# Source code for joy.library

    # -*- coding: utf-8 -*-
    #
    #    Copyright © 2014, 2015, 2017, 2018 Simon Forman
    #
    #    This file is part of Thun
    #
    #    Thun is free software: you can redistribute it and/or modify
    #    it under the terms of the GNU General Public License as published by
    #    the Free Software Foundation, either version 3 of the License, or
    #    (at your option) any later version.
    #
    #    Thun is distributed in the hope that it will be useful,
    #    but WITHOUT ANY WARRANTY; without even the implied warranty of
    #    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    #    GNU General Public License for more details.
    #
    #    You should have received a copy of the GNU General Public License
    #    along with Thun.  If not see <http://www.gnu.org/licenses/>.
    #
    '''
    This module contains the Joy function infrastructure and a library of
    functions.  Its main export is a Python function initialize() that
    returns a dictionary of Joy functions suitable for use with the joy()
    function.
    '''
    from inspect import getdoc
    from functools import wraps
    from inspect import getmembers, isfunction
    import operator, math

    from .parser import text_to_expression, Symbol
    from .utils.stack import list_to_stack, iter_stack, pick, concat
    from .utils.brutal_hackery import rename_code_object

    from .utils import generated_library as genlib

    _dictionary = {}

    def inscribe(function):
      '''A decorator to inscribe functions into the default dictionary.'''
      _dictionary[function.name] = function
      return function

    def initialize():
      '''Return a dictionary of Joy functions for use with joy().'''
      return _dictionary.copy()

    ALIASES = (
      ('add', ['+']),
      ('and', ['&']),
      ('bool', ['truthy']),
      ('mul', ['*']),
      ('truediv', ['/']),
      ('mod', ['%', 'rem', 'remainder', 'modulus']),
      ('eq', ['=']),
      ('ge', ['>=']),
      ('getitem', ['pick', 'at']),
      ('gt', ['>']),
      ('le', ['<=']),
      ('lshift', ['<<']),
      ('lt', ['<']),
      ('ne', ['<>', '!=']),
      ('rshift', ['>>']),
      ('sub', ['-']),
      ('xor', ['^']),
      ('succ', ['++']),
      ('pred', ['--']),
      ('rolldown', ['roll<']),
      ('rollup', ['roll>']),
      ('id', [u'•']),
      )

    def add_aliases(D, A):
      '''
      Given a dict and a iterable of (name, [alias, ...]) pairs, create
      additional entries in the dict mapping each alias to the named function
      if it's in the dict.  Aliases for functions not in the dict are ignored.
      '''
      for name, aliases in A:
        try:
          F = D[name]
        except KeyError:
          continue
        for alias in aliases:
          D[alias] = F

    definitions = ('''\
    of == swap at
    product == 1 swap [*] step
    flatten == [] swap [concat] step
    quoted == [unit] dip
    unquoted == [i] dip
    enstacken == stack [clear] dip
    disenstacken == ? [uncons ?] loop pop
    ? == dup truthy
    dinfrirst == dip infra first
    nullary == [stack] dinfrirst
    unary == nullary popd
    binary == nullary [popop] dip
    ternary == unary [popop] dip
    pam == [i] map
    run == [] swap infra
    sqr == dup mul
    size == 0 swap [pop ++] step
    fork == [i] app2
    cleave == fork [popd] dip
    average == [sum 1.0 *] [size] cleave /
    gcd == 1 [tuck modulus dup 0 >] loop pop
    least_fraction == dup [gcd] infra [div] concat map
    *fraction == [uncons] dip uncons [swap] dip concat [*] infra [*] dip cons
    *fraction0 == concat [[swap] dip * [*] dip] infra
    down_to_zero == [0 >] [dup --] while
    range_to_zero == unit [down_to_zero] infra
    anamorphism == [pop []] swap [dip swons] genrec
    range == [0 <=] [1 - dup] anamorphism
    while == swap [nullary] cons dup dipd concat loop
    dupdipd == dup dipd
    primrec == [i] genrec
    step_zero == 0 roll> step
    codireco == cons dip rest cons
    make_generator == [codireco] ccons
    ifte == [nullary not] dipd branch
    '''
    #
    # ifte == [nullary] dipd swap branch
    # genrec == [[genrec] cons cons cons cons] nullary swons concat ifte

