"itertools" --- Functions creating iterators for efficient looping
******************************************************************

======================================================================

Este módulo implementa un número de piezas básicas *iterator*
inspiradas en *constructs* de APL, Haskell y SML.  Cada pieza ha sido
reconvertida a una forma apropiada para Python.

El módulo estandariza un conjunto base de herramientas rápidas y
eficientes en memoria, útiles por sí mismas o en combinación con
otras.  Juntas, forman un "álgebra de iteradores", haciendo posible la
construcción de herramientas especializadas, sucintas y eficientes, en
Python puro.

Por ejemplo, SML provee una herramienta de tabulación "tabulate(f)",
que produce una secuencia "f(0), f(1), ...".  En Python, se puede
lograr el mismo efecto al combinar "map()" y "count()" para formar
"map(f, count())".

**General iterators:**

+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| Iterador                     | Argumentos                   | Resultados                                        | Ejemplo                                                       |
|==============================|==============================|===================================================|===============================================================|
| "accumulate()"               | p [,func]                    | p0, p0+p1, p0+p1+p2, ...                          | "accumulate([1,2,3,4,5]) → 1 3 6 10 15"                       |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "batched()"                  | p, n                         | (p0, p1, ..., p_n-1), ...                         | "batched('ABCDEFG', n=3) → ABC DEF G"                         |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "chain()"                    | p, q, ...                    | p0, p1, ... plast, q0, q1, ...                    | "chain('ABC', 'DEF') → A B C D E F"                           |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "chain.from_iterable()"      | iterable                     | p0, p1, ... plast, q0, q1, ...                    | "chain.from_iterable(['ABC', 'DEF']) → A B C D E F"           |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "compress()"                 | data, selectors              | (d[0] if s[0]), (d[1] if s[1]), ...               | "compress('ABCDEF', [1,0,1,0,1,1]) → A C E F"                 |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "count()"                    | [start[, step]]              | start, start+step, start+2*step, ...              | "count(10) → 10 11 12 13 14 ..."                              |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "cycle()"                    | p                            | p0, p1, ... plast, p0, p1, ...                    | "cycle('ABCD') → A B C D A B C D ..."                         |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "dropwhile()"                | predicate, seq               | seq[n], seq[n+1], starting when predicate fails   | "dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8"               |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "filterfalse()"              | predicate, seq               | elements of seq where predicate(elem) fails       | "filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8"               |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "groupby()"                  | iterable[, key]              | sub-iteradores agrupados según el valor de key(v) | "groupby(['A','B','DEF'], len) → (1, A B) (3, DEF)"           |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "islice()"                   | seq, [start,] stop [, step]  | elementos de seq[start:stop:step]                 | "islice('ABCDEFG', 2, None) → C D E F G"                      |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "pairwise()"                 | iterable                     | (p[0], p[1]), (p[1], p[2])                        | "pairwise('ABCDEFG') → AB BC CD DE EF FG"                     |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "repeat()"                   | elem [,n]                    | elem, elem, elem, ... indefinidamente o hasta n   | "repeat(10, 3) → 10 10 10"                                    |
|                              |                              | veces                                             |                                                               |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "starmap()"                  | func, seq                    | func(*seq[0]), func(*seq[1]), ...                 | "starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000"            |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "takewhile()"                | predicate, seq               | seq[0], seq[1], until predicate fails             | "takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4"                 |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "tee()"                      | it, n                        | it1, it2, ... itn  divide un iterador en n        | "tee('ABC', 2) → A B C, A B C"                                |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+
| "zip_longest()"              | p, q, ...                    | (p[0], q[0]), (p[1], q[1]), ...                   | "zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-"      |
+------------------------------+------------------------------+---------------------------------------------------+---------------------------------------------------------------+

**Iteradores combinatorios:**

+------------------------------------------------+----------------------+---------------------------------------------------------------+
| Iterador                                       | Argumentos           | Resultados                                                    |
|================================================|======================|===============================================================|
| "product()"                                    | p, q, ... [repeat=1] | producto cartesiano, equivalente a un bucle *for* anidado     |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "permutations()"                               | p[, r]               | tuplas de longitud r, en todas los órdenes posibles, sin      |
|                                                |                      | elementos repetidos                                           |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "combinations()"                               | p, r                 | tuplas de longitud r, ordenadas, sin elementos repetidos      |
+------------------------------------------------+----------------------+---------------------------------------------------------------+
| "combinations_with_replacement()"              | p, r                 | tuplas de longitud r, ordenadas, con elementos repetidos      |
+------------------------------------------------+----------------------+---------------------------------------------------------------+

+------------------------------------------------+---------------------------------------------------------------+
| Ejemplos                                       | Resultados                                                    |
|================================================|===============================================================|
| "product('ABCD', repeat=2)"                    | "AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD"             |
+------------------------------------------------+---------------------------------------------------------------+
| "permutations('ABCD', 2)"                      | "AB AC AD BA BC BD CA CB CD DA DB DC"                         |
+------------------------------------------------+---------------------------------------------------------------+
| "combinations('ABCD', 2)"                      | "AB AC AD BC BD CD"                                           |
+------------------------------------------------+---------------------------------------------------------------+
| "combinations_with_replacement('ABCD', 2)"     | "AA AB AC AD BB BC BD CC CD DD"                               |
+------------------------------------------------+---------------------------------------------------------------+


Itertool Functions
==================

The following functions all construct and return iterators. Some
provide streams of infinite length, so they should only be accessed by
functions or loops that truncate the stream.

itertools.accumulate(iterable[, function, *, initial=None])

   Make an iterator that returns accumulated sums or accumulated
   results from other binary functions.

