Fixpoint first_activations_rec (proof_tree : cbv) : list cbv :=
match proof_tree with
| cbv_constr lp _ _ _ _ => flat_map first_activations_rec lp
| cbv_split lp p _ _ _ _ => first_activations_rec p ++ flat_map first_activations_rec lp
| cbv_update _ _ _ _ _ _ _ as p' => [p']
| cbv_read _ _ _ => []
end.
Definition first_activations (proof_tree : cbv) : list cbv :=
match proof_tree with
| cbv_constr lp _ _ _ _ => []
| cbv_split lp p _ _ _ _ => []
| cbv_update _ _ p _ _ _ _ => first_activations_rec p
| cbv_read _ _ _ => []
end.
Definition strict_activations (proof_tree: cbv) :=
flat_map activations (first_activations proof_tree).

Lemma in_first_activations_rec_nb_judgements_le p proof_tree :
In p (first_activations_rec proof_tree) -> nb_judgements p <= nb_judgements proof_tree.