Beispiel #1
0
def get_all_relevant_facts(proposition, assumptions=True,
    context=global_assumptions, use_known_facts=True, iterations=oo):
    # The relevant facts might introduce new keys, e.g., Q.zero(x*y) will
    # introduce the keys Q.zero(x) and Q.zero(y), so we need to run it until
    # we stop getting new things. Hopefully this strategy won't lead to an
    # infinite loop in the future.
    i = 0
    relevant_facts = set()
    exprs = None
    while exprs != set():
        (relevant_facts, exprs) = get_relevant_facts(proposition, And.make_args(assumptions),
                context, use_known_facts=use_known_facts, exprs=exprs, relevant_facts=relevant_facts)
        i += 1
        if i >= iterations:
            return And(*relevant_facts)

    return And(*relevant_facts)
Beispiel #2
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def ask(proposition, assumptions=True, context=global_assumptions):
    """
    Method for inferring properties about objects.

    **Syntax**

        * ask(proposition)

        * ask(proposition, assumptions)

            where ``proposition`` is any boolean expression

    Examples
    ========
    >>> from sympy import ask, Q, pi
    >>> from sympy.abc import x, y
    >>> ask(Q.rational(pi))
    False
    >>> ask(Q.even(x*y), Q.even(x) & Q.integer(y))
    True
    >>> ask(Q.prime(x*y), Q.integer(x) &  Q.integer(y))
    False

    **Remarks**
        Relations in assumptions are not implemented (yet), so the following
        will not give a meaningful result.

        >>> ask(Q.positive(x), Q.is_true(x > 0)) # doctest: +SKIP

        It is however a work in progress and should be available before
        the official release

    """
    assumptions = And(assumptions, And(*context))
    if isinstance(proposition, AppliedPredicate):
        key, expr = proposition.func, sympify(proposition.arg)
    else:
        key, expr = Q.is_true, sympify(proposition)

    # direct resolution method, no logic
    res = key(expr)._eval_ask(assumptions)
    if res is not None:
        return res

    if assumptions is True:
        return

    if not expr.is_Atom:
        return

    local_facts = _extract_facts(assumptions, expr)
    if local_facts is None or local_facts is True:
        return

    # See if there's a straight-forward conclusion we can make for the inference
    if local_facts.is_Atom:
        if key in known_facts_dict[local_facts]:
            return True
        if Not(key) in known_facts_dict[local_facts]:
            return False
    elif local_facts.func is And and all(k in known_facts_dict
                                         for k in local_facts.args):
        for assum in local_facts.args:
            if assum.is_Atom:
                if key in known_facts_dict[assum]:
                    return True
                if Not(key) in known_facts_dict[assum]:
                    return False
            elif assum.func is Not and assum.args[0].is_Atom:
                if key in known_facts_dict[assum]:
                    return False
                if Not(key) in known_facts_dict[assum]:
                    return True
    elif (isinstance(key, Predicate) and local_facts.func is Not
          and local_facts.args[0].is_Atom):
        if local_facts.args[0] in known_facts_dict[key]:
            return False

    # Failing all else, we do a full logical inference
    return ask_full_inference(key, local_facts)
Beispiel #3
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def test_sympy__logic__boolalg__And():
    from sympy.logic.boolalg import And
    assert _test_args(And(x, y, 2))
Beispiel #4
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 def as_relational(self, other):
     return And(
         Lambda(self.sym, self.condition)(other),
         self.base_set.contains(other))
Beispiel #5
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    def __new__(cls, sym, condition, base_set=S.UniversalSet):
        # nonlinsolve uses ConditionSet to return an unsolved system
        # of equations (see _return_conditionset in solveset) so until
        # that is changed we do minimal checking of the args
        sym = _sympify(sym)
        base_set = _sympify(base_set)
        condition = _sympify(condition)

        if isinstance(condition, FiniteSet):
            condition_orig = condition
            temp = (Eq(lhs, 0) for lhs in condition)
            condition = And(*temp)
            SymPyDeprecationWarning(
                feature="Using {} for condition".format(condition_orig),
                issue=17651,
                deprecated_since_version='1.5',
                useinstead="{} for condition".format(condition)).warn()

        condition = as_Boolean(condition)

        if isinstance(sym, Tuple):  # unsolved eqns syntax
            return Basic.__new__(cls, sym, condition, base_set)

        if not isinstance(base_set, Set):
            raise TypeError('expecting set for base_set')

        if condition is S.false:
            return S.EmptySet
        elif condition is S.true:
            return base_set
        if isinstance(base_set, EmptySet):
            return base_set

        know = None
        if isinstance(base_set, FiniteSet):
            sifted = sift(base_set,
                          lambda _: fuzzy_bool(condition.subs(sym, _)))
            if sifted[None]:
                know = FiniteSet(*sifted[True])
                base_set = FiniteSet(*sifted[None])
            else:
                return FiniteSet(*sifted[True])

        if isinstance(base_set, cls):
            s, c, base_set = base_set.args
            if sym == s:
                condition = And(condition, c)
            elif sym not in c.free_symbols:
                condition = And(condition, c.xreplace({s: sym}))
            elif s not in condition.free_symbols:
                condition = And(condition.xreplace({sym: s}), c)
                sym = s
            else:
                # user will have to use cls.sym to get symbol
                dum = Symbol('lambda')
                if dum in condition.free_symbols or \
                        dum in c.free_symbols:
                    dum = Dummy(str(dum))
                condition = And(condition.xreplace({sym: dum}),
                                c.xreplace({s: dum}))
                sym = dum

        if not isinstance(sym, Symbol):
            s = Dummy('lambda')
            if s not in condition.xreplace({sym: s}).free_symbols:
                raise ValueError(
                    'non-symbol dummy not recognized in condition')

