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urojony 2020-04-21 10:13:02 +02:00
parent e7367cf50f
commit e145a4db9f
5 changed files with 1360 additions and 155 deletions
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6
.ipynb_checkpoints/Untitled-checkpoint.ipynb Normal file
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@ -0,0 +1,6 @@
{
"cells": [],
"metadata": {},
"nbformat": 4,
"nbformat_minor": 2
}

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examples.ipynb
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@ -1,5 +1,12 @@
{ {
"cells": [ "cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this notebook there are presented examples of usage of shiroin, a python library for proving inequalities of multivariate polynomials."
]
},
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 1, "execution_count": 1,
@ -16,7 +23,7 @@
"cell_type": "markdown", "cell_type": "markdown",
"metadata": {}, "metadata": {},
"source": [ "source": [
"The first line obviously loads this package. The second one sets a seed for proving functions. If you don't write it, you can get slightly different proof each time you run a function. The next two lines\n", "The first line obviously loads this package. The second one sets a seed for proving functions. If you don't write it, you can get a slightly different proof each time you run a function. The next two lines provide a nicer display of proofs, i.e. formulas will be shown instead of LaTeX code of these formulas. Note that this works on Jupyter, but not on the git page.\n",
"\n", "\n",
"Now let's make some proofs. We will use problems from https://www.imomath.com/index.php?options=593&lmm=0." "Now let's make some proofs. We will use problems from https://www.imomath.com/index.php?options=593&lmm=0."
] ]
@ -89,16 +96,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -208,7 +205,31 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$2ab \\le a^2+b^2$$$$2ac \\le a^2+c^2$$$$2bc \\le b^2+c^2$$" "$$2ab \\le a^2+b^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2ac \\le a^2+c^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2bc \\le b^2+c^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -312,16 +333,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -436,7 +447,31 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$2ab \\le a^2+b^2$$$$2ac \\le a^2+c^2$$$$2ad \\le a^2+d^2$$" "$$2ab \\le a^2+b^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2ac \\le a^2+c^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2ad \\le a^2+d^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -559,7 +594,31 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$28ab \\le 14a^2+14b^2$$$$28ac \\le 14a^2+14c^2$$$$28ad \\le 14a^2+14d^2$$" "$$28ab \\le 14a^2+14b^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$28ac \\le 14a^2+14c^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$28ad \\le 14a^2+14d^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -661,16 +720,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -710,7 +759,7 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"prove(makesubs(S(\"-x**3*y + 2*x**2*y**2 - x**2 - x*y**3 + 2*x*y - y**2\"),[('y', 'inf')])" "prove(makesubs(S(\"-x**3*y + 2*x**2*y**2 - x**2 - x*y**3 + 2*x*y - y**2\"),[('y', 'oo')])"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -824,7 +873,43 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$12x^2y^2 \\le 6x^3y+6xy^3$$$$3x^2y \\le 2x^3+y^3$$$$3xy^2 \\le x^3+2y^3$$$$6xy \\le 3x^2+3y^2$$" "$$12x^2y^2 \\le 6x^3y+6xy^3$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$3x^2y \\le 2x^3+y^3$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$3xy^2 \\le x^3+2y^3$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$6xy \\le 3x^2+3y^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -984,7 +1069,31 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$2x^3y \\le x^4y^2+x^2$$$$4x^2y \\le x^3y^2+2x^2+xy^2$$$$2xy \\le x^2+y^2$$" "$$2x^3y \\le x^4y^2+x^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$4x^2y \\le x^3y^2+2x^2+xy^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2xy \\le x^2+y^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -1101,7 +1210,31 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$2ab \\le a^2+b^2$$$$2ac \\le a^2+c^2$$$$2bc \\le b^2+c^2$$" "$$2ab \\le a^2+b^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2ac \\le a^2+c^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2bc \\le b^2+c^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -1201,16 +1334,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -1310,16 +1433,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -1356,7 +1469,7 @@
