Prolog - Checking if a binary tree is ordered or not

Question

I want to write a program in Prolog that confirms if a b-tree of integers is ordered or not. The order goes from smaller to greater. This is what I've written so far but I do not reach any solid work. Does someone know how to do that?

Domains
element=integer
tree=a(tree,element,tree);void

Predicates
     nondeterm ordre(tree)

Clauses
    order(a(_,Node,a(_,Node2,_))):-Node<Node2.
    order(a(Esq,Node,Dre)) :- 
        order(Esq), 
        write(Node),nl, 
        order(Dre).

Goal
       order(a(a(void,1,void),2,a(a(void,3,void),4,a(void,6,void)))).

Huge Thanks.

Answer 1

domains
    element = integer
    arbre = a (arbre, element, arbre) ; buit

predicates
    nondeterm ordenat (arbre)
    nondeterm ordenat2 (arbre, element)

clauses
    ordenat2 (a (buit, E, buit), E).
    ordenat2 (a (buit, E, R), MR) :-
        ordenat2 (R, MR),
        E<MR.
    ordenat2 (a (L, E, buit), E) :-
        ordenat2 (L, ML),
        ML<E.
    ordenat2 (a (L, E, R), MR) :-
        ordenat2 (L, ML), ML<E,
        ordenat2 (R, MR), E<MR.

    ordenat (A) :-
        ordenat2 (A, _).

goal
    B=a (a (a (buit, 1, buit), 2, a (buit, 3, buit)), 4, a (a (buit, 5, buit), 6, a (buit, 7, buit))),
    ordenat (B)
    .

Result: yes

Answer 2

Using the same same tree structure as before (the atom btree denoting an empty tree; the structure btree(Key,Left,Right) denoting a non-empty tree, something like this should do you:

• An empty tree is ordered by definition

  is_ordered( btree ).
  

• A non-empty tree is ordered if

  • its left children are ordered and their keys are less than that of the current node

  • its right children are ordered and their keys are greater than that of the current node

    is_ordered( btree( K , L , R ) ) :-
      is_ordered( L < K ) ,
      is_ordered( R > K )
      .
    

• By definition, an empty tree is less than any specified key value and also greater than any specified key value.

  is_ordered( btree < K ).
  is_ordered( btree > K ).
  

• A non-empty tree is less than a specified key value if

  • its key is less than the specified key, and

  • it is itself ordered

    is_ordered( btree(K1,L,R) < K ) :-
    K1 < K ,
    is_ordered( btree(K1,L,R) )
    .
    

• A non-empty tree is greater than a specified key value if

  • its key is greater than the specified key, and

  • it is itself ordered

    is_ordered( btree(K1,L,R) > K ) :-
      K1 > K ,
      is_ordered( btree(K1,L,R) )
      .