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shiroindev/.ipynb_checkpoints/statistics-checkpoint.ipynb

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added statistics and fixed few minor bugs
2020-05-17 23:46:53 +02:00
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this notebook several methods for proving inequalities have been compared.\n",
"\n",
"First of all we need a dataset. 35 inequalities were selected from https://www.imomath.com/index.php?options=592&lmm=0. Each inequality is represented by a tuple: inequality (in LaTeX), inequality constraints (for example: $a\\in [0,1]$, equality constraints (for example: $abc=1$) and a function which converts inequality to a formula which we want to prove it's nonnegativity. In most cases when this function is simply a difference between left and right-hand side, but sometimes it's better to use difference between squares of sides (to get rid of square roots). These tuples are used as parameters in `parser` function which converts them to equivalent inequalities with default constraints (i.e. all variables are positive).\n",
"\n",
"Some tuples have less than 4 elements. In this case `parser` use default arguments."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"from importlib import reload\n",
"from sympy import *\n",
"import shiroindev\n",
"from shiroindev import *\n",
"from sympy.parsing.latex import parse_latex\n",
"shiro.seed=1\n",
"from IPython.display import Latex\n",
"shiro.display=lambda x:display(Latex(x))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`sympy` has a `parse_latex` function which converts LaTeX formula to a `sympy` one. Unfortunately it doesn't deal with missing braces in the way that LaTeX do. For example `\\frac12` generates $\\frac12$ which is the same as `\\frac{1}{2}`, but `parse_latex` accepts only the second version. So here there is a boring function which adds missing braces."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\\frac{ \\sqrt {3}}{2}\n",
"a^2+b^2+c^2\\geq ab+bc+ca\n"
]
}
],
"source": [
"def addbraces(s):\n",
" arg={r'\\frac':2,r'\\sqrt':1}\n",
" s2=''\n",
" p=0\n",
" while 1:\n",
" m=re.search(r\"\\\\[a-zA-Z]+\", s[p:])\n",
" if not m:\n",
" break\n",
" s2+=s[p:p+m.end()]\n",
" #print('a',s[p:m.end()],p)\n",
" p+=m.end()\n",
" if m.group() in arg:\n",
" for i in range(arg[m.group()]):\n",
" sp=re.search('^ *',s[p:])\n",
" s2+=sp.group()\n",
" #print('b',sp.group(),p)\n",
" p+=sp.end()\n",
" if s[p]=='{':\n",
" cb=re.search(r'^\\{.*?\\}',s[p:])\n",
" ab=addbraces(cb.group())\n",
" s2+=ab\n",
" #print('c',ab,p)\n",
" p+=cb.end()\n",
" else:\n",
" s2+='{'+s[p]+'}'\n",
" #print('d','{'+s[p]+'}',p)\n",
" p+=1\n",
" #print('e',p)\n",
" s2+=s[p:]\n",
" return s2\n",
"print(addbraces(r'\\frac{ \\sqrt 3}2'))\n",
"print(addbraces(r'a^2+b^2+c^2\\geq ab+bc+ca'))"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"#(formula,intervals,subs,function)\n",
"dif=lambda b,s:b-s\n",
"dif2=lambda b,s:b*b-s*s\n",
"ineqs=[\n",
" (r'a^2+b^2+c^2\\geq ab+bc+ca',),\n",
" (r'a^2+b^2+c^2+d^2\\geq a(b+c+d)',),\n",
" (r'1\\leq\\frac{a^2b^2}{c^2}+\\frac{b^2c^2}{a^2}+\\frac{c^2a^2}{b^2}',\n",
" '[0,1-f],[0,1]','[a,sqrt(1-e-f)],[b,sqrt(e)],[c,sqrt(f)]',),\n",
" (r'\\frac1{1-x^2}+\\frac1{1-y^2}\\geq \\frac2{1-xy}','[0,1],[0,1]'),\n",
" (r'a^3+b^3\\geq a^2b+ab^2',),\n",
" (r'\\frac a{b+c}+\\frac b{c+a}+\\frac c{a+b}\\geq \\frac32',),\n",
" (r'2a^3+b^3\\geq 3a^2b',),\n",
" (r'a^3+b^3+c^3\\geq a^2b+b^2c+c^2a',),\n",
" (r'\\frac a{b+c}+\\frac b{c+d}+ \\frac c{d+a}+ \\frac d{a+b}\\geq 2',),\n",
" (r'\\frac{a^3}{a^2+ab+b^2}+ \\frac{b^3}{b^2+bc+c^2}+ \\frac{c^3}{c^2+ca+a^2} \\geq \\frac{a+b+c}3',),\n",
" (r'\\sqrt{ab}+\\sqrt{cd}+\\sqrt{ef}\\leq\\sqrt{(a+c+e)(b+d+f)}','','',dif2),\n",
" (r'\\frac{5a^3-ab^2}{a+b}+\\frac{5b^3-bc^2}{b+c}+\\frac{5c^3-ca^2}{c+a}\\geq 2(a^2+b^2+c^2)',),\n",
" #(r'\\frac{a^2}{(1+b)(1-c)}+\\frac{b^3}{(1+c)(1-d)}+\\frac{c^3}{(1+d)(1-a)}+\\frac{d^3}{(1+a)(1-b)}\\geq\\frac1{15}',\n",
" # '[0,1-c-d],[0,1-d],[0,1]','[a,1-b-c-d]'),\n",
" (r'\\frac{x^3}{(1+y)(1+z)}+\\frac{y^3}{(1+z)(1+x)}+\\frac{z^3}{(1+x)(1+y)}\\geq\\frac{3}{4}','','[z,1/(x*y)]'),\n",
" (r'\\frac ab+\\frac bc+\\frac ca\\geq \\frac{(a+b+c)^2}{ab+bc+ca}',),\n",
" (r'\\frac{a^2}b+\\frac{b^2}c+\\frac{c^2}a\\geq \\frac{a^2+b^2+c^2}{a+b+c}',),\n",
" (r'\\frac{a^2}b+\\frac{b^2}c+\\frac{c^2}a\\geq a+b+c+\\frac{4(a-b)^2}{a+b+c}',),\n",
" (r'\\frac1{a^3+b^3+abc}+ \\frac1{b^3+c^3+abc} +\\frac1{c^3+a^3+ abc} \\leq \\frac1{abc}',),\n",
" (r'\\frac1{a^3(b+c)}+ \\frac1{b^3(c+a)}+ \\frac1{c^3(a+b)} \\geq \\frac32','','[c,1/(a*b)]'),\n",