   The *function* defaults to addition.  The *function* should accept
   two arguments, an accumulated total and a value from the
   *iterable*.

   If an *initial* value is provided, the accumulation will start with
   that value and the output will have one more element than the input
   iterable.

   Aproximadamente equivalente a:

      def accumulate(iterable, function=operator.add, *, initial=None):
          'Return running totals'
          # accumulate([1,2,3,4,5]) → 1 3 6 10 15
          # accumulate([1,2,3,4,5], initial=100) → 100 101 103 106 110 115
          # accumulate([1,2,3,4,5], operator.mul) → 1 2 6 24 120

          iterator = iter(iterable)
          total = initial
          if initial is None:
              try:
                  total = next(iterator)
              except StopIteration:
                  return

          yield total
          for element in iterator:
              total = function(total, element)
              yield total

   To compute a running minimum, set *function* to "min()". For a
   running maximum, set *function* to "max()". Or for a running
   product, set *function* to "operator.mul()". To build an
   amortization table, accumulate the interest and apply payments:

      >>> data = [3, 4, 6, 2, 1, 9, 0, 7, 5, 8]
      >>> list(accumulate(data, max))              # running maximum
      [3, 4, 6, 6, 6, 9, 9, 9, 9, 9]
      >>> list(accumulate(data, operator.mul))     # running product
      [3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]

      # Amortize a 5% loan of 1000 with 10 annual payments of 90
      >>> update = lambda balance, payment: round(balance * 1.05) - payment
      >>> list(accumulate(repeat(90, 10), update, initial=1_000))
      [1000, 960, 918, 874, 828, 779, 728, 674, 618, 559, 497]

   Para una función similar que retorne únicamente el valor final
   acumulado, revisa "functools.reduce()".

   Added in version 3.2.

   Distinto en la versión 3.3: Added the optional *function*
   parameter.

   Distinto en la versión 3.8: Adicionó el argumento opcional
   *initial*.

itertools.batched(iterable, n, *, strict=False)

   Batch data from the *iterable* into tuples of length *n*. The last
   batch may be shorter than *n*.

   If *strict* is true, will raise a "ValueError" if the final batch
   is shorter than *n*.

   Loops over the input iterable and accumulates data into tuples up
   to size *n*.  The input is consumed lazily, just enough to fill a
   batch. The result is yielded as soon as the batch is full or when
   the input iterable is exhausted:

      >>> flattened_data = ['roses', 'red', 'violets', 'blue', 'sugar', 'sweet']
      >>> unflattened = list(batched(flattened_data, 2))
      >>> unflattened
      [('roses', 'red'), ('violets', 'blue'), ('sugar', 'sweet')]

   Aproximadamente equivalente a:

      def batched(iterable, n, *, strict=False):
          # batched('ABCDEFG', 3) → ABC DEF G
          if n < 1:
              raise ValueError('n must be at least one')
          iterator = iter(iterable)
          while batch := tuple(islice(iterator, n)):
              if strict and len(batch) != n:
                  raise ValueError('batched(): incomplete batch')
              yield batch

   Added in version 3.12.

   Distinto en la versión 3.13: Added the *strict* option.

itertools.chain(*iterables)

   Make an iterator that returns elements from the first iterable
   until it is exhausted, then proceeds to the next iterable, until
   all of the iterables are exhausted.  This combines multiple data
   sources into a single iterator.  Roughly equivalent to:

      def chain(*iterables):
          # chain('ABC', 'DEF') → A B C D E F
          for iterable in iterables:
              yield from iterable

   Note that unpacking in comprehensions provides similar
   functionality so that "list(chain(p, q))" could be written as "[*s
   for s in (p, q)]".

classmethod chain.from_iterable(iterable)

   Constructor alternativo para "chain()".  Obtiene entradas enlazadas
   de un mismo argumento que se evalúa perezosamente.  Aproximadamente
   equivalente a:

      def from_iterable(iterables):
          # chain.from_iterable(['ABC', 'DEF']) → A B C D E F
          for iterable in iterables:
              yield from iterable

   Note that unpacking in comprehensions provides similar
   functionality so that "list(chain.from_iterable(iterables))" could
   be written as "[*s for s in iterables]".

itertools.combinations(iterable, r)

   Retorna subsecuencias de longitud *r* con elementos del *iterable*
   de entrada.

   The output is a subsequence of "product()" keeping only entries
   that are subsequences of the *iterable*.  The length of the output
   is given by "math.comb()" which computes "n! / r! / (n - r)!" when
   "0 ≤ r ≤ n" or zero when "r > n".

   The combination tuples are emitted in lexicographic order according
   to the order of the input *iterable*. If the input *iterable* is
   sorted, the output tuples will be produced in sorted order.

   Elements are treated as unique based on their position, not on
   their value.  If the input elements are unique, there will be no
   repeated values within each combination.

