Module X86Inj.
(** Injections on registers *)
Definition reg_ren f (r:PregEq.t) (rs rs' : regset) : Prop :=
val_obs f (Pregmap.get r rs) (Pregmap.get r rs').
Definition regset_ren f rs rs' : Prop :=
forall r, reg_ren f r rs rs'.
Definition core_inj f c c' :=
match c, c' with State rs _, State rs' _ => regset_ren f rs rs' end.
Include X86WD.
(* Lemma decode_longs_val_obs_list: forall f (vals vals' : seq val) tys (Hobs_eq: val_obs_list f vals vals'), val_obs_list f (val_casted.decode_longs tys vals) (val_casted.decode_longs tys vals'). *)
(** ** Proof that related states take related steps*)
Proof with eauto with renamings.
intros.
destruct Hge_wd as (? & ?).
unfold Senv.symbol_address in *.
specialize (H0 id ofs _ ltac:(reflexivity)).
destruct (Senv.find_symbol g id) eqn:Hsymb; constructor.
destruct H0 as [b' Hfg'].
assert (Heq := Hfg _ _ Hfg'); subst...
Qed.
Proof with eauto with renamings reg_renamings val_renamings.
intros.
destruct Hmem_obs_eq.
pose proof (injective weak_obs_eq0).
induction Heval; subst;
try by (eexists; split)...
- eapply loadv_val_obs with (vptr2 := Val.offset_ptr (rs' RSP) ofs) in H...
destruct H as (varg' & Hload' & Hval_obs)...
- assert (Hb: val_obs f (Senv.symbol_address g id ofs)
(Senv.symbol_address g id ofs))
by (eapply symb_val_obs; eauto).
eapply loadv_val_obs with (vptr2 := Senv.symbol_address g id ofs) in H;
eauto.
destruct H as (varg' & Hload' & Hval_obs)...
- assert (Hb: val_obs f (Senv.symbol_address g id ofs)
(Senv.symbol_address g id ofs))
by (eapply symb_val_obs; eauto)...
- destruct IHHeval1 as (vhi' & ? & ?), IHHeval2 as (vlo' & ? & ?)...
- destruct IHHeval1 as (vhi' & ? & ?), IHHeval2 as (vlo' & ? & ?).
eexists; split; [constructor; eauto|].
destruct Archi.ptr64...
Qed.

Lemma eval_builtin_args_ren:
forall (g : genv) (rs rs' : regset) (f fg: memren) (m m' : mem)
(args : seq (builtin_arg preg)) (vargs : seq val)
(Hmem_obs_eq: mem_obs_eq f m m')
(Hrs_ren: regset_ren f rs rs')
(Hfg: forall b1 b2, fg b1 = Some b2 -> b1 = b2)
(Hge_wd: ge_wd fg g)
(Hincr: ren_incr fg f)
(Heval: eval_builtin_args g rs (rs RSP) m args vargs),
exists vargs',
eval_builtin_args g rs' (rs' RSP) m' args vargs' /\
val_obs_list f vargs vargs'.