o GZh@sbdZddlZddlmZmZmZddlmZddlm Z ddl m Z m Z m Z mZddlmZmZddlmZmZmZmZdd lmZdd lmZdd lmZdd lmZdd lm Z m!Z!m"Z"ddl#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*ddl+m,Z,ddZ-ddZ.ddZ/d*ddZ0ddZ1ddZ2dej3dfd d!Z4d"d#Z5d+d$d%Z6d&d'Z7gfd(d)Z8dS),z>> from sympy import solve_poly_inequality, Poly >>> from sympy.abc import x >>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==') [{0}] >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '!=') [Interval.open(-oo, -1), Interval.open(-1, 1), Interval.open(1, oo)] >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '==') [{-1}, {1}] See Also ======== solve_poly_inequalities z8For efficiency reasons, `poly` should be a Poly instancer%could not determine truth value of %sF)Zmultiple==!=T)NF><>=)r#T<=)r$Tz'%s' is not a valid relation) isinstancer ValueErroras_expr is_numberrrtrueRealsfalseEmptySetNotImplementedErrorZ real_rootsrappendNegativeInfinityInfinityZLCreversedinsert)Zpolyreltreals intervalsroot_intervalleftrightsignZeq_signequalZ right_openZ multiplicityrCI/var/www/auris/lib/python3.10/site-packages/sympy/solvers/inequalities.pysolve_poly_inequalitysx       0 )         rEcCstdd|DS)aSolve polynomial inequalities with rational coefficients. Examples ======== >>> from sympy import Poly >>> from sympy.solvers.inequalities import solve_poly_inequalities >>> from sympy.abc import x >>> solve_poly_inequalities((( ... Poly(x**2 - 3), ">"), ( ... Poly(-x**2 + 1), ">"))) Union(Interval.open(-oo, -sqrt(3)), Interval.open(-1, 1), Interval.open(sqrt(3), oo)) cSsg|] }t|D]}|qqSrC)rE).0psrCrCrD sz+solve_poly_inequalities..)r)ZpolysrCrCrDsolve_poly_inequalitiesqsrJcCstj}|D]j}|s qttjtjg}|D]P\\}}}t|||}t|d}g} t||D]\} } | | } | tjurA| | q.| }g} |D]} |D]} | | 8} qL| tjur]| | qH| }|sdnq|D]} | | }qgq|S)a3Solve a system of rational inequalities with rational coefficients. Examples ======== >>> from sympy.abc import x >>> from sympy import solve_rational_inequalities, Poly >>> 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)), '>=')]]) Union(Interval.open(-oo, 0), Interval.Lopen(0, 1)) See Also ======== solve_poly_inequality r!) rr1rr4r5rE itertoolsproduct intersectr3union)eqsresult_eqsZglobal_intervalsnumerdenomr8Znumer_intervalsZdenom_intervalsr;Znumer_intervalZglobal_intervalr>Zdenom_intervalrCrCrDsolve_rational_inequalitiess@        rTTc sd}g}tj}|D]}|sq g}tj}|D]} t| tr!| \} } n| jr/| j| j| j} } nd} | tj urAtj tj d} } } n| tj urQtj tj d} } } n| \} } z t| | f\\} } } Wn tysttdw| jjs| | d} } }| j}|js|js| | } t| d| } |t| ddM}q|| | f| fq|r|t|gM}tfdd|Dg}||8}||O}q |s|r|}|r|}|S) a8Reduce a system of rational inequalities with rational coefficients. Examples ======== >>> from sympy import 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) -2 < x >>> reduce_rational_inequalities([[(x + 2, ">")]], x) -2 < x >>> reduce_rational_inequalities([[x + 2]], x) Eq(x, -2) This function find the non-infinite solution set so if the unknown symbol is declared as extended real rather than real then the result may include finiteness conditions: >>> y = Symbol('y', extended_real=True) >>> reduce_rational_inequalities([[y + 2 > 0]], y) (-2 < y) & (y < oo) Tr!z only polynomials and rational functions are supported in this context. Fr) relationalcs6g|]}|D]\\}}}|r||jfdfqqS)r!)hasone)rFindr=genrCrDrIs z0reduce_rational_inequalities..)rr1r/r*tupleZ is_Relationallhsrhsrel_opr.ZeroOner0Ztogetheras_numer_denomrrrdomainZis_ExactZto_exactZ get_exactZis_ZZZis_QQrsolve_univariate_inequalityr3rTZevalf as_relational)exprsr\rUexactrOZsolution_exprsrQZ_solexprr8rRrSoptrdexcluderCr[rDreduce_rational_inequalitiess\             rmcs|jdur ttdfddddd}g}|D]"\}}||vr-t|d|}n t| d||}||g|qt||S) aReduce an inequality with nested absolute values. Examples ======== >>> from sympy import reduce_abs_inequality, Abs, Symbol >>> x = Symbol('x', real=True) >>> reduce_abs_inequality(Abs(x - 5) - 3, '<', x) (2 < x) & (x < 8) >>> reduce_abs_inequality(Abs(x + 2)*3 - 13, '<', x) (-19/3 < x) & (x < 7/3) See Also ======== reduce_abs_inequalities Fzs Cannot solve inequalities with absolute values containing non-real variables. csg}|js|jr)|j|jD]}|}|s|}qfddt||D}q|S|jrG|jjs6t d| fdd|j D|St |t rw|jd}|D]\}}|||t|dgf|| |t|dgfqU|S|gfg}|S)Ncs*g|]\\}}\}}||||fqSrCrC)rFrjcondsZ_exprZ_conds)oprCrDrIEs*zBreduce_abs_inequality.._bottom_up_scan..z'Only Integer Powers are allowed on Abs.c3s |] \}}||fVqdSNrC)rFrjrn)rYrCrD LszAreduce_abs_inequality.._bottom_up_scan..r)Zis_AddZis_MulfuncargsrKrLis_Powexp is_Integerr+extendbaser*rr3r r )rjrgargrirn_bottom_up_scan)rYrorDr{9s2       z.reduce_abs_inequality.._bottom_up_scanr%r'r&r(r)is_extended_real TypeErrorrkeysrr3rm)rjr8r\mapping inequalitiesrnrCrzrDreduce_abs_inequalitys      rcstfdd|DS)aReduce a system of inequalities with nested absolute values. Examples ======== >>> from sympy import reduce_abs_inequalities, Abs, Symbol >>> x = Symbol('x', extended_real=True) >>> reduce_abs_inequalities([(Abs(3*x - 5) - 7, '<'), ... (Abs(x + 25) - 13, '>')], x) (-2/3 < x) & (x < 4) & (((-oo < x) & (x < -38)) | ((-12 < x) & (x < oo))) >>> reduce_abs_inequalities([(Abs(x - 4) + Abs(3*x - 5) - 7, '<')], x) (1/2 < x) & (x < 4) See Also ======== reduce_abs_inequality csg|] \}}t||qSrC)r)rFrjr8r[rCrDrI{sz+reduce_abs_inequalities..r)rgr\rCr[rDreduce_abs_inequalitiesfs rFc) sddlm}|tjdurttd|tjur-td|d|}|r+| }|S }|}j durCtj }|s>|S| |Sj duret ddd z  |iWn tydttd wd}tjurp|}ntjurztj }njj} t| } | tjkrt| } | d} | tjur|}na| tjurtj }nX| durt| |} j} | d vrˈ| jdr|}n#| jdstj }n| d vr| jdr|}n | jdstj }|j|j}}||tjurtd| dd|}|}|dur]| \}}z|jvrt | jd krt!t"| |}|dur&t!Wnt!tfy?ttd#t$dwt| fdd}g}|D] }|%t"||qS|sit&|}djvosjdk}zet'|j(t)|j|j}t)||t*|t|j|j|j|v|j|v}t+dd|Drt,|ddd}n*t-|dd}|drtz|d}t |d krt.|}Wn tytwWn tytdwtj}/tj0}tjkrd}t)}zt1||}t2|ts"|D]}||vr||r|j r|t)|7}q no|j|j} }!t,|t)|!D]S}|| }"| |!kr||}#t3| |}$|$|vr|$j r||$r|"ra|#ra|t| |7}n |"rm|t4| |7}n|#ry|t5| |7}n|t6| |7}|} q1|D] }%|t)|%8}qWntytj}d}Ynw|tj urt!td#||f||}tj g}&|j} | |vr|| r| j7r|&8t)| |D]?}'|'}!|t3| |!r|&8t| |!dd|'|vr|9|'n|'|vr |9|'||'}(n|}(|(r|&8t)|'|!} q|j}!|!|vr1||!r1|!j7r1|&8t)|!|t3| |!rB|&8t6| |!|tjkrQ|rQ||}n t:t;|&||#|}|sb|S| |S)aTSolves a real univariate inequality. Parameters ========== expr : Relational The target inequality gen : Symbol The variable for which the inequality is solved relational : bool A Relational type output is expected or not domain : Set The domain over which the equation is solved continuous: bool True if expr is known to be continuous over the given domain (and so continuous_domain() does not need to be called on it) Raises ====== NotImplementedError The solution of the inequality cannot be determined due to limitation in :func:`sympy.solvers.solveset.solvify`. Notes ===== Currently, we cannot solve all the inequalities due to limitations in :func:`sympy.solvers.solveset.solvify`. Also, the solution returned for trigonometric inequalities are restricted in its periodic