#This Sage code verifies the expansion of the polynomial c(x1,x2,x3,x4,x5,x6,x7)
#as a sum of 218 terms, each of which is a product of symmetric polynomials.
#It uses fairly minimalist Sage functionality: polynomial rings over GF(2).

R.<x1,x2,x3,x4,x5,x6,x7>=PolynomialRing(GF(2))
varlist=[x1,x2,x3,x4,x5,x6,x7]

def productofpairsumsexceptone(L,i,j):
    prod=R(1)
    for u in range(len(L)):
        for v in range(u+1,len(L)):
            if((u!=i) or (v!=j)):
                prod*=(L[u]+L[v])
    return prod
def cpoly(L):
    total=R(0)
    for i in range(len(L)):
        for j in range(i+1,len(L)):
            total+=(L[i]*L[j]*((productofpairsumsexceptone(L,i,j))**2))
    return total

#This gives c(x1,x2,x3,x4,x5,x6,x7) according to the definition of the paper.
#It avoids the use of non-polynomial rational functions.
cfromdef=cpoly(varlist)

def symfunc(L,n):
    if(0==n):
        return R(1)
    if(0==len(L)):
        return R(0)
    Lshortened=L[:-1]
    return symfunc(Lshortened,n)+L[-1]*symfunc(Lshortened,n-1)
def productofsymfuncs(L,indices):
    prod=R(1)
    for j in range(len(indices)):
        prod*=(symfunc(L,j+1)**indices[j])
    return prod

