2014-11-20-112409.sagews

EX7.<P,Q,R,S,V,W> = BooleanPolynomialRing(6,order='lex')
load('ProbRecurse.py')
load('Prob.py')
import numpy
Prob(0)
Prob(1)
Prob(P*Q*R)                                        # P and Q and R
Prob(P*Q + P + Q)                                  # P or Q
Prob(P*Q + P + 1)                                  # if P then Q
Prob(P + Q)                                        # P xor Q
Prob(P + Q + R)                                    # P xor Q xor R
Prob(P + Q + R + S)                                # etc
Pr1 = (P + R + P*R)*(P + Q + P*Q)                  # premise 1 of exercise 7
Pr2 = 1 + Q*R + Q*R*(1 + V + V*W)                  # premise 2 of exercise 7
Pr3 = 1 + P + P*S + (1 +P + P*S)*(S + S*W)         # premise 3 of exercise 7
Pr4 = 1 + W                                        # not W -- premise 4 of exercise 7
Concl = 1 + V + V*S                                # conclusion to be proved of exercise 7
Pr1
Pr2
Pr3
Pr4
Concl
Prob(Pr1)
Prob(Pr2)
Prob(Pr3)
Prob(Pr4)
Prob(Concl)
P.parent()
P.args()
P.base_ring()
Idl = ideal(Pr1 + 1, Pr2 + 1, Pr3 + 1, Pr4 + 1, Concl + 0)     # assume premises true and conclusion false for indirect proof
proof = Idl.groebner_basis()
proof
ProbRing = Prob(P).parent()
ProbRing
Prob(P).args()
Prob(P).base_ring()
PRI = (Prob(P*Q*R + P + Q*R) - 7/10, Prob(Q*R*V*W + Q*R*V + 1) - 1/2, Prob(P*S + P + S*W + S + 1) - 1/3, Prob(W + 1) - 1/2)*ProbRing
PRI.groebner_basis()
f = Prob(P*Q*R + P + Q*R)
x = f.parent().gens()
f(1/2,1/2,1/2,1/2,1/2,1/2)
ff=RR(f(1/2,1/2,1/2,1/2,1/2,1/2))
-(ff*numpy.log2(ff)+(1-ff)*numpy.log2(1-ff))
pqrs=Prob(P + Q + R + S)
pqrs(1/2,1/2,1/2,1/2,0,0)
Pr1*Pr2*Pr3*Pr4*(Concl+1)                         # another indirect proof if 0
Pr1*Pr2*Pr3*Pr4
Prob(Pr1*Pr2*Pr3*Pr4)

0 1 p*q*r -p*q + p + q p*q - p + 1 -2*p*q + p + q 4*p*q*r - 2*p*q - 2*p*r + p - 2*q*r + q + r -8*p*q*r*s + 4*p*q*r + 4*p*q*s - 2*p*q + 4*p*r*s - 2*p*r - 2*p*s + p + 4*q*r*s - 2*q*r - 2*q*s + q - 2*r*s + r + s P*Q*R + P + Q*R Q*R*V*W + Q*R*V + 1 P*S + P + S*W + S + 1 W + 1 S*V + V + 1 -p*q*r + p + q*r q*r*v*w - q*r*v + 1 p*s - p + s*w - s + 1 -w + 1 s*v - v + 1 Boolean PolynomialRing in P, Q, R, S, V, W (P, Q, R, S, V, W) Finite Field of size 2 [1] Multivariate Polynomial Ring in p, q, r, s, v, w over Rational Field (p, q, r, s, v, w) Rational Field [p*q*r - p - q*r + 7/10, p*s - p - 1/2*s + 2/3, p*v - p - 7/10*v + 1, q*r*s - 2/3*q*r - 2/5*s + 1/15, q*r*v - 1, s*v - 5/2*s - 1/6*v + 5/3, w - 1/2] 5/8 0.954434002924965 1/2 0 P*Q*R*S*V*W + P*Q*R*S*V + P*Q*R*S*W + P*Q*R*S + P*Q*R*V*W + P*Q*R*V + P*Q*R*W + P*Q*R + Q*R*S*V*W + Q*R*S*V + Q*R*S*W + Q*R*S + Q*R*V*W + Q*R*V + Q*R*W + Q*R p*q*r*s*v*w - p*q*r*s*v - p*q*r*s*w + p*q*r*s - p*q*r*v*w + p*q*r*v + p*q*r*w - p*q*r - q*r*s*v*w + q*r*s*v + q*r*s*w - q*r*s + q*r*v*w - q*r*v - q*r*w + q*r

(Prob(P + Q + R + S))(1/2,1/2,1/2,1/2,0,0)

1/2

BPR = P.parent#BooleanPolynomialRing(objects^2,'X',order='degrevlex')
KDELTA = lambda N, M: BPR(N == M)
NOT = lambda A: BPR(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: BPR(1) + A + A*B
IFF = lambda A, B: BPR(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: BPR(1) + A*B
NOR = lambda A, B: BPR(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([[mylist[i]+NOT(KDELTA(i,j)) for i in range(len(mylist))] for j in range(len(mylist))][k]) for k in range(len(mylist))])

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'P' is not defined

P*IF(P,Q)
Prob(P*IF(P,Q))/Prob(IF(P,Q))
(Prob(P*IF(P,Q))/Prob(IF(P,Q)))(1/4,1/2,1/2,1/2,1/2,1/2)

P*Q p*q/(p*q - p + 1) 1/7

x, y, z, lam = var('x, y, z, lam')
f = sqrt((x - 1)^2 + (y - 2)^2 +(z - 3)^2)
g = 2*x + y - z - 5
h = f - g * lam
gradh = h.gradient([x, y, z, lam])
critical = solve([gradh[0] == 0, gradh[1] == 0, gradh[2] == 0, gradh[3] == 0,],x, y, z, lam, solution_dict=True)
critical 
gradh

[{lam: 1/6*sqrt(3)*sqrt(2), z: 7/3, x: 7/3, y: 8/3}] (-2*lam + (x - 1)/sqrt((x - 1)^2 + (y - 2)^2 + (z - 3)^2), -lam + (y - 2)/sqrt((x - 1)^2 + (y - 2)^2 + (z - 3)^2), lam + (z - 3)/sqrt((x - 1)^2 + (y - 2)^2 + (z - 3)^2), -2*x - y + z + 5)

h.gradient([x, y, z, lam])

(-2*lam + (x - 1)/sqrt((x - 1)^2 + (y - 2)^2 + (z - 3)^2), -lam + (y - 2)/sqrt((x - 1)^2 + (y - 2)^2 + (z - 3)^2), lam + (z - 3)/sqrt((x - 1)^2 + (y - 2)^2 + (z - 3)^2), -2*x - y + z + 5)

gradh=(ln(1-x)-ln(x)+lam*(y-1),ln(1-y)-ln(y)+lam*x,1-.5-x*(1-y))
critical = solve([gradh[0] == 0, gradh[1] == 0, gradh[2] == 0,],x, y, lam, solution_dict=True)
critical 
gradh

[{lam*(y - 1) - log(x) + log(-x + 1): 0}, {x*(y - 1) + 0.5: 0}, {lam*x - log(y) + log(-y + 1): 0}] (lam*(y - 1) - log(x) + log(-x + 1), lam*x - log(y) + log(-y + 1), x*(y - 1) + 0.500000000000000)

f=-(x*y*ln(x*y)+x*(1-y)*ln(x*(1-y))+(1-x)*y*ln((1-x)*y)+(1-x)*(1-y)*ln((1-x)*(1-y)))
g=1-.5-x+x*y
h=f-g*lam
gradh=h.gradient([x,y,lam])
critical = solve([gradh[0] == 0, gradh[1] == 0, gradh[2] == 0,],x, y, lam, solution_dict=True)
critical

[{-x*y + x - 0.5: 0}, {-lam*x - (x - 1)*log((x - 1)*(y - 1)) + x*log(-x*(y - 1)) + (x - 1)*log(-(x - 1)*y) - x*log(x*y): 0}, {-lam*(y - 1) - (y - 1)*log((x - 1)*(y - 1)) + (y - 1)*log(-x*(y - 1)) + y*log(-(x - 1)*y) - y*log(x*y): 0}]

gradh

(-lam*(y - 1) - (y - 1)*log((x - 1)*(y - 1)) + (y - 1)*log(-x*(y - 1)) + y*log(-(x - 1)*y) - y*log(x*y), -lam*x - (x - 1)*log((x - 1)*(y - 1)) + x*log(-x*(y - 1)) + (x - 1)*log(-(x - 1)*y) - x*log(x*y), -x*y + x - 0.500000000000000)

from operator import add
from operator import mul

x, y, z, lam = var('x, y, z, lam')
nn=3
binn=[Integer(ii+2**nn).binary()[1:nn+1] for ii in range(2**nn)]
binn
rbinn=[[ZZ(binn[jj][ii]) for ii in range(len(binn[jj]))] for jj in range(len(binn))]
rbinn
rrbinn=[reduce(mul,[rbinn[ii][jj]+[x,y,z][jj]*(-1)**rbinn[ii][jj] for jj in range(len(rbinn[ii]))]) for ii in range(len(rbinn))]
rrbinn

['000', '001', '010', '011', '100', '101', '110', '111'] [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]] [x*y*z, -x*y*(z - 1), -x*(y - 1)*z, x*(y - 1)*(z - 1), -(x - 1)*y*z, (x - 1)*y*(z - 1), (x - 1)*(y - 1)*z, -(x - 1)*(y - 1)*(z - 1)]

rrbinn[0][0]*rrbinn[0][1]

x*y

load('maxentrocond.sage')

<string>:8: DeprecationWarning: using 'degrevlex' in Boolean polynomial rings is deprecated. If needed, reverse the order of variables manually and use 'degneglex' See http://trac.sagemath.org/13849 for details.

