Ltac exec_induction base_case :=
intros;
match goal with
| [xs : list _, i : nat, Hexec: internal_execution _ ?Tp ?M _ _ |- _] =>
generalize dependent Tp; generalize dependent M;
induction xs as [| x xs' IHxs]; intros;
[ match goal with
| [Hexec: internal_execution _ _ _ _ _ |- _] =>
try (inversion Hexec; subst;
by Tactics.pf_cleanup)
end
| match goal with
| [Hexec: internal_execution _ _ _ _ _ |- _] =>
simpl in Hexec;
destruct (x == i) eqn:Heq;
move/eqP:Heq=>Heq;
try eauto;
subst;
try (inversion Hexec as [|? ? ? ? ? ? ? ? ? ?
HschedN HschedS Hcomp ? Htrans]; subst;
simpl in Htrans;
simpl in HschedN; inversion HschedN; subst;
Tactics.pf_cleanup;
specialize (IHxs _ _  Htrans);
rewrite <- IHxs;
erewrite base_case; eauto)
end
]
end.

(** [Mem.valid_block] is preserved by [internal_step]*)
Lemma internal_step_valid:
forall tp m tp' m' i (cnt: containsThread tp i)
(Hcomp: mem_compatible tp m)
(Hstep: internal_step cnt Hcomp tp' m') b
(Hvalid: Mem.valid_block m b),
Mem.valid_block m' b.