Proof using.
intros.
unfold in_applied_entries in *. break_exists_exists.
intuition. red. exists x. intuition.
- red. auto.
- unfold applied_entries in *. break_match.
+ find_apply_lem_hyp in_rev.
find_apply_lem_hyp removeAfterIndex_in.
eauto.
+ simpl in *. intuition.
Qed.
Program Instance TR : TraceRelation step_failure :=
{
init := step_failure_init;
T := output_before_input client id client' id';
R := fun s => entries_ordered client id client' id' (snd s)
}.
Next Obligation.
unfold output_before_input.
eapply before_func_dec.
Defined.
Next Obligation.
simpl in *.
unfold entries_ordered in *.
find_apply_lem_hyp step_failure_star_raft_intermediate_reachable.
find_eapply_lem_hyp applied_entries_monotonic'; eauto.
break_exists; repeat find_rewrite.
eauto using before_func_app.
Defined.
Next Obligation.
simpl in *.
find_copy_eapply_lem_hyp output_before_input_not_key_in_input_trace; eauto.
find_copy_apply_lem_hyp output_before_input_key_in_output_trace.
find_eapply_lem_hyp output_implies_applied;
[|eapply refl_trans_n1_1n_trace; econstructor; eauto using refl_trans_1n_n1_trace].
eapply in_applied_entries_entries_ordered; auto.
intuition.
find_false.
find_apply_lem_hyp in_applied_entries_applied_implies_input_state.
break_exists. intuition.
eexists. eapply applied_implies_input; eauto.
eapply refl_trans_n1_1n_trace; econstructor; eauto using refl_trans_1n_n1_trace.
Defined.

Theorem causal_order_preserved :
forall failed net tr,
step_failure_star step_failure_init (failed, net) tr ->
output_before_input client id client' id' tr ->
entries_ordered client id client' id' net.