Cadabra
a field-theory motivated approach to computer algebra

cdb.core.manip

Manipulating equations, inequalities and similar expressions
This package contains various helper functions to manipulate expressions in elementary ways, e.g. moving terms around or multiplying sides of an expression by a factor.

eq_to_subrule(ex: Ex) -> Ex

Create substitution rule from an equation.
Creates a substitution rule from an equation Ex.
eq_to_subrule($a = b + c + d$);
\(\displaystyle{}b+c+d \rightarrow a\)
b + c + d → a

multiply_through(ex: Ex, factor: Ex, auto_distribute=False) -> Ex

Multiply all terms by the same factor.
Multiplies all terms in ex by factor. If auto_distribute is True, distribute is called on the resulting expression.
multiply_through($a+b < c$, $k$);
\(\displaystyle{}k \left(a+b\right) < k c\)
\less(k (a + b) , k c)

get_lhs(ex: Ex) -> Ex

Return left-hand side of an expression
Returns the left hand side of an equation, inequality or expression whose top node contains two children.
get_lhs($a=b$);
\(\displaystyle{}a\)
a

get_rhs(ex: Ex) -> Ex

Return right-hand side of an expression
Returns the right-hand side of an equation, inequality or expression whose top node contains two children.
get_rhs($a=b$);
\(\displaystyle{}b\)
b

swap_sides(ex: Ex) -> Ex

Swap the left and right-hand sides of an expression.
Swaps the two sides of an equation, inequality or expression whose top node contains two children. Inequalities will flip as appropriate.
swap_sides($a=b$);
\(\displaystyle{}b = a\)
b = a
swap_sides($a>b$);
\(\displaystyle{}b < a\)
\less(b , a)

to_rhs(ex: Ex, part: Ex) -> Ex

Move the indicated term of the expression to the right-hand side.
If ex is an equation or inequality which contains part on its left hand side, then moves part to the right hand side. part must contain the multiplier of the term to be moved, including minus signs.
to_rhs($a+b = c+d$, $b$);
\(\displaystyle{}a = c+d-b\)
a = c + d-b
to_rhs($a-b = c$, $-b$);
\(\displaystyle{}a = c+b\)
a = c + b

to_lhs(ex: Ex, part: Ex) -> Ex

Move the indicated term of the expression to the left-hand side.
Complement of to_rhs.
# Moving terms test if __name__ == "__main__": ex0 := a - b = c + d; to_rhs(ex0, $-b$); to_rhs(ex0, $a$); to_lhs(ex0, $d + b$);
\(\displaystyle{}a = c+d+b\)
a = c + d + b
\(\displaystyle{} = c+d+b-a\)
= c + d + b-a
\(\displaystyle{}-d-b = c-a\)
-d-b = c-a
# Equation test if __name__ == "__main__": ex1 := a_{\mu}a^{\mu} + b = c; eq_to_subrule(ex1); get_lhs(ex1); get_rhs(ex1); multiply_through(ex1, $k_{\alpha}$); swap_sides(ex1); swap_sides(ex1);
\(\displaystyle{}c \rightarrow a_{\mu} a^{\mu}+b\)
c → a_{\mu} a^{\mu} + b
\(\displaystyle{}a_{\mu} a^{\mu}+b\)
a_{\mu} a^{\mu} + b
\(\displaystyle{}c\)
c
\(\displaystyle{}k_{\alpha} \left(a_{\mu} a^{\mu}+b\right) = k_{\alpha} c\)
k_{\alpha} (a_{\mu} a^{\mu} + b) = k_{\alpha} c
\(\displaystyle{}k_{\alpha} c = k_{\alpha} \left(a_{\mu} a^{\mu}+b\right)\)
k_{\alpha} c = k_{\alpha} (a_{\mu} a^{\mu} + b)
\(\displaystyle{}k_{\alpha} \left(a_{\mu} a^{\mu}+b\right) = k_{\alpha} c\)
k_{\alpha} (a_{\mu} a^{\mu} + b) = k_{\alpha} c
# Inequality test if __name__ == "__main__": {f, i}::Symbol. ex2 := w_{f} - w_{i} > 2 \pi W_{a b}\beta^{a}\beta^{b}; get_lhs(ex2); get_rhs(ex2); swap_sides(ex2); swap_sides(ex2); multiply_through(ex2, $2n$);
\(\displaystyle{}w_{f}-w_{i}\)
w_{f}-w_{i}
\(\displaystyle{}2\pi W_{a b} \beta^{a} \beta^{b}\)
2\pi W_{a b} \beta^{a} \beta^{b}
\(\displaystyle{}2\pi W_{a b} \beta^{a} \beta^{b} < w_{f}-w_{i}\)
\less(2\pi W_{a b} \beta^{a} \beta^{b} , w_{f}-w_{i})
\(\displaystyle{}w_{f}-w_{i} > 2\pi W_{a b} \beta^{a} \beta^{b}\)
\greater(w_{f}-w_{i} , 2\pi W_{a b} \beta^{a} \beta^{b})
\(\displaystyle{}2n \left(w_{f}-w_{i}\right) > 4n \pi W_{a b} \beta^{a} \beta^{b}\)
\greater(2n (w_{f}-w_{i}) , 4n \pi W_{a b} \beta^{a} \beta^{b})
# Fail tests if __name__ == "__main__": ex3 := a, b, c; try: eq_to_subrule(ex3) except Exception as e: e; try: get_lhs(ex3) except Exception as e: e; try: get_rhs(ex3) except Exception as e: e; try: swap_sides(ex3) except Exception as e: e; try: to_rhs(ex3, $a$) except Exception as e: e; try: to_lhs(ex3, $b$) except Exception as e: e; try: t::Coordinate. {i, j}::Indices(values={t}, position=fixed). ex := a_{i}. evaluate(ex, ${a_{t} = 1}$) get_component(ex, $x$) except Exception as e: e;
eq_to_subrule called on expression which is not an equation
get_lhs called on expression whose top node contains 3 arguments, 2 required
get_lhs called on expression whose top node contains 3 arguments, 2 required
swap_sides called on expression whose top node contains 3 arguments, 2 required
to_lhs called on expression which is not an equation or inequality
swap_sides called on expression whose top node contains 3 arguments, 2 required
name 'get_component' is not defined
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