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"""Tools for solving inequalities and systems of inequalities. """ 

 

from __future__ import print_function, division 

 

from sympy.core import Symbol, Dummy 

from sympy.core.compatibility import iterable, reduce 

from sympy.sets import Interval 

from sympy.core.relational import Relational, Eq, Ge, Lt 

from sympy.sets.sets import FiniteSet, Union 

from sympy.core.singleton import S 

 

from sympy.functions import Abs 

from sympy.logic import And 

from sympy.polys import Poly, PolynomialError, parallel_poly_from_expr 

from sympy.polys.polyutils import _nsort 

from sympy.utilities.misc import filldedent 

 

def solve_poly_inequality(poly, rel): 

    """Solve a polynomial inequality with rational coefficients. 

 

    Examples 

    ======== 

 

    >>> from sympy import Poly 

    >>> from sympy.abc import x 

    >>> from sympy.solvers.inequalities import solve_poly_inequality 

 

    >>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==') 

    [{0}] 

 

    >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '!=') 

    [(-oo, -1), (-1, 1), (1, oo)] 

 

    >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '==') 

    [{-1}, {1}] 

 

    See Also 

    ======== 

    solve_poly_inequalities 

    """ 

    if not isinstance(poly, Poly): 

        raise ValueError( 

            'For efficiency reasons, `poly` should be a Poly instance') 

    if poly.is_number: 

        t = Relational(poly.as_expr(), 0, rel) 

        if t is S.true: 

            return [S.Reals] 

        elif t is S.false: 

            return [S.EmptySet] 

        else: 

            raise NotImplementedError( 

                "could not determine truth value of %s" % t) 

 

    reals, intervals = poly.real_roots(multiple=False), [] 

 

    if rel == '==': 

        for root, _ in reals: 

            interval = Interval(root, root) 

            intervals.append(interval) 

    elif rel == '!=': 

        left = S.NegativeInfinity 

 

        for right, _ in reals + [(S.Infinity, 1)]: 

            interval = Interval(left, right, True, True) 

            intervals.append(interval) 

            left = right 

    else: 

        if poly.LC() > 0: 

            sign = +1 

        else: 

            sign = -1 

 

        eq_sign, equal = None, False 

 

        if rel == '>': 

            eq_sign = +1 

        elif rel == '<': 

            eq_sign = -1 

        elif rel == '>=': 

            eq_sign, equal = +1, True 

        elif rel == '<=': 

            eq_sign, equal = -1, True 

        else: 

            raise ValueError("'%s' is not a valid relation" % rel) 

 

        right, right_open = S.Infinity, True 

 

        for left, multiplicity in reversed(reals): 

            if multiplicity % 2: 

                if sign == eq_sign: 

                    intervals.insert( 

                        0, Interval(left, right, not equal, right_open)) 

 

                sign, right, right_open = -sign, left, not equal 

            else: 

                if sign == eq_sign and not equal: 

                    intervals.insert( 

                        0, Interval(left, right, True, right_open)) 

                    right, right_open = left, True 

                elif sign != eq_sign and equal: 

                    intervals.insert(0, Interval(left, left)) 

 

        if sign == eq_sign: 

            intervals.insert( 

                0, Interval(S.NegativeInfinity, right, True, right_open)) 

 

    return intervals 

 

 

def solve_poly_inequalities(polys): 

    """Solve polynomial inequalities with rational coefficients. 

 

    Examples 

    ======== 

 

    >>> from sympy.solvers.inequalities import solve_poly_inequalities 

    >>> from sympy.polys import Poly 

    >>> from sympy.abc import x 

    >>> solve_poly_inequalities((( 

    ... Poly(x**2 - 3), ">"), ( 

    ... Poly(-x**2 + 1), ">"))) 

    (-oo, -sqrt(3)) U (-1, 1) U (sqrt(3), oo) 

    """ 

    from sympy import Union 

    return Union(*[solve_poly_inequality(*p) for p in polys]) 

 

 

def solve_rational_inequalities(eqs): 

    """Solve a system of rational inequalities with rational coefficients. 