   Aproximadamente equivalente a:

      def combinations(iterable, r):
          # combinations('ABCD', 2) → AB AC AD BC BD CD
          # combinations(range(4), 3) → 012 013 023 123

          pool = tuple(iterable)
          n = len(pool)
          if r > n:
              return
          indices = list(range(r))

          yield tuple(pool[i] for i in indices)
          while True:
              for i in reversed(range(r)):
                  if indices[i] != i + n - r:
                      break
              else:
                  return
              indices[i] += 1
              for j in range(i+1, r):
                  indices[j] = indices[j-1] + 1
              yield tuple(pool[i] for i in indices)

itertools.combinations_with_replacement(iterable, r)

   Retorna subsecuencias, de longitud *r*, con elementos del
   *iterable* de entrada, permitiendo que haya elementos individuales
   repetidos más de una vez.

   The output is a subsequence of "product()" that keeps only entries
   that are subsequences (with possible repeated elements) of the
   *iterable*.  The number of subsequence returned is "(n + r - 1)! /
   r! / (n - 1)!" when "n > 0".

   The combination tuples are emitted in lexicographic order according
   to the order of the input *iterable*. if the input *iterable* is
   sorted, the output tuples will be produced in sorted order.

   Elements are treated as unique based on their position, not on
   their value.  If the input elements are unique, the generated
   combinations will also be unique.

   Aproximadamente equivalente a:

      def combinations_with_replacement(iterable, r):
          # combinations_with_replacement('ABC', 2) → AA AB AC BB BC CC

          pool = tuple(iterable)
          n = len(pool)
          if not n and r:
              return
          indices = [0] * r

          yield tuple(pool[i] for i in indices)
          while True:
              for i in reversed(range(r)):
                  if indices[i] != n - 1:
                      break
              else:
                  return
              indices[i:] = [indices[i] + 1] * (r - i)
              yield tuple(pool[i] for i in indices)

   Added in version 3.1.

itertools.compress(data, selectors)

   Make an iterator that returns elements from *data* where the
   corresponding element in *selectors* is true.  Stops when either
   the *data* or *selectors* iterables have been exhausted.  Roughly
   equivalent to:

      def compress(data, selectors):
          # compress('ABCDEF', [1,0,1,0,1,1]) → A C E F
          return (datum for datum, selector in zip(data, selectors) if selector)

   Added in version 3.1.

itertools.count(start=0, step=1)

   Make an iterator that returns evenly spaced values beginning with
   *start*. Can be used with "map()" to generate consecutive data
   points or with "zip()" to add sequence numbers.  Roughly equivalent
   to:

      def count(start=0, step=1):
          # count(10) → 10 11 12 13 14 ...
          # count(2.5, 0.5) → 2.5 3.0 3.5 ...
          n = start
          while True:
              yield n
              n += step

   When counting with floating-point numbers, better accuracy can
   sometimes be achieved by substituting multiplicative code such as:
   "(start + step * i for i in count())".

   Distinto en la versión 3.1: Se adicionó el argumento *step* y se
   permitieron argumentos diferentes a enteros.

itertools.cycle(iterable)

   Make an iterator returning elements from the *iterable* and saving
   a copy of each.  When the iterable is exhausted, return elements
   from the saved copy.  Repeats indefinitely.  Roughly equivalent to:

      def cycle(iterable):
          # cycle('ABCD') → A B C D A B C D A B C D ...

          saved = []
          for element in iterable:
              yield element
              saved.append(element)

          while saved:
              for element in saved:
                  yield element

   This itertool may require significant auxiliary storage (depending
   on the length of the iterable).

itertools.dropwhile(predicate, iterable)

   Make an iterator that drops elements from the *iterable* while the
   *predicate* is true and afterwards returns every element.  Roughly
   equivalent to:

      def dropwhile(predicate, iterable):
          # dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8

          iterator = iter(iterable)
          for x in iterator:
              if not predicate(x):
                  yield x
                  break

          for x in iterator:
              yield x

   Note this does not produce *any* output until the predicate first
   becomes false, so this itertool may have a lengthy start-up time.

itertools.filterfalse(predicate, iterable)

   Make an iterator that filters elements from the *iterable*
   returning only those for which the *predicate* returns a false
   value.  If *predicate* is "None", returns the items that are false.
   Roughly equivalent to:

      def filterfalse(predicate, iterable):
          # filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8

          if predicate is None:
              predicate = bool

          for x in iterable:
              if not predicate(x):
                  yield x

itertools.groupby(iterable, key=None)

   Crea un iterador que retorna claves consecutivas y grupos del
   *iterable*. *key* es una función que calcula un valor clave para
   cada elemento.  Si no se especifica o es "None", *key* es una
   función de identidad por defecto y retorna el elemento sin cambios.
   Generalmente, el iterable necesita estar ordenado con la misma
   función *key*.

   El funcionamiento de "groupby()" es similar al del filtro "uniq" en
   Unix.  Genera un salto o un nuevo grupo cada vez que el valor de la
   función clave cambia (por lo que usualmente es necesario ordenar
   los datos usando la misma función clave).  Ese comportamiento
   difiere del de GROUP BY de SQL, el cual agrega elementos comunes
   sin importar el orden de entrada.

