From f5e0cc14d4f0efaa388d1f3682daa143155a1d7e Mon Sep 17 00:00:00 2001 From: urojony Date: Sun, 17 May 2020 23:46:53 +0200 Subject: [PATCH] added statistics and fixed few minor bugs --- .ipynb_checkpoints/examples-checkpoint.ipynb | 2045 +++-- .ipynb_checkpoints/sandbox-checkpoint.ipynb | 2706 +++++-- .../statistics-checkpoint.ipynb | 476 ++ __pycache__/shiroindev.cpython-36.pyc | Bin 15679 -> 16636 bytes __pycache__/shiroindev.cpython-37.pyc | Bin 15309 -> 19697 bytes examples.ipynb | 380 +- examples.pdf | Bin 227669 -> 59577 bytes sandbox.ipynb | 7076 ++++++++++++++--- shiroindev.py | 352 +- statistics.ipynb | 1563 ++++ statistics.pdf | Bin 0 -> 71326 bytes 11 files changed, 12183 insertions(+), 2415 deletions(-) create mode 100644 .ipynb_checkpoints/statistics-checkpoint.ipynb create mode 100644 statistics.ipynb create mode 100644 statistics.pdf diff --git a/.ipynb_checkpoints/examples-checkpoint.ipynb b/.ipynb_checkpoints/examples-checkpoint.ipynb index b8a7d20..4e27353 100644 --- a/.ipynb_checkpoints/examples-checkpoint.ipynb +++ b/.ipynb_checkpoints/examples-checkpoint.ipynb @@ -4,7 +4,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this notebook there are presented examples of usage of shiroin, a python library for proving inequalities of multivariate polynomials." + "In this notebook there are presented examples of usage of shiroin, a python library for proving inequalities of multivariate polynomials.\n", + "\n", + "At the beginning we need to load the packages." ] }, { @@ -13,17 +15,20 @@ "metadata": {}, "outputs": [], "source": [ + "from sympy import *\n", "from shiroindev import *\n", - "sVars.seed=1\n", "from IPython.display import Latex\n", - "sVars.display=lambda x:display(Latex(x))" + "shiro.seed=1\n", + "shiro.display=lambda x:display(Latex(x))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "The first line obviously loads this package. The second one sets a seed for proving functions. If you don't write it, you can get a slightly different proof each time you run a function. The next two lines provide a nicer display of proofs, i.e. formulas will be shown instead of LaTeX code of these formulas. Note that this works on Jupyter, but not on the git page.\n", + "`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. \n", + "\n", + "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.\n", "\n", "Now let's make some proofs. We will use problems from https://www.imomath.com/index.php?options=593&lmm=0." ] @@ -51,7 +56,7 @@ { "data": { "text/latex": [ - "numerator: $a^2-ab-ac+b^2-bc+c^2$" + "numerator: $a^{2} - a b - a c + b^{2} - b c + c^{2}$" ], "text/plain": [ "" @@ -111,7 +116,7 @@ { "data": { "text/latex": [ - "$$ ab+ac+bc \\le a^2+b^2+c^2 $$" + "$$ a b+a c+b c \\le a^{2}+b^{2}+c^{2} $$" ], "text/plain": [ "" @@ -139,7 +144,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Function prove prints several things. The first two gives us a formula after expanding it. To proceed, a **numerator** has to be a **polynomial with integer coefficients**. The next one is status, which is the return status of the first use of ```scipy.optimize.linprog```. Possible outputs and explanations are\n", + "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\n", "\n", "* 0 - found a proof with real coefficients,\n", "* 1 - need more time, \n", @@ -157,7 +162,7 @@ { "data": { "text/latex": [ - "numerator: $2a^2-2ab-2ac+2b^2-2bc+2c^2$" + "numerator: $2 a^{2} - 2 a b - 2 a c + 2 b^{2} - 2 b c + 2 c^{2}$" ], "text/plain": [ "" @@ -205,7 +210,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 a b \\le a^{2}+b^{2}$$" ], "text/plain": [ "" @@ -217,7 +222,7 @@ { "data": { "text/latex": [ - "$$2ac \\le a^2+c^2$$" + "$$2 a c \\le a^{2}+c^{2}$$" ], "text/plain": [ "" @@ -229,7 +234,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 b c \\le b^{2}+c^{2}$$" ], "text/plain": [ "" @@ -300,7 +305,7 @@ { "data": { "text/latex": [ - "numerator: $a^2-ab-ac-ad+b^2+c^2+d^2$" + "numerator: $a^{2} - a b - a c - a d + b^{2} + c^{2} + d^{2}$" ], "text/plain": [ "" @@ -348,7 +353,7 @@ { "data": { "text/latex": [ - "$$ ab+ac+ad \\le a^2+b^2+c^2+d^2 $$" + "$$ a b+a c+a d \\le a^{2}+b^{2}+c^{2}+d^{2} $$" ], "text/plain": [ "" @@ -379,9 +384,10 @@ "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.\n", "\n", "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. \n", - "2. If the symmetry group of the formula is transitive (for example, the formula is cyclic or symmetric)\n", + "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.\n", + "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).\n", "\n", - "So let's try $a=2$ and $b=c=d=1$." + "In the formula above $b,c,d$ are in one orbit and the formula is homogenous, so let's try $a=2$ and $b=c=d=1$." ] }, { @@ -392,7 +398,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 2a$" + "Substitute $a\\to 2 e$" ], "text/plain": [ "" @@ -404,7 +410,7 @@ { "data": { "text/latex": [ - "numerator: $4a^2-2ab-2ac-2ad+b^2+c^2+d^2$" + "numerator: $b^{2} - 2 b e + c^{2} - 2 c e + d^{2} - 2 d e + 4 e^{2}$" ], "text/plain": [ "" @@ -452,7 +458,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 b e \\le b^{2}+e^{2}$$" ], "text/plain": [ "" @@ -464,7 +470,7 @@ { "data": { "text/latex": [ - "$$2ac \\le a^2+c^2$$" + "$$2 c e \\le c^{2}+e^{2}$$" ], "text/plain": [ "" @@ -476,7 +482,7 @@ { "data": { "text/latex": [ - "$$2ad \\le a^2+d^2$$" + "$$2 d e \\le d^{2}+e^{2}$$" ], "text/plain": [ "" @@ -488,7 +494,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le a^2 $$" + "$$ 0 \\le e^{2} $$" ], "text/plain": [ "" @@ -528,7 +534,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Function makes a substitution $a\\to 2a$ (which should be understood as $a=2a'$) 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 $a'^2=0$ and eventually $a=0$. We can also try a little bit lower value for $a$." + "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 $a$." ] }, { @@ -539,7 +545,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 7a/4$" + "Substitute $a\\to \\frac{7 f}{4}$" ], "text/plain": [ "" @@ -551,7 +557,7 @@ { "data": { "text/latex": [ - "numerator: $49a^2-28ab-28ac-28ad+16b^2+16c^2+16d^2$" + "numerator: $16 b^{2} - 28 b f + 16 c^{2} - 28 c f + 16 d^{2} - 28 d f + 49 f^{2}$" ], "text/plain": [ "" @@ -599,7 +605,7 @@ { "data": { "text/latex": [ - "$$28ab \\le 14a^2+14b^2$$" + "$$28 b f \\le 14 b^{2}+14 f^{2}$$" ], "text/plain": [ "" @@ -611,7 +617,7 @@ { "data": { "text/latex": [ - "$$28ac \\le 14a^2+14c^2$$" + "$$28 c f \\le 14 c^{2}+14 f^{2}$$" ], "text/plain": [ "" @@ -623,7 +629,7 @@ { "data": { "text/latex": [ - "$$28ad \\le 14a^2+14d^2$$" + "$$28 d f \\le 14 d^{2}+14 f^{2}$$" ], "text/plain": [ "" @@ -635,7 +641,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 7a^2+2b^2+2c^2+2d^2 $$" + "$$ 0 \\le 2 b^{2}+2 c^{2}+2 d^{2}+7 f^{2} $$" ], "text/plain": [ "" @@ -675,24 +681,88 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Now we can see that if $a^2+b^2+c^2+d^2-a(b+c+d)=0$, then $7a'^2+2b^2+2c^2+2d^2=0$ 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.\n", + "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.\n", "\n", - "Let's skip the problem 3 and look solve the problem 4 instead.\n", + "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.\n", "\n", - "#### Problem 4\n", - "If $x$ and $y$ are two positive numbers less than 1, prove that\n", - "$$\\frac{1}{1-x^2}+\\frac{1}{1-y^2}\\ge \\frac{2}{1-xy}.$$" + "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." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Optimization terminated successfully.\n", + " Current function value: 1.154701\n", + " Iterations: 68\n", + " Function evaluations: 127\n" + ] + }, + { + "data": { + "text/plain": [ + "(1.4339109663193974,\n", + " 0.8278441585048405,\n", + " 0.8279027492686561,\n", + " 0.8278930696996669)" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "formula=S('a^2+b^2+c^2+d^2-a*(b+c+d)')\n", + "numvalues=findvalues(formula)\n", + "numvalues" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "numerator: $-x^3y+2x^2y^2-x^2-xy^3+2xy-y^2$" + "$\\displaystyle \\left( \\frac{10}{7}, \\ \\frac{5}{6}, \\ \\frac{5}{6}, \\ \\frac{5}{6}\\right)$" + ], + "text/plain": [ + "(10/7, 5/6, 5/6, 5/6)" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "values=nsimplify(numvalues,tolerance=0.1,rational=True)\n", + "values" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $a\\to \\frac{10 e}{7}$" ], "text/plain": [ "" @@ -704,7 +774,197 @@ { "data": { "text/latex": [ - "denominator: $x^3y^3-x^3y-x^2y^2+x^2-xy^3+xy+y^2-1$" + "Substitute $b\\to \\frac{5 f}{6}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $c\\to \\frac{5 g}{6}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $d\\to \\frac{5 h}{6}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $3600 e^{2} - 2100 e f - 2100 e g - 2100 e h + 1225 f^{2} + 1225 g^{2} + 1225 h^{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1764$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "From weighted AM-GM inequality:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2100 e f \\le 1050 e^{2}+1050 f^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2100 e g \\le 1050 e^{2}+1050 g^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2100 e h \\le 1050 e^{2}+1050 h^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 450 e^{2}+175 f^{2}+175 g^{2}+175 h^{2} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "newproof()\n", + "prove(formula,values)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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.\n", + "\n", + "Let's skip the problem 3 and look solve the problem 4 instead.\n", + "\n", + "#### Problem 4\n", + "If $x$ and $y$ are two positive numbers less than 1, prove that\n", + "$$\\frac{1}{1-x^2}+\\frac{1}{1-y^2}\\ge \\frac{2}{1-xy}.$$" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "numerator: $- x^{3} y + 2 x^{2} y^{2} - x^{2} - x y^{3} + 2 x y - y^{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $x^{3} y^{3} - x^{3} y - x^{2} y^{2} + x^{2} - x y^{3} + x y + y^{2} - 1$" ], "text/plain": [ "" @@ -740,7 +1000,7 @@ { "data": { "text/latex": [ - "$$ x^3y+x^2+xy^3+y^2 \\le 2x^2y^2+2xy $$" + "$$ x^{3} y+x^{2}+x y^{3}+y^{2} \\le 2 x^{2} y^{2}+2 x y $$" ], "text/plain": [ "" @@ -779,7 +1039,7 @@ "2" ] }, - "execution_count": 7, + "execution_count": 10, "metadata": {}, "output_type": "execute_result" } @@ -794,19 +1054,19 @@ "source": [ "`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.\n", "\n", - "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 `formula` is small. `values` should be inside corresponding `intervals`. This argument is optional but it's better to use it.\n", + "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.\n", "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')." ] }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $x\\to 1-1/(x+1)$" + "Substitute $x\\to 1 - \\frac{1}{a + 1}$" ], "text/plain": [ "" @@ -818,7 +1078,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to 1-1/(y+1)$" + "Substitute $y\\to 1 - \\frac{1}{b + 1}$" ], "text/plain": [ "" @@ -830,7 +1090,7 @@ { "data": { "text/latex": [ - "numerator: $6x^3y+3x^3-12x^2y^2-3x^2y+3x^2+6xy^3-3xy^2-6xy+3y^3+3y^2$" + "numerator: $6 a^{3} b + 3 a^{3} - 12 a^{2} b^{2} - 3 a^{2} b + 3 a^{2} + 6 a b^{3} - 3 a b^{2} - 6 a b + 3 b^{3} + 3 b^{2}$" ], "text/plain": [ "" @@ -842,7 +1102,7 @@ { "data": { "text/latex": [ - "denominator: $4x^2y+2x^2+4xy^2+8xy+3x+2y^2+3y+1$" + "denominator: $4 a^{2} b + 2 a^{2} + 4 a b^{2} + 8 a b + 3 a + 2 b^{2} + 3 b + 1$" ], "text/plain": [ "" @@ -878,7 +1138,7 @@ { "data": { "text/latex": [ - "$$12x^2y^2 \\le 6x^3y+6xy^3$$" + "$$12 a^{2} b^{2} \\le 6 a^{3} b+6 a b^{3}$$" ], "text/plain": [ "" @@ -890,7 +1150,7 @@ { "data": { "text/latex": [ - "$$3x^2y \\le 2x^3+y^3$$" + "$$3 a^{2} b \\le 2 a^{3}+b^{3}$$" ], "text/plain": [ "" @@ -902,7 +1162,7 @@ { "data": { "text/latex": [ - "$$3xy^2 \\le x^3+2y^3$$" + "$$3 a b^{2} \\le a^{3}+2 b^{3}$$" ], "text/plain": [ "" @@ -914,7 +1174,7 @@ { "data": { "text/latex": [ - "$$6xy \\le 3x^2+3y^2$$" + "$$6 a b \\le 3 a^{2}+3 b^{2}$$" ], "text/plain": [ "" @@ -953,12 +1213,13 @@ "0" ] }, - "execution_count": 8, + "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ + "newproof()\n", "newformula,newvalues=makesubs('1/(1-x^2)+1/(1-y^2)-2/(1-x*y)','[0,1],[0,1]','1/2,1/2')\n", "prove(newformula*3,newvalues)" ] @@ -978,411 +1239,7 @@ "* constraints for variables are written as intervals,\n", "* there are no \"backwards dependencies\", i.e. there is no $x$ in the interval of $y$.\n", "\n", - "If these two conditions hold, then you can use `makesubs` function.\n", - "**Warning:** at this moment `makesubs` **doesn't warn you if your list of intervals doesn't follow these rules!**\n" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "Substitute $x\\to -y+1+(y-1)/(x+1)$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $y\\to 1-1/(y+1)$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $y\\to y/2$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "numerator: $x^4y^2+x^3y^2-2x^3y-4x^2y+4x^2+xy^2-2xy+y^2$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "denominator: $x^3y^2+2x^3y+2x^2y^2+4x^2y+xy^2+2xy$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "status: 0" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "From weighted AM-GM inequality:" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$2x^3y \\le x^4y^2+x^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$4x^2y \\le x^3y^2+2x^2+xy^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$2xy \\le x^2+y^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le 0 $$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "0" - ] - }, - "execution_count": 9, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "formula=Sm('xy/z+yz/x+zx/y-1').subs('z',S('1-x-y'))\n", - "newformula,values=makesubs(formula,'[0,1-y],[0,1]','1/3,1/3')\n", - "prove(newformula,values)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The proof is found, so the assumption that 1 is the minimum of `xy/z+yz/x+zx/y` was good.\n", - "\n", - "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`. \n", - "\n", - "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. " - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "numerator: $2a^2-2ab-2ac+2b^2-2bc+2c^2$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "denominator: $1$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "status: 0" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "From weighted AM-GM inequality:" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$2ab \\le a^2+b^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$2ac \\le a^2+c^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$2bc \\le b^2+c^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le 0 $$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "0" - ] - }, - "execution_count": 10, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "prove('(a^2+b^2+c^2-a*b-a*c-b*c)*2')\n", - "#prove(S('(a^2+b^2+c^2-a*b-a*c-b*c)*2'))\n", - "#prove(Sm('2(a^2+b^2+c^2-ab-ac-bc)'))\n", - "#prove(cyclize('2*a^2-2*a*b'))\n", - "#prove(symmetrize('a^2-a*b'))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now look at formula $(x-1)^4$. It's quite obvious that it's nonnegative, but `prove` fails to show this!" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "numerator: $x^4-4x^3+6x^2-4x+1$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "denominator: $1$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "status: 2" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Program couldn't find any proof." - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 4x^3+4x \\le x^4+6x^2+1 $$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "2" - ] - }, - "execution_count": 11, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "prove('(x-1)^4')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "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 $(-\\infty,1)$." + "If these two conditions hold, then you can use `makesubs` function." ] }, { @@ -1393,7 +1250,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+1$" + "Substitute $x\\to - y + 1 + \\frac{y - 1}{a + 1}$" ], "text/plain": [ "" @@ -1405,7 +1262,7 @@ { "data": { "text/latex": [ - "numerator: $x^4$" + "Substitute $y\\to 1 - \\frac{1}{b + 1}$" ], "text/plain": [ "" @@ -1417,7 +1274,31 @@ { "data": { "text/latex": [ - "denominator: $1$" + "Substitute $b\\to \\frac{c}{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $a^{4} c^{2} + a^{3} c^{2} - 2 a^{3} c - 4 a^{2} c + 4 a^{2} + a c^{2} - 2 a c + c^{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $a^{3} c^{2} + 2 a^{3} c + 2 a^{2} c^{2} + 4 a^{2} c + a c^{2} + 2 a c$" ], "text/plain": [ "" @@ -1441,7 +1322,55 @@ { "data": { "text/latex": [ - "$$ 0 \\le x^4 $$" + "From weighted AM-GM inequality:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 a^{3} c \\le a^{4} c^{2}+a^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$4 a^{2} c \\le a^{3} c^{2}+2 a^{2}+a c^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 a c \\le a^{2}+c^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 0 $$" ], "text/plain": [ "" @@ -1474,32 +1403,32 @@ } ], "source": [ - "prove(makesubs('(x-1)^4','(1,oo)'))" + "newproof()\n", + "formula=Sm('xy/z+yz/x+zx/y-1').subs('z',S('1-x-y'))\n", + "newformula,values=makesubs(formula,'[0,1-y],[0,1]','1/3,1/3')\n", + "prove(newformula,values)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The proof is found, so the assumption that 1 is the minimum of `xy/z+yz/x+zx/y` was good.\n", + "\n", + "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`. \n", + "\n", + "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. " ] }, { "cell_type": "code", "execution_count": 13, - "metadata": { - "scrolled": true - }, + "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $x\\to 1-x$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "numerator: $x^4$" + "numerator: $2 a^{2} - 2 a b - 2 a c + 2 b^{2} - 2 b c + 2 c^{2}$" ], "text/plain": [ "" @@ -1535,7 +1464,55 @@ { "data": { "text/latex": [ - "$$ 0 \\le x^4 $$" + "From weighted AM-GM inequality:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 a b \\le a^{2}+b^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 a c \\le a^{2}+c^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 b c \\le b^{2}+c^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 0 $$" ], "text/plain": [ "" @@ -1568,19 +1545,18 @@ } ], "source": [ - "prove(makesubs('(x-1)^4','(-oo,1)'))" + "prove('(a^2+b^2+c^2-a*b-a*c-b*c)*2')\n", + "#prove(S('(a^2+b^2+c^2-a*b-a*c-b*c)*2'))\n", + "#prove(Sm('2(a^2+b^2+c^2-ab-ac-bc)'))\n", + "#prove(cyclize('2*a^2-2*a*b'))\n", + "#prove(symmetrize('a^2-a*b'))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Now let's go to the problem 10\n", - "#### Problem 10\n", - "If $a,b,c,d>0$, prove that\n", - "$$\\frac a{b+c}+\\frac b{c+d}+ \\frac c{d+a}+ \\frac d{a+b}\\geq 2.$$\n", - "\n", - "Let's try a simple approach." + "Now look at formula $(x-1)^4$. It's quite obvious that it's nonnegative, but `prove` fails to show this!" ] }, { @@ -1591,31 +1567,7 @@ { "data": { "text/latex": [ - "$\\displaystyle \\frac{a}{b + c} + \\frac{b}{c + d} + \\frac{c}{a + d} + \\frac{d}{a + b} - 2$" - ], - "text/plain": [ - "a/(b + c) + b/(c + d) + c/(a + d) + d/(a + b) - 2" - ] - }, - "execution_count": 14, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "formula=cyclize('a/(b+c)',variables='a,b,c,d')-2\n", - "formula" - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "numerator: $a^3c+a^3d+a^2b^2-a^2bd-2a^2c^2-a^2cd+a^2d^2+ab^3-ab^2c-ab^2d-abc^2+ac^3-acd^2+b^3d+b^2c^2-2b^2d^2+bc^3-bc^2d-bcd^2+bd^3+c^2d^2+cd^3$" + "numerator: $x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$" ], "text/plain": [ "" @@ -1627,7 +1579,7 @@ { "data": { "text/latex": [ - "denominator: $a^2bc+a^2bd+a^2c^2+a^2cd+ab^2c+ab^2d+abc^2+2abcd+abd^2+ac^2d+acd^2+b^2cd+b^2d^2+bc^2d+bcd^2$" + "denominator: $1$" ], "text/plain": [ "" @@ -1663,7 +1615,7 @@ { "data": { "text/latex": [ - "$$ a^2bd+2a^2c^2+a^2cd+ab^2c+ab^2d+abc^2+acd^2+2b^2d^2+bc^2d+bcd^2 \\le a^3c+a^3d+a^2b^2+a^2d^2+ab^3+ac^3+b^3d+b^2c^2+bc^3+bd^3+c^2d^2+cd^3 $$" + "$$ 4 x^{3}+4 x \\le x^{4}+6 x^{2}+1 $$" ], "text/plain": [ "" @@ -1678,11 +1630,322 @@ "2" ] }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "prove('(x-1)^4')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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 $(-\\infty,1)$." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $x\\to a + 1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $a^{4}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le a^{4} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], + "source": [ + "newproof()\n", + "prove(makesubs('(x-1)^4','(1,oo)'))" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $x\\to 1 - a$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $a^{4}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le a^{4} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "newproof()\n", + "prove(makesubs('(x-1)^4','(-oo,1)'))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now let's go to the problem 10\n", + "#### Problem 10\n", + "If $a,b,c,d>0$, prove that\n", + "$$\\frac a{b+c}+\\frac b{c+d}+ \\frac c{d+a}+ \\frac d{a+b}\\geq 2.$$\n", + "\n", + "Let's try a simple approach." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\frac{a}{b + c} + \\frac{b}{c + d} + \\frac{c}{a + d} + \\frac{d}{a + b} - 2$" + ], + "text/plain": [ + "a/(b + c) + b/(c + d) + c/(a + d) + d/(a + b) - 2" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "formula=cyclize('a/(b+c)',variables='a,b,c,d')-2\n", + "formula" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "numerator: $a^{3} c + a^{3} d + a^{2} b^{2} - a^{2} b d - 2 a^{2} c^{2} - a^{2} c d + a^{2} d^{2} + a b^{3} - a b^{2} c - a b^{2} d - a b c^{2} + a c^{3} - a c d^{2} + b^{3} d + b^{2} c^{2} - 2 b^{2} d^{2} + b c^{3} - b c^{2} d - b c d^{2} + b d^{3} + c^{2} d^{2} + c d^{3}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $a^{2} b c + a^{2} b d + a^{2} c^{2} + a^{2} c d + a b^{2} c + a b^{2} d + a b c^{2} + 2 a b c d + a b d^{2} + a c^{2} d + a c d^{2} + b^{2} c d + b^{2} d^{2} + b c^{2} d + b c d^{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 2" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Program couldn't find any proof." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ a^{2} b d+2 a^{2} c^{2}+a^{2} c d+a b^{2} c+a b^{2} d+a b c^{2}+a c d^{2}+2 b^{2} d^{2}+b c^{2} d+b c d^{2} \\le a^{3} c+a^{3} d+a^{2} b^{2}+a^{2} d^{2}+a b^{3}+a c^{3}+b^{3} d+b^{2} c^{2}+b c^{3}+b d^{3}+c^{2} d^{2}+c d^{3} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "prove(formula)" ] @@ -1696,13 +1959,13 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "Substitute $a\\to c + e$" ], "text/plain": [ "" @@ -1714,7 +1977,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+d$" + "Substitute $b\\to d + f$" ], "text/plain": [ "" @@ -1726,7 +1989,7 @@ { "data": { "text/latex": [ - "numerator: $a^3c+a^3d+a^2b^2+a^2bd+a^2c^2+2a^2cd+a^2d^2+ab^3+ab^2c+2ab^2d-abc^2+abd^2+b^3c+b^3d+b^2c^2+2b^2cd+b^2d^2$" + "numerator: $c^{2} e^{2} - c^{2} e f + c^{2} f^{2} + 2 c d e^{2} + 2 c d f^{2} + c e^{3} + c e f^{2} + c f^{3} + d^{2} e^{2} + d^{2} e f + d^{2} f^{2} + d e^{3} + d e^{2} f + 2 d e f^{2} + d f^{3} + e^{2} f^{2} + e f^{3}$" ], "text/plain": [ "" @@ -1738,7 +2001,7 @@ { "data": { "text/latex": [ - "denominator: $a^2bc+a^2bd+a^2c^2+2a^2cd+a^2d^2+ab^2c+ab^2d+3abc^2+6abcd+3abd^2+2ac^3+6ac^2d+6acd^2+2ad^3+b^2c^2+2b^2cd+b^2d^2+2bc^3+6bc^2d+6bcd^2+2bd^3+c^4+4c^3d+6c^2d^2+4cd^3+d^4$" + "denominator: $c^{4} + 4 c^{3} d + 2 c^{3} e + 2 c^{3} f + 6 c^{2} d^{2} + 6 c^{2} d e + 6 c^{2} d f + c^{2} e^{2} + 3 c^{2} e f + c^{2} f^{2} + 4 c d^{3} + 6 c d^{2} e + 6 c d^{2} f + 2 c d e^{2} + 6 c d e f + 2 c d f^{2} + c e^{2} f + c e f^{2} + d^{4} + 2 d^{3} e + 2 d^{3} f + d^{2} e^{2} + 3 d^{2} e f + d^{2} f^{2} + d e^{2} f + d e f^{2}$" ], "text/plain": [ "" @@ -1786,7 +2049,7 @@ { "data": { "text/latex": [ - "$$ abc^2 \\le a^3c+a^3d+a^2b^2+a^2bd+a^2c^2+2a^2cd+a^2d^2+ab^3+ab^2c+2ab^2d+abd^2+b^3c+b^3d+b^2c^2+2b^2cd+b^2d^2 $$" + "$$ c^{2} e f \\le c^{2} e^{2}+c^{2} f^{2}+2 c d e^{2}+2 c d f^{2}+c e^{3}+c e f^{2}+c f^{3}+d^{2} e^{2}+d^{2} e f+d^{2} f^{2}+d e^{3}+d e^{2} f+2 d e f^{2}+d f^{3}+e^{2} f^{2}+e f^{3} $$" ], "text/plain": [ "" @@ -1801,24 +2064,25 @@ "0" ] }, - "execution_count": 16, + "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ + "newproof()\n", "prove(makesubs(formula,'[c,oo],[d,oo]'))" ] }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "Substitute $a\\to c + e$" ], "text/plain": [ "" @@ -1830,7 +2094,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+d$" + "Substitute $b\\to d + f$" ], "text/plain": [ "" @@ -1842,7 +2106,7 @@ { "data": { "text/latex": [ - "numerator: $2a^3c+2a^3d+2a^2b^2+2a^2bd+2a^2c^2+4a^2cd+2a^2d^2+2ab^3+2ab^2c+4ab^2d-2abc^2+2abd^2+2b^3c+2b^3d+2b^2c^2+4b^2cd+2b^2d^2$" + "numerator: $2 c^{2} e^{2} - 2 c^{2} e f + 2 c^{2} f^{2} + 4 c d e^{2} + 4 c d f^{2} + 2 c e^{3} + 2 c e f^{2} + 2 c f^{3} + 2 d^{2} e^{2} + 2 d^{2} e f + 2 d^{2} f^{2} + 2 d e^{3} + 2 d e^{2} f + 4 d e f^{2} + 2 d f^{3} + 2 e^{2} f^{2} + 2 e f^{3}$" ], "text/plain": [ "" @@ -1854,7 +2118,7 @@ { "data": { "text/latex": [ - "denominator: $a^2bc+a^2bd+a^2c^2+2a^2cd+a^2d^2+ab^2c+ab^2d+3abc^2+6abcd+3abd^2+2ac^3+6ac^2d+6acd^2+2ad^3+b^2c^2+2b^2cd+b^2d^2+2bc^3+6bc^2d+6bcd^2+2bd^3+c^4+4c^3d+6c^2d^2+4cd^3+d^4$" + "denominator: $c^{4} + 4 c^{3} d + 2 c^{3} e + 2 c^{3} f + 6 c^{2} d^{2} + 6 c^{2} d e + 6 c^{2} d f + c^{2} e^{2} + 3 c^{2} e f + c^{2} f^{2} + 4 c d^{3} + 6 c d^{2} e + 6 c d^{2} f + 2 c d e^{2} + 6 c d e f + 2 c d f^{2} + c e^{2} f + c e f^{2} + d^{4} + 2 d^{3} e + 2 d^{3} f + d^{2} e^{2} + 3 d^{2} e f + d^{2} f^{2} + d e^{2} f + d e f^{2}$" ], "text/plain": [ "" @@ -1890,7 +2154,7 @@ { "data": { "text/latex": [ - "$$2abc^2 \\le a^2c^2+b^2c^2$$" + "$$2 c^{2} e f \\le c^{2} e^{2}+c^{2} f^{2}$$" ], "text/plain": [ "" @@ -1902,7 +2166,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 2a^3c+2a^3d+2a^2b^2+2a^2bd+a^2c^2+4a^2cd+2a^2d^2+2ab^3+2ab^2c+4ab^2d+2abd^2+2b^3c+2b^3d+b^2c^2+4b^2cd+2b^2d^2 $$" + "$$ 0 \\le c^{2} e^{2}+c^{2} f^{2}+4 c d e^{2}+4 c d f^{2}+2 c e^{3}+2 c e f^{2}+2 c f^{3}+2 d^{2} e^{2}+2 d^{2} e f+2 d^{2} f^{2}+2 d e^{3}+2 d e^{2} f+4 d e f^{2}+2 d f^{3}+2 e^{2} f^{2}+2 e f^{3} $$" ], "text/plain": [ "" @@ -1929,12 +2193,13 @@ "0" ] }, - "execution_count": 17, + "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ + "newproof()\n", "prove(makesubs(formula,'[c,oo],[d,oo]')*2)" ] }, @@ -1947,13 +2212,13 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $a\\to c-c/(a+1)$" + "Substitute $a\\to c - \\frac{c}{e + 1}$" ], "text/plain": [ "" @@ -1965,7 +2230,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to d-d/(b+1)$" + "Substitute $b\\to d - \\frac{d}{f + 1}$" ], "text/plain": [ "" @@ -1977,7 +2242,7 @@ { "data": { "text/latex": [ - "numerator: $2a^3bc^2d^2+4a^3bcd^3+2a^3bd^4+2a^3c^2d^2+2a^3cd^3-2a^2b^2c^3d+2a^2b^2cd^3-4a^2bc^3d+4a^2bc^2d^2+12a^2bcd^3+6a^2bd^4-2a^2c^3d+4a^2c^2d^2+6a^2cd^3+2ab^3c^4+4ab^3c^3d+2ab^3c^2d^2+6ab^2c^4+8ab^2c^3d+4ab^2c^2d^2+4ab^2cd^3+6abc^4+4abc^3d+6abc^2d^2+12abcd^3+6abd^4+2ac^4+4ac^2d^2+6acd^3+2b^3c^3d+2b^3c^2d^2+4b^2c^3d+4b^2c^2d^2+2b^2cd^3+2bc^3d+4bc^2d^2+4bcd^3+2bd^4+2c^2d^2+2cd^3$" + "numerator: $2 c^{4} e f^{3} + 6 c^{4} e f^{2} + 6 c^{4} e f + 2 c^{4} e - 2 c^{3} d e^{2} f^{2} - 4 c^{3} d e^{2} f - 2 c^{3} d e^{2} + 4 c^{3} d e f^{3} + 8 c^{3} d e f^{2} + 4 c^{3} d e f + 2 c^{3} d f^{3} + 4 c^{3} d f^{2} + 2 c^{3} d f + 2 c^{2} d^{2} e^{3} f + 2 c^{2} d^{2} e^{3} + 4 c^{2} d^{2} e^{2} f + 4 c^{2} d^{2} e^{2} + 2 c^{2} d^{2} e f^{3} + 4 c^{2} d^{2} e f^{2} + 6 c^{2} d^{2} e f + 4 c^{2} d^{2} e + 2 c^{2} d^{2} f^{3} + 4 c^{2} d^{2} f^{2} + 4 c^{2} d^{2} f + 2 c^{2} d^{2} + 4 c d^{3} e^{3} f + 2 c d^{3} e^{3} + 2 c d^{3} e^{2} f^{2} + 12 c d^{3} e^{2} f + 6 c d^{3} e^{2} + 4 c d^{3} e f^{2} + 12 c d^{3} e f + 6 c d^{3} e + 2 c d^{3} f^{2} + 4 c d^{3} f + 2 c d^{3} + 2 d^{4} e^{3} f + 6 d^{4} e^{2} f + 6 d^{4} e f + 2 d^{4} f$" ], "text/plain": [ "" @@ -1989,7 +2254,7 @@ { "data": { "text/latex": [ - "denominator: $a^3b^3c^4+4a^3b^3c^3d+6a^3b^3c^2d^2+4a^3b^3cd^3+a^3b^3d^4+3a^3b^2c^4+10a^3b^2c^3d+12a^3b^2c^2d^2+6a^3b^2cd^3+a^3b^2d^4+3a^3bc^4+8a^3bc^3d+7a^3bc^2d^2+2a^3bcd^3+a^3c^4+2a^3c^3d+a^3c^2d^2+a^2b^3c^4+6a^2b^3c^3d+12a^2b^3c^2d^2+10a^2b^3cd^3+3a^2b^3d^4+3a^2b^2c^4+15a^2b^2c^3d+24a^2b^2c^2d^2+15a^2b^2cd^3+3a^2b^2d^4+3a^2bc^4+12a^2bc^3d+14a^2bc^2d^2+5a^2bcd^3+a^2c^4+3a^2c^3d+2a^2c^2d^2+2ab^3c^3d+7ab^3c^2d^2+8ab^3cd^3+3ab^3d^4+5ab^2c^3d+14ab^2c^2d^2+12ab^2cd^3+3ab^2d^4+4abc^3d+8abc^2d^2+4abcd^3+ac^3d+ac^2d^2+b^3c^2d^2+2b^3cd^3+b^3d^4+2b^2c^2d^2+3b^2cd^3+b^2d^4+bc^2d^2+bcd^3$" + "denominator: $c^{4} e^{3} f^{3} + 3 c^{4} e^{3} f^{2} + 3 c^{4} e^{3} f + c^{4} e^{3} + c^{4} e^{2} f^{3} + 3 c^{4} e^{2} f^{2} + 3 c^{4} e^{2} f + c^{4} e^{2} + 4 c^{3} d e^{3} f^{3} + 10 c^{3} d e^{3} f^{2} + 8 c^{3} d e^{3} f + 2 c^{3} d e^{3} + 6 c^{3} d e^{2} f^{3} + 15 c^{3} d e^{2} f^{2} + 12 c^{3} d e^{2} f + 3 c^{3} d e^{2} + 2 c^{3} d e f^{3} + 5 c^{3} d e f^{2} + 4 c^{3} d e f + c^{3} d e + 6 c^{2} d^{2} e^{3} f^{3} + 12 c^{2} d^{2} e^{3} f^{2} + 7 c^{2} d^{2} e^{3} f + c^{2} d^{2} e^{3} + 12 c^{2} d^{2} e^{2} f^{3} + 24 c^{2} d^{2} e^{2} f^{2} + 14 c^{2} d^{2} e^{2} f + 2 c^{2} d^{2} e^{2} + 7 c^{2} d^{2} e f^{3} + 14 c^{2} d^{2} e f^{2} + 8 c^{2} d^{2} e f + c^{2} d^{2} e + c^{2} d^{2} f^{3} + 2 c^{2} d^{2} f^{2} + c^{2} d^{2} f + 4 c d^{3} e^{3} f^{3} + 6 c d^{3} e^{3} f^{2} + 2 c d^{3} e^{3} f + 10 c d^{3} e^{2} f^{3} + 15 c d^{3} e^{2} f^{2} + 5 c d^{3} e^{2} f + 8 c d^{3} e f^{3} + 12 c d^{3} e f^{2} + 4 c d^{3} e f + 2 c d^{3} f^{3} + 3 c d^{3} f^{2} + c d^{3} f + d^{4} e^{3} f^{3} + d^{4} e^{3} f^{2} + 3 d^{4} e^{2} f^{3} + 3 d^{4} e^{2} f^{2} + 3 d^{4} e f^{3} + 3 d^{4} e f^{2} + d^{4} f^{3} + d^{4} f^{2}$" ], "text/plain": [ "" @@ -2025,7 +2290,7 @@ { "data": { "text/latex": [ - "$$2a^2b^2c^3d \\le a^3bc^2d^2+ab^3c^4$$" + "$$2 c^{3} d e^{2} f^{2} \\le c^{4} e f^{3}+c^{2} d^{2} e^{3} f$$" ], "text/plain": [ "" @@ -2037,7 +2302,7 @@ { "data": { "text/latex": [ - "$$2a^2c^3d \\le a^3c^2d^2+ac^4$$" + "$$2 c^{3} d e^{2} \\le c^{4} e+c^{2} d^{2} e^{3}$$" ], "text/plain": [ "" @@ -2049,7 +2314,7 @@ { "data": { "text/latex": [ - "$$4a^2bc^3d \\le a^3bc^2d^2+a^3c^2d^2+ab^3c^4+ac^4$$" + "$$4 c^{3} d e^{2} f \\le c^{4} e f^{3}+c^{4} e+c^{2} d^{2} e^{3} f+c^{2} d^{2} e^{3}$$" ], "text/plain": [ "" @@ -2061,7 +2326,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^3bcd^3+2a^3bd^4+2a^3cd^3+2a^2b^2cd^3+4a^2bc^2d^2+12a^2bcd^3+6a^2bd^4+4a^2c^2d^2+6a^2cd^3+4ab^3c^3d+2ab^3c^2d^2+6ab^2c^4+8ab^2c^3d+4ab^2c^2d^2+4ab^2cd^3+6abc^4+4abc^3d+6abc^2d^2+12abcd^3+6abd^4+4ac^2d^2+6acd^3+2b^3c^3d+2b^3c^2d^2+4b^2c^3d+4b^2c^2d^2+2b^2cd^3+2bc^3d+4bc^2d^2+4bcd^3+2bd^4+2c^2d^2+2cd^3 $$" + "$$ 0 \\le 6 c^{4} e f^{2}+6 c^{4} e f+4 c^{3} d e f^{3}+8 c^{3} d e f^{2}+4 c^{3} d e f+2 c^{3} d f^{3}+4 c^{3} d f^{2}+2 c^{3} d f+4 c^{2} d^{2} e^{2} f+4 c^{2} d^{2} e^{2}+2 c^{2} d^{2} e f^{3}+4 c^{2} d^{2} e f^{2}+6 c^{2} d^{2} e f+4 c^{2} d^{2} e+2 c^{2} d^{2} f^{3}+4 c^{2} d^{2} f^{2}+4 c^{2} d^{2} f+2 c^{2} d^{2}+4 c d^{3} e^{3} f+2 c d^{3} e^{3}+2 c d^{3} e^{2} f^{2}+12 c d^{3} e^{2} f+6 c d^{3} e^{2}+4 c d^{3} e f^{2}+12 c d^{3} e f+6 c d^{3} e+2 c d^{3} f^{2}+4 c d^{3} f+2 c d^{3}+2 d^{4} e^{3} f+6 d^{4} e^{2} f+6 d^{4} e f+2 d^{4} f $$" ], "text/plain": [ "" @@ -2088,12 +2353,13 @@ "0" ] }, - "execution_count": 18, + "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ + "newproof()\n", "prove(makesubs(formula,'[0,c],[0,d]')*2)" ] }, @@ -2106,13 +2372,13 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "numerator: $x^4-4x^3+6x^2-4x+1$" + "numerator: $x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$" ], "text/plain": [ "" @@ -2136,8 +2402,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2149,7 +2414,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to 1+x$" + "Substitute $x\\to 1+a$" ], "text/plain": [ "" @@ -2161,7 +2426,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $x^4$" + "Numerator after substitutions: $a^{4}$" ], "text/plain": [ "" @@ -2185,7 +2450,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le x^4 $$" + "$$ 0 \\le a^{4} $$" ], "text/plain": [ "" @@ -2209,8 +2474,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2222,7 +2486,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to 1/(1+x)$" + "Substitute $x\\to 1/(1+b)$" ], "text/plain": [ "" @@ -2234,7 +2498,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $x^4$" + "Numerator after substitutions: $b^{4}$" ], "text/plain": [ "" @@ -2258,7 +2522,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le x^4 $$" + "$$ 0 \\le b^{4} $$" ], "text/plain": [ "" @@ -2278,21 +2542,32 @@ }, "metadata": {}, "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "Counter({0: 2})" + ] + }, + "execution_count": 22, + "metadata": {}, + "output_type": "execute_result" } ], "source": [ + "newproof()\n", "powerprove('(x-1)^4')" ] }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "numerator: $4a^5+4a^4b+4a^4c-6a^4-4a^3b-2a^3c+4ab^3-9ab^2+4ac^2-18ac+9a+4b^4+4b^3c-6b^3-3b^2c+4bc^2-12bc+10b+4c^3-6c^2+11c$" + "numerator: $4 a^{5} + 4 a^{4} b + 4 a^{4} c - 6 a^{4} - 4 a^{3} b - 2 a^{3} c + 4 a b^{3} - 9 a b^{2} + 4 a c^{2} - 18 a c + 9 a + 4 b^{4} + 4 b^{3} c - 6 b^{3} - 3 b^{2} c + 4 b c^{2} - 12 b c + 10 b + 4 c^{3} - 6 c^{2} + 11 c$" ], "text/plain": [ "" @@ -2316,8 +2591,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2329,7 +2603,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1+a,b\\to 1+b,c\\to 1+c$" + "Substitute $a\\to 1+d,b\\to 1+e,c\\to 1+f$" ], "text/plain": [ "" @@ -2341,7 +2615,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5+4a^4b+4a^4c+22a^4+12a^3b+14a^3c+42a^3+12a^2b+18a^2c+34a^2+4ab^3+3ab^2-2ab+4ac^2+4b^4+4b^3c+18b^3+9b^2c+18b^2+4bc^2+2bc+4c^3+14c^2$" + "Numerator after substitutions: $4 d^{5} + 4 d^{4} e + 4 d^{4} f + 22 d^{4} + 12 d^{3} e + 14 d^{3} f + 42 d^{3} + 12 d^{2} e + 18 d^{2} f + 34 d^{2} + 4 d e^{3} + 3 d e^{2} - 2 d e + 4 d f^{2} + 4 e^{4} + 4 e^{3} f + 18 e^{3} + 9 e^{2} f + 18 e^{2} + 4 e f^{2} + 2 e f + 4 f^{3} + 14 f^{2}$" ], "text/plain": [ "" @@ -2377,7 +2651,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 d e \\le d^{2}+e^{2}$$" ], "text/plain": [ "" @@ -2389,7 +2663,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5+4a^4b+4a^4c+22a^4+12a^3b+14a^3c+42a^3+12a^2b+18a^2c+33a^2+4ab^3+3ab^2+4ac^2+4b^4+4b^3c+18b^3+9b^2c+17b^2+4bc^2+2bc+4c^3+14c^2 $$" + "$$ 0 \\le 4 d^{5}+4 d^{4} e+4 d^{4} f+22 d^{4}+12 d^{3} e+14 d^{3} f+42 d^{3}+12 d^{2} e+18 d^{2} f+33 d^{2}+4 d e^{3}+3 d e^{2}+4 d f^{2}+4 e^{4}+4 e^{3} f+18 e^{3}+9 e^{2} f+17 e^{2}+4 e f^{2}+2 e f+4 f^{3}+14 f^{2} $$" ], "text/plain": [ "" @@ -2413,8 +2687,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2426,7 +2699,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1/(1+a),b\\to 1+b,c\\to 1+c$" + "Substitute $a\\to 1/(1+g),b\\to 1+h,c\\to 1+i$" ], "text/plain": [ "" @@ -2438,7 +2711,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4+4a^5b^3c+14a^5b^3+9a^5b^2c+15a^5b^2+4a^5bc^2+2a^5bc+6a^5b+4a^5c^3+10a^5c^2+8a^5c+10a^5+20a^4b^4+20a^4b^3c+74a^4b^3+45a^4b^2c+78a^4b^2+20a^4bc^2+10a^4bc+24a^4b+20a^4c^3+54a^4c^2+30a^4c+40a^4+40a^3b^4+40a^3b^3c+156a^3b^3+90a^3b^2c+162a^3b^2+40a^3bc^2+20a^3bc+36a^3b+40a^3c^3+116a^3c^2+40a^3c+60a^3+40a^2b^4+40a^2b^3c+164a^2b^3+90a^2b^2c+168a^2b^2+40a^2bc^2+20a^2bc+20a^2b+40a^2c^3+124a^2c^2+18a^2c+34a^2+20ab^4+20ab^3c+86ab^3+45ab^2c+87ab^2+20abc^2+10abc+2ab+20ac^3+66ac^2+4b^4+4b^3c+18b^3+9b^2c+18b^2+4bc^2+2bc+4c^3+14c^2$" + "Numerator after substitutions: $4 g^{5} h^{4} + 4 g^{5} h^{3} i + 14 g^{5} h^{3} + 9 g^{5} h^{2} i + 15 g^{5} h^{2} + 4 g^{5} h i^{2} + 2 g^{5} h i + 6 g^{5} h + 4 g^{5} i^{3} + 10 g^{5} i^{2} + 8 g^{5} i + 10 g^{5} + 20 g^{4} h^{4} + 20 g^{4} h^{3} i + 74 g^{4} h^{3} + 45 g^{4} h^{2} i + 78 g^{4} h^{2} + 20 g^{4} h i^{2} + 10 g^{4} h i + 24 g^{4} h + 20 g^{4} i^{3} + 54 g^{4} i^{2} + 30 g^{4} i + 40 g^{4} + 40 g^{3} h^{4} + 40 g^{3} h^{3} i + 156 g^{3} h^{3} + 90 g^{3} h^{2} i + 162 g^{3} h^{2} + 40 g^{3} h i^{2} + 20 g^{3} h i + 36 g^{3} h + 40 g^{3} i^{3} + 116 g^{3} i^{2} + 40 g^{3} i + 60 g^{3} + 40 g^{2} h^{4} + 40 g^{2} h^{3} i + 164 g^{2} h^{3} + 90 g^{2} h^{2} i + 168 g^{2} h^{2} + 40 g^{2} h i^{2} + 20 g^{2} h i + 20 g^{2} h + 40 g^{2} i^{3} + 124 g^{2} i^{2} + 18 g^{2} i + 34 g^{2} + 20 g h^{4} + 20 g h^{3} i + 86 g h^{3} + 45 g h^{2} i + 87 g h^{2} + 20 g h i^{2} + 10 g h i + 2 g h + 20 g i^{3} + 66 g i^{2} + 4 h^{4} + 4 h^{3} i + 18 h^{3} + 9 h^{2} i + 18 h^{2} + 4 h i^{2} + 2 h i + 4 i^{3} + 14 i^{2}$" ], "text/plain": [ "" @@ -2462,7 +2735,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4+4a^5b^3c+14a^5b^3+9a^5b^2c+15a^5b^2+4a^5bc^2+2a^5bc+6a^5b+4a^5c^3+10a^5c^2+8a^5c+10a^5+20a^4b^4+20a^4b^3c+74a^4b^3+45a^4b^2c+78a^4b^2+20a^4bc^2+10a^4bc+24a^4b+20a^4c^3+54a^4c^2+30a^4c+40a^4+40a^3b^4+40a^3b^3c+156a^3b^3+90a^3b^2c+162a^3b^2+40a^3bc^2+20a^3bc+36a^3b+40a^3c^3+116a^3c^2+40a^3c+60a^3+40a^2b^4+40a^2b^3c+164a^2b^3+90a^2b^2c+168a^2b^2+40a^2bc^2+20a^2bc+20a^2b+40a^2c^3+124a^2c^2+18a^2c+34a^2+20ab^4+20ab^3c+86ab^3+45ab^2c+87ab^2+20abc^2+10abc+2ab+20ac^3+66ac^2+4b^4+4b^3c+18b^3+9b^2c+18b^2+4bc^2+2bc+4c^3+14c^2 $$" + "$$ 0 \\le 4 g^{5} h^{4}+4 g^{5} h^{3} i+14 g^{5} h^{3}+9 g^{5} h^{2} i+15 g^{5} h^{2}+4 g^{5} h i^{2}+2 g^{5} h i+6 g^{5} h+4 g^{5} i^{3}+10 g^{5} i^{2}+8 g^{5} i+10 g^{5}+20 g^{4} h^{4}+20 g^{4} h^{3} i+74 g^{4} h^{3}+45 g^{4} h^{2} i+78 g^{4} h^{2}+20 g^{4} h i^{2}+10 g^{4} h i+24 g^{4} h+20 g^{4} i^{3}+54 g^{4} i^{2}+30 g^{4} i+40 g^{4}+40 g^{3} h^{4}+40 g^{3} h^{3} i+156 g^{3} h^{3}+90 g^{3} h^{2} i+162 g^{3} h^{2}+40 g^{3} h i^{2}+20 g^{3} h i+36 g^{3} h+40 g^{3} i^{3}+116 g^{3} i^{2}+40 g^{3} i+60 g^{3}+40 g^{2} h^{4}+40 g^{2} h^{3} i+164 g^{2} h^{3}+90 g^{2} h^{2} i+168 g^{2} h^{2}+40 g^{2} h i^{2}+20 g^{2} h i+20 g^{2} h+40 g^{2} i^{3}+124 g^{2} i^{2}+18 g^{2} i+34 g^{2}+20 g h^{4}+20 g h^{3} i+86 g h^{3}+45 g h^{2} i+87 g h^{2}+20 g h i^{2}+10 g h i+2 g h+20 g i^{3}+66 g i^{2}+4 h^{4}+4 h^{3} i+18 h^{3}+9 h^{2} i+18 h^{2}+4 h i^{2}+2 h i+4 i^{3}+14 i^{2} $$" ], "text/plain": [ "" @@ -2486,8 +2759,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2499,7 +2771,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1+a,b\\to 1/(1+b),c\\to 1+c$" + "Substitute $a\\to 1+j,b\\to 1/(1+k),c\\to 1+l$" ], "text/plain": [ "" @@ -2511,7 +2783,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4+16a^5b^3+24a^5b^2+16a^5b+4a^5+4a^4b^4c+18a^4b^4+16a^4b^3c+76a^4b^3+24a^4b^2c+120a^4b^2+16a^4bc+84a^4b+4a^4c+22a^4+14a^3b^4c+30a^3b^4+56a^3b^3c+132a^3b^3+84a^3b^2c+216a^3b^2+56a^3bc+156a^3b+14a^3c+42a^3+18a^2b^4c+22a^2b^4+72a^2b^3c+100a^2b^3+108a^2b^2c+168a^2b^2+72a^2bc+124a^2b+18a^2c+34a^2+4ab^4c^2+ab^4+16ab^3c^2+8ab^3+24ab^2c^2+9ab^2+16abc^2+2ab+4ac^2+4b^4c^3+10b^4c^2+3b^4c+4b^4+16b^3c^3+44b^3c^2+8b^3c+18b^3+24b^2c^3+72b^2c^2+3b^2c+18b^2+16bc^3+52bc^2-2bc+4c^3+14c^2$" + "Numerator after substitutions: $4 j^{5} k^{4} + 16 j^{5} k^{3} + 24 j^{5} k^{2} + 16 j^{5} k + 4 j^{5} + 4 j^{4} k^{4} l + 18 j^{4} k^{4} + 16 j^{4} k^{3} l + 76 j^{4} k^{3} + 24 j^{4} k^{2} l + 120 j^{4} k^{2} + 16 j^{4} k l + 84 j^{4} k + 4 j^{4} l + 22 j^{4} + 14 j^{3} k^{4} l + 30 j^{3} k^{4} + 56 j^{3} k^{3} l + 132 j^{3} k^{3} + 84 j^{3} k^{2} l + 216 j^{3} k^{2} + 56 j^{3} k l + 156 j^{3} k + 14 j^{3} l + 42 j^{3} + 18 j^{2} k^{4} l + 22 j^{2} k^{4} + 72 j^{2} k^{3} l + 100 j^{2} k^{3} + 108 j^{2} k^{2} l + 168 j^{2} k^{2} + 72 j^{2} k l + 124 j^{2} k + 18 j^{2} l + 34 j^{2} + 4 j k^{4} l^{2} + j k^{4} + 16 j k^{3} l^{2} + 8 j k^{3} + 24 j k^{2} l^{2} + 9 j k^{2} + 16 j k l^{2} + 2 j k + 4 j l^{2} + 4 k^{4} l^{3} + 10 k^{4} l^{2} + 3 k^{4} l + 4 k^{4} + 16 k^{3} l^{3} + 44 k^{3} l^{2} + 8 k^{3} l + 18 k^{3} + 24 k^{2} l^{3} + 72 k^{2} l^{2} + 3 k^{2} l + 18 k^{2} + 16 k l^{3} + 52 k l^{2} - 2 k l + 4 l^{3} + 14 l^{2}$" ], "text/plain": [ "" @@ -2547,7 +2819,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 k l \\le k^{2}+l^{2}$$" ], "text/plain": [ "" @@ -2559,7 +2831,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4+16a^5b^3+24a^5b^2+16a^5b+4a^5+4a^4b^4c+18a^4b^4+16a^4b^3c+76a^4b^3+24a^4b^2c+120a^4b^2+16a^4bc+84a^4b+4a^4c+22a^4+14a^3b^4c+30a^3b^4+56a^3b^3c+132a^3b^3+84a^3b^2c+216a^3b^2+56a^3bc+156a^3b+14a^3c+42a^3+18a^2b^4c+22a^2b^4+72a^2b^3c+100a^2b^3+108a^2b^2c+168a^2b^2+72a^2bc+124a^2b+18a^2c+34a^2+4ab^4c^2+ab^4+16ab^3c^2+8ab^3+24ab^2c^2+9ab^2+16abc^2+2ab+4ac^2+4b^4c^3+10b^4c^2+3b^4c+4b^4+16b^3c^3+44b^3c^2+8b^3c+18b^3+24b^2c^3+72b^2c^2+3b^2c+17b^2+16bc^3+52bc^2+4c^3+13c^2 $$" + "$$ 0 \\le 4 j^{5} k^{4}+16 j^{5} k^{3}+24 j^{5} k^{2}+16 j^{5} k+4 j^{5}+4 j^{4} k^{4} l+18 j^{4} k^{4}+16 j^{4} k^{3} l+76 j^{4} k^{3}+24 j^{4} k^{2} l+120 j^{4} k^{2}+16 j^{4} k l+84 j^{4} k+4 j^{4} l+22 j^{4}+14 j^{3} k^{4} l+30 j^{3} k^{4}+56 j^{3} k^{3} l+132 j^{3} k^{3}+84 j^{3} k^{2} l+216 j^{3} k^{2}+56 j^{3} k l+156 j^{3} k+14 j^{3} l+42 j^{3}+18 j^{2} k^{4} l+22 j^{2} k^{4}+72 j^{2} k^{3} l+100 j^{2} k^{3}+108 j^{2} k^{2} l+168 j^{2} k^{2}+72 j^{2} k l+124 j^{2} k+18 j^{2} l+34 j^{2}+4 j k^{4} l^{2}+j k^{4}+16 j k^{3} l^{2}+8 j k^{3}+24 j k^{2} l^{2}+9 j k^{2}+16 j k l^{2}+2 j k+4 j l^{2}+4 k^{4} l^{3}+10 k^{4} l^{2}+3 k^{4} l+4 k^{4}+16 k^{3} l^{3}+44 k^{3} l^{2}+8 k^{3} l+18 k^{3}+24 k^{2} l^{3}+72 k^{2} l^{2}+3 k^{2} l+17 k^{2}+16 k l^{3}+52 k l^{2}+4 l^{3}+13 l^{2} $$" ], "text/plain": [ "" @@ -2583,8 +2855,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2596,7 +2867,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1/(1+a),b\\to 1/(1+b),c\\to 1+c$" + "Substitute $a\\to 1/(1+m),b\\to 1/(1+n),c\\to 1+o$" ], "text/plain": [ "" @@ -2608,7 +2879,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4c^3+6a^5b^4c^2+11a^5b^4c+9a^5b^4+16a^5b^3c^3+28a^5b^3c^2+40a^5b^3c+38a^5b^3+24a^5b^2c^3+48a^5b^2c^2+51a^5b^2c+57a^5b^2+16a^5bc^3+36a^5bc^2+30a^5bc+34a^5b+4a^5c^3+10a^5c^2+8a^5c+10a^5+20a^4b^4c^3+34a^4b^4c^2+45a^4b^4c+40a^4b^4+80a^4b^3c^3+156a^4b^3c^2+160a^4b^3c+170a^4b^3+120a^4b^2c^3+264a^4b^2c^2+195a^4b^2c+246a^4b^2+80a^4bc^3+196a^4bc^2+110a^4bc+136a^4b+20a^4c^3+54a^4c^2+30a^4c+40a^4+40a^3b^4c^3+76a^3b^4c^2+70a^3b^4c+70a^3b^4+160a^3b^3c^3+344a^3b^3c^2+240a^3b^3c+300a^3b^3+240a^3b^2c^3+576a^3b^2c^2+270a^3b^2c+414a^3b^2+160a^3bc^3+424a^3bc^2+140a^3bc+204a^3b+40a^3c^3+116a^3c^2+40a^3c+60a^3+40a^2b^4c^3+84a^2b^4c^2+48a^2b^4c+58a^2b^4+160a^2b^3c^3+376a^2b^3c^2+152a^2b^3c+248a^2b^3+240a^2b^2c^3+624a^2b^2c^2+138a^2b^2c+312a^2b^2+160a^2bc^3+456a^2bc^2+52a^2bc+116a^2b+40a^2c^3+124a^2c^2+18a^2c+34a^2+20ab^4c^3+46ab^4c^2+15ab^4c+19ab^4+80ab^3c^3+204ab^3c^2+40ab^3c+82ab^3+120ab^2c^3+336ab^2c^2+15ab^2c+81ab^2+80abc^3+244abc^2-10abc-2ab+20ac^3+66ac^2+4b^4c^3+10b^4c^2+3b^4c+4b^4+16b^3c^3+44b^3c^2+8b^3c+18b^3+24b^2c^3+72b^2c^2+3b^2c+18b^2+16bc^3+52bc^2-2bc+4c^3+14c^2$" + "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}$" ], "text/plain": [ "" @@ -2644,7 +2915,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 m n \\le m^{2}+n^{2}$$" ], "text/plain": [ "" @@ -2656,7 +2927,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 n o \\le n^{2}+o^{2}$$" ], "text/plain": [ "" @@ -2668,7 +2939,7 @@ { "data": { "text/latex": [ - "$$10abc \\le 2a^2+2ab^2c+4ac^2+2b^3$$" + "$$10 m n o \\le 2 m^{2}+2 m n^{2} o+4 m o^{2}+2 n^{3}$$" ], "text/plain": [ "" @@ -2680,7 +2951,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4c^3+6a^5b^4c^2+11a^5b^4c+9a^5b^4+16a^5b^3c^3+28a^5b^3c^2+40a^5b^3c+38a^5b^3+24a^5b^2c^3+48a^5b^2c^2+51a^5b^2c+57a^5b^2+16a^5bc^3+36a^5bc^2+30a^5bc+34a^5b+4a^5c^3+10a^5c^2+8a^5c+10a^5+20a^4b^4c^3+34a^4b^4c^2+45a^4b^4c+40a^4b^4+80a^4b^3c^3+156a^4b^3c^2+160a^4b^3c+170a^4b^3+120a^4b^2c^3+264a^4b^2c^2+195a^4b^2c+246a^4b^2+80a^4bc^3+196a^4bc^2+110a^4bc+136a^4b+20a^4c^3+54a^4c^2+30a^4c+40a^4+40a^3b^4c^3+76a^3b^4c^2+70a^3b^4c+70a^3b^4+160a^3b^3c^3+344a^3b^3c^2+240a^3b^3c+300a^3b^3+240a^3b^2c^3+576a^3b^2c^2+270a^3b^2c+414a^3b^2+160a^3bc^3+424a^3bc^2+140a^3bc+204a^3b+40a^3c^3+116a^3c^2+40a^3c+60a^3+40a^2b^4c^3+84a^2b^4c^2+48a^2b^4c+58a^2b^4+160a^2b^3c^3+376a^2b^3c^2+152a^2b^3c+248a^2b^3+240a^2b^2c^3+624a^2b^2c^2+138a^2b^2c+312a^2b^2+160a^2bc^3+456a^2bc^2+52a^2bc+116a^2b+40a^2c^3+124a^2c^2+18a^2c+31a^2+20ab^4c^3+46ab^4c^2+15ab^4c+19ab^4+80ab^3c^3+204ab^3c^2+40ab^3c+82ab^3+120ab^2c^3+336ab^2c^2+13ab^2c+81ab^2+80abc^3+244abc^2+20ac^3+62ac^2+4b^4c^3+10b^4c^2+3b^4c+4b^4+16b^3c^3+44b^3c^2+8b^3c+16b^3+24b^2c^3+72b^2c^2+3b^2c+16b^2+16bc^3+52bc^2+4c^3+13c^2 $$" + "$$ 0 \\le 4 m^{5} n^{4} o^{3}+6 m^{5} n^{4} o^{2}+11 m^{5} n^{4} o+9 m^{5} n^{4}+16 m^{5} n^{3} o^{3}+28 m^{5} n^{3} o^{2}+40 m^{5} n^{3} o+38 m^{5} n^{3}+24 m^{5} n^{2} o^{3}+48 m^{5} n^{2} o^{2}+51 m^{5} n^{2} o+57 m^{5} n^{2}+16 m^{5} n o^{3}+36 m^{5} n o^{2}+30 m^{5} n o+34 m^{5} n+4 m^{5} o^{3}+10 m^{5} o^{2}+8 m^{5} o+10 m^{5}+20 m^{4} n^{4} o^{3}+34 m^{4} n^{4} o^{2}+45 m^{4} n^{4} o+40 m^{4} n^{4}+80 m^{4} n^{3} o^{3}+156 m^{4} n^{3} o^{2}+160 m^{4} n^{3} o+170 m^{4} n^{3}+120 m^{4} n^{2} o^{3}+264 m^{4} n^{2} o^{2}+195 m^{4} n^{2} o+246 m^{4} n^{2}+80 m^{4} n o^{3}+196 m^{4} n o^{2}+110 m^{4} n o+136 m^{4} n+20 m^{4} o^{3}+54 m^{4} o^{2}+30 m^{4} o+40 m^{4}+40 m^{3} n^{4} o^{3}+76 m^{3} n^{4} o^{2}+70 m^{3} n^{4} o+70 m^{3} n^{4}+160 m^{3} n^{3} o^{3}+344 m^{3} n^{3} o^{2}+240 m^{3} n^{3} o+300 m^{3} n^{3}+240 m^{3} n^{2} o^{3}+576 m^{3} n^{2} o^{2}+270 m^{3} n^{2} o+414 m^{3} n^{2}+160 m^{3} n o^{3}+424 m^{3} n o^{2}+140 m^{3} n o+204 m^{3} n+40 m^{3} o^{3}+116 m^{3} o^{2}+40 m^{3} o+60 m^{3}+40 m^{2} n^{4} o^{3}+84 m^{2} n^{4} o^{2}+48 m^{2} n^{4} o+58 m^{2} n^{4}+160 m^{2} n^{3} o^{3}+376 m^{2} n^{3} o^{2}+152 m^{2} n^{3} o+248 m^{2} n^{3}+240 m^{2} n^{2} o^{3}+624 m^{2} n^{2} o^{2}+138 m^{2} n^{2} o+312 m^{2} n^{2}+160 m^{2} n o^{3}+456 m^{2} n o^{2}+52 m^{2} n o+116 m^{2} n+40 m^{2} o^{3}+124 m^{2} o^{2}+18 m^{2} o+31 m^{2}+20 m n^{4} o^{3}+46 m n^{4} o^{2}+15 m n^{4} o+19 m n^{4}+80 m n^{3} o^{3}+204 m n^{3} o^{2}+40 m n^{3} o+82 m n^{3}+120 m n^{2} o^{3}+336 m n^{2} o^{2}+13 m n^{2} o+81 m n^{2}+80 m n o^{3}+244 m n o^{2}+20 m o^{3}+62 m o^{2}+4 n^{4} o^{3}+10 n^{4} o^{2}+3 n^{4} o+4 n^{4}+16 n^{3} o^{3}+44 n^{3} o^{2}+8 n^{3} o+16 n^{3}+24 n^{2} o^{3}+72 n^{2} o^{2}+3 n^{2} o+16 n^{2}+16 n o^{3}+52 n o^{2}+4 o^{3}+13 o^{2} $$" ], "text/plain": [ "" @@ -2704,8 +2975,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2717,7 +2987,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1+a,b\\to 1+b,c\\to 1/(1+c)$" + "Substitute $a\\to 1+p,b\\to 1+q,c\\to 1/(1+r)$" ], "text/plain": [ "" @@ -2729,7 +2999,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5c^3+12a^5c^2+12a^5c+4a^5+4a^4bc^3+12a^4bc^2+12a^4bc+4a^4b+18a^4c^3+58a^4c^2+62a^4c+22a^4+12a^3bc^3+36a^3bc^2+36a^3bc+12a^3b+28a^3c^3+98a^3c^2+112a^3c+42a^3+12a^2bc^3+36a^2bc^2+36a^2bc+12a^2b+16a^2c^3+66a^2c^2+84a^2c+34a^2+4ab^3c^3+12ab^3c^2+12ab^3c+4ab^3+3ab^2c^3+9ab^2c^2+9ab^2c+3ab^2-2abc^3-6abc^2-6abc-2ab+4ac^3+4ac^2+4b^4c^3+12b^4c^2+12b^4c+4b^4+14b^3c^3+46b^3c^2+50b^3c+18b^3+9b^2c^3+36b^2c^2+45b^2c+18b^2+2bc^3-2bc+10c^3+14c^2$" + "Numerator after substitutions: $4 p^{5} r^{3} + 12 p^{5} r^{2} + 12 p^{5} r + 4 p^{5} + 4 p^{4} q r^{3} + 12 p^{4} q r^{2} + 12 p^{4} q r + 4 p^{4} q + 18 p^{4} r^{3} + 58 p^{4} r^{2} + 62 p^{4} r + 22 p^{4} + 12 p^{3} q r^{3} + 36 p^{3} q r^{2} + 36 p^{3} q r + 12 p^{3} q + 28 p^{3} r^{3} + 98 p^{3} r^{2} + 112 p^{3} r + 42 p^{3} + 12 p^{2} q r^{3} + 36 p^{2} q r^{2} + 36 p^{2} q r + 12 p^{2} q + 16 p^{2} r^{3} + 66 p^{2} r^{2} + 84 p^{2} r + 34 p^{2} + 4 p q^{3} r^{3} + 12 p q^{3} r^{2} + 12 p q^{3} r + 4 p q^{3} + 3 p q^{2} r^{3} + 9 p q^{2} r^{2} + 9 p q^{2} r + 3 p q^{2} - 2 p q r^{3} - 6 p q r^{2} - 6 p q r - 2 p q + 4 p r^{3} + 4 p r^{2} + 4 q^{4} r^{3} + 12 q^{4} r^{2} + 12 q^{4} r + 4 q^{4} + 14 q^{3} r^{3} + 46 q^{3} r^{2} + 50 q^{3} r + 18 q^{3} + 9 q^{2} r^{3} + 36 q^{2} r^{2} + 45 q^{2} r + 18 q^{2} + 2 q r^{3} - 2 q r + 10 r^{3} + 14 r^{2}$" ], "text/plain": [ "" @@ -2765,7 +3035,7 @@ { "data": { "text/latex": [ - "$$2abc^3 \\le a^2bc^3+bc^3$$" + "$$2 p q r^{3} \\le p^{2} q r^{3}+q r^{3}$$" ], "text/plain": [ "" @@ -2777,7 +3047,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 p q \\le p^{2}+q^{2}$$" ], "text/plain": [ "" @@ -2789,7 +3059,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 q r \\le q^{2}+r^{2}$$" ], "text/plain": [ "" @@ -2801,7 +3071,7 @@ { "data": { "text/latex": [ - "$$6abc \\le 2a^3b+2b^2c+2c^2$$" + "$$6 p q r \\le 2 p^{3} q+2 q^{2} r+2 r^{2}$$" ], "text/plain": [ "" @@ -2813,7 +3083,7 @@ { "data": { "text/latex": [ - "$$6abc^2 \\le 2ab^2c^3+ab^2+3ac^2$$" + "$$6 p q r^{2} \\le 2 p q^{2} r^{3}+p q^{2}+3 p r^{2}$$" ], "text/plain": [ "" @@ -2825,7 +3095,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5c^3+12a^5c^2+12a^5c+4a^5+4a^4bc^3+12a^4bc^2+12a^4bc+4a^4b+18a^4c^3+58a^4c^2+62a^4c+22a^4+12a^3bc^3+36a^3bc^2+36a^3bc+10a^3b+28a^3c^3+98a^3c^2+112a^3c+42a^3+11a^2bc^3+36a^2bc^2+36a^2bc+12a^2b+16a^2c^3+66a^2c^2+84a^2c+33a^2+4ab^3c^3+12ab^3c^2+12ab^3c+4ab^3+ab^2c^3+9ab^2c^2+9ab^2c+2ab^2+4ac^3+ac^2+4b^4c^3+12b^4c^2+12b^4c+4b^4+14b^3c^3+46b^3c^2+50b^3c+18b^3+9b^2c^3+36b^2c^2+43b^2c+16b^2+bc^3+10c^3+11c^2 $$" + "$$ 0 \\le 4 p^{5} r^{3}+12 p^{5} r^{2}+12 p^{5} r+4 p^{5}+4 p^{4} q r^{3}+12 p^{4} q r^{2}+12 p^{4} q r+4 p^{4} q+18 p^{4} r^{3}+58 p^{4} r^{2}+62 p^{4} r+22 p^{4}+12 p^{3} q r^{3}+36 p^{3} q r^{2}+36 p^{3} q r+10 p^{3} q+28 p^{3} r^{3}+98 p^{3} r^{2}+112 p^{3} r+42 p^{3}+11 p^{2} q r^{3}+36 p^{2} q r^{2}+36 p^{2} q r+12 p^{2} q+16 p^{2} r^{3}+66 p^{2} r^{2}+84 p^{2} r+33 p^{2}+4 p q^{3} r^{3}+12 p q^{3} r^{2}+12 p q^{3} r+4 p q^{3}+p q^{2} r^{3}+9 p q^{2} r^{2}+9 p q^{2} r+2 p q^{2}+4 p r^{3}+p r^{2}+4 q^{4} r^{3}+12 q^{4} r^{2}+12 q^{4} r+4 q^{4}+14 q^{3} r^{3}+46 q^{3} r^{2}+50 q^{3} r+18 q^{3}+9 q^{2} r^{3}+36 q^{2} r^{2}+43 q^{2} r+16 q^{2}+q r^{3}+10 r^{3}+11 r^{2} $$" ], "text/plain": [ "" @@ -2849,8 +3119,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2862,7 +3131,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1/(1+a),b\\to 1+b,c\\to 1/(1+c)$" + "Substitute $a\\to 1/(1+s),b\\to 1+t,c\\to 1/(1+u)$" ], "text/plain": [ "" @@ -2874,7 +3143,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4c^3+12a^5b^4c^2+12a^5b^4c+4a^5b^4+10a^5b^3c^3+34a^5b^3c^2+38a^5b^3c+14a^5b^3+6a^5b^2c^3+27a^5b^2c^2+36a^5b^2c+15a^5b^2+8a^5bc^3+18a^5bc^2+16a^5bc+6a^5b+8a^5c^3+24a^5c^2+22a^5c+10a^5+20a^4b^4c^3+60a^4b^4c^2+60a^4b^4c+20a^4b^4+54a^4b^3c^3+182a^4b^3c^2+202a^4b^3c+74a^4b^3+33a^4b^2c^3+144a^4b^2c^2+189a^4b^2c+78a^4b^2+34a^4bc^3+72a^4bc^2+62a^4bc+24a^4b+44a^4c^3+114a^4c^2+90a^4c+40a^4+40a^3b^4c^3+120a^3b^4c^2+120a^3b^4c+40a^3b^4+116a^3b^3c^3+388a^3b^3c^2+428a^3b^3c+156a^3b^3+72a^3b^2c^3+306a^3b^2c^2+396a^3b^2c+162a^3b^2+56a^3bc^3+108a^3bc^2+88a^3bc+36a^3b+96a^3c^3+216a^3c^2+140a^3c+60a^3+40a^2b^4c^3+120a^2b^4c^2+120a^2b^4c+40a^2b^4+124a^2b^3c^3+412a^2b^3c^2+452a^2b^3c+164a^2b^3+78a^2b^2c^3+324a^2b^2c^2+414a^2b^2c+168a^2b^2+40a^2bc^3+60a^2bc^2+40a^2bc+20a^2b+100a^2c^3+190a^2c^2+84a^2c+34a^2+20ab^4c^3+60ab^4c^2+60ab^4c+20ab^4+66ab^3c^3+218ab^3c^2+238ab^3c+86ab^3+42ab^2c^3+171ab^2c^2+216ab^2c+87ab^2+12abc^3+6abc^2-4abc+2ab+46ac^3+66ac^2+4b^4c^3+12b^4c^2+12b^4c+4b^4+14b^3c^3+46b^3c^2+50b^3c+18b^3+9b^2c^3+36b^2c^2+45b^2c+18b^2+2bc^3-2bc+10c^3+14c^2$" + "Numerator after substitutions: $4 s^{5} t^{4} u^{3} + 12 s^{5} t^{4} u^{2} + 12 s^{5} t^{4} u + 4 s^{5} t^{4} + 10 s^{5} t^{3} u^{3} + 34 s^{5} t^{3} u^{2} + 38 s^{5} t^{3} u + 14 s^{5} t^{3} + 6 s^{5} t^{2} u^{3} + 27 s^{5} t^{2} u^{2} + 36 s^{5} t^{2} u + 15 s^{5} t^{2} + 8 s^{5} t u^{3} + 18 s^{5} t u^{2} + 16 s^{5} t u + 6 s^{5} t + 8 s^{5} u^{3} + 24 s^{5} u^{2} + 22 s^{5} u + 10 s^{5} + 20 s^{4} t^{4} u^{3} + 60 s^{4} t^{4} u^{2} + 60 s^{4} t^{4} u + 20 s^{4} t^{4} + 54 s^{4} t^{3} u^{3} + 182 s^{4} t^{3} u^{2} + 202 s^{4} t^{3} u + 74 s^{4} t^{3} + 33 s^{4} t^{2} u^{3} + 144 s^{4} t^{2} u^{2} + 189 s^{4} t^{2} u + 78 s^{4} t^{2} + 34 s^{4} t u^{3} + 72 s^{4} t u^{2} + 62 s^{4} t u + 24 s^{4} t + 44 s^{4} u^{3} + 114 s^{4} u^{2} + 90 s^{4} u + 40 s^{4} + 40 s^{3} t^{4} u^{3} + 120 s^{3} t^{4} u^{2} + 120 s^{3} t^{4} u + 40 s^{3} t^{4} + 116 s^{3} t^{3} u^{3} + 388 s^{3} t^{3} u^{2} + 428 s^{3} t^{3} u + 156 s^{3} t^{3} + 72 s^{3} t^{2} u^{3} + 306 s^{3} t^{2} u^{2} + 396 s^{3} t^{2} u + 162 s^{3} t^{2} + 56 s^{3} t u^{3} + 108 s^{3} t u^{2} + 88 s^{3} t u + 36 s^{3} t + 96 s^{3} u^{3} + 216 s^{3} u^{2} + 140 s^{3} u + 60 s^{3} + 40 s^{2} t^{4} u^{3} + 120 s^{2} t^{4} u^{2} + 120 s^{2} t^{4} u + 40 s^{2} t^{4} + 124 s^{2} t^{3} u^{3} + 412 s^{2} t^{3} u^{2} + 452 s^{2} t^{3} u + 164 s^{2} t^{3} + 78 s^{2} t^{2} u^{3} + 324 s^{2} t^{2} u^{2} + 414 s^{2} t^{2} u + 168 s^{2} t^{2} + 40 s^{2} t u^{3} + 60 s^{2} t u^{2} + 40 s^{2} t u + 20 s^{2} t + 100 s^{2} u^{3} + 190 s^{2} u^{2} + 84 s^{2} u + 34 s^{2} + 20 s t^{4} u^{3} + 60 s t^{4} u^{2} + 60 s t^{4} u + 20 s t^{4} + 66 s t^{3} u^{3} + 218 s t^{3} u^{2} + 238 s t^{3} u + 86 s t^{3} + 42 s t^{2} u^{3} + 171 s t^{2} u^{2} + 216 s t^{2} u + 87 s t^{2} + 12 s t u^{3} + 6 s t u^{2} - 4 s t u + 2 s t + 46 s u^{3} + 66 s u^{2} + 4 t^{4} u^{3} + 12 t^{4} u^{2} + 12 t^{4} u + 4 t^{4} + 14 t^{3} u^{3} + 46 t^{3} u^{2} + 50 t^{3} u + 18 t^{3} + 9 t^{2} u^{3} + 36 t^{2} u^{2} + 45 t^{2} u + 18 t^{2} + 2 t u^{3} - 2 t u + 10 u^{3} + 14 u^{2}$" ], "text/plain": [ "" @@ -2910,7 +3179,7 @@ { "data": { "text/latex": [ - "$$4abc \\le a^2c^2+2ab^2+c^2$$" + "$$4 s t u \\le s^{2} u^{2}+2 s t^{2}+u^{2}$$" ], "text/plain": [ "" @@ -2922,7 +3191,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 t u \\le t^{2}+u^{2}$$" ], "text/plain": [ "" @@ -2934,7 +3203,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4c^3+12a^5b^4c^2+12a^5b^4c+4a^5b^4+10a^5b^3c^3+34a^5b^3c^2+38a^5b^3c+14a^5b^3+6a^5b^2c^3+27a^5b^2c^2+36a^5b^2c+15a^5b^2+8a^5bc^3+18a^5bc^2+16a^5bc+6a^5b+8a^5c^3+24a^5c^2+22a^5c+10a^5+20a^4b^4c^3+60a^4b^4c^2+60a^4b^4c+20a^4b^4+54a^4b^3c^3+182a^4b^3c^2+202a^4b^3c+74a^4b^3+33a^4b^2c^3+144a^4b^2c^2+189a^4b^2c+78a^4b^2+34a^4bc^3+72a^4bc^2+62a^4bc+24a^4b+44a^4c^3+114a^4c^2+90a^4c+40a^4+40a^3b^4c^3+120a^3b^4c^2+120a^3b^4c+40a^3b^4+116a^3b^3c^3+388a^3b^3c^2+428a^3b^3c+156a^3b^3+72a^3b^2c^3+306a^3b^2c^2+396a^3b^2c+162a^3b^2+56a^3bc^3+108a^3bc^2+88a^3bc+36a^3b+96a^3c^3+216a^3c^2+140a^3c+60a^3+40a^2b^4c^3+120a^2b^4c^2+120a^2b^4c+40a^2b^4+124a^2b^3c^3+412a^2b^3c^2+452a^2b^3c+164a^2b^3+78a^2b^2c^3+324a^2b^2c^2+414a^2b^2c+168a^2b^2+40a^2bc^3+60a^2bc^2+40a^2bc+20a^2b+100a^2c^3+189a^2c^2+84a^2c+34a^2+20ab^4c^3+60ab^4c^2+60ab^4c+20ab^4+66ab^3c^3+218ab^3c^2+238ab^3c+86ab^3+42ab^2c^3+171ab^2c^2+216ab^2c+85ab^2+12abc^3+6abc^2+2ab+46ac^3+66ac^2+4b^4c^3+12b^4c^2+12b^4c+4b^4+14b^3c^3+46b^3c^2+50b^3c+18b^3+9b^2c^3+36b^2c^2+45b^2c+17b^2+2bc^3+10c^3+12c^2 $$" + "$$ 0 \\le 4 s^{5} t^{4} u^{3}+12 s^{5} t^{4} u^{2}+12 s^{5} t^{4} u+4 s^{5} t^{4}+10 s^{5} t^{3} u^{3}+34 s^{5} t^{3} u^{2}+38 s^{5} t^{3} u+14 s^{5} t^{3}+6 s^{5} t^{2} u^{3}+27 s^{5} t^{2} u^{2}+36 s^{5} t^{2} u+15 s^{5} t^{2}+8 s^{5} t u^{3}+18 s^{5} t u^{2}+16 s^{5} t u+6 s^{5} t+8 s^{5} u^{3}+24 s^{5} u^{2}+22 s^{5} u+10 s^{5}+20 s^{4} t^{4} u^{3}+60 s^{4} t^{4} u^{2}+60 s^{4} t^{4} u+20 s^{4} t^{4}+54 s^{4} t^{3} u^{3}+182 s^{4} t^{3} u^{2}+202 s^{4} t^{3} u+74 s^{4} t^{3}+33 s^{4} t^{2} u^{3}+144 s^{4} t^{2} u^{2}+189 s^{4} t^{2} u+78 s^{4} t^{2}+34 s^{4} t u^{3}+72 s^{4} t u^{2}+62 s^{4} t u+24 s^{4} t+44 s^{4} u^{3}+114 s^{4} u^{2}+90 s^{4} u+40 s^{4}+40 s^{3} t^{4} u^{3}+120 s^{3} t^{4} u^{2}+120 s^{3} t^{4} u+40 s^{3} t^{4}+116 s^{3} t^{3} u^{3}+388 s^{3} t^{3} u^{2}+428 s^{3} t^{3} u+156 s^{3} t^{3}+72 s^{3} t^{2} u^{3}+306 s^{3} t^{2} u^{2}+396 s^{3} t^{2} u+162 s^{3} t^{2}+56 s^{3} t u^{3}+108 s^{3} t u^{2}+88 s^{3} t u+36 s^{3} t+96 s^{3} u^{3}+216 s^{3} u^{2}+140 s^{3} u+60 s^{3}+40 s^{2} t^{4} u^{3}+120 s^{2} t^{4} u^{2}+120 s^{2} t^{4} u+40 s^{2} t^{4}+124 s^{2} t^{3} u^{3}+412 s^{2} t^{3} u^{2}+452 s^{2} t^{3} u+164 s^{2} t^{3}+78 s^{2} t^{2} u^{3}+324 s^{2} t^{2} u^{2}+414 s^{2} t^{2} u+168 s^{2} t^{2}+40 s^{2} t u^{3}+60 s^{2} t u^{2}+40 s^{2} t u+20 s^{2} t+100 s^{2} u^{3}+189 s^{2} u^{2}+84 s^{2} u+34 s^{2}+20 s t^{4} u^{3}+60 s t^{4} u^{2}+60 s t^{4} u+20 s t^{4}+66 s t^{3} u^{3}+218 s t^{3} u^{2}+238 s t^{3} u+86 s t^{3}+42 s t^{2} u^{3}+171 s t^{2} u^{2}+216 s t^{2} u+85 s t^{2}+12 s t u^{3}+6 s t u^{2}+2 s t+46 s u^{3}+66 s u^{2}+4 t^{4} u^{3}+12 t^{4} u^{2}+12 t^{4} u+4 t^{4}+14 t^{3} u^{3}+46 t^{3} u^{2}+50 t^{3} u+18 t^{3}+9 t^{2} u^{3}+36 t^{2} u^{2}+45 t^{2} u+17 t^{2}+2 t u^{3}+10 u^{3}+12 u^{2} $$" ], "text/plain": [ "" @@ -2958,8 +3227,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2971,7 +3239,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1+a,b\\to 1/(1+b),c\\to 1/(1+c)$" + "Substitute $a\\to 1+v,b\\to 1/(1+w),c\\to 1/(1+x)$" ], "text/plain": [ "" @@ -2983,7 +3251,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4c^3+12a^5b^4c^2+12a^5b^4c+4a^5b^4+16a^5b^3c^3+48a^5b^3c^2+48a^5b^3c+16a^5b^3+24a^5b^2c^3+72a^5b^2c^2+72a^5b^2c+24a^5b^2+16a^5bc^3+48a^5bc^2+48a^5bc+16a^5b+4a^5c^3+12a^5c^2+12a^5c+4a^5+14a^4b^4c^3+46a^4b^4c^2+50a^4b^4c+18a^4b^4+60a^4b^3c^3+196a^4b^3c^2+212a^4b^3c+76a^4b^3+96a^4b^2c^3+312a^4b^2c^2+336a^4b^2c+120a^4b^2+68a^4bc^3+220a^4bc^2+236a^4bc+84a^4b+18a^4c^3+58a^4c^2+62a^4c+22a^4+16a^3b^4c^3+62a^3b^4c^2+76a^3b^4c+30a^3b^4+76a^3b^3c^3+284a^3b^3c^2+340a^3b^3c+132a^3b^3+132a^3b^2c^3+480a^3b^2c^2+564a^3b^2c+216a^3b^2+100a^3bc^3+356a^3bc^2+412a^3bc+156a^3b+28a^3c^3+98a^3c^2+112a^3c+42a^3+4a^2b^4c^3+30a^2b^4c^2+48a^2b^4c+22a^2b^4+28a^2b^3c^3+156a^2b^3c^2+228a^2b^3c+100a^2b^3+60a^2b^2c^3+288a^2b^2c^2+396a^2b^2c+168a^2b^2+52a^2bc^3+228a^2bc^2+300a^2bc+124a^2b+16a^2c^3+66a^2c^2+84a^2c+34a^2+5ab^4c^3+7ab^4c^2+3ab^4c+ab^4+24ab^3c^3+40ab^3c^2+24ab^3c+8ab^3+33ab^2c^3+51ab^2c^2+27ab^2c+9ab^2+18abc^3+22abc^2+6abc+2ab+4ac^3+4ac^2+7b^4c^3+16b^4c^2+9b^4c+4b^4+38b^3c^3+82b^3c^2+46b^3c+18b^3+63b^2c^3+120b^2c^2+51b^2c+18b^2+38bc^3+56bc^2+2bc+10c^3+14c^2$" + "Numerator after substitutions: $4 v^{5} w^{4} x^{3} + 12 v^{5} w^{4} x^{2} + 12 v^{5} w^{4} x + 4 v^{5} w^{4} + 16 v^{5} w^{3} x^{3} + 48 v^{5} w^{3} x^{2} + 48 v^{5} w^{3} x + 16 v^{5} w^{3} + 24 v^{5} w^{2} x^{3} + 72 v^{5} w^{2} x^{2} + 72 v^{5} w^{2} x + 24 v^{5} w^{2} + 16 v^{5} w x^{3} + 48 v^{5} w x^{2} + 48 v^{5} w x + 16 v^{5} w + 4 v^{5} x^{3} + 12 v^{5} x^{2} + 12 v^{5} x + 4 v^{5} + 14 v^{4} w^{4} x^{3} + 46 v^{4} w^{4} x^{2} + 50 v^{4} w^{4} x + 18 v^{4} w^{4} + 60 v^{4} w^{3} x^{3} + 196 v^{4} w^{3} x^{2} + 212 v^{4} w^{3} x + 76 v^{4} w^{3} + 96 v^{4} w^{2} x^{3} + 312 v^{4} w^{2} x^{2} + 336 v^{4} w^{2} x + 120 v^{4} w^{2} + 68 v^{4} w x^{3} + 220 v^{4} w x^{2} + 236 v^{4} w x + 84 v^{4} w + 18 v^{4} x^{3} + 58 v^{4} x^{2} + 62 v^{4} x + 22 v^{4} + 16 v^{3} w^{4} x^{3} + 62 v^{3} w^{4} x^{2} + 76 v^{3} w^{4} x + 30 v^{3} w^{4} + 76 v^{3} w^{3} x^{3} + 284 v^{3} w^{3} x^{2} + 340 v^{3} w^{3} x + 132 v^{3} w^{3} + 132 v^{3} w^{2} x^{3} + 480 v^{3} w^{2} x^{2} + 564 v^{3} w^{2} x + 216 v^{3} w^{2} + 100 v^{3} w x^{3} + 356 v^{3} w x^{2} + 412 v^{3} w x + 156 v^{3} w + 28 v^{3} x^{3} + 98 v^{3} x^{2} + 112 v^{3} x + 42 v^{3} + 4 v^{2} w^{4} x^{3} + 30 v^{2} w^{4} x^{2} + 48 v^{2} w^{4} x + 22 v^{2} w^{4} + 28 v^{2} w^{3} x^{3} + 156 v^{2} w^{3} x^{2} + 228 v^{2} w^{3} x + 100 v^{2} w^{3} + 60 v^{2} w^{2} x^{3} + 288 v^{2} w^{2} x^{2} + 396 v^{2} w^{2} x + 168 v^{2} w^{2} + 52 v^{2} w x^{3} + 228 v^{2} w x^{2} + 300 v^{2} w x + 124 v^{2} w + 16 v^{2} x^{3} + 66 v^{2} x^{2} + 84 v^{2} x + 34 v^{2} + 5 v w^{4} x^{3} + 7 v w^{4} x^{2} + 3 v w^{4} x + v w^{4} + 24 v w^{3} x^{3} + 40 v w^{3} x^{2} + 24 v w^{3} x + 8 v w^{3} + 33 v w^{2} x^{3} + 51 v w^{2} x^{2} + 27 v w^{2} x + 9 v w^{2} + 18 v w x^{3} + 22 v w x^{2} + 6 v w x + 2 v w + 4 v x^{3} + 4 v x^{2} + 7 w^{4} x^{3} + 16 w^{4} x^{2} + 9 w^{4} x + 4 w^{4} + 38 w^{3} x^{3} + 82 w^{3} x^{2} + 46 w^{3} x + 18 w^{3} + 63 w^{2} x^{3} + 120 w^{2} x^{2} + 51 w^{2} x + 18 w^{2} + 38 w x^{3} + 56 w x^{2} + 2 w x + 10 x^{3} + 14 x^{2}$" ], "text/plain": [ "" @@ -3007,7 +3275,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4c^3+12a^5b^4c^2+12a^5b^4c+4a^5b^4+16a^5b^3c^3+48a^5b^3c^2+48a^5b^3c+16a^5b^3+24a^5b^2c^3+72a^5b^2c^2+72a^5b^2c+24a^5b^2+16a^5bc^3+48a^5bc^2+48a^5bc+16a^5b+4a^5c^3+12a^5c^2+12a^5c+4a^5+14a^4b^4c^3+46a^4b^4c^2+50a^4b^4c+18a^4b^4+60a^4b^3c^3+196a^4b^3c^2+212a^4b^3c+76a^4b^3+96a^4b^2c^3+312a^4b^2c^2+336a^4b^2c+120a^4b^2+68a^4bc^3+220a^4bc^2+236a^4bc+84a^4b+18a^4c^3+58a^4c^2+62a^4c+22a^4+16a^3b^4c^3+62a^3b^4c^2+76a^3b^4c+30a^3b^4+76a^3b^3c^3+284a^3b^3c^2+340a^3b^3c+132a^3b^3+132a^3b^2c^3+480a^3b^2c^2+564a^3b^2c+216a^3b^2+100a^3bc^3+356a^3bc^2+412a^3bc+156a^3b+28a^3c^3+98a^3c^2+112a^3c+42a^3+4a^2b^4c^3+30a^2b^4c^2+48a^2b^4c+22a^2b^4+28a^2b^3c^3+156a^2b^3c^2+228a^2b^3c+100a^2b^3+60a^2b^2c^3+288a^2b^2c^2+396a^2b^2c+168a^2b^2+52a^2bc^3+228a^2bc^2+300a^2bc+124a^2b+16a^2c^3+66a^2c^2+84a^2c+34a^2+5ab^4c^3+7ab^4c^2+3ab^4c+ab^4+24ab^3c^3+40ab^3c^2+24ab^3c+8ab^3+33ab^2c^3+51ab^2c^2+27ab^2c+9ab^2+18abc^3+22abc^2+6abc+2ab+4ac^3+4ac^2+7b^4c^3+16b^4c^2+9b^4c+4b^4+38b^3c^3+82b^3c^2+46b^3c+18b^3+63b^2c^3+120b^2c^2+51b^2c+18b^2+38bc^3+56bc^2+2bc+10c^3+14c^2 $$" + "$$ 0 \\le 4 v^{5} w^{4} x^{3}+16 v^{5} w^{3} x^{3}+24 v^{5} w^{2} x^{3}+16 v^{5} w x^{3}+4 v^{5} x^{3}+14 v^{4} w^{4} x^{3}+60 v^{4} w^{3} x^{3}+96 v^{4} w^{2} x^{3}+68 v^{4} w x^{3}+18 v^{4} x^{3}+16 v^{3} w^{4} x^{3}+76 v^{3} w^{3} x^{3}+132 v^{3} w^{2} x^{3}+100 v^{3} w x^{3}+28 v^{3} x^{3}+4 v^{2} w^{4} x^{3}+28 v^{2} w^{3} x^{3}+60 v^{2} w^{2} x^{3}+52 v^{2} w x^{3}+16 v^{2} x^{3}+5 v w^{4} x^{3}+24 v w^{3} x^{3}+33 v w^{2} x^{3}+18 v w x^{3}+4 v x^{3}+7 w^{4} x^{3}+38 w^{3} x^{3}+63 w^{2} x^{3}+38 w x^{3}+10 x^{3}+12 v^{5} w^{4} x^{2}+48 v^{5} w^{3} x^{2}+72 v^{5} w^{2} x^{2}+48 v^{5} w x^{2}+12 v^{5} x^{2}+46 v^{4} w^{4} x^{2}+196 v^{4} w^{3} x^{2}+312 v^{4} w^{2} x^{2}+220 v^{4} w x^{2}+58 v^{4} x^{2}+62 v^{3} w^{4} x^{2}+284 v^{3} w^{3} x^{2}+480 v^{3} w^{2} x^{2}+356 v^{3} w x^{2}+98 v^{3} x^{2}+30 v^{2} w^{4} x^{2}+156 v^{2} w^{3} x^{2}+288 v^{2} w^{2} x^{2}+228 v^{2} w x^{2}+66 v^{2} x^{2}+7 v w^{4} x^{2}+40 v w^{3} x^{2}+51 v w^{2} x^{2}+22 v w x^{2}+4 v x^{2}+16 w^{4} x^{2}+82 w^{3} x^{2}+120 w^{2} x^{2}+56 w x^{2}+14 x^{2}+12 v^{5} w^{4} x+48 v^{5} w^{3} x+72 v^{5} w^{2} x+48 v^{5} w x+12 v^{5} x+50 v^{4} w^{4} x+212 v^{4} w^{3} x+336 v^{4} w^{2} x+236 v^{4} w x+62 v^{4} x+76 v^{3} w^{4} x+340 v^{3} w^{3} x+564 v^{3} w^{2} x+412 v^{3} w x+112 v^{3} x+48 v^{2} w^{4} x+228 v^{2} w^{3} x+396 v^{2} w^{2} x+300 v^{2} w x+84 v^{2} x+3 v w^{4} x+24 v w^{3} x+27 v w^{2} x+6 v w x+9 w^{4} x+46 w^{3} x+51 w^{2} x+2 w x+4 v^{5} w^{4}+16 v^{5} w^{3}+24 v^{5} w^{2}+16 v^{5} w+4 v^{5}+18 v^{4} w^{4}+76 v^{4} w^{3}+120 v^{4} w^{2}+84 v^{4} w+22 v^{4}+30 v^{3} w^{4}+132 v^{3} w^{3}+216 v^{3} w^{2}+156 v^{3} w+42 v^{3}+22 v^{2} w^{4}+100 v^{2} w^{3}+168 v^{2} w^{2}+124 v^{2} w+34 v^{2}+v w^{4}+8 v w^{3}+9 v w^{2}+2 v w+4 w^{4}+18 w^{3}+18 w^{2} $$" ], "text/plain": [ "" @@ -3031,8 +3299,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -3044,7 +3311,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1/(1+a),b\\to 1/(1+b),c\\to 1/(1+c)$" + "Substitute $a\\to 1/(1+y),b\\to 1/(1+z),c\\to 1/(1+a_{1})$" ], "text/plain": [ "" @@ -3056,7 +3323,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $11a^5b^4c^2+16a^5b^4c+9a^5b^4+10a^5b^3c^3+62a^5b^3c^2+74a^5b^3c+38a^5b^3+30a^5b^2c^3+117a^5b^2c^2+120a^5b^2c+57a^5b^2+24a^5bc^3+78a^5bc^2+72a^5bc+34a^5b+8a^5c^3+24a^5c^2+22a^5c+10a^5+9a^4b^4c^3+64a^4b^4c^2+75a^4b^4c+40a^4b^4+86a^4b^3c^3+346a^4b^3c^2+350a^4b^3c+170a^4b^3+195a^4b^2c^3+612a^4b^2c^2+543a^4b^2c+246a^4b^2+142a^4bc^3+384a^4bc^2+298a^4bc+136a^4b+44a^4c^3+114a^4c^2+90a^4c+40a^4+36a^3b^4c^3+146a^3b^4c^2+140a^3b^4c+70a^3b^4+244a^3b^3c^3+764a^3b^3c^2+660a^3b^3c+300a^3b^3+480a^3b^2c^3+1278a^3b^2c^2+972a^3b^2c+414a^3b^2+328a^3bc^3+756a^3bc^2+472a^3bc+204a^3b+96a^3c^3+216a^3c^2+140a^3c+60a^3+54a^2b^4c^3+162a^2b^4c^2+126a^2b^4c+58a^2b^4+312a^2b^3c^3+816a^2b^3c^2+592a^2b^3c+248a^2b^3+558a^2b^2c^3+1284a^2b^2c^2+798a^2b^2c+312a^2b^2+360a^2bc^3+700a^2bc^2+296a^2bc+116a^2b+100a^2c^3+190a^2c^2+84a^2c+34a^2+30ab^4c^3+73ab^4c^2+42ab^4c+19ab^4+166ab^3c^3+370ab^3c^2+206ab^3c+82ab^3+282ab^2c^3+549ab^2c^2+228ab^2c+81ab^2+172abc^3+258abc^2+4abc-2ab+46ac^3+66ac^2+7b^4c^3+16b^4c^2+9b^4c+4b^4+38b^3c^3+82b^3c^2+46b^3c+18b^3+63b^2c^3+120b^2c^2+51b^2c+18b^2+38bc^3+56bc^2+2bc+10c^3+14c^2$" + "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}$" ], "text/plain": [ "" @@ -3092,7 +3359,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 y z \\le y^{2}+z^{2}$$" ], "text/plain": [ "" @@ -3104,7 +3371,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 11a^5b^4c^2+16a^5b^4c+9a^5b^4+10a^5b^3c^3+62a^5b^3c^2+74a^5b^3c+38a^5b^3+30a^5b^2c^3+117a^5b^2c^2+120a^5b^2c+57a^5b^2+24a^5bc^3+78a^5bc^2+72a^5bc+34a^5b+8a^5c^3+24a^5c^2+22a^5c+10a^5+9a^4b^4c^3+64a^4b^4c^2+75a^4b^4c+40a^4b^4+86a^4b^3c^3+346a^4b^3c^2+350a^4b^3c+170a^4b^3+195a^4b^2c^3+612a^4b^2c^2+543a^4b^2c+246a^4b^2+142a^4bc^3+384a^4bc^2+298a^4bc+136a^4b+44a^4c^3+114a^4c^2+90a^4c+40a^4+36a^3b^4c^3+146a^3b^4c^2+140a^3b^4c+70a^3b^4+244a^3b^3c^3+764a^3b^3c^2+660a^3b^3c+300a^3b^3+480a^3b^2c^3+1278a^3b^2c^2+972a^3b^2c+414a^3b^2+328a^3bc^3+756a^3bc^2+472a^3bc+204a^3b+96a^3c^3+216a^3c^2+140a^3c+60a^3+54a^2b^4c^3+162a^2b^4c^2+126a^2b^4c+58a^2b^4+312a^2b^3c^3+816a^2b^3c^2+592a^2b^3c+248a^2b^3+558a^2b^2c^3+1284a^2b^2c^2+798a^2b^2c+312a^2b^2+360a^2bc^3+700a^2bc^2+296a^2bc+116a^2b+100a^2c^3+190a^2c^2+84a^2c+33a^2+30ab^4c^3+73ab^4c^2+42ab^4c+19ab^4+166ab^3c^3+370ab^3c^2+206ab^3c+82ab^3+282ab^2c^3+549ab^2c^2+228ab^2c+81ab^2+172abc^3+258abc^2+4abc+46ac^3+66ac^2+7b^4c^3+16b^4c^2+9b^4c+4b^4+38b^3c^3+82b^3c^2+46b^3c+18b^3+63b^2c^3+120b^2c^2+51b^2c+17b^2+38bc^3+56bc^2+2bc+10c^3+14c^2 $$" + "$$ 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} $$" ], "text/plain": [ "" @@ -3124,9 +3391,20 @@ }, "metadata": {}, "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "Counter({0: 8})" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" } ], "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", "powerprove(formula)" ] @@ -3145,13 +3423,13 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "numerator: $21f(-1,4,3)-21f(1,2,3)-21f(2,3,1)+21f(3,-1,4)-21f(3,1,2)+21f(4,3,-1)$" + "numerator: $21 f{\\left(-1,4,3 \\right)} - 21 f{\\left(1,2,3 \\right)} - 21 f{\\left(2,3,1 \\right)} + 21 f{\\left(3,-1,4 \\right)} - 21 f{\\left(3,1,2 \\right)} + 21 f{\\left(4,3,-1 \\right)}$" ], "text/plain": [ "" @@ -3255,18 +3533,575 @@ }, "metadata": {}, "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" } ], "source": [ "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')" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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.\n", + "\n", + "#### Problem\n", + "Prove the inequality\n", + "$$\\left(\\frac{a - b}{c}\\right)^2 + \\left(\\frac{b - c}{a}\\right)^2 + \\left(\\frac{c - a}{b}\\right)^2\\ge 2 \\sqrt{2} \\left(\\frac{a - b}{c} + \\frac{b - c}{a}+ \\frac{c-a}{b}\\right)$$\n", + "for any positive numbers $a,b,c$.\n", + "\n", + "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 $a\\ge c\\ge b$." + ] + }, { "cell_type": "code", - "execution_count": null, + "execution_count": 25, "metadata": {}, "outputs": [], - "source": [] + "source": [ + "shiro.display=lambda x:None #turn off printing of proofs\n", + "newproof()\n", + "formula=cyclize('((a-b)/c)^2-2*sqrt(2)*(a-b)/c')" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "formula1=makesubs(formula,'[b,oo],[c,oo]',variables='a,b') #a>=b>=c\n", + "prove(formula1) " + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "formula2=makesubs(formula,'[c,oo],[b,oo]',variables='a,c') #a>=c>=b\n", + "prove(formula2) " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So the case $a\\ge c\\ge b$ is done, but $a\\ge b\\ge c$ is not. But maybe we can adjust values." + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Optimization terminated successfully.\n", + " Current function value: 1.000000\n", + " Iterations: 137\n", + " Function evaluations: 249\n" + ] + }, + { + "data": { + "text/plain": [ + "(1.7908873553542452e-10, 2.5326984818340415e-10, 7.129450063690368)" + ] + }, + "execution_count": 28, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "values=findvalues(formula1)\n", + "values" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "First and second value is approximately equal to 0, so we can try to replace 0 with 1." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "prove(formula1,values='1,1,7')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The key observation is that the `formula1` is homogenous, so we can scale values." + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(1, 1.4142142855953455, 39809595184.05965)" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "newvalues=(1,values[1]/values[0],values[2]/values[0])\n", + "newvalues" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "2.000002045581953" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "newvalues[1]**2" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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." + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 32, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "prove(formula1,values='1,sqrt(2),1')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So after getting the code all together we have got the following proof." + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "Case $a\\ge c\\ge b$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $a\\to c + d$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $c\\to b + e$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $2 b^{4} d^{2} + 2 b^{4} d e + 2 b^{4} e^{2} + 4 b^{3} d^{3} + 2 \\sqrt{2} b^{3} d^{2} e + 10 b^{3} d^{2} e + 2 \\sqrt{2} b^{3} d e^{2} + 6 b^{3} d e^{2} + 4 b^{3} e^{3} + 2 b^{2} d^{4} + 2 \\sqrt{2} b^{2} d^{3} e + 10 b^{2} d^{3} e + 6 \\sqrt{2} b^{2} d^{2} e^{2} + 12 b^{2} d^{2} e^{2} + 4 b^{2} d e^{3} + 4 \\sqrt{2} b^{2} d e^{3} + 2 b^{2} e^{4} + 2 b d^{4} e + 2 \\sqrt{2} b d^{3} e^{2} + 6 b d^{3} e^{2} + 4 b d^{2} e^{3} + 4 \\sqrt{2} b d^{2} e^{3} + 2 \\sqrt{2} b d e^{4} + d^{4} e^{2} + 2 d^{3} e^{3} + d^{2} e^{4}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $b^{6} + 2 b^{5} d + 4 b^{5} e + b^{4} d^{2} + 6 b^{4} d e + 6 b^{4} e^{2} + 2 b^{3} d^{2} e + 6 b^{3} d e^{2} + 4 b^{3} e^{3} + b^{2} d^{2} e^{2} + 2 b^{2} d e^{3} + b^{2} e^{4}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 2 b^{4} d^{2}+2 b^{4} d e+2 b^{4} e^{2}+4 b^{3} d^{3}+2 \\sqrt{2} b^{3} d^{2} e+10 b^{3} d^{2} e+2 \\sqrt{2} b^{3} d e^{2}+6 b^{3} d e^{2}+4 b^{3} e^{3}+2 b^{2} d^{4}+2 \\sqrt{2} b^{2} d^{3} e+10 b^{2} d^{3} e+6 \\sqrt{2} b^{2} d^{2} e^{2}+12 b^{2} d^{2} e^{2}+4 \\sqrt{2} b^{2} d e^{3}+4 b^{2} d e^{3}+2 b^{2} e^{4}+2 b d^{4} e+2 \\sqrt{2} b d^{3} e^{2}+6 b d^{3} e^{2}+4 \\sqrt{2} b d^{2} e^{3}+4 b d^{2} e^{3}+2 \\sqrt{2} b d e^{4}+d^{4} e^{2}+2 d^{3} e^{3}+d^{2} e^{4} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Case $a\\ge b\\ge c$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $a\\to b + f$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $b\\to c + g$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $f\\to \\sqrt{2} h$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $2 c^{4} g^{2} + 2 \\sqrt{2} c^{4} g h + 4 c^{4} h^{2} + 4 c^{3} g^{3} - 4 c^{3} g^{2} h + 6 \\sqrt{2} c^{3} g^{2} h - 4 \\sqrt{2} c^{3} g h^{2} + 20 c^{3} g h^{2} + 8 \\sqrt{2} c^{3} h^{3} + 2 c^{2} g^{4} - 8 c^{2} g^{3} h + 4 \\sqrt{2} c^{2} g^{3} h - 12 \\sqrt{2} c^{2} g^{2} h^{2} + 24 c^{2} g^{2} h^{2} - 8 c^{2} g h^{3} + 20 \\sqrt{2} c^{2} g h^{3} + 8 c^{2} h^{4} - 4 c g^{4} h - 8 \\sqrt{2} c g^{3} h^{2} + 8 c g^{3} h^{2} - 8 c g^{2} h^{3} + 12 \\sqrt{2} c g^{2} h^{3} + 8 c g h^{4} + 2 g^{4} h^{2} + 4 \\sqrt{2} g^{3} h^{3} + 4 g^{2} h^{4}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $c^{6} + 4 c^{5} g + 2 \\sqrt{2} c^{5} h + 6 c^{4} g^{2} + 6 \\sqrt{2} c^{4} g h + 2 c^{4} h^{2} + 4 c^{3} g^{3} + 6 \\sqrt{2} c^{3} g^{2} h + 4 c^{3} g h^{2} + c^{2} g^{4} + 2 \\sqrt{2} c^{2} g^{3} h + 2 c^{2} g^{2} h^{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "From weighted AM-GM inequality:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$4 c^{3} g^{2} h \\le 2 c^{4} g^{2}+2 c^{2} g^{2} h^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$4 \\sqrt{2} c^{3} g h^{2} \\le 2 \\sqrt{2} c^{4} g h+\\sqrt{2} c^{3} h^{3}+\\sqrt{2} c g^{2} h^{3}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$8 c^{2} g^{3} h \\le 4 c^{3} g^{3}+4 c g^{3} h^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$12 \\sqrt{2} c^{2} g^{2} h^{2} \\le 6 \\sqrt{2} c^{3} g^{2} h+6 \\sqrt{2} c g^{2} h^{3}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$8 c^{2} g h^{3} \\le 4 c^{3} g h^{2}+2 c^{2} h^{4}+2 g^{2} h^{4}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$4 c g^{4} h \\le 2 c^{2} g^{4}+2 g^{4} h^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$8 \\sqrt{2} c g^{3} h^{2} \\le 4 \\sqrt{2} c^{2} g^{3} h+4 \\sqrt{2} g^{3} h^{3}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$8 c g^{2} h^{3} \\le 4 c g^{3} h^{2}+4 c g h^{4}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 4 c^{4} h^{2}+16 c^{3} g h^{2}+7 \\sqrt{2} c^{3} h^{3}+22 c^{2} g^{2} h^{2}+20 \\sqrt{2} c^{2} g h^{3}+6 c^{2} h^{4}+5 \\sqrt{2} c g^{2} h^{3}+4 c g h^{4}+2 g^{2} h^{4} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 33, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "newproof()\n", + "shiro.display=lambda x:display(Latex(x)) #turn on printing proofs \n", + "formula=cyclize('((a-b)/c)^2-2*sqrt(2)*(a-b)/c')\n", + "display(Latex('Case $a\\ge c\\ge b$'))\n", + "formula1=makesubs(formula,'[c,oo],[b,oo]',variables='a,c,b')\n", + "prove(formula1)\n", + "display(Latex('Case $a\\ge b\\ge c$'))\n", + "formula2=makesubs(formula,'[b,oo],[c,oo]')\n", + "prove(formula2,values='1,2**(1/2),1')" + ] } ], "metadata": { diff --git a/.ipynb_checkpoints/sandbox-checkpoint.ipynb b/.ipynb_checkpoints/sandbox-checkpoint.ipynb index 8b7f55a..3e2ac3a 100644 --- a/.ipynb_checkpoints/sandbox-checkpoint.ipynb +++ b/.ipynb_checkpoints/sandbox-checkpoint.ipynb @@ -16,145 +16,39 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "from shiroindev import *\n", "from sympy import *\n", "from itertools import permutations, combinations\n", - "sVars.seed=1\n", + "shiro.seed=1\n", "from IPython.display import Latex\n", - "sVars.display=lambda x:display(Latex(x))" + "shiro.display=lambda x:display(Latex(x))" ] }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "numerator: $12a^3-18a^2b+6a^2c+6ab^2-18ac^2+12b^3-18b^2c+6bc^2+12c^3$" + "__________________________" ], "text/plain": [ "" ] }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "denominator: $4a^2b+2a^2c+2ab^2+9abc+4ac^2+4b^2c+2bc^2$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "status: 0" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "From weighted AM-GM inequality:" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$18a^2b \\le 10a^3+6ab^2+2b^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$18ac^2 \\le 2a^3+6a^2c+10c^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$18b^2c \\le 10b^3+6bc^2+2c^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le 0 $$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "0" - ] - }, - "execution_count": 3, + "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "formula=cyclize('(a+b)/(2*b+c)')-2\n", - "prove(formula*6)" + "Latex(r'__________________________')" ] }, { @@ -165,7 +59,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to s-s/(a+1)$" + "Substitute $a\\to s-\\frac{s}{d+1}$" ], "text/plain": [ "" @@ -177,7 +71,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to s-s/(b+1)$" + "Substitute $b\\to s-\\frac{s}{e+1}$" ], "text/plain": [ "" @@ -189,7 +83,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to s-s/(c+1)$" + "Substitute $c\\to s-\\frac{s}{f+1}$" ], "text/plain": [ "" @@ -201,7 +95,7 @@ { "data": { "text/latex": [ - "numerator: $abcs^2+s^2$" + "numerator: $defs^2+s^2$" ], "text/plain": [ "" @@ -213,7 +107,7 @@ { "data": { "text/latex": [ - "denominator: $abc+ab+ac+a+bc+b+c+1$" + "denominator: $def+de+df+d+ef+e+f+1$" ], "text/plain": [ "" @@ -237,7 +131,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le abcs^2+s^2 $$" + "$$ 0 \\le defs^2+s^2 $$" ], "text/plain": [ "" @@ -317,7 +211,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $x\\to g+y$" ], "text/plain": [ "" @@ -329,7 +223,7 @@ { "data": { "text/latex": [ - "$$y^{2}$$" + "$$g^{2}$$" ], "text/plain": [ "" @@ -372,7 +266,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+z$" + "Substitute $y\\to h+x$" ], "text/plain": [ "" @@ -384,7 +278,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} \\left(x + z\\right)^{t} - x y y^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} + x y^{t} z - x z z^{t} + x z \\left(x + z\\right)^{t} + y^{2} y^{t} - 2 y y^{t} z + y^{t} z^{2}$$" + "4 $$h^{2} \\left(h + x\\right)^{t} - h x x^{t} + h x z^{t} + h x \\left(h + x\\right)^{t} + h x^{t} z - h z z^{t} - h z \\left(h + x\\right)^{t} + x^{2} z^{t} - 2 x z z^{t} + z^{2} z^{t}$$" ], "text/plain": [ "" @@ -396,7 +290,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to y+z$" + "Substitute $z\\to i+x$" ], "text/plain": [ "" @@ -408,7 +302,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} x^{t} - x x^{t} y - 2 x x^{t} z + x y z^{t} - x y \\left(y + z\\right)^{t} + x^{t} y z + x^{t} z^{2} + y^{2} \\left(y + z\\right)^{t} - y z z^{t} + y z \\left(y + z\\right)^{t}$$" + "4 $$i^{2} \\left(i + x\\right)^{t} - i x x^{t} + i x y^{t} + i x \\left(i + x\\right)^{t} + i x^{t} y - i y y^{t} - i y \\left(i + x\\right)^{t} + x^{2} y^{t} - 2 x y y^{t} + y^{2} y^{t}$$" ], "text/plain": [ "" @@ -420,7 +314,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $z\\to j+y$" ], "text/plain": [ "" @@ -432,7 +326,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} z^{t} - x x^{t} y + x y z^{t} + x y \\left(x + y\\right)^{t} - 2 x z z^{t} + x^{t} y z + y^{2} \\left(x + y\\right)^{t} - y z z^{t} - y z \\left(x + y\\right)^{t} + z^{2} z^{t}$$" + "4 $$j^{2} \\left(j + y\\right)^{t} - j x x^{t} + j x y^{t} - j x \\left(j + y\\right)^{t} + j x^{t} y - j y y^{t} + j y \\left(j + y\\right)^{t} + x^{2} x^{t} - 2 x x^{t} y + x^{t} y^{2}$$" ], "text/plain": [ "" @@ -444,7 +338,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+z$" + "Substitute $y\\to k+x$" ], "text/plain": [ "" @@ -456,7 +350,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to y+z$" + "Substitute $z\\to l+x$" ], "text/plain": [ "" @@ -468,7 +362,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} \\left(x + z\\right)^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} - x y \\left(y + z\\right)^{t} + y^{2} \\left(y + z\\right)^{t}$$" + "4 $$k^{2} \\left(k + x\\right)^{t} + k l x^{t} - k l \\left(k + x\\right)^{t} - k l \\left(l + x\\right)^{t} + l^{2} \\left(l + x\\right)^{t}$$" ], "text/plain": [ "" @@ -480,7 +374,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $l\\to k+m$" ], "text/plain": [ "" @@ -492,7 +386,7 @@ { "data": { "text/latex": [ - "3 $$x^{2} z^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} + x y \\left(x + y + z\\right)^{t} + y^{2} \\left(x + y + z\\right)^{t}$$" + "3 $$k^{2} x^{t} + k m x^{t} - k m \\left(k + x\\right)^{t} + k m \\left(k + m + x\\right)^{t} + m^{2} \\left(k + m + x\\right)^{t}$$" ], "text/plain": [ "" @@ -504,7 +398,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to y+z$" + "Substitute $z\\to n+x$" ], "text/plain": [ "" @@ -516,7 +410,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+z$" + "Substitute $y\\to o+x$" ], "text/plain": [ "" @@ -528,7 +422,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} \\left(x + z\\right)^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} - x y \\left(y + z\\right)^{t} + y^{2} \\left(y + z\\right)^{t}$$" + "4 $$n^{2} \\left(n + x\\right)^{t} + n o x^{t} - n o \\left(n + x\\right)^{t} - n o \\left(o + x\\right)^{t} + o^{2} \\left(o + x\\right)^{t}$$" ], "text/plain": [ "" @@ -540,7 +434,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $o\\to n+p$" ], "text/plain": [ "" @@ -552,7 +446,7 @@ { "data": { "text/latex": [ - "3 $$x^{2} z^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} + x y \\left(x + y + z\\right)^{t} + y^{2} \\left(x + y + z\\right)^{t}$$" + "3 $$n^{2} x^{t} + n p x^{t} - n p \\left(n + x\\right)^{t} + n p \\left(n + p + x\\right)^{t} + p^{2} \\left(n + p + x\\right)^{t}$$" ], "text/plain": [ "" @@ -574,7 +468,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+y$" + "Substitute $x\\to q+y$" ], "text/plain": [ "" @@ -586,10 +480,10 @@ { "data": { "text/latex": [ - "$\\displaystyle x^{2} \\left(x + y\\right)^{t} - x y y^{t} + x y z^{t} + x y \\left(x + y\\right)^{t} + x y^{t} z - x z z^{t} - x z \\left(x + y\\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$" + "$\\displaystyle q^{2} \\left(q + y\\right)^{t} - q y y^{t} + q y z^{t} + q y \\left(q + y\\right)^{t} + q y^{t} z - q z z^{t} - q z \\left(q + y\\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$" ], "text/plain": [ - "x**2*(x + y)**t - x*y*y**t + x*y*z**t + x*y*(x + y)**t + x*y**t*z - x*z*z**t - x*z*(x + y)**t + y**2*z**t - 2*y*z*z**t + z**2*z**t" + "q**2*(q + y)**t - q*y*y**t + q*y*z**t + q*y*(q + y)**t + q*y**t*z - q*z*z**t - q*z*(q + y)**t + y**2*z**t - 2*y*z*z**t + z**2*z**t" ] }, "execution_count": 10, @@ -609,7 +503,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+y$" + "Substitute $x\\to r+y$" ], "text/plain": [ "" @@ -621,10 +515,10 @@ { "data": { "text/latex": [ - "$\\displaystyle x^{2} \\left(x + y\\right)^{t} - x y y^{t} + x y z^{t} + x y \\left(x + y\\right)^{t} + x y^{t} z - x z z^{t} - x z \\left(x + y\\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$" + "$\\displaystyle r^{2} \\left(r + y\\right)^{t} - r y y^{t} + r y z^{t} + r y \\left(r + y\\right)^{t} + r y^{t} z - r z z^{t} - r z \\left(r + y\\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$" ], "text/plain": [ - "x**2*(x + y)**t - x*y*y**t + x*y*z**t + x*y*(x + y)**t + x*y**t*z - x*z*z**t - x*z*(x + y)**t + y**2*z**t - 2*y*z*z**t + z**2*z**t" + "r**2*(r + y)**t - r*y*y**t + r*y*z**t + r*y*(r + y)**t + r*y**t*z - r*z*z**t - r*z*(r + y)**t + y**2*z**t - 2*y*z*z**t + z**2*z**t" ] }, "execution_count": 11, @@ -644,7 +538,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+z$" + "Substitute $y\\to u+x$" ], "text/plain": [ "" @@ -656,7 +550,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} \\left(x + z\\right)^{t} - x y y^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} + x y^{t} z - x z z^{t} + x z \\left(x + z\\right)^{t} + y^{2} y^{t} - 2 y y^{t} z + y^{t} z^{2}$$" + "4 $$u^{2} \\left(u + x\\right)^{t} - u x x^{t} + u x z^{t} + u x \\left(u + x\\right)^{t} + u x^{t} z - u z z^{t} - u z \\left(u + x\\right)^{t} + x^{2} z^{t} - 2 x z z^{t} + z^{2} z^{t}$$" ], "text/plain": [ "" @@ -668,7 +562,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to y+z$" + "Substitute $z\\to v+x$" ], "text/plain": [ "" @@ -680,7 +574,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} x^{t} - x x^{t} y - 2 x x^{t} z + x y z^{t} - x y \\left(y + z\\right)^{t} + x^{t} y z + x^{t} z^{2} + y^{2} \\left(y + z\\right)^{t} - y z z^{t} + y z \\left(y + z\\right)^{t}$$" + "4 $$v^{2} \\left(v + x\\right)^{t} - v x x^{t} + v x y^{t} + v x \\left(v + x\\right)^{t} + v x^{t} y - v y y^{t} - v y \\left(v + x\\right)^{t} + x^{2} y^{t} - 2 x y y^{t} + y^{2} y^{t}$$" ], "text/plain": [ "" @@ -692,7 +586,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $z\\to w+y$" ], "text/plain": [ "" @@ -704,7 +598,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} z^{t} - x x^{t} y + x y z^{t} + x y \\left(x + y\\right)^{t} - 2 x z z^{t} + x^{t} y z + y^{2} \\left(x + y\\right)^{t} - y z z^{t} - y z \\left(x + y\\right)^{t} + z^{2} z^{t}$$" + "4 $$w^{2} \\left(w + y\\right)^{t} - w x x^{t} + w x y^{t} - w x \\left(w + y\\right)^{t} + w x^{t} y - w y y^{t} + w y \\left(w + y\\right)^{t} + x^{2} x^{t} - 2 x x^{t} y + x^{t} y^{2}$$" ], "text/plain": [ "" @@ -716,7 +610,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+z$" + "Substitute $y\\to a_{1}+x$" ], "text/plain": [ "" @@ -728,7 +622,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to y+z$" + "Substitute $z\\to b_{1}+x$" ], "text/plain": [ "" @@ -740,7 +634,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} \\left(x + z\\right)^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} - x y \\left(y + z\\right)^{t} + y^{2} \\left(y + z\\right)^{t}$$" + "4 $$a_{1}^{2} \\left(a_{1} + x\\right)^{t} + a_{1} b_{1} x^{t} - a_{1} b_{1} \\left(a_{1} + x\\right)^{t} - a_{1} b_{1} \\left(b_{1} + x\\right)^{t} + b_{1}^{2} \\left(b_{1} + x\\right)^{t}$$" ], "text/plain": [ "" @@ -752,7 +646,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $b_{1}\\to a_{1}+c_{1}$" ], "text/plain": [ "" @@ -764,7 +658,7 @@ { "data": { "text/latex": [ - "3 $$x^{2} z^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} + x y \\left(x + y + z\\right)^{t} + y^{2} \\left(x + y + z\\right)^{t}$$" + "3 $$a_{1}^{2} x^{t} + a_{1} c_{1} x^{t} - a_{1} c_{1} \\left(a_{1} + x\\right)^{t} + a_{1} c_{1} \\left(a_{1} + c_{1} + x\\right)^{t} + c_{1}^{2} \\left(a_{1} + c_{1} + x\\right)^{t}$$" ], "text/plain": [ "" @@ -776,7 +670,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to y+z$" + "Substitute $z\\to d_{1}+x$" ], "text/plain": [ "" @@ -788,7 +682,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+z$" + "Substitute $y\\to e_{1}+x$" ], "text/plain": [ "" @@ -800,7 +694,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} \\left(x + z\\right)^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} - x y \\left(y + z\\right)^{t} + y^{2} \\left(y + z\\right)^{t}$$" + "4 $$d_{1}^{2} \\left(d_{1} + x\\right)^{t} + d_{1} e_{1} x^{t} - d_{1} e_{1} \\left(d_{1} + x\\right)^{t} - d_{1} e_{1} \\left(e_{1} + x\\right)^{t} + e_{1}^{2} \\left(e_{1} + x\\right)^{t}$$" ], "text/plain": [ "" @@ -812,7 +706,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $e_{1}\\to d_{1}+f_{1}$" ], "text/plain": [ "" @@ -824,7 +718,7 @@ { "data": { "text/latex": [ - "3 $$x^{2} z^{t} + x y z^{t} - x y \\left(x + z\\right)^{t} + x y \\left(x + y + z\\right)^{t} + y^{2} \\left(x + y + z\\right)^{t}$$" + "3 $$d_{1}^{2} x^{t} + d_{1} f_{1} x^{t} - d_{1} f_{1} \\left(d_{1} + x\\right)^{t} + d_{1} f_{1} \\left(d_{1} + f_{1} + x\\right)^{t} + f_{1}^{2} \\left(d_{1} + f_{1} + x\\right)^{t}$$" ], "text/plain": [ "" @@ -953,7 +847,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 11a/8$" + "Substitute $a\\to 11g_{1}/8$" ], "text/plain": [ "" @@ -965,7 +859,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to 4b/5$" + "Substitute $b\\to 4h_{1}/5$" ], "text/plain": [ "" @@ -977,7 +871,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to 4c/5$" + "Substitute $c\\to 4i_{1}/5$" ], "text/plain": [ "" @@ -989,7 +883,7 @@ { "data": { "text/latex": [ - "Substitute $d\\to 4d/5$" + "Substitute $d\\to 4j_{1}/5$" ], "text/plain": [ "" @@ -1001,7 +895,7 @@ { "data": { "text/latex": [ - "numerator: $3025a^2-1760ab-1760ac-1760ad+1024b^2+1024c^2+1024d^2$" + "numerator: $3025g_{1}^2-1760g_{1}h_{1}-1760g_{1}i_{1}-1760g_{1}j_{1}+1024h_{1}^2+1024i_{1}^2+1024j_{1}^2$" ], "text/plain": [ "" @@ -1049,7 +943,7 @@ { "data": { "text/latex": [ - "$$1760ab \\le 880a^2+880b^2$$" + "$$1760g_{1}h_{1} \\le 880g_{1}^2+880h_{1}^2$$" ], "text/plain": [ "" @@ -1061,7 +955,7 @@ { "data": { "text/latex": [ - "$$1760ac \\le 880a^2+880c^2$$" + "$$1760g_{1}i_{1} \\le 880g_{1}^2+880i_{1}^2$$" ], "text/plain": [ "" @@ -1073,7 +967,7 @@ { "data": { "text/latex": [ - "$$1760ad \\le 880a^2+880d^2$$" + "$$1760g_{1}j_{1} \\le 880g_{1}^2+880j_{1}^2$$" ], "text/plain": [ "" @@ -1085,7 +979,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 385a^2+144b^2+144c^2+144d^2 $$" + "$$ 0 \\le 385g_{1}^2+144h_{1}^2+144i_{1}^2+144j_{1}^2 $$" ], "text/plain": [ "" @@ -1148,76 +1042,29 @@ "outputs": [ { "data": { - "text/latex": [ - "$\\displaystyle - \\frac{2 \\sqrt{2} \\left(a - b\\right)}{c} + \\frac{\\left(a - b\\right)^{2}}{c^{2}} - \\frac{2 \\sqrt{2} \\left(- a + c\\right)}{b} + \\frac{\\left(- a + c\\right)^{2}}{b^{2}} - \\frac{2 \\sqrt{2} \\left(b - c\\right)}{a} + \\frac{\\left(b - c\\right)^{2}}{a^{2}}$" - ], "text/plain": [ - "-2*sqrt(2)*(a - b)/c + (a - b)**2/c**2 - 2*sqrt(2)*(-a + c)/b + (-a + c)**2/b**2 - 2*sqrt(2)*(b - c)/a + (b - c)**2/a**2" + "\"formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\\ndisplay(formula)\\nfrom scipy.optimize import fmin\\nimport numpy as np\\ndef f(x):\\n num,den=fraction(cancel(newformula))\\n fs=sorted(newformula.free_symbols,key=str)\\n return float(num.subs(list(zip(fs,x))))\\nnewformula=(makesubs(formula,'[b,oo],[c,oo]'))\\nprint(f([2,1,1]))\\nfmin(f,[2,1,1])\\nprint(fmin(f,np.array([2,1,1])))\\ndisplay(simplify(newformula))\"" ] }, + "execution_count": 20, "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $a\\to a+b$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $b\\to b+c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Optimization terminated successfully.\n", - " Current function value: 0.000000\n", - " Iterations: 51\n", - " Function evaluations: 97\n", - "[1.17092486 1.170961 1.17093843]\n" - ] - }, - { - "data": { - "text/latex": [ - "$\\displaystyle \\frac{a^{4} b^{2} + 2 a^{4} b c + 2 a^{4} c^{2} + 2 a^{3} b^{3} - 2 \\sqrt{2} a^{3} b^{2} c + 6 a^{3} b^{2} c - 2 \\sqrt{2} a^{3} b c^{2} + 10 a^{3} b c^{2} + 4 a^{3} c^{3} + a^{2} b^{4} - 4 \\sqrt{2} a^{2} b^{3} c + 4 a^{2} b^{3} c - 6 \\sqrt{2} a^{2} b^{2} c^{2} + 12 a^{2} b^{2} c^{2} - 2 \\sqrt{2} a^{2} b c^{3} + 10 a^{2} b c^{3} + 2 a^{2} c^{4} - 2 \\sqrt{2} a b^{4} c - 4 \\sqrt{2} a b^{3} c^{2} + 4 a b^{3} c^{2} - 2 \\sqrt{2} a b^{2} c^{3} + 6 a b^{2} c^{3} + 2 a b c^{4} + 2 b^{4} c^{2} + 4 b^{3} c^{3} + 2 b^{2} c^{4}}{c^{2} \\left(a^{2} b^{2} + 2 a^{2} b c + a^{2} c^{2} + 2 a b^{3} + 6 a b^{2} c + 6 a b c^{2} + 2 a c^{3} + b^{4} + 4 b^{3} c + 6 b^{2} c^{2} + 4 b c^{3} + c^{4}\\right)}$" - ], - "text/plain": [ - "(a**4*b**2 + 2*a**4*b*c + 2*a**4*c**2 + 2*a**3*b**3 - 2*sqrt(2)*a**3*b**2*c + 6*a**3*b**2*c - 2*sqrt(2)*a**3*b*c**2 + 10*a**3*b*c**2 + 4*a**3*c**3 + a**2*b**4 - 4*sqrt(2)*a**2*b**3*c + 4*a**2*b**3*c - 6*sqrt(2)*a**2*b**2*c**2 + 12*a**2*b**2*c**2 - 2*sqrt(2)*a**2*b*c**3 + 10*a**2*b*c**3 + 2*a**2*c**4 - 2*sqrt(2)*a*b**4*c - 4*sqrt(2)*a*b**3*c**2 + 4*a*b**3*c**2 - 2*sqrt(2)*a*b**2*c**3 + 6*a*b**2*c**3 + 2*a*b*c**4 + 2*b**4*c**2 + 4*b**3*c**3 + 2*b**2*c**4)/(c**2*(a**2*b**2 + 2*a**2*b*c + a**2*c**2 + 2*a*b**3 + 6*a*b**2*c + 6*a*b*c**2 + 2*a*c**3 + b**4 + 4*b**3*c + 6*b**2*c**2 + 4*b*c**3 + c**4))" - ] - }, - "metadata": {}, - "output_type": "display_data" + "output_type": "execute_result" } ], "source": [ - "formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\n", + "\"\"\"formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\n", "display(formula)\n", "from scipy.optimize import fmin\n", "import numpy as np\n", "def f(x):\n", - " num,den=fraction(cancel(formula))\n", + " num,den=fraction(cancel(newformula))\n", " fs=sorted(newformula.free_symbols,key=str)\n", - " return num.subs(zip(fs,x))\n", + " return float(num.subs(list(zip(fs,x))))\n", "newformula=(makesubs(formula,'[b,oo],[c,oo]'))\n", + "print(f([2,1,1]))\n", + "fmin(f,[2,1,1])\n", "print(fmin(f,np.array([2,1,1])))\n", - "display(simplify(newformula))\n", + "display(simplify(newformula))\"\"\"\n", "#prove(newformula)\n", "#prove(makesubs(formula,'[b,oo],[a,oo]',variables='c,b,a'))" ] @@ -1326,13 +1173,13 @@ }, { "cell_type": "code", - "execution_count": 107, + "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "Substitute $a\\to b+k_{1}$" ], "text/plain": [ "" @@ -1344,7 +1191,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "Substitute $b\\to c+l_{1}$" ], "text/plain": [ "" @@ -1354,68 +1201,15 @@ "output_type": "display_data" }, { - "data": { - "text/latex": [ - "$\\displaystyle a^{8} b^{4} + 2 a^{8} b^{2} c^{2} + 2 a^{8} c^{4} + 2 a^{6} b^{6} - 2 \\sqrt{2} a^{6} b^{4} c^{2} + 6 a^{6} b^{4} c^{2} - 2 \\sqrt{2} a^{6} b^{2} c^{4} + 10 a^{6} b^{2} c^{4} + 4 a^{6} c^{6} + a^{4} b^{8} - 4 \\sqrt{2} a^{4} b^{6} c^{2} + 4 a^{4} b^{6} c^{2} - 6 \\sqrt{2} a^{4} b^{4} c^{4} + 12 a^{4} b^{4} c^{4} - 2 \\sqrt{2} a^{4} b^{2} c^{6} + 10 a^{4} b^{2} c^{6} + 2 a^{4} c^{8} - 2 \\sqrt{2} a^{2} b^{8} c^{2} - 4 \\sqrt{2} a^{2} b^{6} c^{4} + 4 a^{2} b^{6} c^{4} - 2 \\sqrt{2} a^{2} b^{4} c^{6} + 6 a^{2} b^{4} c^{6} + 2 a^{2} b^{2} c^{8} + 2 b^{8} c^{4} + 4 b^{6} c^{6} + 2 b^{4} c^{8}$" - ], - "text/plain": [ - "a**8*b**4 + 2*a**8*b**2*c**2 + 2*a**8*c**4 + 2*a**6*b**6 - 2*sqrt(2)*a**6*b**4*c**2 + 6*a**6*b**4*c**2 - 2*sqrt(2)*a**6*b**2*c**4 + 10*a**6*b**2*c**4 + 4*a**6*c**6 + a**4*b**8 - 4*sqrt(2)*a**4*b**6*c**2 + 4*a**4*b**6*c**2 - 6*sqrt(2)*a**4*b**4*c**4 + 12*a**4*b**4*c**4 - 2*sqrt(2)*a**4*b**2*c**6 + 10*a**4*b**2*c**6 + 2*a**4*c**8 - 2*sqrt(2)*a**2*b**8*c**2 - 4*sqrt(2)*a**2*b**6*c**4 + 4*a**2*b**6*c**4 - 2*sqrt(2)*a**2*b**4*c**6 + 6*a**2*b**4*c**6 + 2*a**2*b**2*c**8 + 2*b**8*c**4 + 4*b**6*c**6 + 2*b**4*c**8" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "{a: 2, b: 2, c: 2}\n" + "ename": "NameError", + "evalue": "name 'fs' is not defined", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\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[0mformula\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmakesubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mformula\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m'[b,oo],[c,oo]'\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[0mnum\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mden\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mfraction\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mformula\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mnum\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mnum\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfs\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmap\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;32mlambda\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mfs\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[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 5\u001b[0m \u001b[0mdisplay\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnum\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 6\u001b[0m \u001b[0mnumm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mNameError\u001b[0m: name 'fs' is not defined" ] - }, - { - "data": { - "text/latex": [ - "$\\displaystyle \\frac{19 \\sqrt{2}}{12}$" - ], - "text/plain": [ - "19*sqrt(2)/12" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "x\n", - "Optimization terminated successfully.\n", - " Current function value: 1.000000\n", - " Iterations: 147\n", - " Function evaluations: 267\n" - ] - }, - { - "data": { - "text/plain": [ - "(4.4110810020775736e-12, 28.52020821072961, 3.1191098416324057e-12)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$\\displaystyle \\left( 0, \\ \\frac{57}{2}, \\ 0\\right)$" - ], - "text/plain": [ - "(0, 57/2, 0)" - ] - }, - "execution_count": 107, - "metadata": {}, - "output_type": "execute_result" } ], "source": [ @@ -1470,7 +1264,7 @@ }, { "cell_type": "code", - "execution_count": 65, + "execution_count": 26, "metadata": {}, "outputs": [ { @@ -1479,7 +1273,7 @@ "1.414211498165453" ] }, - "execution_count": 65, + "execution_count": 26, "metadata": {}, "output_type": "execute_result" } @@ -1490,7 +1284,7 @@ }, { "cell_type": "code", - "execution_count": 111, + "execution_count": 28, "metadata": {}, "outputs": [ { @@ -1508,7 +1302,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "Substitute $a\\to c+r_{1}$" ], "text/plain": [ "" @@ -1520,7 +1314,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to b+c$" + "Substitute $c\\to b+s_{1}$" ], "text/plain": [ "" @@ -1532,7 +1326,7 @@ { "data": { "text/latex": [ - "numerator: $2a^4b^2+2a^4bc+a^4c^2+4a^3b^3+2\\sqrt{2}a^3b^2c+10a^3b^2c+2\\sqrt{2}a^3bc^2+6a^3bc^2+2a^3c^3+2a^2b^4+2\\sqrt{2}a^2b^3c+10a^2b^3c+6\\sqrt{2}a^2b^2c^2+12a^2b^2c^2+4a^2bc^3+4\\sqrt{2}a^2bc^3+a^2c^4+2ab^4c+2\\sqrt{2}ab^3c^2+6ab^3c^2+4ab^2c^3+4\\sqrt{2}ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4$" + "numerator: $2b^4r_{1}^2+2b^4r_{1}s_{1}+2b^4s_{1}^2+4b^3r_{1}^3+2\\sqrt{2}b^3r_{1}^2s_{1}+10b^3r_{1}^2s_{1}+2\\sqrt{2}b^3r_{1}s_{1}^2+6b^3r_{1}s_{1}^2+4b^3s_{1}^3+2b^2r_{1}^4+2\\sqrt{2}b^2r_{1}^3s_{1}+10b^2r_{1}^3s_{1}+6\\sqrt{2}b^2r_{1}^2s_{1}^2+12b^2r_{1}^2s_{1}^2+4b^2r_{1}s_{1}^3+4\\sqrt{2}b^2r_{1}s_{1}^3+2b^2s_{1}^4+2br_{1}^4s_{1}+2\\sqrt{2}br_{1}^3s_{1}^2+6br_{1}^3s_{1}^2+4br_{1}^2s_{1}^3+4\\sqrt{2}br_{1}^2s_{1}^3+2\\sqrt{2}br_{1}s_{1}^4+r_{1}^4s_{1}^2+2r_{1}^3s_{1}^3+r_{1}^2s_{1}^4$" ], "text/plain": [ "" @@ -1544,7 +1338,7 @@ { "data": { "text/latex": [ - "denominator: $a^2b^4+2a^2b^3c+a^2b^2c^2+2ab^5+6ab^4c+6ab^3c^2+2ab^2c^3+b^6+4b^5c+6b^4c^2+4b^3c^3+b^2c^4$" + "denominator: $b^6+2b^5r_{1}+4b^5s_{1}+b^4r_{1}^2+6b^4r_{1}s_{1}+6b^4s_{1}^2+2b^3r_{1}^2s_{1}+6b^3r_{1}s_{1}^2+4b^3s_{1}^3+b^2r_{1}^2s_{1}^2+2b^2r_{1}s_{1}^3+b^2s_{1}^4$" ], "text/plain": [ "" @@ -1568,7 +1362,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 2a^4b^2+2a^4bc+a^4c^2+4a^3b^3+2\\sqrt{2}a^3b^2c+10a^3b^2c+2\\sqrt{2}a^3bc^2+6a^3bc^2+2a^3c^3+2a^2b^4+2\\sqrt{2}a^2b^3c+10a^2b^3c+6\\sqrt{2}a^2b^2c^2+12a^2b^2c^2+4\\sqrt{2}a^2bc^3+4a^2bc^3+a^2c^4+2ab^4c+2\\sqrt{2}ab^3c^2+6ab^3c^2+4\\sqrt{2}ab^2c^3+4ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4 $$" + "$$ 0 \\le r_{1}^4s_{1}^2+2br_{1}^4s_{1}+2b^2r_{1}^4+2r_{1}^3s_{1}^3+2\\sqrt{2}br_{1}^3s_{1}^2+6br_{1}^3s_{1}^2+2\\sqrt{2}b^2r_{1}^3s_{1}+10b^2r_{1}^3s_{1}+4b^3r_{1}^3+r_{1}^2s_{1}^4+4\\sqrt{2}br_{1}^2s_{1}^3+4br_{1}^2s_{1}^3+6\\sqrt{2}b^2r_{1}^2s_{1}^2+12b^2r_{1}^2s_{1}^2+2\\sqrt{2}b^3r_{1}^2s_{1}+10b^3r_{1}^2s_{1}+2b^4r_{1}^2+2\\sqrt{2}br_{1}s_{1}^4+4\\sqrt{2}b^2r_{1}s_{1}^3+4b^2r_{1}s_{1}^3+2\\sqrt{2}b^3r_{1}s_{1}^2+6b^3r_{1}s_{1}^2+2b^4r_{1}s_{1}+2b^2s_{1}^4+4b^3s_{1}^3+2b^4s_{1}^2 $$" ], "text/plain": [ "" @@ -1604,7 +1398,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "Substitute $a\\to b+t_{1}$" ], "text/plain": [ "" @@ -1616,7 +1410,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "Substitute $b\\to c+u_{1}$" ], "text/plain": [ "" @@ -1628,7 +1422,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to \\sqrt{2}a$" + "Substitute $t_{1}\\to \\sqrt{2}v_{1}$" ], "text/plain": [ "" @@ -1640,7 +1434,7 @@ { "data": { "text/latex": [ - "numerator: $16a^4b^2+32a^4bc+32a^4c^2+16\\sqrt{2}a^3b^3-32a^3b^2c+48\\sqrt{2}a^3b^2c-32a^3bc^2+80\\sqrt{2}a^3bc^2+32\\sqrt{2}a^3c^3+8a^2b^4-32\\sqrt{2}a^2b^3c+32a^2b^3c-48\\sqrt{2}a^2b^2c^2+96a^2b^2c^2-16\\sqrt{2}a^2bc^3+80a^2bc^3+16a^2c^4-16ab^4c-32ab^3c^2+16\\sqrt{2}ab^3c^2-16ab^2c^3+24\\sqrt{2}ab^2c^3+8\\sqrt{2}abc^4+8b^4c^2+16b^3c^3+8b^2c^4$" + "numerator: $8c^4u_{1}^2+8\\sqrt{2}c^4u_{1}v_{1}+16c^4v_{1}^2+16c^3u_{1}^3-16c^3u_{1}^2v_{1}+24\\sqrt{2}c^3u_{1}^2v_{1}-16\\sqrt{2}c^3u_{1}v_{1}^2+80c^3u_{1}v_{1}^2+32\\sqrt{2}c^3v_{1}^3+8c^2u_{1}^4-32c^2u_{1}^3v_{1}+16\\sqrt{2}c^2u_{1}^3v_{1}-48\\sqrt{2}c^2u_{1}^2v_{1}^2+96c^2u_{1}^2v_{1}^2-32c^2u_{1}v_{1}^3+80\\sqrt{2}c^2u_{1}v_{1}^3+32c^2v_{1}^4-16cu_{1}^4v_{1}-32\\sqrt{2}cu_{1}^3v_{1}^2+32cu_{1}^3v_{1}^2-32cu_{1}^2v_{1}^3+48\\sqrt{2}cu_{1}^2v_{1}^3+32cu_{1}v_{1}^4+8u_{1}^4v_{1}^2+16\\sqrt{2}u_{1}^3v_{1}^3+16u_{1}^2v_{1}^4$" ], "text/plain": [ "" @@ -1652,7 +1446,7 @@ { "data": { "text/latex": [ - "denominator: $2a^2b^2c^2+4a^2bc^3+2a^2c^4+2\\sqrt{2}ab^3c^2+6\\sqrt{2}ab^2c^3+6\\sqrt{2}abc^4+2\\sqrt{2}ac^5+b^4c^2+4b^3c^3+6b^2c^4+4bc^5+c^6$" + "denominator: $c^6+4c^5u_{1}+2\\sqrt{2}c^5v_{1}+6c^4u_{1}^2+6\\sqrt{2}c^4u_{1}v_{1}+2c^4v_{1}^2+4c^3u_{1}^3+6\\sqrt{2}c^3u_{1}^2v_{1}+4c^3u_{1}v_{1}^2+c^2u_{1}^4+2\\sqrt{2}c^2u_{1}^3v_{1}+2c^2u_{1}^2v_{1}^2$" ], "text/plain": [ "" @@ -1688,7 +1482,7 @@ { "data": { "text/latex": [ - "$$32a^3b^2c \\le 16a^4b^2+16a^2b^2c^2$$" + "$$16cu_{1}^4v_{1} \\le 8u_{1}^4v_{1}^2+8c^2u_{1}^4$$" ], "text/plain": [ "" @@ -1700,7 +1494,7 @@ { "data": { "text/latex": [ - "$$32\\sqrt{2}a^2b^3c \\le 16\\sqrt{2}a^3b^3+16\\sqrt{2}ab^3c^2$$" + "$$32\\sqrt{2}cu_{1}^3v_{1}^2 \\le 16\\sqrt{2}u_{1}^3v_{1}^3+16\\sqrt{2}c^2u_{1}^3v_{1}$$" ], "text/plain": [ "" @@ -1712,7 +1506,7 @@ { "data": { "text/latex": [ - "$$48\\sqrt{2}a^2b^2c^2 \\le 24\\sqrt{2}a^3b^2c+24\\sqrt{2}ab^2c^3$$" + "$$32c^2u_{1}^3v_{1} \\le 16cu_{1}^3v_{1}^2+16c^3u_{1}^3$$" ], "text/plain": [ "" @@ -1724,7 +1518,7 @@ { "data": { "text/latex": [ - "$$16\\sqrt{2}a^2bc^3 \\le 8\\sqrt{2}a^3bc^2+8\\sqrt{2}abc^4$$" + "$$32cu_{1}^2v_{1}^3 \\le 16cu_{1}^3v_{1}^2+16cu_{1}v_{1}^4$$" ], "text/plain": [ "" @@ -1736,7 +1530,7 @@ { "data": { "text/latex": [ - "$$16ab^4c \\le 8a^2b^4+8b^4c^2$$" + "$$48\\sqrt{2}c^2u_{1}^2v_{1}^2 \\le 24\\sqrt{2}cu_{1}^2v_{1}^3+24\\sqrt{2}c^3u_{1}^2v_{1}$$" ], "text/plain": [ "" @@ -1748,7 +1542,7 @@ { "data": { "text/latex": [ - "$$32ab^3c^2 \\le 16a^2b^3c+16b^3c^3$$" + "$$16c^3u_{1}^2v_{1} \\le 8c^2u_{1}^2v_{1}^2+8c^4u_{1}^2$$" ], "text/plain": [ "" @@ -1760,7 +1554,7 @@ { "data": { "text/latex": [ - "$$32a^3bc^2 \\le 16a^4bc+16a^2bc^3$$" + "$$32c^2u_{1}v_{1}^3 \\le 16u_{1}^2v_{1}^4+16c^4v_{1}^2$$" ], "text/plain": [ "" @@ -1772,7 +1566,7 @@ { "data": { "text/latex": [ - "$$16ab^2c^3 \\le 8a^2b^2c^2+8b^2c^4$$" + "$$16\\sqrt{2}c^3u_{1}v_{1}^2 \\le 8\\sqrt{2}c^2u_{1}v_{1}^3+8\\sqrt{2}c^4u_{1}v_{1}$$" ], "text/plain": [ "" @@ -1784,7 +1578,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 16a^4bc+32a^4c^2+24\\sqrt{2}a^3b^2c+72\\sqrt{2}a^3bc^2+32\\sqrt{2}a^3c^3+16a^2b^3c+72a^2b^2c^2+64a^2bc^3+16a^2c^4 $$" + "$$ 0 \\le 24\\sqrt{2}cu_{1}^2v_{1}^3+88c^2u_{1}^2v_{1}^2+16cu_{1}v_{1}^4+72\\sqrt{2}c^2u_{1}v_{1}^3+80c^3u_{1}v_{1}^2+32c^2v_{1}^4+32\\sqrt{2}c^3v_{1}^3 $$" ], "text/plain": [ "" @@ -1807,11 +1601,14 @@ }, { "data": { + "text/latex": [ + "$\\displaystyle - \\frac{2 \\sqrt{2} \\left(a - b\\right)}{c} + \\frac{\\left(a - b\\right)^{2}}{c^{2}} - \\frac{2 \\sqrt{2} \\left(- a + c\\right)}{b} + \\frac{\\left(- a + c\\right)^{2}}{b^{2}} - \\frac{2 \\sqrt{2} \\left(b - c\\right)}{a} + \\frac{\\left(b - c\\right)^{2}}{a^{2}}$" + ], "text/plain": [ - "0" + "-2*sqrt(2)*(a - b)/c + (a - b)**2/c**2 - 2*sqrt(2)*(-a + c)/b + (-a + c)**2/b**2 - 2*sqrt(2)*(b - c)/a + (b - c)**2/a**2" ] }, - "execution_count": 111, + "execution_count": 28, "metadata": {}, "output_type": "execute_result" } @@ -1823,18 +1620,19 @@ "prove(formula1)\n", "display(Latex('Case $a\\ge b\\ge c$'))\n", "formula2=makesubs(formula,'[b,oo],[c,oo]')\n", - "prove(formula2*4,values='2**(1/2),1,1')" + "prove(formula2*4,values='1,sqrt(2),1')\n", + "formula" ] }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "Substitute $a\\to b+w_{1}$" ], "text/plain": [ "" @@ -1846,7 +1644,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "Substitute $b\\to c+x_{1}$" ], "text/plain": [ "" @@ -1870,7 +1668,7 @@ { "data": { "text/latex": [ - "numerator: $4a^4b^2+8a^4bc+8a^4c^2+4\\sqrt{2}a^3b^3-8a^3b^2c+12\\sqrt{2}a^3b^2c-8a^3bc^2+20\\sqrt{2}a^3bc^2+8\\sqrt{2}a^3c^3+2a^2b^4-8\\sqrt{2}a^2b^3c+8a^2b^3c-12\\sqrt{2}a^2b^2c^2+24a^2b^2c^2-4\\sqrt{2}a^2bc^3+20a^2bc^3+4a^2c^4-4ab^4c-8ab^3c^2+4\\sqrt{2}ab^3c^2-4ab^2c^3+6\\sqrt{2}ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4$" + "numerator: $2c^4w_{1}^2+2c^4w_{1}x_{1}+2c^4x_{1}^2+4c^3w_{1}^3-2\\sqrt{2}c^3w_{1}^2x_{1}+10c^3w_{1}^2x_{1}-2\\sqrt{2}c^3w_{1}x_{1}^2+6c^3w_{1}x_{1}^2+4c^3x_{1}^3+2c^2w_{1}^4-2\\sqrt{2}c^2w_{1}^3x_{1}+10c^2w_{1}^3x_{1}-6\\sqrt{2}c^2w_{1}^2x_{1}^2+12c^2w_{1}^2x_{1}^2-4\\sqrt{2}c^2w_{1}x_{1}^3+4c^2w_{1}x_{1}^3+2c^2x_{1}^4+2cw_{1}^4x_{1}-2\\sqrt{2}cw_{1}^3x_{1}^2+6cw_{1}^3x_{1}^2-4\\sqrt{2}cw_{1}^2x_{1}^3+4cw_{1}^2x_{1}^3-2\\sqrt{2}cw_{1}x_{1}^4+w_{1}^4x_{1}^2+2w_{1}^3x_{1}^3+w_{1}^2x_{1}^4$" ], "text/plain": [ "" @@ -1894,7 +1692,7 @@ { "data": { "text/latex": [ - "status: 0" + "status: 2" ], "text/plain": [ "" @@ -1906,7 +1704,7 @@ { "data": { "text/latex": [ - "From weighted AM-GM inequality:" + "Program couldn't find any proof." ], "text/plain": [ "" @@ -1918,115 +1716,7 @@ { "data": { "text/latex": [ - "$$8a^3b^2c \\le 4a^4b^2+4a^2b^2c^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$8\\sqrt{2}a^2b^3c \\le 4\\sqrt{2}a^3b^3+4\\sqrt{2}ab^3c^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$12\\sqrt{2}a^2b^2c^2 \\le 6\\sqrt{2}a^3b^2c+6\\sqrt{2}ab^2c^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$4\\sqrt{2}a^2bc^3 \\le 2\\sqrt{2}a^3bc^2+2\\sqrt{2}abc^4$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$4ab^4c \\le 2a^2b^4+2b^4c^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$8ab^3c^2 \\le 4a^2b^3c+4b^3c^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$8a^3bc^2 \\le 4a^4bc+4a^2bc^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$4ab^2c^3 \\le 2a^2b^2c^2+2b^2c^4$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le 4a^4bc+8a^4c^2+6\\sqrt{2}a^3b^2c+18\\sqrt{2}a^3bc^2+8\\sqrt{2}a^3c^3+4a^2b^3c+18a^2b^2c^2+16a^2bc^3+4a^2c^4 $$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." + "$$ 2\\sqrt{2}cw_{1}x_{1}^4+4\\sqrt{2}cw_{1}^2x_{1}^3+4\\sqrt{2}c^2w_{1}x_{1}^3+2\\sqrt{2}cw_{1}^3x_{1}^2+6\\sqrt{2}c^2w_{1}^2x_{1}^2+2\\sqrt{2}c^3w_{1}x_{1}^2+2\\sqrt{2}c^2w_{1}^3x_{1}+2\\sqrt{2}c^3w_{1}^2x_{1} \\le w_{1}^2x_{1}^4+2c^2x_{1}^4+2w_{1}^3x_{1}^3+4cw_{1}^2x_{1}^3+4c^2w_{1}x_{1}^3+4c^3x_{1}^3+w_{1}^4x_{1}^2+6cw_{1}^3x_{1}^2+12c^2w_{1}^2x_{1}^2+6c^3w_{1}x_{1}^2+2c^4x_{1}^2+2cw_{1}^4x_{1}+10c^2w_{1}^3x_{1}+10c^3w_{1}^2x_{1}+2c^4w_{1}x_{1}+2c^2w_{1}^4+4c^3w_{1}^3+2c^4w_{1}^2 $$" ], "text/plain": [ "" @@ -2038,10 +1728,10 @@ { "data": { "text/plain": [ - "0" + "2" ] }, - "execution_count": 27, + "execution_count": 29, "metadata": {}, "output_type": "execute_result" } @@ -2057,7 +1747,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 30, "metadata": {}, "outputs": [ { @@ -2066,7 +1756,7 @@ "(1, (x, y, sqrt(2)))" ] }, - "execution_count": 28, + "execution_count": 30, "metadata": {}, "output_type": "execute_result" } @@ -2077,7 +1767,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 31, "metadata": {}, "outputs": [ { @@ -2086,7 +1776,7 @@ "(1, (x, y, sqrt(2)))" ] }, - "execution_count": 29, + "execution_count": 31, "metadata": {}, "output_type": "execute_result" } @@ -2097,7 +1787,7 @@ }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 32, "metadata": {}, "outputs": [ { @@ -2106,7 +1796,7 @@ "[(1, 1, 1), (0, 0, 0)]" ] }, - "execution_count": 30, + "execution_count": 32, "metadata": {}, "output_type": "execute_result" } @@ -2117,7 +1807,7 @@ }, { "cell_type": "code", - "execution_count": 31, + "execution_count": 33, "metadata": {}, "outputs": [ { @@ -2126,7 +1816,7 @@ "[1, 58]" ] }, - "execution_count": 31, + "execution_count": 33, "metadata": {}, "output_type": "execute_result" } @@ -2137,7 +1827,7 @@ }, { "cell_type": "code", - "execution_count": 32, + "execution_count": 34, "metadata": {}, "outputs": [ { @@ -2146,7 +1836,7 @@ "(x, y, sqrt(2), sqrt(3), sqrt(6))" ] }, - "execution_count": 32, + "execution_count": 34, "metadata": {}, "output_type": "execute_result" } @@ -2157,7 +1847,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 35, "metadata": {}, "outputs": [ { @@ -2166,7 +1856,7 @@ "([0], [(0, 0)], [1, 7], [(2, 0), (0, 1)], (x, y))" ] }, - "execution_count": 33, + "execution_count": 35, "metadata": {}, "output_type": "execute_result" } @@ -2188,7 +1878,7 @@ }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 36, "metadata": {}, "outputs": [ { @@ -2281,7 +1971,7 @@ "0" ] }, - "execution_count": 34, + "execution_count": 36, "metadata": {}, "output_type": "execute_result" } @@ -2292,7 +1982,7 @@ }, { "cell_type": "code", - "execution_count": 35, + "execution_count": 37, "metadata": {}, "outputs": [ { @@ -2301,7 +1991,7 @@ "[[(1, 0), (0, 2), (0, 1), (0, 0)], [1, 1, 1, 1]]" ] }, - "execution_count": 35, + "execution_count": 37, "metadata": {}, "output_type": "execute_result" } @@ -2312,7 +2002,7 @@ }, { "cell_type": "code", - "execution_count": 168, + "execution_count": 38, "metadata": {}, "outputs": [ { @@ -2324,7 +2014,7 @@ "Poly(x + (sqrt(x)), x, sqrt(x), domain='ZZ')" ] }, - "execution_count": 168, + "execution_count": 38, "metadata": {}, "output_type": "execute_result" } @@ -2336,7 +2026,7 @@ }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 39, "metadata": {}, "outputs": [], "source": [ @@ -2345,7 +2035,7 @@ }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 40, "metadata": {}, "outputs": [ { @@ -2357,7 +2047,7 @@ "Poly(x**2 + 1, x, domain='ZZ')" ] }, - "execution_count": 42, + "execution_count": 40, "metadata": {}, "output_type": "execute_result" } @@ -2368,7 +2058,7 @@ }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 41, "metadata": {}, "outputs": [ { @@ -2380,7 +2070,7 @@ "Poly(x*(1/(x + y)), x, 1/(x + y), domain='ZZ')" ] }, - "execution_count": 43, + "execution_count": 41, "metadata": {}, "output_type": "execute_result" } @@ -2391,7 +2081,7 @@ }, { "cell_type": "code", - "execution_count": 44, + "execution_count": 42, "metadata": {}, "outputs": [ { @@ -2403,7 +2093,7 @@ "Poly(x + (sqrt(x)) + (x**(1/3))**2 + (x**(1/3)) + (x**(1/4)) + sqrt(2), x, sqrt(x), x**(1/3), x**(1/4), domain='QQ')" ] }, - "execution_count": 44, + "execution_count": 42, "metadata": {}, "output_type": "execute_result" } @@ -2414,7 +2104,7 @@ }, { "cell_type": "code", - "execution_count": 45, + "execution_count": 43, "metadata": {}, "outputs": [ { @@ -2423,7 +2113,7 @@ "[1, sqrt(6)]" ] }, - "execution_count": 45, + "execution_count": 43, "metadata": {}, "output_type": "execute_result" } @@ -2434,7 +2124,7 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 44, "metadata": {}, "outputs": [ { @@ -2443,7 +2133,7 @@ "[1, x]" ] }, - "execution_count": 47, + "execution_count": 44, "metadata": {}, "output_type": "execute_result" } @@ -2454,9 +2144,22 @@ }, { "cell_type": "code", - "execution_count": 50, + "execution_count": 45, "metadata": {}, "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/decorators.py:38: SymPyDeprecationWarning: \n", + "\n", + "source has been deprecated since SymPy 1.3. Use ?? in IPython/Jupyter\n", + "or inspect.getsource instead. See\n", + "https://github.com/sympy/sympy/issues/14905 for more info.\n", + "\n", + " _warn_deprecation(wrapped, 3)\n" + ] + }, { "name": "stdout", "output_type": "stream", @@ -6569,19 +6272,6 @@ " return isinstance(g, f.__class__) and f.gens == g.gens and f.rep.eq(g.rep, strict=True)\n", "\n" ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/decorators.py:38: SymPyDeprecationWarning: \n", - "\n", - "source has been deprecated since SymPy 1.3. Use ?? in IPython/Jupyter\n", - "or inspect.getsource instead. See\n", - "https://github.com/sympy/sympy/issues/14905 for more info.