Definition embeds : list A -> list A -> Prop := Embeds leA.Definition sublist : list (list A) -> list (list A) -> Prop := Embeds (eq (A:= list A)).Fixpoint firsts (l : list (list A)) : list A :=
match l with 
| nil => nil
| w::ws => match w with 
		| nil => nil
		| a::w' => a :: (firsts ws)
	   end
end.Definition Tree := tree (list (list A) * (list A)).Fact eq_Tree_dec : forall t t' : Tree, {t = t'} + {t <> t'}.Definition root_label (t : Tree) : option A := 
match (root t) with 
  | (l,r) => head r
end.Fixpoint roots_labels (ts : list Tree) : option (list A) :=  
match ts with 
| nil => Some nil
| t :: ts' => match (root_label t) with 
                  | None => None
                  | Some a => match (roots_labels ts') with 
                                       | Some ts'' => Some (a :: ts'')
                                       | None => None
                                       end
                  end
end.Inductive is_insert_forest : list Tree -> list A -> A -> list Tree -> Prop :=
| is_if0 : forall w a, is_insert_forest nil w a nil
| is_if2 : forall vs a' ts f w a f' l, 
             is_insert_forest f w a f' ->
             ~(leA a' a) -> 
             is_insert_forest ((node (vs,a'::l) ts) :: f) w a ((node (vs,a'::l) ts) :: f')
| is_if4 : forall vs a'  ts f w a t' l, 
             leA a' a -> 
             is_insert_tree (node (vs,a'::l) ts) w a t' -> 
             is_insert_forest ((node (vs,a'::l) ts):: f) w a (t' :: f) 
with is_insert_tree : Tree -> list A -> A -> Tree -> Prop :=
| is_it1 : forall vs l ts w a rrts, 
             roots_labels ts = Some rrts ->  
             ~ greater leA a rrts ->
             is_insert_tree (node (vs,l) ts) w a (node (vs,l) ((node (w::vs, a::l) ts) :: ts))
| is_it2 : forall l ts w a rrts f', 
             roots_labels ts = Some rrts ->  
             greater leA a rrts ->
             is_insert_forest ts w a f' ->
             is_insert_tree (node l ts) w a (node l f').Inductive is_insert_forest : list Tree -> list A -> A -> list Tree -> Prop :=
| is_if0 : forall w a, is_insert_forest nil w a nil
| is_if2 : forall vs a' ts f w a f' l, 
             is_insert_forest f w a f' ->
             ~(leA a' a) -> 
             is_insert_forest ((node (vs,a'::l) ts) :: f) w a ((node (vs,a'::l) ts) :: f')
| is_if4 : forall vs a'  ts f w a t' l, 
             leA a' a -> 
             is_insert_tree (node (vs,a'::l) ts) w a t' -> 
             is_insert_forest ((node (vs,a'::l) ts):: f) w a (t' :: f) 
with is_insert_tree : Tree -> list A -> A -> Tree -> Prop :=
| is_it1 : forall vs l ts w a rrts, 
             roots_labels ts = Some rrts ->  
             ~ greater leA a rrts ->
             is_insert_tree (node (vs,l) ts) w a (node (vs,l) ((node (w::vs, a::l) ts) :: ts))
| is_it2 : forall l ts w a rrts f', 
             roots_labels ts = Some rrts ->  
             greater leA a rrts ->
             is_insert_forest ts w a f' ->
             is_insert_tree (node l ts) w a (node l f').Inductive is_insert_forest : list Tree -> list A -> A -> list Tree -> Prop :=
| is_if0 : forall w a, is_insert_forest nil w a nil
| is_if2 : forall vs a' ts f w a f' l, 
             is_insert_forest f w a f' ->
             ~(leA a' a) -> 
             is_insert_forest ((node (vs,a'::l) ts) :: f) w a ((node (vs,a'::l) ts) :: f')
| is_if4 : forall vs a'  ts f w a t' l, 
             leA a' a -> 
             is_insert_tree (node (vs,a'::l) ts) w a t' -> 
             is_insert_forest ((node (vs,a'::l) ts):: f) w a (t' :: f) 
