a tjgS|tj urPtj gSt d||j ddg}}|dkr|D]\}}t ||}||qzn|dkrtj}|tjdfgD]$\} }t || d d }||| }qnJ|dkrd} nd } d \} } |d krd} nD|d kr"d } n4|dkr6d\} } n |dkrJd\} } n td|tjd } } t|D]\}}|dr| | kr|dt || | | | || } } } nT| | kr| s|dt || d | |d } } n"| | krj| rj|dt ||qj| | kr.|dt tj| d | |S)aSolve a polynomial inequality with rational coefficients. Examples ======== >>> from sympy import solve_poly_inequality, Poly >>> from sympy.abc import x >>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==') [{0}] >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '!=') [Interval.open(-oo, -1), Interval.open(-1, 1), Interval.open(1, oo)] >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '==') [{-1}, {1}] See Also ======== solve_poly_inequalities z8For efficiency reasons, `poly` should be a Poly instancer%could not determine truth value of %sF)multiple==!=T)NF><>=)r$T<=)r%Tz'%s' is not a valid relation) isinstancer ValueErroras_expr is_numberr rtrueRealsfalseEmptySetNotImplementedErrorZ real_rootsr appendNegativeInfinityInfinityZLCreversedinsert)Zpolyreltreals intervalsroot_intervalleftrightsignZeq_signequalZ right_openZ multiplicityrDj/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/solvers/inequalities.pysolve_poly_inequalitysr                     rFcCstdd|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]}|qqSrD)rF).0psrDrDrE }z+solve_poly_inequalities..)r)ZpolysrDrDrEsolve_poly_inequalitiesosrLcCstj}|D]}|sq ttjtjg}|D]\\}}}t|||}t|d}g} |D],} |D]"} | | } | tjur\| | q\qT| }g} |D]*} |D] } | | 8} q| tjur| | q| }|s(qq(|D]} || }qq |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") rr2r r5r6rF intersectr4union)eqsresult_eqsZglobal_intervalsnumerdenomr9Znumer_intervalsZdenom_intervalsr<Znumer_intervalZglobal_intervalr?Zdenom_intervalrDrDrEsolve_rational_inequalitiess6      rTTc sd}g}|rtjntj}|D]\}g}|D]>}t|trD|\}} n&|jr`|j|j|j}} n |d}} |tj urtj tj d} } } n0|tj urtj tj d} } } n| \} } zt| | f\\} } } WntyttdYn0| jjs| | d} } }| j} | jsX| jsX| | }t|d| }|t|ddM}q*|| | f| fq*|r||q|r|t|M}tfdd|Dg}||8}|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 ||jfdfq qS)r")hasZone)rGindr>genrDrErJsz0reduce_rational_inequalities..)rr0r2r+tupleZ is_Relationallhsrhsrel_opr/ZeroOner1Ztogetheras_numer_denomrrrdomainZis_ExactZto_exactZ get_exactZis_ZZZis_QQr solve_univariate_inequalityr4rTZevalf as_relational)exprsr[rUexactrOZsolution_exprsrQexprr9rRrSoptrcexcluderDrZrEreduce_rational_inequalitiessV                 rlcs|jdurttdfdd|}ddd}g}|D]D\}}||vr^t|d|}nt| d||}||g|q>> 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. c s&g}|js|jrr|j}|jD]R}|}|s2|}qg}|D].