More update to 4.3.0

[joypy/Thun.git] / docs / 2._Library_Examples.rst

Examples (and some documentation) for the Words in the Library
==============================================================

.. code:: ipython3

    from notebook_preamble import J, V

Stack Chatter
=============

This is what I like to call the functions that just rearrange things on
the stack. (One thing I want to mention is that during a hypothetical
compilation phase these "stack chatter" words effectively disappear,
because we can map the logical stack locations to registers that remain
static for the duration of the computation. This remains to be done but
it's "off the shelf" technology.)

``clear``
~~~~~~~~~

.. code:: ipython3

    J('1 2 3 clear')


.. parsed-literal::

    


``dup`` ``dupd``
~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('1 2 3 dup')


.. parsed-literal::

    1 2 3 3


.. code:: ipython3

    J('1 2 3 dupd')


.. parsed-literal::

    1 2 2 3


``enstacken`` ``disenstacken`` ``stack`` ``unstack``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Replace the stack with a quote of itself.

.. code:: ipython3

    J('1 2 3 enstacken')


.. parsed-literal::

    [3 2 1]


Unpack a list onto the stack.

.. code:: ipython3

    J('4 5 6 [3 2 1] unstack')


.. parsed-literal::

    4 5 6 3 2 1


Get the stack on the stack.

.. code:: ipython3

    J('1 2 3 stack')


.. parsed-literal::

    1 2 3 [3 2 1]


Replace the stack with the list on top. The items appear reversed but
they are not, is on the top of both the list and the stack.

.. code:: ipython3

    J('1 2 3 [4 5 6] disenstacken')


.. parsed-literal::

    6 5 4


``pop`` ``popd`` ``popop``
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('1 2 3 pop')


.. parsed-literal::

    1 2


.. code:: ipython3

    J('1 2 3 popd')


.. parsed-literal::

    1 3


.. code:: ipython3

    J('1 2 3 popop')


.. parsed-literal::

    1


``roll<`` ``rolldown`` ``roll>`` ``rollup``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The "down" and "up" refer to the movement of two of the top three items
(displacing the third.)

.. code:: ipython3

    J('1 2 3 roll<')


.. parsed-literal::

    2 3 1


.. code:: ipython3

    J('1 2 3 roll>')


.. parsed-literal::

    3 1 2


``swap``
~~~~~~~~

.. code:: ipython3

    J('1 2 3 swap')


.. parsed-literal::

    1 3 2


``tuck`` ``over``
~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('1 2 3 tuck')


.. parsed-literal::

    1 3 2 3


.. code:: ipython3

    J('1 2 3 over')


.. parsed-literal::

    1 2 3 2


``unit`` ``quoted`` ``unquoted``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('1 2 3 unit')


.. parsed-literal::

    1 2 [3]


.. code:: ipython3

    J('1 2 3 quoted')


.. parsed-literal::

    1 [2] 3


.. code:: ipython3

    J('1 [2] 3 unquoted')


.. parsed-literal::

    1 2 3


.. code:: ipython3

    V('1 [dup] 3 unquoted')  # Unquoting evaluates.  Be aware.


.. parsed-literal::

                  • 1 [dup] 3 unquoted
                1 • [dup] 3 unquoted
          1 [dup] • 3 unquoted
        1 [dup] 3 • unquoted
        1 [dup] 3 • [i] dip
    1 [dup] 3 [i] • dip
          1 [dup] • i 3
                1 • dup 3
              1 1 • 3
            1 1 3 • 


List words
==========

``concat`` ``swoncat`` ``shunt``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('[1 2 3] [4 5 6] concat')


.. parsed-literal::

    [1 2 3 4 5 6]


.. code:: ipython3

    J('[1 2 3] [4 5 6] swoncat')


.. parsed-literal::

    [4 5 6 1 2 3]


.. code:: ipython3

    J('[1 2 3] [4 5 6] shunt')


.. parsed-literal::

    [6 5 4 1 2 3]


``cons`` ``swons`` ``uncons``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('1 [2 3] cons')


.. parsed-literal::

    [1 2 3]


.. code:: ipython3

    J('[2 3] 1 swons')


.. parsed-literal::

    [1 2 3]


.. code:: ipython3

    J('[1 2 3] uncons')


.. parsed-literal::

    1 [2 3]


``first`` ``second`` ``third`` ``rest``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('[1 2 3 4] first')


.. parsed-literal::

    1


.. code:: ipython3

    J('[1 2 3 4] second')


