example derivation steps for a CAS or theorem prover to validate

in order of increasing complexity, here are a set of derivation steps for a CAS or theorem prover to validate

start with "a = b"
add "2" to both sides
end with "a + 2 = b + 2"


start with "\sin x = f(x)"
multiply both sides by "2"
end with "2 \sin x = 2 f(x)"


start with "\sin x = f(x)"
substitute "2 y" for "x"
end with "\sin (2 y) = f(2 y)"