% =================================================================
:- 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 in the given list [X|Xs].
	quicksort(Bigs, Bs),              % X is NOT in Xs.
  cons(X,Bs,XBs),
	append(Ls, XBs, ResL).

% partition(4,[2,7,4,1],L,B) ==> L(ittles)=[2,1], B(igs)=[7,4] 
:- 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], Bs, [Y | Bs]) :-   % error Bs --> Ls
				constr( (X=<Y)  ),   
  			partition(X, Xs, Ls, Bs).
  			
:- pred append(list(int),list(int),list(int)).
:- mode append(in,in,out).
append([],Y,Y).
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]).

% =================================================================
% catamorphisms

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

listMax([],IsDef,A) :- constr( ((~IsDef) & (A=0)) ).   % 0 can be any int.
listMax([H|T],IsDef,Max) :-
         constr( IsDef & (Max = ite(IsDefT, ite(H>MaxT,H,MaxT), H )) ),
         listMax(T,IsDefT,MaxT).

:- pred lastElem(list(int),bool,int).
:- mode lastElem(in,out,out).
:- cata lastElem/3-1.
 
lastElem([],IsDef,Last) :- constr( ((~IsDef) & (Last=0)) ). % 0 can be any int.
lastElem([H|T],IsDef,Last) :-
         constr( IsDef & (Last=ite(IsDefT,LastT,H)) ),
         lastElem(T,IsDefT,LastT).

% =================================================================
% Verification.
% Property:
:- pred ff1.
ff1 :-         % listMax of unsorted equal lastElem of ascending, quicksorted list 
   constr( ~((    (IsDefLMax = IsDefSMax) & 
                  (IsDefLMax = IsDefSLast)) & 
                  (IsDefLMax => ((MaxL=LastS) & (MaxL=MaxS))) )),
   listMax(L,IsDefLMax,MaxL), listMax(S,IsDefSMax,MaxS), 
   lastElem(S,IsDefSLast,LastS),
   quicksort(L,S).

:- pred ff2.
ff2 :-         % listMax and lastElem on partition
  constr( ~(( ((((  IsDefS &  IsDefB )  => (  IsDefL & ( MaxLs=Max ) )) &
							 ((  IsDefS & ~IsDefB )  => (  IsDefL & ( MaxLs=MSmalls ) ))) &
							 (( ~IsDefS &  IsDefB )  => (  IsDefL & ( MaxLs=MBigs ) ))) &
							 (( ~IsDefS & ~IsDefB )  =  ( ~IsDefL ))
						 )
          &  (  (((~IsDefLSmalls &  IsDefLBigs ) => (  IsDefLastLs & ( LastLs=LastBigs ) )) &
							 ((  IsDefLSmalls & ~IsDefLBigs ) => (  IsDefLastLs & ( LastLs=LastSmalls ) ))) &
							 (( ~IsDefLSmalls & ~IsDefLBigs ) =  ( ~IsDefLastLs ))
							 ) )),
  listMax(Smalls,IsDefS,MSmalls), listMax(Bigs,IsDefB,MBigs),          % for listMax
  listMax(Ls,IsDefL,MaxLs),                                            % for listMax
  constr( Max = ite(MBigs>MSmalls,MBigs,MSmalls) ),                    % for listMax
  lastElem(Smalls,IsDefLSmalls,LastSmalls), lastElem(Bigs,IsDefLBigs,LastBigs),  % for lastElem
	 lastElem(Ls,IsDefLastLs,LastLs),
  partition(P,Ls,Smalls,Bigs).



:- pred ff3.
ff3 :-       % listMax and lastElem on append
   constr( ~(( ((((   IsDefXs &  IsDefYs )  => (  IsDefZs & ( MaxZs=Max ) )) &
                ((    IsDefXs & ~IsDefYs )  => (  IsDefZs & ( MaxZs=MaxXs ) ))) &
                ((   ~IsDefXs &  IsDefYs )  => (  IsDefZs & ( MaxZs=MaxYs ) ))) &
                ((   ~IsDefXs & ~IsDefYs )  =  ( ~IsDefZs ))
               )
            &  ( ((((( IsDefYs_LE )                => (  IsDefZs_LE & ( LastZs=LastYs ) )) &
                 ((    IsDefXs_LE & ~IsDefYs_LE )  => (  IsDefZs_LE & ( LastZs=LastXs ) )))) &
                 ((   ~IsDefXs_LE & ~IsDefYs_LE )  =  ( ~IsDefZs_LE )))
              ))
         ),
   listMax(Xs,IsDefXs,MaxXs), listMax(Ys,IsDefYs,MaxYs),
   listMax(Zs,IsDefZs,MaxZs), 
   constr( Max = ite(MaxXs>MaxYs,MaxXs,MaxYs) ), 
   lastElem(Xs,IsDefXs_LE,LastXs), lastElem(Ys,IsDefYs_LE,LastYs),
   lastElem(Zs,IsDefZs_LE,LastZs),   
   append(Xs,Ys,Zs).

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