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rl/Lib/site-packages/sympy/assumptions/relation/__init__.py

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+"""
+A module to implement finitary relations [1] as predicate.
+
+References
+==========
+
+.. [1] https://en.wikipedia.org/wiki/Finitary_relation
+
+"""
+
+__all__ = ['BinaryRelation', 'AppliedBinaryRelation']
+
+from .binrel import BinaryRelation, AppliedBinaryRelation

rl/Lib/site-packages/sympy/assumptions/relation/binrel.py

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+"""
+General binary relations.
+"""
+from typing import Optional
+
+from sympy.core.singleton import S
+from sympy.assumptions import AppliedPredicate, ask, Predicate, Q  # type: ignore
+from sympy.core.kind import BooleanKind
+from sympy.core.relational import Eq, Ne, Gt, Lt, Ge, Le
+from sympy.logic.boolalg import conjuncts, Not
+
+__all__ = ["BinaryRelation", "AppliedBinaryRelation"]
+
+
+class BinaryRelation(Predicate):
+    """
+    Base class for all binary relational predicates.
+
+    Explanation
+    ===========
+
+    Binary relation takes two arguments and returns ``AppliedBinaryRelation``
+    instance. To evaluate it to boolean value, use :obj:`~.ask()` or
+    :obj:`~.refine()` function.
+
+    You can add support for new types by registering the handler to dispatcher.
+    See :obj:`~.Predicate()` for more information about predicate dispatching.
+
+    Examples
+    ========
+
+    Applying and evaluating to boolean value:
+
+    >>> from sympy import Q, ask, sin, cos
+    >>> from sympy.abc import x
+    >>> Q.eq(sin(x)**2+cos(x)**2, 1)
+    Q.eq(sin(x)**2 + cos(x)**2, 1)
+    >>> ask(_)
+    True
+
+    You can define a new binary relation by subclassing and dispatching.
+    Here, we define a relation $R$ such that $x R y$ returns true if
+    $x = y + 1$.
+
+    >>> from sympy import ask, Number, Q
+    >>> from sympy.assumptions import BinaryRelation
+    >>> class MyRel(BinaryRelation):
+    ...     name = "R"
+    ...     is_reflexive = False
+    >>> Q.R = MyRel()
+    >>> @Q.R.register(Number, Number)
+    ... def _(n1, n2, assumptions):
+    ...     return ask(Q.zero(n1 - n2 - 1), assumptions)
+    >>> Q.R(2, 1)
+    Q.R(2, 1)
+
+    Now, we can use ``ask()`` to evaluate it to boolean value.
+
+    >>> ask(Q.R(2, 1))
+    True
+    >>> ask(Q.R(1, 2))
+    False
+
+    ``Q.R`` returns ``False`` with minimum cost if two arguments have same
+    structure because it is antireflexive relation [1] by
+    ``is_reflexive = False``.
+
+    >>> ask(Q.R(x, x))
+    False
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Reflexive_relation
+    """
+
+    is_reflexive: Optional[bool] = None
+    is_symmetric: Optional[bool] = None
+
+    def __call__(self, *args):
+        if not len(args) == 2:
+            raise ValueError("Binary relation takes two arguments, but got %s." % len(args))
+        return AppliedBinaryRelation(self, *args)
+
+    @property
+    def reversed(self):
+        if self.is_symmetric:
+            return self
+        return None
+
+    @property
+    def negated(self):
+        return None
+
+    def _compare_reflexive(self, lhs, rhs):
+        # quick exit for structurally same arguments
+        # do not check != here because it cannot catch the
+        # equivalent arguments with different structures.
+
+        # reflexivity does not hold to NaN
+        if lhs is S.NaN or rhs is S.NaN:
+            return None
+
+        reflexive = self.is_reflexive
+        if reflexive is None:
+            pass
+        elif reflexive and (lhs == rhs):
+            return True
+        elif not reflexive and (lhs == rhs):
+            return False
+        return None
+
+    def eval(self, args, assumptions=True):
+        # quick exit for structurally same arguments
+        ret = self._compare_reflexive(*args)
+        if ret is not None:
+            return ret
+
+        # don't perform simplify on args here. (done by AppliedBinaryRelation._eval_ask)
+        # evaluate by multipledispatch
+        lhs, rhs = args
+        ret = self.handler(lhs, rhs, assumptions=assumptions)
+        if ret is not None:
+            return ret
+
+        # check reversed order if the relation is reflexive
+        if self.is_reflexive:
+            types = (type(lhs), type(rhs))
+            if self.handler.dispatch(*types) is not self.handler.dispatch(*reversed(types)):
+                ret = self.handler(rhs, lhs, assumptions=assumptions)
+
+        return ret
+
+
+class AppliedBinaryRelation(AppliedPredicate):
+    """
+    The class of expressions resulting from applying ``BinaryRelation``
+    to the arguments.
+
+    """
+
+    @property
+    def lhs(self):
+        """The left-hand side of the relation."""
+        return self.arguments[0]
+
+    @property
+    def rhs(self):
+        """The right-hand side of the relation."""
+        return self.arguments[1]
+
+    @property
+    def reversed(self):
+        """
+        Try to return the relationship with sides reversed.
+        """
+        revfunc = self.function.reversed
+        if revfunc is None:
+            return self
+        return revfunc(self.rhs, self.lhs)
+
+    @property
+    def reversedsign(self):
+        """
+        Try to return the relationship with signs reversed.
+        """
+        revfunc = self.function.reversed
+        if revfunc is None:
+            return self
+        if not any(side.kind is BooleanKind for side in self.arguments):
+            return revfunc(-self.lhs, -self.rhs)
+        return self
+
+    @property
+    def negated(self):
+        neg_rel = self.function.negated
+        if neg_rel is None:
+            return Not(self, evaluate=False)
+        return neg_rel(*self.arguments)
+
+    def _eval_ask(self, assumptions):
+        conj_assumps = set()
+        binrelpreds = {Eq: Q.eq, Ne: Q.ne, Gt: Q.gt, Lt: Q.lt, Ge: Q.ge, Le: Q.le}
+        for a in conjuncts(assumptions):
+            if a.func in binrelpreds:
+                conj_assumps.add(binrelpreds[type(a)](*a.args))
+            else:
+                conj_assumps.add(a)
+
+        # After CNF in assumptions module is modified to take polyadic
+        # predicate, this will be removed
+        if any(rel in conj_assumps for rel in (self, self.reversed)):
+            return True
+        neg_rels = (self.negated, self.reversed.negated, Not(self, evaluate=False),
+            Not(self.reversed, evaluate=False))
+        if any(rel in conj_assumps for rel in neg_rels):
+            return False
+
+        # evaluation using multipledispatching
+        ret = self.function.eval(self.arguments, assumptions)
+        if ret is not None:
+            return ret
+
+        # simplify the args and try again
+        args = tuple(a.simplify() for a in self.arguments)
+        return self.function.eval(args, assumptions)
+
+    def __bool__(self):
+        ret = ask(self)
+        if ret is None:
+            raise TypeError("Cannot determine truth value of %s" % self)
+        return ret