def LambdaList(index):
    if(1==index):
        return [2,2,2,2,3,0,1]
    if(2==index):
        return [2,2,2,3,0,3,0]
    if(3==index):
        return [2,2,2,3,1,1,1]
    if(4==index):
        return [2,2,2,4,0,0,2]
    if(5==index):
        return [2,2,3,0,3,2,0]
    if(6==index):
        return [2,2,3,1,1,3,0]
    if(7==index):
        return [2,2,4,0,0,4,0]
    if(8==index):
        return [2,3,0,3,2,2,0]
    if(9==index):
        return [2,3,0,5,0,0,2]
    if(10==index):
        return [2,3,1,1,3,2,0]
    if(11==index):
        return [2,3,1,3,1,0,2]
    if(12==index):
        return [2,3,2,1,1,2,1]
    if(13==index):
        return [2,4,0,0,4,2,0]
    if(14==index):
        return [2,4,0,0,5,0,1]
    if(15==index):
        return [2,4,0,1,3,1,1]
    if(16==index):
        return [2,4,0,3,0,1,2]
    if(17==index):
        return [2,4,1,0,3,0,2]
    if(18==index):
        return [2,4,1,1,1,1,2]
    if(19==index):
        return [2,5,0,1,1,0,3]
    if(20==index):
        return [2,6,0,0,0,0,4]
    if(21==index):
        return [3,0,3,2,2,2,0]
    if(22==index):
        return [3,0,3,4,0,0,2]
    if(23==index):
        return [3,0,5,0,0,4,0]
    if(24==index):
        return [3,1,1,3,2,2,0]
    if(25==index):
        return [3,1,1,5,0,0,2]
    if(26==index):
        return [3,1,3,1,0,4,0]
    if(27==index):
        return [3,2,1,1,2,3,0]
    if(28==index):
        return [3,2,1,3,0,1,2]
    if(29==index):
        return [3,2,2,0,2,2,1]
    if(30==index):
        return [3,2,2,1,0,3,1]
    if(31==index):
        return [3,3,0,1,2,2,1]
    if(32==index):
        return [3,3,0,3,0,0,3]
    if(33==index):
        return [3,3,1,1,0,2,2]
    if(34==index):
        return [3,4,0,0,2,0,3]
    if(35==index):
        return [3,4,0,1,0,1,3]
    if(36==index):
        return [3,4,1,0,0,0,4]
    if(37==index):
        return [4,0,0,4,2,2,0]
    if(38==index):
        return [4,0,0,4,3,0,1]
    if(39==index):
        return [4,0,0,5,0,3,0]
    if(40==index):
        return [4,0,0,5,1,1,1]
    if(41==index):
        return [4,0,1,3,1,3,0]
    if(42==index):
        return [4,0,1,4,1,0,2]
    if(43==index):
        return [4,0,3,0,1,4,0]
    if(44==index):
        return [4,1,0,3,0,4,0]
    if(45==index):
        return [4,1,0,3,1,2,1]
    if(46==index):
        return [4,1,1,1,1,4,0]
    if(47==index):
        return [4,2,0,1,1,3,1]
    if(48==index):
        return [4,2,0,2,1,0,3]
    if(49==index):
        return [4,2,1,0,1,2,2]
    if(50==index):
        return [5,0,0,3,0,3,1]
    if(51==index):
        return [5,0,0,4,0,0,3]
    if(52==index):
        return [5,0,1,1,0,5,0]
    if(53==index):
        return [5,0,1,2,0,2,2]
    if(54==index):
        return [5,0,2,0,0,4,1]
    if(55==index):
        return [5,1,0,1,0,4,1]
    if(56==index):
        return [5,2,0,0,0,2,3]
    if(57==index):
        return [6,0,0,0,0,6,0]
    if(58==index):
        return [6,0,0,0,1,4,1]
    if(59==index):
        return [0,3,2,2,2,2,0]
    if(60==index):
        return [0,3,2,4,0,0,2]
    if(61==index):
        return [0,3,4,0,0,4,0]
    if(62==index):
        return [0,5,0,0,4,2,0]
    if(63==index):
        return [0,5,0,2,2,0,2]
    if(64==index):
        return [0,7,0,0,0,0,4]
    if(65==index):
        return [1,1,3,2,2,2,0]
    if(66==index):
        return [1,1,3,4,0,0,2]
    if(67==index):
        return [1,1,5,0,0,4,0]
    if(68==index):
        return [1,3,1,0,4,2,0]
    if(69==index):
        return [1,3,1,2,2,0,2]
    if(70==index):
        return [1,5,1,0,0,0,4]
    if(71==index):
        return [2,1,1,2,3,2,0]
    if(72==index):
        return [2,1,1,4,1,0,2]
    if(73==index):
        return [2,1,3,0,1,4,0]
    if(74==index):
        return [2,2,0,2,2,3,0]
    if(75==index):
        return [2,2,0,4,0,1,2]
    if(76==index):
        return [2,2,1,0,3,3,0]
    if(77==index):
        return [2,2,1,2,1,1,2]
    if(78==index):
        return [2,2,2,0,1,3,1]
    if(79==index):
        return [2,3,0,0,3,2,1]
