Source code for diofant.core.relational

from ..logic.boolalg import Boolean, BooleanAtom, false, true
from .compatibility import ordered
from .evalf import EvalfMixin
from .evaluate import global_evaluate
from .expr import Expr
from .function import _coeff_isneg
from .symbol import Dummy, Symbol
from .sympify import sympify


__all__ = (
    'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge',
    'Relational', 'Equality', 'Unequality', 'StrictLessThan', 'LessThan',
    'StrictGreaterThan', 'GreaterThan',
)


# Note, see issue sympy/sympy#4986.  Ideally, we wouldn't want to subclass both Boolean
# and Expr.

[docs]class Relational(Boolean, Expr, EvalfMixin): """Base class for all relation types. Subclasses of Relational should generally be instantiated directly, but Relational can be instantiated with a valid `rop` value to dispatch to the appropriate subclass. Parameters ========== rop : str or None Indicates what subclass to instantiate. Valid values can be found in the keys of Relational.ValidRelationalOperator. Examples ======== >>> Rel(y, x+x**2, '==') Eq(y, x**2 + x) """ is_Relational = True # ValidRelationOperator - Defined below, because the necessary classes # have not yet been defined def __new__(cls, lhs, rhs, rop=None, **assumptions): # If called by a subclass, do nothing special and pass on to Expr. if cls is not Relational: return Expr.__new__(cls, lhs, rhs, **assumptions) # If called directly with an operator, look up the subclass # corresponding to that operator and delegate to it try: new_cls = cls.ValidRelationOperator[rop] return new_cls(lhs, rhs, **assumptions) except KeyError: raise ValueError("Invalid relational operator symbol: %r" % rop) @property def lhs(self): """The left-hand side of the relation.""" return self.args[0] @property def rhs(self): """The right-hand side of the relation.""" return self.args[1] @property def reversed(self): """Return the relationship with sides (and sign) reversed. Examples ======== >>> Eq(x, 1) Eq(x, 1) >>> _.reversed Eq(1, x) >>> x < 1 x < 1 >>> _.reversed 1 > x """ ops = {Gt: Lt, Ge: Le, Lt: Gt, Le: Ge} a, b = self.args return ops.get(self.func, self.func)(b, a, evaluate=False) def _eval_evalf(self, prec): return self.func(*[s._evalf(prec) for s in self.args]) @property def canonical(self): """Return a canonical form of the relational. The rules for the canonical form, in order of decreasing priority are: 1) Number on right if left is not a Number; 2) Symbol on the left; 3) Gt/Ge changed to Lt/Le; 4) Lt/Le are unchanged; 5) Eq and Ne get ordered args. """ r = self if r.func in (Ge, Gt): r = r.reversed elif r.func in (Lt, Le): pass elif r.func in (Eq, Ne): r = r.func(*ordered(r.args), evaluate=False) else: raise NotImplementedError if r.lhs.is_Number and not r.rhs.is_Number: r = r.reversed elif r.rhs.is_Symbol and not r.lhs.is_Symbol: r = r.reversed if _coeff_isneg(r.lhs): r = r.reversed.func(-r.lhs, -r.rhs, evaluate=False) return r
[docs] def equals(self, other, failing_expression=False): """Return True if the sides of the relationship are mathematically identical and the type of relationship is the same. If failing_expression is True, return the expression whose truth value was unknown. """ if isinstance(other, Relational): if self == other or self.reversed == other: return True a, b = self, other if a.func in (Eq, Ne) or b.func in (Eq, Ne): if a.func != b.func: return False l = a.lhs.equals(b.lhs, failing_expression=failing_expression) r = a.rhs.equals(b.rhs, failing_expression=failing_expression) if l is True: return r if r is True: return l lr = a.lhs.equals(b.rhs, failing_expression=failing_expression) rl = a.rhs.equals(b.lhs, failing_expression=failing_expression) if lr is True: return rl if rl is True: return lr e = (l, r, lr, rl) if all(i is False for i in e): return False if failing_expression: return a.lhs - a.rhs - b.lhs + b.rhs else: if b.func != a.func: b = b.reversed if a.func != b.func: return False l = a.lhs.equals(b.lhs, failing_expression=failing_expression) if l is False: return False r = a.rhs.equals(b.rhs, failing_expression=failing_expression) if r is False: return False if l is True: return r return l
