EdPRdZddlZddlmZmZmZddlmZmZm Z ddl m Z ddl m Z mZmZmZddlmZmZmZmZddlmZdd lmZdd lmZmZdd lmZdd lm Z m!Z!m"Z"dd l#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*ddl+m,Z,dZ-dZ.dZ/ddZ0dZ1dZ2dej3dfdZ4dZ5ddZ6dZ7gfdZ8dS) 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)multiple==!=T)NF><>=)r&T<=)r'Tz'%s' is not a valid relation) isinstancer ValueErroras_expr is_numberr rtrueRealsfalseEmptySetNotImplementedError real_rootsrappendNegativeInfinityInfinityLCreversedinsert)polyreltreals intervalsroot_intervalleftrightsigneq_signequal right_open multiplicitys :lib/python3.11/site-packages/sympy/solvers/inequalities.pysolve_poly_inequalityrMsU, dD ! !H FHH H ||~~= t||~~q# . . ; =G9  !'\ =J< %7!;== =669E d{1J ' 'GD!d++H   X & & & & ' -J!!*a 11  HE1eT488H   X & & &DD  7799q= DDD$ #: CGG CZ CGG D[ C%NGUU D[ C%NGUU;cABB BJz"*5// > > D,a >7?I$$8D%UJGGIII,0%5yZe7?>5>$$8D%zBBDDD(,d:EEW_>>$$Qt(<(<=== 7? J   8A.tZHH J J J c(td|DS)aSolve polynomial inequalities with rational coefficients. Examples ======== >>> from sympy import Poly >>> from sympy.solvers.inequalities import solve_poly_inequalities >>> from sympy.abc import x >>> solve_poly_inequalities((( ... Poly(x**2 - 3), ">"), ( ... Poly(-x**2 + 1), ">"))) Union(Interval.open(-oo, -sqrt(3)), Interval.open(-1, 1), Interval.open(sqrt(3), oo)) c*g|]}t|D]}|S)rM).0pss rL z+solve_poly_inequalities..~s+GGG-BA-FGG1GGGGrN)r)polyss rLsolve_poly_inequalitiesrWps GGeGGG HHrNc.tj}|D]}|sttjtjg}|D]\\}}}t ||z|}t |d}g} t j||D]=\} } | | } | tjur| | >| }g} |D]/} |D]} | | z} | tjur| | 0| }|sn|D]} | | }|S)a3Solve a system of rational inequalities with rational coefficients. Examples ======== >>> from sympy.abc import x >>> from sympy import solve_rational_inequalities, Poly >>> solve_rational_inequalities([[ ... ((Poly(-x + 1), Poly(1, x)), '>='), ... ((Poly(-x + 1), Poly(1, x)), '<=')]]) {1} >>> solve_rational_inequalities([[ ... ((Poly(x), Poly(1, x)), '!='), ... ((Poly(-x + 1), Poly(1, x)), '>=')]]) Union(Interval.open(-oo, 0), Interval.Lopen(0, 1)) See Also ======== solve_poly_inequality r$) rr4rr8r9rM itertoolsproduct intersectr7union)eqsresult_eqsglobal_intervalsnumerdenomr>numer_intervalsdenom_intervalsrAnumer_intervalglobal_intervalrDdenom_intervals rLsolve_rational_inequalitiesrhs{.ZF$,$,  $Q%7DDE#'   NUEC3E%KEEO3E4@@OI3<3D#%54747 / //)33ODD1:-/$$X...( I#3 6 6&566N#~5OO"!