        rv = Basic.__new__(cls, sym, condition, base_set)
        return rv if know is None else Union(know, rv)
Beispiel #6
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def test_fcode_Logical():
    x, y, z = symbols("x y z")
    # unary Not
    assert fcode(Not(x), source_format="free") == ".not. x"
    # binary And
    assert fcode(And(x, y), source_format="free") == "x .and. y"
    assert fcode(And(x, Not(y)), source_format="free") == "x .and. .not. y"
    assert fcode(And(Not(x), y), source_format="free") == "y .and. .not. x"
    assert fcode(And(Not(x), Not(y)), source_format="free") == \
        ".not. x .and. .not. y"
    assert fcode(Not(And(x, y), evaluate=False), source_format="free") == \
        ".not. (x .and. y)"
    # binary Or
    assert fcode(Or(x, y), source_format="free") == "x .or. y"
    assert fcode(Or(x, Not(y)), source_format="free") == "x .or. .not. y"
    assert fcode(Or(Not(x), y), source_format="free") == "y .or. .not. x"
    assert fcode(Or(Not(x), Not(y)), source_format="free") == \
        ".not. x .or. .not. y"
    assert fcode(Not(Or(x, y), evaluate=False), source_format="free") == \
        ".not. (x .or. y)"
    # mixed And/Or
    assert fcode(And(Or(y, z), x), source_format="free") == "x .and. (y .or. z)"
    assert fcode(And(Or(z, x), y), source_format="free") == "y .and. (x .or. z)"
    assert fcode(And(Or(x, y), z), source_format="free") == "z .and. (x .or. y)"
    assert fcode(Or(And(y, z), x), source_format="free") == "x .or. y .and. z"
    assert fcode(Or(And(z, x), y), source_format="free") == "y .or. x .and. z"
    assert fcode(Or(And(x, y), z), source_format="free") == "z .or. x .and. y"
    # trinary And
    assert fcode(And(x, y, z), source_format="free") == "x .and. y .and. z"
    assert fcode(And(x, y, Not(z)), source_format="free") == \
        "x .and. y .and. .not. z"
    assert fcode(And(x, Not(y), z), source_format="free") == \
        "x .and. z .and. .not. y"
    assert fcode(And(Not(x), y, z), source_format="free") == \
        "y .and. z .and. .not. x"
    assert fcode(Not(And(x, y, z), evaluate=False), source_format="free") == \
        ".not. (x .and. y .and. z)"
    # trinary Or
    assert fcode(Or(x, y, z), source_format="free") == "x .or. y .or. z"
    assert fcode(Or(x, y, Not(z)), source_format="free") == \
        "x .or. y .or. .not. z"
    assert fcode(Or(x, Not(y), z), source_format="free") == \
        "x .or. z .or. .not. y"
    assert fcode(Or(Not(x), y, z), source_format="free") == \
        "y .or. z .or. .not. x"
    assert fcode(Not(Or(x, y, z), evaluate=False), source_format="free") == \
        ".not. (x .or. y .or. z)"
Beispiel #7
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def test_BooleanFunction_diff():
    assert And(x, y).diff(x) == Piecewise((0, Eq(y, False)), (1, True))
Beispiel #8
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def test_issue_8975():
    assert Or(And(-oo < x, x <= -2), And(2 <= x, x < oo)).as_set() == \
        Interval(-oo, -2) + Interval(2, oo)
Beispiel #9
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def to_NNF(expr):
    """
    Generates the Negation Normal Form of any boolean expression in terms
    of AND, OR, and Literal objects.
    """

    if isinstance(expr, Not):
        arg = expr.args[0]
        tmp = to_NNF(arg)  # Strategy: negate the NNF of expr
        return ~tmp

    if isinstance(expr, Or):
        return OR(*[to_NNF(x) for x in Or.make_args(expr)])

    if isinstance(expr, And):
        return AND(*[to_NNF(x) for x in And.make_args(expr)])

    if isinstance(expr, Nand):
        tmp = AND(*[to_NNF(x) for x in expr.args])
        return ~tmp

    if isinstance(expr, Nor):
        tmp = OR(*[to_NNF(x) for x in expr.args])
        return ~tmp

    if isinstance(expr, Xor):
        cnfs = []
        for i in range(0, len(expr.args) + 1, 2):
            for neg in combinations(expr.args, i):
                clause = [
                    ~to_NNF(s) if s in neg else to_NNF(s) for s in expr.args
                ]
                cnfs.append(OR(*clause))
        return AND(*cnfs)

    if isinstance(expr, Xnor):
        cnfs = []
        for i in range(0, len(expr.args) + 1, 2):
            for neg in combinations(expr.args, i):
                clause = [
                    ~to_NNF(s) if s in neg else to_NNF(s) for s in expr.args
                ]
                cnfs.append(OR(*clause))
        return ~AND(*cnfs)

    if isinstance(expr, Implies):
        L, R = to_NNF(expr.args[0]), to_NNF(expr.args[1])
        return OR(~L, R)

    if isinstance(expr, Equivalent):
        cnfs = []
        for a, b in zip_longest(expr.args,
                                expr.args[1:],
                                fillvalue=expr.args[0]):
            a = to_NNF(a)
            b = to_NNF(b)
            cnfs.append(OR(~a, b))
        return AND(*cnfs)

    if isinstance(expr, ITE):
        L = to_NNF(expr.args[0])
        M = to_NNF(expr.args[1])
        R = to_NNF(expr.args[2])
        return AND(OR(~L, M), OR(L, R))

    else:
        return Literal(expr)
Beispiel #10
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    def _eval_interval(self, sym, a, b):
        """Evaluates the function along the sym in a given interval ab"""
        # FIXME: Currently complex intervals are not supported.  A possible
        # replacement algorithm, discussed in issue 2128, can be found in the
        # following papers;
        #     http://portal.acm.org/citation.cfm?id=281649
        #     http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.4127&rep=rep1&type=pdf

        if a is None or b is None:
            # In this case, it is just simple substitution
            return piecewise_fold(
                super(Piecewise, self)._eval_interval(sym, a, b))

