Section DynamicNetLemmas.
Context {base_params : BaseParams}.
Context {multi_params : MultiParams base_params}.
Context {overlay_params : NameOverlayParams multi_params}.
Context {new_msg_params : NewMsgParams multi_params}.
Context {fail_msg_params : FailMsgParams multi_params}.
Proof using.
move => net failed tr H.
remember step_ordered_dynamic_failure_init as y in *.
have ->: net = snd (failed, net) by [].
move: Heqy.
induction H using refl_trans_1n_trace_n1_ind => H_init /=; first by rewrite H_init.
concludes => {H_init}.
match goal with
| [ H : step_ordered_dynamic_failure _ _ _ |- _ ] => invc H
end; rewrite /=.
- move => n H_in.
rewrite /= in IHrefl_trans_1n_trace1.
rewrite /update /=.
have H_neq: h <> n by move => H_eq; case: H_in; left.
have H_not_in: ~ In n (odnwNodes net0) by move => H_not_in; case: H_in; right.
case name_eq_dec => H_dec; first by rewrite H_dec in H_neq.
exact: IHrefl_trans_1n_trace1.
- move => n H_in.
rewrite /= in IHrefl_trans_1n_trace1.
rewrite /update /=.
have H_neq: n <> to by move => H_eq; rewrite H_eq in H_in.
case name_eq_dec => H_dec //.
exact: IHrefl_trans_1n_trace1.
- move => n H_in.
rewrite /= in IHrefl_trans_1n_trace1.
rewrite /update.
have H_neq: n <> h by move => H_eq; rewrite H_eq in H_in.
case name_eq_dec => H_dec //.
exact: IHrefl_trans_1n_trace1.
- move => n H_in.
rewrite /= in IHrefl_trans_1n_trace1.
exact: IHrefl_trans_1n_trace1.
Qed.
Proof using.
move => net failed tr H.
remember step_ordered_dynamic_failure_init as y in *.
have ->: net = snd (failed, net) by [].
move: Heqy.
induction H using refl_trans_1n_trace_n1_ind => H_init /=; first by rewrite H_init.
repeat find_rewrite.
concludes => {H_init}.
match goal with
| [ H : step_ordered_dynamic_failure _ _ _ |- _ ] => invc H
end; rewrite /=.
- move => n H_in.
case: H_in => H_in.
rewrite -H_in /update.
break_if => //.
by exists (init_handlers h).
have H_neq: n <> h by move => H_eq; rewrite H_eq in H_in.
have [d H_eq] := IHrefl_trans_1n_trace1 _ H_in.
exists d.
rewrite /update /=.
by break_if.
- move => n H_in.
rewrite /update.
break_if; first by exists d'.
have [d0 H_eq] := IHrefl_trans_1n_trace1 _ H_in.
by exists d0.
- move => n H_in.
rewrite /update.
break_if; first by exists d'.
have [d0 H_eq] := IHrefl_trans_1n_trace1 _ H_in.
by exists d0.
- move => n H_in.
exact: IHrefl_trans_1n_trace1.
Qed.
Proof using.
move => net failed tr H.
remember step_ordered_dynamic_failure_init as y in *.
have ->: failed = fst (failed, net) by [].
have H_eq_o: net = snd (failed, net) by [].
rewrite {2}H_eq_o {H_eq_o}.
move: Heqy.
induction H using refl_trans_1n_trace_n1_ind => H_init /=; first by rewrite H_init.
repeat find_rewrite.
concludes => {H_init}.
match goal with
| [ H : step_ordered_dynamic_failure _ _ _ |- _ ] => invc H
end; rewrite /=.
- move => n H_in.
right.
exact: IHrefl_trans_1n_trace1.
- move => n H_in.
exact: IHrefl_trans_1n_trace1.
- move => n H_in.
exact: IHrefl_trans_1n_trace1.
- move => n H_in.
case: H_in => H_in; first by rewrite -H_in.
exact: IHrefl_trans_1n_trace1.
Qed.

Lemma ordered_dynamic_state_not_initialized_not_failed :
forall net failed tr,
step_ordered_dynamic_failure_star step_ordered_dynamic_failure_init (failed, net) tr ->
forall n, ~ In n (odnwNodes net) ->
~ In n failed.