H
g
L
gradH
gradg
gradL
critical

-(x - 1)*(y - 1)*log((x - 1)*(y - 1)) + x*(y - 1)*log(-x*(y - 1)) + (x - 1)*y*log(-(x - 1)*y) - x*y*log(x*y) x*y - x + 0.500000000000000 -(x - 1)*(y - 1)*log((x - 1)*(y - 1)) + x*(y - 1)*log(-x*(y - 1)) + (x - 1)*y*log(-(x - 1)*y) - x*y*log(x*y) + (x*y - x + 0.500000000000000)*lam (-log(x) + log(-x + 1), -log(y) + log(-y + 1), 0) (lam*(y - 1.00000000000000), lam*x, x*y - x + 0.500000000000000) (lam*(y - 1.00000000000000) - log(x) + log(-x + 1), lam*x - log(y) + log(-y + 1), x*y - x + 0.500000000000000) [{lam*(y - 1.0) - log(x) + log(-x + 1): 0}, {x*y - x + 0.5: 0}, {lam*x - log(y) + log(-y + 1): 0}]

gradg
gradH

(lam*(y - 1.00000000000000), lam*x, x*y - x + 0.500000000000000) [-log(x) + log(-x + 1), -log(y) + log(-y + 1), 0]

load('maxentrocond.sage')
H
g
L
gradH
gradg
gradL
critical

-(x - 1)*(y - 1)*log((x - 1)*(y - 1)) + x*(y - 1)*log(-x*(y - 1)) + (x - 1)*y*log(-(x - 1)*y) - x*y*log(x*y) x*y - x + 0.500000000000000 -(x - 1)*(y - 1)*log((x - 1)*(y - 1)) + x*(y - 1)*log(-x*(y - 1)) + (x - 1)*y*log(-(x - 1)*y) - x*y*log(x*y) + (x*y - x + 0.500000000000000)*lam (-log(x) + log(-x + 1), -log(y) + log(-y + 1), 0) (lam*(y - 1.00000000000000), lam*x, x*y - x + 0.500000000000000) (lam*(y - 1.00000000000000) - log(x) + log(-x + 1), lam*x - log(y) + log(-y + 1), x*y - x + 0.500000000000000) [{lam*(y - 1.0) - log(x) + log(-x + 1): 0}, {x*y - x + 0.5: 0}, {lam*x - log(y) + log(-y + 1): 0}]

gradH = [(ln(1-vars[ii])-ln(vars[ii])) for ii in range(len(vars))]
gradH.append(0)
gradH

[-log(x) + log(-x + 1), -log(y) + log(-y + 1), 0]

gradg

(lam*(y - 1.00000000000000), lam*x, x*y - x + 0.500000000000000)

gradg+vector(gradH)

(lam*(y - 1.00000000000000) - log(x) + log(-x + 1), lam*x - log(y) + log(-y + 1), x*y - x + 0.500000000000000)

ln(1)

0

yfun

y^2 - 2*y + 0.500000000000000

y=var('y')
yfun = y**2-2*y+.5
yfun
yroot=find_root(yfun == 0,0,1)
yroot
xroot=.5/(1-yroot)
xroot

y^2 - 2*y + 0.500000000000000 0.2928932188134525 0.707106781186548

xr=[xroot,1-xroot];yr=[yroot,1-yroot]
xr;yr
prsc = [xr[ii]*yr[jj] for ii in range(2) for jj in range(2)];prsc
ent=0
for ii in range(len(prsc)):
    ent = ent - prsc[ii]*ln(prsc[ii])/ln(2)
RR(ent)

[0.707106781186548, 0.292893218813452] [0.2928932188134525, 0.7071067811865475] [0.207106781186548, 0.500000000000000, 0.0857864376269049, 0.207106781186547] 1.74485867971294

hx=-RR((xr[0]*ln(xr[0])+xr[1]*ln(xr[1]))/ln(2));hx
hy=-RR((yr[0]*ln(yr[0])+yr[1]*ln(yr[1]))/ln(2));hy
hx+hy

0.872429339856468 0.872429339856468 1.74485867971294

plot(yfun,(y,0,1))

dy=.05
xroot=.5/(1-yroot-dy)
xr=[xroot,1-xroot];yr=[yroot+dy,1-yroot-dy]
xr;yr
prsc = [xr[ii]*yr[jj] for ii in range(2) for jj in range(2)];prsc
ent=0
for ii in range(len(prsc)):
    ent = ent - prsc[ii]*ln(prsc[ii])/ln(2)
RR(ent)

[0.760911337875927, 0.239088662124073] [0.342893218813452, 0.657106781186547] [0.260911337875927, 0.500000000000000, 0.0819818809375253, 0.157106781186547] 1.72108191409896

import numpy
import scipy.optimize
Pr=.9
def F(s):
    return [numpy.log(1-s[0])-numpy.log(s[0])-s[2]*(1-s[1]),numpy.log(1-s[1])-numpy.log(s[1])+s[2]*s[0],s[0]*(s[1]-1)+1-Pr]
ssoln = scipy.optimize.broyden1(F, [.5,.2,1], f_tol=1e-14)
ssoln

array([ 0.31622777, 0.68377223, 2.4386288 ])

gradL
(gradL[0])(x=.5,y=.5,lam=3)

(lam*(y - 1.00000000000000) - log(x) + log(-x + 1), lam*x - log(y) + log(-y + 1), x*y - x + 0.500000000000000) -1.50000000000000

def F2(s):
    tmp=[]
    for ii in range(len(gradL)):
        tmp.append((gradL[ii])(x=s[0],y=s[1],lam=s[2]))
    return tmp

BPR = BooleanPolynomialRing(2,'x',order='degrevlex')
NewGens = str(BPR.gens()).lower()
NewGens = NewGens[1:len(NewGens) - 1]
NewGens
vars = var(NewGens)
vars
ftest=vars[0]^2+vars[1]+1
ftest(1,2)

'x0, x1' (x0, x1) 4

load('maxentrocond2.sage')
H
g
L
gradH
gradg
gradL
critical
ssoln
htt

<string>:8: DeprecationWarning: using 'degrevlex' in Boolean polynomial rings is deprecated. If needed, reverse the order of variables manually and use 'degneglex' See http://trac.sagemath.org/13849 for details.

-(x0 - 1)*(x1 - 1)*log((x0 - 1)*(x1 - 1)) + x0*(x1 - 1)*log(-x0*(x1 - 1)) + (x0 - 1)*x1*log(-(x0 - 1)*x1) - x0*x1*log(x0*x1) x0*x1 - x0 + 0.500000000000000 -(x0 - 1)*(x1 - 1)*log((x0 - 1)*(x1 - 1)) + x0*(x1 - 1)*log(-x0*(x1 - 1)) + (x0 - 1)*x1*log(-(x0 - 1)*x1) - x0*x1*log(x0*x1) + (x0*x1 - x0 + 0.500000000000000)*lam0 (-log(x0) + log(-x0 + 1), -log(x1) + log(-x1 + 1), 0) (lam0*(x1 - 1.00000000000000), lam0*x0, x0*x1 - x0 + 0.500000000000000) (lam0*(x1 - 1.00000000000000) - log(x0) + log(-x0 + 1), lam0*x0 - log(x1) + log(-x1 + 1), x0*x1 - x0 + 0.500000000000000) [{lam0*(x1 - 1.0) - log(x0) + log(-x0 + 1): 0}, {x0*x1 - x0 + 0.5: 0}, {lam0*x0 - log(x1) + log(-x1 + 1): 0}] [0.70710678118654735, 0.29289321881345232, -1.246450480280461] 1.74485867971294

y=var('y')
yfun = y**2-2*y+.5
yfun
yroot=find_root(yfun == 0,0,1)
yroot
xroot=.5/(1-yroot)
xroot
xr=[xroot,1-xroot];yr=[yroot,1-yroot]
xr;yr
prsc = [xr[ii]*yr[jj] for ii in range(2) for jj in range(2)];prsc
ent=0
for ii in range(len(prsc)):
    ent = ent - prsc[ii]*ln(prsc[ii])/ln(2)
RR(ent)
hx=-RR((xr[0]*ln(xr[0])+xr[1]*ln(xr[1]))/ln(2));hx
hy=-RR((yr[0]*ln(yr[0])+yr[1]*ln(yr[1]))/ln(2));hy
hx+hy
RR(sqrt(2)/2)

y^2 - 2*y + 0.500000000000000 0.2928932188134525 0.707106781186548 [0.707106781186548, 0.292893218813452] [0.2928932188134525, 0.7071067811865475] [0.207106781186548, 0.500000000000000, 0.0857864376269049, 0.207106781186547] 1.74485867971294 0.872429339856468 0.872429339856468 1.74485867971294 0.707106781186548

plot(yfun,(y,0,1))

ssoln

array([ 0.70710678, 0.29289322, -1.24645048])

tuple(x)

