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

% ====================================================
% Program. mergesorting in ascending order.E.g., 3 =< 5 =< 5 =< 8 =< ...
:- pred mergesort(list(int),list(int)).
:- mode mergesort(in,out).
mergesort([], []).
mergesort([H], [H]).
mergesort([H,K|T],S) :-
   cons(K,T,KT),          
   cons(H,KT,HKT),        
   split(HKT,L1,L2),       
   mergesort(L1,S1),        
   mergesort(L2,S2),
   merge(S1,S2,S).        

:- pred cons(int,list(int),list(int)).
:- mode cons(in,in,out).
cons(H,T,[H|T]).           

:- pred split(list(int),list(int),list(int)).
:- mode split(in,out,out).
split(L,L1,L2) :- N1=(N div 2),      % integer division is div in SMTLIB.
     length(L,N),                    
     takedrop(N1,L,L1,L2).

:- pred length(list(int),int).
:- mode length(in,out).
length([],L) :- constr(L=0).
length([H|T],L) :-
  constr( L=(LT+1) ),
  length(T,LT).

:- pred takedrop(int,list(int),list(int),list(int)).
:- mode takedrop(in,in,out,out).
takedrop(N,[],[],[]).
takedrop(N,L,[],L) :- N=<0.
takedrop(N,[H|T],[H|T1],L) :- 
  N>=1, N1=N-1,         
  takedrop(N1,T,T1,L).  

:- pred merge(list(int),list(int),list(int)).
:- mode merge(in,in,out).
merge([],L,L).                   
merge([X|Xs],[],[X|Xs]).
merge([X|Xs],[Y|Ys],[X|Zs]) :-
     constr(X=<Y), cons(Y,Ys,YYs),
     merge(Xs,YYs,Zs).
merge([X|Xs],[Y|Ys],[Y|Zs]) :-
     constr(X>Y), cons(X,Xs,XXs),
     merge(XXs,Ys,Zs).

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

listMax([],IsDef,A) :- constr( ((~IsDef) & (A=0)) ).
listMax([H|T],IsDef,Max) :-
         constr( IsDef & (Max = ite(IsDefT, ite(H>MaxT,H,MaxT), H )) ),
         listMax(T,IsDefT,MaxT).
         
:- pred lengthcata(list(int),int).
:- mode lengthcata(in,out).
:- cata lengthcata/2-1.

lengthcata([],L) :- constr(L=0).
lengthcata([H|T],L) :-
  constr( L=(LT+1) ),
  lengthcata(T,LT).

% ================================================================= 
% verification. 
% Property:
:- pred ff1.
ff1 :-
   constr(~( ((IsDefMaxL = IsDefMaxS) & (IsDefMaxS => (MaxL=MaxS))) & 
     ((~IsDefMaxS) => true) )),
   listMax(L,IsDefMaxL,MaxL), listMax(S,IsDefMaxS,MaxS),
   mergesort(L,S).


:- pred ff2.
ff2 :- 
  constr( ~((((( IsDefYs &  IsDefZs )  => (  IsDefXs & ( MXs=ite(MYs>MZs,MYs,MZs) ) )) &
           ((    IsDefYs & ~IsDefZs )  => (  IsDefXs & ( MXs=MYs ) ))) &
           ((   ~IsDefYs &  IsDefZs )  => (  IsDefXs & ( MXs=MZs ) ))) &
           ((   ~IsDefYs & ~IsDefZs )  =  ( ~IsDefXs ))
         )),
   listMax(Xs,IsDefXs,MXs), listMax(Ys,IsDefYs,MYs), listMax(Zs,IsDefZs,MZs),
   split(Xs,Ys,Zs).

:- pred ff3.
ff3 :- 
  constr( ~((((( IsDefYs &  IsDefZs )  => (  IsDefXs & ( MXs=ite(MYs>MZs,MYs,MZs) ) )) &
           ((    IsDefYs & ~IsDefZs )  => (  IsDefXs & ( MXs=MYs ) ))) &
           ((   ~IsDefYs &  IsDefZs )  => (  IsDefXs & ( MXs=MZs ) ))) &
           ((   ~IsDefYs & ~IsDefZs )  =  ( ~IsDefXs )) 
         )),
   listMax(Xs,IsDefXs,MXs), listMax(Ys,IsDefYs,MYs), listMax(Zs,IsDefZs,MZs),
   constr( N1=(N div 2) ), 
   lengthcata(Xs,N), 
   takedrop(N1,Xs,Ys,Zs).

:- pred ff4.
ff4 :- 
   constr( ~(((((   IsDefXs &  IsDefYs ) => (  IsDefZs & ( MaxZs=Max ) )) &
                ((  IsDefXs & ~IsDefYs ) => (  IsDefZs & ( MaxZs=MaxXs ) ))) &
                (( ~IsDefXs &  IsDefYs ) => (  IsDefZs & ( MaxZs=MaxYs ) ))) &
                (( ~IsDefXs & ~IsDefYs ) =  ( ~IsDefZs & ( MaxZs=0 ) ))
              ) ),
   listMax(Xs,IsDefXs,MaxXs), listMax(Ys,IsDefYs,MaxYs), listMax(Zs,IsDefZs,MaxZs),
   constr(Max = ite(MaxXs>MaxYs,MaxXs,MaxYs)),
   merge(Xs,Ys,Zs).

:- pred ff5.
ff5 :- 
   constr(~( MaxHT = ite(IsDefMaxT,ite(H>MaxT,H,MaxT),H))),
   listMax(T,IsDefMaxT,MaxT), listMax(HT,IsDefMaxHT,MaxHT), 
   cons(H,T,HT).

% ================================================================= 
:- query ff1/0.
:- query ff2/0.
:- query ff3/0.
:- query ff4/0.
:- query ff5/0.
% =================================================================