        mul = 1
        if (a == b) is True:
            return S.Zero
        elif (a > b) is True:
            a, b, mul = b, a, -1
        elif (a <= b) is not True:
            newargs = []
            for e, c in self.args:
                intervals = self._sort_expr_cond(sym, S.NegativeInfinity,
                                                 S.Infinity, c)
                values = []
                for lower, upper, expr in intervals:
                    if (a < lower) is True:
                        mid = lower
                        rep = b
                        val = e._eval_interval(sym, mid, b)
                        val += self._eval_interval(sym, a, mid)
                    elif (a > upper) is True:
                        mid = upper
                        rep = b
                        val = e._eval_interval(sym, mid, b)
                        val += self._eval_interval(sym, a, mid)
                    elif (a >= lower) is True and (a <= upper) is True:
                        rep = b
                        val = e._eval_interval(sym, a, b)
                    elif (b < lower) is True:
                        mid = lower
                        rep = a
                        val = e._eval_interval(sym, a, mid)
                        val += self._eval_interval(sym, mid, b)
                    elif (b > upper) is True:
                        mid = upper
                        rep = a
                        val = e._eval_interval(sym, a, mid)
                        val += self._eval_interval(sym, mid, b)
                    elif ((b >= lower) is True) and ((b <= upper) is True):
                        rep = a
                        val = e._eval_interval(sym, a, b)
                    else:
                        raise NotImplementedError(
                            """The evaluation of a Piecewise interval when both the lower
                            and the upper limit are symbolic is not yet implemented."""
                        )
                    values.append(val)
                if len(set(values)) == 1:
                    try:
                        c = c.subs(sym, rep)
                    except AttributeError:
                        pass
                    e = values[0]
                    newargs.append((e, c))
                else:
                    for i in range(len(values)):
                        newargs.append(
                            (values[i], (c is True and i == len(values) - 1)
                             or And(rep >= intervals[i][0],
                                    rep <= intervals[i][1])))
            return self.func(*newargs)

        # Determine what intervals the expr,cond pairs affect.
        int_expr = self._sort_expr_cond(sym, a, b)

        # Finally run through the intervals and sum the evaluation.
        ret_fun = 0
        for int_a, int_b, expr in int_expr:
            if isinstance(expr, Piecewise):
                # If we still have a Piecewise by now, _sort_expr_cond would
                # already have determined that its conditions are independent
                # of the integration variable, thus we just use substitution.
                ret_fun += piecewise_fold(
                    super(Piecewise,
                          expr)._eval_interval(sym, Max(a, int_a),
                                               Min(b, int_b)))
            else:
                ret_fun += expr._eval_interval(sym, Max(a, int_a),
                                               Min(b, int_b))
        return mul * ret_fun
Beispiel #11
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def ask(proposition, assumptions=True, context=global_assumptions):
    """
    Method for inferring properties about objects.

    **Syntax**

        * ask(proposition)

        * ask(proposition, assumptions)

            where ``proposition`` is any boolean expression

    Examples
    ========

    >>> from sympy import ask, Q, pi
    >>> from sympy.abc import x, y
    >>> ask(Q.rational(pi))
    False
    >>> ask(Q.even(x*y), Q.even(x) & Q.integer(y))
    True
    >>> ask(Q.prime(4*x), Q.integer(x))
    False

    **Remarks**
        Relations in assumptions are not implemented (yet), so the following
        will not give a meaningful result.

        >>> ask(Q.positive(x), Q.is_true(x > 0)) # doctest: +SKIP

        It is however a work in progress.

    """
    from sympy.assumptions.satask import satask

    if not isinstance(proposition, (BooleanFunction, AppliedPredicate, bool, BooleanAtom)):
        raise TypeError("proposition must be a valid logical expression")

    if not isinstance(assumptions, (BooleanFunction, AppliedPredicate, bool, BooleanAtom)):
        raise TypeError("assumptions must be a valid logical expression")

    if isinstance(proposition, AppliedPredicate):
        key, expr = proposition.func, sympify(proposition.arg)
    else:
        key, expr = Q.is_true, sympify(proposition)

    assumptions = And(assumptions, And(*context))
    assumptions = to_cnf(assumptions)

    local_facts = _extract_facts(assumptions, expr)

    known_facts_cnf = get_known_facts_cnf()
    known_facts_dict = get_known_facts_dict()

    if local_facts and satisfiable(And(local_facts, known_facts_cnf)) is False:
        raise ValueError("inconsistent assumptions %s" % assumptions)

    # direct resolution method, no logic
    res = key(expr)._eval_ask(assumptions)
    if res is not None:
        return bool(res)

    if local_facts is None:
        return satask(proposition, assumptions=assumptions, context=context)


    # See if there's a straight-forward conclusion we can make for the inference
    if local_facts.is_Atom:
        if key in known_facts_dict[local_facts]:
            return True
        if Not(key) in known_facts_dict[local_facts]:
            return False
    elif (local_facts.func is And and
            all(k in known_facts_dict for k in local_facts.args)):
        for assum in local_facts.args:
            if assum.is_Atom:
                if key in known_facts_dict[assum]:
                    return True
                if Not(key) in known_facts_dict[assum]:
                    return False
            elif assum.func is Not and assum.args[0].is_Atom:
                if key in known_facts_dict[assum]:
                    return False
                if Not(key) in known_facts_dict[assum]:
                    return True
    elif (isinstance(key, Predicate) and
            local_facts.func is Not and local_facts.args[0].is_Atom):
        if local_facts.args[0] in known_facts_dict[key]:
            return False