   El grupo retornado es un iterador mismo que comparte el iterable
   subyacente con "groupby()".  Al compartir la fuente, cuando el
   objeto "groupby()" se avanza, el grupo previo deja de ser visible.
   En ese caso, si los datos se necesitan posteriormente, se deberían
   almacenar como lista:

      groups = []
      uniquekeys = []
      data = sorted(data, key=keyfunc)
      for k, g in groupby(data, keyfunc):
          groups.append(list(g))      # Store group iterator as a list
          uniquekeys.append(k)

   "groupby()" es aproximadamente equivalente a:

      def groupby(iterable, key=None):
          # [k for k, g in groupby('AAAABBBCCDAABBB')] → A B C D A B
          # [list(g) for k, g in groupby('AAAABBBCCD')] → AAAA BBB CC D

          keyfunc = (lambda x: x) if key is None else key
          iterator = iter(iterable)
          exhausted = False

          def _grouper(target_key):
              nonlocal curr_value, curr_key, exhausted
              yield curr_value
              for curr_value in iterator:
                  curr_key = keyfunc(curr_value)
                  if curr_key != target_key:
                      return
                  yield curr_value
              exhausted = True

          try:
              curr_value = next(iterator)
          except StopIteration:
              return
          curr_key = keyfunc(curr_value)

          while not exhausted:
              target_key = curr_key
              curr_group = _grouper(target_key)
              yield curr_key, curr_group
              if curr_key == target_key:
                  for _ in curr_group:
                      pass

itertools.islice(iterable, stop)
itertools.islice(iterable, start, stop[, step])

   Make an iterator that returns selected elements from the iterable.
   Works like sequence slicing but does not support negative values
   for *start*, *stop*, or *step*.

   If *start* is zero or "None", iteration starts at zero.  Otherwise,
   elements from the iterable are skipped until *start* is reached.

   If *stop* is "None", iteration continues until the input is
   exhausted, if at all.  Otherwise, it stops at the specified
   position.

   If *step* is "None", the step defaults to one.  Elements are
   returned consecutively unless *step* is set higher than one which
   results in items being skipped.

   Aproximadamente equivalente a:

      def islice(iterable, *args):
          # islice('ABCDEFG', 2) → A B
          # islice('ABCDEFG', 2, 4) → C D
          # islice('ABCDEFG', 2, None) → C D E F G
          # islice('ABCDEFG', 0, None, 2) → A C E G

          s = slice(*args)
          start = 0 if s.start is None else s.start
          stop = s.stop
          step = 1 if s.step is None else s.step
          if start < 0 or (stop is not None and stop < 0) or step <= 0:
              raise ValueError

          indices = count() if stop is None else range(max(start, stop))
          next_i = start
          for i, element in zip(indices, iterable):
              if i == next_i:
                  yield element
                  next_i += step

   If the input is an iterator, then fully consuming the *islice*
   advances the input iterator by "max(start, stop)" steps regardless
   of the *step* value.

itertools.pairwise(iterable)

   Retorna los sucesivos pares superpuestos tomados de la entrada
   *iterable*.

   El número de tuplas de 2 elementos en el iterador de salida será
   uno menos que el número de entradas. Estará vacío si la entrada
   iterable tiene menos de dos valores.

   Aproximadamente equivalente a:

      def pairwise(iterable):
          # pairwise('ABCDEFG') → AB BC CD DE EF FG

          iterator = iter(iterable)
          a = next(iterator, None)

          for b in iterator:
              yield a, b
              a = b

   Added in version 3.10.

itertools.permutations(iterable, r=None)

   Return successive *r* length permutations of elements from the
   *iterable*.

   Si *r* no es especificado o si es "None", entonces por defecto *r*
   será igual a la longitud de *iterable* y todas las permutaciones de
   máxima longitud serán generadas.

   The output is a subsequence of "product()" where entries with
   repeated elements have been filtered out.  The length of the output
   is given by "math.perm()" which computes "n! / (n - r)!" when "0 ≤
   r ≤ n" or zero when "r > n".

   The permutation tuples are emitted in lexicographic order according
   to the order of the input *iterable*.  If the input *iterable* is
   sorted, the output tuples will be produced in sorted order.

   Elements are treated as unique based on their position, not on
   their value.  If the input elements are unique, there will be no
   repeated values within a permutation.

   Aproximadamente equivalente a:

      def permutations(iterable, r=None):
          # permutations('ABCD', 2) → AB AC AD BA BC BD CA CB CD DA DB DC
          # permutations(range(3)) → 012 021 102 120 201 210

          pool = tuple(iterable)
          n = len(pool)
          r = n if r is None else r
          if r > n:
              return

          indices = list(range(n))
          cycles = list(range(n, n-r, -1))
          yield tuple(pool[i] for i in indices[:r])

          while n:
              for i in reversed(range(r)):
                  cycles[i] -= 1
                  if cycles[i] == 0:
                      indices[i:] = indices[i+1:] + indices[i:i+1]
                      cycles[i] = n - i
                  else:
                      j = cycles[i]
                      indices[i], indices[-j] = indices[-j], indices[i]
                      yield tuple(pool[i] for i in indices[:r])
                      break
              else:
                  return

itertools.product(*iterables, repeat=1)

   Cartesian product of the input iterables.