    # Another definition for while. FWIW
    # while == over [[i] dip nullary] ccons [nullary] dip loop

    ##ccons == cons cons
    ##unit == [] cons
    ##second == rest first
    ##third == rest rest first
    ##swons == swap cons
    ##swoncat == swap concat

    ##Zipper
    ##z-down == [] swap uncons swap
    ##z-up == swons swap shunt
    ##z-right == [swons] cons dip uncons swap
    ##z-left == swons [uncons swap] dip swap

    ##Quadratic Formula
    ##divisor == popop 2 *
    ##minusb == pop neg
    ##radical == swap dup * rollup * 4 * - sqrt
    ##root1 == + swap /
    ##root2 == - swap /
    ##q0 == [[divisor] [minusb] [radical]] pam
    ##q1 == [[root1] [root2]] pam
    ##quadratic == [q0] ternary i [q1] ternary

    # Project Euler
    ##'''\
    ##PE1.1 == + dup [+] dip
    ##PE1.2 == dup [3 & PE1.1] dip 2 >>
    ##PE1.3 == 14811 swap [PE1.2] times pop
    ##PE1 == 0 0 66 [7 PE1.3] times 4 PE1.3 pop
    ##'''
    #PE1.2 == [PE1.1] step
    #PE1 == 0 0 66 [[3 2 1 3 1 2 3] PE1.2] times [3 2 1 3] PE1.2 pop
    )

    def FunctionWrapper(f):
      '''Set name attribute.'''
      if not f.__doc__:
        raise ValueError('Function %s must have doc string.' % f.__name__)
      f.name = f.__name__.rstrip('_')  # Don't shadow builtins.
      return f

    def SimpleFunctionWrapper(f):
      '''
      Wrap functions that take and return just a stack.
      '''
      @FunctionWrapper
      @wraps(f)
      @rename_code_object(f.__name__)
      def inner(stack, expression, dictionary):
        return f(stack), expression, dictionary
      return inner

    def BinaryBuiltinWrapper(f):
      '''
      Wrap functions that take two arguments and return a single result.
      '''
      @FunctionWrapper
      @wraps(f)
      @rename_code_object(f.__name__)
      def inner(stack, expression, dictionary):
        (a, (b, stack)) = stack
        result = f(b, a)
        return (result, stack), expression, dictionary
      return inner

    def UnaryBuiltinWrapper(f):
      '''
      Wrap functions that take one argument and return a single result.
      '''
      @FunctionWrapper
      @wraps(f)
      @rename_code_object(f.__name__)
      def inner(stack, expression, dictionary):
        (a, stack) = stack
        result = f(a)
        return (result, stack), expression, dictionary
      return inner

    class DefinitionWrapper(object):
      '''
      Provide implementation of defined functions, and some helper methods.
      '''

      def __init__(self, name, body_text, doc=None):
        self.name = self.__name__ = name
        self.body = text_to_expression(body_text)
        self._body = tuple(iter_stack(self.body))
        self.__doc__ = doc or body_text
        self._compiled = None

      def __call__(self, stack, expression, dictionary):
        if self._compiled:
          return self._compiled(stack, expression, dictionary)
        expression = list_to_stack(self._body, expression)
        return stack, expression, dictionary

      @classmethod
      def parse_definition(class_, defi):
        '''
        Given some text describing a Joy function definition parse it and
        return a DefinitionWrapper.
        '''
        name, proper, body_text = (n.strip() for n in defi.partition('=='))
        if not proper:
          raise ValueError('Definition %r failed' % (defi,))
        return class_(name, body_text)

