Fixpoint activations (proof_tree : cbv) : list cbv :=
match proof_tree with
| cbv_constr lp _ _ => flat_map activations lp
| cbv_split lp p _ _ => activations p ++ flat_map activations lp
| cbv_function _ _ p _ _ as p' => p' :: activations p
end.

Lemma cbv_function_in_activations_InCBV proof_tree sub_proof_tree i s p t v:
sub_proof_tree = cbv_function i s p t v ->
InCBV sub_proof_tree proof_tree ->
In sub_proof_tree (activations proof_tree).