   Aproximadamente equivalente a tener bucles *for* anidados en un
   generador. Por ejemplo, "product(A, B)" es equivalente a "((x,y)
   for x in A for y in B)".

   Los bucles anidados hacen ciclos como un cuentapasos o taxímetro,
   con el elemento más hacia la derecha avanzando en cada iteración.
   Este patrón crea un orden lexicográfico en el que, si los iterables
   de entrada están ordenados, las tuplas producidas son emitidas de
   manera ordenada.

   Para calcular el producto de un iterable consigo mismo, especifica
   el número de repeticiones con el argumento opcional *repeat*.  Por
   ejemplo, "product(A, repeat=4)" es equivalente a "product(A, A, A,
   A)".

   Esta función es aproximadamente equivalente al código siguiente,
   exceptuando que la implementación real no acumula resultados
   intermedios en memoria:

      def product(*iterables, repeat=1):
          # product('ABCD', 'xy') → Ax Ay Bx By Cx Cy Dx Dy
          # product(range(2), repeat=3) → 000 001 010 011 100 101 110 111

          if repeat < 0:
              raise ValueError('repeat argument cannot be negative')
          pools = [tuple(pool) for pool in iterables] * repeat

          result = [[]]
          for pool in pools:
              result = [x+[y] for x in result for y in pool]

          for prod in result:
              yield tuple(prod)

   Antes de que "product()" se ejecute, consume completamente los
   iterables de entrada, manteniendo grupos de valores en la memoria
   para generar los productos. En consecuencia, solo es útil con
   entradas finitas.

itertools.repeat(object[, times])

   Crea un iterador que retorna *object* una y otra vez. Se ejecuta
   indefinidamente a menos que se especifique el argumento *times*.

   Aproximadamente equivalente a:

      def repeat(object, times=None):
          # repeat(10, 3) → 10 10 10
          if times is None:
              while True:
                  yield object
          else:
              for i in range(times):
                  yield object

   Un uso común de *repeat* es proporcionar un flujo de valores
   constantes a *map* o *zip*:

      >>> list(map(pow, range(10), repeat(2)))
      [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

itertools.starmap(function, iterable)

   Make an iterator that computes the *function* using arguments
   obtained from the *iterable*.  Used instead of "map()" when
   argument parameters have already been "pre-zipped" into tuples.

   La diferencia entre "map()" y "starmap()" es paralela a la
   distinción entre "function(a,b)" y "function(*c)". Aproximadamente
   equivalente a:

      def starmap(function, iterable):
          # starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000
          for args in iterable:
              yield function(*args)

itertools.takewhile(predicate, iterable)

   Make an iterator that returns elements from the *iterable* as long
   as the *predicate* is true.  Roughly equivalent to:

      def takewhile(predicate, iterable):
          # takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4
          for x in iterable:
              if not predicate(x):
                  break
              yield x

   Note, the element that first fails the predicate condition is
   consumed from the input iterator and there is no way to access it.
   This could be an issue if an application wants to further consume
   the input iterator after *takewhile* has been run to exhaustion.
   To work around this problem, consider using more-itertools
   before_and_after() instead.

itertools.tee(iterable, n=2)

   Retorna *n* iteradores independientes de un mismo iterador.

   Aproximadamente equivalente a:

      def tee(iterable, n=2):
          if n < 0:
              raise ValueError
          if n == 0:
              return ()
          iterator = _tee(iterable)
          result = [iterator]
          for _ in range(n - 1):
              result.append(_tee(iterator))
          return tuple(result)

      class _tee:

          def __init__(self, iterable):
              it = iter(iterable)
              if isinstance(it, _tee):
                  self.iterator = it.iterator
                  self.link = it.link
              else:
                  self.iterator = it
                  self.link = [None, None]

          def __iter__(self):
              return self

          def __next__(self):
              link = self.link
              if link[1] is None:
                  link[0] = next(self.iterator)
                  link[1] = [None, None]
              value, self.link = link
              return value

   When the input *iterable* is already a tee iterator object, all
   members of the return tuple are constructed as if they had been
   produced by the upstream "tee()" call.  This "flattening step"
   allows nested "tee()" calls to share the same underlying data chain
   and to have a single update step rather than a chain of calls.

   The flattening property makes tee iterators efficiently peekable:

      def lookahead(tee_iterator):
           "Return the next value without moving the input forward"
           [forked_iterator] = tee(tee_iterator, 1)
           return next(forked_iterator)

      >>> iterator = iter('abcdef')
      >>> [iterator] = tee(iterator, 1)   # Make the input peekable
      >>> next(iterator)                  # Move the iterator forward
      'a'
      >>> lookahead(iterator)             # Check next value
      'b'
      >>> next(iterator)                  # Continue moving forward
      'b'

   "tee" iterators are not threadsafe. A "RuntimeError" may be raised
   when simultaneously using iterators returned by the same "tee()"
   call, even if the original *iterable* is threadsafe.

   Esta herramienta de iteración puede requerir almacenamiento
   auxiliar significativo (dependiendo de qué tantos datos necesitan
   ser almacenados). En general, si un iterador utiliza todos o la
   mayoría de los datos antes que otro iterador comience, es más
   rápido utilizar "list()" en vez de "tee()".

itertools.zip_longest(*iterables, fillvalue=None)

   Make an iterator that aggregates elements from each of the
   *iterables*.