      @classmethod
      def add_definitions(class_, defs, dictionary):
        '''
        Scan multi-line string defs for definitions and add them to the
        dictionary.
        '''
        for definition in _text_to_defs(defs):
          class_.add_def(definition, dictionary)

      @classmethod
      def add_def(class_, definition, dictionary):
        '''
        Add the definition to the dictionary.
        '''
        F = class_.parse_definition(definition)
        dictionary[F.name] = F

    def _text_to_defs(text):
      return (line.strip() for line in text.splitlines() if '==' in line)

    #
    # Functions
    #

    # Load the auto-generated primitives into the dictionary.
    for name, primitive in getmembers(genlib, isfunction):
      inscribe(SimpleFunctionWrapper(primitive))

    @inscribe
    @SimpleFunctionWrapper
    def parse(stack):
      '''Parse the string on the stack to a Joy expression.'''
      text, stack = stack
      expression = text_to_expression(text)
      return expression, stack

    @inscribe
    @SimpleFunctionWrapper
    def getitem(stack):
      '''
      ::

        getitem == drop first

      Expects an integer and a quote on the stack and returns the item at the
      nth position in the quote counting from 0.
      ::

           [a b c d] 0 getitem
        -------------------------
                    a

      '''
      n, (Q, stack) = stack
      return pick(Q, n), stack

    @inscribe
    @SimpleFunctionWrapper
    def drop(stack):
      '''
      ::

        drop == [rest] times

      Expects an integer and a quote on the stack and returns the quote with
      n items removed off the top.
      ::

           [a b c d] 2 drop
        ----------------------
               [c d]

      '''
      n, (Q, stack) = stack
      while n > 0:
        try:
          _, Q = Q
        except ValueError:
          raise IndexError
        n -= 1
      return Q, stack

    @inscribe
    @SimpleFunctionWrapper
    def take(stack):
      '''
      Expects an integer and a quote on the stack and returns the quote with
      just the top n items in reverse order (because that's easier and you can
      use reverse if needed.)
      ::

           [a b c d] 2 take
        ----------------------
               [b a]

      '''
      n, (Q, stack) = stack
      x = ()
      while n > 0:
        try:
          item, Q = Q
        except ValueError:
          raise IndexError
        x = item, x
        n -= 1
      return x, stack

    @inscribe
    @SimpleFunctionWrapper
    def choice(stack):
      '''
      Use a Boolean value to select one of two items.
      ::

            A B False choice
         ----------------------
                   A

            A B True choice
         ---------------------
                   B

      Currently Python semantics are used to evaluate the "truthiness" of the
      Boolean value (so empty string, zero, etc. are counted as false, etc.)
      '''
      (if_, (then, (else_, stack))) = stack
      return then if if_ else else_, stack

    @inscribe
    @SimpleFunctionWrapper
    def select(stack):
      '''
      Use a Boolean value to select one of two items from a sequence.
      ::

            [A B] False select
         ------------------------
                    A

            [A B] True select
         -----------------------
                   B

      The sequence can contain more than two items but not fewer.
      Currently Python semantics are used to evaluate the "truthiness" of the
      Boolean value (so empty string, zero, etc. are counted as false, etc.)
      '''
      (flag, (choices, stack)) = stack
      (else_, (then, _)) = choices
      return then if flag else else_, stack

    @inscribe
    @SimpleFunctionWrapper
    def max_(S):
      '''Given a list find the maximum.'''
      tos, stack = S
      return max(iter_stack(tos)), stack

    @inscribe
    @SimpleFunctionWrapper
    def min_(S):
      '''Given a list find the minimum.'''
      tos, stack = S
      return min(iter_stack(tos)), stack

    @inscribe
    @SimpleFunctionWrapper
    def sum_(S):
      '''Given a quoted sequence of numbers return the sum.

      sum == 0 swap [+] step
      '''
      tos, stack = S
      return sum(iter_stack(tos)), stack