" (r'\\frac{a^3}{b^2-bc+c^2}+\\frac{b^3}{c^2-ca+a^2} + \\frac{c^3}{a^2-ab+b^2} \\geq 3 \\cdot\\frac{ab+bc+ca}{a+b+c}',),\n",
" (r'\\frac{x^5-x^2}{x^5+y^2+z^2}+\\frac{y^5-y^2}{y^5+z^2+x^2}+\\frac{z^5-z^2}{z^5+x^2+y^2}\\geq0','','[x,1/(y*z)]'),\n",
" (r'(a+b-c)(b+c-a)(c+a-b)\\leq abc',),\n",
" (r'\\frac1{1+xy}+\\frac1{1+yz}+\\frac1{1+zx}\\leq \\frac34','','[x,(y+z)/(y*z-1)]'),\n",
" (r'\\frac{x\\sqrt x}{y+z}+\\frac{y\\sqrt y}{z+x}+\\frac{z\\sqrt z}{x+y}\\geq\\frac{ \\sqrt 3}2',\n",
" '[0,1-z],[0,1]','[x,1-y-z]'),\n",
" (r'a^4+b^4+c^4+d^4\\geq 4abcd',),\n",
" (r'\\frac{ab}{a+b}+\\frac{bc}{b+c}+\\frac{ca}{c+a}\\leq\\frac{3(ab+bc+ca)}{2(a+b+c)}',),\n",
" (r'\\sqrt{a-1}+\\sqrt{b-1}+ \\sqrt{c-1} \\leq \\sqrt{c(ab+1)}','[1,oo],[1,oo],[1,oo]','',dif2),\n",
" (r'(x-1)(y-1)(z-1)\\geq 8','[0,1-b-c],[0,1-c],[0,1]','[x,1/a],[y,1/b],[z,1/c]'),\n",
" (r'ay+bz+cx\\leq s^2','[0,s],[0,s],[0,s]','[x,s-a],[y,s-b],[z,s-c]'),\n",
" (r'x_1^2+x_2^2+x_3^2+x_4^2+x_5^2\\geq 2 (x_1x_2+x_2x_3+x_3x_4+x_4x_5)/\\sqrt{3}',),\n",
" (r' xy+yz+zx - 2xyz \\leq \\frac7{27}', '[0,1-z],[0,1]','[x,1-y-z]'),\n",
" (r'0 \\leq xy+yz+zx - 2xyz', '[0,1-z],[0,1]','[x,1-y-z]'),\n",
" (r'\\sqrt{3+a+b+c}\\geq\\sqrt a+\\sqrt b+\\sqrt c','[1-z,1],[0,1]','[a,1/x-1],[b,1/y-1],[c,1/z-1],[x,2-y-z]',\n",
" dif2),\n",
" (r'\\frac{2a^3}{a^2+b^2}+\\frac{2b^3}{b^2+c^2}+\\frac{2c^3}{c^2+a^2}\\geq a+b+c',),\n",
" (r'\\frac{a^2}{b+c}+\\frac{b^2}{c+a}+\\frac{c^2}{a+b}\\geq \\frac12','[0,1-c],[0,1]','[a,1-b-c]'),\n",
" (r'\\frac{a+b}{2b+c}+\\frac{b+c}{2c+a}+\\frac{c+a}{2a+b}\\geq 2',),\n",
" #(r'\\frac x{5-y^2}+\\frac y{5-z^2}+\\frac z{5-x^2}\\geq \\frac34','[0,sqrt(5)],[0,sqrt(5)]','[x,1/(y*z)]')\n",
"]\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"def parser(formula,intervals='[]',subs='[]',func=dif):\n",
" newproof()\n",
" shiro.display=lambda x:None\n",
" #display=lambda x:None\n",
" if intervals=='':\n",
" intervals='[]'\n",
" if subs=='':\n",
" subs='[]'\n",
" formula=addbraces(formula)\n",
" formula=Sm(str(parse_latex(formula)))\n",
" formula,_=fractioncancel(formula)\n",
" formula=formula.subs(shiroindev._smakeiterable2(subs))\n",
" formula=makesubs(formula,intervals)\n",
" b,s=formula.lhs,formula.rhs\n",
" if type(formula)==LessThan:\n",
" b,s=s,b\n",
" formula=func(b,s)\n",
" formula=simplify(formula)\n",
" num,den=fractioncancel(formula)\n",
" return num"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
" 74%|███████▍ | 26/35 [00:21<00:07, 1.21it/s]\n"
]
},
{
"ename": "KeyboardInterrupt",
"evalue": "",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m<ipython-input-5-099273dd73e0>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0mineqs2\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mineq\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mtqdm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mineqs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mineqs2\u001b[0m\u001b[0;34m+=\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mparser\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mineq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m<ipython-input-4-03e6aa4471bf>\u001b[0m in \u001b[0;36mparser\u001b[0;34m(formula, intervals, subs, func)\u001b[0m\n\u001b[1;32m 16\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 17\u001b[0m \u001b[0mformula\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 18\u001b[0;31m \u001b[0mformula\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mformula\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 19\u001b[0m \u001b[0mnum\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mden\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mfractioncancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mformula\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 20\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mnum\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/simplify/simplify.py\u001b[0m in \u001b[0;36msimplify\u001b[0;34m(expr, ratio, measure, rational, inverse)\u001b[0m\n\u001b[1;32m 560\u001b[0m \u001b[0m_e\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 561\u001b[0m \u001b[0mexpr1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_e\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_mexpand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_e\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# issue 6829\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 562\u001b[0;31m \u001b[0mexpr2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 563\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 564\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mratio\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mInfinity\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/rationaltools.py\u001b[0m in \u001b[0;36mtogether\u001b[0;34m(expr, deep, fraction)\u001b[0m\n\u001b[1;32m 83\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 84\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 85\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0m_together\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