(x0, x1, lam0)

ftest(vector(x))=gradL

Error in lines 1-1 Traceback (most recent call last): File "/projects/5511fe15-8085-4d1d-bdc7-c6bf6c99e693/.sagemathcloud/sage_server.py", line 865, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 SyntaxError: can't assign to function call

ftest=(gradL[2]).function(*x)
ftest
ftest(.5,.5,3)

(x0, x1, lam0) |--> x0*x1 - x0 + 0.500000000000000 0.250000000000000

for ii in range(len(x)):
    gradL[ii]=(gradL[ii]).function(*x)

natoms

['00', '01', '10', '11']

NOR(P,Q)

P*Q + P + Q + 1

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
Prob(XOR(P,Q))
Prob(UNIQUE([P,Q,R]))
Prob(UNIQUE([P,Q,R,S]))

-2*p*q + p + q 3*p*q*r - 2*p*q - 2*p*r + p - 2*q*r + q + r -4*p*q*r*s + 3*p*q*r + 3*p*q*s - 2*p*q + 3*p*r*s - 2*p*r - 2*p*s + p + 3*q*r*s - 2*q*r - 2*q*s + q - 2*r*s + r + s

KDELTA = lambda A, B: A.parent(A == B)

FORALL([P,Q,R])

P*Q*R

P*Q*(P+Q)
Prob(P*Q*(P+Q))
Prob(P+Q)
Prob(P*Q*(P+Q))/Prob(P+Q)
Prob((1+P*Q)*(P+Q))/Prob(P+Q)
(1+P*Q)*(P+Q)
Prob((P+R)*(Q*S+P)*((P+R)+(Q*S+P)))
UNIQUE([P,Q,R])

0 0 -2*p*q + p + q 0 1 P + Q 0 P*Q*R + P + Q + R

myideal=ideal(UNIQUE([P,Q,R])+1)
myQR=EX7.quotient_ring(myideal)
myQR.gens()

(Q + R + 1, Q, R, S, V, W)

p,q=var('p','q')
pcon(p,q)=Prob((P+Q)*P)/Prob(P+Q)
pcon
pcon(.5,.5)
pcon(.25,.7)

(p, q) |--> (p*q - p)/(2*p*q - p - q) 0.500000000000000 0.125000000000000

pcon.gradient()
Prob(NOR(P,Q))
mylist=[P,Q,R]
[FORALL(C) for C in [[A + NOT(KDELTA(A,B)) for A in mylist] for B in mylist]]
[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist]

(p, q) |--> (-(p*q - p)*(2*q - 1)/(2*p*q - p - q)^2 + (q - 1)/(2*p*q - p - q), -(p*q - p)*(2*p - 1)/(2*p*q - p - q)^2 + p/(2*p*q - p - q)) p*q - p - q + 1 [P*Q*R + P*Q + P*R + P, P*Q*R + P*Q + Q*R + Q, P*Q*R + P*R + Q*R + R] [P*Q*R + P*Q + P*R + P, P*Q*R + P*Q + Q*R + Q, P*Q*R + P*R + Q*R + R]

numpy.random.uniform(.1, .9)

0.23296928604967065

load('PuzzlePart.sage')

25 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X24 = R[4,4] REI.gens() = 105

clues;len(clues);QR.gens();len(QR.gens());matrix(BPR, objects, objects, QRG)

[X1*X4*X21 + X4*X21, X14*X22 + X22, X3*X15 + X3, X18*X19*X23 + X19*X23, X17*X22*X23 + X17*X23, X13*X14*X23 + X14*X23, X1*X2*X11 + X2*X11, X1*X3*X16 + X3*X16, X1*X2*X7 + X1*X7, X8*X16 + X16, X13*X14*X19 + X13*X19, X19*X23 + X19, X19*X23*X24 + X19*X23, X2*X10*X12 + X2*X10, X12 + 1, X7*X11 + X11, X10*X15*X17 + X10*X17, X9*X16*X19 + X9*X16, X7*X11 + X7, X2*X3*X13 + X2*X13, X4*X15*X19 + X4*X15, X7*X21*X22 + X7*X21, X7*X8*X13 + X7*X13, X1*X4*X9 + X1*X9, X3*X5*X8 + X3*X5, X10*X20*X22 + X10*X22, X2*X4*X14 + X2*X14, X7*X16*X17 + X7*X16, X13*X17 + X17, X4*X20*X24 + X4*X20, X0 + 1, X1*X3*X8 + X1*X8, X4*X10*X14 + X4*X10, X6*X9*X21 + X9*X21, X4*X20 + X4, X2*X20*X22 + X2*X20, X6*X7*X11 + X7*X11, X5*X10*X11 + X5*X11, X5*X9*X20 + X9*X20, X11*X13*X16 + X13*X16, X8*X9*X19 + X8*X19, X14*X22*X24 + X14*X22, X5*X8*X15 + X8*X15, X4*X5*X9 + X4*X5, X0*X4*X20 + X4*X20, X11*X16*X17 + X11*X17, X2*X3*X17 + X3*X17, X15*X19*X20 + X19*X20, X2*X15*X17 + X2*X15, X13*X17 + X13, X2*X5*X7 + X2*X5, X8*X16 + X8, X9*X21 + X21, X13*X17*X18 + X13*X17, X7*X8*X17 + X8*X17, X6 + 1, X1*X5 + X1, X14*X17*X19 + X14*X17, X12*X14*X22 + X14*X22, X11*X21*X22 + X11*X22, X3*X4*X23 + X4*X23, X3*X15 + X15, X2*X10 + X10, X5*X7*X10 + X7*X10, X7*X9*X22 + X9*X22, X15*X20*X23 + X15*X23, X14*X22 + X14, X4*X20 + X20, X0*X3*X15 + X3*X15, X9*X21*X24 + X9*X21, X1*X10*X11 + X1*X10, X10*X13*X15 + X13*X15, X2*X4*X22 + X4*X22, X7*X9*X14 + X7*X14, X0*X2*X10 + X2*X10, X19*X23 + X23, X8*X16*X18 + X8*X16, X9*X11*X14 + X9*X11, X8*X9*X23 + X9*X23, X16*X19*X21 + X19*X21, X8*X11*X13 + X8*X11, X0*X1*X5 + X1*X5, X3*X20*X23 + X3*X20, X3*X15*X18 + X3*X15, X5*X20*X21 + X5*X21, X7*X11*X12 + X7*X11, X3*X10*X13 + X3*X10, X3*X4*X19 + X3*X19, X10*X14*X20 + X14*X20, X1*X5 + X5, X17*X19*X22 + X19*X22, X2*X10 + X2, X16*X21*X23 + X16*X23, X8*X21*X23 + X8*X21, X1*X20*X21 + X1*X20, X13*X22*X23 + X13*X22, X24 + 1, X18 + 1, X1*X15*X16 + X1*X15, X1*X5*X6 + X1*X5, X11*X14*X21 + X14*X21, X12*X13*X17 + X13*X17, X5*X15*X16 + X5*X16, X9*X21 + X9, X6*X8*X16 + X8*X16] 105 (1, X1, X2, X3, X4, X1, 1, X7, X8, X9, X2, X7, 1, X13, X14, X3, X8, X13, 1, X19, X4, X9, X14, X19, 1) 25 [ 1 X1 X2 X3 X4] [ X1 1 X7 X8 X9] [ X2 X7 1 X13 X14] [ X3 X8 X13 1 X19] [ X4 X9 X14 X19 1]

load('PuzzlePart2.sage')

25 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X24 = R[4,4] REI2.gens() = 105

