Proof with try congruence.
assert (SIZE: forall v sz, do_alloc_size v = Some sz -> v = Vptrofs sz).
{ intros until sz; unfold Vptrofs; destruct v; simpl; destruct Archi.ptr64 eqn:SF;
intros EQ; inv EQ; f_equal; symmetry; eauto with ptrofs. }
intros until m'.
destruct ef; simpl.
(* EF_external *)
eapply do_external_function_sound; eauto.
(* EF_builtin *)
eapply do_external_function_sound; eauto.
(* EF_runtime *)
eapply do_external_function_sound; eauto.
(* EF_vload *)
unfold do_ef_volatile_load. destruct vargs... destruct v... destruct vargs...
mydestr. destruct p as [[w'' t''] v]; mydestr.
exploit do_volatile_load_sound; eauto. intuition. econstructor; eauto.
auto.
(* EF_vstore *)
unfold do_ef_volatile_store. destruct vargs... destruct v... destruct vargs... destruct vargs...
mydestr. destruct p as [[w'' t''] m'']. mydestr.
exploit do_volatile_store_sound; eauto. intuition. econstructor; eauto.
auto.
(* EF_malloc *)
unfold do_ef_malloc. destruct vargs... destruct vargs... mydestr.
destruct (Mem.alloc m (- size_chunk Mptr) (Ptrofs.unsigned i)) as [m1 b] eqn:?. mydestr.
split. apply SIZE in Heqo. subst v. econstructor; eauto. constructor.
(* EF_free *)
unfold do_ef_free. destruct vargs... destruct v... destruct vargs...
mydestr. split. apply SIZE in Heqo0. econstructor; eauto. congruence. omega. constructor.
(* EF_memcpy *)
unfold do_ef_memcpy. destruct vargs... destruct v... destruct vargs...
destruct v... destruct vargs... mydestr.
apply Decidable_sound in Heqb1. red in Heqb1.
split. econstructor; eauto; tauto. constructor.
(* EF_annot *)
unfold do_ef_annot. mydestr.
split. constructor. apply list_eventval_of_val_sound; auto.
econstructor. constructor; eauto. constructor.
(* EF_annot_val *)
unfold do_ef_annot_val. destruct vargs... destruct vargs... mydestr.
split. constructor. apply eventval_of_val_sound; auto.
econstructor. constructor; eauto. constructor.
(* EF_inline_asm *)
eapply do_inline_assembly_sound; eauto.
(* EF_debug *)
unfold do_ef_debug. mydestr. split; constructor.
Qed.

(* Specifications: *)
Lemma do_ef_external_complete:
forall ef w vargs m w' t vres m',
external_call ef ge vargs m t vres m' -> possible_trace w t w' ->
do_external ef w vargs m = Some(w', t, vres, m').