 

    Examples 

    ======== 

 

    >>> from sympy.abc import x 

    >>> from sympy import Poly 

    >>> from sympy.solvers.inequalities import solve_rational_inequalities 

 

    >>> solve_rational_inequalities([[ 

    ... ((Poly(-x + 1), Poly(1, x)), '>='), 

    ... ((Poly(-x + 1), Poly(1, x)), '<=')]]) 

    {1} 

 

    >>> solve_rational_inequalities([[ 

    ... ((Poly(x), Poly(1, x)), '!='), 

    ... ((Poly(-x + 1), Poly(1, x)), '>=')]]) 

    (-oo, 0) U (0, 1] 

 

    See Also 

    ======== 

    solve_poly_inequality 

    """ 

    result = S.EmptySet 

 

    for _eqs in eqs: 

        if not _eqs: 

            continue 

 

        global_intervals = [Interval(S.NegativeInfinity, S.Infinity)] 

 

        for (numer, denom), rel in _eqs: 

            numer_intervals = solve_poly_inequality(numer*denom, rel) 

            denom_intervals = solve_poly_inequality(denom, '==') 

 

            intervals = [] 

 

            for numer_interval in numer_intervals: 

                for global_interval in global_intervals: 

                    interval = numer_interval.intersect(global_interval) 

 

                    if interval is not S.EmptySet: 

                        intervals.append(interval) 

 

            global_intervals = intervals 

 

            intervals = [] 

 

            for global_interval in global_intervals: 

                for denom_interval in denom_intervals: 

                    global_interval -= denom_interval 

 

                if global_interval is not S.EmptySet: 

                    intervals.append(global_interval) 

 

            global_intervals = intervals 

 

            if not global_intervals: 

                break 

 

        for interval in global_intervals: 

            result = result.union(interval) 

 

    return result 

 

 

def reduce_rational_inequalities(exprs, gen, relational=True): 

    """Reduce a system of rational inequalities with rational coefficients. 

 

    Examples 

    ======== 

 

    >>> from sympy import Poly, Symbol 

    >>> from sympy.solvers.inequalities import reduce_rational_inequalities 

 

    >>> x = Symbol('x', real=True) 

 

    >>> reduce_rational_inequalities([[x**2 <= 0]], x) 

    Eq(x, 0) 

 

    >>> reduce_rational_inequalities([[x + 2 > 0]], x) 

    And(-2 < x, x < oo) 

    >>> reduce_rational_inequalities([[(x + 2, ">")]], x) 

    And(-2 < x, x < oo) 

    >>> reduce_rational_inequalities([[x + 2]], x) 

    Eq(x, -2) 

    """ 

    exact = True 

    eqs = [] 

    solution = S.EmptySet 

    for _exprs in exprs: 

        _eqs = [] 

 

        for expr in _exprs: 

            if isinstance(expr, tuple): 

                expr, rel = expr 

            else: 

                if expr.is_Relational: 

                    expr, rel = expr.lhs - expr.rhs, expr.rel_op 

                else: 

                    expr, rel = expr, '==' 

 

            if expr is S.true: 

                numer, denom, rel = S.Zero, S.One, '==' 

            elif expr is S.false: 

                numer, denom, rel = S.One, S.One, '==' 

            else: 

                numer, denom = expr.together().as_numer_denom() 

 

            try: 

                (numer, denom), opt = parallel_poly_from_expr( 

                    (numer, denom), gen) 

            except PolynomialError: 

                raise PolynomialError(filldedent(''' 

                    only polynomials and 

                    rational functions are supported in this context''')) 

 

            if not opt.domain.is_Exact: 

                numer, denom, exact = numer.to_exact(), denom.to_exact(), False 

 

            domain = opt.domain.get_exact() 

 

            if not (domain.is_ZZ or domain.is_QQ): 

                expr = numer/denom 

                expr = Relational(expr, 0, rel) 

                solution = Union(solution, solve_univariate_inequality(expr, gen, relational=False)) 

            else: 

                _eqs.append(((numer, denom), rel)) 

 

        eqs.append(_eqs) 

 

    solution = Union(solution, solve_rational_inequalities(eqs)) 

 

    if not exact: 

        solution = solution.evalf() 

 

    if relational: 

        solution = solution.as_relational(gen) 

 

    return solution 

 

 

def reduce_abs_inequality(expr, rel, gen): 

    """Reduce an inequality with nested absolute values. 

 

    Examples 

    ======== 

 

    >>> from sympy import Abs, Symbol 

    >>> from sympy.solvers.inequalities import reduce_abs_inequality 

    >>> x = Symbol('x', real=True) 

 

    >>> reduce_abs_inequality(Abs(x - 5) - 3, '<', x) 

    And(2 < x, x < 8) 

 

    >>> reduce_abs_inequality(Abs(x + 2)*3 - 13, '<', x) 

    And(-19/3 < x, x < 7/3) 

 

    See Also 

    ======== 

 

    reduce_abs_inequalities 

    """ 

    if gen.is_real is False: 

         raise TypeError(filldedent(''' 

            can't solve inequalities with absolute 

            values containing non-real variables''')) 

 

    def _bottom_up_scan(expr): 

        exprs = [] 

 

        if expr.is_Add or expr.is_Mul: 

            op = expr.func 

 

            for arg in expr.args: 

                _exprs = _bottom_up_scan(arg) 

