#!/usr/bin/env python
# coding: utf-8

# # Advanced Expression Manipulation Solutions

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from sympy import *
x, y, z = symbols('x y z')
init_printing()
from IPython.display import display


# For each exercise, fill in the function according to its docstring. 

# ## Creating expressions from classes

# 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)`.
# 
# $$x^2 + 4xyz$$
# $$x^{(x^y)}$$
# $$x - \frac{y}{z}$$
# 

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def explicit_classes1():
    """
    Returns the expression x**2 + 4*x*y*z, built using SymPy classes explicitly.

    >>> explicit_classes1()
    x**2 + 4*x*y*z
    """
    return Add(Pow(x, 2), Mul(4, x, y, z))


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explicit_classes1()


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def explicit_classes2():
    """
    Returns the expression x**(x**y), built using SymPy classes explicitly.

    >>> explicit_classes2()
    x**(x**y)
    """
    return Pow(x, Pow(x, y))


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explicit_classes2()


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def explicit_classes3():
    """
    Returns the expression x - y/z, built using SymPy classes explicitly.

    >>> explicit_classes3()
    x - y/z
    """
    return Add(x, Mul(-1, Mul(y, Pow(z, -1))))


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explicit_classes3()


# ## Nested args

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expr = x**2 - y*(2**(x + 3) + z)


# Use nested `.args` calls to get the 3 in expr.

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def nested_args():
    """
    Get the 3 in the above expression.

    >>> nested_args()
    3
    """
    expr = x**2 - y*(2**(x + 3) + z)
    return expr.args[0].args[2].args[1].args[1].args[0]


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nested_args()


# ## Traversal 

# Write a post-order traversal function that prints each node.

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def post(expr):
    """
    Post-order traversal

    >>> expr = x**2 - y*(2**(x + 3) + z)
    >>> post(expr)
    -1
    y
    2
    3
    x
    x + 3
    2**(x + 3)
    z
    2**(x + 3) + z
    -y*(2**(x + 3) + z)
    x
    2
    x**2
    x**2 - y*(2**(x + 3) + z)
    """
    for arg in expr.args:
        post(arg)
    display(expr)


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expr = x**2 - y*(2**(x + 3) + z)


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post(expr)