    # Failing all else, we do a full logical inference
    res = ask_full_inference(key, local_facts, known_facts_cnf)
    if res is None:
        return satask(proposition, assumptions=assumptions, context=context)
    return res
Beispiel #12
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def test_to_anf():
    x, y, z = symbols('x,y,z')
    assert to_anf(And(x, y)) == And(x, y)
    assert to_anf(Or(x, y)) == Xor(x, y, And(x, y))
    assert to_anf(Or(Implies(x, y), And(x, y), y)) == \
            Xor(x, True, x & y, remove_true=False)
    assert to_anf(Or(Nand(x, y), Nor(x, y), Xnor(x, y), Implies(x, y))) == True
    assert to_anf(Or(x, Not(y), Nor(x,z), And(x, y), Nand(y, z))) == \
            Xor(True, And(y, z), And(x, y, z), remove_true=False)
    assert to_anf(Xor(x, y)) == Xor(x, y)
    assert to_anf(Not(x)) == Xor(x, True, remove_true=False)
    assert to_anf(Nand(x, y)) == Xor(True, And(x, y), remove_true=False)
    assert to_anf(Nor(x, y)) == Xor(x, y, True, And(x, y), remove_true=False)
    assert to_anf(Implies(x, y)) == Xor(x, True, And(x, y), remove_true=False)
    assert to_anf(Equivalent(x, y)) == Xor(x, y, True, remove_true=False)
    assert to_anf(Nand(x | y, x >> y), deep=False) == \
            Xor(True, And(Or(x, y), Implies(x, y)), remove_true=False)
    assert to_anf(Nor(x ^ y, x & y), deep=False) == \
            Xor(True, Or(Xor(x, y), And(x, y)), remove_true=False)
Beispiel #13
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def test_And():
    assert And() is true
    assert And(A) == A
    assert And(True) is true
    assert And(False) is false
    assert And(True, True) is true
    assert And(True, False) is false
    assert And(False, False) is false
    assert And(True, A) == A
    assert And(False, A) is false
    assert And(True, True, True) is true
    assert And(True, True, A) == A
    assert And(True, False, A) is false
    assert And(1, A) == A
    raises(TypeError, lambda: And(2, A))
    raises(TypeError, lambda: And(A < 2, A))
    assert And(A < 1, A >= 1) is false
    e = A > 1
    assert And(e, e.canonical) == e.canonical
    g, l, ge, le = A > B, B < A, A >= B, B <= A
    assert And(g, l, ge, le) == And(ge, g)
    assert {And(*i) for i in permutations((l,g,le,ge))} == {And(ge, g)}
    assert And(And(Eq(a, 0), Eq(b, 0)), And(Ne(a, 0), Eq(c, 0))) is false
Beispiel #14
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def test_simplification():
    """
    Test working of simplification methods.
    """
    set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]]
    set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]]
    assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x))
    assert Not(SOPform([x, y, z], set2)) == \
        Not(Or(And(Not(x), Not(z)), And(x, z)))
    assert POSform([x, y, z], set1 + set2) is true
    assert SOPform([x, y, z], set1 + set2) is true
    assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true

    minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
                [1, 1, 1, 1]]
    dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
    assert (
        SOPform([w, x, y, z], minterms, dontcares) ==
        Or(And(y, z), And(Not(w), Not(x))))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)

    minterms = [1, 3, 7, 11, 15]
    dontcares = [0, 2, 5]
    assert (
        SOPform([w, x, y, z], minterms, dontcares) ==
        Or(And(y, z), And(Not(w), Not(x))))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)

    minterms = [1, [0, 0, 1, 1], 7, [1, 0, 1, 1],
                [1, 1, 1, 1]]
    dontcares = [0, [0, 0, 1, 0], 5]
    assert (
        SOPform([w, x, y, z], minterms, dontcares) ==
        Or(And(y, z), And(Not(w), Not(x))))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)

    minterms = [1, {y: 1, z: 1}]
    dontcares = [0, [0, 0, 1, 0], 5]
    assert (
        SOPform([w, x, y, z], minterms, dontcares) ==
        Or(And(y, z), And(Not(w), Not(x))))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)


    minterms = [{y: 1, z: 1}, 1]
    dontcares = [[0, 0, 0, 0]]

    minterms = [[0, 0, 0]]
    raises(ValueError, lambda: SOPform([w, x, y, z], minterms))
    raises(ValueError, lambda: POSform([w, x, y, z], minterms))

    raises(TypeError, lambda: POSform([w, x, y, z], ["abcdefg"]))

    # test simplification
    ans = And(A, Or(B, C))
    assert simplify_logic(A & (B | C)) == ans
    assert simplify_logic((A & B) | (A & C)) == ans
    assert simplify_logic(Implies(A, B)) == Or(Not(A), B)
    assert simplify_logic(Equivalent(A, B)) == \
        Or(And(A, B), And(Not(A), Not(B)))
    assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C)
    assert simplify_logic(And(Equality(A, 2), A)) is S.false
    assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A)
    assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C)
    assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \
        == And(Equality(A, 3), Or(B, C))
    b = (~x & ~y & ~z) | (~x & ~y & z)
    e = And(A, b)
    assert simplify_logic(e) == A & ~x & ~y
    raises(ValueError, lambda: simplify_logic(A & (B | C), form='blabla'))

    # Check that expressions with nine variables or more are not simplified
    # (without the force-flag)
    a, b, c, d, e, f, g, h, j = symbols('a b c d e f g h j')
    expr = a & b & c & d & e & f & g & h & j | \
        a & b & c & d & e & f & g & h & ~j
    # This expression can be simplified to get rid of the j variables
    assert simplify_logic(expr) == expr

    # check input
    ans = SOPform([x, y], [[1, 0]])
    assert SOPform([x, y], [[1, 0]]) == ans
    assert POSform([x, y], [[1, 0]]) == ans

    raises(ValueError, lambda: SOPform([x], [[1]], [[1]]))
    assert SOPform([x], [[1]], [[0]]) is true
    assert SOPform([x], [[0]], [[1]]) is true
    assert SOPform([x], [], []) is false

    raises(ValueError, lambda: POSform([x], [[1]], [[1]]))
    assert POSform([x], [[1]], [[0]]) is true
    assert POSform([x], [[0]], [[1]]) is true
    assert POSform([x], [], []) is false