with is_insert_tree : Tree -> list A -> A -> Tree -> Prop :=
| is_it1 : forall vs l ts w a rrts, 
             roots_labels ts = Some rrts ->  
             ~ greater leA a rrts ->
             is_insert_tree (node (vs,l) ts) w a (node (vs,l) ((node (w::vs, a::l) ts) :: ts))
| is_it2 : forall l ts w a rrts f', 
             roots_labels ts = Some rrts ->  
             greater leA a rrts ->
             is_insert_forest ts w a f' ->
             is_insert_tree (node l ts) w a (node l f').Inductive is_insert_forest : list Tree -> list A -> A -> list Tree -> Prop :=
| is_if0 : forall w a, is_insert_forest nil w a nil
| is_if2 : forall vs a' ts f w a f' l, 
             is_insert_forest f w a f' ->
             ~(leA a' a) -> 
             is_insert_forest ((node (vs,a'::l) ts) :: f) w a ((node (vs,a'::l) ts) :: f')
| is_if4 : forall vs a'  ts f w a t' l, 
             leA a' a -> 
             is_insert_tree (node (vs,a'::l) ts) w a t' -> 
             is_insert_forest ((node (vs,a'::l) ts):: f) w a (t' :: f) 
with is_insert_tree : Tree -> list A -> A -> Tree -> Prop :=
| is_it1 : forall vs l ts w a rrts, 
             roots_labels ts = Some rrts ->  
             ~ greater leA a rrts ->
             is_insert_tree (node (vs,l) ts) w a (node (vs,l) ((node (w::vs, a::l) ts) :: ts))
| is_it2 : forall l ts w a rrts f', 
             roots_labels ts = Some rrts ->  
             greater leA a rrts ->
             is_insert_forest ts w a f' ->
             is_insert_tree (node l ts) w a (node l f').Inductive is_insert_forest : list Tree -> list A -> A -> list Tree -> Prop :=
| is_if0 : forall w a, is_insert_forest nil w a nil
| is_if2 : forall vs a' ts f w a f' l, 
             is_insert_forest f w a f' ->
             ~(leA a' a) -> 
             is_insert_forest ((node (vs,a'::l) ts) :: f) w a ((node (vs,a'::l) ts) :: f')
| is_if4 : forall vs a'  ts f w a t' l, 
             leA a' a -> 
             is_insert_tree (node (vs,a'::l) ts) w a t' -> 
             is_insert_forest ((node (vs,a'::l) ts):: f) w a (t' :: f) 
with is_insert_tree : Tree -> list A -> A -> Tree -> Prop :=
| is_it1 : forall vs l ts w a rrts, 
             roots_labels ts = Some rrts ->  
             ~ greater leA a rrts ->
             is_insert_tree (node (vs,l) ts) w a (node (vs,l) ((node (w::vs, a::l) ts) :: ts))
| is_it2 : forall l ts w a rrts f', 
             roots_labels ts = Some rrts ->  
             greater leA a rrts ->
             is_insert_forest ts w a f' ->
             is_insert_tree (node l ts) w a (node l f').Inductive is_insert_forest : list Tree -> list A -> A -> list Tree -> Prop :=
| is_if0 : forall w a, is_insert_forest nil w a nil
| is_if2 : forall vs a' ts f w a f' l, 
             is_insert_forest f w a f' ->
             ~(leA a' a) -> 
             is_insert_forest ((node (vs,a'::l) ts) :: f) w a ((node (vs,a'::l) ts) :: f')
| is_if4 : forall vs a'  ts f w a t' l, 
             leA a' a -> 
             is_insert_tree (node (vs,a'::l) ts) w a t' -> 
             is_insert_forest ((node (vs,a'::l) ts):: f) w a (t' :: f) 
with is_insert_tree : Tree -> list A -> A -> Tree -> Prop :=
| is_it1 : forall vs l ts w a rrts, 
             roots_labels ts = Some rrts ->  
             ~ greater leA a rrts ->
             is_insert_tree (node (vs,l) ts) w a (node (vs,l) ((node (w::vs, a::l) ts) :: ts))
| is_it2 : forall l ts w a rrts f', 
             roots_labels ts = Some rrts ->  
             greater leA a rrts ->