len(constraints2);constraints2;constraints2[0].parent();QRG2

105 [x2*x10*x12 - x2*x10 + 62203649/193057324, x5*x20*x21 - x5*x21 + 145251550/176756029, x2*x5*x7 - x2*x5 + 13294602/167964109, x13*x17 - x13 + 53752795/116314207, x0*x1*x5 - x1*x5 + 24801404/73211043, x0*x3*x15 - x3*x15 + 22827353/141536662, x3*x4*x19 - x3*x19 + 90536883/153915703, x8*x16*x18 - x8*x16 + 16391305/109369072, x1*x3*x16 - x3*x16 + 1532997/24131747, x14*x17*x19 - x14*x17 + 31242773/119837697, x18*x19*x23 - x19*x23 + 426167851/641636391, x7*x11 - x11 + 7557854/46014365, x10*x15*x17 - x10*x17 + 81770998/164278037, x11*x21*x22 - x11*x22 + 37578567/53745559, x7*x9*x14 - x7*x14 + 15731608/78113873, x8*x9*x19 - x8*x19 + 25425294/105569273, x3*x15 - x15 + 3231321/41224288, x16*x21*x23 - x16*x23 + 70266841/147014939, x10*x20*x22 - x10*x22 + 6873531/23189599, x3*x10*x13 - x3*x10 + 12747995/180417161, x7*x16*x17 - x7*x16 + 72569386/110897241, x7*x9*x22 - x9*x22 + 126605039/241223591, x17*x22*x23 - x17*x23 + 200590648/246390539, x1*x20*x21 - x1*x20 + 42292682/84010945, x14*x22 - x14 + 43757447/229960415, x6 - 69535386/185052941, x12*x14*x22 - x14*x22 + 74882460/194990881, x0*x4*x20 - x4*x20 + 1209941651/1542345415, x3*x4*x23 - x4*x23 + 80457933/112290076, x0*x2*x10 - x2*x10 + 210808211/272813755, x11*x13*x16 - x13*x16 + 43101815/137474056, x11*x14*x21 - x14*x21 + 83581164/157946179, x9*x21 - x9 + 107228941/204234132, x7*x21*x22 - x7*x21 + 132095151/242046925, x11*x16*x17 - x11*x17 + 79038230/98122057, x5*x9*x20 - x9*x20 + 335599798/421644351, x5*x10*x11 - x5*x11 + 66242059/167141062, x19*x23 - x23 + 49039437/126222658, x12 - 268701550/339362021, x8*x9*x23 - x9*x23 + 83354562/127238057, x8*x16 - x16 + 50354431/168064135, x4*x5*x9 - x4*x5 + 67570886/101655503, x8*x16 - x8 + 6724800/79890731, x8*x11*x13 - x8*x11 + 45583496/120448297, x2*x15*x17 - x2*x15 + 9741908/99895239, x15*x19*x20 - x19*x20 + 183435827/258564398, x7*x8*x13 - x7*x13 + 133294783/195762397, x6*x7*x11 - x7*x11 + 274971476/337687067, x10*x13*x15 - x13*x15 + 102094315/197500217, x3*x15 - x3 + 28661302/61727057, x14*x22*x24 - x14*x22 + 42153293/121148350, x13*x17 - x17 + 13153103/112313285, x3*x5*x8 - x3*x5 + 125666262/215356303, x1*x3*x8 - x1*x8 + 2799551/49427694, x4*x20 - x20 + 32491907/244592185, x1*x5*x6 - x1*x5 + 107971821/134584024, x6*x9*x21 - x9*x21 + 115861933/315983587, x18 - 33127973/147643308, x13*x14*x23 - x14*x23 + 121216620/152651173, x1*x5 - x1 + 96817268/144121465, x16*x19*x21 - x19*x21 + 32019023/100189280, x9*x21 - x21 + 25591873/154413949, x13*x14*x19 - x13*x19 + 23330422/114516395, x7*x11*x12 - x7*x11 + 6973063/74296605, x19*x23 - x19 + 42829018/203820483, x1*x5 - x5 + 53894501/196846215, x5*x15*x16 - x5*x16 + 120276253/237142128, x9*x21*x24 - x9*x21 + 80214453/143263319, x3*x20*x23 - x3*x20 + 154620210/260192879, x1*x4*x9 - x1*x9 + 11874187/215284671, x4*x10*x14 - x4*x10 + 75835979/144476735, x17*x19*x22 - x19*x22 + 29299815/116793706, x14*x22 - x22 + 76591445/347969086, x5*x8*x15 - x8*x15 + 29107204/225541941, x7*x11 - x7 + 265870287/418334620, x13*x17*x18 - x13*x17 + 70443531/175528207, x2*x4*x14 - x2*x14 + 14366107/164308973, x5*x7*x10 - x7*x10 + 14187049/138409097, x10*x14*x20 - x14*x20 + 84580334/150319449, x6*x8*x16 - x8*x16 + 56634376/131432223, x1*x10*x11 - x1*x10 + 71072583/147044227, x3*x15*x18 - x3*x15 + 31868003/48399562, x4*x20 - x4 + 47395399/139814408, x1*x2*x11 - x2*x11 + 18518707/123781877, x1*x2*x7 - x1*x7 + 13716435/70088324, x2*x20*x22 - x2*x20 + 143386372/219854821, x9*x11*x14 - x9*x11 + 74236774/133750331, x19*x23*x24 - x19*x23 + 5530867/100656898, x9*x16*x19 - x9*x16 + 10706477/32252722, x24 - 11330596/38802581, x2*x4*x22 - x4*x22 + 45044410/65069701, x13*x22*x23 - x13*x22 + 285205056/341276053, x12*x13*x17 - x13*x17 + 93473574/149323807, x4*x20*x24 - x4*x20 + 48236487/287809139, x1*x15*x16 - x1*x15 + 180928591/251220634, x15*x20*x23 - x15*x23 + 27884830/113267253, x0 - 32632973/39059746, x4*x15*x19 - x4*x15 + 58730901/120827245, x2*x10 - x2 + 40380811/79503558, x2*x3*x17 - x3*x17 + 14295370/174213847, x7*x8*x17 - x8*x17 + 235975366/405753989, x8*x21*x23 - x8*x21 + 2189977952/2593090641, x2*x10 - x10 + 19137668/66803295, x1*x4*x21 - x4*x21 + 39580717/116644552, x2*x3*x13 - x2*x13 + 45759724/102990237] Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24 over Rational Field (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

GB

[1]

load('Puzzle7.sage')

objects = 24 types = 4 rel_classes = 6 576 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X575 = R[23,23] REI.gens() = 13414 A 1st pass type 0 to type 1 [ 0 X7 X7 + 1 0 0 0] [ X30 X31 0 0 0 0] [ X54 X55 X56 X57 X58 0] [ 0 X79 X80 0 X82 0] [ X102 X103 X104 X105 X106 0] [ 0 0 0 0 0 1] type 0 to type 2 [ X12 X13 X14 X15 0 X17] [ 0 0 0 0 1 0] [ X60 X61 X62 X63 0 X65] [ X84 0 0 0 0 X89] [X108 X109 X110 0 0 X113] [X132 X133 X134 X135 0 X137] type 0 to type 3 [ X18 0 X13 0 0 X23] [ 0 0 X30 + 1 X30 0 0] [ X66 X67 X68 X54 0 X71] [X30 + 1 X30 0 0 0 0] [ X114 X115 X116 X102 0 0] [ 0 0 0 0 1 0] type 1 to type 2 [ 0 X30 + 1 0 0 X30 0] [ X180 X181 X182 X183 X31 X185] [ X204 X205 0 X207 0 0] [ X228 0 0 0 0 0] [ X252 0 0 X255 0 0] [ X132 X133 X134 X135 0 X137] type 1 to type 3 [ 0 0 0 1 0 0] [ X186 X187 X181 + X31 0 0 X191] [ X210 X211 X205 0 0 X215] [X133 + X30 X30 + 1 X133 0 0 0] [ X258 X259 0 0 0 X263] [ 0 0 0 0 1 0] type 2 to type 3 [ X306 X307 X308 0 X132 X311] [ 0 0 X133 + X30 X30 + 1 X133 0] [ X354 0 0 0 X134 X359] [ 0 0 0 0 X135 X135 + 1] [ 0 0 X30 + 1 X30 0 0] [ X426 X427 0 0 X137 X431] number of variables left to be determined: 75 REI.gens() = 13469 A 2nd pass type 0 to type 1 [0 0 1 0 0 0] [1 0 0 0 0 0] [0 0 0 0 1 0] [0 1 0 0 0 0] [0 0 0 1 0 0] [0 0 0 0 0 1] type 0 to type 2 [0 1 0 0 0 0] [0 0 0 0 1 0] [0 0 0 1 0 0] [0 0 0 0 0 1] [1 0 0 0 0 0] [0 0 1 0 0 0] type 0 to type 3 [0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 0 1] [0 1 0 0 0 0] [1 0 0 0 0 0] [0 0 0 0 1 0] type 1 to type 2 [0 0 0 0 1 0] [0 0 0 0 0 1] [0 1 0 0 0 0] [1 0 0 0 0 0] [0 0 0 1 0 0] [0 0 1 0 0 0] type 1 to type 3 [0 0 0 1 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [1 0 0 0 0 0] [0 0 0 0 0 1] [0 0 0 0 1 0] type 2 to type 3 [1 0 0 0 0 0] [0 0 1 0 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1] [0 0 0 1 0 0] [0 1 0 0 0 0]

load('PuzzlePart3.sage')

objects = 12 types = 3 rel_classes = 4 144 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X143 = R[11,11] REI.gens() = 1656 A 1st pass type 0 to type 1 [X61 + X62 + X63 + X73 + X74 + X75 + X85 + X86 + X87 X61 + X62 + X63 + 1 X73 + X74 + X75 + 1 X85 + X86 + X87 + 1] [ X61 + X73 + X85 + 1 X61 X73 X85] [ X62 + X74 + X86 + 1 X62 X74 X86] [ X63 + X75 + X87 + 1 X63 X75 X87] type 0 to type 2 [X109 + X110 + X111 + X121 + X122 + X123 + X133 + X134 + X135 X109 + X110 + X111 + 1 X121 + X122 + X123 + 1 X133 + X134 + X135 + 1] [ X109 + X121 + X133 + 1 X109 X121 X133] [ X110 + X122 + X134 + 1 X110 X122 X134] [ X111 + X123 + X135 + 1 X111 X123 X135] type 1 to type 2 [X113 + X114 + X115 + X125 + X126 + X127 + X137 + X138 + X139 X113 + X114 + X115 + 1 X125 + X126 + X127 + 1 X137 + X138 + X139 + 1] [ X113 + X125 + X137 + 1 X113 X125 X137] [ X114 + X126 + X138 + 1 X114 X126 X138] [ X115 + X127 + X139 + 1 X115 X127 X139] number of variables left to be determined: 27

