From 8e2f98b0c0e15cc3de1b16fa1955c8e329ead11e Mon Sep 17 00:00:00 2001 From: urojony Date: Mon, 18 May 2020 20:49:46 +0200 Subject: [PATCH] replace pdfs with htmls --- .ipynb_checkpoints/examples-checkpoint.ipynb | 12 + .../statistics-checkpoint.ipynb | 1190 +- examples.html | 17915 ++++++++++++++++ examples.ipynb | 12 + examples.pdf | Bin 59577 -> 0 bytes statistics.html | 14995 +++++++++++++ statistics.ipynb | 7 - statistics.pdf | Bin 71326 -> 0 bytes 8 files changed, 34069 insertions(+), 62 deletions(-) create mode 100644 examples.html delete mode 100644 examples.pdf create mode 100644 statistics.html delete mode 100644 statistics.pdf diff --git a/.ipynb_checkpoints/examples-checkpoint.ipynb b/.ipynb_checkpoints/examples-checkpoint.ipynb index 4e27353..7453a7b 100644 --- a/.ipynb_checkpoints/examples-checkpoint.ipynb +++ b/.ipynb_checkpoints/examples-checkpoint.ipynb @@ -2556,6 +2556,9 @@ ], "source": [ "newproof()\n", + "#this is equivalent to\n", + "#prove(makesubs('(x-1)^4','[1,oo]'))\n", + "#prove(makesubs('(x-1)^4','[-oo,1]'))\n", "powerprove('(x-1)^4')" ] }, @@ -3406,6 +3409,15 @@ "source": [ "newproof()\n", "formula=Sm('-(3a + 2b + c)(2a^3 + 3b^2 + 6c + 1) + (4a + 4b + 4c)(a^4 + b^3 + c^2 + 3)')\n", + "#this is equivalent to\n", + "#prove(makesubs(formula,'[1,oo],[1,oo],[1,oo]'))\n", + "#prove(makesubs(formula,'[-1,oo],[1,oo],[1,oo]'))\n", + "#prove(makesubs(formula,'[1,oo],[-1,oo],[1,oo]'))\n", + "#prove(makesubs(formula,'[-1,oo],[-1,oo],[1,oo]'))\n", + "#prove(makesubs(formula,'[1,oo],[1,oo],[-1,oo]'))\n", + "#prove(makesubs(formula,'[-1,oo],[1,oo],[-1,oo]'))\n", + "#prove(makesubs(formula,'[1,oo],[-1,oo],[-1,oo]'))\n", + "#prove(makesubs(formula,'[-1,oo],[-1,oo],[-1,oo]'))\n", "powerprove(formula)" ] }, diff --git a/.ipynb_checkpoints/statistics-checkpoint.ipynb b/.ipynb_checkpoints/statistics-checkpoint.ipynb index ad6b447..d65b5bc 100644 --- a/.ipynb_checkpoints/statistics-checkpoint.ipynb +++ b/.ipynb_checkpoints/statistics-checkpoint.ipynb @@ -17,7 +17,6 @@ "metadata": {}, "outputs": [], "source": [ - "from importlib import reload\n", "from sympy import *\n", "import shiroindev\n", "from shiroindev import *\n", @@ -58,25 +57,20 @@ " 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", @@ -147,7 +141,6 @@ "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", @@ -175,34 +168,7 @@ "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\u001b[0m in \u001b[0;36m\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\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 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\u001b[0mfraction\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;36mquo\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 872\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 873\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mquo\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# Term\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 874\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mself\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[0minv\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[0m\u001b[1;32m 875\u001b[0m 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\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 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\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: " + "100%|██████████| 35/35 [00:46<00:00, 1.32s/it]\n" ] } ], @@ -222,9 +188,675 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 6, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{2} - ab - ac + b^{2} - bc + c^{2}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**2 - a*b - a*c + b**2 - b*c + c**2, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{2} - ab - ac - ad + b^{2} + c^{2} + d^{2}, a, b, c, d, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**2 - a*b - a*c - a*d + b**2 + c**2 + d**2, a, b, c, d, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "2\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{4}b^{2} + a^{3}b^{2} - a^{3}b - 2 a^{2}b + a^{2} + ab^{2} - ab + b^{2}, a, b, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**4*b**2 + a**3*b**2 - a**3*b - 2*a**2*b + a**2 + a*b**2 - a*b + b**2, a, b, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "3\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 a^{3}b + a^{3} - 4 a^{2}b^{2} - a^{2}b + a^{2} + 2 ab^{3} - ab^{2} - 2 ab + b^{3} + b^{2}, a, b, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*a**3*b + a**3 - 4*a**2*b**2 - a**2*b + a**2 + 2*a*b**3 - a*b**2 - 2*a*b + b**3 + b**2, a, b, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "4\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{3} - a^{2}b - ab^{2} + b^{3}, a, b, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**3 - a**2*b - a*b**2 + b**3, a, b, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "5\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 a^{3} - a^{2}b - a^{2}c - ab^{2} - ac^{2} + 2 b^{3} - b^{2}c - bc^{2} + 2 c^{3}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*a**3 - a**2*b - a**2*c - a*b**2 - a*c**2 + 2*b**3 - b**2*c - b*c**2 + 2*c**3, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "6\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 a^{3} - 3 a^{2}b + b^{3}, a, b, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*a**3 - 3*a**2*b + b**3, a, b, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "7\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{3} - a^{2}b - ac^{2} + b^{3} - b^{2}c + c^{3}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**3 - a**2*b - a*c**2 + b**3 - b**2*c + c**3, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "8\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{3}c + a^{3}d + a^{2}b^{2} - a^{2}bd - 2 a^{2}c^{2} - a^{2}cd + a^{2}d^{2} + ab^{3} - ab^{2}c - ab^{2}d - abc^{2} + ac^{3} - acd^{2} + b^{3}d + b^{2}c^{2} - 2 b^{2}d^{2} + bc^{3} - bc^{2}d - bcd^{2} + bd^{3} + c^{2}d^{2} + cd^{3}, a, b, c, d, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**3*c + a**3*d + a**2*b**2 - a**2*b*d - 2*a**2*c**2 - a**2*c*d + a**2*d**2 + a*b**3 - a*b**2*c - a*b**2*d - a*b*c**2 + a*c**3 - a*c*d**2 + b**3*d + b**2*c**2 - 2*b**2*d**2 + b*c**3 - b*c**2*d - b*c*d**2 + b*d**3 + c**2*d**2 + c*d**3, a, b, c, d, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "9\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 a^{5}b^{2} + 2 a^{5}bc + 2 a^{5}c^{2} + a^{4}b^{3} - a^{4}b^{2}c - a^{4}bc^{2} + a^{4}c^{3} + a^{3}b^{4} - 2 a^{3}b^{3}c - 4 a^{3}b^{2}c^{2} - 2 a^{3}bc^{3} + a^{3}c^{4} + 2 a^{2}b^{5} - a^{2}b^{4}c - 4 a^{2}b^{3}c^{2} - 4 a^{2}b^{2}c^{3} - a^{2}bc^{4} + 2 a^{2}c^{5} + 2 ab^{5}c - ab^{4}c^{2} - 2 ab^{3}c^{3} - ab^{2}c^{4} + 2 abc^{5} + 2 b^{5}c^{2} + b^{4}c^{3} + b^{3}c^{4} + 2 b^{2}c^{5}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*a**5*b**2 + 2*a**5*b*c + 2*a**5*c**2 + a**4*b**3 - a**4*b**2*c - a**4*b*c**2 + a**4*c**3 + a**3*b**4 - 2*a**3*b**3*c - 4*a**3*b**2*c**2 - 2*a**3*b*c**3 + a**3*c**4 + 2*a**2*b**5 - a**2*b**4*c - 4*a**2*b**3*c**2 - 4*a**2*b**2*c**3 - a**2*b*c**4 + 2*a**2*c**5 + 2*a*b**5*c - a*b**4*c**2 - 2*a*b**3*c**3 - a*b**2*c**4 + 2*a*b*c**5 + 2*b**5*c**2 + b**4*c**3 + b**3*c**4 + 2*b**2*c**5, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "10\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( ad + af - 2 \\sqrt{a}\\sqrt{b}\\sqrt{c}\\sqrt{d} - 2 \\sqrt{a}\\sqrt{b}\\sqrt{e}\\sqrt{f} + bc + be + cf - 2 \\sqrt{c}\\sqrt{d}\\sqrt{e}\\sqrt{f} + de, \\sqrt{a}, \\sqrt{b}, \\sqrt{c}, \\sqrt{d}, \\sqrt{e}, \\sqrt{f}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "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')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "11\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 3 a^{4}b + 3 a^{4}c - 2 a^{3}b^{2} + 2 a^{3}c^{2} + 2 a^{2}b^{3} - 6 a^{2}b^{2}c - 6 a^{2}bc^{2} - 2 a^{2}c^{3} + 3 ab^{4} - 6 ab^{2}c^{2} + 3 ac^{4} + 3 b^{4}c - 2 b^{3}c^{2} + 2 b^{2}c^{3} + 3 bc^{4}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(3*a**4*b + 3*a**4*c - 2*a**3*b**2 + 2*a**3*c**2 + 2*a**2*b**3 - 6*a**2*b**2*c - 6*a**2*b*c**2 - 2*a**2*c**3 + 3*a*b**4 - 6*a*b**2*c**2 + 3*a*c**4 + 3*b**4*c - 2*b**3*c**2 + 2*b**2*c**3 + 3*b*c**4, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "12\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 4 x^{8}y^{4} + 4 x^{7}y^{4} - 3 x^{5}y^{5} - 3 x^{5}y^{4} + 4 x^{4}y^{8} + 4 x^{4}y^{7} - 3 x^{4}y^{5} - 6 x^{4}y^{4} - 3 x^{4}y^{3} - 3 x^{3}y^{4} - 3 x^{3}y^{3} + 4 xy + 4, x, y, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(4*x**8*y**4 + 4*x**7*y**4 - 3*x**5*y**5 - 3*x**5*y**4 + 4*x**4*y**8 + 4*x**4*y**7 - 3*x**4*y**5 - 6*x**4*y**4 - 3*x**4*y**3 - 3*x**3*y**4 - 3*x**3*y**3 + 4*x*y + 4, x, y, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "13\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{3}c^{2} + a^{2}b^{3} - a^{2}b^{2}c - a^{2}bc^{2} - ab^{2}c^{2} + b^{2}c^{3}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**3*c**2 + a**2*b**3 - a**2*b**2*c - a**2*b*c**2 - a*b**2*c**2 + b**2*c**3, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "14\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{4}c + a^{3}c^{2} + a^{2}b^{3} + ab^{4} + b^{2}c^{3} + bc^{4}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**4*c + a**3*c**2 + a**2*b**3 + a*b**4 + b**2*c**3 + b*c**4, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "15\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{4}c - 4 a^{3}bc + a^{3}c^{2} + a^{2}b^{3} + 6 a^{2}b^{2}c - 2 a^{2}bc^{2} + ab^{4} - 4 ab^{3}c - 2 ab^{2}c^{2} + b^{2}c^{3} + bc^{4}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**4*c - 4*a**3*b*c + a**3*c**2 + a**2*b**3 + 6*a**2*b**2*c - 2*a**2*b*c**2 + a*b**4 - 4*a*b**3*c - 2*a*b**2*c**2 + b**2*c**3 + b*c**4, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "16\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{6}b^{3} + a^{6}c^{3} - 2 a^{5}b^{2}c^{2} + a^{3}b^{6} + a^{3}c^{6} - 2 a^{2}b^{5}c^{2} - 2 a^{2}b^{2}c^{5} + b^{6}c^{3} + b^{3}c^{6}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**6*b**3 + a**6*c**3 - 2*a**5*b**2*c**2 + a**3*b**6 + a**3*c**6 - 2*a**2*b**5*c**2 - 2*a**2*b**2*c**5 + b**6*c**3 + b**3*c**6, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "17\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 a^{8}b^{8} + 2 a^{7}b^{6} + 2 a^{6}b^{7} - 3 a^{6}b^{5} - 3 a^{5}b^{6} + 2 a^{5}b^{5} - 3 a^{5}b^{3} + 2 a^{5}b^{2} - 6 a^{4}b^{4} + 2 a^{4}b^{3} + 2 a^{4} - 3 a^{3}b^{5} + 2 a^{3}b^{4} - 3 a^{3}b^{2} + 2 a^{3}b + 2 a^{2}b^{5} - 3 a^{2}b^{3} + 2 ab^{3} + 2 b^{4}, a, b, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*a**8*b**8 + 2*a**7*b**6 + 2*a**6*b**7 - 3*a**6*b**5 - 3*a**5*b**6 + 2*a**5*b**5 - 3*a**5*b**3 + 2*a**5*b**2 - 6*a**4*b**4 + 2*a**4*b**3 + 2*a**4 - 3*a**3*b**5 + 2*a**3*b**4 - 3*a**3*b**2 + 2*a**3*b + 2*a**2*b**5 - 3*a**2*b**3 + 2*a*b**3 + 2*b**4, a, b, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "18\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{8} - a^{6}bc - 2 a^{5}b^{3} + a^{5}b^{2}c + a^{5}bc^{2} - 2 a^{5}c^{3} + 3 a^{4}b^{4} - a^{4}b^{3}c + 2 a^{4}b^{2}c^{2} - a^{4}bc^{3} + 3 a^{4}c^{4} - 2 a^{3}b^{5} - a^{3}b^{4}c - a^{3}b^{3}c^{2} - a^{3}b^{2}c^{3} - a^{3}bc^{4} - 2 a^{3}c^{5} + a^{2}b^{5}c + 2 a^{2}b^{4}c^{2} - a^{2}b^{3}c^{3} + 2 a^{2}b^{2}c^{4} + a^{2}bc^{5} - ab^{6}c + ab^{5}c^{2} - ab^{4}c^{3} - ab^{3}c^{4} + ab^{2}c^{5} - abc^{6} + b^{8} - 2 b^{5}c^{3} + 3 b^{4}c^{4} - 2 b^{3}c^{5} + c^{8}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**8 - a**6*b*c - 2*a**5*b**3 + a**5*b**2*c + a**5*b*c**2 - 2*a**5*c**3 + 3*a**4*b**4 - a**4*b**3*c + 2*a**4*b**2*c**2 - a**4*b*c**3 + 3*a**4*c**4 - 2*a**3*b**5 - a**3*b**4*c - a**3*b**3*c**2 - a**3*b**2*c**3 - a**3*b*c**4 - 2*a**3*c**5 + a**2*b**5*c + 2*a**2*b**4*c**2 - a**2*b**3*c**3 + 2*a**2*b**2*c**4 + a**2*b*c**5 - a*b**6*c + a*b**5*c**2 - a*b**4*c**3 - a*b**3*c**4 + a*b**2*c**5 - a*b*c**6 + b**8 - 2*b**5*c**3 + 3*b**4*c**4 - 2*b**3*c**5 + c**8, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "19\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( y^{18}z^{9} + 2 y^{16}z^{14} - y^{15}z^{9} + 2 y^{14}z^{16} - y^{14}z^{13} - y^{13}z^{14} - y^{13}z^{11} - y^{12}z^{12} + y^{12}z^{9} - y^{11}z^{13} - y^{11}z^{7} + 2 y^{11}z^{4} - y^{10}z^{5} + y^{9}z^{18} - y^{9}z^{15} + y^{9}z^{12} - y^{9}z^{6} - y^{8}z^{4} - y^{7}z^{11} - y^{7}z^{5} + 2 y^{7}z^{2} - y^{6}z^{9} + y^{6}z^{6} - y^{5}z^{10} - y^{5}z^{7} + 2 y^{4}z^{11} - y^{4}z^{8} - y^{3}z^{3} + 2 y^{2}z^{7} + 1, y, z, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(y**18*z**9 + 2*y**16*z**14 - y**15*z**9 + 2*y**14*z**16 - y**14*z**13 - y**13*z**14 - y**13*z**11 - y**12*z**12 + y**12*z**9 - y**11*z**13 - y**11*z**7 + 2*y**11*z**4 - y**10*z**5 + y**9*z**18 - y**9*z**15 + y**9*z**12 - y**9*z**6 - y**8*z**4 - y**7*z**11 - y**7*z**5 + 2*y**7*z**2 - y**6*z**9 + y**6*z**6 - y**5*z**10 - y**5*z**7 + 2*y**4*z**11 - y**4*z**8 - y**3*z**3 + 2*y**2*z**7 + 1, y, z, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "20\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{3} - a^{2}b - a^{2}c - ab^{2} + 3 abc - ac^{2} + b^{3} - b^{2}c - bc^{2} + c^{3}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**3 - a**2*b - a**2*c - a*b**2 + 3*a*b*c - a*c**2 + b**3 - b**2*c - b*c**2 + c**3, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "21\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 y^{4}z^{2} - y^{3}z^{3} - 5 y^{3}z + 2 y^{2}z^{4} - 9 y^{2}z^{2} + 5 y^{2} - 5 yz^{3} + 23 yz + 5 z^{2} - 9, y, z, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*y**4*z**2 - y**3*z**3 - 5*y**3*z + 2*y**2*z**4 - 9*y**2*z**2 + 5*y**2 - 5*y*z**3 + 23*y*z + 5*z**2 - 9, y, z, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "22\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 a^{\\frac{5}{2}}b\\frac{1}{\\sqrt{a b + a + b + 1}} + 2 a^{\\frac{5}{2}}\\frac{1}{\\sqrt{a b + a + b + 1}} + 2 a^{2}b^{\\frac{7}{2}}\\frac{1}{\\sqrt{b + 1}} + 2 a^{2}b^{\\frac{5}{2}}\\frac{1}{\\sqrt{b + 1}} + 2 a^{2}b^{2}\\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}} - a^{2}b^{2}\\sqrt{3} + 2 a^{2}b\\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}} - a^{2}b\\sqrt{3} + 2 a^{\\frac{3}{2}}b\\frac{1}{\\sqrt{a b + a + b + 1}} + 4 ab^{\\frac{7}{2}}\\frac{1}{\\sqrt{b + 1}} + 4 ab^{\\frac{5}{2}}\\frac{1}{\\sqrt{b + 1}} + 4 ab^{2}\\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}} - 2 ab^{2}\\sqrt{3} + 2 ab^{\\frac{3}{2}}\\frac{1}{\\sqrt{b + 1}} + 6 ab\\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}} - 2 ab\\sqrt{3} + 2 a\\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}} - a\\sqrt{3} + 2 b^{\\frac{7}{2}}\\frac{1}{\\sqrt{b + 1}} + 2 b^{\\frac{5}{2}}\\frac{1}{\\sqrt{b + 1}} + 2 b^{2}\\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}} - b^{2}\\sqrt{3} + 4 b\\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}} - b\\sqrt{3} + 2 \\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}}, \\sqrt{a}, \\sqrt{b}, \\frac{1}{a b \\sqrt{a b + a + b + 1} + a \\sqrt{a b + a + b + 1} + b \\sqrt{a b + a + b + 1} + \\sqrt{a b + a + b + 1}}, \\frac{1}{\\sqrt{a b + a + b + 1}}, \\frac{1}{\\sqrt{b + 1}}, \\sqrt{3}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*(sqrt(a))**5*(sqrt(b))**2*(1/sqrt(a*b + a + b + 1)) + 2*(sqrt(a))**5*(1/sqrt(a*b + a + b + 1)) + 2*(sqrt(a))**4*(sqrt(b))**7*(1/sqrt(b + 1)) + 2*(sqrt(a))**4*(sqrt(b))**5*(1/sqrt(b + 1)) + 2*(sqrt(a))**4*(sqrt(b))**4*(1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1))) - (sqrt(a))**4*(sqrt(b))**4*(sqrt(3)) + 2*(sqrt(a))**4*(sqrt(b))**2*(1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1))) - (sqrt(a))**4*(sqrt(b))**2*(sqrt(3)) + 2*(sqrt(a))**3*(sqrt(b))**2*(1/sqrt(a*b + a + b + 1)) + 4*(sqrt(a))**2*(sqrt(b))**7*(1/sqrt(b + 1)) + 4*(sqrt(a))**2*(sqrt(b))**5*(1/sqrt(b + 1)) + 4*(sqrt(a))**2*(sqrt(b))**4*(1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1))) - 2*(sqrt(a))**2*(sqrt(b))**4*(sqrt(3)) + 2*(sqrt(a))**2*(sqrt(b))**3*(1/sqrt(b + 1)) + 6*(sqrt(a))**2*(sqrt(b))**2*(1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1))) - 2*(sqrt(a))**2*(sqrt(b))**2*(sqrt(3)) + 2*(sqrt(a))**2*(1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1))) - (sqrt(a))**2*(sqrt(3)) + 2*(sqrt(b))**7*(1/sqrt(b + 1)) + 2*(sqrt(b))**5*(1/sqrt(b + 1)) + 2*(sqrt(b))**4*(1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1))) - (sqrt(b))**4*(sqrt(3)) + 4*(sqrt(b))**2*(1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1))) - (sqrt(b))**2*(sqrt(3)) + 2*(1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1))), sqrt(a), sqrt(b), 1/(a*b*sqrt(a*b + a + b + 1) + a*sqrt(a*b + a + b + 1) + b*sqrt(a*b + a + b + 1) + sqrt(a*b + a + b + 1)), 1/sqrt(a*b + a + b + 1), 1/sqrt(b + 1), sqrt(3), domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "23\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{4} - 4 abcd + b^{4} + c^{4} + d^{4}, a, b, c, d, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**4 - 4*a*b*c*d + b**4 + c**4 + d**4, a, b, c, d, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "24\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{3}b^{2} + a^{3}c^{2} + a^{2}b^{3} - 2 a^{2}b^{2}c - 2 a^{2}bc^{2} + a^{2}c^{3} - 2 ab^{2}c^{2} + b^{3}c^{2} + b^{2}c^{3}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**3*b**2 + a**3*c**2 + a**2*b**3 - 2*a**2*b**2*c - 2*a**2*b*c**2 + a**2*c**3 - 2*a*b**2*c**2 + b**3*c**2 + b**2*c**3, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "25\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( def + de + df - 2 \\sqrt{d}\\sqrt{e} - 2 \\sqrt{d}\\sqrt{f} + ef - 2 \\sqrt{e}\\sqrt{f} + f + 2, \\sqrt{d}, \\sqrt{e}, \\sqrt{f}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly((sqrt(d))**2*(sqrt(e))**2*(sqrt(f))**2 + (sqrt(d))**2*(sqrt(e))**2 + (sqrt(d))**2*(sqrt(f))**2 - 2*(sqrt(d))*(sqrt(e)) - 2*(sqrt(d))*(sqrt(f)) + (sqrt(e))**2*(sqrt(f))**2 - 2*(sqrt(e))*(sqrt(f)) + (sqrt(f))**2 + 2, sqrt(d), sqrt(e), sqrt(f), domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "26\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( de^{2}f^{2} + de^{2}f + 2 def^{2} - 6 def + de + df^{2} + df + e^{2}f^{2} + e^{2}f + 2 ef^{2} + 3 ef + e + f^{2} + 2 f + 1, d, e, f, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(d*e**2*f**2 + d*e**2*f + 2*d*e*f**2 - 6*d*e*f + d*e + d*f**2 + d*f + e**2*f**2 + e**2*f + 2*e*f**2 + 3*e*f + e + f**2 + 2*f + 1, d, e, f, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "27\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( s^{2}def + s^{2}, s, d, e, f, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(s**2*d*e*f + s**2, s, d, e, f, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "28\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( -2 \\sqrt{3}x_{1}x_{2} - 2 \\sqrt{3}x_{2}x_{3} - 2 \\sqrt{3}x_{3}x_{4} - 2 \\sqrt{3}x_{4}x_{5} + 3 x_{1}^{2} + 3 x_{2}^{2} + 3 x_{3}^{2} + 3 x_{4}^{2} + 3 x_{5}^{2}, \\sqrt{3}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(-2*(sqrt(3))*x_1*x_2 - 2*(sqrt(3))*x_2*x_3 - 2*(sqrt(3))*x_3*x_4 - 2*(sqrt(3))*x_4*x_5 + 3*x_1**2 + 3*x_2**2 + 3*x_3**2 + 3*x_4**2 + 3*x_5**2, sqrt(3), x_1, x_2, x_3, x_4, x_5, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "29\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 7 a^{2}b^{3} - 6 a^{2}b^{2} - 6 a^{2}b + 7 a^{2} + 14 ab^{3} - 12 ab^{2} + 15 ab - 13 a + 7 b^{3} - 6 b^{2} - 6 b + 7, a, b, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(7*a**2*b**3 - 6*a**2*b**2 - 6*a**2*b + 7*a**2 + 14*a*b**3 - 12*a*b**2 + 15*a*b - 13*a + 7*b**3 - 6*b**2 - 6*b + 7, a, b, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "30\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{2}b^{2} + a^{2}b + 2 ab^{2} + ab + a + b^{2} + b, a, b, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**2*b**2 + a**2*b + 2*a*b**2 + a*b + a + b**2 + b, a, b, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "31\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( -2 \\frac{1}{\\sqrt{b}}\\sqrt{\\frac{a^{2} b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a^{2} b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1}} - 2 \\frac{1}{\\sqrt{b}}\\sqrt{\\frac{a b}{a^{2} b + a^{2} + a b + 2 a + 1} + \\frac{b}{a^{2} b + a^{2} + a b + 2 a + 1}} - 2 \\sqrt{\\frac{a^{2} b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a^{2} b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1}}\\sqrt{\\frac{a b}{a^{2} b + a^{2} + a b + 2 a + 1} + \\frac{b}{a^{2} b + a^{2} + a b + 2 a + 1}} + 3, \\frac{1}{\\sqrt{b}}, \\sqrt{\\frac{a^{2} b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a^{2} b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \\frac{a b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1}}, \\sqrt{\\frac{a b}{a^{2} b + a^{2} + a b + 2 a + 1} + \\frac{b}{a^{2} b + a^{2} + a b + 2 a + 1}}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(-2*(1/sqrt(b))*(sqrt(a**2*b**2/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a**2*b/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a*b**2/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a*b/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1))) - 2*(1/sqrt(b))*(sqrt(a*b/(a**2*b + a**2 + a*b + 2*a + 1) + b/(a**2*b + a**2 + a*b + 2*a + 1))) - 2*(sqrt(a**2*b**2/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a**2*b/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a*b**2/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a*b/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1)))*(sqrt(a*b/(a**2*b + a**2 + a*b + 2*a + 1) + b/(a**2*b + a**2 + a*b + 2*a + 1))) + 3, 1/sqrt(b), sqrt(a**2*b**2/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a**2*b/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a*b**2/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1) + a*b/(a**2*b + a**2 + a*b**2 + 3*a*b + 2*a + b**2 + 2*b + 1)), sqrt(a*b/(a**2*b + a**2 + a*b + 2*a + 1) + b/(a**2*b + a**2 + a*b + 2*a + 1)), domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "32\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{5}b^{2} + a^{5}c^{2} + a^{4}b^{3} - a^{4}b^{2}c - a^{4}bc^{2} - a^{4}c^{3} - a^{3}b^{4} + a^{3}c^{4} + a^{2}b^{5} - a^{2}b^{4}c - a^{2}bc^{4} + a^{2}c^{5} - ab^{4}c^{2} - ab^{2}c^{4} + b^{5}c^{2} + b^{4}c^{3} - b^{3}c^{4} + b^{2}c^{5}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(a**5*b**2 + a**5*c**2 + a**4*b**3 - a**4*b**2*c - a**4*b*c**2 - a**4*c**3 - a**3*b**4 + a**3*c**4 + a**2*b**5 - a**2*b**4*c - a**2*b*c**4 + a**2*c**5 - a*b**4*c**2 - a*b**2*c**4 + b**5*c**2 + b**4*c**3 - b**3*c**4 + b**2*c**5, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "33\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 a^{2}d^{3} - a^{2}d^{2} - a^{2}d + 2 a^{2} + 4 ad^{3} - 2 ad^{2} - 3 a + 2 d^{3} - d^{2} - d + 2, a, d, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*a**2*d**3 - a**2*d**2 - a**2*d + 2*a**2 + 4*a*d**3 - 2*a*d**2 - 3*a + 2*d**3 - d**2 - d + 2, a, d, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "34\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 2 a^{3} - 3 a^{2}b + a^{2}c + ab^{2} - 3 ac^{2} + 2 b^{3} - 3 b^{2}c + bc^{2} + 2 c^{3}, a, b, c, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(2*a**3 - 3*a**2*b + a**2*c + a*b**2 - 3*a*c**2 + 2*b**3 - 3*b**2*c + b*c**2 + 2*c**3, a, b, c, domain='ZZ')" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "for i,ineq in zip(range(len(ineqs2)),ineqs2):\n", " print(i)\n", @@ -242,7 +874,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, "metadata": {}, "outputs": [], "source": [ @@ -251,9 +883,28 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0,2,2,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,2,2,0,0,2,2,0,2,2,0,2,0,2,0,\n", + "20.714609994000057\n" + ] + }, + { + "data": { + "text/plain": [ + "Counter({0: 21, 2: 14})" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "from collections import Counter\n", "from timeit import default_timer as timer\n", @@ -278,9 +929,28 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 9, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,2,2,0,0,2,0,0,2,2,0,2,0,2,0,\n", + "21.832711064998875\n" + ] + }, + { + "data": { + "text/plain": [ + "Counter({0: 24, 2: 11})" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "def cut(a):\n", " if a<=0 or a>=100 or (a is None):\n", @@ -308,9 +978,28 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 10, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,2,2,0,0,2,0,0,2,2,0,2,0,2,0,\n", + "21.697120987002563\n" + ] + }, + { + "data": { + "text/plain": [ + "Counter({0: 23, 2: 12})" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "def cut(a):\n", " if a<=0 or a>=1000 or (a is None):\n", @@ -341,9 +1030,28 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 11, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,2,2,0,0,2,0,0,0,2,0,2,0,2,0,\n", + "23.187820409999404\n" + ] + }, + { + "data": { + "text/plain": [ + "Counter({0: 24, 2: 11})" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "def betw(a):\n", " return a>0.001 and a<1000 and a!=None\n", @@ -381,9 +1089,20 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 17, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "image/png": 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SxSQ3JvlAkg1JPpLk6Um+kOTmJMcl2b97rsSXuzHpT0nyYOCPgRd3N7m9uFvlUUmuTHJLktcu2c7ru/VvSPK6Je1vSnJTks8CR872p5d+mjeUqUndiKbfYDKq6fVM7r79GnAmcDJwBvB14OtV9eFuuIYvdfO/EFhVVWd16zoXOBF4CnAAcBPw88DRTJ5LcTwQJqOnns7kH7ALgCcweW74V4C/rqp3DfkzS9szysPrpT3EN6vqOoAk1wNrq6qSXAcsMhm18+Ql5+8fAqzczro+XVX3APck2QQcBvwGcGlV/bDbxiXAk5gEwaVVdXfXftkgP53Uk0Gglt2z5PP9S6bvZ/K7cR/w/Kq6aelCSZ4wZV33dctnmW17KK49htcIpO37DPC73UioJDm2a9/C5BTQNFcBz0vysCT7A6cCn+/aT03y0CQHAM/d/aVL/RkE0va9nckjHq9NsqGbBljH5OLw0ovFP6N71OYFTK4tfBH4QFV9tWv/GHANk+ctfH64H0GazovFktQ4jwgkqXEGgSQ1ziCQpMYZBJLUOINAkhpnEEhS4wwCSWqcQSBJjfs/ZA6/DVKt+QAAAAAASUVORK5CYII=\n", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import matplotlib.pyplot as plt\n", "plt.bar(['1','2','3','4'],tm)\n", @@ -394,7 +1113,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 13, "metadata": {}, "outputs": [], "source": [ @@ -404,9 +1123,20 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 14, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "image/png": 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In this notebook there are presented examples of usage of shiroin, a python library for proving inequalities of multivariate polynomials.