    @inscribe
    @SimpleFunctionWrapper
    def remove(S):
      '''
      Expects an item on the stack and a quote under it and removes that item
      from the the quote.  The item is only removed once.
      ::

           [1 2 3 1] 1 remove
        ------------------------
                [2 3 1]

      '''
      (tos, (second, stack)) = S
      l = list(iter_stack(second))
      l.remove(tos)
      return list_to_stack(l), stack

    @inscribe
    @SimpleFunctionWrapper
    def unique(S):
      '''Given a list remove duplicate items.'''
      tos, stack = S
      I = list(iter_stack(tos))
      list_to_stack(sorted(set(I), key=I.index))
      return list_to_stack(sorted(set(I), key=I.index)), stack

    @inscribe
    @SimpleFunctionWrapper
    def sort_(S):
      '''Given a list return it sorted.'''
      tos, stack = S
      return list_to_stack(sorted(iter_stack(tos))), stack

    @inscribe
    @SimpleFunctionWrapper
    def clear(stack):
      '''Clear everything from the stack.
      ::

        clear == stack [pop stack] loop

           ... clear
        ---------------

      '''
      return ()

    @inscribe
    @SimpleFunctionWrapper
    def unstack(stack):
      '''
      The unstack operator expects a list on top of the stack and makes that
      the stack discarding the rest of the stack.
      '''
      return stack[0]

    @inscribe
    @SimpleFunctionWrapper
    def reverse(S):
      '''Reverse the list on the top of the stack.
      ::

        reverse == [] swap shunt
      '''
      (tos, stack) = S
      res = ()
      for term in iter_stack(tos):
        res = term, res
      return res, stack

    @inscribe
    @SimpleFunctionWrapper
    def concat_(S):
      '''Concatinate the two lists on the top of the stack.
      ::

           [a b c] [d e f] concat
        ----------------------------
               [a b c d e f]

    '''
      (tos, (second, stack)) = S
      return concat(second, tos), stack

    @inscribe
    @SimpleFunctionWrapper
    def shunt(stack):
      '''Like concat but reverses the top list into the second.
      ::

        shunt == [swons] step == reverse swap concat

           [a b c] [d e f] shunt
        ---------------------------
             [f e d a b c]

      '''
      (tos, (second, stack)) = stack
      while tos:
        term, tos = tos
        second = term, second
      return second, stack

    @inscribe
    @SimpleFunctionWrapper
    def zip_(S):
      '''
      Replace the two lists on the top of the stack with a list of the pairs
      from each list.  The smallest list sets the length of the result list.
      '''
      (tos, (second, stack)) = S
      accumulator = [
        (a, (b, ()))
        for a, b in zip(iter_stack(tos), iter_stack(second))
        ]
      return list_to_stack(accumulator), stack

    @inscribe
    @SimpleFunctionWrapper
    def succ(S):
      '''Increment TOS.'''
      (tos, stack) = S
      return tos + 1, stack

    @inscribe
    @SimpleFunctionWrapper
    def pred(S):
      '''Decrement TOS.'''
      (tos, stack) = S
      return tos - 1, stack

    @inscribe
    @SimpleFunctionWrapper
    def pm(stack):
      '''
      Plus or minus
      ::

           a b pm
        -------------
           a+b a-b

      '''
      a, (b, stack) = stack
      p, m, = b + a, b - a
      return m, (p, stack)

    def floor(n):
      return int(math.floor(n))

    floor.__doc__ = math.floor.__doc__

    @inscribe
    @SimpleFunctionWrapper
    def divmod_(S):
      '''
      divmod(x, y) -> (quotient, remainder)

      Return the tuple (x//y, x%y).  Invariant: div*y + mod == x.
      '''
      a, (b, stack) = S
      d, m = divmod(a, b)
      return d, (m, stack)

    def sqrt(a):
      '''
      Return the square root of the number a.
      Negative numbers return complex roots.
      '''
      try:
        r = math.sqrt(a)
      except ValueError:
        assert a < 0, repr(a)
        r = math.sqrt(-a) * 1j
      return r

