MZd*XddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z mZddlmZmZmZmZmZmZmZddlmZdd lmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%dd l&m'Z'dd l(m)Z)m*Z*m+Z+m,Z,dd l-m.Z.dd l/m0Z0ddl1m2Z2m3Z3m4Z4m5Z5ddl6m7Z7ddl8m9Z9ddl:m;Z;mm?Z?m@Z@ddlAmBZBddlCmDZDmEZEmFZFddlGmHZHmIZIddlJmKZKddlLmMZMddlNmOZOmPZPddlQmRZRddlSmTZTddlUmVZVmWZWmXZXddlYmZZZddl[m\Z\ddl]m^Z^ddl_m`Z`dd lambZbdd!lcmdZddd"lemfZfmgZgdd#lhmiZidd$ljmkZkmlZldd%lmmnZndd&lompZpdd'lqmrZrdd(lsmtZtmuZudd)lvmwZwmxZxmyZymzZzm{Z{m|Z|m}Z}m~Z~mZe3d*e+\ZZZGd,d-ZGd.d/eZGd0d1eZeZeZe d2ed3e}e~ee}e~e~zgZed4d5eegzZd6Zd7Zd8ZGd9d:ZGd;dZGd?d@eZdAZdBZdCZdDZdEZdFZdGZdHZdIZdJZdKZdLZdMZdNZdOZdPZdQZdRZdSZdTZdUZdVZdWZdXZdYZdZZd[Zd\Zd]Zd^Zeud_Zd`ZdaZdbZdcZddZdeZdfZdgZdhZdiZdjZdkZdlZdmZdnZdoZdpZdqZdrZdsZdtZduZdvZdwZdxZdyZdzZd{Zd|Zd}Zd~ZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZeudZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZy))refine)Sum)Add)Basic)Tuple) ExprBuilder unchangedExprUnevaluatedExpr)Functionexpand WildFunction AppliedUndef DerivativediffSubs)Mul) NumberSymbolEzooooFloatIRationalnanIntegerNumberpi_illegal)Pow)GeLtGtLe)S)default_sort_key)SymbolsymbolsDummyWild)sympify) factorial) exp_polarexplog)sqrtMax) Piecewise)tansincos) Heaviside DiracDelta)Si)gamma) integrateIntegral) FockState)apart)factorcancelPoly)together)O) FiniteSet)combsimp) gammasimp)powsimp)collectradsimp)ratsimp)simplify nsimplify)trigsimp)Indexed)meter)raisesXFAIL) abcntuxyzzf,g,h)clscXeZdZdZdZdZdZdZdZdZ dZ d Z d Z d Z d Zd Zy) DummyNumbera Minimal implementation of a number that works with SymPy. If one has a Number class (e.g. Sage Integer, or some other custom class) that one wants to work well with SymPy, one has to implement at least the methods of this class DummyNumber, resp. its subclasses I5 and F1_1. Basically, one just needs to implement either __int__() or __float__() and then one needs to make sure that the class works with Python integers and with itself. cXt|ttfr||jzStSN isinstanceintfloatnumberNotImplementedselfrQs .jsE  B B E E aC aC qD rjr)rs rg test_basicrs  !99rjc$d}t|sJy)Nc<|}||z }|}||z}|}||z}|}||z}yr^rrs rgrztest_ibasic..ss:  Q  Q  Q  Qrjrrs rg test_ibasicrs !99rjceZdZdZdZdZdZdZdZdZ dZ d Z d Z d Z d Zd ZdZdZdZdZdZdZdZdZy)NonBasicaVThis class represents an object that knows how to implement binary operations like +, -, etc with Expr but is not a subclass of Basic itself. The NonExpr subclass below does subclass Basic but not Expr. For both NonBasic and NonExpr it should be possible for them to override Expr.