*46$$_555( #  ) , ,H\\(++FF , MrNTcd}g}|r tjn tj}|D]}g}|D]}t|tr|\}} n"|jr|j|jz |j} }n|d} }|tj urtj tj d} } } nR|tj urtj tj d} } } n)| \} } t| | f\\} } } n*#t $rt!t#dwxYw| jjs*| | d}} } | j} | js4| js-| | z }t1|d| }|t3|dz}|| | f| f|r|||r3|t7|z}t7fd|Dg}||z}|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) relationalcfg|]-}|D](\\}}}|||jfdf).S)r$)hasone)rRindrCgens rLrUz0reduce_rational_inequalities..s^0=0=0=A0=0=(fq!as0=!QUT0B0=0=0=0=rN)rr2r4r-tuple is_Relationallhsrhsrel_opr1ZeroOner3togetheras_numer_denomrrr domainis_Exactto_exact get_exactis_ZZis_QQr solve_univariate_inequalityr7rhevalf as_relational)exprsrqrjexactr]solution_exprsr_exprr>rarboptr{excludes ` rLreduce_rational_inequalitiesrs: E C/qwwQZH))# 3# 3D$&& + cc%+ $48 3T[#DD $d#Dqv~ @$%FAE4cu @$%E15$cu#}}==?? u &=ENC')')#"   %j2''  :& P&+nn&6&68H8H%euZ))++FL 3FL 3U{!$3//7ceTTTT eU^S12222   JJt    /444-0=0=0=0=0=0=0=/>??G $X$>>##/))#.. Os %C>>'D%cZ|jdurttdfdddd}g}|D]^\}}||vrt |d|}nt | d||}||g|z_t ||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. c rg}|js|jrC|j|jD]3}|}|s|}fdt j||D}4n|jrM|jjstd| fd|j Dnt|trr|jd}|D]X\}}|||t|dgzf|| |t!|dgzfYn|gfg}|S)NcDg|]\\}}\}}||||zfSrQrQ)rRrconds_expr_condsops rLrUzBreduce_abs_inequality.._bottom_up_scan..CsH>>>Ca=D%RaSXZ`bbuoouv~>>>>rNz'Only Integer Powers are allowed on Abs.c3,K|]\}}|z|fVdSNrQ)rRrrros rL zAreduce_abs_inequality.._bottom_up_scan..Js0XXkdE$'5)XXXXXXrNr)is_Addis_MulfuncargsrYrZis_Powexp is_Integerr.extendbaser-rr7r r )rrargrrror_bottom_up_scans @@rLrz.reduce_abs_inequality.._bottom_up_scan7s ; !$+ !By > >(-->"EE>>>>%-eV<<>>>EE  >[ !A< L !JKKK LLXXXX__TY=W=WXXX X X X X c " " !$_TYq\22F% = = e tUbqkk]%:;<<< teUbqkk]%:;<<<< =BZLE rNr(r*r)r+r)is_extended_real TypeErrorr keysr r7r)rr>rqmapping inequalitiesrrs @rLreduce_abs_inequalityrs( u$%  >t$$GL&t,,,, e gllnn $ 6tQ,,DDteQ 55DTFUN++++ ' c : ::rNc.tfd|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 c8g|]\}}t||SrQ)r)rRrr>rqs rLrUz+reduce_abs_inequalities..ys9!!! D#(c377!!!rNr)rrqs `rLreduce_abs_inequalitiesrds8* !!!!!!! ""rNFc N(ddlm}|tjdurt t d|tjur?td||}|r| }|S }|}j dur%tj }|s|n| |Sj Ttdd  |in*#t$rtt d wxYwd}tjur|}n؉tjurtj }njjz } t'| } | tjkrRt+| } | d} | tjur|}n| tjur tj }n| t/| |} j} | d vrF| jdr|}nq| jds tj }nI| d vrE| jdr|}n'| jds tj }|j|j}}||z tjur't9d| dd|}|}|E| \}} |jvrtA| jd krtBtE| |}|tBnU#tBt f$rAt t d#tIdzwxYwt+| ((fd}g}|D]&}|%tE||'|stM(|}djvo jdk} tO|j(tS|j|jz }tS||ztU|zt9|j|j|j|v|j|v}tWd|DrtY|dd}nnt[|d}|drt  |d}tA|d krtUt]|}n#t$rt wxYwn#t $rt dwxYwtj}t_(tjkrd}tS} tat_(|}tc|t8s.