Section VarDSerializedCorrect.
Variable n : nat.
Instance raft_params : RaftParams VarD.vard_base_params :=
raft_params n.
Instance base_params : BaseParams :=
transformed_base_params n.
Instance multi_params : MultiParams _ :=
transformed_multi_params n.
Instance failure_params : FailureParams _ :=
transformed_failure_params n.
Proof using.
induction tr; simpl; intro H_inp; auto.
destruct a, s; simpl.
- assert (H_inp': input_correct tr).
intros client id i0 i1 h h' H_in H_in'.
eapply H_inp; right; eauto.
concludes.
intros client id i0 i1 h h' H_in H_in'.
simpl in *.
break_or_hyp; break_or_hyp.
* find_injection; find_injection; auto.
* find_injection.
eapply H_inp.
+ right.
find_apply_lem_hyp In_filterMap.
break_exists_name e.
break_and.
destruct e, s; simpl in *; [ find_injection; eauto | destruct l; congruence ].
+ left; eauto.
* find_injection.
eapply H_inp.
+ left; eauto.
+ right.
find_apply_lem_hyp In_filterMap.
break_exists_name e.
break_and.
destruct e, s; simpl in *; [ find_injection; eauto | destruct l; congruence ].
* eapply IHtr; eauto.
- destruct l.
* apply IHtr.
intros client id i0 i1 h h' H_in H_in'.
eapply H_inp; right; eauto.
* assert (H_inp': input_correct tr).
intros client id i0 i1 h h' H_in H_in'.
eapply H_inp; right; eauto.
concludes.
intros client id i0 i1 h h' H_in H_in'.
simpl in *.
break_or_hyp; [ find_inversion | idtac ].
break_or_hyp; [ find_inversion | idtac ].
eapply IHtr; eauto.
Qed.
Proof using.
induction tr; simpl; auto.
destruct a, s; simpl; intro H_inp.
- assert (H_inp': input_correct (filterMap trace_non_empty_out tr)).
intros client id i0 i1 h h' H_in H_in'.
eapply H_inp; right; eauto.
concludes.
intros client id i0 i1 h h' H_in H_in'.
simpl in *.
break_or_hyp; break_or_hyp.
* find_injection; find_injection; auto.
* find_injection.
eapply H_inp; [ idtac | left; eauto ].
right.
eapply filterMap_In; eauto.
simpl; eauto.
* find_injection.
eapply H_inp; [ left; eauto | idtac ].
right.
eapply filterMap_In; eauto.
simpl; eauto.
* eapply IHtr; eauto.
- destruct l.
* intros client id i0 i1 h h' H_in H_in'.
simpl in *.
break_or_hyp; [ find_inversion | idtac ].
break_or_hyp; [ find_inversion | idtac ].
eapply IHtr; eauto.
* assert (H_inp': input_correct (filterMap trace_non_empty_out tr)).
intros client id i0 i1 h h' H_in H_in'.
eapply H_inp; right; eauto.
concludes.
intros client id i0 i1 h h' H_in H_in'.
simpl in *.
break_or_hyp; [ find_inversion | idtac ].
break_or_hyp; [ find_inversion | idtac ].
eapply IHtr; eauto.
Qed.
Proof using.
intros tr tr' H_in H_eq.
apply correct_filterMap_trace_non_empty_out_input_correct.
rewrite <- H_eq.
apply correct_input_correct_filterMap_trace_non_empty_out; auto.
Qed.
Proof using.
induction tr; simpl; auto.
destruct a, s; simpl.
- rewrite IHtr; auto.
- destruct l; auto.
Qed.
Proof using.
induction tr; simpl; auto.
destruct a, s; simpl.
- rewrite IHtr; auto.
- destruct l; auto.
rewrite IHtr; auto.
Qed.
Proof using.
intros tr tr' l tr1 H_exp H_eq.
rewrite get_input_tr_filterMap_trace_non_empty_out in H_exp.
rewrite get_output_tr_filterMap_trace_non_empty_out in H_exp.
rewrite <- H_eq in H_exp.
rewrite <- get_input_tr_filterMap_trace_non_empty_out in H_exp.
rewrite <- get_output_tr_filterMap_trace_non_empty_out in H_exp.
auto.
Qed.
Proof using.
induction tr; simpl; auto.
destruct a, s; simpl.
- rewrite IHtr; auto.
- rewrite IHtr.
destruct l; auto.
Qed.

Lemma equivalent_filterMap_trace_non_empty_out :
forall tr tr' l,
equivalent key (import tr') l ->
filterMap trace_non_empty_out tr = filterMap trace_non_empty_out tr' ->
equivalent key (import tr) l.