 

                if not exprs: 

                    exprs = _exprs 

                else: 

                    args = [] 

 

                    for expr, conds in exprs: 

                        for _expr, _conds in _exprs: 

                            args.append((op(expr, _expr), conds + _conds)) 

 

                    exprs = args 

        elif expr.is_Pow: 

            n = expr.exp 

 

            if not n.is_Integer or n < 0: 

                raise ValueError( 

                    "only non-negative integer powers are allowed") 

 

            _exprs = _bottom_up_scan(expr.base) 

 

            for expr, conds in _exprs: 

                exprs.append((expr**n, conds)) 

        elif isinstance(expr, Abs): 

            _exprs = _bottom_up_scan(expr.args[0]) 

 

            for expr, conds in _exprs: 

                exprs.append(( expr, conds + [Ge(expr, 0)])) 

                exprs.append((-expr, conds + [Lt(expr, 0)])) 

        else: 

            exprs = [(expr, [])] 

 

        return exprs 

 

    exprs = _bottom_up_scan(expr) 

 

    mapping = {'<': '>', '<=': '>='} 

    inequalities = [] 

 

    for expr, conds in exprs: 

        if rel not in mapping.keys(): 

            expr = Relational( expr, 0, rel) 

        else: 

            expr = Relational(-expr, 0, mapping[rel]) 

 

        inequalities.append([expr] + conds) 

 

    return reduce_rational_inequalities(inequalities, gen) 

 

 

def reduce_abs_inequalities(exprs, gen): 

    """Reduce a system of inequalities with nested absolute values. 

 

    Examples 

    ======== 

 

    >>> from sympy import Abs, Symbol 

    >>> from sympy.abc import x 

    >>> from sympy.solvers.inequalities import reduce_abs_inequalities 

    >>> x = Symbol('x', real=True) 

 

    >>> reduce_abs_inequalities([(Abs(3*x - 5) - 7, '<'), 

    ... (Abs(x + 25) - 13, '>')], x) 

    And(-2/3 < x, Or(And(-12 < x, x < oo), And(-oo < x, x < -38)), x < 4) 

 

    >>> reduce_abs_inequalities([(Abs(x - 4) + Abs(3*x - 5) - 7, '<')], x) 

    And(1/2 < x, x < 4) 

 

    See Also 

    ======== 

 

    reduce_abs_inequality 

    """ 

    return And(*[ reduce_abs_inequality(expr, rel, gen) 

        for expr, rel in exprs ]) 

 

 

def solve_univariate_inequality(expr, gen, relational=True): 

    """Solves a real univariate inequality. 

 

    Examples 

    ======== 

 

    >>> from sympy.solvers.inequalities import solve_univariate_inequality 

    >>> from sympy.core.symbol import Symbol 

    >>> x = Symbol('x', real=True) 

 

    >>> solve_univariate_inequality(x**2 >= 4, x) 

    Or(And(-oo < x, x <= -2), And(2 <= x, x < oo)) 

 

    >>> solve_univariate_inequality(x**2 >= 4, x, relational=False) 

    (-oo, -2] U [2, oo) 

 

    """ 

 

    from sympy.solvers.solvers import solve, denoms 

 

    e = expr.lhs - expr.rhs 

    parts = n, d = e.as_numer_denom() 

    if all(i.is_polynomial(gen) for i in parts): 

        solns = solve(n, gen, check=False) 

        singularities = solve(d, gen, check=False) 

    else: 

        solns = solve(e, gen, check=False) 

        singularities = [] 

        for d in denoms(e): 

            singularities.extend(solve(d, gen)) 

 

    include_x = expr.func(0, 0) 

 

    def valid(x): 

        v = e.subs(gen, x) 

        try: 

            r = expr.func(v, 0) 

        except TypeError: 

            r = S.false 

        if r in (S.true, S.false): 

            return r 

        if v.is_real is False: 

            return S.false 

        else: 

            v = v.n(2) 

            if v.is_comparable: 

                return expr.func(v, 0) 

            return S.false 

 

    start = S.NegativeInfinity 

    sol_sets = [S.EmptySet] 

    try: 

        reals = _nsort(set(solns + singularities), separated=True)[0] 

    except NotImplementedError: 

        raise NotImplementedError('sorting of these roots is not supported') 

    for x in reals: 

        end = x 

 

        if end in [S.NegativeInfinity, S.Infinity]: 

            if valid(S(0)): 

                sol_sets.append(Interval(start, S.Infinity, True, True)) 

                break 

 

        if valid((start + end)/2 if start != S.NegativeInfinity else end - 1): 

            sol_sets.append(Interval(start, end, True, True)) 

 

        if x in singularities: 

            singularities.remove(x) 

        elif include_x: 

            sol_sets.append(FiniteSet(x)) 