    #def execute(S):
    #  (text, stack) = S
    #  if isinstance(text, str):
    #    return run(text, stack)
    #  return stack

    @inscribe
    @SimpleFunctionWrapper
    def id_(stack):
      '''The identity function.'''
      return stack

    @inscribe
    @SimpleFunctionWrapper
    def void(stack):
      '''True if the form on TOS is void otherwise False.'''
      form, stack = stack
      return _void(form), stack

    def _void(form):
      return any(not _void(i) for i in iter_stack(form))

    ##  transpose
    ##  sign
    ##  take

    @inscribe
    @FunctionWrapper
    def words(stack, expression, dictionary):
      '''Print all the words in alphabetical order.'''
      print(' '.join(sorted(dictionary)))
      return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def sharing(stack, expression, dictionary):
      '''Print redistribution information.'''
      print("You may convey verbatim copies of the Program's source code as"
            ' you receive it, in any medium, provided that you conspicuously'
            ' and appropriately publish on each copy an appropriate copyright'
            ' notice; keep intact all notices stating that this License and'
            ' any non-permissive terms added in accord with section 7 apply'
            ' to the code; keep intact all notices of the absence of any'
            ' warranty; and give all recipients a copy of this License along'
            ' with the Program.'
            ' You should have received a copy of the GNU General Public License'
            ' along with Thun.  If not see <http://www.gnu.org/licenses/>.')
      return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def warranty(stack, expression, dictionary):
      '''Print warranty information.'''
      print('THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY'
            ' APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE'
            ' COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM'
            ' "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR'
            ' IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES'
            ' OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE'
            ' ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS'
            ' WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE'
            ' COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION.')
      return stack, expression, dictionary

    # def simple_manual(stack):
    #   '''
    #   Print words and help for each word.
    #   '''
    #   for name, f in sorted(FUNCTIONS.items()):
    #     d = getdoc(f)
    #     boxline = '+%s+' % ('-' * (len(name) + 2))
    #     print('\n'.join((
    #       boxline,
    #       '| %s |' % (name,),
    #       boxline,
    #       d if d else '   ...',
    #       '',
    #       '--' * 40,
    #       '',
    #       )))
    #   return stack

    @inscribe
    @FunctionWrapper
    def help_(S, expression, dictionary):
      '''Accepts a quoted symbol on the top of the stack and prints its docs.'''
      ((symbol, _), stack) = S
      word = dictionary[symbol]
      print(getdoc(word))
      return stack, expression, dictionary

    #
    # § Combinators
    #

    # Several combinators depend on other words in their definitions,
    # we use symbols to prevent hard-coding these, so in theory, you
    # could change the word in the dictionary to use different semantics.
    S_choice = Symbol('choice')
    S_first = Symbol('first')
    S_getitem = Symbol('getitem')
    S_genrec = Symbol('genrec')
    S_loop = Symbol('loop')
    S_i = Symbol('i')
    S_ifte = Symbol('ifte')
    S_infra = Symbol('infra')
    S_step = Symbol('step')
    S_times = Symbol('times')
    S_swaack = Symbol('swaack')
    S_truthy = Symbol('truthy')

    @inscribe
    @FunctionWrapper
    def i(stack, expression, dictionary):
      '''
      The i combinator expects a quoted program on the stack and unpacks it
      onto the pending expression for evaluation.
      ::

           [Q] i
        -----------
            Q

      '''
      quote, stack = stack
      return stack, concat(quote, expression), dictionary

    @inscribe
    @FunctionWrapper
    def x(stack, expression, dictionary):
      '''
      ::

        x == dup i

        ... [Q] x = ... [Q] dup i
        ... [Q] x = ... [Q] [Q] i
        ... [Q] x = ... [Q]  Q

      '''
      quote, _ = stack
      return stack, concat(quote, expression), dictionary

