% =================================================================
:- pred constr(bool).
:- mode constr(in).
:- ignore constr/1.
:- type bintree_int ---> leaf ; node(int,bintree_int,bintree_int).
% This is a binary tree supporting a heap. 
% leaf is the EMPTY bintree_int. Nodes may have DUPLICATE integer values.
% ================================================================
% Program: descending sorting.
:- pred heapsort(list(int),list(int)).
:- mode heapsort(in,out).

heapsort(L,SL) :-
  list_to_heap(L,H), heap_to_list(H,SL).
% ===============================================================
% making a heap out of a list.
% Program predicate: not be a catamorphism.
:- pred list_to_heap(list(int),bintree_int).
:- mode list_to_heap(in,out).

list_to_heap([], leaf).
list_to_heap([X|Xs], Heap) :-
  list_to_heap(Xs, HeapXs), 
  insert_heap(X, HeapXs, Heap).

% -------
:- pred insert_heap(int,bintree_int,bintree_int).
:- mode insert_heap(in,in,out).

insert_heap(X, leaf, node(X,leaf,leaf)).
insert_heap(X, node(Top,L,R), node(Top,R,L1)) :- % Torsion! node(Top,R,L1), not node(Top,L1,R), and
          constr(X=<Top), insert_heap(X,L,L1).   % top unchanged. Insert X in the left son-node.
insert_heap(X, node(Top,L,R), node(X,R,L1)) :-   % Torsion! node(Top,R,L1), not node(Top,L1,R), and
          constr(X>Top), insert_heap(Top,L,L1).  % X on top. Insert Top in the left son-node.
% ----------------------------------

:- pred mkbtree(int,bintree_int,bintree_int,bintree_int).
:- mode mkbtree(in,in,in,out).

mkbtree(Top,Left,Right, node(Top,Left,Right)).

% ----------------------------------
% Making a weakly-descending list out of a heap.
% Program predicate: need not be a catamorphism.
:- pred heap_to_list(bintree_int,list(int)).
:- mode heap_to_list(in,out).   

heap_to_list(leaf, []).
heap_to_list(node(Top,leaf,leaf), [Top]).                                    
heap_to_list(node(Top,node(LTop,LL,LR),leaf), [Top|Tail]) :-                   
    mkbtree(LTop,LL,LR,NewHeap),
    heap_to_list(NewHeap,Tail).
heap_to_list(node(Top,leaf,node(RTop,RL,RR)), [Top|Tail]) :-                  
    mkbtree(RTop,RL,RR,NewHeap),
    heap_to_list(NewHeap,Tail).
heap_to_list(node(Top,node(LT,LL,LR),node(RT,RL,RR)), [Top|Tail]) :-        
     mkbtree(LT,LL,LR,NewLHeap), mkbtree(RT,RL,RR,NewRHeap),
     heap_merge(NewLHeap,NewRHeap,MergedHeap),
     heap_to_list(MergedHeap,Tail).

:- pred heap_merge(bintree_int,bintree_int,bintree_int).
:- mode heap_merge(in,in,out).                                % heap_merge is a total function

heap_merge(leaf,leaf, leaf).                           
heap_merge(node(LTop,LL,LR),leaf, node(LTop,LL,LR)).   
heap_merge(leaf,node(RTop,RL,RR), node(RTop,RL,RR)).   
heap_merge(node(LTop,LL,LR),node(RTop,RL,RR), node(RTop,node(LTop,LL,LR),MergedHeap)) :- 
          LTop =< RTop, heap_merge(RL,RR,MergedHeap).  
heap_merge(node(LTop,LL,LR),node(RTop,RL,RR), node(LTop,MergedHeap,node(RTop,RL,RR))) :- 
          LTop > RTop, heap_merge(LL,LR,MergedHeap).  

% ===============================================================
% Catamorphisms.
% size of a binary-tree.
:- pred btreesize(bintree_int,int).
:- mode btreesize(in,out).
:- cata btreesize/2-1.

btreesize(leaf,S) :- S=1.
btreesize(node(T,L,R),S) :- S=SL+SR+1, btreesize(L,SL), btreesize(R,SR).

% --- size of a list
:- pred listsize(list(int),int).
:- mode listsize(in,out).
:- cata listsize/2-1.

listsize([],Res) :- Res=1.
listsize([H|T],Res) :-
    Res=ResT+1,
    listsize(T,ResT).

% ===============================================================
% ff1. Contract on heapsort: listsize         
:- pred ff1.        
ff1 :-
   constr(~(N1=N2)),
   listsize(L,N1), listsize(S,N2),
   heapsort(L,S).
              
:- spec list_to_heap(L,H) ==>
        listsize(L,NL), btreesize(H,NH) => constr(true).
               
:- spec heap_to_list(H,L) ==> 
        btreesize(H,SH), listsize(L,LL) => constr(true).
                 
:- spec insert_heap(X,H,H1) ==>
    btreesize(H,SH), btreesize(H1,SH1) => constr(true).
                         
:- spec heap_merge(HL,HR,H1) ==>
   btreesize(HL,SHL), btreesize(HR,SHR), btreesize(H1,SH1) => constr(true).

:- spec mkbtree(T,HL,HR,H1) ==> 
   btreesize(HL,SHL), btreesize(HR,SHR), btreesize(H1,SH1) => constr(true).

% ===============================================================
:- query ff1/0.
% ===============================================================