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

% =================================================================
% Program. ascending sorting
:- pred insertionsort(list(int),list(int)).
:- mode insertionsort(in,out).

insertionsort([],[]).
insertionsort([X],[Y]). % error: Y --> X
insertionsort([X|Xs],S) :- ord_ins(X,TS,S),
         insertionsort(Xs,TS).

:- pred ord_ins(int,list(int),list(int)).   
:- mode ord_ins(in,in,out).

ord_ins(A,[],[A]).
ord_ins(A,[B|Xs],[A,B|Xs]) :- constr( A=<B ).
ord_ins(A,[B|Xs],[B|AXs]) :- constr( A>B ), 
     ord_ins(A,Xs,AXs).

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

%-----  listsum --- sum of integers in nodes of the given list.
:- pred listsum(list(int),int).
:- mode listsum(in,out).
:- cata listsum/2-1.

listsum([],S) :- S=0.
listsum([H|T],S) :-
  S=(H+ST),
  listsum(T,ST).
  
% =================================================================
% Verification. 
% Property:
:- pred ff1. 
ff1 :- 
     constr( ~(LS=SS)), 
     listsum(L,LS),  
     listsum(S,SS), 
     insertionsort(L,S).

% contracts (postcondition: true)

:- spec ord_ins(A,L1,L2) ==> listsum(L1,SL1) , listsum(L2,SL2).

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