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"\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/exprtools.py\u001b[0m in \u001b[0;36mmul\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 864\u001b[0m \u001b[0mdenom\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdenom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdenom\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 865\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 866\u001b[0;31m \u001b[0mnumer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdenom\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnumer\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnormal\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdenom\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 867\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 868\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mTerm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnumer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdenom\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/exprtools.py\u001b[0m in \u001b[0;36mnormal\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 556\u001b[0m \u001b[0;32mdel\u001b[0m \u001b[0mother_factors\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfactor\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 557\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 558\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mFactors\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself_factors\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mFactors\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother_factors\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 559\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 560\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mdiv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;31m# Factors\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/exprtools.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, factors)\u001b[0m\n\u001b[1;32m 367\u001b[0m \u001b[0mhandle\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 368\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mfactors\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 369\u001b[0;31m \u001b[0;32mif\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mI\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 370\u001b[0m \u001b[0mhandle\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 371\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mhandle\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36m__eq__\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 333\u001b[0m \u001b[0;31m# (https://github.com/sympy/sympy/issues/4269), we only compare\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 334\u001b[0m \u001b[0;31m# types in Python 2 directly if they actually have __ne__.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 335\u001b[0;31m \u001b[0;32mif\u001b[0m \u001b[0mPY3\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__ne__\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__ne__\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 336\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mtself\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mtother\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 337\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mKeyboardInterrupt\u001b[0m: "
]
}
],
"source": [
"from tqdm import tqdm\n",
"ineqs2=[]\n",
"for ineq in tqdm(ineqs):\n",
" ineqs2+=[parser(*ineq)]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's look at the formulas when converted to polynomials."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"for i,ineq in zip(range(len(ineqs2)),ineqs2):\n",
" print(i)\n",
" display(reducegens(assumeall(ineq,positive=True)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Most formulas was converted to polynomials of independent variables. However formulas No. 22 and No. 31 were not. For this reason it's very unlikely that any method of proving these ones will succeed.\n",
