Inductive tree : Set := 
      | node : A -> list tree -> tree.Inductive tree : Set := 
      | node : A -> list tree -> tree.Fixpoint tree_size (t : tree) : nat :=
      match t with 
	| node _ l => S ((fix l_size (l : list tree) : nat :=
			    match l with 
			      | nil => 0
			      | t' :: l' => (tree_size t') + (l_size l')
			    end) l)
      end.Definition root (t : tree) : A :=
	match t with 
	| node a l => a
	end.Definition subtrees (t : tree) : list tree :=
	match t with 
	| node _ l => l
	end.Inductive tree_in_forest : tree -> list tree -> Prop :=
| tif0 : forall t t' l, In t' l -> subtree t t' -> tree_in_forest t l
with subtree : tree -> tree -> Prop :=
| sub0 : forall t, subtree t t
| sub1 : forall t l ts, tree_in_forest t ts -> subtree t (node l ts).Inductive tree_in_forest : tree -> list tree -> Prop :=
| tif0 : forall t t' l, In t' l -> subtree t t' -> tree_in_forest t l
with subtree : tree -> tree -> Prop :=
| sub0 : forall t, subtree t t
| sub1 : forall t l ts, tree_in_forest t ts -> subtree t (node l ts).Inductive tree_in_forest : tree -> list tree -> Prop :=
| tif0 : forall t t' l, In t' l -> subtree t t' -> tree_in_forest t l
with subtree : tree -> tree -> Prop :=
| sub0 : forall t, subtree t t
| sub1 : forall t l ts, tree_in_forest t ts -> subtree t (node l ts).Inductive tree_in_forest : tree -> list tree -> Prop :=
| tif0 : forall t t' l, In t' l -> subtree t t' -> tree_in_forest t l
with subtree : tree -> tree -> Prop :=
| sub0 : forall t, subtree t t
| sub1 : forall t l ts, tree_in_forest t ts -> subtree t (node l ts).Inductive tree_in_forest : tree -> list tree -> Prop :=
| tif0 : forall t t' l, In t' l -> subtree t t' -> tree_in_forest t l
with subtree : tree -> tree -> Prop :=
| sub0 : forall t, subtree t t
| sub1 : forall t l ts, tree_in_forest t ts -> subtree t (node l ts).Fixpoint tree_rect_aux t : P t :=
      match t as t return P t with
	| node f v => H2 f
	  ((fix vt_rect (v : list tree) : Q v :=
	    match v as v return Q v with
	      | nil => H3
	      | cons t' v' => H4 (tree_rect_aux t') (vt_rect v')
	    end) v)
      end.Inductive lforall (P : tree -> Type) : list tree -> Type :=
| lforall_nil : lforall P nil
| lforall_cons : forall a l, lforall P l -> P a -> lforall P (a::l).Inductive lforall (P : tree -> Type) : list tree -> Type :=
| lforall_nil : lforall P nil
| lforall_cons : forall a l, lforall P l -> P a -> lforall P (a::l).Inductive lforall (P : tree -> Type) : list tree -> Type :=
| lforall_nil : lforall P nil
| lforall_cons : forall a l, lforall P l -> P a -> lforall P (a::l).

(* Specifications: *)
Lemma tree_rect : forall P : tree -> Type, (forall x, P (node x nil)) -> (forall f v, lforall P v -> P (node f v)) -> forall t, P t.
Lemma tree_ind : forall P : tree -> Prop, (forall x, P (node x nil)) -> (forall f v, (forall u, In u v -> P u) -> P (node f v)) -> forall t, P t.