.. parsed-literal::

    2


.. code:: ipython3

    J('[1 2 3 4] third')


.. parsed-literal::

    3


.. code:: ipython3

    J('[1 2 3 4] rest')


.. parsed-literal::

    [2 3 4]


``flatten``
~~~~~~~~~~~

.. code:: ipython3

    J('[[1] [2 [3] 4] [5 6]] flatten')


.. parsed-literal::

    [1 2 [3] 4 5 6]


``getitem`` ``at`` ``of`` ``drop`` ``take``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

``at`` and ``getitem`` are the same function. ``of == swap at``

.. code:: ipython3

    J('[10 11 12 13 14] 2 getitem')


.. parsed-literal::

    12


.. code:: ipython3

    J('[1 2 3 4] 0 at')


.. parsed-literal::

    1


.. code:: ipython3

    J('2 [1 2 3 4] of')


.. parsed-literal::

    3


.. code:: ipython3

    J('[1 2 3 4] 2 drop')


.. parsed-literal::

    [3 4]


.. code:: ipython3

    J('[1 2 3 4] 2 take')  # reverses the order


.. parsed-literal::

    [2 1]


``reverse`` could be defines as ``reverse == dup size take``

``remove``
~~~~~~~~~~

.. code:: ipython3

    J('[1 2 3 1 4] 1 remove')


.. parsed-literal::

    [2 3 1 4]


``reverse``
~~~~~~~~~~~

.. code:: ipython3

    J('[1 2 3 4] reverse')


.. parsed-literal::

    [4 3 2 1]


``size``
~~~~~~~~

.. code:: ipython3

    J('[1 1 1 1] size')


.. parsed-literal::

    4


``swaack``
~~~~~~~~~~

"Swap stack" swap the list on the top of the stack for the stack, and
put the old stack on top of the new one. Think of it as a context
switch. Niether of the lists/stacks change their order.

.. code:: ipython3

    J('1 2 3 [4 5 6] swaack')


.. parsed-literal::

    6 5 4 [3 2 1]


``choice`` ``select``
~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('23 9 1 choice')


.. parsed-literal::

    9


.. code:: ipython3

    J('23 9 0 choice')


.. parsed-literal::

    23


.. code:: ipython3

    J('[23 9 7] 1 select')  # select is basically getitem, should retire it?


.. parsed-literal::

    9


.. code:: ipython3

    J('[23 9 7] 0 select')


.. parsed-literal::

    23


``zip``
~~~~~~~

.. code:: ipython3

    J('[1 2 3] [6 5 4] zip')


.. parsed-literal::

    [[6 1] [5 2] [4 3]]


.. code:: ipython3

    J('[1 2 3] [6 5 4] zip [sum] map')


.. parsed-literal::

    [7 7 7]


Math words
==========

``+`` ``add``
~~~~~~~~~~~~~

.. code:: ipython3

    J('23 9 +')


.. parsed-literal::

    32


``-`` ``sub``
~~~~~~~~~~~~~

.. code:: ipython3

    J('23 9 -')


.. parsed-literal::

    14


``*`` ``mul``
~~~~~~~~~~~~~

.. code:: ipython3

    J('23 9 *')


.. parsed-literal::

    207


``/`` ``div`` ``floordiv`` ``truediv``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('23 9 /')


.. parsed-literal::

    2.5555555555555554


.. code:: ipython3

    J('23 -9 truediv')


.. parsed-literal::

    -2.5555555555555554


.. code:: ipython3

    J('23 9 div')


.. parsed-literal::

    2.5555555555555554


.. code:: ipython3

    J('23 9 floordiv')


.. parsed-literal::

    2


.. code:: ipython3

    J('23 -9 div')


.. parsed-literal::

    -2.5555555555555554


.. code:: ipython3

    J('23 -9 floordiv')


.. parsed-literal::

    -3


``%`` ``mod`` ``modulus`` ``rem`` ``remainder``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('23 9 %')


.. parsed-literal::

    5


``neg``
~~~~~~~

.. code:: ipython3

    J('23 neg -5 neg')


.. parsed-literal::