\n", - "\n", - " _warn_deprecation(wrapped, 3)\n" - ] } ], "source": [ @@ -6590,7 +6280,7 @@ }, { "cell_type": "code", - "execution_count": 126, + "execution_count": 46, "metadata": {}, "outputs": [ { @@ -6602,7 +6292,7 @@ "Poly((sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')" ] }, - "execution_count": 126, + "execution_count": 46, "metadata": {}, "output_type": "execute_result" } @@ -6613,7 +6303,7 @@ }, { "cell_type": "code", - "execution_count": 115, + "execution_count": 47, "metadata": {}, "outputs": [ { @@ -6625,7 +6315,7 @@ "-x + sqrt(x**2)" ] }, - "execution_count": 115, + "execution_count": 47, "metadata": {}, "output_type": "execute_result" } @@ -6637,7 +6327,7 @@ }, { "cell_type": "code", - "execution_count": 130, + "execution_count": 48, "metadata": {}, "outputs": [ { @@ -6646,7 +6336,7 @@ "\"Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')\"" ] }, - "execution_count": 130, + "execution_count": 48, "metadata": {}, "output_type": "execute_result" } @@ -6657,7 +6347,7 @@ }, { "cell_type": "code", - "execution_count": 131, + "execution_count": 49, "metadata": {}, "outputs": [ { @@ -6666,7 +6356,7 @@ "\"Poly(x**2 + x + (sqrt(x)), x, sqrt(x), domain='ZZ')\"" ] }, - "execution_count": 131, + "execution_count": 49, "metadata": {}, "output_type": "execute_result" } @@ -6677,13 +6367,13 @@ }, { "cell_type": "code", - "execution_count": 164, + "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "Substitute $a\\to b+y_{1}$" ], "text/plain": [ "" @@ -6695,7 +6385,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "Substitute $b\\to c+z_{1}$" ], "text/plain": [ "" @@ -6717,7 +6407,7 @@ { "data": { "text/plain": [ - "(2.5326984818340415e-10, 7.129450063690368, 1.7908873553542452e-10)" + "(1.7908873553542452e-10, 2.5326984818340415e-10, 7.129450063690368)" ] }, "metadata": {}, @@ -6726,20 +6416,21 @@ { "data": { "text/plain": [ - "1.4142142855953455" + "2.5119572187973787e-11" ] }, - "execution_count": 164, + "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def findvalues(formula,values=None,variables=None):\n", + " formula=S(formula)\n", " num,den=fractioncancel(formula)\n", " if variables==None:\n", " variables=sorted(num.free_symbols,key=str)\n", - " num=num.subs(zip(variables,list(map(lambda x:x**2,fs))))\n", + " num=num.subs(zip(variables,list(map(lambda x:x**2,variables))))\n", " num=Poly(num)\n", " newformula=S((num.abs()+num)/(num.abs()-num))\n", " f=lambdify(variables,newformula)\n", @@ -6758,13 +6449,13 @@ }, { "cell_type": "code", - "execution_count": 156, + "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "Substitute $a\\to b+e_{2}$" ], "text/plain": [ "" @@ -6776,7 +6467,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "Substitute $b\\to c+f_{2}$" ], "text/plain": [ "" @@ -6790,29 +6481,19 @@ "output_type": "stream", "text": [ "ok\n", - "100 loops, best of 5: 9.06 ms per loop\n" + "100 loops, best of 5: 8.66 ms per loop\n", + "1000 loops, best of 5: 1.31 ms per loop\n" ] }, { - "ename": "TypeError", - "evalue": "int() argument must be a string, a bytes-like object or a number, not 'dict'", - "output_type": "error", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mTypeError\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 16\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'ok'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 17\u001b[0m \u001b[0mget_ipython\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmagic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'timeit evaluate((random(),random(),random()))'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 18\u001b[0;31m \u001b[0mget_ipython\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmagic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'timeit evaluate2((random(),random(),random()))'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 19\u001b[0m \u001b[0mPoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x^2+sqrt(x)'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meval\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/IPython/core/interactiveshell.py\u001b[0m in \u001b[0;36mmagic\u001b[0;34m(self, arg_s)\u001b[0m\n\u001b[1;32m 2158\u001b[0m \u001b[0mmagic_name\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_\u001b[0m\u001b[0;34m,\u001b[0m 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'chop': chop,\n\u001b[1;32m 1436\u001b[0m 'strict': strict, 'verbose': verbose}\n", - "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/mpmath/libmp/libmpf.py\u001b[0m in \u001b[0;36mdps_to_prec\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m 65\u001b[0m \"\"\"Return the number of bits required to represent n decimals\n\u001b[1;32m 66\u001b[0m accurately.\"\"\"\n\u001b[0;32m---> 67\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mmax\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mround\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;36m3.3219280948873626\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 68\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 69\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mrepr_dps\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\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;31mTypeError\u001b[0m: int() argument must be a string, a bytes-like object or a number, not 'dict'" - ] + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 54, + "metadata": {}, + "output_type": "execute_result" } ], "source": [ @@ -6829,7 +6510,7 @@ "def evaluate(x):\n", " return newformula.evalf(subs=dict(zip(fs,(1,1,1))))\n", "def evaluate2(x):\n", - " return num1.evalf(dict(zip(fs,x)))/num2.eval(dict(zip(fs,x)))\n", + " return num1.eval(dict(zip(fs,x)))/num2.eval(dict(zip(fs,x)))\n", "#display(num2.eval(dict(zip(fs,(1,1,1)))))\n", "print('ok')\n", "%timeit evaluate((random(),random(),random()))\n", @@ -6839,18 +6520,9 @@ }, { "cell_type": "code", - "execution_count": 162, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "100000 loops, best of 5: 6.56 µs per loop\n", - "100000 loops, best of 5: 6.39 µs per loop\n" - ] - } - ], + "outputs": [], "source": [ "evaluate4=lambdify(fs,newformula)\n", "evaluate5=lambda x:evaluate4(*x)\n", @@ -6860,19 +6532,9 @@ }, { "cell_type": "code", - "execution_count": 172, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "first attempt: Poly(x**2 + x + (sqrt(x)), x, sqrt(x), domain='ZZ')\n", - "second attempt: Poly(x**2 + x + (sqrt(x)), x, sqrt(x), domain='ZZ')\n", - "third attempt: Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')\n" - ] - } - ], + "outputs": [], "source": [ "from sympy import *\n", "print('first attempt:',Poly('x^2+x+sqrt(x)'))\n", @@ -6883,7 +6545,7 @@ }, { "cell_type": "code", - "execution_count": 176, + "execution_count": 52, "metadata": {}, "outputs": [ { @@ -6895,7 +6557,7 @@ "sqrt(x) + x" ] }, - "execution_count": 176, + "execution_count": 52, "metadata": {}, "output_type": "execute_result" } @@ -6906,7 +6568,7 @@ }, { "cell_type": "code", - "execution_count": 178, + "execution_count": 55, "metadata": {}, "outputs": [ { @@ -6915,7 +6577,7 @@ "(x*(1 - x), [])" ] }, - "execution_count": 178, + "execution_count": 55, "metadata": {}, "output_type": "execute_result" } @@ -6927,7 +6589,7 @@ }, { "cell_type": "code", - "execution_count": 206, + "execution_count": 56, "metadata": {}, "outputs": [ { @@ -6936,7 +6598,7 @@ "True" ] }, - "execution_count": 206, + "execution_count": 56, "metadata": {}, "output_type": "execute_result" } @@ -6948,7 +6610,7 @@ }, { "cell_type": "code", - "execution_count": 223, + "execution_count": 57, "metadata": {}, "outputs": [ { @@ -6957,7 +6619,7 @@ "(x, 1/2)" ] }, - "execution_count": 223, + "execution_count": 57, "metadata": {}, "output_type": "execute_result" } @@ -6968,21 +6630,16 @@ }, { "cell_type": "code", - "execution_count": 222, + "execution_count": 58, "metadata": {}, "outputs": [ { - "data": { - "text/latex": [ - "$\\displaystyle \\operatorname{Poly}{\\left( x^{2} + y^{2} + \\sqrt{x + y} + \\sqrt{x} + \\sqrt[3]{x_{12}}, x, y, \\sqrt{x + y}, \\sqrt{x}, \\sqrt[3]{x_{12}}, domain=\\mathbb{Z} \\right)}$" - ], - "text/plain": [ - "Poly(x**2 + y**2 + (sqrt(x + y)) + (sqrt(x)) + (x12**(1/3)), x, y, sqrt(x + y), sqrt(x), x12**(1/3), domain='ZZ')" - ] - }, - "execution_count": 222, - "metadata": {}, - "output_type": "execute_result" + "ename": "SyntaxError", + "evalue": "invalid syntax (, line 12)", + "output_type": "error", + "traceback": [ + "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m12\u001b[0m\n\u001b[0;31m newgens[gen[0]]=\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" + ] } ], "source": [ @@ -6992,11 +6649,11 @@ "\telse:\n", "\t\treturn [formula,S('1')]\n", "pol=Poly('x**2+y**2+sqrt(x+y)+x**(1/2)+x**(1/3)')\n", - "genes=pol.gens\n", "newgens={}\n", - "for gen in genes:\n", - " gen=_powr(gen):\n", - " if gen[0] in newgens:\n", + "for gen in pol.gens:\n", + " base,pw=_powr(gen)\n", + " num,den=fraction(pw)\n", + " if (gen[0],num) in newgens:\n", " newgens[gen[0]]=\n", " else:\n", " newgens[gen[0]]=gen[1]" @@ -7004,27 +6661,27 @@ }, { "cell_type": "code", - "execution_count": 227, + "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "sympy.core.add.Add" + "True" ] }, - "execution_count": 227, + "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "S('x+y*6').func==" + "isinstance(S('1'),Rational)" ] }, { "cell_type": "code", - "execution_count": 229, + "execution_count": 60, "metadata": {}, "outputs": [ { @@ -7033,7 +6690,7 @@ "False" ] }, - "execution_count": 229, + "execution_count": 60, "metadata": {}, "output_type": "execute_result" } @@ -7042,6 +6699,1925 @@ "6 in {3:4,5:6}" ] }, + { + "cell_type": "code", + "execution_count": 61, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle 1$" + ], + "text/plain": [ + "1" + ] + }, + "execution_count": 61, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from math import *\n", + "from sympy import *\n", + "gcd(S('a/b'),S('c/d'))" + ] + }, + { + "cell_type": "code", + "execution_count": 62, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x^{2 \\pi}, x^{\\pi}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly((x**pi)**2, x**pi, domain='ZZ')" + ] + }, + "execution_count": 62, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('x^(2*pi)')" + ] + }, + { + "cell_type": "code", + "execution_count": 63, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\frac{1}{8}$" + ], + "text/plain": [ + "1/8" + ] + }, + "execution_count": 63, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "gcd(S('1/8'),S('1/4'))" + ] + }, + { + "cell_type": "code", + "execution_count": 64, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(3, (x, sqrt(2)))" + ] + }, + "execution_count": 64, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "S('sqrt(2)*3*x').as_coeff_mul()" + ] + }, + { + "cell_type": "code", + "execution_count": 65, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')\n", + "Poly((x**(1/6))**12 + (x**(1/6))**3 + (x**(1/6))**2 + y**2, x**(1/6), y, domain='ZZ')\n", + "Poly((sqrt(x)) + (sqrt(x*z)) + (sqrt(z)), sqrt(x), sqrt(x*z), sqrt(z), domain='ZZ')\n", + "Poly((sqrt(x))*(sqrt(z)) + (sqrt(x)) + (sqrt(z)), sqrt(x), sqrt(z), domain='ZZ')\n", + "Poly((sqrt(t + z)) + (sqrt(t*x + t*y + x*z + y*z)) + (sqrt(x + y)), sqrt(t + z), sqrt(t*x + t*y + x*z + y*z), sqrt(x + y), domain='ZZ')\n", + "Poly((sqrt(2)) + (sqrt(3)) + (sqrt(6)), sqrt(2), sqrt(3), sqrt(6), domain='ZZ')\n" + ] + } + ], + "source": [ + "def _powr(formula):\n", + "\tif formula.func==Pow:\n", + "\t\treturn formula.args\n", + "\telse:\n", + "\t\treturn [formula,S('1')]\n", + "def reducegens(formula):\n", + "\tpol=Poly(formula)\n", + "\tnewgens={}\n", + "\tind={}\n", + "\tfor gen in pol.gens:\n", + "\t\tbase,pw=_powr(gen)\n", + "\t\tcoef,_=pw.as_coeff_mul()\n", + "\t\tml=pw/coef\n", + "\t\tif base**ml in newgens:\n", + "\t\t\tnewgens[base**ml]=gcd(newgens[base**ml],coef)\n", + "\t\telse:\n", + "\t\t\tnewgens[base**ml]=coef\n", + "\t\t\tind[base**ml]=S('tmp'+str(len(ind)))\n", + "\tfor gen in pol.gens:\n", + "\t\tbase,pw=_powr(gen)\n", + "\t\tcoef,_=pw.as_coeff_mul()\n", + "\t\tml=pw/coef\n", + "\t\tpol=pol.replace(gen,ind[base**ml]**(coef/newgens[base**ml]))\n", + "\tnewpol=Poly(pol.as_expr())\n", + "\tfor gen in newgens:\n", + "\t\tnewpol=newpol.replace(ind[gen],gen**newgens[gen])\n", + "\treturn newpol\n", + "print(reducegens(S('x^2+x+sqrt(x)')))\n", + "print(reducegens(S('x**2+y**2+x**(1/2)+x**(1/3)')))\n", + "print(reducegens(S('sqrt(x)+sqrt(z)+sqrt(x*z)')))\n", + "print(reducegens(S('sqrt(x)+sqrt(z)+sqrt(x)*sqrt(z)')))\n", + "print(reducegens(S('sqrt(x+y)+sqrt(z+t)+sqrt((x+y)*(z+t))')))\n", + "print(reducegens(S('sqrt(2)+sqrt(3)+sqrt(6)')))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "S('-2').as_coeff_mul()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "x=Symbol('x', positive=True)\n", + "y=Symbol('y', positive=True)\n", + "Poly(sqrt(x)+sqrt(y)+sqrt(x*y))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "sqrt(x*y)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "simplify(S('sqrt(x*y)-sqrt(x)*sqrt(y)'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "x=symbols('x',positive=True)\n", + "ask(Q.positive(x))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "Poly('x^2+x')+Poly('-x+7')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "print(reducegens(S('sqrt(8)')))\n", + "sqrt(-1*-1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "Poly('sqrt(2*x)+sqrt(3*x)+sqrt(6*x)')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "Poly('sqrt(2*pi)+sqrt(3*pi)')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "solve(S('sqrt(6*x)')-S('sqrt(2*y)'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "Poly('sqrt(6)+sqrt(10)+sqrt(14)+sqrt(15)+sqrt(21)+sqrt(35)+sqrt()')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from shiroindev import *\n", + "formula=S('(a+b-c)^2')\n", + "prove(makesubs(formula,'[c,oo]'))\n", + "prove(makesubs(formula,'[a,oo]',variables='c'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "formula=S('(a+b-d)^2+(a+b-c)^2')\n", + "prove(makesubs(formula,'[a+b,oo],[a+b,oo]',variables='c,d'))\n", + "prove(makesubs(formula,'[0,a+b],[a+b,oo]',variables='c,d'))\n", + "prove(makesubs(formula,'[0,a+b],[0,a+b]',variables='c,d'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "findvalues(formula)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "display(cyclize('((a-b)/c)^2'))\n", + "display(S('sqrt(8)')*cyclize('(a-b)/c'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "formula=cyclize('((a-b)/c)^2-8**(1/2)*(a-b)/c')\n", + "display(Latex('Case $a\\ge c\\ge b$'))\n", + "formula1=makesubs(formula,'[c,oo],[b,oo]',variables='a,c,b')\n", + "prove(formula1)\n", + "display(Latex('Case $a\\ge b\\ge c$'))\n", + "formula2=makesubs(formula,'[b,oo],[c,oo]')\n", + "prove(formula2*4,values='2**(1/2),1,1')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "formula=cyclize('((a-b)/c)^2-8**(1/2)*(a-b)/c')\n", + "formula2=makesubs(formula,'[c,oo],[c,oo]')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "findvalues(formula2,values=[5,2,7])" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "x,y=symbols('x,y')\n", + "ask(Q.positive(x**2+1),Q.real(x))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "formula=Sm('(a^2+b^2+c^2)^2- 3(a^3b+b^3c+c^3a)')\n", + "formula1=makesubs(formula,'[b,oo],[c,oo]',variables='a,b')\n", + "formula2=makesubs(formula,'[c,oo],[b,oo]',variables='a,c')\n", + "prove(formula2)\n", + "print(findvalues(formula1))\n", + "#prove(formula1,values='4*b*(1-b)-1,4*c*(1-c)-1,1')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "makesubs(Sm('a^2-2ab+bc-c^2+ca'),'[c,oo],[a,oo]',variables='b,c')" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle - \\sqrt{x} \\sqrt{y} + \\sqrt{x y}$" + ], + "text/plain": [ + "-sqrt(x)*sqrt(y) + sqrt(x*y)" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle 0$" + ], + "text/plain": [ + "0" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "\n", + "formula=S('sqrt(x*y)-sqrt(x)*sqrt(y)')\n", + "display(formula)\n", + "def assumeall(formula,**kwargs):\n", + " formula=S(formula)\n", + " fs=formula.free_symbols\n", + " for x in fs:\n", + " y=Symbol(str(x),**kwargs)\n", + " formula=formula.subs(x,y)\n", + " return formula\n", + "display(assumeall(formula,positive=True))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "Poly(assumeall('x+sqrt(x)',positive=True))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "str(type(S('f1(x,y,z)')))=='f1'" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "ReprPrinter?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "srepr(S('x^2+f(x)+f(x,y)+f()+sqrt(y)'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy import Function,srepr,S\n", + "import re\n", + "f=Function('f')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def allsymbols(formula):\n", + " formula=S(formula)\n", + " funcsymbols=[x[10:-2] for x in re.findall(r\"Function\\(\\'.*?\\'\\)\",srepr(formula))]\n", + " return set(funcsymbols)|set(map(str,formula.free_symbols))\n", + "def vargen(n):\n", + " x='abcdefghijklmnopqrstuvwxyz'[n%26]\n", + " if n>=26:\n", + " x+=str(n//26)\n", + " return x\n", + "allsymbols(S('f(5,7)+f(5)+g(8)+sqrt(x*y)'))\n", + "vargen(36)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy import *\n", + "from sympy.printing.repr import *\n", + "ReprPrinter??" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "{5,7} | {8,7}" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "[x[10:-2] for x in re.findall(r\"Function\\(\\'.*?\\'\\)\",srepr(S('x^2')))]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "S('f(5,7)+f(5)+g(8)+sqrt(x*y)')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "srepr(S('f(5,7)+f(5)+g(8)+sqrt(x*y)'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "display(Latex('$'+str(S('z12345'))+'$'))" + ] + }, + { + "cell_type": "code", + "execution_count": 68, + "metadata": {}, + "outputs": [ + { + "ename": "ValueError", + "evalue": "invalid literal for int() with base 10: ''", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mValueError\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 1\u001b[0m \u001b[0;34m'xyz'\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m2\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[0;32m----> 2\u001b[0;31m \u001b[0mint\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[0;31mValueError\u001b[0m: invalid literal for int() with base 10: ''" + ] + } + ], + "source": [ + "'xyz'[-2::]" + ] + }, + { + "cell_type": "code", + "execution_count": 84, + "metadata": {}, + "outputs": [], + "source": [ + "def sortkey(x):\n", + " x=str(x)\n", + " i=len(x)-1\n", + " while i>=0:\n", + " if x[i] not in '0123456789':\n", + " break\n", + " i-=1\n", + " if i==len(x)-1:\n", + " return (-1,x)\n", + " return (int(x[i+1:]),x[:i+1])" + ] + }, + { + "cell_type": "code", + "execution_count": 85, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[x, y, x0, qq1, z1, t2]" + ] + }, + "execution_count": 85, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sorted(S('x,y,z1,t2,qq1,x0'),key=sortkey)" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Optimization terminated successfully.\n", + " Current function value: 1.154701\n", + " Iterations: 85\n", + " Function evaluations: 152\n" + ] + }, + { + "data": { + "text/plain": [ + "[1.0,\n", + " 3.000209557356011,\n", + " 3.9998018341331227,\n", + " 2.9997236704279775,\n", + " 0.999903278712487]" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2')\n", + "formula=S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2- (x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')\n", + "values=findvalues(formula)\n", + "values=[value/values[0] for value in values]\n", + "[value**2 for value in values]" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\frac{2 \\sqrt{3}}{3}$" + ], + "text/plain": [ + "2*sqrt(3)/3" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "formula1=S('(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2)/(x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')\n", + "formula1.subs(S('[[x_1,1],[x_2,sqrt(3)],[x_3,2],[x_4,sqrt(3)],[x_5,1]]'))" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "formula2=S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2- (2/sqrt(3))*(x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $x_{2}\\to \\sqrt{3} g$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $x_{3}\\to 2 h$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $x_{4}\\to \\sqrt{3} i$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $3 g^{2} - 4 g h - 2 g x_{1} + 4 h^{2} - 4 h i + 3 i^{2} - 2 i x_{5} + x_{1}^{2} + x_{5}^{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "From weighted AM-GM inequality:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$4 g h \\le 2 g^{2}+2 h^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 g x_{1} \\le g^{2}+x_{1}^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$4 h i \\le 2 h^{2}+2 i^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 i x_{5} \\le i^{2}+x_{5}^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 0 $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "prove(formula2,values='1,sqrt(3),2,sqrt(3),1')" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{k}a^{n}, a^{k}, a^{n}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly((a**k)*(a**n), a**k, a**n, domain='ZZ')" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('a^(n+k)')" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "ename": "AttributeError", + "evalue": "'FiniteSet' object has no attribute 'as_coeff_Mul'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'a^{n+k}/{b^n}+{b^{n+k}}/{c^n}+{c^{n+k}}/{a^n}-( a^k+b^k+c^k)'\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/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/sympify.py\u001b[0m in \u001b[0;36msympify\u001b[0;34m(a, locals, convert_xor, strict, rational, evaluate)\u001b[0m\n\u001b[1;32m 382\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 383\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'\\n'\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[0;32m--> 384\u001b[0;31m \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mparse_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlocal_dict\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mlocals\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtransformations\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtransformations\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mevaluate\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 385\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mTokenError\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mSyntaxError\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mexc\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 386\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mSympifyError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'could not parse %r'\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexc\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/parsing/sympy_parser.py\u001b[0m in \u001b[0;36mparse_expr\u001b[0;34m(s, local_dict, transformations, global_dict, evaluate)\u001b[0m\n\u001b[1;32m 964\u001b[0m \u001b[0mcode\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcompile\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mevaluateFalse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcode\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'eval'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 965\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 966\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0meval_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcode\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlocal_dict\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mglobal_dict\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 967\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 968\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/parsing/sympy_parser.py\u001b[0m in \u001b[0;36meval_expr\u001b[0;34m(code, local_dict, global_dict)\u001b[0m\n\u001b[1;32m 877\u001b[0m \"\"\"\n\u001b[1;32m 878\u001b[0m expr = eval(\n\u001b[0;32m--> 879\u001b[0;31m code, global_dict, local_dict) # take local objects in preference\n\u001b[0m\u001b[1;32m 880\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 881\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n", + "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m__pow__\u001b[0;34m(self, other, mod)\u001b[0m\n\u001b[1;32m 163\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__pow__\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[0mmod\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\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 164\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mmod\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 165\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_pow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\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 166\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 167\u001b[0m \u001b[0m_self\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmod\u001b[0m \u001b[0;34m=\u001b[0m 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\u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mstrict\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 91\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\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 92\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0mSympifyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 93\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mretval\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/decorators.py\u001b[0m in \u001b[0;36mbinary_op_wrapper\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 127\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mf\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 128\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 129\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mfunc\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;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 130\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mbinary_op_wrapper\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 131\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mpriority_decorator\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/expr.py\u001b[0m in \u001b[0;36m_pow\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 159\u001b[0m \u001b[0;34m@\u001b[0m\u001b[0mcall_highest_priority\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'__rpow__'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 160\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_pow\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;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 161\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mPow\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;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 162\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 163\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__pow__\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[0mmod\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\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/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/cache.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 92\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\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 93\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 94\u001b[0;31m \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\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 95\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 96\u001b[0m \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\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/power.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, b, e, evaluate)\u001b[0m\n\u001b[1;32m 274\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Atom\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mExp1\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexp_polar\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 275\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mnumer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdenom\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlog\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mim\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfactor_terms\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 276\u001b[0;31m \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mex\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfactor_terms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_coeff_Mul\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 277\u001b[0m \u001b[0mden\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdenom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 278\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mden\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlog\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mden\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mAttributeError\u001b[0m: 'FiniteSet' object has no attribute 'as_coeff_Mul'" + ] + } + ], + "source": [ + "S('a^{n+k}/{b^n}+{b^(n+k)/(c^n)+c^(n+k)/{a^n}-( a^k+b^k+c^k)')" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'767980716'" + ] + }, + "execution_count": 25, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "import re\n", + "s=r\"\"\"99-my-name-is-John-Smith-6376827-%^-1-2-767980716\"\"\"\n", + "re.compile(r\"^(.*?)-\").search(str(s[::-1])).group(1)[::-1]" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n", + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n" + ] + }, + { + "data": { + "text/latex": [ + "numerator: $a^{k} a^{2 n} c^{n} - a^{k} a^{n} b^{n} c^{n} + a^{n} b^{k} b^{2 n} - a^{n} b^{k} b^{n} c^{n} - a^{n} b^{n} c^{k} c^{n} + b^{n} c^{k} c^{2 n}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 2" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Program couldn't find any proof." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ a^{k} a^{n} b^{n} c^{n}+a^{n} b^{k} b^{n} c^{n}+a^{n} b^{n} c^{k} c^{n} \\le a^{k} a^{2 n} c^{n}+a^{n} b^{k} b^{2 n}+b^{n} c^{k} c^{2 n} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $n\\to d + k$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $a^{2 d} a^{3 k} c^{d} c^{k} - a^{d} a^{2 k} b^{d} b^{k} c^{d} c^{k} + a^{d} a^{k} b^{2 d} b^{3 k} - a^{d} a^{k} b^{d} b^{2 k} c^{d} c^{k} - a^{d} a^{k} b^{d} b^{k} c^{d} c^{2 k} + b^{d} b^{k} c^{2 d} c^{3 k}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 2" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Program couldn't find any proof." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ a^{d} a^{2 k} b^{d} b^{k} c^{d} c^{k}+a^{d} a^{k} b^{d} b^{2 k} c^{d} c^{k}+a^{d} a^{k} b^{d} b^{k} c^{d} c^{2 k} \\le a^{2 d} a^{3 k} c^{d} c^{k}+a^{d} a^{k} b^{2 d} b^{3 k}+b^{d} b^{k} c^{2 d} c^{3 k} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from sympy import *\n", + "from sympy.parsing.latex import parse_latex\n", + "from shiroindev import *\n", + "shiro.seed=1\n", + "from IPython.display import Latex\n", + "shiro.display=lambda x:display(Latex(x))\n", + "formula=expand(parse_latex(r'(\\frac{a^{n+k}}{b^n}+\\frac{b^{n+k}}{c^n}+\\frac{c^{n+k}}{a^n}-( a^k+b^k+c^k))*a^nb^nc^n'))\n", + "newproof()\n", + "prove(formula)\n", + "prove(makesubs(formula,['k',oo],variables='n'))" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 6 x^{3}y^{9} + 5 x^{2}y^{7} + 4 xy^{8}, x, y, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(6*x**3*y**9 + 5*x**2*y**7 + 4*x*y**8, x, y, domain='ZZ')" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly(dict(zip([(1,8),(2,7),(3,9)],[4,5,6])),gens=S('x,y'))" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sqrt{x} + x^{2} + x$" + ], + "text/plain": [ + "sqrt(x) + x**2 + x" + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly({1:1,2:1,4:1},gens=S('sqrt(x)')).as_expr()" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [], + "source": [ + "def _writ2(coef,fun,variables):\n", + "\treturn latex(Poly(dict(zip(fun,coef)),gens=variables).as_expr())" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'8 x^{3} y^{4} + 7 x y^{2}'" + ] + }, + "execution_count": 32, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "_writ2([7,8],[(1,2),(3,4)],S('x,y'))" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n", + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle 2 a b$" + ], + "text/plain": [ + "2*(a*b)" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "parse_latex(r' a^2+b^2\\geq 2ab').rhs" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n", + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle a^{2} + b^{2}$" + ], + "text/plain": [ + "a**2 + b**2" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "parse_latex(r'2ab \\geq a^2+b^2').rhs" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "numerator: $a d + a f + b c + b e + c f + d e - 2 \\sqrt{a b} \\sqrt{c d} - 2 \\sqrt{a b} \\sqrt{e f} - 2 \\sqrt{c d} \\sqrt{e f}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "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')\n", + "Poly((sqrt(a))**2*(sqrt(d))**2 + (sqrt(a))**2*(sqrt(f))**2 + (sqrt(b))**2*(sqrt(c))**2 + (sqrt(b))**2*(sqrt(e))**2 + (sqrt(c))**2*(sqrt(f))**2 + (sqrt(d))**2*(sqrt(e))**2, sqrt(a), sqrt(b), sqrt(c), sqrt(d), sqrt(e), sqrt(f), domain='QQ')\n", + "Poly(2*(sqrt(a))*(sqrt(b))*(sqrt(c))*(sqrt(d)) + 2*(sqrt(a))*(sqrt(b))*(sqrt(e))*(sqrt(f)) + 2*(sqrt(c))*(sqrt(d))*(sqrt(e))*(sqrt(f)), sqrt(a), sqrt(b), sqrt(c), sqrt(d), sqrt(e), sqrt(f), domain='QQ')\n" + ] + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "From weighted AM-GM inequality:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 \\sqrt{a} \\sqrt{b} \\sqrt{c} \\sqrt{d} \\le a d+b c$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 \\sqrt{a} \\sqrt{b} \\sqrt{e} \\sqrt{f} \\le a f+b e$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 \\sqrt{c} \\sqrt{d} \\sqrt{e} \\sqrt{f} \\le c f+d e$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 0 $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 45, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "fr=parse_latex(r'\\sqrt{ab}+\\sqrt{cd}+\\sqrt{ef}\\leq\\sqrt{(a+c+e)(b+d+f)}')\n", + "prove(fr.rhs**2-fr.lhs**2)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle 0$" + ], + "text/plain": [ + "0" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "assumeall(S('sqrt(x*y)-sqrt(x)*sqrt(y)'),positive=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "ename": "GeneratorsError", + "evalue": "duplicated generators: [sqrt(x), sqrt(x)]", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mGeneratorsError\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 1\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mreducegens\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x+sqrt(x)'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0mq\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x+sqrt(x)'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mPoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'sqrt(x)'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'sqrt(x)'\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[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 4\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mPoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'sqrt(x)'\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[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, rep, *gens, **args)\u001b[0m\n\u001b[1;32m 107\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0margs\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 108\u001b[0m \u001b[0;34m\"\"\"Create a new polynomial instance out of something useful. \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 109\u001b[0;31m \u001b[0mopt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbuild_options\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\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 110\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 111\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;34m'order'\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py\u001b[0m in \u001b[0;36mbuild_options\u001b[0;34m(gens, args)\u001b[0m\n\u001b[1;32m 729\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 730\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0;36m1\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;34m'opt'\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margs\u001b[0m \u001b[0;32mor\u001b[0m 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\u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpreprocess\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 153\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 154\u001b[0;31m \u001b[0mpreprocess_options\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\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 155\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 156\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mvalue\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mdict\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdefaults\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mitems\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[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py\u001b[0m in \u001b[0;36mpreprocess_options\u001b[0;34m(args)\u001b[0m\n\u001b[1;32m 150\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 151\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mvalue\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 152\u001b[0;31m \u001b[0mself\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0moption\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpreprocess\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\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 153\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 154\u001b[0m \u001b[0mpreprocess_options\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py\u001b[0m in \u001b[0;36mpreprocess\u001b[0;34m(cls, gens)\u001b[0m\n\u001b[1;32m 289\u001b[0m \u001b[0mgens\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 290\u001b[0m \u001b[0;32melif\u001b[0m \u001b[0mhas_dups\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\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--> 291\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mGeneratorsError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"duplicated generators: %s\"\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\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 292\u001b[0m \u001b[0;32melif\u001b[0m \u001b[0many\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgen\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_commutative\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mFalse\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mgen\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mgens\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 293\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mGeneratorsError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"non-commutative generators: %s\"\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\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;31mGeneratorsError\u001b[0m: duplicated generators: [sqrt(x), sqrt(x)]" + ] + } + ], + "source": [ + "p=reducegens('x+sqrt(x)')\n", + "q=S('x+sqrt(x)')\n", + "print(Poly(p,gens=[S('sqrt(x)'),S('sqrt(x)')]))\n", + "print(Poly(q,gens=S('sqrt(x)')))" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": {}, + "outputs": [], + "source": [ + "from importlib import reload\n", + "from sympy import *\n", + "import shiroindev\n", + "reload(shiroindev)\n", + "from shiroindev import *\n", + "from sympy.parsing.latex import parse_latex\n", + "from itertools import permutations, combinations\n", + "shiro.seed=1\n", + "from IPython.display import Latex\n", + "shiro.display=lambda x:display(Latex(x))" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x, \\sqrt{x}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly((sqrt(x))**2, sqrt(x), domain='ZZ')" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('sqrt(x)')*Poly('sqrt(x)')" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 0, x, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(0, x, domain='ZZ')" + ] + }, + "execution_count": 48, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('x')-Poly('x').abs()" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x, x, y, domain=\\mathbb{Q} \\right)}$" + ], + "text/plain": [ + "Poly(x, x, y, domain='QQ')" + ] + }, + "execution_count": 49, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "fr=Poly('x-y')\n", + "(fr+fr.abs())*(S('1/2'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from itertools import permutations \n", + "t5=[]\n", + "for ineq in ineqs2:\n", + " u=0\n", + " for vars in permutations(ineq.free_symbols):\n", + " ineqp=makesubs(ineq,list(zip(vars[1:],[oo]*len(vars[1:]))),variables=vars[:-1])\n", + " u=max(u,prove(ineqp))\n", + " print(u,end=',')\n", + " t5+=[u]\n", + "Counter(t5)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "pvalue 4.43444926375551e-06\n", + "statistic 59.0\n" + ] + } + ], + "source": [ + "from statsmodels.stats.contingency_tables import mcnemar\n", + "print(mcnemar([[101,121],[59,33]]))" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [], + "source": [ + "nsimplify?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "shiro.display=lambda x:display(Latex(x))\n", + "newproof()\n", + "fr=makesubs(ineqs2[15],'[c,oo],[a,oo]',variables='b,c')\n", + "display(fr)\n", + "fr2=fractioncancel(fr.subs(S('[[e,(sqrt(5)-1)/2*f]]')))[0]\n", + "display(fr2)\n", + "prove(makesubs(fr2,'11/5,9/4',variables=[S('sqrt(5)')]))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def makesubs(formula,intervals,values=None,variables=None,numden=False):\n", + "#This function generates a new formula which satisfies this condition:\n", + "#for all positive variables new formula is nonnegative iff\n", + "#for all variables in corresponding intervals old formula is nonnegative\n", + "\tformula=S(formula)\n", + "\taddsymbols(formula)\n", + "\tintervals=_smakeiterable2(intervals)\n", + "\tif variables: variables=_smakeiterable(variables)\n", + "\telse: variables=sorted(formula.free_symbols,key=str)\n", + "\tif values!=None:\n", + "\t\tvalues=_smakeiterable(values)\n", + "\t\tequations=[var-value for var,value in zip(variables,values)]\n", + "\telse:\n", + "\t\tequations=[]\n", + "\tnewvars=[]\n", + "\tusedvars=set()\n", + "\tfor var,interval in zip(variables,intervals):\n", + "\t\tend1,end2=interval\n", + "\t\tz=newvar()\n", + "\t\tnewvars+=[z]\n", + "\t\tusedvars|={z}\n", + "\t\tif (end1.free_symbols|end2.free_symbols)&usedvars:\n", + "\t\t\tshiro.warning(shiro.translation[\n", + "\t\t\t'Warning: intervals contain backwards dependencies. Consider changing order of variables'])\n", + "\t\tif end1 in {S('-oo'),S('oo')}:\n", + "\t\t\tend1,end2=end2,end1\n", + "\t\tif {end1,end2}=={S('-oo'),S('oo')}:\n", + "\t\t\tsub1=sub2=(z-1/z)\n", + "\t\telif end2==S('oo'):\n", + "\t\t\tsub1=sub2=(end1+z)\n", + "\t\telif end2==S('-oo'):\n", + "\t\t\tsub1=sub2=end1-z\n", + "\t\telse:\n", + "\t\t\tsub1=end2+(end1-end2)/z\n", + "\t\t\tsub2=end2+(end1-end2)/(1+z)\n", + "\t\tformula=formula.subs(var,sub1)\n", + "\t\tshiro.display(shiro.translation['Substitute']+\" $\"+latex(var)+'\\\\to '+latex(sub2)+'$')\n", + "\t\tequations=[equation.subs(var,sub1) for equation in equations]\n", + "\tnum,den=fractioncancel(formula)\n", + "\tfor var,interval in zip(newvars,intervals):\n", + "\t\tif {interval[0],interval[1]} & {S('oo'),S('-oo')}==set():\n", + "\t\t\tnum=num.subs(var,var+1)\n", + "\t\t\tden=den.subs(var,var+1)\n", + "\t\t\tequations=[equation.subs(var,var+1) for equation in equations]\n", + "\tif values:\n", + "\t\tvalues=ssolve(equations,newvars)\n", + "\t\tif len(values):\n", + "\t\t\tvalues=values[0]\n", + "\tnum,den=expand(num),expand(den)\n", + "\t#shiro.display(shiro.translation[\"Formula after substitution:\"],\"$$\",latex(num/den),'$$')\n", + "\tif values and numden:\n", + "\t\treturn num,den,values\n", + "\telif values:\n", + "\t\treturn num/den,values\n", + "\telif numden:\n", + "\t\treturn num,den\n", + "\telse:\n", + "\t\treturn num/den" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "{3, 5, 7, 8}" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "x={5,7,8}\n", + "x|={3}\n", + "x" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "{x}" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from shiroindev import *\n", + "S('x+7').free_symbols" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "False" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "bool(set())" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $x\\to - y + 1 + \\frac{y - 1}{a + 1}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $y\\to 1 - \\frac{1}{b + 1}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $b\\to \\frac{c}{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $a^{4} c^{2} + a^{3} c^{2} - 2 a^{3} c - 4 a^{2} c + 4 a^{2} + a c^{2} - 2 a c + c^{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $a^{3} c^{2} + 2 a^{3} c + 2 a^{2} c^{2} + 4 a^{2} c + a c^{2} + 2 a c$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "From weighted AM-GM inequality:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 a^{3} c \\le a^{4} c^{2}+a^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$4 a^{2} c \\le a^{3} c^{2}+2 a^{2}+a c^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 a c \\le a^{2}+c^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 0 $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from importlib import reload\n", + "newproof()\n", + "import shiroindev\n", + "reload(shiroindev)\n", + "from shiroindev import *\n", + "shiro.display=lambda x:display(Latex(x))\n", + "formula=Sm('xy/z+yz/x+zx/y-1').subs('z',S('1-x-y'))\n", + "newformula,values=makesubs(formula,'[0,1-y],[0,1]','1/3,1/3')\n", + "prove(newformula,values)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": {}, + "outputs": [], + "source": [ + "def findvalues(formula,values=None,variables=None,**kwargs):\n", + "\t\"\"\"finds a candidate for parameter \"values\" in \"prove\" function\n", + "\tblabla\"\"\"\n", + "\tformula=S(formula)\n", + "\taddsymbols(formula)\n", + "\tnum,den=fractioncancel(formula)\n", + "\tif variables==None:\n", + "\t\tvariables=sorted(formula.free_symbols,key=str)\n", + "\tnum=num.subs(zip(variables,list(map(lambda x:x**2,variables))))\n", + "\tnum=Poly(num)\n", + "\tnewformula=S((num.abs()+num)/(num.abs()-num))\n", + "\tf=lambdify(variables,newformula)\n", + "\tf2=lambda x:f(*x)\n", + "\tif values==None:\n", + "\t\tvalues=[1.0]*len(variables)\n", + "\telse:\n", + "\t\tvalues=S(values)\n", + "\ttup=tuple(fmin(f2,values,**kwargs))\n", + "\treturn tuple([x*x for x in tup])" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Optimization terminated successfully.\n", + " Current function value: 1.000000\n", + " Iterations: 51\n", + " Function evaluations: 109\n" + ] + }, + { + "data": { + "text/plain": [ + "(1.9999999989778676, 0.6689493176765962)" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "findvalues(Sm('(x^2-4x+4)/y'))" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [], + "source": [ + "findvalues?" + ] + }, { "cell_type": "code", "execution_count": null, diff --git a/.ipynb_checkpoints/statistics-checkpoint.ipynb b/.ipynb_checkpoints/statistics-checkpoint.ipynb new file mode 100644 index 0000000..ad6b447 --- /dev/null +++ b/.ipynb_checkpoints/statistics-checkpoint.ipynb @@ -0,0 +1,476 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this notebook several methods for proving inequalities have been compared.\n", + "\n", + "First of all we need a dataset. 35 inequalities were selected from https://www.imomath.com/index.php?options=592&lmm=0. Each inequality is represented by a tuple: inequality (in LaTeX), inequality constraints (for example: $a\\in [0,1]$, equality constraints (for example: $abc=1$) and a function which converts inequality to a formula which we want to prove it's nonnegativity. In most cases when this function is simply a difference between left and right-hand side, but sometimes it's better to use difference between squares of sides (to get rid of square roots). These tuples are used as parameters in `parser` function which converts them to equivalent inequalities with default constraints (i.e. all variables are positive).\n", + "\n", + "Some tuples have less than 4 elements. In this case `parser` use default arguments." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "from importlib import reload\n", + "from sympy import *\n", + "import shiroindev\n", + "from shiroindev import *\n", + "from sympy.parsing.latex import parse_latex\n", + "shiro.seed=1\n", + "from IPython.display import Latex\n", + "shiro.display=lambda x:display(Latex(x))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "`sympy` has a `parse_latex` function which converts LaTeX formula to a `sympy` one. Unfortunately it doesn't deal with missing braces in the way that LaTeX do. For example `\\frac12` generates $\\frac12$ which is the same as `\\frac{1}{2}`, but `parse_latex` accepts only the second version. So here there is a boring function which adds missing braces." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\\frac{ \\sqrt {3}}{2}\n", + "a^2+b^2+c^2\\geq ab+bc+ca\n" + ] + } + ], + "source": [ + "def addbraces(s):\n", + " arg={r'\\frac':2,r'\\sqrt':1}\n", + " s2=''\n", + " p=0\n", + " while 1:\n", + " m=re.search(r\"\\\\[a-zA-Z]+\", s[p:])\n", + " if not m:\n", + " break\n", + " s2+=s[p:p+m.end()]\n", + " #print('a',s[p:m.end()],p)\n", + " p+=m.end()\n", + " if m.group() in arg:\n", + " for i in range(arg[m.group()]):\n", + " sp=re.search('^ *',s[p:])\n", + " s2+=sp.group()\n", + " #print('b',sp.group(),p)\n", + " p+=sp.end()\n", + " if s[p]=='{':\n", + " cb=re.search(r'^\\{.*?\\}',s[p:])\n", + " ab=addbraces(cb.group())\n", + " s2+=ab\n", + " #print('c',ab,p)\n", + " p+=cb.end()\n", + " else:\n", + " s2+='{'+s[p]+'}'\n", + " #print('d','{'+s[p]+'}',p)\n", + " p+=1\n", + " #print('e',p)\n", + " s2+=s[p:]\n", + " return s2\n", + "print(addbraces(r'\\frac{ \\sqrt 3}2'))\n", + "print(addbraces(r'a^2+b^2+c^2\\geq ab+bc+ca'))" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "#(formula,intervals,subs,function)\n", + "dif=lambda b,s:b-s\n", + "dif2=lambda b,s:b*b-s*s\n", + "ineqs=[\n", + " (r'a^2+b^2+c^2\\geq ab+bc+ca',),\n", + " (r'a^2+b^2+c^2+d^2\\geq a(b+c+d)',),\n", + " (r'1\\leq\\frac{a^2b^2}{c^2}+\\frac{b^2c^2}{a^2}+\\frac{c^2a^2}{b^2}',\n", + " '[0,1-f],[0,1]','[a,sqrt(1-e-f)],[b,sqrt(e)],[c,sqrt(f)]',),\n", + " (r'\\frac1{1-x^2}+\\frac1{1-y^2}\\geq \\frac2{1-xy}','[0,1],[0,1]'),\n", + " (r'a^3+b^3\\geq a^2b+ab^2',),\n", + " (r'\\frac a{b+c}+\\frac b{c+a}+\\frac c{a+b}\\geq \\frac32',),\n", + " (r'2a^3+b^3\\geq 3a^2b',),\n", + " (r'a^3+b^3+c^3\\geq a^2b+b^2c+c^2a',),\n", + " (r'\\frac a{b+c}+\\frac b{c+d}+ \\frac c{d+a}+ \\frac d{a+b}\\geq 2',),\n", + " (r'\\frac{a^3}{a^2+ab+b^2}+ \\frac{b^3}{b^2+bc+c^2}+ \\frac{c^3}{c^2+ca+a^2} \\geq \\frac{a+b+c}3',),\n", + " (r'\\sqrt{ab}+\\sqrt{cd}+\\sqrt{ef}\\leq\\sqrt{(a+c+e)(b+d+f)}','','',dif2),\n", + " (r'\\frac{5a^3-ab^2}{a+b}+\\frac{5b^3-bc^2}{b+c}+\\frac{5c^3-ca^2}{c+a}\\geq 2(a^2+b^2+c^2)',),\n", + " #(r'\\frac{a^2}{(1+b)(1-c)}+\\frac{b^3}{(1+c)(1-d)}+\\frac{c^3}{(1+d)(1-a)}+\\frac{d^3}{(1+a)(1-b)}\\geq\\frac1{15}',\n", + " # '[0,1-c-d],[0,1-d],[0,1]','[a,1-b-c-d]'),\n", + " (r'\\frac{x^3}{(1+y)(1+z)}+\\frac{y^3}{(1+z)(1+x)}+\\frac{z^3}{(1+x)(1+y)}\\geq\\frac{3}{4}','','[z,1/(x*y)]'),\n", + " (r'\\frac ab+\\frac bc+\\frac ca\\geq \\frac{(a+b+c)^2}{ab+bc+ca}',),\n", + " (r'\\frac{a^2}b+\\frac{b^2}c+\\frac{c^2}a\\geq \\frac{a^2+b^2+c^2}{a+b+c}',),\n", + " (r'\\frac{a^2}b+\\frac{b^2}c+\\frac{c^2}a\\geq a+b+c+\\frac{4(a-b)^2}{a+b+c}',),\n", + " (r'\\frac1{a^3+b^3+abc}+ \\frac1{b^3+c^3+abc} +\\frac1{c^3+a^3+ abc} \\leq \\frac1{abc}',),\n", + " (r'\\frac1{a^3(b+c)}+ \\frac1{b^3(c+a)}+ \\frac1{c^3(a+b)} \\geq \\frac32','','[c,1/(a*b)]'),\n", + " (r'\\frac{a^3}{b^2-bc+c^2}+\\frac{b^3}{c^2-ca+a^2} + \\frac{c^3}{a^2-ab+b^2} \\geq 3 \\cdot\\frac{ab+bc+ca}{a+b+c}',),\n", + " (r'\\frac{x^5-x^2}{x^5+y^2+z^2}+\\frac{y^5-y^2}{y^5+z^2+x^2}+\\frac{z^5-z^2}{z^5+x^2+y^2}\\geq0','','[x,1/(y*z)]'),\n", + " (r'(a+b-c)(b+c-a)(c+a-b)\\leq abc',),\n", + " (r'\\frac1{1+xy}+\\frac1{1+yz}+\\frac1{1+zx}\\leq \\frac34','','[x,(y+z)/(y*z-1)]'),\n", + " (r'\\frac{x\\sqrt x}{y+z}+\\frac{y\\sqrt y}{z+x}+\\frac{z\\sqrt z}{x+y}\\geq\\frac{ \\sqrt 3}2',\n", + " '[0,1-z],[0,1]','[x,1-y-z]'),\n", + " (r'a^4+b^4+c^4+d^4\\geq 4abcd',),\n", + " (r'\\frac{ab}{a+b}+\\frac{bc}{b+c}+\\frac{ca}{c+a}\\leq\\frac{3(ab+bc+ca)}{2(a+b+c)}',),\n", + " (r'\\sqrt{a-1}+\\sqrt{b-1}+ \\sqrt{c-1} \\leq \\sqrt{c(ab+1)}','[1,oo],[1,oo],[1,oo]','',dif2),\n", + " (r'(x-1)(y-1)(z-1)\\geq 8','[0,1-b-c],[0,1-c],[0,1]','[x,1/a],[y,1/b],[z,1/c]'),\n", + " (r'ay+bz+cx\\leq s^2','[0,s],[0,s],[0,s]','[x,s-a],[y,s-b],[z,s-c]'),\n", + " (r'x_1^2+x_2^2+x_3^2+x_4^2+x_5^2\\geq 2 (x_1x_2+x_2x_3+x_3x_4+x_4x_5)/\\sqrt{3}',),\n", + " (r' xy+yz+zx - 2xyz \\leq \\frac7{27}', '[0,1-z],[0,1]','[x,1-y-z]'),\n", + " (r'0 \\leq xy+yz+zx - 2xyz', '[0,1-z],[0,1]','[x,1-y-z]'),\n", + " (r'\\sqrt{3+a+b+c}\\geq\\sqrt a+\\sqrt b+\\sqrt c','[1-z,1],[0,1]','[a,1/x-1],[b,1/y-1],[c,1/z-1],[x,2-y-z]',\n", + " dif2),\n", + " (r'\\frac{2a^3}{a^2+b^2}+\\frac{2b^3}{b^2+c^2}+\\frac{2c^3}{c^2+a^2}\\geq a+b+c',),\n", + " (r'\\frac{a^2}{b+c}+\\frac{b^2}{c+a}+\\frac{c^2}{a+b}\\geq \\frac12','[0,1-c],[0,1]','[a,1-b-c]'),\n", + " (r'\\frac{a+b}{2b+c}+\\frac{b+c}{2c+a}+\\frac{c+a}{2a+b}\\geq 2',),\n", + " #(r'\\frac x{5-y^2}+\\frac y{5-z^2}+\\frac z{5-x^2}\\geq \\frac34','[0,sqrt(5)],[0,sqrt(5)]','[x,1/(y*z)]')\n", + "]\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "def parser(formula,intervals='[]',subs='[]',func=dif):\n", + " newproof()\n", + " shiro.display=lambda x:None\n", + " #display=lambda x:None\n", + " if intervals=='':\n", + " intervals='[]'\n", + " if subs=='':\n", + " subs='[]'\n", + " formula=addbraces(formula)\n", + " formula=Sm(str(parse_latex(formula)))\n", + " formula,_=fractioncancel(formula)\n", + " formula=formula.subs(shiroindev._smakeiterable2(subs))\n", + " formula=makesubs(formula,intervals)\n", + " b,s=formula.lhs,formula.rhs\n", + " if type(formula)==LessThan:\n", + " b,s=s,b\n", + " formula=func(b,s)\n", + " formula=simplify(formula)\n", + " num,den=fractioncancel(formula)\n", + " return num" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + " 74%|███████▍ | 26/35 [00:21<00:07, 1.21it/s]\n" + ] + }, + { + "ename": "KeyboardInterrupt", + "evalue": "", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\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 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\u001b[0;36m__eq__\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 333\u001b[0m \u001b[0;31m# (https://github.com/sympy/sympy/issues/4269), we only compare\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 334\u001b[0m \u001b[0;31m# types in Python 2 directly if they actually have __ne__.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 335\u001b[0;31m \u001b[0;32mif\u001b[0m \u001b[0mPY3\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__ne__\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__ne__\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 336\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mtself\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mtother\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 337\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mKeyboardInterrupt\u001b[0m: " + ] + } + ], + "source": [ + "from tqdm import tqdm\n", + "ineqs2=[]\n", + "for ineq in tqdm(ineqs):\n", + " ineqs2+=[parser(*ineq)]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now let's look at the formulas when converted to polynomials." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "for i,ineq in zip(range(len(ineqs2)),ineqs2):\n", + " print(i)\n", + " display(reducegens(assumeall(ineq,positive=True)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Most formulas was converted to polynomials of independent variables. However formulas No. 22 and No. 31 were not. For this reason it's very unlikely that any method of proving these ones will succeed.\n", + "\n", + "Now let's try some methods of proving these inequalities. The first one would be the simple `prove`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "tm=[0]*4" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from collections import Counter\n", + "from timeit import default_timer as timer\n", + "start=timer()\n", + "t=[]\n", + "for ineq in ineqs2:\n", + " t+=[prove(ineq)]\n", + " print(t[-1],end=',')\n", + "tm[0]=timer()-start\n", + "print('\\n',tm[0],sep='')\n", + "Counter(t)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Code 0 means that the proof was found, all other codes means that proof wasn't found. So this method has proved 21 inequalities.