Turing fixed-point combinator ($\Theta$)

Put $D=\lambda xf.f(xxf)$ and $\Theta =DD$. Then $\Theta f{\to }_{\beta }^{\ast }f(\Theta f)$. Unlike the Curry fixed-point combinator, every finite reduct retains a closed subterm $DD$. Contracting this subterm recreates one; every other generated redex substitutes a variable and preserves it. The Church-Rosser theorem therefore proves $Y/{\equiv }_{\beta }\Theta$.