    if(80==index):
        return [2,3,0,2,1,0,3]
    if(81==index):
        return [2,3,1,0,1,2,2]
    if(82==index):
        return [2,4,0,0,1,1,3]
    if(83==index):
        return [3,0,1,2,2,3,0]
    if(84==index):
        return [3,0,1,4,0,1,2]
    if(85==index):
        return [3,0,3,0,0,5,0]
    if(86==index):
        return [3,1,0,2,2,2,1]
    if(87==index):
        return [3,1,0,4,0,0,3]
    if(88==index):
        return [3,1,1,0,2,4,0]
    if(89==index):
        return [3,1,1,2,0,2,2]
    if(90==index):
        return [3,1,2,0,0,4,1]
    if(91==index):
        return [3,2,0,0,2,3,1]
    if(92==index):
        return [3,2,0,2,0,1,3]
    if(93==index):
        return [3,2,1,0,0,3,2]
    if(94==index):
        return [3,3,0,0,0,2,3]
    if(95==index):
        return [4,0,0,2,0,5,0]
    if(96==index):
        return [4,0,0,2,1,3,1]
    if(97==index):
        return [4,0,1,0,1,5,0]
    if(98==index):
        return [4,1,0,0,0,6,0]
    if(99==index):
        return [4,1,0,0,1,4,1]
    if(100==index):
        return [5,0,0,0,0,5,1]
    if(101==index):
        return [0,0,4,2,2,2,0]
    if(102==index):
        return [0,0,4,2,3,0,1]
    if(103==index):
        return [0,0,4,3,0,3,0]
    if(104==index):
        return [0,0,4,3,1,1,1]
    if(105==index):
        return [0,0,5,0,3,2,0]
    if(106==index):
        return [0,0,5,1,1,3,0]
    if(107==index):
        return [0,1,3,1,3,2,0]
    if(108==index):
        return [0,1,3,3,1,0,2]
    if(109==index):
        return [0,1,4,1,0,4,0]
    if(110==index):
        return [0,1,4,1,1,2,1]
    if(111==index):
        return [0,2,2,0,5,0,1]
    if(112==index):
        return [0,2,2,1,3,1,1]
    if(113==index):
        return [0,2,2,2,2,0,2]
    if(114==index):
        return [0,2,2,3,0,1,2]
    if(115==index):
        return [0,2,3,0,3,0,2]
    if(116==index):
        return [0,2,3,1,1,1,2]
    if(117==index):
        return [0,3,0,1,4,2,0]
    if(118==index):
        return [0,3,0,3,2,0,2]
    if(119==index):
        return [0,3,2,1,1,0,3]
    if(120==index):
        return [0,5,0,1,0,0,4]
    if(121==index):
        return [1,0,3,0,4,2,0]
    if(122==index):
        return [1,0,3,1,2,3,0]
    if(123==index):
        return [1,0,3,2,2,0,2]
    if(124==index):
        return [1,0,3,3,0,1,2]
    if(125==index):
        return [1,0,4,0,2,2,1]
    if(126==index):
        return [1,0,4,1,0,3,1]
    if(127==index):
        return [1,1,1,1,4,2,0]
    if(128==index):
        return [1,1,1,3,2,0,2]
    if(129==index):
        return [1,1,2,1,2,2,1]
    if(130==index):
        return [1,1,2,3,0,0,3]
    if(131==index):
        return [1,1,3,1,0,2,2]
    if(132==index):
        return [1,2,2,0,2,0,3]
    if(133==index):
        return [1,2,2,1,0,1,3]
    if(134==index):
        return [1,3,1,1,0,0,4]
    if(135==index):
        return [2,0,1,1,3,3,0]
    if(136==index):
        return [2,0,1,3,1,1,2]
    if(137==index):
        return [2,0,2,1,0,5,0]
    if(138==index):
        return [2,0,2,1,1,3,1]
    if(139==index):
        return [2,0,2,2,0,2,2]
    if(140==index):
        return [2,0,2,2,1,0,3]
    if(141==index):
        return [2,0,3,0,1,2,2]
    if(142==index):
        return [2,1,0,1,2,4,0]
    if(143==index):
        return [2,1,0,1,3,2,1]
    if(144==index):
        return [2,1,0,3,0,2,2]
    if(145==index):
        return [2,1,0,3,1,0,3]
    if(146==index):
        return [2,2,0,0,3,0,3]
    if(147==index):
        return [2,2,0,1,0,3,2]
    if(148==index):
        return [2,2,0,1,1,1,3]
    if(149==index):
        return [2,2,0,2,0,0,4]
    if(150==index):
        return [2,2,1,0,1,0,4]
    if(151==index):
        return [3,0,0,1,2,3,1]
    if(152==index):
        return [3,0,0,3,0,1,3]
    if(153==index):
        return [3,0,1,0,2,2,2]
    if(154==index):
        return [3,0,1,2,0,0,4]
    if(155==index):
        return [3,0,2,0,0,2,3]
    if(156==index):
        return [3,2,0,0,0,0,5]
    if(157==index):
        return [0,0,3,0,3,3,0]
    if(158==index):