\}}|D] \}}||||||fqFq:|}qn|jr|j} | jstd|j }|D]\}}||| |fqnjt |t r|jd}|D]>\}}|||t |dgf|| |t |dgfqn |gfg}|S)Nz'Only Integer Powers are allowed on Abs.r)Zis_AddZis_Mulfuncargsr4is_Powexp is_Integerr,baser+rr r ) rirfopargrhrncondsZ_exprZ_condsrX_bottom_up_scanrDrErw6s4          z.reduce_abs_inequality.._bottom_up_scanr&r(r'r)r)is_extended_real TypeErrorrkeysr r4rl)rir9r[rfmapping inequalitiesrurDrvrEreduce_abs_inequalitys   '   r~cstfdd|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||qSrD)r~)rGrir9rZrDrErJsz+reduce_abs_inequalities..r)rfr[rDrZrEreduce_abs_inequalitiesms rFc( sddlm}|tjdur*ttdn2|tjur\td|d|}|rX| }|S}|}j durtj }|s||S| |Sj durt ddd z |iWntyttd Yn0d}tjur|}ntjurtj }njj} t| } | tjkrTt| } | d} | tjur@|}n| tjurtj }n| durt| |} j} | d vr| jdr|}n| jdstj }n6| d vrވ| jdr|}n| jdstj }|j|j}}||tjurtd| dd|}|}|dur| \}}z>|jvrLt | jd krLt!t"| |}|durft!Wn4t!tfyttd#t$dYn0t| fdd}g}|D]}|%t"||q|st&|}djvojdk}zt'|j(t)|j|j}t)||t*|t|j|j|j|v|j|v}t+dd|Drzt,|ddd}n\t-|dd}|drtz&|d}t |d krt*t.|}WntytYn0WntytdYn0tj}t/tjkrd}t)}z4t0t/|}t1|tsl|D].}||vr:||r:|j r:|t)|7}q:n|j|j}} t,|t)| D]}||}!|| kr*||}"t2||}#|#|vr*|#j r*||#r*|!r|"r|t||7}n@|!r|t3||7}n(|"r|t4||7}n|t5||7}|}q|D]}$|t)|$8}q6Wntyjtj}d}Yn0|tj urt!td#||f||}tj g}%|j}||vr||r|j6r|%7t)||D]~}&|&} |t2|| r|%7t|| dd|&|vr|8|&n6|&|vr:|8|&||&}'n|}'|'rR|%7t)|&| }q|j} | |vr|| r| j6r|%7t)| |t2|| r|%7t5|| t/tjkr|r||}nt9t:|%||#|}|s|S| |S)aSSolves a real univariate inequality. Parameters ========== expr : Relational The target inequality gen : Symbol The variable for which the inequality is solved relational : bool A Relational type output is expected or not domain : Set The domain over which the equation is solved continuous: bool True if expr is known to be continuous over the given domain (and so continuous_domain() doesn't need to be called on it) Raises ====== NotImplementedError The solution of the inequality cannot be determined due to limitation in :func:`sympy.solvers.solveset.solvify`. Notes ===== Currently, we cannot solve all the inequalities due to limitations in :func:`sympy.solvers.solveset.solvify`. Also, the solution returned for trigonometric inequalities are restricted in its periodic interval. See Also ======== sympy.solvers.solveset.solvify: solver returning solveset solutions with solve's output API Examples ======== >>> from sympy import solve_univariate_inequality, Symbol, sin, Interval, S >>> x = Symbol('x') >>> solve_univariate_inequality(x**2 >= 4, x) ((2 <= x) & (x < oo)) | ((-oo < x) & (x <= -2)) >>> solve_univariate_inequality(x**2 >= 4, x, relational=False) Union(Interval(-oo, -2), Interval(2, oo)) >>> domain = Interval(0, S.Infinity) >>> solve_univariate_inequality(x**2 >= 4, x, False, domain) Interval(2, oo) >>> solve_univariate_inequality(sin(x) > 0, x, relational=False) Interval.open(0, pi) rdenomsFz| Inequalities in the complex domain are not supported. Try the real domain by setting domain=S.Reals)rU continuousNr[TZ extended_realz When gen is real, the relational has a complex part which leads to an invalid comparison like I < 0. rx)r&r(r$z The inequality, %s, cannot be solved using solve_univariate_inequality. xcst|}z|d}Wnty8tj}Yn0|tjtjfvrN|S|jdur^tjS|d}|j rz|dSt d|dS)NrFr*z!relationship did not evaluate: %s) subsrrmrzrr1r/ryrXZ is_comparabler3)rvrZ expanded_erir[rDrEvalids      z*solve_univariate_inequality..valid=r#css|] }|jVqdSN)r.)rGrrDrDrE