320 KiB
320 KiB
def extrema(fr):
t=[]
for x in fr.free_symbols:
t+=[fr.diff(x)]
print(t)
return solve(t)from shiroindev import *
from sympy import *
from itertools import permutations, combinations
shiro.seed=1
from IPython.display import Latex
shiro.display=lambda x:display(Latex(x))Latex(r'__________________________')__________________________
prove(makesubs(Sm('-a(s-b)-b(s-c)-c(s-a)+ s^2'),'[0,s],[0,s],[0,s]'))Substitute $a\to s-\frac{s}{d+1}$
Substitute $b\to s-\frac{s}{e+1}$
Substitute $c\to s-\frac{s}{f+1}$
numerator: $defs^2+s^2$
denominator: $def+de+df+d+ef+e+f+1$
status: 0
$$ 0 \le defs^2+s^2 $$
The sum of all inequalities gives us a proof of the inequality.
0
def nonneg(formula):
formula=expand(formula)
for addend in formula.as_ordered_terms():
coef,facts=addend.as_coeff_mul()
if coef<0:
return False
return Truedef symprove(formula,n):
formula=S(formula)
if n==0:
return
ls=list(formula.free_symbols)
for i in range(len(ls)):
a=ls[i]
for j in range(i+1,len(ls)):
b=ls[j]
if expand(formula-formula.subs({a:b, b:a}, simultaneous=True))==S(0):
formula=makesubs(formula,[[b,S('oo')]],variables=[a,b])
sVars.display('$$'+latex(formula)+'$$')
symprove(formula,n-1)symprove('x^2-2*x*y+y^2',4)Substitute $x\to g+y$
$$g^{2}$$
def provesym(formula,n):
formula=S(formula)
if n==0:
return
fs=list(formula.free_symbols)
print
for i in range(2,len(fs)+1):
for fs2 in combinations(fs,i):
for fsp in permutations(fs2[1:]):
if expand(formula-formula.subs(zip((fs2[0],)+fsp,fsp+(fs2[0],)), simultaneous=True))==S(0):
newformula=makesubs(formula,[[fs2[0],oo]]*(len(fsp)),variables=fsp)
sVars.display(str(n)+' $$'+latex(newformula)+'$$')
provesym(newformula,n-1)provesym(Sm('x^t(x-y)(x-z) + y^t (y-z)(y-x) + z^t (z-x)(z-y)'),4)Substitute $y\to h+x$
4 $$h^{2} \left(h + x\right)^{t} - h x x^{t} + h x z^{t} + h x \left(h + x\right)^{t} + h x^{t} z - h z z^{t} - h z \left(h + x\right)^{t} + x^{2} z^{t} - 2 x z z^{t} + z^{2} z^{t}$$
Substitute $z\to i+x$
4 $$i^{2} \left(i + x\right)^{t} - i x x^{t} + i x y^{t} + i x \left(i + x\right)^{t} + i x^{t} y - i y y^{t} - i y \left(i + x\right)^{t} + x^{2} y^{t} - 2 x y y^{t} + y^{2} y^{t}$$
Substitute $z\to j+y$
4 $$j^{2} \left(j + y\right)^{t} - j x x^{t} + j x y^{t} - j x \left(j + y\right)^{t} + j x^{t} y - j y y^{t} + j y \left(j + y\right)^{t} + x^{2} x^{t} - 2 x x^{t} y + x^{t} y^{2}$$
Substitute $y\to k+x$
Substitute $z\to l+x$
4 $$k^{2} \left(k + x\right)^{t} + k l x^{t} - k l \left(k + x\right)^{t} - k l \left(l + x\right)^{t} + l^{2} \left(l + x\right)^{t}$$
Substitute $l\to k+m$
3 $$k^{2} x^{t} + k m x^{t} - k m \left(k + x\right)^{t} + k m \left(k + m + x\right)^{t} + m^{2} \left(k + m + x\right)^{t}$$
Substitute $z\to n+x$
Substitute $y\to o+x$
4 $$n^{2} \left(n + x\right)^{t} + n o x^{t} - n o \left(n + x\right)^{t} - n o \left(o + x\right)^{t} + o^{2} \left(o + x\right)^{t}$$
Substitute $o\to n+p$
3 $$n^{2} x^{t} + n p x^{t} - n p \left(n + x\right)^{t} + n p \left(n + p + x\right)^{t} + p^{2} \left(n + p + x\right)^{t}$$
makesubs(Sm('x^t(x-y)(x-z) + y^t (y-z)(y-x) + z^t (z-x)(z-y)'),'[y,oo]',variables='x')Substitute $x\to q+y$
$\displaystyle q^{2} \left(q + y\right)^{t} - q y y^{t} + q y z^{t} + q y \left(q + y\right)^{t} + q y^{t} z - q z z^{t} - q z \left(q + y\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$
makesubs(Sm('x^t*(x-y)(x-z) + y^t*(y-z)(y-x) + z^t*(z-x)(z-y)'),'[y,oo]',variables='x')Substitute $x\to r+y$
$\displaystyle r^{2} \left(r + y\right)^{t} - r y y^{t} + r y z^{t} + r y \left(r + y\right)^{t} + r y^{t} z - r z z^{t} - r z \left(r + y\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$
provesym(Sm('x^t(x-y)(x-z) + y^t (y-z)(y-x) + z^t (z-x)(z-y)'),4)Substitute $y\to u+x$
4 $$u^{2} \left(u + x\right)^{t} - u x x^{t} + u x z^{t} + u x \left(u + x\right)^{t} + u x^{t} z - u z z^{t} - u z \left(u + x\right)^{t} + x^{2} z^{t} - 2 x z z^{t} + z^{2} z^{t}$$
Substitute $z\to v+x$
4 $$v^{2} \left(v + x\right)^{t} - v x x^{t} + v x y^{t} + v x \left(v + x\right)^{t} + v x^{t} y - v y y^{t} - v y \left(v + x\right)^{t} + x^{2} y^{t} - 2 x y y^{t} + y^{2} y^{t}$$
Substitute $z\to w+y$
4 $$w^{2} \left(w + y\right)^{t} - w x x^{t} + w x y^{t} - w x \left(w + y\right)^{t} + w x^{t} y - w y y^{t} + w y \left(w + y\right)^{t} + x^{2} x^{t} - 2 x x^{t} y + x^{t} y^{2}$$
Substitute $y\to a_{1}+x$
Substitute $z\to b_{1}+x$
4 $$a_{1}^{2} \left(a_{1} + x\right)^{t} + a_{1} b_{1} x^{t} - a_{1} b_{1} \left(a_{1} + x\right)^{t} - a_{1} b_{1} \left(b_{1} + x\right)^{t} + b_{1}^{2} \left(b_{1} + x\right)^{t}$$
Substitute $b_{1}\to a_{1}+c_{1}$
3 $$a_{1}^{2} x^{t} + a_{1} c_{1} x^{t} - a_{1} c_{1} \left(a_{1} + x\right)^{t} + a_{1} c_{1} \left(a_{1} + c_{1} + x\right)^{t} + c_{1}^{2} \left(a_{1} + c_{1} + x\right)^{t}$$
Substitute $z\to d_{1}+x$
Substitute $y\to e_{1}+x$
4 $$d_{1}^{2} \left(d_{1} + x\right)^{t} + d_{1} e_{1} x^{t} - d_{1} e_{1} \left(d_{1} + x\right)^{t} - d_{1} e_{1} \left(e_{1} + x\right)^{t} + e_{1}^{2} \left(e_{1} + x\right)^{t}$$
Substitute $e_{1}\to d_{1}+f_{1}$
3 $$d_{1}^{2} x^{t} + d_{1} f_{1} x^{t} - d_{1} f_{1} \left(d_{1} + x\right)^{t} + d_{1} f_{1} \left(d_{1} + f_{1} + x\right)^{t} + f_{1}^{2} \left(d_{1} + f_{1} + x\right)^{t}$$
Sm('x^t(x-y)(x-z) + y^t (y-z)(y-x) + z^t (z-x)(z-y)')$\displaystyle x^{t} \left(x - y\right) \left(x - z\right) + y^{t} \left(- x + y\right) \left(y - z\right) + z^{t} \left(- x + z\right) \left(- y + z\right)$
S('x+2*y+3*z').subs(zip(S('[x,y,z]'),S('[y,z,x]')),simultaneous=True)$\displaystyle 3 x + y + 2 z$
S('[x,y,z]')[x, y, z]
formula=Sm('x^2(x+y)^t-xyy^t+xyz^t+xy(x+y)^t+xy^tz-xzzt-xz(x+y)^t+y^2z^t-2yzz^t+z^2z^t')
expand(formula.subs(S('[y,z],[z,y]'),simultaneous=True)-formula)$\displaystyle - t x y^{2} + t x z^{2} - x^{2} \left(x + y\right)^{t} + x^{2} \left(x + z\right)^{t} + x y y^{t} - x y \left(x + y\right)^{t} - x y \left(x + z\right)^{t} - x z z^{t} + x z \left(x + y\right)^{t} + x z \left(x + z\right)^{t} + y^{2} y^{t} - y^{2} z^{t} - 2 y y^{t} z + 2 y z z^{t} + y^{t} z^{2} - z^{2} z^{t}$
def point(formula):
fs=list(formula.free_symbols)
il=1
for s in fs[1:]:
il*=s
fr=formula.subs(fs[0],1/il)
print(fr)
return(extrema(fr))
#point(Sm('(a^2+b^2+c^2+d^2)-a*(b+c+d)'))prove(Sm('(a^2+b^2+c^2+d^2)-a*(b+c+d)'),'11/8,4/5,4/5,4/5')Substitute $a\to 11g_{1}/8$
Substitute $b\to 4h_{1}/5$
Substitute $c\to 4i_{1}/5$
Substitute $d\to 4j_{1}/5$
numerator: $3025g_{1}^2-1760g_{1}h_{1}-1760g_{1}i_{1}-1760g_{1}j_{1}+1024h_{1}^2+1024i_{1}^2+1024j_{1}^2$
denominator: $1600$
status: 0
From weighted AM-GM inequality:
$$1760g_{1}h_{1} \le 880g_{1}^2+880h_{1}^2$$
$$1760g_{1}i_{1} \le 880g_{1}^2+880i_{1}^2$$
$$1760g_{1}j_{1} \le 880g_{1}^2+880j_{1}^2$$
$$ 0 \le 385g_{1}^2+144h_{1}^2+144i_{1}^2+144j_{1}^2 $$
The sum of all inequalities gives us a proof of the inequality.
0
216/125-5/30.06133333333333324
"""formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))
display(formula)
from scipy.optimize import fmin
import numpy as np
def f(x):
num,den=fraction(cancel(newformula))
fs=sorted(newformula.free_symbols,key=str)
return float(num.subs(list(zip(fs,x))))
newformula=(makesubs(formula,'[b,oo],[c,oo]'))
print(f([2,1,1]))
fmin(f,[2,1,1])
print(fmin(f,np.array([2,1,1])))
display(simplify(newformula))"""
#prove(newformula)
#prove(makesubs(formula,'[b,oo],[a,oo]',variables='c,b,a'))"formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\ndisplay(formula)\nfrom scipy.optimize import fmin\nimport numpy as np\ndef f(x):\n num,den=fraction(cancel(newformula))\n fs=sorted(newformula.free_symbols,key=str)\n return float(num.subs(list(zip(fs,x))))\nnewformula=(makesubs(formula,'[b,oo],[c,oo]'))\nprint(f([2,1,1]))\nfmin(f,[2,1,1])\nprint(fmin(f,np.array([2,1,1])))\ndisplay(simplify(newformula))"def g(x):
return x[0]+x[1]
fmin(g,[0,0])Warning: Maximum number of function evaluations has been exceeded.
array([-1.62831265e+41, -1.93382892e+41])
formula=Sm('(a^2+b^2+c^2+d^2)/(a*(b+c+d))')
display(formula)
def f(x):
fs=sorted(formula.free_symbols,key=str)
return formula.subs(zip(fs,x))
nsimplify(tuple(fmin(f,(1,1,1,1))),tolerance=0.1,rational=True)$\displaystyle \frac{a^{2} + b^{2} + c^{2} + d^{2}}{a \left(b + c + d\right)}$
Optimization terminated successfully.
Current function value: 1.154701
Iterations: 80
Function evaluations: 144
$\displaystyle \left( \frac{11}{8}, \ \frac{4}{5}, \ \frac{4}{5}, \ \frac{4}{5}\right)$
_[0]/_[1]$\displaystyle \frac{55}{32}$
formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))
formula=(makesubs(formula,'[b,oo],[c,oo]'))
num,den=fraction(cancel(formula))
num=num.subs(zip(fs,list(map(lambda x:x**2,fs))))
display(num)
numm=0
nump=0
for addend in num.as_ordered_terms():
coef,facts=addend.as_coeff_mul()
if coef<0:
numm-=addend
else:
nump+=addend
num=nump/numm
fs=sorted(num.free_symbols,key=str)
numm,nump=Poly(numm),Poly(nump)
def f(x):
return nump.eval(dict(zip(fs,x)))/numm.eval(dict(zip(fs,x)))
print(dict(zip(fs,(2,2,2))))
display(f((2,2,2)))
print('x')
tup=tuple(fmin(f,(2,2,2)))
display(tuple([x*x for x in tup]))
nsimplify(tuple([x*x for x in tup]),tolerance=0.1,rational=True)Substitute $a\to b+k_{1}$
Substitute $b\to c+l_{1}$
[0;31m---------------------------------------------------------------------------[0m [0;31mNameError[0m Traceback (most recent call last) [0;32m<ipython-input-24-10599ede4c20>[0m in [0;36m<module>[0;34m()[0m [1;32m 2[0m [0mformula[0m[0;34m=[0m[0;34m([0m[0mmakesubs[0m[0;34m([0m[0mformula[0m[0;34m,[0m[0;34m'[b,oo],[c,oo]'[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 3[0m [0mnum[0m[0;34m,[0m[0mden[0m[0;34m=[0m[0mfraction[0m[0;34m([0m[0mcancel[0m[0;34m([0m[0mformula[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m----> 4[0;31m [0mnum[0m[0;34m=[0m[0mnum[0m[0;34m.[0m[0msubs[0m[0;34m([0m[0mzip[0m[0;34m([0m[0mfs[0m[0;34m,[0m[0mlist[0m[0;34m([0m[0mmap[0m[0;34m([0m[0;32mlambda[0m [0mx[0m[0;34m:[0m[0mx[0m[0;34m**[0m[0;36m2[0m[0;34m,[0m[0mfs[0m[0;34m)[0m[0;34m)[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 5[0m [0mdisplay[0m[0;34m([0m[0mnum[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 6[0m [0mnumm[0m[0;34m=[0m[0;36m0[0m[0;34m[0m[0;34m[0m[0m [0;31mNameError[0m: name 'fs' is not defined
nsimplify((2.1002573656763053, 10.340431462974655, 1.7661001788212371),tolerance=0.3,rational=True)$\displaystyle \left( 2, \ \frac{31}{3}, \ \frac{7}{4}\right)$
(2.1002573656763053/1.7661001788212371)**21.414211498165453
formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))
display(Latex('Case $a\ge c\ge b$'))
formula1=makesubs(formula,'[c,oo],[b,oo]',variables='a,c,b')
prove(formula1)
display(Latex('Case $a\ge b\ge c$'))
formula2=makesubs(formula,'[b,oo],[c,oo]')
prove(formula2*4,values='1,sqrt(2),1')
formulaCase $a\ge c\ge b$
Substitute $a\to c+r_{1}$
Substitute $c\to b+s_{1}$
numerator: $2b^4r_{1}^2+2b^4r_{1}s_{1}+2b^4s_{1}^2+4b^3r_{1}^3+2\sqrt{2}b^3r_{1}^2s_{1}+10b^3r_{1}^2s_{1}+2\sqrt{2}b^3r_{1}s_{1}^2+6b^3r_{1}s_{1}^2+4b^3s_{1}^3+2b^2r_{1}^4+2\sqrt{2}b^2r_{1}^3s_{1}+10b^2r_{1}^3s_{1}+6\sqrt{2}b^2r_{1}^2s_{1}^2+12b^2r_{1}^2s_{1}^2+4b^2r_{1}s_{1}^3+4\sqrt{2}b^2r_{1}s_{1}^3+2b^2s_{1}^4+2br_{1}^4s_{1}+2\sqrt{2}br_{1}^3s_{1}^2+6br_{1}^3s_{1}^2+4br_{1}^2s_{1}^3+4\sqrt{2}br_{1}^2s_{1}^3+2\sqrt{2}br_{1}s_{1}^4+r_{1}^4s_{1}^2+2r_{1}^3s_{1}^3+r_{1}^2s_{1}^4$
denominator: $b^6+2b^5r_{1}+4b^5s_{1}+b^4r_{1}^2+6b^4r_{1}s_{1}+6b^4s_{1}^2+2b^3r_{1}^2s_{1}+6b^3r_{1}s_{1}^2+4b^3s_{1}^3+b^2r_{1}^2s_{1}^2+2b^2r_{1}s_{1}^3+b^2s_{1}^4$
status: 0
$$ 0 \le r_{1}^4s_{1}^2+2br_{1}^4s_{1}+2b^2r_{1}^4+2r_{1}^3s_{1}^3+2\sqrt{2}br_{1}^3s_{1}^2+6br_{1}^3s_{1}^2+2\sqrt{2}b^2r_{1}^3s_{1}+10b^2r_{1}^3s_{1}+4b^3r_{1}^3+r_{1}^2s_{1}^4+4\sqrt{2}br_{1}^2s_{1}^3+4br_{1}^2s_{1}^3+6\sqrt{2}b^2r_{1}^2s_{1}^2+12b^2r_{1}^2s_{1}^2+2\sqrt{2}b^3r_{1}^2s_{1}+10b^3r_{1}^2s_{1}+2b^4r_{1}^2+2\sqrt{2}br_{1}s_{1}^4+4\sqrt{2}b^2r_{1}s_{1}^3+4b^2r_{1}s_{1}^3+2\sqrt{2}b^3r_{1}s_{1}^2+6b^3r_{1}s_{1}^2+2b^4r_{1}s_{1}+2b^2s_{1}^4+4b^3s_{1}^3+2b^4s_{1}^2 $$
The sum of all inequalities gives us a proof of the inequality.
Case $a\ge b\ge c$
Substitute $a\to b+t_{1}$
Substitute $b\to c+u_{1}$
Substitute $t_{1}\to \sqrt{2}v_{1}$
numerator: $8c^4u_{1}^2+8\sqrt{2}c^4u_{1}v_{1}+16c^4v_{1}^2+16c^3u_{1}^3-16c^3u_{1}^2v_{1}+24\sqrt{2}c^3u_{1}^2v_{1}-16\sqrt{2}c^3u_{1}v_{1}^2+80c^3u_{1}v_{1}^2+32\sqrt{2}c^3v_{1}^3+8c^2u_{1}^4-32c^2u_{1}^3v_{1}+16\sqrt{2}c^2u_{1}^3v_{1}-48\sqrt{2}c^2u_{1}^2v_{1}^2+96c^2u_{1}^2v_{1}^2-32c^2u_{1}v_{1}^3+80\sqrt{2}c^2u_{1}v_{1}^3+32c^2v_{1}^4-16cu_{1}^4v_{1}-32\sqrt{2}cu_{1}^3v_{1}^2+32cu_{1}^3v_{1}^2-32cu_{1}^2v_{1}^3+48\sqrt{2}cu_{1}^2v_{1}^3+32cu_{1}v_{1}^4+8u_{1}^4v_{1}^2+16\sqrt{2}u_{1}^3v_{1}^3+16u_{1}^2v_{1}^4$
denominator: $c^6+4c^5u_{1}+2\sqrt{2}c^5v_{1}+6c^4u_{1}^2+6\sqrt{2}c^4u_{1}v_{1}+2c^4v_{1}^2+4c^3u_{1}^3+6\sqrt{2}c^3u_{1}^2v_{1}+4c^3u_{1}v_{1}^2+c^2u_{1}^4+2\sqrt{2}c^2u_{1}^3v_{1}+2c^2u_{1}^2v_{1}^2$
status: 0
From weighted AM-GM inequality:
$$16cu_{1}^4v_{1} \le 8u_{1}^4v_{1}^2+8c^2u_{1}^4$$
$$32\sqrt{2}cu_{1}^3v_{1}^2 \le 16\sqrt{2}u_{1}^3v_{1}^3+16\sqrt{2}c^2u_{1}^3v_{1}$$
$$32c^2u_{1}^3v_{1} \le 16cu_{1}^3v_{1}^2+16c^3u_{1}^3$$
$$32cu_{1}^2v_{1}^3 \le 16cu_{1}^3v_{1}^2+16cu_{1}v_{1}^4$$
$$48\sqrt{2}c^2u_{1}^2v_{1}^2 \le 24\sqrt{2}cu_{1}^2v_{1}^3+24\sqrt{2}c^3u_{1}^2v_{1}$$
$$16c^3u_{1}^2v_{1} \le 8c^2u_{1}^2v_{1}^2+8c^4u_{1}^2$$
$$32c^2u_{1}v_{1}^3 \le 16u_{1}^2v_{1}^4+16c^4v_{1}^2$$
$$16\sqrt{2}c^3u_{1}v_{1}^2 \le 8\sqrt{2}c^2u_{1}v_{1}^3+8\sqrt{2}c^4u_{1}v_{1}$$
$$ 0 \le 24\sqrt{2}cu_{1}^2v_{1}^3+88c^2u_{1}^2v_{1}^2+16cu_{1}v_{1}^4+72\sqrt{2}c^2u_{1}v_{1}^3+80c^3u_{1}v_{1}^2+32c^2v_{1}^4+32\sqrt{2}c^3v_{1}^3 $$
The sum of all inequalities gives us a proof of the inequality.
$\displaystyle - \frac{2 \sqrt{2} \left(a - b\right)}{c} + \frac{\left(a - b\right)^{2}}{c^{2}} - \frac{2 \sqrt{2} \left(- a + c\right)}{b} + \frac{\left(- a + c\right)^{2}}{b^{2}} - \frac{2 \sqrt{2} \left(b - c\right)}{a} + \frac{\left(b - c\right)^{2}}{a^{2}}$
formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))
formula=(makesubs(formula,'[b,oo],[c,oo]'))
formula,_=fraction(cancel(formula))
formula=formula.subs('a','2**(1/2)*a')
sVars.display(r'Substitute $a\to\sqrt{2}a$')
prove(formula)Substitute $a\to b+w_{1}$
Substitute $b\to c+x_{1}$
Substitute $a\to\sqrt{2}a$
numerator: $2c^4w_{1}^2+2c^4w_{1}x_{1}+2c^4x_{1}^2+4c^3w_{1}^3-2\sqrt{2}c^3w_{1}^2x_{1}+10c^3w_{1}^2x_{1}-2\sqrt{2}c^3w_{1}x_{1}^2+6c^3w_{1}x_{1}^2+4c^3x_{1}^3+2c^2w_{1}^4-2\sqrt{2}c^2w_{1}^3x_{1}+10c^2w_{1}^3x_{1}-6\sqrt{2}c^2w_{1}^2x_{1}^2+12c^2w_{1}^2x_{1}^2-4\sqrt{2}c^2w_{1}x_{1}^3+4c^2w_{1}x_{1}^3+2c^2x_{1}^4+2cw_{1}^4x_{1}-2\sqrt{2}cw_{1}^3x_{1}^2+6cw_{1}^3x_{1}^2-4\sqrt{2}cw_{1}^2x_{1}^3+4cw_{1}^2x_{1}^3-2\sqrt{2}cw_{1}x_{1}^4+w_{1}^4x_{1}^2+2w_{1}^3x_{1}^3+w_{1}^2x_{1}^4$
denominator: $1$
status: 2
Program couldn't find any proof.
$$ 2\sqrt{2}cw_{1}x_{1}^4+4\sqrt{2}cw_{1}^2x_{1}^3+4\sqrt{2}c^2w_{1}x_{1}^3+2\sqrt{2}cw_{1}^3x_{1}^2+6\sqrt{2}c^2w_{1}^2x_{1}^2+2\sqrt{2}c^3w_{1}x_{1}^2+2\sqrt{2}c^2w_{1}^3x_{1}+2\sqrt{2}c^3w_{1}^2x_{1} \le w_{1}^2x_{1}^4+2c^2x_{1}^4+2w_{1}^3x_{1}^3+4cw_{1}^2x_{1}^3+4c^2w_{1}x_{1}^3+4c^3x_{1}^3+w_{1}^4x_{1}^2+6cw_{1}^3x_{1}^2+12c^2w_{1}^2x_{1}^2+6c^3w_{1}x_{1}^2+2c^4x_{1}^2+2cw_{1}^4x_{1}+10c^2w_{1}^3x_{1}+10c^3w_{1}^2x_{1}+2c^4w_{1}x_{1}+2c^2w_{1}^4+4c^3w_{1}^3+2c^4w_{1}^2 $$
2
S('2**(1/2)*x*y').as_coeff_mul()(1, (x, y, sqrt(2)))
S('sqrt(2)*x*y').as_coeff_mul()(1, (x, y, sqrt(2)))
Poly('sqrt(2)*x*y+8').monoms()[(1, 1, 1), (0, 0, 0)]
Poly('sqrt(2)*x*y+58').coeffs()[1, 58]
Poly('sqrt(2)*x*y+sqrt(3)+sqrt(2)*sqrt(3)').gens(x, y, sqrt(2), sqrt(3), sqrt(6))
def _formula2list(formula):
neg=pos=0
for addend in formula.as_ordered_terms():
coef,facts=addend.as_coeff_mul()
if coef<0:
neg-=addend
else:
pos+=addend
neg=Poly(neg,gens=Poly(formula).gens)
pos=Poly(pos,gens=Poly(formula).gens)
return neg.coeffs(),neg.monoms(),pos.coeffs(),pos.monoms(),Poly(formula).gens
_formula2list(S('x^2+7*y'))([0], [(0, 0)], [1, 7], [(2, 0), (0, 1)], (x, y))
prove(S('sqrt(2)*x^2-sqrt(8)*x*y+sqrt(2)*y^2'))numerator: $\sqrt{2}x^2-2\sqrt{2}xy+\sqrt{2}y^2$
denominator: $1$
status: 0
From weighted AM-GM inequality:
$$2\sqrt{2}xy \le \sqrt{2}x^2+\sqrt{2}y^2$$
$$ 0 \le 0 $$
The sum of all inequalities gives us a proof of the inequality.