} }
], ],
"source": [ "source": [
"prove(makesubs('(x-1)^4','(1,inf)'))" "prove(makesubs('(x-1)^4','(1,oo)'))"
] ]
}, },
{ {
@ -1414,16 +1527,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -1460,7 +1563,7 @@
} }
], ],
"source": [ "source": [
"prove(makesubs('(x-1)^4','(-inf,1)'))" "prove(makesubs('(x-1)^4','(-oo,1)'))"
] ]
}, },
{ {
@ -1540,16 +1643,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -1673,16 +1766,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -1719,7 +1802,7 @@
} }
], ],
"source": [ "source": [
"prove(makesubs(formula,'[c,inf],[d,inf]'))" "prove(makesubs(formula,'[c,oo],[d,oo]'))"
] ]
}, },
{ {
@ -1847,7 +1930,7 @@
} }
], ],
"source": [ "source": [
"prove(makesubs(formula,'[c,inf],[d,inf]')*2)" "prove(makesubs(formula,'[c,oo],[d,oo]')*2)"
] ]
}, },
{ {
@ -1937,7 +2020,19 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$2a^2b^2c^3d \\le a^3bc^2d^2+ab^3c^4$$$$2a^2c^3d \\le a^3c^2d^2+ac^4$$" "$$2a^2b^2c^3d \\le a^3bc^2d^2+ab^3c^4$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2a^2c^3d \\le a^3c^2d^2+ac^4$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -2082,16 +2177,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -2165,16 +2250,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -2379,16 +2454,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -2574,7 +2639,19 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$2ab \\le a^2+b^2$$$$2bc \\le b^2+c^2$$" "$$2ab \\le a^2+b^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2bc \\le b^2+c^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -2683,7 +2760,31 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$2abc^3 \\le a^2bc^3+bc^3$$$$2ab \\le a^2+b^2$$$$2bc \\le b^2+c^2$$" "$$2abc^3 \\le a^2bc^3+bc^3$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2ab \\le a^2+b^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2bc \\le b^2+c^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -2804,7 +2905,19 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$4abc \\le a^2c^2+2ab^2+c^2$$$$2bc \\le b^2+c^2$$" "$$4abc \\le a^2c^2+2ab^2+c^2$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$2bc \\le b^2+c^2$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"
@ -2886,16 +2999,6 @@
"metadata": {}, "metadata": {},
"output_type": "display_data" "output_type": "display_data"
}, },
{
"data": {
"text/latex": [],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
@ -3091,7 +3194,31 @@
{ {
"data": { "data": {
"text/latex": [ "text/latex": [
"$$21f(1, 2, 3) \\le 11f(-1, 4, 3)+8f(3, -1, 4)+2f(4, 3, -1)$$$$21f(2, 3, 1) \\le 8f(-1, 4, 3)+2f(3, -1, 4)+11f(4, 3, -1)$$$$21f(3, 1, 2) \\le 2f(-1, 4, 3)+11f(3, -1, 4)+8f(4, 3, -1)$$" "$$21f(1, 2, 3) \\le 11f(-1, 4, 3)+8f(3, -1, 4)+2f(4, 3, -1)$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$21f(2, 3, 1) \\le 8f(-1, 4, 3)+2f(3, -1, 4)+11f(4, 3, -1)$$"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$$21f(3, 1, 2) \\le 2f(-1, 4, 3)+11f(3, -1, 4)+8f(4, 3, -1)$$"
], ],
"text/plain": [ "text/plain": [
"<IPython.core.display.Latex object>" "<IPython.core.display.Latex object>"

38
shiroindev.py
View File

@ -26,7 +26,7 @@ for phrase in translationList:
sVars.translation[phrase]=phrase sVars.translation[phrase]=phrase
from scipy.optimize import linprog from scipy.optimize import linprog
import random import random
from sympy import S,cancel,fraction,Pow,expand,solve,latex from sympy import S,cancel,fraction,Pow,expand,solve,latex,oo
import re import re
def _remzero(coef,fun): def _remzero(coef,fun):
#coef, fun represents an expression. #coef, fun represents an expression.
