Inductive lift_relation {state: Type} (R: state -> MetaState state -> Prop):
  MetaState state-> MetaState state -> Prop :=
| lift_relation_Error:
    lift_relation R Error Error
| lift_relation_NonTerminating:
    lift_relation R NonTerminating NonTerminating
| lift_relation_Terminating:
    forall s ms, R s ms -> lift_relation R (Terminating s) ms.Inductive lift_relation {state: Type} (R: state -> MetaState state -> Prop):
  MetaState state-> MetaState state -> Prop :=
| lift_relation_Error:
    lift_relation R Error Error
| lift_relation_NonTerminating:
    lift_relation R NonTerminating NonTerminating
| lift_relation_Terminating:
    forall s ms, R s ms -> lift_relation R (Terminating s) ms.Inductive lift_relation {state: Type} (R: state -> MetaState state -> Prop):
  MetaState state-> MetaState state -> Prop :=
| lift_relation_Error:
    lift_relation R Error Error
| lift_relation_NonTerminating:
    lift_relation R NonTerminating NonTerminating
| lift_relation_Terminating:
    forall s ms, R s ms -> lift_relation R (Terminating s) ms.Inductive lift_relation {state: Type} (R: state -> MetaState state -> Prop):
  MetaState state-> MetaState state -> Prop :=
| lift_relation_Error:
    lift_relation R Error Error
| lift_relation_NonTerminating:
    lift_relation R NonTerminating NonTerminating
| lift_relation_Terminating:
    forall s ms, R s ms -> lift_relation R (Terminating s) ms.Definition lift_Krelation {state: Type} {state_R: Relation state}: MetaState state -> MetaState state -> Prop := Total.forward.Inductive lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| lift_join_Error1:
    forall mx my, lift_join Error mx my
| lift_join_Error2:
    forall mx my, lift_join mx Error my
| lift_join_NonTerminating1:
    forall x, lift_join NonTerminating (Terminating x) NonTerminating
| lift_join_NonTerminating2:
    forall x, lift_join (Terminating x) NonTerminating NonTerminating
| lift_join_NonTerminating3:
    lift_join NonTerminating NonTerminating NonTerminating
| lift_join_Terminating:
    forall x y z,
      join x y z ->
      lift_join (Terminating x) (Terminating y) (Terminating z).Inductive lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| lift_join_Error1:
    forall mx my, lift_join Error mx my
| lift_join_Error2:
    forall mx my, lift_join mx Error my
| lift_join_NonTerminating1:
    forall x, lift_join NonTerminating (Terminating x) NonTerminating
| lift_join_NonTerminating2:
    forall x, lift_join (Terminating x) NonTerminating NonTerminating
| lift_join_NonTerminating3:
    lift_join NonTerminating NonTerminating NonTerminating
| lift_join_Terminating:
    forall x y z,
      join x y z ->
      lift_join (Terminating x) (Terminating y) (Terminating z).Inductive lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| lift_join_Error1:
    forall mx my, lift_join Error mx my
| lift_join_Error2:
    forall mx my, lift_join mx Error my
| lift_join_NonTerminating1:
    forall x, lift_join NonTerminating (Terminating x) NonTerminating
| lift_join_NonTerminating2:
    forall x, lift_join (Terminating x) NonTerminating NonTerminating
| lift_join_NonTerminating3:
    lift_join NonTerminating NonTerminating NonTerminating
| lift_join_Terminating:
    forall x y z,
      join x y z ->
      lift_join (Terminating x) (Terminating y) (Terminating z).Inductive lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| lift_join_Error1:
    forall mx my, lift_join Error mx my
| lift_join_Error2:
    forall mx my, lift_join mx Error my
| lift_join_NonTerminating1:
    forall x, lift_join NonTerminating (Terminating x) NonTerminating
| lift_join_NonTerminating2:
    forall x, lift_join (Terminating x) NonTerminating NonTerminating
| lift_join_NonTerminating3:
    lift_join NonTerminating NonTerminating NonTerminating
| lift_join_Terminating:
    forall x y z,
      join x y z ->
      lift_join (Terminating x) (Terminating y) (Terminating z).Inductive lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| lift_join_Error1:
    forall mx my, lift_join Error mx my
| lift_join_Error2:
    forall mx my, lift_join mx Error my
| lift_join_NonTerminating1:
    forall x, lift_join NonTerminating (Terminating x) NonTerminating
| lift_join_NonTerminating2:
    forall x, lift_join (Terminating x) NonTerminating NonTerminating
| lift_join_NonTerminating3:
    lift_join NonTerminating NonTerminating NonTerminating
| lift_join_Terminating:
    forall x y z,
      join x y z ->
      lift_join (Terminating x) (Terminating y) (Terminating z).Inductive lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| lift_join_Error1:
    forall mx my, lift_join Error mx my
| lift_join_Error2:
    forall mx my, lift_join mx Error my
| lift_join_NonTerminating1:
    forall x, lift_join NonTerminating (Terminating x) NonTerminating
| lift_join_NonTerminating2:
    forall x, lift_join (Terminating x) NonTerminating NonTerminating
| lift_join_NonTerminating3:
    lift_join NonTerminating NonTerminating NonTerminating
| lift_join_Terminating:
    forall x y z,
      join x y z ->
      lift_join (Terminating x) (Terminating y) (Terminating z).Inductive lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| lift_join_Error1:
    forall mx my, lift_join Error mx my
| lift_join_Error2:
    forall mx my, lift_join mx Error my
| lift_join_NonTerminating1:
    forall x, lift_join NonTerminating (Terminating x) NonTerminating
| lift_join_NonTerminating2:
    forall x, lift_join (Terminating x) NonTerminating NonTerminating
| lift_join_NonTerminating3:
