,
<a class="reference external" href="https://github.com/python/cpython/tree/3.6/Lib/test/test_generators.py">Lib/test/test_generators.py</a>, contains
a number of more interesting examples. Here’s one generator that implements an
in-order traversal of a tree using generators recursively.
<div class="highlight-python3 notranslate"><div class="highlight"><pre># 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
</pre></div>
</div>
Two other examples in <code class="docutils literal notranslate">test_generators.py</code> 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).
<div class="section" id="passing-values-into-a-generator">
<h3>Passing values into a generator<a class="headerlink" href="#passing-values-into-a-generator" title="Permalink to this headline">¶</a></h3>
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.
<a class="reference internal" href="../reference/simple_stmts.html#yield"><code class="xref std std-keyword docutils literal notranslate">yield</code></a> became an expression, returning a value that can be assigned to
a variable or otherwise operated on:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>val = (yield i)
</pre></div>
</div>
I recommend that you always put parentheses around a <code class="docutils literal notranslate">yield</code> 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.
(<a class="pep reference external" href="https://www.python.org/dev/peps/pep-0342">PEP 342</a> explains the exact rules, which are that a <code class="docutils literal notranslate">yield</code>-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 <code class="docutils literal notranslate">val = yield i</code>
but have to use parentheses when there’s an operation, as in <code class="docutils literal notranslate">val = (yield i)
+ 12</code>.)
Values are sent into a generator by calling its <a class="reference internal" href="../reference/expressions.html#generator.send" title="generator.send"><code class="xref py py-meth docutils literal notranslate">send(value)</code></a> method. This method resumes the generator’s code and the
<code class="docutils literal notranslate">yield</code> expression returns the specified value. If the regular
<a class="reference internal" href="../reference/expressions.html#generator.__next__" title="generator.__next__"><code class="xref py py-meth docutils literal notranslate">__next__()</code></a> method is called, the <code class="docutils literal notranslate">yield</code> returns <code class="docutils literal notranslate">None</code>.
Here’s a simple counter that increments by 1 and allows changing the value of
the internal counter.
<div class="highlight-python notranslate"><div class="highlight"><pre>def counter(maximum):
 i = 0
 while i &lt; maximum:
 val = (yield i)
 # If value provided, change counter
 if val is not None:
 i = val
 else:
 i += 1
</pre></div>
</div>
And here’s an example of changing the counter:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; it = counter(10) 
&gt;&gt;&gt; next(it) 
0
&gt;&gt;&gt; next(it) 
1
&gt;&gt;&gt; it.send(8) 
8
&gt;&gt;&gt; next(it) 
9
&gt;&gt;&gt; next(it) 
Traceback (most recent call last):
 File &quot;t.py&quot;, line 15, in &lt;module&gt;
 it.next()
StopIteration
</pre></div>
</div>
Because <code class="docutils literal notranslate">yield</code> will often be returning <code class="docutils literal notranslate">None</code>, you should always check for
this case. Don’t just use its value in expressions unless you’re sure that the
<a class="reference internal" href="../reference/expressions.html#generator.send" title="generator.send"><code class="xref py py-meth docutils literal notranslate">send()</code></a> method will be the only method used to resume your
generator function.
In addition to <a class="reference internal" href="../reference/expressions.html#generator.send" title="generator.send"><code class="xref py py-meth docutils literal notranslate">send()</code></a>, there are two other methods on
generators:
<ul>
<li><a class="reference internal" href="../reference/expressions.html#generator.throw" title="generator.throw"><code class="xref py py-meth docutils literal notranslate">throw(type, value=None, traceback=None)</code></a> is used to
raise an exception inside the generator; the exception is raised by the
<code class="docutils literal notranslate">yield</code> expression where the generator’s execution is paused.
</li>
<li><a class="reference internal" href="../reference/expressions.html#generator.close" title="generator.close"><code class="xref py py-meth docutils literal notranslate">close()</code></a> raises a <a class="reference internal" href="../library/exceptions.html#GeneratorExit" title="GeneratorExit"><code class="xref py py-exc docutils literal notranslate">GeneratorExit</code></a> exception inside the
generator to terminate the iteration. On receiving this exception, the
generator’s code must either raise <a class="reference internal" href="../library/exceptions.html#GeneratorExit" title="GeneratorExit"><code class="xref py py-exc docutils literal notranslate">GeneratorExit</code></a> or
<a class="reference internal" href="../library/exceptions.html#StopIteration" title="StopIteration"><code class="xref py py-exc docutils literal notranslate">StopIteration</code></a>; catching the exception and doing anything else is
illegal and will trigger a <a class="reference internal" href="../library/exceptions.html#RuntimeError" title="RuntimeError"><code class="xref py py-exc docutils literal notranslate">RuntimeError</code></a>. <a class="reference internal" href="../reference/expressions.html#generator.close" title="generator.close"><code class="xref py py-meth docutils literal notranslate">close()</code></a>
