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Edit File : functools.py \| Size : 28.27 KB
"""functools.py - Tools for working with functions and callable objects """ # Python module wrapper for _functools C module # to allow utilities written in Python to be added # to the functools module. # Written by Nick Coghlan <ncoghlan at gmail.com>, # Raymond Hettinger <python at rcn.com>, # and Łukasz Langa <lukasz at langa.pl>. # Copyright (C) 2006-2013 Python Software Foundation. # See C source code for _functools credits/copyright __all__ = ['update_wrapper', 'wraps', 'WRAPPER_ASSIGNMENTS', 'WRAPPER_UPDATES', 'total_ordering', 'cmp_to_key', 'lru_cache', 'reduce', 'partial', 'partialmethod', 'singledispatch'] try: from _functools import reduce except ImportError: pass from abc import get_cache_token from collections import namedtuple from types import MappingProxyType from weakref import WeakKeyDictionary try: from _thread import RLock except ImportError: class RLock: 'Dummy reentrant lock for builds without threads' def __enter__(self): pass def __exit__(self, exctype, excinst, exctb): pass ################################################################################ ### update_wrapper() and wraps() decorator ################################################################################ # update_wrapper() and wraps() are tools to help write # wrapper functions that can handle naive introspection WRAPPER_ASSIGNMENTS = ('__module__', '__name__', '__qualname__', '__doc__', '__annotations__') WRAPPER_UPDATES = ('__dict__',) def update_wrapper(wrapper, wrapped, assigned = WRAPPER_ASSIGNMENTS, updated = WRAPPER_UPDATES): """Update a wrapper function to look like the wrapped function wrapper is the function to be updated wrapped is the original function assigned is a tuple naming the attributes assigned directly from the wrapped function to the wrapper function (defaults to functools.WRAPPER_ASSIGNMENTS) updated is a tuple naming the attributes of the wrapper that are updated with the corresponding attribute from the wrapped function (defaults to functools.WRAPPER_UPDATES) """ for attr in assigned: try: value = getattr(wrapped, attr) except AttributeError: pass else: setattr(wrapper, attr, value) for attr in updated: getattr(wrapper, attr).update(getattr(wrapped, attr, {})) # Issue #17482: set __wrapped__ last so we don't inadvertently copy it # from the wrapped function when updating __dict__ wrapper.__wrapped__ = wrapped # Return the wrapper so this can be used as a decorator via partial() return wrapper def wraps(wrapped, assigned = WRAPPER_ASSIGNMENTS, updated = WRAPPER_UPDATES): """Decorator factory to apply update_wrapper() to a wrapper function Returns a decorator that invokes update_wrapper() with the decorated function as the wrapper argument and the arguments to wraps() as the remaining arguments. Default arguments are as for update_wrapper(). This is a convenience function to simplify applying partial() to update_wrapper(). """ return partial(update_wrapper, wrapped=wrapped, assigned=assigned, updated=updated) ################################################################################ ### total_ordering class decorator ################################################################################ # The total ordering functions all invoke the root magic method directly # rather than using the corresponding operator. This avoids possible # infinite recursion that could occur when the operator dispatch logic # detects a NotImplemented result and then calls a reflected method. def _gt_from_lt(self, other, NotImplemented=NotImplemented): 'Return a > b. Computed by @total_ordering from (not a < b) and (a != b).' op_result = self.__lt__(other) if op_result is NotImplemented: return op_result return not op_result and self != other def _le_from_lt(self, other, NotImplemented=NotImplemented): 'Return a <= b. Computed by @total_ordering from (a < b) or (a == b).' op_result = self.__lt__(other) return op_result or self == other def _ge_from_lt(self, other, NotImplemented=NotImplemented): 'Return a >= b. Computed by @total_ordering from (not a < b).' op_result = self.__lt__(other) if op_result is NotImplemented: return op_result return not op_result def _ge_from_le(self, other, NotImplemented=NotImplemented): 'Return a >= b. Computed by @total_ordering from (not a <= b) or (a == b).' op_result = self.__le__(other) if op_result is NotImplemented: return op_result return not op_result or self == other def _lt_from_le(self, other, NotImplemented=NotImplemented): 'Return a < b. Computed by @total_ordering from (a <= b) and (a != b).' op_result = self.__le__(other) if op_result is NotImplemented: return op_result return op_result and self != other def _gt_from_le(self, other, NotImplemented=NotImplemented): 'Return a > b. Computed by @total_ordering from (not a <= b).' op_result = self.__le__(other) if op_result is NotImplemented: return op_result return not op_result def _lt_from_gt(self, other, NotImplemented=NotImplemented): 'Return a < b. Computed by @total_ordering from (not a > b) and (a != b).' op_result = self.__gt__(other) if op_result is NotImplemented: return op_result return not op_result and self != other def _ge_from_gt(self, other, NotImplemented=NotImplemented): 'Return a >= b. Computed by @total_ordering from (a > b) or (a == b).' op_result = self.__gt__(other) return op_result or self == other def _le_from_gt(self, other, NotImplemented=NotImplemented): 'Return a <= b. Computed by @total_ordering from (not a > b).' op_result = self.__gt__(other) if op_result is NotImplemented: return op_result return not op_result def _le_from_ge(self, other, NotImplemented=NotImplemented): 'Return a <= b. Computed by @total_ordering from (not a >= b) or (a == b).' op_result = self.__ge__(other) if op_result is NotImplemented: return op_result return not op_result or self == other def _gt_from_ge(self, other, NotImplemented=NotImplemented): 'Return a > b. Computed by @total_ordering from (a >= b) and (a != b).' op_result = self.