__add__ etc because Expr.__add__ should be returning NotImplemented for non Expr classes. Otherwise Expr.__add__ would create meaningless objects like Add(Integer(1), FiniteSet(2)) and it wouldn't be possible for other classes to override these operations when interacting with Expr. ctd||SN+ SpecialOprfothers rgrmzNonBasic.__add__dE**rjctd||Srrrs rgrhzNonBasic.__radd__eT**rjctd||SN-rrs rgrrzNonBasic.__sub__rrjctd||Srrrs rgrpzNonBasic.__rsub__rrjctd||SN*rrs rgrvzNonBasic.__mul__rrjctd||Srrrs rgrtzNonBasic.__rmul__rrjctd||SN/rrs rgrzzNonBasic.__truediv__rrjctd||Srrrs rgrxzNonBasic.__rtruediv__rrjctd||SN//rrs rg __floordiv__zNonBasic.__floordiv__tU++rjctd||Srrrs rg __rfloordiv__zNonBasic.__rfloordiv__ud++rjctd||SN%rrs rg__mod__zNonBasic.__mod__rrjctd||Srrrs rg__rmod__zNonBasic.__rmod__rrjctd||SNdivmodrrs rg __divmod__zNonBasic.__divmod__s4//rjctd||Srrrs rg __rdivmod__zNonBasic.__rdivmod__s5$//rjctd||SN**rrs rgr~zNonBasic.__pow__rrjctd||Srrrs rgr|zNonBasic.__rpow__rrjctd||S)Nrrs rg__gt__zNonBasic.__gt__rrjctd||S)N<=rrs rg__le__zNonBasic.__le__rrjctd||S)N>=rrs rg__ge__zNonBasic.__ge__rrjN)rrrrrmrhrrrprvrtrzrxrrrrrrr~r|rrrrrrjrgrrsk ++++++++,,++00,,++,,rjrceZdZdZy)NonExprzCLike NonBasic above except this is a subclass of Basic but not ExprNrrrrrrjrgrrsMrjrceZdZdZdZy)rz>Represents the results of operations with NonBasic and NonExprcDtj|}|||f|_|Sr^)object__new__args)rZoparg1arg2objs rgrzSpecialOp.__new__s#nnS!d# rjN)rrrrrrrjrgrr s HrjrceZdZdZy) NonArithmeticzGRepresents a Basic subclass that does not support arithmetic operationsNrrrjrgrrsQrjrcttjtjtjtj tj tjtdtdtdtdtdtdtdtdtdg }|D]-ttfD]}|zd|ff|zd|ff|z d|ff|z d|ff|zd|ff|zd|ff|z d|ff|z d|ff|zd|ff|zd|ff|zd |ff|zd |fft!|d |fft!