\n", + "\n", + "The second method uses `findvalues` function, rationalizes the result numbers and gives them as additional parameter to `prove` function." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def cut(a):\n", + " if a<=0 or a>=100 or (a is None):\n", + " return 1\n", + " return a\n", + "start=timer()\n", + "t2=[]\n", + "for ineq in ineqs2:\n", + " numvalues=findvalues(ineq,disp=0)\n", + " values=nsimplify(numvalues,tolerance=0.1,rational=True)\n", + " values=list(map(cut,values))\n", + " t2+=[prove(ineq,values=values)]\n", + " print(t2[-1],end=',')\n", + "tm[1]=timer()-start\n", + "print('\\n',tm[1],sep='')\n", + "Counter(t2)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The third method is similar to the second one, but instead of rationalize values it squares, rationalizes and makes square roots of these values." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def cut(a):\n", + " if a<=0 or a>=1000 or (a is None):\n", + " return S(1)\n", + " return a\n", + " \n", + "start=timer()\n", + "t3=[]\n", + "for ineq in ineqs2:\n", + " numvalues=findvalues(ineq,disp=0)\n", + " numvalues=tuple([x**2 for x in numvalues])\n", + " values=nsimplify(numvalues,tolerance=0.1,rational=True)\n", + " values=[sqrt(x) for x in values]\n", + " values=list(map(cut,values))\n", + " t3+=[prove(ineq,values=values)]\n", + " print(t3[-1],end=',')\n", + "tm[2]=timer()-start\n", + "print('\\n',tm[2],sep='')\n", + "Counter(t3)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finally, the fourth method is a slight modification to the third method. It does the same \"findvalues, square, rationalize and make square roots\" thing, but then it scales the values and runs it again. It can sometimes help with uniform formulas." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def betw(a):\n", + " return a>0.001 and a<1000 and a!=None\n", + "def cut(a):\n", + " if betw(a):\n", + " return a\n", + " return S(1)\n", + "\n", + "start=timer()\n", + "t4=[]\n", + "for ineq in ineqs2:\n", + " numvalues=findvalues(ineq,disp=0)\n", + " n=1\n", + " numvalues2=[]\n", + " for i in numvalues:\n", + " if betw(i):\n", + " n=1/i\n", + " break\n", + " for i in numvalues:\n", + " if betw(i):\n", + " numvalues2+=[i*n]\n", + " else:\n", + " numvalues2+=[1]\n", + " numvalues3=findvalues(ineq,values=numvalues2,disp=0)\n", + " numvalues4=tuple([x**2 for x in numvalues3])\n", + " values=nsimplify(numvalues4,tolerance=0.1,rational=True)\n", + " values=[sqrt(x) for x in values]\n", + " values=list(map(cut,values))\n", + " t4+=[prove(ineq,values=values)]\n", + " print(t4[-1],end=',')\n", + "tm[3]=timer()-start\n", + "print('\\n',tm[3],sep='')\n", + "Counter(t4)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "plt.bar(['1','2','3','4'],tm)\n", + "plt.ylabel('time [seconds]')\n", + "plt.xlabel('method')\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "import pandas as pd\n", + "u=pd.DataFrame(zip(['']*len(ineqs),t,t2,t3,t4))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "plt.bar(['1','2','3','4'],[sum(u[i]==0)/len(ineqs2)*100 for i in range(1,5)])\n", + "plt.ylabel('inequalities proven [%]')\n", + "plt.xlabel('method')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Down below there are contingency tables and McNemar test for every pair of methods." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "for i in range(4):\n", + " for j in range(i+1,4):\n", + " display(pd.crosstab(u[i+1]==0, u[j+1]==0))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from statsmodels.stats.contingency_tables import mcnemar\n", + "for i in range(4):\n", + " for j in range(i+1,4):\n", + " print(mcnemar(pd.crosstab(u[i+1]==0, u[j+1]==0)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In all cases p-value is greater than 0.05, so there is no statistical difference between the methods." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", 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zLdVHKe&>b&VKNR3_S0im=l~NEG^Qk%;4Tj8&BL)w6e<-of>J40_oFm*UVdp(yzn;| z=HJl(hA4jk6yGcAH%0k_l;VZEW))kHNai8CNa9Y&J)RKr=1?2YD?;SE`22g!rvC() zB|P}|Z%=mzrb$a7BFh;=rgMcH%n4-m*;4VkenLF+A4w=26DIbtgI9+0Ecu}WLIJ|X zv{&}C@+FoD7Yjd!{G4JUhguBP3q?n6uz?>!Y0AfrLS1I15Oz1rR!HNnVf(fb!fo@o zJ-o5j*`^%Ziv^SJ!TVQS2Iptwo_aLq`WeVyev$sc`G1Cb{7Pg&{|d+W)w{p!g@ubS z`g_&Z!b1K7{MlT2BiPXR*_G&x3d7c}KZlzO$O>y`QvY%t0#{@N|dmMc}55S<$N zZCreIh+rX40n6;(WWr4@4ENn@dyz+oiM3_Tdze|jR9V}a<9T%ra!yVU1%dIYq6?IjOv}Q" @@ -116,7 +116,7 @@ { "data": { "text/latex": [ - "$$ ab+ac+bc \\le a^2+b^2+c^2 $$" + "$$ a b+a c+b c \\le a^{2}+b^{2}+c^{2} $$" ], "text/plain": [ "" @@ -162,7 +162,7 @@ { "data": { "text/latex": [ - "numerator: $2a^2-2ab-2ac+2b^2-2bc+2c^2$" + "numerator: $2 a^{2} - 2 a b - 2 a c + 2 b^{2} - 2 b c + 2 c^{2}$" ], "text/plain": [ "" @@ -210,7 +210,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 a b \\le a^{2}+b^{2}$$" ], "text/plain": [ "" @@ -222,7 +222,7 @@ { "data": { "text/latex": [ - "$$2ac \\le a^2+c^2$$" + "$$2 a c \\le a^{2}+c^{2}$$" ], "text/plain": [ "" @@ -234,7 +234,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 b c \\le b^{2}+c^{2}$$" ], "text/plain": [ "" @@ -305,7 +305,7 @@ { "data": { "text/latex": [ - "numerator: $a^2-ab-ac-ad+b^2+c^2+d^2$" + "numerator: $a^{2} - a b - a c - a d + b^{2} + c^{2} + d^{2}$" ], "text/plain": [ "" @@ -353,7 +353,7 @@ { "data": { "text/latex": [ - "$$ ab+ac+ad \\le a^2+b^2+c^2+d^2 $$" + "$$ a b+a c+a d \\le a^{2}+b^{2}+c^{2}+d^{2} $$" ], "text/plain": [ "" @@ -398,7 +398,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 2a$" + "Substitute $a\\to 2 e$" ], "text/plain": [ "" @@ -410,7 +410,7 @@ { "data": { "text/latex": [ - "numerator: $4a^2-2ab-2ac-2ad+b^2+c^2+d^2$" + "numerator: $b^{2} - 2 b e + c^{2} - 2 c e + d^{2} - 2 d e + 4 e^{2}$" ], "text/plain": [ "" @@ -458,7 +458,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 b e \\le b^{2}+e^{2}$$" ], "text/plain": [ "" @@ -470,7 +470,7 @@ { "data": { "text/latex": [ - "$$2ac \\le a^2+c^2$$" + "$$2 c e \\le c^{2}+e^{2}$$" ], "text/plain": [ "" @@ -482,7 +482,7 @@ { "data": { "text/latex": [ - "$$2ad \\le a^2+d^2$$" + "$$2 d e \\le d^{2}+e^{2}$$" ], "text/plain": [ "" @@ -494,7 +494,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le a^2 $$" + "$$ 0 \\le e^{2} $$" ], "text/plain": [ "" @@ -534,7 +534,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Function makes a substitution $a\\to 2a$ (which should be understood as $a=2a'$) 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 $a'^2=0$ and eventually $a=0$. We can also try a little bit lower value for $a$." + "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 $a$." ] }, { @@ -545,7 +545,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 7a/4$" + "Substitute $a\\to \\frac{7 f}{4}$" ], "text/plain": [ "" @@ -557,7 +557,7 @@ { "data": { "text/latex": [ - "numerator: $49a^2-28ab-28ac-28ad+16b^2+16c^2+16d^2$" + "numerator: $16 b^{2} - 28 b f + 16 c^{2} - 28 c f + 16 d^{2} - 28 d f + 49 f^{2}$" ], "text/plain": [ "" @@ -605,7 +605,7 @@ { "data": { "text/latex": [ - "$$28ab \\le 14a^2+14b^2$$" + "$$28 b f \\le 14 b^{2}+14 f^{2}$$" ], "text/plain": [ "" @@ -617,7 +617,7 @@ { "data": { "text/latex": [ - "$$28ac \\le 14a^2+14c^2$$" + "$$28 c f \\le 14 c^{2}+14 f^{2}$$" ], "text/plain": [ "" @@ -629,7 +629,7 @@ { "data": { "text/latex": [ - "$$28ad \\le 14a^2+14d^2$$" + "$$28 d f \\le 14 d^{2}+14 f^{2}$$" ], "text/plain": [ "" @@ -641,7 +641,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 7a^2+2b^2+2c^2+2d^2 $$" + "$$ 0 \\le 2 b^{2}+2 c^{2}+2 d^{2}+7 f^{2} $$" ], "text/plain": [ "" @@ -681,7 +681,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Now we can see that if $a^2+b^2+c^2+d^2-a(b+c+d)=0$, then $7a'^2+2b^2+2c^2+2d^2=0$ 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.\n", + "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.\n", + "\n", + "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.\n", "\n", "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." ] @@ -760,7 +762,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 10a/7$" + "Substitute $a\\to \\frac{10 e}{7}$" ], "text/plain": [ "" @@ -772,7 +774,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to 5b/6$" + "Substitute $b\\to \\frac{5 f}{6}$" ], "text/plain": [ "" @@ -784,7 +786,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to 5c/6$" + "Substitute $c\\to \\frac{5 g}{6}$" ], "text/plain": [ "" @@ -796,7 +798,7 @@ { "data": { "text/latex": [ - "Substitute $d\\to 5d/6$" + "Substitute $d\\to \\frac{5 h}{6}$" ], "text/plain": [ "" @@ -808,7 +810,7 @@ { "data": { "text/latex": [ - "numerator: $3600a^2-2100ab-2100ac-2100ad+1225b^2+1225c^2+1225d^2$" + "numerator: $3600 e^{2} - 2100 e f - 2100 e g - 2100 e h + 1225 f^{2} + 1225 g^{2} + 1225 h^{2}$" ], "text/plain": [ "" @@ -856,7 +858,7 @@ { "data": { "text/latex": [ - "$$2100ab \\le 1050a^2+1050b^2$$" + "$$2100 e f \\le 1050 e^{2}+1050 f^{2}$$" ], "text/plain": [ "" @@ -868,7 +870,7 @@ { "data": { "text/latex": [ - "$$2100ac \\le 1050a^2+1050c^2$$" + "$$2100 e g \\le 1050 e^{2}+1050 g^{2}$$" ], "text/plain": [ "" @@ -880,7 +882,7 @@ { "data": { "text/latex": [ - "$$2100ad \\le 1050a^2+1050d^2$$" + "$$2100 e h \\le 1050 e^{2}+1050 h^{2}$$" ], "text/plain": [ "" @@ -892,7 +894,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 450a^2+175b^2+175c^2+175d^2 $$" + "$$ 0 \\le 450 e^{2}+175 f^{2}+175 g^{2}+175 h^{2} $$" ], "text/plain": [ "" @@ -925,6 +927,7 @@ } ], "source": [ + "newproof()\n", "prove(formula,values)" ] }, @@ -949,7 +952,7 @@ { "data": { "text/latex": [ - "numerator: $-x^3y+2x^2y^2-x^2-xy^3+2xy-y^2$" + "numerator: $- x^{3} y + 2 x^{2} y^{2} - x^{2} - x y^{3} + 2 x y - y^{2}$" ], "text/plain": [ "" @@ -961,7 +964,7 @@ { "data": { "text/latex": [ - "denominator: $x^3y^3-x^3y-x^2y^2+x^2-xy^3+xy+y^2-1$" + "denominator: $x^{3} y^{3} - x^{3} y - x^{2} y^{2} + x^{2} - x y^{3} + x y + y^{2} - 1$" ], "text/plain": [ "" @@ -997,7 +1000,7 @@ { "data": { "text/latex": [ - "$$ x^3y+x^2+xy^3+y^2 \\le 2x^2y^2+2xy $$" + "$$ x^{3} y+x^{2}+x y^{3}+y^{2} \\le 2 x^{2} y^{2}+2 x y $$" ], "text/plain": [ "" @@ -1063,7 +1066,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to 1-1/(x+1)$" + "Substitute $x\\to 1 - \\frac{1}{a + 1}$" ], "text/plain": [ "" @@ -1075,7 +1078,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to 1-1/(y+1)$" + "Substitute $y\\to 1 - \\frac{1}{b + 1}$" ], "text/plain": [ "" @@ -1087,7 +1090,7 @@ { "data": { "text/latex": [ - "numerator: $6x^3y+3x^3-12x^2y^2-3x^2y+3x^2+6xy^3-3xy^2-6xy+3y^3+3y^2$" + "numerator: $6 a^{3} b + 3 a^{3} - 12 a^{2} b^{2} - 3 a^{2} b + 3 a^{2} + 6 a b^{3} - 3 a b^{2} - 6 a b + 3 b^{3} + 3 b^{2}$" ], "text/plain": [ "" @@ -1099,7 +1102,7 @@ { "data": { "text/latex": [ - "denominator: $4x^2y+2x^2+4xy^2+8xy+3x+2y^2+3y+1$" + "denominator: $4 a^{2} b + 2 a^{2} + 4 a b^{2} + 8 a b + 3 a + 2 b^{2} + 3 b + 1$" ], "text/plain": [ "" @@ -1135,7 +1138,7 @@ { "data": { "text/latex": [ - "$$12x^2y^2 \\le 6x^3y+6xy^3$$" + "$$12 a^{2} b^{2} \\le 6 a^{3} b+6 a b^{3}$$" ], "text/plain": [ "" @@ -1147,7 +1150,7 @@ { "data": { "text/latex": [ - "$$3x^2y \\le 2x^3+y^3$$" + "$$3 a^{2} b \\le 2 a^{3}+b^{3}$$" ], "text/plain": [ "" @@ -1159,7 +1162,7 @@ { "data": { "text/latex": [ - "$$3xy^2 \\le x^3+2y^3$$" + "$$3 a b^{2} \\le a^{3}+2 b^{3}$$" ], "text/plain": [ "" @@ -1171,7 +1174,7 @@ { "data": { "text/latex": [ - "$$6xy \\le 3x^2+3y^2$$" + "$$6 a b \\le 3 a^{2}+3 b^{2}$$" ], "text/plain": [ "" @@ -1216,6 +1219,7 @@ } ], "source": [ + "newproof()\n", "newformula,newvalues=makesubs('1/(1-x^2)+1/(1-y^2)-2/(1-x*y)','[0,1],[0,1]','1/2,1/2')\n", "prove(newformula*3,newvalues)" ] @@ -1235,8 +1239,7 @@ "* constraints for variables are written as intervals,\n", "* there are no \"backwards dependencies\", i.e. there is no $x$ in the interval of $y$.\n", "\n", - "If these two conditions hold, then you can use `makesubs` function.\n", - "**Warning:** at this moment `makesubs` **doesn't warn you if your list of intervals doesn't follow these rules!**\n" + "If these two conditions hold, then you can use `makesubs` function." ] }, { @@ -1247,7 +1250,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to -y+1+(y-1)/(x+1)$" + "Substitute $x\\to - y + 1 + \\frac{y - 1}{a + 1}$" ], "text/plain": [ "" @@ -1259,7 +1262,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to 1-1/(y+1)$" + "Substitute $y\\to 1 - \\frac{1}{b + 1}$" ], "text/plain": [ "" @@ -1271,7 +1274,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to y/2$" + "Substitute $b\\to \\frac{c}{2}$" ], "text/plain": [ "" @@ -1283,7 +1286,7 @@ { "data": { "text/latex": [ - "numerator: $x^4y^2+x^3y^2-2x^3y-4x^2y+4x^2+xy^2-2xy+y^2$" + "numerator: $a^{4} c^{2} + a^{3} c^{2} - 2 a^{3} c - 4 a^{2} c + 4 a^{2} + a c^{2} - 2 a c + c^{2}$" ], "text/plain": [ "" @@ -1295,7 +1298,7 @@ { "data": { "text/latex": [ - "denominator: $x^3y^2+2x^3y+2x^2y^2+4x^2y+xy^2+2xy$" + "denominator: $a^{3} c^{2} + 2 a^{3} c + 2 a^{2} c^{2} + 4 a^{2} c + a c^{2} + 2 a c$" ], "text/plain": [ "" @@ -1331,7 +1334,7 @@ { "data": { "text/latex": [ - "$$2x^3y \\le x^4y^2+x^2$$" + "$$2 a^{3} c \\le a^{4} c^{2}+a^{2}$$" ], "text/plain": [ "" @@ -1343,7 +1346,7 @@ { "data": { "text/latex": [ - "$$4x^2y \\le x^3y^2+2x^2+xy^2$$" + "$$4 a^{2} c \\le a^{3} c^{2}+2 a^{2}+a c^{2}$$" ], "text/plain": [ "" @@ -1355,7 +1358,7 @@ { "data": { "text/latex": [ - "$$2xy \\le x^2+y^2$$" + "$$2 a c \\le a^{2}+c^{2}$$" ], "text/plain": [ "" @@ -1400,6 +1403,7 @@ } ], "source": [ + "newproof()\n", "formula=Sm('xy/z+yz/x+zx/y-1').subs('z',S('1-x-y'))\n", "newformula,values=makesubs(formula,'[0,1-y],[0,1]','1/3,1/3')\n", "prove(newformula,values)" @@ -1424,7 +1428,7 @@ { "data": { "text/latex": [ - "numerator: $2a^2-2ab-2ac+2b^2-2bc+2c^2$" + "numerator: $2 a^{2} - 2 a b - 2 a c + 2 b^{2} - 2 b c + 2 c^{2}$" ], "text/plain": [ "" @@ -1472,7 +1476,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 a b \\le a^{2}+b^{2}$$" ], "text/plain": [ "" @@ -1484,7 +1488,7 @@ { "data": { "text/latex": [ - "$$2ac \\le a^2+c^2$$" + "$$2 a c \\le a^{2}+c^{2}$$" ], "text/plain": [ "" @@ -1496,7 +1500,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 b c \\le b^{2}+c^{2}$$" ], "text/plain": [ "" @@ -1563,7 +1567,7 @@ { "data": { "text/latex": [ - "numerator: $x^4-4x^3+6x^2-4x+1$" + "numerator: $x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$" ], "text/plain": [ "" @@ -1611,7 +1615,7 @@ { "data": { "text/latex": [ - "$$ 4x^3+4x \\le x^4+6x^2+1 $$" + "$$ 4 x^{3}+4 x \\le x^{4}+6 x^{2}+1 $$" ], "text/plain": [ "" @@ -1650,7 +1654,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+1$" + "Substitute $x\\to a + 1$" ], "text/plain": [ "" @@ -1662,7 +1666,7 @@ { "data": { "text/latex": [ - "numerator: $x^4$" + "numerator: $a^{4}$" ], "text/plain": [ "" @@ -1698,7 +1702,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le x^4 $$" + "$$ 0 \\le a^{4} $$" ], "text/plain": [ "" @@ -1731,6 +1735,7 @@ } ], "source": [ + "newproof()\n", "prove(makesubs('(x-1)^4','(1,oo)'))" ] }, @@ -1744,7 +1749,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to 1-x$" + "Substitute $x\\to 1 - a$" ], "text/plain": [ "" @@ -1756,7 +1761,7 @@ { "data": { "text/latex": [ - "numerator: $x^4$" + "numerator: $a^{4}$" ], "text/plain": [ "" @@ -1792,7 +1797,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le x^4 $$" + "$$ 0 \\le a^{4} $$" ], "text/plain": [ "" @@ -1825,6 +1830,7 @@ } ], "source": [ + "newproof()\n", "prove(makesubs('(x-1)^4','(-oo,1)'))" ] }, @@ -1872,7 +1878,7 @@ { "data": { "text/latex": [ - "numerator: $a^3c+a^3d+a^2b^2-a^2bd-2a^2c^2-a^2cd+a^2d^2+ab^3-ab^2c-ab^2d-abc^2+ac^3-acd^2+b^3d+b^2c^2-2b^2d^2+bc^3-bc^2d-bcd^2+bd^3+c^2d^2+cd^3$" + "numerator: $a^{3} c + a^{3} d + a^{2} b^{2} - a^{2} b d - 2 a^{2} c^{2} - a^{2} c d + a^{2} d^{2} + a b^{3} - a b^{2} c - a b^{2} d - a b c^{2} + a c^{3} - a c d^{2} + b^{3} d + b^{2} c^{2} - 2 b^{2} d^{2} + b c^{3} - b c^{2} d - b c d^{2} + b d^{3} + c^{2} d^{2} + c d^{3}$" ], "text/plain": [ "" @@ -1884,7 +1890,7 @@ { "data": { "text/latex": [ - "denominator: $a^2bc+a^2bd+a^2c^2+a^2cd+ab^2c+ab^2d+abc^2+2abcd+abd^2+ac^2d+acd^2+b^2cd+b^2d^2+bc^2d+bcd^2$" + "denominator: $a^{2} b c + a^{2} b d + a^{2} c^{2} + a^{2} c d + a b^{2} c + a b^{2} d + a b c^{2} + 2 a b c d + a b d^{2} + a c^{2} d + a c d^{2} + b^{2} c d + b^{2} d^{2} + b c^{2} d + b c d^{2}$" ], "text/plain": [ "" @@ -1920,7 +1926,7 @@ { "data": { "text/latex": [ - "$$ a^2bd+2a^2c^2+a^2cd+ab^2c+ab^2d+abc^2+acd^2+2b^2d^2+bc^2d+bcd^2 \\le a^3c+a^3d+a^2b^2+a^2d^2+ab^3+ac^3+b^3d+b^2c^2+bc^3+bd^3+c^2d^2+cd^3 $$" + "$$ a^{2} b d+2 a^{2} c^{2}+a^{2} c d+a b^{2} c+a b^{2} d+a b c^{2}+a c d^{2}+2 b^{2} d^{2}+b c^{2} d+b c d^{2} \\le a^{3} c+a^{3} d+a^{2} b^{2}+a^{2} d^{2}+a b^{3}+a c^{3}+b^{3} d+b^{2} c^{2}+b c^{3}+b d^{3}+c^{2} d^{2}+c d^{3} $$" ], "text/plain": [ "" @@ -1959,7 +1965,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "Substitute $a\\to c + e$" ], "text/plain": [ "" @@ -1971,7 +1977,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+d$" + "Substitute $b\\to d + f$" ], "text/plain": [ "" @@ -1983,7 +1989,7 @@ { "data": { "text/latex": [ - "numerator: $a^3c+a^3d+a^2b^2+a^2bd+a^2c^2+2a^2cd+a^2d^2+ab^3+ab^2c+2ab^2d-abc^2+abd^2+b^3c+b^3d+b^2c^2+2b^2cd+b^2d^2$" + "numerator: $c^{2} e^{2} - c^{2} e f + c^{2} f^{2} + 2 c d e^{2} + 2 c d f^{2} + c e^{3} + c e f^{2} + c f^{3} + d^{2} e^{2} + d^{2} e f + d^{2} f^{2} + d e^{3} + d e^{2} f + 2 d e f^{2} + d f^{3} + e^{2} f^{2} + e f^{3}$" ], "text/plain": [ "" @@ -1995,7 +2001,7 @@ { "data": { "text/latex": [ - "denominator: $a^2bc+a^2bd+a^2c^2+2a^2cd+a^2d^2+ab^2c+ab^2d+3abc^2+6abcd+3abd^2+2ac^3+6ac^2d+6acd^2+2ad^3+b^2c^2+2b^2cd+b^2d^2+2bc^3+6bc^2d+6bcd^2+2bd^3+c^4+4c^3d+6c^2d^2+4cd^3+d^4$" + "denominator: $c^{4} + 4 c^{3} d + 2 c^{3} e + 2 c^{3} f + 6 c^{2} d^{2} + 6 c^{2} d e + 6 c^{2} d f + c^{2} e^{2} + 3 c^{2} e f + c^{2} f^{2} + 4 c d^{3} + 6 c d^{2} e + 6 c d^{2} f + 2 c d e^{2} + 6 c d e f + 2 c d f^{2} + c e^{2} f + c e f^{2} + d^{4} + 2 d^{3} e + 2 d^{3} f + d^{2} e^{2} + 3 d^{2} e f + d^{2} f^{2} + d e^{2} f + d e f^{2}$" ], "text/plain": [ "" @@ -2043,7 +2049,7 @@ { "data": { "text/latex": [ - "$$ abc^2 \\le a^3c+a^3d+a^2b^2+a^2bd+a^2c^2+2a^2cd+a^2d^2+ab^3+ab^2c+2ab^2d+abd^2+b^3c+b^3d+b^2c^2+2b^2cd+b^2d^2 $$" + "$$ c^{2} e f \\le c^{2} e^{2}+c^{2} f^{2}+2 c d e^{2}+2 c d f^{2}+c e^{3}+c e f^{2}+c f^{3}+d^{2} e^{2}+d^{2} e f+d^{2} f^{2}+d e^{3}+d e^{2} f+2 d e f^{2}+d f^{3}+e^{2} f^{2}+e f^{3} $$" ], "text/plain": [ "" @@ -2064,6 +2070,7 @@ } ], "source": [ + "newproof()\n", "prove(makesubs(formula,'[c,oo],[d,oo]'))" ] }, @@ -2075,7 +2082,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "Substitute $a\\to c + e$" ], "text/plain": [ "" @@ -2087,7 +2094,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+d$" + "Substitute $b\\to d + f$" ], "text/plain": [ "" @@ -2099,7 +2106,7 @@ { "data": { "text/latex": [ - "numerator: $2a^3c+2a^3d+2a^2b^2+2a^2bd+2a^2c^2+4a^2cd+2a^2d^2+2ab^3+2ab^2c+4ab^2d-2abc^2+2abd^2+2b^3c+2b^3d+2b^2c^2+4b^2cd+2b^2d^2$" + "numerator: $2 c^{2} e^{2} - 2 c^{2} e f + 2 c^{2} f^{2} + 4 c d e^{2} + 4 c d f^{2} + 2 c e^{3} + 2 c e f^{2} + 2 c f^{3} + 2 d^{2} e^{2} + 2 d^{2} e f + 2 d^{2} f^{2} + 2 d e^{3} + 2 d e^{2} f + 4 d e f^{2} + 2 d f^{3} + 2 e^{2} f^{2} + 2 e f^{3}$" ], "text/plain": [ "" @@ -2111,7 +2118,7 @@ { "data": { "text/latex": [ - "denominator: $a^2bc+a^2bd+a^2c^2+2a^2cd+a^2d^2+ab^2c+ab^2d+3abc^2+6abcd+3abd^2+2ac^3+6ac^2d+6acd^2+2ad^3+b^2c^2+2b^2cd+b^2d^2+2bc^3+6bc^2d+6bcd^2+2bd^3+c^4+4c^3d+6c^2d^2+4cd^3+d^4$" + "denominator: $c^{4} + 4 c^{3} d + 2 c^{3} e + 2 c^{3} f + 6 c^{2} d^{2} + 6 c^{2} d e + 6 c^{2} d f + c^{2} e^{2} + 3 c^{2} e f + c^{2} f^{2} + 4 c d^{3} + 6 c d^{2} e + 6 c d^{2} f + 2 c d e^{2} + 6 c d e f + 2 c d f^{2} + c e^{2} f + c e f^{2} + d^{4} + 2 d^{3} e + 2 d^{3} f + d^{2} e^{2} + 3 d^{2} e f + d^{2} f^{2} + d e^{2} f + d e f^{2}$" ], "text/plain": [ "" @@ -2147,7 +2154,7 @@ { "data": { "text/latex": [ - "$$2abc^2 \\le a^2c^2+b^2c^2$$" + "$$2 c^{2} e f \\le c^{2} e^{2}+c^{2} f^{2}$$" ], "text/plain": [ "" @@ -2159,7 +2166,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 2a^3c+2a^3d+2a^2b^2+2a^2bd+a^2c^2+4a^2cd+2a^2d^2+2ab^3+2ab^2c+4ab^2d+2abd^2+2b^3c+2b^3d+b^2c^2+4b^2cd+2b^2d^2 $$" + "$$ 0 \\le c^{2} e^{2}+c^{2} f^{2}+4 c d e^{2}+4 c d f^{2}+2 c e^{3}+2 c e f^{2}+2 c f^{3}+2 d^{2} e^{2}+2 d^{2} e f+2 d^{2} f^{2}+2 d e^{3}+2 d e^{2} f+4 d e f^{2}+2 d f^{3}+2 e^{2} f^{2}+2 e f^{3} $$" ], "text/plain": [ "" @@ -2192,6 +2199,7 @@ } ], "source": [ + "newproof()\n", "prove(makesubs(formula,'[c,oo],[d,oo]')*2)" ] }, @@ -2210,7 +2218,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to c-c/(a+1)$" + "Substitute $a\\to c - \\frac{c}{e + 1}$" ], "text/plain": [ "" @@ -2222,7 +2230,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to d-d/(b+1)$" + "Substitute $b\\to d - \\frac{d}{f + 1}$" ], "text/plain": [ "" @@ -2234,7 +2242,7 @@ { "data": { "text/latex": [ - "numerator: $2a^3bc^2d^2+4a^3bcd^3+2a^3bd^4+2a^3c^2d^2+2a^3cd^3-2a^2b^2c^3d+2a^2b^2cd^3-4a^2bc^3d+4a^2bc^2d^2+12a^2bcd^3+6a^2bd^4-2a^2c^3d+4a^2c^2d^2+6a^2cd^3+2ab^3c^4+4ab^3c^3d+2ab^3c^2d^2+6ab^2c^4+8ab^2c^3d+4ab^2c^2d^2+4ab^2cd^3+6abc^4+4abc^3d+6abc^2d^2+12abcd^3+6abd^4+2ac^4+4ac^2d^2+6acd^3+2b^3c^3d+2b^3c^2d^2+4b^2c^3d+4b^2c^2d^2+2b^2cd^3+2bc^3d+4bc^2d^2+4bcd^3+2bd^4+2c^2d^2+2cd^3$" + "numerator: $2 c^{4} e f^{3} + 6 c^{4} e f^{2} + 6 c^{4} e f + 2 c^{4} e - 2 c^{3} d e^{2} f^{2} - 4 c^{3} d e^{2} f - 2 c^{3} d e^{2} + 4 c^{3} d e f^{3} + 8 c^{3} d e f^{2} + 4 c^{3} d e f + 2 c^{3} d f^{3} + 4 c^{3} d f^{2} + 2 c^{3} d f + 2 c^{2} d^{2} e^{3} f + 2 c^{2} d^{2} e^{3} + 4 c^{2} d^{2} e^{2} f + 4 c^{2} d^{2} e^{2} + 2 c^{2} d^{2} e f^{3} + 4 c^{2} d^{2} e f^{2} + 6 c^{2} d^{2} e f + 4 c^{2} d^{2} e + 2 c^{2} d^{2} f^{3} + 4 c^{2} d^{2} f^{2} + 4 c^{2} d^{2} f + 2 c^{2} d^{2} + 4 c d^{3} e^{3} f + 2 c d^{3} e^{3} + 2 c d^{3} e^{2} f^{2} + 12 c d^{3} e^{2} f + 6 c d^{3} e^{2} + 4 c d^{3} e f^{2} + 12 c d^{3} e f + 6 c d^{3} e + 2 c d^{3} f^{2} + 4 c d^{3} f + 2 c d^{3} + 2 d^{4} e^{3} f + 6 d^{4} e^{2} f + 6 d^{4} e f + 2 d^{4} f$" ], "text/plain": [ "" @@ -2246,7 +2254,7 @@ { "data": { "text/latex": [ - "denominator: $a^3b^3c^4+4a^3b^3c^3d+6a^3b^3c^2d^2+4a^3b^3cd^3+a^3b^3d^4+3a^3b^2c^4+10a^3b^2c^3d+12a^3b^2c^2d^2+6a^3b^2cd^3+a^3b^2d^4+3a^3bc^4+8a^3bc^3d+7a^3bc^2d^2+2a^3bcd^3+a^3c^4+2a^3c^3d+a^3c^2d^2+a^2b^3c^4+6a^2b^3c^3d+12a^2b^3c^2d^2+10a^2b^3cd^3+3a^2b^3d^4+3a^2b^2c^4+15a^2b^2c^3d+24a^2b^2c^2d^2+15a^2b^2cd^3+3a^2b^2d^4+3a^2bc^4+12a^2bc^3d+14a^2bc^2d^2+5a^2bcd^3+a^2c^4+3a^2c^3d+2a^2c^2d^2+2ab^3c^3d+7ab^3c^2d^2+8ab^3cd^3+3ab^3d^4+5ab^2c^3d+14ab^2c^2d^2+12ab^2cd^3+3ab^2d^4+4abc^3d+8abc^2d^2+4abcd^3+ac^3d+ac^2d^2+b^3c^2d^2+2b^3cd^3+b^3d^4+2b^2c^2d^2+3b^2cd^3+b^2d^4+bc^2d^2+bcd^3$" + "denominator: $c^{4} e^{3} f^{3} + 3 c^{4} e^{3} f^{2} + 3 c^{4} e^{3} f + c^{4} e^{3} + c^{4} e^{2} f^{3} + 3 c^{4} e^{2} f^{2} + 3 c^{4} e^{2} f + c^{4} e^{2} + 4 c^{3} d e^{3} f^{3} + 10 c^{3} d e^{3} f^{2} + 8 c^{3} d e^{3} f + 2 c^{3} d e^{3} + 6 c^{3} d e^{2} f^{3} + 15 c^{3} d e^{2} f^{2} + 12 c^{3} d e^{2} f + 3 c^{3} d e^{2} + 2 c^{3} d e f^{3} + 5 c^{3} d e f^{2} + 4 c^{3} d e f + c^{3} d e + 6 c^{2} d^{2} e^{3} f^{3} + 12 c^{2} d^{2} e^{3} f^{2} + 7 c^{2} d^{2} e^{3} f + c^{2} d^{2} e^{3} + 12 c^{2} d^{2} e^{2} f^{3} + 24 c^{2} d^{2} e^{2} f^{2} + 14 c^{2} d^{2} e^{2} f + 2 c^{2} d^{2} e^{2} + 7 c^{2} d^{2} e f^{3} + 14 c^{2} d^{2} e f^{2} + 8 c^{2} d^{2} e f + c^{2} d^{2} e + c^{2} d^{2} f^{3} + 2 c^{2} d^{2} f^{2} + c^{2} d^{2} f + 4 c d^{3} e^{3} f^{3} + 6 c d^{3} e^{3} f^{2} + 2 c d^{3} e^{3} f + 10 c d^{3} e^{2} f^{3} + 15 c d^{3} e^{2} f^{2} + 5 c d^{3} e^{2} f + 8 c d^{3} e f^{3} + 12 c d^{3} e f^{2} + 4 c d^{3} e f + 2 c d^{3} f^{3} + 3 c d^{3} f^{2} + c d^{3} f + d^{4} e^{3} f^{3} + d^{4} e^{3} f^{2} + 3 d^{4} e^{2} f^{3} + 3 d^{4} e^{2} f^{2} + 3 d^{4} e f^{3} + 3 d^{4} e f^{2} + d^{4} f^{3} + d^{4} f^{2}$" ], "text/plain": [ "" @@ -2282,7 +2290,7 @@ { "data": { "text/latex": [ - "$$2a^2b^2c^3d \\le a^3bc^2d^2+ab^3c^4$$" + "$$2 c^{3} d e^{2} f^{2} \\le c^{4} e f^{3}+c^{2} d^{2} e^{3} f$$" ], "text/plain": [ "" @@ -2294,7 +2302,7 @@ { "data": { "text/latex": [ - "$$2a^2c^3d \\le a^3c^2d^2+ac^4$$" + "$$2 c^{3} d e^{2} \\le c^{4} e+c^{2} d^{2} e^{3}$$" ], "text/plain": [ "" @@ -2306,7 +2314,7 @@ { "data": { "text/latex": [ - "$$4a^2bc^3d \\le a^3bc^2d^2+a^3c^2d^2+ab^3c^4+ac^4$$" + "$$4 c^{3} d e^{2} f \\le c^{4} e f^{3}+c^{4} e+c^{2} d^{2} e^{3} f+c^{2} d^{2} e^{3}$$" ], "text/plain": [ "" @@ -2318,7 +2326,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^3bcd^3+2a^3bd^4+2a^3cd^3+2a^2b^2cd^3+4a^2bc^2d^2+12a^2bcd^3+6a^2bd^4+4a^2c^2d^2+6a^2cd^3+4ab^3c^3d+2ab^3c^2d^2+6ab^2c^4+8ab^2c^3d+4ab^2c^2d^2+4ab^2cd^3+6abc^4+4abc^3d+6abc^2d^2+12abcd^3+6abd^4+4ac^2d^2+6acd^3+2b^3c^3d+2b^3c^2d^2+4b^2c^3d+4b^2c^2d^2+2b^2cd^3+2bc^3d+4bc^2d^2+4bcd^3+2bd^4+2c^2d^2+2cd^3 $$" + "$$ 0 \\le 6 c^{4} e f^{2}+6 c^{4} e f+4 c^{3} d e f^{3}+8 c^{3} d e f^{2}+4 c^{3} d e f+2 c^{3} d f^{3}+4 c^{3} d f^{2}+2 c^{3} d f+4 c^{2} d^{2} e^{2} f+4 c^{2} d^{2} e^{2}+2 c^{2} d^{2} e f^{3}+4 c^{2} d^{2} e f^{2}+6 c^{2} d^{2} e f+4 c^{2} d^{2} e+2 c^{2} d^{2} f^{3}+4 c^{2} d^{2} f^{2}+4 c^{2} d^{2} f+2 c^{2} d^{2}+4 c d^{3} e^{3} f+2 c d^{3} e^{3}+2 c d^{3} e^{2} f^{2}+12 c d^{3} e^{2} f+6 c d^{3} e^{2}+4 c d^{3} e f^{2}+12 c d^{3} e f+6 c d^{3} e+2 c d^{3} f^{2}+4 c d^{3} f+2 c d^{3}+2 d^{4} e^{3} f+6 d^{4} e^{2} f+6 d^{4} e f+2 d^{4} f $$" ], "text/plain": [ "" @@ -2351,6 +2359,7 @@ } ], "source": [ + "newproof()\n", "prove(makesubs(formula,'[0,c],[0,d]')*2)" ] }, @@ -2369,7 +2378,7 @@ { "data": { "text/latex": [ - "numerator: $x^4-4x^3+6x^2-4x+1$" + "numerator: $x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$" ], "text/plain": [ "" @@ -2393,8 +2402,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2406,7 +2414,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to 1+x$" + "Substitute $x\\to 1+a$" ], "text/plain": [ "" @@ -2418,7 +2426,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $x^4$" + "Numerator after substitutions: $a^{4}$" ], "text/plain": [ "" @@ -2442,7 +2450,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le x^4 $$" + "$$ 0 \\le a^{4} $$" ], "text/plain": [ "" @@ -2466,8 +2474,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2479,7 +2486,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to 1/(1+x)$" + "Substitute $x\\to 1/(1+b)$" ], "text/plain": [ "" @@ -2491,7 +2498,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $x^4$" + "Numerator after substitutions: $b^{4}$" ], "text/plain": [ "" @@ -2515,7 +2522,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le x^4 $$" + "$$ 0 \\le b^{4} $$" ], "text/plain": [ "" @@ -2548,6 +2555,7 @@ } ], "source": [ + "newproof()\n", "powerprove('(x-1)^4')" ] }, @@ -2559,7 +2567,7 @@ { "data": { "text/latex": [ - "numerator: $4a^5+4a^4b+4a^4c-6a^4-4a^3b-2a^3c+4ab^3-9ab^2+4ac^2-18ac+9a+4b^4+4b^3c-6b^3-3b^2c+4bc^2-12bc+10b+4c^3-6c^2+11c$" + "numerator: $4 a^{5} + 4 a^{4} b + 4 a^{4} c - 6 a^{4} - 4 a^{3} b - 2 a^{3} c + 4 a b^{3} - 9 a b^{2} + 4 a c^{2} - 18 a c + 9 a + 4 b^{4} + 4 b^{3} c - 6 b^{3} - 3 b^{2} c + 4 b c^{2} - 12 b c + 10 b + 4 c^{3} - 6 c^{2} + 11 c$" ], "text/plain": [ "" @@ -2583,8 +2591,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2596,7 +2603,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1+a,b\\to 1+b,c\\to 1+c$" + "Substitute $a\\to 1+d,b\\to 1+e,c\\to 1+f$" ], "text/plain": [ "" @@ -2608,7 +2615,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5+4a^4b+4a^4c+22a^4+12a^3b+14a^3c+42a^3+12a^2b+18a^2c+34a^2+4ab^3+3ab^2-2ab+4ac^2+4b^4+4b^3c+18b^3+9b^2c+18b^2+4bc^2+2bc+4c^3+14c^2$" + "Numerator after substitutions: $4 d^{5} + 4 d^{4} e + 4 d^{4} f + 22 d^{4} + 12 d^{3} e + 14 d^{3} f + 42 d^{3} + 12 d^{2} e + 18 d^{2} f + 34 d^{2} + 4 d e^{3} + 3 d e^{2} - 2 d e + 4 d f^{2} + 4 e^{4} + 4 e^{3} f + 18 e^{3} + 9 e^{2} f + 18 e^{2} + 4 e f^{2} + 2 e f + 4 f^{3} + 14 f^{2}$" ], "text/plain": [ "" @@ -2644,7 +2651,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 d e \\le d^{2}+e^{2}$$" ], "text/plain": [ "" @@ -2656,7 +2663,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5+4a^4b+4a^4c+22a^4+12a^3b+14a^3c+42a^3+12a^2b+18a^2c+33a^2+4ab^3+3ab^2+4ac^2+4b^4+4b^3c+18b^3+9b^2c+17b^2+4bc^2+2bc+4c^3+14c^2 $$" + "$$ 0 \\le 4 d^{5}+4 d^{4} e+4 d^{4} f+22 d^{4}+12 d^{3} e+14 d^{3} f+42 d^{3}+12 d^{2} e+18 d^{2} f+33 d^{2}+4 d e^{3}+3 d e^{2}+4 d f^{2}+4 e^{4}+4 e^{3} f+18 e^{3}+9 e^{2} f+17 e^{2}+4 e f^{2}+2 e f+4 f^{3}+14 f^{2} $$" ], "text/plain": [ "" @@ -2680,8 +2687,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2693,7 +2699,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1/(1+a),b\\to 1+b,c\\to 1+c$" + "Substitute $a\\to 1/(1+g),b\\to 1+h,c\\to 1+i$" ], "text/plain": [ "" @@ -2705,7 +2711,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4+4a^5b^3c+14a^5b^3+9a^5b^2c+15a^5b^2+4a^5bc^2+2a^5bc+6a^5b+4a^5c^3+10a^5c^2+8a^5c+10a^5+20a^4b^4+20a^4b^3c+74a^4b^3+45a^4b^2c+78a^4b^2+20a^4bc^2+10a^4bc+24a^4b+20a^4c^3+54a^4c^2+30a^4c+40a^4+40a^3b^4+40a^3b^3c+156a^3b^3+90a^3b^2c+162a^3b^2+40a^3bc^2+20a^3bc+36a^3b+40a^3c^3+116a^3c^2+40a^3c+60a^3+40a^2b^4+40a^2b^3c+164a^2b^3+90a^2b^2c+168a^2b^2+40a^2bc^2+20a^2bc+20a^2b+40a^2c^3+124a^2c^2+18a^2c+34a^2+20ab^4+20ab^3c+86ab^3+45ab^2c+87ab^2+20abc^2+10abc+2ab+20ac^3+66ac^2+4b^4+4b^3c+18b^3+9b^2c+18b^2+4bc^2+2bc+4c^3+14c^2$" + "Numerator after substitutions: $4 g^{5} h^{4} + 4 g^{5} h^{3} i + 14 g^{5} h^{3} + 9 g^{5} h^{2} i + 15 g^{5} h^{2} + 4 g^{5} h i^{2} + 2 g^{5} h i + 6 g^{5} h + 4 g^{5} i^{3} + 10 g^{5} i^{2} + 8 g^{5} i + 10 g^{5} + 20 g^{4} h^{4} + 20 g^{4} h^{3} i + 74 g^{4} h^{3} + 45 g^{4} h^{2} i + 78 g^{4} h^{2} + 20 g^{4} h i^{2} + 10 g^{4} h i + 24 g^{4} h + 20 g^{4} i^{3} + 54 g^{4} i^{2} + 30 g^{4} i + 40 g^{4} + 40 g^{3} h^{4} + 40 g^{3} h^{3} i + 156 g^{3} h^{3} + 90 g^{3} h^{2} i + 162 g^{3} h^{2} + 40 g^{3} h i^{2} + 20 g^{3} h i + 36 g^{3} h + 40 g^{3} i^{3} + 116 g^{3} i^{2} + 40 g^{3} i + 60 g^{3} + 40 g^{2} h^{4} + 40 g^{2} h^{3} i + 164 g^{2} h^{3} + 90 g^{2} h^{2} i + 168 g^{2} h^{2} + 40 g^{2} h i^{2} + 20 g^{2} h i + 20 g^{2} h + 40 g^{2} i^{3} + 124 g^{2} i^{2} + 18 g^{2} i + 34 g^{2} + 20 g h^{4} + 20 g h^{3} i + 86 g h^{3} + 45 g h^{2} i + 87 g h^{2} + 20 g h i^{2} + 10 g h i + 2 g h + 20 g i^{3} + 66 g i^{2} + 4 h^{4} + 4 h^{3} i + 18 h^{3} + 9 h^{2} i + 18 h^{2} + 4 h i^{2} + 2 h i + 4 i^{3} + 14 i^{2}$" ], "text/plain": [ "" @@ -2729,7 +2735,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4+4a^5b^3c+14a^5b^3+9a^5b^2c+15a^5b^2+4a^5bc^2+2a^5bc+6a^5b+4a^5c^3+10a^5c^2+8a^5c+10a^5+20a^4b^4+20a^4b^3c+74a^4b^3+45a^4b^2c+78a^4b^2+20a^4bc^2+10a^4bc+24a^4b+20a^4c^3+54a^4c^2+30a^4c+40a^4+40a^3b^4+40a^3b^3c+156a^3b^3+90a^3b^2c+162a^3b^2+40a^3bc^2+20a^3bc+36a^3b+40a^3c^3+116a^3c^2+40a^3c+60a^3+40a^2b^4+40a^2b^3c+164a^2b^3+90a^2b^2c+168a^2b^2+40a^2bc^2+20a^2bc+20a^2b+40a^2c^3+124a^2c^2+18a^2c+34a^2+20ab^4+20ab^3c+86ab^3+45ab^2c+87ab^2+20abc^2+10abc+2ab+20ac^3+66ac^2+4b^4+4b^3c+18b^3+9b^2c+18b^2+4bc^2+2bc+4c^3+14c^2 $$" + "$$ 0 \\le 4 g^{5} h^{4}+4 g^{5} h^{3} i+14 g^{5} h^{3}+9 g^{5} h^{2} i+15 g^{5} h^{2}+4 g^{5} h i^{2}+2 g^{5} h i+6 g^{5} h+4 g^{5} i^{3}+10 g^{5} i^{2}+8 g^{5} i+10 g^{5}+20 g^{4} h^{4}+20 g^{4} h^{3} i+74 g^{4} h^{3}+45 g^{4} h^{2} i+78 g^{4} h^{2}+20 g^{4} h i^{2}+10 g^{4} h i+24 g^{4} h+20 g^{4} i^{3}+54 g^{4} i^{2}+30 g^{4} i+40 g^{4}+40 g^{3} h^{4}+40 g^{3} h^{3} i+156 g^{3} h^{3}+90 g^{3} h^{2} i+162 g^{3} h^{2}+40 g^{3} h i^{2}+20 g^{3} h i+36 g^{3} h+40 g^{3} i^{3}+116 g^{3} i^{2}+40 g^{3} i+60 g^{3}+40 g^{2} h^{4}+40 g^{2} h^{3} i+164 g^{2} h^{3}+90 g^{2} h^{2} i+168 g^{2} h^{2}+40 g^{2} h i^{2}+20 g^{2} h i+20 g^{2} h+40 g^{2} i^{3}+124 g^{2} i^{2}+18 g^{2} i+34 g^{2}+20 g h^{4}+20 g h^{3} i+86 g h^{3}+45 g h^{2} i+87 g h^{2}+20 g h i^{2}+10 g h i+2 g h+20 g i^{3}+66 g i^{2}+4 h^{4}+4 h^{3} i+18 h^{3}+9 h^{2} i+18 h^{2}+4 h i^{2}+2 h i+4 i^{3}+14 i^{2} $$" ], "text/plain": [ "" @@ -2753,8 +2759,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2766,7 +2771,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1+a,b\\to 1/(1+b),c\\to 1+c$" + "Substitute $a\\to 1+j,b\\to 1/(1+k),c\\to 1+l$" ], "text/plain": [ "" @@ -2778,7 +2783,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4+16a^5b^3+24a^5b^2+16a^5b+4a^5+4a^4b^4c+18a^4b^4+16a^4b^3c+76a^4b^3+24a^4b^2c+120a^4b^2+16a^4bc+84a^4b+4a^4c+22a^4+14a^3b^4c+30a^3b^4+56a^3b^3c+132a^3b^3+84a^3b^2c+216a^3b^2+56a^3bc+156a^3b+14a^3c+42a^3+18a^2b^4c+22a^2b^4+72a^2b^3c+100a^2b^3+108a^2b^2c+168a^2b^2+72a^2bc+124a^2b+18a^2c+34a^2+4ab^4c^2+ab^4+16ab^3c^2+8ab^3+24ab^2c^2+9ab^2+16abc^2+2ab+4ac^2+4b^4c^3+10b^4c^2+3b^4c+4b^4+16b^3c^3+44b^3c^2+8b^3c+18b^3+24b^2c^3+72b^2c^2+3b^2c+18b^2+16bc^3+52bc^2-2bc+4c^3+14c^2$" + "Numerator after substitutions: $4 j^{5} k^{4} + 16 j^{5} k^{3} + 24 j^{5} k^{2} + 16 j^{5} k + 4 j^{5} + 4 j^{4} k^{4} l + 18 j^{4} k^{4} + 16 j^{4} k^{3} l + 76 j^{4} k^{3} + 24 j^{4} k^{2} l + 120 j^{4} k^{2} + 16 j^{4} k l + 84 j^{4} k + 4 j^{4} l + 22 j^{4} + 14 j^{3} k^{4} l + 30 j^{3} k^{4} + 56 j^{3} k^{3} l + 132 j^{3} k^{3} + 84 j^{3} k^{2} l + 216 j^{3} k^{2} + 56 j^{3} k l + 156 j^{3} k + 14 j^{3} l + 42 j^{3} + 18 j^{2} k^{4} l + 22 j^{2} k^{4} + 72 j^{2} k^{3} l + 100 j^{2} k^{3} + 108 j^{2} k^{2} l + 168 j^{2} k^{2} + 72 j^{2} k l + 124 j^{2} k + 18 j^{2} l + 34 j^{2} + 4 j k^{4} l^{2} + j k^{4} + 16 j k^{3} l^{2} + 8 j k^{3} + 24 j k^{2} l^{2} + 9 j k^{2} + 16 j k l^{2} + 2 j k + 4 j l^{2} + 4 k^{4} l^{3} + 10 k^{4} l^{2} + 3 k^{4} l + 4 k^{4} + 16 k^{3} l^{3} + 44 k^{3} l^{2} + 8 k^{3} l + 18 k^{3} + 24 k^{2} l^{3} + 72 k^{2} l^{2} + 3 k^{2} l + 18 k^{2} + 16 k l^{3} + 52 k l^{2} - 2 k l + 4 l^{3} + 14 l^{2}$" ], "text/plain": [ "" @@ -2814,7 +2819,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 k l \\le k^{2}+l^{2}$$" ], "text/plain": [ "" @@ -2826,7 +2831,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4+16a^5b^3+24a^5b^2+16a^5b+4a^5+4a^4b^4c+18a^4b^4+16a^4b^3c+76a^4b^3+24a^4b^2c+120a^4b^2+16a^4bc+84a^4b+4a^4c+22a^4+14a^3b^4c+30a^3b^4+56a^3b^3c+132a^3b^3+84a^3b^2c+216a^3b^2+56a^3bc+156a^3b+14a^3c+42a^3+18a^2b^4c+22a^2b^4+72a^2b^3c+100a^2b^3+108a^2b^2c+168a^2b^2+72a^2bc+124a^2b+18a^2c+34a^2+4ab^4c^2+ab^4+16ab^3c^2+8ab^3+24ab^2c^2+9ab^2+16abc^2+2ab+4ac^2+4b^4c^3+10b^4c^2+3b^4c+4b^4+16b^3c^3+44b^3c^2+8b^3c+18b^3+24b^2c^3+72b^2c^2+3b^2c+17b^2+16bc^3+52bc^2+4c^3+13c^2 $$" + "$$ 0 \\le 4 j^{5} k^{4}+16 j^{5} k^{3}+24 j^{5} k^{2}+16 j^{5} k+4 j^{5}+4 j^{4} k^{4} l+18 j^{4} k^{4}+16 j^{4} k^{3} l+76 j^{4} k^{3}+24 j^{4} k^{2} l+120 j^{4} k^{2}+16 j^{4} k l+84 j^{4} k+4 j^{4} l+22 j^{4}+14 j^{3} k^{4} l+30 j^{3} k^{4}+56 j^{3} k^{3} l+132 j^{3} k^{3}+84 j^{3} k^{2} l+216 j^{3} k^{2}+56 j^{3} k l+156 j^{3} k+14 j^{3} l+42 j^{3}+18 j^{2} k^{4} l+22 j^{2} k^{4}+72 j^{2} k^{3} l+100 j^{2} k^{3}+108 j^{2} k^{2} l+168 j^{2} k^{2}+72 j^{2} k l+124 j^{2} k+18 j^{2} l+34 j^{2}+4 j k^{4} l^{2}+j k^{4}+16 j k^{3} l^{2}+8 j k^{3}+24 j k^{2} l^{2}+9 j k^{2}+16 j k l^{2}+2 j k+4 j l^{2}+4 k^{4} l^{3}+10 k^{4} l^{2}+3 k^{4} l+4 k^{4}+16 k^{3} l^{3}+44 k^{3} l^{2}+8 k^{3} l+18 k^{3}+24 k^{2} l^{3}+72 k^{2} l^{2}+3 k^{2} l+17 k^{2}+16 k l^{3}+52 k l^{2}+4 l^{3}+13 l^{2} $$" ], "text/plain": [ "" @@ -2850,8 +2855,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2863,7 +2867,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1/(1+a),b\\to 1/(1+b),c\\to 1+c$" + "Substitute $a\\to 1/(1+m),b\\to 1/(1+n),c\\to 1+o$" ], "text/plain": [ "" @@ -2875,7 +2879,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4c^3+6a^5b^4c^2+11a^5b^4c+9a^5b^4+16a^5b^3c^3+28a^5b^3c^2+40a^5b^3c+38a^5b^3+24a^5b^2c^3+48a^5b^2c^2+51a^5b^2c+57a^5b^2+16a^5bc^3+36a^5bc^2+30a^5bc+34a^5b+4a^5c^3+10a^5c^2+8a^5c+10a^5+20a^4b^4c^3+34a^4b^4c^2+45a^4b^4c+40a^4b^4+80a^4b^3c^3+156a^4b^3c^2+160a^4b^3c+170a^4b^3+120a^4b^2c^3+264a^4b^2c^2+195a^4b^2c+246a^4b^2+80a^4bc^3+196a^4bc^2+110a^4bc+136a^4b+20a^4c^3+54a^4c^2+30a^4c+40a^4+40a^3b^4c^3+76a^3b^4c^2+70a^3b^4c+70a^3b^4+160a^3b^3c^3+344a^3b^3c^2+240a^3b^3c+300a^3b^3+240a^3b^2c^3+576a^3b^2c^2+270a^3b^2c+414a^3b^2+160a^3bc^3+424a^3bc^2+140a^3bc+204a^3b+40a^3c^3+116a^3c^2+40a^3c+60a^3+40a^2b^4c^3+84a^2b^4c^2+48a^2b^4c+58a^2b^4+160a^2b^3c^3+376a^2b^3c^2+152a^2b^3c+248a^2b^3+240a^2b^2c^3+624a^2b^2c^2+138a^2b^2c+312a^2b^2+160a^2bc^3+456a^2bc^2+52a^2bc+116a^2b+40a^2c^3+124a^2c^2+18a^2c+34a^2+20ab^4c^3+46ab^4c^2+15ab^4c+19ab^4+80ab^3c^3+204ab^3c^2+40ab^3c+82ab^3+120ab^2c^3+336ab^2c^2+15ab^2c+81ab^2+80abc^3+244abc^2-10abc-2ab+20ac^3+66ac^2+4b^4c^3+10b^4c^2+3b^4c+4b^4+16b^3c^3+44b^3c^2+8b^3c+18b^3+24b^2c^3+72b^2c^2+3b^2c+18b^2+16bc^3+52bc^2-2bc+4c^3+14c^2$" + "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}$" ], "text/plain": [ "" @@ -2911,7 +2915,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 m n \\le m^{2}+n^{2}$$" ], "text/plain": [ "" @@ -2923,7 +2927,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 n o \\le n^{2}+o^{2}$$" ], "text/plain": [ "" @@ -2935,7 +2939,7 @@ { "data": { "text/latex": [ - "$$10abc \\le 2a^2+2ab^2c+4ac^2+2b^3$$" + "$$10 m n o \\le 2 m^{2}+2 m n^{2} o+4 m o^{2}+2 n^{3}$$" ], "text/plain": [ "" @@ -2947,7 +2951,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4c^3+6a^5b^4c^2+11a^5b^4c+9a^5b^4+16a^5b^3c^3+28a^5b^3c^2+40a^5b^3c+38a^5b^3+24a^5b^2c^3+48a^5b^2c^2+51a^5b^2c+57a^5b^2+16a^5bc^3+36a^5bc^2+30a^5bc+34a^5b+4a^5c^3+10a^5c^2+8a^5c+10a^5+20a^4b^4c^3+34a^4b^4c^2+45a^4b^4c+40a^4b^4+80a^4b^3c^3+156a^4b^3c^2+160a^4b^3c+170a^4b^3+120a^4b^2c^3+264a^4b^2c^2+195a^4b^2c+246a^4b^2+80a^4bc^3+196a^4bc^2+110a^4bc+136a^4b+20a^4c^3+54a^4c^2+30a^4c+40a^4+40a^3b^4c^3+76a^3b^4c^2+70a^3b^4c+70a^3b^4+160a^3b^3c^3+344a^3b^3c^2+240a^3b^3c+300a^3b^3+240a^3b^2c^3+576a^3b^2c^2+270a^3b^2c+414a^3b^2+160a^3bc^3+424a^3bc^2+140a^3bc+204a^3b+40a^3c^3+116a^3c^2+40a^3c+60a^3+40a^2b^4c^3+84a^2b^4c^2+48a^2b^4c+58a^2b^4+160a^2b^3c^3+376a^2b^3c^2+152a^2b^3c+248a^2b^3+240a^2b^2c^3+624a^2b^2c^2+138a^2b^2c+312a^2b^2+160a^2bc^3+456a^2bc^2+52a^2bc+116a^2b+40a^2c^3+124a^2c^2+18a^2c+31a^2+20ab^4c^3+46ab^4c^2+15ab^4c+19ab^4+80ab^3c^3+204ab^3c^2+40ab^3c+82ab^3+120ab^2c^3+336ab^2c^2+13ab^2c+81ab^2+80abc^3+244abc^2+20ac^3+62ac^2+4b^4c^3+10b^4c^2+3b^4c+4b^4+16b^3c^3+44b^3c^2+8b^3c+16b^3+24b^2c^3+72b^2c^2+3b^2c+16b^2+16bc^3+52bc^2+4c^3+13c^2 $$" + "$$ 0 \\le 4 m^{5} n^{4} o^{3}+6 m^{5} n^{4} o^{2}+11 m^{5} n^{4} o+9 m^{5} n^{4}+16 m^{5} n^{3} o^{3}+28 m^{5} n^{3} o^{2}+40 m^{5} n^{3} o+38 m^{5} n^{3}+24 m^{5} n^{2} o^{3}+48 m^{5} n^{2} o^{2}+51 m^{5} n^{2} o+57 m^{5} n^{2}+16 m^{5} n o^{3}+36 m^{5} n o^{2}+30 m^{5} n o+34 m^{5} n+4 m^{5} o^{3}+10 m^{5} o^{2}+8 m^{5} o+10 m^{5}+20 m^{4} n^{4} o^{3}+34 m^{4} n^{4} o^{2}+45 m^{4} n^{4} o+40 m^{4} n^{4}+80 m^{4} n^{3} o^{3}+156 m^{4} n^{3} o^{2}+160 m^{4} n^{3} o+170 m^{4} n^{3}+120 m^{4} n^{2} o^{3}+264 m^{4} n^{2} o^{2}+195 m^{4} n^{2} o+246 m^{4} n^{2}+80 m^{4} n o^{3}+196 m^{4} n o^{2}+110 m^{4} n o+136 m^{4} n+20 m^{4} o^{3}+54 m^{4} o^{2}+30 m^{4} o+40 m^{4}+40 m^{3} n^{4} o^{3}+76 m^{3} n^{4} o^{2}+70 m^{3} n^{4} o+70 m^{3} n^{4}+160 m^{3} n^{3} o^{3}+344 m^{3} n^{3} o^{2}+240 m^{3} n^{3} o+300 m^{3} n^{3}+240 m^{3} n^{2} o^{3}+576 m^{3} n^{2} o^{2}+270 m^{3} n^{2} o+414 m^{3} n^{2}+160 m^{3} n o^{3}+424 m^{3} n o^{2}+140 m^{3} n o+204 m^{3} n+40 m^{3} o^{3}+116 m^{3} o^{2}+40 m^{3} o+60 m^{3}+40 m^{2} n^{4} o^{3}+84 m^{2} n^{4} o^{2}+48 m^{2} n^{4} o+58 m^{2} n^{4}+160 m^{2} n^{3} o^{3}+376 m^{2} n^{3} o^{2}+152 m^{2} n^{3} o+248 m^{2} n^{3}+240 m^{2} n^{2} o^{3}+624 m^{2} n^{2} o^{2}+138 m^{2} n^{2} o+312 m^{2} n^{2}+160 m^{2} n o^{3}+456 m^{2} n o^{2}+52 m^{2} n o+116 m^{2} n+40 m^{2} o^{3}+124 m^{2} o^{2}+18 m^{2} o+31 m^{2}+20 m n^{4} o^{3}+46 m n^{4} o^{2}+15 m n^{4} o+19 m n^{4}+80 m n^{3} o^{3}+204 m n^{3} o^{2}+40 m n^{3} o+82 m n^{3}+120 m n^{2} o^{3}+336 m n^{2} o^{2}+13 m n^{2} o+81 m n^{2}+80 m n o^{3}+244 m n o^{2}+20 m o^{3}+62 m o^{2}+4 n^{4} o^{3}+10 n^{4} o^{2}+3 n^{4} o+4 n^{4}+16 n^{3} o^{3}+44 n^{3} o^{2}+8 n^{3} o+16 n^{3}+24 n^{2} o^{3}+72 n^{2} o^{2}+3 n^{2} o+16 n^{2}+16 n o^{3}+52 n o^{2}+4 o^{3}+13 o^{2} $$" ], "text/plain": [ "" @@ -2971,8 +2975,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -2984,7 +2987,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1+a,b\\to 1+b,c\\to 1/(1+c)$" + "Substitute $a\\to 1+p,b\\to 1+q,c\\to 1/(1+r)$" ], "text/plain": [ "" @@ -2996,7 +2999,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5c^3+12a^5c^2+12a^5c+4a^5+4a^4bc^3+12a^4bc^2+12a^4bc+4a^4b+18a^4c^3+58a^4c^2+62a^4c+22a^4+12a^3bc^3+36a^3bc^2+36a^3bc+12a^3b+28a^3c^3+98a^3c^2+112a^3c+42a^3+12a^2bc^3+36a^2bc^2+36a^2bc+12a^2b+16a^2c^3+66a^2c^2+84a^2c+34a^2+4ab^3c^3+12ab^3c^2+12ab^3c+4ab^3+3ab^2c^3+9ab^2c^2+9ab^2c+3ab^2-2abc^3-6abc^2-6abc-2ab+4ac^3+4ac^2+4b^4c^3+12b^4c^2+12b^4c+4b^4+14b^3c^3+46b^3c^2+50b^3c+18b^3+9b^2c^3+36b^2c^2+45b^2c+18b^2+2bc^3-2bc+10c^3+14c^2$" + "Numerator after substitutions: $4 p^{5} r^{3} + 12 p^{5} r^{2} + 12 p^{5} r + 4 p^{5} + 4 p^{4} q r^{3} + 12 p^{4} q r^{2} + 12 p^{4} q r + 4 p^{4} q + 18 p^{4} r^{3} + 58 p^{4} r^{2} + 62 p^{4} r + 22 p^{4} + 12 p^{3} q r^{3} + 36 p^{3} q r^{2} + 36 p^{3} q r + 12 p^{3} q + 28 p^{3} r^{3} + 98 p^{3} r^{2} + 112 p^{3} r + 42 p^{3} + 12 p^{2} q r^{3} + 36 p^{2} q r^{2} + 36 p^{2} q r + 12 p^{2} q + 16 p^{2} r^{3} + 66 p^{2} r^{2} + 84 p^{2} r + 34 p^{2} + 4 p q^{3} r^{3} + 12 p q^{3} r^{2} + 12 p q^{3} r + 4 p q^{3} + 3 p q^{2} r^{3} + 9 p q^{2} r^{2} + 9 p q^{2} r + 3 p q^{2} - 2 p q r^{3} - 6 p q r^{2} - 6 p q r - 2 p q + 4 p r^{3} + 4 p r^{2} + 4 q^{4} r^{3} + 12 q^{4} r^{2} + 12 q^{4} r + 4 q^{4} + 14 q^{3} r^{3} + 46 q^{3} r^{2} + 50 q^{3} r + 18 q^{3} + 9 q^{2} r^{3} + 36 q^{2} r^{2} + 45 q^{2} r + 18 q^{2} + 2 q r^{3} - 2 q r + 10 r^{3} + 14 r^{2}$" ], "text/plain": [ "" @@ -3032,7 +3035,7 @@ { "data": { "text/latex": [ - "$$2abc^3 \\le a^2bc^3+bc^3$$" + "$$2 p q r^{3} \\le p^{2} q r^{3}+q r^{3}$$" ], "text/plain": [ "" @@ -3044,7 +3047,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 p q \\le p^{2}+q^{2}$$" ], "text/plain": [ "" @@ -3056,7 +3059,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 q r \\le q^{2}+r^{2}$$" ], "text/plain": [ "" @@ -3068,7 +3071,7 @@ { "data": { "text/latex": [ - "$$6abc \\le 2a^3b+2b^2c+2c^2$$" + "$$6 p q r \\le 2 p^{3} q+2 q^{2} r+2 r^{2}$$" ], "text/plain": [ "" @@ -3080,7 +3083,7 @@ { "data": { "text/latex": [ - "$$6abc^2 \\le 2ab^2c^3+ab^2+3ac^2$$" + "$$6 p q r^{2} \\le 2 p q^{2} r^{3}+p q^{2}+3 p r^{2}$$" ], "text/plain": [ "" @@ -3092,7 +3095,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5c^3+12a^5c^2+12a^5c+4a^5+4a^4bc^3+12a^4bc^2+12a^4bc+4a^4b+18a^4c^3+58a^4c^2+62a^4c+22a^4+12a^3bc^3+36a^3bc^2+36a^3bc+10a^3b+28a^3c^3+98a^3c^2+112a^3c+42a^3+11a^2bc^3+36a^2bc^2+36a^2bc+12a^2b+16a^2c^3+66a^2c^2+84a^2c+33a^2+4ab^3c^3+12ab^3c^2+12ab^3c+4ab^3+ab^2c^3+9ab^2c^2+9ab^2c+2ab^2+4ac^3+ac^2+4b^4c^3+12b^4c^2+12b^4c+4b^4+14b^3c^3+46b^3c^2+50b^3c+18b^3+9b^2c^3+36b^2c^2+43b^2c+16b^2+bc^3+10c^3+11c^2 $$" + "$$ 0 \\le 4 p^{5} r^{3}+12 p^{5} r^{2}+12 p^{5} r+4 p^{5}+4 p^{4} q r^{3}+12 p^{4} q r^{2}+12 p^{4} q r+4 p^{4} q+18 p^{4} r^{3}+58 p^{4} r^{2}+62 p^{4} r+22 p^{4}+12 p^{3} q r^{3}+36 p^{3} q r^{2}+36 p^{3} q r+10 p^{3} q+28 p^{3} r^{3}+98 p^{3} r^{2}+112 p^{3} r+42 p^{3}+11 p^{2} q r^{3}+36 p^{2} q r^{2}+36 p^{2} q r+12 p^{2} q+16 p^{2} r^{3}+66 p^{2} r^{2}+84 p^{2} r+33 p^{2}+4 p q^{3} r^{3}+12 p q^{3} r^{2}+12 p q^{3} r+4 p q^{3}+p q^{2} r^{3}+9 p q^{2} r^{2}+9 p q^{2} r+2 p q^{2}+4 p r^{3}+p r^{2}+4 q^{4} r^{3}+12 q^{4} r^{2}+12 q^{4} r+4 q^{4}+14 q^{3} r^{3}+46 q^{3} r^{2}+50 q^{3} r+18 q^{3}+9 q^{2} r^{3}+36 q^{2} r^{2}+43 q^{2} r+16 q^{2}+q r^{3}+10 r^{3}+11 r^{2} $$" ], "text/plain": [ "" @@ -3116,8 +3119,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -3129,7 +3131,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1/(1+a),b\\to 1+b,c\\to 1/(1+c)$" + "Substitute $a\\to 1/(1+s),b\\to 1+t,c\\to 1/(1+u)$" ], "text/plain": [ "" @@ -3141,7 +3143,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4c^3+12a^5b^4c^2+12a^5b^4c+4a^5b^4+10a^5b^3c^3+34a^5b^3c^2+38a^5b^3c+14a^5b^3+6a^5b^2c^3+27a^5b^2c^2+36a^5b^2c+15a^5b^2+8a^5bc^3+18a^5bc^2+16a^5bc+6a^5b+8a^5c^3+24a^5c^2+22a^5c+10a^5+20a^4b^4c^3+60a^4b^4c^2+60a^4b^4c+20a^4b^4+54a^4b^3c^3+182a^4b^3c^2+202a^4b^3c+74a^4b^3+33a^4b^2c^3+144a^4b^2c^2+189a^4b^2c+78a^4b^2+34a^4bc^3+72a^4bc^2+62a^4bc+24a^4b+44a^4c^3+114a^4c^2+90a^4c+40a^4+40a^3b^4c^3+120a^3b^4c^2+120a^3b^4c+40a^3b^4+116a^3b^3c^3+388a^3b^3c^2+428a^3b^3c+156a^3b^3+72a^3b^2c^3+306a^3b^2c^2+396a^3b^2c+162a^3b^2+56a^3bc^3+108a^3bc^2+88a^3bc+36a^3b+96a^3c^3+216a^3c^2+140a^3c+60a^3+40a^2b^4c^3+120a^2b^4c^2+120a^2b^4c+40a^2b^4+124a^2b^3c^3+412a^2b^3c^2+452a^2b^3c+164a^2b^3+78a^2b^2c^3+324a^2b^2c^2+414a^2b^2c+168a^2b^2+40a^2bc^3+60a^2bc^2+40a^2bc+20a^2b+100a^2c^3+190a^2c^2+84a^2c+34a^2+20ab^4c^3+60ab^4c^2+60ab^4c+20ab^4+66ab^3c^3+218ab^3c^2+238ab^3c+86ab^3+42ab^2c^3+171ab^2c^2+216ab^2c+87ab^2+12abc^3+6abc^2-4abc+2ab+46ac^3+66ac^2+4b^4c^3+12b^4c^2+12b^4c+4b^4+14b^3c^3+46b^3c^2+50b^3c+18b^3+9b^2c^3+36b^2c^2+45b^2c+18b^2+2bc^3-2bc+10c^3+14c^2$" + "Numerator after substitutions: $4 s^{5} t^{4} u^{3} + 12 s^{5} t^{4} u^{2} + 12 s^{5} t^{4} u + 4 s^{5} t^{4} + 10 s^{5} t^{3} u^{3} + 34 s^{5} t^{3} u^{2} + 38 s^{5} t^{3} u + 14 s^{5} t^{3} + 6 s^{5} t^{2} u^{3} + 27 s^{5} t^{2} u^{2} + 36 s^{5} t^{2} u + 15 s^{5} t^{2} + 8 s^{5} t u^{3} + 18 s^{5} t u^{2} + 16 s^{5} t u + 6 s^{5} t + 8 s^{5} u^{3} + 24 s^{5} u^{2} + 22 s^{5} u + 10 s^{5} + 20 s^{4} t^{4} u^{3} + 60 s^{4} t^{4} u^{2} + 60 s^{4} t^{4} u + 20 s^{4} t^{4} + 54 s^{4} t^{3} u^{3} + 182 s^{4} t^{3} u^{2} + 202 s^{4} t^{3} u + 74 s^{4} t^{3} + 33 s^{4} t^{2} u^{3} + 144 s^{4} t^{2} u^{2} + 189 s^{4} t^{2} u + 78 s^{4} t^{2} + 34 s^{4} t u^{3} + 72 s^{4} t u^{2} + 62 s^{4} t u + 24 s^{4} t + 44 s^{4} u^{3} + 114 s^{4} u^{2} + 90 s^{4} u + 40 s^{4} + 40 s^{3} t^{4} u^{3} + 120 s^{3} t^{4} u^{2} + 120 s^{3} t^{4} u + 40 s^{3} t^{4} + 116 s^{3} t^{3} u^{3} + 388 s^{3} t^{3} u^{2} + 428 s^{3} t^{3} u + 156 s^{3} t^{3} + 72 s^{3} t^{2} u^{3} + 306 s^{3} t^{2} u^{2} + 396 s^{3} t^{2} u + 162 s^{3} t^{2} + 56 s^{3} t u^{3} + 108 s^{3} t u^{2} + 88 s^{3} t u + 36 s^{3} t + 96 s^{3} u^{3} + 216 s^{3} u^{2} + 140 s^{3} u + 60 s^{3} + 40 s^{2} t^{4} u^{3} + 120 s^{2} t^{4} u^{2} + 120 s^{2} t^{4} u + 40 s^{2} t^{4} + 124 s^{2} t^{3} u^{3} + 412 s^{2} t^{3} u^{2} + 452 s^{2} t^{3} u + 164 s^{2} t^{3} + 78 s^{2} t^{2} u^{3} + 324 s^{2} t^{2} u^{2} + 414 s^{2} t^{2} u + 168 s^{2} t^{2} + 40 s^{2} t u^{3} + 60 s^{2} t u^{2} + 40 s^{2} t u + 20 s^{2} t + 100 s^{2} u^{3} + 190 s^{2} u^{2} + 84 s^{2} u + 34 s^{2} + 20 s t^{4} u^{3} + 60 s t^{4} u^{2} + 60 s t^{4} u + 20 s t^{4} + 66 s t^{3} u^{3} + 218 s t^{3} u^{2} + 238 s t^{3} u + 86 s t^{3} + 42 s t^{2} u^{3} + 171 s t^{2} u^{2} + 216 s t^{2} u + 87 s t^{2} + 12 s t u^{3} + 6 s t u^{2} - 4 s t u + 2 s t + 46 s u^{3} + 66 s u^{2} + 4 t^{4} u^{3} + 12 t^{4} u^{2} + 12 t^{4} u + 4 t^{4} + 14 t^{3} u^{3} + 46 t^{3} u^{2} + 50 t^{3} u + 18 t^{3} + 9 t^{2} u^{3} + 36 t^{2} u^{2} + 45 t^{2} u + 18 t^{2} + 2 t u^{3} - 2 t u + 10 u^{3} + 14 u^{2}$" ], "text/plain": [ "" @@ -3177,7 +3179,7 @@ { "data": { "text/latex": [ - "$$4abc \\le a^2c^2+2ab^2+c^2$$" + "$$4 s t u \\le s^{2} u^{2}+2 s t^{2}+u^{2}$$" ], "text/plain": [ "" @@ -3189,7 +3191,7 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$2 t u \\le t^{2}+u^{2}$$" ], "text/plain": [ "" @@ -3201,7 +3203,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4c^3+12a^5b^4c^2+12a^5b^4c+4a^5b^4+10a^5b^3c^3+34a^5b^3c^2+38a^5b^3c+14a^5b^3+6a^5b^2c^3+27a^5b^2c^2+36a^5b^2c+15a^5b^2+8a^5bc^3+18a^5bc^2+16a^5bc+6a^5b+8a^5c^3+24a^5c^2+22a^5c+10a^5+20a^4b^4c^3+60a^4b^4c^2+60a^4b^4c+20a^4b^4+54a^4b^3c^3+182a^4b^3c^2+202a^4b^3c+74a^4b^3+33a^4b^2c^3+144a^4b^2c^2+189a^4b^2c+78a^4b^2+34a^4bc^3+72a^4bc^2+62a^4bc+24a^4b+44a^4c^3+114a^4c^2+90a^4c+40a^4+40a^3b^4c^3+120a^3b^4c^2+120a^3b^4c+40a^3b^4+116a^3b^3c^3+388a^3b^3c^2+428a^3b^3c+156a^3b^3+72a^3b^2c^3+306a^3b^2c^2+396a^3b^2c+162a^3b^2+56a^3bc^3+108a^3bc^2+88a^3bc+36a^3b+96a^3c^3+216a^3c^2+140a^3c+60a^3+40a^2b^4c^3+120a^2b^4c^2+120a^2b^4c+40a^2b^4+124a^2b^3c^3+412a^2b^3c^2+452a^2b^3c+164a^2b^3+78a^2b^2c^3+324a^2b^2c^2+414a^2b^2c+168a^2b^2+40a^2bc^3+60a^2bc^2+40a^2bc+20a^2b+100a^2c^3+189a^2c^2+84a^2c+34a^2+20ab^4c^3+60ab^4c^2+60ab^4c+20ab^4+66ab^3c^3+218ab^3c^2+238ab^3c+86ab^3+42ab^2c^3+171ab^2c^2+216ab^2c+85ab^2+12abc^3+6abc^2+2ab+46ac^3+66ac^2+4b^4c^3+12b^4c^2+12b^4c+4b^4+14b^3c^3+46b^3c^2+50b^3c+18b^3+9b^2c^3+36b^2c^2+45b^2c+17b^2+2bc^3+10c^3+12c^2 $$" + "$$ 0 \\le 4 s^{5} t^{4} u^{3}+12 s^{5} t^{4} u^{2}+12 s^{5} t^{4} u+4 s^{5} t^{4}+10 s^{5} t^{3} u^{3}+34 s^{5} t^{3} u^{2}+38 s^{5} t^{3} u+14 s^{5} t^{3}+6 s^{5} t^{2} u^{3}+27 s^{5} t^{2} u^{2}+36 s^{5} t^{2} u+15 s^{5} t^{2}+8 s^{5} t u^{3}+18 s^{5} t u^{2}+16 s^{5} t u+6 s^{5} t+8 s^{5} u^{3}+24 s^{5} u^{2}+22 s^{5} u+10 s^{5}+20 s^{4} t^{4} u^{3}+60 s^{4} t^{4} u^{2}+60 s^{4} t^{4} u+20 s^{4} t^{4}+54 s^{4} t^{3} u^{3}+182 s^{4} t^{3} u^{2}+202 s^{4} t^{3} u+74 s^{4} t^{3}+33 s^{4} t^{2} u^{3}+144 s^{4} t^{2} u^{2}+189 s^{4} t^{2} u+78 s^{4} t^{2}+34 s^{4} t u^{3}+72 s^{4} t u^{2}+62 s^{4} t u+24 s^{4} t+44 s^{4} u^{3}+114 s^{4} u^{2}+90 s^{4} u+40 s^{4}+40 s^{3} t^{4} u^{3}+120 s^{3} t^{4} u^{2}+120 s^{3} t^{4} u+40 s^{3} t^{4}+116 s^{3} t^{3} u^{3}+388 s^{3} t^{3} u^{2}+428 s^{3} t^{3} u+156 s^{3} t^{3}+72 s^{3} t^{2} u^{3}+306 s^{3} t^{2} u^{2}+396 s^{3} t^{2} u+162 s^{3} t^{2}+56 s^{3} t u^{3}+108 s^{3} t u^{2}+88 s^{3} t u+36 s^{3} t+96 s^{3} u^{3}+216 s^{3} u^{2}+140 s^{3} u+60 s^{3}+40 s^{2} t^{4} u^{3}+120 s^{2} t^{4} u^{2}+120 s^{2} t^{4} u+40 s^{2} t^{4}+124 s^{2} t^{3} u^{3}+412 s^{2} t^{3} u^{2}+452 s^{2} t^{3} u+164 s^{2} t^{3}+78 s^{2} t^{2} u^{3}+324 s^{2} t^{2} u^{2}+414 s^{2} t^{2} u+168 s^{2} t^{2}+40 s^{2} t u^{3}+60 s^{2} t u^{2}+40 s^{2} t u+20 s^{2} t+100 s^{2} u^{3}+189 s^{2} u^{2}+84 s^{2} u+34 s^{2}+20 s t^{4} u^{3}+60 s t^{4} u^{2}+60 s t^{4} u+20 s t^{4}+66 s t^{3} u^{3}+218 s t^{3} u^{2}+238 s t^{3} u+86 s t^{3}+42 s t^{2} u^{3}+171 s t^{2} u^{2}+216 s t^{2} u+85 s t^{2}+12 s t u^{3}+6 s t u^{2}+2 s t+46 s u^{3}+66 s u^{2}+4 t^{4} u^{3}+12 t^{4} u^{2}+12 t^{4} u+4 t^{4}+14 t^{3} u^{3}+46 t^{3} u^{2}+50 t^{3} u+18 t^{3}+9 t^{2} u^{3}+36 t^{2} u^{2}+45 t^{2} u+17 t^{2}+2 t u^{3}+10 u^{3}+12 u^{2} $$" ], "text/plain": [ "" @@ -3225,8 +3227,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -3238,7 +3239,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1+a,b\\to 1/(1+b),c\\to 1/(1+c)$" + "Substitute $a\\to 1+v,b\\to 1/(1+w),c\\to 1/(1+x)$" ], "text/plain": [ "" @@ -3250,7 +3251,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $4a^5b^4c^3+12a^5b^4c^2+12a^5b^4c+4a^5b^4+16a^5b^3c^3+48a^5b^3c^2+48a^5b^3c+16a^5b^3+24a^5b^2c^3+72a^5b^2c^2+72a^5b^2c+24a^5b^2+16a^5bc^3+48a^5bc^2+48a^5bc+16a^5b+4a^5c^3+12a^5c^2+12a^5c+4a^5+14a^4b^4c^3+46a^4b^4c^2+50a^4b^4c+18a^4b^4+60a^4b^3c^3+196a^4b^3c^2+212a^4b^3c+76a^4b^3+96a^4b^2c^3+312a^4b^2c^2+336a^4b^2c+120a^4b^2+68a^4bc^3+220a^4bc^2+236a^4bc+84a^4b+18a^4c^3+58a^4c^2+62a^4c+22a^4+16a^3b^4c^3+62a^3b^4c^2+76a^3b^4c+30a^3b^4+76a^3b^3c^3+284a^3b^3c^2+340a^3b^3c+132a^3b^3+132a^3b^2c^3+480a^3b^2c^2+564a^3b^2c+216a^3b^2+100a^3bc^3+356a^3bc^2+412a^3bc+156a^3b+28a^3c^3+98a^3c^2+112a^3c+42a^3+4a^2b^4c^3+30a^2b^4c^2+48a^2b^4c+22a^2b^4+28a^2b^3c^3+156a^2b^3c^2+228a^2b^3c+100a^2b^3+60a^2b^2c^3+288a^2b^2c^2+396a^2b^2c+168a^2b^2+52a^2bc^3+228a^2bc^2+300a^2bc+124a^2b+16a^2c^3+66a^2c^2+84a^2c+34a^2+5ab^4c^3+7ab^4c^2+3ab^4c+ab^4+24ab^3c^3+40ab^3c^2+24ab^3c+8ab^3+33ab^2c^3+51ab^2c^2+27ab^2c+9ab^2+18abc^3+22abc^2+6abc+2ab+4ac^3+4ac^2+7b^4c^3+16b^4c^2+9b^4c+4b^4+38b^3c^3+82b^3c^2+46b^3c+18b^3+63b^2c^3+120b^2c^2+51b^2c+18b^2+38bc^3+56bc^2+2bc+10c^3+14c^2$" + "Numerator after substitutions: $4 v^{5} w^{4} x^{3} + 12 v^{5} w^{4} x^{2} + 12 v^{5} w^{4} x + 4 v^{5} w^{4} + 16 v^{5} w^{3} x^{3} + 48 v^{5} w^{3} x^{2} + 48 v^{5} w^{3} x + 16 v^{5} w^{3} + 24 v^{5} w^{2} x^{3} + 72 v^{5} w^{2} x^{2} + 72 v^{5} w^{2} x + 24 v^{5} w^{2} + 16 v^{5} w x^{3} + 48 v^{5} w x^{2} + 48 v^{5} w x + 16 v^{5} w + 4 v^{5} x^{3} + 12 v^{5} x^{2} + 12 v^{5} x + 4 v^{5} + 14 v^{4} w^{4} x^{3} + 46 v^{4} w^{4} x^{2} + 50 v^{4} w^{4} x + 18 v^{4} w^{4} + 60 v^{4} w^{3} x^{3} + 196 v^{4} w^{3} x^{2} + 212 v^{4} w^{3} x + 76 v^{4} w^{3} + 96 v^{4} w^{2} x^{3} + 312 v^{4} w^{2} x^{2} + 336 v^{4} w^{2} x + 120 v^{4} w^{2} + 68 v^{4} w x^{3} + 220 v^{4} w x^{2} + 236 v^{4} w x + 84 v^{4} w + 18 v^{4} x^{3} + 58 v^{4} x^{2} + 62 v^{4} x + 22 v^{4} + 16 v^{3} w^{4} x^{3} + 62 v^{3} w^{4} x^{2} + 76 v^{3} w^{4} x + 30 v^{3} w^{4} + 76 v^{3} w^{3} x^{3} + 284 v^{3} w^{3} x^{2} + 340 v^{3} w^{3} x + 132 v^{3} w^{3} + 132 v^{3} w^{2} x^{3} + 480 v^{3} w^{2} x^{2} + 564 v^{3} w^{2} x + 216 v^{3} w^{2} + 100 v^{3} w x^{3} + 356 v^{3} w x^{2} + 412 v^{3} w x + 156 v^{3} w + 28 v^{3} x^{3} + 98 v^{3} x^{2} + 112 v^{3} x + 42 v^{3} + 4 v^{2} w^{4} x^{3} + 30 v^{2} w^{4} x^{2} + 48 v^{2} w^{4} x + 22 v^{2} w^{4} + 28 v^{2} w^{3} x^{3} + 156 v^{2} w^{3} x^{2} + 228 v^{2} w^{3} x + 100 v^{2} w^{3} + 60 v^{2} w^{2} x^{3} + 288 v^{2} w^{2} x^{2} + 396 v^{2} w^{2} x + 168 v^{2} w^{2} + 52 v^{2} w x^{3} + 228 v^{2} w x^{2} + 300 v^{2} w x + 124 v^{2} w + 16 v^{2} x^{3} + 66 v^{2} x^{2} + 84 v^{2} x + 34 v^{2} + 5 v w^{4} x^{3} + 7 v w^{4} x^{2} + 3 v w^{4} x + v w^{4} + 24 v w^{3} x^{3} + 40 v w^{3} x^{2} + 24 v w^{3} x + 8 v w^{3} + 33 v w^{2} x^{3} + 51 v w^{2} x^{2} + 27 v w^{2} x + 9 v w^{2} + 18 v w x^{3} + 22 v w x^{2} + 6 v w x + 2 v w + 4 v x^{3} + 4 v x^{2} + 7 w^{4} x^{3} + 16 w^{4} x^{2} + 9 w^{4} x + 4 w^{4} + 38 w^{3} x^{3} + 82 w^{3} x^{2} + 46 w^{3} x + 18 w^{3} + 63 w^{2} x^{3} + 120 w^{2} x^{2} + 51 w^{2} x + 18 w^{2} + 38 w x^{3} + 56 w x^{2} + 2 w x + 10 x^{3} + 14 x^{2}$" ], "text/plain": [ "" @@ -3274,7 +3275,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^5b^4c^3+12a^5b^4c^2+12a^5b^4c+4a^5b^4+16a^5b^3c^3+48a^5b^3c^2+48a^5b^3c+16a^5b^3+24a^5b^2c^3+72a^5b^2c^2+72a^5b^2c+24a^5b^2+16a^5bc^3+48a^5bc^2+48a^5bc+16a^5b+4a^5c^3+12a^5c^2+12a^5c+4a^5+14a^4b^4c^3+46a^4b^4c^2+50a^4b^4c+18a^4b^4+60a^4b^3c^3+196a^4b^3c^2+212a^4b^3c+76a^4b^3+96a^4b^2c^3+312a^4b^2c^2+336a^4b^2c+120a^4b^2+68a^4bc^3+220a^4bc^2+236a^4bc+84a^4b+18a^4c^3+58a^4c^2+62a^4c+22a^4+16a^3b^4c^3+62a^3b^4c^2+76a^3b^4c+30a^3b^4+76a^3b^3c^3+284a^3b^3c^2+340a^3b^3c+132a^3b^3+132a^3b^2c^3+480a^3b^2c^2+564a^3b^2c+216a^3b^2+100a^3bc^3+356a^3bc^2+412a^3bc+156a^3b+28a^3c^3+98a^3c^2+112a^3c+42a^3+4a^2b^4c^3+30a^2b^4c^2+48a^2b^4c+22a^2b^4+28a^2b^3c^3+156a^2b^3c^2+228a^2b^3c+100a^2b^3+60a^2b^2c^3+288a^2b^2c^2+396a^2b^2c+168a^2b^2+52a^2bc^3+228a^2bc^2+300a^2bc+124a^2b+16a^2c^3+66a^2c^2+84a^2c+34a^2+5ab^4c^3+7ab^4c^2+3ab^4c+ab^4+24ab^3c^3+40ab^3c^2+24ab^3c+8ab^3+33ab^2c^3+51ab^2c^2+27ab^2c+9ab^2+18abc^3+22abc^2+6abc+2ab+4ac^3+4ac^2+7b^4c^3+16b^4c^2+9b^4c+4b^4+38b^3c^3+82b^3c^2+46b^3c+18b^3+63b^2c^3+120b^2c^2+51b^2c+18b^2+38bc^3+56bc^2+2bc+10c^3+14c^2 $$" + "$$ 0 \\le 4 v^{5} w^{4} x^{3}+16 v^{5} w^{3} x^{3}+24 v^{5} w^{2} x^{3}+16 v^{5} w x^{3}+4 v^{5} x^{3}+14 v^{4} w^{4} x^{3}+60 v^{4} w^{3} x^{3}+96 v^{4} w^{2} x^{3}+68 v^{4} w x^{3}+18 v^{4} x^{3}+16 v^{3} w^{4} x^{3}+76 v^{3} w^{3} x^{3}+132 v^{3} w^{2} x^{3}+100 v^{3} w x^{3}+28 v^{3} x^{3}+4 v^{2} w^{4} x^{3}+28 v^{2} w^{3} x^{3}+60 v^{2} w^{2} x^{3}+52 v^{2} w x^{3}+16 v^{2} x^{3}+5 v w^{4} x^{3}+24 v w^{3} x^{3}+33 v w^{2} x^{3}+18 v w x^{3}+4 v x^{3}+7 w^{4} x^{3}+38 w^{3} x^{3}+63 w^{2} x^{3}+38 w x^{3}+10 x^{3}+12 v^{5} w^{4} x^{2}+48 v^{5} w^{3} x^{2}+72 v^{5} w^{2} x^{2}+48 v^{5} w x^{2}+12 v^{5} x^{2}+46 v^{4} w^{4} x^{2}+196 v^{4} w^{3} x^{2}+312 v^{4} w^{2} x^{2}+220 v^{4} w x^{2}+58 v^{4} x^{2}+62 v^{3} w^{4} x^{2}+284 v^{3} w^{3} x^{2}+480 v^{3} w^{2} x^{2}+356 v^{3} w x^{2}+98 v^{3} x^{2}+30 v^{2} w^{4} x^{2}+156 v^{2} w^{3} x^{2}+288 v^{2} w^{2} x^{2}+228 v^{2} w x^{2}+66 v^{2} x^{2}+7 v w^{4} x^{2}+40 v w^{3} x^{2}+51 v w^{2} x^{2}+22 v w x^{2}+4 v x^{2}+16 w^{4} x^{2}+82 w^{3} x^{2}+120 w^{2} x^{2}+56 w x^{2}+14 x^{2}+12 v^{5} w^{4} x+48 v^{5} w^{3} x+72 v^{5} w^{2} x+48 v^{5} w x+12 v^{5} x+50 v^{4} w^{4} x+212 v^{4} w^{3} x+336 v^{4} w^{2} x+236 v^{4} w x+62 v^{4} x+76 v^{3} w^{4} x+340 v^{3} w^{3} x+564 v^{3} w^{2} x+412 v^{3} w x+112 v^{3} x+48 v^{2} w^{4} x+228 v^{2} w^{3} x+396 