{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Advanced Expression Manipulation Solutions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "x, y, z = symbols('x y z')\n",
    "init_printing()\n",
    "from IPython.display import display"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For each exercise, fill in the function according to its docstring. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Creating expressions from classes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create the following objects without using any mathematical operators like `+`, `-`, `*`, `/`, or `**` by explicitly using the classes `Add`, `Mul`, and `Pow`.  You may use `x` instead of `Symbol('x')` and `4` instead of `Integer(4)`.\n",
    "\n",
    "$$x^2 + 4xyz$$\n",
    "$$x^{(x^y)}$$\n",
    "$$x - \\frac{y}{z}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def explicit_classes1():\n",
    "    \"\"\"\n",
    "    Returns the expression x**2 + 4*x*y*z, built using SymPy classes explicitly.\n",
    "\n",
    "    >>> explicit_classes1()\n",
    "    x**2 + 4*x*y*z\n",
    "    \"\"\"\n",
    "    return Add(Pow(x, 2), Mul(4, x, y, z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "explicit_classes1()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def explicit_classes2():\n",
    "    \"\"\"\n",
    "    Returns the expression x**(x**y), built using SymPy classes explicitly.\n",
    "\n",
    "    >>> explicit_classes2()\n",
    "    x**(x**y)\n",
    "    \"\"\"\n",
    "    return Pow(x, Pow(x, y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "explicit_classes2()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def explicit_classes3():\n",
    "    \"\"\"\n",
    "    Returns the expression x - y/z, built using SymPy classes explicitly.\n",
    "\n",
    "    >>> explicit_classes3()\n",
    "    x - y/z\n",
    "    \"\"\"\n",
    "    return Add(x, Mul(-1, Mul(y, Pow(z, -1))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "explicit_classes3()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Nested args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x**2 - y*(2**(x + 3) + z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use nested `.args` calls to get the 3 in expr."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def nested_args():\n",
    "    \"\"\"\n",
    "    Get the 3 in the above expression.\n",
    "\n",
    "    >>> nested_args()\n",
    "    3\n",
    "    \"\"\"\n",
    "    expr = x**2 - y*(2**(x + 3) + z)\n",
    "    return expr.args[0].args[2].args[1].args[1].args[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nested_args()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Traversal "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write a post-order traversal function that prints each node."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def post(expr):\n",
    "    \"\"\"\n",
    "    Post-order traversal\n",
    "\n",
    "    >>> expr = x**2 - y*(2**(x + 3) + z)\n",
    "    >>> post(expr)\n",
    "    -1\n",
    "    y\n",
    "    2\n",
    "    3\n",
    "    x\n",
    "    x + 3\n",
    "    2**(x + 3)\n",
    "    z\n",
    "    2**(x + 3) + z\n",
    "    -y*(2**(x + 3) + z)\n",
    "    x\n",
    "    2\n",
    "    x**2\n",
    "    x**2 - y*(2**(x + 3) + z)\n",
    "    \"\"\"\n",
    "    for arg in expr.args:\n",
    "        post(arg)\n",
    "    display(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x**2 - y*(2**(x + 3) + z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "post(expr)"
   ]
  }
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