0
[Poly('1+2**(1/3)+4**(1/3)+x').monoms(),Poly('1+2**(1/3)+4**(1/3)+x').coeffs()][[(1, 0), (0, 2), (0, 1), (0, 0)], [1, 1, 1, 1]]
x=Symbol('x', positive=True)
Poly(x+sqrt(x))$\displaystyle \operatorname{Poly}{\left( x + \sqrt{x}, x, \sqrt{x}, domain=\mathbb{Z} \right)}$
prove?Poly('x^2-1').abs()$\displaystyle \operatorname{Poly}{\left( x^{2} + 1, x, domain=\mathbb{Z} \right)}$
Poly(Poly('x')/Poly('x+y'))$\displaystyle \operatorname{Poly}{\left( x\frac{1}{x + y}, x, \frac{1}{x + y}, domain=\mathbb{Z} \right)}$
Poly('sqrt(2)+x+sqrt(x)+x**(1/4)+x**(1/3)+x**(2/3)',extension=1)$\displaystyle \operatorname{Poly}{\left( x + \sqrt{x} + x^{\frac{2}{3}} + \sqrt[3]{x} + \sqrt[4]{x} + \sqrt{2}, x, \sqrt{x}, \sqrt[3]{x}, \sqrt[4]{x}, domain=QQ \right)}$
Poly('x+sqrt(6)',S('x'),S('sqrt(2)'),S('sqrt(3)')).coeffs()[1, sqrt(6)]
Poly('x+sqrt(x)',S('sqrt(x)')).coeffs()[1, x]
source(Poly)/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/decorators.py:38: SymPyDeprecationWarning: source has been deprecated since SymPy 1.3. Use ?? in IPython/Jupyter or inspect.getsource instead. See https://github.com/sympy/sympy/issues/14905 for more info. _warn_deprecation(wrapped, 3)
In file: /home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polytools.py
class Poly(Expr):
"""
Generic class for representing and operating on polynomial expressions.
Subclasses Expr class.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
Create a univariate polynomial:
>>> Poly(x*(x**2 + x - 1)**2)
Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ')
Create a univariate polynomial with specific domain:
>>> from sympy import sqrt
>>> Poly(x**2 + 2*x + sqrt(3), domain='R')
Poly(1.0*x**2 + 2.0*x + 1.73205080756888, x, domain='RR')
Create a multivariate polynomial:
>>> Poly(y*x**2 + x*y + 1)
Poly(x**2*y + x*y + 1, x, y, domain='ZZ')
Create a univariate polynomial, where y is a constant:
>>> Poly(y*x**2 + x*y + 1,x)
Poly(y*x**2 + y*x + 1, x, domain='ZZ[y]')
You can evaluate the above polynomial as a function of y:
>>> Poly(y*x**2 + x*y + 1,x).eval(2)
6*y + 1
See Also
========
sympy.core.expr.Expr
"""
__slots__ = ['rep', 'gens']
is_commutative = True
is_Poly = True
_op_priority = 10.001
def __new__(cls, rep, *gens, **args):
"""Create a new polynomial instance out of something useful. """
opt = options.build_options(gens, args)
if 'order' in opt:
raise NotImplementedError("'order' keyword is not implemented yet")
if iterable(rep, exclude=str):
if isinstance(rep, dict):
return cls._from_dict(rep, opt)
else:
return cls._from_list(list(rep), opt)
else:
rep = sympify(rep)
if rep.is_Poly:
return cls._from_poly(rep, opt)
else:
return cls._from_expr(rep, opt)
@classmethod
def new(cls, rep, *gens):
"""Construct :class:`Poly` instance from raw representation. """
if not isinstance(rep, DMP):
raise PolynomialError(
"invalid polynomial representation: %s" % rep)
elif rep.lev != len(gens) - 1:
raise PolynomialError("invalid arguments: %s, %s" % (rep, gens))
obj = Basic.__new__(cls)
obj.rep = rep
obj.gens = gens
return obj
@classmethod
def from_dict(cls, rep, *gens, **args):
"""Construct a polynomial from a ``dict``. """
opt = options.build_options(gens, args)
return cls._from_dict(rep, opt)
@classmethod
def from_list(cls, rep, *gens, **args):
"""Construct a polynomial from a ``list``. """
opt = options.build_options(gens, args)
return cls._from_list(rep, opt)
@classmethod
def from_poly(cls, rep, *gens, **args):
"""Construct a polynomial from a polynomial. """
opt = options.build_options(gens, args)
return cls._from_poly(rep, opt)
@classmethod
def from_expr(cls, rep, *gens, **args):
"""Construct a polynomial from an expression. """
opt = options.build_options(gens, args)
return cls._from_expr(rep, opt)
@classmethod
def _from_dict(cls, rep, opt):
"""Construct a polynomial from a ``dict``. """
gens = opt.gens
if not gens:
raise GeneratorsNeeded(
"can't initialize from 'dict' without generators")
level = len(gens) - 1
domain = opt.domain
if domain is None:
domain, rep = construct_domain(rep, opt=opt)
else:
for monom, coeff in rep.items():
rep[monom] = domain.convert(coeff)
return cls.new(DMP.from_dict(rep, level, domain), *gens)
@classmethod
def _from_list(cls, rep, opt):
"""Construct a polynomial from a ``list``. """
gens = opt.gens
if not gens:
raise GeneratorsNeeded(
"can't initialize from 'list' without generators")
elif len(gens) != 1:
raise MultivariatePolynomialError(
"'list' representation not supported")
level = len(gens) - 1
domain = opt.domain
if domain is None:
domain, rep = construct_domain(rep, opt=opt)
else:
rep = list(map(domain.convert, rep))
return cls.new(DMP.from_list(rep, level, domain), *gens)
@classmethod
def _from_poly(cls, rep, opt):
"""Construct a polynomial from a polynomial. """
if cls != rep.__class__:
rep = cls.new(rep.rep, *rep.gens)
gens = opt.gens
field = opt.field
domain = opt.domain
if gens and rep.gens != gens:
if set(rep.gens) != set(gens):
return cls._from_expr(rep.as_expr(), opt)
else:
rep = rep.reorder(*gens)
if 'domain' in opt and domain:
rep = rep.set_domain(domain)
elif field is True:
rep = rep.to_field()
return rep
@classmethod
def _from_expr(cls, rep, opt):
"""Construct a polynomial from an expression. """
rep, opt = _dict_from_expr(rep, opt)
return cls._from_dict(rep, opt)
def _hashable_content(self):
"""Allow SymPy to hash Poly instances. """
return (self.rep, self.gens)
def __hash__(self):
return super(Poly, self).__hash__()
@property
def free_symbols(self):
"""
Free symbols of a polynomial expression.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> Poly(x**2 + 1).free_symbols
{x}
>>> Poly(x**2 + y).free_symbols
{x, y}
>>> Poly(x**2 + y, x).free_symbols
{x, y}
>>> Poly(x**2 + y, x, z).free_symbols
{x, y}
"""
symbols = set()
gens = self.gens
for i in range(len(gens)):
for monom in self.monoms():
if monom[i]:
symbols |= gens[i].free_symbols
break
return symbols | self.free_symbols_in_domain
@property
def free_symbols_in_domain(self):
"""
Free symbols of the domain of ``self``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 1).free_symbols_in_domain
set()
>>> Poly(x**2 + y).free_symbols_in_domain
set()
>>> Poly(x**2 + y, x).free_symbols_in_domain
{y}
"""
domain, symbols = self.rep.dom, set()
if domain.is_Composite:
for gen in domain.symbols:
symbols |= gen.free_symbols
elif domain.is_EX:
for coeff in self.coeffs():
symbols |= coeff.free_symbols
return symbols
@property
def args(self):
"""
Don't mess up with the core.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).args
(x**2 + 1,)
"""
return (self.as_expr(),)
@property
def gen(self):
"""
Return the principal generator.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).gen
x
"""
return self.gens[0]
@property
def domain(self):
"""Get the ground domain of ``self``. """
return self.get_domain()
@property
def zero(self):
"""Return zero polynomial with ``self``'s properties. """
return self.new(self.rep.zero(self.rep.lev, self.rep.dom), *self.gens)
@property
def one(self):
"""Return one polynomial with ``self``'s properties. """
return self.new(self.rep.one(self.rep.lev, self.rep.dom), *self.gens)
@property
def unit(self):
"""Return unit polynomial with ``self``'s properties. """
return self.new(self.rep.unit(self.rep.lev, self.rep.dom), *self.gens)
def unify(f, g):
"""
Make ``f`` and ``g`` belong to the same domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f, g = Poly(x/2 + 1), Poly(2*x + 1)
>>> f
Poly(1/2*x + 1, x, domain='QQ')
>>> g
Poly(2*x + 1, x, domain='ZZ')
>>> F, G = f.unify(g)
>>> F
Poly(1/2*x + 1, x, domain='QQ')
>>> G
Poly(2*x + 1, x, domain='QQ')
"""
_, per, F, G = f._unify(g)
return per(F), per(G)
def _unify(f, g):
g = sympify(g)
if not g.is_Poly:
try:
return f.rep.dom, f.per, f.rep, f.rep.per(f.rep.dom.from_sympy(g))
except CoercionFailed:
raise UnificationFailed("can't unify %s with %s" % (f, g))
if isinstance(f.rep, DMP) and isinstance(g.rep, DMP):
gens = _unify_gens(f.gens, g.gens)
dom, lev = f.rep.dom.unify(g.rep.dom, gens), len(gens) - 1
if f.gens != gens:
f_monoms, f_coeffs = _dict_reorder(
f.rep.to_dict(), f.gens, gens)
if f.rep.dom != dom:
f_coeffs = [dom.convert(c, f.rep.dom) for c in f_coeffs]
F = DMP(dict(list(zip(f_monoms, f_coeffs))), dom, lev)
else:
F = f.rep.convert(dom)
if g.gens != gens:
g_monoms, g_coeffs = _dict_reorder(
g.rep.to_dict(), g.gens, gens)
if g.rep.dom != dom:
g_coeffs = [dom.convert(c, g.rep.dom) for c in g_coeffs]
G = DMP(dict(list(zip(g_monoms, g_coeffs))), dom, lev)
else:
G = g.rep.convert(dom)
else:
raise UnificationFailed("can't unify %s with %s" % (f, g))
cls = f.__class__
def per(rep, dom=dom, gens=gens, remove=None):
if remove is not None:
gens = gens[:remove] + gens[remove + 1:]
if not gens:
return dom.to_sympy(rep)
return cls.new(rep, *gens)
return dom, per, F, G
def per(f, rep, gens=None, remove=None):
"""
Create a Poly out of the given representation.
Examples
========
>>> from sympy import Poly, ZZ
>>> from sympy.abc import x, y
>>> from sympy.polys.polyclasses import DMP
>>> a = Poly(x**2 + 1)
>>> a.per(DMP([ZZ(1), ZZ(1)], ZZ), gens=[y])
Poly(y + 1, y, domain='ZZ')
"""
if gens is None:
gens = f.gens
if remove is not None:
gens = gens[:remove] + gens[remove + 1:]
if not gens:
return f.rep.dom.to_sympy(rep)
return f.__class__.new(rep, *gens)
def set_domain(f, domain):
"""Set the ground domain of ``f``. """
opt = options.build_options(f.gens, {'domain': domain})
return f.per(f.rep.convert(opt.domain))
def get_domain(f):
"""Get the ground domain of ``f``. """
return f.rep.dom
def set_modulus(f, modulus):
"""
Set the modulus of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(5*x**2 + 2*x - 1, x).set_modulus(2)
Poly(x**2 + 1, x, modulus=2)
"""
modulus = options.Modulus.preprocess(modulus)
return f.set_domain(FF(modulus))
def get_modulus(f):
"""
Get the modulus of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, modulus=2).get_modulus()
2
"""
domain = f.get_domain()
if domain.is_FiniteField:
return Integer(domain.characteristic())
else:
raise PolynomialError("not a polynomial over a Galois field")
def _eval_subs(f, old, new):
"""Internal implementation of :func:`subs`. """
if old in f.gens:
if new.is_number:
return f.eval(old, new)
else:
try:
return f.replace(old, new)
except PolynomialError:
pass
return f.as_expr().subs(old, new)
def exclude(f):
"""
Remove unnecessary generators from ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import a, b, c, d, x
>>> Poly(a + x, a, b, c, d, x).exclude()
Poly(a + x, a, x, domain='ZZ')
"""
J, new = f.rep.exclude()
gens = []
for j in range(len(f.gens)):
if j not in J:
gens.append(f.gens[j])
return f.per(new, gens=gens)
def replace(f, x, y=None, *_ignore):
# XXX this does not match Basic's signature
"""
Replace ``x`` with ``y`` in generators list.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 1, x).replace(x, y)
Poly(y**2 + 1, y, domain='ZZ')
"""
if y is None:
if f.is_univariate:
x, y = f.gen, x
else:
raise PolynomialError(
"syntax supported only in univariate case")
if x == y or x not in f.gens:
return f
if x in f.gens and y not in f.gens:
dom = f.get_domain()
if not dom.is_Composite or y not in dom.symbols:
gens = list(f.gens)
gens[gens.index(x)] = y
return f.per(f.rep, gens=gens)
raise PolynomialError("can't replace %s with %s in %s" % (x, y, f))
def reorder(f, *gens, **args):
"""
Efficiently apply new order of generators.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + x*y**2, x, y).reorder(y, x)
Poly(y**2*x + x**2, y, x, domain='ZZ')
"""
opt = options.Options((), args)
if not gens:
gens = _sort_gens(f.gens, opt=opt)
elif set(f.gens) != set(gens):
raise PolynomialError(
"generators list can differ only up to order of elements")
rep = dict(list(zip(*_dict_reorder(f.rep.to_dict(), f.gens, gens))))
return f.per(DMP(rep, f.rep.dom, len(gens) - 1), gens=gens)
def ltrim(f, gen):
"""
Remove dummy generators from ``f`` that are to the left of
specified ``gen`` in the generators as ordered. When ``gen``
is an integer, it refers to the generator located at that
position within the tuple of generators of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> Poly(y**2 + y*z**2, x, y, z).ltrim(y)
Poly(y**2 + y*z**2, y, z, domain='ZZ')
>>> Poly(z, x, y, z).ltrim(-1)
Poly(z, z, domain='ZZ')
"""
rep = f.as_dict(native=True)
j = f._gen_to_level(gen)
terms = {}
for monom, coeff in rep.items():
if any(i for i in monom[:j]):
# some generator is used in the portion to be trimmed
raise PolynomialError("can't left trim %s" % f)
terms[monom[j:]] = coeff
gens = f.gens[j:]
return f.new(DMP.from_dict(terms, len(gens) - 1, f.rep.dom), *gens)
def has_only_gens(f, *gens):
"""
Return ``True`` if ``Poly(f, *gens)`` retains ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> Poly(x*y + 1, x, y, z).has_only_gens(x, y)
True
>>> Poly(x*y + z, x, y, z).has_only_gens(x, y)
False
"""
indices = set()
for gen in gens:
try:
index = f.gens.index(gen)
except ValueError:
raise GeneratorsError(
"%s doesn't have %s as generator" % (f, gen))
else:
indices.add(index)
for monom in f.monoms():
for i, elt in enumerate(monom):
if i not in indices and elt:
return False
return True
def to_ring(f):
"""
Make the ground domain a ring.
Examples
========
>>> from sympy import Poly, QQ
>>> from sympy.abc import x
>>> Poly(x**2 + 1, domain=QQ).to_ring()
Poly(x**2 + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'to_ring'):
result = f.rep.to_ring()
else: # pragma: no cover
raise OperationNotSupported(f, 'to_ring')
return f.per(result)
def to_field(f):
"""
Make the ground domain a field.
Examples
========
>>> from sympy import Poly, ZZ
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x, domain=ZZ).to_field()
Poly(x**2 + 1, x, domain='QQ')
"""
if hasattr(f.rep, 'to_field'):
result = f.rep.to_field()
else: # pragma: no cover
raise OperationNotSupported(f, 'to_field')
return f.per(result)
def to_exact(f):
"""
Make the ground domain exact.
Examples
========
>>> from sympy import Poly, RR
>>> from sympy.abc import x
>>> Poly(x**2 + 1.0, x, domain=RR).to_exact()
Poly(x**2 + 1, x, domain='QQ')
"""
if hasattr(f.rep, 'to_exact'):
result = f.rep.to_exact()
else: # pragma: no cover
raise OperationNotSupported(f, 'to_exact')
return f.per(result)
def retract(f, field=None):
"""
Recalculate the ground domain of a polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = Poly(x**2 + 1, x, domain='QQ[y]')
>>> f
Poly(x**2 + 1, x, domain='QQ[y]')
>>> f.retract()
Poly(x**2 + 1, x, domain='ZZ')
>>> f.retract(field=True)
Poly(x**2 + 1, x, domain='QQ')
"""
dom, rep = construct_domain(f.as_dict(zero=True),
field=field, composite=f.domain.is_Composite or None)
return f.from_dict(rep, f.gens, domain=dom)
def slice(f, x, m, n=None):
"""Take a continuous subsequence of terms of ``f``. """
if n is None:
j, m, n = 0, x, m
else:
j = f._gen_to_level(x)
m, n = int(m), int(n)
if hasattr(f.rep, 'slice'):
result = f.rep.slice(m, n, j)
else: # pragma: no cover
raise OperationNotSupported(f, 'slice')
return f.per(result)
def coeffs(f, order=None):
"""
Returns all non-zero coefficients from ``f`` in lex order.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x + 3, x).coeffs()
[1, 2, 3]
See Also
========
all_coeffs
coeff_monomial
nth
"""
return [f.rep.dom.to_sympy(c) for c in f.rep.coeffs(order=order)]
def monoms(f, order=None):
"""
Returns all non-zero monomials from ``f`` in lex order.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).monoms()
[(2, 0), (1, 2), (1, 1), (0, 1)]
See Also
========
all_monoms
"""
return f.rep.monoms(order=order)
def terms(f, order=None):
"""
Returns all non-zero terms from ``f`` in lex order.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).terms()
[((2, 0), 1), ((1, 2), 2), ((1, 1), 1), ((0, 1), 3)]
See Also
========
all_terms
"""
return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.terms(order=order)]
def all_coeffs(f):
"""
Returns all coefficients from a univariate polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x - 1, x).all_coeffs()
[1, 0, 2, -1]
"""
return [f.rep.dom.to_sympy(c) for c in f.rep.all_coeffs()]
def all_monoms(f):
"""
Returns all monomials from a univariate polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x - 1, x).all_monoms()
[(3,), (2,), (1,), (0,)]
See Also
========
all_terms
"""
return f.rep.all_monoms()
def all_terms(f):
"""
Returns all terms from a univariate polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x - 1, x).all_terms()
[((3,), 1), ((2,), 0), ((1,), 2), ((0,), -1)]
"""
return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.all_terms()]
def termwise(f, func, *gens, **args):
"""
Apply a function to all terms of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> def func(k, coeff):
... k = k[0]
... return coeff//10**(2-k)
>>> Poly(x**2 + 20*x + 400).termwise(func)
Poly(x**2 + 2*x + 4, x, domain='ZZ')
"""
terms = {}
for monom, coeff in f.terms():
result = func(monom, coeff)
if isinstance(result, tuple):
monom, coeff = result
else:
coeff = result
if coeff:
if monom not in terms:
terms[monom] = coeff
else:
raise PolynomialError(
"%s monomial was generated twice" % monom)
return f.from_dict(terms, *(gens or f.gens), **args)
def length(f):
"""
Returns the number of non-zero terms in ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 2*x - 1).length()
3
"""
return len(f.as_dict())
def as_dict(f, native=False, zero=False):
"""
Switch to a ``dict`` representation.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x*y**2 - y, x, y).as_dict()
{(0, 1): -1, (1, 2): 2, (2, 0): 1}
"""
if native:
return f.rep.to_dict(zero=zero)
else:
return f.rep.to_sympy_dict(zero=zero)
def as_list(f, native=False):
"""Switch to a ``list`` representation. """
if native:
return f.rep.to_list()
else:
return f.rep.to_sympy_list()
def as_expr(f, *gens):
"""
Convert a Poly instance to an Expr instance.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**2 + 2*x*y**2 - y, x, y)
>>> f.as_expr()
x**2 + 2*x*y**2 - y
>>> f.as_expr({x: 5})
10*y**2 - y + 25
>>> f.as_expr(5, 6)
379
"""
if not gens:
gens = f.gens
elif len(gens) == 1 and isinstance(gens[0], dict):
mapping = gens[0]
gens = list(f.gens)
for gen, value in mapping.items():
try:
index = gens.index(gen)
except ValueError:
raise GeneratorsError(
"%s doesn't have %s as generator" % (f, gen))
else:
gens[index] = value
return basic_from_dict(f.rep.to_sympy_dict(), *gens)
def lift(f):
"""
Convert algebraic coefficients to rationals.
Examples
========
>>> from sympy import Poly, I
>>> from sympy.abc import x
>>> Poly(x**2 + I*x + 1, x, extension=I).lift()
Poly(x**4 + 3*x**2 + 1, x, domain='QQ')
"""
if hasattr(f.rep, 'lift'):
result = f.rep.lift()
else: # pragma: no cover
raise OperationNotSupported(f, 'lift')
return f.per(result)
def deflate(f):
"""
Reduce degree of ``f`` by mapping ``x_i**m`` to ``y_i``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**6*y**2 + x**3 + 1, x, y).deflate()
((3, 2), Poly(x**2*y + x + 1, x, y, domain='ZZ'))
"""
if hasattr(f.rep, 'deflate'):
J, result = f.rep.deflate()
else: # pragma: no cover
raise OperationNotSupported(f, 'deflate')
return J, f.per(result)
def inject(f, front=False):
"""
Inject ground domain generators into ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**2*y + x*y**3 + x*y + 1, x)
>>> f.inject()
Poly(x**2*y + x*y**3 + x*y + 1, x, y, domain='ZZ')
>>> f.inject(front=True)
Poly(y**3*x + y*x**2 + y*x + 1, y, x, domain='ZZ')
"""
dom = f.rep.dom
if dom.is_Numerical:
return f
elif not dom.is_Poly:
raise DomainError("can't inject generators over %s" % dom)
if hasattr(f.rep, 'inject'):
result = f.rep.inject(front=front)
else: # pragma: no cover
raise OperationNotSupported(f, 'inject')
if front:
gens = dom.symbols + f.gens
else:
gens = f.gens + dom.symbols
return f.new(result, *gens)
def eject(f, *gens):
"""
Eject selected generators into the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**2*y + x*y**3 + x*y + 1, x, y)
>>> f.eject(x)
Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]')
>>> f.eject(y)
Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]')
"""
dom = f.rep.dom
if not dom.is_Numerical:
raise DomainError("can't eject generators over %s" % dom)
k = len(gens)
if f.gens[:k] == gens:
_gens, front = f.gens[k:], True
elif f.gens[-k:] == gens:
_gens, front = f.gens[:-k], False
else:
raise NotImplementedError(
"can only eject front or back generators")
dom = dom.inject(*gens)
if hasattr(f.rep, 'eject'):
result = f.rep.eject(dom, front=front)
else: # pragma: no cover
raise OperationNotSupported(f, 'eject')
return f.new(result, *_gens)
def terms_gcd(f):
"""
Remove GCD of terms from the polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**6*y**2 + x**3*y, x, y).terms_gcd()
((3, 1), Poly(x**3*y + 1, x, y, domain='ZZ'))
"""
if hasattr(f.rep, 'terms_gcd'):
J, result = f.rep.terms_gcd()
else: # pragma: no cover
raise OperationNotSupported(f, 'terms_gcd')
return J, f.per(result)
def add_ground(f, coeff):
"""
Add an element of the ground domain to ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x + 1).add_ground(2)
Poly(x + 3, x, domain='ZZ')
"""
if hasattr(f.rep, 'add_ground'):
result = f.rep.add_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'add_ground')
return f.per(result)
def sub_ground(f, coeff):
"""
Subtract an element of the ground domain from ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x + 1).sub_ground(2)
Poly(x - 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'sub_ground'):
result = f.rep.sub_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'sub_ground')
return f.per(result)
def mul_ground(f, coeff):
"""
Multiply ``f`` by a an element of the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x + 1).mul_ground(2)
Poly(2*x + 2, x, domain='ZZ')
"""
if hasattr(f.rep, 'mul_ground'):
result = f.rep.mul_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'mul_ground')
return f.per(result)
def quo_ground(f, coeff):
"""
Quotient of ``f`` by a an element of the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x + 4).quo_ground(2)
Poly(x + 2, x, domain='ZZ')
>>> Poly(2*x + 3).quo_ground(2)
Poly(x + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'quo_ground'):
result = f.rep.quo_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'quo_ground')
return f.per(result)
def exquo_ground(f, coeff):
"""
Exact quotient of ``f`` by a an element of the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x + 4).exquo_ground(2)
Poly(x + 2, x, domain='ZZ')
>>> Poly(2*x + 3).exquo_ground(2)
Traceback (most recent call last):
...