rl/Lib/site-packages/sympy/assumptions/relation/equality.py

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+"""
+Module for mathematical equality [1] and inequalities [2].
+
+The purpose of this module is to provide the instances which represent the
+binary predicates in order to combine the relationals into logical inference
+system. Objects such as ``Q.eq``, ``Q.lt`` should remain internal to
+assumptions module, and user must use the classes such as :obj:`~.Eq()`,
+:obj:`~.Lt()` instead to construct the relational expressions.
+
+References
+==========
+
+.. [1] https://en.wikipedia.org/wiki/Equality_(mathematics)
+.. [2] https://en.wikipedia.org/wiki/Inequality_(mathematics)
+"""
+from sympy.assumptions import Q
+from sympy.core.relational import is_eq, is_neq, is_gt, is_ge, is_lt, is_le
+
+from .binrel import BinaryRelation
+
+__all__ = ['EqualityPredicate', 'UnequalityPredicate', 'StrictGreaterThanPredicate',
+    'GreaterThanPredicate', 'StrictLessThanPredicate', 'LessThanPredicate']
+
+
+class EqualityPredicate(BinaryRelation):
+    """
+    Binary predicate for $=$.
+
+    The purpose of this class is to provide the instance which represent
+    the equality predicate in order to allow the logical inference.
+    This class must remain internal to assumptions module and user must
+    use :obj:`~.Eq()` instead to construct the equality expression.
+
+    Evaluating this predicate to ``True`` or ``False`` is done by
+    :func:`~.core.relational.is_eq()`
+
+    Examples
+    ========
+
+    >>> from sympy import ask, Q
+    >>> Q.eq(0, 0)
+    Q.eq(0, 0)
+    >>> ask(_)
+    True
+
+    See Also
+    ========
+
+    sympy.core.relational.Eq
+
+    """
+    is_reflexive = True
+    is_symmetric = True
+
+    name = 'eq'
+    handler = None  # Do not allow dispatching by this predicate
+
+    @property
+    def negated(self):
+        return Q.ne
+
+    def eval(self, args, assumptions=True):
+        if assumptions == True:
+            # default assumptions for is_eq is None
+            assumptions = None
+        return is_eq(*args, assumptions)
+
+
+class UnequalityPredicate(BinaryRelation):
+    r"""
+    Binary predicate for $\neq$.
+
+    The purpose of this class is to provide the instance which represent
+    the inequation predicate in order to allow the logical inference.
+    This class must remain internal to assumptions module and user must
+    use :obj:`~.Ne()` instead to construct the inequation expression.
+
+    Evaluating this predicate to ``True`` or ``False`` is done by
+    :func:`~.core.relational.is_neq()`
+
+    Examples
+    ========
+
+    >>> from sympy import ask, Q
+    >>> Q.ne(0, 0)
+    Q.ne(0, 0)
+    >>> ask(_)
+    False
+
+    See Also
+    ========
+
+    sympy.core.relational.Ne
+
+    """
+    is_reflexive = False
+    is_symmetric = True
+
+    name = 'ne'
+    handler = None
+
+    @property
+    def negated(self):
+        return Q.eq
+
+    def eval(self, args, assumptions=True):
+        if assumptions == True:
+            # default assumptions for is_neq is None
+            assumptions = None
+        return is_neq(*args, assumptions)
+
+
+class StrictGreaterThanPredicate(BinaryRelation):
+    """
+    Binary predicate for $>$.
+
+    The purpose of this class is to provide the instance which represent
+    the ">" predicate in order to allow the logical inference.
+    This class must remain internal to assumptions module and user must
+    use :obj:`~.Gt()` instead to construct the equality expression.
+
+    Evaluating this predicate to ``True`` or ``False`` is done by
+    :func:`~.core.relational.is_gt()`
+
+    Examples
+    ========
+
+    >>> from sympy import ask, Q
+    >>> Q.gt(0, 0)
+    Q.gt(0, 0)
+    >>> ask(_)
+    False
+
+    See Also