def _eval_simplify(self, ratio, measure): r = self.func(self.lhs.simplify(ratio=ratio, measure=measure), self.rhs.simplify(ratio=ratio, measure=measure)) if r.is_Relational: dif = r.lhs - r.rhs # We want a Number to compare with zero and be sure to get a # True/False answer. Check if we can deduce that dif is # definitively zero or non-zero. if not dif.has(Dummy, Symbol): know = dif.equals(0) if know is False: if isinstance(r, Eq): return False elif isinstance(r, Ne): return True r = r.canonical if measure(r) < ratio*measure(self): return r else: return self def __bool__(self): raise TypeError("cannot determine truth value of Relational")
[docs] def as_set(self): """ Rewrites univariate inequality in terms of real sets Examples ======== >>> x = Symbol('x', real=True) >>> (x > 0).as_set() (0, oo) >>> Eq(x, 0).as_set() {0} """ from ..solvers.inequalities import solve_univariate_inequality syms = self.free_symbols if len(syms) == 1: sym = syms.pop() else: raise NotImplementedError("Sorry, Relational.as_set procedure" " is not yet implemented for" " multivariate expressions") return solve_univariate_inequality(self, sym, relational=False)
Rel = Relational
[docs]class Equality(Relational): """An equal relation between two objects. Represents that two objects are equal. If they can be easily shown to be definitively equal (or unequal), this will reduce to True (or False). Otherwise, the relation is maintained as an unevaluated Equality object. Use the ``simplify`` function on this object for more nontrivial evaluation of the equality relation. As usual, the keyword argument ``evaluate=False`` can be used to prevent any evaluation. Examples ======== >>> Eq(y, x + x**2) Eq(y, x**2 + x) >>> Eq(2, 5) false >>> Eq(2, 5, evaluate=False) Eq(2, 5) >>> _.doit() false >>> Eq(exp(x), exp(x).rewrite(cos)) Eq(E**x, sinh(x) + cosh(x)) >>> simplify(_) true See Also ======== diofant.logic.boolalg.Equivalent : for representing equality between two boolean expressions Notes ===== This class is not the same as the == operator. The == operator tests for exact structural equality between two expressions; this class compares expressions mathematically. If either object defines an `_eval_Eq` method, it can be used in place of the default algorithm. If `lhs._eval_Eq(rhs)` or `rhs._eval_Eq(lhs)` returns anything other than None, that return value will be substituted for the Equality. If None is returned by `_eval_Eq`, an Equality object will be created as usual. """ rel_op = '==' is_Equality = True def __new__(cls, lhs, rhs, **options): lhs = sympify(lhs, strict=True) rhs = sympify(rhs, strict=True) evaluate = options.pop('evaluate', global_evaluate[0]) if evaluate: # If one expression has an _eval_Eq, return its results. if hasattr(lhs, '_eval_Eq'): r = lhs._eval_Eq(rhs) if r is not None: return r if hasattr(rhs, '_eval_Eq'): r = rhs._eval_Eq(lhs) if r is not None: return r # If expressions have the same structure, they must be equal. if lhs == rhs: return true elif all(isinstance(i, BooleanAtom) for i in (rhs, lhs)): return false # equal args already evaluated # If appropriate, check if the difference evaluates. Detect # incompatibility such as lhs real and rhs not real. if isinstance(lhs, Expr) and isinstance(rhs, Expr): r = (lhs - rhs).is_zero if r is not None: return sympify(r, strict=True) return Relational.__new__(cls, lhs, rhs, **options)
Eq = Equality
[docs]class Unequality(Relational): """An unequal relation between two objects. Represents that two objects are not equal. If they can be shown to be definitively equal, this will reduce to False; if definitively unequal, this will reduce to True. Otherwise, the relation is maintained as an Unequality object. Examples ======== >>> Ne(y, x+x**2) Ne(y, x**2 + x) See Also ======== Equality Notes ===== This class is not the same as the != operator. The != operator tests for exact structural equality between two expressions; this class compares expressions mathematically. This class is effectively the inverse of Equality. As such, it uses the same algorithms, including any available `_eval_Eq` methods. """ rel_op = '!=' def __new__(cls, lhs, rhs, **options): lhs = sympify(lhs, strict=True) rhs = sympify(rhs, strict=True) evaluate = options.pop('evaluate', global_evaluate[0]) if evaluate: is_equal = Equality(lhs, rhs) if is_equal == true or is_equal == false: return ~is_equal return Relational.__new__(cls, lhs, rhs, **options)