   If the iterables are of uneven length, missing values are filled-in
   with *fillvalue*.  If not specified, *fillvalue* defaults to
   "None".

   Iteration continues until the longest iterable is exhausted.

   Aproximadamente equivalente a:

      def zip_longest(*iterables, fillvalue=None):
          # zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-

          iterators = list(map(iter, iterables))
          num_active = len(iterators)
          if not num_active:
              return

          while True:
              values = []
              for i, iterator in enumerate(iterators):
                  try:
                      value = next(iterator)
                  except StopIteration:
                      num_active -= 1
                      if not num_active:
                          return
                      iterators[i] = repeat(fillvalue)
                      value = fillvalue
                  values.append(value)
              yield tuple(values)

   If one of the iterables is potentially infinite, then the
   "zip_longest()" function should be wrapped with something that
   limits the number of calls (for example "islice()" or
   "takewhile()").


Fórmulas con itertools
======================

Esta sección muestra fórmulas para crear un conjunto de herramientas
extendido usando las herramientas de itertools como piezas básicas.

The primary purpose of the itertools recipes is educational.  The
recipes show various ways of thinking about individual tools — for
example, that "chain.from_iterable" is related to the concept of
flattening.  The recipes also give ideas about ways that the tools can
be combined — for example, how "starmap()" and "repeat()" can work
together.  The recipes also show patterns for using itertools with the
"operator" and "collections" modules as well as with the built-in
itertools such as "map()", "filter()", "reversed()", and
"enumerate()".

A secondary purpose of the recipes is to serve as an incubator.  The
"accumulate()", "compress()", and "pairwise()" itertools started out
as recipes.  Currently, the "sliding_window()", "derangements()", and
"sieve()" recipes are being tested to see whether they prove their
worth.

Substantially all of these recipes and many, many others can be
installed from the more-itertools project found on the Python Package
Index:

   python -m pip install more-itertools

Many of the recipes offer the same high performance as the underlying
toolset. Superior memory performance is kept by processing elements
one at a time rather than bringing the whole iterable into memory all
at once. Code volume is kept small by linking the tools together in a
functional style.  High speed is retained by preferring "vectorized"
building blocks over the use of for-loops and *generators* which incur
interpreter overhead.

   from itertools import (accumulate, batched, chain, combinations, compress,
        count, cycle, filterfalse, groupby, islice, permutations, product,
        repeat, starmap, tee, zip_longest)
   from collections import Counter, deque
   from contextlib import suppress
   from functools import reduce
   from heapq import heappush, heappushpop, heappush_max, heappushpop_max
   from math import comb, isqrt, prod, sumprod
   from operator import getitem, is_not, itemgetter, mul, neg, truediv


   # ==== Basic one liners ====

   def take(n, iterable):
       "Return first n items of the iterable as a list."
       return list(islice(iterable, n))

   def prepend(value, iterable):
       "Prepend a single value in front of an iterable."
       # prepend(1, [2, 3, 4]) → 1 2 3 4
       return chain([value], iterable)

   def repeatfunc(function, times=None, *args):
       "Repeat calls to a function with specified arguments."
       if times is None:
           return starmap(function, repeat(args))
       return starmap(function, repeat(args, times))

   def flatten(list_of_lists):
       "Flatten one level of nesting."
       return chain.from_iterable(list_of_lists)

   def ncycles(iterable, n):
       "Returns the sequence elements n times."
       return chain.from_iterable(repeat(tuple(iterable), n))

   def loops(n):
       "Loop n times. Like range(n) but without creating integers."
       # for _ in loops(100): ...
       return repeat(None, n)

   def tail(n, iterable):
       "Return an iterator over the last n items."
       # tail(3, 'ABCDEFG') → E F G
       return iter(deque(iterable, maxlen=n))

   def consume(iterator, n=None):
       "Advance the iterator n-steps ahead. If n is None, consume entirely."
       # Use functions that consume iterators at C speed.
       if n is None:
           deque(iterator, maxlen=0)
       else:
           next(islice(iterator, n, n), None)

   def nth(iterable, n, default=None):
       "Returns the nth item or a default value."
       return next(islice(iterable, n, None), default)

   def quantify(iterable, predicate=bool):
       "Given a predicate that returns True or False, count the True results."
       return sum(map(predicate, iterable))

   def first_true(iterable, default=False, predicate=None):
       "Returns the first true value or the *default* if there is no true value."
       # first_true([a, b, c], x) → a or b or c or x
       # first_true([a, b], x, f) → a if f(a) else b if f(b) else x
       return next(filter(predicate, iterable), default)

   def all_equal(iterable, key=None):
       "Returns True if all the elements are equal to each other."
       # all_equal('4٤௪౪໔', key=int) → True
       return len(take(2, groupby(iterable, key))) <= 1