    @inscribe
    @FunctionWrapper
    def b(stack, expression, dictionary):
      '''
      ::

        b == [i] dip i

        ... [P] [Q] b == ... [P] i [Q] i
        ... [P] [Q] b == ... P Q

      '''
      q, (p, (stack)) = stack
      return stack, concat(p, concat(q, expression)), dictionary

    @inscribe
    @FunctionWrapper
    def dupdip(stack, expression, dictionary):
      '''
      ::

        [F] dupdip == dup [F] dip

        ... a [F] dupdip
        ... a dup [F] dip
        ... a a   [F] dip
        ... a F a

      '''
      F, stack = stack
      a = stack[0]
      return stack, concat(F, (a,  expression)), dictionary

    @inscribe
    @FunctionWrapper
    def infra(stack, expression, dictionary):
      '''
      Accept a quoted program and a list on the stack and run the program
      with the list as its stack.
      ::

           ... [a b c] [Q] . infra
        -----------------------------
           c b a . Q [...] swaack

      '''
      (quote, (aggregate, stack)) = stack
      return aggregate, concat(quote, (stack, (S_swaack, expression))), dictionary

    @inscribe
    @FunctionWrapper
    def genrec(stack, expression, dictionary):
      '''
      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] primrec
            == [I] [T] [R [P] i] ifte
            == [I] [T] [R P] ifte

      '''
      (rec2, (rec1, stack)) = stack
      (then, (if_, _)) = stack
      F = (if_, (then, (rec1, (rec2, (S_genrec, ())))))
      else_ = concat(rec1, (F, rec2))
      return (else_, stack), (S_ifte, expression), dictionary

    @inscribe
    @FunctionWrapper
    def map_(S, expression, dictionary):
      '''
      Run the quoted program on TOS on the items in the list under it, push a
      new list with the results (in place of the program and original list.
      '''
    #  (quote, (aggregate, stack)) = S
    #  results = list_to_stack([
    #    joy((term, stack), quote, dictionary)[0][0]
    #    for term in iter_stack(aggregate)
    #    ])
    #  return (results, stack), expression, dictionary
      (quote, (aggregate, stack)) = S
      if not aggregate:
        return (aggregate, stack), expression, dictionary
      batch = ()
      for term in iter_stack(aggregate):
        s = term, stack
        batch = (s, (quote, (S_infra, (S_first, batch))))
      stack = (batch, ((), stack))
      return stack, (S_infra, expression), dictionary

    #def cleave(S, expression, dictionary):
    #  '''
    #  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).
    #  '''
    #  (Q, (P, (x, stack))) = S
    #  p = joy((x, stack), P, dictionary)[0][0]
    #  q = joy((x, stack), Q, dictionary)[0][0]
    #  return (q, (p, stack)), expression, dictionary

    @inscribe
    @FunctionWrapper
    def branch(stack, expression, dictionary):
      '''
      Use a Boolean value to select one of two quoted programs to run.

      ::

          branch == roll< choice i

      ::

            False [F] [T] branch
         --------------------------
                   F

            True [F] [T] branch
         -------------------------
                      T

      '''
      (then, (else_, (flag, stack))) = stack
      return stack, concat(then if flag else else_, expression), dictionary

    ##@inscribe
    ##@FunctionWrapper
    ##def ifte(stack, expression, dictionary):
    ##  '''
    ##  If-Then-Else Combinator
    ##  ::
    ##
    ##                  ... [if] [then] [else] ifte
    ##       ---------------------------------------------------
    ##          ... [[else] [then]] [...] [if] infra select i
    ##
    ##
    ##
    ##
    ##                ... [if] [then] [else] ifte
    ##       -------------------------------------------------------
    ##          ... [else] [then] [...] [if] infra first choice i
    ##
    ##
    ##  Has the effect of grabbing a copy of the stack on which to run the
    ##  if-part using infra.
    ##  '''
    ##  (else_, (then, (if_, stack))) = stack
    ##  expression = (S_infra, (S_first, (S_choice, (S_i, expression))))
    ##  stack = (if_, (stack, (then, (else_, stack))))
    ##  return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def cond(stack, expression, dictionary):
      '''
      This combinator works like a case statement.  It expects a single quote
      on the stack that must contain zero or more condition quotes and a
      default quote.  Each condition clause should contain a quoted predicate
      followed by the function expression to run if that predicate returns
      true.  If no predicates return true the default function runs.