v^{2} w^{2} x+300 v^{2} w x+84 v^{2} x+3 v w^{4} x+24 v w^{3} x+27 v w^{2} x+6 v w x+9 w^{4} x+46 w^{3} x+51 w^{2} x+2 w x+4 v^{5} w^{4}+16 v^{5} w^{3}+24 v^{5} w^{2}+16 v^{5} w+4 v^{5}+18 v^{4} w^{4}+76 v^{4} w^{3}+120 v^{4} w^{2}+84 v^{4} w+22 v^{4}+30 v^{3} w^{4}+132 v^{3} w^{3}+216 v^{3} w^{2}+156 v^{3} w+42 v^{3}+22 v^{2} w^{4}+100 v^{2} w^{3}+168 v^{2} w^{2}+124 v^{2} w+34 v^{2}+v w^{4}+8 v w^{3}+9 v w^{2}+2 v w+4 w^{4}+18 w^{3}+18 w^{2} $$" ], "text/plain": [ "" @@ -3298,8 +3299,7 @@ { "data": { "text/latex": [ - "\n", - "\\hline\n" + "_______________________" ], "text/plain": [ "" @@ -3311,7 +3311,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 1/(1+a),b\\to 1/(1+b),c\\to 1/(1+c)$" + "Substitute $a\\to 1/(1+y),b\\to 1/(1+z),c\\to 1/(1+a_{1})$" ], "text/plain": [ "" @@ -3323,7 +3323,7 @@ { "data": { "text/latex": [ - "Numerator after substitutions: $11a^5b^4c^2+16a^5b^4c+9a^5b^4+10a^5b^3c^3+62a^5b^3c^2+74a^5b^3c+38a^5b^3+30a^5b^2c^3+117a^5b^2c^2+120a^5b^2c+57a^5b^2+24a^5bc^3+78a^5bc^2+72a^5bc+34a^5b+8a^5c^3+24a^5c^2+22a^5c+10a^5+9a^4b^4c^3+64a^4b^4c^2+75a^4b^4c+40a^4b^4+86a^4b^3c^3+346a^4b^3c^2+350a^4b^3c+170a^4b^3+195a^4b^2c^3+612a^4b^2c^2+543a^4b^2c+246a^4b^2+142a^4bc^3+384a^4bc^2+298a^4bc+136a^4b+44a^4c^3+114a^4c^2+90a^4c+40a^4+36a^3b^4c^3+146a^3b^4c^2+140a^3b^4c+70a^3b^4+244a^3b^3c^3+764a^3b^3c^2+660a^3b^3c+300a^3b^3+480a^3b^2c^3+1278a^3b^2c^2+972a^3b^2c+414a^3b^2+328a^3bc^3+756a^3bc^2+472a^3bc+204a^3b+96a^3c^3+216a^3c^2+140a^3c+60a^3+54a^2b^4c^3+162a^2b^4c^2+126a^2b^4c+58a^2b^4+312a^2b^3c^3+816a^2b^3c^2+592a^2b^3c+248a^2b^3+558a^2b^2c^3+1284a^2b^2c^2+798a^2b^2c+312a^2b^2+360a^2bc^3+700a^2bc^2+296a^2bc+116a^2b+100a^2c^3+190a^2c^2+84a^2c+34a^2+30ab^4c^3+73ab^4c^2+42ab^4c+19ab^4+166ab^3c^3+370ab^3c^2+206ab^3c+82ab^3+282ab^2c^3+549ab^2c^2+228ab^2c+81ab^2+172abc^3+258abc^2+4abc-2ab+46ac^3+66ac^2+7b^4c^3+16b^4c^2+9b^4c+4b^4+38b^3c^3+82b^3c^2+46b^3c+18b^3+63b^2c^3+120b^2c^2+51b^2c+18b^2+38bc^3+56bc^2+2bc+10c^3+14c^2$" + "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}$" ], "text/plain": [ "" @@ -3359,7 +3359,7 @@ { "data": { "text/latex": [ - "$$2ab \\le a^2+b^2$$" + "$$2 y z \\le y^{2}+z^{2}$$" ], "text/plain": [ "" @@ -3371,7 +3371,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 11a^5b^4c^2+16a^5b^4c+9a^5b^4+10a^5b^3c^3+62a^5b^3c^2+74a^5b^3c+38a^5b^3+30a^5b^2c^3+117a^5b^2c^2+120a^5b^2c+57a^5b^2+24a^5bc^3+78a^5bc^2+72a^5bc+34a^5b+8a^5c^3+24a^5c^2+22a^5c+10a^5+9a^4b^4c^3+64a^4b^4c^2+75a^4b^4c+40a^4b^4+86a^4b^3c^3+346a^4b^3c^2+350a^4b^3c+170a^4b^3+195a^4b^2c^3+612a^4b^2c^2+543a^4b^2c+246a^4b^2+142a^4bc^3+384a^4bc^2+298a^4bc+136a^4b+44a^4c^3+114a^4c^2+90a^4c+40a^4+36a^3b^4c^3+146a^3b^4c^2+140a^3b^4c+70a^3b^4+244a^3b^3c^3+764a^3b^3c^2+660a^3b^3c+300a^3b^3+480a^3b^2c^3+1278a^3b^2c^2+972a^3b^2c+414a^3b^2+328a^3bc^3+756a^3bc^2+472a^3bc+204a^3b+96a^3c^3+216a^3c^2+140a^3c+60a^3+54a^2b^4c^3+162a^2b^4c^2+126a^2b^4c+58a^2b^4+312a^2b^3c^3+816a^2b^3c^2+592a^2b^3c+248a^2b^3+558a^2b^2c^3+1284a^2b^2c^2+798a^2b^2c+312a^2b^2+360a^2bc^3+700a^2bc^2+296a^2bc+116a^2b+100a^2c^3+190a^2c^2+84a^2c+33a^2+30ab^4c^3+73ab^4c^2+42ab^4c+19ab^4+166ab^3c^3+370ab^3c^2+206ab^3c+82ab^3+282ab^2c^3+549ab^2c^2+228ab^2c+81ab^2+172abc^3+258abc^2+4abc+46ac^3+66ac^2+7b^4c^3+16b^4c^2+9b^4c+4b^4+38b^3c^3+82b^3c^2+46b^3c+18b^3+63b^2c^3+120b^2c^2+51b^2c+17b^2+38bc^3+56bc^2+2bc+10c^3+14c^2 $$" + "$$ 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} $$" ], "text/plain": [ "" @@ -3404,6 +3404,7 @@ } ], "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", "powerprove(formula)" ] @@ -3428,7 +3429,7 @@ { "data": { "text/latex": [ - "numerator: $21f(-1,4,3)-21f(1,2,3)-21f(2,3,1)+21f(3,-1,4)-21f(3,1,2)+21f(4,3,-1)$" + "numerator: $21 f{\\left(-1,4,3 \\right)} - 21 f{\\left(1,2,3 \\right)} - 21 f{\\left(2,3,1 \\right)} + 21 f{\\left(3,-1,4 \\right)} - 21 f{\\left(3,1,2 \\right)} + 21 f{\\left(4,3,-1 \\right)}$" ], "text/plain": [ "" @@ -3568,7 +3569,8 @@ "metadata": {}, "outputs": [], "source": [ - "sVars.display=lambda x:None #turn off printing of proofs\n", + "shiro.display=lambda x:None #turn off printing of proofs\n", + "newproof()\n", "formula=cyclize('((a-b)/c)^2-2*sqrt(2)*(a-b)/c')" ] }, @@ -3639,7 +3641,7 @@ { "data": { "text/plain": [ - "(2.5326984818340415e-10, 7.129450063690368, 1.7908873553542452e-10)" + "(1.7908873553542452e-10, 2.5326984818340415e-10, 7.129450063690368)" ] }, "execution_count": 28, @@ -3656,7 +3658,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "First and third value is approximately equal to 0, so we can try to replace 0 with 1." + "First and second value is approximately equal to 0, so we can try to replace 0 with 1." ] }, { @@ -3676,7 +3678,7 @@ } ], "source": [ - "prove(formula1,values='1,7,1')" + "prove(formula1,values='1,1,7')" ] }, { @@ -3694,7 +3696,7 @@ { "data": { "text/plain": [ - "(1.4142142855953455, 39809595184.05965, 1)" + "(1, 1.4142142855953455, 39809595184.05965)" ] }, "execution_count": 30, @@ -3703,7 +3705,7 @@ } ], "source": [ - "newvalues=(values[0]/values[2],values[1]/values[2],1)\n", + "newvalues=(1,values[1]/values[0],values[2]/values[0])\n", "newvalues" ] }, @@ -3724,14 +3726,14 @@ } ], "source": [ - "newvalues[0]**2" + "newvalues[1]**2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Now the second value is very big. Technically we could use it, but it would run into overflow error, so we will use 1 instead of it. First value is very close to $\\sqrt{2}$, so this value will be our next try." + "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." ] }, { @@ -3751,7 +3753,7 @@ } ], "source": [ - "prove(formula1,values='sqrt(2),1,1')" + "prove(formula1,values='1,sqrt(2),1')" ] }, { @@ -3781,7 +3783,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "Substitute $a\\to c + d$" ], "text/plain": [ "" @@ -3793,7 +3795,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to b+c$" + "Substitute $c\\to b + e$" ], "text/plain": [ "" @@ -3805,7 +3807,7 @@ { "data": { "text/latex": [ - "numerator: $2a^4b^2+2a^4bc+a^4c^2+4a^3b^3+2\\sqrt{2}a^3b^2c+10a^3b^2c+2\\sqrt{2}a^3bc^2+6a^3bc^2+2a^3c^3+2a^2b^4+2\\sqrt{2}a^2b^3c+10a^2b^3c+6\\sqrt{2}a^2b^2c^2+12a^2b^2c^2+4a^2bc^3+4\\sqrt{2}a^2bc^3+a^2c^4+2ab^4c+2\\sqrt{2}ab^3c^2+6ab^3c^2+4ab^2c^3+4\\sqrt{2}ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4$" + "numerator: $2 b^{4} d^{2} + 2 b^{4} d e + 2 b^{4} e^{2} + 4 b^{3} d^{3} + 2 \\sqrt{2} b^{3} d^{2} e + 10 b^{3} d^{2} e + 2 \\sqrt{2} b^{3} d e^{2} + 6 b^{3} d e^{2} + 4 b^{3} e^{3} + 2 b^{2} d^{4} + 2 \\sqrt{2} b^{2} d^{3} e + 10 b^{2} d^{3} e + 6 \\sqrt{2} b^{2} d^{2} e^{2} + 12 b^{2} d^{2} e^{2} + 4 b^{2} d e^{3} + 4 \\sqrt{2} b^{2} d e^{3} + 2 b^{2} e^{4} + 2 b d^{4} e + 2 \\sqrt{2} b d^{3} e^{2} + 6 b d^{3} e^{2} + 4 b d^{2} e^{3} + 4 \\sqrt{2} b d^{2} e^{3} + 2 \\sqrt{2} b d e^{4} + d^{4} e^{2} + 2 d^{3} e^{3} + d^{2} e^{4}$" ], "text/plain": [ "" @@ -3817,7 +3819,7 @@ { "data": { "text/latex": [ - "denominator: $a^2b^4+2a^2b^3c+a^2b^2c^2+2ab^5+6ab^4c+6ab^3c^2+2ab^2c^3+b^6+4b^5c+6b^4c^2+4b^3c^3+b^2c^4$" + "denominator: $b^{6} + 2 b^{5} d + 4 b^{5} e + b^{4} d^{2} + 6 b^{4} d e + 6 b^{4} e^{2} + 2 b^{3} d^{2} e + 6 b^{3} d e^{2} + 4 b^{3} e^{3} + b^{2} d^{2} e^{2} + 2 b^{2} d e^{3} + b^{2} e^{4}$" ], "text/plain": [ "" @@ -3841,7 +3843,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 2a^4b^2+2a^4bc+a^4c^2+4a^3b^3+2\\sqrt{2}a^3b^2c+10a^3b^2c+2\\sqrt{2}a^3bc^2+6a^3bc^2+2a^3c^3+2a^2b^4+2\\sqrt{2}a^2b^3c+10a^2b^3c+6\\sqrt{2}a^2b^2c^2+12a^2b^2c^2+4\\sqrt{2}a^2bc^3+4a^2bc^3+a^2c^4+2ab^4c+2\\sqrt{2}ab^3c^2+6ab^3c^2+4\\sqrt{2}ab^2c^3+4ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4 $$" + "$$ 0 \\le 2 b^{4} d^{2}+2 b^{4} d e+2 b^{4} e^{2}+4 b^{3} d^{3}+2 \\sqrt{2} b^{3} d^{2} e+10 b^{3} d^{2} e+2 \\sqrt{2} b^{3} d e^{2}+6 b^{3} d e^{2}+4 b^{3} e^{3}+2 b^{2} d^{4}+2 \\sqrt{2} b^{2} d^{3} e+10 b^{2} d^{3} e+6 \\sqrt{2} b^{2} d^{2} e^{2}+12 b^{2} d^{2} e^{2}+4 \\sqrt{2} b^{2} d e^{3}+4 b^{2} d e^{3}+2 b^{2} e^{4}+2 b d^{4} e+2 \\sqrt{2} b d^{3} e^{2}+6 b d^{3} e^{2}+4 \\sqrt{2} b d^{2} e^{3}+4 b d^{2} e^{3}+2 \\sqrt{2} b d e^{4}+d^{4} e^{2}+2 d^{3} e^{3}+d^{2} e^{4} $$" ], "text/plain": [ "" @@ -3877,7 +3879,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "Substitute $a\\to b + f$" ], "text/plain": [ "" @@ -3889,7 +3891,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "Substitute $b\\to c + g$" ], "text/plain": [ "" @@ -3901,7 +3903,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to \\sqrt{2}a$" + "Substitute $f\\to \\sqrt{2} h$" ], "text/plain": [ "" @@ -3913,7 +3915,7 @@ { "data": { "text/latex": [ - "numerator: $4a^4b^2+8a^4bc+8a^4c^2+4\\sqrt{2}a^3b^3-8a^3b^2c+12\\sqrt{2}a^3b^2c-8a^3bc^2+20\\sqrt{2}a^3bc^2+8\\sqrt{2}a^3c^3+2a^2b^4-8\\sqrt{2}a^2b^3c+8a^2b^3c-12\\sqrt{2}a^2b^2c^2+24a^2b^2c^2-4\\sqrt{2}a^2bc^3+20a^2bc^3+4a^2c^4-4ab^4c-8ab^3c^2+4\\sqrt{2}ab^3c^2-4ab^2c^3+6\\sqrt{2}ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4$" + "numerator: $2 c^{4} g^{2} + 2 \\sqrt{2} c^{4} g h + 4 c^{4} h^{2} + 4 c^{3} g^{3} - 4 c^{3} g^{2} h + 6 \\sqrt{2} c^{3} g^{2} h - 4 \\sqrt{2} c^{3} g h^{2} + 20 c^{3} g h^{2} + 8 \\sqrt{2} c^{3} h^{3} + 2 c^{2} g^{4} - 8 c^{2} g^{3} h + 4 \\sqrt{2} c^{2} g^{3} h - 12 \\sqrt{2} c^{2} g^{2} h^{2} + 24 c^{2} g^{2} h^{2} - 8 c^{2} g h^{3} + 20 \\sqrt{2} c^{2} g h^{3} + 8 c^{2} h^{4} - 4 c g^{4} h - 8 \\sqrt{2} c g^{3} h^{2} + 8 c g^{3} h^{2} - 8 c g^{2} h^{3} + 12 \\sqrt{2} c g^{2} h^{3} + 8 c g h^{4} + 2 g^{4} h^{2} + 4 \\sqrt{2} g^{3} h^{3} + 4 g^{2} h^{4}$" ], "text/plain": [ "" @@ -3925,7 +3927,7 @@ { "data": { "text/latex": [ - "denominator: $2a^2b^2c^2+4a^2bc^3+2a^2c^4+2\\sqrt{2}ab^3c^2+6\\sqrt{2}ab^2c^3+6\\sqrt{2}abc^4+2\\sqrt{2}ac^5+b^4c^2+4b^3c^3+6b^2c^4+4bc^5+c^6$" + "denominator: $c^{6} + 4 c^{5} g + 2 \\sqrt{2} c^{5} h + 6 c^{4} g^{2} + 6 \\sqrt{2} c^{4} g h + 2 c^{4} h^{2} + 4 c^{3} g^{3} + 6 \\sqrt{2} c^{3} g^{2} h + 4 c^{3} g h^{2} + c^{2} g^{4} + 2 \\sqrt{2} c^{2} g^{3} h + 2 c^{2} g^{2} h^{2}$" ], "text/plain": [ "" @@ -3961,7 +3963,7 @@ { "data": { "text/latex": [ - "$$8a^3b^2c \\le 4a^4b^2+4a^2b^2c^2$$" + "$$4 c^{3} g^{2} h \\le 2 c^{4} g^{2}+2 c^{2} g^{2} h^{2}$$" ], "text/plain": [ "" @@ -3973,7 +3975,7 @@ { "data": { "text/latex": [ - "$$8\\sqrt{2}a^2b^3c \\le 4\\sqrt{2}a^3b^3+4\\sqrt{2}ab^3c^2$$" + "$$4 \\sqrt{2} c^{3} g h^{2} \\le 2 \\sqrt{2} c^{4} g h+\\sqrt{2} c^{3} h^{3}+\\sqrt{2} c g^{2} h^{3}$$" ], "text/plain": [ "" @@ -3985,7 +3987,7 @@ { "data": { "text/latex": [ - "$$12\\sqrt{2}a^2b^2c^2 \\le 6\\sqrt{2}a^3b^2c+6\\sqrt{2}ab^2c^3$$" + "$$8 c^{2} g^{3} h \\le 4 c^{3} g^{3}+4 c g^{3} h^{2}$$" ], "text/plain": [ "" @@ -3997,7 +3999,7 @@ { "data": { "text/latex": [ - "$$4\\sqrt{2}a^2bc^3 \\le 2\\sqrt{2}a^3bc^2+2\\sqrt{2}abc^4$$" + "$$12 \\sqrt{2} c^{2} g^{2} h^{2} \\le 6 \\sqrt{2} c^{3} g^{2} h+6 \\sqrt{2} c g^{2} h^{3}$$" ], "text/plain": [ "" @@ -4009,7 +4011,7 @@ { "data": { "text/latex": [ - "$$4ab^4c \\le 2a^2b^4+2b^4c^2$$" + "$$8 c^{2} g h^{3} \\le 4 c^{3} g h^{2}+2 c^{2} h^{4}+2 g^{2} h^{4}$$" ], "text/plain": [ "" @@ -4021,7 +4023,7 @@ { "data": { "text/latex": [ - "$$8ab^3c^2 \\le 4a^2b^3c+4b^3c^3$$" + "$$4 c g^{4} h \\le 2 c^{2} g^{4}+2 g^{4} h^{2}$$" ], "text/plain": [ "" @@ -4033,7 +4035,7 @@ { "data": { "text/latex": [ - "$$8a^3bc^2 \\le 4a^4bc+4a^2bc^3$$" + "$$8 \\sqrt{2} c g^{3} h^{2} \\le 4 \\sqrt{2} c^{2} g^{3} h+4 \\sqrt{2} g^{3} h^{3}$$" ], "text/plain": [ "" @@ -4045,7 +4047,7 @@ { "data": { "text/latex": [ - "$$4ab^2c^3 \\le 2a^2b^2c^2+2b^2c^4$$" + "$$8 c g^{2} h^{3} \\le 4 c g^{3} h^{2}+4 c g h^{4}$$" ], "text/plain": [ "" @@ -4057,7 +4059,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 4a^4bc+8a^4c^2+6\\sqrt{2}a^3b^2c+18\\sqrt{2}a^3bc^2+8\\sqrt{2}a^3c^3+4a^2b^3c+18a^2b^2c^2+16a^2bc^3+4a^2c^4 $$" + "$$ 0 \\le 4 c^{4} h^{2}+16 c^{3} g h^{2}+7 \\sqrt{2} c^{3} h^{3}+22 c^{2} g^{2} h^{2}+20 \\sqrt{2} c^{2} g h^{3}+6 c^{2} h^{4}+5 \\sqrt{2} c g^{2} h^{3}+4 c g h^{4}+2 g^{2} h^{4} $$" ], "text/plain": [ "" @@ -4090,22 +4092,16 @@ } ], "source": [ - "sVars.display=lambda x:display(Latex(x)) #turn on printing proofs \n", + "newproof()\n", + "shiro.display=lambda x:display(Latex(x)) #turn on printing proofs \n", "formula=cyclize('((a-b)/c)^2-2*sqrt(2)*(a-b)/c')\n", "display(Latex('Case $a\\ge c\\ge b$'))\n", "formula1=makesubs(formula,'[c,oo],[b,oo]',variables='a,c,b')\n", "prove(formula1)\n", "display(Latex('Case $a\\ge b\\ge c$'))\n", "formula2=makesubs(formula,'[b,oo],[c,oo]')\n", - "prove(formula2,values='2**(1/2),1,1')" + "prove(formula2,values='1,2**(1/2),1')" ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] } ], "metadata": { diff --git a/examples.pdf b/examples.pdf index 71c7234f4aefd958513db3209d7541198c746231..f90f158f16c5ce450ee3b0594c67779a0ed6b2f4 100644 GIT binary patch literal 59577 zcma&NL$EMPu&ueRf7`Zg+qP}nwr$(CZQHhOqfbZl;C8$>xRZ*?h#XZ8)|Z*9NaTe@ zX&7i(ph%AIZ{DF881d=x?F=oUxVh;>Ev%hQ9O*=@4V+DcO^ob}P3WXeY|Wg_@flf| znRt1joSYp^3~Zp>H)k}j;*MGo{&xEY^E@9mI{y&{*7#JIER2Wz^rY=Tlc&9ItX@pc5BsR~sD^ebhoEL1;=4%aWx8Ah zPp))JR*`)*Qd3P0%4+eJ8=9nY3?WFGgUCg)OHDT_6ZqLJX|qg@4)E{;T7Ia?4$endW#+UmpN+CdI&dGiR!`1A@ZBS}xpU7-(GT|h zf5g9BsYpN`F1Rpvn7uqS!RVmb52bgQ+-?*^p(ns22!KbUR_>l&oWvrEh_m^m9%XSM z$_+vYtYXanf&{!Z{M#ldM8Fc(G>YXpL(7!vqk=w#Npj%9O&TdkS|qA+A?}lh##St2 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"metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "denominator: $4a^2b+2a^2c+2ab^2+9abc+4ac^2+4b^2c+2bc^2$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "status: 0" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "From weighted AM-GM inequality:" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$18a^2b \\le 10a^3+6ab^2+2b^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$18ac^2 \\le 2a^3+6a^2c+10c^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$18b^2c \\le 10b^3+6bc^2+2c^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le 0 $$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "0" - ] - }, - "execution_count": 3, + "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "formula=cyclize('(a+b)/(2*b+c)')-2\n", - "prove(formula*6)" + "Latex(r'__________________________')" ] }, { @@ -165,7 +59,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to s-s/(a+1)$" + "Substitute $a\\to s-\\frac{s}{d+1}$" ], "text/plain": [ "" @@ -177,7 +71,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to s-s/(b+1)$" + "Substitute $b\\to s-\\frac{s}{e+1}$" ], "text/plain": [ "" @@ -189,7 +83,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to s-s/(c+1)$" + "Substitute $c\\to s-\\frac{s}{f+1}$" ], "text/plain": [ "" @@ -201,7 +95,7 @@ { "data": { "text/latex": [ - "numerator: $abcs^2+s^2$" + "numerator: $defs^2+s^2$" ], "text/plain": [ "" @@ -213,7 +107,7 @@ { "data": { "text/latex": [ - "denominator: $abc+ab+ac+a+bc+b+c+1$" + "denominator: $def+de+df+d+ef+e+f+1$" ], "text/plain": [ "" @@ -237,7 +131,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le abcs^2+s^2 $$" + "$$ 0 \\le defs^2+s^2 $$" ], "text/plain": [ "" @@ -317,7 +211,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+y$" + "Substitute $x\\to g+y$" ], "text/plain": [ "" @@ -329,7 +223,7 @@ { "data": { "text/latex": [ - "$$x^{2}$$" + "$$g^{2}$$" ], "text/plain": [ "" @@ -372,7 +266,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $y\\to h+x$" ], "text/plain": [ "" @@ -384,7 +278,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} z^{t} - x x^{t} y + x y z^{t} + x y \\left(x + y\\right)^{t} - 2 x z z^{t} + x^{t} y z + y^{2} \\left(x + y\\right)^{t} - y z z^{t} - y z \\left(x + y\\right)^{t} + z^{2} z^{t}$$" + "4 $$h^{2} \\left(h + x\\right)^{t} - h x x^{t} + h x z^{t} + h x \\left(h + x\\right)^{t} + h x^{t} z - h z z^{t} - h z \\left(h + x\\right)^{t} + x^{2} z^{t} - 2 x z z^{t} + z^{2} z^{t}$$" ], "text/plain": [ "" @@ -396,7 +290,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to x+z$" + "Substitute $z\\to i+x$" ], "text/plain": [ "" @@ -408,7 +302,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} y^{t} - x x^{t} z - 2 x y y^{t} + x y^{t} z + x z \\left(x + z\\right)^{t} + x^{t} y z + y^{2} y^{t} - y y^{t} z - y z \\left(x + z\\right)^{t} + z^{2} \\left(x + z\\right)^{t}$$" + "4 $$i^{2} \\left(i + x\\right)^{t} - i x x^{t} + i x y^{t} + i x \\left(i + x\\right)^{t} + i x^{t} y - i y y^{t} - i y \\left(i + x\\right)^{t} + x^{2} y^{t} - 2 x y y^{t} + y^{2} y^{t}$$" ], "text/plain": [ "" @@ -420,7 +314,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to y+z$" + "Substitute $z\\to j+y$" ], "text/plain": [ "" @@ -432,7 +326,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} x^{t} - 2 x x^{t} y - x x^{t} z + x y^{t} z - x z \\left(y + z\\right)^{t} + x^{t} y^{2} + x^{t} y z - y y^{t} z + y z \\left(y + z\\right)^{t} + z^{2} \\left(y + z\\right)^{t}$$" + "4 $$j^{2} \\left(j + y\\right)^{t} - j x x^{t} + j x y^{t} - j x \\left(j + y\\right)^{t} + j x^{t} y - j y y^{t} + j y \\left(j + y\\right)^{t} + x^{2} x^{t} - 2 x x^{t} y + x^{t} y^{2}$$" ], "text/plain": [ "" @@ -444,7 +338,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $y\\to k+x$" ], "text/plain": [ "" @@ -456,7 +350,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to x+z$" + "Substitute $z\\to l+x$" ], "text/plain": [ "" @@ -468,7 +362,7 @@ { "data": { "text/latex": [ - "4 $$x^{t} y z + y^{2} \\left(x + y\\right)^{t} - y z \\left(x + y\\right)^{t} - y z \\left(x + z\\right)^{t} + z^{2} \\left(x + z\\right)^{t}$$" + "4 $$k^{2} \\left(k + x\\right)^{t} + k l x^{t} - k l \\left(k + x\\right)^{t} - k l \\left(l + x\\right)^{t} + l^{2} \\left(l + x\\right)^{t}$$" ], "text/plain": [ "" @@ -480,7 +374,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to y+z$" + "Substitute $l\\to k+m$" ], "text/plain": [ "" @@ -492,7 +386,7 @@ { "data": { "text/latex": [ - "3 $$x^{t} y^{2} + x^{t} y z - y z \\left(x + y\\right)^{t} + y z \\left(x + y + z\\right)^{t} + z^{2} \\left(x + y + z\\right)^{t}$$" + "3 $$k^{2} x^{t} + k m x^{t} - k m \\left(k + x\\right)^{t} + k m \\left(k + m + x\\right)^{t} + m^{2} \\left(k + m + x\\right)^{t}$$" ], "text/plain": [ "" @@ -504,7 +398,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to x+z$" + "Substitute $z\\to n+x$" ], "text/plain": [ "" @@ -516,7 +410,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $y\\to o+x$" ], "text/plain": [ "" @@ -528,7 +422,7 @@ { "data": { "text/latex": [ - "4 $$x^{t} y z + y^{2} \\left(x + y\\right)^{t} - y z \\left(x + y\\right)^{t} - y z \\left(x + z\\right)^{t} + z^{2} \\left(x + z\\right)^{t}$$" + "4 $$n^{2} \\left(n + x\\right)^{t} + n o x^{t} - n o \\left(n + x\\right)^{t} - n o \\left(o + x\\right)^{t} + o^{2} \\left(o + x\\right)^{t}$$" ], "text/plain": [ "" @@ -540,7 +434,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to y+z$" + "Substitute $o\\to n+p$" ], "text/plain": [ "" @@ -552,7 +446,7 @@ { "data": { "text/latex": [ - "3 $$x^{t} y^{2} + x^{t} y z - y z \\left(x + y\\right)^{t} + y z \\left(x + y + z\\right)^{t} + z^{2} \\left(x + y + z\\right)^{t}$$" + "3 $$n^{2} x^{t} + n p x^{t} - n p \\left(n + x\\right)^{t} + n p \\left(n + p + x\\right)^{t} + p^{2} \\left(n + p + x\\right)^{t}$$" ], "text/plain": [ "" @@ -574,7 +468,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+y$" + "Substitute $x\\to q+y$" ], "text/plain": [ "" @@ -586,10 +480,10 @@ { "data": { "text/latex": [ - "$\\displaystyle x^{2} \\left(x + y\\right)^{t} - x y y^{t} + x y z^{t} + x y \\left(x + y\\right)^{t} + x y^{t} z - x z z^{t} - x z \\left(x + y\\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$" + "$\\displaystyle q^{2} \\left(q + y\\right)^{t} - q y y^{t} + q y z^{t} + q y \\left(q + y\\right)^{t} + q y^{t} z - q z z^{t} - q z \\left(q + y\\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$" ], "text/plain": [ - "x**2*(x + y)**t - x*y*y**t + x*y*z**t + x*y*(x + y)**t + x*y**t*z - x*z*z**t - x*z*(x + y)**t + y**2*z**t - 2*y*z*z**t + z**2*z**t" + "q**2*(q + y)**t - q*y*y**t + q*y*z**t + q*y*(q + y)**t + q*y**t*z - q*z*z**t - q*z*(q + y)**t + y**2*z**t - 2*y*z*z**t + z**2*z**t" ] }, "execution_count": 10, @@ -609,7 +503,7 @@ { "data": { "text/latex": [ - "Substitute $x\\to x+y$" + "Substitute $x\\to r+y$" ], "text/plain": [ "" @@ -621,10 +515,10 @@ { "data": { "text/latex": [ - "$\\displaystyle x^{2} \\left(x + y\\right)^{t} - x y y^{t} + x y z^{t} + x y \\left(x + y\\right)^{t} + x y^{t} z - x z z^{t} - x z \\left(x + y\\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$" + "$\\displaystyle r^{2} \\left(r + y\\right)^{t} - r y y^{t} + r y z^{t} + r y \\left(r + y\\right)^{t} + r y^{t} z - r z z^{t} - r z \\left(r + y\\right)^{t} + y^{2} z^{t} - 2 y z z^{t} + z^{2} z^{t}$" ], "text/plain": [ - "x**2*(x + y)**t - x*y*y**t + x*y*z**t + x*y*(x + y)**t + x*y**t*z - x*z*z**t - x*z*(x + y)**t + y**2*z**t - 2*y*z*z**t + z**2*z**t" + "r**2*(r + y)**t - r*y*y**t + r*y*z**t + r*y*(r + y)**t + r*y**t*z - r*z*z**t - r*z*(r + y)**t + y**2*z**t - 2*y*z*z**t + z**2*z**t" ] }, "execution_count": 11, @@ -644,7 +538,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $y\\to u+x$" ], "text/plain": [ "" @@ -656,7 +550,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} z^{t} - x x^{t} y + x y z^{t} + x y \\left(x + y\\right)^{t} - 2 x z z^{t} + x^{t} y z + y^{2} \\left(x + y\\right)^{t} - y z z^{t} - y z \\left(x + y\\right)^{t} + z^{2} z^{t}$$" + "4 $$u^{2} \\left(u + x\\right)^{t} - u x x^{t} + u x z^{t} + u x \\left(u + x\\right)^{t} + u x^{t} z - u z z^{t} - u z \\left(u + x\\right)^{t} + x^{2} z^{t} - 2 x z z^{t} + z^{2} z^{t}$$" ], "text/plain": [ "" @@ -668,7 +562,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to x+z$" + "Substitute $z\\to v+x$" ], "text/plain": [ "" @@ -680,7 +574,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} y^{t} - x x^{t} z - 2 x y y^{t} + x y^{t} z + x z \\left(x + z\\right)^{t} + x^{t} y z + y^{2} y^{t} - y y^{t} z - y z \\left(x + z\\right)^{t} + z^{2} \\left(x + z\\right)^{t}$$" + "4 $$v^{2} \\left(v + x\\right)^{t} - v x x^{t} + v x y^{t} + v x \\left(v + x\\right)^{t} + v x^{t} y - v y y^{t} - v y \\left(v + x\\right)^{t} + x^{2} y^{t} - 2 x y y^{t} + y^{2} y^{t}$$" ], "text/plain": [ "" @@ -692,7 +586,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to y+z$" + "Substitute $z\\to w+y$" ], "text/plain": [ "" @@ -704,7 +598,7 @@ { "data": { "text/latex": [ - "4 $$x^{2} x^{t} - 2 x x^{t} y - x x^{t} z + x y^{t} z - x z \\left(y + z\\right)^{t} + x^{t} y^{2} + x^{t} y z - y y^{t} z + y z \\left(y + z\\right)^{t} + z^{2} \\left(y + z\\right)^{t}$$" + "4 $$w^{2} \\left(w + y\\right)^{t} - w x x^{t} + w x y^{t} - w x \\left(w + y\\right)^{t} + w x^{t} y - w y y^{t} + w y \\left(w + y\\right)^{t} + x^{2} x^{t} - 2 x x^{t} y + x^{t} y^{2}$$" ], "text/plain": [ "" @@ -716,7 +610,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $y\\to a_{1}+x$" ], "text/plain": [ "" @@ -728,7 +622,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to x+z$" + "Substitute $z\\to b_{1}+x$" ], "text/plain": [ "" @@ -740,7 +634,7 @@ { "data": { "text/latex": [ - "4 $$x^{t} y z + y^{2} \\left(x + y\\right)^{t} - y z \\left(x + y\\right)^{t} - y z \\left(x + z\\right)^{t} + z^{2} \\left(x + z\\right)^{t}$$" + "4 $$a_{1}^{2} \\left(a_{1} + x\\right)^{t} + a_{1} b_{1} x^{t} - a_{1} b_{1} \\left(a_{1} + x\\right)^{t} - a_{1} b_{1} \\left(b_{1} + x\\right)^{t} + b_{1}^{2} \\left(b_{1} + x\\right)^{t}$$" ], "text/plain": [ "" @@ -752,7 +646,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to y+z$" + "Substitute $b_{1}\\to a_{1}+c_{1}$" ], "text/plain": [ "" @@ -764,7 +658,7 @@ { "data": { "text/latex": [ - "3 $$x^{t} y^{2} + x^{t} y z - y z \\left(x + y\\right)^{t} + y z \\left(x + y + z\\right)^{t} + z^{2} \\left(x + y + z\\right)^{t}$$" + "3 $$a_{1}^{2} x^{t} + a_{1} c_{1} x^{t} - a_{1} c_{1} \\left(a_{1} + x\\right)^{t} + a_{1} c_{1} \\left(a_{1} + c_{1} + x\\right)^{t} + c_{1}^{2} \\left(a_{1} + c_{1} + x\\right)^{t}$$" ], "text/plain": [ "" @@ -776,7 +670,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to x+z$" + "Substitute $z\\to d_{1}+x$" ], "text/plain": [ "" @@ -788,7 +682,7 @@ { "data": { "text/latex": [ - "Substitute $y\\to x+y$" + "Substitute $y\\to e_{1}+x$" ], "text/plain": [ "" @@ -800,7 +694,7 @@ { "data": { "text/latex": [ - "4 $$x^{t} y z + y^{2} \\left(x + y\\right)^{t} - y z \\left(x + y\\right)^{t} - y z \\left(x + z\\right)^{t} + z^{2} \\left(x + z\\right)^{t}$$" + "4 $$d_{1}^{2} \\left(d_{1} + x\\right)^{t} + d_{1} e_{1} x^{t} - d_{1} e_{1} \\left(d_{1} + x\\right)^{t} - d_{1} e_{1} \\left(e_{1} + x\\right)^{t} + e_{1}^{2} \\left(e_{1} + x\\right)^{t}$$" ], "text/plain": [ "" @@ -812,7 +706,7 @@ { "data": { "text/latex": [ - "Substitute $z\\to y+z$" + "Substitute $e_{1}\\to d_{1}+f_{1}$" ], "text/plain": [ "" @@ -824,7 +718,7 @@ { "data": { "text/latex": [ - "3 $$x^{t} y^{2} + x^{t} y z - y z \\left(x + y\\right)^{t} + y z \\left(x + y + z\\right)^{t} + z^{2} \\left(x + y + z\\right)^{t}$$" + "3 $$d_{1}^{2} x^{t} + d_{1} f_{1} x^{t} - d_{1} f_{1} \\left(d_{1} + x\\right)^{t} + d_{1} f_{1} \\left(d_{1} + f_{1} + x\\right)^{t} + f_{1}^{2} \\left(d_{1} + f_{1} + x\\right)^{t}$$" ], "text/plain": [ "" @@ -953,7 +847,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to 11a/8$" + "Substitute $a\\to 11g_{1}/8$" ], "text/plain": [ "" @@ -965,7 +859,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to 4b/5$" + "Substitute $b\\to 4h_{1}/5$" ], "text/plain": [ "" @@ -977,7 +871,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to 4c/5$" + "Substitute $c\\to 4i_{1}/5$" ], "text/plain": [ "" @@ -989,7 +883,7 @@ { "data": { "text/latex": [ - "Substitute $d\\to 4d/5$" + "Substitute $d\\to 4j_{1}/5$" ], "text/plain": [ "" @@ -1001,7 +895,7 @@ { "data": { "text/latex": [ - "numerator: $3025a^2-1760ab-1760ac-1760ad+1024b^2+1024c^2+1024d^2$" + "numerator: $3025g_{1}^2-1760g_{1}h_{1}-1760g_{1}i_{1}-1760g_{1}j_{1}+1024h_{1}^2+1024i_{1}^2+1024j_{1}^2$" ], "text/plain": [ "" @@ -1049,7 +943,7 @@ { "data": { "text/latex": [ - "$$1760ab \\le 880a^2+880b^2$$" + "$$1760g_{1}h_{1} \\le 880g_{1}^2+880h_{1}^2$$" ], "text/plain": [ "" @@ -1061,7 +955,7 @@ { "data": { "text/latex": [ - "$$1760ac \\le 880a^2+880c^2$$" + "$$1760g_{1}i_{1} \\le 880g_{1}^2+880i_{1}^2$$" ], "text/plain": [ "" @@ -1073,7 +967,7 @@ { "data": { "text/latex": [ - "$$1760ad \\le 880a^2+880d^2$$" + "$$1760g_{1}j_{1} \\le 880g_{1}^2+880j_{1}^2$$" ], "text/plain": [ "" @@ -1085,7 +979,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 385a^2+144b^2+144c^2+144d^2 $$" + "$$ 0 \\le 385g_{1}^2+144h_{1}^2+144i_{1}^2+144j_{1}^2 $$" ], "text/plain": [ "" @@ -1148,76 +1042,29 @@ "outputs": [ { "data": { - "text/latex": [ - "$\\displaystyle - \\frac{2 \\sqrt{2} \\left(a - b\\right)}{c} + \\frac{\\left(a - b\\right)^{2}}{c^{2}} - \\frac{2 \\sqrt{2} \\left(- a + c\\right)}{b} + \\frac{\\left(- a + c\\right)^{2}}{b^{2}} - \\frac{2 \\sqrt{2} \\left(b - c\\right)}{a} + \\frac{\\left(b - c\\right)^{2}}{a^{2}}$" - ], "text/plain": [ - "-2*sqrt(2)*(a - b)/c + (a - b)**2/c**2 - 2*sqrt(2)*(-a + c)/b + (-a + c)**2/b**2 - 2*sqrt(2)*(b - c)/a + (b - c)**2/a**2" + "\"formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\\ndisplay(formula)\\nfrom scipy.optimize import fmin\\nimport numpy as np\\ndef f(x):\\n num,den=fraction(cancel(newformula))\\n fs=sorted(newformula.free_symbols,key=str)\\n return float(num.subs(list(zip(fs,x))))\\nnewformula=(makesubs(formula,'[b,oo],[c,oo]'))\\nprint(f([2,1,1]))\\nfmin(f,[2,1,1])\\nprint(fmin(f,np.array([2,1,1])))\\ndisplay(simplify(newformula))\"" ] }, + "execution_count": 20, "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $a\\to a+b$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $b\\to b+c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Optimization terminated successfully.\n", - " Current function value: 0.000000\n", - " Iterations: 51\n", - " Function evaluations: 97\n", - "[1.17092486 1.170961 1.17093843]\n" - ] - }, - { - "data": { - "text/latex": [ - "$\\displaystyle \\frac{a^{4} b^{2} + 2 a^{4} b c + 2 a^{4} c^{2} + 2 a^{3} b^{3} - 2 \\sqrt{2} a^{3} b^{2} c + 6 a^{3} b^{2} c - 2 \\sqrt{2} a^{3} b c^{2} + 10 a^{3} b c^{2} + 4 a^{3} c^{3} + a^{2} b^{4} - 4 \\sqrt{2} a^{2} b^{3} c + 4 a^{2} b^{3} c - 6 \\sqrt{2} a^{2} b^{2} c^{2} + 12 a^{2} b^{2} c^{2} - 2 \\sqrt{2} a^{2} b c^{3} + 10 a^{2} b c^{3} + 2 a^{2} c^{4} - 2 \\sqrt{2} a b^{4} c - 4 \\sqrt{2} a b^{3} c^{2} + 4 a b^{3} c^{2} - 2 \\sqrt{2} a b^{2} c^{3} + 6 a b^{2} c^{3} + 2 a b c^{4} + 2 b^{4} c^{2} + 4 b^{3} c^{3} + 2 b^{2} c^{4}}{c^{2} \\left(a^{2} b^{2} + 2 a^{2} b c + a^{2} c^{2} + 2 a b^{3} + 6 a b^{2} c + 6 a b c^{2} + 2 a c^{3} + b^{4} + 4 b^{3} c + 6 b^{2} c^{2} + 4 b c^{3} + c^{4}\\right)}$" - ], - "text/plain": [ - "(a**4*b**2 + 2*a**4*b*c + 2*a**4*c**2 + 2*a**3*b**3 - 2*sqrt(2)*a**3*b**2*c + 6*a**3*b**2*c - 2*sqrt(2)*a**3*b*c**2 + 10*a**3*b*c**2 + 4*a**3*c**3 + a**2*b**4 - 4*sqrt(2)*a**2*b**3*c + 4*a**2*b**3*c - 6*sqrt(2)*a**2*b**2*c**2 + 12*a**2*b**2*c**2 - 2*sqrt(2)*a**2*b*c**3 + 10*a**2*b*c**3 + 2*a**2*c**4 - 2*sqrt(2)*a*b**4*c - 4*sqrt(2)*a*b**3*c**2 + 4*a*b**3*c**2 - 2*sqrt(2)*a*b**2*c**3 + 6*a*b**2*c**3 + 2*a*b*c**4 + 2*b**4*c**2 + 4*b**3*c**3 + 2*b**2*c**4)/(c**2*(a**2*b**2 + 2*a**2*b*c + a**2*c**2 + 2*a*b**3 + 6*a*b**2*c + 6*a*b*c**2 + 2*a*c**3 + b**4 + 4*b**3*c + 6*b**2*c**2 + 4*b*c**3 + c**4))" - ] - }, - "metadata": {}, - "output_type": "display_data" + "output_type": "execute_result" } ], "source": [ - "formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\n", + "\"\"\"formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\n", "display(formula)\n", "from scipy.optimize import fmin\n", "import numpy as np\n", "def f(x):\n", - " num,den=fraction(cancel(formula))\n", + " num,den=fraction(cancel(newformula))\n", " fs=sorted(newformula.free_symbols,key=str)\n", - " return num.subs(zip(fs,x))\n", + " return float(num.subs(list(zip(fs,x))))\n", "newformula=(makesubs(formula,'[b,oo],[c,oo]'))\n", + "print(f([2,1,1]))\n", + "fmin(f,[2,1,1])\n", "print(fmin(f,np.array([2,1,1])))\n", - "display(simplify(newformula))\n", + "display(simplify(newformula))\"\"\"\n", "#prove(newformula)\n", "#prove(makesubs(formula,'[b,oo],[a,oo]',variables='c,b,a'))" ] @@ -1332,7 +1179,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "Substitute $a\\to b+k_{1}$" ], "text/plain": [ "" @@ -1344,7 +1191,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "Substitute $b\\to c+l_{1}$" ], "text/plain": [ "" @@ -1394,25 +1241,50 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 25, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\left( 2, \\ \\frac{31}{3}, \\ \\frac{7}{4}\\right)$" + ], + "text/plain": [ + "(2, 31/3, 7/4)" + ] + }, + "execution_count": 25, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "nsimplify((2.1002573656763053, 10.340431462974655, 1.7661001788212371),tolerance=0.3,rational=True)" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 26, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "1.414211498165453" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "(2.1002573656763053/1.7661001788212371)**2" ] }, { "cell_type": "code", - "execution_count": 39, + "execution_count": 28, "metadata": {}, "outputs": [ { @@ -1430,7 +1302,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "Substitute $a\\to c+r_{1}$" ], "text/plain": [ "" @@ -1442,7 +1314,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to b+c$" + "Substitute $c\\to b+s_{1}$" ], "text/plain": [ "" @@ -1454,7 +1326,7 @@ { "data": { "text/latex": [ - "numerator: $2a^4b^2+2a^4bc+a^4c^2+4a^3b^3+2\\sqrt{2}a^3b^2c+10a^3b^2c+2\\sqrt{2}a^3bc^2+6a^3bc^2+2a^3c^3+2a^2b^4+2\\sqrt{2}a^2b^3c+10a^2b^3c+6\\sqrt{2}a^2b^2c^2+12a^2b^2c^2+4a^2bc^3+4\\sqrt{2}a^2bc^3+a^2c^4+2ab^4c+2\\sqrt{2}ab^3c^2+6ab^3c^2+4ab^2c^3+4\\sqrt{2}ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4$" + "numerator: $2b^4r_{1}^2+2b^4r_{1}s_{1}+2b^4s_{1}^2+4b^3r_{1}^3+2\\sqrt{2}b^3r_{1}^2s_{1}+10b^3r_{1}^2s_{1}+2\\sqrt{2}b^3r_{1}s_{1}^2+6b^3r_{1}s_{1}^2+4b^3s_{1}^3+2b^2r_{1}^4+2\\sqrt{2}b^2r_{1}^3s_{1}+10b^2r_{1}^3s_{1}+6\\sqrt{2}b^2r_{1}^2s_{1}^2+12b^2r_{1}^2s_{1}^2+4b^2r_{1}s_{1}^3+4\\sqrt{2}b^2r_{1}s_{1}^3+2b^2s_{1}^4+2br_{1}^4s_{1}+2\\sqrt{2}br_{1}^3s_{1}^2+6br_{1}^3s_{1}^2+4br_{1}^2s_{1}^3+4\\sqrt{2}br_{1}^2s_{1}^3+2\\sqrt{2}br_{1}s_{1}^4+r_{1}^4s_{1}^2+2r_{1}^3s_{1}^3+r_{1}^2s_{1}^4$" ], "text/plain": [ "" @@ -1466,7 +1338,7 @@ { "data": { "text/latex": [ - "denominator: $a^2b^4+2a^2b^3c+a^2b^2c^2+2ab^5+6ab^4c+6ab^3c^2+2ab^2c^3+b^6+4b^5c+6b^4c^2+4b^3c^3+b^2c^4$" + "denominator: $b^6+2b^5r_{1}+4b^5s_{1}+b^4r_{1}^2+6b^4r_{1}s_{1}+6b^4s_{1}^2+2b^3r_{1}^2s_{1}+6b^3r_{1}s_{1}^2+4b^3s_{1}^3+b^2r_{1}^2s_{1}^2+2b^2r_{1}s_{1}^3+b^2s_{1}^4$" ], "text/plain": [ "" @@ -1490,7 +1362,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 2a^4b^2+2a^4bc+a^4c^2+4a^3b^3+2\\sqrt{2}a^3b^2c+10a^3b^2c+2\\sqrt{2}a^3bc^2+6a^3bc^2+2a^3c^3+2a^2b^4+2\\sqrt{2}a^2b^3c+10a^2b^3c+6\\sqrt{2}a^2b^2c^2+12a^2b^2c^2+4\\sqrt{2}a^2bc^3+4a^2bc^3+a^2c^4+2ab^4c+2\\sqrt{2}ab^3c^2+6ab^3c^2+4\\sqrt{2}ab^2c^3+4ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4 $$" + "$$ 0 \\le r_{1}^4s_{1}^2+2br_{1}^4s_{1}+2b^2r_{1}^4+2r_{1}^3s_{1}^3+2\\sqrt{2}br_{1}^3s_{1}^2+6br_{1}^3s_{1}^2+2\\sqrt{2}b^2r_{1}^3s_{1}+10b^2r_{1}^3s_{1}+4b^3r_{1}^3+r_{1}^2s_{1}^4+4\\sqrt{2}br_{1}^2s_{1}^3+4br_{1}^2s_{1}^3+6\\sqrt{2}b^2r_{1}^2s_{1}^2+12b^2r_{1}^2s_{1}^2+2\\sqrt{2}b^3r_{1}^2s_{1}+10b^3r_{1}^2s_{1}+2b^4r_{1}^2+2\\sqrt{2}br_{1}s_{1}^4+4\\sqrt{2}b^2r_{1}s_{1}^3+4b^2r_{1}s_{1}^3+2\\sqrt{2}b^3r_{1}s_{1}^2+6b^3r_{1}s_{1}^2+2b^4r_{1}s_{1}+2b^2s_{1}^4+4b^3s_{1}^3+2b^4s_{1}^2 $$" ], "text/plain": [ "" @@ -1526,7 +1398,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "Substitute $a\\to b+t_{1}$" ], "text/plain": [ "" @@ -1538,7 +1410,7 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "Substitute $b\\to c+u_{1}$" ], "text/plain": [ "" @@ -1550,7 +1422,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to \\sqrt{2}a$" + "Substitute $t_{1}\\to \\sqrt{2}v_{1}$" ], "text/plain": [ "" @@ -1562,7 +1434,7 @@ { "data": { "text/latex": [ - "numerator: $16a^4b^2+32a^4bc+32a^4c^2+16\\sqrt{2}a^3b^3-32a^3b^2c+48\\sqrt{2}a^3b^2c-32a^3bc^2+80\\sqrt{2}a^3bc^2+32\\sqrt{2}a^3c^3+8a^2b^4-32\\sqrt{2}a^2b^3c+32a^2b^3c-48\\sqrt{2}a^2b^2c^2+96a^2b^2c^2-16\\sqrt{2}a^2bc^3+80a^2bc^3+16a^2c^4-16ab^4c-32ab^3c^2+16\\sqrt{2}ab^3c^2-16ab^2c^3+24\\sqrt{2}ab^2c^3+8\\sqrt{2}abc^4+8b^4c^2+16b^3c^3+8b^2c^4$" + "numerator: $8c^4u_{1}^2+8\\sqrt{2}c^4u_{1}v_{1}+16c^4v_{1}^2+16c^3u_{1}^3-16c^3u_{1}^2v_{1}+24\\sqrt{2}c^3u_{1}^2v_{1}-16\\sqrt{2}c^3u_{1}v_{1}^2+80c^3u_{1}v_{1}^2+32\\sqrt{2}c^3v_{1}^3+8c^2u_{1}^4-32c^2u_{1}^3v_{1}+16\\sqrt{2}c^2u_{1}^3v_{1}-48\\sqrt{2}c^2u_{1}^2v_{1}^2+96c^2u_{1}^2v_{1}^2-32c^2u_{1}v_{1}^3+80\\sqrt{2}c^2u_{1}v_{1}^3+32c^2v_{1}^4-16cu_{1}^4v_{1}-32\\sqrt{2}cu_{1}^3v_{1}^2+32cu_{1}^3v_{1}^2-32cu_{1}^2v_{1}^3+48\\sqrt{2}cu_{1}^2v_{1}^3+32cu_{1}v_{1}^4+8u_{1}^4v_{1}^2+16\\sqrt{2}u_{1}^3v_{1}^3+16u_{1}^2v_{1}^4$" ], "text/plain": [ "" @@ -1574,7 +1446,7 @@ { "data": { "text/latex": [ - "denominator: $2a^2b^2c^2+4a^2bc^3+2a^2c^4+2\\sqrt{2}ab^3c^2+6\\sqrt{2}ab^2c^3+6\\sqrt{2}abc^4+2\\sqrt{2}ac^5+b^4c^2+4b^3c^3+6b^2c^4+4bc^5+c^6$" + "denominator: $c^6+4c^5u_{1}+2\\sqrt{2}c^5v_{1}+6c^4u_{1}^2+6\\sqrt{2}c^4u_{1}v_{1}+2c^4v_{1}^2+4c^3u_{1}^3+6\\sqrt{2}c^3u_{1}^2v_{1}+4c^3u_{1}v_{1}^2+c^2u_{1}^4+2\\sqrt{2}c^2u_{1}^3v_{1}+2c^2u_{1}^2v_{1}^2$" ], "text/plain": [ "" @@ -1610,7 +1482,7 @@ { "data": { "text/latex": [ - "$$32a^3b^2c \\le 16a^4b^2+16a^2b^2c^2$$" + "$$16cu_{1}^4v_{1} \\le 8u_{1}^4v_{1}^2+8c^2u_{1}^4$$" ], "text/plain": [ "" @@ -1622,7 +1494,7 @@ { "data": { "text/latex": [ - "$$32\\sqrt{2}a^2b^3c \\le 16\\sqrt{2}a^3b^3+16\\sqrt{2}ab^3c^2$$" + "$$32\\sqrt{2}cu_{1}^3v_{1}^2 \\le 16\\sqrt{2}u_{1}^3v_{1}^3+16\\sqrt{2}c^2u_{1}^3v_{1}$$" ], "text/plain": [ "" @@ -1634,7 +1506,7 @@ { "data": { "text/latex": [ - "$$48\\sqrt{2}a^2b^2c^2 \\le 24\\sqrt{2}a^3b^2c+24\\sqrt{2}ab^2c^3$$" + "$$32c^2u_{1}^3v_{1} \\le 16cu_{1}^3v_{1}^2+16c^3u_{1}^3$$" ], "text/plain": [ "" @@ -1646,7 +1518,7 @@ { "data": { "text/latex": [ - "$$16\\sqrt{2}a^2bc^3 \\le 8\\sqrt{2}a^3bc^2+8\\sqrt{2}abc^4$$" + "$$32cu_{1}^2v_{1}^3 \\le 16cu_{1}^3v_{1}^2+16cu_{1}v_{1}^4$$" ], "text/plain": [ "" @@ -1658,7 +1530,7 @@ { "data": { "text/latex": [ - "$$16ab^4c \\le 8a^2b^4+8b^4c^2$$" + "$$48\\sqrt{2}c^2u_{1}^2v_{1}^2 \\le 24\\sqrt{2}cu_{1}^2v_{1}^3+24\\sqrt{2}c^3u_{1}^2v_{1}$$" ], "text/plain": [ "" @@ -1670,7 +1542,7 @@ { "data": { "text/latex": [ - "$$32ab^3c^2 \\le 16a^2b^3c+16b^3c^3$$" + "$$16c^3u_{1}^2v_{1} \\le 8c^2u_{1}^2v_{1}^2+8c^4u_{1}^2$$" ], "text/plain": [ "" @@ -1682,7 +1554,7 @@ { "data": { "text/latex": [ - "$$32a^3bc^2 \\le 16a^4bc+16a^2bc^3$$" + "$$32c^2u_{1}v_{1}^3 \\le 16u_{1}^2v_{1}^4+16c^4v_{1}^2$$" ], "text/plain": [ "" @@ -1694,7 +1566,7 @@ { "data": { "text/latex": [ - "$$16ab^2c^3 \\le 8a^2b^2c^2+8b^2c^4$$" + "$$16\\sqrt{2}c^3u_{1}v_{1}^2 \\le 8\\sqrt{2}c^2u_{1}v_{1}^3+8\\sqrt{2}c^4u_{1}v_{1}$$" ], "text/plain": [ "" @@ -1706,7 +1578,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le 16a^4bc+32a^4c^2+24\\sqrt{2}a^3b^2c+72\\sqrt{2}a^3bc^2+32\\sqrt{2}a^3c^3+16a^2b^3c+72a^2b^2c^2+64a^2bc^3+16a^2c^4 $$" + "$$ 0 \\le 24\\sqrt{2}cu_{1}^2v_{1}^3+88c^2u_{1}^2v_{1}^2+16cu_{1}v_{1}^4+72\\sqrt{2}c^2u_{1}v_{1}^3+80c^3u_{1}v_{1}^2+32c^2v_{1}^4+32\\sqrt{2}c^3v_{1}^3 $$" ], "text/plain": [ "" @@ -1736,7 +1608,7 @@ "-2*sqrt(2)*(a - b)/c + (a - b)**2/c**2 - 2*sqrt(2)*(-a + c)/b + (-a + c)**2/b**2 - 2*sqrt(2)*(b - c)/a + (b - c)**2/a**2" ] }, - "execution_count": 39, + "execution_count": 28, "metadata": {}, "output_type": "execute_result" } @@ -1748,15 +1620,122 @@ "prove(formula1)\n", "display(Latex('Case $a\\ge b\\ge c$'))\n", "formula2=makesubs(formula,'[b,oo],[c,oo]')\n", - "prove(formula2*4,values='2**(1/2),1,1')\n", + "prove(formula2*4,values='1,sqrt(2),1')\n", "formula" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 29, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $a\\to b+w_{1}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $b\\to c+x_{1}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $a\\to\\sqrt{2}a$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "numerator: $2c^4w_{1}^2+2c^4w_{1}x_{1}+2c^4x_{1}^2+4c^3w_{1}^3-2\\sqrt{2}c^3w_{1}^2x_{1}+10c^3w_{1}^2x_{1}-2\\sqrt{2}c^3w_{1}x_{1}^2+6c^3w_{1}x_{1}^2+4c^3x_{1}^3+2c^2w_{1}^4-2\\sqrt{2}c^2w_{1}^3x_{1}+10c^2w_{1}^3x_{1}-6\\sqrt{2}c^2w_{1}^2x_{1}^2+12c^2w_{1}^2x_{1}^2-4\\sqrt{2}c^2w_{1}x_{1}^3+4c^2w_{1}x_{1}^3+2c^2x_{1}^4+2cw_{1}^4x_{1}-2\\sqrt{2}cw_{1}^3x_{1}^2+6cw_{1}^3x_{1}^2-4\\sqrt{2}cw_{1}^2x_{1}^3+4cw_{1}^2x_{1}^3-2\\sqrt{2}cw_{1}x_{1}^4+w_{1}^4x_{1}^2+2w_{1}^3x_{1}^3+w_{1}^2x_{1}^4$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 2" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Program couldn't find any proof." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 2\\sqrt{2}cw_{1}x_{1}^4+4\\sqrt{2}cw_{1}^2x_{1}^3+4\\sqrt{2}c^2w_{1}x_{1}^3+2\\sqrt{2}cw_{1}^3x_{1}^2+6\\sqrt{2}c^2w_{1}^2x_{1}^2+2\\sqrt{2}c^3w_{1}x_{1}^2+2\\sqrt{2}c^2w_{1}^3x_{1}+2\\sqrt{2}c^3w_{1}^2x_{1} \\le w_{1}^2x_{1}^4+2c^2x_{1}^4+2w_{1}^3x_{1}^3+4cw_{1}^2x_{1}^3+4c^2w_{1}x_{1}^3+4c^3x_{1}^3+w_{1}^4x_{1}^2+6cw_{1}^3x_{1}^2+12c^2w_{1}^2x_{1}^2+6c^3w_{1}x_{1}^2+2c^4x_{1}^2+2cw_{1}^4x_{1}+10c^2w_{1}^3x_{1}+10c^3w_{1}^2x_{1}+2c^4w_{1}x_{1}+2c^2w_{1}^4+4c^3w_{1}^3+2c^4w_{1}^2 $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\n", "formula=(makesubs(formula,'[b,oo],[c,oo]'))\n", @@ -1768,54 +1747,120 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 30, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "(1, (x, y, sqrt(2)))" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "S('2**(1/2)*x*y').as_coeff_mul()" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 31, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "(1, (x, y, sqrt(2)))" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "S('sqrt(2)*x*y').as_coeff_mul()" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 32, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "[(1, 1, 1), (0, 0, 0)]" + ] + }, + "execution_count": 32, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "Poly('sqrt(2)*x*y+8').monoms()" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 33, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "[1, 58]" + ] + }, + "execution_count": 33, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "Poly('sqrt(2)*x*y+58').coeffs()" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 34, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "(x, y, sqrt(2), sqrt(3), sqrt(6))" + ] + }, + "execution_count": 34, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "Poly('sqrt(2)*x*y+sqrt(3)+sqrt(2)*sqrt(3)').gens" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 35, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "([0], [(0, 0)], [1, 7], [(2, 0), (0, 1)], (x, y))" + ] + }, + "execution_count": 35, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "def _formula2list(formula):\n", " neg=pos=0\n", @@ -1831,134 +1876,6 @@ "_formula2list(S('x^2+7*y'))" ] }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "prove(S('sqrt(2)*x^2-sqrt(8)*x*y+sqrt(2)*y^2'))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "[Poly('1+2**(1/3)+4**(1/3)+x').monoms(),Poly('1+2**(1/3)+4**(1/3)+x').coeffs()]" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "x=Symbol('x', positive=True)\n", - "Poly(x+sqrt(x))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "prove?" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "Poly('x^2-1').abs()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "Poly(Poly('x')/Poly('x+y'))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "Poly('sqrt(2)+x+sqrt(x)+x**(1/4)+x**(1/3)+x**(2/3)',extension=1)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "Poly('x+sqrt(6)',S('x'),S('sqrt(2)'),S('sqrt(3)')).coeffs()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "Poly('x+sqrt(x)',S('sqrt(x)')).coeffs()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "source(Poly)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "Poly({1:1,2:1},gens=S('sqrt(x)'))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "x=symbols('x', positive=True)\n", - "S('sqrt(x^2)-x')" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "str(Poly({1:1,2:1,4:1},gens=S('sqrt(x)')))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [ - "str(Poly('x^2+x+sqrt(x)'))" - ] - }, { "cell_type": "code", "execution_count": 36, @@ -1967,7 +1884,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+b$" + "numerator: $\\sqrt{2}x^2-2\\sqrt{2}xy+\\sqrt{2}y^2$" ], "text/plain": [ "" @@ -1979,7 +1896,4496 @@ { "data": { "text/latex": [ - "Substitute $b\\to b+c$" + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "status: 0" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "From weighted AM-GM inequality:" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2\\sqrt{2}xy \\le \\sqrt{2}x^2+\\sqrt{2}y^2$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ 0 \\le 0 $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "The sum of all inequalities gives us a proof of the inequality." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 36, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "prove(S('sqrt(2)*x^2-sqrt(8)*x*y+sqrt(2)*y^2'))" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[[(1, 0), (0, 2), (0, 1), (0, 0)], [1, 1, 1, 1]]" + ] + }, + "execution_count": 37, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "[Poly('1+2**(1/3)+4**(1/3)+x').monoms(),Poly('1+2**(1/3)+4**(1/3)+x').coeffs()]" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x + \\sqrt{x}, x, \\sqrt{x}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(x + (sqrt(x)), x, sqrt(x), domain='ZZ')" + ] + }, + "execution_count": 38, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "x=Symbol('x', positive=True)\n", + "Poly(x+sqrt(x))" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": {}, + "outputs": [], + "source": [ + "prove?" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x^{2} + 1, x, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(x**2 + 1, x, domain='ZZ')" + ] + }, + "execution_count": 40, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('x^2-1').abs()" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x\\frac{1}{x + y}, x, \\frac{1}{x + y}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(x*(1/(x + y)), x, 1/(x + y), domain='ZZ')" + ] + }, + "execution_count": 41, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly(Poly('x')/Poly('x+y'))" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x + \\sqrt{x} + x^{\\frac{2}{3}} + \\sqrt[3]{x} + \\sqrt[4]{x} + \\sqrt{2}, x, \\sqrt{x}, \\sqrt[3]{x}, \\sqrt[4]{x}, domain=QQ \\right)}$" + ], + "text/plain": [ + "Poly(x + (sqrt(x)) + (x**(1/3))**2 + (x**(1/3)) + (x**(1/4)) + sqrt(2), x, sqrt(x), x**(1/3), x**(1/4), domain='QQ')" + ] + }, + "execution_count": 42, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('sqrt(2)+x+sqrt(x)+x**(1/4)+x**(1/3)+x**(2/3)',extension=1)" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[1, sqrt(6)]" + ] + }, + "execution_count": 43, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('x+sqrt(6)',S('x'),S('sqrt(2)'),S('sqrt(3)')).coeffs()" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[1, x]" + ] + }, + "execution_count": 44, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('x+sqrt(x)',S('sqrt(x)')).coeffs()" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/decorators.py:38: SymPyDeprecationWarning: \n", + "\n", + "source has been deprecated since SymPy 1.3. Use ?? in IPython/Jupyter\n", + "or inspect.getsource instead. See\n", + "https://github.com/sympy/sympy/issues/14905 for more info.\n", + "\n", + " _warn_deprecation(wrapped, 3)\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "In file: /home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polytools.py\n", + "class Poly(Expr):\n", + " \"\"\"\n", + " Generic class for representing and operating on polynomial expressions.\n", + " Subclasses Expr class.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " Create a univariate polynomial:\n", + "\n", + " >>> Poly(x*(x**2 + x - 1)**2)\n", + " Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain='ZZ')\n", + "\n", + " Create a univariate polynomial with specific domain:\n", + "\n", + " >>> from sympy import sqrt\n", + " >>> Poly(x**2 + 2*x + sqrt(3), domain='R')\n", + " Poly(1.0*x**2 + 2.0*x + 1.73205080756888, x, domain='RR')\n", + "\n", + " Create a multivariate polynomial:\n", + "\n", + " >>> Poly(y*x**2 + x*y + 1)\n", + " Poly(x**2*y + x*y + 1, x, y, domain='ZZ')\n", + "\n", + " Create a univariate polynomial, where y is a constant:\n", + "\n", + " >>> Poly(y*x**2 + x*y + 1,x)\n", + " Poly(y*x**2 + y*x + 1, x, domain='ZZ[y]')\n", + "\n", + " You can evaluate the above polynomial as a function of y:\n", + "\n", + " >>> Poly(y*x**2 + x*y + 1,x).eval(2)\n", + " 6*y + 1\n", + "\n", + " See Also\n", + " ========\n", + "\n", + " sympy.core.expr.Expr\n", + "\n", + " \"\"\"\n", + "\n", + " __slots__ = ['rep', 'gens']\n", + "\n", + " is_commutative = True\n", + " is_Poly = True\n", + " _op_priority = 10.001\n", + "\n", + " def __new__(cls, rep, *gens, **args):\n", + " \"\"\"Create a new polynomial instance out of something useful. \"\"\"\n", + " opt = options.build_options(gens, args)\n", + "\n", + " if 'order' in opt:\n", + " raise NotImplementedError(\"'order' keyword is not implemented yet\")\n", + "\n", + " if iterable(rep, exclude=str):\n", + " if isinstance(rep, dict):\n", + " return cls._from_dict(rep, opt)\n", + " else:\n", + " return cls._from_list(list(rep), opt)\n", + " else:\n", + " rep = sympify(rep)\n", + "\n", + " if rep.is_Poly:\n", + " return cls._from_poly(rep, opt)\n", + " else:\n", + " return cls._from_expr(rep, opt)\n", + "\n", + " @classmethod\n", + " def new(cls, rep, *gens):\n", + " \"\"\"Construct :class:`Poly` instance from raw representation. \"\"\"\n", + " if not isinstance(rep, DMP):\n", + " raise PolynomialError(\n", + " \"invalid polynomial representation: %s\" % rep)\n", + " elif rep.lev != len(gens) - 1:\n", + " raise PolynomialError(\"invalid arguments: %s, %s\" % (rep, gens))\n", + "\n", + " obj = Basic.__new__(cls)\n", + "\n", + " obj.rep = rep\n", + " obj.gens = gens\n", + "\n", + " return obj\n", + "\n", + " @classmethod\n", + " def from_dict(cls, rep, *gens, **args):\n", + " \"\"\"Construct a polynomial from a ``dict``. \"\"\"\n", + " opt = options.build_options(gens, args)\n", + " return cls._from_dict(rep, opt)\n", + "\n", + " @classmethod\n", + " def from_list(cls, rep, *gens, **args):\n", + " \"\"\"Construct a polynomial from a ``list``. \"\"\"\n", + " opt = options.build_options(gens, args)\n", + " return cls._from_list(rep, opt)\n", + "\n", + " @classmethod\n", + " def from_poly(cls, rep, *gens, **args):\n", + " \"\"\"Construct a polynomial from a polynomial. \"\"\"\n", + " opt = options.build_options(gens, args)\n", + " return cls._from_poly(rep, opt)\n", + "\n", + " @classmethod\n", + " def from_expr(cls, rep, *gens, **args):\n", + " \"\"\"Construct a polynomial from an expression. \"\"\"\n", + " opt = options.build_options(gens, args)\n", + " return cls._from_expr(rep, opt)\n", + "\n", + " @classmethod\n", + " def _from_dict(cls, rep, opt):\n", + " \"\"\"Construct a polynomial from a ``dict``. \"\"\"\n", + " gens = opt.gens\n", + "\n", + " if not gens:\n", + " raise GeneratorsNeeded(\n", + " \"can't initialize from 'dict' without generators\")\n", + "\n", + " level = len(gens) - 1\n", + " domain = opt.domain\n", + "\n", + " if domain is None:\n", + " domain, rep = construct_domain(rep, opt=opt)\n", + " else:\n", + " for monom, coeff in rep.items():\n", + " rep[monom] = domain.convert(coeff)\n", + "\n", + " return cls.new(DMP.from_dict(rep, level, domain), *gens)\n", + "\n", + " @classmethod\n", + " def _from_list(cls, rep, opt):\n", + " \"\"\"Construct a polynomial from a ``list``. \"\"\"\n", + " gens = opt.gens\n", + "\n", + " if not gens:\n", + " raise GeneratorsNeeded(\n", + " \"can't initialize from 'list' without generators\")\n", + " elif len(gens) != 1:\n", + " raise MultivariatePolynomialError(\n", + " \"'list' representation not supported\")\n", + "\n", + " level = len(gens) - 1\n", + " domain = opt.domain\n", + "\n", + " if domain is None:\n", + " domain, rep = construct_domain(rep, opt=opt)\n", + " else:\n", + " rep = list(map(domain.convert, rep))\n", + "\n", + " return cls.new(DMP.from_list(rep, level, domain), *gens)\n", + "\n", + " @classmethod\n", + " def _from_poly(cls, rep, opt):\n", + " \"\"\"Construct a polynomial from a polynomial. \"\"\"\n", + " if cls != rep.__class__:\n", + " rep = cls.new(rep.rep, *rep.gens)\n", + "\n", + " gens = opt.gens\n", + " field = opt.field\n", + " domain = opt.domain\n", + "\n", + " if gens and rep.gens != gens:\n", + " if set(rep.gens) != set(gens):\n", + " return cls._from_expr(rep.as_expr(), opt)\n", + " else:\n", + " rep = rep.reorder(*gens)\n", + "\n", + " if 'domain' in opt and domain:\n", + " rep = rep.set_domain(domain)\n", + " elif field is True:\n", + " rep = rep.to_field()\n", + "\n", + " return rep\n", + "\n", + " @classmethod\n", + " def _from_expr(cls, rep, opt):\n", + " \"\"\"Construct a polynomial from an expression. \"\"\"\n", + " rep, opt = _dict_from_expr(rep, opt)\n", + " return cls._from_dict(rep, opt)\n", + "\n", + " def _hashable_content(self):\n", + " \"\"\"Allow SymPy to hash Poly instances. \"\"\"\n", + " return (self.rep, self.gens)\n", + "\n", + " def __hash__(self):\n", + " return super(Poly, self).__hash__()\n", + "\n", + " @property\n", + " def free_symbols(self):\n", + " \"\"\"\n", + " Free symbols of a polynomial expression.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y, z\n", + "\n", + " >>> Poly(x**2 + 1).free_symbols\n", + " {x}\n", + " >>> Poly(x**2 + y).free_symbols\n", + " {x, y}\n", + " >>> Poly(x**2 + y, x).free_symbols\n", + " {x, y}\n", + " >>> Poly(x**2 + y, x, z).free_symbols\n", + " {x, y}\n", + "\n", + " \"\"\"\n", + " symbols = set()\n", + " gens = self.gens\n", + " for i in range(len(gens)):\n", + " for monom in self.monoms():\n", + " if monom[i]:\n", + " symbols |= gens[i].free_symbols\n", + " break\n", + "\n", + " return symbols | self.free_symbols_in_domain\n", + "\n", + " @property\n", + " def free_symbols_in_domain(self):\n", + " \"\"\"\n", + " Free symbols of the domain of ``self``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + 1).free_symbols_in_domain\n", + " set()\n", + " >>> Poly(x**2 + y).free_symbols_in_domain\n", + " set()\n", + " >>> Poly(x**2 + y, x).free_symbols_in_domain\n", + " {y}\n", + "\n", + " \"\"\"\n", + " domain, symbols = self.rep.dom, set()\n", + "\n", + " if domain.is_Composite:\n", + " for gen in domain.symbols:\n", + " symbols |= gen.free_symbols\n", + " elif domain.is_EX:\n", + " for coeff in self.coeffs():\n", + " symbols |= coeff.free_symbols\n", + "\n", + " return symbols\n", + "\n", + " @property\n", + " def args(self):\n", + " \"\"\"\n", + " Don't mess up with the core.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).args\n", + " (x**2 + 1,)\n", + "\n", + " \"\"\"\n", + " return (self.as_expr(),)\n", + "\n", + " @property\n", + " def gen(self):\n", + " \"\"\"\n", + " Return the principal generator.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).gen\n", + " x\n", + "\n", + " \"\"\"\n", + " return self.gens[0]\n", + "\n", + " @property\n", + " def domain(self):\n", + " \"\"\"Get the ground domain of ``self``. \"\"\"\n", + " return self.get_domain()\n", + "\n", + " @property\n", + " def zero(self):\n", + " \"\"\"Return zero polynomial with ``self``'s properties. \"\"\"\n", + " return self.new(self.rep.zero(self.rep.lev, self.rep.dom), *self.gens)\n", + "\n", + " @property\n", + " def one(self):\n", + " \"\"\"Return one polynomial with ``self``'s properties. \"\"\"\n", + " return self.new(self.rep.one(self.rep.lev, self.rep.dom), *self.gens)\n", + "\n", + " @property\n", + " def unit(self):\n", + " \"\"\"Return unit polynomial with ``self``'s properties. \"\"\"\n", + " return self.new(self.rep.unit(self.rep.lev, self.rep.dom), *self.gens)\n", + "\n", + " def unify(f, g):\n", + " \"\"\"\n", + " Make ``f`` and ``g`` belong to the same domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f, g = Poly(x/2 + 1), Poly(2*x + 1)\n", + "\n", + " >>> f\n", + " Poly(1/2*x + 1, x, domain='QQ')\n", + " >>> g\n", + " Poly(2*x + 1, x, domain='ZZ')\n", + "\n", + " >>> F, G = f.unify(g)\n", + "\n", + " >>> F\n", + " Poly(1/2*x + 1, x, domain='QQ')\n", + " >>> G\n", + " Poly(2*x + 1, x, domain='QQ')\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + " return per(F), per(G)\n", + "\n", + " def _unify(f, g):\n", + " g = sympify(g)\n", + "\n", + " if not g.is_Poly:\n", + " try:\n", + " return f.rep.dom, f.per, f.rep, f.rep.per(f.rep.dom.from_sympy(g))\n", + " except CoercionFailed:\n", + " raise UnificationFailed(\"can't unify %s with %s\" % (f, g))\n", + "\n", + " if isinstance(f.rep, DMP) and isinstance(g.rep, DMP):\n", + " gens = _unify_gens(f.gens, g.gens)\n", + "\n", + " dom, lev = f.rep.dom.unify(g.rep.dom, gens), len(gens) - 1\n", + "\n", + " if f.gens != gens:\n", + " f_monoms, f_coeffs = _dict_reorder(\n", + " f.rep.to_dict(), f.gens, gens)\n", + "\n", + " if f.rep.dom != dom:\n", + " f_coeffs = [dom.convert(c, f.rep.dom) for c in f_coeffs]\n", + "\n", + " F = DMP(dict(list(zip(f_monoms, f_coeffs))), dom, lev)\n", + " else:\n", + " F = f.rep.convert(dom)\n", + "\n", + " if g.gens != gens:\n", + " g_monoms, g_coeffs = _dict_reorder(\n", + " g.rep.to_dict(), g.gens, gens)\n", + "\n", + " if g.rep.dom != dom:\n", + " g_coeffs = [dom.convert(c, g.rep.dom) for c in g_coeffs]\n", + "\n", + " G = DMP(dict(list(zip(g_monoms, g_coeffs))), dom, lev)\n", + " else:\n", + " G = g.rep.convert(dom)\n", + " else:\n", + " raise UnificationFailed(\"can't unify %s with %s\" % (f, g))\n", + "\n", + " cls = f.__class__\n", + "\n", + " def per(rep, dom=dom, gens=gens, remove=None):\n", + " if remove is not None:\n", + " gens = gens[:remove] + gens[remove + 1:]\n", + "\n", + " if not gens:\n", + " return dom.to_sympy(rep)\n", + "\n", + " return cls.new(rep, *gens)\n", + "\n", + " return dom, per, F, G\n", + "\n", + " def per(f, rep, gens=None, remove=None):\n", + " \"\"\"\n", + " Create a Poly out of the given representation.