Context {G C Z:Type}.
Context {genv_symb: G -> injective_PTree block}.
Context (Hcore:@CoreSemantics C mem).
Variable (Hspec : juicy_ext_spec Z).
Variable ge : G.
Definition Hrel n' m m' :=
n' = level m' /\
(level m' < level m)%nat /\
pures_eq (m_phi m) (m_phi m').
Inductive jsafeN_:
nat -> Z -> C -> juicy_mem -> Prop :=
| jsafeN_0: forall z c m, jsafeN_ O z c m
| jsafeN_step:
forall n z c m c' m',
jstep Hcore c m c' m' ->
jm_bupd z (jsafeN_ n z c') m' ->
jsafeN_ (S n) z c m
| jsafeN_external:
forall n z c m e args x,
j_at_external Hcore c m = Some (e,args) ->
ext_spec_pre Hspec e x (genv_symb ge) (sig_args (ef_sig e)) args z m ->
(forall ret m' z' n'
(Hargsty : Val.has_type_list args (sig_args (ef_sig e)))
(Hretty : has_opttyp ret (sig_res (ef_sig e))),
(n' <= n)%nat ->
Hrel n' m m' ->
ext_spec_post Hspec e x (genv_symb ge) (sig_res (ef_sig e)) ret z' m' ->
exists c',
semantics.after_external Hcore ret c (m_dry m') = Some c' /\
jm_bupd z' (jsafeN_ n' z' c') m') ->
jsafeN_ (S n) z c m
| jsafeN_halted:
forall n z c m i,
semantics.halted Hcore c i ->
ext_spec_exit Hspec (Some (Vint i)) z m ->
jsafeN_ n z c m.

(* Specifications: *)
Lemma jsafe_corestep_backward:
forall c m c' m' n z,
jstep Hcore c m c' m' ->
jsafeN_ n z c' m' -> jsafeN_ (S n) z c m.
Lemma jsafe_downward1 :
forall n c m z,
jsafeN_ (S n) z c m -> jsafeN_ n z c m.
Lemma jsafe_downward :
forall n n' c m z,
le n' n ->
jsafeN_ n z c m -> jsafeN_ n' z c m.