will also be called by Python’s garbage collector when the generator is
garbage-collected.
If you need to run cleanup code when a <a class="reference internal" href="../library/exceptions.html#GeneratorExit" title="GeneratorExit"><code class="xref py py-exc docutils literal notranslate">GeneratorExit</code></a> occurs, I suggest
using a <code class="docutils literal notranslate">try: ... finally:</code> suite instead of catching <a class="reference internal" href="../library/exceptions.html#GeneratorExit" title="GeneratorExit"><code class="xref py py-exc docutils literal notranslate">GeneratorExit</code></a>.
</li>
</ul>
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 <code class="docutils literal notranslate">return</code> statement), but coroutines can be entered, exited,
and resumed at many different points (the <code class="docutils literal notranslate">yield</code> statements).
</div>
</div>
<div class="section" id="built-in-functions">
<h2>Built-in functions<a class="headerlink" href="#built-in-functions" title="Permalink to this headline">¶</a></h2>
Let’s look in more detail at built-in functions often used with iterators.
Two of Python’s built-in functions, <a class="reference internal" href="../library/functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate">map()</code></a> and <a class="reference internal" href="../library/functions.html#filter" title="filter"><code class="xref py py-func docutils literal notranslate">filter()</code></a> duplicate the
features of generator expressions:
<dl class="docutils">
<dt><a class="reference internal" href="../library/functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate">map(f, iterA, iterB, ...)</code></a> returns an iterator over the sequence</dt>
<dd><code class="docutils literal notranslate">f(iterA[0], iterB[0]), f(iterA[1], iterB[1]), f(iterA[2], iterB[2]), ...</code>.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; def upper(s):
... return s.upper()
</pre></div>
</div>
<div class="last highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; list(map(upper, [&#39;sentence&#39;, &#39;fragment&#39;]))
[&#39;SENTENCE&#39;, &#39;FRAGMENT&#39;]
&gt;&gt;&gt; [upper(s) for s in [&#39;sentence&#39;, &#39;fragment&#39;]]
[&#39;SENTENCE&#39;, &#39;FRAGMENT&#39;]
</pre></div>
</div>
</dd>
</dl>
You can of course achieve the same effect with a list comprehension.
<a class="reference internal" href="../library/functions.html#filter" title="filter"><code class="xref py py-func docutils literal notranslate">filter(predicate, iter)</code></a> returns an iterator over 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 <a class="reference internal" href="../library/functions.html#filter" title="filter"><code class="xref py py-func docutils literal notranslate">filter()</code></a>, the predicate must take a
single value.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; def is_even(x):
... return (x % 2) == 0
</pre></div>
</div>
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; list(filter(is_even, range(10)))
[0, 2, 4, 6, 8]
</pre></div>
</div>
This can also be written as a list comprehension:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; list(x for x in range(10) if is_even(x))
[0, 2, 4, 6, 8]
</pre></div>
</div>
<a class="reference internal" href="../library/functions.html#enumerate" title="enumerate"><code class="xref py py-func docutils literal notranslate">enumerate(iter, start=0)</code></a> counts off the elements in the
iterable returning 2-tuples containing the count (from start) and
each element.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; for item in enumerate([&#39;subject&#39;, &#39;verb&#39;, &#39;object&#39;]):
... print(item)
(0, &#39;subject&#39;)
(1, &#39;verb&#39;)
(2, &#39;object&#39;)
</pre></div>
</div>
<a class="reference internal" href="../library/functions.html#enumerate" title="enumerate"><code class="xref py py-func docutils literal notranslate">enumerate()</code></a> is often used when looping through a list and recording the
indexes at which certain conditions are met:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>f = open(&#39;data.txt&#39;, &#39;r&#39;)
for i, line in enumerate(f):
 if line.strip() == &#39;&#39;:
 print(&#39;Blank line at line #%i&#39; % i)
</pre></div>
</div>
<a class="reference internal" href="../library/functions.html#sorted" title="sorted"><code class="xref py py-func docutils literal notranslate">sorted(iterable, key=None, reverse=False)</code></a> collects all the
elements of the iterable into a list, sorts the list, and returns the sorted
result. The key and reverse arguments are passed through to the
constructed list’s <a class="reference internal" href="../library/stdtypes.html#list.sort" title="list.sort"><code class="xref py py-meth docutils literal notranslate">sort()</code></a> method.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; import random
&gt;&gt;&gt; # Generate 8 random numbers between [0, 10000)
&gt;&gt;&gt; rand_list = random.sample(range(10000), 8)
&gt;&gt;&gt; rand_list 
[769, 7953, 9828, 6431, 8442, 9878, 6213, 2207]
&gt;&gt;&gt; sorted(rand_list) 
[769, 2207, 6213, 6431, 7953, 8442, 9828, 9878]
&gt;&gt;&gt; sorted(rand_list, reverse=True) 
[9878, 9828, 8442, 7953, 6431, 6213, 2207, 769]
</pre></div>
</div>
(For a more detailed discussion of sorting, see the <a class="reference internal" href="sorting.html#sortinghowto">Sorting HOW TO</a>.)