ExactQuotientFailed: 2 does not divide 3 in ZZ
"""
if hasattr(f.rep, 'exquo_ground'):
result = f.rep.exquo_ground(coeff)
else: # pragma: no cover
raise OperationNotSupported(f, 'exquo_ground')
return f.per(result)
def abs(f):
"""
Make all coefficients in ``f`` positive.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).abs()
Poly(x**2 + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'abs'):
result = f.rep.abs()
else: # pragma: no cover
raise OperationNotSupported(f, 'abs')
return f.per(result)
def neg(f):
"""
Negate all coefficients in ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).neg()
Poly(-x**2 + 1, x, domain='ZZ')
>>> -Poly(x**2 - 1, x)
Poly(-x**2 + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'neg'):
result = f.rep.neg()
else: # pragma: no cover
raise OperationNotSupported(f, 'neg')
return f.per(result)
def add(f, g):
"""
Add two polynomials ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).add(Poly(x - 2, x))
Poly(x**2 + x - 1, x, domain='ZZ')
>>> Poly(x**2 + 1, x) + Poly(x - 2, x)
Poly(x**2 + x - 1, x, domain='ZZ')
"""
g = sympify(g)
if not g.is_Poly:
return f.add_ground(g)
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'add'):
result = F.add(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'add')
return per(result)
def sub(f, g):
"""
Subtract two polynomials ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).sub(Poly(x - 2, x))
Poly(x**2 - x + 3, x, domain='ZZ')
>>> Poly(x**2 + 1, x) - Poly(x - 2, x)
Poly(x**2 - x + 3, x, domain='ZZ')
"""
g = sympify(g)
if not g.is_Poly:
return f.sub_ground(g)
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'sub'):
result = F.sub(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'sub')
return per(result)
def mul(f, g):
"""
Multiply two polynomials ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).mul(Poly(x - 2, x))
Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ')
>>> Poly(x**2 + 1, x)*Poly(x - 2, x)
Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ')
"""
g = sympify(g)
if not g.is_Poly:
return f.mul_ground(g)
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'mul'):
result = F.mul(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'mul')
return per(result)
def sqr(f):
"""
Square a polynomial ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x - 2, x).sqr()
Poly(x**2 - 4*x + 4, x, domain='ZZ')
>>> Poly(x - 2, x)**2
Poly(x**2 - 4*x + 4, x, domain='ZZ')
"""
if hasattr(f.rep, 'sqr'):
result = f.rep.sqr()
else: # pragma: no cover
raise OperationNotSupported(f, 'sqr')
return f.per(result)
def pow(f, n):
"""
Raise ``f`` to a non-negative power ``n``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x - 2, x).pow(3)
Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ')
>>> Poly(x - 2, x)**3
Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ')
"""
n = int(n)
if hasattr(f.rep, 'pow'):
result = f.rep.pow(n)
else: # pragma: no cover
raise OperationNotSupported(f, 'pow')
return f.per(result)
def pdiv(f, g):
"""
Polynomial pseudo-division of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).pdiv(Poly(2*x - 4, x))
(Poly(2*x + 4, x, domain='ZZ'), Poly(20, x, domain='ZZ'))
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'pdiv'):
q, r = F.pdiv(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'pdiv')
return per(q), per(r)
def prem(f, g):
"""
Polynomial pseudo-remainder of ``f`` by ``g``.
Caveat: The function prem(f, g, x) can be safely used to compute
in Z[x] _only_ subresultant polynomial remainder sequences (prs's).
To safely compute Euclidean and Sturmian prs's in Z[x]
employ anyone of the corresponding functions found in
the module sympy.polys.subresultants_qq_zz. The functions
in the module with suffix _pg compute prs's in Z[x] employing
rem(f, g, x), whereas the functions with suffix _amv
compute prs's in Z[x] employing rem_z(f, g, x).
The function rem_z(f, g, x) differs from prem(f, g, x) in that
to compute the remainder polynomials in Z[x] it premultiplies
the divident times the absolute value of the leading coefficient
of the divisor raised to the power degree(f, x) - degree(g, x) + 1.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).prem(Poly(2*x - 4, x))
Poly(20, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'prem'):
result = F.prem(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'prem')
return per(result)
def pquo(f, g):
"""
Polynomial pseudo-quotient of ``f`` by ``g``.
See the Caveat note in the function prem(f, g).
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).pquo(Poly(2*x - 4, x))
Poly(2*x + 4, x, domain='ZZ')
>>> Poly(x**2 - 1, x).pquo(Poly(2*x - 2, x))
Poly(2*x + 2, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'pquo'):
result = F.pquo(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'pquo')
return per(result)
def pexquo(f, g):
"""
Polynomial exact pseudo-quotient of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).pexquo(Poly(2*x - 2, x))
Poly(2*x + 2, x, domain='ZZ')
>>> Poly(x**2 + 1, x).pexquo(Poly(2*x - 4, x))
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'pexquo'):
try:
result = F.pexquo(G)
except ExactQuotientFailed as exc:
raise exc.new(f.as_expr(), g.as_expr())
else: # pragma: no cover
raise OperationNotSupported(f, 'pexquo')
return per(result)
def div(f, g, auto=True):
"""
Polynomial division with remainder of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x))
(Poly(1/2*x + 1, x, domain='QQ'), Poly(5, x, domain='QQ'))
>>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x), auto=False)
(Poly(0, x, domain='ZZ'), Poly(x**2 + 1, x, domain='ZZ'))
"""
dom, per, F, G = f._unify(g)
retract = False
if auto and dom.is_Ring and not dom.is_Field:
F, G = F.to_field(), G.to_field()
retract = True
if hasattr(f.rep, 'div'):
q, r = F.div(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'div')
if retract:
try:
Q, R = q.to_ring(), r.to_ring()
except CoercionFailed:
pass
else:
q, r = Q, R
return per(q), per(r)
def rem(f, g, auto=True):
"""
Computes the polynomial remainder of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x))
Poly(5, x, domain='ZZ')
>>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x), auto=False)
Poly(x**2 + 1, x, domain='ZZ')
"""
dom, per, F, G = f._unify(g)
retract = False
if auto and dom.is_Ring and not dom.is_Field:
F, G = F.to_field(), G.to_field()
retract = True
if hasattr(f.rep, 'rem'):
r = F.rem(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'rem')
if retract:
try:
r = r.to_ring()
except CoercionFailed:
pass
return per(r)
def quo(f, g, auto=True):
"""
Computes polynomial quotient of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).quo(Poly(2*x - 4, x))
Poly(1/2*x + 1, x, domain='QQ')
>>> Poly(x**2 - 1, x).quo(Poly(x - 1, x))
Poly(x + 1, x, domain='ZZ')
"""
dom, per, F, G = f._unify(g)
retract = False
if auto and dom.is_Ring and not dom.is_Field:
F, G = F.to_field(), G.to_field()
retract = True
if hasattr(f.rep, 'quo'):
q = F.quo(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'quo')
if retract:
try:
q = q.to_ring()
except CoercionFailed:
pass
return per(q)
def exquo(f, g, auto=True):
"""
Computes polynomial exact quotient of ``f`` by ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).exquo(Poly(x - 1, x))
Poly(x + 1, x, domain='ZZ')
>>> Poly(x**2 + 1, x).exquo(Poly(2*x - 4, x))
Traceback (most recent call last):
...
ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1
"""
dom, per, F, G = f._unify(g)
retract = False
if auto and dom.is_Ring and not dom.is_Field:
F, G = F.to_field(), G.to_field()
retract = True
if hasattr(f.rep, 'exquo'):
try:
q = F.exquo(G)
except ExactQuotientFailed as exc:
raise exc.new(f.as_expr(), g.as_expr())
else: # pragma: no cover
raise OperationNotSupported(f, 'exquo')
if retract:
try:
q = q.to_ring()
except CoercionFailed:
pass
return per(q)
def _gen_to_level(f, gen):
"""Returns level associated with the given generator. """
if isinstance(gen, int):
length = len(f.gens)
if -length <= gen < length:
if gen < 0:
return length + gen
else:
return gen
else:
raise PolynomialError("-%s <= gen < %s expected, got %s" %
(length, length, gen))
else:
try:
return f.gens.index(sympify(gen))
except ValueError:
raise PolynomialError(
"a valid generator expected, got %s" % gen)
def degree(f, gen=0):
"""
Returns degree of ``f`` in ``x_j``.
The degree of 0 is negative infinity.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + y*x + 1, x, y).degree()
2
>>> Poly(x**2 + y*x + y, x, y).degree(y)
1
>>> Poly(0, x).degree()
-oo
"""
j = f._gen_to_level(gen)
if hasattr(f.rep, 'degree'):
return f.rep.degree(j)
else: # pragma: no cover
raise OperationNotSupported(f, 'degree')
def degree_list(f):
"""
Returns a list of degrees of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + y*x + 1, x, y).degree_list()
(2, 1)
"""
if hasattr(f.rep, 'degree_list'):
return f.rep.degree_list()
else: # pragma: no cover
raise OperationNotSupported(f, 'degree_list')
def total_degree(f):
"""
Returns the total degree of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + y*x + 1, x, y).total_degree()
2
>>> Poly(x + y**5, x, y).total_degree()
5
"""
if hasattr(f.rep, 'total_degree'):
return f.rep.total_degree()
else: # pragma: no cover
raise OperationNotSupported(f, 'total_degree')
def homogenize(f, s):
"""
Returns the homogeneous polynomial of ``f``.
A homogeneous polynomial is a polynomial whose all monomials with
non-zero coefficients have the same total degree. If you only
want to check if a polynomial is homogeneous, then use
:func:`Poly.is_homogeneous`. If you want not only to check if a
polynomial is homogeneous but also compute its homogeneous order,
then use :func:`Poly.homogeneous_order`.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> f = Poly(x**5 + 2*x**2*y**2 + 9*x*y**3)
>>> f.homogenize(z)
Poly(x**5 + 2*x**2*y**2*z + 9*x*y**3*z, x, y, z, domain='ZZ')
"""
if not isinstance(s, Symbol):
raise TypeError("``Symbol`` expected, got %s" % type(s))
if s in f.gens:
i = f.gens.index(s)
gens = f.gens
else:
i = len(f.gens)
gens = f.gens + (s,)
if hasattr(f.rep, 'homogenize'):
return f.per(f.rep.homogenize(i), gens=gens)
raise OperationNotSupported(f, 'homogeneous_order')
def homogeneous_order(f):
"""
Returns the homogeneous order of ``f``.
A homogeneous polynomial is a polynomial whose all monomials with
non-zero coefficients have the same total degree. This degree is
the homogeneous order of ``f``. If you only want to check if a
polynomial is homogeneous, then use :func:`Poly.is_homogeneous`.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**5 + 2*x**3*y**2 + 9*x*y**4)
>>> f.homogeneous_order()
5
"""
if hasattr(f.rep, 'homogeneous_order'):
return f.rep.homogeneous_order()
else: # pragma: no cover
raise OperationNotSupported(f, 'homogeneous_order')
def LC(f, order=None):
"""
Returns the leading coefficient of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(4*x**3 + 2*x**2 + 3*x, x).LC()
4
"""
if order is not None:
return f.coeffs(order)[0]
if hasattr(f.rep, 'LC'):
result = f.rep.LC()
else: # pragma: no cover
raise OperationNotSupported(f, 'LC')
return f.rep.dom.to_sympy(result)
def TC(f):
"""
Returns the trailing coefficient of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x**2 + 3*x, x).TC()
0
"""
if hasattr(f.rep, 'TC'):
result = f.rep.TC()
else: # pragma: no cover
raise OperationNotSupported(f, 'TC')
return f.rep.dom.to_sympy(result)
def EC(f, order=None):
"""
Returns the last non-zero coefficient of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 + 2*x**2 + 3*x, x).EC()
3
"""
if hasattr(f.rep, 'coeffs'):
return f.coeffs(order)[-1]
else: # pragma: no cover
raise OperationNotSupported(f, 'EC')
def coeff_monomial(f, monom):
"""
Returns the coefficient of ``monom`` in ``f`` if there, else None.
Examples
========
>>> from sympy import Poly, exp
>>> from sympy.abc import x, y
>>> p = Poly(24*x*y*exp(8) + 23*x, x, y)
>>> p.coeff_monomial(x)
23
>>> p.coeff_monomial(y)
0
>>> p.coeff_monomial(x*y)
24*exp(8)
Note that ``Expr.coeff()`` behaves differently, collecting terms
if possible; the Poly must be converted to an Expr to use that
method, however:
>>> p.as_expr().coeff(x)
24*y*exp(8) + 23
>>> p.as_expr().coeff(y)
24*x*exp(8)
>>> p.as_expr().coeff(x*y)
24*exp(8)
See Also
========
nth: more efficient query using exponents of the monomial's generators
"""
return f.nth(*Monomial(monom, f.gens).exponents)
def nth(f, *N):
"""
Returns the ``n``-th coefficient of ``f`` where ``N`` are the
exponents of the generators in the term of interest.
Examples
========
>>> from sympy import Poly, sqrt
>>> from sympy.abc import x, y
>>> Poly(x**3 + 2*x**2 + 3*x, x).nth(2)
2
>>> Poly(x**3 + 2*x*y**2 + y**2, x, y).nth(1, 2)
2
>>> Poly(4*sqrt(x)*y)
Poly(4*y*(sqrt(x)), y, sqrt(x), domain='ZZ')
>>> _.nth(1, 1)
4
See Also
========
coeff_monomial
"""
if hasattr(f.rep, 'nth'):
if len(N) != len(f.gens):
raise ValueError('exponent of each generator must be specified')
result = f.rep.nth(*list(map(int, N)))
else: # pragma: no cover
raise OperationNotSupported(f, 'nth')
return f.rep.dom.to_sympy(result)
def coeff(f, x, n=1, right=False):
# the semantics of coeff_monomial and Expr.coeff are different;
# if someone is working with a Poly, they should be aware of the
# differences and chose the method best suited for the query.
# Alternatively, a pure-polys method could be written here but
# at this time the ``right`` keyword would be ignored because Poly
# doesn't work with non-commutatives.
raise NotImplementedError(
'Either convert to Expr with `as_expr` method '
'to use Expr\'s coeff method or else use the '
'`coeff_monomial` method of Polys.')
def LM(f, order=None):
"""
Returns the leading monomial of ``f``.
The Leading monomial signifies the monomial having
the highest power of the principal generator in the
expression f.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LM()
x**2*y**0
"""
return Monomial(f.monoms(order)[0], f.gens)
def EM(f, order=None):
"""
Returns the last non-zero monomial of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).EM()
x**0*y**1
"""
return Monomial(f.monoms(order)[-1], f.gens)
def LT(f, order=None):
"""
Returns the leading term of ``f``.
The Leading term signifies the term having
the highest power of the principal generator in the
expression f along with its coefficient.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LT()
(x**2*y**0, 4)
"""
monom, coeff = f.terms(order)[0]
return Monomial(monom, f.gens), coeff
def ET(f, order=None):
"""
Returns the last non-zero term of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).ET()
(x**0*y**1, 3)
"""
monom, coeff = f.terms(order)[-1]
return Monomial(monom, f.gens), coeff
def max_norm(f):
"""
Returns maximum norm of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(-x**2 + 2*x - 3, x).max_norm()
3
"""
if hasattr(f.rep, 'max_norm'):
result = f.rep.max_norm()
else: # pragma: no cover
raise OperationNotSupported(f, 'max_norm')
return f.rep.dom.to_sympy(result)
def l1_norm(f):
"""
Returns l1 norm of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(-x**2 + 2*x - 3, x).l1_norm()
6
"""
if hasattr(f.rep, 'l1_norm'):
result = f.rep.l1_norm()
else: # pragma: no cover
raise OperationNotSupported(f, 'l1_norm')
return f.rep.dom.to_sympy(result)
def clear_denoms(self, convert=False):
"""
Clear denominators, but keep the ground domain.
Examples
========
>>> from sympy import Poly, S, QQ
>>> from sympy.abc import x
>>> f = Poly(x/2 + S(1)/3, x, domain=QQ)
>>> f.clear_denoms()
(6, Poly(3*x + 2, x, domain='QQ'))
>>> f.clear_denoms(convert=True)
(6, Poly(3*x + 2, x, domain='ZZ'))
"""
f = self
if not f.rep.dom.is_Field:
return S.One, f
dom = f.get_domain()
if dom.has_assoc_Ring:
dom = f.rep.dom.get_ring()
if hasattr(f.rep, 'clear_denoms'):
coeff, result = f.rep.clear_denoms()
else: # pragma: no cover
raise OperationNotSupported(f, 'clear_denoms')
coeff, f = dom.to_sympy(coeff), f.per(result)
if not convert or not dom.has_assoc_Ring:
return coeff, f
else:
return coeff, f.to_ring()
def rat_clear_denoms(self, g):
"""
Clear denominators in a rational function ``f/g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = Poly(x**2/y + 1, x)
>>> g = Poly(x**3 + y, x)
>>> p, q = f.rat_clear_denoms(g)
>>> p
Poly(x**2 + y, x, domain='ZZ[y]')
>>> q
Poly(y*x**3 + y**2, x, domain='ZZ[y]')
"""
f = self
dom, per, f, g = f._unify(g)
f = per(f)
g = per(g)
if not (dom.is_Field and dom.has_assoc_Ring):
return f, g
a, f = f.clear_denoms(convert=True)
b, g = g.clear_denoms(convert=True)
f = f.mul_ground(b)
g = g.mul_ground(a)
return f, g
def integrate(self, *specs, **args):
"""
Computes indefinite integral of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x + 1, x).integrate()
Poly(1/3*x**3 + x**2 + x, x, domain='QQ')
>>> Poly(x*y**2 + x, x, y).integrate((0, 1), (1, 0))
Poly(1/2*x**2*y**2 + 1/2*x**2, x, y, domain='QQ')
"""
f = self
if args.get('auto', True) and f.rep.dom.is_Ring:
f = f.to_field()
if hasattr(f.rep, 'integrate'):
if not specs:
return f.per(f.rep.integrate(m=1))
rep = f.rep
for spec in specs:
if type(spec) is tuple:
gen, m = spec
else:
gen, m = spec, 1
rep = rep.integrate(int(m), f._gen_to_level(gen))
return f.per(rep)
else: # pragma: no cover
raise OperationNotSupported(f, 'integrate')
def diff(f, *specs, **kwargs):
"""
Computes partial derivative of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + 2*x + 1, x).diff()
Poly(2*x + 2, x, domain='ZZ')
>>> Poly(x*y**2 + x, x, y).diff((0, 0), (1, 1))
Poly(2*x*y, x, y, domain='ZZ')
"""
if not kwargs.get('evaluate', True):
return Derivative(f, *specs, **kwargs)
if hasattr(f.rep, 'diff'):
if not specs:
return f.per(f.rep.diff(m=1))
rep = f.rep
for spec in specs:
if type(spec) is tuple:
gen, m = spec
else:
gen, m = spec, 1
rep = rep.diff(int(m), f._gen_to_level(gen))
return f.per(rep)
else: # pragma: no cover
raise OperationNotSupported(f, 'diff')
_eval_derivative = diff
def eval(self, x, a=None, auto=True):
"""
Evaluate ``f`` at ``a`` in the given variable.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> Poly(x**2 + 2*x + 3, x).eval(2)
11
>>> Poly(2*x*y + 3*x + y + 2, x, y).eval(x, 2)
Poly(5*y + 8, y, domain='ZZ')
>>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)
>>> f.eval({x: 2})
Poly(5*y + 2*z + 6, y, z, domain='ZZ')
>>> f.eval({x: 2, y: 5})
Poly(2*z + 31, z, domain='ZZ')
>>> f.eval({x: 2, y: 5, z: 7})
45
>>> f.eval((2, 5))
Poly(2*z + 31, z, domain='ZZ')
>>> f(2, 5)
Poly(2*z + 31, z, domain='ZZ')
"""
f = self
if a is None:
if isinstance(x, dict):
mapping = x
for gen, value in mapping.items():
f = f.eval(gen, value)
return f
elif isinstance(x, (tuple, list)):
values = x
if len(values) > len(f.gens):
raise ValueError("too many values provided")
for gen, value in zip(f.gens, values):
f = f.eval(gen, value)
return f
else:
j, a = 0, x
else:
j = f._gen_to_level(x)
if not hasattr(f.rep, 'eval'): # pragma: no cover
raise OperationNotSupported(f, 'eval')
try:
result = f.rep.eval(a, j)
except CoercionFailed:
if not auto:
raise DomainError("can't evaluate at %s in %s" % (a, f.rep.dom))
else:
a_domain, [a] = construct_domain([a])
new_domain = f.get_domain().unify_with_symbols(a_domain, f.gens)
f = f.set_domain(new_domain)
a = new_domain.convert(a, a_domain)
result = f.rep.eval(a, j)
return f.per(result, remove=j)
def __call__(f, *values):
"""
Evaluate ``f`` at the give values.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)
>>> f(2)
Poly(5*y + 2*z + 6, y, z, domain='ZZ')
>>> f(2, 5)
Poly(2*z + 31, z, domain='ZZ')
>>> f(2, 5, 7)
45
"""
return f.eval(values)
def half_gcdex(f, g, auto=True):
"""
Half extended Euclidean algorithm of ``f`` and ``g``.
Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> Poly(f).half_gcdex(Poly(g))
(Poly(-1/5*x + 3/5, x, domain='QQ'), Poly(x + 1, x, domain='QQ'))
"""
dom, per, F, G = f._unify(g)
if auto and dom.is_Ring:
F, G = F.to_field(), G.to_field()
if hasattr(f.rep, 'half_gcdex'):
s, h = F.half_gcdex(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'half_gcdex')
return per(s), per(h)
def gcdex(f, g, auto=True):
"""
Extended Euclidean algorithm of ``f`` and ``g``.
Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
>>> g = x**3 + x**2 - 4*x - 4
>>> Poly(f).gcdex(Poly(g))
(Poly(-1/5*x + 3/5, x, domain='QQ'),
Poly(1/5*x**2 - 6/5*x + 2, x, domain='QQ'),
Poly(x + 1, x, domain='QQ'))
"""
dom, per, F, G = f._unify(g)
if auto and dom.is_Ring:
F, G = F.to_field(), G.to_field()
if hasattr(f.rep, 'gcdex'):
s, t, h = F.gcdex(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'gcdex')
return per(s), per(t), per(h)
def invert(f, g, auto=True):
"""
Invert ``f`` modulo ``g`` when possible.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).invert(Poly(2*x - 1, x))
Poly(-4/3, x, domain='QQ')
>>> Poly(x**2 - 1, x).invert(Poly(x - 1, x))
Traceback (most recent call last):
...
NotInvertible: zero divisor
"""
dom, per, F, G = f._unify(g)
if auto and dom.is_Ring:
F, G = F.to_field(), G.to_field()
if hasattr(f.rep, 'invert'):
result = F.invert(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'invert')
return per(result)
def revert(f, n):
"""
Compute ``f**(-1)`` mod ``x**n``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(1, x).revert(2)
Poly(1, x, domain='ZZ')
>>> Poly(1 + x, x).revert(1)
Poly(1, x, domain='ZZ')
>>> Poly(x**2 - 1, x).revert(1)
Traceback (most recent call last):
...
NotReversible: only unity is reversible in a ring
>>> Poly(1/x, x).revert(1)
Traceback (most recent call last):
...
PolynomialError: 1/x contains an element of the generators set
"""
if hasattr(f.rep, 'revert'):
result = f.rep.revert(int(n))
else: # pragma: no cover
raise OperationNotSupported(f, 'revert')
return f.per(result)
def subresultants(f, g):
"""
Computes the subresultant PRS of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 1, x).subresultants(Poly(x**2 - 1, x))
[Poly(x**2 + 1, x, domain='ZZ'),
Poly(x**2 - 1, x, domain='ZZ'),
Poly(-2, x, domain='ZZ')]
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'subresultants'):
result = F.subresultants(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'subresultants')
return list(map(per, result))
def resultant(f, g, includePRS=False):
"""
Computes the resultant of ``f`` and ``g`` via PRS.