__ge__(other) if op_result is NotImplemented: return op_result return op_result and self != other def _lt_from_ge(self, other, NotImplemented=NotImplemented): 'Return a < b. Computed by @total_ordering from (not a >= b).' op_result = self.__ge__(other) if op_result is NotImplemented: return op_result return not op_result _convert = { '__lt__': [('__gt__', _gt_from_lt), ('__le__', _le_from_lt), ('__ge__', _ge_from_lt)], '__le__': [('__ge__', _ge_from_le), ('__lt__', _lt_from_le), ('__gt__', _gt_from_le)], '__gt__': [('__lt__', _lt_from_gt), ('__ge__', _ge_from_gt), ('__le__', _le_from_gt)], '__ge__': [('__le__', _le_from_ge), ('__gt__', _gt_from_ge), ('__lt__', _lt_from_ge)] } def total_ordering(cls): """Class decorator that fills in missing ordering methods""" # Find user-defined comparisons (not those inherited from object). roots = [op for op in _convert if getattr(cls, op, None) is not getattr(object, op, None)] if not roots: raise ValueError('must define at least one ordering operation: < > <= >=') root = max(roots) # prefer __lt__ to __le__ to __gt__ to __ge__ for opname, opfunc in _convert[root]: if opname not in roots: opfunc.__name__ = opname setattr(cls, opname, opfunc) return cls ################################################################################ ### cmp_to_key() function converter ################################################################################ def cmp_to_key(mycmp): """Convert a cmp= function into a key= function""" class K(object): __slots__ = ['obj'] def __init__(self, obj): self.obj = obj def __lt__(self, other): return mycmp(self.obj, other.obj) < 0 def __gt__(self, other): return mycmp(self.obj, other.obj) > 0 def __eq__(self, other): return mycmp(self.obj, other.obj) == 0 def __le__(self, other): return mycmp(self.obj, other.obj) <= 0 def __ge__(self, other): return mycmp(self.obj, other.obj) >= 0 __hash__ = None return K try: from _functools import cmp_to_key except ImportError: pass ################################################################################ ### partial() argument application ################################################################################ # Purely functional, no descriptor behaviour def partial(func, args, keywords): """New function with partial application of the given arguments and keywords. """ if hasattr(func, 'func'): args = func.args + args tmpkw = func.keywords.copy() tmpkw.update(keywords) keywords = tmpkw del tmpkw func = func.func def newfunc(fargs, *fkeywords): newkeywords = keywords.copy() newkeywords.update(fkeywords) return func((args + fargs), *newkeywords) newfunc.func = func newfunc.args = args newfunc.keywords = keywords return newfunc try: from _functools import partial except ImportError: pass # Descriptor version class partialmethod(object): """Method descriptor with partial application of the given arguments and keywords. Supports wrapping existing descriptors and handles non-descriptor callables as instance methods. """ def __init__(self, func, args, *keywords): if not callable(func) and not hasattr(func, "__get__"): raise TypeError("{!r} is not callable or a descriptor" .format(func)) # func could be a descriptor like classmethod which isn't callable, # so we can't inherit from partial (it verifies func is callable) if isinstance(func, partialmethod): # flattening is mandatory in order to place cls/self before all # other arguments # it's also more efficient since only one function will be called self.func = func.func self.args = func.args + args self.keywords = func.keywords.copy() self.keywords.update(keywords) else: self.func = func self.args = args self.keywords = keywords def __repr__(self): args = ", ".join(map(repr, self.args)) keywords = ", ".join("{}={!r}".format(k, v) for k, v in self.keywords.items()) format_string = "{module}.{cls}({func}, {args}, {keywords})" return format_string.format(module=self.__class__.__module__, cls=self.__class__.__qualname__, func=self.func, args=args, keywords=keywords) def _make_unbound_method(self): def _method(args, *keywords): call_keywords = self.keywords.copy() call_keywords.update(keywords) cls_or_self, rest = args call_args = (cls_or_self,) + self.args + tuple(rest) return self.func(call_args, call_keywords) _method.__isabstractmethod__ = self.__isabstractmethod__ _method._partialmethod = self return _method def __get__(self, obj, cls): get = getattr(self.func, "__get__", None) result = None if get is not None: new_func = get(obj, cls) if new_func is not self.func: # Assume __get__ returning something new indicates the # creation of an appropriate callable result = partial(new_func, self.args, *self.keywords) try: result.__self__ = new_func.__self__ except AttributeError: pass if result is None: # If the underlying descriptor didn't do anything, treat this # like an instance method result = self._make_unbound_method().__get__(obj, cls) return result @property def __isabstractmethod__(self): return getattr(self.func, "__isabstractmethod__", False) ################################################################################ ### LRU Cache function decorator ################################################################################ _CacheInfo = namedtuple("CacheInfo", ["hits", "misses", "maxsize", "currsize"]) class _HashedSeq(list): """ This class guarantees that hash() will be called no more than once per element. This is important because the lru_cache() will hash the key multiple times on a cache miss. """ __slots__ = 'hashvalue' def __init__(self, tup, hash=hash): self[:] = tup self.hashvalue = hash(tup) def __hash__(self): return self.hashvalue def _make_key(args, kwds, typed, kwd_mark = (object(),), fasttypes = {int, str, frozenset, type(None)}, sorted=sorted, tuple=tuple, type=type, len=len): """Make a cache key from optionally typed