   # ==== Data pipelines ====

   def unique_justseen(iterable, key=None):
       "Yield unique elements, preserving order. Remember only the element just seen."
       # unique_justseen('AAAABBBCCDAABBB') → A B C D A B
       # unique_justseen('ABBcCAD', str.casefold) → A B c A D
       if key is None:
           return map(itemgetter(0), groupby(iterable))
       return map(next, map(itemgetter(1), groupby(iterable, key)))

   def unique_everseen(iterable, key=None):
       "Yield unique elements, preserving order. Remember all elements ever seen."
       # unique_everseen('AAAABBBCCDAABBB') → A B C D
       # unique_everseen('ABBcCAD', str.casefold) → A B c D
       seen = set()
       if key is None:
           for element in filterfalse(seen.__contains__, iterable):
               seen.add(element)
               yield element
       else:
           for element in iterable:
               k = key(element)
               if k not in seen:
                   seen.add(k)
                   yield element

   def unique(iterable, key=None, reverse=False):
       "Yield unique elements in sorted order. Supports unhashable inputs."
       # unique([[1, 2], [3, 4], [1, 2]]) → [1, 2] [3, 4]
       sequenced = sorted(iterable, key=key, reverse=reverse)
       return unique_justseen(sequenced, key=key)

   def sliding_window(iterable, n):
       "Collect data into overlapping fixed-length chunks or blocks."
       # sliding_window('ABCDEFG', 3) → ABC BCD CDE DEF EFG
       iterator = iter(iterable)
       window = deque(islice(iterator, n - 1), maxlen=n)
       for x in iterator:
           window.append(x)
           yield tuple(window)

   def grouper(iterable, n, *, incomplete='fill', fillvalue=None):
       "Collect data into non-overlapping fixed-length chunks or blocks."
       # grouper('ABCDEFG', 3, fillvalue='x')       → ABC DEF Gxx
       # grouper('ABCDEFG', 3, incomplete='strict') → ABC DEF ValueError
       # grouper('ABCDEFG', 3, incomplete='ignore') → ABC DEF
       iterators = [iter(iterable)] * n
       match incomplete:
           case 'fill':
               return zip_longest(*iterators, fillvalue=fillvalue)
           case 'strict':
               return zip(*iterators, strict=True)
           case 'ignore':
               return zip(*iterators)
           case _:
               raise ValueError('Expected fill, strict, or ignore')

   def roundrobin(*iterables):
       "Visit input iterables in a cycle until each is exhausted."
       # roundrobin('ABC', 'D', 'EF') → A D E B F C
       # Algorithm credited to George Sakkis
       iterators = map(iter, iterables)
       for num_active in range(len(iterables), 0, -1):
           iterators = cycle(islice(iterators, num_active))
           yield from map(next, iterators)

   def subslices(seq):
       "Return all contiguous non-empty subslices of a sequence."
       # subslices('ABCD') → A AB ABC ABCD B BC BCD C CD D
       slices = starmap(slice, combinations(range(len(seq) + 1), 2))
       return map(getitem, repeat(seq), slices)

   def derangements(iterable, r=None):
       "Produce r length permutations without fixed points."
       # derangements('ABCD') → BADC BCDA BDAC CADB CDAB CDBA DABC DCAB DCBA
       # Algorithm credited to Stefan Pochmann
       seq = tuple(iterable)
       pos = tuple(range(len(seq)))
       have_moved = map(map, repeat(is_not), repeat(pos), permutations(pos, r=r))
       valid_derangements = map(all, have_moved)
       return compress(permutations(seq, r=r), valid_derangements)

   def iter_index(iterable, value, start=0, stop=None):
       "Return indices where a value occurs in a sequence or iterable."
       # iter_index('AABCADEAF', 'A') → 0 1 4 7
       seq_index = getattr(iterable, 'index', None)
       if seq_index is None:
           iterator = islice(iterable, start, stop)
           for i, element in enumerate(iterator, start):
               if element is value or element == value:
                   yield i
       else:
           stop = len(iterable) if stop is None else stop
           i = start
           with suppress(ValueError):
               while True:
                   yield (i := seq_index(value, i, stop))
                   i += 1

   def iter_except(function, exception, first=None):
       "Convert a call-until-exception interface to an iterator interface."
       # iter_except(d.popitem, KeyError) → non-blocking dictionary iterator
       with suppress(exception):
           if first is not None:
               yield first()
           while True:
               yield function()


   # ==== Mathematical operations ====

   def multinomial(*counts):
       "Number of distinct arrangements of a multiset."
       # Counter('abracadabra').values() → 5 2 2 1 1
       # multinomial(5, 2, 2, 1, 1) → 83160
       return prod(map(comb, accumulate(counts), counts))

   def powerset(iterable):
       "Subsequences of the iterable from shortest to longest."
       # powerset([1,2,3]) → () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)
       s = list(iterable)
       return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

   def sum_of_squares(iterable):
       "Add up the squares of the input values."
       # sum_of_squares([10, 20, 30]) → 1400
       return sumprod(*tee(iterable))


   # ==== Matrix operations ====

   def reshape(matrix, columns):
       "Reshape a 2-D matrix to have a given number of columns."
       # reshape([(0, 1), (2, 3), (4, 5)], 3) →  (0, 1, 2) (3, 4, 5)
       return batched(chain.from_iterable(matrix), columns, strict=True)

   def transpose(matrix):
       "Swap the rows and columns of a 2-D matrix."
       # transpose([(1, 2, 3), (11, 22, 33)]) → (1, 11) (2, 22) (3, 33)
       return zip(*matrix, strict=True)

   def matmul(m1, m2):
       "Multiply two matrices."
       # matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]) → (49, 80) (41, 60)
       n = len(m2[0])
       return batched(starmap(sumprod, product(m1, transpose(m2))), n)


   # ==== Polynomial arithmetic ====

   def convolve(signal, kernel):
       """Discrete linear convolution of two iterables.
       Equivalent to polynomial multiplication.

