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

Lemma assoc_cache_hd (f: function) (lv: list term) (v: value) (C: cache) : assoc_cache (fapply f lv) ((fapply f lv, v) :: C) = Some v.