% =================================================================
:- 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([],[158]). % error: this clause should be removed
perm([X|Xs],[H|Perm]):-
      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. The element X need not be the minimum.
  constr(~(X=Y)),                    % delete(X,L,LD) succeeds **IFF** X is a member of L.
  delete(Y,T,TD).  % error: X should be Y

:- pred cons(int,list(int),list(int)).
:- mode cons(in,in,out).
cons(H,T,[H|T]).
% ====================================================
% catamorphisms
:- pred listMin(list(int),bool,int).
:- mode listMin(in,out,out).
:- cata listMin/3-1.

listMin([],IsDef,A) :- constr( ((~IsDef) & A=0) ).
listMin([H|T],IsDef,Min) :-
         constr( (IsDef & ( Min = ite(IsDefT, ite(H<MinT,H,MinT), H )) ) ),
         listMin(T,IsDefT,MinT).

% ====================================================
% verification.
% Property:
:- pred ff1.
ff1 :-                              % perm
   constr( ~( (IsDefMinL=IsDefMinLP) &  (IsDefMinL => (MinL=MinLP))  )),
   listMin(L,IsDefMinL,MinL), listMin(LP,IsDefMinLP,MinLP),
   perm(L,LP).

:- pred ff2.             % delete(X,L,LD) succeeds iff X is a member of L
ff2 :-                              % delete
   constr( ~( IsDefMinL & (((X=MinL) & IsDefMinLD) => (MinLD >= MinL)) ) ),
  % cons(H,T,L), 
   listMin(L,IsDefMinL,MinL), listMin(LD,IsDefMinLD,MinLD),
   delete(X,L,LD).

:- pred ff3.
ff3 :-                              % cons
  constr( ~( IsDefMinHT &
            (IsDefMinT => (MinHT=ite(H<MinT,H,MinT ) )) ) ),
  listMin(T,IsDefMinT,MinT), listMin(HT,IsDefMinHT,MinHT),
  cons(H,T,HT).
% =================================================================
:- query ff1/0.
:- query ff2/0.
:- query ff3/0.
% =================================================================