positional and keyword arguments The key is constructed in a way that is flat as possible rather than as a nested structure that would take more memory. If there is only a single argument and its data type is known to cache its hash value, then that argument is returned without a wrapper. This saves space and improves lookup speed. """ key = args if kwds: sorted_items = sorted(kwds.items()) key += kwd_mark for item in sorted_items: key += item if typed: key += tuple(type(v) for v in args) if kwds: key += tuple(type(v) for k, v in sorted_items) elif len(key) == 1 and type(key[0]) in fasttypes: return key[0] return _HashedSeq(key) def lru_cache(maxsize=128, typed=False): """Least-recently-used cache decorator. If maxsize* is set to None, the LRU features are disabled and the cache can grow without bound. If typed is True, arguments of different types will be cached separately. For example, f(3.0) and f(3) will be treated as distinct calls with distinct results. Arguments to the cached function must be hashable. View the cache statistics named tuple (hits, misses, maxsize, currsize) with f.cache_info(). Clear the cache and statistics with f.cache_clear(). Access the underlying function with f.__wrapped__. See: http://en.wikipedia.org/wiki/Cache_algorithms#Least_Recently_Used """ # Users should only access the lru_cache through its public API: # cache_info, cache_clear, and f.__wrapped__ # The internals of the lru_cache are encapsulated for thread safety and # to allow the implementation to change (including a possible C version). # Early detection of an erroneous call to @lru_cache without any arguments # resulting in the inner function being passed to maxsize instead of an # integer or None. if maxsize is not None and not isinstance(maxsize, int): raise TypeError('Expected maxsize to be an integer or None') def decorating_function(user_function): wrapper = _lru_cache_wrapper(user_function, maxsize, typed, _CacheInfo) return update_wrapper(wrapper, user_function) return decorating_function def _lru_cache_wrapper(user_function, maxsize, typed, _CacheInfo): # Constants shared by all lru cache instances: sentinel = object() # unique object used to signal cache misses make_key = _make_key # build a key from the function arguments PREV, NEXT, KEY, RESULT = 0, 1, 2, 3 # names for the link fields cache = {} hits = misses = 0 full = False cache_get = cache.get # bound method to lookup a key or return None lock = RLock() # because linkedlist updates aren't threadsafe root = [] # root of the circular doubly linked list root[:] = [root, root, None, None] # initialize by pointing to self if maxsize == 0: def wrapper(args, kwds): # No caching -- just a statistics update after a successful call nonlocal misses result = user_function(args, *kwds) misses += 1 return result elif maxsize is None: def wrapper(args, *kwds): # Simple caching without ordering or size limit nonlocal hits, misses key = make_key(args, kwds, typed) result = cache_get(key, sentinel) if result is not sentinel: hits += 1 return result result = user_function(args, *kwds) cache[key] = result misses += 1 return result else: def wrapper(args, *kwds): # Size limited caching that tracks accesses by recency nonlocal root, hits, misses, full key = make_key(args, kwds, typed) with lock: link = cache_get(key) if link is not None: # Move the link to the front of the circular queue link_prev, link_next, _key, result = link link_prev[NEXT] = link_next link_next[PREV] = link_prev last = root[PREV] last[NEXT] = root[PREV] = link link[PREV] = last link[NEXT] = root hits += 1 return result result = user_function(args, *kwds) with lock: if key in cache: # Getting here means that this same key was added to the # cache while the lock was released. Since the link # update is already done, we need only return the # computed result and update the count of misses. pass elif full: # Use the old root to store the new key and result. oldroot = root oldroot[KEY] = key oldroot[RESULT] = result # Empty the oldest link and make it the new root. # Keep a reference to the old key and old result to # prevent their ref counts from going to zero during the # update. That will prevent potentially arbitrary object # clean-up code (i.e. __del__) from running while we're # still adjusting the links. root = oldroot[NEXT] oldkey = root[KEY] oldresult = root[RESULT] root[KEY] = root[RESULT] = None # Now update the cache dictionary. del cache[oldkey] # Save the potentially reentrant cache[key] assignment # for last, after the root and links have been put in # a consistent state. cache[key] = oldroot else: # Put result in a new link at the front of the queue. last = root[PREV] link = [last, root, key, result] last[NEXT] = root[PREV] = cache[key] = link # Use the __len__() method instead of the len() function # which could potentially be wrapped in an lru_cache itself. full = (cache.__len__() >= maxsize) misses += 1 return result def cache_info(): """Report cache statistics""" with lock: return _CacheInfo(hits, misses, maxsize, cache.__len__()) def cache_clear(): """Clear the cache and cache statistics""" nonlocal hits, misses, full with lock: cache.clear() root[:] = [root, root, None, None] hits = misses = 0 full = False wrapper.cache_info = cache_info wrapper.cache_clear = cache_clear return wrapper try: from _functools import _lru_cache_wrapper except ImportError: pass ################################################################################ ### singledispatch() - single-dispatch generic function decorator ################################################################################ def _c3_merge(sequences): """Merges MROs in sequences* to a single MRO using the C3 algorithm. Adapted from http://www.python.org/download/releases/2.3/mro/. """ result = [] while True: sequences = [s for s in sequences if s] # purge empty sequences if not sequences: return result for s1 in sequences: # find merge