The <a class="reference internal" href="../library/functions.html#any" title="any"><code class="xref py py-func docutils literal notranslate">any(iter)</code></a> and <a class="reference internal" href="../library/functions.html#all" title="all"><code class="xref py py-func docutils literal notranslate">all(iter)</code></a> built-ins look at the
truth values of an iterable’s contents. <a class="reference internal" href="../library/functions.html#any" title="any"><code class="xref py py-func docutils literal notranslate">any()</code></a> returns <code class="docutils literal notranslate">True</code> if any element
in the iterable is a true value, and <a class="reference internal" href="../library/functions.html#all" title="all"><code class="xref py py-func docutils literal notranslate">all()</code></a> returns <code class="docutils literal notranslate">True</code> if all of the
elements are true values:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; any([0, 1, 0])
True
&gt;&gt;&gt; any([0, 0, 0])
False
&gt;&gt;&gt; any([1, 1, 1])
True
&gt;&gt;&gt; all([0, 1, 0])
False
&gt;&gt;&gt; all([0, 0, 0])
False
&gt;&gt;&gt; all([1, 1, 1])
True
</pre></div>
</div>
<a class="reference internal" href="../library/functions.html#zip" title="zip"><code class="xref py py-func docutils literal notranslate">zip(iterA, iterB, ...)</code></a> takes one element from each iterable and
returns them in a tuple:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>zip([&#39;a&#39;, &#39;b&#39;, &#39;c&#39;], (1, 2, 3)) =&gt;
 (&#39;a&#39;, 1), (&#39;b&#39;, 2), (&#39;c&#39;, 3)
</pre></div>
</div>
It 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 <a class="reference external" href="https://en.wikipedia.org/wiki/Lazy_evaluation">lazy evaluation</a>.)
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.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>zip([&#39;a&#39;, &#39;b&#39;], (1, 2, 3)) =&gt;
 (&#39;a&#39;, 1), (&#39;b&#39;, 2)
</pre></div>
</div>
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.
</div>
<div class="section" id="the-itertools-module">
<h2>The itertools module<a class="headerlink" href="#the-itertools-module" title="Permalink to this headline">¶</a></h2>
The <a class="reference internal" href="../library/itertools.html#module-itertools" title="itertools: Functions creating iterators for efficient looping."><code class="xref py py-mod docutils literal notranslate">itertools</code></a> 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:
<ul class="simple">
<li>Functions that create a new iterator based on an existing iterator.</li>
<li>Functions for treating an iterator’s elements as function arguments.</li>
<li>Functions for selecting portions of an iterator’s output.</li>
<li>A function for grouping an iterator’s output.</li>
</ul>
<div class="section" id="creating-new-iterators">
<h3>Creating new iterators<a class="headerlink" href="#creating-new-iterators" title="Permalink to this headline">¶</a></h3>
<a class="reference internal" href="../library/itertools.html#itertools.count" title="itertools.count"><code class="xref py py-func docutils literal notranslate">itertools.count(start, step)</code></a> returns an infinite
stream of evenly spaced values. You can optionally supply the starting number,
which defaults to 0, and the interval between numbers, which defaults to 1:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.count() =&gt;
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
itertools.count(10) =&gt;
 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...
itertools.count(10, 5) =&gt;
 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, ...
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.cycle" title="itertools.cycle"><code class="xref py py-func docutils literal notranslate">itertools.cycle(iter)</code></a> 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.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.cycle([1, 2, 3, 4, 5]) =&gt;
 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, ...
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.repeat" title="itertools.repeat"><code class="xref py py-func docutils literal notranslate">itertools.repeat(elem, [n])</code></a> returns the provided
element n times, or returns the element endlessly if n is not provided.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.repeat(&#39;abc&#39;) =&gt;
 abc, abc, abc, abc, abc, abc, abc, abc, abc, abc, ...
itertools.repeat(&#39;abc&#39;, 5) =&gt;
 abc, abc, abc, abc, abc
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.chain" title="itertools.chain"><code class="xref py py-func docutils literal notranslate">itertools.chain(iterA, iterB, ...)</code></a> 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.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.chain([&#39;a&#39;, &#39;b&#39;, &#39;c&#39;], (1, 2, 3)) =&gt;
 a, b, c, 1, 2, 3
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.islice" title="itertools.islice"><code class="xref py py-func docutils literal notranslate">itertools.islice(iter, [start], stop, [step])</code></a> 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.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.islice(range(10), 8) =&gt;
 0, 1, 2, 3, 4, 5, 6, 7
itertools.islice(range(10), 2, 8) =&gt;
 2, 3, 4, 5, 6, 7
itertools.islice(range(10), 2, 8, 2) =&gt;
 2, 4, 6
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.tee" title="itertools.tee"><code class="xref py py-func docutils literal notranslate">itertools.tee(iter, [n])</code></a> 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.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.tee( itertools.count() ) =&gt;
 iterA, iterB

