off echo, nat$
out "rcontrolrootsE6.r"$

%1) positive roots of E6.
dim:=36$

x(1):=(e1-e2-e3-e4-e5+sqrt(3)*e6)/2$
x(2):=e1+e2$
x(3):=e2-e1$
x(4):=e3-e2$
x(5):=e4-e3$
x(6):=e5-e4$

x(7):=e1+e3$
x(8):=e1+e4$
x(9):=e1+e5$
x(10):=e2+e3$
x(11):=e2+e4$
x(12):=e2+e5$
x(13):=e3+e4$
x(14):=e3+e5$
x(15):=e4+e5$

x(16):=e3-e1$
x(17):=e4-e1$
x(18):=e5-e1$

x(19):=e4-e2$
x(20):=e5-e2$
x(21):=e5-e3$

x(22):=(e1+e2+e3+e4+e5+sqrt(3)*e6)/2$    % + + + + +
x(23):=(-e1-e2+e3+e4+e5+sqrt(3)*e6)/2$   % - - + + +
x(24):=(-e1+e2-e3+e4+e5+sqrt(3)*e6)/2$   % - + - + +
x(25):=(-e1+e2+e3-e4+e5+sqrt(3)*e6)/2$   % - + + - +
x(26):=(-e1+e2+e3+e4-e5+sqrt(3)*e6)/2$   % - + + + -
x(27):=(+e1-e2-e3+e4+e5+sqrt(3)*e6)/2$   % + - - + +
x(28):=(+e1-e2+e3-e4+e5+sqrt(3)*e6)/2$   % + - + - +
x(29):=(+e1-e2+e3+e4-e5+sqrt(3)*e6)/2$   % + - + + -
x(30):=(+e1+e2-e3-e4+e5+sqrt(3)*e6)/2$   % + + - - +
x(31):=(+e1+e2-e3+e4-e5+sqrt(3)*e6)/2$   % + + - + -
x(32):=(+e1+e2+e3-e4-e5+sqrt(3)*e6)/2$   % + + + - -
x(33):=(-e1+e2-e3-e4-e5+sqrt(3)*e6)/2$   % - + - - -
x(34):=(-e1-e2+e3-e4-e5+sqrt(3)*e6)/2$   % - - + - -
x(35):=(-e1-e2-e3+e4-e5+sqrt(3)*e6)/2$   % - - - + -
x(36):=(-e1-e2-e3-e4+e5+sqrt(3)*e6)/2$   % - - - - +



%2) sum_of_positive_roots_list.
write "The following list collects {i,x(i)} where x(i) is the i-th positive root."$
laliste:=for i:=1:DIM join {{i,x(i)}};
lalisteprime :=for i:=1:DIM collect x(i);


write "The following list sum_of_positive_roots_list collects"$
write "{{i,j},k} such that x(i)+x(j) =x(k)."$


on time$
sum_of_positive_roots_list:=
for i:=1:DIM-1 join for j:=i+1:DIM join
 if x(i)+x(j) member lalisteprime then
           for k:=1:DIM join    if x(k)=x(i)+x(j) then {{{i,j},k}}
                                else {}
 else {}$


write "sum_of_positive_roots_list:=",sum_of_positive_roots_list$



%3) verification of Property (P).
listesverif:= for i:=1:DIM-1 join for j:=i+1:DIM join
                                        for k:=i+1:DIM join
    if   x(i)+x(j) member lalisteprime
    and  x(i)+x(k) member lalisteprime
    and  x(j)+x(k) member lalisteprime
 then

           for l:=1:DIM join
                   if x(l)=x(j)+x(k) then {{{i,j,k},x(j)+x(k), l}}
                   else {}
 else {}
 $




write "listesverif:=",ws$
 IF listesverif={} then  DO <<write "Property (P) OK">>
 ELSE  DO <<write "Property (P) not verified for",listesverif >>$

bye$