A;stuff2;clues;clues2;len(clues2);R

[ 1 0 0 0 X61 + X62 + X63 + X73 + X74 + X75 + X85 + X86 + X87 X61 + X62 + X63 + 1 X73 + X74 + X75 + 1 X85 + X86 + X87 + 1 X109 + X110 + X111 + X121 + X122 + X123 + X133 + X134 + X135 X109 + X110 + X111 + 1 X121 + X122 + X123 + 1 X133 + X134 + X135 + 1] [ 0 1 0 0 X61 + X73 + X85 + 1 X61 X73 X85 X109 + X121 + X133 + 1 X109 X121 X133] [ 0 0 1 0 X62 + X74 + X86 + 1 X62 X74 X86 X110 + X122 + X134 + 1 X110 X122 X134] [ 0 0 0 1 X63 + X75 + X87 + 1 X63 X75 X87 X111 + X123 + X135 + 1 X111 X123 X135] [ X61 + X62 + X63 + X73 + X74 + X75 + X85 + X86 + X87 X61 + X73 + X85 + 1 X62 + X74 + X86 + 1 X63 + X75 + X87 + 1 1 0 0 0 X113 + X114 + X115 + X125 + X126 + X127 + X137 + X138 + X139 X113 + X114 + X115 + 1 X125 + X126 + X127 + 1 X137 + X138 + X139 + 1] [ X61 + X62 + X63 + 1 X61 X62 X63 0 1 0 0 X113 + X125 + X137 + 1 X113 X125 X137] [ X73 + X74 + X75 + 1 X73 X74 X75 0 0 1 0 X114 + X126 + X138 + 1 X114 X126 X138] [ X85 + X86 + X87 + 1 X85 X86 X87 0 0 0 1 X115 + X127 + X139 + 1 X115 X127 X139] [X109 + X110 + X111 + X121 + X122 + X123 + X133 + X134 + X135 X109 + X121 + X133 + 1 X110 + X122 + X134 + 1 X111 + X123 + X135 + 1 X113 + X114 + X115 + X125 + X126 + X127 + X137 + X138 + X139 X113 + X125 + X137 + 1 X114 + X126 + X138 + 1 X115 + X127 + X139 + 1 1 0 0 0] [ X109 + X110 + X111 + 1 X109 X110 X111 X113 + X114 + X115 + 1 X113 X114 X115 0 1 0 0] [ X121 + X122 + X123 + 1 X121 X122 X123 X125 + X126 + X127 + 1 X125 X126 X127 0 0 1 0] [ X133 + X134 + X135 + 1 X133 X134 X135 X137 + X138 + X139 + 1 X137 X138 X139 0 0 0 1] set([X87, X86, X85, X73, X74, X75, X137, X138, X139, X126, X125, X127, X121, X122, X123, X135, X134, X133, X62, X61, X63, X110, X109, X111, X115, X113, X114]) [X88*X55 + X55, X49*X4*X1 + X49*X4, X109*X21 + X21, X71*X64*X59 + X64*X59, X118*X22*X21 + X118*X21, X64*X53 + X53, X64, X70*X63*X46 + X63*X46, X43*X36*X7 + X36*X7, X105*X33*X32 + X105*X32, X74*X18*X14 + X74*X18, X58*X46*X40 + X58*X40, X119*X111*X47 + X111*X47, X75*X66*X63 + X75*X66, X34*X25*X22 + X25*X22, X139*X138*X90 + X139*X90, X57*X9*X4 + X57*X4, X52*X50*X28 + X50*X28, X81*X76*X57 + X76*X57, X74*X42*X38 + X74*X42, X106*X10*X8 + X106*X8, X115*X114*X90 + X115*X90, X122*X70*X62 + X122*X70, X59*X23*X16 + X59*X16, X108*X21*X12 + X108*X21, X54*X48*X6 + X48*X6, X51*X40 + X51, X118*X110*X34 + X110*X34, X113*X33*X29 + X113*X33, X92*X86*X32 + X86*X32, X142*X71*X70 + X142*X71, X121*X22 + X121, X100*X98*X50 + X100*X50, X45*X36*X9 + X36*X9, X119*X110*X35 + X110*X35, X139*X133*X19 + X133*X19, X60*X41*X36 + X60*X41, X102*X92*X90 + X102*X92, X140*X83*X80 + X140*X83, X77*X74*X62 + X77*X62, X142*X140*X128 + X142*X128, X70*X62*X34 + X62*X34, X141*X11*X9 + X141*X11, X114*X111*X42 + X111*X42, X106*X103*X94 + X103*X94, X58*X50*X34 + X50*X34, X136*X23*X16 + X136*X23, X127*X123*X87 + X127*X87, X41*X38*X29 + X38*X29, X115*X108*X84 + X115*X84, X43*X37*X19 + X37*X19, X143*X142*X131 + X142*X131, X132*X71*X60 + X132*X71, X66*X18*X17 + X66*X17, X141*X136*X57 + X136*X57, X121*X10*X1 + X121*X10, X102*X101*X100 + X103*X101*X100 + X103*X102*X100 + X103*X102*X101 + X100 + X101 + X102 + X103 + 1, X91 + 1, X114*X113*X77 + X114*X77, X127*X58*X55 + X127*X58, X89*X67 + X89, X69*X9*X5 + X69*X5, X31*X7*X2 + X31*X2, X94*X70*X67 + X94*X67, X128*X46*X44 + X128*X46, X113*X69 + X69, X128*X126*X102 + X128*X102, X32*X8*X2 + X32*X2, X79*X74*X31 + X74*X31, X105*X99*X45 + X99*X45, X109*X108*X1 + X108*X1, X127*X70*X67 + X127*X70, X114*X108*X72 + X114*X72, X126*X46*X42 + X126*X46, X73*X42*X37 + X73*X42, X83*X76*X59 + X76*X59, X102*X8*X6 + X102*X8, X23*X12*X11 + X12*X11, X126*X120*X6 + X120*X6, X79*X77*X67 + X77*X67, X37*X36*X12 + X37*X12, X141*X119*X117 + X141*X119, X82*X70*X66 + X82*X66, X87*X79*X75 + X87*X79, X79*X76*X55 + X76*X55, X65 + 1, X69*X45*X41 + X69*X41, X141*X47*X45 + X141*X47, X62*X29*X26 + X62*X29, X41*X29*X27 + X41*X27, X101*X68 + X101, X139*X133*X85 + X139*X85, X38*X15*X14 + X38*X15, X115*X113*X67 + X113*X67, X139*X136*X55 + X136*X55, X99*X92*X87 + X99*X92, X120*X10 + X120, X95*X87*X47 + X87*X47, 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X12*X1, X138*X135*X42 + X135*X42, X138*X23*X18 + X138*X23, X102*X97*X73 + X102*X73, X75*X72*X36 + X75*X36, X141*X131*X129 + X141*X131, X122*X94*X86 + X122*X94, X124*X94*X88 + X124*X94, X126*X58*X54 + X126*X58, X115*X111*X43 + X111*X43, X120*X46*X36 + X120*X46, X97*X8*X1 + X97*X8, X114*X81 + X114, X134*X35 + X134, X140*X11*X8 + X140*X11, X124*X118*X112 + X124*X118, X18*X6*X1 + X18*X1, X137*X136*X64 + X137*X64, X139*X138*X79 + X138*X79, X140*X139*X103 + X140*X103, X125*X10*X5 + X125*X10, X112*X111*X40 + X111*X40, X68*X44*X41 + X68*X41, X113*X111*X63 + X113*X63, X22*X21*X20 + X23*X21*X20 + X23*X22*X20 + X23*X22*X21 + X20 + X21 + X22 + X23 + 1, X118*X113*X70 + X113*X70, X69*X62*X33 + X62*X33, X49*X16 + X16, X45*X38*X33 + X38*X33, X115*X93*X91 + X115*X93, X42*X6*X3 + X42*X3, X133*X119*X109 + X133*X119, X106*X34*X32 + X106*X32, X79*X31*X30 + X79*X30, X27*X15*X14 + X27*X14, X113*X105*X101 + X113*X105, X41*X5*X3 + X41*X3, X107*X97*X23 + X97*X23, X132*X119*X108 + X132*X119, X41*X37*X17 + 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+ 1, X127*X126*X90 + X127*X90, X141*X138*X81 + X138*X81, X141*X119 + X119, X142*X137*X125 + X142*X125, X142*X83*X82 + X142*X83, X119*X59*X57 + X119*X57, X51*X50*X38 + X51*X38, X119*X109*X23 + X109*X23, X140*X132*X8 + X132*X8, X121*X58*X49 + X121*X58, X103*X97*X19 + X97*X19, X21*X9*X1 + X21*X1, X82*X22*X18 + X82*X18, X128*X94*X92 + X128*X94, X102*X98*X74 + X102*X74, X84*X31*X24 + X84*X31, X18*X17*X16 + X19*X17*X16 + X19*X18*X16 + X19*X18*X17 + X16 + X17 + X18 + X19 + 1, X114*X33*X30 + X114*X33, X39*X38*X27 + X38*X27, X107*X71*X68 + X107*X68, X69*X63*X45 + X63*X45, X117*X109*X21 + X109*X21, X73*X66*X61 + X73*X66, X89*X67*X65 + X89*X67, X114*X109*X18 + X109*X18, X88*X86*X50 + X88*X50, X115*X109*X85 + X115*X85, X138*X47*X42 + X138*X47, X104*X96*X8 + X96*X8, X117*X116*X105 + X116*X105, X89*X55*X53 + X89*X55, X105*X45*X44 + X105*X44, X24*X2 + X2, X110*X33 + X33, X76*X54 + X76, X112*X111*X51 + X112*X51, X58*X10*X4 + X58*X4, X53*X48*X5 + X48*X5, X27, X55*X19*X16 + X55*X16, X78*X74*X30 + X74*X30, X57*X50*X33 + X50*X33, X49*X28*X25 + X49*X28, X64*X61*X49 + X64*X49, X84*X55*X48 + X84*X55, X112*X108*X4 + X108*X4, X77, X103*X92*X91 + X103*X92, X83*X23*X18 + X83*X18, X90*X79*X78 + X90*X79, X103*X44*X43 + X103*X44, X84*X7 + X7, X129*X118 + X129, X135*X107*X99 + X135*X107, X126*X124*X54 + X124*X54, X101*X56*X53 + X101*X56, X44*X38*X32 + X38*X32, X142*X131 + X142, X131*X123*X47 + X123*X47, X136*X95*X88 + X136*X95, X131*X122*X35 + X122*X35, X132*X83*X72 + X132*X83, X129*X127*X93 + X127*X93, X37*X36*X1 + X36*X1, X76*X75*X51 + X76*X51, X42*X37*X18 + X37*X18, X142, X77*X72*X5 + X72*X5, X87*X84*X3 + X84*X3, X125*X121*X17 + X121*X17, X139*X132*X7 + X132*X7, X125*X46*X41 + X125*X46, X105*X21*X20 + X105*X20, X89*X84*X60 + X89*X60, X133*X132*X12 + X133*X12, X129*X122*X33 + X122*X33, X135*X83*X75 + X135*X83, X33*X24*X9 + X24*X9, X101*X8*X5 + X101*X8, X66*X60*X6 + X60*X6, X93*X87*X45 + X87*X45, X104*X98*X32 + X98*X32, X107*X47*X44 + X107*X44, X38, X140*X132*X96 + X140*X96, X42*X41*X40 + X43*X41*X40 + X43*X42*X40 + X43*X42*X41 + X40 + X41 + X42 + X43 + 1, X34*X10*X2 + X34*X2, X133*X23 + X23, X106*X46*X44 + X106*X44, X82*X74*X34 + X74*X34, X70*X61*X22 + X61*X22, X87*X84*X36 + X87*X36, X37, X142*X23*X22 + X142*X23, X130*X129*X118 + X129*X118, X118*X82*X81 + X118*X81, X110*X105*X98 + X110*X105, X109*X69*X61 + X109*X69, X135*X134*X27 + X134*X27, X71*X63*X47 + X63*X47, X36*X3 + X36, X115*X114*X79 + X114*X79, X140*X134*X32 + X134*X32, X76*X75*X40 + X75*X40, X141*X107*X105 + X141*X107, X75*X30*X27 + X75*X30, X85*X84*X1 + X84*X1, X111*X109*X37 + X111*X37, X84*X7 + X84, X46*X34*X27 + X46*X27, X131*X121*X23 + X121*X23, X125*X123*X41 + X123*X41, X38*X3*X2 + X38*X3, X126*X123*X42 + X123*X42, X112*X109*X49 + X112*X49, X55*X31*X28 + X55*X28, X64*X41*X40 + X64*X41, X128*X34*X32 + X128*X34, X137*X95*X89 + X137*X95, X63*X53*X51 + X63*X53, X88*X84*X4 + X84*X4, X141*X140*X105 + X140*X105, X78*X72*X6 + X72*X6, X138*X137*X77 + X138*X77, X141*X119 + X141, X122*X34 + X122, X128*X127*X103 + X128*X103, X51*X49*X37 + X51*X37, X113*X81*X77 + X113*X81, X99*X80*X75 + X99*X80, X101*X99*X63 + X101*X63, X31*X19*X14 + X31*X14, X16*X12*X4 + X12*X4, X128*X123*X44 + X123*X44, X125*X120*X5 + X120*X5, X138*X137*X66 + X137*X66, X116*X108*X8 + X108*X8, X107*X11*X8 + X107*X8, X79*X75*X43 + X75*X43, X88*X43*X40 + X88*X43, X55*X43*X40 + X55*X40, X118*X58*X57 + X118*X57, X36*X27*X24 + X36*X27, X137*X134*X29 + X134*X29, X128*X123*X99 + X128*X99, X82*X73*X22 + X73*X22, X140, X142*X134*X34 + X134*X34, X141*X35*X33 + X141*X35, X61*X41*X37 + X61*X41, X141*X95*X93 + X141*X95, X85*X19 + X19, X124*X46*X40 + X124*X46, X45*X33*X27 + X45*X27, X80*X75*X44 + X75*X44, X109*X33*X25 + X109*X33, X55*X48*X7 + X48*X7, X137*X133*X17 + X133*X17, X68*X62*X32 + X62*X32, X134*X23*X14 + X134*X23, X143*X140*X107 + X140*X107, X143*X132*X11 + X132*X11, X58*X22*X16 + X58*X16, X128*X122*X98 + X128*X98, X107*X59*X56 + X107*X56, X109*X21*X13 + X109*X21, X134*X47*X38 + X134*X47, X87*X19*X15 + X87*X19, X22*X12*X10 + X12*X10, X104*X102*X80 + X102*X80, X97*X96*X12 + X97*X12, X109*X81*X73 + X109*X81, X108*X105*X96 + X108*X105, X127*X124*X55 + X124*X55, X99*X56*X51 + X99*X56, X50*X28 + X28, X115*X112*X88 + X115*X88, X138*X135*X75 + X138*X75, X77*X54*X53 + X77*X54, X15, X118*X112*X58 + X112*X58, X99*X32*X27 + X99*X32, X114*X112*X54 + X112*X54, X93*X84*X9 + X84*X9, X124*X58 + X58, X95*X11*X7 + X95*X7, X136*X133*X16 + X133*X16, X56*X20*X16 + X56*X16, X88*X55*X52 + X88*X55, X75*X54*X51 + X75*X54, X14, X111*X110*X27 + X110*X27, X131*X119*X118 + X131*X118, X140*X137*X101 + X140*X101, X110*X109*X14 + X109*X14, X64*X53 + X64, X125*X122*X62 + X125*X62, X116*X115*X103 + X116*X103, X142*X135*X46 + X135*X46, X122*X22*X14 + X122*X22, X66*X30*X29 + X66*X29, X40*X38*X28 + X38*X28, X114*X9*X6 + X114*X9, X119*X35*X33 + X119*X33, X112*X69*X64 + X112*X69, X100*X97*X49 + X100*X49, X140*X107*X104 + X140*X107, X114*X113*X66 + X113*X66, X98*X80*X74 + X98*X80, X125*X82*X77 + X125*X82, X38*X27*X26 + X38*X27, X126*X124*X76 + X126*X76, X19*X12*X7 + X12*X7, X140*X119*X116 + X140*X119, X136*X135*X51 + X136*X51, X116*X112*X56 + X112*X56, X131*X124*X59 + X124*X59, X129*X124*X112 + X129*X112, X129*X125*X113 + X129*X113, X125*X118*X113 + X125*X118, X107*X102*X83 + X102*X83, X86*X43*X38 + X86*X43, X67*X55*X53 + X67*X53, X121*X106*X97 + X121*X106, X99*X98*X27 + X98*X27, X115*X57*X55 + X115*X57, X18*X12*X6 + X12*X6, X139*X59*X55 + X139*X59, X123*X120*X36 + X123*X36, X115*X93 + X115, X121*X82*X73 + X121*X82, X136*X59 + X136, X28*X25*X16 + X25*X16, X81*X72*X9 + X72*X9, X35*X25*X23 + X25*X23, X62*X53*X50 + X62*X53, X41*X36*X5 + X36*X5, X131*X128*X107 + X128*X107, X128*X22*X20 + X128*X22, X65*X61*X17 + X61*X17, X81*X75*X45 + X75*X45, X69*X21*X17 + X69*X17, X106*X98*X34 + X98*X34, X86*X85*X84 + X87*X85*X84 + X87*X86*X84 + X87*X86*X85 + X84 + X85 + X86 + X87 + 1, X96*X44*X36 + X96*X44, X137*X59*X53 + X137*X59, X93*X33*X31 + X93*X31, X115*X111*X87 + X115*X87, X99*X97*X37 + X99*X37, X33*X9*X2 + X33*X2, X111*X69*X63 + X111*X69, X118*X109*X22 + X109*X22, X125*X94*X89 + X125*X94, X102*X96*X6 + X96*X6, X110*X93*X86 + X110*X93, X102*X80*X78 + X102*X80, X116*X105 + X116, X119*X107*X105 + X119*X105, X120*X10*X0 + X120*X10, X73*X72*X1 + X72*X1, X37*X27*X25 + X37*X27, X25, X90, X128*X127*X92 + X127*X92, X78*X77*X66 + X77*X66, X101*X32*X29 + X101*X32, X131*X126*X83 + X126*X83, X137*X132*X60 + X137*X60, X128*X58*X56 + X128*X58, X142*X134*X122 + X142*X122, X82*X77*X70 + X77*X70, X113*X110*X29 + X110*X29, X141*X139*X93 + X139*X93, X142*X139*X94 + X139*X94, X38*X27 + X27, X126*X123*X75 + X126*X75, X114*X57*X54 + X114*X57, X68*X64*X56 + X64*X56, X108*X57*X48 + X108*X57, X47*X37*X23 + X37*X23, X40*X28*X27 + X40*X27, X86*X67*X62 + X86*X67, X138*X83 + X138, X33*X25*X21 + X25*X21, X102*X97*X18 + X97*X18, X124*X120*X4 + X120*X4, X89*X85*X17 + X85*X17, X140*X131*X128 + X140*X131, X38*X27 + X38, X143 + 1, X136*X132*X48 + X136*X48, X121*X70*X61 + X121*X70, X82*X10*X6 + X82*X6, X61*X5*X1 + X61*X5, X90*X89*X66 + X89*X66, X47*X36*X11 + X36*X11, X125*X120*X60 + X125*X60, X137*X71 + X71, X74*X72*X24 + X74*X24, X74*X73*X14 + X73*X14, X122*X10*X2 + X122*X10, X30*X6*X2 + X30*X2, X110*X33 + X110, X107*X96*X11 + X96*X11, X97*X80*X73 + X97*X80, X134*X133*X25 + X134*X25, X94*X84*X10 + X84*X10, X74*X54*X50 + X74*X54, X65*X60*X5 + X60*X5, X59*X51*X47 + X51*X47, X142*X132*X120 + X142*X120, X47*X38*X35 + X38*X35, X48*X4 + X4, X99*X44 + X99, X131*X83*X82 + X131*X82, X137*X47*X41 + X137*X47, X42*X38*X30 + X38*X30, X103*X80*X79 + X103*X80, X119*X112*X59 + X112*X59, X111*X45 + X45, X110*X9*X2 + X110*X9, X71*X47*X41 + X71*X41, X126*X82 + X82, X105, X119*X71*X69 + X119*X69, X12*X1 + X1, X141, X80*X77*X68 + X77*X68, X34*X24*X10 + X24*X10, X89*X88*X64 + X89*X64, X58*X51*X46 + X51*X46, X111*X33*X27 + X111*X33, X67*X61*X19 + X61*X19, X123*X34*X27 + X123*X34, X50*X49*X25 + X50*X25, X129*X70*X69 + X129*X70, X101*X68 + X68, X72*X6 + X6, X135*X131*X123 + X135*X131, X131*X127*X95 + X127*X95, X90*X79 + X79, X129*X123*X45 + X123*X45, X96*X68*X60 + X96*X68, X107*X100*X59 + X100*X59, X107*X23*X20 + X107*X20, X137*X107*X101 + X137*X107, X66*X61*X18 + X61*X18, X134*X133*X14 + X133*X14, X13*X12*X1 + X12*X1, X102*X56*X54 + X102*X56, X56*X49*X20 + X49*X20, X96*X32*X24 + X96*X32, X79*X73*X19 + X73*X19, X122*X34 + X34, X132*X11 + X11, X51*X16*X15 + X51*X16, X129*X128*X105 + X128*X105, X90*X86*X30 + X86*X30, X92*X88*X56 + X88*X56] [X73*X61 + X73*X62 + X73*X63 + X74*X61 + X74*X62 + X74*X63 + X75*X61 + X75*X62 + X75*X63 + X85*X61 + X85*X62 + X85*X63 + X85*X73 + X85*X74 + X85*X75 + X86*X61 + X86*X62 + X86*X63 + X86*X73 + X86*X74 + X86*X75 + X87*X61 + X87*X62 + X87*X63 + X87*X73 + X87*X74 + X87*X75 + 1, X62*X61 + X63*X61 + X63*X62 + X73*X62 + X73*X63 + X74*X61 + X74*X63 + X74*X73 + X75*X61 + X75*X62 + X75*X73 + X75*X74 + X85*X62 + X85*X63 + X85*X74 + X85*X75 + X86*X61 + X86*X63 + X86*X73 + X86*X75 + X86*X85 + X87*X61 + X87*X62 + X87*X73 + X87*X74 + X87*X85 + X87*X86 + 1, X73*X61 + X85*X61 + X85*X73, X62*X61 + X63*X61 + X63*X62, X74*X62 + X86*X62 + X86*X74, X74*X73 + X75*X73 + X75*X74, X75*X63 + X87*X63 + X87*X75, X86*X85 + X87*X85 + X87*X86, X121*X109 + X121*X110 + X121*X111 + X122*X109 + X122*X110 + X122*X111 + X123*X109 + X123*X110 + X123*X111 + X133*X109 + X133*X110 + X133*X111 + X133*X121 + X133*X122 + X133*X123 + X134*X109 + X134*X110 + X134*X111 + X134*X121 + X134*X122 + X134*X123 + X135*X109 + X135*X110 + X135*X111 + X135*X121 + X135*X122 + X135*X123 + 1, X110*X109 + X111*X109 + X111*X110 + X121*X110 + X121*X111 + X122*X109 + X122*X111 + X122*X121 + X123*X109 + X123*X110 + X123*X121 + X123*X122 + X133*X110 + X133*X111 + X133*X122 + X133*X123 + X134*X109 + X134*X111 + X134*X121 + X134*X123 + X134*X133 + X135*X109 + X135*X110 + X135*X121 + X135*X122 + X135*X133 + X135*X134 + 1, X121*X109 + X133*X109 + X133*X121, X110*X109 + X111*X109 + X111*X110, X122*X110 + X134*X110 + X134*X122, X122*X121 + X123*X121 + X123*X122, X123*X111 + X135*X111 + X135*X123, X134*X133 + X135*X133 + X135*X134, X125*X113 + X125*X114 + X125*X115 + X126*X113 + X126*X114 + X126*X115 + X127*X113 + X127*X114 + X127*X115 + X137*X113 + X137*X114 + X137*X115 + X137*X125 + X137*X126 + X137*X127 + X138*X113 + X138*X114 + X138*X115 + X138*X125 + X138*X126 + X138*X127 + X139*X113 + X139*X114 + X139*X115 + X139*X125 + X139*X126 + X139*X127 + 1, X114*X113 + X115*X113 + X115*X114 + X125*X114 + X125*X115 + X126*X113 + X126*X115 + X126*X125 + X127*X113 + X127*X114 + X127*X125 + X127*X126 + X137*X114 + X137*X115 + X137*X126 + X137*X127 + X138*X113 + X138*X115 + X138*X125 + X138*X127 + X138*X137 + X139*X113 + X139*X114 + X139*X125 + X139*X126 + X139*X137 + X139*X138 + 1, X125*X113 + X137*X113 + X137*X125, X114*X113 + X115*X113 + X115*X114, X126*X114 + X138*X114 + X138*X126, X126*X125 + X127*X125 + X127*X126, X127*X115 + X139*X115 + X139*X127, X138*X137 + X139*X137 + X139*X138] 24 [ X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11] [ X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23] [ X24 X25 X26 X27 X28 X29 X30 X31 X32 X33 X34 X35] [ X36 X37 X38 X39 X40 X41 X42 X43 X44 X45 X46 X47] [ X48 X49 X50 X51 X52 X53 X54 X55 X56 X57 X58 X59] [ X60 X61 X62 X63 X64 X65 X66 X67 X68 X69 X70 X71] [ X72 X73 X74 X75 X76 X77 X78 X79 X80 X81 X82 X83] [ X84 X85 X86 X87 X88 X89 X90 X91 X92 X93 X94 X95] [ X96 X97 X98 X99 X100 X101 X102 X103 X104 X105 X106 X107] [X108 X109 X110 X111 X112 X113 X114 X115 X116 X117 X118 X119] [X120 X121 X122 X123 X124 X125 X126 X127 X128 X129 X130 X131] [X132 X133 X134 X135 X136 X137 X138 X139 X140 X141 X142 X143]