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, ZZ\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> from sympy.polys.polyclasses import DMP\n", + "\n", + " >>> a = Poly(x**2 + 1)\n", + "\n", + " >>> a.per(DMP([ZZ(1), ZZ(1)], ZZ), gens=[y])\n", + " Poly(y + 1, y, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if gens is None:\n", + " gens = f.gens\n", + "\n", + " if remove is not None:\n", + " gens = gens[:remove] + gens[remove + 1:]\n", + "\n", + " if not gens:\n", + " return f.rep.dom.to_sympy(rep)\n", + "\n", + " return f.__class__.new(rep, *gens)\n", + "\n", + " def set_domain(f, domain):\n", + " \"\"\"Set the ground domain of ``f``. \"\"\"\n", + " opt = options.build_options(f.gens, {'domain': domain})\n", + " return f.per(f.rep.convert(opt.domain))\n", + "\n", + " def get_domain(f):\n", + " \"\"\"Get the ground domain of ``f``. \"\"\"\n", + " return f.rep.dom\n", + "\n", + " def set_modulus(f, modulus):\n", + " \"\"\"\n", + " Set the modulus of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(5*x**2 + 2*x - 1, x).set_modulus(2)\n", + " Poly(x**2 + 1, x, modulus=2)\n", + "\n", + " \"\"\"\n", + " modulus = options.Modulus.preprocess(modulus)\n", + " return f.set_domain(FF(modulus))\n", + "\n", + " def get_modulus(f):\n", + " \"\"\"\n", + " Get the modulus of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, modulus=2).get_modulus()\n", + " 2\n", + "\n", + " \"\"\"\n", + " domain = f.get_domain()\n", + "\n", + " if domain.is_FiniteField:\n", + " return Integer(domain.characteristic())\n", + " else:\n", + " raise PolynomialError(\"not a polynomial over a Galois field\")\n", + "\n", + " def _eval_subs(f, old, new):\n", + " \"\"\"Internal implementation of :func:`subs`. \"\"\"\n", + " if old in f.gens:\n", + " if new.is_number:\n", + " return f.eval(old, new)\n", + " else:\n", + " try:\n", + " return f.replace(old, new)\n", + " except PolynomialError:\n", + " pass\n", + "\n", + " return f.as_expr().subs(old, new)\n", + "\n", + " def exclude(f):\n", + " \"\"\"\n", + " Remove unnecessary generators from ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import a, b, c, d, x\n", + "\n", + " >>> Poly(a + x, a, b, c, d, x).exclude()\n", + " Poly(a + x, a, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " J, new = f.rep.exclude()\n", + " gens = []\n", + "\n", + " for j in range(len(f.gens)):\n", + " if j not in J:\n", + " gens.append(f.gens[j])\n", + "\n", + " return f.per(new, gens=gens)\n", + "\n", + " def replace(f, x, y=None, *_ignore):\n", + " # XXX this does not match Basic's signature\n", + " \"\"\"\n", + " Replace ``x`` with ``y`` in generators list.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + 1, x).replace(x, y)\n", + " Poly(y**2 + 1, y, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if y is None:\n", + " if f.is_univariate:\n", + " x, y = f.gen, x\n", + " else:\n", + " raise PolynomialError(\n", + " \"syntax supported only in univariate case\")\n", + "\n", + " if x == y or x not in f.gens:\n", + " return f\n", + "\n", + " if x in f.gens and y not in f.gens:\n", + " dom = f.get_domain()\n", + "\n", + " if not dom.is_Composite or y not in dom.symbols:\n", + " gens = list(f.gens)\n", + " gens[gens.index(x)] = y\n", + " return f.per(f.rep, gens=gens)\n", + "\n", + " raise PolynomialError(\"can't replace %s with %s in %s\" % (x, y, f))\n", + "\n", + " def reorder(f, *gens, **args):\n", + " \"\"\"\n", + " Efficiently apply new order of generators.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + x*y**2, x, y).reorder(y, x)\n", + " Poly(y**2*x + x**2, y, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " opt = options.Options((), args)\n", + "\n", + " if not gens:\n", + " gens = _sort_gens(f.gens, opt=opt)\n", + " elif set(f.gens) != set(gens):\n", + " raise PolynomialError(\n", + " \"generators list can differ only up to order of elements\")\n", + "\n", + " rep = dict(list(zip(*_dict_reorder(f.rep.to_dict(), f.gens, gens))))\n", + "\n", + " return f.per(DMP(rep, f.rep.dom, len(gens) - 1), gens=gens)\n", + "\n", + " def ltrim(f, gen):\n", + " \"\"\"\n", + " Remove dummy generators from ``f`` that are to the left of\n", + " specified ``gen`` in the generators as ordered. When ``gen``\n", + " is an integer, it refers to the generator located at that\n", + " position within the tuple of generators of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y, z\n", + "\n", + " >>> Poly(y**2 + y*z**2, x, y, z).ltrim(y)\n", + " Poly(y**2 + y*z**2, y, z, domain='ZZ')\n", + " >>> Poly(z, x, y, z).ltrim(-1)\n", + " Poly(z, z, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " rep = f.as_dict(native=True)\n", + " j = f._gen_to_level(gen)\n", + "\n", + " terms = {}\n", + "\n", + " for monom, coeff in rep.items():\n", + "\n", + " if any(i for i in monom[:j]):\n", + " # some generator is used in the portion to be trimmed\n", + " raise PolynomialError(\"can't left trim %s\" % f)\n", + "\n", + " terms[monom[j:]] = coeff\n", + "\n", + " gens = f.gens[j:]\n", + "\n", + " return f.new(DMP.from_dict(terms, len(gens) - 1, f.rep.dom), *gens)\n", + "\n", + " def has_only_gens(f, *gens):\n", + " \"\"\"\n", + " Return ``True`` if ``Poly(f, *gens)`` retains ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y, z\n", + "\n", + " >>> Poly(x*y + 1, x, y, z).has_only_gens(x, y)\n", + " True\n", + " >>> Poly(x*y + z, x, y, z).has_only_gens(x, y)\n", + " False\n", + "\n", + " \"\"\"\n", + " indices = set()\n", + "\n", + " for gen in gens:\n", + " try:\n", + " index = f.gens.index(gen)\n", + " except ValueError:\n", + " raise GeneratorsError(\n", + " \"%s doesn't have %s as generator\" % (f, gen))\n", + " else:\n", + " indices.add(index)\n", + "\n", + " for monom in f.monoms():\n", + " for i, elt in enumerate(monom):\n", + " if i not in indices and elt:\n", + " return False\n", + "\n", + " return True\n", + "\n", + " def to_ring(f):\n", + " \"\"\"\n", + " Make the ground domain a ring.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, QQ\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, domain=QQ).to_ring()\n", + " Poly(x**2 + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'to_ring'):\n", + " result = f.rep.to_ring()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'to_ring')\n", + "\n", + " return f.per(result)\n", + "\n", + " def to_field(f):\n", + " \"\"\"\n", + " Make the ground domain a field.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, ZZ\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x, domain=ZZ).to_field()\n", + " Poly(x**2 + 1, x, domain='QQ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'to_field'):\n", + " result = f.rep.to_field()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'to_field')\n", + "\n", + " return f.per(result)\n", + "\n", + " def to_exact(f):\n", + " \"\"\"\n", + " Make the ground domain exact.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, RR\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1.0, x, domain=RR).to_exact()\n", + " Poly(x**2 + 1, x, domain='QQ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'to_exact'):\n", + " result = f.rep.to_exact()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'to_exact')\n", + "\n", + " return f.per(result)\n", + "\n", + " def retract(f, field=None):\n", + " \"\"\"\n", + " Recalculate the ground domain of a polynomial.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = Poly(x**2 + 1, x, domain='QQ[y]')\n", + " >>> f\n", + " Poly(x**2 + 1, x, domain='QQ[y]')\n", + "\n", + " >>> f.retract()\n", + " Poly(x**2 + 1, x, domain='ZZ')\n", + " >>> f.retract(field=True)\n", + " Poly(x**2 + 1, x, domain='QQ')\n", + "\n", + " \"\"\"\n", + " dom, rep = construct_domain(f.as_dict(zero=True),\n", + " field=field, composite=f.domain.is_Composite or None)\n", + " return f.from_dict(rep, f.gens, domain=dom)\n", + "\n", + " def slice(f, x, m, n=None):\n", + " \"\"\"Take a continuous subsequence of terms of ``f``. \"\"\"\n", + " if n is None:\n", + " j, m, n = 0, x, m\n", + " else:\n", + " j = f._gen_to_level(x)\n", + "\n", + " m, n = int(m), int(n)\n", + "\n", + " if hasattr(f.rep, 'slice'):\n", + " result = f.rep.slice(m, n, j)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'slice')\n", + "\n", + " return f.per(result)\n", + "\n", + " def coeffs(f, order=None):\n", + " \"\"\"\n", + " Returns all non-zero coefficients from ``f`` in lex order.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**3 + 2*x + 3, x).coeffs()\n", + " [1, 2, 3]\n", + "\n", + " See Also\n", + " ========\n", + " all_coeffs\n", + " coeff_monomial\n", + " nth\n", + "\n", + " \"\"\"\n", + " return [f.rep.dom.to_sympy(c) for c in f.rep.coeffs(order=order)]\n", + "\n", + " def monoms(f, order=None):\n", + " \"\"\"\n", + " Returns all non-zero monomials from ``f`` in lex order.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).monoms()\n", + " [(2, 0), (1, 2), (1, 1), (0, 1)]\n", + "\n", + " See Also\n", + " ========\n", + " all_monoms\n", + "\n", + " \"\"\"\n", + " return f.rep.monoms(order=order)\n", + "\n", + " def terms(f, order=None):\n", + " \"\"\"\n", + " Returns all non-zero terms from ``f`` in lex order.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).terms()\n", + " [((2, 0), 1), ((1, 2), 2), ((1, 1), 1), ((0, 1), 3)]\n", + "\n", + " See Also\n", + " ========\n", + " all_terms\n", + "\n", + " \"\"\"\n", + " return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.terms(order=order)]\n", + "\n", + " def all_coeffs(f):\n", + " \"\"\"\n", + " Returns all coefficients from a univariate polynomial ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**3 + 2*x - 1, x).all_coeffs()\n", + " [1, 0, 2, -1]\n", + "\n", + " \"\"\"\n", + " return [f.rep.dom.to_sympy(c) for c in f.rep.all_coeffs()]\n", + "\n", + " def all_monoms(f):\n", + " \"\"\"\n", + " Returns all monomials from a univariate polynomial ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**3 + 2*x - 1, x).all_monoms()\n", + " [(3,), (2,), (1,), (0,)]\n", + "\n", + " See Also\n", + " ========\n", + " all_terms\n", + "\n", + " \"\"\"\n", + " return f.rep.all_monoms()\n", + "\n", + " def all_terms(f):\n", + " \"\"\"\n", + " Returns all terms from a univariate polynomial ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**3 + 2*x - 1, x).all_terms()\n", + " [((3,), 1), ((2,), 0), ((1,), 2), ((0,), -1)]\n", + "\n", + " \"\"\"\n", + " return [(m, f.rep.dom.to_sympy(c)) for m, c in f.rep.all_terms()]\n", + "\n", + " def termwise(f, func, *gens, **args):\n", + " \"\"\"\n", + " Apply a function to all terms of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> def func(k, coeff):\n", + " ... k = k[0]\n", + " ... return coeff//10**(2-k)\n", + "\n", + " >>> Poly(x**2 + 20*x + 400).termwise(func)\n", + " Poly(x**2 + 2*x + 4, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " terms = {}\n", + "\n", + " for monom, coeff in f.terms():\n", + " result = func(monom, coeff)\n", + "\n", + " if isinstance(result, tuple):\n", + " monom, coeff = result\n", + " else:\n", + " coeff = result\n", + "\n", + " if coeff:\n", + " if monom not in terms:\n", + " terms[monom] = coeff\n", + " else:\n", + " raise PolynomialError(\n", + " \"%s monomial was generated twice\" % monom)\n", + "\n", + " return f.from_dict(terms, *(gens or f.gens), **args)\n", + "\n", + " def length(f):\n", + " \"\"\"\n", + " Returns the number of non-zero terms in ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 2*x - 1).length()\n", + " 3\n", + "\n", + " \"\"\"\n", + " return len(f.as_dict())\n", + "\n", + " def as_dict(f, native=False, zero=False):\n", + " \"\"\"\n", + " Switch to a ``dict`` representation.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + 2*x*y**2 - y, x, y).as_dict()\n", + " {(0, 1): -1, (1, 2): 2, (2, 0): 1}\n", + "\n", + " \"\"\"\n", + " if native:\n", + " return f.rep.to_dict(zero=zero)\n", + " else:\n", + " return f.rep.to_sympy_dict(zero=zero)\n", + "\n", + " def as_list(f, native=False):\n", + " \"\"\"Switch to a ``list`` representation. \"\"\"\n", + " if native:\n", + " return f.rep.to_list()\n", + " else:\n", + " return f.rep.to_sympy_list()\n", + "\n", + " def as_expr(f, *gens):\n", + " \"\"\"\n", + " Convert a Poly instance to an Expr instance.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> f = Poly(x**2 + 2*x*y**2 - y, x, y)\n", + "\n", + " >>> f.as_expr()\n", + " x**2 + 2*x*y**2 - y\n", + " >>> f.as_expr({x: 5})\n", + " 10*y**2 - y + 25\n", + " >>> f.as_expr(5, 6)\n", + " 379\n", + "\n", + " \"\"\"\n", + " if not gens:\n", + " gens = f.gens\n", + " elif len(gens) == 1 and isinstance(gens[0], dict):\n", + " mapping = gens[0]\n", + " gens = list(f.gens)\n", + "\n", + " for gen, value in mapping.items():\n", + " try:\n", + " index = gens.index(gen)\n", + " except ValueError:\n", + " raise GeneratorsError(\n", + " \"%s doesn't have %s as generator\" % (f, gen))\n", + " else:\n", + " gens[index] = value\n", + "\n", + " return basic_from_dict(f.rep.to_sympy_dict(), *gens)\n", + "\n", + " def lift(f):\n", + " \"\"\"\n", + " Convert algebraic coefficients to rationals.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, I\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + I*x + 1, x, extension=I).lift()\n", + " Poly(x**4 + 3*x**2 + 1, x, domain='QQ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'lift'):\n", + " result = f.rep.lift()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'lift')\n", + "\n", + " return f.per(result)\n", + "\n", + " def deflate(f):\n", + " \"\"\"\n", + " Reduce degree of ``f`` by mapping ``x_i**m`` to ``y_i``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**6*y**2 + x**3 + 1, x, y).deflate()\n", + " ((3, 2), Poly(x**2*y + x + 1, x, y, domain='ZZ'))\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'deflate'):\n", + " J, result = f.rep.deflate()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'deflate')\n", + "\n", + " return J, f.per(result)\n", + "\n", + " def inject(f, front=False):\n", + " \"\"\"\n", + " Inject ground domain generators into ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> f = Poly(x**2*y + x*y**3 + x*y + 1, x)\n", + "\n", + " >>> f.inject()\n", + " Poly(x**2*y + x*y**3 + x*y + 1, x, y, domain='ZZ')\n", + " >>> f.inject(front=True)\n", + " Poly(y**3*x + y*x**2 + y*x + 1, y, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " dom = f.rep.dom\n", + "\n", + " if dom.is_Numerical:\n", + " return f\n", + " elif not dom.is_Poly:\n", + " raise DomainError(\"can't inject generators over %s\" % dom)\n", + "\n", + " if hasattr(f.rep, 'inject'):\n", + " result = f.rep.inject(front=front)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'inject')\n", + "\n", + " if front:\n", + " gens = dom.symbols + f.gens\n", + " else:\n", + " gens = f.gens + dom.symbols\n", + "\n", + " return f.new(result, *gens)\n", + "\n", + " def eject(f, *gens):\n", + " \"\"\"\n", + " Eject selected generators into the ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> f = Poly(x**2*y + x*y**3 + x*y + 1, x, y)\n", + "\n", + " >>> f.eject(x)\n", + " Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]')\n", + " >>> f.eject(y)\n", + " Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]')\n", + "\n", + " \"\"\"\n", + " dom = f.rep.dom\n", + "\n", + " if not dom.is_Numerical:\n", + " raise DomainError(\"can't eject generators over %s\" % dom)\n", + "\n", + " k = len(gens)\n", + "\n", + " if f.gens[:k] == gens:\n", + " _gens, front = f.gens[k:], True\n", + " elif f.gens[-k:] == gens:\n", + " _gens, front = f.gens[:-k], False\n", + " else:\n", + " raise NotImplementedError(\n", + " \"can only eject front or back generators\")\n", + "\n", + " dom = dom.inject(*gens)\n", + "\n", + " if hasattr(f.rep, 'eject'):\n", + " result = f.rep.eject(dom, front=front)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'eject')\n", + "\n", + " return f.new(result, *_gens)\n", + "\n", + " def terms_gcd(f):\n", + " \"\"\"\n", + " Remove GCD of terms from the polynomial ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**6*y**2 + x**3*y, x, y).terms_gcd()\n", + " ((3, 1), Poly(x**3*y + 1, x, y, domain='ZZ'))\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'terms_gcd'):\n", + " J, result = f.rep.terms_gcd()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'terms_gcd')\n", + "\n", + " return J, f.per(result)\n", + "\n", + " def add_ground(f, coeff):\n", + " \"\"\"\n", + " Add an element of the ground domain to ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x + 1).add_ground(2)\n", + " Poly(x + 3, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'add_ground'):\n", + " result = f.rep.add_ground(coeff)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'add_ground')\n", + "\n", + " return f.per(result)\n", + "\n", + " def sub_ground(f, coeff):\n", + " \"\"\"\n", + " Subtract an element of the ground domain from ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x + 1).sub_ground(2)\n", + " Poly(x - 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'sub_ground'):\n", + " result = f.rep.sub_ground(coeff)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'sub_ground')\n", + "\n", + " return f.per(result)\n", + "\n", + " def mul_ground(f, coeff):\n", + " \"\"\"\n", + " Multiply ``f`` by a an element of the ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x + 1).mul_ground(2)\n", + " Poly(2*x + 2, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'mul_ground'):\n", + " result = f.rep.mul_ground(coeff)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'mul_ground')\n", + "\n", + " return f.per(result)\n", + "\n", + " def quo_ground(f, coeff):\n", + " \"\"\"\n", + " Quotient of ``f`` by a an element of the ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(2*x + 4).quo_ground(2)\n", + " Poly(x + 2, x, domain='ZZ')\n", + "\n", + " >>> Poly(2*x + 3).quo_ground(2)\n", + " Poly(x + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'quo_ground'):\n", + " result = f.rep.quo_ground(coeff)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'quo_ground')\n", + "\n", + " return f.per(result)\n", + "\n", + " def exquo_ground(f, coeff):\n", + " \"\"\"\n", + " Exact quotient of ``f`` by a an element of the ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(2*x + 4).exquo_ground(2)\n", + " Poly(x + 2, x, domain='ZZ')\n", + "\n", + " >>> Poly(2*x + 3).exquo_ground(2)\n", + " Traceback (most recent call last):\n", + " ...\n", + " ExactQuotientFailed: 2 does not divide 3 in ZZ\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'exquo_ground'):\n", + " result = f.rep.exquo_ground(coeff)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'exquo_ground')\n", + "\n", + " return f.per(result)\n", + "\n", + " def abs(f):\n", + " \"\"\"\n", + " Make all coefficients in ``f`` positive.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 1, x).abs()\n", + " Poly(x**2 + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'abs'):\n", + " result = f.rep.abs()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'abs')\n", + "\n", + " return f.per(result)\n", + "\n", + " def neg(f):\n", + " \"\"\"\n", + " Negate all coefficients in ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 1, x).neg()\n", + " Poly(-x**2 + 1, x, domain='ZZ')\n", + "\n", + " >>> -Poly(x**2 - 1, x)\n", + " Poly(-x**2 + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'neg'):\n", + " result = f.rep.neg()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'neg')\n", + "\n", + " return f.per(result)\n", + "\n", + " def add(f, g):\n", + " \"\"\"\n", + " Add two polynomials ``f`` and ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).add(Poly(x - 2, x))\n", + " Poly(x**2 + x - 1, x, domain='ZZ')\n", + "\n", + " >>> Poly(x**2 + 1, x) + Poly(x - 2, x)\n", + " Poly(x**2 + x - 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " g = sympify(g)\n", + "\n", + " if not g.is_Poly:\n", + " return f.add_ground(g)\n", + "\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'add'):\n", + " result = F.add(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'add')\n", + "\n", + " return per(result)\n", + "\n", + " def sub(f, g):\n", + " \"\"\"\n", + " Subtract two polynomials ``f`` and ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).sub(Poly(x - 2, x))\n", + " Poly(x**2 - x + 3, x, domain='ZZ')\n", + "\n", + " >>> Poly(x**2 + 1, x) - Poly(x - 2, x)\n", + " Poly(x**2 - x + 3, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " g = sympify(g)\n", + "\n", + " if not g.is_Poly:\n", + " return f.sub_ground(g)\n", + "\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'sub'):\n", + " result = F.sub(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'sub')\n", + "\n", + " return per(result)\n", + "\n", + " def mul(f, g):\n", + " \"\"\"\n", + " Multiply two polynomials ``f`` and ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).mul(Poly(x - 2, x))\n", + " Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ')\n", + "\n", + " >>> Poly(x**2 + 1, x)*Poly(x - 2, x)\n", + " Poly(x**3 - 2*x**2 + x - 2, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " g = sympify(g)\n", + "\n", + " if not g.is_Poly:\n", + " return f.mul_ground(g)\n", + "\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'mul'):\n", + " result = F.mul(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'mul')\n", + "\n", + " return per(result)\n", + "\n", + " def sqr(f):\n", + " \"\"\"\n", + " Square a polynomial ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x - 2, x).sqr()\n", + " Poly(x**2 - 4*x + 4, x, domain='ZZ')\n", + "\n", + " >>> Poly(x - 2, x)**2\n", + " Poly(x**2 - 4*x + 4, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'sqr'):\n", + " result = f.rep.sqr()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'sqr')\n", + "\n", + " return f.per(result)\n", + "\n", + " def pow(f, n):\n", + " \"\"\"\n", + " Raise ``f`` to a non-negative power ``n``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x - 2, x).pow(3)\n", + " Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ')\n", + "\n", + " >>> Poly(x - 2, x)**3\n", + " Poly(x**3 - 6*x**2 + 12*x - 8, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " n = int(n)\n", + "\n", + " if hasattr(f.rep, 'pow'):\n", + " result = f.rep.pow(n)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'pow')\n", + "\n", + " return f.per(result)\n", + "\n", + " def pdiv(f, g):\n", + " \"\"\"\n", + " Polynomial pseudo-division of ``f`` by ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).pdiv(Poly(2*x - 4, x))\n", + " (Poly(2*x + 4, x, domain='ZZ'), Poly(20, x, domain='ZZ'))\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'pdiv'):\n", + " q, r = F.pdiv(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'pdiv')\n", + "\n", + " return per(q), per(r)\n", + "\n", + " def prem(f, g):\n", + " \"\"\"\n", + " Polynomial pseudo-remainder of ``f`` by ``g``.\n", + "\n", + " Caveat: The function prem(f, g, x) can be safely used to compute\n", + " in Z[x] _only_ subresultant polynomial remainder sequences (prs's).\n", + "\n", + " To safely compute Euclidean and Sturmian prs's in Z[x]\n", + " employ anyone of the corresponding functions found in\n", + " the module sympy.polys.subresultants_qq_zz. The functions\n", + " in the module with suffix _pg compute prs's in Z[x] employing\n", + " rem(f, g, x), whereas the functions with suffix _amv\n", + " compute prs's in Z[x] employing rem_z(f, g, x).\n", + "\n", + " The function rem_z(f, g, x) differs from prem(f, g, x) in that\n", + " to compute the remainder polynomials in Z[x] it premultiplies\n", + " the divident times the absolute value of the leading coefficient\n", + " of the divisor raised to the power degree(f, x) - degree(g, x) + 1.\n", + "\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).prem(Poly(2*x - 4, x))\n", + " Poly(20, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'prem'):\n", + " result = F.prem(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'prem')\n", + "\n", + " return per(result)\n", + "\n", + " def pquo(f, g):\n", + " \"\"\"\n", + " Polynomial pseudo-quotient of ``f`` by ``g``.\n", + "\n", + " See the Caveat note in the function prem(f, g).\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).pquo(Poly(2*x - 4, x))\n", + " Poly(2*x + 4, x, domain='ZZ')\n", + "\n", + " >>> Poly(x**2 - 1, x).pquo(Poly(2*x - 2, x))\n", + " Poly(2*x + 2, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'pquo'):\n", + " result = F.pquo(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'pquo')\n", + "\n", + " return per(result)\n", + "\n", + " def pexquo(f, g):\n", + " \"\"\"\n", + " Polynomial exact pseudo-quotient of ``f`` by ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 1, x).pexquo(Poly(2*x - 2, x))\n", + " Poly(2*x + 2, x, domain='ZZ')\n", + "\n", + " >>> Poly(x**2 + 1, x).pexquo(Poly(2*x - 4, x))\n", + " Traceback (most recent call last):\n", + " ...\n", + " ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'pexquo'):\n", + " try:\n", + " result = F.pexquo(G)\n", + " except ExactQuotientFailed as exc:\n", + " raise exc.new(f.as_expr(), g.as_expr())\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'pexquo')\n", + "\n", + " return per(result)\n", + "\n", + " def div(f, g, auto=True):\n", + " \"\"\"\n", + " Polynomial division with remainder of ``f`` by ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x))\n", + " (Poly(1/2*x + 1, x, domain='QQ'), Poly(5, x, domain='QQ'))\n", + "\n", + " >>> Poly(x**2 + 1, x).div(Poly(2*x - 4, x), auto=False)\n", + " (Poly(0, x, domain='ZZ'), Poly(x**2 + 1, x, domain='ZZ'))\n", + "\n", + " \"\"\"\n", + " dom, per, F, G = f._unify(g)\n", + " retract = False\n", + "\n", + " if auto and dom.is_Ring and not dom.is_Field:\n", + " F, G = F.to_field(), G.to_field()\n", + " retract = True\n", + "\n", + " if hasattr(f.rep, 'div'):\n", + " q, r = F.div(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'div')\n", + "\n", + " if retract:\n", + " try:\n", + " Q, R = q.to_ring(), r.to_ring()\n", + " except CoercionFailed:\n", + " pass\n", + " else:\n", + " q, r = Q, R\n", + "\n", + " return per(q), per(r)\n", + "\n", + " def rem(f, g, auto=True):\n", + " \"\"\"\n", + " Computes the polynomial remainder of ``f`` by ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x))\n", + " Poly(5, x, domain='ZZ')\n", + "\n", + " >>> Poly(x**2 + 1, x).rem(Poly(2*x - 4, x), auto=False)\n", + " Poly(x**2 + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " dom, per, F, G = f._unify(g)\n", + " retract = False\n", + "\n", + " if auto and dom.is_Ring and not dom.is_Field:\n", + " F, G = F.to_field(), G.to_field()\n", + " retract = True\n", + "\n", + " if hasattr(f.rep, 'rem'):\n", + " r = F.rem(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'rem')\n", + "\n", + " if retract:\n", + " try:\n", + " r = r.to_ring()\n", + " except CoercionFailed:\n", + " pass\n", + "\n", + " return per(r)\n", + "\n", + " def quo(f, g, auto=True):\n", + " \"\"\"\n", + " Computes polynomial quotient of ``f`` by ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).quo(Poly(2*x - 4, x))\n", + " Poly(1/2*x + 1, x, domain='QQ')\n", + "\n", + " >>> Poly(x**2 - 1, x).quo(Poly(x - 1, x))\n", + " Poly(x + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " dom, per, F, G = f._unify(g)\n", + " retract = False\n", + "\n", + " if auto and dom.is_Ring and not dom.is_Field:\n", + " F, G = F.to_field(), G.to_field()\n", + " retract = True\n", + "\n", + " if hasattr(f.rep, 'quo'):\n", + " q = F.quo(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'quo')\n", + "\n", + " if retract:\n", + " try:\n", + " q = q.to_ring()\n", + " except CoercionFailed:\n", + " pass\n", + "\n", + " return per(q)\n", + "\n", + " def exquo(f, g, auto=True):\n", + " \"\"\"\n", + " Computes polynomial exact quotient of ``f`` by ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 1, x).exquo(Poly(x - 1, x))\n", + " Poly(x + 1, x, domain='ZZ')\n", + "\n", + " >>> Poly(x**2 + 1, x).exquo(Poly(2*x - 4, x))\n", + " Traceback (most recent call last):\n", + " ...\n", + " ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1\n", + "\n", + " \"\"\"\n", + " dom, per, F, G = f._unify(g)\n", + " retract = False\n", + "\n", + " if auto and dom.is_Ring and not dom.is_Field:\n", + " F, G = F.to_field(), G.to_field()\n", + " retract = True\n", + "\n", + " if hasattr(f.rep, 'exquo'):\n", + " try:\n", + " q = F.exquo(G)\n", + " except ExactQuotientFailed as exc:\n", + " raise exc.new(f.as_expr(), g.as_expr())\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'exquo')\n", + "\n", + " if retract:\n", + " try:\n", + " q = q.to_ring()\n", + " except CoercionFailed:\n", + " pass\n", + "\n", + " return per(q)\n", + "\n", + " def _gen_to_level(f, gen):\n", + " \"\"\"Returns level associated with the given generator. \"\"\"\n", + " if isinstance(gen, int):\n", + " length = len(f.gens)\n", + "\n", + " if -length <= gen < length:\n", + " if gen < 0:\n", + " return length + gen\n", + " else:\n", + " return gen\n", + " else:\n", + " raise PolynomialError(\"-%s <= gen < %s expected, got %s\" %\n", + " (length, length, gen))\n", + " else:\n", + " try:\n", + " return f.gens.index(sympify(gen))\n", + " except ValueError:\n", + " raise PolynomialError(\n", + " \"a valid generator expected, got %s\" % gen)\n", + "\n", + " def degree(f, gen=0):\n", + " \"\"\"\n", + " Returns degree of ``f`` in ``x_j``.\n", + "\n", + " The degree of 0 is negative infinity.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + y*x + 1, x, y).degree()\n", + " 2\n", + " >>> Poly(x**2 + y*x + y, x, y).degree(y)\n", + " 1\n", + " >>> Poly(0, x).degree()\n", + " -oo\n", + "\n", + " \"\"\"\n", + " j = f._gen_to_level(gen)\n", + "\n", + " if hasattr(f.rep, 'degree'):\n", + " return f.rep.degree(j)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'degree')\n", + "\n", + " def degree_list(f):\n", + " \"\"\"\n", + " Returns a list of degrees of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + y*x + 1, x, y).degree_list()\n", + " (2, 1)\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'degree_list'):\n", + " return f.rep.degree_list()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'degree_list')\n", + "\n", + " def total_degree(f):\n", + " \"\"\"\n", + " Returns the total degree of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + y*x + 1, x, y).total_degree()\n", + " 2\n", + " >>> Poly(x + y**5, x, y).total_degree()\n", + " 5\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'total_degree'):\n", + " return f.rep.total_degree()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'total_degree')\n", + "\n", + " def homogenize(f, s):\n", + " \"\"\"\n", + " Returns the homogeneous polynomial of ``f``.\n", + "\n", + " A homogeneous polynomial is a polynomial whose all monomials with\n", + " non-zero coefficients have the same total degree. If you only\n", + " want to check if a polynomial is homogeneous, then use\n", + " :func:`Poly.is_homogeneous`. If you want not only to check if a\n", + " polynomial is homogeneous but also compute its homogeneous order,\n", + " then use :func:`Poly.homogeneous_order`.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y, z\n", + "\n", + " >>> f = Poly(x**5 + 2*x**2*y**2 + 9*x*y**3)\n", + " >>> f.homogenize(z)\n", + " Poly(x**5 + 2*x**2*y**2*z + 9*x*y**3*z, x, y, z, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if not isinstance(s, Symbol):\n", + " raise TypeError(\"``Symbol`` expected, got %s\" % type(s))\n", + " if s in f.gens:\n", + " i = f.gens.index(s)\n", + " gens = f.gens\n", + " else:\n", + " i = len(f.gens)\n", + " gens = f.gens + (s,)\n", + " if hasattr(f.rep, 'homogenize'):\n", + " return f.per(f.rep.homogenize(i), gens=gens)\n", + " raise OperationNotSupported(f, 'homogeneous_order')\n", + "\n", + " def homogeneous_order(f):\n", + " \"\"\"\n", + " Returns the homogeneous order of ``f``.\n", + "\n", + " A homogeneous polynomial is a polynomial whose all monomials with\n", + " non-zero coefficients have the same total degree. This degree is\n", + " the homogeneous order of ``f``. If you only want to check if a\n", + " polynomial is homogeneous, then use :func:`Poly.is_homogeneous`.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> f = Poly(x**5 + 2*x**3*y**2 + 9*x*y**4)\n", + " >>> f.homogeneous_order()\n", + " 5\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'homogeneous_order'):\n", + " return f.rep.homogeneous_order()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'homogeneous_order')\n", + "\n", + " def LC(f, order=None):\n", + " \"\"\"\n", + " Returns the leading coefficient of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(4*x**3 + 2*x**2 + 3*x, x).LC()\n", + " 4\n", + "\n", + " \"\"\"\n", + " if order is not None:\n", + " return f.coeffs(order)[0]\n", + "\n", + " if hasattr(f.rep, 'LC'):\n", + " result = f.rep.LC()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'LC')\n", + "\n", + " return f.rep.dom.to_sympy(result)\n", + "\n", + " def TC(f):\n", + " \"\"\"\n", + " Returns the trailing coefficient of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**3 + 2*x**2 + 3*x, x).TC()\n", + " 0\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'TC'):\n", + " result = f.rep.TC()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'TC')\n", + "\n", + " return f.rep.dom.to_sympy(result)\n", + "\n", + " def EC(f, order=None):\n", + " \"\"\"\n", + " Returns the last non-zero coefficient of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**3 + 2*x**2 + 3*x, x).EC()\n", + " 3\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'coeffs'):\n", + " return f.coeffs(order)[-1]\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'EC')\n", + "\n", + " def coeff_monomial(f, monom):\n", + " \"\"\"\n", + " Returns the coefficient of ``monom`` in ``f`` if there, else None.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, exp\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> p = Poly(24*x*y*exp(8) + 23*x, x, y)\n", + "\n", + " >>> p.coeff_monomial(x)\n", + " 23\n", + " >>> p.coeff_monomial(y)\n", + " 0\n", + " >>> p.coeff_monomial(x*y)\n", + " 24*exp(8)\n", + "\n", + " Note that ``Expr.coeff()`` behaves differently, collecting terms\n", + " if possible; the Poly must be converted to an Expr to use that\n", + " method, however:\n", + "\n", + " >>> p.as_expr().coeff(x)\n", + " 24*y*exp(8) + 23\n", + " >>> p.as_expr().coeff(y)\n", + " 24*x*exp(8)\n", + " >>> p.as_expr().coeff(x*y)\n", + " 24*exp(8)\n", + "\n", + " See Also\n", + " ========\n", + " nth: more efficient query using exponents of the monomial's generators\n", + "\n", + " \"\"\"\n", + " return f.nth(*Monomial(monom, f.gens).exponents)\n", + "\n", + " def nth(f, *N):\n", + " \"\"\"\n", + " Returns the ``n``-th coefficient of ``f`` where ``N`` are the\n", + " exponents of the generators in the term of interest.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, sqrt\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**3 + 2*x**2 + 3*x, x).nth(2)\n", + " 2\n", + " >>> Poly(x**3 + 2*x*y**2 + y**2, x, y).nth(1, 2)\n", + " 2\n", + " >>> Poly(4*sqrt(x)*y)\n", + " Poly(4*y*(sqrt(x)), y, sqrt(x), domain='ZZ')\n", + " >>> _.nth(1, 1)\n", + " 4\n", + "\n", + " See Also\n", + " ========\n", + " coeff_monomial\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'nth'):\n", + " if len(N) != len(f.gens):\n", + " raise ValueError('exponent of each generator must be specified')\n", + " result = f.rep.nth(*list(map(int, N)))\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'nth')\n", + "\n", + " return f.rep.dom.to_sympy(result)\n", + "\n", + " def coeff(f, x, n=1, right=False):\n", + " # the semantics of coeff_monomial and Expr.coeff are different;\n", + " # if someone is working with a Poly, they should be aware of the\n", + " # differences and chose the method best suited for the query.\n", + " # Alternatively, a pure-polys method could be written here but\n", + " # at this time the ``right`` keyword would be ignored because Poly\n", + " # doesn't work with non-commutatives.\n", + " raise NotImplementedError(\n", + " 'Either convert to Expr with `as_expr` method '\n", + " 'to use Expr\\'s coeff method or else use the '\n", + " '`coeff_monomial` method of Polys.')\n", + "\n", + " def LM(f, order=None):\n", + " \"\"\"\n", + " Returns the leading monomial of ``f``.\n", + "\n", + " The Leading monomial signifies the monomial having\n", + " the highest power of the principal generator in the\n", + " expression f.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LM()\n", + " x**2*y**0\n", + "\n", + " \"\"\"\n", + " return Monomial(f.monoms(order)[0], f.gens)\n", + "\n", + " def EM(f, order=None):\n", + " \"\"\"\n", + " Returns the last non-zero monomial of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).EM()\n", + " x**0*y**1\n", + "\n", + " \"\"\"\n", + " return Monomial(f.monoms(order)[-1], f.gens)\n", + "\n", + " def LT(f, order=None):\n", + " \"\"\"\n", + " Returns the leading term of ``f``.\n", + "\n", + " The Leading term signifies the term having\n", + " the highest power of the principal generator in the\n", + " expression f along with its coefficient.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LT()\n", + " (x**2*y**0, 4)\n", + "\n", + " \"\"\"\n", + " monom, coeff = f.terms(order)[0]\n", + " return Monomial(monom, f.gens), coeff\n", + "\n", + " def ET(f, order=None):\n", + " \"\"\"\n", + " Returns the last non-zero term of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).ET()\n", + " (x**0*y**1, 3)\n", + "\n", + " \"\"\"\n", + " monom, coeff = f.terms(order)[-1]\n", + " return Monomial(monom, f.gens), coeff\n", + "\n", + " def max_norm(f):\n", + " \"\"\"\n", + " Returns maximum norm of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(-x**2 + 2*x - 3, x).max_norm()\n", + " 3\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'max_norm'):\n", + " result = f.rep.max_norm()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'max_norm')\n", + "\n", + " return f.rep.dom.to_sympy(result)\n", + "\n", + " def l1_norm(f):\n", + " \"\"\"\n", + " Returns l1 norm of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(-x**2 + 2*x - 3, x).l1_norm()\n", + " 6\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'l1_norm'):\n", + " result = f.rep.l1_norm()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'l1_norm')\n", + "\n", + " return f.rep.dom.to_sympy(result)\n", + "\n", + " def clear_denoms(self, convert=False):\n", + " \"\"\"\n", + " Clear denominators, but keep the ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, S, QQ\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = Poly(x/2 + S(1)/3, x, domain=QQ)\n", + "\n", + " >>> f.clear_denoms()\n", + " (6, Poly(3*x + 2, x, domain='QQ'))\n", + " >>> f.clear_denoms(convert=True)\n", + " (6, Poly(3*x + 2, x, domain='ZZ'))\n", + "\n", + " \"\"\"\n", + " f = self\n", + "\n", + " if not f.rep.dom.is_Field:\n", + " return S.One, f\n", + "\n", + " dom = f.get_domain()\n", + " if dom.has_assoc_Ring:\n", + " dom = f.rep.dom.get_ring()\n", + "\n", + " if hasattr(f.rep, 'clear_denoms'):\n", + " coeff, result = f.rep.clear_denoms()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'clear_denoms')\n", + "\n", + " coeff, f = dom.to_sympy(coeff), f.per(result)\n", + "\n", + " if not convert or not dom.has_assoc_Ring:\n", + " return coeff, f\n", + " else:\n", + " return coeff, f.to_ring()\n", + "\n", + " def rat_clear_denoms(self, g):\n", + " \"\"\"\n", + " Clear denominators in a rational function ``f/g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> f = Poly(x**2/y + 1, x)\n", + " >>> g = Poly(x**3 + y, x)\n", + "\n", + " >>> p, q = f.rat_clear_denoms(g)\n", + "\n", + " >>> p\n", + " Poly(x**2 + y, x, domain='ZZ[y]')\n", + " >>> q\n", + " Poly(y*x**3 + y**2, x, domain='ZZ[y]')\n", + "\n", + " \"\"\"\n", + " f = self\n", + "\n", + " dom, per, f, g = f._unify(g)\n", + "\n", + " f = per(f)\n", + " g = per(g)\n", + "\n", + " if not (dom.is_Field and dom.has_assoc_Ring):\n", + " return f, g\n", + "\n", + " a, f = f.clear_denoms(convert=True)\n", + " b, g = g.clear_denoms(convert=True)\n", + "\n", + " f = f.mul_ground(b)\n", + " g = g.mul_ground(a)\n", + "\n", + " return f, g\n", + "\n", + " def integrate(self, *specs, **args):\n", + " \"\"\"\n", + " Computes indefinite integral of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + 2*x + 1, x).integrate()\n", + " Poly(1/3*x**3 + x**2 + x, x, domain='QQ')\n", + "\n", + " >>> Poly(x*y**2 + x, x, y).integrate((0, 1), (1, 0))\n", + " Poly(1/2*x**2*y**2 + 1/2*x**2, x, y, domain='QQ')\n", + "\n", + " \"\"\"\n", + " f = self\n", + "\n", + " if args.get('auto', True) and f.rep.dom.is_Ring:\n", + " f = f.to_field()\n", + "\n", + " if hasattr(f.rep, 'integrate'):\n", + " if not specs:\n", + " return f.per(f.rep.integrate(m=1))\n", + "\n", + " rep = f.rep\n", + "\n", + " for spec in specs:\n", + " if type(spec) is tuple:\n", + " gen, m = spec\n", + " else:\n", + " gen, m = spec, 1\n", + "\n", + " rep = rep.integrate(int(m), f._gen_to_level(gen))\n", + "\n", + " return f.per(rep)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'integrate')\n", + "\n", + " def diff(f, *specs, **kwargs):\n", + " \"\"\"\n", + " Computes partial derivative of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + 2*x + 1, x).diff()\n", + " Poly(2*x + 2, x, domain='ZZ')\n", + "\n", + " >>> Poly(x*y**2 + x, x, y).diff((0, 0), (1, 1))\n", + " Poly(2*x*y, x, y, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if not kwargs.get('evaluate', True):\n", + " return Derivative(f, *specs, **kwargs)\n", + "\n", + " if hasattr(f.rep, 'diff'):\n", + " if not specs:\n", + " return f.per(f.rep.diff(m=1))\n", + "\n", + " rep = f.rep\n", + "\n", + " for spec in specs:\n", + " if type(spec) is tuple:\n", + " gen, m = spec\n", + " else:\n", + " gen, m = spec, 1\n", + "\n", + " rep = rep.diff(int(m), f._gen_to_level(gen))\n", + "\n", + " return f.per(rep)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'diff')\n", + "\n", + " _eval_derivative = diff\n", + "\n", + " def eval(self, x, a=None, auto=True):\n", + " \"\"\"\n", + " Evaluate ``f`` at ``a`` in the given variable.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y, z\n", + "\n", + " >>> Poly(x**2 + 2*x + 3, x).eval(2)\n", + " 11\n", + "\n", + " >>> Poly(2*x*y + 3*x + y + 2, x, y).eval(x, 2)\n", + " Poly(5*y + 8, y, domain='ZZ')\n", + "\n", + " >>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)\n", + "\n", + " >>> f.eval({x: 2})\n", + " Poly(5*y + 2*z + 6, y, z, domain='ZZ')\n", + " >>> f.eval({x: 2, y: 5})\n", + " Poly(2*z + 31, z, domain='ZZ')\n", + " >>> f.eval({x: 2, y: 5, z: 7})\n", + " 45\n", + "\n", + " >>> f.eval((2, 5))\n", + " Poly(2*z + 31, z, domain='ZZ')\n", + " >>> f(2, 5)\n", + " Poly(2*z + 31, z, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " f = self\n", + "\n", + " if a is None:\n", + " if isinstance(x, dict):\n", + " mapping = x\n", + "\n", + " for gen, value in mapping.items():\n", + " f = f.eval(gen, value)\n", + "\n", + " return f\n", + " elif isinstance(x, (tuple, list)):\n", + " values = x\n", + "\n", + " if len(values) > len(f.gens):\n", + " raise ValueError(\"too many values provided\")\n", + "\n", + " for gen, value in zip(f.gens, values):\n", + " f = f.eval(gen, value)\n", + "\n", + " return f\n", + " else:\n", + " j, a = 0, x\n", + " else:\n", + " j = f._gen_to_level(x)\n", + "\n", + " if not hasattr(f.rep, 'eval'): # pragma: no cover\n", + " raise OperationNotSupported(f, 'eval')\n", + "\n", + " try:\n", + " result = f.rep.eval(a, j)\n", + " except CoercionFailed:\n", + " if not auto:\n", + " raise DomainError(\"can't evaluate at %s in %s\" % (a, f.rep.dom))\n", + " else:\n", + " a_domain, [a] = construct_domain([a])\n", + " new_domain = f.get_domain().unify_with_symbols(a_domain, f.gens)\n", + "\n", + " f = f.set_domain(new_domain)\n", + " a = new_domain.convert(a, a_domain)\n", + "\n", + " result = f.rep.eval(a, j)\n", + "\n", + " return f.per(result, remove=j)\n", + "\n", + " def __call__(f, *values):\n", + " \"\"\"\n", + " Evaluate ``f`` at the give values.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y, z\n", + "\n", + " >>> f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)\n", + "\n", + " >>> f(2)\n", + " Poly(5*y + 2*z + 6, y, z, domain='ZZ')\n", + " >>> f(2, 5)\n", + " Poly(2*z + 31, z, domain='ZZ')\n", + " >>> f(2, 5, 7)\n", + " 45\n", + "\n", + " \"\"\"\n", + " return f.eval(values)\n", + "\n", + " def half_gcdex(f, g, auto=True):\n", + " \"\"\"\n", + " Half extended Euclidean algorithm of ``f`` and ``g``.\n", + "\n", + " Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15\n", + " >>> g = x**3 + x**2 - 4*x - 4\n", + "\n", + " >>> Poly(f).half_gcdex(Poly(g))\n", + " (Poly(-1/5*x + 3/5, x, domain='QQ'), Poly(x + 1, x, domain='QQ'))\n", + "\n", + " \"\"\"\n", + " dom, per, F, G = f._unify(g)\n", + "\n", + " if auto and dom.is_Ring:\n", + " F, G = F.to_field(), G.to_field()\n", + "\n", + " if hasattr(f.rep, 'half_gcdex'):\n", + " s, h = F.half_gcdex(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'half_gcdex')\n", + "\n", + " return per(s), per(h)\n", + "\n", + " def gcdex(f, g, auto=True):\n", + " \"\"\"\n", + " Extended Euclidean algorithm of ``f`` and ``g``.\n", + "\n", + " Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15\n", + " >>> g = x**3 + x**2 - 4*x - 4\n", + "\n", + " >>> Poly(f).gcdex(Poly(g))\n", + " (Poly(-1/5*x + 3/5, x, domain='QQ'),\n", + " Poly(1/5*x**2 - 6/5*x + 2, x, domain='QQ'),\n", + " Poly(x + 1, x, domain='QQ'))\n", + "\n", + " \"\"\"\n", + " dom, per, F, G = f._unify(g)\n", + "\n", + " if auto and dom.is_Ring:\n", + " F, G = F.to_field(), G.to_field()\n", + "\n", + " if hasattr(f.rep, 'gcdex'):\n", + " s, t, h = F.gcdex(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'gcdex')\n", + "\n", + " return per(s), per(t), per(h)\n", + "\n", + " def invert(f, g, auto=True):\n", + " \"\"\"\n", + " Invert ``f`` modulo ``g`` when possible.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 1, x).invert(Poly(2*x - 1, x))\n", + " Poly(-4/3, x, domain='QQ')\n", + "\n", + " >>> Poly(x**2 - 1, x).invert(Poly(x - 1, x))\n", + " Traceback (most recent call last):\n", + " ...\n", + " NotInvertible: zero divisor\n", + "\n", + " \"\"\"\n", + " dom, per, F, G = f._unify(g)\n", + "\n", + " if auto and dom.is_Ring:\n", + " F, G = F.to_field(), G.to_field()\n", + "\n", + " if hasattr(f.rep, 'invert'):\n", + " result = F.invert(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'invert')\n", + "\n", + " return per(result)\n", + "\n", + " def revert(f, n):\n", + " \"\"\"\n", + " Compute ``f**(-1)`` mod ``x**n``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(1, x).revert(2)\n", + " Poly(1, x, domain='ZZ')\n", + "\n", + " >>> Poly(1 + x, x).revert(1)\n", + " Poly(1, x, domain='ZZ')\n", + "\n", + " >>> Poly(x**2 - 1, x).revert(1)\n", + " Traceback (most recent call last):\n", + " ...\n", + " NotReversible: only unity is reversible in a ring\n", + "\n", + " >>> Poly(1/x, x).revert(1)\n", + " Traceback (most recent call last):\n", + " ...\n", + " PolynomialError: 1/x contains an element of the generators set\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'revert'):\n", + " result = f.rep.revert(int(n))\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'revert')\n", + "\n", + " return f.per(result)\n", + "\n", + " def subresultants(f, g):\n", + " \"\"\"\n", + " Computes the subresultant PRS of ``f`` and ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 1, x).subresultants(Poly(x**2 - 1, x))\n", + " [Poly(x**2 + 1, x, domain='ZZ'),\n", + " Poly(x**2 - 1, x, domain='ZZ'),\n", + " Poly(-2, x, domain='ZZ')]\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'subresultants'):\n", + " result = F.subresultants(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'subresultants')\n", + "\n", + " return list(map(per, result))\n", + "\n", + " def resultant(f, g, includePRS=False):\n", + " \"\"\"\n", + " Computes the resultant of ``f`` and ``g`` via PRS.\n", + "\n", + " If includePRS=True, it includes the subresultant PRS in the result.\n", + " Because the PRS is used to calculate the resultant, this is more\n", + " efficient than calling :func:`subresultants` separately.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = Poly(x**2 + 1, x)\n", + "\n", + " >>> f.resultant(Poly(x**2 - 1, x))\n", + " 4\n", + " >>> f.resultant(Poly(x**2 - 1, x), includePRS=True)\n", + " (4, [Poly(x**2 + 1, x, domain='ZZ'), Poly(x**2 - 1, x, domain='ZZ'),\n", + " Poly(-2, x, domain='ZZ')])\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'resultant'):\n", + " if includePRS:\n", + " result, R = F.resultant(G, includePRS=includePRS)\n", + " else:\n", + " result = F.resultant(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'resultant')\n", + "\n", + " if includePRS:\n", + " return (per(result, remove=0), list(map(per, R)))\n", + " return per(result, remove=0)\n", + "\n", + " def discriminant(f):\n", + " \"\"\"\n", + " Computes the discriminant of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + 2*x + 3, x).discriminant()\n", + " -8\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'discriminant'):\n", + " result = f.rep.discriminant()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'discriminant')\n", + "\n", + " return f.per(result, remove=0)\n", + "\n", + " def dispersionset(f, g=None):\n", + " r\"\"\"Compute the *dispersion set* of two polynomials.\n", + "\n", + " For two polynomials `f(x)` and `g(x)` with `\\deg f > 0`\n", + " and `\\deg g > 0` the dispersion set `\\operatorname{J}(f, g)` is defined as:\n", + "\n", + " .. math::\n", + " \\operatorname{J}(f, g)\n", + " & := \\{a \\in \\mathbb{N}_0 | \\gcd(f(x), g(x+a)) \\neq 1\\} \\\\\n", + " & = \\{a \\in \\mathbb{N}_0 | \\deg \\gcd(f(x), g(x+a)) \\geq 1\\}\n", + "\n", + " For a single polynomial one defines `\\operatorname{J}(f) := \\operatorname{J}(f, f)`.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import poly\n", + " >>> from sympy.polys.dispersion import dispersion, dispersionset\n", + " >>> from sympy.abc import x\n", + "\n", + " Dispersion set and dispersion of a simple polynomial:\n", + "\n", + " >>> fp = poly((x - 3)*(x + 3), x)\n", + " >>> sorted(dispersionset(fp))\n", + " [0, 6]\n", + " >>> dispersion(fp)\n", + " 6\n", + "\n", + " Note that the definition of the dispersion is not symmetric:\n", + "\n", + " >>> fp = poly(x**4 - 3*x**2 + 1, x)\n", + " >>> gp = fp.shift(-3)\n", + " >>> sorted(dispersionset(fp, gp))\n", + " [2, 3, 4]\n", + " >>> dispersion(fp, gp)\n", + " 4\n", + " >>> sorted(dispersionset(gp, fp))\n", + " []\n", + " >>> dispersion(gp, fp)\n", + " -oo\n", + "\n", + " Computing the dispersion also works over field extensions:\n", + "\n", + " >>> from sympy import sqrt\n", + " >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ')\n", + " >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ')\n", + " >>> sorted(dispersionset(fp, gp))\n", + " [2]\n", + " >>> sorted(dispersionset(gp, fp))\n", + " [1, 4]\n", + "\n", + " We can even perform the computations for polynomials\n", + " having symbolic coefficients:\n", + "\n", + " >>> from sympy.abc import a\n", + " >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)\n", + " >>> sorted(dispersionset(fp))\n", + " [0, 1]\n", + "\n", + " See Also\n", + " ========\n", + "\n", + " dispersion\n", + "\n", + " References\n", + " ==========\n", + "\n", + " 1. [ManWright94]_\n", + " 2. [Koepf98]_\n", + " 3. [Abramov71]_\n", + " 4. [Man93]_\n", + " \"\"\"\n", + " from sympy.polys.dispersion import dispersionset\n", + " return dispersionset(f, g)\n", + "\n", + " def dispersion(f, g=None):\n", + " r\"\"\"Compute the *dispersion* of polynomials.\n", + "\n", + " For two polynomials `f(x)` and `g(x)` with `\\deg f > 0`\n", + " and `\\deg g > 0` the dispersion `\\operatorname{dis}(f, g)` is defined as:\n", + "\n", + " .. math::\n", + " \\operatorname{dis}(f, g)\n", + " & := \\max\\{ J(f,g) \\cup \\{0\\} \\} \\\\\n", + " & = \\max\\{ \\{a \\in \\mathbb{N} | \\gcd(f(x), g(x+a)) \\neq 1\\} \\cup \\{0\\} \\}\n", + "\n", + " and for a single polynomial `\\operatorname{dis}(f) := \\operatorname{dis}(f, f)`.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import poly\n", + " >>> from sympy.polys.dispersion import dispersion, dispersionset\n", + " >>> from sympy.abc import x\n", + "\n", + " Dispersion set and dispersion of a simple polynomial:\n", + "\n", + " >>> fp = poly((x - 3)*(x + 3), x)\n", + " >>> sorted(dispersionset(fp))\n", + " [0, 6]\n", + " >>> dispersion(fp)\n", + " 6\n", + "\n", + " Note that the definition of the dispersion is not symmetric:\n", + "\n", + " >>> fp = poly(x**4 - 3*x**2 + 1, x)\n", + " >>> gp = fp.shift(-3)\n", + " >>> sorted(dispersionset(fp, gp))\n", + " [2, 3, 4]\n", + " >>> dispersion(fp, gp)\n", + " 4\n", + " >>> sorted(dispersionset(gp, fp))\n", + " []\n", + " >>> dispersion(gp, fp)\n", + " -oo\n", + "\n", + " Computing the dispersion also works over field extensions:\n", + "\n", + " >>> from sympy import sqrt\n", + " >>> fp = poly(x**2 + sqrt(5)*x - 1, x, domain='QQ')\n", + " >>> gp = poly(x**2 + (2 + sqrt(5))*x + sqrt(5), x, domain='QQ')\n", + " >>> sorted(dispersionset(fp, gp))\n", + " [2]\n", + " >>> sorted(dispersionset(gp, fp))\n", + " [1, 4]\n", + "\n", + " We can even perform the computations for polynomials\n", + " having symbolic coefficients:\n", + "\n", + " >>> from sympy.abc import a\n", + " >>> fp = poly(4*x**4 + (4*a + 8)*x**3 + (a**2 + 6*a + 4)*x**2 + (a**2 + 2*a)*x, x)\n", + " >>> sorted(dispersionset(fp))\n", + " [0, 1]\n", + "\n", + " See Also\n", + " ========\n", + "\n", + " dispersionset\n", + "\n", + " References\n", + " ==========\n", + "\n", + " 1. [ManWright94]_\n", + " 2. [Koepf98]_\n", + " 3. [Abramov71]_\n", + " 4. [Man93]_\n", + " \"\"\"\n", + " from sympy.polys.dispersion import dispersion\n", + " return dispersion(f, g)\n", + "\n", + " def cofactors(f, g):\n", + " \"\"\"\n", + " Returns the GCD of ``f`` and ``g`` and their cofactors.\n", + "\n", + " Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and\n", + " ``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors\n", + " of ``f`` and ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 1, x).cofactors(Poly(x**2 - 3*x + 2, x))\n", + " (Poly(x - 1, x, domain='ZZ'),\n", + " Poly(x + 1, x, domain='ZZ'),\n", + " Poly(x - 2, x, domain='ZZ'))\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'cofactors'):\n", + " h, cff, cfg = F.cofactors(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'cofactors')\n", + "\n", + " return per(h), per(cff), per(cfg)\n", + "\n", + " def gcd(f, g):\n", + " \"\"\"\n", + " Returns the polynomial GCD of ``f`` and ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 1, x).gcd(Poly(x**2 - 3*x + 2, x))\n", + " Poly(x - 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'gcd'):\n", + " result = F.gcd(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'gcd')\n", + "\n", + " return per(result)\n", + "\n", + " def lcm(f, g):\n", + " \"\"\"\n", + " Returns polynomial LCM of ``f`` and ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 1, x).lcm(Poly(x**2 - 3*x + 2, x))\n", + " Poly(x**3 - 2*x**2 - x + 2, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'lcm'):\n", + " result = F.lcm(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'lcm')\n", + "\n", + " return per(result)\n", + "\n", + " def trunc(f, p):\n", + " \"\"\"\n", + " Reduce ``f`` modulo a constant ``p``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(2*x**3 + 3*x**2 + 5*x + 7, x).trunc(3)\n", + " Poly(-x**3 - x + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " p = f.rep.dom.convert(p)\n", + "\n", + " if hasattr(f.rep, 'trunc'):\n", + " result = f.rep.trunc(p)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'trunc')\n", + "\n", + " return f.per(result)\n", + "\n", + " def monic(self, auto=True):\n", + " \"\"\"\n", + " Divides all coefficients by ``LC(f)``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, ZZ\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(3*x**2 + 6*x + 9, x, domain=ZZ).monic()\n", + " Poly(x**2 + 2*x + 3, x, domain='QQ')\n", + "\n", + " >>> Poly(3*x**2 + 4*x + 2, x, domain=ZZ).monic()\n", + " Poly(x**2 + 4/3*x + 2/3, x, domain='QQ')\n", + "\n", + " \"\"\"\n", + " f = self\n", + "\n", + " if auto and f.rep.dom.is_Ring:\n", + " f = f.to_field()\n", + "\n", + " if hasattr(f.rep, 'monic'):\n", + " result = f.rep.monic()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'monic')\n", + "\n", + " return f.per(result)\n", + "\n", + " def content(f):\n", + " \"\"\"\n", + " Returns the GCD of polynomial coefficients.