Local Arguments update {_} {_} _ _ _ _ _ : simpl never.
Section InputBeforeOutput.
Context {orig_base_params : BaseParams}.
Context {one_node_params : OneNodeParams orig_base_params}.
Context {raft_params : RaftParams orig_base_params}.
Context {oiai : output_implies_applied_interface}.
Context {aiii : applied_implies_input_interface}.
Context {si : sorted_interface}.
Context {lacimi : lastApplied_commitIndex_match_interface}.
Context {lmi : log_matching_interface}.
Context {smsi : state_machine_safety_interface}.
Context {misi : max_index_sanity_interface}.
Context {uii : unique_indices_interface}.
Section inner.
Variable client : clientId.
Variable id : nat.
Fixpoint client_id_in l :=
match l with
| [] => false
| e :: l' =>
if (andb (if clientId_eq_dec (eClient e) client then true else false)
(eId e =? id)) then
true
else
client_id_in l'
end.

Lemma client_id_in_true_in_applied_entries :
forall net,
client_id_in (applied_entries (nwState net)) = true ->
in_applied_entries client id net.