If includePRS=True, it includes the subresultant PRS in the result.
Because the PRS is used to calculate the resultant, this is more
efficient than calling :func:`subresultants` separately.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = Poly(x**2 + 1, x)
>>> f.resultant(Poly(x**2 - 1, x))
4
>>> f.resultant(Poly(x**2 - 1, x), includePRS=True)
(4, [Poly(x**2 + 1, x, domain='ZZ'), Poly(x**2 - 1, x, domain='ZZ'),
Poly(-2, x, domain='ZZ')])
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'resultant'):
if includePRS:
result, R = F.resultant(G, includePRS=includePRS)
else:
result = F.resultant(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'resultant')
if includePRS:
return (per(result, remove=0), list(map(per, R)))
return per(result, remove=0)
def discriminant(f):
"""
Computes the discriminant of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + 2*x + 3, x).discriminant()
-8
"""
if hasattr(f.rep, 'discriminant'):
result = f.rep.discriminant()
else: # pragma: no cover
raise OperationNotSupported(f, 'discriminant')
return f.per(result, remove=0)
def dispersionset(f, g=None):
r"""Compute the *dispersion set* of two polynomials.
For two polynomials `f(x)` and `g(x)` with `\deg f > 0`
and `\deg g > 0` the dispersion set `\operatorname{J}(f, g)` is defined as:
.. math::
\operatorname{J}(f, g)
& := \\{a \in \mathbb{N}_0 | \gcd(f(x), g(x+a)) \neq 1\\} \\\\
& = \\{a \in \mathbb{N}_0 | \deg \gcd(f(x), g(x+a)) \geq 1\\}
For a single polynomial one defines `\operatorname{J}(f) := \operatorname{J}(f, f)`.
Examples
========
>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x
Dispersion set and dispersion of a simple polynomial:
>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6
Note that the definition of the dispersion is not symmetric:
>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo
Computing the dispersion also works over field extensions:
>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]
We can even perform the computations for polynomials
having symbolic coefficients:
>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]
See Also
========
dispersion
References
==========
1. [ManWright94]_
2. [Koepf98]_
3. [Abramov71]_
4. [Man93]_
"""
from sympy.polys.dispersion import dispersionset
return dispersionset(f, g)
def dispersion(f, g=None):
r"""Compute the *dispersion* of polynomials.
For two polynomials `f(x)` and `g(x)` with `\deg f > 0`
and `\deg g > 0` the dispersion `\operatorname{dis}(f, g)` is defined as:
.. math::
\operatorname{dis}(f, g)
& := \max\\{ J(f,g) \cup \\{0\\} \\} \\\\
& = \max\\{ \\{a \in \mathbb{N} | \gcd(f(x), g(x+a)) \neq 1\\} \cup \\{0\\} \\}
and for a single polynomial `\operatorname{dis}(f) := \operatorname{dis}(f, f)`.
Examples
========
>>> from sympy import poly
>>> from sympy.polys.dispersion import dispersion, dispersionset
>>> from sympy.abc import x
Dispersion set and dispersion of a simple polynomial:
>>> fp = poly((x - 3)*(x + 3), x)
>>> sorted(dispersionset(fp))
[0, 6]
>>> dispersion(fp)
6
Note that the definition of the dispersion is not symmetric:
>>> fp = poly(x**4 - 3*x**2 + 1, x)
>>> gp = fp.shift(-3)
>>> sorted(dispersionset(fp, gp))
[2, 3, 4]
>>> dispersion(fp, gp)
4
>>> sorted(dispersionset(gp, fp))
[]
>>> dispersion(gp, fp)
-oo
Computing the dispersion also works over field extensions:
>>> from sympy import sqrt
>>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ<sqrt(5)>')
>>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ<sqrt(5)>')
>>> sorted(dispersionset(fp, gp))
[2]
>>> sorted(dispersionset(gp, fp))
[1, 4]
We can even perform the computations for polynomials
having symbolic coefficients:
>>> from sympy.abc import a
>>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)
>>> sorted(dispersionset(fp))
[0, 1]
See Also
========
dispersionset
References
==========
1. [ManWright94]_
2. [Koepf98]_
3. [Abramov71]_
4. [Man93]_
"""
from sympy.polys.dispersion import dispersion
return dispersion(f, g)
def cofactors(f, g):
"""
Returns the GCD of ``f`` and ``g`` and their cofactors.
Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and
``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors
of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).cofactors(Poly(x**2 - 3*x + 2, x))
(Poly(x - 1, x, domain='ZZ'),
Poly(x + 1, x, domain='ZZ'),
Poly(x - 2, x, domain='ZZ'))
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'cofactors'):
h, cff, cfg = F.cofactors(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'cofactors')
return per(h), per(cff), per(cfg)
def gcd(f, g):
"""
Returns the polynomial GCD of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).gcd(Poly(x**2 - 3*x + 2, x))
Poly(x - 1, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'gcd'):
result = F.gcd(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'gcd')
return per(result)
def lcm(f, g):
"""
Returns polynomial LCM of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 1, x).lcm(Poly(x**2 - 3*x + 2, x))
Poly(x**3 - 2*x**2 - x + 2, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'lcm'):
result = F.lcm(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'lcm')
return per(result)
def trunc(f, p):
"""
Reduce ``f`` modulo a constant ``p``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**3 + 3*x**2 + 5*x + 7, x).trunc(3)
Poly(-x**3 - x + 1, x, domain='ZZ')
"""
p = f.rep.dom.convert(p)
if hasattr(f.rep, 'trunc'):
result = f.rep.trunc(p)
else: # pragma: no cover
raise OperationNotSupported(f, 'trunc')
return f.per(result)
def monic(self, auto=True):
"""
Divides all coefficients by ``LC(f)``.
Examples
========
>>> from sympy import Poly, ZZ
>>> from sympy.abc import x
>>> Poly(3*x**2 + 6*x + 9, x, domain=ZZ).monic()
Poly(x**2 + 2*x + 3, x, domain='QQ')
>>> Poly(3*x**2 + 4*x + 2, x, domain=ZZ).monic()
Poly(x**2 + 4/3*x + 2/3, x, domain='QQ')
"""
f = self
if auto and f.rep.dom.is_Ring:
f = f.to_field()
if hasattr(f.rep, 'monic'):
result = f.rep.monic()
else: # pragma: no cover
raise OperationNotSupported(f, 'monic')
return f.per(result)
def content(f):
"""
Returns the GCD of polynomial coefficients.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(6*x**2 + 8*x + 12, x).content()
2
"""
if hasattr(f.rep, 'content'):
result = f.rep.content()
else: # pragma: no cover
raise OperationNotSupported(f, 'content')
return f.rep.dom.to_sympy(result)
def primitive(f):
"""
Returns the content and a primitive form of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**2 + 8*x + 12, x).primitive()
(2, Poly(x**2 + 4*x + 6, x, domain='ZZ'))
"""
if hasattr(f.rep, 'primitive'):
cont, result = f.rep.primitive()
else: # pragma: no cover
raise OperationNotSupported(f, 'primitive')
return f.rep.dom.to_sympy(cont), f.per(result)
def compose(f, g):
"""
Computes the functional composition of ``f`` and ``g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + x, x).compose(Poly(x - 1, x))
Poly(x**2 - x, x, domain='ZZ')
"""
_, per, F, G = f._unify(g)
if hasattr(f.rep, 'compose'):
result = F.compose(G)
else: # pragma: no cover
raise OperationNotSupported(f, 'compose')
return per(result)
def decompose(f):
"""
Computes a functional decomposition of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**4 + 2*x**3 - x - 1, x, domain='ZZ').decompose()
[Poly(x**2 - x - 1, x, domain='ZZ'), Poly(x**2 + x, x, domain='ZZ')]
"""
if hasattr(f.rep, 'decompose'):
result = f.rep.decompose()
else: # pragma: no cover
raise OperationNotSupported(f, 'decompose')
return list(map(f.per, result))
def shift(f, a):
"""
Efficiently compute Taylor shift ``f(x + a)``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 2*x + 1, x).shift(2)
Poly(x**2 + 2*x + 1, x, domain='ZZ')
"""
if hasattr(f.rep, 'shift'):
result = f.rep.shift(a)
else: # pragma: no cover
raise OperationNotSupported(f, 'shift')
return f.per(result)
def transform(f, p, q):
"""
Efficiently evaluate the functional transformation ``q**n * f(p/q)``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1, x), Poly(x - 1, x))
Poly(4, x, domain='ZZ')
"""
P, Q = p.unify(q)
F, P = f.unify(P)
F, Q = F.unify(Q)
if hasattr(F.rep, 'transform'):
result = F.rep.transform(P.rep, Q.rep)
else: # pragma: no cover
raise OperationNotSupported(F, 'transform')
return F.per(result)
def sturm(self, auto=True):
"""
Computes the Sturm sequence of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 - 2*x**2 + x - 3, x).sturm()
[Poly(x**3 - 2*x**2 + x - 3, x, domain='QQ'),
Poly(3*x**2 - 4*x + 1, x, domain='QQ'),
Poly(2/9*x + 25/9, x, domain='QQ'),
Poly(-2079/4, x, domain='QQ')]
"""
f = self
if auto and f.rep.dom.is_Ring:
f = f.to_field()
if hasattr(f.rep, 'sturm'):
result = f.rep.sturm()
else: # pragma: no cover
raise OperationNotSupported(f, 'sturm')
return list(map(f.per, result))
def gff_list(f):
"""
Computes greatest factorial factorization of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = x**5 + 2*x**4 - x**3 - 2*x**2
>>> Poly(f).gff_list()
[(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)]
"""
if hasattr(f.rep, 'gff_list'):
result = f.rep.gff_list()
else: # pragma: no cover
raise OperationNotSupported(f, 'gff_list')
return [(f.per(g), k) for g, k in result]
def norm(f):
"""
Computes the product, ``Norm(f)``, of the conjugates of
a polynomial ``f`` defined over a number field ``K``.
Examples
========
>>> from sympy import Poly, sqrt
>>> from sympy.abc import x
>>> a, b = sqrt(2), sqrt(3)
A polynomial over a quadratic extension.
Two conjugates x - a and x + a.
>>> f = Poly(x - a, x, extension=a)
>>> f.norm()
Poly(x**2 - 2, x, domain='QQ')
A polynomial over a quartic extension.
Four conjugates x - a, x - a, x + a and x + a.
>>> f = Poly(x - a, x, extension=(a, b))
>>> f.norm()
Poly(x**4 - 4*x**2 + 4, x, domain='QQ')
"""
if hasattr(f.rep, 'norm'):
r = f.rep.norm()
else: # pragma: no cover
raise OperationNotSupported(f, 'norm')
return f.per(r)
def sqf_norm(f):
"""
Computes square-free norm of ``f``.
Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and
``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``,
where ``a`` is the algebraic extension of the ground domain.
Examples
========
>>> from sympy import Poly, sqrt
>>> from sympy.abc import x
>>> s, f, r = Poly(x**2 + 1, x, extension=[sqrt(3)]).sqf_norm()
>>> s
1
>>> f
Poly(x**2 - 2*sqrt(3)*x + 4, x, domain='QQ<sqrt(3)>')
>>> r
Poly(x**4 - 4*x**2 + 16, x, domain='QQ')
"""
if hasattr(f.rep, 'sqf_norm'):
s, g, r = f.rep.sqf_norm()
else: # pragma: no cover
raise OperationNotSupported(f, 'sqf_norm')
return s, f.per(g), f.per(r)
def sqf_part(f):
"""
Computes square-free part of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**3 - 3*x - 2, x).sqf_part()
Poly(x**2 - x - 2, x, domain='ZZ')
"""
if hasattr(f.rep, 'sqf_part'):
result = f.rep.sqf_part()
else: # pragma: no cover
raise OperationNotSupported(f, 'sqf_part')
return f.per(result)
def sqf_list(f, all=False):
"""
Returns a list of square-free factors of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16
>>> Poly(f).sqf_list()
(2, [(Poly(x + 1, x, domain='ZZ'), 2),
(Poly(x + 2, x, domain='ZZ'), 3)])
>>> Poly(f).sqf_list(all=True)
(2, [(Poly(1, x, domain='ZZ'), 1),
(Poly(x + 1, x, domain='ZZ'), 2),
(Poly(x + 2, x, domain='ZZ'), 3)])
"""
if hasattr(f.rep, 'sqf_list'):
coeff, factors = f.rep.sqf_list(all)
else: # pragma: no cover
raise OperationNotSupported(f, 'sqf_list')
return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors]
def sqf_list_include(f, all=False):
"""
Returns a list of square-free factors of ``f``.
Examples
========
>>> from sympy import Poly, expand
>>> from sympy.abc import x
>>> f = expand(2*(x + 1)**3*x**4)
>>> f
2*x**7 + 6*x**6 + 6*x**5 + 2*x**4
>>> Poly(f).sqf_list_include()
[(Poly(2, x, domain='ZZ'), 1),
(Poly(x + 1, x, domain='ZZ'), 3),
(Poly(x, x, domain='ZZ'), 4)]
>>> Poly(f).sqf_list_include(all=True)
[(Poly(2, x, domain='ZZ'), 1),
(Poly(1, x, domain='ZZ'), 2),
(Poly(x + 1, x, domain='ZZ'), 3),
(Poly(x, x, domain='ZZ'), 4)]
"""
if hasattr(f.rep, 'sqf_list_include'):
factors = f.rep.sqf_list_include(all)
else: # pragma: no cover
raise OperationNotSupported(f, 'sqf_list_include')
return [(f.per(g), k) for g, k in factors]
def factor_list(f):
"""
Returns a list of irreducible factors of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y
>>> Poly(f).factor_list()
(2, [(Poly(x + y, x, y, domain='ZZ'), 1),
(Poly(x**2 + 1, x, y, domain='ZZ'), 2)])
"""
if hasattr(f.rep, 'factor_list'):
try:
coeff, factors = f.rep.factor_list()
except DomainError:
return S.One, [(f, 1)]
else: # pragma: no cover
raise OperationNotSupported(f, 'factor_list')
return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors]
def factor_list_include(f):
"""
Returns a list of irreducible factors of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y
>>> Poly(f).factor_list_include()
[(Poly(2*x + 2*y, x, y, domain='ZZ'), 1),
(Poly(x**2 + 1, x, y, domain='ZZ'), 2)]
"""
if hasattr(f.rep, 'factor_list_include'):
try:
factors = f.rep.factor_list_include()
except DomainError:
return [(f, 1)]
else: # pragma: no cover
raise OperationNotSupported(f, 'factor_list_include')
return [(f.per(g), k) for g, k in factors]
def intervals(f, all=False, eps=None, inf=None, sup=None, fast=False, sqf=False):
"""
Compute isolating intervals for roots of ``f``.
For real roots the Vincent-Akritas-Strzebonski (VAS) continued fractions method is used.
References
==========
.. [#] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root
Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005.
.. [#] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the
Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear
Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 3, x).intervals()
[((-2, -1), 1), ((1, 2), 1)]
>>> Poly(x**2 - 3, x).intervals(eps=1e-2)
[((-26/15, -19/11), 1), ((19/11, 26/15), 1)]
"""
if eps is not None:
eps = QQ.convert(eps)
if eps <= 0:
raise ValueError("'eps' must be a positive rational")
if inf is not None:
inf = QQ.convert(inf)
if sup is not None:
sup = QQ.convert(sup)
if hasattr(f.rep, 'intervals'):
result = f.rep.intervals(
all=all, eps=eps, inf=inf, sup=sup, fast=fast, sqf=sqf)
else: # pragma: no cover
raise OperationNotSupported(f, 'intervals')
if sqf:
def _real(interval):
s, t = interval
return (QQ.to_sympy(s), QQ.to_sympy(t))
if not all:
return list(map(_real, result))
def _complex(rectangle):
(u, v), (s, t) = rectangle
return (QQ.to_sympy(u) + I*QQ.to_sympy(v),
QQ.to_sympy(s) + I*QQ.to_sympy(t))
real_part, complex_part = result
return list(map(_real, real_part)), list(map(_complex, complex_part))
else:
def _real(interval):
(s, t), k = interval
return ((QQ.to_sympy(s), QQ.to_sympy(t)), k)
if not all:
return list(map(_real, result))
def _complex(rectangle):
((u, v), (s, t)), k = rectangle
return ((QQ.to_sympy(u) + I*QQ.to_sympy(v),
QQ.to_sympy(s) + I*QQ.to_sympy(t)), k)
real_part, complex_part = result
return list(map(_real, real_part)), list(map(_complex, complex_part))
def refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False):
"""
Refine an isolating interval of a root to the given precision.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 3, x).refine_root(1, 2, eps=1e-2)
(19/11, 26/15)
"""
if check_sqf and not f.is_sqf:
raise PolynomialError("only square-free polynomials supported")
s, t = QQ.convert(s), QQ.convert(t)
if eps is not None:
eps = QQ.convert(eps)
if eps <= 0:
raise ValueError("'eps' must be a positive rational")
if steps is not None:
steps = int(steps)
elif eps is None:
steps = 1
if hasattr(f.rep, 'refine_root'):
S, T = f.rep.refine_root(s, t, eps=eps, steps=steps, fast=fast)
else: # pragma: no cover
raise OperationNotSupported(f, 'refine_root')
return QQ.to_sympy(S), QQ.to_sympy(T)
def count_roots(f, inf=None, sup=None):
"""
Return the number of roots of ``f`` in ``[inf, sup]`` interval.
Examples
========
>>> from sympy import Poly, I
>>> from sympy.abc import x
>>> Poly(x**4 - 4, x).count_roots(-3, 3)
2
>>> Poly(x**4 - 4, x).count_roots(0, 1 + 3*I)
1
"""
inf_real, sup_real = True, True
if inf is not None:
inf = sympify(inf)
if inf is S.NegativeInfinity:
inf = None
else:
re, im = inf.as_real_imag()
if not im:
inf = QQ.convert(inf)
else:
inf, inf_real = list(map(QQ.convert, (re, im))), False
if sup is not None:
sup = sympify(sup)
if sup is S.Infinity:
sup = None
else:
re, im = sup.as_real_imag()
if not im:
sup = QQ.convert(sup)
else:
sup, sup_real = list(map(QQ.convert, (re, im))), False
if inf_real and sup_real:
if hasattr(f.rep, 'count_real_roots'):
count = f.rep.count_real_roots(inf=inf, sup=sup)
else: # pragma: no cover
raise OperationNotSupported(f, 'count_real_roots')
else:
if inf_real and inf is not None:
inf = (inf, QQ.zero)
if sup_real and sup is not None:
sup = (sup, QQ.zero)
if hasattr(f.rep, 'count_complex_roots'):
count = f.rep.count_complex_roots(inf=inf, sup=sup)
else: # pragma: no cover
raise OperationNotSupported(f, 'count_complex_roots')
return Integer(count)
def root(f, index, radicals=True):
"""
Get an indexed root of a polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = Poly(2*x**3 - 7*x**2 + 4*x + 4)
>>> f.root(0)
-1/2
>>> f.root(1)
2
>>> f.root(2)
2
>>> f.root(3)
Traceback (most recent call last):
...
IndexError: root index out of [-3, 2] range, got 3
>>> Poly(x**5 + x + 1).root(0)
CRootOf(x**3 - x**2 + 1, 0)
"""
return sympy.polys.rootoftools.rootof(f, index, radicals=radicals)
def real_roots(f, multiple=True, radicals=True):
"""
Return a list of real roots with multiplicities.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**3 - 7*x**2 + 4*x + 4).real_roots()
[-1/2, 2, 2]
>>> Poly(x**3 + x + 1).real_roots()
[CRootOf(x**3 + x + 1, 0)]
"""
reals = sympy.polys.rootoftools.CRootOf.real_roots(f, radicals=radicals)
if multiple:
return reals
else:
return group(reals, multiple=False)
def all_roots(f, multiple=True, radicals=True):
"""
Return a list of real and complex roots with multiplicities.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**3 - 7*x**2 + 4*x + 4).all_roots()
[-1/2, 2, 2]
>>> Poly(x**3 + x + 1).all_roots()
[CRootOf(x**3 + x + 1, 0),
CRootOf(x**3 + x + 1, 1),
CRootOf(x**3 + x + 1, 2)]
"""
roots = sympy.polys.rootoftools.CRootOf.all_roots(f, radicals=radicals)
if multiple:
return roots
else:
return group(roots, multiple=False)
def nroots(f, n=15, maxsteps=50, cleanup=True):
"""
Compute numerical approximations of roots of ``f``.
Parameters
==========
n ... the number of digits to calculate
maxsteps ... the maximum number of iterations to do
If the accuracy `n` cannot be reached in `maxsteps`, it will raise an
exception. You need to rerun with higher maxsteps.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 3).nroots(n=15)
[-1.73205080756888, 1.73205080756888]
>>> Poly(x**2 - 3).nroots(n=30)
[-1.73205080756887729352744634151, 1.73205080756887729352744634151]
"""
from sympy.functions.elementary.complexes import sign
if f.is_multivariate:
raise MultivariatePolynomialError(
"can't compute numerical roots of %s" % f)
if f.degree() <= 0:
return []
# For integer and rational coefficients, convert them to integers only
# (for accuracy). Otherwise just try to convert the coefficients to
# mpmath.mpc and raise an exception if the conversion fails.
if f.rep.dom is ZZ:
coeffs = [int(coeff) for coeff in f.all_coeffs()]
elif f.rep.dom is QQ:
denoms = [coeff.q for coeff in f.all_coeffs()]
from sympy.core.numbers import ilcm
fac = ilcm(*denoms)
coeffs = [int(coeff*fac) for coeff in f.all_coeffs()]
else:
coeffs = [coeff.evalf(n=n).as_real_imag()
for coeff in f.all_coeffs()]
try:
coeffs = [mpmath.mpc(*coeff) for coeff in coeffs]
except TypeError:
raise DomainError("Numerical domain expected, got %s" % \
f.rep.dom)
dps = mpmath.mp.dps
mpmath.mp.dps = n
try:
# We need to add extra precision to guard against losing accuracy.
# 10 times the degree of the polynomial seems to work well.
roots = mpmath.polyroots(coeffs, maxsteps=maxsteps,
cleanup=cleanup, error=False, extraprec=f.degree()*10)
# Mpmath puts real roots first, then complex ones (as does all_roots)
# so we make sure this convention holds here, too.
roots = list(map(sympify,
sorted(roots, key=lambda r: (1 if r.imag else 0, r.real, abs(r.imag), sign(r.imag)))))
except NoConvergence:
raise NoConvergence(
'convergence to root failed; try n < %s or maxsteps > %s' % (
n, maxsteps))
finally:
mpmath.mp.dps = dps
return roots
def ground_roots(f):
"""
Compute roots of ``f`` by factorization in the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**6 - 4*x**4 + 4*x**3 - x**2).ground_roots()
{0: 2, 1: 2}
"""
if f.is_multivariate:
raise MultivariatePolynomialError(
"can't compute ground roots of %s" % f)
roots = {}
for factor, k in f.factor_list()[1]:
if factor.is_linear:
a, b = factor.all_coeffs()
roots[-b/a] = k
return roots
def nth_power_roots_poly(f, n):
"""
Construct a polynomial with n-th powers of roots of ``f``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = Poly(x**4 - x**2 + 1)
>>> f.nth_power_roots_poly(2)
Poly(x**4 - 2*x**3 + 3*x**2 - 2*x + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(3)
Poly(x**4 + 2*x**2 + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(4)
Poly(x**4 + 2*x**3 + 3*x**2 + 2*x + 1, x, domain='ZZ')
>>> f.nth_power_roots_poly(12)
Poly(x**4 - 4*x**3 + 6*x**2 - 4*x + 1, x, domain='ZZ')
"""
if f.is_multivariate:
raise MultivariatePolynomialError(
"must be a univariate polynomial")
N = sympify(n)
if N.is_Integer and N >= 1:
n = int(N)
else:
raise ValueError("'n' must an integer and n >= 1, got %s" % n)
x = f.gen
t = Dummy('t')
r = f.resultant(f.__class__.from_expr(x**n - t, x, t))
return r.replace(t, x)
def cancel(f, g, include=False):
"""
Cancel common factors in a rational function ``f/g``.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x))
(1, Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ'))
>>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x), include=True)
(Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ'))
"""
dom, per, F, G = f._unify(g)
if hasattr(F, 'cancel'):
result = F.cancel(G, include=include)
else: # pragma: no cover
raise OperationNotSupported(f, 'cancel')
if not include:
if dom.has_assoc_Ring:
dom = dom.get_ring()
cp, cq, p, q = result
cp = dom.to_sympy(cp)
cq = dom.to_sympy(cq)
return cp/cq, per(p), per(q)
else:
return tuple(map(per, result))
@property
def is_zero(f):
"""
Returns ``True`` if ``f`` is a zero polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(0, x).is_zero
True
>>> Poly(1, x).is_zero
False
"""
return f.rep.is_zero
@property
def is_one(f):
"""
Returns ``True`` if ``f`` is a unit polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(0, x).is_one
False
>>> Poly(1, x).is_one
True
"""
return f.rep.is_one
@property
def is_sqf(f):
"""
Returns ``True`` if ``f`` is a square-free polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 - 2*x + 1, x).is_sqf
False
>>> Poly(x**2 - 1, x).is_sqf
True
"""
return f.rep.is_sqf
@property
def is_monic(f):
"""
Returns ``True`` if the leading coefficient of ``f`` is one.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x + 2, x).is_monic
True
>>> Poly(2*x + 2, x).is_monic
False
"""
return f.rep.is_monic
@property
def is_primitive(f):
"""
Returns ``True`` if GCD of the coefficients of ``f`` is one.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(2*x**2 + 6*x + 12, x).is_primitive
False
>>> Poly(x**2 + 3*x + 6, x).is_primitive
True
"""
return f.rep.is_primitive
@property
def is_ground(f):
"""
Returns ``True`` if ``f`` is an element of the ground domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x, x).is_ground
False
>>> Poly(2, x).is_ground
True
>>> Poly(y, x).is_ground
True
"""
return f.rep.is_ground
@property
def is_linear(f):
"""
Returns ``True`` if ``f`` is linear in all its variables.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x + y + 2, x, y).is_linear
True
>>> Poly(x*y + 2, x, y).is_linear
False
"""
return f.rep.is_linear
@property
def is_quadratic(f):
"""
Returns ``True`` if ``f`` is quadratic in all its variables.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x*y + 2, x, y).is_quadratic
True
>>> Poly(x*y**2 + 2, x, y).is_quadratic
False
"""
return f.rep.is_quadratic
@property
def is_monomial(f):
"""
Returns ``True`` if ``f`` is zero or has only one term.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(3*x**2, x).is_monomial
True
>>> Poly(3*x**2 + 1, x).is_monomial
False
"""
return f.rep.is_monomial
@property
def is_homogeneous(f):
"""
Returns ``True`` if ``f`` is a homogeneous polynomial.