|d |ff|zd |ff|zd |ff|kd |ff|kd |ff|kDd |ff|kDd |ff|kd|ff|kd|ff|k\d|ff|k\d|ffg}|D]'\}}t#|t$ur|j&|k(r'Jt)t+fD]t-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fdt-t.fd t-t.fd!t-t.fd"t-t.fd#t-t.fd$t-t.fd%t-t.fd&t-t.fd'0y())a}Tests that Expr uses binary operations cooperatively. In particular it should be possible for non-Expr classes to override binary operators like +, - etc when used with Expr instances. This should work for non-Expr classes whether they are Basic subclasses or not. Also non-Expr classes that do not define binary operators with Expr should give TypeError. ?rrWrrrrrrrrrrrrczSr^renasrgz-test_cooperative_operations..Z q2vrjczSr^rrsrgrz-test_cooperative_operations..[ rAvrjcz Sr^rrsrgrz-test_cooperative_operations..\rrjcz Sr^rrsrgrz-test_cooperative_operations..]rrjczSr^rrsrgrz-test_cooperative_operations..^rrjczSr^rrsrgrz-test_cooperative_operations.._rrjcz Sr^rrsrgrz-test_cooperative_operations..`rrjcz Sr^rrsrgrz-test_cooperative_operations..arrjczSr^rrsrgrz-test_cooperative_operations..b qBwrjczSr^rrsrgrz-test_cooperative_operations..c rQwrjczSr^rrsrgrz-test_cooperative_operations..drrjczSr^rrsrgrz-test_cooperative_operations..errjctSr^rrsrgrz-test_cooperative_operations..fs va}rjctSr^rrsrgrz-test_cooperative_operations..gs vb!}rjczSr^rrsrgrz-test_cooperative_operations..hrrjczSr^rrsrgrz-test_cooperative_operations..irrjckDSr^rrsrgrz-test_cooperative_operations..jrrjckDSr^rrsrgrz-test_cooperative_operations..krrjckSr^rrsrgrz-test_cooperative_operations..lrrjckSr^rrsrgrz-test_cooperative_operations..mrrjck\Sr^rrsrgrz-test_cooperative_operations..nrrjck\Sr^rrsrgrz-test_cooperative_operations..orrjckSr^rrsrgrz-test_cooperative_operations..prrjckSr^rrsrgrz-test_cooperative_operations..qrrjN)r r%ZeroOneInfinityNegativeInfinityComplexInfinityHalfrrr'rrr rrrtyperrrrrO TypeError)exprsneresultsresrrrs @@rgtest_cooperative_operationsrs        c  s  Avc{ Avc{ Avc{ E >0:wy) CBa#r1&R#q"&a#r1&R#q"&a#r1&R#q"&a#r1&R#q"&q4Q-(b4B-(a#r1&R#q"&A2q 1221b 12q4Q-(b4B-(R#r1&a#r1&R#r1&a#r1&b4Q-(q4Q-(b4Q-(q4Q-(1G6% C TCyI-#((d2BBB C; CD!?FH- 0B 9o . 9o . 9o . 9o . 9o . 9o . 9o . 9o . 9. / 9. / 9o . 9o . 94 5 94 5 9. / 9. / 9o . 9o . 9o . 9o . 