where iterA -&gt;
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

and iterB -&gt;
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
</pre></div>
</div>
</div>
<div class="section" id="calling-functions-on-elements">
<h3>Calling functions on elements<a class="headerlink" href="#calling-functions-on-elements" title="Permalink to this headline">¶</a></h3>
The <a class="reference internal" href="../library/operator.html#module-operator" title="operator: Functions corresponding to the standard operators."><code class="xref py py-mod docutils literal notranslate">operator</code></a> module contains a set of functions corresponding to Python’s
operators. Some examples are <a class="reference internal" href="../library/operator.html#operator.add" title="operator.add"><code class="xref py py-func docutils literal notranslate">operator.add(a, b)</code></a> (adds
two values), <a class="reference internal" href="../library/operator.html#operator.ne" title="operator.ne"><code class="xref py py-func docutils literal notranslate">operator.ne(a, b)</code></a> (same as <code class="docutils literal notranslate">a != b</code>), and
<a class="reference internal" href="../library/operator.html#operator.attrgetter" title="operator.attrgetter"><code class="xref py py-func docutils literal notranslate">operator.attrgetter('id')</code></a>
(returns a callable that fetches the <code class="docutils literal notranslate">.id</code> attribute).
<a class="reference internal" href="../library/itertools.html#itertools.starmap" title="itertools.starmap"><code class="xref py py-func docutils literal notranslate">itertools.starmap(func, iter)</code></a> assumes that the
iterable will return a stream of tuples, and calls func using these tuples as
the arguments:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.starmap(os.path.join,
 [(&#39;/bin&#39;, &#39;python&#39;), (&#39;/usr&#39;, &#39;bin&#39;, &#39;java&#39;),
 (&#39;/usr&#39;, &#39;bin&#39;, &#39;perl&#39;), (&#39;/usr&#39;, &#39;bin&#39;, &#39;ruby&#39;)])
=&gt;
 /bin/python, /usr/bin/java, /usr/bin/perl, /usr/bin/ruby
</pre></div>
</div>
</div>
<div class="section" id="selecting-elements">
<h3>Selecting elements<a class="headerlink" href="#selecting-elements" title="Permalink to this headline">¶</a></h3>
Another group of functions chooses a subset of an iterator’s elements based on a
predicate.
<a class="reference internal" href="../library/itertools.html#itertools.filterfalse" title="itertools.filterfalse"><code class="xref py py-func docutils literal notranslate">itertools.filterfalse(predicate, iter)</code></a> is the
opposite of <a class="reference internal" href="../library/functions.html#filter" title="filter"><code class="xref py py-func docutils literal notranslate">filter()</code></a>, returning all elements for which the predicate
returns false:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.filterfalse(is_even, itertools.count()) =&gt;
 1, 3, 5, 7, 9, 11, 13, 15, ...
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.takewhile" title="itertools.takewhile"><code class="xref py py-func docutils literal notranslate">itertools.takewhile(predicate, iter)</code></a> returns
elements for as long as the predicate returns true. Once the predicate returns
false, the iterator will signal the end of its results.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def less_than_10(x):
 return x &lt; 10

itertools.takewhile(less_than_10, itertools.count()) =&gt;
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

itertools.takewhile(is_even, itertools.count()) =&gt;
 0
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.dropwhile" title="itertools.dropwhile"><code class="xref py py-func docutils literal notranslate">itertools.dropwhile(predicate, iter)</code></a> discards
elements while the predicate returns true, and then returns the rest of the
iterable’s results.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.dropwhile(less_than_10, itertools.count()) =&gt;
 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ...

itertools.dropwhile(is_even, itertools.count()) =&gt;
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.compress" title="itertools.compress"><code class="xref py py-func docutils literal notranslate">itertools.compress(data, selectors)</code></a> takes two
iterators and returns only those elements of data for which the corresponding
element of selectors is true, stopping whenever either one is exhausted:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.compress([1, 2, 3, 4, 5], [True, True, False, False, True]) =&gt;
 1, 2, 5
</pre></div>
</div>
</div>
<div class="section" id="combinatoric-functions">
<h3>Combinatoric functions<a class="headerlink" href="#combinatoric-functions" title="Permalink to this headline">¶</a></h3>
The <a class="reference internal" href="../library/itertools.html#itertools.combinations" title="itertools.combinations"><code class="xref py py-func docutils literal notranslate">itertools.combinations(iterable, r)</code></a>
returns an iterator giving all possible r-tuple combinations of the
elements contained in iterable.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.combinations([1, 2, 3, 4, 5], 2) =&gt;
 (1, 2), (1, 3), (1, 4), (1, 5),
 (2, 3), (2, 4), (2, 5),
 (3, 4), (3, 5),
 (4, 5)

itertools.combinations([1, 2, 3, 4, 5], 3) =&gt;
 (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), (1, 4, 5),
 (2, 3, 4), (2, 3, 5), (2, 4, 5),
 (3, 4, 5)
</pre></div>
</div>
The elements within each tuple remain in the same order as
iterable returned them. For example, the number 1 is always before
2, 3, 4, or 5 in the examples above. A similar function,
<a class="reference internal" href="../library/itertools.html#itertools.permutations" title="itertools.permutations"><code class="xref py py-func docutils literal notranslate">itertools.permutations(iterable, r=None)</code></a>,
removes this constraint on the order, returning all possible
arrangements of length r:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.permutations([1, 2, 3, 4, 5], 2) =&gt;
 (1, 2), (1, 3), (1, 4), (1, 5),
 (2, 1), (2, 3), (2, 4), (2, 5),
 (3, 1), (3, 2), (3, 4), (3, 5),
 (4, 1), (4, 2), (4, 3), (4, 5),
 (5, 1), (5, 2), (5, 3), (5, 4)

