% =================================================================
:- pred constr(bool).
:- mode constr(in).
:- ignore constr/1.
% ====================================================
% Program.
:- pred perm(list(int),list(int)).
:- mode perm(in,in).               % in,in: perm is not functional
perm([],[]).
perm([X|Xs],[H|Perm]):-            % a permutation
      cons(X,Xs,XXs),
      delete(H,XXs,Rest),
      perm(Rest,Perm).

:- pred delete(int,list(int),list(int)).
:- mode delete(in,in,in).          % delete is not total
delete(X,[Y|T],T) :- constr(X=Y).  % Taking X away from [Y|T]. Deleting exactly one copy of X.
delete(X,[Y|T],[Y|TD]) :-          % The list need not be ordered. Element X need not be the minimum.
  constr(~(X=Y)),                  % delete(X,L,LD) succeeds **IFF** X is a member of L.
  delete(X,TD,T).  % error: delete(X,TD,T) ---> delete(X,T,TD).

:- pred cons(int,list(int),list(int)).
:- mode cons(in,in,out).
cons(H,T,[H|T]).
% ====================================================
% catamorphisms       
:- pred listmember(int,list(int),bool).
:- mode listmember(in,in,out).
:- cata listmember/3-2.

listmember(X,[],Res) :- constr(~Res).
listmember(X,[Y|Ys],B) :-
      constr( B=ite(X=Y,true,B1) ) ,
      listmember(X,Ys,B1).
  
% ====================================================
% verification.
% Property:
:- pred ff1.
ff1 :- 
    constr( ~(LS=LPS)),                             
    listmember(X,L,LS),  
    listmember(X,LP,LPS),
    perm(L,LP).
   
% contracts (postcondition: true)

:- spec delete(H,L,LD) ==> listmember(X,L,SL), listmember(X,LD,SLD).

:- spec cons(H,T,HT) ==> listmember(X,T,ST), listmember(X,HT,SHT).

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