      It works by rewriting into a chain of nested `ifte` expressions, e.g.::

                [[[B0] T0] [[B1] T1] [D]] cond
          -----------------------------------------
             [B0] [T0] [[B1] [T1] [D] ifte] ifte

      '''
      conditions, stack = stack
      if conditions:
        expression = _cond(conditions, expression)
        try:
          # Attempt to preload the args to first ifte.
          (P, (T, (E, expression))) = expression
        except ValueError:
          # If, for any reason, the argument to cond should happen to contain
          # only the default clause then this optimization will fail.
          pass
        else:
          stack = (E, (T, (P, stack)))
      return stack, expression, dictionary

    def _cond(conditions, expression):
      (clause, rest) = conditions
      if not rest:  # clause is [D]
        return clause
      P, T = clause
      return (P, (T, (_cond(rest, ()), (S_ifte, expression))))

    @inscribe
    @FunctionWrapper
    def dip(stack, expression, dictionary):
      '''
      The dip combinator expects a quoted program on the stack and below it
      some item, it hoists the item into the expression and runs the program
      on the rest of the stack.
      ::

           ... x [Q] dip
        -------------------
             ... Q x

      '''
      (quote, (x, stack)) = stack
      expression = (x, expression)
      return stack, concat(quote, expression), dictionary

    @inscribe
    @FunctionWrapper
    def dipd(S, expression, dictionary):
      '''
      Like dip but expects two items.
      ::

           ... y x [Q] dip
        ---------------------
             ... Q y x

      '''
      (quote, (x, (y, stack))) = S
      expression = (y, (x, expression))
      return stack, concat(quote, expression), dictionary

    @inscribe
    @FunctionWrapper
    def dipdd(S, expression, dictionary):
      '''
      Like dip but expects three items.
      ::

           ... z y x [Q] dip
        -----------------------
             ... Q z y x

      '''
      (quote, (x, (y, (z, stack)))) = S
      expression = (z, (y, (x, expression)))
      return stack, concat(quote, expression), dictionary

    @inscribe
    @FunctionWrapper
    def app1(S, expression, dictionary):
      '''
      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
      '''
      (quote, (x, stack)) = S
      stack = (quote, ((x, stack), stack))
      expression = (S_infra, (S_first, expression))
      return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def app2(S, expression, dictionary):
      '''Like app1 with two items.
      ::

                ... y x [Q] . app2
         -----------------------------------
            ... [y ...] [Q] . infra first
                [x ...] [Q]   infra first

      '''
      (quote, (x, (y, stack))) = S
      expression = (S_infra, (S_first,
        ((x, stack), (quote, (S_infra, (S_first,
          expression))))))
      stack = (quote, ((y, stack), stack))
      return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def app3(S, expression, dictionary):
      '''Like app1 with three items.
      ::

                ... z y x [Q] . app3
         -----------------------------------
            ... [z ...] [Q] . infra first
                [y ...] [Q]   infra first
                [x ...] [Q]   infra first

      '''
      (quote, (x, (y, (z, stack)))) = S
      expression = (S_infra, (S_first,
        ((y, stack), (quote, (S_infra, (S_first,
        ((x, stack), (quote, (S_infra, (S_first,
          expression))))))))))
      stack = (quote, ((z, stack), stack))
      return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def step(S, expression, dictionary):
      '''
      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.
      '''
      (quote, (aggregate, stack)) = S
      if not aggregate:
        return stack, expression, dictionary
      head, tail = aggregate
      stack = quote, (head, stack)
      if tail:
        expression = tail, (quote, (S_step, expression))
      expression = S_i, expression
      return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def times(stack, expression, dictionary):
      '''
      times == [-- dip] cons [swap] infra [0 >] swap while pop
      ::

           ... n [Q] . times
        ---------------------  w/ n <= 0
                 ... .