candidates among seq heads candidate = s1[0] for s2 in sequences: if candidate in s2[1:]: candidate = None break # reject the current head, it appears later else: break if candidate is None: raise RuntimeError("Inconsistent hierarchy") result.append(candidate) # remove the chosen candidate for seq in sequences: if seq[0] == candidate: del seq[0] def _c3_mro(cls, abcs=None): """Computes the method resolution order using extended C3 linearization. If no abcs are given, the algorithm works exactly like the built-in C3 linearization used for method resolution. If given, abcs is a list of abstract base classes that should be inserted into the resulting MRO. Unrelated ABCs are ignored and don't end up in the result. The algorithm inserts ABCs where their functionality is introduced, i.e. issubclass(cls, abc) returns True for the class itself but returns False for all its direct base classes. Implicit ABCs for a given class (either registered or inferred from the presence of a special method like __len__) are inserted directly after the last ABC explicitly listed in the MRO of said class. If two implicit ABCs end up next to each other in the resulting MRO, their ordering depends on the order of types in abcs. """ for i, base in enumerate(reversed(cls.__bases__)): if hasattr(base, '__abstractmethods__'): boundary = len(cls.__bases__) - i break # Bases up to the last explicit ABC are considered first. else: boundary = 0 abcs = list(abcs) if abcs else [] explicit_bases = list(cls.__bases__[:boundary]) abstract_bases = [] other_bases = list(cls.__bases__[boundary:]) for base in abcs: if issubclass(cls, base) and not any( issubclass(b, base) for b in cls.__bases__ ): # If cls is the class that introduces behaviour described by # an ABC base, insert said ABC to its MRO. abstract_bases.append(base) for base in abstract_bases: abcs.remove(base) explicit_c3_mros = [_c3_mro(base, abcs=abcs) for base in explicit_bases] abstract_c3_mros = [_c3_mro(base, abcs=abcs) for base in abstract_bases] other_c3_mros = [_c3_mro(base, abcs=abcs) for base in other_bases] return _c3_merge( [[cls]] + explicit_c3_mros + abstract_c3_mros + other_c3_mros + [explicit_bases] + [abstract_bases] + [other_bases] ) def _compose_mro(cls, types): """Calculates the method resolution order for a given class cls. Includes relevant abstract base classes (with their respective bases) from the types iterable. Uses a modified C3 linearization algorithm. """ bases = set(cls.__mro__) # Remove entries which are already present in the __mro__ or unrelated. def is_related(typ): return (typ not in bases and hasattr(typ, '__mro__') and issubclass(cls, typ)) types = [n for n in types if is_related(n)] # Remove entries which are strict bases of other entries (they will end up # in the MRO anyway. def is_strict_base(typ): for other in types: if typ != other and typ in other.__mro__: return True return False types = [n for n in types if not is_strict_base(n)] # Subclasses of the ABCs in types which are also implemented by # cls can be used to stabilize ABC ordering. type_set = set(types) mro = [] for typ in types: found = [] for sub in typ.__subclasses__(): if sub not in bases and issubclass(cls, sub): found.append([s for s in sub.__mro__ if s in type_set]) if not found: mro.append(typ) continue # Favor subclasses with the biggest number of useful bases found.sort(key=len, reverse=True) for sub in found: for subcls in sub: if subcls not in mro: mro.append(subcls) return _c3_mro(cls, abcs=mro) def _find_impl(cls, registry): """Returns the best matching implementation from registry for type cls. Where there is no registered implementation for a specific type, its method resolution order is used to find a more generic implementation. Note: if registry does not contain an implementation for the base object type, this function may return None. """ mro = _compose_mro(cls, registry.keys()) match = None for t in mro: if match is not None: # If match is an implicit ABC but there is another unrelated, # equally matching implicit ABC, refuse the temptation to guess. if (t in registry and t not in cls.__mro__ and match not in cls.__mro__ and not issubclass(match, t)): raise RuntimeError("Ambiguous dispatch: {} or {}".format( match, t)) break if t in registry: match = t return registry.get(match) def singledispatch(func): """Single-dispatch generic function decorator. Transforms a function into a generic function, which can have different behaviours depending upon the type of its first argument. The decorated function acts as the default implementation, and additional implementations can be registered using the register() attribute of the generic function. """ registry = {} dispatch_cache = WeakKeyDictionary() cache_token = None def dispatch(cls): """generic_func.dispatch(cls) -> <function implementation> Runs the dispatch algorithm to return the best available implementation for the given cls registered on generic_func. """ nonlocal cache_token if cache_token is not None: current_token = get_cache_token() if cache_token != current_token: dispatch_cache.clear() cache_token = current_token try: impl = dispatch_cache[cls] except KeyError: try: impl = registry[cls] except KeyError: impl = _find_impl(cls, registry) dispatch_cache[cls] = impl return impl def register(cls, func=None): """generic_func.register(cls, func) -> func Registers a new implementation for the given cls on a generic_func. """ nonlocal cache_token if func is None: return lambda f: register(cls, f) registry[cls] = func if cache_token is None and hasattr(cls, '__abstractmethods__'): cache_token = get_cache_token() dispatch_cache.clear() return func def wrapper(args, kw): return dispatch(args[0].__class__)(args, **kw) registry[object] = func wrapper.register = register wrapper.dispatch = dispatch wrapper.registry = MappingProxyType(registry) wrapper._clear_cache = dispatch_cache.clear update_wrapper(wrapper, func) return wrapper Back