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(6*x**2 + 8*x + 12, x).content()\n", + " 2\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'content'):\n", + " result = f.rep.content()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'content')\n", + "\n", + " return f.rep.dom.to_sympy(result)\n", + "\n", + " def primitive(f):\n", + " \"\"\"\n", + " Returns the content and a primitive form of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(2*x**2 + 8*x + 12, x).primitive()\n", + " (2, Poly(x**2 + 4*x + 6, x, domain='ZZ'))\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'primitive'):\n", + " cont, result = f.rep.primitive()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'primitive')\n", + "\n", + " return f.rep.dom.to_sympy(cont), f.per(result)\n", + "\n", + " def compose(f, g):\n", + " \"\"\"\n", + " Computes the functional composition of ``f`` and ``g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + x, x).compose(Poly(x - 1, x))\n", + " Poly(x**2 - x, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " _, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(f.rep, 'compose'):\n", + " result = F.compose(G)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'compose')\n", + "\n", + " return per(result)\n", + "\n", + " def decompose(f):\n", + " \"\"\"\n", + " Computes a functional decomposition of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**4 + 2*x**3 - x - 1, x, domain='ZZ').decompose()\n", + " [Poly(x**2 - x - 1, x, domain='ZZ'), Poly(x**2 + x, x, domain='ZZ')]\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'decompose'):\n", + " result = f.rep.decompose()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'decompose')\n", + "\n", + " return list(map(f.per, result))\n", + "\n", + " def shift(f, a):\n", + " \"\"\"\n", + " Efficiently compute Taylor shift ``f(x + a)``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 2*x + 1, x).shift(2)\n", + " Poly(x**2 + 2*x + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'shift'):\n", + " result = f.rep.shift(a)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'shift')\n", + "\n", + " return f.per(result)\n", + "\n", + " def transform(f, p, q):\n", + " \"\"\"\n", + " Efficiently evaluate the functional transformation ``q**n * f(p/q)``.\n", + "\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1, x), Poly(x - 1, x))\n", + " Poly(4, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " P, Q = p.unify(q)\n", + " F, P = f.unify(P)\n", + " F, Q = F.unify(Q)\n", + "\n", + " if hasattr(F.rep, 'transform'):\n", + " result = F.rep.transform(P.rep, Q.rep)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(F, 'transform')\n", + "\n", + " return F.per(result)\n", + "\n", + " def sturm(self, auto=True):\n", + " \"\"\"\n", + " Computes the Sturm sequence of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**3 - 2*x**2 + x - 3, x).sturm()\n", + " [Poly(x**3 - 2*x**2 + x - 3, x, domain='QQ'),\n", + " Poly(3*x**2 - 4*x + 1, x, domain='QQ'),\n", + " Poly(2/9*x + 25/9, x, domain='QQ'),\n", + " Poly(-2079/4, x, domain='QQ')]\n", + "\n", + " \"\"\"\n", + " f = self\n", + "\n", + " if auto and f.rep.dom.is_Ring:\n", + " f = f.to_field()\n", + "\n", + " if hasattr(f.rep, 'sturm'):\n", + " result = f.rep.sturm()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'sturm')\n", + "\n", + " return list(map(f.per, result))\n", + "\n", + " def gff_list(f):\n", + " \"\"\"\n", + " Computes greatest factorial factorization of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = x**5 + 2*x**4 - x**3 - 2*x**2\n", + "\n", + " >>> Poly(f).gff_list()\n", + " [(Poly(x, x, domain='ZZ'), 1), (Poly(x + 2, x, domain='ZZ'), 4)]\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'gff_list'):\n", + " result = f.rep.gff_list()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'gff_list')\n", + "\n", + " return [(f.per(g), k) for g, k in result]\n", + "\n", + " def norm(f):\n", + " \"\"\"\n", + " Computes the product, ``Norm(f)``, of the conjugates of\n", + " a polynomial ``f`` defined over a number field ``K``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, sqrt\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> a, b = sqrt(2), sqrt(3)\n", + "\n", + " A polynomial over a quadratic extension.\n", + " Two conjugates x - a and x + a.\n", + "\n", + " >>> f = Poly(x - a, x, extension=a)\n", + " >>> f.norm()\n", + " Poly(x**2 - 2, x, domain='QQ')\n", + "\n", + " A polynomial over a quartic extension.\n", + " Four conjugates x - a, x - a, x + a and x + a.\n", + "\n", + " >>> f = Poly(x - a, x, extension=(a, b))\n", + " >>> f.norm()\n", + " Poly(x**4 - 4*x**2 + 4, x, domain='QQ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'norm'):\n", + " r = f.rep.norm()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'norm')\n", + "\n", + " return f.per(r)\n", + "\n", + " def sqf_norm(f):\n", + " \"\"\"\n", + " Computes square-free norm of ``f``.\n", + "\n", + " Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and\n", + " ``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``,\n", + " where ``a`` is the algebraic extension of the ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, sqrt\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> s, f, r = Poly(x**2 + 1, x, extension=[sqrt(3)]).sqf_norm()\n", + "\n", + " >>> s\n", + " 1\n", + " >>> f\n", + " Poly(x**2 - 2*sqrt(3)*x + 4, x, domain='QQ')\n", + " >>> r\n", + " Poly(x**4 - 4*x**2 + 16, x, domain='QQ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'sqf_norm'):\n", + " s, g, r = f.rep.sqf_norm()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'sqf_norm')\n", + "\n", + " return s, f.per(g), f.per(r)\n", + "\n", + " def sqf_part(f):\n", + " \"\"\"\n", + " Computes square-free part of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**3 - 3*x - 2, x).sqf_part()\n", + " Poly(x**2 - x - 2, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'sqf_part'):\n", + " result = f.rep.sqf_part()\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'sqf_part')\n", + "\n", + " return f.per(result)\n", + "\n", + " def sqf_list(f, all=False):\n", + " \"\"\"\n", + " Returns a list of square-free factors of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16\n", + "\n", + " >>> Poly(f).sqf_list()\n", + " (2, [(Poly(x + 1, x, domain='ZZ'), 2),\n", + " (Poly(x + 2, x, domain='ZZ'), 3)])\n", + "\n", + " >>> Poly(f).sqf_list(all=True)\n", + " (2, [(Poly(1, x, domain='ZZ'), 1),\n", + " (Poly(x + 1, x, domain='ZZ'), 2),\n", + " (Poly(x + 2, x, domain='ZZ'), 3)])\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'sqf_list'):\n", + " coeff, factors = f.rep.sqf_list(all)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'sqf_list')\n", + "\n", + " return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors]\n", + "\n", + " def sqf_list_include(f, all=False):\n", + " \"\"\"\n", + " Returns a list of square-free factors of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, expand\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = expand(2*(x + 1)**3*x**4)\n", + " >>> f\n", + " 2*x**7 + 6*x**6 + 6*x**5 + 2*x**4\n", + "\n", + " >>> Poly(f).sqf_list_include()\n", + " [(Poly(2, x, domain='ZZ'), 1),\n", + " (Poly(x + 1, x, domain='ZZ'), 3),\n", + " (Poly(x, x, domain='ZZ'), 4)]\n", + "\n", + " >>> Poly(f).sqf_list_include(all=True)\n", + " [(Poly(2, x, domain='ZZ'), 1),\n", + " (Poly(1, x, domain='ZZ'), 2),\n", + " (Poly(x + 1, x, domain='ZZ'), 3),\n", + " (Poly(x, x, domain='ZZ'), 4)]\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'sqf_list_include'):\n", + " factors = f.rep.sqf_list_include(all)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'sqf_list_include')\n", + "\n", + " return [(f.per(g), k) for g, k in factors]\n", + "\n", + " def factor_list(f):\n", + " \"\"\"\n", + " Returns a list of irreducible factors of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y\n", + "\n", + " >>> Poly(f).factor_list()\n", + " (2, [(Poly(x + y, x, y, domain='ZZ'), 1),\n", + " (Poly(x**2 + 1, x, y, domain='ZZ'), 2)])\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'factor_list'):\n", + " try:\n", + " coeff, factors = f.rep.factor_list()\n", + " except DomainError:\n", + " return S.One, [(f, 1)]\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'factor_list')\n", + "\n", + " return f.rep.dom.to_sympy(coeff), [(f.per(g), k) for g, k in factors]\n", + "\n", + " def factor_list_include(f):\n", + " \"\"\"\n", + " Returns a list of irreducible factors of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y\n", + "\n", + " >>> Poly(f).factor_list_include()\n", + " [(Poly(2*x + 2*y, x, y, domain='ZZ'), 1),\n", + " (Poly(x**2 + 1, x, y, domain='ZZ'), 2)]\n", + "\n", + " \"\"\"\n", + " if hasattr(f.rep, 'factor_list_include'):\n", + " try:\n", + " factors = f.rep.factor_list_include()\n", + " except DomainError:\n", + " return [(f, 1)]\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'factor_list_include')\n", + "\n", + " return [(f.per(g), k) for g, k in factors]\n", + "\n", + " def intervals(f, all=False, eps=None, inf=None, sup=None, fast=False, sqf=False):\n", + " \"\"\"\n", + " Compute isolating intervals for roots of ``f``.\n", + "\n", + " For real roots the Vincent-Akritas-Strzebonski (VAS) continued fractions method is used.\n", + "\n", + " References\n", + " ==========\n", + " .. [#] Alkiviadis G. Akritas and Adam W. Strzebonski: A Comparative Study of Two Real Root\n", + " Isolation Methods . Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 297-304, 2005.\n", + " .. [#] Alkiviadis G. Akritas, Adam W. Strzebonski and Panagiotis S. Vigklas: Improving the\n", + " Performance of the Continued Fractions Method Using new Bounds of Positive Roots. Nonlinear\n", + " Analysis: Modelling and Control, Vol. 13, No. 3, 265-279, 2008.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 3, x).intervals()\n", + " [((-2, -1), 1), ((1, 2), 1)]\n", + " >>> Poly(x**2 - 3, x).intervals(eps=1e-2)\n", + " [((-26/15, -19/11), 1), ((19/11, 26/15), 1)]\n", + "\n", + " \"\"\"\n", + " if eps is not None:\n", + " eps = QQ.convert(eps)\n", + "\n", + " if eps <= 0:\n", + " raise ValueError(\"'eps' must be a positive rational\")\n", + "\n", + " if inf is not None:\n", + " inf = QQ.convert(inf)\n", + " if sup is not None:\n", + " sup = QQ.convert(sup)\n", + "\n", + " if hasattr(f.rep, 'intervals'):\n", + " result = f.rep.intervals(\n", + " all=all, eps=eps, inf=inf, sup=sup, fast=fast, sqf=sqf)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'intervals')\n", + "\n", + " if sqf:\n", + " def _real(interval):\n", + " s, t = interval\n", + " return (QQ.to_sympy(s), QQ.to_sympy(t))\n", + "\n", + " if not all:\n", + " return list(map(_real, result))\n", + "\n", + " def _complex(rectangle):\n", + " (u, v), (s, t) = rectangle\n", + " return (QQ.to_sympy(u) + I*QQ.to_sympy(v),\n", + " QQ.to_sympy(s) + I*QQ.to_sympy(t))\n", + "\n", + " real_part, complex_part = result\n", + "\n", + " return list(map(_real, real_part)), list(map(_complex, complex_part))\n", + " else:\n", + " def _real(interval):\n", + " (s, t), k = interval\n", + " return ((QQ.to_sympy(s), QQ.to_sympy(t)), k)\n", + "\n", + " if not all:\n", + " return list(map(_real, result))\n", + "\n", + " def _complex(rectangle):\n", + " ((u, v), (s, t)), k = rectangle\n", + " return ((QQ.to_sympy(u) + I*QQ.to_sympy(v),\n", + " QQ.to_sympy(s) + I*QQ.to_sympy(t)), k)\n", + "\n", + " real_part, complex_part = result\n", + "\n", + " return list(map(_real, real_part)), list(map(_complex, complex_part))\n", + "\n", + " def refine_root(f, s, t, eps=None, steps=None, fast=False, check_sqf=False):\n", + " \"\"\"\n", + " Refine an isolating interval of a root to the given precision.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 3, x).refine_root(1, 2, eps=1e-2)\n", + " (19/11, 26/15)\n", + "\n", + " \"\"\"\n", + " if check_sqf and not f.is_sqf:\n", + " raise PolynomialError(\"only square-free polynomials supported\")\n", + "\n", + " s, t = QQ.convert(s), QQ.convert(t)\n", + "\n", + " if eps is not None:\n", + " eps = QQ.convert(eps)\n", + "\n", + " if eps <= 0:\n", + " raise ValueError(\"'eps' must be a positive rational\")\n", + "\n", + " if steps is not None:\n", + " steps = int(steps)\n", + " elif eps is None:\n", + " steps = 1\n", + "\n", + " if hasattr(f.rep, 'refine_root'):\n", + " S, T = f.rep.refine_root(s, t, eps=eps, steps=steps, fast=fast)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'refine_root')\n", + "\n", + " return QQ.to_sympy(S), QQ.to_sympy(T)\n", + "\n", + " def count_roots(f, inf=None, sup=None):\n", + " \"\"\"\n", + " Return the number of roots of ``f`` in ``[inf, sup]`` interval.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly, I\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**4 - 4, x).count_roots(-3, 3)\n", + " 2\n", + " >>> Poly(x**4 - 4, x).count_roots(0, 1 + 3*I)\n", + " 1\n", + "\n", + " \"\"\"\n", + " inf_real, sup_real = True, True\n", + "\n", + " if inf is not None:\n", + " inf = sympify(inf)\n", + "\n", + " if inf is S.NegativeInfinity:\n", + " inf = None\n", + " else:\n", + " re, im = inf.as_real_imag()\n", + "\n", + " if not im:\n", + " inf = QQ.convert(inf)\n", + " else:\n", + " inf, inf_real = list(map(QQ.convert, (re, im))), False\n", + "\n", + " if sup is not None:\n", + " sup = sympify(sup)\n", + "\n", + " if sup is S.Infinity:\n", + " sup = None\n", + " else:\n", + " re, im = sup.as_real_imag()\n", + "\n", + " if not im:\n", + " sup = QQ.convert(sup)\n", + " else:\n", + " sup, sup_real = list(map(QQ.convert, (re, im))), False\n", + "\n", + " if inf_real and sup_real:\n", + " if hasattr(f.rep, 'count_real_roots'):\n", + " count = f.rep.count_real_roots(inf=inf, sup=sup)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'count_real_roots')\n", + " else:\n", + " if inf_real and inf is not None:\n", + " inf = (inf, QQ.zero)\n", + "\n", + " if sup_real and sup is not None:\n", + " sup = (sup, QQ.zero)\n", + "\n", + " if hasattr(f.rep, 'count_complex_roots'):\n", + " count = f.rep.count_complex_roots(inf=inf, sup=sup)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'count_complex_roots')\n", + "\n", + " return Integer(count)\n", + "\n", + " def root(f, index, radicals=True):\n", + " \"\"\"\n", + " Get an indexed root of a polynomial.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = Poly(2*x**3 - 7*x**2 + 4*x + 4)\n", + "\n", + " >>> f.root(0)\n", + " -1/2\n", + " >>> f.root(1)\n", + " 2\n", + " >>> f.root(2)\n", + " 2\n", + " >>> f.root(3)\n", + " Traceback (most recent call last):\n", + " ...\n", + " IndexError: root index out of [-3, 2] range, got 3\n", + "\n", + " >>> Poly(x**5 + x + 1).root(0)\n", + " CRootOf(x**3 - x**2 + 1, 0)\n", + "\n", + " \"\"\"\n", + " return sympy.polys.rootoftools.rootof(f, index, radicals=radicals)\n", + "\n", + " def real_roots(f, multiple=True, radicals=True):\n", + " \"\"\"\n", + " Return a list of real roots with multiplicities.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(2*x**3 - 7*x**2 + 4*x + 4).real_roots()\n", + " [-1/2, 2, 2]\n", + " >>> Poly(x**3 + x + 1).real_roots()\n", + " [CRootOf(x**3 + x + 1, 0)]\n", + "\n", + " \"\"\"\n", + " reals = sympy.polys.rootoftools.CRootOf.real_roots(f, radicals=radicals)\n", + "\n", + " if multiple:\n", + " return reals\n", + " else:\n", + " return group(reals, multiple=False)\n", + "\n", + " def all_roots(f, multiple=True, radicals=True):\n", + " \"\"\"\n", + " Return a list of real and complex roots with multiplicities.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(2*x**3 - 7*x**2 + 4*x + 4).all_roots()\n", + " [-1/2, 2, 2]\n", + " >>> Poly(x**3 + x + 1).all_roots()\n", + " [CRootOf(x**3 + x + 1, 0),\n", + " CRootOf(x**3 + x + 1, 1),\n", + " CRootOf(x**3 + x + 1, 2)]\n", + "\n", + " \"\"\"\n", + " roots = sympy.polys.rootoftools.CRootOf.all_roots(f, radicals=radicals)\n", + "\n", + " if multiple:\n", + " return roots\n", + " else:\n", + " return group(roots, multiple=False)\n", + "\n", + " def nroots(f, n=15, maxsteps=50, cleanup=True):\n", + " \"\"\"\n", + " Compute numerical approximations of roots of ``f``.\n", + "\n", + " Parameters\n", + " ==========\n", + "\n", + " n ... the number of digits to calculate\n", + " maxsteps ... the maximum number of iterations to do\n", + "\n", + " If the accuracy `n` cannot be reached in `maxsteps`, it will raise an\n", + " exception. You need to rerun with higher maxsteps.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 3).nroots(n=15)\n", + " [-1.73205080756888, 1.73205080756888]\n", + " >>> Poly(x**2 - 3).nroots(n=30)\n", + " [-1.73205080756887729352744634151, 1.73205080756887729352744634151]\n", + "\n", + " \"\"\"\n", + " from sympy.functions.elementary.complexes import sign\n", + " if f.is_multivariate:\n", + " raise MultivariatePolynomialError(\n", + " \"can't compute numerical roots of %s\" % f)\n", + "\n", + " if f.degree() <= 0:\n", + " return []\n", + "\n", + " # For integer and rational coefficients, convert them to integers only\n", + " # (for accuracy). Otherwise just try to convert the coefficients to\n", + " # mpmath.mpc and raise an exception if the conversion fails.\n", + " if f.rep.dom is ZZ:\n", + " coeffs = [int(coeff) for coeff in f.all_coeffs()]\n", + " elif f.rep.dom is QQ:\n", + " denoms = [coeff.q for coeff in f.all_coeffs()]\n", + " from sympy.core.numbers import ilcm\n", + " fac = ilcm(*denoms)\n", + " coeffs = [int(coeff*fac) for coeff in f.all_coeffs()]\n", + " else:\n", + " coeffs = [coeff.evalf(n=n).as_real_imag()\n", + " for coeff in f.all_coeffs()]\n", + " try:\n", + " coeffs = [mpmath.mpc(*coeff) for coeff in coeffs]\n", + " except TypeError:\n", + " raise DomainError(\"Numerical domain expected, got %s\" % \\\n", + " f.rep.dom)\n", + "\n", + " dps = mpmath.mp.dps\n", + " mpmath.mp.dps = n\n", + "\n", + " try:\n", + " # We need to add extra precision to guard against losing accuracy.\n", + " # 10 times the degree of the polynomial seems to work well.\n", + " roots = mpmath.polyroots(coeffs, maxsteps=maxsteps,\n", + " cleanup=cleanup, error=False, extraprec=f.degree()*10)\n", + "\n", + " # Mpmath puts real roots first, then complex ones (as does all_roots)\n", + " # so we make sure this convention holds here, too.\n", + " roots = list(map(sympify,\n", + " sorted(roots, key=lambda r: (1 if r.imag else 0, r.real, abs(r.imag), sign(r.imag)))))\n", + " except NoConvergence:\n", + " raise NoConvergence(\n", + " 'convergence to root failed; try n < %s or maxsteps > %s' % (\n", + " n, maxsteps))\n", + " finally:\n", + " mpmath.mp.dps = dps\n", + "\n", + " return roots\n", + "\n", + " def ground_roots(f):\n", + " \"\"\"\n", + " Compute roots of ``f`` by factorization in the ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**6 - 4*x**4 + 4*x**3 - x**2).ground_roots()\n", + " {0: 2, 1: 2}\n", + "\n", + " \"\"\"\n", + " if f.is_multivariate:\n", + " raise MultivariatePolynomialError(\n", + " \"can't compute ground roots of %s\" % f)\n", + "\n", + " roots = {}\n", + "\n", + " for factor, k in f.factor_list()[1]:\n", + " if factor.is_linear:\n", + " a, b = factor.all_coeffs()\n", + " roots[-b/a] = k\n", + "\n", + " return roots\n", + "\n", + " def nth_power_roots_poly(f, n):\n", + " \"\"\"\n", + " Construct a polynomial with n-th powers of roots of ``f``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = Poly(x**4 - x**2 + 1)\n", + "\n", + " >>> f.nth_power_roots_poly(2)\n", + " Poly(x**4 - 2*x**3 + 3*x**2 - 2*x + 1, x, domain='ZZ')\n", + " >>> f.nth_power_roots_poly(3)\n", + " Poly(x**4 + 2*x**2 + 1, x, domain='ZZ')\n", + " >>> f.nth_power_roots_poly(4)\n", + " Poly(x**4 + 2*x**3 + 3*x**2 + 2*x + 1, x, domain='ZZ')\n", + " >>> f.nth_power_roots_poly(12)\n", + " Poly(x**4 - 4*x**3 + 6*x**2 - 4*x + 1, x, domain='ZZ')\n", + "\n", + " \"\"\"\n", + " if f.is_multivariate:\n", + " raise MultivariatePolynomialError(\n", + " \"must be a univariate polynomial\")\n", + "\n", + " N = sympify(n)\n", + "\n", + " if N.is_Integer and N >= 1:\n", + " n = int(N)\n", + " else:\n", + " raise ValueError(\"'n' must an integer and n >= 1, got %s\" % n)\n", + "\n", + " x = f.gen\n", + " t = Dummy('t')\n", + "\n", + " r = f.resultant(f.__class__.from_expr(x**n - t, x, t))\n", + "\n", + " return r.replace(t, x)\n", + "\n", + " def cancel(f, g, include=False):\n", + " \"\"\"\n", + " Cancel common factors in a rational function ``f/g``.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x))\n", + " (1, Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ'))\n", + "\n", + " >>> Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x), include=True)\n", + " (Poly(2*x + 2, x, domain='ZZ'), Poly(x - 1, x, domain='ZZ'))\n", + "\n", + " \"\"\"\n", + " dom, per, F, G = f._unify(g)\n", + "\n", + " if hasattr(F, 'cancel'):\n", + " result = F.cancel(G, include=include)\n", + " else: # pragma: no cover\n", + " raise OperationNotSupported(f, 'cancel')\n", + "\n", + " if not include:\n", + " if dom.has_assoc_Ring:\n", + " dom = dom.get_ring()\n", + "\n", + " cp, cq, p, q = result\n", + "\n", + " cp = dom.to_sympy(cp)\n", + " cq = dom.to_sympy(cq)\n", + "\n", + " return cp/cq, per(p), per(q)\n", + " else:\n", + " return tuple(map(per, result))\n", + "\n", + " @property\n", + " def is_zero(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is a zero polynomial.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(0, x).is_zero\n", + " True\n", + " >>> Poly(1, x).is_zero\n", + " False\n", + "\n", + " \"\"\"\n", + " return f.rep.is_zero\n", + "\n", + " @property\n", + " def is_one(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is a unit polynomial.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(0, x).is_one\n", + " False\n", + " >>> Poly(1, x).is_one\n", + " True\n", + "\n", + " \"\"\"\n", + " return f.rep.is_one\n", + "\n", + " @property\n", + " def is_sqf(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is a square-free polynomial.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 - 2*x + 1, x).is_sqf\n", + " False\n", + " >>> Poly(x**2 - 1, x).is_sqf\n", + " True\n", + "\n", + " \"\"\"\n", + " return f.rep.is_sqf\n", + "\n", + " @property\n", + " def is_monic(f):\n", + " \"\"\"\n", + " Returns ``True`` if the leading coefficient of ``f`` is one.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x + 2, x).is_monic\n", + " True\n", + " >>> Poly(2*x + 2, x).is_monic\n", + " False\n", + "\n", + " \"\"\"\n", + " return f.rep.is_monic\n", + "\n", + " @property\n", + " def is_primitive(f):\n", + " \"\"\"\n", + " Returns ``True`` if GCD of the coefficients of ``f`` is one.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(2*x**2 + 6*x + 12, x).is_primitive\n", + " False\n", + " >>> Poly(x**2 + 3*x + 6, x).is_primitive\n", + " True\n", + "\n", + " \"\"\"\n", + " return f.rep.is_primitive\n", + "\n", + " @property\n", + " def is_ground(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is an element of the ground domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x, x).is_ground\n", + " False\n", + " >>> Poly(2, x).is_ground\n", + " True\n", + " >>> Poly(y, x).is_ground\n", + " True\n", + "\n", + " \"\"\"\n", + " return f.rep.is_ground\n", + "\n", + " @property\n", + " def is_linear(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is linear in all its variables.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x + y + 2, x, y).is_linear\n", + " True\n", + " >>> Poly(x*y + 2, x, y).is_linear\n", + " False\n", + "\n", + " \"\"\"\n", + " return f.rep.is_linear\n", + "\n", + " @property\n", + " def is_quadratic(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is quadratic in all its variables.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x*y + 2, x, y).is_quadratic\n", + " True\n", + " >>> Poly(x*y**2 + 2, x, y).is_quadratic\n", + " False\n", + "\n", + " \"\"\"\n", + " return f.rep.is_quadratic\n", + "\n", + " @property\n", + " def is_monomial(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is zero or has only one term.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(3*x**2, x).is_monomial\n", + " True\n", + " >>> Poly(3*x**2 + 1, x).is_monomial\n", + " False\n", + "\n", + " \"\"\"\n", + " return f.rep.is_monomial\n", + "\n", + " @property\n", + " def is_homogeneous(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is a homogeneous polynomial.\n", + "\n", + " A homogeneous polynomial is a polynomial whose all monomials with\n", + " non-zero coefficients have the same total degree. If you want not\n", + " only to check if a polynomial is homogeneous but also compute its\n", + " homogeneous order, then use :func:`Poly.homogeneous_order`.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + x*y, x, y).is_homogeneous\n", + " True\n", + " >>> Poly(x**3 + x*y, x, y).is_homogeneous\n", + " False\n", + "\n", + " \"\"\"\n", + " return f.rep.is_homogeneous\n", + "\n", + " @property\n", + " def is_irreducible(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` has no factors over its domain.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> Poly(x**2 + x + 1, x, modulus=2).is_irreducible\n", + " True\n", + " >>> Poly(x**2 + 1, x, modulus=2).is_irreducible\n", + " False\n", + "\n", + " \"\"\"\n", + " return f.rep.is_irreducible\n", + "\n", + " @property\n", + " def is_univariate(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is a univariate polynomial.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + x + 1, x).is_univariate\n", + " True\n", + " >>> Poly(x*y**2 + x*y + 1, x, y).is_univariate\n", + " False\n", + " >>> Poly(x*y**2 + x*y + 1, x).is_univariate\n", + " True\n", + " >>> Poly(x**2 + x + 1, x, y).is_univariate\n", + " False\n", + "\n", + " \"\"\"\n", + " return len(f.gens) == 1\n", + "\n", + " @property\n", + " def is_multivariate(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is a multivariate polynomial.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x, y\n", + "\n", + " >>> Poly(x**2 + x + 1, x).is_multivariate\n", + " False\n", + " >>> Poly(x*y**2 + x*y + 1, x, y).is_multivariate\n", + " True\n", + " >>> Poly(x*y**2 + x*y + 1, x).is_multivariate\n", + " False\n", + " >>> Poly(x**2 + x + 1, x, y).is_multivariate\n", + " True\n", + "\n", + " \"\"\"\n", + " return len(f.gens) != 1\n", + "\n", + " @property\n", + " def is_cyclotomic(f):\n", + " \"\"\"\n", + " Returns ``True`` if ``f`` is a cyclotomic polnomial.\n", + "\n", + " Examples\n", + " ========\n", + "\n", + " >>> from sympy import Poly\n", + " >>> from sympy.abc import x\n", + "\n", + " >>> f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1\n", + "\n", + " >>> Poly(f).is_cyclotomic\n", + " False\n", + "\n", + " >>> g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1\n", + "\n", + " >>> Poly(g).is_cyclotomic\n", + " True\n", + "\n", + " \"\"\"\n", + " return f.rep.is_cyclotomic\n", + "\n", + " def __abs__(f):\n", + " return f.abs()\n", + "\n", + " def __neg__(f):\n", + " return f.neg()\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __add__(f, g):\n", + " if not g.is_Poly:\n", + " try:\n", + " g = f.__class__(g, *f.gens)\n", + " except PolynomialError:\n", + " return f.as_expr() + g\n", + "\n", + " return f.add(g)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __radd__(f, g):\n", + " if not g.is_Poly:\n", + " try:\n", + " g = f.__class__(g, *f.gens)\n", + " except PolynomialError:\n", + " return g + f.as_expr()\n", + "\n", + " return g.add(f)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __sub__(f, g):\n", + " if not g.is_Poly:\n", + " try:\n", + " g = f.__class__(g, *f.gens)\n", + " except PolynomialError:\n", + " return f.as_expr() - g\n", + "\n", + " return f.sub(g)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __rsub__(f, g):\n", + " if not g.is_Poly:\n", + " try:\n", + " g = f.__class__(g, *f.gens)\n", + " except PolynomialError:\n", + " return g - f.as_expr()\n", + "\n", + " return g.sub(f)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __mul__(f, g):\n", + " if not g.is_Poly:\n", + " try:\n", + " g = f.__class__(g, *f.gens)\n", + " except PolynomialError:\n", + " return f.as_expr()*g\n", + "\n", + " return f.mul(g)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __rmul__(f, g):\n", + " if not g.is_Poly:\n", + " try:\n", + " g = f.__class__(g, *f.gens)\n", + " except PolynomialError:\n", + " return g*f.as_expr()\n", + "\n", + " return g.mul(f)\n", + "\n", + " @_sympifyit('n', NotImplemented)\n", + " def __pow__(f, n):\n", + " if n.is_Integer and n >= 0:\n", + " return f.pow(n)\n", + " else:\n", + " return f.as_expr()**n\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __divmod__(f, g):\n", + " if not g.is_Poly:\n", + " g = f.__class__(g, *f.gens)\n", + "\n", + " return f.div(g)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __rdivmod__(f, g):\n", + " if not g.is_Poly:\n", + " g = f.__class__(g, *f.gens)\n", + "\n", + " return g.div(f)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __mod__(f, g):\n", + " if not g.is_Poly:\n", + " g = f.__class__(g, *f.gens)\n", + "\n", + " return f.rem(g)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __rmod__(f, g):\n", + " if not g.is_Poly:\n", + " g = f.__class__(g, *f.gens)\n", + "\n", + " return g.rem(f)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __floordiv__(f, g):\n", + " if not g.is_Poly:\n", + " g = f.__class__(g, *f.gens)\n", + "\n", + " return f.quo(g)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __rfloordiv__(f, g):\n", + " if not g.is_Poly:\n", + " g = f.__class__(g, *f.gens)\n", + "\n", + " return g.quo(f)\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __div__(f, g):\n", + " return f.as_expr()/g.as_expr()\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __rdiv__(f, g):\n", + " return g.as_expr()/f.as_expr()\n", + "\n", + " __truediv__ = __div__\n", + " __rtruediv__ = __rdiv__\n", + "\n", + " @_sympifyit('other', NotImplemented)\n", + " def __eq__(self, other):\n", + " f, g = self, other\n", + "\n", + " if not g.is_Poly:\n", + " try:\n", + " g = f.__class__(g, f.gens, domain=f.get_domain())\n", + " except (PolynomialError, DomainError, CoercionFailed):\n", + " return False\n", + "\n", + " if f.gens != g.gens:\n", + " return False\n", + "\n", + " if f.rep.dom != g.rep.dom:\n", + " try:\n", + " dom = f.rep.dom.unify(g.rep.dom, f.gens)\n", + " except UnificationFailed:\n", + " return False\n", + "\n", + " f = f.set_domain(dom)\n", + " g = g.set_domain(dom)\n", + "\n", + " return f.rep == g.rep\n", + "\n", + " @_sympifyit('g', NotImplemented)\n", + " def __ne__(f, g):\n", + " return not f == g\n", + "\n", + " def __nonzero__(f):\n", + " return not f.is_zero\n", + "\n", + " __bool__ = __nonzero__\n", + "\n", + " def eq(f, g, strict=False):\n", + " if not strict:\n", + " return f == g\n", + " else:\n", + " return f._strict_eq(sympify(g))\n", + "\n", + " def ne(f, g, strict=False):\n", + " return not f.eq(g, strict=strict)\n", + "\n", + " def _strict_eq(f, g):\n", + " return isinstance(g, f.__class__) and f.gens == g.gens and f.rep.eq(g.rep, strict=True)\n", + "\n" + ] + } + ], + "source": [ + "source(Poly)" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x + \\sqrt{x}, \\sqrt{x}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly((sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')" + ] + }, + "execution_count": 46, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly({1:1,2:1},gens=S('sqrt(x)'))" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle - x + \\sqrt{x^{2}}$" + ], + "text/plain": [ + "-x + sqrt(x**2)" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "x=symbols('x', positive=True)\n", + "S('sqrt(x^2)-x')" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "\"Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')\"" + ] + }, + "execution_count": 48, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "str(Poly({1:1,2:1,4:1},gens=S('sqrt(x)')))" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "\"Poly(x**2 + x + (sqrt(x)), x, sqrt(x), domain='ZZ')\"" + ] + }, + "execution_count": 49, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "str(Poly('x^2+x+sqrt(x)'))" + ] + }, + { + "cell_type": "code", + "execution_count": 50, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $a\\to b+y_{1}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $b\\to c+z_{1}$" ], "text/plain": [ "" @@ -2001,7 +6407,7 @@ { "data": { "text/plain": [ - "(2.5326984818340415e-10, 7.129450063690368, 1.7908873553542452e-10)" + "(1.7908873553542452e-10, 2.5326984818340415e-10, 7.129450063690368)" ] }, "metadata": {}, @@ -2010,10 +6416,10 @@ { "data": { "text/plain": [ - "1.4142142855953455" + "2.5119572187973787e-11" ] }, - "execution_count": 36, + "execution_count": 50, "metadata": {}, "output_type": "execute_result" } @@ -2043,9 +6449,53 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 54, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $a\\to b+e_{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "Substitute $b\\to c+f_{2}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ok\n", + "100 loops, best of 5: 8.66 ms per loop\n", + "1000 loops, best of 5: 1.31 ms per loop\n" + ] + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 54, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "from random import random\n", "formula=cyclize(Sm('((a-b)/c)^2-8**(1/2)*(a-b)/c'))\n", @@ -2060,7 +6510,7 @@ "def evaluate(x):\n", " return newformula.evalf(subs=dict(zip(fs,(1,1,1))))\n", "def evaluate2(x):\n", - " return num1.evalf(dict(zip(fs,x)))/num2.eval(dict(zip(fs,x)))\n", + " return num1.eval(dict(zip(fs,x)))/num2.eval(dict(zip(fs,x)))\n", "#display(num2.eval(dict(zip(fs,(1,1,1)))))\n", "print('ok')\n", "%timeit evaluate((random(),random(),random()))\n", @@ -2095,18 +6545,43 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 52, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sqrt{x} + x$" + ], + "text/plain": [ + "sqrt(x) + x" + ] + }, + "execution_count": 52, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "(Poly({1:1,2:1,4:1},gens=S('sqrt(x)'))-Poly('x^2')).as_expr()" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 55, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "(x*(1 - x), [])" + ] + }, + "execution_count": 55, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "from sympy.solvers.solvers import unrad\n", "unrad('sqrt(x)+x')" @@ -2114,9 +6589,20 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 56, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "True" + ] + }, + "execution_count": 56, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "formula=('x^(2/3)+x^(3/4)+sqrt(x+y)')\n", "type((Poly(formula).gens[1]).args[0])==Symbol" @@ -2124,18 +6610,38 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 57, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "(x, 1/2)" + ] + }, + "execution_count": 57, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "S('sqrt(x)').args" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 58, "metadata": {}, - "outputs": [], + "outputs": [ + { + "ename": "SyntaxError", + "evalue": "invalid syntax (, line 12)", + "output_type": "error", + "traceback": [ + "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m12\u001b[0m\n\u001b[0;31m newgens[gen[0]]=\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" + ] + } + ], "source": [ "def _powr(formula):\n", "\tif formula.func==Pow:\n", @@ -2155,27 +6661,63 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 59, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "True" + ] + }, + "execution_count": 59, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "isinstance(S('1'),Rational)" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 60, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "False" + ] + }, + "execution_count": 60, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "6 in {3:4,5:6}" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 61, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle 1$" + ], + "text/plain": [ + "1" + ] + }, + "execution_count": 61, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "from math import *\n", "from sympy import *\n", @@ -2184,36 +6726,88 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 62, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x^{2 \\pi}, x^{\\pi}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly((x**pi)**2, x**pi, domain='ZZ')" + ] + }, + "execution_count": 62, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "Poly('x^(2*pi)')" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 63, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\frac{1}{8}$" + ], + "text/plain": [ + "1/8" + ] + }, + "execution_count": 63, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "gcd(S('1/8'),S('1/4'))" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 64, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "(3, (x, sqrt(2)))" + ] + }, + "execution_count": 64, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "S('sqrt(2)*3*x').as_coeff_mul()" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 65, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Poly((sqrt(x))**4 + (sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ')\n", + "Poly((x**(1/6))**12 + (x**(1/6))**3 + (x**(1/6))**2 + y**2, x**(1/6), y, domain='ZZ')\n", + "Poly((sqrt(x)) + (sqrt(x*z)) + (sqrt(z)), sqrt(x), sqrt(x*z), sqrt(z), domain='ZZ')\n", + "Poly((sqrt(x))*(sqrt(z)) + (sqrt(x)) + (sqrt(z)), sqrt(x), sqrt(z), domain='ZZ')\n", + "Poly((sqrt(t + z)) + (sqrt(t*x + t*y + x*z + y*z)) + (sqrt(x + y)), sqrt(t + z), sqrt(t*x + t*y + x*z + y*z), sqrt(x + y), domain='ZZ')\n", + "Poly((sqrt(2)) + (sqrt(3)) + (sqrt(6)), sqrt(2), sqrt(3), sqrt(6), domain='ZZ')\n" + ] + } + ], "source": [ "def _powr(formula):\n", "\tif formula.func==Pow:\n", @@ -2355,13 +6949,419 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from shiroindev import *\n", + "formula=S('(a+b-c)^2')\n", + "prove(makesubs(formula,'[c,oo]'))\n", + "prove(makesubs(formula,'[a,oo]',variables='c'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "formula=S('(a+b-d)^2+(a+b-c)^2')\n", + "prove(makesubs(formula,'[a+b,oo],[a+b,oo]',variables='c,d'))\n", + "prove(makesubs(formula,'[0,a+b],[a+b,oo]',variables='c,d'))\n", + "prove(makesubs(formula,'[0,a+b],[0,a+b]',variables='c,d'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "findvalues(formula)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "display(cyclize('((a-b)/c)^2'))\n", + "display(S('sqrt(8)')*cyclize('(a-b)/c'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "formula=cyclize('((a-b)/c)^2-8**(1/2)*(a-b)/c')\n", + "display(Latex('Case $a\\ge c\\ge b$'))\n", + "formula1=makesubs(formula,'[c,oo],[b,oo]',variables='a,c,b')\n", + "prove(formula1)\n", + "display(Latex('Case $a\\ge b\\ge c$'))\n", + "formula2=makesubs(formula,'[b,oo],[c,oo]')\n", + "prove(formula2*4,values='2**(1/2),1,1')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "formula=cyclize('((a-b)/c)^2-8**(1/2)*(a-b)/c')\n", + "formula2=makesubs(formula,'[c,oo],[c,oo]')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "findvalues(formula2,values=[5,2,7])" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "x,y=symbols('x,y')\n", + "ask(Q.positive(x**2+1),Q.real(x))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "formula=Sm('(a^2+b^2+c^2)^2- 3(a^3b+b^3c+c^3a)')\n", + "formula1=makesubs(formula,'[b,oo],[c,oo]',variables='a,b')\n", + "formula2=makesubs(formula,'[c,oo],[b,oo]',variables='a,c')\n", + "prove(formula2)\n", + "print(findvalues(formula1))\n", + "#prove(formula1,values='4*b*(1-b)-1,4*c*(1-c)-1,1')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "makesubs(Sm('a^2-2ab+bc-c^2+ca'),'[c,oo],[a,oo]',variables='b,c')" + ] + }, + { + "cell_type": "code", + "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "$\\displaystyle - \\sqrt{x} \\sqrt{y} + \\sqrt{x y}$" + ], + "text/plain": [ + "-sqrt(x)*sqrt(y) + sqrt(x*y)" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$\\displaystyle 0$" + ], + "text/plain": [ + "0" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "\n", + "formula=S('sqrt(x*y)-sqrt(x)*sqrt(y)')\n", + "display(formula)\n", + "def assumeall(formula,**kwargs):\n", + " formula=S(formula)\n", + " fs=formula.free_symbols\n", + " for x in fs:\n", + " y=Symbol(str(x),**kwargs)\n", + " formula=formula.subs(x,y)\n", + " return formula\n", + "display(assumeall(formula,positive=True))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "Poly(assumeall('x+sqrt(x)',positive=True))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "str(type(S('f1(x,y,z)')))=='f1'" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "ReprPrinter?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "srepr(S('x^2+f(x)+f(x,y)+f()+sqrt(y)'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy import Function,srepr,S\n", + "import re\n", + "f=Function('f')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def allsymbols(formula):\n", + " formula=S(formula)\n", + " funcsymbols=[x[10:-2] for x in re.findall(r\"Function\\(\\'.*?\\'\\)\",srepr(formula))]\n", + " return set(funcsymbols)|set(map(str,formula.free_symbols))\n", + "def vargen(n):\n", + " x='abcdefghijklmnopqrstuvwxyz'[n%26]\n", + " if n>=26:\n", + " x+=str(n//26)\n", + " return x\n", + "allsymbols(S('f(5,7)+f(5)+g(8)+sqrt(x*y)'))\n", + "vargen(36)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from sympy import *\n", + "from sympy.printing.repr import *\n", + "ReprPrinter??" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "{5,7} | {8,7}" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "[x[10:-2] for x in re.findall(r\"Function\\(\\'.*?\\'\\)\",srepr(S('x^2')))]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "S('f(5,7)+f(5)+g(8)+sqrt(x*y)')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "srepr(S('f(5,7)+f(5)+g(8)+sqrt(x*y)'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "display(Latex('$'+str(S('z12345'))+'$'))" + ] + }, + { + "cell_type": "code", + "execution_count": 68, + "metadata": {}, + "outputs": [ + { + "ename": "ValueError", + "evalue": "invalid literal for int() with base 10: ''", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mValueError\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 1\u001b[0m \u001b[0;34m'xyz'\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m2\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[0;32m----> 2\u001b[0;31m \u001b[0mint\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[0;31mValueError\u001b[0m: invalid literal for int() with base 10: ''" + ] + } + ], + "source": [ + "'xyz'[-2::]" + ] + }, + { + "cell_type": "code", + "execution_count": 84, + "metadata": {}, + "outputs": [], + "source": [ + "def sortkey(x):\n", + " x=str(x)\n", + " i=len(x)-1\n", + " while i>=0:\n", + " if x[i] not in '0123456789':\n", + " break\n", + " i-=1\n", + " if i==len(x)-1:\n", + " return (-1,x)\n", + " return (int(x[i+1:]),x[:i+1])" + ] + }, + { + "cell_type": "code", + "execution_count": 85, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[x, y, x0, qq1, z1, t2]" + ] + }, + "execution_count": 85, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sorted(S('x,y,z1,t2,qq1,x0'),key=sortkey)" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Optimization terminated successfully.\n", + " Current function value: 1.154701\n", + " Iterations: 85\n", + " Function evaluations: 152\n" + ] + }, + { + "data": { + "text/plain": [ + "[1.0,\n", + " 3.000209557356011,\n", + " 3.9998018341331227,\n", + " 2.9997236704279775,\n", + " 0.999903278712487]" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2')\n", + "formula=S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2- (x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')\n", + "values=findvalues(formula)\n", + "values=[value/values[0] for value in values]\n", + "[value**2 for value in values]" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\frac{2 \\sqrt{3}}{3}$" + ], + "text/plain": [ + "2*sqrt(3)/3" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "formula1=S('(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2)/(x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')\n", + "formula1.subs(S('[[x_1,1],[x_2,sqrt(3)],[x_3,2],[x_4,sqrt(3)],[x_5,1]]'))" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "formula2=S('x_1^2+x_2^2+x_3^2+x_4^2+x_5^2- (2/sqrt(3))*(x_1*x_2+x_2*x_3+x_3*x_4+x_4*x_5)')" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "Substitute $x_{2}\\to \\sqrt{3} g$" ], "text/plain": [ "" @@ -2373,7 +7373,7 @@ { "data": { "text/latex": [ - "numerator: $a^2+2ab+b^2$" + "Substitute $x_{3}\\to 2 h$" ], "text/plain": [ "" @@ -2385,7 +7385,7 @@ { "data": { "text/latex": [ - "denominator: $1$" + "Substitute $x_{4}\\to \\sqrt{3} i$" ], "text/plain": [ "" @@ -2397,55 +7397,7 @@ { "data": { "text/latex": [ - "status: 0" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le a^2+2ab+b^2 $$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $c\\to a+c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "numerator: $b^2-2bc+c^2$" + "numerator: $3 g^{2} - 4 g h - 2 g x_{1} + 4 h^{2} - 4 h i + 3 i^{2} - 2 i x_{5} + x_{1}^{2} + x_{5}^{2}$" ], "text/plain": [ "" @@ -2493,7 +7445,43 @@ { "data": { "text/latex": [ - "$$2bc \\le b^2+c^2$$" + "$$4 g h \\le 2 g^{2}+2 h^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 g x_{1} \\le g^{2}+x_{1}^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$4 h i \\le 2 h^{2}+2 i^{2}$$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$2 i x_{5} \\le i^{2}+x_{5}^{2}$$" ], "text/plain": [ "" @@ -2532,51 +7520,108 @@ "0" ] }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "prove(formula2,values='1,sqrt(3),2,sqrt(3),1')" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( a^{k}a^{n}, a^{k}, a^{n}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly((a**k)*(a**n), a**k, a**n, domain='ZZ')" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('a^(n+k)')" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "ename": "AttributeError", + "evalue": "'FiniteSet' object has no attribute 'as_coeff_Mul'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'a^{n+k}/{b^n}+{b^{n+k}}/{c^n}+{c^{n+k}}/{a^n}-( a^k+b^k+c^k)'\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/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/sympify.py\u001b[0m in \u001b[0;36msympify\u001b[0;34m(a, locals, convert_xor, strict, rational, evaluate)\u001b[0m\n\u001b[1;32m 382\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 383\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'\\n'\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[0;32m--> 384\u001b[0;31m \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mparse_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m 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"\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/parsing/sympy_parser.py\u001b[0m in \u001b[0;36mparse_expr\u001b[0;34m(s, local_dict, transformations, global_dict, evaluate)\u001b[0m\n\u001b[1;32m 964\u001b[0m \u001b[0mcode\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcompile\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mevaluateFalse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcode\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'eval'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 965\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 966\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0meval_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcode\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlocal_dict\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mglobal_dict\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 967\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 968\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/parsing/sympy_parser.py\u001b[0m in \u001b[0;36meval_expr\u001b[0;34m(code, local_dict, global_dict)\u001b[0m\n\u001b[1;32m 877\u001b[0m \"\"\"\n\u001b[1;32m 878\u001b[0m expr = eval(\n\u001b[0;32m--> 879\u001b[0;31m code, global_dict, local_dict) # take local objects in preference\n\u001b[0m\u001b[1;32m 880\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 881\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n", + "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m__pow__\u001b[0;34m(self, other, mod)\u001b[0m\n\u001b[1;32m 163\u001b[0m \u001b[0;32mdef\u001b[0m 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"\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_pow\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 159\u001b[0m \u001b[0;34m@\u001b[0m\u001b[0mcall_highest_priority\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'__rpow__'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 160\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_pow\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;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 161\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mPow\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;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 162\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 163\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__pow__\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[0mmod\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\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/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/core/cache.py\u001b[0m in \u001b[0;36mwrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 92\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mwrapper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\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 93\u001b[0m \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 94\u001b[0;31m \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\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 95\u001b[0m \u001b[0;32mexcept\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 96\u001b[0m \u001b[0mretval\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\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/power.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, b, e, evaluate)\u001b[0m\n\u001b[1;32m 274\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Atom\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mExp1\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexp_polar\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 275\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mnumer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdenom\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlog\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mim\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfactor_terms\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 276\u001b[0;31m \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mex\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfactor_terms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_coeff_Mul\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 277\u001b[0m \u001b[0mden\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdenom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mex\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 278\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mden\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlog\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mden\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mAttributeError\u001b[0m: 'FiniteSet' object has no attribute 'as_coeff_Mul'" + ] + } + ], + "source": [ + "S('a^{n+k}/{b^n}+{b^(n+k)/(c^n)+c^(n+k)/{a^n}-( a^k+b^k+c^k)')" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'767980716'" + ] + }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "from shiroindev import *\n", - "formula=S('(a+b-c)^2')\n", - "prove(makesubs(formula,'[c,oo]'))\n", - "prove(makesubs(formula,'[a,oo]',variables='c'))" + "import re\n", + "s=r\"\"\"99-my-name-is-John-Smith-6376827-%^-1-2-767980716\"\"\"\n", + "re.compile(r\"^(.*?)-\").search(str(s[::-1])).group(1)[::-1]" ] }, { "cell_type": "code", - "execution_count": 34, + "execution_count": 3, "metadata": {}, "outputs": [ { - "data": { - "text/latex": [ - "Substitute $c\\to a+b+c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" + "name": "stdout", + "output_type": "stream", + "text": [ + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n", + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n" + ] }, { "data": { "text/latex": [ - "Substitute $d\\to a+b+d$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "numerator: $c^2+d^2$" + "numerator: $a^{k} a^{2 n} c^{n} - a^{k} a^{n} b^{n} c^{n} + a^{n} b^{k} b^{2 n} - a^{n} b^{k} b^{n} c^{n} - a^{n} b^{n} c^{k} c^{n} + b^{n} c^{k} c^{2 n}$" ], "text/plain": [ "" @@ -2600,7 +7645,7 @@ { "data": { "text/latex": [ - "status: 0" + "status: 2" ], "text/plain": [ "" @@ -2612,7 +7657,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le c^2+d^2 $$" + "Program couldn't find any proof." ], "text/plain": [ "" @@ -2624,7 +7669,7 @@ { "data": { "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." + "$$ a^{k} a^{n} b^{n} c^{n}+a^{n} b^{k} b^{n} c^{n}+a^{n} b^{n} c^{k} c^{n} \\le a^{k} a^{2 n} c^{n}+a^{n} b^{k} b^{2 n}+b^{n} c^{k} c^{2 n} $$" ], "text/plain": [ "" @@ -2636,7 +7681,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to a+b+(-a-b)/(c+1)$" + "Substitute $n\\to d + k$" ], "text/plain": [ "" @@ -2648,7 +7693,7 @@ { "data": { "text/latex": [ - "Substitute $d\\to a+b+d$" + "numerator: $a^{2 d} a^{3 k} c^{d} c^{k} - a^{d} a^{2 k} b^{d} b^{k} c^{d} c^{k} + a^{d} a^{k} b^{2 d} b^{3 k} - a^{d} a^{k} b^{d} b^{2 k} c^{d} c^{k} - a^{d} a^{k} b^{d} b^{k} c^{d} c^{2 k} + b^{d} b^{k} c^{2 d} c^{3 k}$" ], "text/plain": [ "" @@ -2660,7 +7705,7 @@ { "data": { "text/latex": [ - "numerator: $a^2+2ab+b^2+c^2d^2+2cd^2+d^2$" + "denominator: $1$" ], "text/plain": [ "" @@ -2672,7 +7717,7 @@ { "data": { "text/latex": [ - "denominator: $c^2+2c+1$" + "status: 2" ], "text/plain": [ "" @@ -2681,6 +7726,230 @@ "metadata": {}, "output_type": "display_data" }, + { + "data": { + "text/latex": [ + "Program couldn't find any proof." + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "$$ a^{d} a^{2 k} b^{d} b^{k} c^{d} c^{k}+a^{d} a^{k} b^{d} b^{2 k} c^{d} c^{k}+a^{d} a^{k} b^{d} b^{k} c^{d} c^{2 k} \\le a^{2 d} a^{3 k} c^{d} c^{k}+a^{d} a^{k} b^{2 d} b^{3 k}+b^{d} b^{k} c^{2 d} c^{3 k} $$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "2" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from sympy import *\n", + "from sympy.parsing.latex import parse_latex\n", + "from shiroindev import *\n", + "shiro.seed=1\n", + "from IPython.display import Latex\n", + "shiro.display=lambda x:display(Latex(x))\n", + "formula=expand(parse_latex(r'(\\frac{a^{n+k}}{b^n}+\\frac{b^{n+k}}{c^n}+\\frac{c^{n+k}}{a^n}-( a^k+b^k+c^k))*a^nb^nc^n'))\n", + "newproof()\n", + "prove(formula)\n", + "prove(makesubs(formula,['k',oo],variables='n'))" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 6 x^{3}y^{9} + 5 x^{2}y^{7} + 4 xy^{8}, x, y, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(6*x**3*y**9 + 5*x**2*y**7 + 4*x*y**8, x, y, domain='ZZ')" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly(dict(zip([(1,8),(2,7),(3,9)],[4,5,6])),gens=S('x,y'))" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\sqrt{x} + x^{2} + x$" + ], + "text/plain": [ + "sqrt(x) + x**2 + x" + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly({1:1,2:1,4:1},gens=S('sqrt(x)')).as_expr()" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [], + "source": [ + "def _writ2(coef,fun,variables):\n", + "\treturn latex(Poly(dict(zip(fun,coef)),gens=variables).as_expr())" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'8 x^{3} y^{4} + 7 x y^{2}'" + ] + }, + "execution_count": 32, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "_writ2([7,8],[(1,2),(3,4)],S('x,y'))" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n", + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle 2 a b$" + ], + "text/plain": [ + "2*(a*b)" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "parse_latex(r' a^2+b^2\\geq 2ab').rhs" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n", + "ANTLR runtime and generated code versions disagree: 4.8!