interval. See Also ======== sympy.solvers.solveset.solvify: solver returning solveset solutions with solve's output API Examples ======== >>> from sympy import solve_univariate_inequality, Symbol, sin, Interval, S >>> x = Symbol('x') >>> solve_univariate_inequality(x**2 >= 4, x) ((2 <= x) & (x < oo)) | ((-oo < x) & (x <= -2)) >>> solve_univariate_inequality(x**2 >= 4, x, relational=False) Union(Interval(-oo, -2), Interval(2, oo)) >>> domain = Interval(0, S.Infinity) >>> solve_univariate_inequality(x**2 >= 4, x, False, domain) Interval(2, oo) >>> solve_univariate_inequality(sin(x) > 0, x, relational=False) Interval.open(0, pi) rdenomsFz| Inequalities in the complex domain are not supported. Try the real domain by setting domain=S.Reals)rU continuousNr\TZ extended_realz When gen is real, the relational has a complex part which leads to an invalid comparison like I < 0. r|)r%r'r#z The inequality, %s, cannot be solved using solve_univariate_inequality. xcst|}z|d}Wn tytj}Ynw|tjtjfvr'|S|jdur/tjS|d}|j r=|dSt d|)NrFr)z!relationship did not evaluate: %s) subsrrrr~rr0r.r}rYZ is_comparabler2)rvrZ expanded_erjr\rCrDvalids       z*solve_univariate_inequality..valid=r"css|]}|jVqdSrp)r-)rFrrCrCrDrq@sz.solve_univariate_inequality..) separatedcSs|jSrpr})rrCrCrDCsz-solve_univariate_inequality..z'sorting of these roots is not supportedz %s contains imaginary parts which cannot be made 0 for any value of %s satisfying the inequality, leading to relations like I < 0. )<sympy.solvers.solversrZ is_subsetrr/r2rre intersectionrfr}r1r xreplacer~r.r0r^r_rrarrrrr`supinfr5rrMrc free_symbolslenr+rrr rwrsetboundaryrlistallrrsortedZcoeffZ ImaginaryUnitrr*_ptZRopenZLopenopen is_finiter3removerr))rjr\rUrdrrrvZ_genZ_domaineZperiodconstZfranger8rrrYrZZsolnsrZ singularitiesZ include_xZdiscontinuitiesZcritical_pointsr:ZsiftedZ make_realZcoeffIcheckZim_solazstartendZ valid_startZvalid_zptrHZsol_setsr_validrCrrDresf 9                                          recCs|js|js||d}|S|jr|jrtj}|S|jr!|jdus)|jr-|jdur-td|jr3|js9|jr>|jr>||}}|jrZ|jrJ|d}|S|jrT|tj}|S|d}|S|jrt|jrg|tj}|S|jrp|d}|S|d}|S)z$Return a point between start and endr)Nz,cannot proceed with unsigned infinite valuesr#) is_infiniterraZis_extended_positiver+Zis_extended_negativeZHalf)rrrrCrCrDrsF          rc Cshddlm}||jvr |S|j|kr|j}|j|kr"||jjvr"|Sdd}d}tj}|j|j}z t||}| dkrF| | d}n |sP| dkrPt Wnt t fy|sz t|gg|}Wnt yst||}Ynw||||} | tjur||||tjur|||kd}|||| } | tjur|||| tjur|| |kd}||| kd}|tjur| tjur||kn||k}| tjurt| |k|}nt|}Ynwg} |dur| } d} | j|dd\}}| |8} | |8} t| }|j|d d\}} |jd ks&|j|jkrdur+nn |jd vr+|} tj}| |} |jr:| | | }n|j | | }||j||jB}||}||D]*}tt|d||d }t|tr||j|kr|||||jtjur|| |qS| |fD]'}||||tjur||||tjur| ||ur||kn||kq| |t| S) aReturn the inequality with s isolated on the left, if possible. If the relationship is non-linear, a solution involving And or Or may be returned. False or True are returned if the relationship is never True or always True, respectively. If `linear` is True (default is False) an `s`-dependent expression will be isolated on the left, if possible but it will not be solved for `s` unless the expression is linear in `s`. Furthermore, only "safe" operations which do not change the sense of the relationship are applied: no division by an unsigned value is attempted unless the relationship involves Eq or Ne and no division by a value not known to be nonzero is ever attempted. Examples ======== >>> from sympy import Eq, Symbol >>> from