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At the beginning we need to load the packages.

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In [1]:
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from sympy import *
+from shiroindev import *
+from IPython.display import Latex
+shiro.seed=1
+shiro.display=lambda x:display(Latex(x))
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shiro.seed=1 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. This line is here only for the sake of reproducibility.

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The next line provides 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.

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Now let's make some proofs. We will use problems from https://www.imomath.com/index.php?options=593&lmm=0.

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Problem 1

Prove the inequality $a^2+b^2+c^2\ge ab+bc+ca$, if $a,b,c$ are real numbers.

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Function prove tries to prove that given formula is nonnegative, assuming all variables are nonnegative. In this case the nonnegativity assumption is not a problem, since all powers on the left side are even, so if $|a|^2+|b|^2+|c|^2 \ge |ab|+|ac|+|bc|,$ then $a^2+b^2+c^2= |a|^2+|b|^2+|c|^2 \ge |ab|+|ac|+|bc| \ge ab+ac+bc$.

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In [2]:
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prove('(a^2+b^2+c^2-a*b-a*c-b*c)')
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+ +
+ + + + +
+numerator: $a^{2} - a b - a c + b^{2} - b c + c^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+Program couldn't find a solution with integer coefficients. Try to multiple the formula by some integer and run this function again. +
+ +
+ +
+ +
+ + + + +
+$$ a b+a c+b c \le a^{2}+b^{2}+c^{2} $$ +
+ +
+ +
+ +
Out[2]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Function prove prints several things. The first two gives us a formula after expanding it. The next one is status, which is the return status of the first use of scipy.optimize.linprog. Possible outputs and explanations are

+
    +
  • 0 - found a proof with real coefficients,
    • +
  • 1 - need more time,
    • +
  • 2 - function didn't find a proof,
    • +
  • 3,4 - loss of precision (which may happen if it has to work with big numbers).
    • +
+

Then we've got a hint. So let's use it!

+ +
+
+
+
+
+
In [3]:
+
+
+
prove('(a^2+b^2+c^2-a*b-a*c-b*c)*2')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $2 a^{2} - 2 a b - 2 a c + 2 b^{2} - 2 b c + 2 c^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$2 a b \le a^{2}+b^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$2 a c \le a^{2}+c^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$2 b c \le b^{2}+c^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 0 $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[3]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Problem 2

Find all real numbers such that $a^2+b^2+c^2+d^2=a(b+c+d)$.

+ +
+
+
+
+
+
+

At first glance it doesn't look like an inequality problem, but actually it is one. If you try to calculate both sides for different values, you can see that the left side of the equation is never less than the right one. So let's try

+ +
+
+
+
+
+
In [4]:
+
+
+
prove('a^2+b^2+c^2+d^2-a*(b+c+d)')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $a^{2} - a b - a c - a d + b^{2} + c^{2} + d^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 2 +
+ +
+ +
+ +
+ + + + +
+Program couldn't find any proof. +
+ +
+ +
+ +
+ + + + +
+$$ a b+a c+a d \le a^{2}+b^{2}+c^{2}+d^{2} $$ +
+ +
+ +
+ +
Out[4]:
+ + + + +
+
2
+
+ +
+ +
+
+ +
+
+
+
+

This time prove didn't found the proof. But it doesn't mean that the inequality is not true! prove uses a list of $n$ positive values, where $n$ is a number of variables in the formula. List of values should correspond to the list of variables in alphabetical order. Here are a few tips how to choose the right values.

+
    +
  1. Consider a function $pos(values)$ which is the sum of the positive addends in the formula after substitution of values to variables. Analogically, let $neg(values)$ be the sum of the negative addends. We should choose such values for which $quotient=pos(values)/|neg(values)|$ is small.
    2. +
  2. The symmetry group of the formula splits set of variables into orbits. Using the same values for variables in one orbit is recommended. In particular, if the symmetry group of the formula is transitive (for example, when the formula is cyclic), then all values (probably) should be the same.
    3. +
  3. If the formula is homogeneous, then $values=(a_1,a_2,...,a_n)$ provide a proof iff $values=(ka_1,ka_2,...,ka_n)$ provides a proof for any $k\in Q_+$ (as long as you don't run into overflow error).
    4. +
+

In the formula above $b,c,d$ are in one orbit and the formula is homogenous, so let's try $a=2$ and $b=c=d=1$.

+ +
+
+
+
+
+
In [5]:
+
+
+
prove('a^2+b^2+c^2+d^2-a*(b+c+d)','2,1,1,1')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $a\to 2 e$ +
+ +
+ +
+ +
+ + + + +
+numerator: $b^{2} - 2 b e + c^{2} - 2 c e + d^{2} - 2 d e + 4 e^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$2 b e \le b^{2}+e^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$2 c e \le c^{2}+e^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$2 d e \le d^{2}+e^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le e^{2} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[5]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Function makes a substitution $a\to 2e$ and try to prove new inequality. This time it succeeded. Moreover, if starting formula is equal to 0, then all these inequalities have to be equalities, so $e^2=0$ and eventually $a=0$. We can also try a little bit lower value for $a$.

+ +
+
+
+
+
+
In [6]:
+
+
+
prove('a^2+b^2+c^2+d^2-a*(b+c+d)','7/4,1,1,1')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $a\to \frac{7 f}{4}$ +
+ +
+ +
+ +
+ + + + +
+numerator: $16 b^{2} - 28 b f + 16 c^{2} - 28 c f + 16 d^{2} - 28 d f + 49 f^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $16$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$28 b f \le 14 b^{2}+14 f^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$28 c f \le 14 c^{2}+14 f^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$28 d f \le 14 d^{2}+14 f^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 2 b^{2}+2 c^{2}+2 d^{2}+7 f^{2} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[6]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Now we can see that if $a^2+b^2+c^2+d^2-a(b+c+d)=0$, then $7f^2+2b^2+2c^2+2d^2=0$ and eventually $a=b=c=d=0$. Note that inequality is proved only for positive numbers (which, by continuity, can be expanded to nonnegative numbers). But using similar argumentation to the one in previous problem, if $(a,b,c,d)=(x,y,z,t)$ is the solution of $a^2+b^2+c^2+d^2-a(b+c+d)=0$, then $(a,b,c,d)=(|x|,|y|,|z|,|t|)$ is a solution, too. Since the only nonnegative solution is $(0,0,0,0)$, it means that it is the only solution.

+

It is worth noting that this time function prove used $f$ as a new variable instead of $e$. If you want to start a new proof and you don't care about the collision of variables from previous proofs, you can use newproof function, which clears the set of used variables.

+

We can also use the findvalues function to find values for the proof more automatically. It looks for (local) minimum of the $quotient$ value defined above.

+ +
+
+
+
+
+
In [7]:
+
+
+
formula=S('a^2+b^2+c^2+d^2-a*(b+c+d)')
+numvalues=findvalues(formula)
+numvalues
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
Optimization terminated successfully.
+         Current function value: 1.154701
+         Iterations: 68
+         Function evaluations: 127
+
+
+
+ +
+ +
Out[7]:
+ + + + +
+
(1.4339109663193974,
+ 0.8278441585048405,
+ 0.8279027492686561,
+ 0.8278930696996669)
+
+ +
+ +
+
+ +
+
+
+
+

If the $quotient$ value were less than 1, that would mean that the formula is negative for given values. If $quotient$ were equal to 1, then we have to choose exactly these values (or other values for which the $quotient$ is equal to 1. But here $quotient$ is greater than 1, so we can take a point near it and (probably) still have a proof. The values given to the prove function must not be floating point numbers, so we can rationalize them.

+ +
+
+
+
+
+
In [8]:
+
+
+
values=nsimplify(numvalues,tolerance=0.1,rational=True)
+values
+
+ +
+
+
+ +
+
+ + +
+ +
Out[8]:
+ + + + +
+$\displaystyle \left( \frac{10}{7}, \ \frac{5}{6}, \ \frac{5}{6}, \ \frac{5}{6}\right)$ +
+ +
+ +
+
+ +
+
+
+
In [9]:
+
+
+
newproof()
+prove(formula,values)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $a\to \frac{10 e}{7}$ +
+ +
+ +
+ +
+ + + + +
+Substitute $b\to \frac{5 f}{6}$ +
+ +
+ +
+ +
+ + + + +
+Substitute $c\to \frac{5 g}{6}$ +
+ +
+ +
+ +
+ + + + +
+Substitute $d\to \frac{5 h}{6}$ +
+ +
+ +
+ +
+ + + + +
+numerator: $3600 e^{2} - 2100 e f - 2100 e g - 2100 e h + 1225 f^{2} + 1225 g^{2} + 1225 h^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1764$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$2100 e f \le 1050 e^{2}+1050 f^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$2100 e g \le 1050 e^{2}+1050 g^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$2100 e h \le 1050 e^{2}+1050 h^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 450 e^{2}+175 f^{2}+175 g^{2}+175 h^{2} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[9]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

If you set the tolerance bigger, then the values will have smaller numerators and denominators, so coefficients in the proof will be smaller, too. But if the tolerance is too big, then proof will not be found.

+

Let's skip the problem 3 and look solve the problem 4 instead.

+

Problem 4

If $x$ and $y$ are two positive numbers less than 1, prove that +$$\frac{1}{1-x^2}+\frac{1}{1-y^2}\ge \frac{2}{1-xy}.$$

+ +
+
+
+
+
+
In [10]:
+
+
+
prove('1/(1-x^2)+1/(1-y^2)-2/(1-x*y)')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $- x^{3} y + 2 x^{2} y^{2} - x^{2} - x y^{3} + 2 x y - y^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $x^{3} y^{3} - x^{3} y - x^{2} y^{2} + x^{2} - x y^{3} + x y + y^{2} - 1$ +
+ +
+ +
+ +
+ + + + +
+status: 2 +
+ +
+ +
+ +
+ + + + +
+Program couldn't find any proof. +
+ +
+ +
+ +
+ + + + +
+$$ x^{3} y+x^{2}+x y^{3}+y^{2} \le 2 x^{2} y^{2}+2 x y $$ +
+ +
+ +
+ +
+ + + + +
+It looks like the formula is symmetric. You can assume without loss of generality that x >= y. Try +
+ +
+ +
+ +
+ + + + +
+prove(makesubs(S("-x**3*y + 2*x**2*y**2 - x**2 - x*y**3 + 2*x*y - y**2"),[('y', 'oo')]) +
+ +
+ +
+ +
Out[10]:
+ + + + +
+
2
+
+ +
+ +
+
+ +
+
+
+
+

prove assumes that formula is well-defined if all variables are positive, so it doesn't have to analyze the denominator (except of choosing the right sign). In this case it is not true, since if $x=1$, then $1-x^2=0$. Also denominator is equal to $(x^2-1)(y^2-1)(xy-1)$ which is negative for $x,y\in (0,1)$. So we need to make some substitution after which new variables can have all positive values, not just these inside (0,1) interval.

+

We will use a function makesubs to generate these substitutions. It has three basic parameters: formula, intervals and values. intervals are current limitations of variables, values are values of variables for which $quotient$ of formula is small. values should be inside corresponding intervals. This argument is optional but it's better to use it. +Let's go back to our problem. If $x=y$, then $\frac{1}{1-x^2}+\frac{1}{1-y^2}\ge \frac{2}{1-xy}$, so it's the minimum value of the formula. So let values=(1/2,1/2) (warning: do not use decimal point, for example '0.5,0.5').

+ +
+
+
+
+
+
In [11]:
+
+
+
newproof()
+newformula,newvalues=makesubs('1/(1-x^2)+1/(1-y^2)-2/(1-x*y)','[0,1],[0,1]','1/2,1/2')
+prove(newformula*3,newvalues)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $x\to 1 - \frac{1}{a + 1}$ +
+ +
+ +
+ +
+ + + + +
+Substitute $y\to 1 - \frac{1}{b + 1}$ +
+ +
+ +
+ +
+ + + + +
+numerator: $6 a^{3} b + 3 a^{3} - 12 a^{2} b^{2} - 3 a^{2} b + 3 a^{2} + 6 a b^{3} - 3 a b^{2} - 6 a b + 3 b^{3} + 3 b^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $4 a^{2} b + 2 a^{2} + 4 a b^{2} + 8 a b + 3 a + 2 b^{2} + 3 b + 1$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$12 a^{2} b^{2} \le 6 a^{3} b+6 a b^{3}$$ +
+ +
+ +
+ +
+ + + + +
+$$3 a^{2} b \le 2 a^{3}+b^{3}$$ +
+ +
+ +
+ +
+ + + + +
+$$3 a b^{2} \le a^{3}+2 b^{3}$$ +
+ +
+ +
+ +
+ + + + +
+$$6 a b \le 3 a^{2}+3 b^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 0 $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[11]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Now let's get back to problem 3.

+

Problem 3

If $a,b,c$ are positive real numbers that satisfy $a^2+b^2+c^2=1$, find the minimal value of +$$\frac{a^2b^2}{c^2}+\frac{b^2c^2}{a^2}+\frac{c^2a^2}{b^2}$$

+

The problem is equivalent to finding minimum of $xy/z+yz/x+zx/y$ assuming $x+y+z=1$ and $x,y,z>0$. The first idea is to suppose that the minimum is reached when $x=y=z$. In that case, $x=y=z=1/3$ and formula is equal to 1. Now we can substitute $z\to 1-x-y$. Constraints for variables are $x>0$, $y>0$, $x+y<1$. We can rewrite it as $x \in (0,1-y)$, $y \in (0,1)$. These two conditions have two important properties:

+
    +
  • constraints for variables are written as intervals,
    • +
  • there are no "backwards dependencies", i.e. there is no $x$ in the interval of $y$.
    • +
+

If these two conditions hold, then you can use makesubs function.

+ +
+
+
+
+
+
In [12]:
+
+
+
newproof()
+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. +
+ +
+ +
+ +
Out[12]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

The proof is found, so the assumption that 1 is the minimum of xy/z+yz/x+zx/y was good.

+

Functions S and Sm creates a SymPy object from a string. The only difference is that Sm assumes that there are no multi-letter variables and adds a multiplication sign between every two terms which has no operator sign, so object Sm(xy/z+yz/x+zx/y) has 3 variables x,y,z and S('xy/z+yz/x+zx/y') has 6 variables x,y,z,xy,yz,zx.

+

As you may have noticed, formulas are often cyclic or symmetric. Therefore you can use cyclize or symmetrize function to reduce the length of the written formula. Here are a few commands which will do the same as each other.

+ +
+
+
+
+
+
In [13]:
+
+
+
prove('(a^2+b^2+c^2-a*b-a*c-b*c)*2')
+#prove(S('(a^2+b^2+c^2-a*b-a*c-b*c)*2'))
+#prove(Sm('2(a^2+b^2+c^2-ab-ac-bc)'))
+#prove(cyclize('2*a^2-2*a*b'))
+#prove(symmetrize('a^2-a*b'))
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $2 a^{2} - 2 a b - 2 a c + 2 b^{2} - 2 b c + 2 c^{2}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$2 a b \le a^{2}+b^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$2 a c \le a^{2}+c^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$2 b c \le b^{2}+c^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 0 $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[13]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Now look at formula $(x-1)^4$. It's quite obvious that it's nonnegative, but prove fails to show this!

+ +
+
+
+
+
+
In [14]:
+
+
+
prove('(x-1)^4')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 2 +
+ +
+ +
+ +
+ + + + +
+Program couldn't find any proof. +
+ +
+ +
+ +
+ + + + +
+$$ 4 x^{3}+4 x \le x^{4}+6 x^{2}+1 $$ +
+ +
+ +
+ +
Out[14]:
+ + + + +
+
2
+
+ +
+ +
+
+ +
+
+
+
+

But there is a relatively simple method to generate a proof using this library. We will make to proofs: one for $x\in (1,\infty)$ and the second one for $(-\infty,1)$.