    -23 5


pow
~~~

.. code:: ipython3

    J('2 10 pow')


.. parsed-literal::

    1024


``sqr`` ``sqrt``
~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('23 sqr')


.. parsed-literal::

    529


.. code:: ipython3

    J('23 sqrt')


.. parsed-literal::

    4.795831523312719


``++`` ``succ`` ``--`` ``pred``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('1 ++')


.. parsed-literal::

    2


.. code:: ipython3

    J('1 --')


.. parsed-literal::

    0


``<<`` ``lshift`` ``>>`` ``rshift``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('8 1 <<')


.. parsed-literal::

    16


.. code:: ipython3

    J('8 1 >>')


.. parsed-literal::

    4


``average``
~~~~~~~~~~~

.. code:: ipython3

    J('[1 2 3 5] average')


.. parsed-literal::

    2.75


``range`` ``range_to_zero`` ``down_to_zero``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('5 range')


.. parsed-literal::

    [4 3 2 1 0]


.. code:: ipython3

    J('5 range_to_zero')


.. parsed-literal::

    [0 1 2 3 4 5]


.. code:: ipython3

    J('5 down_to_zero')


.. parsed-literal::

    5 4 3 2 1 0


``product``
~~~~~~~~~~~

.. code:: ipython3

    J('[1 2 3 5] product')


.. parsed-literal::

    30


``sum``
~~~~~~~

.. code:: ipython3

    J('[1 2 3 5] sum')


.. parsed-literal::

    11


``min``
~~~~~~~

.. code:: ipython3

    J('[1 2 3 5] min')


.. parsed-literal::

    1


``gcd``
~~~~~~~

.. code:: ipython3

    J('45 30 gcd')


.. parsed-literal::

    15


``least_fraction``
~~~~~~~~~~~~~~~~~~

If we represent fractions as a quoted pair of integers [q d] this word
reduces them to their ... least common factors or whatever.

.. code:: ipython3

    J('[45 30] least_fraction')


.. parsed-literal::

    [3.0 2.0]


.. code:: ipython3

    J('[23 12] least_fraction')


.. parsed-literal::

    [23.0 12.0]


Logic and Comparison
====================

``?`` ``truthy``
~~~~~~~~~~~~~~~~

Get the Boolean value of the item on the top of the stack.

.. code:: ipython3

    J('23 truthy')


.. parsed-literal::

    True


.. code:: ipython3

    J('[] truthy')  # Python semantics.


.. parsed-literal::

    False


.. code:: ipython3

    J('0 truthy')


.. parsed-literal::

    False


::

    ? == dup truthy

.. code:: ipython3

    V('23 ?')


.. parsed-literal::

            • 23 ?
         23 • ?
         23 • dup truthy
      23 23 • truthy
    23 True • 


.. code:: ipython3

    J('[] ?')


.. parsed-literal::

    [] False


.. code:: ipython3

    J('0 ?')


.. parsed-literal::

    0 False


``&`` ``and``
~~~~~~~~~~~~~

.. code:: ipython3

    J('23 9 &')


.. parsed-literal::

    1


``!=`` ``<>`` ``ne``
~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('23 9 !=')


.. parsed-literal::

    True


| The usual suspects: - ``<`` ``lt`` - ``<=`` ``le``
| - ``=`` ``eq`` - ``>`` ``gt`` - ``>=`` ``ge`` - ``not`` - ``or``

``^`` ``xor``
~~~~~~~~~~~~~

.. code:: ipython3

    J('1 1 ^')


.. parsed-literal::

    0


.. code:: ipython3

    J('1 0 ^')


.. parsed-literal::

    1


Miscellaneous
=============

``help``
~~~~~~~~

.. code:: ipython3

    J('[help] help')


.. parsed-literal::

    
    ==== Help on help ====
    
    Accepts a quoted symbol on the top of the stack and prints its docs.
    
    ---- end (help)
    
    


``parse``
~~~~~~~~~

.. code:: ipython3

    J('[parse] help')


.. parsed-literal::

    
    ==== Help on parse ====
    
    Parse the string on the stack to a Joy expression.
    
    ---- end (parse)
    
    


.. code:: ipython3

    J('1 "2 [3] dup" parse')


.. parsed-literal::

    1 [2 [3] dup]