    # check working of simplify
    assert simplify((A & B) | (A & C)) == And(A, Or(B, C))
    assert simplify(And(x, Not(x))) == False
    assert simplify(Or(x, Not(x))) == True
    assert simplify(And(Eq(x, 0), Eq(x, y))) == And(Eq(x, 0), Eq(y, 0))
    assert And(Eq(x - 1, 0), Eq(x, y)).simplify() == And(Eq(x, 1), Eq(y, 1))
    assert And(Ne(x - 1, 0), Ne(x, y)).simplify() == And(Ne(x, 1), Ne(x, y))
    assert And(Eq(x - 1, 0), Ne(x, y)).simplify() == And(Eq(x, 1), Ne(y, 1))
    assert And(Eq(x - 1, 0), Eq(x, z + y), Eq(y + x, 0)).simplify(
        ) == And(Eq(x, 1), Eq(y, -1), Eq(z, 2))
    assert And(Eq(x - 1, 0), Eq(x + 2, 3)).simplify() == Eq(x, 1)
    assert And(Ne(x - 1, 0), Ne(x + 2, 3)).simplify() == Ne(x, 1)
    assert And(Eq(x - 1, 0), Eq(x + 2, 2)).simplify() == False
    assert And(Ne(x - 1, 0), Ne(x + 2, 2)).simplify(
        ) == And(Ne(x, 1), Ne(x, 0))
Beispiel #15
0
def test_convert_to_varsSOP():
    assert _convert_to_varsSOP([0, 1, 0], [x, y, z]) ==  And(Not(x), y, Not(z))
    assert _convert_to_varsSOP([3, 1, 0], [x, y, z]) ==  And(y, Not(z))
Beispiel #16
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def test_issue_8777():
    assert And(x > 2, x < oo).as_set() == Interval(2, oo, left_open=True)
    assert And(x >= 1, x < oo).as_set() == Interval(1, oo)
    assert (x < oo).as_set() == Interval(-oo, oo)
    assert (x > -oo).as_set() == Interval(-oo, oo)
Beispiel #17
0
 def _contains(self, other):
     from sympy.logic.boolalg import And
     return And(*[set.contains(other) for set in self.args])
Beispiel #18
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def test_truth_table():
    assert list(truth_table(And(x, y), [x, y], input=False)) == [False, False, False, True]
    assert list(truth_table(x | y, [x, y], input=False)) == [False, True, True, True]
    assert list(truth_table(x >> y, [x, y], input=False)) == [True, True, False, True]
Beispiel #19
0
 def as_relational(self, symbol):
     """Rewrite an Intersection in terms of equalities and logic operators"""
     return And(*[set.as_relational(symbol) for set in self.args])
Beispiel #20
0
import pprint
import re
from copy import deepcopy

from PLA_functions import *

_input = {"(~(a & b))", "(a|b)", "a", "b"}

a, b, c, d, p, q, r = symbols("a, b, c, d, p, q, r")

props = {a, b, c, d}

form1 = (~(a & d) | (b & c))
form2 = a | b | d

res = And(form1, form2)

res = to_cnf(res)
print(res)

regular = pre_cnf_to_cnf(res, props)
print(regular)

flist = build_formula_list(_input)
print("f list")
for f in flist:
    print(f)

basis = build_res_set(flist, props)
print("f set")
for s in basis:
Beispiel #21
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def test_equals():
    assert Not(Or(A, B)).equals( And(Not(A), Not(B)) ) is True
    assert Equivalent(A, B).equals((A >> B) & (B >> A)) is True
    assert ((A | ~B) & (~A | B)).equals((~A & ~B) | (A & B)) is True
    assert (A >> B).equals(~A >> ~B) is False
    assert (A >> (B >> A)).equals(A >> (C >> A)) is False
Beispiel #22
0
def test_fcode_Xlogical():
    x, y, z = symbols("x y z")
    # binary Xor
    assert fcode(Xor(x, y, evaluate=False), source_format="free") == \
        "x .neqv. y"
    assert fcode(Xor(x, Not(y), evaluate=False), source_format="free") == \
        "x .neqv. .not. y"
    assert fcode(Xor(Not(x), y, evaluate=False), source_format="free") == \
        "y .neqv. .not. x"
    assert fcode(Xor(Not(x), Not(y), evaluate=False),
        source_format="free") == ".not. x .neqv. .not. y"
    assert fcode(Not(Xor(x, y, evaluate=False), evaluate=False),
        source_format="free") == ".not. (x .neqv. y)"
    # binary Equivalent
    assert fcode(Equivalent(x, y), source_format="free") == "x .eqv. y"
    assert fcode(Equivalent(x, Not(y)), source_format="free") == \
        "x .eqv. .not. y"
    assert fcode(Equivalent(Not(x), y), source_format="free") == \
        "y .eqv. .not. x"
    assert fcode(Equivalent(Not(x), Not(y)), source_format="free") == \
        ".not. x .eqv. .not. y"
    assert fcode(Not(Equivalent(x, y), evaluate=False),
        source_format="free") == ".not. (x .eqv. y)"
    # mixed And/Equivalent
    assert fcode(Equivalent(And(y, z), x), source_format="free") == \
        "x .eqv. y .and. z"
    assert fcode(Equivalent(And(z, x), y), source_format="free") == \
        "y .eqv. x .and. z"
    assert fcode(Equivalent(And(x, y), z), source_format="free") == \
        "z .eqv. x .and. y"
    assert fcode(And(Equivalent(y, z), x), source_format="free") == \
        "x .and. (y .eqv. z)"
    assert fcode(And(Equivalent(z, x), y), source_format="free") == \
        "y .and. (x .eqv. z)"
    assert fcode(And(Equivalent(x, y), z), source_format="free") == \
        "z .and. (x .eqv. y)"
    # mixed Or/Equivalent
    assert fcode(Equivalent(Or(y, z), x), source_format="free") == \
        "x .eqv. y .or. z"
    assert fcode(Equivalent(Or(z, x), y), source_format="free") == \
        "y .eqv. x .or. z"
    assert fcode(Equivalent(Or(x, y), z), source_format="free") == \
        "z .eqv. x .or. y"
    assert fcode(Or(Equivalent(y, z), x), source_format="free") == \
        "x .or. (y .eqv. z)"
    assert fcode(Or(Equivalent(z, x), y), source_format="free") == \
        "y .or. (x .eqv. z)"
    assert fcode(Or(Equivalent(x, y), z), source_format="free") == \
        "z .or. (x .eqv. y)"
    # mixed Xor/Equivalent
    assert fcode(Equivalent(Xor(y, z, evaluate=False), x),
        source_format="free") == "x .eqv. (y .neqv. z)"
    assert fcode(Equivalent(Xor(z, x, evaluate=False), y),
        source_format="free") == "y .eqv. (x .neqv. z)"
    assert fcode(Equivalent(Xor(x, y, evaluate=False), z),
        source_format="free") == "z .eqv. (x .neqv. y)"
    assert fcode(Xor(Equivalent(y, z), x, evaluate=False),
        source_format="free") == "x .neqv. (y .eqv. z)"
    assert fcode(Xor(Equivalent(z, x), y, evaluate=False),
        source_format="free") == "y .neqv. (x .eqv. z)"
    assert fcode(Xor(Equivalent(x, y), z, evaluate=False),
        source_format="free") == "z .neqv. (x .eqv. y)"
    # mixed And/Xor
    assert fcode(Xor(And(y, z), x, evaluate=False), source_format="free") == \
        "x .neqv. y .and. z"
    assert fcode(Xor(And(z, x), y, evaluate=False), source_format="free") == \
        "y .neqv. x .and. z"
    assert fcode(Xor(And(x, y), z, evaluate=False), source_format="free") == \
        "z .neqv. x .and. y"
    assert fcode(And(Xor(y, z, evaluate=False), x), source_format="free") == \
        "x .and. (y .neqv. z)"
    assert fcode(And(Xor(z, x, evaluate=False), y), source_format="free") == \
        "y .and. (x .neqv. z)"
    assert fcode(And(Xor(x, y, evaluate=False), z), source_format="free") == \
        "z .and. (x .neqv. y)"
    # mixed Or/Xor
    assert fcode(Xor(Or(y, z), x, evaluate=False), source_format="free") == \
        "x .neqv. y .or. z"
    assert fcode(Xor(Or(z, x), y, evaluate=False), source_format="free") == \
        "y .neqv. x .or. z"
    assert fcode(Xor(Or(x, y), z, evaluate=False), source_format="free") == \
        "z .neqv. x .or. y"
    assert fcode(Or(Xor(y, z, evaluate=False), x), source_format="free") == \
        "x .or. (y .neqv. z)"
    assert fcode(Or(Xor(z, x, evaluate=False), y), source_format="free") == \
        "y .or. (x .neqv. z)"
    assert fcode(Or(Xor(x, y, evaluate=False), z), source_format="free") == \
        "z .or. (x .neqv. y)"
    # trinary Xor
    assert fcode(Xor(x, y, z, evaluate=False), source_format="free") == \
        "x .neqv. y .neqv. z"
    assert fcode(Xor(x, y, Not(z), evaluate=False), source_format="free") == \
        "x .neqv. y .neqv. .not. z"
    assert fcode(Xor(x, Not(y), z, evaluate=False), source_format="free") == \
        "x .neqv. z .neqv. .not. y"
    assert fcode(Xor(Not(x), y, z, evaluate=False), source_format="free") == \
        "y .neqv. z .neqv. .not. x"
Beispiel #23
0
def test_simplification():
    """
    Test working of simplification methods.
    """
    set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]]
    set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]]
    assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x))
    assert Not(SOPform([x, y, z], set2)) == Not(Or(And(Not(x), Not(z)), And(x, z)))
    assert POSform([x, y, z], set1 + set2) is true
    assert SOPform([x, y, z], set1 + set2) is true
    assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true

    minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
        [1, 1, 1, 1]]
    dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
    assert (
        SOPform([w, x, y, z], minterms, dontcares) ==
        Or(And(Not(w), z), And(y, z)))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)

    # test simplification
    ans = And(A, Or(B, C))
    assert simplify_logic(A & (B | C)) == ans
    assert simplify_logic((A & B) | (A & C)) == ans
    assert simplify_logic(Implies(A, B)) == Or(Not(A), B)
    assert simplify_logic(Equivalent(A, B)) == \
           Or(And(A, B), And(Not(A), Not(B)))
    assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C)
    assert simplify_logic(And(Equality(A, 2), A)) is S.false
    assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A)
    assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C)
    assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \
           == And(Equality(A, 3), Or(B, C))
    b = (~x & ~y & ~z) | ( ~x & ~y & z)
    e = And(A, b)
    assert simplify_logic(e) == A & ~x & ~y

    # check input
    ans = SOPform([x, y], [[1, 0]])
    assert SOPform([x, y], [[1, 0]]) == ans
    assert POSform([x, y], [[1, 0]]) == ans

    raises(ValueError, lambda: SOPform([x], [[1]], [[1]]))
    assert SOPform([x], [[1]], [[0]]) is true
    assert SOPform([x], [[0]], [[1]]) is true
    assert SOPform([x], [], []) is false

    raises(ValueError, lambda: POSform([x], [[1]], [[1]]))
    assert POSform([x], [[1]], [[0]]) is true
    assert POSform([x], [[0]], [[1]]) is true
    assert POSform([x], [], []) is false