itertools.permutations([1, 2, 3, 4, 5]) =&gt;
 (1, 2, 3, 4, 5), (1, 2, 3, 5, 4), (1, 2, 4, 3, 5),
 ...
 (5, 4, 3, 2, 1)
</pre></div>
</div>
If you don’t supply a value for r the length of the iterable is used,
meaning that all the elements are permuted.
Note that these functions produce all of the possible combinations by
position and don’t require that the contents of iterable are unique:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.permutations(&#39;aba&#39;, 3) =&gt;
 (&#39;a&#39;, &#39;b&#39;, &#39;a&#39;), (&#39;a&#39;, &#39;a&#39;, &#39;b&#39;), (&#39;b&#39;, &#39;a&#39;, &#39;a&#39;),
 (&#39;b&#39;, &#39;a&#39;, &#39;a&#39;), (&#39;a&#39;, &#39;a&#39;, &#39;b&#39;), (&#39;a&#39;, &#39;b&#39;, &#39;a&#39;)
</pre></div>
</div>
The identical tuple <code class="docutils literal notranslate">('a', 'a', 'b')</code> occurs twice, but the two ‘a’
strings came from different positions.
The <a class="reference internal" href="../library/itertools.html#itertools.combinations_with_replacement" title="itertools.combinations_with_replacement"><code class="xref py py-func docutils literal notranslate">itertools.combinations_with_replacement(iterable, r)</code></a>
function relaxes a different constraint: elements can be repeated
within a single tuple. Conceptually an element is selected for the
first position of each tuple and then is replaced before the second
element is selected.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.combinations_with_replacement([1, 2, 3, 4, 5], 2) =&gt;
 (1, 1), (1, 2), (1, 3), (1, 4), (1, 5),
 (2, 2), (2, 3), (2, 4), (2, 5),
 (3, 3), (3, 4), (3, 5),
 (4, 4), (4, 5),
 (5, 5)
</pre></div>
</div>
</div>
<div class="section" id="grouping-elements">
<h3>Grouping elements<a class="headerlink" href="#grouping-elements" title="Permalink to this headline">¶</a></h3>
The last function I’ll discuss, <a class="reference internal" href="../library/itertools.html#itertools.groupby" title="itertools.groupby"><code class="xref py py-func docutils literal notranslate">itertools.groupby(iter, key_func=None)</code></a>, is the most complicated. <code class="docutils literal notranslate">key_func(elem)</code> 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.
<a class="reference internal" href="../library/itertools.html#itertools.groupby" title="itertools.groupby"><code class="xref py py-func docutils literal notranslate">groupby()</code></a> 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.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>city_list = [(&#39;Decatur&#39;, &#39;AL&#39;), (&#39;Huntsville&#39;, &#39;AL&#39;), (&#39;Selma&#39;, &#39;AL&#39;),
 (&#39;Anchorage&#39;, &#39;AK&#39;), (&#39;Nome&#39;, &#39;AK&#39;),
 (&#39;Flagstaff&#39;, &#39;AZ&#39;), (&#39;Phoenix&#39;, &#39;AZ&#39;), (&#39;Tucson&#39;, &#39;AZ&#39;),
 ...
 ]

def get_state(city_state):
 return city_state[1]

itertools.groupby(city_list, get_state) =&gt;
 (&#39;AL&#39;, iterator-1),
 (&#39;AK&#39;, iterator-2),
 (&#39;AZ&#39;, iterator-3), ...

where
iterator-1 =&gt;
 (&#39;Decatur&#39;, &#39;AL&#39;), (&#39;Huntsville&#39;, &#39;AL&#39;), (&#39;Selma&#39;, &#39;AL&#39;)
iterator-2 =&gt;
 (&#39;Anchorage&#39;, &#39;AK&#39;), (&#39;Nome&#39;, &#39;AK&#39;)
iterator-3 =&gt;
 (&#39;Flagstaff&#39;, &#39;AZ&#39;), (&#39;Phoenix&#39;, &#39;AZ&#39;), (&#39;Tucson&#39;, &#39;AZ&#39;)
</pre></div>
</div>
<a class="reference internal" href="../library/itertools.html#itertools.groupby" title="itertools.groupby"><code class="xref py py-func docutils literal notranslate">groupby()</code></a> 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.
</div>
</div>
<div class="section" id="the-functools-module">
<h2>The functools module<a class="headerlink" href="#the-functools-module" title="Permalink to this headline">¶</a></h2>
The <a class="reference internal" href="../library/functools.html#module-functools" title="functools: Higher-order functions and operations on callable objects."><code class="xref py py-mod docutils literal notranslate">functools</code></a> 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
<a class="reference internal" href="../library/functools.html#functools.partial" title="functools.partial"><code class="xref py py-func docutils literal notranslate">functools.partial()</code></a> 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 <code class="docutils literal notranslate">f(a, b, c)</code>; you may wish to create a new function
<code class="docutils literal notranslate">g(b, c)</code> that’s equivalent to <code class="docutils literal notranslate">f(1, b, c)</code>; you’re filling in a value for
one of <code class="docutils literal notranslate">f()</code>’s parameters. This is called “partial function application”.
The constructor for <a class="reference internal" href="../library/functools.html#functools.partial" title="functools.partial"><code class="xref py py-func docutils literal notranslate">partial()</code></a> takes the arguments
<code class="docutils literal notranslate">(function, arg1, arg2, ..., kwarg1=value1, kwarg2=value2)</code>. The resulting
object is callable, so you can just call it to invoke <code class="docutils literal notranslate">function</code> with the
filled-in arguments.
Here’s a small but realistic example:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>import functools