Ne = Unequality class _Inequality(Relational): """Internal base class for all *Than types. Each subclass must implement _eval_relation to provide the method for comparing two real numbers. """ def __new__(cls, lhs, rhs, **options): lhs = sympify(lhs, strict=True) rhs = sympify(rhs, strict=True) evaluate = options.pop('evaluate', global_evaluate[0]) if evaluate: # First we invoke the appropriate inequality method of `lhs` # (e.g., `lhs.__lt__`). That method will try to reduce to # boolean or raise an exception. It may keep calling # superclasses until it reaches `Expr` (e.g., `Expr.__lt__`). # In some cases, `Expr` will just invoke us again (if neither it # nor a subclass was able to reduce to boolean or raise an # exception). In that case, it must call us with # `evaluate=False` to prevent infinite recursion. return cls._eval_relation(lhs, rhs) # make a "non-evaluated" Expr for the inequality return Relational.__new__(cls, lhs, rhs, **options) class _Greater(_Inequality): """Not intended for general use _Greater is only used so that GreaterThan and StrictGreaterThan may subclass it for the .gts and .lts properties. """ @property def gts(self): """Greater than side argument.""" return self.args[0] @property def lts(self): """Less than side argument.""" return self.args[1] class _Less(_Inequality): """Not intended for general use. _Less is only used so that LessThan and StrictLessThan may subclass it for the .gts and .lts properties. """ @property def gts(self): """Greater than side argument.""" return self.args[1] @property def lts(self): """Less than side argument.""" return self.args[0]
[docs]class GreaterThan(_Greater): r"""Class representations of inequalities. The ``*Than`` classes represent unequal relationships, where the left-hand side is generally bigger or smaller than the right-hand side. For example, the GreaterThan class represents an unequal relationship where the left-hand side is at least as big as the right side, if not bigger. In mathematical notation: lhs >= rhs In total, there are four ``*Than`` classes, to represent the four inequalities: +-----------------+--------+ |Class Name | Symbol | +=================+========+ |GreaterThan | (>=) | +-----------------+--------+ |LessThan | (<=) | +-----------------+--------+ |StrictGreaterThan| (>) | +-----------------+--------+ |StrictLessThan | (<) | +-----------------+--------+ All classes take two arguments, lhs and rhs. +----------------------------+-----------------+ |Signature Example | Math equivalent | +============================+=================+ |GreaterThan(lhs, rhs) | lhs >= rhs | +----------------------------+-----------------+ |LessThan(lhs, rhs) | lhs <= rhs | +----------------------------+-----------------+ |StrictGreaterThan(lhs, rhs) | lhs > rhs | +----------------------------+-----------------+ |StrictLessThan(lhs, rhs) | lhs < rhs | +----------------------------+-----------------+ In addition to the normal .lhs and .rhs of Relations, ``*Than`` inequality objects also have the .lts and .gts properties, which represent the "less than side" and "greater than side" of the operator. Use of .lts and .gts in an algorithm rather than .lhs and .rhs as an assumption of inequality direction will make more explicit the intent of a certain section of code, and will make it similarly more robust to client code changes: >>> e = GreaterThan(x, 1) >>> e x >= 1 >>> '%s >= %s is the same as %s <= %s' % (e.gts, e.lts, e.lts, e.gts) 'x >= 1 is the same as 1 <= x' Examples ======== One generally does not instantiate these classes directly, but uses various convenience methods: >>> e1 = Ge(x, 2) # Ge is a convenience wrapper >>> print(e1) x >= 2 >>> rels = Ge(x, 2), Gt(x, 2), Le(x, 2), Lt(x, 2) >>> print('%s\n%s\n%s\n%s' % rels) x >= 2 x > 2 x <= 2 x < 2 Another option is to use the Python inequality