``run``
~~~~~~~

Evaluate a quoted Joy sequence.

.. code:: ipython3

    J('[1 2 dup + +] run')


.. parsed-literal::

    [5]


Combinators
===========

``app1`` ``app2`` ``app3``
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('[app1] help')


.. parsed-literal::

    
    ==== Help on app1 ====
    
    Given a quoted program on TOS and anything as the second stack item run
    the program and replace the two args with the first result of the
    program.
    ::
    
                     ... x [Q] . app1
            -----------------------------------
               ... [x ...] [Q] . infra first
    
    ---- end (app1)
    
    


.. code:: ipython3

    J('10 4 [sqr *] app1')


.. parsed-literal::

    10 160


.. code:: ipython3

    J('10 3 4 [sqr *] app2')


.. parsed-literal::

    10 90 160


.. code:: ipython3

    J('[app2] help')


.. parsed-literal::

    
    ==== Help on app2 ====
    
    Like app1 with two items.
    ::
    
                   ... y x [Q] . app2
            -----------------------------------
               ... [y ...] [Q] . infra first
                   [x ...] [Q]   infra first
    
    ---- end (app2)
    
    


.. code:: ipython3

    J('10 2 3 4 [sqr *] app3')


.. parsed-literal::

    10 40 90 160


``anamorphism``
~~~~~~~~~~~~~~~

Given an initial value, a predicate function ``[P]``, and a generator
function ``[G]``, the ``anamorphism`` combinator creates a sequence.

::

       n [P] [G] anamorphism
    ---------------------------
              [...]

Example, ``range``:

::

    range == [0 <=] [1 - dup] anamorphism

.. code:: ipython3

    J('3 [0 <=] [1 - dup] anamorphism')


.. parsed-literal::

    [2 1 0]


``branch``
~~~~~~~~~~

.. code:: ipython3

    J('3 4 1 [+] [*] branch')


.. parsed-literal::

    12


.. code:: ipython3

    J('3 4 0 [+] [*] branch')


.. parsed-literal::

    7


``cleave``
~~~~~~~~~~

::

    ... x [P] [Q] cleave

From the original Joy docs: "The cleave combinator expects two
quotations, and below that an item ``x`` It first executes ``[P]``, with
``x`` on top, and saves the top result element. Then it executes
``[Q]``, again with ``x``, and saves the top result. Finally it restores
the stack to what it was below ``x`` and pushes the two results P(X) and
Q(X)."

Note that ``P`` and ``Q`` can use items from the stack freely, since the
stack (below ``x``) is restored. ``cleave`` is a kind of *parallel*
primitive, and it would make sense to create a version that uses, e.g.
Python threads or something, to actually run ``P`` and ``Q``
concurrently. The current implementation of ``cleave`` is a definition
in terms of ``app2``:

::

    cleave == [i] app2 [popd] dip

.. code:: ipython3

    J('10 2 [+] [-] cleave')


.. parsed-literal::

    10 12 8


``dip`` ``dipd`` ``dipdd``
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('1 2 3 4 5 [+] dip')


.. parsed-literal::

    1 2 7 5


.. code:: ipython3

    J('1 2 3 4 5 [+] dipd')


.. parsed-literal::

    1 5 4 5


.. code:: ipython3

    J('1 2 3 4 5 [+] dipdd')


.. parsed-literal::

    3 3 4 5


``dupdip``
~~~~~~~~~~

Expects a quoted program ``[Q]`` on the stack and some item under it,
``dup`` the item and ``dip`` the quoted program under it.

::

    n [Q] dupdip == n Q n

.. code:: ipython3

    V('23 [++] dupdip *')  # N(N + 1)


.. parsed-literal::

            • 23 [++] dupdip *
         23 • [++] dupdip *
    23 [++] • dupdip *
         23 • ++ 23 *
         24 • 23 *
      24 23 • *
        552 • 


``genrec`` ``primrec``
~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython3

    J('[genrec] help')


.. parsed-literal::

    
    ==== Help on genrec ====
    
    General Recursion Combinator.
    ::
    
                              [if] [then] [rec1] [rec2] genrec
        ---------------------------------------------------------------------
           [if] [then] [rec1 [[if] [then] [rec1] [rec2] genrec] rec2] ifte
    
    From "Recursion Theory and Joy" (j05cmp.html) by Manfred von Thun:
    "The genrec combinator takes four program parameters in addition to
    whatever data parameters it needs. Fourth from the top is an if-part,
    followed by a then-part. If the if-part yields true, then the then-part
    is executed and the combinator terminates. The other two parameters are
    the rec1-part and the rec2-part. If the if-part yields false, the
    rec1-part is executed. Following that the four program parameters and
    the combinator are again pushed onto the stack bundled up in a quoted
    form. Then the rec2-part is executed, where it will find the bundled
    form. Typically it will then execute the bundled form, either with i or
    with app2, or some other combinator."
    
    The way to design one of these is to fix your base case [then] and the
    test [if], and then treat rec1 and rec2 as an else-part "sandwiching"
    a quotation of the whole function.
    
    For example, given a (general recursive) function 'F':
    ::
    
            F == [I] [T] [R1] [R2] genrec
    
    If the [I] if-part fails you must derive R1 and R2 from:
    ::
    
            ... R1 [F] R2
    
    Just set the stack arguments in front, and figure out what R1 and R2
    have to do to apply the quoted [F] in the proper way.  In effect, the
    genrec combinator turns into an ifte combinator with a quoted copy of
    the original definition in the else-part:
    ::
    
            F == [I] [T] [R1]   [R2] genrec
              == [I] [T] [R1 [F] R2] ifte
    
    Primitive recursive functions are those where R2 == i.
    ::
    
            P == [I] [T] [R] tailrec
              == [I] [T] [R [P] i] ifte
              == [I] [T] [R P] ifte
    
    ---- end (genrec)
    
    


.. code:: ipython3

    J('3 [1 <=] [] [dup --] [i *] genrec')


.. parsed-literal::

    6


``i``
~~~~~

