% =================================================================
:- pred constr(bool).    
:- mode constr(in).
:- ignore constr/1.

% =================================================================
% Program
:- pred quicksort(list(int),list(int)).
:- mode quicksort(in,out).
quicksort([], []).
quicksort([X | Xs], ResL) :-
	partition(X, Xs, Smalls, Bigs),   % partition of the tail Xs 
	quicksort(Smalls, Ls),            % with respect to the pivot X.
	quicksort(Bigs, Bs),              % X is NOT in Xs.
  cons(X,Bs,XBs),
	append(Ls, XBs, ResL).

:- pred partition(int,list(int),list(int),list(int)).
:- mode partition(in,in,out,out).
partition(X, [], [], []).
partition(X, [Y | Ys], [Y | Ls], Bs) :- 
      	constr( (X>Y) ),    
      	partition(X, Ys, Ls, Bs).
partition(X, [Y | Xs], Ls, [Y | Bs]) :-  
				constr( (X=<Y)  ),   
  			partition(X, Xs, Ls, Bs).

:- pred append(list(int),list(int),list(int)).
:- mode append(in,in,out).
append([],Ys,Ys).
append([X|Xs],Ys,[X|Zs]) :- append(Xs,Ys,Zs).

:- pred cons(int,list(int),list(int)).
:- mode cons(in,in,out).
cons(H,T,[H|T]).

% =================================================================
% catamorphism

:- pred length(list(int),int).
:- mode length(in,out).
:- cata length/2-1.

length([],L) :- constr((L=0)).
length([H|T],L) :-
  constr( (L=LT+1) ),
  length(T,LT).

% =================================================================
% Verification. 
% Property:
:- pred ff1.
ff1 :- 
   constr(~(LL=LS)), 
   length(L,LL), length(S,LS),  
   quicksort(L,S).

:- pred ff2.
ff2 :- 
   constr( ~(LLs = (LSmalls+LBigs)) ), 
   length(Smalls,LSmalls), length(Bigs,LBigs), length(Ls,LLs), 
   partition(X,Ls,Smalls,Bigs).

:- pred ff3.
ff3 :- 
   constr( ~(LZs = (LXs+LYs)) ), 
   length(Xs,LXs), length(Ys,LYs), length(Zs,LZs), 
   append(Xs,Ys,Zs).

% =================================================================
:- query ff1/0.
:- query ff2/0.
:- query ff3/0.
% =================================================================