|D]*}||vr$||r|j r|tS|z }+n|j|j} }tY|tS| zD]}||}!|| kr||}"te||}#|#|vrx|#j rq||#rf|!r|"r|t9||z }nN|!r|t9j3||z }n3|"r|t9j4||z }n|t9j5||z }|}|D]}$|tS|$z}n#t$rtj}d}YnwxYw|tj ur7tCt d#|d|d||}tj g}%|j}||vr4||r)|j6r"|%7tS||D]}&|&} |te|| r%|%7t9|| dd|&|vr|8|&nK|&|vr!|8|&||&}'n|}'|'r"|%7tS|&| }|j} | |vr4|| r)| j6r"|%7tS| |te|| r(|%7t9j5|| t_(tjkr|r||}n,tstu|%||#|}|s|n| |S)aTSolves a real univariate inequality. Parameters ========== expr : Relational The target inequality gen : Symbol The variable for which the inequality is solved relational : bool A Relational type output is expected or not domain : Set The domain over which the equation is solved continuous: bool True if expr is known to be continuous over the given domain (and so continuous_domain() does not need to be called on it) Raises ====== NotImplementedError The solution of the inequality cannot be determined due to limitation in :func:`sympy.solvers.solveset.solvify`. Notes ===== Currently, we cannot solve all the inequalities due to limitations in :func:`sympy.solvers.solveset.solvify`. Also, the solution returned for trigonometric inequalities are restricted in its periodic interval. See Also ======== sympy.solvers.solveset.solvify: solver returning solveset solutions with solve's output API Examples ======== >>> from sympy import solve_univariate_inequality, Symbol, sin, Interval, S >>> x = Symbol('x') >>> solve_univariate_inequality(x**2 >= 4, x) ((2 <= x) & (x < oo)) | ((-oo < x) & (x <= -2)) >>> solve_univariate_inequality(x**2 >= 4, x, relational=False) Union(Interval(-oo, -2), Interval(2, oo)) >>> domain = Interval(0, S.Infinity) >>> solve_univariate_inequality(x**2 >= 4, x, False, domain) Interval(2, oo) >>> solve_univariate_inequality(sin(x) > 0, x, relational=False) Interval.open(0, pi) rdenomsFz| Inequalities in the complex domain are not supported. Try the real domain by setting domain=S.Reals)rj continuousNrqT extended_realz When gen is real, the relational has a complex part which leads to an invalid comparison like I < 0. r)r(r*r&z The inequality, %s, cannot be solved using solve_univariate_inequality. xct|} |d}n#t$rtj}YnwxYw|tjtjfvr|S|jdur tjS|d}|j r|dStd|z)NrFr,z!relationship did not evaluate: %s) subsrrrrr3r1rro is_comparabler5)rvr expanded_errqs rLvalidz*solve_univariate_inequality..validsOOCA77  !QAA    AAA ))H%.A7NAA/#yyA.-;a?AAAs=AA=r%c3$K|] }|jV dSr)r0)rRrs rLrz.solve_univariate_inequality..