:- pred constr(bool). 
:- mode constr(in). 
:- ignore constr/1.
:- type tree(D) ---> leaf ; node( D,   tree(D),     tree(D) ). 
%                 empty tree     key  left-subtree right-subtree
% tree(d) is the type of the trees whose keys have type D. 
% Here we assume that "D" is "int". 
% Note: duplicates are NOT allowed.   
% ==========================================================
% Program. 
% Binary Search Tree - duplicates not allowed.

:- pred bstinsert(int,tree(int),tree(int)).
:- mode bstinsert(in,in,out).

bstinsert(X,leaf,node(X,leaf,leaf)).
bstinsert(X,node(A,L,R), node(A,L,R)) :- constr(X=A).
bstinsert(X,node(A,L,R), node(A,L1,R)) :- constr(X<A), bstinsert(X,L,L1).
bstinsert(X,node(A,L,R), node(A,L,R1)) :- constr(A<X), bstinsert(X,R,R1).

% ==========================================================
% catamorphisms
:- pred bstree(tree(int),bool).
:- mode bstree(in,out).
:- cata bstree/2-1.

bstree(leaf,B) :- constr(B).
bstree(node(N,L,R),B) :- 
  bstree(L,BL), 
  bstree(R,BR),
  treemax(L,IsDefL,MaxL), 
  treemin(R,IsDefR,MinR),
  constr( ( (GeqLeft=(IsDefL => N>MaxL)) &
          (LeqRight=(IsDefR => N<MinR)) ) &
          (B= ((GeqLeft & BL) & (LeqRight & BR)))).


:- pred treemax(tree(int),bool,int).  
:- mode treemax(in,out,out).
:- cata treemax/3-1.

treemax(leaf,IsDef,Max) :- constr(~IsDef & Max=0).
treemax(node(N,L,R),IsDef,Max) :- 
  treemax(L,IsDefL,MaxL),
  treemax(R,IsDefR,MaxR),
  constr(IsDef & 
         (Max= ite(IsDefL & IsDefR, 
                     ite(MaxL>MaxR, 
                          ite(MaxL>N,MaxL,N), 
                          ite(MaxR>N,MaxR,N) ),            % max(MaxL,max(MaxR,N))
                     ite(IsDefL,
                          ite(MaxL>N,MaxL,N),              % max(MaxL,N)
                          ite(IsDefR & MaxR>N,MaxR,N))))). % max(MaxR,N)

:- pred treemin(tree(int),bool,int). 
:- mode treemin(in,out,out).
:- cata treemin/3-1.

treemin(leaf,IsDef,Min) :- constr(~IsDef & Min=0). 
treemin(node(N,L,R),IsDef,Min) :- 
    treemin(L,IsDefL,MinL),
    treemin(R,IsDefR,MinR),
  constr(IsDef & 
         (Min= ite(IsDefL & IsDefR, 
                     ite(MinL<MinR, 
                          ite(MinL<N,MinL,N), 
                          ite(MinR<N,MinR,N) ),            % min(MinL,min(MinR,N))
                     ite(IsDefL,
                          ite(MinL<N,MinL,N),              % min(MinL,N)
                          ite(IsDefR & MinR<N,MinR,N))))). % min(MinR,N)

% ==========================================================
% Contract on bstinsert:
:- spec bstinsert(X,T1,T2) ==>
       bstree(T1,B1), 
       treemin(T1,IsDef1,Min1),treemin(T2,IsDef2,Min2),
       treemax(T1,IsDef1,Max1),treemax(T2,IsDef2,Max2)
       =>
       constr(IsDef2 & (B1 => (Min2=ite(IsDef1 & Min1<X,Min1,X) &  
              Max2=ite(IsDef1 & Max1>X,Max1,X)) ) ).
% ==========================================================
% Verification

:- pred ff1.
ff1 :- 
   constr(~ (IsDef2 & (B1 => (Min2=ite(IsDef1 & Min1<X,Min1,X) &  Max2=ite(IsDef1 & Max1>X,Max1,X)) )  )),
   bstinsert(X,T1,T2),
   bstree(T1,B1), 
   treemin(T1,IsDef1,Min1),treemin(T2,IsDef2,Min2),
   treemax(T1,IsDef1,Max1),treemax(T2,IsDef2,Max2).
% ==========================================================% NON-CATA predicates: bstinsert constr