A homogeneous polynomial is a polynomial whose all monomials with
non-zero coefficients have the same total degree. If you want not
only to check if a polynomial is homogeneous but also compute its
homogeneous order, then use :func:`Poly.homogeneous_order`.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + x*y, x, y).is_homogeneous
True
>>> Poly(x**3 + x*y, x, y).is_homogeneous
False
"""
return f.rep.is_homogeneous
@property
def is_irreducible(f):
"""
Returns ``True`` if ``f`` has no factors over its domain.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> Poly(x**2 + x + 1, x, modulus=2).is_irreducible
True
>>> Poly(x**2 + 1, x, modulus=2).is_irreducible
False
"""
return f.rep.is_irreducible
@property
def is_univariate(f):
"""
Returns ``True`` if ``f`` is a univariate polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + x + 1, x).is_univariate
True
>>> Poly(x*y**2 + x*y + 1, x, y).is_univariate
False
>>> Poly(x*y**2 + x*y + 1, x).is_univariate
True
>>> Poly(x**2 + x + 1, x, y).is_univariate
False
"""
return len(f.gens) == 1
@property
def is_multivariate(f):
"""
Returns ``True`` if ``f`` is a multivariate polynomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x, y
>>> Poly(x**2 + x + 1, x).is_multivariate
False
>>> Poly(x*y**2 + x*y + 1, x, y).is_multivariate
True
>>> Poly(x*y**2 + x*y + 1, x).is_multivariate
False
>>> Poly(x**2 + x + 1, x, y).is_multivariate
True
"""
return len(f.gens) != 1
@property
def is_cyclotomic(f):
"""
Returns ``True`` if ``f`` is a cyclotomic polnomial.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1
>>> Poly(f).is_cyclotomic
False
>>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1
>>> Poly(g).is_cyclotomic
True
"""
return f.rep.is_cyclotomic
def __abs__(f):
return f.abs()
def __neg__(f):
return f.neg()
@_sympifyit('g', NotImplemented)
def __add__(f, g):
if not g.is_Poly:
try:
g = f.__class__(g, *f.gens)
except PolynomialError:
return f.as_expr() + g
return f.add(g)
@_sympifyit('g', NotImplemented)
def __radd__(f, g):
if not g.is_Poly:
try:
g = f.__class__(g, *f.gens)
except PolynomialError:
return g + f.as_expr()
return g.add(f)
@_sympifyit('g', NotImplemented)
def __sub__(f, g):
if not g.is_Poly:
try:
g = f.__class__(g, *f.gens)
except PolynomialError:
return f.as_expr() - g
return f.sub(g)
@_sympifyit('g', NotImplemented)
def __rsub__(f, g):
if not g.is_Poly:
try:
g = f.__class__(g, *f.gens)
except PolynomialError:
return g - f.as_expr()
return g.sub(f)
@_sympifyit('g', NotImplemented)
def __mul__(f, g):
if not g.is_Poly:
try:
g = f.__class__(g, *f.gens)
except PolynomialError:
return f.as_expr()*g
return f.mul(g)
@_sympifyit('g', NotImplemented)
def __rmul__(f, g):
if not g.is_Poly:
try:
g = f.__class__(g, *f.gens)
except PolynomialError:
return g*f.as_expr()
return g.mul(f)
@_sympifyit('n', NotImplemented)
def __pow__(f, n):
if n.is_Integer and n >= 0:
return f.pow(n)
else:
return f.as_expr()**n
@_sympifyit('g', NotImplemented)
def __divmod__(f, g):
if not g.is_Poly:
g = f.__class__(g, *f.gens)
return f.div(g)
@_sympifyit('g', NotImplemented)
def __rdivmod__(f, g):
if not g.is_Poly:
g = f.__class__(g, *f.gens)
return g.div(f)
@_sympifyit('g', NotImplemented)
def __mod__(f, g):
if not g.is_Poly:
g = f.__class__(g, *f.gens)
return f.rem(g)
@_sympifyit('g', NotImplemented)
def __rmod__(f, g):
if not g.is_Poly:
g = f.__class__(g, *f.gens)
return g.rem(f)
@_sympifyit('g', NotImplemented)
def __floordiv__(f, g):
if not g.is_Poly:
g = f.__class__(g, *f.gens)
return f.quo(g)
@_sympifyit('g', NotImplemented)
def __rfloordiv__(f, g):
if not g.is_Poly:
g = f.__class__(g, *f.gens)
return g.quo(f)
@_sympifyit('g', NotImplemented)
def __div__(f, g):
return f.as_expr()/g.as_expr()
@_sympifyit('g', NotImplemented)
def __rdiv__(f, g):
return g.as_expr()/f.as_expr()
__truediv__ = __div__
__rtruediv__ = __rdiv__
@_sympifyit('other', NotImplemented)
def __eq__(self, other):
f, g = self, other
if not g.is_Poly:
try:
g = f.__class__(g, f.gens, domain=f.get_domain())
except (PolynomialError, DomainError, CoercionFailed):
return False
if f.gens != g.gens:
return False
if f.rep.dom != g.rep.dom:
try:
dom = f.rep.dom.unify(g.rep.dom, f.gens)
except UnificationFailed:
return False
f = f.set_domain(dom)
g = g.set_domain(dom)
return f.rep == g.rep
@_sympifyit('g', NotImplemented)
def __ne__(f, g):
return not f == g
def __nonzero__(f):
return not f.is_zero
__bool__ = __nonzero__
def eq(f, g, strict=False):
if not strict:
return f == g
else:
return f._strict_eq(sympify(g))
def ne(f, g, strict=False):
return not f.eq(g, strict=strict)
def _strict_eq(f, g):
return isinstance(g, f.__class__) and f.gens == g.gens and f.rep.eq(g.rep, strict=True)
Poly({1:1,2:1},gens=S('sqrt(x)'))$\displaystyle \operatorname{Poly}{\left( x + \sqrt{x}, \sqrt{x}, domain=\mathbb{Z} \right)}$
x=symbols('x', positive=True)
S('sqrt(x^2)-x')$\displaystyle - x + \sqrt{x^{2}}$
str(Poly({1:1,2:1,4:1},gens=S('sqrt(x)')))"Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')"
str(Poly('x^2+x+sqrt(x)'))"Poly(x**2 + x + (sqrt(x)), x, sqrt(x), domain='ZZ')"
def findvalues(formula,values=None,variables=None):
formula=S(formula)
num,den=fractioncancel(formula)
if variables==None:
variables=sorted(num.free_symbols,key=str)
num=num.subs(zip(variables,list(map(lambda x:x**2,variables))))
num=Poly(num)
newformula=S((num.abs()+num)/(num.abs()-num))
f=lambdify(variables,newformula)
f2=lambda x:f(*x)
if values==None:
values=[1.0]*len(variables)
tup=tuple(fmin(f2,values))
return tuple([x*x for x in tup])
formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))
formula=(makesubs(formula,'[b,oo],[c,oo]'))
values=findvalues(formula)
display(values)
nsimplify(values,tolerance=0.1,rational=True)
values[0]/values[2]Substitute $a\to b+y_{1}$
Substitute $b\to c+z_{1}$
Optimization terminated successfully.
Current function value: 1.000000
Iterations: 137
Function evaluations: 249
(1.7908873553542452e-10, 2.5326984818340415e-10, 7.129450063690368)
2.5119572187973787e-11
from random import random
formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))
formula=(makesubs(formula,'[b,oo],[c,oo]'))
num,den=fractioncancel(formula)
fs=sorted(num.free_symbols,key=str)
num=num.subs(zip(fs,list(map(lambda x:x**2,fs))))
num=Poly(num,domain='RR')
num1=(num.abs()+num)
num2=(num.abs()-num)
newformula=S(num1/num2)
def evaluate(x):
return newformula.evalf(subs=dict(zip(fs,(1,1,1))))
def evaluate2(x):
return num1.eval(dict(zip(fs,x)))/num2.eval(dict(zip(fs,x)))
#display(num2.eval(dict(zip(fs,(1,1,1)))))
print('ok')
%timeit evaluate((random(),random(),random()))
%timeit evaluate2((random(),random(),random()))
Poly('x^2+sqrt(x)').eval
Substitute $a\to b+e_{2}$
Substitute $b\to c+f_{2}$
ok 100 loops, best of 5: 8.66 ms per loop 1000 loops, best of 5: 1.31 ms per loop
<bound method Poly.eval of Poly(x**2 + (sqrt(x)), x, sqrt(x), domain='ZZ')>
evaluate4=lambdify(fs,newformula)
evaluate5=lambda x:evaluate4(*x)
%timeit evaluate4(*(random(),random(),random()))
%timeit evaluate5((random(),random(),random()))from sympy import *
print('first attempt:',Poly('x^2+x+sqrt(x)'))
x=Symbol('x', positive=True)
print('second attempt:',Poly(x**2+x+sqrt(x)))
print('third attempt:',str(Poly({1:1,2:1,4:1},gens=S('sqrt(x)'))))(Poly({1:1,2:1,4:1},gens=S('sqrt(x)'))-Poly('x^2')).as_expr()$\displaystyle \sqrt{x} + x$
from sympy.solvers.solvers import unrad
unrad('sqrt(x)+x')(x*(1 - x), [])
formula=('x^(2/3)+x^(3/4)+sqrt(x+y)')
type((Poly(formula).gens[1]).args[0])==SymbolTrue
S('sqrt(x)').args(x, 1/2)
def _powr(formula):
if formula.func==Pow:
return formula.args
else:
return [formula,S('1')]
pol=Poly('x**2+y**2+sqrt(x+y)+x**(1/2)+x**(1/3)')
newgens={}
for gen in pol.gens:
base,pw=_powr(gen)
num,den=fraction(pw)
if (gen[0],num) in newgens:
newgens[gen[0]]=
else:
newgens[gen[0]]=gen[1][0;36m File [0;32m"<ipython-input-58-578d4067f943>"[0;36m, line [0;32m12[0m [0;31m newgens[gen[0]]=[0m [0m ^[0m [0;31mSyntaxError[0m[0;31m:[0m invalid syntax
isinstance(S('1'),Rational)True
6 in {3:4,5:6}False
from math import *
from sympy import *
gcd(S('a/b'),S('c/d'))$\displaystyle 1$
Poly('x^(2*pi)')$\displaystyle \operatorname{Poly}{\left( x^{2 \pi}, x^{\pi}, domain=\mathbb{Z} \right)}$
gcd(S('1/8'),S('1/4'))$\displaystyle \frac{1}{8}$
S('sqrt(2)*3*x').as_coeff_mul()(3, (x, sqrt(2)))
def _powr(formula):
if formula.func==Pow:
return formula.args
else:
return [formula,S('1')]
def reducegens(formula):
pol=Poly(formula)
newgens={}
ind={}
for gen in pol.gens:
base,pw=_powr(gen)
coef,_=pw.as_coeff_mul()
ml=pw/coef
if base**ml in newgens:
newgens[base**ml]=gcd(newgens[base**ml],coef)
else:
newgens[base**ml]=coef
ind[base**ml]=S('tmp'+str(len(ind)))
for gen in pol.gens:
base,pw=_powr(gen)
coef,_=pw.as_coeff_mul()
ml=pw/coef
pol=pol.replace(gen,ind[base**ml]**(coef/newgens[base**ml]))
newpol=Poly(pol.as_expr())
for gen in newgens:
newpol=newpol.replace(ind[gen],gen**newgens[gen])
return newpol
print(reducegens(S('x^2+x+sqrt(x)')))
print(reducegens(S('x**2+y**2+x**(1/2)+x**(1/3)')))
print(reducegens(S('sqrt(x)+sqrt(z)+sqrt(x*z)')))
print(reducegens(S('sqrt(x)+sqrt(z)+sqrt(x)*sqrt(z)')))
print(reducegens(S('sqrt(x+y)+sqrt(z+t)+sqrt((x+y)*(z+t))')))
print(reducegens(S('sqrt(2)+sqrt(3)+sqrt(6)')))Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ') Poly((x**(1/6))**12 + (x**(1/6))**3 + (x**(1/6))**2 + y**2, x**(1/6), y, domain='ZZ') Poly((sqrt(x)) + (sqrt(x*z)) + (sqrt(z)), sqrt(x), sqrt(x*z), sqrt(z), domain='ZZ') Poly((sqrt(x))*(sqrt(z)) + (sqrt(x)) + (sqrt(z)), sqrt(x), sqrt(z), domain='ZZ') Poly((sqrt(t + z)) + (sqrt(t*x + t*y + x*z + y*z)) + (sqrt(x + y)), sqrt(t + z), sqrt(t*x + t*y + x*z + y*z), sqrt(x + y), domain='ZZ') Poly((sqrt(2)) + (sqrt(3)) + (sqrt(6)), sqrt(2), sqrt(3), sqrt(6), domain='ZZ')
S('-2').as_coeff_mul()x=Symbol('x', positive=True)
y=Symbol('y', positive=True)
Poly(sqrt(x)+sqrt(y)+sqrt(x*y))sqrt(x*y)simplify(S('sqrt(x*y)-sqrt(x)*sqrt(y)'))x=symbols('x',positive=True)
ask(Q.positive(x))Poly('x^2+x')+Poly('-x+7')print(reducegens(S('sqrt(8)')))
sqrt(-1*-1)Poly('sqrt(2*x)+sqrt(3*x)+sqrt(6*x)')Poly('sqrt(2*pi)+sqrt(3*pi)')solve(S('sqrt(6*x)')-S('sqrt(2*y)'))Poly('sqrt(6)+sqrt(10)+sqrt(14)+sqrt(15)+sqrt(21)+sqrt(35)+sqrt()')from shiroindev import *
formula=S('(a+b-c)^2')
prove(makesubs(formula,'[c,oo]'))
prove(makesubs(formula,'[a,oo]',variables='c'))formula=S('(a+b-d)^2+(a+b-c)^2')
prove(makesubs(formula,'[a+b,oo],[a+b,oo]',variables='c,d'))
prove(makesubs(formula,'[0,a+b],[a+b,oo]',variables='c,d'))
prove(makesubs(formula,'[0,a+b],[0,a+b]',variables='c,d'))findvalues(formula)display(cyclize('((a-b)/c)^2'))
display(S('sqrt(8)')*cyclize('(a-b)/c'))formula=cyclize('((a-b)/c)^2-8**(1/2)*(a-b)/c')
display(Latex('Case $a\ge c\ge b$'))
formula1=makesubs(formula,'[c,oo],[b,oo]',variables='a,c,b')
prove(formula1)
display(Latex('Case $a\ge b\ge c$'))
formula2=makesubs(formula,'[b,oo],[c,oo]')
prove(formula2*4,values='2**(1/2),1,1')formula=cyclize('((a-b)/c)^2-8**(1/2)*(a-b)/c')
formula2=makesubs(formula,'[c,oo],[c,oo]')findvalues(formula2,values=[5,2,7])x,y=symbols('x,y')
ask(Q.positive(x**2+1),Q.real(x))formula=Sm('(a^2+b^2+c^2)^2- 3(a^3b+b^3c+c^3a)')
formula1=makesubs(formula,'[b,oo],[c,oo]',variables='a,b')
formula2=makesubs(formula,'[c,oo],[b,oo]',variables='a,c')
prove(formula2)
print(findvalues(formula1))
#prove(formula1,values='4*b*(1-b)-1,4*c*(1-c)-1,1')makesubs(Sm('a^2-2ab+bc-c^2+ca'),'[c,oo],[a,oo]',variables='b,c')
formula=S('sqrt(x*y)-sqrt(x)*sqrt(y)')
display(formula)
def assumeall(formula,**kwargs):
formula=S(formula)
fs=formula.free_symbols
for x in fs:
y=Symbol(str(x),**kwargs)
formula=formula.subs(x,y)
return formula
display(assumeall(formula,positive=True))$\displaystyle - \sqrt{x} \sqrt{y} + \sqrt{x y}$
$\displaystyle 0$
Poly(assumeall('x+sqrt(x)',positive=True))str(type(S('f1(x,y,z)')))=='f1'ReprPrinter?srepr(S('x^2+f(x)+f(x,y)+f()+sqrt(y)'))from sympy import Function,srepr,S
import re
f=Function('f')def allsymbols(formula):
formula=S(formula)
funcsymbols=[x[10:-2] for x in re.findall(r"Function\(\'.*?\'\)",srepr(formula))]
return set(funcsymbols)|set(map(str,formula.free_symbols))
def vargen(n):
x='abcdefghijklmnopqrstuvwxyz'[n%26]
if n>=26:
x+=str(n//26)
return x
allsymbols(S('f(5,7)+f(5)+g(8)+sqrt(x*y)'))
vargen(36)from sympy import *
from sympy.printing.repr import *
ReprPrinter??{5,7} | {8,7}[x[10:-2] for x in re.findall(r"Function\(\'.*?\'\)",srepr(S('x^2')))]S('f(5,7)+f(5)+g(8)+sqrt(x*y)')srepr(S('f(5,7)+f(5)+g(8)+sqrt(x*y)'))display(Latex('$'+str(S('z12345'))+'$'))'xyz'[-2::][0;31m---------------------------------------------------------------------------[0m [0;31mValueError[0m Traceback (most recent call last) [0;32m<ipython-input-68-247357fa9171>[0m in [0;36m<module>[0;34m()[0m [1;32m 1[0m [0;34m'xyz'[0m[0;34m[[0m[0;34m-[0m[0;36m2[0m[0;34m:[0m[0;34m:[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [0;32m----> 2[0;31m [0mint[0m[0;34m([0m[0;34m''[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m [0;31mValueError[0m: invalid literal for int() with base 10: ''
def sortkey(x):
x=str(x)
i=len(x)-1
while i>=0:
if x[i] not in '0123456789':
break
i-=1
if i==len(x)-1:
return (-1,x)
return (int(x[i+1:]),x[:i+1])sorted(S('x,y,z1,t2,qq1,x0'),key=sortkey)[x, y, x0, qq1, z1, t2]
S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2')
formula=S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2- (x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')
values=findvalues(formula)
values=[value/values[0] for value in values]
[value**2 for value in values]Optimization terminated successfully.
Current function value: 1.154701
Iterations: 85
Function evaluations: 152
[1.0, 3.000209557356011, 3.9998018341331227, 2.9997236704279775, 0.999903278712487]
formula1=S('(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2)/(x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')
formula1.subs(S('[[x_1,1],[x_2,sqrt(3)],[x_3,2],[x_4,sqrt(3)],[x_5,1]]'))$\displaystyle \frac{2 \sqrt{3}}{3}$
formula2=S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2- (2/sqrt(3))*(x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')prove(formula2,values='1,sqrt(3),2,sqrt(3),1')Substitute $x_{2}\to \sqrt{3} g$
Substitute $x_{3}\to 2 h$
Substitute $x_{4}\to \sqrt{3} i$
numerator: $3 g^{2} - 4 g h - 2 g x_{1} + 4 h^{2} - 4 h i + 3 i^{2} - 2 i x_{5} + x_{1}^{2} + x_{5}^{2}$
denominator: $1$
status: 0
From weighted AM-GM inequality:
$$4 g h \le 2 g^{2}+2 h^{2}$$
$$2 g x_{1} \le g^{2}+x_{1}^{2}$$
$$4 h i \le 2 h^{2}+2 i^{2}$$
$$2 i x_{5} \le i^{2}+x_{5}^{2}$$
$$ 0 \le 0 $$
The sum of all inequalities gives us a proof of the inequality.