9. / 9. / 9. / 9. /3 0K>0rjc~ddlm}tdktjusJtdktjusJtdkDtj usJtdk\tj usJt dktj usJt dktj usJt dkDtjusJt dk\tjusJt dd}|dz |dz ktjusJ|ttzttzdzj|k(sJy)Nr)r"rTrealr) sympy.core.relationalr"rr%falsetruer'rWrfunc)r"r"s rgtest_relationalr)ts( Fqww   !G   Fqvv   !G   C!G   C1H   C!G   C1H  sA EAEMagg %% % a!eQUQY  $ $ ** *rjc|tdd}tdd}tdd}tdd }|d kt|d k(sJ|d kt|d k(sJ|d kDt|d k(sJ|d k\t |d k(sJtddd }tddd }tddd }tddd }|d kt j usJ|d kt j usJ|d kDt j usJ|d k\t j usJtdd }tdd }tdd }tdd }|d kt j usJ|d kt j usJ|d kDt j usJ|d k\t j usJtddd }tddd }tddd }tddd }|d kt jusJ|d kt jusJ|d kDt jusJ|d k\t jusJy)Nm1F) nonnegativem2positivem3) nonpositivem4)negativerT)r,r$r/r$)r1r$)r3r$)r'r"r$r#r!r%r'r&)r+r-r0r2s rgtest_relational_assumptionsr5s: % (B u %B % (B u %B Fr"ay  !G2q !! ! Fr"ay  !G2q !! ! %d 3B u4 0B %d 3B u4 0B Fqvv   !G   Fqvv   !G   t $B $ 'B t $B $ 'B Fqvv   !G   Fqvv   !G   u4 0B %d 3B u4 0B %d 3B Fqww   !G   Fqww   !G  rjctD]pttfdttfddk(r dzdk(sJttfdttfdttfdry) NcdzSN1rrsrgrz"test_basic_nostr.. #)rjcdz Sr8rr:srgrz"test_basic_nostr..r;rjrr911cdzSr8rr:srgrz"test_basic_nostr..s cCirjcdz Sr8rr:srgrz"test_basic_nostr..r;rjcdzSr8rr:srgrz"test_basic_nostr..s #*rj) basic_objsrOrr:s@rgtest_basic_nostrrBsb.y+,y+, !89$ $$ 9/ 0y+,y,-.rjc8dtz tztzjtdddtz tdtzk(sJdtz tztzjtdddtz tzttzk(sJdtz dztzjtdddtz tdtzk(sJdtz dztzjtdddtz dzttzk(sJdtz tzjtdddtz tdtzk(sJdtz tzttzztzjtdddtz tdtzk(sJdtz tzttzztzjtdddtz tzttzk(sJyNr)rWrXseriesrBrrjrg'test_series_expansion_for_uniform_orderrGs aC!GaK  1a (AaC!Aq'M 99 9 aC!GaK  1a (AaC!GadN :: : aC!GaK  1a (AaC!Aq'M 99 9 aC!GaK  1a (AaC!GadN :: : aC!G  Aq! $!a1g 55 5 aC!GacMA  % %aA .!A#!Q- ?? ? aC!GacMA  % %aA .!A#'AaD. @@ @rjcddttdtdz dz zzzjtdk(sJdtdzz dztztdzzjtddk(sJdtz dztztdzzjtddk(sJtdzdtz zjtddk(sJdtdzzjtddk(sJtdzjtddk(sJttdzzjtddk(sJtdzjtddk(sJy)Nr!rrEr!rr)rWr/leadtermrrjrg test_leadtermrMs] !c!fSVma'(( ( 2 21 5 ?? ? adFQJNQT ! + +A .q 1R 77 7 aC!GaK!Q$  ( ( +A ." 44 4 qD1Q3J  #A &" ,, , 1H  q !! $ )) ) E  A q !Q && & 1H  q !! $ )) ) qD??1 a A %% %rjcRddttdtdz dz zzzjtdk(sJdtdzz dztztdzzjtdtdzz k(sJdtz dztztdzzjtdtz k(sJtdzdtz zjtdtz k(sJdtdzzjtdk(sJtdzjtdk(sJttdzzjttk(sJtdzjttdzk(sJttzjttusJt t ddtztt ddztdtjzdz z dzzt ddz dtdztjzdz z}|jtdtdztzdtjztzz dtdzdtjzzz k(sJdtz tdzz }dtdzdz }|jttd|z zjtd tdztzd tzz d tdzd zz k(sJy) Nr!rrEc2tdzjdSNrErWas_leading_termrrjrgrz&test_as_leading_term..sA66q9rj riii@) rWr/rRrrO ValueErrorrr0r% ImaginaryUnitsubsrX)rds rgtest_as_leading_termr[s !c!fSVma'(( ( 9 9! < AA A adFQJNQT ! 