"\n",
"Now let's try some methods of proving these inequalities. The first one would be the simple `prove`."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"tm=[0]*4"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"from collections import Counter\n",
"from timeit import default_timer as timer\n",
"start=timer()\n",
"t=[]\n",
"for ineq in ineqs2:\n",
" t+=[prove(ineq)]\n",
" print(t[-1],end=',')\n",
"tm[0]=timer()-start\n",
"print('\\n',tm[0],sep='')\n",
"Counter(t)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Code 0 means that the proof was found, all other codes means that proof wasn't found. So this method has proved 21 inequalities.\n",
"\n",
"The second method uses `findvalues` function, rationalizes the result numbers and gives them as additional parameter to `prove` function."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def cut(a):\n",
" if a<=0 or a>=100 or (a is None):\n",
" return 1\n",
" return a\n",
"start=timer()\n",
"t2=[]\n",
"for ineq in ineqs2:\n",
" numvalues=findvalues(ineq,disp=0)\n",
" values=nsimplify(numvalues,tolerance=0.1,rational=True)\n",
" values=list(map(cut,values))\n",
" t2+=[prove(ineq,values=values)]\n",
" print(t2[-1],end=',')\n",
"tm[1]=timer()-start\n",
"print('\\n',tm[1],sep='')\n",
"Counter(t2)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The third method is similar to the second one, but instead of rationalize values it squares, rationalizes and makes square roots of these values."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def cut(a):\n",
" if a<=0 or a>=1000 or (a is None):\n",
" return S(1)\n",
" return a\n",
" \n",
"start=timer()\n",
"t3=[]\n",
"for ineq in ineqs2:\n",
" numvalues=findvalues(ineq,disp=0)\n",
" numvalues=tuple([x**2 for x in numvalues])\n",
" values=nsimplify(numvalues,tolerance=0.1,rational=True)\n",
" values=[sqrt(x) for x in values]\n",
" values=list(map(cut,values))\n",
" t3+=[prove(ineq,values=values)]\n",
" print(t3[-1],end=',')\n",
"tm[2]=timer()-start\n",
"print('\\n',tm[2],sep='')\n",
"Counter(t3)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Finally, the fourth method is a slight modification to the third method. It does the same \"findvalues, square, rationalize and make square roots\" thing, but then it scales the values and runs it again. It can sometimes help with uniform formulas."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def betw(a):\n",
" return a>0.001 and a<1000 and a!=None\n",
"def cut(a):\n",
" if betw(a):\n",
" return a\n",
" return S(1)\n",
"\n",
"start=timer()\n",
"t4=[]\n",
"for ineq in ineqs2:\n",
" numvalues=findvalues(ineq,disp=0)\n",
" n=1\n",
" numvalues2=[]\n",
" for i in numvalues:\n",
" if betw(i):\n",
" n=1/i\n",
" break\n",
" for i in numvalues:\n",
" if betw(i):\n",
" numvalues2+=[i*n]\n",
" else:\n",
" numvalues2+=[1]\n",
" numvalues3=findvalues(ineq,values=numvalues2,disp=0)\n",
" numvalues4=tuple([x**2 for x in numvalues3])\n",
" values=nsimplify(numvalues4,tolerance=0.1,rational=True)\n",
" values=[sqrt(x) for x in values]\n",
" values=list(map(cut,values))\n",
" t4+=[prove(ineq,values=values)]\n",
" print(t4[-1],end=',')\n",
"tm[3]=timer()-start\n",
"print('\\n',tm[3],sep='')\n",
"Counter(t4)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"plt.bar(['1','2','3','4'],tm)\n",
"plt.ylabel('time [seconds]')\n",
"plt.xlabel('method')\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import pandas as pd\n",
"u=pd.DataFrame(zip(['']*len(ineqs),t,t2,t3,t4))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"plt.bar(['1','2','3','4'],[sum(u[i]==0)/len(ineqs2)*100 for i in range(1,5)])\n",
"plt.ylabel('inequalities proven [%]')\n",
"plt.xlabel('method')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Down below there are contingency tables and McNemar test for every pair of methods."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"for i in range(4):\n",
" for j in range(i+1,4):\n",
" display(pd.crosstab(u[i+1]==0, u[j+1]==0))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"from statsmodels.stats.contingency_tables import mcnemar\n",
"for i in range(4):\n",
" for j in range(i+1,4):\n",
" print(mcnemar(pd.crosstab(u[i+1]==0, u[j+1]==0)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In all cases p-value is greater than 0.05, so there is no statistical difference between the methods."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.3"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
Reference in New Issue Copy Permalink