# Equality of lambda terms which do not have normal form

In the context of lambda calculus, how should one prove $$\beta$$-equality of terms that do not have normal form?

In particular, how to prove that these are different combinators: $$Y = λf.(λx.f(xx))(λx.f(xx)) \\ Θ = (λxf.f(xxf))(λxf.f(xxf))$$ These work the same if we reduce them, but Wikipedia states that they are different, and that there are in fact infinitely many combinators that work the same as these.

• Syntactically they are obviously different terms. Is there some other notion of equality under which you want to prove them different? Jun 10 at 6:20