Edit File : functools.py | Size : 28.27 KB

"""functools.py - Tools for working with functions and callable objects
"""
# Python module wrapper for _functools C module
# to allow utilities written in Python to be added
# to the functools module.
# Written by Nick Coghlan <ncoghlan at gmail.com>,
# Raymond Hettinger <python at rcn.com>,
# and Łukasz Langa <lukasz at langa.pl>.
#   Copyright (C) 2006-2013 Python Software Foundation.
# See C source code for _functools credits/copyright

__all__ = ['update_wrapper', 'wraps', 'WRAPPER_ASSIGNMENTS', 'WRAPPER_UPDATES',
           'total_ordering', 'cmp_to_key', 'lru_cache', 'reduce', 'partial',
           'partialmethod', 'singledispatch']

try:
    from _functools import reduce
except ImportError:
    pass
from abc import get_cache_token
from collections import namedtuple
from types import MappingProxyType
from weakref import WeakKeyDictionary
try:
    from _thread import RLock
except ImportError:
    class RLock:
        'Dummy reentrant lock for builds without threads'
        def __enter__(self): pass
        def __exit__(self, exctype, excinst, exctb): pass

################################################################################
### update_wrapper() and wraps() decorator
################################################################################

# update_wrapper() and wraps() are tools to help write
# wrapper functions that can handle naive introspection

WRAPPER_ASSIGNMENTS = ('__module__', '__name__', '__qualname__', '__doc__',
                       '__annotations__')
WRAPPER_UPDATES = ('__dict__',)
def update_wrapper(wrapper,
                   wrapped,
                   assigned = WRAPPER_ASSIGNMENTS,
                   updated = WRAPPER_UPDATES):
    """Update a wrapper function to look like the wrapped function

wrapper is the function to be updated
       wrapped is the original function
       assigned is a tuple naming the attributes assigned directly
       from the wrapped function to the wrapper function (defaults to
       functools.WRAPPER_ASSIGNMENTS)
       updated is a tuple naming the attributes of the wrapper that
       are updated with the corresponding attribute from the wrapped
       function (defaults to functools.WRAPPER_UPDATES)
    """
    for attr in assigned:
        try:
            value = getattr(wrapped, attr)
        except AttributeError:
            pass
        else:
            setattr(wrapper, attr, value)
    for attr in updated:
        getattr(wrapper, attr).update(getattr(wrapped, attr, {}))
    # Issue #17482: set __wrapped__ last so we don't inadvertently copy it
    # from the wrapped function when updating __dict__
    wrapper.__wrapped__ = wrapped
    # Return the wrapper so this can be used as a decorator via partial()
    return wrapper

def wraps(wrapped,
          assigned = WRAPPER_ASSIGNMENTS,
          updated = WRAPPER_UPDATES):
    """Decorator factory to apply update_wrapper() to a wrapper function

Returns a decorator that invokes update_wrapper() with the decorated
       function as the wrapper argument and the arguments to wraps() as the
       remaining arguments. Default arguments are as for update_wrapper().
       This is a convenience function to simplify applying partial() to
       update_wrapper().
    """
    return partial(update_wrapper, wrapped=wrapped,
                   assigned=assigned, updated=updated)

################################################################################
### total_ordering class decorator
################################################################################

# The total ordering functions all invoke the root magic method directly
# rather than using the corresponding operator.  This avoids possible
# infinite recursion that could occur when the operator dispatch logic
# detects a NotImplemented result and then calls a reflected method.

def _gt_from_lt(self, other, NotImplemented=NotImplemented):
    'Return a > b.  Computed by @total_ordering from (not a < b) and (a != b).'
    op_result = self.__lt__(other)
    if op_result is NotImplemented:
        return op_result
    return not op_result and self != other

def _le_from_lt(self, other, NotImplemented=NotImplemented):
    'Return a <= b.  Computed by @total_ordering from (a < b) or (a == b).'
    op_result = self.__lt__(other)
    return op_result or self == other

def _ge_from_lt(self, other, NotImplemented=NotImplemented):
    'Return a >= b.  Computed by @total_ordering from (not a < b).'
    op_result = self.__lt__(other)
    if op_result is NotImplemented:
        return op_result
    return not op_result

def _ge_from_le(self, other, NotImplemented=NotImplemented):
    'Return a >= b.  Computed by @total_ordering from (not a <= b) or (a == b).'
    op_result = self.__le__(other)
    if op_result is NotImplemented:
        return op_result
    return not op_result or self == other

def _lt_from_le(self, other, NotImplemented=NotImplemented):
    'Return a < b.  Computed by @total_ordering from (a <= b) and (a != b).'
    op_result = self.__le__(other)
    if op_result is NotImplemented:
        return op_result
    return op_result and self != other

def _gt_from_le(self, other, NotImplemented=NotImplemented):
    'Return a > b.  Computed by @total_ordering from (not a <= b).'
    op_result = self.__le__(other)
    if op_result is NotImplemented:
        return op_result
    return not op_result

def _lt_from_gt(self, other, NotImplemented=NotImplemented):
    'Return a < b.  Computed by @total_ordering from (not a > b) and (a != b).'
    op_result = self.__gt__(other)
    if op_result is NotImplemented:
        return op_result
    return not op_result and self != other

def _ge_from_gt(self, other, NotImplemented=NotImplemented):
    'Return a >= b.  Computed by @total_ordering from (a > b) or (a == b).'
    op_result = self.__gt__(other)
    return op_result or self == other

def _le_from_gt(self, other, NotImplemented=NotImplemented):
    'Return a <= b.  Computed by @total_ordering from (not a > b).'
    op_result = self.__gt__(other)
    if op_result is NotImplemented:
        return op_result
    return not op_result

def _le_from_ge(self, other, NotImplemented=NotImplemented):
    'Return a <= b.  Computed by @total_ordering from (not a >= b) or (a == b).'
    op_result = self.__ge__(other)
    if op_result is NotImplemented:
        return op_result
    return not op_result or self == other

def _gt_from_ge(self, other, NotImplemented=NotImplemented):
    'Return a > b.  Computed by @total_ordering from (a >= b) and (a != b).'
    op_result = self.__ge__(other)
    if op_result is NotImplemented:
        return op_result
    return op_result and self != other

def _lt_from_ge(self, other, NotImplemented=NotImplemented):
    'Return a < b.  Computed by @total_ordering from (not a >= b).'
    op_result = self.__ge__(other)
    if op_result is NotImplemented:
        return op_result
    return not op_result

_convert = {
    '__lt__': [('__gt__', _gt_from_lt),
               ('__le__', _le_from_lt),
               ('__ge__', _ge_from_lt)],
    '__le__': [('__ge__', _ge_from_le),
               ('__lt__', _lt_from_le),
               ('__gt__', _gt_from_le)],
    '__gt__': [('__lt__', _lt_from_gt),
               ('__ge__', _ge_from_gt),
               ('__le__', _le_from_gt)],
    '__ge__': [('__le__', _le_from_ge),
               ('__gt__', _gt_from_ge),
               ('__lt__', _lt_from_ge)]
}

def total_ordering(cls):
    """Class decorator that fills in missing ordering methods"""
    # Find user-defined comparisons (not those inherited from object).
    roots = [op for op in _convert if getattr(cls, op, None) is not getattr(object, op, None)]
    if not roots:
        raise ValueError('must define at least one ordering operation: < > <= >=')
    root = max(roots)       # prefer __lt__ to __le__ to __gt__ to __ge__
    for opname, opfunc in _convert[root]:
        if opname not in roots:
            opfunc.__name__ = opname
            setattr(cls, opname, opfunc)
    return cls

