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.

(* Specifications: *)
Lemma incl_first_activations_semi p : incl (first_activations p) (first_activations_and_semi p).
Lemma first_activations_and_semi_rec_incl_sub_trees p : incl (first_activations_and_semi_rec p) (sub_trees p).
Lemma first_activations_and_semi_incl_sub_trees p : incl (first_activations_and_semi p) (sub_trees p).
Lemma cbv_big_induction : forall (P : cbv -> Prop), (forall i s p c1 t c2 v, (forall p', In p' (first_activations (cbv_update i s p c1 t c2 v)) -> P p') -> P (cbv_update i s p c1 t c2 v)) -> forall i s p c1 t c2 v, P (cbv_update i s p c1 t c2 v).
Lemma activations_first_activations i s p' c1 t c2 v : let p := cbv_update i s p' c1 t c2 v in activations p = p :: flat_map activations (first_activations p).
Lemma cache_consistent p : wf p -> exists C, cache_beq (cache_right p) (C ++ cache_left p) = true /\ (forall t, assoc_cache t C <> None -> assoc_cache t (cache_left p) = None).
Lemma cache_path_consistent c1 c2 lp : andl (map wf lp) -> cache_path c1 c2 lp = true -> forall p, In p lp -> exists c3, c2 = c3 ++ cache_right p /\ (forall t, assoc_cache t c3 <> None -> assoc_cache t (cache_right p) = None).
Lemma cache_path_consistent_head C p lp : wf p -> andl (map wf lp) -> cache_path (cache_right p) C lp = true -> exists C', C = C' ++ cache_right p /\ (forall t, assoc_cache t C' <> None -> assoc_cache t (cache_right p) = None).
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.