sympy.solvers.inequalities import _solve_inequality as f >>> from sympy.abc import x, y For linear expressions, the symbol can be isolated: >>> f(x - 2 < 0, x) x < 2 >>> f(-x - 6 < x, x) x > -3 Sometimes nonlinear relationships will be False >>> f(x**2 + 4 < 0, x) False Or they may involve more than one region of values: >>> f(x**2 - 4 < 0, x) (-2 < x) & (x < 2) To restrict the solution to a relational, set linear=True and only the x-dependent portion will be isolated on the left: >>> f(x**2 - 4 < 0, x, linear=True) x**2 < 4 Division of only nonzero quantities is allowed, so x cannot be isolated by dividing by y: >>> y.is_nonzero is None # it is unknown whether it is 0 or not True >>> f(x*y < 1, x) x*y < 1 And while an equality (or inequality) still holds after dividing by a non-zero quantity >>> nz = Symbol('nz', nonzero=True) >>> f(Eq(x*nz, 1), x) Eq(x, 1/nz) the sign must be known for other inequalities involving > or <: >>> f(x*nz <= 1, x) nz*x <= 1 >>> p = Symbol('p', positive=True) >>> f(x*p <= 1, x) x <= 1/p When there are denominators in the original expression that are removed by expansion, conditions for them will be returned as part of the result: >>> f(x < x*(2/x - 1), x) (x < 1) & Ne(x, 0) rrcSsJz|||}|tjur|WS|dvrWdS|WSty$tjYSw)NTF)rrNaNr~)ierHrXrrCrCrDclassifys    z#_solve_inequality..classifyNr#T)Zas_AddF)r"r!)linear)rrrr_r6r^rr5rZdegreerrr,r2rrmrer.r0rrZas_independentris_zeroZ is_negativeZ is_positiver`rb_solve_inequalityr r*r3)rrHrrrrZoorjrGZokooZoknoornrr_baxZefrZbeginning_denomsZcurrent_denomsrZcrXrCrCrDrs J                  rc s8ii}}g}|D]x}|j|j}}|t}t|dkr"|n"|j|@} t| dkr>| |tt |d|q t t d| rU| g||fq |fdd} | rutdd| Dru| g||fq |tt |d|q dd |D} d d |D} t| | |S) Nr#rzZ inequality has more than one symbol of interest. cs |o|jp|jo|jj Srp)rVZ is_Functionrtrurv)ur[rCrDrs z&_reduce_inequalities..css|]}t|tVqdSrp)r*rrFrXrCrCrDrqz'_reduce_inequalities..cSsg|] \}}t|g|qSrC)rmrFr\rgrCrCrDrIsz(_reduce_inequalities..cSsg|] \}}t||qSrC)rrrCrCrDrIs)r^r`Zatomsr rpoprr3rrr2rZ is_polynomial setdefaultfindritemsr) rsymbolsZ poly_partZabs_partotherZ inequalityrjr8genscommon componentsZ poly_reducedZ abs_reducedrCr[rD_reduce_inequalitiests,        rcsJt|s|g}dd|D}tjdd|D}t|s |g}t|p%||@}tdd|Dr7ttddd|Dfd d|D}fd d |D}g}|D]<}t|trj||j |j d }n |d vrst |d }|dkrxqT|dkrt jS|j jrtd|||qT|}~t||}|ddDS)aEReduce a system of inequalities with rational coefficients. Examples ======== >>> from sympy.abc import x, y >>> from sympy import reduce_inequalities >>> reduce_inequalities(0 <= x + 3, []) (-3 <= x) & (x < oo) >>> reduce_inequalities(0 <= x + y*2 - 1, [x]) (x < oo) & (x >= 1 - 2*y) cSsg|]}t|qSrCrrrCrCrDrIsz'reduce_inequalities..cSsg|]}|jqSrC)rrrCrCrDrIscss|]}|jduVqdS)FNrrrCrCrDrqrz&reduce_inequalities..zP inequalities cannot contain symbols that are not real. cSs&i|]}|jdur|t|jddqS)NTr)r}r namerrCrCrD s z'reduce_inequalities..csg|]}|qSrCrrZrecastrCrDrIcsh|]}|qSrCrrrrCrD rz&reduce_inequalities..rrTFr cSsi|]\}}||qSrCrC)rFkrrCrCrDrr)rrrNanyr~rr*rrrr^r,r_r rr0r-r2r3rrr)rrrZkeeprXrrCrrDreduce_inequalitiessB      r)T)F)9__doc__rKZsympy.calculus.utilrrrZ sympy.corerZsympy.core.exprtoolsrZsympy.core.relationalrr r r Zsympy.core.symbolr r Zsympy.sets.setsrrrrZsympy.core.singletonrZsympy.core.functionrZ$sympy.functions.elementary.complexesrZ sympy.logicrZ sympy.polysrrrZsympy.polys.polyutilsrZsympy.solvers.solvesetrrZsympy.utilities.iterablesrrZsympy.utilities.miscrrErJrTrmrrr/rerrrrrCrCrCrDs>        [ B[G) !.-