################################################################################
### cmp_to_key() function converter
################################################################################

def cmp_to_key(mycmp):
    """Convert a cmp= function into a key= function"""
    class K(object):
        __slots__ = ['obj']
        def __init__(self, obj):
            self.obj = obj
        def __lt__(self, other):
            return mycmp(self.obj, other.obj) < 0
        def __gt__(self, other):
            return mycmp(self.obj, other.obj) > 0
        def __eq__(self, other):
            return mycmp(self.obj, other.obj) == 0
        def __le__(self, other):
            return mycmp(self.obj, other.obj) <= 0
        def __ge__(self, other):
            return mycmp(self.obj, other.obj) >= 0
        __hash__ = None
    return K

try:
    from _functools import cmp_to_key
except ImportError:
    pass

################################################################################
### partial() argument application
################################################################################

# Purely functional, no descriptor behaviour
def partial(func, *args, **keywords):
    """New function with partial application of the given arguments
    and keywords.
    """
    if hasattr(func, 'func'):
        args = func.args + args
        tmpkw = func.keywords.copy()
        tmpkw.update(keywords)
        keywords = tmpkw
        del tmpkw
        func = func.func

def newfunc(*fargs, **fkeywords):
        newkeywords = keywords.copy()
        newkeywords.update(fkeywords)
        return func(*(args + fargs), **newkeywords)
    newfunc.func = func
    newfunc.args = args
    newfunc.keywords = keywords
    return newfunc

try:
    from _functools import partial
except ImportError:
    pass

# Descriptor version
class partialmethod(object):
    """Method descriptor with partial application of the given arguments
    and keywords.

Supports wrapping existing descriptors and handles non-descriptor
    callables as instance methods.
    """

def __init__(self, func, *args, **keywords):
        if not callable(func) and not hasattr(func, "__get__"):
            raise TypeError("{!r} is not callable or a descriptor"
                                 .format(func))

# func could be a descriptor like classmethod which isn't callable,
        # so we can't inherit from partial (it verifies func is callable)
        if isinstance(func, partialmethod):
            # flattening is mandatory in order to place cls/self before all
            # other arguments
            # it's also more efficient since only one function will be called
            self.func = func.func
            self.args = func.args + args
            self.keywords = func.keywords.copy()
            self.keywords.update(keywords)
        else:
            self.func = func
            self.args = args
            self.keywords = keywords

def __repr__(self):
        args = ", ".join(map(repr, self.args))
        keywords = ", ".join("{}={!r}".format(k, v)
                                 for k, v in self.keywords.items())
        format_string = "{module}.{cls}({func}, {args}, {keywords})"
        return format_string.format(module=self.__class__.__module__,
                                    cls=self.__class__.__qualname__,
                                    func=self.func,
                                    args=args,
                                    keywords=keywords)

def _make_unbound_method(self):
        def _method(*args, **keywords):
            call_keywords = self.keywords.copy()
            call_keywords.update(keywords)
            cls_or_self, *rest = args
            call_args = (cls_or_self,) + self.args + tuple(rest)
            return self.func(*call_args, **call_keywords)
        _method.__isabstractmethod__ = self.__isabstractmethod__
        _method._partialmethod = self
        return _method

def __get__(self, obj, cls):
        get = getattr(self.func, "__get__", None)
        result = None
        if get is not None:
            new_func = get(obj, cls)
            if new_func is not self.func:
                # Assume __get__ returning something new indicates the
                # creation of an appropriate callable
                result = partial(new_func, *self.args, **self.keywords)
                try:
                    result.__self__ = new_func.__self__
                except AttributeError:
                    pass
        if result is None:
            # If the underlying descriptor didn't do anything, treat this
            # like an instance method
            result = self._make_unbound_method().__get__(obj, cls)
        return result

@property
    def __isabstractmethod__(self):
        return getattr(self.func, "__isabstractmethod__", False)

################################################################################
### LRU Cache function decorator
################################################################################

_CacheInfo = namedtuple("CacheInfo", ["hits", "misses", "maxsize", "currsize"])

class _HashedSeq(list):
    """ This class guarantees that hash() will be called no more than once
        per element.  This is important because the lru_cache() will hash
        the key multiple times on a cache miss.

"""

__slots__ = 'hashvalue'

def __init__(self, tup, hash=hash):
        self[:] = tup
        self.hashvalue = hash(tup)

def __hash__(self):
        return self.hashvalue

def _make_key(args, kwds, typed,
             kwd_mark = (object(),),
             fasttypes = {int, str, frozenset, type(None)},
             sorted=sorted, tuple=tuple, type=type, len=len):
    """Make a cache key from optionally typed positional and keyword arguments

The key is constructed in a way that is flat as possible rather than
    as a nested structure that would take more memory.

If there is only a single argument and its data type is known to cache
    its hash value, then that argument is returned without a wrapper.  This
    saves space and improves lookup speed.

"""
    key = args
    if kwds:
        sorted_items = sorted(kwds.items())
        key += kwd_mark
        for item in sorted_items:
            key += item
    if typed:
        key += tuple(type(v) for v in args)
        if kwds:
            key += tuple(type(v) for k, v in sorted_items)
    elif len(key) == 1 and type(key[0]) in fasttypes:
        return key[0]
    return _HashedSeq(key)

def lru_cache(maxsize=128, typed=False):
    """Least-recently-used cache decorator.

If *maxsize* is set to None, the LRU features are disabled and the cache
    can grow without bound.

