Inductive refl_trans_n1_trace (step : step_relation) : step_relation :=
  | RTn1TBase : forall x, refl_trans_n1_trace step x x []
  | RTn1TStep : forall x x' x'' cs cs',
                  refl_trans_n1_trace step x x' cs ->
                  step x' x'' cs' ->
                  refl_trans_n1_trace step x x'' (cs ++ cs').Inductive refl_trans_n1_trace (step : step_relation) : step_relation :=
  | RTn1TBase : forall x, refl_trans_n1_trace step x x []
  | RTn1TStep : forall x x' x'' cs cs',
                  refl_trans_n1_trace step x x' cs ->
                  step x' x'' cs' ->
                  refl_trans_n1_trace step x x'' (cs ++ cs').Inductive refl_trans_n1_trace (step : step_relation) : step_relation :=
  | RTn1TBase : forall x, refl_trans_n1_trace step x x []
  | RTn1TStep : forall x x' x'' cs cs',
                  refl_trans_n1_trace step x x' cs ->
                  step x' x'' cs' ->
                  refl_trans_n1_trace step x x'' (cs ++ cs').

Lemma RT1n_step : forall (step : step_relation) x y z l l', refl_trans_1n_trace step x y l -> step y z l' -> refl_trans_1n_trace step x z (l ++ l').