>s$<z-solve_univariate_inequality..As Q=OrNz'sorting of these roots is not supportedz zZ contains imaginary parts which cannot be made 0 for any value of zm satisfying the inequality, leading to relations like I < 0. );sympy.solvers.solversr is_subsetrr2r5r r intersectionrrr4rxreplacerr1r3rtrurrwrrrrvsupinfr9rr[rz free_symbolslenr.rrrrrsetboundaryrlistallrrsortedrrr-_ptRopenLopenopen is_finiter7removerr))rrqrjr{rrrv_gen_domaineperiodconstfranger>rrrorpsolnsr singularities include_xdiscontinuitiescritical_pointsr@sifted make_realcheckim_solazstartend valid_startvalid_zptrTsol_setsr_validrs)`` @rLrr}s r-,,,,,   E)  !*."####$$ $ qw  ( ce <<< >$$WQV%<%<==== D*:sFCCt{*Bt{d/BI U"%fofj&*55'6#7#7#,em.Cd#G%G%/%#''3|VZJf,fj.FHH(I(I <8$($7$7E$222112& U U U)*STTT U I*~~'' 8"" Jf==A%a223!"77A 57%%((7qGY7 &)A,, 67&'UAEs!')C..(H!I!I&&A*/%,,K$| J*/%((%(]]#%]#:!Jr?R!JW\W\]_W`W`!J'2%Jw%J(.(5!2D2D(D)4%J(.(.2J2J(J)0%J(.(.2J2J(J(.(-q2I2I(I$%EE!.33A"ill2FF!"""WF!EEE"QZ'>$ZZ!% #t 4 4 4 4ddd 1<&=&=>>> &//77  |HJE 255<< 2EO 2 % 0 0111  5UC))FOOHUCt$D$DEEE % 6!((++++O++'..q111!&q!*6  ! 555*Cf} 0s 0  0 #///uS__%% ; eS 9 9:::*~~' LE L**733!H% 7<<<@DdOO ;22R%5%5d%;%;;sR C88'D!AL$$AM6,CT7S<;T<TTT,*EZ==[[c|js|js ||zdz }n|jr|jr tj}n|jr|j|jr|jt d|jr|js|jr |jr||}}|jr*|jr|dz}nM|jr|tjz}n6|dz}n0|jr)|jr|tjz}n|jr|dz}n|dz }|S)z$Return a point between start and endr,Nz,cannot proceed with unsigned infinite valuesr&) is_infiniterrwis_extended_positiver.is_extended_negativeHalf)rrrs rLrrs>  S_ck1_  s V   M%"< M M$'$< MKLL L O $ 8 $! $&+&@ $e3E ? ) 1W+ 16\QY   ' Z) U1W IrNctddlm}||jvr|S|j|kr|j}|j|kr||jjvr|Sd}d}t j}|j|jz } t||}| dkr)| | d}n!|s| dkrtnd#ttf$rO|s: t|gg|}n #t$rt||}YnwxYw||||} | t jur3||||t jur|||kd}|||| } | t jurP|||| t jur6|| |kd}||| kd}|t jur=| t jur||kn||k}| t jurt'| |k|}nt|}YnwxYwg} || } d} | |d\}}| |z} | |z} t+| }||d\}} |jdks |j|jcxkr nn|jd vr|} t j}| |z} |jr| | | }n|j | | }||j||jz}||}||z D]v}t7t9|d|| }t;|t8r?|j|kr4||||jt jur| |w| |fD]W}||||t jur<||||t jur#| ||ur||kn||kX| |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) rrc |||}|tjur|S|dvrdS|S#t$rtjcYSwxYw)NTF)rrNaNr)ierTrnrs rLclassifyz#_solve_inequality..classifysh 1 AAEz -' H   5LLL s%000A A Nr&T)as_AddF)r%r$)linear)rrrrur;rtrr9rdegreerr/r5rrrr1r3rras_independentr is_zero is_negative is_positivervrx_solve_inequalityr r-r7)rrTrrrroorrSokoooknoorrrubaxefrbeginning_denomscurrent_denomsrpcrns rLrrsT-,,,,,  v{ [ v{q 33     B B 6BF?D qMM 88::? &a((BB &AHHJJN &% % 0 1  81B4&!