.. code:: ipython3

    V('1 2 3 [+ +] i')


.. parsed-literal::

                • 1 2 3 [+ +] i
              1 • 2 3 [+ +] i
            1 2 • 3 [+ +] i
          1 2 3 • [+ +] i
    1 2 3 [+ +] • i
          1 2 3 • + +
            1 5 • +
              6 • 


``ifte``
~~~~~~~~

::

    [predicate] [then] [else] ifte

.. code:: ipython3

    J('1 2 [1] [+] [*] ifte')


.. parsed-literal::

    3


.. code:: ipython3

    J('1 2 [0] [+] [*] ifte')


.. parsed-literal::

    2


``infra``
~~~~~~~~~

.. code:: ipython3

    V('1 2 3 [4 5 6] [* +] infra')


.. parsed-literal::

                        • 1 2 3 [4 5 6] [* +] infra
                      1 • 2 3 [4 5 6] [* +] infra
                    1 2 • 3 [4 5 6] [* +] infra
                  1 2 3 • [4 5 6] [* +] infra
          1 2 3 [4 5 6] • [* +] infra
    1 2 3 [4 5 6] [* +] • infra
                  6 5 4 • * + [3 2 1] swaack
                   6 20 • + [3 2 1] swaack
                     26 • [3 2 1] swaack
             26 [3 2 1] • swaack
             1 2 3 [26] • 


``loop``
~~~~~~~~

.. code:: ipython3

    J('[loop] help')


.. parsed-literal::

    
    ==== Help on loop ====
    
    Basic loop combinator.
    ::
    
               ... True [Q] loop
            -----------------------
                  ... Q [Q] loop
    
               ... False [Q] loop
            ------------------------
                      ...
    
    ---- end (loop)
    
    


.. code:: ipython3

    V('3 dup [1 - dup] loop')


.. parsed-literal::

                  • 3 dup [1 - dup] loop
                3 • dup [1 - dup] loop
              3 3 • [1 - dup] loop
    3 3 [1 - dup] • loop
                3 • 1 - dup [1 - dup] loop
              3 1 • - dup [1 - dup] loop
                2 • dup [1 - dup] loop
              2 2 • [1 - dup] loop
    2 2 [1 - dup] • loop
                2 • 1 - dup [1 - dup] loop
              2 1 • - dup [1 - dup] loop
                1 • dup [1 - dup] loop
              1 1 • [1 - dup] loop
    1 1 [1 - dup] • loop
                1 • 1 - dup [1 - dup] loop
              1 1 • - dup [1 - dup] loop
                0 • dup [1 - dup] loop
              0 0 • [1 - dup] loop
    0 0 [1 - dup] • loop
                0 • 


``map`` ``pam``
~~~~~~~~~~~~~~~

.. code:: ipython3

    J('10 [1 2 3] [*] map')


.. parsed-literal::

    10 [10 20 30]


.. code:: ipython3

    J('10 5 [[*][/][+][-]] pam')


.. parsed-literal::

    10 5 [50 2.0 15 5]


``nullary`` ``unary`` ``binary`` ``ternary``
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Run a quoted program enforcing
`arity <https://en.wikipedia.org/wiki/Arity>`__.

.. code:: ipython3

    J('1 2 3 4 5 [+] nullary')


.. parsed-literal::

    1 2 3 4 5 9


.. code:: ipython3

    J('1 2 3 4 5 [+] unary')


.. parsed-literal::

    1 2 3 4 9


.. code:: ipython3

    J('1 2 3 4 5 [+] binary')  # + has arity 2 so this is technically pointless...