If *typed* is True, arguments of different types will be cached separately.
    For example, f(3.0) and f(3) will be treated as distinct calls with
    distinct results.

Arguments to the cached function must be hashable.

View the cache statistics named tuple (hits, misses, maxsize, currsize)
    with f.cache_info().  Clear the cache and statistics with f.cache_clear().
    Access the underlying function with f.__wrapped__.

See:  http://en.wikipedia.org/wiki/Cache_algorithms#Least_Recently_Used

"""

# Users should only access the lru_cache through its public API:
    #       cache_info, cache_clear, and f.__wrapped__
    # The internals of the lru_cache are encapsulated for thread safety and
    # to allow the implementation to change (including a possible C version).

# Early detection of an erroneous call to @lru_cache without any arguments
    # resulting in the inner function being passed to maxsize instead of an
    # integer or None.
    if maxsize is not None and not isinstance(maxsize, int):
        raise TypeError('Expected maxsize to be an integer or None')

def decorating_function(user_function):
        wrapper = _lru_cache_wrapper(user_function, maxsize, typed, _CacheInfo)
        return update_wrapper(wrapper, user_function)

return decorating_function

def _lru_cache_wrapper(user_function, maxsize, typed, _CacheInfo):
    # Constants shared by all lru cache instances:
    sentinel = object()          # unique object used to signal cache misses
    make_key = _make_key         # build a key from the function arguments
    PREV, NEXT, KEY, RESULT = 0, 1, 2, 3   # names for the link fields

cache = {}
    hits = misses = 0
    full = False
    cache_get = cache.get    # bound method to lookup a key or return None
    lock = RLock()           # because linkedlist updates aren't threadsafe
    root = []                # root of the circular doubly linked list
    root[:] = [root, root, None, None]     # initialize by pointing to self

if maxsize == 0:

def wrapper(*args, **kwds):
            # No caching -- just a statistics update after a successful call
            nonlocal misses
            result = user_function(*args, **kwds)
            misses += 1
            return result

elif maxsize is None:

def wrapper(*args, **kwds):
            # Simple caching without ordering or size limit
            nonlocal hits, misses
            key = make_key(args, kwds, typed)
            result = cache_get(key, sentinel)
            if result is not sentinel:
                hits += 1
                return result
            result = user_function(*args, **kwds)
            cache[key] = result
            misses += 1
            return result

else:

def wrapper(*args, **kwds):
            # Size limited caching that tracks accesses by recency
            nonlocal root, hits, misses, full
            key = make_key(args, kwds, typed)
            with lock:
                link = cache_get(key)
                if link is not None:
                    # Move the link to the front of the circular queue
                    link_prev, link_next, _key, result = link
                    link_prev[NEXT] = link_next
                    link_next[PREV] = link_prev
                    last = root[PREV]
                    last[NEXT] = root[PREV] = link
                    link[PREV] = last
                    link[NEXT] = root
                    hits += 1
                    return result
            result = user_function(*args, **kwds)
            with lock:
                if key in cache:
                    # Getting here means that this same key was added to the
                    # cache while the lock was released.  Since the link
                    # update is already done, we need only return the
                    # computed result and update the count of misses.
                    pass
                elif full:
                    # Use the old root to store the new key and result.
                    oldroot = root
                    oldroot[KEY] = key
                    oldroot[RESULT] = result
                    # Empty the oldest link and make it the new root.
                    # Keep a reference to the old key and old result to
                    # prevent their ref counts from going to zero during the
                    # update. That will prevent potentially arbitrary object
                    # clean-up code (i.e. __del__) from running while we're
                    # still adjusting the links.
                    root = oldroot[NEXT]
                    oldkey = root[KEY]
                    oldresult = root[RESULT]
                    root[KEY] = root[RESULT] = None
                    # Now update the cache dictionary.
                    del cache[oldkey]
                    # Save the potentially reentrant cache[key] assignment
                    # for last, after the root and links have been put in
                    # a consistent state.
                    cache[key] = oldroot
                else:
                    # Put result in a new link at the front of the queue.
                    last = root[PREV]
                    link = [last, root, key, result]
                    last[NEXT] = root[PREV] = cache[key] = link
                    # Use the __len__() method instead of the len() function
                    # which could potentially be wrapped in an lru_cache itself.
                    full = (cache.__len__() >= maxsize)
                misses += 1
            return result

def cache_info():
        """Report cache statistics"""
        with lock:
            return _CacheInfo(hits, misses, maxsize, cache.__len__())

def cache_clear():
        """Clear the cache and cache statistics"""
        nonlocal hits, misses, full
        with lock:
            cache.clear()
            root[:] = [root, root, None, None]
            hits = misses = 0
            full = False

wrapper.cache_info = cache_info
    wrapper.cache_clear = cache_clear
    return wrapper

try:
    from _functools import _lru_cache_wrapper
except ImportError:
    pass

################################################################################
### singledispatch() - single-dispatch generic function decorator
################################################################################

def _c3_merge(sequences):
    """Merges MROs in *sequences* to a single MRO using the C3 algorithm.

Adapted from http://www.python.org/download/releases/2.3/mro/.

"""
    result = []
    while True:
        sequences = [s for s in sequences if s]   # purge empty sequences
        if not sequences:
            return result
        for s1 in sequences:   # find merge candidates among seq heads
            candidate = s1[0]
            for s2 in sequences:
                if candidate in s2[1:]:
                    candidate = None
                    break      # reject the current head, it appears later
            else:
                break
        if candidate is None:
            raise RuntimeError("Inconsistent hierarchy")
        result.append(candidate)
        # remove the chosen candidate
        for seq in sequences:
            if seq[0] == candidate:
                del seq[0]

def _c3_mro(cls, abcs=None):
    """Computes the method resolution order using extended C3 linearization.