+ +
+
+
+
+
+
In [15]:
+
+
+
newproof()
+prove(makesubs('(x-1)^4','(1,oo)'))
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $x\to a + 1$ +
+ +
+ +
+ +
+ + + + +
+numerator: $a^{4}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le a^{4} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[15]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
In [16]:
+
+
+
newproof()
+prove(makesubs('(x-1)^4','(-oo,1)'))
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $x\to 1 - a$ +
+ +
+ +
+ +
+ + + + +
+numerator: $a^{4}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le a^{4} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[16]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Now let's go to the problem 10

+

Problem 10

If $a,b,c,d>0$, prove that +$$\frac a{b+c}+\frac b{c+d}+ \frac c{d+a}+ \frac d{a+b}\geq 2.$$

+

Let's try a simple approach.

+ +
+
+
+
+
+
In [17]:
+
+
+
formula=cyclize('a/(b+c)',variables='a,b,c,d')-2
+formula
+
+ +
+
+
+ +
+
+ + +
+ +
Out[17]:
+ + + + +
+$\displaystyle \frac{a}{b + c} + \frac{b}{c + d} + \frac{c}{a + d} + \frac{d}{a + b} - 2$ +
+ +
+ +
+
+ +
+
+
+
In [18]:
+
+
+
prove(formula)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $a^{3} c + a^{3} d + a^{2} b^{2} - a^{2} b d - 2 a^{2} c^{2} - a^{2} c d + a^{2} d^{2} + a b^{3} - a b^{2} c - a b^{2} d - a b c^{2} + a c^{3} - a c d^{2} + b^{3} d + b^{2} c^{2} - 2 b^{2} d^{2} + b c^{3} - b c^{2} d - b c d^{2} + b d^{3} + c^{2} d^{2} + c d^{3}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $a^{2} b c + a^{2} b d + a^{2} c^{2} + a^{2} c d + a b^{2} c + a b^{2} d + a b c^{2} + 2 a b c d + a b d^{2} + a c^{2} d + a c d^{2} + b^{2} c d + b^{2} d^{2} + b c^{2} d + b c d^{2}$ +
+ +
+ +
+ +
+ + + + +
+status: 2 +
+ +
+ +
+ +
+ + + + +
+Program couldn't find any proof. +
+ +
+ +
+ +
+ + + + +
+$$ a^{2} b d+2 a^{2} c^{2}+a^{2} c d+a b^{2} c+a b^{2} d+a b c^{2}+a c d^{2}+2 b^{2} d^{2}+b c^{2} d+b c d^{2} \le a^{3} c+a^{3} d+a^{2} b^{2}+a^{2} d^{2}+a b^{3}+a c^{3}+b^{3} d+b^{2} c^{2}+b c^{3}+b d^{3}+c^{2} d^{2}+c d^{3} $$ +
+ +
+ +
+ +
Out[18]:
+ + + + +
+
2
+
+ +
+ +
+
+ +
+
+
+
+

This problem, like the previous one, can be solved by splitting the domain of variables to several subdomains. But we can also use the symmetry of this inequality. For example, without loss of generality we can assume that $a\ge c$ and $b\ge d$, so $a\in [c,\infty)$, $b\in [d,\infty)$.

+ +
+
+
+
+
+
In [19]:
+
+
+
newproof()
+prove(makesubs(formula,'[c,oo],[d,oo]'))
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $a\to c + e$ +
+ +
+ +
+ +
+ + + + +
+Substitute $b\to d + f$ +
+ +
+ +
+ +
+ + + + +
+numerator: $c^{2} e^{2} - c^{2} e f + c^{2} f^{2} + 2 c d e^{2} + 2 c d f^{2} + c e^{3} + c e f^{2} + c f^{3} + d^{2} e^{2} + d^{2} e f + d^{2} f^{2} + d e^{3} + d e^{2} f + 2 d e f^{2} + d f^{3} + e^{2} f^{2} + e f^{3}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $c^{4} + 4 c^{3} d + 2 c^{3} e + 2 c^{3} f + 6 c^{2} d^{2} + 6 c^{2} d e + 6 c^{2} d f + c^{2} e^{2} + 3 c^{2} e f + c^{2} f^{2} + 4 c d^{3} + 6 c d^{2} e + 6 c d^{2} f + 2 c d e^{2} + 6 c d e f + 2 c d f^{2} + c e^{2} f + c e f^{2} + d^{4} + 2 d^{3} e + 2 d^{3} f + d^{2} e^{2} + 3 d^{2} e f + d^{2} f^{2} + d e^{2} f + d e f^{2}$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+Program couldn't find a solution with integer coefficients. Try to multiple the formula by some integer and run this function again. +
+ +
+ +
+ +
+ + + + +
+$$ c^{2} e f \le c^{2} e^{2}+c^{2} f^{2}+2 c d e^{2}+2 c d f^{2}+c e^{3}+c e f^{2}+c f^{3}+d^{2} e^{2}+d^{2} e f+d^{2} f^{2}+d e^{3}+d e^{2} f+2 d e f^{2}+d f^{3}+e^{2} f^{2}+e f^{3} $$ +
+ +
+ +
+ +
Out[19]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
In [20]:
+
+
+
newproof()
+prove(makesubs(formula,'[c,oo],[d,oo]')*2)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $a\to c + e$ +
+ +
+ +
+ +
+ + + + +
+Substitute $b\to d + f$ +
+ +
+ +
+ +
+ + + + +
+numerator: $2 c^{2} e^{2} - 2 c^{2} e f + 2 c^{2} f^{2} + 4 c d e^{2} + 4 c d f^{2} + 2 c e^{3} + 2 c e f^{2} + 2 c f^{3} + 2 d^{2} e^{2} + 2 d^{2} e f + 2 d^{2} f^{2} + 2 d e^{3} + 2 d e^{2} f + 4 d e f^{2} + 2 d f^{3} + 2 e^{2} f^{2} + 2 e f^{3}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $c^{4} + 4 c^{3} d + 2 c^{3} e + 2 c^{3} f + 6 c^{2} d^{2} + 6 c^{2} d e + 6 c^{2} d f + c^{2} e^{2} + 3 c^{2} e f + c^{2} f^{2} + 4 c d^{3} + 6 c d^{2} e + 6 c d^{2} f + 2 c d e^{2} + 6 c d e f + 2 c d f^{2} + c e^{2} f + c e f^{2} + d^{4} + 2 d^{3} e + 2 d^{3} f + d^{2} e^{2} + 3 d^{2} e f + d^{2} f^{2} + d e^{2} f + d e f^{2}$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$2 c^{2} e f \le c^{2} e^{2}+c^{2} f^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le c^{2} e^{2}+c^{2} f^{2}+4 c d e^{2}+4 c d f^{2}+2 c e^{3}+2 c e f^{2}+2 c f^{3}+2 d^{2} e^{2}+2 d^{2} e f+2 d^{2} f^{2}+2 d e^{3}+2 d e^{2} f+4 d e f^{2}+2 d f^{3}+2 e^{2} f^{2}+2 e f^{3} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[20]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

It's a good idea to use intervals that are unbounded from one side (i.e. those which contain $\pm\infty$). In this problem we could assume that $a\in (0,c]$, $b\in (0,d]$ as well. But as you can see, in this case the proof is several times longer.

+ +
+
+
+
+
+
In [21]:
+
+
+
newproof()
+prove(makesubs(formula,'[0,c],[0,d]')*2)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Substitute $a\to c - \frac{c}{e + 1}$ +
+ +
+ +
+ +
+ + + + +
+Substitute $b\to d - \frac{d}{f + 1}$ +
+ +
+ +
+ +
+ + + + +
+numerator: $2 c^{4} e f^{3} + 6 c^{4} e f^{2} + 6 c^{4} e f + 2 c^{4} e - 2 c^{3} d e^{2} f^{2} - 4 c^{3} d e^{2} f - 2 c^{3} d e^{2} + 4 c^{3} d e f^{3} + 8 c^{3} d e f^{2} + 4 c^{3} d e f + 2 c^{3} d f^{3} + 4 c^{3} d f^{2} + 2 c^{3} d f + 2 c^{2} d^{2} e^{3} f + 2 c^{2} d^{2} e^{3} + 4 c^{2} d^{2} e^{2} f + 4 c^{2} d^{2} e^{2} + 2 c^{2} d^{2} e f^{3} + 4 c^{2} d^{2} e f^{2} + 6 c^{2} d^{2} e f + 4 c^{2} d^{2} e + 2 c^{2} d^{2} f^{3} + 4 c^{2} d^{2} f^{2} + 4 c^{2} d^{2} f + 2 c^{2} d^{2} + 4 c d^{3} e^{3} f + 2 c d^{3} e^{3} + 2 c d^{3} e^{2} f^{2} + 12 c d^{3} e^{2} f + 6 c d^{3} e^{2} + 4 c d^{3} e f^{2} + 12 c d^{3} e f + 6 c d^{3} e + 2 c d^{3} f^{2} + 4 c d^{3} f + 2 c d^{3} + 2 d^{4} e^{3} f + 6 d^{4} e^{2} f + 6 d^{4} e f + 2 d^{4} f$ +
+ +
+ +
+ +
+ + + + +
+denominator: $c^{4} e^{3} f^{3} + 3 c^{4} e^{3} f^{2} + 3 c^{4} e^{3} f + c^{4} e^{3} + c^{4} e^{2} f^{3} + 3 c^{4} e^{2} f^{2} + 3 c^{4} e^{2} f + c^{4} e^{2} + 4 c^{3} d e^{3} f^{3} + 10 c^{3} d e^{3} f^{2} + 8 c^{3} d e^{3} f + 2 c^{3} d e^{3} + 6 c^{3} d e^{2} f^{3} + 15 c^{3} d e^{2} f^{2} + 12 c^{3} d e^{2} f + 3 c^{3} d e^{2} + 2 c^{3} d e f^{3} + 5 c^{3} d e f^{2} + 4 c^{3} d e f + c^{3} d e + 6 c^{2} d^{2} e^{3} f^{3} + 12 c^{2} d^{2} e^{3} f^{2} + 7 c^{2} d^{2} e^{3} f + c^{2} d^{2} e^{3} + 12 c^{2} d^{2} e^{2} f^{3} + 24 c^{2} d^{2} e^{2} f^{2} + 14 c^{2} d^{2} e^{2} f + 2 c^{2} d^{2} e^{2} + 7 c^{2} d^{2} e f^{3} + 14 c^{2} d^{2} e f^{2} + 8 c^{2} d^{2} e f + c^{2} d^{2} e + c^{2} d^{2} f^{3} + 2 c^{2} d^{2} f^{2} + c^{2} d^{2} f + 4 c d^{3} e^{3} f^{3} + 6 c d^{3} e^{3} f^{2} + 2 c d^{3} e^{3} f + 10 c d^{3} e^{2} f^{3} + 15 c d^{3} e^{2} f^{2} + 5 c d^{3} e^{2} f + 8 c d^{3} e f^{3} + 12 c d^{3} e f^{2} + 4 c d^{3} e f + 2 c d^{3} f^{3} + 3 c d^{3} f^{2} + c d^{3} f + d^{4} e^{3} f^{3} + d^{4} e^{3} f^{2} + 3 d^{4} e^{2} f^{3} + 3 d^{4} e^{2} f^{2} + 3 d^{4} e f^{3} + 3 d^{4} e f^{2} + d^{4} f^{3} + d^{4} f^{2}$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$2 c^{3} d e^{2} f^{2} \le c^{4} e f^{3}+c^{2} d^{2} e^{3} f$$ +
+ +
+ +
+ +
+ + + + +
+$$2 c^{3} d e^{2} \le c^{4} e+c^{2} d^{2} e^{3}$$ +
+ +
+ +
+ +
+ + + + +
+$$4 c^{3} d e^{2} f \le c^{4} e f^{3}+c^{4} e+c^{2} d^{2} e^{3} f+c^{2} d^{2} e^{3}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 6 c^{4} e f^{2}+6 c^{4} e f+4 c^{3} d e f^{3}+8 c^{3} d e f^{2}+4 c^{3} d e f+2 c^{3} d f^{3}+4 c^{3} d f^{2}+2 c^{3} d f+4 c^{2} d^{2} e^{2} f+4 c^{2} d^{2} e^{2}+2 c^{2} d^{2} e f^{3}+4 c^{2} d^{2} e f^{2}+6 c^{2} d^{2} e f+4 c^{2} d^{2} e+2 c^{2} d^{2} f^{3}+4 c^{2} d^{2} f^{2}+4 c^{2} d^{2} f+2 c^{2} d^{2}+4 c d^{3} e^{3} f+2 c d^{3} e^{3}+2 c d^{3} e^{2} f^{2}+12 c d^{3} e^{2} f+6 c d^{3} e^{2}+4 c d^{3} e f^{2}+12 c d^{3} e f+6 c d^{3} e+2 c d^{3} f^{2}+4 c d^{3} f+2 c d^{3}+2 d^{4} e^{3} f+6 d^{4} e^{2} f+6 d^{4} e f+2 d^{4} f $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[21]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Function powerprove is a shortcut for splitting domain $R_+^n$ on several subdomains and proving the inequality for each of them. This function uses $2^n$ of $n$-dimensional intervals with a common point (by default it's $(1,1,...,1)$), where $n$ is a number of variables. Here there are two examples of using it. As you can see, proofs are very long.

