Otj. UddlmZddlmZddlmZmZmZmZm Z m Z m Z ddl m Z ddlmZddlmZmZddZd Zd Zd Zd Zd ZdZdZdZdZeeeeeeeedZded<dS)) annotations)Callable)SAddExprBasicMulPowRational) fuzzy_not)Boolean)askQTct|ts|S|jsfd|jD}|j|}t |dr|}||S|jj}t |d}||S||}|||kr|St|ts|St|S)a Simplify an expression using assumptions. Explanation =========== Unlike :func:`~.simplify` which performs structural simplification without any assumption, this function transforms the expression into the form which is only valid under certain assumptions. Note that ``simplify()`` is generally not done in refining process. Refining boolean expression involves reducing it to ``S.true`` or ``S.false``. Unlike :func:`~.ask`, the expression will not be reduced if the truth value cannot be determined. Examples ======== >>> from sympy import refine, sqrt, Q >>> from sympy.abc import x >>> refine(sqrt(x**2), Q.real(x)) Abs(x) >>> refine(sqrt(x**2), Q.positive(x)) x >>> refine(Q.real(x), Q.positive(x)) True >>> refine(Q.positive(x), Q.real(x)) Q.positive(x) See Also ======== sympy.simplify.simplify.simplify : Structural simplification without assumptions. sympy.assumptions.ask.ask : Query for boolean expressions using assumptions. c0g|]}t|S)refine).0arg assumptionss V/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/assumptions/refine.py zrefine..4s#>>>SsK((>>> _eval_refineN) isinstanceris_Atomargsfunchasattrr __class____name__ handlers_dictgetrr)exprrrref_exprnamehandlernew_exprs ` rrr sJ dE " " < >>>>DI>>>ty$t^$$$$[11  O > "Dd++G wt[))Hdh.. h % % (K ( ((rcDddlm}|jd}tt j|r1t tt j|r|Stt j|r| St|tr~fd|jD}g}g}|D]H}t||r!| |jd3| |It||t|zSdS)aF Handler for the absolute value. Examples ======== >>> from sympy import Q, Abs >>> from sympy.assumptions.refine import refine_abs >>> from sympy.abc import x >>> refine_abs(Abs(x), Q.real(x)) >>> refine_abs(Abs(x), Q.positive(x)) x >>> refine_abs(Abs(x), Q.negative(x)) -x rAbscJg|]}tt| Sr)rabs)rars rrzrefine_abs..bs) ; ; ;QVCFFK ( ( ; ; ;rN) $sympy.functions.elementary.complexesr+rrrrealr negativerr append)r$rr+rrnon_absin_absis ` r refine_absr7Gs5"988888 )A,C 16#;; $$ c!*S//;77 8 8  1:c??K((t #s 1 ; ; ; ;#( ; ; ; " "A!S!! " afQi((((q!!!!G}ss3<0000 1 1rcddlm}ddlm}t |j|rst tj|jj d|rAt tj |j |r|jj d|j zSt tj|j|r|jj rt tj |j |rt|j|j zSt tj|j |r-||jt|j|j zzSt |j trHt |jt r.t|jj|jj |j zzS|jt"jur|j jrx|}|j \}}t+|}t+}t+}t-|} |D]q} t tj | |r|| :t tj| |r|| r||z}t-|dzr||z}|t"jzdz} n ||z}|dz} | |kst-|| kr&|| |jt3|z}d|j z} t tj | |r| r | |jz} | jr| \} }|jr|jt"jurt tj|j |r| dzdz } t tj | |r|j|j zSt tj| |r|j|j dzzS|j|j | zzS||kr|SdSdSdSdS)as Handler for instances of Pow. Examples ======== >>> from sympy import Q >>> from sympy.assumptions.refine import refine_Pow >>> from sympy.abc import x,y,z >>> refine_Pow((-1)**x, Q.real(x)) >>> refine_Pow((-1)**x, Q.even(x)) 1 >>> refine_Pow((-1)**x, Q.odd(x)) -1 For powers of -1, even parts of the exponent can be simplified: >>> refine_Pow((-1)**(x+y), Q.even(x)) (-1)**y >>> refine_Pow((-1)**(x+y+z), Q.odd(x) & Q.odd(z)) (-1)**y >>> refine_Pow((-1)**(x+y+2), Q.odd(x)) (-1)**(y + 1) >>> refine_Pow((-1)**(x+3), True) (-1)**(x + 1) rr*)signN)r/r+sympy.functionsr9rbaserrr0revenexp is_numberr-oddr r r NegativeOneis_Add as_coeff_addsetlenaddOnercould_extract_minus_sign as_two_termsis_Powinteger)r$rr+r9oldcoeffterms even_terms odd_termsinitial_number_of_termst new_coeffe2r6ps r refine_PowrWms8988888$$$$$$$)S!!1 qvdinQ'((+ 6 6 1AF48$$k22 19>!$0 0 16$)  k**? 9  D16$(##[11 249~~1115??K00 DtDITY48)CCC dh ) ) I$)S)) I49>**ty}tx/GHH 9 % %x4  $x4466 uE  UU EE *-e**'))A16!99k22)"q))))QU1XX{33)! a(((#y>>A%*Y&E!&! 3IIY&E % I%%U6M)M)MIIi(((9sE{3DtxZqvbzz;//(2244(di9 >??,,DAqx>AFam$;$;qy//==>!"Q A"16!99k::>'+y!%'7 7!$QU1XX{!;!;>'+y1519'= ='+y1519'= =$;;K? ?  & %4 4 f;rcddlm}|j\}}tt j|t j|z|r|||z Stt j|t j|z|r|||z tj z Stt j|t j|z|r|||z tj zStt j |t j|z|r tj Stt j|t j |z|rtj dz Stt j|t j |z|rtj dz Stt j |t j |z|r tj S|S)a Handler for the atan2 function. Examples ======== >>> from sympy import Q, atan2 >>> from sympy.assumptions.refine import refine_atan2 >>> from sympy.abc import x, y >>> refine_atan2(atan2(y,x), Q.real(y) & Q.positive(x)) atan(y/x) >>> refine_atan2(atan2(y,x), Q.negative(y) & Q.negative(x)) atan(y/x) - pi >>> refine_atan2(atan2(y,x), Q.positive(y) & Q.negative(x)) atan(y/x) + pi >>> refine_atan2(atan2(y,x), Q.zero(y) & Q.negative(x)) pi >>> refine_atan2(atan2(y,x), Q.positive(y) & Q.zero(x)) pi/2 >>> refine_atan2(atan2(y,x), Q.negative(y) & Q.zero(x)) -pi/2 >>> refine_atan2(atan2(y,x), Q.zero(y) & Q.zero(x)) nan r)atanr:) (sympy.functions.elementary.trigonometricrYrrrr0positiver1rPizeroNaN)r$rrYyxs r refine_atan2ras2>===== 9DAq 16!99qz!}} $k22tAE{{ QZ]]QZ]] *K 8 8 tAE{{QT!! QZ]]QZ]] *K 8 8 tAE{{QT!! QVAYYA & 4 4 t QZ]]QVAYY & 4 4tAv QZ]]QVAYY & 4 4uQw QVAYY "K 0 0u  rc|jd}ttj||r|Sttj||r t jSt||S)a  Handler for real part. Examples ======== >>> from sympy.assumptions.refine import refine_re >>> from sympy import Q, re >>> from sympy.abc import x >>> refine_re(re(x), Q.real(x)) x >>> refine_re(re(x), Q.imaginary(x)) 0 r)rrrr0 imaginaryrZero _refine_reimr$rrs r refine_rergsa )A,C 16#;; $$  1;s  [))v k * **rc|jd}ttj||r tjSttj||rtj |zSt||S)a Handler for imaginary part. Explanation =========== >>> from sympy.assumptions.refine import refine_im >>> from sympy import Q, im >>> from sympy.abc import x >>> refine_im(im(x), Q.real(x)) 0 >>> refine_im(im(x), Q.imaginary(x)) -I*x r) rrrr0rrdrc ImaginaryUnitrerfs r refine_imrjsl )A,C 16#;; $$v  1;s  [))' 3&& k * **rc|jd}ttj||r tjSttj||r tjSdS)a" Handler for complex argument Explanation =========== >>> from sympy.assumptions.refine import refine_arg >>> from sympy import Q, arg >>> from sympy.abc import x >>> refine_arg(arg(x), Q.positive(x)) 0 >>> refine_arg(arg(x), Q.negative(x)) pi rN)rrrr[rrdr1r\)r$rrgs r refine_argrm+sV 1B 1:b>>;''v  1:b>>;''t 4rcn|d}||krt||}||kr|SdS)NT)complex)expandr)r$rexpandedrefineds rrereBsD{{T{**H4;// h  N 4rc|jd}ttj||r tjSttj|r\ttj||r tjSttj ||r tj Sttj |rt| \}}ttj||r tj Sttj ||r tj S|S)a* Handler for sign. Examples ======== >>> from sympy.assumptions.refine import refine_sign >>> from sympy import Symbol, Q, sign, im >>> x = Symbol('x', real = True) >>> expr = sign(x) >>> refine_sign(expr, Q.positive(x) & Q.nonzero(x)) 1 >>> refine_sign(expr, Q.negative(x) & Q.nonzero(x)) -1 >>> refine_sign(expr, Q.zero(x)) 0 >>> y = Symbol('y', imaginary = True) >>> expr = sign(y) >>> refine_sign(expr, Q.positive(im(y))) I >>> refine_sign(expr, Q.negative(im(y))) -I r)rrrr]rrdr0r[rHr1rBrc as_real_imagri)r$rrarg_rearg_ims r refine_signrwMs 0 )A,C 16#;; $$v  16#;;! qz# , , 5L qz# , , !=  1;s  $))++ qz&!!; / / #? " qz&!!; / / $O# # Krcddlm}|j\}}}tt j||r&||z r|S||||SdS)aU Handler for symmetric part. Examples ======== >>> from sympy.assumptions.refine import refine_matrixelement >>> from sympy import MatrixSymbol, Q >>> X = MatrixSymbol('X', 3, 3) >>> refine_matrixelement(X[0, 1], Q.symmetric(X)) X[0, 1] >>> refine_matrixelement(X[1, 0], Q.symmetric(X)) X[0, 1] r) MatrixElementN)"sympy.matrices.expressions.matexprryrrr symmetricrI)r$rrymatrixr6js rrefine_matrixelementr~vs{A@@@@@9LFAq 1;v   ,,+ E + + - - K}VQ***++r)r+r atan2reimrr9ryz*dict[str, Callable[[Expr, Boolean], Expr]]r"N)T) __future__rtypingr sympy.corerrrrr r r sympy.core.logicr sympy.logic.boolalgr sympy.assumptionsrrrr7rWrargrjrmrerwr~r"__annotations__rrrrs""""""">>>>>>>>>>>>>>>>>>&&&&&&''''''$$$$$$$$9)9)9)9)x#1#1#1La a a H***Z+++.+++,.&&&R+++.       ) = =      r