@ -76,7 +76,7 @@ def Sm(formula):
#Adds multiplication signs and sympifies a formula. #Adds multiplication signs and sympifies a formula.
#For example, Sm('(2x+y)(7+5xz)') -> S('(2*x+y)*(7+5*x*z)') #For example, Sm('(2x+y)(7+5xz)') -> S('(2*x+y)*(7+5*x*z)')
if type(formula)==str: if type(formula)==str:
formula.replace(' ','') formula=formula.replace(' ','')
for i in range(2): for i in range(2):
formula=re.sub(r'([0-9a-zA-Z)])([(a-zA-Z])',r'\1*\2',formula) formula=re.sub(r'([0-9a-zA-Z)])([(a-zA-Z])',r'\1*\2',formula)
formula=S(formula) formula=S(formula)
@ -156,7 +156,7 @@ def _list2proof(lcoef,lfun,rcoef,rfun,variables,itermax,linprogiter,_writ2=_writ
#them from the original one. #them from the original one.
#If LHS is empty, then break. #If LHS is empty, then break.
localseed=sVars.seed localseed=sVars.seed
bufer='' bufer=[]
itern=0 itern=0
if len(lcoef)==0: #if LHS is empty if len(lcoef)==0: #if LHS is empty
sVars.print(sVars.translation['status:']+' 0') sVars.print(sVars.translation['status:']+' 0')
@ -210,13 +210,13 @@ def _list2proof(lcoef,lfun,rcoef,rfun,variables,itermax,linprogiter,_writ2=_writ
#step back #step back
lcoef,lfun=oldlcoef,oldlfun lcoef,lfun=oldlcoef,oldlfun
rcoef,rfun=oldrcoef,oldrfun rcoef,rfun=oldrcoef,oldrfun
bufer='' bufer=[]
continue continue
if status==0:#if found a solution with real coefficients if status==0:#if found a solution with real coefficients
if bufer: for ineq in bufer:
sVars.print(bufer) sVars.print(ineq)
foundreal=1 foundreal=1
bufer='' bufer=[]
oldlfun,oldrfun=lfun,rfun oldlfun,oldrfun=lfun,rfun
oldlcoef,oldrcoef=lcoef[:],rcoef[:] oldlcoef,oldrcoef=lcoef[:],rcoef[:]
for i in range(m): for i in range(m):
@ -231,20 +231,22 @@ def _list2proof(lcoef,lfun,rcoef,rfun,variables,itermax,linprogiter,_writ2=_writ
isok=_check(lcoef[i],lfun[i],res.x[i*n:i*n+n],rfun) isok=_check(lcoef[i],lfun[i],res.x[i*n:i*n+n],rfun)
if not isok: if not isok:
continue continue
bufer+='$$'+_writ2(lcoef[i],lfun[i],variables)+' \\le ' bufer+=['']
bufer[-1]+='$$'+_writ2(lcoef[i],lfun[i],variables)+' \\le '
lcoef[i]=0 lcoef[i]=0
for j in range(n): for j in range(n):
rcoef[j]-=int(round(res.x[i*n+j])) rcoef[j]-=int(round(res.x[i*n+j]))
for j,k in zip(res.x[i*n:i*n+n],rfun): for j,k in zip(res.x[i*n:i*n+n],rfun):
if j<0.0001: if j<0.0001:
continue continue
if(c):bufer+='+' if(c):bufer[-1]+='+'
else:c=1 else:c=1
bufer+=_writ2(int(round(j)),k,variables) bufer[-1]+=_writ2(int(round(j)),k,variables)
bufer+='$$' bufer[-1]+='$$'
lcoef,lfun=_remzero(lcoef,lfun) lcoef,lfun=_remzero(lcoef,lfun)
rcoef,rfun=_remzero(rcoef,rfun) rcoef,rfun=_remzero(rcoef,rfun)
sVars.print(bufer) for ineq in bufer:
sVars.print(ineq)
lhs='+'.join([_writ2(c,f,variables) for c,f in zip(lcoef,lfun)]) lhs='+'.join([_writ2(c,f,variables) for c,f in zip(lcoef,lfun)])
if lhs=='': if lhs=='':
lhs='0' lhs='0'
@ -295,7 +297,7 @@ def prove(formula,values=None,variables=None,niter=200,linprogiter=10000):
"You can assume without loss of generality that "]+ "You can assume without loss of generality that "]+
' >= '.join([str(x) for x in fs])+'. '+sVars.translation['Try']) ' >= '.join([str(x) for x in fs])+'. '+sVars.translation['Try'])
sVars.print('prove(makesubs(S("'+str(num)+'"),'+ sVars.print('prove(makesubs(S("'+str(num)+'"),'+
str([(str(x),'inf') for x in variables[1:]])+')') str([(str(x),'oo') for x in variables[1:]])+')')
return st return st
def powerprove(formula,values=None,variables=None,niter=200,linprogiter=10000): def powerprove(formula,values=None,variables=None,niter=200,linprogiter=10000):
#This is a bruteforce and ineffective function for proving inequalities. #This is a bruteforce and ineffective function for proving inequalities.