def log(message, subsystem):
 &quot;&quot;&quot;Write the contents of &#39;message&#39; to the specified subsystem.&quot;&quot;&quot;
 print(&#39;%s: %s&#39; % (subsystem, message))
 ...

server_log = functools.partial(log, subsystem=&#39;server&#39;)
server_log(&#39;Unable to open socket&#39;)
</pre></div>
</div>
<a class="reference internal" href="../library/functools.html#functools.reduce" title="functools.reduce"><code class="xref py py-func docutils literal notranslate">functools.reduce(func, iter, [initial_value])</code></a>
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. <a class="reference internal" href="../library/functools.html#functools.reduce" title="functools.reduce"><code class="xref py py-func docutils literal notranslate">functools.reduce()</code></a>
takes the first two elements A and B returned by the iterator and calculates
<code class="docutils literal notranslate">func(A, B)</code>. It then requests the third element, C, calculates
<code class="docutils literal notranslate">func(func(A, B), C)</code>, combines this result with the fourth element returned,
and continues until the iterable is exhausted. If the iterable returns no
values at all, a <a class="reference internal" href="../library/exceptions.html#TypeError" title="TypeError"><code class="xref py py-exc docutils literal notranslate">TypeError</code></a> exception is raised. If the initial value is
supplied, it’s used as a starting point and <code class="docutils literal notranslate">func(initial_value, A)</code> is the
first calculation.
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; import operator, functools
&gt;&gt;&gt; functools.reduce(operator.concat, [&#39;A&#39;, &#39;BB&#39;, &#39;C&#39;])
&#39;ABBC&#39;
&gt;&gt;&gt; functools.reduce(operator.concat, [])
Traceback (most recent call last):
 ...
TypeError: reduce() of empty sequence with no initial value
&gt;&gt;&gt; functools.reduce(operator.mul, [1, 2, 3], 1)
6
&gt;&gt;&gt; functools.reduce(operator.mul, [], 1)
1
</pre></div>
</div>
If you use <a class="reference internal" href="../library/operator.html#operator.add" title="operator.add"><code class="xref py py-func docutils literal notranslate">operator.add()</code></a> with <a class="reference internal" href="../library/functools.html#functools.reduce" title="functools.reduce"><code class="xref py py-func docutils literal notranslate">functools.reduce()</code></a>, you’ll add up all the
elements of the iterable. This case is so common that there’s a special
built-in called <a class="reference internal" href="../library/functions.html#sum" title="sum"><code class="xref py py-func docutils literal notranslate">sum()</code></a> to compute it:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; import functools, operator
&gt;&gt;&gt; functools.reduce(operator.add, [1, 2, 3, 4], 0)
10
&gt;&gt;&gt; sum([1, 2, 3, 4])
10
&gt;&gt;&gt; sum([])
0
</pre></div>
</div>
For many uses of <a class="reference internal" href="../library/functools.html#functools.reduce" title="functools.reduce"><code class="xref py py-func docutils literal notranslate">functools.reduce()</code></a>, though, it can be clearer to just
write the obvious <a class="reference internal" href="../reference/compound_stmts.html#for"><code class="xref std std-keyword docutils literal notranslate">for</code></a> loop:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>import functools
# Instead of:
product = functools.reduce(operator.mul, [1, 2, 3], 1)

# You can write:
product = 1
for i in [1, 2, 3]:
 product *= i
</pre></div>
</div>
A related function is <a class="reference internal" href="../library/itertools.html#itertools.accumulate" title="itertools.accumulate"><code class="xref py py-func docutils literal notranslate">itertools.accumulate(iterable, func=operator.add)</code></a>. It performs the same calculation, but instead of
returning only the final result, <code class="xref py py-func docutils literal notranslate">accumulate()</code> returns an iterator that
also yields each partial result:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>itertools.accumulate([1, 2, 3, 4, 5]) =&gt;
 1, 3, 6, 10, 15

