I am working on a proof of the theorem that every LEFT-RESET Turing machine has an equivalent conventional, single tape Turing machine. The transitions for the LEFT-RESET Turing machine are {Right, RESET}, and the transitions for the conventional Turing machine are {Left, Right}. Here is the general outline of my proof:
Given a LEFT-RESET Turing machine $T_1$ and a conventional, I will prove that $T_1$ has an equivalent conventional Turing machine $T_2$. There are two possible transitions for $T_1$: right and reset. The right transition of $T_1$ is identical to the right transition of $T_2$. When $T_1$ performs a reset, $T_1$ moves its read-write head left until it reaches the leftmost cell on the tape. Therefore, a LEFT-RESET Turing machine is equivalent to a conventional Turing machine. Q.E.D.
Do I have to prove that there is a series of transitions on $T_1$ that are equivalent to a single left transition on $T_2$? Other write-ups I have seen include this, but if the LEFT-RESET Turing machine cannot perform a traditional left transition as an operation, I do not understand why it is part of the proof.