        return [0,0,3,2,1,1,2]
    if(159==index):
        return [0,0,4,0,0,5,0]
    if(160==index):
        return [0,0,4,0,1,3,1]
    if(161==index):
        return [0,1,1,0,5,2,0]
    if(162==index):
        return [0,1,1,2,3,0,2]
    if(163==index):
        return [0,1,2,0,2,4,0]
    if(164==index):
        return [0,1,2,0,3,2,1]
    if(165==index):
        return [0,1,2,2,0,2,2]
    if(166==index):
        return [0,1,2,2,1,0,3]
    if(167==index):
        return [0,1,3,0,1,2,2]
    if(168==index):
        return [0,2,0,0,4,3,0]
    if(169==index):
        return [0,2,0,2,2,1,2]
    if(170==index):
        return [0,2,2,0,1,1,3]
    if(171==index):
        return [0,3,0,0,2,2,2]
    if(172==index):
        return [0,3,0,2,0,0,4]
    if(173==index):
        return [0,3,1,0,1,0,4]
    if(174==index):
        return [0,4,0,0,0,1,4]
    if(175==index):
        return [1,0,1,0,4,3,0]
    if(176==index):
        return [1,0,1,2,2,1,2]
    if(177==index):
        return [1,0,2,0,2,3,1]
    if(178==index):
        return [1,0,2,2,0,1,3]
    if(179==index):
        return [1,0,3,0,0,3,2]
    if(180==index):
        return [1,1,0,0,4,2,1]
    if(181==index):
        return [1,1,0,2,2,0,3]
    if(182==index):
        return [1,1,1,0,2,2,2]
    if(183==index):
        return [1,1,1,2,0,0,4]
    if(184==index):
        return [1,1,2,0,0,2,3]
    if(185==index):
        return [1,2,1,0,0,1,4]
    if(186==index):
        return [1,3,0,0,0,0,5]
    if(187==index):
        return [2,0,0,0,2,5,0]
    if(188==index):
        return [2,0,0,0,3,3,1]
    if(189==index):
        return [2,0,0,2,0,3,2]
    if(190==index):
        return [2,0,0,2,1,1,3]
    if(191==index):
        return [0,0,0,0,6,2,0]
    if(192==index):
        return [0,0,0,0,7,0,1]
    if(193==index):
        return [0,0,0,1,4,3,0]
    if(194==index):
        return [0,0,0,1,5,1,1]
    if(195==index):
        return [0,0,1,0,5,0,2]
    if(196==index):
        return [0,0,1,1,3,1,2]
    if(197==index):
        return [0,0,2,0,2,2,2]
    if(198==index):
        return [0,0,2,0,3,0,3]
    if(199==index):
        return [0,0,2,1,0,3,2]
    if(200==index):
        return [0,0,2,1,1,1,3]
    if(201==index):
        return [0,1,0,1,2,2,2]
    if(202==index):
        return [0,1,0,1,3,0,3]
    if(203==index):
        return [0,1,1,1,1,0,4]
    if(204==index):
        return [0,2,0,0,2,0,4]
    if(205==index):
        return [0,2,0,1,0,1,4]
    if(206==index):
        return [1,0,0,0,4,0,3]
    if(207==index):
        return [1,0,0,1,2,1,3]
    if(208==index):
        return [1,0,1,0,2,0,4]
    if(209==index):
        return [1,0,1,1,0,1,4]
    if(210==index):
        return [1,1,0,1,0,0,5]
    if(211==index):
        return [2,0,0,0,0,2,4]
    if(212==index):
        return [2,0,0,0,1,0,5]
    if(213==index):
        return [0,0,0,0,2,3,2]
    if(214==index):
        return [0,0,0,0,3,1,3]
    if(215==index):
        return [0,0,1,0,1,1,4]
    if(216==index):
        return [0,1,0,0,0,2,4]
    if(217==index):
        return [0,1,0,0,1,0,5]
    if(218==index):
        return [1,0,0,0,0,1,5]

#This gives our proposed expression for c(x1,x2,x3,x4,x5,x6,x7) as a sum of 218 products of symmetric functions.
cfromsymfuncs=sum(productofsymfuncs(varlist,LambdaList(j)) for j in range(1,219))

#This verifies that our proposed expression for c(x1,x2,x3,x4,x5,x6,x7) matches the original definition.
print "The difference between the original definition and the expansion is:", cfromdef-cfromsymfuncs

#This verifies that each of the 218 terms has at least one factor of the 1st, 2nd, 5th, or 6th symmetric function.
for j in range(1,219):
    verified=True
    index=LambdaList(j)
    if((index[0]+index[1]+index[4]+index[5])<=0):
        print "Term ", j, "has no factor that is a 1st, 2nd, 5th, or 6th symmetric function"
        verified=False
print "It is", verified, "that each of the terms has at least one factor of the 1st, 2nd, 5th, or 6th symmetric function."