itertools.accumulate([1, 2, 3, 4, 5], operator.mul) =&gt;
 1, 2, 6, 24, 120
</pre></div>
</div>
<div class="section" id="the-operator-module">
<h3>The operator module<a class="headerlink" href="#the-operator-module" title="Permalink to this headline">¶</a></h3>
The <a class="reference internal" href="../library/operator.html#module-operator" title="operator: Functions corresponding to the standard operators."><code class="xref py py-mod docutils literal notranslate">operator</code></a> 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:
<ul class="simple">
<li>Math operations: <code class="docutils literal notranslate">add()</code>, <code class="docutils literal notranslate">sub()</code>, <code class="docutils literal notranslate">mul()</code>, <code class="docutils literal notranslate">floordiv()</code>, <code class="docutils literal notranslate">abs()</code>, …</li>
<li>Logical operations: <code class="docutils literal notranslate">not_()</code>, <code class="docutils literal notranslate">truth()</code>.</li>
<li>Bitwise operations: <code class="docutils literal notranslate">and_()</code>, <code class="docutils literal notranslate">or_()</code>, <code class="docutils literal notranslate">invert()</code>.</li>
<li>Comparisons: <code class="docutils literal notranslate">eq()</code>, <code class="docutils literal notranslate">ne()</code>, <code class="docutils literal notranslate">lt()</code>, <code class="docutils literal notranslate">le()</code>, <code class="docutils literal notranslate">gt()</code>, and <code class="docutils literal notranslate">ge()</code>.</li>
<li>Object identity: <code class="docutils literal notranslate">is_()</code>, <code class="docutils literal notranslate">is_not()</code>.</li>
</ul>
Consult the operator module’s documentation for a complete list.
</div>
</div>
<div class="section" id="small-functions-and-the-lambda-expression">
<h2>Small functions and the lambda expression<a class="headerlink" href="#small-functions-and-the-lambda-expression" title="Permalink to this headline">¶</a></h2>
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:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>stripped_lines = [line.strip() for line in lines]
existing_files = filter(os.path.exists, file_list)
</pre></div>
</div>
If the function you need doesn’t exist, you need to write it. One way to write
small functions is to use the <a class="reference internal" href="../reference/expressions.html#lambda"><code class="xref std std-keyword docutils literal notranslate">lambda</code></a> statement. <code class="docutils literal notranslate">lambda</code> takes a
number of parameters and an expression combining these parameters, and creates
an anonymous function that returns the value of the expression:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>adder = lambda x, y: x+y

print_assign = lambda name, value: name + &#39;=&#39; + str(value)
</pre></div>
</div>
An alternative is to just use the <code class="docutils literal notranslate">def</code> statement and define a function in the
usual way:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def adder(x, y):
 return x + y

def print_assign(name, value):
 return name + &#39;=&#39; + str(value)
</pre></div>
</div>
Which alternative is preferable? That’s a style question; my usual course is to
avoid using <code class="docutils literal notranslate">lambda</code>.
One reason for my preference is that <code class="docutils literal notranslate">lambda</code> 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 <code class="docutils literal notranslate">if... elif... else</code>
comparisons or <code class="docutils literal notranslate">try... except</code> statements. If you try to do too much in a
<code class="docutils literal notranslate">lambda</code> statement, you’ll end up with an overly complicated expression that’s
hard to read. Quick, what’s the following code doing?
<div class="highlight-python3 notranslate"><div class="highlight"><pre>import functools
total = functools.reduce(lambda a, b: (0, a[1] + b[1]), items)[1]
</pre></div>
</div>
You can figure it out, but it takes time to disentangle the expression to figure
out what’s going on. Using a short nested <code class="docutils literal notranslate">def</code> statements makes things a
little bit better:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>import functools
def combine(a, b):
 return 0, a[1] + b[1]