+ +
+
+
+
+
+
In [22]:
+
+
+
newproof()
+#this is equivalent to
+#prove(makesubs('(x-1)^4','[1,oo]'))
+#prove(makesubs('(x-1)^4','[-oo,1]'))
+powerprove('(x-1)^4')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+_______________________ +
+ +
+ +
+ +
+ + + + +
+Substitute $x\to 1+a$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $a^{4}$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le a^{4} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
+ + + + +
+_______________________ +
+ +
+ +
+ +
+ + + + +
+Substitute $x\to 1/(1+b)$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $b^{4}$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le b^{4} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[22]:
+ + + + +
+
Counter({0: 2})
+
+ +
+ +
+
+ +
+
+
+
In [23]:
+
+
+
newproof()
+formula=Sm('-(3a + 2b + c)(2a^3 + 3b^2 + 6c + 1) + (4a + 4b + 4c)(a^4 + b^3 + c^2 + 3)')
+#this is equivalent to
+#prove(makesubs(formula,'[1,oo],[1,oo],[1,oo]'))
+#prove(makesubs(formula,'[-1,oo],[1,oo],[1,oo]'))
+#prove(makesubs(formula,'[1,oo],[-1,oo],[1,oo]'))
+#prove(makesubs(formula,'[-1,oo],[-1,oo],[1,oo]'))
+#prove(makesubs(formula,'[1,oo],[1,oo],[-1,oo]'))
+#prove(makesubs(formula,'[-1,oo],[1,oo],[-1,oo]'))
+#prove(makesubs(formula,'[1,oo],[-1,oo],[-1,oo]'))
+#prove(makesubs(formula,'[-1,oo],[-1,oo],[-1,oo]'))
+powerprove(formula)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $4 a^{5} + 4 a^{4} b + 4 a^{4} c - 6 a^{4} - 4 a^{3} b - 2 a^{3} c + 4 a b^{3} - 9 a b^{2} + 4 a c^{2} - 18 a c + 9 a + 4 b^{4} + 4 b^{3} c - 6 b^{3} - 3 b^{2} c + 4 b c^{2} - 12 b c + 10 b + 4 c^{3} - 6 c^{2} + 11 c$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+_______________________ +
+ +
+ +
+ +
+ + + + +
+Substitute $a\to 1+d,b\to 1+e,c\to 1+f$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $4 d^{5} + 4 d^{4} e + 4 d^{4} f + 22 d^{4} + 12 d^{3} e + 14 d^{3} f + 42 d^{3} + 12 d^{2} e + 18 d^{2} f + 34 d^{2} + 4 d e^{3} + 3 d e^{2} - 2 d e + 4 d f^{2} + 4 e^{4} + 4 e^{3} f + 18 e^{3} + 9 e^{2} f + 18 e^{2} + 4 e f^{2} + 2 e f + 4 f^{3} + 14 f^{2}$ +
+ +
+ +
+ +
+ + + + +
+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 $a\to 1/(1+g),b\to 1+h,c\to 1+i$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $4 g^{5} h^{4} + 4 g^{5} h^{3} i + 14 g^{5} h^{3} + 9 g^{5} h^{2} i + 15 g^{5} h^{2} + 4 g^{5} h i^{2} + 2 g^{5} h i + 6 g^{5} h + 4 g^{5} i^{3} + 10 g^{5} i^{2} + 8 g^{5} i + 10 g^{5} + 20 g^{4} h^{4} + 20 g^{4} h^{3} i + 74 g^{4} h^{3} + 45 g^{4} h^{2} i + 78 g^{4} h^{2} + 20 g^{4} h i^{2} + 10 g^{4} h i + 24 g^{4} h + 20 g^{4} i^{3} + 54 g^{4} i^{2} + 30 g^{4} i + 40 g^{4} + 40 g^{3} h^{4} + 40 g^{3} h^{3} i + 156 g^{3} h^{3} + 90 g^{3} h^{2} i + 162 g^{3} h^{2} + 40 g^{3} h i^{2} + 20 g^{3} h i + 36 g^{3} h + 40 g^{3} i^{3} + 116 g^{3} i^{2} + 40 g^{3} i + 60 g^{3} + 40 g^{2} h^{4} + 40 g^{2} h^{3} i + 164 g^{2} h^{3} + 90 g^{2} h^{2} i + 168 g^{2} h^{2} + 40 g^{2} h i^{2} + 20 g^{2} h i + 20 g^{2} h + 40 g^{2} i^{3} + 124 g^{2} i^{2} + 18 g^{2} i + 34 g^{2} + 20 g h^{4} + 20 g h^{3} i + 86 g h^{3} + 45 g h^{2} i + 87 g h^{2} + 20 g h i^{2} + 10 g h i + 2 g h + 20 g i^{3} + 66 g i^{2} + 4 h^{4} + 4 h^{3} i + 18 h^{3} + 9 h^{2} i + 18 h^{2} + 4 h i^{2} + 2 h i + 4 i^{3} + 14 i^{2}$ +
+ +
+ +
+ +
+ + + + +
+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 $a\to 1+j,b\to 1/(1+k),c\to 1+l$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $4 j^{5} k^{4} + 16 j^{5} k^{3} + 24 j^{5} k^{2} + 16 j^{5} k + 4 j^{5} + 4 j^{4} k^{4} l + 18 j^{4} k^{4} + 16 j^{4} k^{3} l + 76 j^{4} k^{3} + 24 j^{4} k^{2} l + 120 j^{4} k^{2} + 16 j^{4} k l + 84 j^{4} k + 4 j^{4} l + 22 j^{4} + 14 j^{3} k^{4} l + 30 j^{3} k^{4} + 56 j^{3} k^{3} l + 132 j^{3} k^{3} + 84 j^{3} k^{2} l + 216 j^{3} k^{2} + 56 j^{3} k l + 156 j^{3} k + 14 j^{3} l + 42 j^{3} + 18 j^{2} k^{4} l + 22 j^{2} k^{4} + 72 j^{2} k^{3} l + 100 j^{2} k^{3} + 108 j^{2} k^{2} l + 168 j^{2} k^{2} + 72 j^{2} k l + 124 j^{2} k + 18 j^{2} l + 34 j^{2} + 4 j k^{4} l^{2} + j k^{4} + 16 j k^{3} l^{2} + 8 j k^{3} + 24 j k^{2} l^{2} + 9 j k^{2} + 16 j k l^{2} + 2 j k + 4 j l^{2} + 4 k^{4} l^{3} + 10 k^{4} l^{2} + 3 k^{4} l + 4 k^{4} + 16 k^{3} l^{3} + 44 k^{3} l^{2} + 8 k^{3} l + 18 k^{3} + 24 k^{2} l^{3} + 72 k^{2} l^{2} + 3 k^{2} l + 18 k^{2} + 16 k l^{3} + 52 k l^{2} - 2 k l + 4 l^{3} + 14 l^{2}$ +
+ +
+ +
+ +
+ + + + +
+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 $a\to 1/(1+m),b\to 1/(1+n),c\to 1+o$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $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}$ +
+ +
+ +
+ +
+ + + + +
+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 $a\to 1+p,b\to 1+q,c\to 1/(1+r)$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $4 p^{5} r^{3} + 12 p^{5} r^{2} + 12 p^{5} r + 4 p^{5} + 4 p^{4} q r^{3} + 12 p^{4} q r^{2} + 12 p^{4} q r + 4 p^{4} q + 18 p^{4} r^{3} + 58 p^{4} r^{2} + 62 p^{4} r + 22 p^{4} + 12 p^{3} q r^{3} + 36 p^{3} q r^{2} + 36 p^{3} q r + 12 p^{3} q + 28 p^{3} r^{3} + 98 p^{3} r^{2} + 112 p^{3} r + 42 p^{3} + 12 p^{2} q r^{3} + 36 p^{2} q r^{2} + 36 p^{2} q r + 12 p^{2} q + 16 p^{2} r^{3} + 66 p^{2} r^{2} + 84 p^{2} r + 34 p^{2} + 4 p q^{3} r^{3} + 12 p q^{3} r^{2} + 12 p q^{3} r + 4 p q^{3} + 3 p q^{2} r^{3} + 9 p q^{2} r^{2} + 9 p q^{2} r + 3 p q^{2} - 2 p q r^{3} - 6 p q r^{2} - 6 p q r - 2 p q + 4 p r^{3} + 4 p r^{2} + 4 q^{4} r^{3} + 12 q^{4} r^{2} + 12 q^{4} r + 4 q^{4} + 14 q^{3} r^{3} + 46 q^{3} r^{2} + 50 q^{3} r + 18 q^{3} + 9 q^{2} r^{3} + 36 q^{2} r^{2} + 45 q^{2} r + 18 q^{2} + 2 q r^{3} - 2 q r + 10 r^{3} + 14 r^{2}$ +
+ +
+ +
+ +
+ + + + +
+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 $a\to 1/(1+s),b\to 1+t,c\to 1/(1+u)$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $4 s^{5} t^{4} u^{3} + 12 s^{5} t^{4} u^{2} + 12 s^{5} t^{4} u + 4 s^{5} t^{4} + 10 s^{5} t^{3} u^{3} + 34 s^{5} t^{3} u^{2} + 38 s^{5} t^{3} u + 14 s^{5} t^{3} + 6 s^{5} t^{2} u^{3} + 27 s^{5} t^{2} u^{2} + 36 s^{5} t^{2} u + 15 s^{5} t^{2} + 8 s^{5} t u^{3} + 18 s^{5} t u^{2} + 16 s^{5} t u + 6 s^{5} t + 8 s^{5} u^{3} + 24 s^{5} u^{2} + 22 s^{5} u + 10 s^{5} + 20 s^{4} t^{4} u^{3} + 60 s^{4} t^{4} u^{2} + 60 s^{4} t^{4} u + 20 s^{4} t^{4} + 54 s^{4} t^{3} u^{3} + 182 s^{4} t^{3} u^{2} + 202 s^{4} t^{3} u + 74 s^{4} t^{3} + 33 s^{4} t^{2} u^{3} + 144 s^{4} t^{2} u^{2} + 189 s^{4} t^{2} u + 78 s^{4} t^{2} + 34 s^{4} t u^{3} + 72 s^{4} t u^{2} + 62 s^{4} t u + 24 s^{4} t + 44 s^{4} u^{3} + 114 s^{4} u^{2} + 90 s^{4} u + 40 s^{4} + 40 s^{3} t^{4} u^{3} + 120 s^{3} t^{4} u^{2} + 120 s^{3} t^{4} u + 40 s^{3} t^{4} + 116 s^{3} t^{3} u^{3} + 388 s^{3} t^{3} u^{2} + 428 s^{3} t^{3} u + 156 s^{3} t^{3} + 72 s^{3} t^{2} u^{3} + 306 s^{3} t^{2} u^{2} + 396 s^{3} t^{2} u + 162 s^{3} t^{2} + 56 s^{3} t u^{3} + 108 s^{3} t u^{2} + 88 s^{3} t u + 36 s^{3} t + 96 s^{3} u^{3} + 216 s^{3} u^{2} + 140 s^{3} u + 60 s^{3} + 40 s^{2} t^{4} u^{3} + 120 s^{2} t^{4} u^{2} + 120 s^{2} t^{4} u + 40 s^{2} t^{4} + 124 s^{2} t^{3} u^{3} + 412 s^{2} t^{3} u^{2} + 452 s^{2} t^{3} u + 164 s^{2} t^{3} + 78 s^{2} t^{2} u^{3} + 324 s^{2} t^{2} u^{2} + 414 s^{2} t^{2} u + 168 s^{2} t^{2} + 40 s^{2} t u^{3} + 60 s^{2} t u^{2} + 40 s^{2} t u + 20 s^{2} t + 100 s^{2} u^{3} + 190 s^{2} u^{2} + 84 s^{2} u + 34 s^{2} + 20 s t^{4} u^{3} + 60 s t^{4} u^{2} + 60 s t^{4} u + 20 s t^{4} + 66 s t^{3} u^{3} + 218 s t^{3} u^{2} + 238 s t^{3} u + 86 s t^{3} + 42 s t^{2} u^{3} + 171 s t^{2} u^{2} + 216 s t^{2} u + 87 s t^{2} + 12 s t u^{3} + 6 s t u^{2} - 4 s t u + 2 s t + 46 s u^{3} + 66 s u^{2} + 4 t^{4} u^{3} + 12 t^{4} u^{2} + 12 t^{4} u + 4 t^{4} + 14 t^{3} u^{3} + 46 t^{3} u^{2} + 50 t^{3} u + 18 t^{3} + 9 t^{2} u^{3} + 36 t^{2} u^{2} + 45 t^{2} u + 18 t^{2} + 2 t u^{3} - 2 t u + 10 u^{3} + 14 u^{2}$ +
+ +
+ +
+ +
+ + + + +
+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 $a\to 1+v,b\to 1/(1+w),c\to 1/(1+x)$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $4 v^{5} w^{4} x^{3} + 12 v^{5} w^{4} x^{2} + 12 v^{5} w^{4} x + 4 v^{5} w^{4} + 16 v^{5} w^{3} x^{3} + 48 v^{5} w^{3} x^{2} + 48 v^{5} w^{3} x + 16 v^{5} w^{3} + 24 v^{5} w^{2} x^{3} + 72 v^{5} w^{2} x^{2} + 72 v^{5} w^{2} x + 24 v^{5} w^{2} + 16 v^{5} w x^{3} + 48 v^{5} w x^{2} + 48 v^{5} w x + 16 v^{5} w + 4 v^{5} x^{3} + 12 v^{5} x^{2} + 12 v^{5} x + 4 v^{5} + 14 v^{4} w^{4} x^{3} + 46 v^{4} w^{4} x^{2} + 50 v^{4} w^{4} x + 18 v^{4} w^{4} + 60 v^{4} w^{3} x^{3} + 196 v^{4} w^{3} x^{2} + 212 v^{4} w^{3} x + 76 v^{4} w^{3} + 96 v^{4} w^{2} x^{3} + 312 v^{4} w^{2} x^{2} + 336 v^{4} w^{2} x + 120 v^{4} w^{2} + 68 v^{4} w x^{3} + 220 v^{4} w x^{2} + 236 v^{4} w x + 84 v^{4} w + 18 v^{4} x^{3} + 58 v^{4} x^{2} + 62 v^{4} x + 22 v^{4} + 16 v^{3} w^{4} x^{3} + 62 v^{3} w^{4} x^{2} + 76 v^{3} w^{4} x + 30 v^{3} w^{4} + 76 v^{3} w^{3} x^{3} + 284 v^{3} w^{3} x^{2} + 340 v^{3} w^{3} x + 132 v^{3} w^{3} + 132 v^{3} w^{2} x^{3} + 480 v^{3} w^{2} x^{2} + 564 v^{3} w^{2} x + 216 v^{3} w^{2} + 100 v^{3} w x^{3} + 356 v^{3} w x^{2} + 412 v^{3} w x + 156 v^{3} w + 28 v^{3} x^{3} + 98 v^{3} x^{2} + 112 v^{3} x + 42 v^{3} + 4 v^{2} w^{4} x^{3} + 30 v^{2} w^{4} x^{2} + 48 v^{2} w^{4} x + 22 v^{2} w^{4} + 28 v^{2} w^{3} x^{3} + 156 v^{2} w^{3} x^{2} + 228 v^{2} w^{3} x + 100 v^{2} w^{3} + 60 v^{2} w^{2} x^{3} + 288 v^{2} w^{2} x^{2} + 396 v^{2} w^{2} x + 168 v^{2} w^{2} + 52 v^{2} w x^{3} + 228 v^{2} w x^{2} + 300 v^{2} w x + 124 v^{2} w + 16 v^{2} x^{3} + 66 v^{2} x^{2} + 84 v^{2} x + 34 v^{2} + 5 v w^{4} x^{3} + 7 v w^{4} x^{2} + 3 v w^{4} x + v w^{4} + 24 v w^{3} x^{3} + 40 v w^{3} x^{2} + 24 v w^{3} x + 8 v w^{3} + 33 v w^{2} x^{3} + 51 v w^{2} x^{2} + 27 v w^{2} x + 9 v w^{2} + 18 v w x^{3} + 22 v w x^{2} + 6 v w x + 2 v w + 4 v x^{3} + 4 v x^{2} + 7 w^{4} x^{3} + 16 w^{4} x^{2} + 9 w^{4} x + 4 w^{4} + 38 w^{3} x^{3} + 82 w^{3} x^{2} + 46 w^{3} x + 18 w^{3} + 63 w^{2} x^{3} + 120 w^{2} x^{2} + 51 w^{2} x + 18 w^{2} + 38 w x^{3} + 56 w x^{2} + 2 w x + 10 x^{3} + 14 x^{2}$ +
+ +
+ +
+ +
+ + + + +
+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 $a\to 1/(1+y),b\to 1/(1+z),c\to 1/(1+a_{1})$ +
+ +
+ +
+ +
+ + + + +
+Numerator after substitutions: $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}$ +
+ +
+ +
+ +
+ + + + +
+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. +
+ +
+ +
+ +
Out[23]:
+ + + + +
+
Counter({0: 8})
+
+ +
+ +
+
+ +
+
+
+
+

Now let's take a look at slightly another kind of the problem.

+

Problem

Let $f:R^3\to R$ be a convex function. Prove that +$$f(1,2,3)+f(2,3,1)+f(3,1,2)\le f(4,3,-1)+f(3,-1,4)+f(-1,4,3).$$

+

To create a proof, we will use provef function. It assumes that $f$ is convex and nonnegative, then it tries to find a proof. However, if the last inequality is $0\le 0$, then the proof works for any convex function.

+ +
+
+
+
+
+
In [24]:
+
+
+
provef('(-f(1,2,3)-f(2,3,1)-f(3,1,2)+f(4,3,-1)+f(3,-1,4)+f(-1,4,3))*21')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+numerator: $21 f{\left(-1,4,3 \right)} - 21 f{\left(1,2,3 \right)} - 21 f{\left(2,3,1 \right)} + 21 f{\left(3,-1,4 \right)} - 21 f{\left(3,1,2 \right)} + 21 f{\left(4,3,-1 \right)}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $1$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From Jensen inequality: +
+ +
+ +
+ +
+ + + + +
+$$21f(1, 2, 3) \le 11f(-1, 4, 3)+8f(3, -1, 4)+2f(4, 3, -1)$$ +
+ +
+ +
+ +
+ + + + +
+$$21f(2, 3, 1) \le 8f(-1, 4, 3)+2f(3, -1, 4)+11f(4, 3, -1)$$ +
+ +
+ +
+ +
+ + + + +
+$$21f(3, 1, 2) \le 2f(-1, 4, 3)+11f(3, -1, 4)+8f(4, 3, -1)$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 0 $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[24]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

Let's try to solve problem 6 from the finals of LXIII Polish Mathematical Olympiad. It was one of the hardest inequality in the history of this contest, solved only by 3 finalists.