If no *abcs* are given, the algorithm works exactly like the built-in C3
    linearization used for method resolution.

If given, *abcs* is a list of abstract base classes that should be inserted
    into the resulting MRO. Unrelated ABCs are ignored and don't end up in the
    result. The algorithm inserts ABCs where their functionality is introduced,
    i.e. issubclass(cls, abc) returns True for the class itself but returns
    False for all its direct base classes. Implicit ABCs for a given class
    (either registered or inferred from the presence of a special method like
    __len__) are inserted directly after the last ABC explicitly listed in the
    MRO of said class. If two implicit ABCs end up next to each other in the
    resulting MRO, their ordering depends on the order of types in *abcs*.

"""
    for i, base in enumerate(reversed(cls.__bases__)):
        if hasattr(base, '__abstractmethods__'):
            boundary = len(cls.__bases__) - i
            break   # Bases up to the last explicit ABC are considered first.
    else:
        boundary = 0
    abcs = list(abcs) if abcs else []
    explicit_bases = list(cls.__bases__[:boundary])
    abstract_bases = []
    other_bases = list(cls.__bases__[boundary:])
    for base in abcs:
        if issubclass(cls, base) and not any(
                issubclass(b, base) for b in cls.__bases__
            ):
            # If *cls* is the class that introduces behaviour described by
            # an ABC *base*, insert said ABC to its MRO.
            abstract_bases.append(base)
    for base in abstract_bases:
        abcs.remove(base)
    explicit_c3_mros = [_c3_mro(base, abcs=abcs) for base in explicit_bases]
    abstract_c3_mros = [_c3_mro(base, abcs=abcs) for base in abstract_bases]
    other_c3_mros = [_c3_mro(base, abcs=abcs) for base in other_bases]
    return _c3_merge(
        [[cls]] +
        explicit_c3_mros + abstract_c3_mros + other_c3_mros +
        [explicit_bases] + [abstract_bases] + [other_bases]
    )

def _compose_mro(cls, types):
    """Calculates the method resolution order for a given class *cls*.

Includes relevant abstract base classes (with their respective bases) from
    the *types* iterable. Uses a modified C3 linearization algorithm.

"""
    bases = set(cls.__mro__)
    # Remove entries which are already present in the __mro__ or unrelated.
    def is_related(typ):
        return (typ not in bases and hasattr(typ, '__mro__')
                                 and issubclass(cls, typ))
    types = [n for n in types if is_related(n)]
    # Remove entries which are strict bases of other entries (they will end up
    # in the MRO anyway.
    def is_strict_base(typ):
        for other in types:
            if typ != other and typ in other.__mro__:
                return True
        return False
    types = [n for n in types if not is_strict_base(n)]
    # Subclasses of the ABCs in *types* which are also implemented by
    # *cls* can be used to stabilize ABC ordering.
    type_set = set(types)
    mro = []
    for typ in types:
        found = []
        for sub in typ.__subclasses__():
            if sub not in bases and issubclass(cls, sub):
                found.append([s for s in sub.__mro__ if s in type_set])
        if not found:
            mro.append(typ)
            continue
        # Favor subclasses with the biggest number of useful bases
        found.sort(key=len, reverse=True)
        for sub in found:
            for subcls in sub:
                if subcls not in mro:
                    mro.append(subcls)
    return _c3_mro(cls, abcs=mro)

def _find_impl(cls, registry):
    """Returns the best matching implementation from *registry* for type *cls*.

Where there is no registered implementation for a specific type, its method
    resolution order is used to find a more generic implementation.

Note: if *registry* does not contain an implementation for the base
    *object* type, this function may return None.

"""
    mro = _compose_mro(cls, registry.keys())
    match = None
    for t in mro:
        if match is not None:
            # If *match* is an implicit ABC but there is another unrelated,
            # equally matching implicit ABC, refuse the temptation to guess.
            if (t in registry and t not in cls.__mro__
                              and match not in cls.__mro__
                              and not issubclass(match, t)):
                raise RuntimeError("Ambiguous dispatch: {} or {}".format(
                    match, t))
            break
        if t in registry:
            match = t
    return registry.get(match)

def singledispatch(func):
    """Single-dispatch generic function decorator.

Transforms a function into a generic function, which can have different
    behaviours depending upon the type of its first argument. The decorated
    function acts as the default implementation, and additional
    implementations can be registered using the register() attribute of the
    generic function.

"""
    registry = {}
    dispatch_cache = WeakKeyDictionary()
    cache_token = None

def dispatch(cls):
        """generic_func.dispatch(cls) -> <function implementation>

Runs the dispatch algorithm to return the best available implementation
        for the given *cls* registered on *generic_func*.

"""
        nonlocal cache_token
        if cache_token is not None:
            current_token = get_cache_token()
            if cache_token != current_token:
                dispatch_cache.clear()
                cache_token = current_token
        try:
            impl = dispatch_cache[cls]
        except KeyError:
            try:
                impl = registry[cls]
            except KeyError:
                impl = _find_impl(cls, registry)
            dispatch_cache[cls] = impl
        return impl

def register(cls, func=None):
        """generic_func.register(cls, func) -> func

Registers a new implementation for the given *cls* on a *generic_func*.

"""
        nonlocal cache_token
        if func is None:
            return lambda f: register(cls, f)
        registry[cls] = func
        if cache_token is None and hasattr(cls, '__abstractmethods__'):
            cache_token = get_cache_token()
        dispatch_cache.clear()
        return func

def wrapper(*args, **kw):
        return dispatch(args[0].__class__)(*args, **kw)

registry[object] = func
    wrapper.register = register
    wrapper.dispatch = dispatch
    wrapper.registry = MappingProxyType(registry)
    wrapper._clear_cache = dispatch_cache.clear
    update_wrapper(wrapper, func)
    return wrapper

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