       Convolutions are mathematically commutative; however, the inputs are
       evaluated differently.  The signal is consumed lazily and can be
       infinite. The kernel is fully consumed before the calculations begin.

       Article:  https://betterexplained.com/articles/intuitive-convolution/
       Video:    https://www.youtube.com/watch?v=KuXjwB4LzSA
       """
       # convolve([1, -1, -20], [1, -3]) → 1 -4 -17 60
       # convolve(data, [0.25, 0.25, 0.25, 0.25]) → Moving average (blur)
       # convolve(data, [1/2, 0, -1/2]) → 1st derivative estimate
       # convolve(data, [1, -2, 1]) → 2nd derivative estimate
       kernel = tuple(kernel)[::-1]
       n = len(kernel)
       padded_signal = chain(repeat(0, n-1), signal, repeat(0, n-1))
       windowed_signal = sliding_window(padded_signal, n)
       return map(sumprod, repeat(kernel), windowed_signal)

   def polynomial_from_roots(roots):
       """Compute a polynomial's coefficients from its roots.

          (x - 5) (x + 4) (x - 3)  expands to:   x³ -4x² -17x + 60
       """
       # polynomial_from_roots([5, -4, 3]) → [1, -4, -17, 60]
       factors = zip(repeat(1), map(neg, roots))
       return list(reduce(convolve, factors, [1]))

   def polynomial_eval(coefficients, x):
       """Evaluate a polynomial at a specific value.

       Computes with better numeric stability than Horner's method.
       """
       # Evaluate x³ -4x² -17x + 60 at x = 5
       # polynomial_eval([1, -4, -17, 60], x=5) → 0
       n = len(coefficients)
       if not n:
           return type(x)(0)
       powers = map(pow, repeat(x), reversed(range(n)))
       return sumprod(coefficients, powers)

   def polynomial_derivative(coefficients):
       """Compute the first derivative of a polynomial.

          f(x)  =  x³ -4x² -17x + 60
          f'(x) = 3x² -8x  -17
       """
       # polynomial_derivative([1, -4, -17, 60]) → [3, -8, -17]
       n = len(coefficients)
       powers = reversed(range(1, n))
       return list(map(mul, coefficients, powers))


   # ==== Number theory ====

   def sieve(n):
       "Primes less than n."
       # sieve(30) → 2 3 5 7 11 13 17 19 23 29
       if n > 2:
           yield 2
       data = bytearray((0, 1)) * (n // 2)
       for p in iter_index(data, 1, start=3, stop=isqrt(n) + 1):
           data[p*p : n : p+p] = bytes(len(range(p*p, n, p+p)))
       yield from iter_index(data, 1, start=3)

   def factor(n):
       "Prime factors of n."
       # factor(99) → 3 3 11
       # factor(1_000_000_000_000_007) → 47 59 360620266859
       # factor(1_000_000_000_000_403) → 1000000000000403
       for prime in sieve(isqrt(n) + 1):
           while not n % prime:
               yield prime
               n //= prime
               if n == 1:
                   return
       if n > 1:
           yield n

   def is_prime(n):
       "Return True if n is prime."
       # is_prime(1_000_000_000_000_403) → True
       return n > 1 and next(factor(n)) == n

   def totient(n):
       "Count of natural numbers up to n that are coprime to n."
       # https://mathworld.wolfram.com/TotientFunction.html
       # totient(12) → 4 because len([1, 5, 7, 11]) == 4
       for prime in set(factor(n)):
           n -= n // prime
       return n


   # ==== Running statistics ====

   def running_mean(iterable):
       "Average of values seen so far."
       # running_mean([37, 33, 38, 28]) → 37 35 36 34
       return map(truediv, accumulate(iterable), count(1))

   def running_min(iterable):
       "Smallest of values seen so far."
       # running_min([37, 33, 38, 28]) → 37 33 33 28
       return accumulate(iterable, func=min)

   def running_max(iterable):
       "Largest of values seen so far."
       # running_max([37, 33, 38, 28]) → 37 37 38 38
       return accumulate(iterable, func=max)

   def running_median(iterable):
       "Median of values seen so far."
       # running_median([37, 33, 38, 28]) → 37 35 37 35
       read = iter(iterable).__next__
       lo = []  # max-heap
       hi = []  # min-heap the same size as or one smaller than lo
       with suppress(StopIteration):
           while True:
               heappush_max(lo, heappushpop(hi, read()))
               yield lo[0]
               heappush(hi, heappushpop_max(lo, read()))
               yield (lo[0] + hi[0]) / 2

   def running_statistics(iterable):
       "Aggregate statistics for values seen so far."
       # Generate tuples:  (size, minimum, median, maximum, mean)
       t0, t1, t2, t3 = tee(iterable, 4)
       return zip(
           count(1),
           running_min(t0),
           running_median(t1),
           running_max(t2),
           running_mean(t3),
       )