    # check working of simplify
    assert simplify((A & B) | (A & C)) == And(A, Or(B, C))
    assert simplify(And(x, Not(x))) == False
    assert simplify(Or(x, Not(x))) == True
    assert simplify(And(Eq(x, 0), Eq(x, y))) == And(Eq(x, 0), Eq(y, 0))
    assert And(Eq(x - 1, 0), Eq(x, y)).simplify() == And(Eq(x, 1), Eq(y, 1))
    assert And(Ne(x - 1, 0), Ne(x, y)).simplify() == And(Ne(x, 1), Ne(x, y))
    assert And(Eq(x - 1, 0), Ne(x, y)).simplify() == And(Eq(x, 1), Ne(y, 1))
    assert And(Eq(x - 1, 0), Eq(x, z + y), Eq(y + x, 0)).simplify(
        ) == And(Eq(x, 1), Eq(y, -1), Eq(z, 2))
    assert And(Eq(x - 1, 0), Eq(x + 2, 3)).simplify() == Eq(x, 1)
    assert And(Ne(x - 1, 0), Ne(x + 2, 3)).simplify() == Ne(x, 1)
    assert And(Eq(x - 1, 0), Eq(x + 2, 2)).simplify() == False
    assert And(Ne(x - 1, 0), Ne(x + 2, 2)).simplify(
        ) == And(Ne(x, 1), Ne(x, 0))
Beispiel #24
0
 def _contains(self, other):
     return And(Contains(other, self.base_set),
                Lambda(self.sym, self.condition)(other))
Beispiel #25
0
def test_And():
    assert And() is true
    assert And(A) == A
    assert And(True) is true
    assert And(False) is false
    assert And(True, True ) is true
    assert And(True, False) is false
    assert And(False, False) is false
    assert And(True, A) == A
    assert And(False, A) is false
    assert And(True, True, True) is true
    assert And(True, True, A) == A
    assert And(True, False, A) is false
    assert And(1, A) == A
    raises(TypeError, lambda: And(2, A))
    raises(TypeError, lambda: And(A < 2, A))
    assert And(A < 1, A >= 1) is false
    e = A > 1
    assert And(e, e.canonical) == e.canonical
    g, l, ge, le = A > B, B < A, A >= B, B <= A
    assert And(g, l, ge, le) == And(l, le)
Beispiel #26
0
 def _handle_irel(self, x, handler):
     """Return either None (if the conditions of self depend only on x) else
     a Piecewise expression whose expressions (handled by the handler that
     was passed) are paired with the governing x-independent relationals,
     e.g. Piecewise((A, a(x) & b(y)), (B, c(x) | c(y)) ->
     Piecewise(
         (handler(Piecewise((A, a(x) & True), (B, c(x) | True)), b(y) & c(y)),
         (handler(Piecewise((A, a(x) & True), (B, c(x) | False)), b(y)),
         (handler(Piecewise((A, a(x) & False), (B, c(x) | True)), c(y)),
         (handler(Piecewise((A, a(x) & False), (B, c(x) | False)), True))
     """
     # identify governing relationals
     rel = self.atoms(Relational)
     irel = list(ordered([r for r in rel if x not in r.free_symbols
         and r not in (S.true, S.false)]))
     if irel:
         args = {}
         exprinorder = []
         for truth in product((1, 0), repeat=len(irel)):
             reps = dict(zip(irel, truth))
             # only store the true conditions since the false are implied
             # when they appear lower in the Piecewise args
             if 1 not in truth:
                 cond = None  # flag this one so it doesn't get combined
             else:
                 andargs = Tuple(*[i for i in reps if reps[i]])
                 free = list(andargs.free_symbols)
                 if len(free) == 1:
                     from sympy.solvers.inequalities import (
                         reduce_inequalities, _solve_inequality)
                     try:
                         t = reduce_inequalities(andargs, free[0])
                         # ValueError when there are potentially
                         # nonvanishing imaginary parts
                     except (ValueError, NotImplementedError):
                         # at least isolate free symbol on left
                         t = And(*[_solve_inequality(
                             a, free[0], linear=True)
                             for a in andargs])
                 else:
                     t = And(*andargs)
                 if t is S.false:
                     continue  # an impossible combination
                 cond = t
             expr = handler(self.xreplace(reps))
             if isinstance(expr, self.func) and len(expr.args) == 1:
                 expr, econd = expr.args[0]
                 cond = And(econd, True if cond is None else cond)
             # the ec pairs are being collected since all possibilities
             # are being enumerated, but don't put the last one in since
             # its expr might match a previous expression and it
             # must appear last in the args
             if cond is not None:
                 args.setdefault(expr, []).append(cond)
                 # but since we only store the true conditions we must maintain
                 # the order so that the expression with the most true values
                 # comes first
                 exprinorder.append(expr)
         # convert collected conditions as args of Or
         for k in args:
             args[k] = Or(*args[k])
         # take them in the order obtained
         args = [(e, args[e]) for e in uniq(exprinorder)]
         # add in the last arg
         args.append((expr, True))
         # if any condition reduced to True, it needs to go last
         # and there should only be one of them or else the exprs
         # should agree
         trues = [i for i in range(len(args)) if args[i][1] is S.true]
         if not trues:
             # make the last one True since all cases were enumerated
             e, c = args[-1]
             args[-1] = (e, S.true)
         else:
             assert len(set([e for e, c in [args[i] for i in trues]])) == 1
             args.append(args.pop(trues.pop()))
             while trues:
                 args.pop(trues.pop())
         return Piecewise(*args)
Beispiel #27
0
def test_distribute():
    assert distribute_and_over_or(Or(And(A, B), C)) == And(Or(A, C), Or(B, C))
    assert distribute_or_over_and(And(A, Or(B, C))) == Or(And(A, B), And(A, C))
Beispiel #28
0
    'algebraic': ['sympy.assumptions.handlers.sets.AskAlgebraicHandler'],
    'is_true': ['sympy.assumptions.handlers.TautologicalHandler']
}
for name, value in _handlers_dict.iteritems():
    register_handler(name, value[0])


known_facts_keys = [getattr(Q, attr) for attr in Q.__dict__ \
                                                if not attr.startswith('__')]
known_facts = And(Implies(Q.real, Q.complex),
                  Equivalent(Q.even, Q.integer & ~Q.odd),
                  Equivalent(Q.extended_real, Q.real | Q.infinity),
                  Equivalent(Q.odd, Q.integer & ~Q.even),
                  Equivalent(Q.prime, Q.integer & Q.positive & ~Q.composite),
                  Implies(Q.integer, Q.rational),
                  Implies(Q.imaginary, Q.complex & ~Q.real),
                  Equivalent(Q.negative, Q.nonzero & ~Q.positive),
                  Equivalent(Q.positive, Q.nonzero & ~Q.negative),
                  Equivalent(Q.rational, Q.real & ~Q.irrational),
                  Equivalent(Q.real, Q.rational | Q.irrational),
                  Implies(Q.nonzero, Q.real),
                  Equivalent(Q.nonzero, Q.positive | Q.negative))

################################################################################
# Note: The following facts are generated by the compute_known_facts function. #
################################################################################
# -{ Known facts in CNF }-
known_facts_cnf = And(
    Or(Not(Q.integer), Q.even, Q.odd),
    Or(Not(Q.extended_real), Q.infinity, Q.real),
    Or(Not(Q.real), Q.irrational, Q.rational), Or(Not(Q.real), Q.complex),
Beispiel #29
0
def test_true_false():
    assert true is S.true
    assert false is S.false
    assert true is not True
    assert false is not False
    assert true
    assert not false
    assert true == True
    assert false == False
    assert not (true == False)
    assert not (false == True)
    assert not (true == false)

    assert hash(true) == hash(True)
    assert hash(false) == hash(False)
    assert len({true, True}) == len({false, False}) == 1

    assert isinstance(true, BooleanAtom)
    assert isinstance(false, BooleanAtom)
    # We don't want to subclass from bool, because bool subclasses from
    # int. But operators like &, |, ^, <<, >>, and ~ act differently on 0 and
    # 1 then we want them to on true and false.  See the docstrings of the
    # various And, Or, etc. functions for examples.
    assert not isinstance(true, bool)
    assert not isinstance(false, bool)