+

Problem

Prove the inequality +$$\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2\ge 2 \sqrt{2} \left(\frac{a - b}{c} + \frac{b - c}{a}+ \frac{c-a}{b}\right)$$ +for any positive numbers $a,b,c$.

+

The first observation is that the formula is cyclic, so without loss of generality we may assume that $a\ge b,c$. We can go a step further and divide it into two cases: $a\ge b\ge c$ and $a\ge c\ge b$.

+ +
+
+
+
+
+
In [25]:
+
+
+
shiro.display=lambda x:None #turn off printing of proofs
+newproof()
+formula=cyclize('((a-b)/c)^2-2*sqrt(2)*(a-b)/c')
+
+ +
+
+
+ +
+
+
+
In [26]:
+
+
+
formula1=makesubs(formula,'[b,oo],[c,oo]',variables='a,b') #a>=b>=c
+prove(formula1) 
+
+ +
+
+
+ +
+
+ + +
+ +
Out[26]:
+ + + + +
+
2
+
+ +
+ +
+
+ +
+
+
+
In [27]:
+
+
+
formula2=makesubs(formula,'[c,oo],[b,oo]',variables='a,c') #a>=c>=b
+prove(formula2) 
+
+ +
+
+
+ +
+
+ + +
+ +
Out[27]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

So the case $a\ge c\ge b$ is done, but $a\ge b\ge c$ is not. But maybe we can adjust values.

+ +
+
+
+
+
+
In [28]:
+
+
+
values=findvalues(formula1)
+values
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
Optimization terminated successfully.
+         Current function value: 1.000000
+         Iterations: 137
+         Function evaluations: 249
+
+
+
+ +
+ +
Out[28]:
+ + + + +
+
(1.7908873553542452e-10, 2.5326984818340415e-10, 7.129450063690368)
+
+ +
+ +
+
+ +
+
+
+
+

First and second value is approximately equal to 0, so we can try to replace 0 with 1.

+ +
+
+
+
+
+
In [29]:
+
+
+
prove(formula1,values='1,1,7')
+
+ +
+
+
+ +
+
+ + +
+ +
Out[29]:
+ + + + +
+
2
+
+ +
+ +
+
+ +
+
+
+
+

The key observation is that the formula1 is homogenous, so we can scale values.

+ +
+
+
+
+
+
In [30]:
+
+
+
newvalues=(1,values[1]/values[0],values[2]/values[0])
+newvalues
+
+ +
+
+
+ +
+
+ + +
+ +
Out[30]:
+ + + + +
+
(1, 1.4142142855953455, 39809595184.05965)
+
+ +
+ +
+
+ +
+
+
+
In [31]:
+
+
+
newvalues[1]**2
+
+ +
+
+
+ +
+
+ + +
+ +
Out[31]:
+ + + + +
+
2.000002045581953
+
+ +
+ +
+
+ +
+
+
+
+

Now the third value is very big. Technically we could use it, but it would run into overflow error, so we will use 1 instead of it. Second value is very close to $\sqrt{2}$, so this value will be our next try.

+ +
+
+
+
+
+
In [32]:
+
+
+
prove(formula1,values='1,sqrt(2),1')
+
+ +
+
+
+ +
+
+ + +
+ +
Out[32]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+
+

So after getting the code all together we have got the following proof.

+ +
+
+
+
+
+
In [33]:
+
+
+
newproof()
+shiro.display=lambda x:display(Latex(x)) #turn on printing proofs 
+formula=cyclize('((a-b)/c)^2-2*sqrt(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,values='1,2**(1/2),1')
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+Case $a\ge c\ge b$ +
+ +
+ +
+ +
+ + + + +
+Substitute $a\to c + d$ +
+ +
+ +
+ +
+ + + + +
+Substitute $c\to b + e$ +
+ +
+ +
+ +
+ + + + +
+numerator: $2 b^{4} d^{2} + 2 b^{4} d e + 2 b^{4} e^{2} + 4 b^{3} d^{3} + 2 \sqrt{2} b^{3} d^{2} e + 10 b^{3} d^{2} e + 2 \sqrt{2} b^{3} d e^{2} + 6 b^{3} d e^{2} + 4 b^{3} e^{3} + 2 b^{2} d^{4} + 2 \sqrt{2} b^{2} d^{3} e + 10 b^{2} d^{3} e + 6 \sqrt{2} b^{2} d^{2} e^{2} + 12 b^{2} d^{2} e^{2} + 4 b^{2} d e^{3} + 4 \sqrt{2} b^{2} d e^{3} + 2 b^{2} e^{4} + 2 b d^{4} e + 2 \sqrt{2} b d^{3} e^{2} + 6 b d^{3} e^{2} + 4 b d^{2} e^{3} + 4 \sqrt{2} b d^{2} e^{3} + 2 \sqrt{2} b d e^{4} + d^{4} e^{2} + 2 d^{3} e^{3} + d^{2} e^{4}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $b^{6} + 2 b^{5} d + 4 b^{5} e + b^{4} d^{2} + 6 b^{4} d e + 6 b^{4} e^{2} + 2 b^{3} d^{2} e + 6 b^{3} d e^{2} + 4 b^{3} e^{3} + b^{2} d^{2} e^{2} + 2 b^{2} d e^{3} + b^{2} e^{4}$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 2 b^{4} d^{2}+2 b^{4} d e+2 b^{4} e^{2}+4 b^{3} d^{3}+2 \sqrt{2} b^{3} d^{2} e+10 b^{3} d^{2} e+2 \sqrt{2} b^{3} d e^{2}+6 b^{3} d e^{2}+4 b^{3} e^{3}+2 b^{2} d^{4}+2 \sqrt{2} b^{2} d^{3} e+10 b^{2} d^{3} e+6 \sqrt{2} b^{2} d^{2} e^{2}+12 b^{2} d^{2} e^{2}+4 \sqrt{2} b^{2} d e^{3}+4 b^{2} d e^{3}+2 b^{2} e^{4}+2 b d^{4} e+2 \sqrt{2} b d^{3} e^{2}+6 b d^{3} e^{2}+4 \sqrt{2} b d^{2} e^{3}+4 b d^{2} e^{3}+2 \sqrt{2} b d e^{4}+d^{4} e^{2}+2 d^{3} e^{3}+d^{2} e^{4} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
+ + + + +
+Case $a\ge b\ge c$ +
+ +
+ +
+ +
+ + + + +
+Substitute $a\to b + f$ +
+ +
+ +
+ +
+ + + + +
+Substitute $b\to c + g$ +
+ +
+ +
+ +
+ + + + +
+Substitute $f\to \sqrt{2} h$ +
+ +
+ +
+ +
+ + + + +
+numerator: $2 c^{4} g^{2} + 2 \sqrt{2} c^{4} g h + 4 c^{4} h^{2} + 4 c^{3} g^{3} - 4 c^{3} g^{2} h + 6 \sqrt{2} c^{3} g^{2} h - 4 \sqrt{2} c^{3} g h^{2} + 20 c^{3} g h^{2} + 8 \sqrt{2} c^{3} h^{3} + 2 c^{2} g^{4} - 8 c^{2} g^{3} h + 4 \sqrt{2} c^{2} g^{3} h - 12 \sqrt{2} c^{2} g^{2} h^{2} + 24 c^{2} g^{2} h^{2} - 8 c^{2} g h^{3} + 20 \sqrt{2} c^{2} g h^{3} + 8 c^{2} h^{4} - 4 c g^{4} h - 8 \sqrt{2} c g^{3} h^{2} + 8 c g^{3} h^{2} - 8 c g^{2} h^{3} + 12 \sqrt{2} c g^{2} h^{3} + 8 c g h^{4} + 2 g^{4} h^{2} + 4 \sqrt{2} g^{3} h^{3} + 4 g^{2} h^{4}$ +
+ +
+ +
+ +
+ + + + +
+denominator: $c^{6} + 4 c^{5} g + 2 \sqrt{2} c^{5} h + 6 c^{4} g^{2} + 6 \sqrt{2} c^{4} g h + 2 c^{4} h^{2} + 4 c^{3} g^{3} + 6 \sqrt{2} c^{3} g^{2} h + 4 c^{3} g h^{2} + c^{2} g^{4} + 2 \sqrt{2} c^{2} g^{3} h + 2 c^{2} g^{2} h^{2}$ +
+ +
+ +
+ +
+ + + + +
+status: 0 +
+ +
+ +
+ +
+ + + + +
+From weighted AM-GM inequality: +
+ +
+ +
+ +
+ + + + +
+$$4 c^{3} g^{2} h \le 2 c^{4} g^{2}+2 c^{2} g^{2} h^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$4 \sqrt{2} c^{3} g h^{2} \le 2 \sqrt{2} c^{4} g h+\sqrt{2} c^{3} h^{3}+\sqrt{2} c g^{2} h^{3}$$ +
+ +
+ +
+ +
+ + + + +
+$$8 c^{2} g^{3} h \le 4 c^{3} g^{3}+4 c g^{3} h^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$12 \sqrt{2} c^{2} g^{2} h^{2} \le 6 \sqrt{2} c^{3} g^{2} h+6 \sqrt{2} c g^{2} h^{3}$$ +
+ +
+ +
+ +
+ + + + +
+$$8 c^{2} g h^{3} \le 4 c^{3} g h^{2}+2 c^{2} h^{4}+2 g^{2} h^{4}$$ +
+ +
+ +
+ +
+ + + + +
+$$4 c g^{4} h \le 2 c^{2} g^{4}+2 g^{4} h^{2}$$ +
+ +
+ +
+ +
+ + + + +
+$$8 \sqrt{2} c g^{3} h^{2} \le 4 \sqrt{2} c^{2} g^{3} h+4 \sqrt{2} g^{3} h^{3}$$ +
+ +
+ +
+ +
+ + + + +
+$$8 c g^{2} h^{3} \le 4 c g^{3} h^{2}+4 c g h^{4}$$ +
+ +
+ +
+ +
+ + + + +
+$$ 0 \le 4 c^{4} h^{2}+16 c^{3} g h^{2}+7 \sqrt{2} c^{3} h^{3}+22 c^{2} g^{2} h^{2}+20 \sqrt{2} c^{2} g h^{3}+6 c^{2} h^{4}+5 \sqrt{2} c g^{2} h^{3}+4 c g h^{4}+2 g^{2} h^{4} $$ +
+ +
+ +
+ +
+ + + + +
+The sum of all inequalities gives us a proof of the inequality. +
+ +
+ +
+ +
Out[33]:
+ + + + +
+
0
+
+ +
+ +
+
+ +
+
+
+ + + + + + diff --git a/examples.ipynb b/examples.ipynb index 4e27353..7453a7b 100644 --- a/examples.ipynb +++ b/examples.ipynb @@ -2556,6 +2556,9 @@ ], "source": [ "newproof()\n", + "#this is equivalent to\n", + "#prove(makesubs('(x-1)^4','[1,oo]'))\n", + "#prove(makesubs('(x-1)^4','[-oo,1]'))\n", "powerprove('(x-1)^4')" ] }, @@ -3406,6 +3409,15 @@ "source": [ "newproof()\n", "formula=Sm('-(3a + 2b + c)(2a^3 + 3b^2 + 6c + 1) + (4a + 4b + 4c)(a^4 + b^3 + c^2 + 3)')\n", + "#this is equivalent to\n", + "#prove(makesubs(formula,'[1,oo],[1,oo],[1,oo]'))\n", + "#prove(makesubs(formula,'[-1,oo],[1,oo],[1,oo]'))\n", + "#prove(makesubs(formula,'[1,oo],[-1,oo],[1,oo]'))\n", + "#prove(makesubs(formula,'[-1,oo],[-1,oo],[1,oo]'))\n", + "#prove(makesubs(formula,'[1,oo],[1,oo],[-1,oo]'))\n", + "#prove(makesubs(formula,'[-1,oo],[1,oo],[-1,oo]'))\n", + "#prove(makesubs(formula,'[1,oo],[-1,oo],[-1,oo]'))\n", + "#prove(makesubs(formula,'[-1,oo],[-1,oo],[-1,oo]'))\n", "powerprove(formula)" ] }, diff 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zdzYNFmF4lJf5)MjZ`R}u48$x|d(y02n}Vwj2Wtf3Az+l(aP-0~qml4SlIB%yc+^@x=7ilbNE;BTZEl&%OA2W}FJ}@9U1e2kZ(+D`0iL<5FT*O-F z?_-vh*YoM#{Jw66H&`*ln=$*MXaPZLQZZxcvlx&b?(ocS3?4fbTnek$>A1f5{*>LZ zmxmdr!;44E^jk0?A@r&!_Brop87pH&I0i_z%Y_Pv$GX{j(8^l+z(qC*I~y05cUsd- zXy1*IJ*i(jpE)ccW&K(EH=w~w6}`|+eJ(%cc9-g~a*pkZV` zi6Y)46WUt^=EsB^CLaLH-B0Le^+8s=sb_w)nWK~l_ zo)~#5?h16_{o2bj>-cRjGGaHG?)FY$7jB15o-VC3TS)#&qGrneD{F*FoSbOs36fej z{1}YowhJ0mWf7odEV;JIjQ8ygMZw^}LwW&@n^#C*L;j9$+N#l6i<;s*ZSUuXN8b9ThX9|eEnVM)xEA(iCQ&SYt&3@*q&obB-k!R2&52nu(>GW#BfOQwQg4L4=^iLe+B!G)gkbr(iRA1Xm2CyXW#5tpI^`s+)bQ;2<7A&x8peic}}0m6RvmQa%YY} zlBZ_?%d0T8WiUkV-(x4h@n5l%wR3Q@a%GowFn6<(wz6=xv|?AbaFE3V6PPe^oy1x7&`B}*@1XW?hG3x5*YI@!|@1wsq zd#0RioG*X8k+LoFH?lW9Fy(80`pV|PGvOY%otDQ!w&-C5{3dO+lv{^SWUQKS06C7j-69P)+e;pv>=-|u=moA=qH z&RwN$eZPJdt&e(w9b$0U{`Yg@{FjEvU&~e5)j;-IZjS7#l)U_}YgS!9xlz8hKmMDA zt6TuSztyf|1`yXJegePK>h)Lux4PQ5lqEkjcX&QhpI-l5VH2Bt%kqgjGEUy=;a(u^ zx1?sWvJmu;O%6_e-3b|)kT6P}{3hfoWHaZdn5n0x&oG0Up;~xnNr+8I56Ogm`vwsE zjT*C>OO>#rFcsc=FUfoLO)0E~eHh0tX|_Dlh27Z2N-+-u;??vq=pvCHaeK2u&=PJo z6k)s&bpkPjFYyXBD9UqP0Mgb2KG= zkfQvYglVP}4~V01lQ3k$C4#*OR`n6OH-cn*NxvFWu(sE8u;8Q=zrTc2o_PaTKve_B zB-%WQ`9M%$Nm0F_3xiC#8%{-yX-*DB=BO~^K8uMjg`)t5oS{vC0YI3BA-8=46l{1` zBs5z7EGox9n^;8t-g+u($w^qg5TxUG8xmUL@(hS$llb|Ks2ld|+xiFy#HhDeVp5?W zrE%@oZk}(0Pv{F`-hIIYW!3uq)NsR-91#*P-KUJnhyRmL<4O_j1SS9ByjGX=aa!av zghvRnVnYb^8NdwFSUbTR^Oz!CZufgwGQ`?tQYuy=*aIq)JzDjB*fP34+&Sh!oDJ#q zRgmty?pp7Y^kKHy#k<{EpC@vX^n-9L@ikNc@m}6nW}1mZHS8D1KF^Ew27Q?xShMnS zatU)4cV)w(lUxs27CG{J$*jb{wMnCZ^IAT zbvv=Z67e(1;st80apl_Fd)Gcp6ZW{Kq|^D!MY9Wtx+*?2#w3|;OD}uB*TidW^LVu_ zJoeARiWio~4YzVfvBPAh+sf^1N`Zkd?v)-Wdn&7bQT&wD`$6r#+;I9*s*c` zc|0J%WI@2+rQkc8N@q^-9e>*pjexRuoJld%GZAAG2X3S<6E+dK9oK4#@<3{)aijE< zAMs&X%&v^T=xx1cljryvqmHk7FR^yeTgqwf#!KakCEr5ReLCX%eP7t2C@^(W_+4py z;^L+~_LpT;w5s!&c$Md@yViE(mBC4CQc>}vj!upBIJCg&2V@!jx@C%}k4rceMa1?*?$& z_c0O5m1M=;GRk@J`Y3WpKK(nIR!LB&(4FtgxiP9Jf~Z$vwJ?v z_6})$dFQW7!mT6gow8#ML85xQ0IC?9aG--Ss&R&jrPFN5uaQIVp3|RVz`SLL zXs7>Dn)P^o4nB$7ceA~ur5N)4ShL(sphxa>ymY^WsA_%ogKf;aDqNK|$#4hf-zwgJ zO4YhzCx4zWxVJWStaPpV2?u=gx;~(Nhn^42a^oFOG*4bybhrCVDwg*ioEv(+B;TLb z&p#X#e!~V`v$<5lT2gG&mN&2?8UsIvp(6;1pJ0D(mVFAm;1`-$KI;PJL7piOOLB(X zowOVqsA8y4?nP9mIV+vcd!9VkYb2t}LxcJ{_VL@IAuC^6Pgf)WmZt9&cDczwjsa=cruXeAf`1C88g>77N`w%H>&Z&fn!|^P1kZ_hpB#|p2 zYj{ui|M=LO_|*c^9Ab52v9~}LrO|ReTMKv|*Sx{!5}w=iI#pJdFF-t&D3j6CsW_w`vOzv=wAd!Hl$0Oo^jRivn_0fo?@ zFqe|Zu=fsmVA|YgC|6dyKrWN_Z!TS0VmkvH?zPc!3XT4f8$^(@#GLSSdjDEGQ^&LaN6tMYCJZ zvm2o=)*^rA9%Iv&l>VosuN9-`S!GG8*Y{0)~PF**$L(v6K+2f#B zvj#ElD9{k5pE@{ZpPIBo1cZgLeNk*z$ZdJ(KbI1UqX{Pmu3~#99WgHIs`fOshwxSn zkhr;H8wc--vIjd#;~xhq%|M9tf~wwX3$-8WU_{J#R?a^hSv>h_S*tqn7yZ(ySh5oXBAil9dD^Z`4l~|x%R3cjh-A1{4$_?tctA@_ru7n6 zHhN^|(on$)>eXj8p;ECViqa|y#v$Y51r;SPp%sJ;V_ajL1XI}9 z%mSSxYbzTGMD8ZutkG*7Z&v!N4f)ixxp09k3h$n=KC1Ws6K4Jwx!nK7g8G_P*6f zjxm?Ac7LGsOeJjoFGel4{Wv3a_i9+I611lC&-{ZNV$f?|aDA5*^QCzVfGHz)TMG0uD@r z^bouBP%r(EAW^m_C|9K + + + +statistics + + + + + + + + + + + + + + + + + + + + + + + +