0
Poly('a^(n+k)')$\displaystyle \operatorname{Poly}{\left( a^{k}a^{n}, a^{k}, a^{n}, domain=\mathbb{Z} \right)}$
S('a^{n+k}/{b^n}+{b^(n+k)/(c^n)+c^(n+k)/{a^n}-( a^k+b^k+c^k)')[0;31m---------------------------------------------------------------------------[0m [0;31mAttributeError[0m Traceback (most recent call last) [0;32m<ipython-input-13-ca88f614d264>[0m in [0;36m<module>[0;34m()[0m [0;32m----> 1[0;31m [0mS[0m[0;34m([0m[0;34m'a^{n+k}/{b^n}+{b^{n+k}}/{c^n}+{c^{n+k}}/{a^n}-( a^k+b^k+c^k)'[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/sympify.py[0m in [0;36msympify[0;34m(a, locals, convert_xor, strict, rational, evaluate)[0m [1;32m 382[0m [0;32mtry[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 383[0m [0ma[0m [0;34m=[0m [0ma[0m[0;34m.[0m[0mreplace[0m[0;34m([0m[0;34m'\n'[0m[0;34m,[0m [0;34m''[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 384[0;31m [0mexpr[0m [0;34m=[0m [0mparse_expr[0m[0;34m([0m[0ma[0m[0;34m,[0m [0mlocal_dict[0m[0;34m=[0m[0mlocals[0m[0;34m,[0m [0mtransformations[0m[0;34m=[0m[0mtransformations[0m[0;34m,[0m [0mevaluate[0m[0;34m=[0m[0mevaluate[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 385[0m [0;32mexcept[0m [0;34m([0m[0mTokenError[0m[0;34m,[0m [0mSyntaxError[0m[0;34m)[0m [0;32mas[0m [0mexc[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 386[0m [0;32mraise[0m [0mSympifyError[0m[0;34m([0m[0;34m'could not parse %r'[0m [0;34m%[0m [0ma[0m[0;34m,[0m [0mexc[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/parsing/sympy_parser.py[0m in [0;36mparse_expr[0;34m(s, local_dict, transformations, global_dict, evaluate)[0m [1;32m 964[0m [0mcode[0m [0;34m=[0m [0mcompile[0m[0;34m([0m[0mevaluateFalse[0m[0;34m([0m[0mcode[0m[0;34m)[0m[0;34m,[0m [0;34m'<string>'[0m[0;34m,[0m [0;34m'eval'[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 965[0m [0;34m[0m[0m [0;32m--> 966[0;31m [0;32mreturn[0m [0meval_expr[0m[0;34m([0m[0mcode[0m[0;34m,[0m [0mlocal_dict[0m[0;34m,[0m [0mglobal_dict[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 967[0m [0;34m[0m[0m [1;32m 968[0m [0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/parsing/sympy_parser.py[0m in [0;36meval_expr[0;34m(code, local_dict, global_dict)[0m [1;32m 877[0m """ [1;32m 878[0m expr = eval( [0;32m--> 879[0;31m code, global_dict, local_dict) # take local objects in preference [0m[1;32m 880[0m [0;34m[0m[0m [1;32m 881[0m [0;32mreturn[0m [0mexpr[0m[0;34m[0m[0;34m[0m[0m [0;32m<string>[0m in [0;36m<module>[0;34m()[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/expr.py[0m in [0;36m__pow__[0;34m(self, other, mod)[0m [1;32m 163[0m [0;32mdef[0m [0m__pow__[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mother[0m[0;34m,[0m [0mmod[0m[0;34m=[0m[0;32mNone[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 164[0m [0;32mif[0m [0mmod[0m [0;32mis[0m [0;32mNone[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 165[0;31m [0;32mreturn[0m [0mself[0m[0;34m.[0m[0m_pow[0m[0;34m([0m[0mother[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 166[0m [0;32mtry[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 167[0m [0m_self[0m[0;34m,[0m [0mother[0m[0;34m,[0m [0mmod[0m [0;34m=[0m [0mas_int[0m[0;34m([0m[0mself[0m[0;34m)[0m[0;34m,[0m [0mas_int[0m[0;34m([0m[0mother[0m[0;34m)[0m[0;34m,[0m [0mas_int[0m[0;34m([0m[0mmod[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/decorators.py[0m in [0;36m__sympifyit_wrapper[0;34m(a, b)[0m [1;32m 89[0m [0;32mif[0m [0;32mnot[0m [0mhasattr[0m[0;34m([0m[0mb[0m[0;34m,[0m [0;34m'_op_priority'[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 90[0m [0mb[0m [0;34m=[0m [0msympify[0m[0;34m([0m[0mb[0m[0;34m,[0m [0mstrict[0m[0;34m=[0m[0;32mTrue[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m---> 91[0;31m [0;32mreturn[0m [0mfunc[0m[0;34m([0m[0ma[0m[0;34m,[0m [0mb[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 92[0m [0;32mexcept[0m [0mSympifyError[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 93[0m [0;32mreturn[0m [0mretval[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/decorators.py[0m in [0;36mbinary_op_wrapper[0;34m(self, other)[0m [1;32m 127[0m [0;32mif[0m [0mf[0m [0;32mis[0m [0;32mnot[0m [0;32mNone[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 128[0m [0;32mreturn[0m [0mf[0m[0;34m([0m[0mself[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 129[0;31m [0;32mreturn[0m [0mfunc[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mother[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 130[0m [0;32mreturn[0m [0mbinary_op_wrapper[0m[0;34m[0m[0;34m[0m[0m [1;32m 131[0m [0;32mreturn[0m [0mpriority_decorator[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/expr.py[0m in [0;36m_pow[0;34m(self, other)[0m [1;32m 159[0m [0;34m@[0m[0mcall_highest_priority[0m[0;34m([0m[0;34m'__rpow__'[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 160[0m [0;32mdef[0m [0m_pow[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mother[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 161[0;31m [0;32mreturn[0m [0mPow[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mother[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 162[0m [0;34m[0m[0m [1;32m 163[0m [0;32mdef[0m [0m__pow__[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mother[0m[0;34m,[0m [0mmod[0m[0;34m=[0m[0;32mNone[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/cache.py[0m in [0;36mwrapper[0;34m(*args, **kwargs)[0m [1;32m 92[0m [0;32mdef[0m [0mwrapper[0m[0;34m([0m[0;34m*[0m[0margs[0m[0;34m,[0m [0;34m**[0m[0mkwargs[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 93[0m [0;32mtry[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m---> 94[0;31m [0mretval[0m [0;34m=[0m [0mcfunc[0m[0;34m([0m[0;34m*[0m[0margs[0m[0;34m,[0m [0;34m**[0m[0mkwargs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 95[0m [0;32mexcept[0m [0mTypeError[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 96[0m [0mretval[0m [0;34m=[0m [0mfunc[0m[0;34m([0m[0;34m*[0m[0margs[0m[0;34m,[0m [0;34m**[0m[0mkwargs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/power.py[0m in [0;36m__new__[0;34m(cls, b, e, evaluate)[0m [1;32m 274[0m [0;32mif[0m [0;32mnot[0m [0me[0m[0;34m.[0m[0mis_Atom[0m [0;32mand[0m [0mb[0m [0;32mis[0m [0;32mnot[0m [0mS[0m[0;34m.[0m[0mExp1[0m [0;32mand[0m [0;32mnot[0m [0misinstance[0m[0;34m([0m[0mb[0m[0;34m,[0m [0mexp_polar[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 275[0m [0;32mfrom[0m [0msympy[0m [0;32mimport[0m [0mnumer[0m[0;34m,[0m [0mdenom[0m[0;34m,[0m [0mlog[0m[0;34m,[0m [0msign[0m[0;34m,[0m [0mim[0m[0;34m,[0m [0mfactor_terms[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 276[0;31m [0mc[0m[0;34m,[0m [0mex[0m [0;34m=[0m [0mfactor_terms[0m[0;34m([0m[0me[0m[0;34m,[0m [0msign[0m[0;34m=[0m[0;32mFalse[0m[0;34m)[0m[0;34m.[0m[0mas_coeff_Mul[0m[0;34m([0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 277[0m [0mden[0m [0;34m=[0m [0mdenom[0m[0;34m([0m[0mex[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 278[0m [0;32mif[0m [0misinstance[0m[0;34m([0m[0mden[0m[0;34m,[0m [0mlog[0m[0;34m)[0m [0;32mand[0m [0mden[0m[0;34m.[0m[0margs[0m[0;34m[[0m[0;36m0[0m[0;34m][0m [0;34m==[0m [0mb[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;31mAttributeError[0m: 'FiniteSet' object has no attribute 'as_coeff_Mul'
import re
s=r"""99-my-name-is-John-Smith-6376827-%^-1-2-767980716"""
re.compile(r"^(.*?)-").search(str(s[::-1])).group(1)[::-1]'767980716'
from sympy import *
from sympy.parsing.latex import parse_latex
from shiroindev import *
shiro.seed=1
from IPython.display import Latex
shiro.display=lambda x:display(Latex(x))
formula=expand(parse_latex(r'(\frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}-( a^k+b^k+c^k))*a^nb^nc^n'))
newproof()
prove(formula)
prove(makesubs(formula,['k',oo],variables='n'))ANTLR runtime and generated code versions disagree: 4.8!=4.7.1 ANTLR runtime and generated code versions disagree: 4.8!=4.7.1
numerator: $a^{k} a^{2 n} c^{n} - a^{k} a^{n} b^{n} c^{n} + a^{n} b^{k} b^{2 n} - a^{n} b^{k} b^{n} c^{n} - a^{n} b^{n} c^{k} c^{n} + b^{n} c^{k} c^{2 n}$
denominator: $1$
status: 2
Program couldn't find any proof.
$$ a^{k} a^{n} b^{n} c^{n}+a^{n} b^{k} b^{n} c^{n}+a^{n} b^{n} c^{k} c^{n} \le a^{k} a^{2 n} c^{n}+a^{n} b^{k} b^{2 n}+b^{n} c^{k} c^{2 n} $$
Substitute $n\to d + k$
numerator: $a^{2 d} a^{3 k} c^{d} c^{k} - a^{d} a^{2 k} b^{d} b^{k} c^{d} c^{k} + a^{d} a^{k} b^{2 d} b^{3 k} - a^{d} a^{k} b^{d} b^{2 k} c^{d} c^{k} - a^{d} a^{k} b^{d} b^{k} c^{d} c^{2 k} + b^{d} b^{k} c^{2 d} c^{3 k}$
denominator: $1$
status: 2
Program couldn't find any proof.
$$ a^{d} a^{2 k} b^{d} b^{k} c^{d} c^{k}+a^{d} a^{k} b^{d} b^{2 k} c^{d} c^{k}+a^{d} a^{k} b^{d} b^{k} c^{d} c^{2 k} \le a^{2 d} a^{3 k} c^{d} c^{k}+a^{d} a^{k} b^{2 d} b^{3 k}+b^{d} b^{k} c^{2 d} c^{3 k} $$
2
Poly(dict(zip([(1,8),(2,7),(3,9)],[4,5,6])),gens=S('x,y'))$\displaystyle \operatorname{Poly}{\left( 6 x^{3}y^{9} + 5 x^{2}y^{7} + 4 xy^{8}, x, y, domain=\mathbb{Z} \right)}$
Poly({1:1,2:1,4:1},gens=S('sqrt(x)')).as_expr()$\displaystyle \sqrt{x} + x^{2} + x$
def _writ2(coef,fun,variables):
return latex(Poly(dict(zip(fun,coef)),gens=variables).as_expr())_writ2([7,8],[(1,2),(3,4)],S('x,y'))'8 x^{3} y^{4} + 7 x y^{2}'parse_latex(r' a^2+b^2\geq 2ab').rhsANTLR runtime and generated code versions disagree: 4.8!=4.7.1 ANTLR runtime and generated code versions disagree: 4.8!=4.7.1
$\displaystyle 2 a b$
parse_latex(r'2ab \geq a^2+b^2').rhsANTLR runtime and generated code versions disagree: 4.8!=4.7.1 ANTLR runtime and generated code versions disagree: 4.8!=4.7.1
$\displaystyle a^{2} + b^{2}$
fr=parse_latex(r'\sqrt{ab}+\sqrt{cd}+\sqrt{ef}\leq\sqrt{(a+c+e)(b+d+f)}')
prove(fr.rhs**2-fr.lhs**2)numerator: $a d + a f + b c + b e + c f + d e - 2 \sqrt{a b} \sqrt{c d} - 2 \sqrt{a b} \sqrt{e f} - 2 \sqrt{c d} \sqrt{e f}$
denominator: $1$
Poly((sqrt(a))**2*(sqrt(d))**2 + (sqrt(a))**2*(sqrt(f))**2 - 2*(sqrt(a))*(sqrt(b))*(sqrt(c))*(sqrt(d)) - 2*(sqrt(a))*(sqrt(b))*(sqrt(e))*(sqrt(f)) + (sqrt(b))**2*(sqrt(c))**2 + (sqrt(b))**2*(sqrt(e))**2 + (sqrt(c))**2*(sqrt(f))**2 - 2*(sqrt(c))*(sqrt(d))*(sqrt(e))*(sqrt(f)) + (sqrt(d))**2*(sqrt(e))**2, sqrt(a), sqrt(b), sqrt(c), sqrt(d), sqrt(e), sqrt(f), domain='ZZ') Poly((sqrt(a))**2*(sqrt(d))**2 + (sqrt(a))**2*(sqrt(f))**2 + (sqrt(b))**2*(sqrt(c))**2 + (sqrt(b))**2*(sqrt(e))**2 + (sqrt(c))**2*(sqrt(f))**2 + (sqrt(d))**2*(sqrt(e))**2, sqrt(a), sqrt(b), sqrt(c), sqrt(d), sqrt(e), sqrt(f), domain='QQ') Poly(2*(sqrt(a))*(sqrt(b))*(sqrt(c))*(sqrt(d)) + 2*(sqrt(a))*(sqrt(b))*(sqrt(e))*(sqrt(f)) + 2*(sqrt(c))*(sqrt(d))*(sqrt(e))*(sqrt(f)), sqrt(a), sqrt(b), sqrt(c), sqrt(d), sqrt(e), sqrt(f), domain='QQ')
status: 0
From weighted AM-GM inequality:
$$2 \sqrt{a} \sqrt{b} \sqrt{c} \sqrt{d} \le a d+b c$$
$$2 \sqrt{a} \sqrt{b} \sqrt{e} \sqrt{f} \le a f+b e$$
$$2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{f} \le c f+d e$$
$$ 0 \le 0 $$
The sum of all inequalities gives us a proof of the inequality.
0
assumeall(S('sqrt(x*y)-sqrt(x)*sqrt(y)'),positive=True)$\displaystyle 0$
p=reducegens('x+sqrt(x)')
q=S('x+sqrt(x)')
print(Poly(p,gens=[S('sqrt(x)'),S('sqrt(x)')]))
print(Poly(q,gens=S('sqrt(x)')))[0;31m---------------------------------------------------------------------------[0m [0;31mGeneratorsError[0m Traceback (most recent call last) [0;32m<ipython-input-31-b2d05a6575cd>[0m in [0;36m<module>[0;34m()[0m [1;32m 1[0m [0mp[0m[0;34m=[0m[0mreducegens[0m[0;34m([0m[0;34m'x+sqrt(x)'[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 2[0m [0mq[0m[0;34m=[0m[0mS[0m[0;34m([0m[0;34m'x+sqrt(x)'[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m----> 3[0;31m [0mprint[0m[0;34m([0m[0mPoly[0m[0;34m([0m[0mp[0m[0;34m,[0m[0mgens[0m[0;34m=[0m[0;34m[[0m[0mS[0m[0;34m([0m[0;34m'sqrt(x)'[0m[0;34m)[0m[0;34m,[0m[0mS[0m[0;34m([0m[0;34m'sqrt(x)'[0m[0;34m)[0m[0;34m][0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 4[0m [0mprint[0m[0;34m([0m[0mPoly[0m[0;34m([0m[0mq[0m[0;34m,[0m[0mgens[0m[0;34m=[0m[0mS[0m[0;34m([0m[0;34m'sqrt(x)'[0m[0;34m)[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polytools.py[0m in [0;36m__new__[0;34m(cls, rep, *gens, **args)[0m [1;32m 107[0m [0;32mdef[0m [0m__new__[0m[0;34m([0m[0mcls[0m[0;34m,[0m [0mrep[0m[0;34m,[0m [0;34m*[0m[0mgens[0m[0;34m,[0m [0;34m**[0m[0margs[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 108[0m [0;34m"""Create a new polynomial instance out of something useful. """[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 109[0;31m [0mopt[0m [0;34m=[0m [0moptions[0m[0;34m.[0m[0mbuild_options[0m[0;34m([0m[0mgens[0m[0;34m,[0m [0margs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 110[0m [0;34m[0m[0m [1;32m 111[0m [0;32mif[0m [0;34m'order'[0m [0;32min[0m [0mopt[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py[0m in [0;36mbuild_options[0;34m(gens, args)[0m [1;32m 729[0m [0;34m[0m[0m [1;32m 730[0m [0;32mif[0m [0mlen[0m[0;34m([0m[0margs[0m[0;34m)[0m [0;34m!=[0m [0;36m1[0m [0;32mor[0m [0;34m'opt'[0m [0;32mnot[0m [0;32min[0m [0margs[0m [0;32mor[0m [0mgens[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 731[0;31m [0;32mreturn[0m [0mOptions[0m[0;34m([0m[0mgens[0m[0;34m,[0m [0margs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 732[0m [0;32melse[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 733[0m [0;32mreturn[0m [0margs[0m[0;34m[[0m[0;34m'opt'[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py[0m in [0;36m__init__[0;34m(self, gens, args, flags, strict)[0m [1;32m 152[0m [0mself[0m[0;34m[[0m[0moption[0m[0;34m][0m [0;34m=[0m [0mcls[0m[0;34m.[0m[0mpreprocess[0m[0;34m([0m[0mvalue[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 153[0m [0;34m[0m[0m [0;32m--> 154[0;31m [0mpreprocess_options[0m[0;34m([0m[0margs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 155[0m [0;34m[0m[0m [1;32m 156[0m [0;32mfor[0m [0mkey[0m[0;34m,[0m [0mvalue[0m [0;32min[0m [0mdict[0m[0;34m([0m[0mdefaults[0m[0;34m)[0m[0;34m.[0m[0mitems[0m[0;34m([0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py[0m in [0;36mpreprocess_options[0;34m(args)[0m [1;32m 150[0m [0;34m[0m[0m [1;32m 151[0m [0;32mif[0m [0mvalue[0m [0;32mis[0m [0;32mnot[0m [0;32mNone[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 152[0;31m [0mself[0m[0;34m[[0m[0moption[0m[0;34m][0m [0;34m=[0m [0mcls[0m[0;34m.[0m[0mpreprocess[0m[0;34m([0m[0mvalue[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 153[0m [0;34m[0m[0m [1;32m 154[0m [0mpreprocess_options[0m[0;34m([0m[0margs[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py[0m in [0;36mpreprocess[0;34m(cls, gens)[0m [1;32m 289[0m [0mgens[0m [0;34m=[0m [0;34m([0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 290[0m [0;32melif[0m [0mhas_dups[0m[0;34m([0m[0mgens[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 291[0;31m [0;32mraise[0m [0mGeneratorsError[0m[0;34m([0m[0;34m"duplicated generators: %s"[0m [0;34m%[0m [0mstr[0m[0;34m([0m[0mgens[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 292[0m [0;32melif[0m [0many[0m[0;34m([0m[0mgen[0m[0;34m.[0m[0mis_commutative[0m [0;32mis[0m [0;32mFalse[0m [0;32mfor[0m [0mgen[0m [0;32min[0m [0mgens[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 293[0m [0;32mraise[0m [0mGeneratorsError[0m[0;34m([0m[0;34m"non-commutative generators: %s"[0m [0;34m%[0m [0mstr[0m[0;34m([0m[0mgens[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;31mGeneratorsError[0m: duplicated generators: [sqrt(x), sqrt(x)]
from importlib import reload
from sympy import *
import shiroindev
reload(shiroindev)
from shiroindev import *
from sympy.parsing.latex import parse_latex
from itertools import permutations, combinations
shiro.seed=1
from IPython.display import Latex
shiro.display=lambda x:display(Latex(x))Poly('sqrt(x)')*Poly('sqrt(x)')$\displaystyle \operatorname{Poly}{\left( x, \sqrt{x}, domain=\mathbb{Z} \right)}$
Poly('x')-Poly('x').abs()$\displaystyle \operatorname{Poly}{\left( 0, x, domain=\mathbb{Z} \right)}$
fr=Poly('x-y')
(fr+fr.abs())*(S('1/2'))$\displaystyle \operatorname{Poly}{\left( x, x, y, domain=\mathbb{Q} \right)}$
from itertools import permutations
t5=[]
for ineq in ineqs2:
u=0
for vars in permutations(ineq.free_symbols):
ineqp=makesubs(ineq,list(zip(vars[1:],[oo]*len(vars[1:]))),variables=vars[:-1])
u=max(u,prove(ineqp))
print(u,end=',')
t5+=[u]
Counter(t5)from statsmodels.stats.contingency_tables import mcnemar
print(mcnemar([[101,121],[59,33]]))pvalue 4.43444926375551e-06 statistic 59.0
nsimplify?shiro.display=lambda x:display(Latex(x))
newproof()
fr=makesubs(ineqs2[15],'[c,oo],[a,oo]',variables='b,c')
display(fr)
fr2=fractioncancel(fr.subs(S('[[e,(sqrt(5)-1)/2*f]]')))[0]
display(fr2)
prove(makesubs(fr2,'11/5,9/4',variables=[S('sqrt(5)')]))def makesubs(formula,intervals,values=None,variables=None,numden=False):
#This function generates a new formula which satisfies this condition:
#for all positive variables new formula is nonnegative iff
#for all variables in corresponding intervals old formula is nonnegative
formula=S(formula)
addsymbols(formula)
intervals=_smakeiterable2(intervals)
if variables: variables=_smakeiterable(variables)
else: variables=sorted(formula.free_symbols,key=str)
if values!=None:
values=_smakeiterable(values)
equations=[var-value for var,value in zip(variables,values)]
else:
equations=[]
newvars=[]
usedvars=set()
for var,interval in zip(variables,intervals):
end1,end2=interval
z=newvar()
newvars+=[z]
usedvars|={z}
if (end1.free_symbols|end2.free_symbols)&usedvars:
shiro.warning(shiro.translation[
'Warning: intervals contain backwards dependencies. Consider changing order of variables'])
if end1 in {S('-oo'),S('oo')}:
end1,end2=end2,end1
if {end1,end2}=={S('-oo'),S('oo')}:
sub1=sub2=(z-1/z)
elif end2==S('oo'):
sub1=sub2=(end1+z)
elif end2==S('-oo'):
sub1=sub2=end1-z
else:
sub1=end2+(end1-end2)/z
sub2=end2+(end1-end2)/(1+z)
formula=formula.subs(var,sub1)
shiro.display(shiro.translation['Substitute']+" $"+latex(var)+'\\\\to '+latex(sub2)+'$')
equations=[equation.subs(var,sub1) for equation in equations]
num,den=fractioncancel(formula)
for var,interval in zip(newvars,intervals):
if {interval[0],interval[1]} & {S('oo'),S('-oo')}==set():
num=num.subs(var,var+1)
den=den.subs(var,var+1)
equations=[equation.subs(var,var+1) for equation in equations]
if values:
values=ssolve(equations,newvars)
if len(values):
values=values[0]
num,den=expand(num),expand(den)
#shiro.display(shiro.translation["Formula after substitution:"],"$$",latex(num/den),'$$')
if values and numden:
return num,den,values
elif values:
return num/den,values
elif numden:
return num,den
else:
return num/denx={5,7,8}
x|={3}
x{3, 5, 7, 8}from shiroindev import *
S('x+7').free_symbols{x}bool(set())False
from importlib import reload
newproof()
import shiroindev
reload(shiroindev)
from shiroindev import *
shiro.display=lambda x:display(Latex(x))
formula=Sm('xy/z+yz/x+zx/y-1').subs('z',S('1-x-y'))
newformula,values=makesubs(formula,'[0,1-y],[0,1]','1/3,1/3')
prove(newformula,values)Substitute $x\to - y + 1 + \frac{y - 1}{a + 1}$
Substitute $y\to 1 - \frac{1}{b + 1}$
Substitute $b\to \frac{c}{2}$
numerator: $a^{4} c^{2} + a^{3} c^{2} - 2 a^{3} c - 4 a^{2} c + 4 a^{2} + a c^{2} - 2 a c + c^{2}$
denominator: $a^{3} c^{2} + 2 a^{3} c + 2 a^{2} c^{2} + 4 a^{2} c + a c^{2} + 2 a c$
status: 0
From weighted AM-GM inequality:
$$2 a^{3} c \le a^{4} c^{2}+a^{2}$$
$$4 a^{2} c \le a^{3} c^{2}+2 a^{2}+a c^{2}$$
$$2 a c \le a^{2}+c^{2}$$
$$ 0 \le 0 $$
The sum of all inequalities gives us a proof of the inequality.
0
def findvalues(formula,values=None,variables=None,**kwargs):
"""finds a candidate for parameter "values" in "prove" function
blabla"""
formula=S(formula)
addsymbols(formula)
num,den=fractioncancel(formula)
if variables==None:
variables=sorted(formula.free_symbols,key=str)
num=num.subs(zip(variables,list(map(lambda x:x**2,variables))))
num=Poly(num)
newformula=S((num.abs()+num)/(num.abs()-num))
f=lambdify(variables,newformula)
f2=lambda x:f(*x)
if values==None:
values=[1.0]*len(variables)
else:
values=S(values)
tup=tuple(fmin(f2,values,**kwargs))
return tuple([x*x for x in tup])findvalues(Sm('(x^2-4x+4)/y'))Optimization terminated successfully.
Current function value: 1.000000
Iterations: 51
Function evaluations: 109
(1.9999999989778676, 0.6689493176765962)
shiroindev._remzero?import shiroindev
reload(shiroindev)
from shiroindev import *str(cyclize('((a-b)/c)^2-2*sqrt(2)*(a-b)/c'))'-2*sqrt(2)*(a - b)/c + (a - b)**2/c**2 - 2*sqrt(2)*(-a + c)/b + (-a + c)**2/b**2 - 2*sqrt(2)*(b - c)/a + (b - c)**2/a**2'
newproof()
formula=Sm('-(3a+2b+c)(2a^3+3b^2+6c+1)+(4a+4b+4c)(a^4+b^3+c^2+3)')
prove(makesubs(formula,'[1,oo],[1,oo],[1,oo]'))
prove(makesubs(formula,'[1,0],[1,oo],[1,oo]'))
prove(makesubs(formula,'[1,oo],[1,0],[1,oo]'))
prove(makesubs(formula,'[1,0],[1,0],[1,oo]'))
prove(makesubs(formula,'[1,oo],[1,oo],[1,0]'))
prove(makesubs(formula,'[1,0],[1,oo],[1,0]'))
prove(makesubs(formula,'[1,oo],[1,0],[1,0]'))
prove(makesubs(formula,'[1,0],[1,0],[1,0]'))
Substitute $a\to d + 1$
Substitute $b\to e + 1$
Substitute $c\to f + 1$
numerator: $4 d^{5} + 4 d^{4} e + 4 d^{4} f + 22 d^{4} + 12 d^{3} e + 14 d^{3} f + 42 d^{3} + 12 d^{2} e + 18 d^{2} f + 34 d^{2} + 4 d e^{3} + 3 d e^{2} - 2 d e + 4 d f^{2} + 4 e^{4} + 4 e^{3} f + 18 e^{3} + 9 e^{2} f + 18 e^{2} + 4 e f^{2} + 2 e f + 4 f^{3} + 14 f^{2}$
denominator: $1$
status: 0
From weighted AM-GM inequality:
$$2 d e \le d^{2}+e^{2}$$
$$ 0 \le 4 d^{5}+4 d^{4} e+4 d^{4} f+22 d^{4}+12 d^{3} e+14 d^{3} f+42 d^{3}+12 d^{2} e+18 d^{2} f+33 d^{2}+4 d e^{3}+3 d e^{2}+4 d f^{2}+4 e^{4}+4 e^{3} f+18 e^{3}+9 e^{2} f+17 e^{2}+4 e f^{2}+2 e f+4 f^{3}+14 f^{2} $$
The sum of all inequalities gives us a proof of the inequality.