           ... 1 [Q] . times
        ---------------------------------
                 ... . Q

           ... n [Q] . times
        ---------------------------------  w/ n > 1
                 ... . Q (n - 1) [Q] times

      '''
      # times == [-- dip] cons [swap] infra [0 >] swap while pop
      (quote, (n, stack)) = stack
      if n <= 0:
        return stack, expression, dictionary
      n -= 1
      if n:
        expression = n, (quote, (S_times, expression))
      expression = concat(quote, expression)
      return stack, expression, dictionary

    # The current definition above works like this:

    #             [P] [Q] while
    # --------------------------------------
    #    [P] nullary [Q [P] nullary] loop

    #   while == [pop i not] [popop] [dudipd] primrec

    #def while_(S, expression, dictionary):
    #  '''[if] [body] while'''
    #  (body, (if_, stack)) = S
    #  while joy(stack, if_, dictionary)[0][0]:
    #    stack = joy(stack, body, dictionary)[0]
    #  return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def loop(stack, expression, dictionary):
      '''
      Basic loop combinator.
      ::

           ... True [Q] loop
        -----------------------
              ... Q [Q] loop

           ... False [Q] loop
        ------------------------
                  ...

      '''
      quote, (flag, stack) = stack
      if flag:
        expression = concat(quote, (quote, (S_loop, expression)))
      return stack, expression, dictionary

    @inscribe
    @FunctionWrapper
    def cmp_(stack, expression, dictionary):
      '''
      cmp takes two values and three quoted programs on the stack and runs
      one of the three depending on the results of comparing the two values:
      ::

               a b [G] [E] [L] cmp
            ------------------------- a > b
                    G

               a b [G] [E] [L] cmp
            ------------------------- a = b
                        E

               a b [G] [E] [L] cmp
            ------------------------- a < b
                            L
      '''
      L, (E, (G, (b, (a, stack)))) = stack
      expression = concat(G if a > b else L if a < b else E, expression)
      return stack, expression, dictionary

    #  FunctionWrapper(cleave),
    #  FunctionWrapper(while_),

    for F in (
      BinaryBuiltinWrapper(operator.add),
      BinaryBuiltinWrapper(operator.and_),
      BinaryBuiltinWrapper(operator.div),
      BinaryBuiltinWrapper(operator.eq),
      BinaryBuiltinWrapper(operator.floordiv),
      BinaryBuiltinWrapper(operator.ge),
      BinaryBuiltinWrapper(operator.gt),
      BinaryBuiltinWrapper(operator.le),
      BinaryBuiltinWrapper(operator.lshift),
      BinaryBuiltinWrapper(operator.lt),
      BinaryBuiltinWrapper(operator.mod),
      BinaryBuiltinWrapper(operator.mul),
      BinaryBuiltinWrapper(operator.ne),
      BinaryBuiltinWrapper(operator.or_),
      BinaryBuiltinWrapper(operator.pow),
      BinaryBuiltinWrapper(operator.rshift),
      BinaryBuiltinWrapper(operator.sub),
      BinaryBuiltinWrapper(operator.truediv),
      BinaryBuiltinWrapper(operator.xor),

      UnaryBuiltinWrapper(abs),
      UnaryBuiltinWrapper(bool),
      UnaryBuiltinWrapper(floor),
      UnaryBuiltinWrapper(operator.neg),
      UnaryBuiltinWrapper(operator.not_),
      UnaryBuiltinWrapper(sqrt),
      ):
      inscribe(F)
    del F  # Otherwise Sphinx autodoc will pick it up.

    add_aliases(_dictionary, ALIASES)

    DefinitionWrapper.add_definitions(definitions, _dictionary)