Axiom determinism:
    forall x y y',
      valid x y  -> int_step x y y' ->
      forall x',
        valid x' y -> int_step x' y y'.Axiom valid_step:
    forall x y y',
      valid x y  -> int_step x y y' ->
      valid x y'.(** *Normal explicit safety *)
    CoInductive exp_safety (x:X) (y:Y): Prop:=
    | halted_safety : halted x y -> exp_safety x y
    | internal_safety y': int_step x y y'  ->
                          (forall x',  valid x' y' -> exp_safety x' y') ->
                          exp_safety x y
    | external_safety x' y': ext_step x y x' y' ->
                             (forall x',  valid x' y' -> exp_safety x' y') ->
                             exp_safety x y.(** *Normal explicit safety *)
    CoInductive exp_safety (x:X) (y:Y): Prop:=
    | halted_safety : halted x y -> exp_safety x y
    | internal_safety y': int_step x y y'  ->
                          (forall x',  valid x' y' -> exp_safety x' y') ->
                          exp_safety x y
    | external_safety x' y': ext_step x y x' y' ->
                             (forall x',  valid x' y' -> exp_safety x' y') ->
                             exp_safety x y.(** *Normal explicit safety *)
    CoInductive exp_safety (x:X) (y:Y): Prop:=
    | halted_safety : halted x y -> exp_safety x y
    | internal_safety y': int_step x y y'  ->
                          (forall x',  valid x' y' -> exp_safety x' y') ->
                          exp_safety x y
    | external_safety x' y': ext_step x y x' y' ->
                             (forall x',  valid x' y' -> exp_safety x' y') ->
                             exp_safety x y.(** *Normal explicit safety *)
    CoInductive exp_safety (x:X) (y:Y): Prop:=
    | halted_safety : halted x y -> exp_safety x y
    | internal_safety y': int_step x y y'  ->
                          (forall x',  valid x' y' -> exp_safety x' y') ->
                          exp_safety x y
    | external_safety x' y': ext_step x y x' y' ->
                             (forall x',  valid x' y' -> exp_safety x' y') ->
                             exp_safety x y.(*Parametrized version*)
    Inductive exp_safety_gen {_exp_safety} (x:X) (y:Y): Prop:=
    | paco_halted_safety : halted x y -> exp_safety_gen x y
    | paco_internal_safety y': int_step x y y'  ->
                               (forall x',  valid x' y' -> _exp_safety x' y') ->
                               exp_safety_gen x y
    | paco_external_safety x' y': ext_step x y x' y' ->
                                  (forall x',  valid x' y' -> _exp_safety x' y') ->
                                  exp_safety_gen x y.(*Parametrized version*)
    Inductive exp_safety_gen {_exp_safety} (x:X) (y:Y): Prop:=
    | paco_halted_safety : halted x y -> exp_safety_gen x y
    | paco_internal_safety y': int_step x y y'  ->
                               (forall x',  valid x' y' -> _exp_safety x' y') ->
                               exp_safety_gen x y
    | paco_external_safety x' y': ext_step x y x' y' ->
                                  (forall x',  valid x' y' -> _exp_safety x' y') ->
                                  exp_safety_gen x y.(*Parametrized version*)
    Inductive exp_safety_gen {_exp_safety} (x:X) (y:Y): Prop:=
    | paco_halted_safety : halted x y -> exp_safety_gen x y
    | paco_internal_safety y': int_step x y y'  ->
                               (forall x',  valid x' y' -> _exp_safety x' y') ->
                               exp_safety_gen x y
    | paco_external_safety x' y': ext_step x y x' y' ->
                                  (forall x',  valid x' y' -> _exp_safety x' y') ->
                                  exp_safety_gen x y.(*Parametrized version*)
    Inductive exp_safety_gen {_exp_safety} (x:X) (y:Y): Prop:=
    | paco_halted_safety : halted x y -> exp_safety_gen x y
    | paco_internal_safety y': int_step x y y'  ->
                               (forall x',  valid x' y' -> _exp_safety x' y') ->
                               exp_safety_gen x y
    | paco_external_safety x' y': ext_step x y x' y' ->
                                  (forall x',  valid x' y' -> _exp_safety x' y') ->
                                  exp_safety_gen x y.Definition paco_exp_safety := paco2 (@exp_safety_gen).

Lemma exp_safety_paco_correct: forall x y, exp_safety x y <-> paco_exp_safety bot2 x y.