 

        start = end 

 

    end = S.Infinity 

 

    if valid(start + 1): 

        sol_sets.append(Interval(start, end, True, True)) 

 

    rv = Union(*sol_sets) 

    return rv if not relational else rv.as_relational(gen) 

 

 

def _solve_inequality(ie, s): 

    """ A hacky replacement for solve, since the latter only works for 

        univariate inequalities. """ 

    if not ie.rel_op in ('>', '>=', '<', '<='): 

        raise NotImplementedError 

    expr = ie.lhs - ie.rhs 

    try: 

        p = Poly(expr, s) 

        if p.degree() != 1: 

            raise NotImplementedError 

    except (PolynomialError, NotImplementedError): 

        try: 

            n, d = expr.as_numer_denom() 

            return reduce_rational_inequalities([[ie]], s) 

        except PolynomialError: 

            return solve_univariate_inequality(ie, s) 

    a, b = p.all_coeffs() 

    if a.is_positive: 

        return ie.func(s, -b/a) 

    elif a.is_negative: 

        return ie.func(-b/a, s) 

    else: 

        raise NotImplementedError 

 

 

def _reduce_inequalities(inequalities, symbols): 

    # helper for reduce_inequalities 

 

    poly_part, abs_part = {}, {} 

    other = [] 

 

    for inequality in inequalities: 

 

        expr, rel = inequality.lhs, inequality.rel_op  # rhs is 0 

 

        # check for gens using atoms which is more strict than free_symbols to 

        # guard against EX domain which won't be handled by 

        # reduce_rational_inequalities 

        gens = expr.atoms(Symbol) 

 

        if len(gens) == 1: 

            gen = gens.pop() 

        else: 

            common = expr.free_symbols & symbols 

            if len(common) == 1: 

                gen = common.pop() 

                other.append(_solve_inequality(Relational(expr, 0, rel), gen)) 

                continue 

            else: 

                raise NotImplementedError(filldedent(''' 

                    inequality has more than one 

                    symbol of interest''')) 

 

        if expr.is_polynomial(gen): 

            poly_part.setdefault(gen, []).append((expr, rel)) 

        else: 

            components = expr.find(lambda u: 

                u.has(gen) and ( 

                u.is_Function or u.is_Pow and not u.exp.is_Integer)) 

            if components and all(isinstance(i, Abs) for i in components): 

                abs_part.setdefault(gen, []).append((expr, rel)) 

            else: 

                other.append(_solve_inequality(Relational(expr, 0, rel), gen)) 

 

    poly_reduced = [] 

    abs_reduced = [] 

 

    for gen, exprs in poly_part.items(): 

        poly_reduced.append(reduce_rational_inequalities([exprs], gen)) 

 

    for gen, exprs in abs_part.items(): 

        abs_reduced.append(reduce_abs_inequalities(exprs, gen)) 

 

    return And(*(poly_reduced + abs_reduced + other)) 

 

 

def reduce_inequalities(inequalities, symbols=[]): 

    """Reduce a system of inequalities with rational coefficients. 

 

    Examples 

    ======== 

 

    >>> from sympy import sympify as S, Symbol 

    >>> from sympy.abc import x, y 

    >>> from sympy.solvers.inequalities import reduce_inequalities 

 

    >>> reduce_inequalities(0 <= x + 3, []) 

    And(-3 <= x, x < oo) 

 

    >>> reduce_inequalities(0 <= x + y*2 - 1, [x]) 

    x >= -2*y + 1 

    """ 

    if not iterable(inequalities): 

        inequalities = [inequalities] 

 

    # prefilter 

    keep = [] 

    for i in inequalities: 

        if isinstance(i, Relational): 

            i = i.func(i.lhs.as_expr() - i.rhs.as_expr(), 0) 

        elif i not in (True, False): 

            i = Eq(i, 0) 

        if i == True: 

            continue 

        elif i == False: 

            return S.false 

        if i.lhs.is_number: 

            raise NotImplementedError( 

                "could not determine truth value of %s" % i) 

        keep.append(i) 

    inequalities = keep 

    del keep 

 

    gens = reduce(set.union, [i.free_symbols for i in inequalities], set()) 

 

    if not iterable(symbols): 

        symbols = [symbols] 

    symbols = set(symbols) or gens 

 

    # make vanilla symbol real 

    recast = dict([(i, Dummy(i.name, real=True)) 

        for i in gens if i.is_real is None]) 

    inequalities = [i.xreplace(recast) for i in inequalities] 

    symbols = set([i.xreplace(recast) for i in symbols]) 

 

    # solve system 

    rv = _reduce_inequalities(inequalities, symbols) 

 

    # restore original symbols and return 

    return rv.xreplace(dict([(v, k) for k, v in recast.items()]))