+
+ +
+
+
+

In this notebook several methods for proving inequalities have been compared.

+

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).

+

Some tuples have less than 4 elements. In this case parser use default arguments.

+ +
+
+
+
+
+
In [1]:
+
+
+
from sympy import *
+import shiroindev
+from shiroindev import *
+from sympy.parsing.latex import parse_latex
+shiro.seed=1
+from IPython.display import Latex
+shiro.display=lambda x:display(Latex(x))
+
+ +
+
+
+ +
+
+
+
+

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.

+ +
+
+
+
+
+
In [2]:
+
+
+
def addbraces(s):
+    arg={r'\frac':2,r'\sqrt':1}
+    s2=''
+    p=0
+    while 1:
+        m=re.search(r"\\[a-zA-Z]+", s[p:])
+        if not m:
+            break
+        s2+=s[p:p+m.end()]
+        p+=m.end()
+        if m.group() in arg:
+            for i in range(arg[m.group()]):
+                sp=re.search('^ *',s[p:])
+                s2+=sp.group()
+                p+=sp.end()
+                if s[p]=='{':
+                    cb=re.search(r'^\{.*?\}',s[p:])
+                    ab=addbraces(cb.group())
+                    s2+=ab
+                    p+=cb.end()
+                else:
+                    s2+='{'+s[p]+'}'
+                    p+=1
+    s2+=s[p:]
+    return s2
+print(addbraces(r'\frac{ \sqrt 3}2'))
+print(addbraces(r'a^2+b^2+c^2\geq ab+bc+ca'))
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
\frac{ \sqrt {3}}{2}
+a^2+b^2+c^2\geq ab+bc+ca
+
+
+
+ +
+
+ +
+
+
+
In [3]:
+
+
+
#(formula,intervals,subs,function)
+dif=lambda b,s:b-s
+dif2=lambda b,s:b*b-s*s
+ineqs=[
+    (r'a^2+b^2+c^2\geq ab+bc+ca',),
+    (r'a^2+b^2+c^2+d^2\geq a(b+c+d)',),
+    (r'1\leq\frac{a^2b^2}{c^2}+\frac{b^2c^2}{a^2}+\frac{c^2a^2}{b^2}',
+     '[0,1-f],[0,1]','[a,sqrt(1-e-f)],[b,sqrt(e)],[c,sqrt(f)]',),
+    (r'\frac1{1-x^2}+\frac1{1-y^2}\geq \frac2{1-xy}','[0,1],[0,1]'),
+    (r'a^3+b^3\geq a^2b+ab^2',),
+    (r'\frac a{b+c}+\frac b{c+a}+\frac c{a+b}\geq \frac32',),
+    (r'2a^3+b^3\geq 3a^2b',),
+    (r'a^3+b^3+c^3\geq a^2b+b^2c+c^2a',),
+    (r'\frac a{b+c}+\frac b{c+d}+ \frac c{d+a}+ \frac d{a+b}\geq 2',),
+    (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',),
+    (r'\sqrt{ab}+\sqrt{cd}+\sqrt{ef}\leq\sqrt{(a+c+e)(b+d+f)}','','',dif2),
+    (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)',),
+    #(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}',
+    # '[0,1-c-d],[0,1-d],[0,1]','[a,1-b-c-d]'),
+    (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)]'),
+    (r'\frac ab+\frac bc+\frac ca\geq \frac{(a+b+c)^2}{ab+bc+ca}',),
+    (r'\frac{a^2}b+\frac{b^2}c+\frac{c^2}a\geq \frac{a^2+b^2+c^2}{a+b+c}',),
+    (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}',),
+    (r'\frac1{a^3+b^3+abc}+ \frac1{b^3+c^3+abc} +\frac1{c^3+a^3+ abc} \leq \frac1{abc}',),
+    (r'\frac1{a^3(b+c)}+ \frac1{b^3(c+a)}+ \frac1{c^3(a+b)} \geq \frac32','','[c,1/(a*b)]'),
+    (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}',),
+    (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)]'),
+    (r'(a+b-c)(b+c-a)(c+a-b)\leq abc',),
+    (r'\frac1{1+xy}+\frac1{1+yz}+\frac1{1+zx}\leq \frac34','','[x,(y+z)/(y*z-1)]'),
+    (r'\frac{x\sqrt x}{y+z}+\frac{y\sqrt y}{z+x}+\frac{z\sqrt z}{x+y}\geq\frac{ \sqrt 3}2',
+     '[0,1-z],[0,1]','[x,1-y-z]'),
+    (r'a^4+b^4+c^4+d^4\geq 4abcd',),
+    (r'\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\leq\frac{3(ab+bc+ca)}{2(a+b+c)}',),
+    (r'\sqrt{a-1}+\sqrt{b-1}+ \sqrt{c-1} \leq \sqrt{c(ab+1)}','[1,oo],[1,oo],[1,oo]','',dif2),
+    (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]'),
+    (r'ay+bz+cx\leq s^2','[0,s],[0,s],[0,s]','[x,s-a],[y,s-b],[z,s-c]'),
+    (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}',),
+    (r' xy+yz+zx - 2xyz \leq \frac7{27}', '[0,1-z],[0,1]','[x,1-y-z]'),
+    (r'0 \leq xy+yz+zx - 2xyz', '[0,1-z],[0,1]','[x,1-y-z]'),
+    (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]',
+     dif2),
+    (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',),
+    (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]'),
+    (r'\frac{a+b}{2b+c}+\frac{b+c}{2c+a}+\frac{c+a}{2a+b}\geq 2',),
+    #(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)]')
+]
+
+ +
+
+
+ +
+
+
+
In [4]:
+
+
+
def parser(formula,intervals='[]',subs='[]',func=dif):
+    newproof()
+    shiro.display=lambda x:None
+    if intervals=='':
+        intervals='[]'
+    if subs=='':
+        subs='[]'
+    formula=addbraces(formula)
+    formula=Sm(str(parse_latex(formula)))
+    formula,_=fractioncancel(formula)
+    formula=formula.subs(shiroindev._smakeiterable2(subs))
+    formula=makesubs(formula,intervals)
+    b,s=formula.lhs,formula.rhs
+    if type(formula)==LessThan:
+        b,s=s,b
+    formula=func(b,s)
+    formula=simplify(formula)
+    num,den=fractioncancel(formula)
+    return num
+
+ +
+
+
+ +
+
+
+
In [5]:
+
+
+
from tqdm import tqdm
+ineqs2=[]
+for ineq in tqdm(ineqs):
+    ineqs2+=[parser(*ineq)]
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
100%|██████████| 35/35 [00:46<00:00,  1.32s/it]
+
+
+
+ +
+
+ +
+
+
+
+

Now let's look at the formulas when converted to polynomials.