.. parsed-literal::

    1 2 3 9


.. code:: ipython3

    J('1 2 3 4 5 [+] ternary')


.. parsed-literal::

    1 2 9


``step``
~~~~~~~~

.. code:: ipython3

    J('[step] help')


.. parsed-literal::

    
    ==== Help on step ====
    
    Run a quoted program on each item in a sequence.
    ::
    
               ... [] [Q] . step
            -----------------------
                      ... .
    
    
               ... [a] [Q] . step
            ------------------------
                     ... a . Q
    
    
               ... [a b c] [Q] . step
            ----------------------------------------
                         ... a . Q [b c] [Q] step
    
    The step combinator executes the quotation on each member of the list
    on top of the stack.
    
    ---- end (step)
    
    


.. code:: ipython3

    V('0 [1 2 3] [+] step')


.. parsed-literal::

                  • 0 [1 2 3] [+] step
                0 • [1 2 3] [+] step
        0 [1 2 3] • [+] step
    0 [1 2 3] [+] • step
          0 1 [+] • i [2 3] [+] step
              0 1 • + [2 3] [+] step
                1 • [2 3] [+] step
          1 [2 3] • [+] step
      1 [2 3] [+] • step
          1 2 [+] • i [3] [+] step
              1 2 • + [3] [+] step
                3 • [3] [+] step
            3 [3] • [+] step
        3 [3] [+] • step
          3 3 [+] • i
              3 3 • +
                6 • 


``times``
~~~~~~~~~

.. code:: ipython3

    V('3 2 1 2 [+] times')


.. parsed-literal::

                • 3 2 1 2 [+] times
              3 • 2 1 2 [+] times
            3 2 • 1 2 [+] times
          3 2 1 • 2 [+] times
        3 2 1 2 • [+] times
    3 2 1 2 [+] • times
          3 2 1 • + 1 [+] times
            3 3 • 1 [+] times
          3 3 1 • [+] times
      3 3 1 [+] • times
            3 3 • +
              6 • 


``b``
~~~~~

.. code:: ipython3

    J('[b] help')


.. parsed-literal::

    
    ==== Help on b ====
    
    ::
    
            b == [i] dip i
    
            ... [P] [Q] b == ... [P] i [Q] i
            ... [P] [Q] b == ... P Q
    
    ---- end (b)
    
    


.. code:: ipython3

    V('1 2 [3] [4] b')


.. parsed-literal::

                • 1 2 [3] [4] b
              1 • 2 [3] [4] b
            1 2 • [3] [4] b
        1 2 [3] • [4] b
    1 2 [3] [4] • b
            1 2 • 3 4
          1 2 3 • 4
        1 2 3 4 • 


``while``
~~~~~~~~~

::

    [predicate] [body] while

.. code:: ipython3

    J('3 [0 >] [dup --] while')


.. parsed-literal::

    3 2 1 0


``x``
~~~~~

.. code:: ipython3

    J('[x] help')


.. parsed-literal::

    
    ==== Help on x ====
    
    ::
    
            x == dup i
    
            ... [Q] x = ... [Q] dup i
            ... [Q] x = ... [Q] [Q] i
            ... [Q] x = ... [Q]  Q
    
    ---- end (x)
    
    


.. code:: ipython3

    V('1 [2] [i 3] x')  # Kind of a pointless example.


.. parsed-literal::

                • 1 [2] [i 3] x
              1 • [2] [i 3] x
          1 [2] • [i 3] x
    1 [2] [i 3] • x
    1 [2] [i 3] • i 3
          1 [2] • i 3 3
              1 • 2 3 3
            1 2 • 3 3
          1 2 3 • 3
        1 2 3 3 • 


``void``
========

Implements `**Laws of Form**
*arithmetic* <https://en.wikipedia.org/wiki/Laws_of_Form#The_primary_arithmetic_.28Chapter_4.29>`__
over quote-only datastructures (that is, datastructures that consist
soley of containers, without strings or numbers or anything else.)

.. code:: ipython3

    J('[] void')


.. parsed-literal::

    False


.. code:: ipython3

    J('[[]] void')


.. parsed-literal::

    True


.. code:: ipython3

    J('[[][[]]] void')


.. parsed-literal::

    True


.. code:: ipython3

    J('[[[]][[][]]] void')


.. parsed-literal::

    False