             is_insert_forest ts w a f' ->
             is_insert_tree (node l ts) w a (node l f').Inductive is_insert_forest : list Tree -> list A -> A -> list Tree -> Prop :=
| is_if0 : forall w a, is_insert_forest nil w a nil
| is_if2 : forall vs a' ts f w a f' l, 
             is_insert_forest f w a f' ->
             ~(leA a' a) -> 
             is_insert_forest ((node (vs,a'::l) ts) :: f) w a ((node (vs,a'::l) ts) :: f')
| is_if4 : forall vs a'  ts f w a t' l, 
             leA a' a -> 
             is_insert_tree (node (vs,a'::l) ts) w a t' -> 
             is_insert_forest ((node (vs,a'::l) ts):: f) w a (t' :: f) 
with is_insert_tree : Tree -> list A -> A -> Tree -> Prop :=
| is_it1 : forall vs l ts w a rrts, 
             roots_labels ts = Some rrts ->  
             ~ greater leA a rrts ->
             is_insert_tree (node (vs,l) ts) w a (node (vs,l) ((node (w::vs, a::l) ts) :: ts))
| is_it2 : forall l ts w a rrts f', 
             roots_labels ts = Some rrts ->  
             greater leA a rrts ->
             is_insert_forest ts w a f' ->
             is_insert_tree (node l ts) w a (node l f').Inductive is_forest : list (list A) -> list Tree -> Prop :=
| is_f0 : is_forest nil nil
| is_f1 : forall a w ws f f',  is_forest ws f ->
            greater leA a (bad_subsequence leA leA_dec (firsts ws)) ->
            is_insert_forest f w a f' -> is_forest ((a::w)::ws) f' 
| is_f2 : forall a w ws f,  is_forest ws f ->
            ~ greater leA a (bad_subsequence leA leA_dec (firsts ws)) ->
            is_forest ((a::w)::ws) ((node (w::ws, a::nil) f)::f).Inductive is_forest : list (list A) -> list Tree -> Prop :=
| is_f0 : is_forest nil nil
| is_f1 : forall a w ws f f',  is_forest ws f ->
            greater leA a (bad_subsequence leA leA_dec (firsts ws)) ->
            is_insert_forest f w a f' -> is_forest ((a::w)::ws) f' 
| is_f2 : forall a w ws f,  is_forest ws f ->
            ~ greater leA a (bad_subsequence leA leA_dec (firsts ws)) ->
            is_forest ((a::w)::ws) ((node (w::ws, a::nil) f)::f).Inductive is_forest : list (list A) -> list Tree -> Prop :=
| is_f0 : is_forest nil nil
| is_f1 : forall a w ws f f',  is_forest ws f ->
            greater leA a (bad_subsequence leA leA_dec (firsts ws)) ->
            is_insert_forest f w a f' -> is_forest ((a::w)::ws) f' 
| is_f2 : forall a w ws f,  is_forest ws f ->
            ~ greater leA a (bad_subsequence leA leA_dec (firsts ws)) ->
            is_forest ((a::w)::ws) ((node (w::ws, a::nil) f)::f).Inductive is_forest : list (list A) -> list Tree -> Prop :=
| is_f0 : is_forest nil nil
| is_f1 : forall a w ws f f',  is_forest ws f ->
            greater leA a (bad_subsequence leA leA_dec (firsts ws)) ->
            is_insert_forest f w a f' -> is_forest ((a::w)::ws) f' 
| is_f2 : forall a w ws f,  is_forest ws f ->
            ~ greater leA a (bad_subsequence leA leA_dec (firsts ws)) ->
            is_forest ((a::w)::ws) ((node (w::ws, a::nil) f)::f).Definition P_on_tree (ws : list (list A)) (t : Tree) : Prop := forall t', subtree t' t -> P ws t'.Definition P_on_forest (ws : list (list A)) (f : list Tree) : Prop := 
  forall t, tree_in_forest t f -> P ws t.Fact P_on_node : forall a ts ws, P_on_tree ws (node a ts) -> 
forall t, In t ts -> P_on_tree ws t.Definition from_insert_forest (t : Tree) (a : A) : Prop := 
forall a' l vs ts, t = node (vs, a'::l) ts -> leA a' a.

Lemma P_on_split : forall ws a t f, P_on_tree ws (node a (t::f)) -> P_on_tree ws t -> P_on_tree ws (node a f).