Inductive reach (M: sTM) : mconfig (states M) ->  mconfig (states M) -> Prop :=
  |reachI c : reach c c
  |reachS c d: reach (step c) d -> (halt (cstate c) = false) -> reach c d.Inductive reach (M: sTM) : mconfig (states M) ->  mconfig (states M) -> Prop :=
  |reachI c : reach c c
  |reachS c d: reach (step c) d -> (halt (cstate c) = false) -> reach c d.Inductive reach (M: sTM) : mconfig (states M) ->  mconfig (states M) -> Prop :=
  |reachI c : reach c c
  |reachS c d: reach (step c) d -> (halt (cstate c) = false) -> reach c d.Definition Halt' (M: sTM) (start: mconfig (states M)) :=
    exists (f: mconfig (states M)), halt (cstate f)=true /\ reach start f.

Lemma TM_terminates_Halt (M:sTM) (start: mconfig (states M)) : TMterminates start <-> Halt' start.