operators (>=, >, <=, <) directly. Their main advantage over the Ge, Gt, Le, and Lt counterparts, is that one can write a more "mathematical looking" statement rather than littering the math with oddball function calls. However there are certain (minor) caveats of which to be aware (search for 'gotcha', below). >>> e2 = x >= 2 >>> print(e2) x >= 2 >>> print("e1: %s, e2: %s" % (e1, e2)) e1: x >= 2, e2: x >= 2 >>> e1 == e2 True However, it is also perfectly valid to instantiate a ``*Than`` class less succinctly and less conveniently: >>> rels = Rel(x, 1, '>='), Relational(x, 1, '>='), GreaterThan(x, 1) >>> print('%s\n%s\n%s' % rels) x >= 1 x >= 1 x >= 1 >>> rels = Rel(x, 1, '>'), Relational(x, 1, '>'), StrictGreaterThan(x, 1) >>> print('%s\n%s\n%s' % rels) x > 1 x > 1 x > 1 >>> rels = Rel(x, 1, '<='), Relational(x, 1, '<='), LessThan(x, 1) >>> print("%s\n%s\n%s" % rels) x <= 1 x <= 1 x <= 1 >>> rels = Rel(x, 1, '<'), Relational(x, 1, '<'), StrictLessThan(x, 1) >>> print('%s\n%s\n%s' % rels) x < 1 x < 1 x < 1 Notes ===== There are a couple of "gotchas" when using Python's operators. The first enters the mix when comparing against a literal number as the lhs argument. Due to the order that Python decides to parse a statement, it may not immediately find two objects comparable. For example, to evaluate the statement (1 < x), Python will first recognize the number 1 as a native number, and then that x is *not* a native number. At this point, because a native Python number does not know how to compare itself with a Diofant object Python will try the reflective operation, (x > 1). Unfortunately, there is no way available to Diofant to recognize this has happened, so the statement (1 < x) will turn silently into (x > 1). >>> e1 = x > 1 >>> e2 = x >= 1 >>> e3 = x < 1 >>> e4 = x <= 1 >>> e5 = 1 > x >>> e6 = 1 >= x >>> e7 = 1 < x >>> e8 = 1 <= x >>> print("%s %s\n"*4 % (e1, e2, e3, e4, e5, e6, e7, e8)) x > 1 x >= 1 x < 1 x <= 1 x < 1 x <= 1 x > 1 x >= 1 If the order of the statement is important (for visual output to the console, perhaps), one can work around this annoyance in a couple ways: (1) "sympify" the literal before comparison, (2) use one of the wrappers, or (3) use the less succinct methods described above: >>> e1 = Integer(1) > x >>> e2 = Integer(1) >= x >>> e3 = Integer(1) < x >>> e4 = Integer(1) <= x >>> e5 = Gt(1, x) >>> e6 = Ge(1, x) >>> e7 = Lt(1, x) >>> e8 = Le(1, x) >>> print("%s %s\n"*4 % (e1, e2, e3, e4, e5, e6, e7, e8)) 1 > x 1 >= x 1 < x 1 <= x 1 > x 1 >= x 1 < x 1 <= x The other gotcha is with chained inequalities. Occasionally, one may be tempted to write statements like: >>> x < y < z Traceback (most recent call last): ... TypeError: symbolic boolean expression has no truth value. Due to an implementation detail or decision of Python, to create a chained inequality, the only method currently available is to make use of And: >>> And(x < y, y < z) (x < y) & (y < z) """ rel_op = '>=' @classmethod def _eval_relation(cls, lhs, rhs): return sympify(lhs >= rhs, strict=True)
Ge = GreaterThan
[docs]class LessThan(_Less): __doc__ = GreaterThan.__doc__ rel_op = '<=' @classmethod def _eval_relation(cls, lhs, rhs): return sympify(lhs <= rhs, strict=True)
Le = LessThan
[docs]class StrictGreaterThan(_Greater): __doc__ = GreaterThan.__doc__ rel_op = '>' @classmethod def _eval_relation(cls, lhs, rhs): return sympify(lhs > rhs, strict=True)
Gt = StrictGreaterThan
[docs]class StrictLessThan(_Less): __doc__ = GreaterThan.__doc__ rel_op = '<' @classmethod def _eval_relation(cls, lhs, rhs): return sympify(lhs < rhs, strict=True)
Lt = StrictLessThan # A class-specific (not object-specific) data item used for a minor speedup. It # is defined here, rather than directly in the class, because the classes that # it references have not been defined until now (e.g. StrictLessThan). Relational.ValidRelationOperator = { None: Equality, '==': Equality, 'eq': Equality, '!=': Unequality, '<>': Unequality, 'ne': Unequality, '>=': GreaterThan, 'ge': GreaterThan, '<=': LessThan, 'le': LessThan, '>': StrictGreaterThan, 'gt': StrictGreaterThan, '<': StrictLessThan, 'lt': StrictLessThan, }