% ================================================================= 
:- pred constr(bool). 
:- mode constr(in).
:- ignore constr/1.
% ================================================================= 
% Program. ascending sorting  (ascending because we take the minimum)  
:- pred selectionsort(list(int),list(int)).
:- mode selectionsort(in,out).
selectionsort([],[]).
selectionsort([X],[Y]). % error: erase clause
selectionsort([H|T],[Min|T1]) :-
 cons(H,T,HT),        
 listMin(HT,IsDef,Min),              % HT is not empty. IsDef is true.     
 delete(Min,HT,L1),                  % Taking the minimum away from list HT.
 selectionsort(L1,T1).               

:- pred delete(int,list(int),list(int)).
:- mode delete(in,in,in).
%delete(X,[],[]).                    % delete is not total
delete(X,[Y|T],T) :- constr(X=Y).    % taking X away from [Y|T]. 
delete(X,[Y|T],[Y|TD]) :-            % Deleting exactly one copy of X only. 
  constr(~(X=Y)),                    % The list is not ordered. 
  delete(X,T,TD).                    % The element X need not be the minimum.
  
:- pred listMin(list(int),bool,int).     
:- mode listMin(in,out,out).

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

:- pred cons(int,list(int),list(int)).
:- mode cons(in,in,out).             
cons(H,T,[H|T]).
% ====================================================
% catamorphisms       
% --- min of a list
:- 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 :- 
    constr( ~((IsDefLMin = IsDefSMin) & 
    ((IsDefLMin & IsDefSMin) => (LMin=SMin)) )), 
    listmin(L,IsDefLMin,LMin), listmin(S,IsDefSMin,SMin),
    selectionsort(L,S).

% contracts (postcondition: true)

:- spec listMin(HT,IsDef,Min) ==> listmin(HT,B1,SHT).

:- spec delete(X,YT,T) ==> listmin(YT,B1,SYT) , listmin(T,B2,ST).

:- spec cons(H,T,HT) ==> listmin(T,B1,ST), listmin(HT,B2,SHT).

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