<<" 8 8 80Q77 88B2&&Dqv~ +((2q""5"5"@ +WWQVT**HRRC((E ,HRRC((AG3 ,WWbS1Wd++WWQ"Wd++QV| *"&!&.>a2ggq2v&*bS1Wb))BT A+. E %: IIKK   4 002 Q q !__  5 111 I        !%     - AA q = *CBB!!!S))B"6"&>>FF26NN:!N2 % %A!"Q((Af===A!R   %QUaZ %8B15))QV3%LL!$$$#r : :AQ""af, :HRA&&af4 : a2g8QUU1q5999 LL ;s8 A2CH5*C=<H5=DH5DDH54H5c  ii}}g}|D]}|j|j}}|t}t |dkr| n|j|z} t | dkrG|  |tt|d| ttd| r-| g||f| fd} | rFtd| Dr-| g||fu|tt|d| d|D} d|D} t#| | z|zS)Nr&rzZ inequality has more than one symbol of interest. cd|o|jp|jo |jj Sr)rl is_Functionrrr)urqs rLrz&_reduce_inequalities..s4c D B!B!%2B.BrNc3@K|]}t|tVdSr)r-rrRrns rLrz'_reduce_inequalities..s,!I!I*Q"4"4!I!I!I!I!I!IrNc6g|]\}}t|g|SrQ)rrRrqrs rLrUz(_reduce_inequalities..s)ccc:30%#>>cccrNc4g|]\}}t||SrQ)rrs rLrUz(_reduce_inequalities..s'ZZZ:3*5#66ZZZrN)rtrvatomsrrpoprr7rr r5r is_polynomial setdefaultfindritemsr)rsymbols poly_partabs_partother inequalityrr>genscommon components poly_reduced abs_reducedrqs @rL_reduce_inequalitiesr rs&bxI E"OO NJ$5c zz&!! t99> ((**CC&0F6{{a jjll .z$3/G/GMMNNN)*6++   c " " O  b ) ) 0 0$ = = = =$D$D$D$DEEJ Oc!I!Ij!I!I!III O##C,,33T3K@@@@ .z$3/G/GMMNNNNccQZQ`Q`QbQbcccLZZIYIYZZZK  +e3 55rNct|s|g}d|D}tjd|D}t|s|g}t|p||z}td|Drt t dd|Dfd|D}fd|D}g}|D]}t |trH||j |j z d}n|d vrt|d}|d krz|d krtjcS|j jrt!d |z|||}~t%||}|d DS)aEReduce a system of inequalities with rational coefficients. Examples ======== >>> from sympy.abc import x, y >>> from sympy import reduce_inequalities >>> reduce_inequalities(0 <= x + 3, []) (-3 <= x) & (x < oo) >>> reduce_inequalities(0 <= x + y*2 - 1, [x]) (x < oo) & (x >= 1 - 2*y) c,g|]}t|SrQ)rr s rLrUz'reduce_inequalities..s5551GAJJ555rNcg|] }|j SrQ)rr s rLrUz'reduce_inequalities..s>>>A>>>rNc3(K|] }|jduVdS)FNrr s rLrz&reduce_inequalities..s* 8 811  & 8 8 8 8 8 8rNzP inequalities cannot contain symbols that are not real. cJi|] }|j |t|jd!S)NTr)rrnamer s rL z'reduce_inequalities..s@555 +5aqvT222555rNc:g|]}|SrQrrRrnrecasts rLrUz'reduce_inequalities..s%===1AJJv&&===rNc:h|]}|SrQr)r*s rL z&reduce_inequalities..s%333aqzz&!!333rNrrTFr"ci|]\}}|| SrQrQ)rRkrs rLr'z'reduce_inequalities..s888A1888rN)rrr\anyrr r-r rrtr/rur rr3r0r5r7r rr)rrrkeeprnrr+s @rLreduce_inequalitiesr2s L ! !&$~ 55 555L 355;>>>>> ?D G  )7||#tt+G 8 8 8 8 888 $  55555F==== ===L33337333G D    a $ $ qu}}8!<rBsrBB----------------888888888888DDDDDDDDDDDD""""""******88888888FFFFFFFFFF((((((4444444444444444++++++XXXvIII"???DWWWWtD;D;D;N"""27;17W\d<d<d<d