=4.7.1\n" + ] + }, + { + "data": { + "text/latex": [ + "$\\displaystyle a^{2} + b^{2}$" + ], + "text/plain": [ + "a**2 + b**2" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "parse_latex(r'2ab \\geq a^2+b^2').rhs" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "numerator: $a d + a f + b c + b e + c f + d e - 2 \\sqrt{a b} \\sqrt{c d} - 2 \\sqrt{a b} \\sqrt{e f} - 2 \\sqrt{c d} \\sqrt{e f}$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/latex": [ + "denominator: $1$" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "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')\n", + "Poly((sqrt(a))**2*(sqrt(d))**2 + (sqrt(a))**2*(sqrt(f))**2 + (sqrt(b))**2*(sqrt(c))**2 + (sqrt(b))**2*(sqrt(e))**2 + (sqrt(c))**2*(sqrt(f))**2 + (sqrt(d))**2*(sqrt(e))**2, sqrt(a), sqrt(b), sqrt(c), sqrt(d), sqrt(e), sqrt(f), domain='QQ')\n", + "Poly(2*(sqrt(a))*(sqrt(b))*(sqrt(c))*(sqrt(d)) + 2*(sqrt(a))*(sqrt(b))*(sqrt(e))*(sqrt(f)) + 2*(sqrt(c))*(sqrt(d))*(sqrt(e))*(sqrt(f)), sqrt(a), sqrt(b), sqrt(c), sqrt(d), sqrt(e), sqrt(f), domain='QQ')\n" + ] + }, { "data": { "text/latex": [ @@ -2696,7 +7965,7 @@ { "data": { "text/latex": [ - "$$ 0 \\le a^2+2ab+b^2+c^2d^2+2cd^2+d^2 $$" + "From weighted AM-GM inequality:" ], "text/plain": [ "" @@ -2708,7 +7977,7 @@ { "data": { "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." + "$$2 \\sqrt{a} \\sqrt{b} \\sqrt{c} \\sqrt{d} \\le a d+b c$$" ], "text/plain": [ "" @@ -2720,7 +7989,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to a+b+(-a-b)/(c+1)$" + "$$2 \\sqrt{a} \\sqrt{b} \\sqrt{e} \\sqrt{f} \\le a f+b e$$" ], "text/plain": [ "" @@ -2732,7 +8001,7 @@ { "data": { "text/latex": [ - "Substitute $d\\to a+b+(-a-b)/(d+1)$" + "$$2 \\sqrt{c} \\sqrt{d} \\sqrt{e} \\sqrt{f} \\le c f+d e$$" ], "text/plain": [ "" @@ -2744,43 +8013,7 @@ { "data": { "text/latex": [ - "numerator: $a^2c^2+2a^2c+a^2d^2+2a^2d+2a^2+2abc^2+4abc+2abd^2+4abd+4ab+b^2c^2+2b^2c+b^2d^2+2b^2d+2b^2$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "denominator: $c^2d^2+2c^2d+c^2+2cd^2+4cd+2c+d^2+2d+1$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "status: 0" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le a^2c^2+2a^2c+a^2d^2+2a^2d+2a^2+2abc^2+4abc+2abd^2+4abd+4ab+b^2c^2+2b^2c+b^2d^2+2b^2d+2b^2 $$" + "$$ 0 \\le 0 $$" ], "text/plain": [ "" @@ -2807,95 +8040,355 @@ "0" ] }, - "execution_count": 34, + "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "formula=S('(a+b-d)^2+(a+b-c)^2')\n", - "prove(makesubs(formula,'[a+b,oo],[a+b,oo]',variables='c,d'))\n", - "prove(makesubs(formula,'[0,a+b],[a+b,oo]',variables='c,d'))\n", - "prove(makesubs(formula,'[0,a+b],[0,a+b]',variables='c,d'))" + "fr=parse_latex(r'\\sqrt{ab}+\\sqrt{cd}+\\sqrt{ef}\\leq\\sqrt{(a+c+e)(b+d+f)}')\n", + "prove(fr.rhs**2-fr.lhs**2)" ] }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle 0$" + ], + "text/plain": [ + "0" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "assumeall(S('sqrt(x*y)-sqrt(x)*sqrt(y)'),positive=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "ename": "GeneratorsError", + "evalue": "duplicated generators: [sqrt(x), sqrt(x)]", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mGeneratorsError\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 1\u001b[0m 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\u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mPoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'sqrt(x)'\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[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, rep, *gens, **args)\u001b[0m\n\u001b[1;32m 107\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0margs\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 108\u001b[0m \u001b[0;34m\"\"\"Create a new polynomial instance out of something useful. \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 109\u001b[0;31m \u001b[0mopt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbuild_options\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\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 110\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 111\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;34m'order'\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py\u001b[0m in \u001b[0;36mbuild_options\u001b[0;34m(gens, args)\u001b[0m\n\u001b[1;32m 729\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 730\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0;36m1\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;34m'opt'\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margs\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mgens\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 731\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mOptions\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\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 732\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 733\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'opt'\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;32m/home/grzegorz/Pobrane/SageMath/local/lib/python3.7/site-packages/sympy/polys/polyoptions.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, gens, args, flags, strict)\u001b[0m\n\u001b[1;32m 152\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0moption\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpreprocess\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 153\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 154\u001b[0;31m \u001b[0mpreprocess_options\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\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 155\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 156\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mvalue\u001b[0m \u001b[0;32min\u001b[0m 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\u001b[0mhas_dups\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\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--> 291\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mGeneratorsError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"duplicated generators: %s\"\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\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 292\u001b[0m \u001b[0;32melif\u001b[0m \u001b[0many\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgen\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_commutative\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mFalse\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mgen\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mgens\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 293\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mGeneratorsError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"non-commutative generators: %s\"\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\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;31mGeneratorsError\u001b[0m: duplicated generators: [sqrt(x), sqrt(x)]" + ] + } + ], + "source": [ + "p=reducegens('x+sqrt(x)')\n", + "q=S('x+sqrt(x)')\n", + "print(Poly(p,gens=[S('sqrt(x)'),S('sqrt(x)')]))\n", + "print(Poly(q,gens=S('sqrt(x)')))" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": {}, + "outputs": [], + "source": [ + "from importlib import reload\n", + "from sympy import *\n", + "import shiroindev\n", + "reload(shiroindev)\n", + "from shiroindev import *\n", + "from sympy.parsing.latex import parse_latex\n", + "from itertools import permutations, combinations\n", + "shiro.seed=1\n", + "from IPython.display import Latex\n", + "shiro.display=lambda x:display(Latex(x))" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x, \\sqrt{x}, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly((sqrt(x))**2, sqrt(x), domain='ZZ')" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('sqrt(x)')*Poly('sqrt(x)')" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( 0, x, domain=\\mathbb{Z} \\right)}$" + ], + "text/plain": [ + "Poly(0, x, domain='ZZ')" + ] + }, + "execution_count": 48, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Poly('x')-Poly('x').abs()" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": {}, + "outputs": [ + { + "data": { + "text/latex": [ + "$\\displaystyle \\operatorname{Poly}{\\left( x, x, y, domain=\\mathbb{Q} \\right)}$" + ], + "text/plain": [ + "Poly(x, x, y, domain='QQ')" + ] + }, + "execution_count": 49, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "fr=Poly('x-y')\n", + "(fr+fr.abs())*(S('1/2'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "from itertools import permutations \n", + "t5=[]\n", + "for ineq in ineqs2:\n", + " u=0\n", + " for vars in permutations(ineq.free_symbols):\n", + " ineqp=makesubs(ineq,list(zip(vars[1:],[oo]*len(vars[1:]))),variables=vars[:-1])\n", + " u=max(u,prove(ineqp))\n", + " print(u,end=',')\n", + " t5+=[u]\n", + "Counter(t5)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "Optimization terminated successfully.\n", - " Current function value: 1.000000\n", - " Iterations: 65\n", - " Function evaluations: 124\n" + "pvalue 4.43444926375551e-06\n", + "statistic 59.0\n" ] - }, + } + ], + "source": [ + "from statsmodels.stats.contingency_tables import mcnemar\n", + "print(mcnemar([[101,121],[59,33]]))" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [], + "source": [ + "nsimplify?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "shiro.display=lambda x:display(Latex(x))\n", + "newproof()\n", + "fr=makesubs(ineqs2[15],'[c,oo],[a,oo]',variables='b,c')\n", + "display(fr)\n", + "fr2=fractioncancel(fr.subs(S('[[e,(sqrt(5)-1)/2*f]]')))[0]\n", + "display(fr2)\n", + "prove(makesubs(fr2,'11/5,9/4',variables=[S('sqrt(5)')]))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def makesubs(formula,intervals,values=None,variables=None,numden=False):\n", + "#This function generates a new formula which satisfies this condition:\n", + "#for all positive variables new formula is nonnegative iff\n", + "#for all variables in corresponding intervals old formula is nonnegative\n", + "\tformula=S(formula)\n", + "\taddsymbols(formula)\n", + "\tintervals=_smakeiterable2(intervals)\n", + "\tif variables: variables=_smakeiterable(variables)\n", + "\telse: variables=sorted(formula.free_symbols,key=str)\n", + "\tif values!=None:\n", + "\t\tvalues=_smakeiterable(values)\n", + "\t\tequations=[var-value for var,value in zip(variables,values)]\n", + "\telse:\n", + "\t\tequations=[]\n", + "\tnewvars=[]\n", + "\tusedvars=set()\n", + "\tfor var,interval in zip(variables,intervals):\n", + "\t\tend1,end2=interval\n", + "\t\tz=newvar()\n", + "\t\tnewvars+=[z]\n", + "\t\tusedvars|={z}\n", + "\t\tif (end1.free_symbols|end2.free_symbols)&usedvars:\n", + "\t\t\tshiro.warning(shiro.translation[\n", + "\t\t\t'Warning: intervals contain backwards dependencies. Consider changing order of variables'])\n", + "\t\tif end1 in {S('-oo'),S('oo')}:\n", + "\t\t\tend1,end2=end2,end1\n", + "\t\tif {end1,end2}=={S('-oo'),S('oo')}:\n", + "\t\t\tsub1=sub2=(z-1/z)\n", + "\t\telif end2==S('oo'):\n", + "\t\t\tsub1=sub2=(end1+z)\n", + "\t\telif end2==S('-oo'):\n", + "\t\t\tsub1=sub2=end1-z\n", + "\t\telse:\n", + "\t\t\tsub1=end2+(end1-end2)/z\n", + "\t\t\tsub2=end2+(end1-end2)/(1+z)\n", + "\t\tformula=formula.subs(var,sub1)\n", + "\t\tshiro.display(shiro.translation['Substitute']+\" $\"+latex(var)+'\\\\to '+latex(sub2)+'$')\n", + "\t\tequations=[equation.subs(var,sub1) for equation in equations]\n", + "\tnum,den=fractioncancel(formula)\n", + "\tfor var,interval in zip(newvars,intervals):\n", + "\t\tif {interval[0],interval[1]} & {S('oo'),S('-oo')}==set():\n", + "\t\t\tnum=num.subs(var,var+1)\n", + "\t\t\tden=den.subs(var,var+1)\n", + "\t\t\tequations=[equation.subs(var,var+1) for equation in equations]\n", + "\tif values:\n", + "\t\tvalues=ssolve(equations,newvars)\n", + "\t\tif len(values):\n", + "\t\t\tvalues=values[0]\n", + "\tnum,den=expand(num),expand(den)\n", + "\t#shiro.display(shiro.translation[\"Formula after substitution:\"],\"$$\",latex(num/den),'$$')\n", + "\tif values and numden:\n", + "\t\treturn num,den,values\n", + "\telif values:\n", + "\t\treturn num/den,values\n", + "\telif numden:\n", + "\t\treturn num,den\n", + "\telse:\n", + "\t\treturn num/den" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ { "data": { "text/plain": [ - "(0.8742272651582828,\n", - " 0.45912361618095604,\n", - " 1.3333325089656345,\n", - " 1.3333771594633597)" + "{3, 5, 7, 8}" ] }, - "execution_count": 27, + "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "findvalues(formula)" + "x={5,7,8}\n", + "x|={3}\n", + "x" ] }, { "cell_type": "code", - "execution_count": 41, + "execution_count": 12, "metadata": {}, "outputs": [ { "data": { - "text/latex": [ - "$\\displaystyle \\frac{\\left(a - b\\right)^{2}}{c^{2}} + \\frac{\\left(- a + c\\right)^{2}}{b^{2}} + \\frac{\\left(b - c\\right)^{2}}{a^{2}}$" - ], "text/plain": [ - "(a - b)**2/c**2 + (-a + c)**2/b**2 + (b - c)**2/a**2" + "{x}" ] }, + "execution_count": 12, "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$\\displaystyle 2 \\sqrt{2} \\left(\\frac{a - b}{c} + \\frac{- a + c}{b} + \\frac{b - c}{a}\\right)$" - ], - "text/plain": [ - "2*sqrt(2)*((a - b)/c + (-a + c)/b + (b - c)/a)" - ] - }, - "metadata": {}, - "output_type": "display_data" + "output_type": "execute_result" } ], "source": [ - "display(cyclize('((a-b)/c)^2'))\n", - "display(S('sqrt(8)')*cyclize('(a-b)/c'))" + "from shiroindev import *\n", + "S('x+7').free_symbols" ] }, { "cell_type": "code", - "execution_count": 42, + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "False" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "bool(set())" + ] + }, + { + "cell_type": "code", + "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/latex": [ - "Case $a\\ge c\\ge b$" + "Substitute $x\\to - y + 1 + \\frac{y - 1}{a + 1}$" ], "text/plain": [ "" @@ -2907,7 +8400,7 @@ { "data": { "text/latex": [ - "Substitute $a\\to a+c$" + "Substitute $y\\to 1 - \\frac{1}{b + 1}$" ], "text/plain": [ "" @@ -2919,7 +8412,7 @@ { "data": { "text/latex": [ - "Substitute $c\\to b+c$" + "Substitute $b\\to \\frac{c}{2}$" ], "text/plain": [ "" @@ -2931,7 +8424,7 @@ { "data": { "text/latex": [ - "numerator: $2a^4b^2+2a^4bc+a^4c^2+4a^3b^3+2\\sqrt{2}a^3b^2c+10a^3b^2c+2\\sqrt{2}a^3bc^2+6a^3bc^2+2a^3c^3+2a^2b^4+2\\sqrt{2}a^2b^3c+10a^2b^3c+6\\sqrt{2}a^2b^2c^2+12a^2b^2c^2+4a^2bc^3+4\\sqrt{2}a^2bc^3+a^2c^4+2ab^4c+2\\sqrt{2}ab^3c^2+6ab^3c^2+4ab^2c^3+4\\sqrt{2}ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4$" + "numerator: $a^{4} c^{2} + a^{3} c^{2} - 2 a^{3} c - 4 a^{2} c + 4 a^{2} + a c^{2} - 2 a c + c^{2}$" ], "text/plain": [ "" @@ -2943,115 +8436,7 @@ { "data": { "text/latex": [ - "denominator: $a^2b^4+2a^2b^3c+a^2b^2c^2+2ab^5+6ab^4c+6ab^3c^2+2ab^2c^3+b^6+4b^5c+6b^4c^2+4b^3c^3+b^2c^4$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "status: 0" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le 2a^4b^2+2a^4bc+a^4c^2+4a^3b^3+2\\sqrt{2}a^3b^2c+10a^3b^2c+2\\sqrt{2}a^3bc^2+6a^3bc^2+2a^3c^3+2a^2b^4+2\\sqrt{2}a^2b^3c+10a^2b^3c+6\\sqrt{2}a^2b^2c^2+12a^2b^2c^2+4\\sqrt{2}a^2bc^3+4a^2bc^3+a^2c^4+2ab^4c+2\\sqrt{2}ab^3c^2+6ab^3c^2+4\\sqrt{2}ab^2c^3+4ab^2c^3+2\\sqrt{2}abc^4+2b^4c^2+4b^3c^3+2b^2c^4 $$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "The sum of all inequalities gives us a proof of the inequality." - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Case $a\\ge b\\ge c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $a\\to a+b$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $b\\to b+c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $a\\to \\sqrt{2}a$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "numerator: $16a^4b^2+32a^4bc+32a^4c^2+16\\sqrt{2}a^3b^3-32a^3b^2c+48\\sqrt{2}a^3b^2c-32a^3bc^2+80\\sqrt{2}a^3bc^2+32\\sqrt{2}a^3c^3+8a^2b^4-32\\sqrt{2}a^2b^3c+32a^2b^3c-48\\sqrt{2}a^2b^2c^2+96a^2b^2c^2-16\\sqrt{2}a^2bc^3+80a^2bc^3+16a^2c^4-16ab^4c-32ab^3c^2+16\\sqrt{2}ab^3c^2-16ab^2c^3+24\\sqrt{2}ab^2c^3+8\\sqrt{2}abc^4+8b^4c^2+16b^3c^3+8b^2c^4$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "denominator: $2a^2b^2c^2+4a^2bc^3+2a^2c^4+2\\sqrt{2}ab^3c^2+6\\sqrt{2}ab^2c^3+6\\sqrt{2}abc^4+2\\sqrt{2}ac^5+b^4c^2+4b^3c^3+6b^2c^4+4bc^5+c^6$" + "denominator: $a^{3} c^{2} + 2 a^{3} c + 2 a^{2} c^{2} + 4 a^{2} c + a c^{2} + 2 a c$" ], "text/plain": [ "" @@ -3087,7 +8472,7 @@ { "data": { "text/latex": [ - "$$32a^3b^2c \\le 16a^4b^2+16a^2b^2c^2$$" + "$$2 a^{3} c \\le a^{4} c^{2}+a^{2}$$" ], "text/plain": [ "" @@ -3099,7 +8484,7 @@ { "data": { "text/latex": [ - "$$32\\sqrt{2}a^2b^3c \\le 16\\sqrt{2}a^3b^3+16\\sqrt{2}ab^3c^2$$" + "$$4 a^{2} c \\le a^{3} c^{2}+2 a^{2}+a c^{2}$$" ], "text/plain": [ "" @@ -3111,7 +8496,7 @@ { "data": { "text/latex": [ - "$$48\\sqrt{2}a^2b^2c^2 \\le 24\\sqrt{2}a^3b^2c+24\\sqrt{2}ab^2c^3$$" + "$$2 a c \\le a^{2}+c^{2}$$" ], "text/plain": [ "" @@ -3123,67 +8508,7 @@ { "data": { "text/latex": [ - "$$16\\sqrt{2}a^2bc^3 \\le 8\\sqrt{2}a^3bc^2+8\\sqrt{2}abc^4$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$16ab^4c \\le 8a^2b^4+8b^4c^2$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$32ab^3c^2 \\le 16a^2b^3c+16b^3c^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$32a^3bc^2 \\le 16a^4bc+16a^2bc^3$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$16ab^2c^3 \\le 8a^2b^2c^2+8b^2c^4$$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ 0 \\le 16a^4bc+32a^4c^2+24\\sqrt{2}a^3b^2c+72\\sqrt{2}a^3bc^2+32\\sqrt{2}a^3c^3+16a^2b^3c+72a^2b^2c^2+64a^2bc^3+16a^2c^4 $$" + "$$ 0 \\le 0 $$" ], "text/plain": [ "" @@ -3206,102 +8531,8 @@ }, { "data": { - "text/latex": [ - "$\\displaystyle - \\frac{2 \\sqrt{2} \\left(a - b\\right)}{c} + \\frac{\\left(a - b\\right)^{2}}{c^{2}} - \\frac{2 \\sqrt{2} \\left(- a + c\\right)}{b} + \\frac{\\left(- a + c\\right)^{2}}{b^{2}} - \\frac{2 \\sqrt{2} \\left(b - c\\right)}{a} + \\frac{\\left(b - c\\right)^{2}}{a^{2}}$" - ], "text/plain": [ - "-2*sqrt(2)*(a - b)/c + (a - b)**2/c**2 - 2*sqrt(2)*(-a + c)/b + (-a + c)**2/b**2 - 2*sqrt(2)*(b - c)/a + (b - c)**2/a**2" - ] - }, - "execution_count": 42, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "formula=cyclize('((a-b)/c)^2-8**(1/2)*(a-b)/c')\n", - "display(Latex('Case $a\\ge c\\ge b$'))\n", - "formula1=makesubs(formula,'[c,oo],[b,oo]',variables='a,c,b')\n", - "prove(formula1)\n", - "display(Latex('Case $a\\ge b\\ge c$'))\n", - "formula2=makesubs(formula,'[b,oo],[c,oo]')\n", - "prove(formula2*4,values='2**(1/2),1,1')" - ] - }, - { - "cell_type": "code", - "execution_count": 44, - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "Substitute $a\\to a+c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "Substitute $b\\to b+c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "formula=cyclize('((a-b)/c)^2-8**(1/2)*(a-b)/c')\n", - "formula2=makesubs(formula,'[c,oo],[c,oo]')" - ] - }, - { - "cell_type": "code", - "execution_count": 49, - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Optimization terminated successfully.\n", - " Current function value: 1.000000\n", - " Iterations: 271\n", - " Function evaluations: 485\n" - ] - }, - { - "data": { - "text/plain": [ - "(12.42960536989153, 12.429310393555546, 0.00020878153950706037)" - ] - }, - "execution_count": 49, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "findvalues(formula2,values=[5,2,7])" - ] - }, - { - "cell_type": "code", - "execution_count": 16, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "True" + "0" ] }, "execution_count": 16, @@ -3310,81 +8541,92 @@ } ], "source": [ - "x,y=symbols('x,y')\n", - "ask(Q.positive(x**2+1),Q.real(x))" + "from importlib import reload\n", + "newproof()\n", + "import shiroindev\n", + "reload(shiroindev)\n", + "from shiroindev import *\n", + "shiro.display=lambda x:display(Latex(x))\n", + "formula=Sm('xy/z+yz/x+zx/y-1').subs('z',S('1-x-y'))\n", + "newformula,values=makesubs(formula,'[0,1-y],[0,1]','1/3,1/3')\n", + "prove(newformula,values)" ] }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 33, "metadata": {}, - "outputs": [ - { - "ename": "NameError", - "evalue": "name 'Sm' is not defined", - "output_type": "error", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mformula\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mSm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'(a^2+b^2+c^2)^2- 3(a^3b+b^3c+c^3a)'\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 2\u001b[0m \u001b[0mformula1\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmakesubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mformula\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m'[b,oo],[c,oo]'\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mvariables\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'a,b'\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[0mformula2\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmakesubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mformula\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m'[c,oo],[b,oo]'\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mvariables\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'a,c'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mprove\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mformula2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfindvalues\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mformula1\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;31mNameError\u001b[0m: name 'Sm' is not defined" - ] - } - ], + "outputs": [], "source": [ - "formula=Sm('(a^2+b^2+c^2)^2- 3(a^3b+b^3c+c^3a)')\n", - "formula1=makesubs(formula,'[b,oo],[c,oo]',variables='a,b')\n", - "formula2=makesubs(formula,'[c,oo],[b,oo]',variables='a,c')\n", - "prove(formula2)\n", - "print(findvalues(formula1))\n", - "#prove(formula1,values='4*b*(1-b)-1,4*c*(1-c)-1,1')" + "def findvalues(formula,values=None,variables=None,**kwargs):\n", + "\t\"\"\"finds a candidate for parameter \"values\" in \"prove\" function\n", + "\tblabla\"\"\"\n", + "\tformula=S(formula)\n", + "\taddsymbols(formula)\n", + "\tnum,den=fractioncancel(formula)\n", + "\tif variables==None:\n", + "\t\tvariables=sorted(formula.free_symbols,key=str)\n", + "\tnum=num.subs(zip(variables,list(map(lambda x:x**2,variables))))\n", + "\tnum=Poly(num)\n", + "\tnewformula=S((num.abs()+num)/(num.abs()-num))\n", + "\tf=lambdify(variables,newformula)\n", + "\tf2=lambda x:f(*x)\n", + "\tif values==None:\n", + "\t\tvalues=[1.0]*len(variables)\n", + "\telse:\n", + "\t\tvalues=S(values)\n", + "\ttup=tuple(fmin(f2,values,**kwargs))\n", + "\treturn tuple([x*x for x in tup])" ] }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 34, "metadata": {}, "outputs": [ { - "data": { - "text/latex": [ - "Substitute $b\\to b+c$" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" + "name": "stdout", + "output_type": "stream", + "text": [ + "Optimization terminated successfully.\n", + " Current function value: 1.000000\n", + " Iterations: 51\n", + " Function evaluations: 109\n" + ] }, { "data": { - "text/latex": [ - "Substitute $c\\to a+c$" - ], "text/plain": [ - "" + "(1.9999999989778676, 0.6689493176765962)" ] }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$\\displaystyle - a b - a c + b c$" - ], - "text/plain": [ - "-a*b - a*c + b*c" - ] - }, - "execution_count": 28, + "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "makesubs(Sm('a^2-2ab+bc-c^2+ca'),'[c,oo],[a,oo]',variables='b,c')" + "findvalues(Sm('(x^2-4x+4)/y'))" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": {}, + "outputs": [], + "source": [ + "shiroindev._remzero?" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": {}, + "outputs": [], + "source": [ + "import shiroindev\n", + "reload(shiroindev)\n", + "from shiroindev import *" ] }, { diff --git a/shiroindev.py b/shiroindev.py index 106ed66..0a4dddb 100644 --- a/shiroindev.py +++ b/shiroindev.py @@ -2,14 +2,28 @@ from __future__ import (absolute_import, division, print_function, unicode_literals) import warnings,operator warnings.filterwarnings("ignore") -#Seed is needed to select the weights in linprog function. -#None means that the seed is random. +def vargen(n): + """default function generating names for variables""" + q=len(shiro.alphabet) + x=shiro.alphabet[n%q] + if n>=q: + x+=str(n//q) + return x class Vars: pass -sVars=Vars() -sVars.display=print -sVars.seed=None -sVars.translation={} +shiro=Vars() +shiro.display=shiro.warning=print +shiro.seed=None +"""Seed is needed to select the weights in linprog function. +None means that the seed is random""" +shiro.translation={} +shiro.varind=0 +shiro.varset=set() +"""set of used symbols in the proof""" +shiro.vargen=vargen +"""function generating names for variables""" +shiro.alphabet='abcdefghijklmnopqrstuvwxyz' +"""list of names for variables""" translationList=['numerator:','denominator:','status:', 'Substitute',"Formula after substitution:", "Numerator after substitutions:","From weighted AM-GM inequality:", @@ -19,22 +33,38 @@ translationList=['numerator:','denominator:','status:', "Program couldn't find any proof.", "Try to set higher linprogiter parameter.", "It looks like the formula is symmetric. You can assume without loss of"+ -" generality that ","Try", 'From Jensen inequality:' +" generality that ","Try", 'From Jensen inequality:', +'Warning: intervals contain backwards dependencies. Consider changing order of variables and intervals.' ] #Initialize english-english dictionary. for phrase in translationList: - sVars.translation[phrase]=phrase + shiro.translation[phrase]=phrase from scipy.optimize import linprog,fmin import random -from sympy import S,cancel,fraction,Pow,expand,solve,latex,oo,Poly,lambdify +from sympy import S,cancel,fraction,Pow,expand,solve,latex,oo,Poly,lambdify,srepr,gcd,Symbol +from sympy.parsing.sympy_parser import parse_expr, standard_transformations,\ +implicit_multiplication_application, convert_xor from collections import Counter import re +def addsymbols(formula): + formula=S(formula) + funcsymbols=[x[10:-2] for x in re.findall(r"Function\(\'.*?\'\)",srepr(formula))] + shiro.varset|=set(funcsymbols)|set(map(str,formula.free_symbols)) +def newvar(): + while 1: + x=shiro.vargen(shiro.varind) + shiro.varind+=1 + if x not in shiro.varset: + return S(x) +def newproof(): + shiro.varset=set() + shiro.varind=0 def _remzero(coef,fun): -#coef, fun represents an expression. -#For example, if expression=5f(2,3)+0f(4,6)+8f(1,4) -#then coef=[5,0,8], fun=[[2,3],[4,6],[1,4]] -#_remzero removes addends with coefficient equal to zero. -#In this example ncoef=[5,8], nfun=[[2,3],[1,4]] + """coef, fun represents an expression. + For example, if expression=5f(2,3)+0f(4,6)+8f(1,4) + then coef=[5,0,8], fun=[[2,3],[4,6],[1,4]] + _remzero removes addends with coefficient equal to zero. + In this example ncoef=[5,8], nfun=[[2,3],[1,4]]""" ncoef=[] nfun=[] for c,f in zip(coef,fun): @@ -42,11 +72,11 @@ def _remzero(coef,fun): ncoef+=[c] nfun+=[f] return ncoef,nfun -def slatex(formula): #fancy function which makes latex code more readable, but still correct - formula=re.sub(r'\^{(.)}',r'^\1',latex(formula,fold_short_frac=True).replace(' ','').replace('\\left(','(').replace('\\right)',')')) - return re.sub(r'\{(\(.+?\))\}',r'\1',formula) +# ~ def slatex(formula): #fancy function which makes latex code more readable, but still correct + # ~ formula=re.sub(r'\^{(.)}',r'^\1',latex(formula,fold_short_frac=True).replace(' ','').replace('\\left(','(').replace('\\right)',')')) + # ~ return re.sub(r'\{(\(.+?\))\}',r'\1',formula) def _writ2(coef,fun,variables): - return slatex(S((str(coef)+'*'+'*'.join([str(x)+'^'+str(y) for x,y in zip(variables,fun)])))) + return latex(Poly({fun:coef},gens=variables).as_expr()) def _writ(coef,fun,nullvar): return str(coef)+'f('+str(fun)[1:-1-(len(fun)==1)]+')' def _check(coef,fun,res,rfun): @@ -61,19 +91,35 @@ def _powr(formula): else: return [formula,S('1')] def fractioncancel(formula): -#workaround for buggy cancel function + """workaround for buggy cancel function""" num,den=fraction(cancel(formula/S('tmp'))) den=den.subs(S('tmp'),S('1')) return num,den def ssolve(formula,variables): -#workaround for inconsistent solve function + """workaround for inconsistent solve function""" result=solve(formula,variables) if type(result)==dict: result=[[result[var] for var in variables]] return result -def sstr(formula): - return str(formula).replace('**','^').replace('*','').replace(' ','') +#def sstr(formula): +# return str(formula).replace('**','^').replace('*','').replace(' ','') +def assumeall(formula,**kwargs): + """Adds assumptions to all free symbols in formula. + >>> assumeall('sqrt(x*y)-sqrt(x)*sqrt(y)',positive=True) + 0 + """ + formula=S(formula) + fs=formula.free_symbols + for x in fs: + y=Symbol(str(x),**kwargs) + formula=formula.subs(x,y) + return formula def reducegens(formula): + """Reduces size of the generator of the polynomial + >>>Poly('x+sqrt(x)') + Poly(x + (sqrt(x)), x, sqrt(x), domain='ZZ') + >>>reducegens('x+sqrt(x)') + Poly((sqrt(x))**2 + (sqrt(x)), sqrt(x), domain='ZZ') """ pol=Poly(formula) newgens={} ind={} @@ -96,82 +142,90 @@ def reducegens(formula): newpol=newpol.replace(ind[gen],gen**newgens[gen]) return newpol def Sm(formula): -#Adds multiplication signs and sympifies a formula. -#For example, Sm('(2x+y)(7+5xz)') -> S('(2*x+y)*(7+5*x*z)') - if type(formula)==str: - formula=formula.replace(' ','') - for i in range(2): - formula=re.sub(r'([0-9a-zA-Z)])([(a-zA-Z])',r'\1*\2',formula) - formula=S(formula) - return formula + """Adds multiplication signs and sympifies a formula. + For example, Sm('(2x+y)(7+5xz)') -> S('(2*x+y)*(7+5*x*z)')""" + if type(formula)!=str: + if _isiterable(formula): return type(formula)(map(Sm,formula)) + else: return S(formula) + formula=formula.translate({ord('{'):None,ord('}'):' '}) + transformations = (standard_transformations +(implicit_multiplication_application,convert_xor)) + return parse_expr(formula, transformations=transformations) +# ~ def Sm(formula): +# ~ #Adds multiplication signs and sympifies a formula. +# ~ #For example, Sm('(2x+y)(7+5xz)') -> S('(2*x+y)*(7+5*x*z)') + # ~ if type(formula)==str: + # ~ formula=formula.replace(' ','') + # ~ for i in range(2): + # ~ formula=re.sub(r'([0-9a-zA-Z)])([(a-zA-Z])',r'\1*\2',formula) + # ~ formula=S(formula) + # ~ return formula def _input2fraction(formula,variables,values): -#makes some substitutions and converts formula to a fraction -#with expanded numerator and denominator + """makes some substitutions and converts formula to a fraction + with expanded numerator and denominator""" formula=S(formula) subst=[] for x,y in zip(variables,values): if y!=1: - sVars.display(sVars.translation['Substitute']+' $'+str(x)+'\\to '+slatex(S(y)*S(x))+'$') - subst+=[(x,x*y)] + z=newvar() + shiro.display(shiro.translation['Substitute']+' $'+latex(x)+'\\to '+latex(S(y)*S(z))+'$') + subst+=[(x,z*y)] formula=formula.subs(subst) numerator,denominator=fractioncancel(formula) - sVars.display(sVars.translation['numerator:']+' $'+slatex(numerator)+'$') - sVars.display(sVars.translation['denominator:']+' $'+slatex(denominator)+'$') + shiro.display(shiro.translation['numerator:']+' $'+latex(numerator)+'$') + shiro.display(shiro.translation['denominator:']+' $'+latex(denominator)+'$') return (numerator,denominator) def _formula2list(formula): -#Splits a polynomial to a difference of two polynomials with positive -#coefficients and extracts coefficients and powers of both polynomials. -#'variables' is used to set order of powers -#For example, If formula=5x^2-4xy+8y^3, variables=[x,y], then -#the program tries to prove that -#0<=5x^2-4xy+8y^3 -#4xy<=5x^2+8y^3 -#returns [4],[(1,1)], [5,8],[(2,0),(0,3)], (x,y) - formula=reducegens(formula) - neg=(formula.abs()-formula)/2 - pos=(formula.abs()+formula)/2 - neg=Poly(neg,gens=formula.gens) - pos=Poly(pos,gens=formula.gens) + """Splits a polynomial to a difference of two polynomials with positive + coefficients and extracts coefficients and powers of both polynomials. + 'variables' is used to set order of powers + For example, If formula=5x^2-4xy+8y^3, variables=[x,y], then + the program tries to prove that + 0<=5x^2-4xy+8y^3 + 4xy<=5x^2+8y^3 + returns [4],[(1,1)], [5,8],[(2,0),(0,3)], (x,y)""" + formula=reducegens(assumeall(formula,positive=True)) + neg=(formula.abs()-formula)*S('1/2') + pos=(formula.abs()+formula)*S('1/2') return neg.coeffs(),neg.monoms(),pos.coeffs(),pos.monoms(),Poly(formula).gens def _list2proof(lcoef,lfun,rcoef,rfun,variables,itermax,linprogiter,_writ2=_writ2,theorem="From weighted AM-GM inequality:"): -#Now the formula is splitted on two polynomials with positive coefficients. -#we will call them LHS and RHS and our inequality to prove would -#be LHS<=RHS (instead of 0<=RHS-LHS). + """Now the formula is splitted on two polynomials with positive coefficients. + we will call them LHS and RHS and our inequality to prove would + be LHS<=RHS (instead of 0<=RHS-LHS). + + Suppose we are trying to prove that + 30x^2y^2+60xy^4<=48x^3+56y^6 (assuming x,y>0). + Program will try to find some a,b,c,d such that + 30x^2y^2<=ax^3+by^6 + 60xy^4<=cx^3+dy^6 + where a+c<=48 and b+d<=56 (assumption 1). + We need some additional equalities to meet assumptions + of the weighted AM-GM inequality. + a+b=30 and c+d=60 (assumption 2) + 3a+0b=30*2, 0a+6b=30*2, 3c+0d=60*1, 0c+6d=60*4 (assumption 3) -#suppose we are trying to prove that -#30x^2y^2+60xy^4<=48x^3+56y^6 (assuming x,y>0) -#program will try to find some a,b,c,d such that -#30x^2y^2<=ax^3+by^6 -#60xy^4<=cx^3+dy^6 -#where a+c<=48 and b+d<=56 (assumption 1) -#We need some additional equalities to meet assumptions -#of the weighted AM-GM inequality. -#a+b=30 and c+d=60 (assumption 2) -#3a+0b=30*2, 0a+6b=30*2, 3c+0d=60*1, 0c+6d=60*4 (assumption 3) - -#The sketch of the algorithm. -# for i in range(itermax): - #1. Create a vector of random numbers (weights). - #2. Try to find real solution of the problem (with linprog). - #3. If there is no solution (status: 2) - #3a. If the solution was never found, break. - #3b. Else, step back (to the bigger inequality) - #4. If the soltuion was found (status: 0) - #Check out which of variables (in example: a,b,c,d) looks like integer. - #If there are some inequalities with all integer coefficients, subtract - #them from the original one. - #If LHS is empty, then break. - localseed=sVars.seed + The sketch of the algorithm. + for i in range(itermax): + 1. Create a vector of random numbers (weights). + 2. Try to find real solution of the problem (with linprog). + 3. If there is no solution (status: 2) + 3a. If the solution was never found, break. + 3b. Else, step back (to the bigger inequality) + 4. If the soltuion was found (status: 0) + Check out which of variables (in example: a,b,c,d) looks like integer. + If there are some inequalities with all integer coefficients, subtract + them from the original one. + If LHS is empty, then break.""" + localseed=shiro.seed bufer=[] lcoef,lfun=_remzero(lcoef,lfun) rcoef,rfun=_remzero(rcoef,rfun) itern=0 if len(lcoef)==0: #if LHS is empty - sVars.display(sVars.translation['status:']+' 0') + shiro.display(shiro.translation['status:']+' 0') status=0 elif len(rcoef)==0: #if RHS is empty, but LHS is not - sVars.display(sVars.translation['status:']+' 2') + shiro.display(shiro.translation['status:']+' 2') status=2 itermax=0 foundreal=0 @@ -208,9 +262,9 @@ def _list2proof(lcoef,lfun,rcoef,rfun,variables,itermax,linprogiter,_writ2=_writ res=linprog(vecc,A_eq=A,b_eq=b,A_ub=A_ub,b_ub=b_ub,options={'maxiter':linprogiter}) status=res.status if itern==1: - sVars.display(sVars.translation['status:']+' '+str(status)) + shiro.display(shiro.translation['status:']+' '+str(status)) if status==0: - sVars.display(sVars.translation[theorem]) + shiro.display(shiro.translation[theorem]) if status==2: #if real solution of current inequality doesn't exist if foundreal==0: #if this is the first inequality, then break break @@ -222,7 +276,7 @@ def _list2proof(lcoef,lfun,rcoef,rfun,variables,itermax,linprogiter,_writ2=_writ continue if status==0:#if found a solution with real coefficients for ineq in bufer: - sVars.display(ineq) + shiro.display(ineq) foundreal=1 bufer=[] oldlfun,oldrfun=lfun,rfun @@ -254,25 +308,25 @@ def _list2proof(lcoef,lfun,rcoef,rfun,variables,itermax,linprogiter,_writ2=_writ lcoef,lfun=_remzero(lcoef,lfun) rcoef,rfun=_remzero(rcoef,rfun) for ineq in bufer: - sVars.display(ineq) + shiro.display(ineq) lhs='+'.join([_writ2(c,f,variables) for c,f in zip(lcoef,lfun)]) if lhs=='': lhs='0' elif status==0: - sVars.display(sVars.translation[ + shiro.display(shiro.translation[ "Program couldn't find a solution with integer coefficients. Try "+ "to multiple the formula by some integer and run this function again."]) elif(status==2): - sVars.display(sVars.translation["Program couldn't find any proof."]) + shiro.display(shiro.translation["Program couldn't find any proof."]) #return res.status elif status==1: - sVars.display(sVars.translation["Try to set higher linprogiter parameter."]) + shiro.display(shiro.translation["Try to set higher linprogiter parameter."]) rhs='+'.join([_writ2(c,f,variables) for c,f in zip(rcoef,rfun)]) if rhs=='': rhs='0' - sVars.display('$$ '+slatex(lhs)+' \\le '+slatex(rhs)+' $$') + shiro.display('$$ '+latex(lhs)+' \\le '+latex(rhs)+' $$') if lhs=='0': - sVars.display(sVars.translation['The sum of all inequalities gives us a proof of the inequality.']) + shiro.display(shiro.translation['The sum of all inequalities gives us a proof of the inequality.']) return status def _isiterable(obj): try: @@ -281,18 +335,25 @@ def _isiterable(obj): except TypeError: return False def _smakeiterable(x): + if x=='': + return [] x=S(x) if _isiterable(x): return x return (x,) def _smakeiterable2(x): + if x=='': + return [] x=S(x) + if len(x)==0: + return [] if _isiterable(x[0]): return x return (x,) def prove(formula,values=None,variables=None,niter=200,linprogiter=10000): -#tries to prove that formula>=0 assuming all variables are positive + """tries to prove that formula>=0 assuming all variables are positive""" formula=S(formula) + addsymbols(formula) if variables: variables=_smakeiterable(variables) else: variables=sorted(formula.free_symbols,key=str) if values: values=_smakeiterable(values) @@ -301,46 +362,50 @@ def prove(formula,values=None,variables=None,niter=200,linprogiter=10000): st=_list2proof(*(_formula2list(num)+(niter,linprogiter))) if st==2 and issymetric(num): fs=sorted([str(x) for x in num.free_symbols]) - sVars.display(sVars.translation["It looks like the formula is symmetric. "+ + shiro.display(shiro.translation["It looks like the formula is symmetric. "+ "You can assume without loss of generality that "]+ - ' >= '.join([str(x) for x in fs])+'. '+sVars.translation['Try']) - sVars.display('prove(makesubs(S("'+str(num)+'"),'+ + ' >= '.join([str(x) for x in fs])+'. '+shiro.translation['Try']) + shiro.display('prove(makesubs(S("'+str(num)+'"),'+ str([(str(x),'oo') for x in variables[1:]])+')') return st def powerprove(formula,values=None,variables=None,niter=200,linprogiter=10000): -#This is a bruteforce and ineffective function for proving inequalities. -#It can be used as the last resort. + """This is a bruteforce and ineffective function for proving inequalities. + It can be used as the last resort.""" formula=S(formula) + addsymbols(formula) if variables: variables=_smakeiterable(variables) else: variables=sorted(formula.free_symbols,key=str) if values: values=_smakeiterable(values) else: values=[1]*len(variables) num,den=_input2fraction(formula,variables,values) - subst2=[] statusses=[] - for j in range(len(variables)): - subst2+=[(variables[j],1+variables[j])] + for i in range(1<R instead of a polynomial. provef checks if a formula -#is nonnegative for any nonnegative and convex function f. If so, it -#provides a proof of nonnegativity. + """This function is similar to prove, formula is a linear combination of + values of f:R^k->R instead of a polynomial. provef checks if a formula + is nonnegative for any nonnegative and convex function f. If so, it + provides a proof of nonnegativity.""" formula=S(formula) + addsymbols(formula) num,den=_input2fraction(formula,[],[]) return _list2proof(*(_formula2listf(num)+(niter,linprogiter,_writ,'From Jensen inequality:'))) -def issymetric(formula): #checks if formula is symmetric -#and has at least two variables +def issymetric(formula): + """checks if formula is symmetric + and has at least two variables""" + formula=S(formula) + addsymbols(formula) if len(formula.free_symbols)<2: return False ls=list(formula.free_symbols) @@ -425,9 +501,10 @@ def issymetric(formula): #checks if formula is symmetric return False return True def cyclize(formula,oper=operator.add,variables=None,init=None): -#cyclize('a^2*b')=S('a^2*b+b^2*a') -#cyclize('a^2*b',variables='a,b,c')=S('a^2*b+b^2*c+c^2*a') + """cyclize('a^2*b')=S('a^2*b+b^2*a') + cyclize('a^2*b',variables='a,b,c')=S('a^2*b+b^2*c+c^2*a')""" formula=S(formula) + addsymbols(formula) if variables==None: variables=sorted(formula.free_symbols,key=str) else: @@ -446,10 +523,11 @@ def cyclize(formula,oper=operator.add,variables=None,init=None): init=oper(init,formula) return init def symmetrize(formula,oper=operator.add,variables=None,init=None): -#symmetrize('a^2*b')=S('a^2*b+b^2*a') -#symmetrize('a^2*b',variables='a,b,c')= -#=S('a^2*b+a^2*c+b^2*a+b^2*c+c^2*a+c^2*b') + """symmetrize('a^2*b')=S('a^2*b+b^2*a') + symmetrize('a^2*b',variables='a,b,c')= + =S('a^2*b+a^2*c+b^2*a+b^2*c+c^2*a+c^2*b')""" formula=S(formula) + addsymbols(formula) if variables==None: variables=sorted(formula.free_symbols,key=str) else: @@ -457,8 +535,10 @@ def symmetrize(formula,oper=operator.add,variables=None,init=None): for i in range(1,len(variables)): formula=cyclize(formula,oper,variables[:i+1]) return formula -def findvalues(formula,values=None,variables=None): +def findvalues(formula,values=None,variables=None,**kwargs): + """finds a candidate for parameter "values" in "prove" function""" formula=S(formula) + addsymbols(formula) num,den=fractioncancel(formula) if variables==None: variables=sorted(num.free_symbols,key=str) @@ -471,5 +551,5 @@ def findvalues(formula,values=None,variables=None): values=[1.0]*len(variables) else: values=S(values) - tup=tuple(fmin(f2,values)) + tup=tuple(fmin(f2,values,**kwargs)) return tuple([x*x for x in tup]) diff --git a/statistics.ipynb b/statistics.ipynb new file mode 100644 index 0000000..fe47221 --- /dev/null +++ b/statistics.ipynb @@ -0,0 +1,1563 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this notebook several methods for proving inequalities have been compared.\n", + "\n", + "First of all we need a dataset. 35 inequalities were selected from https://www.imomath.com/index.php?options=592&lmm=0. Each inequality is represented by a tuple: inequality (in LaTeX), inequality constraints (for example: $a\\in [0,1]$, equality constraints (for example: $abc=1$) and a function which converts inequality to a formula which we want to prove it's nonnegativity. In most cases when this function is simply a difference between left and right-hand side, but sometimes it's better to use difference between squares of sides (to get rid of square roots). These tuples are used as parameters in `parser` function which converts them to equivalent inequalities with default constraints (i.e. all variables are positive).\n", + "\n", + "Some tuples have less than 4 elements. In this case `parser` use default arguments." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "from importlib import reload\n", + "from sympy import *\n", + "import shiroindev\n", + "from shiroindev import *\n", + "from sympy.parsing.latex import parse_latex\n", + "shiro.seed=1\n", + "from IPython.display import Latex\n", + "shiro.display=lambda x:display(Latex(x))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "`sympy` has a `parse_latex` function which converts LaTeX formula to a `sympy` one. Unfortunately it doesn't deal with missing braces in the way that LaTeX do. For example `\\frac12` generates $\\frac12$ which is the same as `\\frac{1}{2}`, but `parse_latex` accepts only the second version. So here there is a boring function which adds missing braces." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\\frac{ \\sqrt {3}}{2}\n", + "a^2+b^2+c^2\\geq ab+bc+ca\n" + ] + } + ], + "source": [ + "def addbraces(s):\n", + " arg={r'\\frac':2,r'\\sqrt':1}\n", + " s2=''\n", + " p=0\n", + " while 1:\n", + " m=re.search(r\"\\\\[a-zA-Z]+\", s[p:])\n", + " if not m:\n", + " break\n", + " s2+=s[p:p+m.end()]\n", + " #print('a',s[p:m.end()],p)\n", + " p+=m.end()\n", + " if m.group() in arg:\n", + " for i in range(arg[m.group()]):\n", + " sp=re.search('^ *',s[p:])\n", + " s2+=sp.group()\n", + " #print('b',sp.group(),p)\n", + " p+=sp.end()\n", + " if s[p]=='{':\n", + " cb=re.search(r'^\\{.*?\\}',s[p:])\n", + " ab=addbraces(cb.group())\n", + " s2+=ab\n", + " #print('c',ab,p)\n", + " p+=cb.end()\n", + " else:\n", + " s2+='{'+s[p]+'}'\n", + " #print('d','{'+s[p]+'}',p)\n", + " p+=1\n", + " #print('e',p)\n", + " s2+=s[p:]\n", + " return s2\n", + "print(addbraces(r'\\frac{ \\sqrt 3}2'))\n", + "print(addbraces(r'a^2+b^2+c^2\\geq ab+bc+ca'))" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "#(formula,intervals,subs,function)\n", + "dif=lambda b,s:b-s\n", + "dif2=lambda b,s:b*b-s*s\n", + "ineqs=[\n", + " (r'a^2+b^2+c^2\\geq ab+bc+ca',),\n", + " (r'a^2+b^2+c^2+d^2\\geq a(b+c+d)',),\n", + " (r'1\\leq\\frac{a^2b^2}{c^2}+\\frac{b^2c^2}{a^2}+\\frac{c^2a^2}{b^2}',\n", + " '[0,1-f],[0,1]','[a,sqrt(1-e-f)],[b,sqrt(e)],[c,sqrt(f)]',),\n", + " (r'\\frac1{1-x^2}+\\frac1{1-y^2}\\geq \\frac2{1-xy}','[0,1],[0,1]'),\n", + " (r'a^3+b^3\\geq a^2b+ab^2',),\n", + " (r'\\frac a{b+c}+\\frac b{c+a}+\\frac c{a+b}\\geq \\frac32',),\n", + " (r'2a^3+b^3\\geq 3a^2b',),\n", + " (r'a^3+b^3+c^3\\geq a^2b+b^2c+c^2a',),\n", + " (r'\\frac a{b+c}+\\frac b{c+d}+ \\frac c{d+a}+ \\frac d{a+b}\\geq 2',),\n", + " (r'\\frac{a^3}{a^2+ab+b^2}+ \\frac{b^3}{b^2+bc+c^2}+ \\frac{c^3}{c^2+ca+a^2} \\geq \\frac{a+b+c}3',),\n", + " (r'\\sqrt{ab}+\\sqrt{cd}+\\sqrt{ef}\\leq\\sqrt{(a+c+e)(b+d+f)}','','',dif2),\n", + " (r'\\frac{5a^3-ab^2}{a+b}+\\frac{5b^3-bc^2}{b+c}+\\frac{5c^3-ca^2}{c+a}\\geq 2(a^2+b^2+c^2)',),\n", + " #(r'\\frac{a^2}{(1+b)(1-c)}+\\frac{b^3}{(1+c)(1-d)}+\\frac{c^3}{(1+d)(1-a)}+\\frac{d^3}{(1+a)(1-b)}\\geq\\frac1{15}',\n", + " # '[0,1-c-d],[0,1-d],[0,1]','[a,1-b-c-d]'),\n", + " (r'\\frac{x^3}{(1+y)(1+z)}+\\frac{y^3}{(1+z)(1+x)}+\\frac{z^3}{(1+x)(1+y)}\\geq\\frac{3}{4}','','[z,1/(x*y)]'),\n", + " (r'\\frac ab+\\frac bc+\\frac ca\\geq \\frac{(a+b+c)^2}{ab+bc+ca}',),\n", + " (r'\\frac{a^2}b+\\frac{b^2}c+\\frac{c^2}a\\geq \\frac{a^2+b^2+c^2}{a+b+c}',),\n", + " (r'\\frac{a^2}b+\\frac{b^2}c+\\frac{c^2}a\\geq a+b+c+\\frac{4(a-b)^2}{a+b+c}',),\n", + " (r'\\frac1{a^3+b^3+abc}+ \\frac1{b^3+c^3+abc} +\\frac1{c^3+a^3+ abc} \\leq \\frac1{abc}',),\n", + " (r'\\frac1{a^3(b+c)}+ \\frac1{b^3(c+a)}+ \\frac1{c^3(a+b)} \\geq \\frac32','','[c,1/(a*b)]'),\n", + " (r'\\frac{a^3}{b^2-bc+c^2}+\\frac{b^3}{c^2-ca+a^2} + \\frac{c^3}{a^2-ab+b^2} \\geq 3 \\cdot\\frac{ab+bc+ca}{a+b+c}',),\n", + " (r'\\frac{x^5-x^2}{x^5+y^2+z^2}+\\frac{y^5-y^2}{y^5+z^2+x^2}+\\frac{z^5-z^2}{z^5+x^2+y^2}\\geq0','','[x,1/(y*z)]'),\n", + " (r'(a+b-c)(b+c-a)(c+a-b)\\leq abc',),\n", + " (r'\\frac1{1+xy}+\\frac1{1+yz}+\\frac1{1+zx}\\leq \\frac34','','[x,(y+z)/(y*z-1)]'),\n", + " (r'\\frac{x\\sqrt x}{y+z}+\\frac{y\\sqrt y}{z+x}+\\frac{z\\sqrt z}{x+y}\\geq\\frac{ \\sqrt 3}2',\n", + " '[0,1-z],[0,1]','[x,1-y-z]'),\n", + " (r'a^4+b^4+c^4+d^4\\geq 4abcd',),\n", + " (r'\\frac{ab}{a+b}+\\frac{bc}{b+c}+\\frac{ca}{c+a}\\leq\\frac{3(ab+bc+ca)}{2(a+b+c)}',),\n", + " (r'\\sqrt{a-1}+\\sqrt{b-1}+ \\sqrt{c-1} \\leq \\sqrt{c(ab+1)}','[1,oo],[1,oo],[1,oo]','',dif2),\n", + " (r'(x-1)(y-1)(z-1)\\geq 8','[0,1-b-c],[0,1-c],[0,1]','[x,1/a],[y,1/b],[z,1/c]'),\n", + " (r'ay+bz+cx\\leq s^2','[0,s],[0,s],[0,s]','[x,s-a],[y,s-b],[z,s-c]'),\n", + " (r'x_1^2+x_2^2+x_3^2+x_4^2+x_5^2\\geq 2 (x_1x_2+x_2x_3+x_3x_4+x_4x_5)/\\sqrt{3}',),\n", + " (r' xy+yz+zx - 2xyz \\leq \\frac7{27}', '[0,1-z],[0,1]','[x,1-y-z]'),\n", + " (r'0 \\leq xy+yz+zx - 2xyz', '[0,1-z],[0,1]','[x,1-y-z]'),\n", + " (r'\\sqrt{3+a+b+c}\\geq\\sqrt a+\\sqrt b+\\sqrt c','[1-z,1],[0,1]','[a,1/x-1],[b,1/y-1],[c,1/z-1],[x,2-y-z]',\n", + " dif2),\n", + " (r'\\frac{2a^3}{a^2+b^2}+\\frac{2b^3}{b^2+c^2}+\\frac{2c^3}{c^2+a^2}\\geq a+b+c',),\n", + " (r'\\frac{a^2}{b+c}+\\frac{b^2}{c+a}+\\frac{c^2}{a+b}\\geq \\frac12','[0,1-c],[0,1]','[a,1-b-c]'),\n", + " (r'\\frac{a+b}{2b+c}+\\frac{b+c}{2c+a}+\\frac{c+a}{2a+b}\\geq 2',),\n", + " #(r'\\frac x{5-y^2}+\\frac y{5-z^2}+\\frac z{5-x^2}\\geq \\frac34','[0,sqrt(5)],[0,sqrt(5)]','[x,1/(y*z)]')\n", + "]\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "def parser(formula,intervals='[]',subs='[]',func=dif):\n", + " newproof()\n", + " shiro.display=lambda x:None\n", + " #display=lambda x:None\n", + " if intervals=='':\n", + " intervals='[]'\n", + " if subs=='':\n", + " subs='[]'\n", + " formula=addbraces(formula)\n", + " formula=Sm(str(parse_latex(formula)))\n", + " formula,_=fractioncancel(formula)\n", + " formula=formula.subs(shiroindev._smakeiterable2(subs))\n", + " formula=makesubs(formula,intervals)\n", + " b,s=formula.lhs,formula.rhs\n", + " if type(formula)==LessThan:\n", + " b,s=s,b\n", + " formula=func(b,s)\n", + " formula=simplify(formula)\n", + " num,den=fractioncancel(formula)\n", + " return num" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 35/35 [00:46<00:00, 1.32s/it]\n" + ] + } + ], + "source": [ + "from tqdm import tqdm\n", + "ineqs2=[]\n", + "for ineq in tqdm(ineqs):\n", + " ineqs2+=[parser(*ineq)]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now let's look at the formulas when converted to polynomials." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "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", + " display(reducegens(assumeall(ineq,positive=True)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Most formulas was converted to polynomials of independent variables. However formulas No. 22 and No. 31 were not. For this reason it's very unlikely that any method of proving these ones will succeed.\n", + "\n", + "Now let's try some methods of proving these inequalities. The first one would be the simple `prove`." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "tm=[0]*4" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "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", + "start=timer()\n", + "t=[]\n", + "for ineq in ineqs2:\n", + " t+=[prove(ineq)]\n", + " print(t[-1],end=',')\n", + "tm[0]=timer()-start\n", + "print('\\n',tm[0],sep='')\n", + "Counter(t)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Code 0 means that the proof was found, all other codes means that proof wasn't found. So this method has proved 21 inequalities.\n", + "\n", + "The second method uses `findvalues` function, rationalizes the result numbers and gives them as additional parameter to `prove` function." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "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", + " return 1\n", + " return a\n", + "start=timer()\n", + "t2=[]\n", + "for ineq in ineqs2:\n", + " numvalues=findvalues(ineq,disp=0)\n", + " values=nsimplify(numvalues,tolerance=0.1,rational=True)\n", + " values=list(map(cut,values))\n", + " t2+=[prove(ineq,values=values)]\n", + " print(t2[-1],end=',')\n", + "tm[1]=timer()-start\n", + "print('\\n',tm[1],sep='')\n", + "Counter(t2)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The third method is similar to the second one, but instead of rationalize values it squares, rationalizes and makes square roots of these values." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "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", + " return S(1)\n", + " return a\n", + " \n", + "start=timer()\n", + "t3=[]\n", + "for ineq in ineqs2:\n", + " numvalues=findvalues(ineq,disp=0)\n", + " numvalues=tuple([x**2 for x in numvalues])\n", + " values=nsimplify(numvalues,tolerance=0.1,rational=True)\n", + " values=[sqrt(x) for x in values]\n", + " values=list(map(cut,values))\n", + " t3+=[prove(ineq,values=values)]\n", + " print(t3[-1],end=',')\n", + "tm[2]=timer()-start\n", + "print('\\n',tm[2],sep='')\n", + "Counter(t3)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finally, the fourth method is a slight modification to the third method. It does the same \"findvalues, square, rationalize and make square roots\" thing, but then it scales the values and runs it again. It can sometimes help with uniform formulas." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "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", + "def cut(a):\n", + " if betw(a):\n", + " return a\n", + " return S(1)\n", + "\n", + "start=timer()\n", + "t4=[]\n", + "for ineq in ineqs2:\n", + " numvalues=findvalues(ineq,disp=0)\n", + " n=1\n", + " numvalues2=[]\n", + " for i in numvalues:\n", + " if betw(i):\n", + " n=1/i\n", + " break\n", + " for i in numvalues:\n", + " if betw(i):\n", + " numvalues2+=[i*n]\n", + " else:\n", + " numvalues2+=[1]\n", + " numvalues3=findvalues(ineq,values=numvalues2,disp=0)\n", + " numvalues4=tuple([x**2 for x in numvalues3])\n", + " values=nsimplify(numvalues4,tolerance=0.1,rational=True)\n", + " values=[sqrt(x) for x in values]\n", + " values=list(map(cut,values))\n", + " t4+=[prove(ineq,values=values)]\n", + " print(t4[-1],end=',')\n", + "tm[3]=timer()-start\n", + "print('\\n',tm[3],sep='')\n", + "Counter(t4)" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "

" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt\n", + "plt.bar(['1','2','3','4'],tm)\n", + "plt.ylabel('time [seconds]')\n", + "plt.xlabel('method')\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "import pandas as pd\n", + "u=pd.DataFrame(zip(['']*len(ineqs),t,t2,t3,t4))" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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" + ], + "text/plain": [ + "4 False True \n", + "3 \n", + "False 11 1\n", + "True 0 23" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "for i in range(4):\n", + " for j in range(i+1,4):\n", + " display(pd.crosstab(u[i+1]==0, u[j+1]==0))" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "pvalue 0.25\n", + "statistic 0.0\n", + "pvalue 0.5\n", + "statistic 0.0\n", + "pvalue 0.25\n", + "statistic 0.0\n", + "pvalue 1.0\n", + "statistic 0.0\n", + "pvalue 1.0\n", + "statistic 1.0\n", + "pvalue 1.0\n", + "statistic 0.0\n" + ] + } + ], + "source": [ + "from statsmodels.stats.contingency_tables import mcnemar\n", + "for i in range(4):\n", + " for j in range(i+1,4):\n", + " print(mcnemar(pd.crosstab(u[i+1]==0, u[j+1]==0)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In all cases p-value is greater than 0.05, so there is no statistical difference between the methods." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/statistics.pdf b/statistics.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5d30eec2fcbcf58b5ba85cb5f95d94225f6e4e95 GIT binary patch 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=> 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