GrKz.solve_univariate_inequality..) separatedcSs|jSrry)rrDrDrEJrKz-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_subsetrr0r3rrd intersectionreryr2rxreplacerzr/r1r]r^rr`rrmrr_supinfr6r rMrb free_symbolslenr,rrrextendrsetboundaryrlistallrrsortedrrr+_ptZRopenZLopenopen is_finiter4removerr)(rir[rUrcrrrvZ_genZ_domaineZperiodconstZfranger9rrrXrYZsolnsrZ singularitiesZ include_xZdiscontinuitiesZcritical_pointsr;ZsiftedZ make_realcheckZim_solazstartendZ valid_startZvalid_zptrIZsol_setsr_validrDrrErdsF9                                           rdcCs|js|js||d}n|jr.|jr.tj}n|jr>|jdusN|jrV|jdurVtd|jrb|jsn|jrx|jrx||}}|jr|jr|d}q|jr|tj}q|d}n0|jr|jr|tj}n|jr|d}n|d}|S)z$Return a point between start and endr*Nz,cannot proceed with unsigned infinite valuesr$) is_infiniterr`Zis_extended_positiver,Zis_extended_negativeZHalf)rrrrDrDrErs:         rc Cs|ddlm}||jvr|S|j|kr*|j}|j|krD||jjvrD|Sdd}d}tj}|j|j}zBt||}| dkr| | d}n|s| dkrt Wn,t t fy|szt|gg|}Wnt yt||}Yn0||||} | tjur*||||tjur*|||kd}|||| } | tjur~|||| tjur~|| |kd}||| kd}|tjur| tjur||kn||k}| tjurt| |k|}nt|}Yn0g} |durj| } d} | j|dd\}}| |8} | |8} t| }|j|d d\}} |jd ks`|j|jkrPdurjnn|jd vrj|} tj}| |} |jr| | | }n|j | | }||j||jB}||}||D]T}tt|d||d }t|tr|j|kr||||jtjur| |q| |fD]N}||||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) rrcSsNz0|||}|tjur|WS|dvr,WdS|WStyHtjYS0dS)NTF)rrNaNrz)ierIrWrrDrDrEclassify s   z#_solve_inequality..classifyNr$T)Zas_AddF)r#r")linear)rrrr^r7r]rr6rZdegreermr-r3rrlrdr/r1rrZas_independentris_zeroZ is_negativeZ is_positiver_ra_solve_inequalityr r+r4)rrIrrrrZoorirHZokooZoknoorurr^baxZefrZbeginning_denomsZcurrent_denomsrYcrWrDrDrErsJ                  $ rcshii}}g}|D]}|j|j}}|t}t|dkrD|nF|j|@} t| dkr~| |tt |d|qn t t d| r| g||fq|fdd} | rtdd| Dr| g||fq|tt |d|qg} g} |D]\} | t| gq|D]\} | t| q.css|]}t|tVqdSr)r+rrGrWrDrDrErrKz'_reduce_inequalities..)r]r_Zatomsrrpoprr4rr r3rZ is_polynomial setdefaultfindritemsrlrr)r}symbolsZ poly_partZabs_partotherZ inequalityrir9genscommon componentsZ poly_reducedZ abs_reducedrfrDrZrE_reduce_inequalities{s4        rcsPt|s|g}dd|D}tjdd|D}t|s@|g}t|pJ||@}tdd|Drnttddd|Dfd d|D}fd d |D}g}|D]~}t|tr||j |j d }n|d vrt |d }|dkrqn|dkrt jS|j jrtd|||q|}~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|qSrD)rrrDrDrErJrKz'reduce_inequalities..cSsg|] }|jqSrD)rrrDrDrErJrKcss|]}|jduVqdS)FNrrrDrDrErrKz&reduce_inequalities..zP inequalities cannot contain symbols that are not real. cSs&i|]}|jdur|t|jddqS)NTr)ryrnamerrDrDrE s z'reduce_inequalities..csg|]}|qSrDrrZrecastrDrErJrKcsh|]}|qSrDrrrrDrE rKz&reduce_inequalities..rrTFr cSsi|]\}}||qSrDrD)rGkrrDrDrErrK)rrrNanyrzrr+r rmr]r-r^r rr1r.r3r4rrr)r}rrZkeeprWrrDrrEreduce_inequalitiessB        rN)T)F)8__doc__Zsympy.calculus.utilrrrZ sympy.corerrrZsympy.core.exprtoolsrZsympy.core.relationalr r r r Zsympy.sets.setsr rrrZsympy.core.singletonrZsympy.core.functionrZ$sympy.functions.elementary.complexesrrZ sympy.logicrZ sympy.polysrrrZsympy.polys.polyutilsrZsympy.solvers.solvesetrrZsympy.utilities.iterablesrrZsympy.utilities.miscrrFrLrTrlr~rr0rdrrrrrDrDrDrEs8      [B ZQ)! .3