2 21 51a4 ?? ? aC!GaK!Q$  / / 2ac 99 9 qD1Q3J ' ' *ac 11 1 1H % %a (A -- - E " "1 % ** * 1H % %a (A -- - qD ! !! $1 ,, , F # #A &" ,, , :9: 1HQN"T!WQ__%>BB 1a. T!W9Q__4Q6 7A  Q  DGA1??*1, ,qay2aoo;M/MN OO O A1 A T!WaA 66!Q1W  - -a 0 d1ga$q& 3tAw;#45 66 6rjcttdztdtdzztdtdzzjttdtdzdfk(sJyrD)rWr5r4rLrrjrgtest_leadterm2r]s] c!fHSSV_ $s1s1v: 6 @ @ C CF OQ    rjcvttztzjtttzdfk(sJy)Nr)rXrYrWrLrrjrgtest_leadterm3r_s+ EAI   "q1uaj 00 0rjcttdztdtdzztdtdzzjttdtdzk(sJyrP)rWr5r4rRrrjrgtest_as_leading_term2rasX c!fHSSV_ $s1s1v: 6 G G J AAJ  rjcdtztzjtdtzk(sJdtzttzztdzzjtdtzttzzk(sJyNr)rrWrRrrjrgtest_as_leading_term3rdsa FQJ ' ' *a"f 44 4 aC"Q$JA  . .q 1QqS2a4Z ?? ?rjctddd}|dz d|dzzd|zzdzz |dz|dzd|zzdzz z |dz|dzz z|d|dzzd|zzdzz z ||tztzz zd|z|dzz zdz d|tztzz zd|dzz zdtz z }|jtj|dz k(sJy)NrTT)integerr/r!rrUrE)r'rWrRr?)rTr"s rgtest_as_leading_term4rgssD40A AqAv!|a 1a4A!a#88 1a!e  !AqD&1Q3,*+ ,./1qk :<=aCQK H  qsQwK QU) $&'c *A  Q  & & (AaC // /rjcGddtdtz jtdtz k(sJdjtdk(sJttfdy)Nc eZdZy)&test_as_leading_term_stub..fooNrrrrrjrgfoorj  rjrlrEcBtjtSr^rQrlsrgrz+test_as_leading_term_stub.. sA(>(>q(Arj)r rWrRrONotImplementedErrorros@rgtest_as_leading_term_stubrqse h qs8 # #A &#ac( 22 2 q6 ! !! $A .. .  ABrjcttdztjtdtdzzk(sJttdztjtdk(sJt tdztjttdzdz k(sJt tdztjtttdzzk(sJtt ttjtdk(sJtt ttjtdtz jtk(sJy)Nr!rrrUrE)rrWrRrXr;r.r/rrjrg#test_as_leading_term_deriv_integralrss a1fa 0 0 3qAv == = a1fa 0 0 3q 88 8 AFA  . .q 1QT!V ;; ; AFA  . .q 1Qq!tV ;; ; c!fa 0 0 3q 88 8 c!fa 0 0 3!7L7LQ7O OO Orjc tjthk(sJdtzjttjhk(sJddt tzzjt thk(sJddt tzzjt t tjtdthk(sJdtttzzzjtdtthk(sJtjjtjhk(sJtjjt tk(sJttjttk(sJtdtjtjthk(sJtdtjtjthk(sJtttjthk(sJttttjtthk(sJtttzttjtthk(sJtttztttjttthk(sJttttzztttjtttthk(sJt t"zjt$t"hk(sJt t"zjt$t t t"zjt t$cxk(rt"t hk(sJJt't'tjt&t't'tt'thk(sJdtdtzzzt'dtzzjt(dtdtzzzt'dtzzdtzdtzhk(sJt+tttzdz}|jt,t+thk(sJ|jt,t.t+ttthk(sJ|jt.t+ttthk(sJ|jt,t t+ttdhk(sJ|jt.t tdttt+thk(sJy)NrErrr!)rWatomsr%rr5r'rrXrsetr4rr@rrYrUrrrr.rfrr rs rg test_atomsrys 779   E==?q!%%j (( ( #a&L   'A3 .. . #a&L   /AEE1Q43C CC C q1a4yM "qtQl 22 2 66<<>affX %% % 66<< 35 (( ( r7==  %% % 1:   !&&! ,, , 1:   !%% ++ + 1:   ! $$ $ 1a=   QF ** * Aq!  " " $A .. . Aq!Q  % % 'Aq!9 44 4 AaCAq ! ' ' )aAq\ 99 9 bD<< %" -- - bD<< a ( 2 Q % 1*,a 11 11 1 s1v;  S !c#a&k3q6%: :: : 1q5 MCAJ & - -c 2 QAYQU #QUAE2 33 3 1A A 77< 1   77< * 1s1v   778  1s1v   77< ( 1qt    778V $ 1s1vqt  rjc6 tddd}tdjttt dusJt jjttt dusJtjtdusJtjtdusJtdzjtdusJtdzjtdusJtdzjtdusJtdzjtdusJdtzjtdusJdtzjtdusJt|zjtdusJt|zj|dusJttzjtdusJ||zj|dusJ|tzj|dusJt| zjtdusJdtz|zjtdusJtdzdtzzdz jtdusJtdzdtzzdz jtdusJtdzdtzzdz jdusJttjtdusJttdzjtdusJtdzdtzttzzdz jtdusJtdzdtzttzzdz jtdusJtdztdzzttdzzzttzztdzjdusJtdztdzzttdzzzttzzttzjdusJtdztdzzttdzzzttzztdzjttdusJtdztdzzttdzzzttzzttzjttdusJd ttz d zjttdusJy) NkT)r,rfrrJFr!rE) r'r is_polynomialrWrXrYr%Pir0r.rw)r{s rgtest_is_polynomialrIsisd3A A; $ $Q1 - 55 5 DD  1a (D 00 0 ??1  %% % ??1  %% % qD   "d ** * qD   "d ** * G " "1 % .. . G " "1 % -- - qD   "e ++ + qD   "d ** * qD   "e ++ + qD   "e ++ + qD   "e ++ + qD   "e ++ + qD   "e ++ + G " "1 % .. . qS1H # #A &% // / qD1Q3JN ) )! , 44 4 qD1Q3JN ) )! , 44 4 qD1Q3JN ) ) +t 33 3 7  #u ,, , GQJ % %a (E 11 1 qD1Q3tAw;  " 1 1! 4 << < qD1Q3tAw;  " 1 1! 4 == = TAqDMAq!tH $qs *SV 3 B B D LL L TAqDMAq!tH $qs *SV 3 B B D MM M A1 1a4 1Q3&Q/q!1D MM M A1 1a4 1Q3&Q/q!1D NN N adFQJ % %ad +u 44 4rjc^tdjdusJtdjtdusJtddjdusJtddjtdusJdtz jdusJdtz jtdusJttz jdusJttz jtdusJttz jttdusJtdzdtz tz zjdusJtdzdtz tz zjtdusJtdzdtz tz zjttdusJt ttz jdusJt ttz jtdusJt ttz jtdusJt ttz jttdusJt D]6}|jrJdtfD]}||z jsJ8y)NrET6rTrF)ris_rational_functionrWrrXr4r)irZs rgtest_is_rational_functionrzsP 1: * * , 44 4 1: * *1 - 55 5 B  0 0 2d :: : B  0 0 3t ;; ; qD & & (D 00 0 qD & &q )T 11 1 aC % % '4 // / aC % %a (D 00 0 aC % %a +t 33 3 qD1Q3q5L . . 0D 88 8 qD1Q3q5L . .q 1T 99 9 qD1Q3q5L . .q! 4 << < F1H * * , 55 5 F1H * *1 - 66 6 F1H * *1 - 55 5 F1H * *1a 0E 99 9 4))+++Q 4A!113 33 44rjcttdzz tztzttdzzz}|j tddusJ|j tddusJ|j tt dusJddtt dt dz dz zzz}|j tddusJ|j tddusJ|j tt dusJtdd}tdtz |ztz}|j tddusJ|j tddusJ|j tt dusJt ttz}|j tddusJ|j tddusJ|j tddusJ|j tt dusJt ttzj tddusJt ttzj tddusJtt tzj tdJdt tzj tddusJdt tzj tddusJy) NrrTrEr!FrTrf) rQrWrRrSis_meromorphicrr/r'r4r)rwgrTrs rgtest_is_meromorphicrs !Q$ Q1a4A  Aq !T )) )  Aq !T )) )  As #t ++ + Aa#a&Q-!#$ $$A  Aq !U ** *  Aq !T )) )  As #u ,, ,sD!A AaC! 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