@ -339,17 +341,17 @@ def makesubs(formula,intervals,values=None,variables=None,numden=False):
equations=[] equations=[]
for var,interval in zip(variables,intervals): for var,interval in zip(variables,intervals):
end1,end2=interval end1,end2=interval
if end1 in {S('-inf'),S('inf')}: if end1 in {S('-oo'),S('oo')}:
end1,end2=end2,end1 end1,end2=end2,end1
if {end1,end2}=={S('-inf'),S('inf')}: if {end1,end2}=={S('-oo'),S('oo')}:
formula=formula.subs(var,var-1/var) formula=formula.subs(var,var-1/var)
equations=[equation.subs(var,var-1/var) for equation in equations] equations=[equation.subs(var,var-1/var) for equation in equations]
sVars.print(sVars.translation['Substitute']+' $'+str(var)+'\\to '+var-1/var+'$') sVars.print(sVars.translation['Substitute']+' $'+str(var)+'\\to '+var-1/var+'$')
elif end2==S('inf'): elif end2==S('oo'):
formula=formula.subs(var,end1+var) formula=formula.subs(var,end1+var)
equations=[equation.subs(var,end1+var) for equation in equations] equations=[equation.subs(var,end1+var) for equation in equations]
sVars.print(sVars.translation['Substitute']+' $'+str(var)+'\\to '+sstr(end1+var)+'$') sVars.print(sVars.translation['Substitute']+' $'+str(var)+'\\to '+sstr(end1+var)+'$')
elif end2==S('-inf'): elif end2==S('-oo'):
formula=formula.subs(var,end1-var) formula=formula.subs(var,end1-var)
equations=[equation.subs(var,end1-var) for equation in equations] equations=[equation.subs(var,end1-var) for equation in equations]
sVars.print(sVars.translation['Substitute']+" $"+str(var)+'\\to '+sstr(end1-var)+'$') sVars.print(sVars.translation['Substitute']+" $"+str(var)+'\\to '+sstr(end1-var)+'$')
@ -359,7 +361,7 @@ def makesubs(formula,intervals,values=None,variables=None,numden=False):
equations=[equation.subs(var,end2+(end1-end2)/var) for equation in equations] equations=[equation.subs(var,end2+(end1-end2)/var) for equation in equations]
num,den=fractioncancel(formula) num,den=fractioncancel(formula)
for var,interval in zip(variables,intervals): for var,interval in zip(variables,intervals):
if {interval[0],interval[1]} & {S('inf'),S('-inf')}==set(): if {interval[0],interval[1]} & {S('oo'),S('-oo')}==set():
num=num.subs(var,var+1) num=num.subs(var,var+1)
den=den.subs(var,var+1) den=den.subs(var,var+1)
equations=[equation.subs(var,var+1) for equation in equations] equations=[equation.subs(var,var+1) for equation in equations]