    # Note: using 'is' comparison is important here. We want these to return
    # true and false, not True and False

    assert Not(true) is false
    assert Not(True) is false
    assert Not(false) is true
    assert Not(False) is true
    assert ~true is false
    assert ~false is true

    for T, F in cartes([True, true], [False, false]):
        assert And(T, F) is false
        assert And(F, T) is false
        assert And(F, F) is false
        assert And(T, T) is true
        assert And(T, x) == x
        assert And(F, x) is false
        if not (T is True and F is False):
            assert T & F is false
            assert F & T is false
        if not F is False:
            assert F & F is false
        if not T is True:
            assert T & T is true

        assert Or(T, F) is true
        assert Or(F, T) is true
        assert Or(F, F) is false
        assert Or(T, T) is true
        assert Or(T, x) is true
        assert Or(F, x) == x
        if not (T is True and F is False):
            assert T | F is true
            assert F | T is true
        if not F is False:
            assert F | F is false
        if not T is True:
            assert T | T is true

        assert Xor(T, F) is true
        assert Xor(F, T) is true
        assert Xor(F, F) is false
        assert Xor(T, T) is false
        assert Xor(T, x) == ~x
        assert Xor(F, x) == x
        if not (T is True and F is False):
            assert T ^ F is true
            assert F ^ T is true
        if not F is False:
            assert F ^ F is false
        if not T is True:
            assert T ^ T is false

        assert Nand(T, F) is true
        assert Nand(F, T) is true
        assert Nand(F, F) is true
        assert Nand(T, T) is false
        assert Nand(T, x) == ~x
        assert Nand(F, x) is true

        assert Nor(T, F) is false
        assert Nor(F, T) is false
        assert Nor(F, F) is true
        assert Nor(T, T) is false
        assert Nor(T, x) is false
        assert Nor(F, x) == ~x

        assert Implies(T, F) is false
        assert Implies(F, T) is true
        assert Implies(F, F) is true
        assert Implies(T, T) is true
        assert Implies(T, x) == x
        assert Implies(F, x) is true
        assert Implies(x, T) is true
        assert Implies(x, F) == ~x
        if not (T is True and F is False):
            assert T >> F is false
            assert F << T is false
            assert F >> T is true
            assert T << F is true
        if not F is False:
            assert F >> F is true
            assert F << F is true
        if not T is True:
            assert T >> T is true
            assert T << T is true

        assert Equivalent(T, F) is false
        assert Equivalent(F, T) is false
        assert Equivalent(F, F) is true
        assert Equivalent(T, T) is true
        assert Equivalent(T, x) == x
        assert Equivalent(F, x) == ~x
        assert Equivalent(x, T) == x
        assert Equivalent(x, F) == ~x

        assert ITE(T, T, T) is true
        assert ITE(T, T, F) is true
        assert ITE(T, F, T) is false
        assert ITE(T, F, F) is false
        assert ITE(F, T, T) is true
        assert ITE(F, T, F) is false
        assert ITE(F, F, T) is true
        assert ITE(F, F, F) is false

    assert all(i.simplify(1, 2) is i for i in (S.true, S.false))
Beispiel #30
0
def heurisch_wrapper(f,
                     x,
                     rewrite=False,
                     hints=None,
                     mappings=None,
                     retries=3,
                     degree_offset=0,
                     unnecessary_permutations=None):
    """
    A wrapper around the heurisch integration algorithm.

    This method takes the result from heurisch and checks for poles in the
    denominator. For each of these poles, the integral is reevaluated, and
    the final integration result is given in terms of a Piecewise.

    Examples
    ========

    >>> from sympy.core import symbols
    >>> from sympy.functions import cos
    >>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper
    >>> n, x = symbols('n x')
    >>> heurisch(cos(n*x), x)
    sin(n*x)/n
    >>> heurisch_wrapper(cos(n*x), x)
    Piecewise((x, n == 0), (sin(n*x)/n, True))

    See Also
    ========

    heurisch
    """
    f = sympify(f)
    if x not in f.free_symbols:
        return f * x

    res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
                   unnecessary_permutations)
    if not isinstance(res, Basic):
        return res
    # We consider each denominator in the expression, and try to find
    # cases where one or more symbolic denominator might be zero. The
    # conditions for these cases are stored in the list slns.
    slns = []
    for d in denoms(res):
        try:
            slns += solve(d, dict=True, exclude=(x, ))
        except NotImplementedError:
            pass
    if not slns:
        return res
    slns = list(uniq(slns))
    # Remove the solutions corresponding to poles in the original expression.
    slns0 = []
    for d in denoms(f):
        try:
            slns0 += solve(d, dict=True, exclude=(x, ))
        except NotImplementedError:
            pass
    slns = [s for s in slns if s not in slns0]
    if not slns:
        return res
    if len(slns) > 1:
        eqs = []
        for sub_dict in slns:
            eqs.extend([Eq(key, value) for key, value in sub_dict.items()])
        slns = solve(eqs, dict=True, exclude=(x, )) + slns
    # For each case listed in the list slns, we reevaluate the integral.
    pairs = []
    for sub_dict in slns:
        expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
                        degree_offset, unnecessary_permutations)
        cond = And(*[Eq(key, value) for key, value in sub_dict.items()])
        pairs.append((expr, cond))
    pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
                           degree_offset, unnecessary_permutations), True))
    return Piecewise(*pairs)
Beispiel #31
0
def test_multivariate_bool_as_set():
    x, y = symbols('x,y')

    assert And(x >= 0, y >= 0).as_set() == Interval(0, oo)*Interval(0, oo)
    assert Or(x >= 0, y >= 0).as_set() == S.Reals*S.Reals - \
        Interval(-oo, 0, True, True)*Interval(-oo, 0, True, True)