total = functools.reduce(combine, items)[1]
</pre></div>
</div>
But it would be best of all if I had simply used a <code class="docutils literal notranslate">for</code> loop:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>total = 0
for a, b in items:
 total += b
</pre></div>
</div>
Or the <a class="reference internal" href="../library/functions.html#sum" title="sum"><code class="xref py py-func docutils literal notranslate">sum()</code></a> built-in and a generator expression:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>total = sum(b for a, b in items)
</pre></div>
</div>
Many uses of <a class="reference internal" href="../library/functools.html#functools.reduce" title="functools.reduce"><code class="xref py py-func docutils literal notranslate">functools.reduce()</code></a> are clearer when written as <code class="docutils literal notranslate">for</code> loops.
Fredrik Lundh once suggested the following set of rules for refactoring uses of
<code class="docutils literal notranslate">lambda</code>:
<ol class="arabic simple">
<li>Write a lambda function.</li>
<li>Write a comment explaining what the heck that lambda does.</li>
<li>Study the comment for a while, and think of a name that captures the essence
of the comment.</li>
<li>Convert the lambda to a def statement, using that name.</li>
<li>Remove the comment.</li>
</ol>
I really like these rules, but you’re free to disagree
about whether this lambda-free style is better.
</div>
<div class="section" id="revision-history-and-acknowledgements">
<h2>Revision History and Acknowledgements<a class="headerlink" href="#revision-history-and-acknowledgements" title="Permalink to this headline">¶</a></h2>
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 <code class="docutils literal notranslate">functional</code> module written by Collin
Winter; adds short section on the operator module; a few other edits.
</div>
<div class="section" id="references">
<h2>References<a class="headerlink" href="#references" title="Permalink to this headline">¶</a></h2>
<div class="section" id="general">
<h3>General<a class="headerlink" href="#general" title="Permalink to this headline">¶</a></h3>
Structure and Interpretation of Computer Programs, by Harold Abelson and
Gerald Jay Sussman with Julie Sussman. Full text at
<a class="reference external" href="https://mitpress.mit.edu/sicp/">https://mitpress.mit.edu/sicp/</a>. 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.
<a class="reference external" href="http://www.defmacro.org/ramblings/fp.html">http://www.defmacro.org/ramblings/fp.html</a>: A general introduction to functional
programming that uses Java examples and has a lengthy historical introduction.
<a class="reference external" href="https://en.wikipedia.org/wiki/Functional_programming">https://en.wikipedia.org/wiki/Functional_programming</a>: General Wikipedia entry
describing functional programming.
<a class="reference external" href="https://en.wikipedia.org/wiki/Coroutine">https://en.wikipedia.org/wiki/Coroutine</a>: Entry for coroutines.
<a class="reference external" href="https://en.wikipedia.org/wiki/Currying">https://en.wikipedia.org/wiki/Currying</a>: Entry for the concept of currying.
</div>
<div class="section" id="python-specific">
<h3>Python-specific<a class="headerlink" href="#python-specific" title="Permalink to this headline">¶</a></h3>
<a class="reference external" href="http://gnosis.cx/TPiP/">http://gnosis.cx/TPiP/</a>: The first chapter of David Mertz’s book
<cite>Text Processing in Python</cite> 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
<a class="reference external" href="https://www.ibm.com/developerworks/linux/library/l-prog/index.html">part 1</a>,
<a class="reference external" href="https://www.ibm.com/developerworks/linux/library/l-prog2/index.html">part 2</a>, and
<a class="reference external" href="https://www.ibm.com/developerworks/linux/library/l-prog3/index.html">part 3</a>,
</div>
<div class="section" id="python-documentation">
<h3>Python documentation<a class="headerlink" href="#python-documentation" title="Permalink to this headline">¶</a></h3>
Documentation for the <a class="reference internal" href="../library/itertools.html#module-itertools" title="itertools: Functions creating iterators for efficient looping."><code class="xref py py-mod docutils literal notranslate">itertools</code></a> module.
Documentation for the <a class="reference internal" href="../library/functools.html#module-functools" title="functools: Higher-order functions and operations on callable objects."><code class="xref py py-mod docutils literal notranslate">functools</code></a> module.
Documentation for the <a class="reference internal" href="../library/operator.html#module-operator" title="operator: Functions corresponding to the standard operators."><code class="xref py py-mod docutils literal notranslate">operator</code></a> module.
<a class="pep reference external" href="https://www.python.org/dev/peps/pep-0289">PEP 289</a>: “Generator Expressions”
<a class="pep reference external" href="https://www.python.org/dev/peps/pep-0342">PEP 342</a>: “Coroutines via Enhanced Generators” describes the new generator
features in Python 2.5.
</div>
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 <h3><a href="../contents.html">Table Of Contents</a></h3>
 <ul>
<li><a class="reference internal" href="#">Functional Programming HOWTO</a><ul>
<li><a class="reference internal" href="#introduction">Introduction</a><ul>
<li><a class="reference internal" href="#formal-provability">Formal provability</a></li>
<li><a class="reference internal" href="#modularity">Modularity</a></li>
<li><a class="reference internal" href="#ease-of-debugging-and-testing">Ease of debugging and testing</a></li>
<li><a class="reference internal" href="#composability">Composability</a></li>
</ul>
</li>
<li><a class="reference internal" href="#iterators">Iterators</a><ul>
<li><a class="reference internal" href="#data-types-that-support-iterators">Data Types That Support Iterators</a></li>
</ul>
</li>
<li><a class="reference internal" href="#generator-expressions-and-list-comprehensions">Generator expressions and list comprehensions</a></li>
<li><a class="reference internal" href="#generators">Generators</a><ul>
<li><a class="reference internal" href="#passing-values-into-a-generator">Passing values into a generator</a></li>
</ul>
</li>
<li><a class="reference internal" href="#built-in-functions">Built-in functions</a></li>
<li><a class="reference internal" href="#the-itertools-module">The itertools module</a><ul>
<li><a class="reference internal" href="#creating-new-iterators">Creating new iterators</a></li>
<li><a class="reference internal" href="#calling-functions-on-elements">Calling functions on elements</a></li>
<li><a class="reference internal" href="#selecting-elements">Selecting elements</a></li>
<li><a class="reference internal" href="#combinatoric-functions">Combinatoric functions</a></li>
<li><a class="reference internal" href="#grouping-elements">Grouping elements</a></li>
</ul>
</li>
<li><a class="reference internal" href="#the-functools-module">The functools module</a><ul>
<li><a class="reference internal" href="#the-operator-module">The operator module</a></li>
</ul>
</li>
<li><a class="reference internal" href="#small-functions-and-the-lambda-expression">Small functions and the lambda expression</a></li>
<li><a class="reference internal" href="#revision-history-and-acknowledgements">Revision History and Acknowledgements</a></li>
<li><a class="reference internal" href="#references">References</a><ul>
<li><a class="reference internal" href="#general">General</a></li>
<li><a class="reference internal" href="#python-specific">Python-specific</a></li>
<li><a class="reference internal" href="#python-documentation">Python documentation</a></li>
</ul>
</li>
</ul>
</li>
</ul>

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