Substitute $a\to \frac{1}{g + 1}$
Substitute $b\to h + 1$
Substitute $c\to i + 1$
numerator: $4 g^{5} h^{4} + 4 g^{5} h^{3} i + 14 g^{5} h^{3} + 9 g^{5} h^{2} i + 15 g^{5} h^{2} + 4 g^{5} h i^{2} + 2 g^{5} h i + 6 g^{5} h + 4 g^{5} i^{3} + 10 g^{5} i^{2} + 8 g^{5} i + 10 g^{5} + 20 g^{4} h^{4} + 20 g^{4} h^{3} i + 74 g^{4} h^{3} + 45 g^{4} h^{2} i + 78 g^{4} h^{2} + 20 g^{4} h i^{2} + 10 g^{4} h i + 24 g^{4} h + 20 g^{4} i^{3} + 54 g^{4} i^{2} + 30 g^{4} i + 40 g^{4} + 40 g^{3} h^{4} + 40 g^{3} h^{3} i + 156 g^{3} h^{3} + 90 g^{3} h^{2} i + 162 g^{3} h^{2} + 40 g^{3} h i^{2} + 20 g^{3} h i + 36 g^{3} h + 40 g^{3} i^{3} + 116 g^{3} i^{2} + 40 g^{3} i + 60 g^{3} + 40 g^{2} h^{4} + 40 g^{2} h^{3} i + 164 g^{2} h^{3} + 90 g^{2} h^{2} i + 168 g^{2} h^{2} + 40 g^{2} h i^{2} + 20 g^{2} h i + 20 g^{2} h + 40 g^{2} i^{3} + 124 g^{2} i^{2} + 18 g^{2} i + 34 g^{2} + 20 g h^{4} + 20 g h^{3} i + 86 g h^{3} + 45 g h^{2} i + 87 g h^{2} + 20 g h i^{2} + 10 g h i + 2 g h + 20 g i^{3} + 66 g i^{2} + 4 h^{4} + 4 h^{3} i + 18 h^{3} + 9 h^{2} i + 18 h^{2} + 4 h i^{2} + 2 h i + 4 i^{3} + 14 i^{2}$
denominator: $g^{5} + 5 g^{4} + 10 g^{3} + 10 g^{2} + 5 g + 1$
status: 0
$$ 0 \le 4 g^{5} h^{4}+4 g^{5} h^{3} i+14 g^{5} h^{3}+9 g^{5} h^{2} i+15 g^{5} h^{2}+4 g^{5} h i^{2}+2 g^{5} h i+6 g^{5} h+4 g^{5} i^{3}+10 g^{5} i^{2}+8 g^{5} i+10 g^{5}+20 g^{4} h^{4}+20 g^{4} h^{3} i+74 g^{4} h^{3}+45 g^{4} h^{2} i+78 g^{4} h^{2}+20 g^{4} h i^{2}+10 g^{4} h i+24 g^{4} h+20 g^{4} i^{3}+54 g^{4} i^{2}+30 g^{4} i+40 g^{4}+40 g^{3} h^{4}+40 g^{3} h^{3} i+156 g^{3} h^{3}+90 g^{3} h^{2} i+162 g^{3} h^{2}+40 g^{3} h i^{2}+20 g^{3} h i+36 g^{3} h+40 g^{3} i^{3}+116 g^{3} i^{2}+40 g^{3} i+60 g^{3}+40 g^{2} h^{4}+40 g^{2} h^{3} i+164 g^{2} h^{3}+90 g^{2} h^{2} i+168 g^{2} h^{2}+40 g^{2} h i^{2}+20 g^{2} h i+20 g^{2} h+40 g^{2} i^{3}+124 g^{2} i^{2}+18 g^{2} i+34 g^{2}+20 g h^{4}+20 g h^{3} i+86 g h^{3}+45 g h^{2} i+87 g h^{2}+20 g h i^{2}+10 g h i+2 g h+20 g i^{3}+66 g i^{2}+4 h^{4}+4 h^{3} i+18 h^{3}+9 h^{2} i+18 h^{2}+4 h i^{2}+2 h i+4 i^{3}+14 i^{2} $$
The sum of all inequalities gives us a proof of the inequality.
Substitute $a\to j + 1$
Substitute $b\to \frac{1}{k + 1}$
Substitute $c\to l + 1$
numerator: $4 j^{5} k^{4} + 16 j^{5} k^{3} + 24 j^{5} k^{2} + 16 j^{5} k + 4 j^{5} + 4 j^{4} k^{4} l + 18 j^{4} k^{4} + 16 j^{4} k^{3} l + 76 j^{4} k^{3} + 24 j^{4} k^{2} l + 120 j^{4} k^{2} + 16 j^{4} k l + 84 j^{4} k + 4 j^{4} l + 22 j^{4} + 14 j^{3} k^{4} l + 30 j^{3} k^{4} + 56 j^{3} k^{3} l + 132 j^{3} k^{3} + 84 j^{3} k^{2} l + 216 j^{3} k^{2} + 56 j^{3} k l + 156 j^{3} k + 14 j^{3} l + 42 j^{3} + 18 j^{2} k^{4} l + 22 j^{2} k^{4} + 72 j^{2} k^{3} l + 100 j^{2} k^{3} + 108 j^{2} k^{2} l + 168 j^{2} k^{2} + 72 j^{2} k l + 124 j^{2} k + 18 j^{2} l + 34 j^{2} + 4 j k^{4} l^{2} + j k^{4} + 16 j k^{3} l^{2} + 8 j k^{3} + 24 j k^{2} l^{2} + 9 j k^{2} + 16 j k l^{2} + 2 j k + 4 j l^{2} + 4 k^{4} l^{3} + 10 k^{4} l^{2} + 3 k^{4} l + 4 k^{4} + 16 k^{3} l^{3} + 44 k^{3} l^{2} + 8 k^{3} l + 18 k^{3} + 24 k^{2} l^{3} + 72 k^{2} l^{2} + 3 k^{2} l + 18 k^{2} + 16 k l^{3} + 52 k l^{2} - 2 k l + 4 l^{3} + 14 l^{2}$
denominator: $k^{4} + 4 k^{3} + 6 k^{2} + 4 k + 1$
status: 0
From weighted AM-GM inequality:
$$2 k l \le k^{2}+l^{2}$$
$$ 0 \le 4 j^{5} k^{4}+16 j^{5} k^{3}+24 j^{5} k^{2}+16 j^{5} k+4 j^{5}+4 j^{4} k^{4} l+18 j^{4} k^{4}+16 j^{4} k^{3} l+76 j^{4} k^{3}+24 j^{4} k^{2} l+120 j^{4} k^{2}+16 j^{4} k l+84 j^{4} k+4 j^{4} l+22 j^{4}+14 j^{3} k^{4} l+30 j^{3} k^{4}+56 j^{3} k^{3} l+132 j^{3} k^{3}+84 j^{3} k^{2} l+216 j^{3} k^{2}+56 j^{3} k l+156 j^{3} k+14 j^{3} l+42 j^{3}+18 j^{2} k^{4} l+22 j^{2} k^{4}+72 j^{2} k^{3} l+100 j^{2} k^{3}+108 j^{2} k^{2} l+168 j^{2} k^{2}+72 j^{2} k l+124 j^{2} k+18 j^{2} l+34 j^{2}+4 j k^{4} l^{2}+j k^{4}+16 j k^{3} l^{2}+8 j k^{3}+24 j k^{2} l^{2}+9 j k^{2}+16 j k l^{2}+2 j k+4 j l^{2}+4 k^{4} l^{3}+10 k^{4} l^{2}+3 k^{4} l+4 k^{4}+16 k^{3} l^{3}+44 k^{3} l^{2}+8 k^{3} l+18 k^{3}+24 k^{2} l^{3}+72 k^{2} l^{2}+3 k^{2} l+17 k^{2}+16 k l^{3}+52 k l^{2}+4 l^{3}+13 l^{2} $$
The sum of all inequalities gives us a proof of the inequality.
Substitute $a\to \frac{1}{m + 1}$
Substitute $b\to \frac{1}{n + 1}$
Substitute $c\to o + 1$
numerator: $4 m^{5} n^{4} o^{3} + 6 m^{5} n^{4} o^{2} + 11 m^{5} n^{4} o + 9 m^{5} n^{4} + 16 m^{5} n^{3} o^{3} + 28 m^{5} n^{3} o^{2} + 40 m^{5} n^{3} o + 38 m^{5} n^{3} + 24 m^{5} n^{2} o^{3} + 48 m^{5} n^{2} o^{2} + 51 m^{5} n^{2} o + 57 m^{5} n^{2} + 16 m^{5} n o^{3} + 36 m^{5} n o^{2} + 30 m^{5} n o + 34 m^{5} n + 4 m^{5} o^{3} + 10 m^{5} o^{2} + 8 m^{5} o + 10 m^{5} + 20 m^{4} n^{4} o^{3} + 34 m^{4} n^{4} o^{2} + 45 m^{4} n^{4} o + 40 m^{4} n^{4} + 80 m^{4} n^{3} o^{3} + 156 m^{4} n^{3} o^{2} + 160 m^{4} n^{3} o + 170 m^{4} n^{3} + 120 m^{4} n^{2} o^{3} + 264 m^{4} n^{2} o^{2} + 195 m^{4} n^{2} o + 246 m^{4} n^{2} + 80 m^{4} n o^{3} + 196 m^{4} n o^{2} + 110 m^{4} n o + 136 m^{4} n + 20 m^{4} o^{3} + 54 m^{4} o^{2} + 30 m^{4} o + 40 m^{4} + 40 m^{3} n^{4} o^{3} + 76 m^{3} n^{4} o^{2} + 70 m^{3} n^{4} o + 70 m^{3} n^{4} + 160 m^{3} n^{3} o^{3} + 344 m^{3} n^{3} o^{2} + 240 m^{3} n^{3} o + 300 m^{3} n^{3} + 240 m^{3} n^{2} o^{3} + 576 m^{3} n^{2} o^{2} + 270 m^{3} n^{2} o + 414 m^{3} n^{2} + 160 m^{3} n o^{3} + 424 m^{3} n o^{2} + 140 m^{3} n o + 204 m^{3} n + 40 m^{3} o^{3} + 116 m^{3} o^{2} + 40 m^{3} o + 60 m^{3} + 40 m^{2} n^{4} o^{3} + 84 m^{2} n^{4} o^{2} + 48 m^{2} n^{4} o + 58 m^{2} n^{4} + 160 m^{2} n^{3} o^{3} + 376 m^{2} n^{3} o^{2} + 152 m^{2} n^{3} o + 248 m^{2} n^{3} + 240 m^{2} n^{2} o^{3} + 624 m^{2} n^{2} o^{2} + 138 m^{2} n^{2} o + 312 m^{2} n^{2} + 160 m^{2} n o^{3} + 456 m^{2} n o^{2} + 52 m^{2} n o + 116 m^{2} n + 40 m^{2} o^{3} + 124 m^{2} o^{2} + 18 m^{2} o + 34 m^{2} + 20 m n^{4} o^{3} + 46 m n^{4} o^{2} + 15 m n^{4} o + 19 m n^{4} + 80 m n^{3} o^{3} + 204 m n^{3} o^{2} + 40 m n^{3} o + 82 m n^{3} + 120 m n^{2} o^{3} + 336 m n^{2} o^{2} + 15 m n^{2} o + 81 m n^{2} + 80 m n o^{3} + 244 m n o^{2} - 10 m n o - 2 m n + 20 m o^{3} + 66 m o^{2} + 4 n^{4} o^{3} + 10 n^{4} o^{2} + 3 n^{4} o + 4 n^{4} + 16 n^{3} o^{3} + 44 n^{3} o^{2} + 8 n^{3} o + 18 n^{3} + 24 n^{2} o^{3} + 72 n^{2} o^{2} + 3 n^{2} o + 18 n^{2} + 16 n o^{3} + 52 n o^{2} - 2 n o + 4 o^{3} + 14 o^{2}$
denominator: $m^{5} n^{4} + 4 m^{5} n^{3} + 6 m^{5} n^{2} + 4 m^{5} n + m^{5} + 5 m^{4} n^{4} + 20 m^{4} n^{3} + 30 m^{4} n^{2} + 20 m^{4} n + 5 m^{4} + 10 m^{3} n^{4} + 40 m^{3} n^{3} + 60 m^{3} n^{2} + 40 m^{3} n + 10 m^{3} + 10 m^{2} n^{4} + 40 m^{2} n^{3} + 60 m^{2} n^{2} + 40 m^{2} n + 10 m^{2} + 5 m n^{4} + 20 m n^{3} + 30 m n^{2} + 20 m n + 5 m + n^{4} + 4 n^{3} + 6 n^{2} + 4 n + 1$
status: 0
From weighted AM-GM inequality:
$$2 m n \le m^{2}+n^{2}$$
$$2 n o \le n^{2}+o^{2}$$
$$10 m n o \le 2 m^{2}+2 m n^{2} o+4 m o^{2}+2 n^{3}$$
$$ 0 \le 4 m^{5} n^{4} o^{3}+6 m^{5} n^{4} o^{2}+11 m^{5} n^{4} o+9 m^{5} n^{4}+16 m^{5} n^{3} o^{3}+28 m^{5} n^{3} o^{2}+40 m^{5} n^{3} o+38 m^{5} n^{3}+24 m^{5} n^{2} o^{3}+48 m^{5} n^{2} o^{2}+51 m^{5} n^{2} o+57 m^{5} n^{2}+16 m^{5} n o^{3}+36 m^{5} n o^{2}+30 m^{5} n o+34 m^{5} n+4 m^{5} o^{3}+10 m^{5} o^{2}+8 m^{5} o+10 m^{5}+20 m^{4} n^{4} o^{3}+34 m^{4} n^{4} o^{2}+45 m^{4} n^{4} o+40 m^{4} n^{4}+80 m^{4} n^{3} o^{3}+156 m^{4} n^{3} o^{2}+160 m^{4} n^{3} o+170 m^{4} n^{3}+120 m^{4} n^{2} o^{3}+264 m^{4} n^{2} o^{2}+195 m^{4} n^{2} o+246 m^{4} n^{2}+80 m^{4} n o^{3}+196 m^{4} n o^{2}+110 m^{4} n o+136 m^{4} n+20 m^{4} o^{3}+54 m^{4} o^{2}+30 m^{4} o+40 m^{4}+40 m^{3} n^{4} o^{3}+76 m^{3} n^{4} o^{2}+70 m^{3} n^{4} o+70 m^{3} n^{4}+160 m^{3} n^{3} o^{3}+344 m^{3} n^{3} o^{2}+240 m^{3} n^{3} o+300 m^{3} n^{3}+240 m^{3} n^{2} o^{3}+576 m^{3} n^{2} o^{2}+270 m^{3} n^{2} o+414 m^{3} n^{2}+160 m^{3} n o^{3}+424 m^{3} n o^{2}+140 m^{3} n o+204 m^{3} n+40 m^{3} o^{3}+116 m^{3} o^{2}+40 m^{3} o+60 m^{3}+40 m^{2} n^{4} o^{3}+84 m^{2} n^{4} o^{2}+48 m^{2} n^{4} o+58 m^{2} n^{4}+160 m^{2} n^{3} o^{3}+376 m^{2} n^{3} o^{2}+152 m^{2} n^{3} o+248 m^{2} n^{3}+240 m^{2} n^{2} o^{3}+624 m^{2} n^{2} o^{2}+138 m^{2} n^{2} o+312 m^{2} n^{2}+160 m^{2} n o^{3}+456 m^{2} n o^{2}+52 m^{2} n o+116 m^{2} n+40 m^{2} o^{3}+124 m^{2} o^{2}+18 m^{2} o+31 m^{2}+20 m n^{4} o^{3}+46 m n^{4} o^{2}+15 m n^{4} o+19 m n^{4}+80 m n^{3} o^{3}+204 m n^{3} o^{2}+40 m n^{3} o+82 m n^{3}+120 m n^{2} o^{3}+336 m n^{2} o^{2}+13 m n^{2} o+81 m n^{2}+80 m n o^{3}+244 m n o^{2}+20 m o^{3}+62 m o^{2}+4 n^{4} o^{3}+10 n^{4} o^{2}+3 n^{4} o+4 n^{4}+16 n^{3} o^{3}+44 n^{3} o^{2}+8 n^{3} o+16 n^{3}+24 n^{2} o^{3}+72 n^{2} o^{2}+3 n^{2} o+16 n^{2}+16 n o^{3}+52 n o^{2}+4 o^{3}+13 o^{2} $$
The sum of all inequalities gives us a proof of the inequality.
Substitute $a\to p + 1$
Substitute $b\to q + 1$
Substitute $c\to \frac{1}{r + 1}$
numerator: $4 p^{5} r^{3} + 12 p^{5} r^{2} + 12 p^{5} r + 4 p^{5} + 4 p^{4} q r^{3} + 12 p^{4} q r^{2} + 12 p^{4} q r + 4 p^{4} q + 18 p^{4} r^{3} + 58 p^{4} r^{2} + 62 p^{4} r + 22 p^{4} + 12 p^{3} q r^{3} + 36 p^{3} q r^{2} + 36 p^{3} q r + 12 p^{3} q + 28 p^{3} r^{3} + 98 p^{3} r^{2} + 112 p^{3} r + 42 p^{3} + 12 p^{2} q r^{3} + 36 p^{2} q r^{2} + 36 p^{2} q r + 12 p^{2} q + 16 p^{2} r^{3} + 66 p^{2} r^{2} + 84 p^{2} r + 34 p^{2} + 4 p q^{3} r^{3} + 12 p q^{3} r^{2} + 12 p q^{3} r + 4 p q^{3} + 3 p q^{2} r^{3} + 9 p q^{2} r^{2} + 9 p q^{2} r + 3 p q^{2} - 2 p q r^{3} - 6 p q r^{2} - 6 p q r - 2 p q + 4 p r^{3} + 4 p r^{2} + 4 q^{4} r^{3} + 12 q^{4} r^{2} + 12 q^{4} r + 4 q^{4} + 14 q^{3} r^{3} + 46 q^{3} r^{2} + 50 q^{3} r + 18 q^{3} + 9 q^{2} r^{3} + 36 q^{2} r^{2} + 45 q^{2} r + 18 q^{2} + 2 q r^{3} - 2 q r + 10 r^{3} + 14 r^{2}$
denominator: $r^{3} + 3 r^{2} + 3 r + 1$
status: 0
From weighted AM-GM inequality:
$$2 p q r^{3} \le p^{2} q r^{3}+q r^{3}$$
$$2 p q \le p^{2}+q^{2}$$
$$2 q r \le q^{2}+r^{2}$$
$$6 p q r \le 2 p^{3} q+2 q^{2} r+2 r^{2}$$
$$6 p q r^{2} \le 2 p q^{2} r^{3}+p q^{2}+3 p r^{2}$$
$$ 0 \le 4 p^{5} r^{3}+12 p^{5} r^{2}+12 p^{5} r+4 p^{5}+4 p^{4} q r^{3}+12 p^{4} q r^{2}+12 p^{4} q r+4 p^{4} q+18 p^{4} r^{3}+58 p^{4} r^{2}+62 p^{4} r+22 p^{4}+12 p^{3} q r^{3}+36 p^{3} q r^{2}+36 p^{3} q r+10 p^{3} q+28 p^{3} r^{3}+98 p^{3} r^{2}+112 p^{3} r+42 p^{3}+11 p^{2} q r^{3}+36 p^{2} q r^{2}+36 p^{2} q r+12 p^{2} q+16 p^{2} r^{3}+66 p^{2} r^{2}+84 p^{2} r+33 p^{2}+4 p q^{3} r^{3}+12 p q^{3} r^{2}+12 p q^{3} r+4 p q^{3}+p q^{2} r^{3}+9 p q^{2} r^{2}+9 p q^{2} r+2 p q^{2}+4 p r^{3}+p r^{2}+4 q^{4} r^{3}+12 q^{4} r^{2}+12 q^{4} r+4 q^{4}+14 q^{3} r^{3}+46 q^{3} r^{2}+50 q^{3} r+18 q^{3}+9 q^{2} r^{3}+36 q^{2} r^{2}+43 q^{2} r+16 q^{2}+q r^{3}+10 r^{3}+11 r^{2} $$
The sum of all inequalities gives us a proof of the inequality.
Substitute $a\to \frac{1}{s + 1}$
Substitute $b\to t + 1$
Substitute $c\to \frac{1}{u + 1}$
numerator: $4 s^{5} t^{4} u^{3} + 12 s^{5} t^{4} u^{2} + 12 s^{5} t^{4} u + 4 s^{5} t^{4} + 10 s^{5} t^{3} u^{3} + 34 s^{5} t^{3} u^{2} + 38 s^{5} t^{3} u + 14 s^{5} t^{3} + 6 s^{5} t^{2} u^{3} + 27 s^{5} t^{2} u^{2} + 36 s^{5} t^{2} u + 15 s^{5} t^{2} + 8 s^{5} t u^{3} + 18 s^{5} t u^{2} + 16 s^{5} t u + 6 s^{5} t + 8 s^{5} u^{3} + 24 s^{5} u^{2} + 22 s^{5} u + 10 s^{5} + 20 s^{4} t^{4} u^{3} + 60 s^{4} t^{4} u^{2} + 60 s^{4} t^{4} u + 20 s^{4} t^{4} + 54 s^{4} t^{3} u^{3} + 182 s^{4} t^{3} u^{2} + 202 s^{4} t^{3} u + 74 s^{4} t^{3} + 33 s^{4} t^{2} u^{3} + 144 s^{4} t^{2} u^{2} + 189 s^{4} t^{2} u + 78 s^{4} t^{2} + 34 s^{4} t u^{3} + 72 s^{4} t u^{2} + 62 s^{4} t u + 24 s^{4} t + 44 s^{4} u^{3} + 114 s^{4} u^{2} + 90 s^{4} u + 40 s^{4} + 40 s^{3} t^{4} u^{3} + 120 s^{3} t^{4} u^{2} + 120 s^{3} t^{4} u + 40 s^{3} t^{4} + 116 s^{3} t^{3} u^{3} + 388 s^{3} t^{3} u^{2} + 428 s^{3} t^{3} u + 156 s^{3} t^{3} + 72 s^{3} t^{2} u^{3} + 306 s^{3} t^{2} u^{2} + 396 s^{3} t^{2} u + 162 s^{3} t^{2} + 56 s^{3} t u^{3} + 108 s^{3} t u^{2} + 88 s^{3} t u + 36 s^{3} t + 96 s^{3} u^{3} + 216 s^{3} u^{2} + 140 s^{3} u + 60 s^{3} + 40 s^{2} t^{4} u^{3} + 120 s^{2} t^{4} u^{2} + 120 s^{2} t^{4} u + 40 s^{2} t^{4} + 124 s^{2} t^{3} u^{3} + 412 s^{2} t^{3} u^{2} + 452 s^{2} t^{3} u + 164 s^{2} t^{3} + 78 s^{2} t^{2} u^{3} + 324 s^{2} t^{2} u^{2} + 414 s^{2} t^{2} u + 168 s^{2} t^{2} + 40 s^{2} t u^{3} + 60 s^{2} t u^{2} + 40 s^{2} t u + 20 s^{2} t + 100 s^{2} u^{3} + 190 s^{2} u^{2} + 84 s^{2} u + 34 s^{2} + 20 s t^{4} u^{3} + 60 s t^{4} u^{2} + 60 s t^{4} u + 20 s t^{4} + 66 s t^{3} u^{3} + 218 s t^{3} u^{2} + 238 s t^{3} u + 86 s t^{3} + 42 s t^{2} u^{3} + 171 s t^{2} u^{2} + 216 s t^{2} u + 87 s t^{2} + 12 s t u^{3} + 6 s t u^{2} - 4 s t u + 2 s t + 46 s u^{3} + 66 s u^{2} + 4 t^{4} u^{3} + 12 t^{4} u^{2} + 12 t^{4} u + 4 t^{4} + 14 t^{3} u^{3} + 46 t^{3} u^{2} + 50 t^{3} u + 18 t^{3} + 9 t^{2} u^{3} + 36 t^{2} u^{2} + 45 t^{2} u + 18 t^{2} + 2 t u^{3} - 2 t u + 10 u^{3} + 14 u^{2}$
denominator: $s^{5} u^{3} + 3 s^{5} u^{2} + 3 s^{5} u + s^{5} + 5 s^{4} u^{3} + 15 s^{4} u^{2} + 15 s^{4} u + 5 s^{4} + 10 s^{3} u^{3} + 30 s^{3} u^{2} + 30 s^{3} u + 10 s^{3} + 10 s^{2} u^{3} + 30 s^{2} u^{2} + 30 s^{2} u + 10 s^{2} + 5 s u^{3} + 15 s u^{2} + 15 s u + 5 s + u^{3} + 3 u^{2} + 3 u + 1$
status: 0
From weighted AM-GM inequality:
$$4 s t u \le s^{2} u^{2}+2 s t^{2}+u^{2}$$
$$2 t u \le t^{2}+u^{2}$$
$$ 0 \le 4 s^{5} t^{4} u^{3}+12 s^{5} t^{4} u^{2}+12 s^{5} t^{4} u+4 s^{5} t^{4}+10 s^{5} t^{3} u^{3}+34 s^{5} t^{3} u^{2}+38 s^{5} t^{3} u+14 s^{5} t^{3}+6 s^{5} t^{2} u^{3}+27 s^{5} t^{2} u^{2}+36 s^{5} t^{2} u+15 s^{5} t^{2}+8 s^{5} t u^{3}+18 s^{5} t u^{2}+16 s^{5} t u+6 s^{5} t+8 s^{5} u^{3}+24 s^{5} u^{2}+22 s^{5} u+10 s^{5}+20 s^{4} t^{4} u^{3}+60 s^{4} t^{4} u^{2}+60 s^{4} t^{4} u+20 s^{4} t^{4}+54 s^{4} t^{3} u^{3}+182 s^{4} t^{3} u^{2}+202 s^{4} t^{3} u+74 s^{4} t^{3}+33 s^{4} t^{2} u^{3}+144 s^{4} t^{2} u^{2}+189 s^{4} t^{2} u+78 s^{4} t^{2}+34 s^{4} t u^{3}+72 s^{4} t u^{2}+62 s^{4} t u+24 s^{4} t+44 s^{4} u^{3}+114 s^{4} u^{2}+90 s^{4} u+40 s^{4}+40 s^{3} t^{4} u^{3}+120 s^{3} t^{4} u^{2}+120 s^{3} t^{4} u+40 s^{3} t^{4}+116 s^{3} t^{3} u^{3}+388 s^{3} t^{3} u^{2}+428 s^{3} t^{3} u+156 s^{3} t^{3}+72 s^{3} t^{2} u^{3}+306 s^{3} t^{2} u^{2}+396 s^{3} t^{2} u+162 s^{3} t^{2}+56 s^{3} t u^{3}+108 s^{3} t u^{2}+88 s^{3} t u+36 s^{3} t+96 s^{3} u^{3}+216 s^{3} u^{2}+140 s^{3} u+60 s^{3}+40 s^{2} t^{4} u^{3}+120 s^{2} t^{4} u^{2}+120 s^{2} t^{4} u+40 s^{2} t^{4}+124 s^{2} t^{3} u^{3}+412 s^{2} t^{3} u^{2}+452 s^{2} t^{3} u+164 s^{2} t^{3}+78 s^{2} t^{2} u^{3}+324 s^{2} t^{2} u^{2}+414 s^{2} t^{2} u+168 s^{2} t^{2}+40 s^{2} t u^{3}+60 s^{2} t u^{2}+40 s^{2} t u+20 s^{2} t+100 s^{2} u^{3}+189 s^{2} u^{2}+84 s^{2} u+34 s^{2}+20 s t^{4} u^{3}+60 s t^{4} u^{2}+60 s t^{4} u+20 s t^{4}+66 s t^{3} u^{3}+218 s t^{3} u^{2}+238 s t^{3} u+86 s t^{3}+42 s t^{2} u^{3}+171 s t^{2} u^{2}+216 s t^{2} u+85 s t^{2}+12 s t u^{3}+6 s t u^{2}+2 s t+46 s u^{3}+66 s u^{2}+4 t^{4} u^{3}+12 t^{4} u^{2}+12 t^{4} u+4 t^{4}+14 t^{3} u^{3}+46 t^{3} u^{2}+50 t^{3} u+18 t^{3}+9 t^{2} u^{3}+36 t^{2} u^{2}+45 t^{2} u+17 t^{2}+2 t u^{3}+10 u^{3}+12 u^{2} $$
The sum of all inequalities gives us a proof of the inequality.
