% =================================================================
:- 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([Z],leaf). %error: [Z] ---> []
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.                   
%-----  treecount --- how many X's in the given binary tree.
:- pred treecount(int,bintree_int,int).
:- mode treecount(in,in,out).
:- cata treecount/3-2.

treecount(X,leaf,Res) :- constr( Res=0 ).
treecount(X,node(A,L,R),Res) :- 
    constr( Res=ite(X=A,ResL+ResR+1,ResL+ResR) ),
    treecount(X,L,ResL),
    treecount(X,R,ResR). 

%-----  listcount --- how many X's in the given list.
:- pred listcount(int,list(int),int).
:- mode listcount(in,in,out).
:- cata listcount/3-2.

listcount(X,[],N) :- N=0.
listcount(X,[H|T],N) :-
  constr( (N=ite(X=H,(NT+1),NT)) ),
  listcount(X,T,NT).

% ===============================================================
%ff1. Contract on heapsort        
:- pred ff1.
ff1 :- 
   constr(~(LB=SB)), 
   listcount(X,L,LB), listcount(X,S,SB),  
   heapsort(L,S).
  
%%% ---------------
%% Contract on list_to_heap:    (postcondition: true)            
%:- spec list_to_heap(L,H) ==>
%  listcount(X,L,NL), treecount(X,H,NH).
%  
%% ------------------
%% Contract on heap_to_list: (postcondition: true)
%:- spec heap_to_list(H,L) ==> 
%  treecount(X,H,SH), listcount(X,L,LL).
%  
%% ------------------ 
%% Contract on insert_heap:         (postcondition: true)       
%:- spec insert_heap(X,H,H1) ==>
%  treecount(Y,H,SH), treecount(Y,H1,SH1).
%  
%% ----------------------------------------------------------------
%% Contract on heap_merge:               (postcondition: true)         
%:- spec heap_merge(HL,HR,H1) ==>
%  treecount(X,HL,SHL), treecount(X,HR,SHR), treecount(X,H1,SH1).
%%----------------------------------------------------------------
%% Contract on mkbtree:         (postcondition: true)            
%:- spec mkbtree(T,HL,HR,H1) ==> 
%  treecount(X,HL,BHL), treecount(X,HR,BHR), treecount(X,H1,BH1).

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

%  Catamorphic abstraction

:- cata_abs list(int) ==> listcount(X,L,LB).
:- cata_abs bintree_int ==> treecount(X,HL,SHL).

%% =================================================================