+ +
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+
+
+
+
In [6]:
+
+
+
for i,ineq in zip(range(len(ineqs2)),ineqs2):
+    print(i)
+    display(reducegens(assumeall(ineq,positive=True)))
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
0
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{2} - ab - ac + b^{2} - bc + c^{2}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
1
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{2} - ab - ac - ad + b^{2} + c^{2} + d^{2}, a, b, c, d, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
2
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{4}b^{2} + a^{3}b^{2} - a^{3}b - 2 a^{2}b + a^{2} + ab^{2} - ab + b^{2}, a, b, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
3
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 a^{3}b + a^{3} - 4 a^{2}b^{2} - a^{2}b + a^{2} + 2 ab^{3} - ab^{2} - 2 ab + b^{3} + b^{2}, a, b, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
4
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{3} - a^{2}b - ab^{2} + b^{3}, a, b, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
5
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 a^{3} - a^{2}b - a^{2}c - ab^{2} - ac^{2} + 2 b^{3} - b^{2}c - bc^{2} + 2 c^{3}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
6
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 a^{3} - 3 a^{2}b + b^{3}, a, b, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
7
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{3} - a^{2}b - ac^{2} + b^{3} - b^{2}c + c^{3}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
8
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{3}c + a^{3}d + a^{2}b^{2} - a^{2}bd - 2 a^{2}c^{2} - a^{2}cd + a^{2}d^{2} + ab^{3} - ab^{2}c - ab^{2}d - abc^{2} + ac^{3} - acd^{2} + b^{3}d + b^{2}c^{2} - 2 b^{2}d^{2} + bc^{3} - bc^{2}d - bcd^{2} + bd^{3} + c^{2}d^{2} + cd^{3}, a, b, c, d, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
9
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 a^{5}b^{2} + 2 a^{5}bc + 2 a^{5}c^{2} + a^{4}b^{3} - a^{4}b^{2}c - a^{4}bc^{2} + a^{4}c^{3} + a^{3}b^{4} - 2 a^{3}b^{3}c - 4 a^{3}b^{2}c^{2} - 2 a^{3}bc^{3} + a^{3}c^{4} + 2 a^{2}b^{5} - a^{2}b^{4}c - 4 a^{2}b^{3}c^{2} - 4 a^{2}b^{2}c^{3} - a^{2}bc^{4} + 2 a^{2}c^{5} + 2 ab^{5}c - ab^{4}c^{2} - 2 ab^{3}c^{3} - ab^{2}c^{4} + 2 abc^{5} + 2 b^{5}c^{2} + b^{4}c^{3} + b^{3}c^{4} + 2 b^{2}c^{5}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
10
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( ad + af - 2 \sqrt{a}\sqrt{b}\sqrt{c}\sqrt{d} - 2 \sqrt{a}\sqrt{b}\sqrt{e}\sqrt{f} + bc + be + cf - 2 \sqrt{c}\sqrt{d}\sqrt{e}\sqrt{f} + de, \sqrt{a}, \sqrt{b}, \sqrt{c}, \sqrt{d}, \sqrt{e}, \sqrt{f}, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
11
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 3 a^{4}b + 3 a^{4}c - 2 a^{3}b^{2} + 2 a^{3}c^{2} + 2 a^{2}b^{3} - 6 a^{2}b^{2}c - 6 a^{2}bc^{2} - 2 a^{2}c^{3} + 3 ab^{4} - 6 ab^{2}c^{2} + 3 ac^{4} + 3 b^{4}c - 2 b^{3}c^{2} + 2 b^{2}c^{3} + 3 bc^{4}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
12
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 4 x^{8}y^{4} + 4 x^{7}y^{4} - 3 x^{5}y^{5} - 3 x^{5}y^{4} + 4 x^{4}y^{8} + 4 x^{4}y^{7} - 3 x^{4}y^{5} - 6 x^{4}y^{4} - 3 x^{4}y^{3} - 3 x^{3}y^{4} - 3 x^{3}y^{3} + 4 xy + 4, x, y, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
13
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{3}c^{2} + a^{2}b^{3} - a^{2}b^{2}c - a^{2}bc^{2} - ab^{2}c^{2} + b^{2}c^{3}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
14
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{4}c + a^{3}c^{2} + a^{2}b^{3} + ab^{4} + b^{2}c^{3} + bc^{4}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
15
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{4}c - 4 a^{3}bc + a^{3}c^{2} + a^{2}b^{3} + 6 a^{2}b^{2}c - 2 a^{2}bc^{2} + ab^{4} - 4 ab^{3}c - 2 ab^{2}c^{2} + b^{2}c^{3} + bc^{4}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
16
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{6}b^{3} + a^{6}c^{3} - 2 a^{5}b^{2}c^{2} + a^{3}b^{6} + a^{3}c^{6} - 2 a^{2}b^{5}c^{2} - 2 a^{2}b^{2}c^{5} + b^{6}c^{3} + b^{3}c^{6}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
17
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 a^{8}b^{8} + 2 a^{7}b^{6} + 2 a^{6}b^{7} - 3 a^{6}b^{5} - 3 a^{5}b^{6} + 2 a^{5}b^{5} - 3 a^{5}b^{3} + 2 a^{5}b^{2} - 6 a^{4}b^{4} + 2 a^{4}b^{3} + 2 a^{4} - 3 a^{3}b^{5} + 2 a^{3}b^{4} - 3 a^{3}b^{2} + 2 a^{3}b + 2 a^{2}b^{5} - 3 a^{2}b^{3} + 2 ab^{3} + 2 b^{4}, a, b, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
18
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{8} - a^{6}bc - 2 a^{5}b^{3} + a^{5}b^{2}c + a^{5}bc^{2} - 2 a^{5}c^{3} + 3 a^{4}b^{4} - a^{4}b^{3}c + 2 a^{4}b^{2}c^{2} - a^{4}bc^{3} + 3 a^{4}c^{4} - 2 a^{3}b^{5} - a^{3}b^{4}c - a^{3}b^{3}c^{2} - a^{3}b^{2}c^{3} - a^{3}bc^{4} - 2 a^{3}c^{5} + a^{2}b^{5}c + 2 a^{2}b^{4}c^{2} - a^{2}b^{3}c^{3} + 2 a^{2}b^{2}c^{4} + a^{2}bc^{5} - ab^{6}c + ab^{5}c^{2} - ab^{4}c^{3} - ab^{3}c^{4} + ab^{2}c^{5} - abc^{6} + b^{8} - 2 b^{5}c^{3} + 3 b^{4}c^{4} - 2 b^{3}c^{5} + c^{8}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
19
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( y^{18}z^{9} + 2 y^{16}z^{14} - y^{15}z^{9} + 2 y^{14}z^{16} - y^{14}z^{13} - y^{13}z^{14} - y^{13}z^{11} - y^{12}z^{12} + y^{12}z^{9} - y^{11}z^{13} - y^{11}z^{7} + 2 y^{11}z^{4} - y^{10}z^{5} + y^{9}z^{18} - y^{9}z^{15} + y^{9}z^{12} - y^{9}z^{6} - y^{8}z^{4} - y^{7}z^{11} - y^{7}z^{5} + 2 y^{7}z^{2} - y^{6}z^{9} + y^{6}z^{6} - y^{5}z^{10} - y^{5}z^{7} + 2 y^{4}z^{11} - y^{4}z^{8} - y^{3}z^{3} + 2 y^{2}z^{7} + 1, y, z, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
20
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{3} - a^{2}b - a^{2}c - ab^{2} + 3 abc - ac^{2} + b^{3} - b^{2}c - bc^{2} + c^{3}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
21
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 y^{4}z^{2} - y^{3}z^{3} - 5 y^{3}z + 2 y^{2}z^{4} - 9 y^{2}z^{2} + 5 y^{2} - 5 yz^{3} + 23 yz + 5 z^{2} - 9, y, z, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
22
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 a^{\frac{5}{2}}b\frac{1}{\sqrt{a b + a + b + 1}} + 2 a^{\frac{5}{2}}\frac{1}{\sqrt{a b + a + b + 1}} + 2 a^{2}b^{\frac{7}{2}}\frac{1}{\sqrt{b + 1}} + 2 a^{2}b^{\frac{5}{2}}\frac{1}{\sqrt{b + 1}} + 2 a^{2}b^{2}\frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}} - a^{2}b^{2}\sqrt{3} + 2 a^{2}b\frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}} - a^{2}b\sqrt{3} + 2 a^{\frac{3}{2}}b\frac{1}{\sqrt{a b + a + b + 1}} + 4 ab^{\frac{7}{2}}\frac{1}{\sqrt{b + 1}} + 4 ab^{\frac{5}{2}}\frac{1}{\sqrt{b + 1}} + 4 ab^{2}\frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}} - 2 ab^{2}\sqrt{3} + 2 ab^{\frac{3}{2}}\frac{1}{\sqrt{b + 1}} + 6 ab\frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}} - 2 ab\sqrt{3} + 2 a\frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}} - a\sqrt{3} + 2 b^{\frac{7}{2}}\frac{1}{\sqrt{b + 1}} + 2 b^{\frac{5}{2}}\frac{1}{\sqrt{b + 1}} + 2 b^{2}\frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}} - b^{2}\sqrt{3} + 4 b\frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}} - b\sqrt{3} + 2 \frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}}, \sqrt{a}, \sqrt{b}, \frac{1}{a b \sqrt{a b + a + b + 1} + a \sqrt{a b + a + b + 1} + b \sqrt{a b + a + b + 1} + \sqrt{a b + a + b + 1}}, \frac{1}{\sqrt{a b + a + b + 1}}, \frac{1}{\sqrt{b + 1}}, \sqrt{3}, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
23
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{4} - 4 abcd + b^{4} + c^{4} + d^{4}, a, b, c, d, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
24
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{3}b^{2} + a^{3}c^{2} + a^{2}b^{3} - 2 a^{2}b^{2}c - 2 a^{2}bc^{2} + a^{2}c^{3} - 2 ab^{2}c^{2} + b^{3}c^{2} + b^{2}c^{3}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
25
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( def + de + df - 2 \sqrt{d}\sqrt{e} - 2 \sqrt{d}\sqrt{f} + ef - 2 \sqrt{e}\sqrt{f} + f + 2, \sqrt{d}, \sqrt{e}, \sqrt{f}, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
26
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( de^{2}f^{2} + de^{2}f + 2 def^{2} - 6 def + de + df^{2} + df + e^{2}f^{2} + e^{2}f + 2 ef^{2} + 3 ef + e + f^{2} + 2 f + 1, d, e, f, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
27
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( s^{2}def + s^{2}, s, d, e, f, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
28
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( -2 \sqrt{3}x_{1}x_{2} - 2 \sqrt{3}x_{2}x_{3} - 2 \sqrt{3}x_{3}x_{4} - 2 \sqrt{3}x_{4}x_{5} + 3 x_{1}^{2} + 3 x_{2}^{2} + 3 x_{3}^{2} + 3 x_{4}^{2} + 3 x_{5}^{2}, \sqrt{3}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
29
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 7 a^{2}b^{3} - 6 a^{2}b^{2} - 6 a^{2}b + 7 a^{2} + 14 ab^{3} - 12 ab^{2} + 15 ab - 13 a + 7 b^{3} - 6 b^{2} - 6 b + 7, a, b, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
30
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{2}b^{2} + a^{2}b + 2 ab^{2} + ab + a + b^{2} + b, a, b, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
31
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( -2 \frac{1}{\sqrt{b}}\sqrt{\frac{a^{2} b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a^{2} b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1}} - 2 \frac{1}{\sqrt{b}}\sqrt{\frac{a b}{a^{2} b + a^{2} + a b + 2 a + 1} + \frac{b}{a^{2} b + a^{2} + a b + 2 a + 1}} - 2 \sqrt{\frac{a^{2} b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a^{2} b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1}}\sqrt{\frac{a b}{a^{2} b + a^{2} + a b + 2 a + 1} + \frac{b}{a^{2} b + a^{2} + a b + 2 a + 1}} + 3, \frac{1}{\sqrt{b}}, \sqrt{\frac{a^{2} b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a^{2} b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a b^{2}}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1} + \frac{a b}{a^{2} b + a^{2} + a b^{2} + 3 a b + 2 a + b^{2} + 2 b + 1}}, \sqrt{\frac{a b}{a^{2} b + a^{2} + a b + 2 a + 1} + \frac{b}{a^{2} b + a^{2} + a b + 2 a + 1}}, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
32
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( a^{5}b^{2} + a^{5}c^{2} + a^{4}b^{3} - a^{4}b^{2}c - a^{4}bc^{2} - a^{4}c^{3} - a^{3}b^{4} + a^{3}c^{4} + a^{2}b^{5} - a^{2}b^{4}c - a^{2}bc^{4} + a^{2}c^{5} - ab^{4}c^{2} - ab^{2}c^{4} + b^{5}c^{2} + b^{4}c^{3} - b^{3}c^{4} + b^{2}c^{5}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
33
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 a^{2}d^{3} - a^{2}d^{2} - a^{2}d + 2 a^{2} + 4 ad^{3} - 2 ad^{2} - 3 a + 2 d^{3} - d^{2} - d + 2, a, d, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+ +
+ + +
+
34
+
+
+
+ +
+ +
+ + + + +
+$\displaystyle \operatorname{Poly}{\left( 2 a^{3} - 3 a^{2}b + a^{2}c + ab^{2} - 3 ac^{2} + 2 b^{3} - 3 b^{2}c + bc^{2} + 2 c^{3}, a, b, c, domain=\mathbb{Z} \right)}$ +
+ +
+ +
+
+ +
+
+
+
+

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.

+

Now let's try some methods of proving these inequalities. The first one would be the simple prove.

+ +
+
+
+
+
+
In [7]:
+
+
+
tm=[0]*4
+
+ +
+
+
+ +
+
+
+
In [8]:
+
+
+
from collections import Counter
+from timeit import default_timer as timer
+start=timer()
+t=[]
+for ineq in ineqs2:
+    t+=[prove(ineq)]
+    print(t[-1],end=',')
+tm[0]=timer()-start
+print('\n',tm[0],sep='')
+Counter(t)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
0,2,2,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,2,2,0,0,2,2,0,2,2,0,2,0,2,0,
+20.714609994000057
+
+
+
+ +
+ +
Out[8]:
+ + + + +
+
Counter({0: 21, 2: 14})
+
+ +
+ +
+
+ +
+
+
+
+

Code 0 means that the proof was found, all other codes means that proof wasn't found. So this method has proved 21 inequalities.

+

The second method uses findvalues function, rationalizes the result numbers and gives them as additional parameter to prove function.

+ +
+
+
+
+
+
In [9]:
+
+
+
def cut(a):
+    if a<=0 or a>=100 or (a is None):
+        return 1
+    return a
+start=timer()
+t2=[]
+for ineq in ineqs2:
+    numvalues=findvalues(ineq,disp=0)
+    values=nsimplify(numvalues,tolerance=0.1,rational=True)
+    values=list(map(cut,values))
+    t2+=[prove(ineq,values=values)]
+    print(t2[-1],end=',')
+tm[1]=timer()-start
+print('\n',tm[1],sep='')
+Counter(t2)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,2,2,0,0,2,0,0,2,2,0,2,0,2,0,
+21.832711064998875
+
+
+
+ +
+ +
Out[9]:
+ + + + +
+
Counter({0: 24, 2: 11})
+
+ +
+ +
+
+ +
+
+
+
+

The third method is similar to the second one, but instead of rationalize values it squares, rationalizes and makes square roots of these values.

+ +
+
+
+
+
+
In [10]:
+
+
+
def cut(a):
+    if a<=0 or a>=1000 or (a is None):
+        return S(1)
+    return a
+    
+start=timer()
+t3=[]
+for ineq in ineqs2:
+    numvalues=findvalues(ineq,disp=0)
+    numvalues=tuple([x**2 for x in numvalues])
+    values=nsimplify(numvalues,tolerance=0.1,rational=True)
+    values=[sqrt(x) for x in values]
+    values=list(map(cut,values))
+    t3+=[prove(ineq,values=values)]
+    print(t3[-1],end=',')
+tm[2]=timer()-start
+print('\n',tm[2],sep='')
+Counter(t3)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,2,2,0,0,2,0,0,2,2,0,2,0,2,0,
+21.697120987002563
+
+
+
+ +
+ +
Out[10]:
+ + + + +
+
Counter({0: 23, 2: 12})
+
+ +
+ +
+
+ +
+
+
+
+

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.

+ +
+
+
+
+
+
In [11]:
+
+
+
def betw(a):
+    return a>0.001 and a<1000 and a!=None
+def cut(a):
+    if betw(a):
+        return a
+    return S(1)
+
+start=timer()
+t4=[]
+for ineq in ineqs2:
+    numvalues=findvalues(ineq,disp=0)
+    n=1
+    numvalues2=[]
+    for i in numvalues:
+        if betw(i):
+            n=1/i
+            break
+    for i in numvalues:
+        if betw(i):
+            numvalues2+=[i*n]
+        else:
+            numvalues2+=[1]
+    numvalues3=findvalues(ineq,values=numvalues2,disp=0)
+    numvalues4=tuple([x**2 for x in numvalues3])
+    values=nsimplify(numvalues4,tolerance=0.1,rational=True)
+    values=[sqrt(x) for x in values]
+    values=list(map(cut,values))
+    t4+=[prove(ineq,values=values)]
+    print(t4[-1],end=',')
+tm[3]=timer()-start
+print('\n',tm[3],sep='')
+Counter(t4)
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,2,2,0,0,2,0,0,0,2,0,2,0,2,0,
+23.187820409999404
+
+
+
+ +
+ +
Out[11]:
+ + + + +
+
Counter({0: 24, 2: 11})
+
+ +
+ +
+
+ +
+
+
+
In [17]:
+
+
+
import matplotlib.pyplot as plt
+plt.bar(['1','2','3','4'],tm)
+plt.ylabel('time [seconds]')
+plt.xlabel('method')
+plt.show()
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+ +
+ +
+ +
+
+ +
+
+
+
In [13]:
+
+
+
import pandas as pd
+u=pd.DataFrame(zip(['']*len(ineqs),t,t2,t3,t4))
+
+ +
+
+
+ +
+
+
+
In [14]:
+
+
+
plt.bar(['1','2','3','4'],[sum(u[i]==0)/len(ineqs2)*100 for i in range(1,5)])
+plt.ylabel('inequalities proven [%]')
+plt.xlabel('method')
+plt.show()
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + + +
+ +
+ +
+ +
+
+ +
+
+
+
+

Down below there are contingency tables and McNemar test for every pair of methods.

+ +
+
+
+
+
+
In [15]:
+
+
+
for i in range(4):
+    for j in range(i+1,4):
+        display(pd.crosstab(u[i+1]==0, u[j+1]==0))
+
+ +
+
+
+ +
+
+ + +
+ +
+ + + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
2FalseTrue
1
False113
True021
+
+
+ +
+ +
+ +
+ + + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
3FalseTrue
1
False122
True021
+
+
+ +
+ +
+ +
+ + + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
4FalseTrue
1
False113
True021
+
+
+ +
+ +
+ +
+ + + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
3FalseTrue
2
False110
True123
+
+
+ +
+ +
+ +
+ + + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
4FalseTrue
2
False101
True123
+
+
+ +
+ +
+ +
+ + + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
4FalseTrue
3
False111
True023
+
+
+ +
+ +
+
+ +
+
+
+
In [16]:
+
+
+
from statsmodels.stats.contingency_tables import mcnemar
+for i in range(4):
+    for j in range(i+1,4):
+        print(mcnemar(pd.crosstab(u[i+1]==0, u[j+1]==0)))
+
+ +
+
+
+ +
+
+ + +
+ +
+ + +
+
pvalue      0.25
+statistic   0.0
+pvalue      0.5
+statistic   0.0
+pvalue      0.25
+statistic   0.0
+pvalue      1.0
+statistic   0.0
+pvalue      1.0
+statistic   1.0
+pvalue      1.0
+statistic   0.0
+
+
+
+ +
+
+ +
+
+
+
+

In all cases p-value is greater than 0.05, so there is no statistical difference between the methods.

+ +
+
+
+
+
+ + + + + + diff --git a/statistics.ipynb b/statistics.ipynb index fe47221..d65b5bc 100644 --- a/statistics.ipynb +++ b/statistics.ipynb @@ -17,7 +17,6 @@ "metadata": {}, "outputs": [], "source": [ - "from importlib import reload\n", "from sympy import *\n", "import shiroindev\n", "from shiroindev import *\n", @@ -58,25 +57,20 @@ " 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", @@ -147,7 +141,6 @@ "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", diff --git a/statistics.pdf b/statistics.pdf deleted file mode 100644 index 5d30eec2fcbcf58b5ba85cb5f95d94225f6e4e95..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 71326 zcma%iQ;;q&v+W-2v2EM7ZQHgzv&S=gY}>YN+qV6U&AacZ`*2R3`|v*`ol3eY4_)c3 z)mb4`5D}wgq+^95J-P+FLNPKEFc8=qSwZpe(2H5xxR^T8i`f{un2MMh+nbou%b41k zyI2q~v9UAr@j*GeIGGySLV0X7YyFAcY(xC%%^he1j@}xAkyNbxjV21M6G2_mr`z~D zVkA^6x=hcfP;S;CiEeJJeS-?XdgaP1m}BUoGAAb zV0Qlvl35q#fEG4BZ8bniOcYMuC=ka0Nya`YueV(!tH!=wkh8P%IjF#p01Bjz!Yp=> zW*JB!!_v>cSb81>zGOVl&AgYi$je9yDVs(Z!wHGn{ws}P<%l0Oq-@=!857l%_-5js z-C2DST(el)IdRfVnGVpQs+rzjlhH6njo)Adi%fLaAmrO4->R7?+3A+ZC&Ap2xX|OM zc+#noI6*r9sbnxu9-ufBT2+yG9(BiXH<656v{{YIE5#euEi_U5$EtR+Nz!^9LxIkl zs11G(Y8n12gDw_r!J*KeIL@0IT^72BN$CX3Wkgu&lYkKhx)C|EWPMrvW|yCjPY(Gc 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