Substitute $a\to v + 1$
Substitute $b\to \frac{1}{w + 1}$
Substitute $c\to \frac{1}{x + 1}$
numerator: $4 v^{5} w^{4} x^{3} + 12 v^{5} w^{4} x^{2} + 12 v^{5} w^{4} x + 4 v^{5} w^{4} + 16 v^{5} w^{3} x^{3} + 48 v^{5} w^{3} x^{2} + 48 v^{5} w^{3} x + 16 v^{5} w^{3} + 24 v^{5} w^{2} x^{3} + 72 v^{5} w^{2} x^{2} + 72 v^{5} w^{2} x + 24 v^{5} w^{2} + 16 v^{5} w x^{3} + 48 v^{5} w x^{2} + 48 v^{5} w x + 16 v^{5} w + 4 v^{5} x^{3} + 12 v^{5} x^{2} + 12 v^{5} x + 4 v^{5} + 14 v^{4} w^{4} x^{3} + 46 v^{4} w^{4} x^{2} + 50 v^{4} w^{4} x + 18 v^{4} w^{4} + 60 v^{4} w^{3} x^{3} + 196 v^{4} w^{3} x^{2} + 212 v^{4} w^{3} x + 76 v^{4} w^{3} + 96 v^{4} w^{2} x^{3} + 312 v^{4} w^{2} x^{2} + 336 v^{4} w^{2} x + 120 v^{4} w^{2} + 68 v^{4} w x^{3} + 220 v^{4} w x^{2} + 236 v^{4} w x + 84 v^{4} w + 18 v^{4} x^{3} + 58 v^{4} x^{2} + 62 v^{4} x + 22 v^{4} + 16 v^{3} w^{4} x^{3} + 62 v^{3} w^{4} x^{2} + 76 v^{3} w^{4} x + 30 v^{3} w^{4} + 76 v^{3} w^{3} x^{3} + 284 v^{3} w^{3} x^{2} + 340 v^{3} w^{3} x + 132 v^{3} w^{3} + 132 v^{3} w^{2} x^{3} + 480 v^{3} w^{2} x^{2} + 564 v^{3} w^{2} x + 216 v^{3} w^{2} + 100 v^{3} w x^{3} + 356 v^{3} w x^{2} + 412 v^{3} w x + 156 v^{3} w + 28 v^{3} x^{3} + 98 v^{3} x^{2} + 112 v^{3} x + 42 v^{3} + 4 v^{2} w^{4} x^{3} + 30 v^{2} w^{4} x^{2} + 48 v^{2} w^{4} x + 22 v^{2} w^{4} + 28 v^{2} w^{3} x^{3} + 156 v^{2} w^{3} x^{2} + 228 v^{2} w^{3} x + 100 v^{2} w^{3} + 60 v^{2} w^{2} x^{3} + 288 v^{2} w^{2} x^{2} + 396 v^{2} w^{2} x + 168 v^{2} w^{2} + 52 v^{2} w x^{3} + 228 v^{2} w x^{2} + 300 v^{2} w x + 124 v^{2} w + 16 v^{2} x^{3} + 66 v^{2} x^{2} + 84 v^{2} x + 34 v^{2} + 5 v w^{4} x^{3} + 7 v w^{4} x^{2} + 3 v w^{4} x + v w^{4} + 24 v w^{3} x^{3} + 40 v w^{3} x^{2} + 24 v w^{3} x + 8 v w^{3} + 33 v w^{2} x^{3} + 51 v w^{2} x^{2} + 27 v w^{2} x + 9 v w^{2} + 18 v w x^{3} + 22 v w x^{2} + 6 v w x + 2 v w + 4 v x^{3} + 4 v x^{2} + 7 w^{4} x^{3} + 16 w^{4} x^{2} + 9 w^{4} x + 4 w^{4} + 38 w^{3} x^{3} + 82 w^{3} x^{2} + 46 w^{3} x + 18 w^{3} + 63 w^{2} x^{3} + 120 w^{2} x^{2} + 51 w^{2} x + 18 w^{2} + 38 w x^{3} + 56 w x^{2} + 2 w x + 10 x^{3} + 14 x^{2}$
denominator: $w^{4} x^{3} + 3 w^{4} x^{2} + 3 w^{4} x + w^{4} + 4 w^{3} x^{3} + 12 w^{3} x^{2} + 12 w^{3} x + 4 w^{3} + 6 w^{2} x^{3} + 18 w^{2} x^{2} + 18 w^{2} x + 6 w^{2} + 4 w x^{3} + 12 w x^{2} + 12 w x + 4 w + x^{3} + 3 x^{2} + 3 x + 1$
status: 0
$$ 0 \le 4 v^{5} w^{4} x^{3}+16 v^{5} w^{3} x^{3}+24 v^{5} w^{2} x^{3}+16 v^{5} w x^{3}+4 v^{5} x^{3}+14 v^{4} w^{4} x^{3}+60 v^{4} w^{3} x^{3}+96 v^{4} w^{2} x^{3}+68 v^{4} w x^{3}+18 v^{4} x^{3}+16 v^{3} w^{4} x^{3}+76 v^{3} w^{3} x^{3}+132 v^{3} w^{2} x^{3}+100 v^{3} w x^{3}+28 v^{3} x^{3}+4 v^{2} w^{4} x^{3}+28 v^{2} w^{3} x^{3}+60 v^{2} w^{2} x^{3}+52 v^{2} w x^{3}+16 v^{2} x^{3}+5 v w^{4} x^{3}+24 v w^{3} x^{3}+33 v w^{2} x^{3}+18 v w x^{3}+4 v x^{3}+7 w^{4} x^{3}+38 w^{3} x^{3}+63 w^{2} x^{3}+38 w x^{3}+10 x^{3}+12 v^{5} w^{4} x^{2}+48 v^{5} w^{3} x^{2}+72 v^{5} w^{2} x^{2}+48 v^{5} w x^{2}+12 v^{5} x^{2}+46 v^{4} w^{4} x^{2}+196 v^{4} w^{3} x^{2}+312 v^{4} w^{2} x^{2}+220 v^{4} w x^{2}+58 v^{4} x^{2}+62 v^{3} w^{4} x^{2}+284 v^{3} w^{3} x^{2}+480 v^{3} w^{2} x^{2}+356 v^{3} w x^{2}+98 v^{3} x^{2}+30 v^{2} w^{4} x^{2}+156 v^{2} w^{3} x^{2}+288 v^{2} w^{2} x^{2}+228 v^{2} w x^{2}+66 v^{2} x^{2}+7 v w^{4} x^{2}+40 v w^{3} x^{2}+51 v w^{2} x^{2}+22 v w x^{2}+4 v x^{2}+16 w^{4} x^{2}+82 w^{3} x^{2}+120 w^{2} x^{2}+56 w x^{2}+14 x^{2}+12 v^{5} w^{4} x+48 v^{5} w^{3} x+72 v^{5} w^{2} x+48 v^{5} w x+12 v^{5} x+50 v^{4} w^{4} x+212 v^{4} w^{3} x+336 v^{4} w^{2} x+236 v^{4} w x+62 v^{4} x+76 v^{3} w^{4} x+340 v^{3} w^{3} x+564 v^{3} w^{2} x+412 v^{3} w x+112 v^{3} x+48 v^{2} w^{4} x+228 v^{2} w^{3} x+396 v^{2} w^{2} x+300 v^{2} w x+84 v^{2} x+3 v w^{4} x+24 v w^{3} x+27 v w^{2} x+6 v w x+9 w^{4} x+46 w^{3} x+51 w^{2} x+2 w x+4 v^{5} w^{4}+16 v^{5} w^{3}+24 v^{5} w^{2}+16 v^{5} w+4 v^{5}+18 v^{4} w^{4}+76 v^{4} w^{3}+120 v^{4} w^{2}+84 v^{4} w+22 v^{4}+30 v^{3} w^{4}+132 v^{3} w^{3}+216 v^{3} w^{2}+156 v^{3} w+42 v^{3}+22 v^{2} w^{4}+100 v^{2} w^{3}+168 v^{2} w^{2}+124 v^{2} w+34 v^{2}+v w^{4}+8 v w^{3}+9 v w^{2}+2 v w+4 w^{4}+18 w^{3}+18 w^{2} $$
The sum of all inequalities gives us a proof of the inequality.
Substitute $a\to \frac{1}{y + 1}$
Substitute $b\to \frac{1}{z + 1}$
Substitute $c\to \frac{1}{a_{1} + 1}$
numerator: $10 a_{1}^{3} y^{5} z^{3} + 30 a_{1}^{3} y^{5} z^{2} + 24 a_{1}^{3} y^{5} z + 8 a_{1}^{3} y^{5} + 9 a_{1}^{3} y^{4} z^{4} + 86 a_{1}^{3} y^{4} z^{3} + 195 a_{1}^{3} y^{4} z^{2} + 142 a_{1}^{3} y^{4} z + 44 a_{1}^{3} y^{4} + 36 a_{1}^{3} y^{3} z^{4} + 244 a_{1}^{3} y^{3} z^{3} + 480 a_{1}^{3} y^{3} z^{2} + 328 a_{1}^{3} y^{3} z + 96 a_{1}^{3} y^{3} + 54 a_{1}^{3} y^{2} z^{4} + 312 a_{1}^{3} y^{2} z^{3} + 558 a_{1}^{3} y^{2} z^{2} + 360 a_{1}^{3} y^{2} z + 100 a_{1}^{3} y^{2} + 30 a_{1}^{3} y z^{4} + 166 a_{1}^{3} y z^{3} + 282 a_{1}^{3} y z^{2} + 172 a_{1}^{3} y z + 46 a_{1}^{3} y + 7 a_{1}^{3} z^{4} + 38 a_{1}^{3} z^{3} + 63 a_{1}^{3} z^{2} + 38 a_{1}^{3} z + 10 a_{1}^{3} + 11 a_{1}^{2} y^{5} z^{4} + 62 a_{1}^{2} y^{5} z^{3} + 117 a_{1}^{2} y^{5} z^{2} + 78 a_{1}^{2} y^{5} z + 24 a_{1}^{2} y^{5} + 64 a_{1}^{2} y^{4} z^{4} + 346 a_{1}^{2} y^{4} z^{3} + 612 a_{1}^{2} y^{4} z^{2} + 384 a_{1}^{2} y^{4} z + 114 a_{1}^{2} y^{4} + 146 a_{1}^{2} y^{3} z^{4} + 764 a_{1}^{2} y^{3} z^{3} + 1278 a_{1}^{2} y^{3} z^{2} + 756 a_{1}^{2} y^{3} z + 216 a_{1}^{2} y^{3} + 162 a_{1}^{2} y^{2} z^{4} + 816 a_{1}^{2} y^{2} z^{3} + 1284 a_{1}^{2} y^{2} z^{2} + 700 a_{1}^{2} y^{2} z + 190 a_{1}^{2} y^{2} + 73 a_{1}^{2} y z^{4} + 370 a_{1}^{2} y z^{3} + 549 a_{1}^{2} y z^{2} + 258 a_{1}^{2} y z + 66 a_{1}^{2} y + 16 a_{1}^{2} z^{4} + 82 a_{1}^{2} z^{3} + 120 a_{1}^{2} z^{2} + 56 a_{1}^{2} z + 14 a_{1}^{2} + 16 a_{1} y^{5} z^{4} + 74 a_{1} y^{5} z^{3} + 120 a_{1} y^{5} z^{2} + 72 a_{1} y^{5} z + 22 a_{1} y^{5} + 75 a_{1} y^{4} z^{4} + 350 a_{1} y^{4} z^{3} + 543 a_{1} y^{4} z^{2} + 298 a_{1} y^{4} z + 90 a_{1} y^{4} + 140 a_{1} y^{3} z^{4} + 660 a_{1} y^{3} z^{3} + 972 a_{1} y^{3} z^{2} + 472 a_{1} y^{3} z + 140 a_{1} y^{3} + 126 a_{1} y^{2} z^{4} + 592 a_{1} y^{2} z^{3} + 798 a_{1} y^{2} z^{2} + 296 a_{1} y^{2} z + 84 a_{1} y^{2} + 42 a_{1} y z^{4} + 206 a_{1} y z^{3} + 228 a_{1} y z^{2} + 4 a_{1} y z + 9 a_{1} z^{4} + 46 a_{1} z^{3} + 51 a_{1} z^{2} + 2 a_{1} z + 9 y^{5} z^{4} + 38 y^{5} z^{3} + 57 y^{5} z^{2} + 34 y^{5} z + 10 y^{5} + 40 y^{4} z^{4} + 170 y^{4} z^{3} + 246 y^{4} z^{2} + 136 y^{4} z + 40 y^{4} + 70 y^{3} z^{4} + 300 y^{3} z^{3} + 414 y^{3} z^{2} + 204 y^{3} z + 60 y^{3} + 58 y^{2} z^{4} + 248 y^{2} z^{3} + 312 y^{2} z^{2} + 116 y^{2} z + 34 y^{2} + 19 y z^{4} + 82 y z^{3} + 81 y z^{2} - 2 y z + 4 z^{4} + 18 z^{3} + 18 z^{2}$
denominator: $a_{1}^{3} y^{5} z^{4} + 4 a_{1}^{3} y^{5} z^{3} + 6 a_{1}^{3} y^{5} z^{2} + 4 a_{1}^{3} y^{5} z + a_{1}^{3} y^{5} + 5 a_{1}^{3} y^{4} z^{4} + 20 a_{1}^{3} y^{4} z^{3} + 30 a_{1}^{3} y^{4} z^{2} + 20 a_{1}^{3} y^{4} z + 5 a_{1}^{3} y^{4} + 10 a_{1}^{3} y^{3} z^{4} + 40 a_{1}^{3} y^{3} z^{3} + 60 a_{1}^{3} y^{3} z^{2} + 40 a_{1}^{3} y^{3} z + 10 a_{1}^{3} y^{3} + 10 a_{1}^{3} y^{2} z^{4} + 40 a_{1}^{3} y^{2} z^{3} + 60 a_{1}^{3} y^{2} z^{2} + 40 a_{1}^{3} y^{2} z + 10 a_{1}^{3} y^{2} + 5 a_{1}^{3} y z^{4} + 20 a_{1}^{3} y z^{3} + 30 a_{1}^{3} y z^{2} + 20 a_{1}^{3} y z + 5 a_{1}^{3} y + a_{1}^{3} z^{4} + 4 a_{1}^{3} z^{3} + 6 a_{1}^{3} z^{2} + 4 a_{1}^{3} z + a_{1}^{3} + 3 a_{1}^{2} y^{5} z^{4} + 12 a_{1}^{2} y^{5} z^{3} + 18 a_{1}^{2} y^{5} z^{2} + 12 a_{1}^{2} y^{5} z + 3 a_{1}^{2} y^{5} + 15 a_{1}^{2} y^{4} z^{4} + 60 a_{1}^{2} y^{4} z^{3} + 90 a_{1}^{2} y^{4} z^{2} + 60 a_{1}^{2} y^{4} z + 15 a_{1}^{2} y^{4} + 30 a_{1}^{2} y^{3} z^{4} + 120 a_{1}^{2} y^{3} z^{3} + 180 a_{1}^{2} y^{3} z^{2} + 120 a_{1}^{2} y^{3} z + 30 a_{1}^{2} y^{3} + 30 a_{1}^{2} y^{2} z^{4} + 120 a_{1}^{2} y^{2} z^{3} + 180 a_{1}^{2} y^{2} z^{2} + 120 a_{1}^{2} y^{2} z + 30 a_{1}^{2} y^{2} + 15 a_{1}^{2} y z^{4} + 60 a_{1}^{2} y z^{3} + 90 a_{1}^{2} y z^{2} + 60 a_{1}^{2} y z + 15 a_{1}^{2} y + 3 a_{1}^{2} z^{4} + 12 a_{1}^{2} z^{3} + 18 a_{1}^{2} z^{2} + 12 a_{1}^{2} z + 3 a_{1}^{2} + 3 a_{1} y^{5} z^{4} + 12 a_{1} y^{5} z^{3} + 18 a_{1} y^{5} z^{2} + 12 a_{1} y^{5} z + 3 a_{1} y^{5} + 15 a_{1} y^{4} z^{4} + 60 a_{1} y^{4} z^{3} + 90 a_{1} y^{4} z^{2} + 60 a_{1} y^{4} z + 15 a_{1} y^{4} + 30 a_{1} y^{3} z^{4} + 120 a_{1} y^{3} z^{3} + 180 a_{1} y^{3} z^{2} + 120 a_{1} y^{3} z + 30 a_{1} y^{3} + 30 a_{1} y^{2} z^{4} + 120 a_{1} y^{2} z^{3} + 180 a_{1} y^{2} z^{2} + 120 a_{1} y^{2} z + 30 a_{1} y^{2} + 15 a_{1} y z^{4} + 60 a_{1} y z^{3} + 90 a_{1} y z^{2} + 60 a_{1} y z + 15 a_{1} y + 3 a_{1} z^{4} + 12 a_{1} z^{3} + 18 a_{1} z^{2} + 12 a_{1} z + 3 a_{1} + y^{5} z^{4} + 4 y^{5} z^{3} + 6 y^{5} z^{2} + 4 y^{5} z + y^{5} + 5 y^{4} z^{4} + 20 y^{4} z^{3} + 30 y^{4} z^{2} + 20 y^{4} z + 5 y^{4} + 10 y^{3} z^{4} + 40 y^{3} z^{3} + 60 y^{3} z^{2} + 40 y^{3} z + 10 y^{3} + 10 y^{2} z^{4} + 40 y^{2} z^{3} + 60 y^{2} z^{2} + 40 y^{2} z + 10 y^{2} + 5 y z^{4} + 20 y z^{3} + 30 y z^{2} + 20 y z + 5 y + z^{4} + 4 z^{3} + 6 z^{2} + 4 z + 1$
status: 0
From weighted AM-GM inequality:
$$2 y z \le y^{2}+z^{2}$$
$$ 0 \le 11 a_{1}^{2} y^{5} z^{4}+16 a_{1} y^{5} z^{4}+9 y^{5} z^{4}+10 a_{1}^{3} y^{5} z^{3}+62 a_{1}^{2} y^{5} z^{3}+74 a_{1} y^{5} z^{3}+38 y^{5} z^{3}+30 a_{1}^{3} y^{5} z^{2}+117 a_{1}^{2} y^{5} z^{2}+120 a_{1} y^{5} z^{2}+57 y^{5} z^{2}+24 a_{1}^{3} y^{5} z+78 a_{1}^{2} y^{5} z+72 a_{1} y^{5} z+34 y^{5} z+8 a_{1}^{3} y^{5}+24 a_{1}^{2} y^{5}+22 a_{1} y^{5}+10 y^{5}+9 a_{1}^{3} y^{4} z^{4}+64 a_{1}^{2} y^{4} z^{4}+75 a_{1} y^{4} z^{4}+40 y^{4} z^{4}+86 a_{1}^{3} y^{4} z^{3}+346 a_{1}^{2} y^{4} z^{3}+350 a_{1} y^{4} z^{3}+170 y^{4} z^{3}+195 a_{1}^{3} y^{4} z^{2}+612 a_{1}^{2} y^{4} z^{2}+543 a_{1} y^{4} z^{2}+246 y^{4} z^{2}+142 a_{1}^{3} y^{4} z+384 a_{1}^{2} y^{4} z+298 a_{1} y^{4} z+136 y^{4} z+44 a_{1}^{3} y^{4}+114 a_{1}^{2} y^{4}+90 a_{1} y^{4}+40 y^{4}+36 a_{1}^{3} y^{3} z^{4}+146 a_{1}^{2} y^{3} z^{4}+140 a_{1} y^{3} z^{4}+70 y^{3} z^{4}+244 a_{1}^{3} y^{3} z^{3}+764 a_{1}^{2} y^{3} z^{3}+660 a_{1} y^{3} z^{3}+300 y^{3} z^{3}+480 a_{1}^{3} y^{3} z^{2}+1278 a_{1}^{2} y^{3} z^{2}+972 a_{1} y^{3} z^{2}+414 y^{3} z^{2}+328 a_{1}^{3} y^{3} z+756 a_{1}^{2} y^{3} z+472 a_{1} y^{3} z+204 y^{3} z+96 a_{1}^{3} y^{3}+216 a_{1}^{2} y^{3}+140 a_{1} y^{3}+60 y^{3}+54 a_{1}^{3} y^{2} z^{4}+162 a_{1}^{2} y^{2} z^{4}+126 a_{1} y^{2} z^{4}+58 y^{2} z^{4}+312 a_{1}^{3} y^{2} z^{3}+816 a_{1}^{2} y^{2} z^{3}+592 a_{1} y^{2} z^{3}+248 y^{2} z^{3}+558 a_{1}^{3} y^{2} z^{2}+1284 a_{1}^{2} y^{2} z^{2}+798 a_{1} y^{2} z^{2}+312 y^{2} z^{2}+360 a_{1}^{3} y^{2} z+700 a_{1}^{2} y^{2} z+296 a_{1} y^{2} z+116 y^{2} z+100 a_{1}^{3} y^{2}+190 a_{1}^{2} y^{2}+84 a_{1} y^{2}+33 y^{2}+30 a_{1}^{3} y z^{4}+73 a_{1}^{2} y z^{4}+42 a_{1} y z^{4}+19 y z^{4}+166 a_{1}^{3} y z^{3}+370 a_{1}^{2} y z^{3}+206 a_{1} y z^{3}+82 y z^{3}+282 a_{1}^{3} y z^{2}+549 a_{1}^{2} y z^{2}+228 a_{1} y z^{2}+81 y z^{2}+172 a_{1}^{3} y z+258 a_{1}^{2} y z+4 a_{1} y z+46 a_{1}^{3} y+66 a_{1}^{2} y+7 a_{1}^{3} z^{4}+16 a_{1}^{2} z^{4}+9 a_{1} z^{4}+4 z^{4}+38 a_{1}^{3} z^{3}+82 a_{1}^{2} z^{3}+46 a_{1} z^{3}+18 z^{3}+63 a_{1}^{3} z^{2}+120 a_{1}^{2} z^{2}+51 a_{1} z^{2}+17 z^{2}+38 a_{1}^{3} z+56 a_{1}^{2} z+2 a_{1} z+10 a_{1}^{3}+14 a_{1}^{2} $$
The sum of all inequalities gives us a proof of the inequality.
0