+    ========
+
+    sympy.core.relational.Gt
+
+    """
+    is_reflexive = False
+    is_symmetric = False
+
+    name = 'gt'
+    handler = None
+
+    @property
+    def reversed(self):
+        return Q.lt
+
+    @property
+    def negated(self):
+        return Q.le
+
+    def eval(self, args, assumptions=True):
+        if assumptions == True:
+            # default assumptions for is_gt is None
+            assumptions = None
+        return is_gt(*args, assumptions)
+
+
+class GreaterThanPredicate(BinaryRelation):
+    """
+    Binary predicate for $>=$.
+
+    The purpose of this class is to provide the instance which represent
+    the ">=" predicate in order to allow the logical inference.
+    This class must remain internal to assumptions module and user must
+    use :obj:`~.Ge()` instead to construct the equality expression.
+
+    Evaluating this predicate to ``True`` or ``False`` is done by
+    :func:`~.core.relational.is_ge()`
+
+    Examples
+    ========
+
+    >>> from sympy import ask, Q
+    >>> Q.ge(0, 0)
+    Q.ge(0, 0)
+    >>> ask(_)
+    True
+
+    See Also
+    ========
+
+    sympy.core.relational.Ge
+
+    """
+    is_reflexive = True
+    is_symmetric = False
+
+    name = 'ge'
+    handler = None
+
+    @property
+    def reversed(self):
+        return Q.le
+
+    @property
+    def negated(self):
+        return Q.lt
+
+    def eval(self, args, assumptions=True):
+        if assumptions == True:
+            # default assumptions for is_ge is None
+            assumptions = None
+        return is_ge(*args, assumptions)
+
+
+class StrictLessThanPredicate(BinaryRelation):
+    """
+    Binary predicate for $<$.
+
+    The purpose of this class is to provide the instance which represent
+    the "<" predicate in order to allow the logical inference.
+    This class must remain internal to assumptions module and user must
+    use :obj:`~.Lt()` instead to construct the equality expression.
+
+    Evaluating this predicate to ``True`` or ``False`` is done by
+    :func:`~.core.relational.is_lt()`
+
+    Examples
+    ========
+
+    >>> from sympy import ask, Q
+    >>> Q.lt(0, 0)
+    Q.lt(0, 0)
+    >>> ask(_)
+    False
+
+    See Also
+    ========
+
+    sympy.core.relational.Lt
+
+    """
+    is_reflexive = False
+    is_symmetric = False
+
+    name = 'lt'
+    handler = None
+
+    @property
+    def reversed(self):
+        return Q.gt
+
+    @property
+    def negated(self):
+        return Q.ge
+
+    def eval(self, args, assumptions=True):
+        if assumptions == True:
+            # default assumptions for is_lt is None
+            assumptions = None
+        return is_lt(*args, assumptions)
+
+
+class LessThanPredicate(BinaryRelation):
+    """
+    Binary predicate for $<=$.
+
+    The purpose of this class is to provide the instance which represent
+    the "<=" predicate in order to allow the logical inference.
+    This class must remain internal to assumptions module and user must
+    use :obj:`~.Le()` instead to construct the equality expression.
+
+    Evaluating this predicate to ``True`` or ``False`` is done by
+    :func:`~.core.relational.is_le()`
+
+    Examples
+    ========
+
+    >>> from sympy import ask, Q
+    >>> Q.le(0, 0)
+    Q.le(0, 0)
+    >>> ask(_)
+    True
+
+    See Also
+    ========
+
+    sympy.core.relational.Le
+
+    """
+    is_reflexive = True
+    is_symmetric = False
+
+    name = 'le'
+    handler = None
+
+    @property
+    def reversed(self):
+        return Q.ge
+
+    @property
+    def negated(self):
+        return Q.gt
+
+    def eval(self, args, assumptions=True):
+        if assumptions == True:
+            # default assumptions for is_le is None
+            assumptions = None
+        return is_le(*args, assumptions)