Proof using misi smsi lmi.
intros.
invcs H0; simpl in *;
try match goal with
| _ : key_in_output_trace _ _ [] |- _ =>
unfold key_in_output_trace in *; break_exists; simpl in *; intuition
end.
- unfold key_in_output_trace in *.
break_exists; simpl in *; intuition.
find_inversion.
unfold in_applied_entries in *. simpl in *.
unfold RaftNetHandler in *.
repeat break_let. repeat find_inversion. simpl in *.
find_eapply_lem_hyp RIR_handleMessage; eauto.
find_apply_lem_hyp key_in_output_list_split.
find_copy_apply_lem_hyp doLeader_key_in_output_list.
intuition.
match goal with
| H : doLeader ?st ?h = _ |- _ =>
replace st with ((update name_eq_dec (nwState net) h st) h) in H;
[|rewrite_update; auto]
end.
find_eapply_lem_hyp RIR_doLeader; eauto.
simpl in *.
match goal with
| _ : raft_intermediate_reachable ?net' |- context [update _ (nwState net) ?h ?d] =>
remember net' as n;
assert ((update name_eq_dec (nwState net) h d) = (update name_eq_dec (nwState n) h d)) by
(subst; simpl in *; repeat rewrite update_overwrite; auto)
end.
unfold raft_data in *. simpl in *.
unfold raft_data in *. simpl in *.
repeat find_rewrite.
eapply doGenericServer_key_in_output_list; eauto.
simpl in *. rewrite_update; eauto.
- unfold key_in_output_trace in *.
break_exists; simpl in *; intuition.
find_inversion.
unfold in_applied_entries in *. simpl in *.
unfold RaftInputHandler in *.
repeat break_let. repeat find_inversion. simpl in *.
find_copy_eapply_lem_hyp RIR_handleInput; eauto.
find_apply_lem_hyp key_in_output_list_split.
intuition; [exfalso; eapply handleInput_key_in_output_list; eauto|].
find_apply_lem_hyp key_in_output_list_split.
intuition; [exfalso; eapply doLeader_key_in_output_list; eauto|].
match goal with
| H : doLeader ?st ?h = _ |- _ =>
replace st with ((update name_eq_dec (nwState net) h st) h) in H;
[|rewrite_update; auto]
end.
find_eapply_lem_hyp RIR_doLeader; eauto.
simpl in *.
match goal with
| _ : raft_intermediate_reachable ?net' |- context [update _ (nwState net) ?h ?d] =>
remember net' as n;
assert ((update name_eq_dec (nwState net) h d) = (update name_eq_dec (nwState n) h d)) by
(subst; simpl in *; repeat rewrite update_overwrite; auto)
end.
unfold raft_data in *. simpl in *.
unfold raft_data in *. simpl in *.
repeat find_rewrite.
eapply doGenericServer_key_in_output_list; eauto.
simpl in *. rewrite_update; eauto.
Qed.
Program Instance TR : TraceRelation step_failure :=
{
init := step_failure_init;
T := key_in_output_trace client id ;
T_dec := key_in_output_trace_dec client id ;
R := fun s => in_applied_entries client id (snd s)
}.
Next Obligation.
unfold in_applied_entries in *.
break_exists; eexists; intuition eauto.
eapply applied_entries_monotonic; eauto.
eauto using refl_trans_1n_n1_trace, step_failure_star_raft_intermediate_reachable.
Defined.
Next Obligation.
unfold key_in_output_trace in *. intuition.
break_exists; intuition.
Defined.
Next Obligation.
simpl in *.
find_apply_lem_hyp step_failure_star_raft_intermediate_reachable.
find_apply_lem_hyp in_output_changed; auto.
eauto using output_implies_in_applied_entries.
Defined.

Theorem output_implies_applied :
forall failed net tr,
step_failure_star step_failure_init (failed, net) tr ->
key_in_output_trace client id tr ->
in_applied_entries client id net.