    lift_join NonTerminating NonTerminating NonTerminating
| lift_join_Terminating:
    forall x y z,
      join x y z ->
      lift_join (Terminating x) (Terminating y) (Terminating z).Inductive strong_lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| strong_lift_join_Error1:
    forall mx, strong_lift_join Error mx Error
| strong_lift_join_Error2:
    forall mx, strong_lift_join mx Error Error
| strong_lift_join_NonTerminating1:
    forall x, strong_lift_join NonTerminating (Terminating x) NonTerminating
| strong_lift_join_NonTerminating2:
    forall x, strong_lift_join (Terminating x) NonTerminating NonTerminating
| strong_lift_join_NonTerminating3:
    strong_lift_join NonTerminating NonTerminating NonTerminating
| strong_lift_join_Terminating:
    forall x y z,
      join x y z ->
      strong_lift_join (Terminating x) (Terminating y) (Terminating z).Inductive strong_lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| strong_lift_join_Error1:
    forall mx, strong_lift_join Error mx Error
| strong_lift_join_Error2:
    forall mx, strong_lift_join mx Error Error
| strong_lift_join_NonTerminating1:
    forall x, strong_lift_join NonTerminating (Terminating x) NonTerminating
| strong_lift_join_NonTerminating2:
    forall x, strong_lift_join (Terminating x) NonTerminating NonTerminating
| strong_lift_join_NonTerminating3:
    strong_lift_join NonTerminating NonTerminating NonTerminating
| strong_lift_join_Terminating:
    forall x y z,
      join x y z ->
      strong_lift_join (Terminating x) (Terminating y) (Terminating z).Inductive strong_lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| strong_lift_join_Error1:
    forall mx, strong_lift_join Error mx Error
| strong_lift_join_Error2:
    forall mx, strong_lift_join mx Error Error
| strong_lift_join_NonTerminating1:
    forall x, strong_lift_join NonTerminating (Terminating x) NonTerminating
| strong_lift_join_NonTerminating2:
    forall x, strong_lift_join (Terminating x) NonTerminating NonTerminating
| strong_lift_join_NonTerminating3:
    strong_lift_join NonTerminating NonTerminating NonTerminating
| strong_lift_join_Terminating:
    forall x y z,
      join x y z ->
      strong_lift_join (Terminating x) (Terminating y) (Terminating z).Inductive strong_lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| strong_lift_join_Error1:
    forall mx, strong_lift_join Error mx Error
| strong_lift_join_Error2:
    forall mx, strong_lift_join mx Error Error
| strong_lift_join_NonTerminating1:
    forall x, strong_lift_join NonTerminating (Terminating x) NonTerminating
| strong_lift_join_NonTerminating2:
    forall x, strong_lift_join (Terminating x) NonTerminating NonTerminating
| strong_lift_join_NonTerminating3:
    strong_lift_join NonTerminating NonTerminating NonTerminating
| strong_lift_join_Terminating:
    forall x y z,
      join x y z ->
      strong_lift_join (Terminating x) (Terminating y) (Terminating z).Inductive strong_lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| strong_lift_join_Error1:
    forall mx, strong_lift_join Error mx Error
| strong_lift_join_Error2:
    forall mx, strong_lift_join mx Error Error
| strong_lift_join_NonTerminating1:
    forall x, strong_lift_join NonTerminating (Terminating x) NonTerminating
| strong_lift_join_NonTerminating2:
    forall x, strong_lift_join (Terminating x) NonTerminating NonTerminating
| strong_lift_join_NonTerminating3:
    strong_lift_join NonTerminating NonTerminating NonTerminating
| strong_lift_join_Terminating:
    forall x y z,
      join x y z ->
      strong_lift_join (Terminating x) (Terminating y) (Terminating z).Inductive strong_lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| strong_lift_join_Error1:
    forall mx, strong_lift_join Error mx Error
| strong_lift_join_Error2:
    forall mx, strong_lift_join mx Error Error
| strong_lift_join_NonTerminating1:
    forall x, strong_lift_join NonTerminating (Terminating x) NonTerminating
| strong_lift_join_NonTerminating2:
    forall x, strong_lift_join (Terminating x) NonTerminating NonTerminating
| strong_lift_join_NonTerminating3:
    strong_lift_join NonTerminating NonTerminating NonTerminating
| strong_lift_join_Terminating:
    forall x y z,
      join x y z ->
      strong_lift_join (Terminating x) (Terminating y) (Terminating z).Inductive strong_lift_join
          {state: Type}
          {J_state: Join state}:
  MetaState state -> MetaState state -> MetaState state -> Prop :=
| strong_lift_join_Error1:
    forall mx, strong_lift_join Error mx Error
| strong_lift_join_Error2:
    forall mx, strong_lift_join mx Error Error
| strong_lift_join_NonTerminating1:
    forall x, strong_lift_join NonTerminating (Terminating x) NonTerminating
| strong_lift_join_NonTerminating2:
    forall x, strong_lift_join (Terminating x) NonTerminating NonTerminating
| strong_lift_join_NonTerminating3:
    strong_lift_join NonTerminating NonTerminating NonTerminating
| strong_lift_join_Terminating:
    forall x y z,
      join x y z ->
      strong_lift_join (Terminating x) (Terminating y) (Terminating z).Definition lift_function {A B: Type} (f: A -> B): MetaState A -> MetaState B :=
  fun ma =>
  match ma with
  | NonTerminating => NonTerminating
  | Error => Error
  | Terminating a => Terminating (f a)
  end.

Lemma clos_refl_trans_lift_relation_forward {state: Type} (R: state -> MetaState state -> Prop): forall x y, clos_refl_trans _ (lift_relation R) x y -> match x with | Error => y = Error | NonTerminating => y = NonTerminating | _ => True end.