Section TM.
Hypothesis Σ:Set.
Hypothesis Σ0:Σ.
Definition HaltsAt(tm:TM Σ)(n:nat)(st:ExecState Σ): Prop :=
exists st', Steps Σ tm n st st' /\ step Σ tm st' = None.
Definition Halts(tm:TM Σ)(st:ExecState Σ): Prop :=
exists n, HaltsAt tm n st.
Definition HaltsFromInit(tm:TM Σ): Prop :=
Halts tm (InitES Σ Σ0).

Lemma Steps_NonHalt {tm m n st st0}:
m<n ->
Steps Σ tm n st st0 ->
~HaltsAt tm m st.