R[11,0];A[11,0]
[[Prob(A[x,y])-Prob(R[x,y]) for y in range(12)] for x in range(12)]

X132 X133 + X134 + X135 + 1 [[-x0 + 1, -x1, -x2, -x3, 256*x61*x62*x63*x73*x74*x75*x85*x86*x87 - 128*x61*x62*x63*x73*x74*x75*x85*x86 - 128*x61*x62*x63*x73*x74*x75*x85*x87 - 128*x61*x62*x63*x73*x74*x75*x86*x87 - 128*x61*x62*x63*x73*x74*x85*x86*x87 - 128*x61*x62*x63*x73*x75*x85*x86*x87 - 128*x61*x62*x63*x74*x75*x85*x86*x87 - 128*x61*x62*x73*x74*x75*x85*x86*x87 - 128*x61*x63*x73*x74*x75*x85*x86*x87 - 128*x62*x63*x73*x74*x75*x85*x86*x87 + 64*x61*x62*x63*x73*x74*x75*x85 + 64*x61*x62*x63*x73*x74*x75*x86 + 64*x61*x62*x63*x73*x74*x85*x86 + 64*x61*x62*x63*x73*x75*x85*x86 + 64*x61*x62*x63*x74*x75*x85*x86 + 64*x61*x62*x73*x74*x75*x85*x86 + 64*x61*x63*x73*x74*x75*x85*x86 + 64*x62*x63*x73*x74*x75*x85*x86 + 64*x61*x62*x63*x73*x74*x75*x87 + 64*x61*x62*x63*x73*x74*x85*x87 + 64*x61*x62*x63*x73*x75*x85*x87 + 64*x61*x62*x63*x74*x75*x85*x87 + 64*x61*x62*x73*x74*x75*x85*x87 + 64*x61*x63*x73*x74*x75*x85*x87 + 64*x62*x63*x73*x74*x75*x85*x87 + 64*x61*x62*x63*x73*x74*x86*x87 + 64*x61*x62*x63*x73*x75*x86*x87 + 64*x61*x62*x63*x74*x75*x86*x87 + 64*x61*x62*x73*x74*x75*x86*x87 + 64*x61*x63*x73*x74*x75*x86*x87 + 64*x62*x63*x73*x74*x75*x86*x87 + 64*x61*x62*x63*x73*x85*x86*x87 + 64*x61*x62*x63*x74*x85*x86*x87 + 64*x61*x62*x73*x74*x85*x86*x87 + 64*x61*x63*x73*x74*x85*x86*x87 + 64*x62*x63*x73*x74*x85*x86*x87 + 64*x61*x62*x63*x75*x85*x86*x87 + 64*x61*x62*x73*x75*x85*x86*x87 + 64*x61*x63*x73*x75*x85*x86*x87 + 64*x62*x63*x73*x75*x85*x86*x87 + 64*x61*x62*x74*x75*x85*x86*x87 + 64*x61*x63*x74*x75*x85*x86*x87 + 64*x62*x63*x74*x75*x85*x86*x87 + 64*x61*x73*x74*x75*x85*x86*x87 + 64*x62*x73*x74*x75*x85*x86*x87 + 64*x63*x73*x74*x75*x85*x86*x87 - 32*x61*x62*x63*x73*x74*x75 - 32*x61*x62*x63*x73*x74*x85 - 32*x61*x62*x63*x73*x75*x85 - 32*x61*x62*x63*x74*x75*x85 - 32*x61*x62*x73*x74*x75*x85 - 32*x61*x63*x73*x74*x75*x85 - 32*x62*x63*x73*x74*x75*x85 - 32*x61*x62*x63*x73*x74*x86 - 32*x61*x62*x63*x73*x75*x86 - 32*x61*x62*x63*x74*x75*x86 - 32*x61*x62*x73*x74*x75*x86 - 32*x61*x63*x73*x74*x75*x86 - 32*x62*x63*x73*x74*x75*x86 - 32*x61*x62*x63*x73*x85*x86 - 32*x61*x62*x63*x74*x85*x86 - 32*x61*x62*x73*x74*x85*x86 - 32*x61*x63*x73*x74*x85*x86 - 32*x62*x63*x73*x74*x85*x86 - 32*x61*x62*x63*x75*x85*x86 - 32*x61*x62*x73*x75*x85*x86 - 32*x61*x63*x73*x75*x85*x86 - 32*x62*x63*x73*x75*x85*x86 - 32*x61*x62*x74*x75*x85*x86 - 32*x61*x63*x74*x75*x85*x86 - 32*x62*x63*x74*x75*x85*x86 - 32*x61*x73*x74*x75*x85*x86 - 32*x62*x73*x74*x75*x85*x86 - 32*x63*x73*x74*x75*x85*x86 - 32*x61*x62*x63*x73*x74*x87 - 32*x61*x62*x63*x73*x75*x87 - 32*x61*x62*x63*x74*x75*x87 - 32*x61*x62*x73*x74*x75*x87 - 32*x61*x63*x73*x74*x75*x87 - 32*x62*x63*x73*x74*x75*x87 - 32*x61*x62*x63*x73*x85*x87 - 32*x61*x62*x63*x74*x85*x87 - 32*x61*x62*x73*x74*x85*x87 - 32*x61*x63*x73*x74*x85*x87 - 32*x62*x63*x73*x74*x85*x87 - 32*x61*x62*x63*x75*x85*x87 - 32*x61*x62*x73*x75*x85*x87 - 32*x61*x63*x73*x75*x85*x87 - 32*x62*x63*x73*x75*x85*x87 - 32*x61*x62*x74*x75*x85*x87 - 32*x61*x63*x74*x75*x85*x87 - 32*x62*x63*x74*x75*x85*x87 - 32*x61*x73*x74*x75*x85*x87 - 32*x62*x73*x74*x75*x85*x87 - 32*x63*x73*x74*x75*x85*x87 - 32*x61*x62*x63*x73*x86*x87 - 32*x61*x62*x63*x74*x86*x87 - 32*x61*x62*x73*x74*x86*x87 - 32*x61*x63*x73*x74*x86*x87 - 32*x62*x63*x73*x74*x86*x87 - 32*x61*x62*x63*x75*x86*x87 - 32*x61*x62*x73*x75*x86*x87 - 32*x61*x63*x73*x75*x86*x87 - 32*x62*x63*x73*x75*x86*x87 - 32*x61*x62*x74*x75*x86*x87 - 32*x61*x63*x74*x75*x86*x87 - 32*x62*x63*x74*x75*x86*x87 - 32*x61*x73*x74*x75*x86*x87 - 32*x62*x73*x74*x75*x86*x87 - 32*x63*x73*x74*x75*x86*x87 - 32*x61*x62*x63*x85*x86*x87 - 32*x61*x62*x73*x85*x86*x87 - 32*x61*x63*x73*x85*x86*x87 - 32*x62*x63*x73*x85*x86*x87 - 32*x61*x62*x74*x85*x86*x87 - 32*x61*x63*x74*x85*x86*x87 - 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x112 - x113 - x114 - x115 + 1, 0, 0, 0, -x116, -x117 + 1, -x118, -x119], [-4*x121*x122*x123 + 2*x121*x122 + 2*x121*x123 + 2*x122*x123 - x120 - x121 - x122 - x123 + 1, 0, 0, 0, -4*x125*x126*x127 + 2*x125*x126 + 2*x125*x127 + 2*x126*x127 - x124 - x125 - x126 - x127 + 1, 0, 0, 0, -x128, -x129, -x130 + 1, -x131], [-4*x133*x134*x135 + 2*x133*x134 + 2*x133*x135 + 2*x134*x135 - x132 - x133 - x134 - x135 + 1, 0, 0, 0, -4*x137*x138*x139 + 2*x137*x138 + 2*x137*x139 + 2*x138*x139 - x136 - x137 - x138 - x139 + 1, 0, 0, 0, -x140, -x141, -x142, -x143 + 1]]

load('Puzzle8b.sage')

objects = 12 types = 3 rel_classes = 4 144 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X143 = R[11,11] REI.gens() = 1589 A 1st pass type 0 to type 1 [ X138 X124 + X126 + X128 0 X90 + X115 + X139] [ X49 X61 0 X85] [ X138 + 1 X124 + X125 + X126 + X129 + 1 X124 + X129 + 1 X88 + X90 + X115] [ X51 X63 X75 X87] type 0 to type 2 [ X138 + X140 X138 X126 + X129 + X142 X138 + 1] [ X97 X138 X121 X133] [X124 + X126 + X128 + X129 + 1 X124 + X129 + X138 + 1 X124 + X126 + X129 X138] [ X99 X111 X123 X135] type 1 to type 2 [X124 + X126 + X128 + X129 + X138 + 1 X124 + X129 + 1 X124 0] [ X101 X125 + X129 + X137 + 1 X125 X137] [ X116 + X138 X138 + 1 X126 X138] [ X103 X115 X127 X139] number of variables left to be determined: 31 REI.gens() = 1613 A 2nd pass type 0 to type 1 [0 1 0 0] [0 0 0 1] [1 0 0 0] [0 0 1 0] type 0 to type 2 [0 0 0 1] [1 0 0 0] [0 0 1 0] [0 1 0 0] type 1 to type 2 [0 0 1 0] [0 0 0 1] [0 1 0 0] [1 0 0 0]

<string>:18: DeprecationWarning: using 'degrevlex' in Boolean polynomial rings is deprecated. If needed, reverse the order of variables manually and use 'degneglex' See http://trac.sagemath.org/13849 for details.

if 0 in clues2: clues2.remove(0)
A;A3

[ 1 X133 + X138 0 X75 + X111 + X135 X138 X124 + X126 + X128 0 X90 + X115 + X139 X138 + X140 X138 X126 + X129 + X142 X138 + 1] [ X133 + X138 X13 X49 + X138 X37 X49 X61 0 X85 X97 X138 X121 X133] [ 0 X49 + X138 1 X51 + X75 + X111 X138 + 1 X124 + X125 + X126 + X129 + 1 X124 + X129 + 1 X88 + X90 + X115 X124 + X126 + X128 + X129 + 1 X124 + X129 + X138 + 1 X124 + X126 + X129 X138] [ X75 + X111 + X135 X37 X51 + X75 + X111 X39 X51 X63 X75 X87 X99 X111 X123 X135] [ X138 X49 X138 + 1 X51 1 X125 + X128 + X129 + X137 + 1 X124 + X129 + X138 + 1 X88 X124 + X126 + X128 + X129 + X138 + 1 X124 + X129 + 1 X124 0] [ X124 + X126 + X128 X61 X124 + X125 + X126 + X129 + 1 X63 X125 + X128 + X129 + X137 + 1 1 X124 + X125 + X126 + X128 + X129 + 1 X89 X101 X125 + X129 + X137 + 1 X125 X137] [ 0 0 X124 + X129 + 1 X75 X124 + X129 + X138 + 1 X124 + X125 + X126 + X128 + X129 + 1 1 X90 X116 + X138 X138 + 1 X126 X138] [ X90 + X115 + X139 X85 X88 + X90 + X115 X87 X88 X89 X90 X91 X103 X115 X127 X139] [ X138 + X140 X97 X124 + X126 + X128 + X129 + 1 X99 X124 + X126 + X128 + X129 + X138 + 1 X101 X116 + X138 X103 X104 X116 X128 X140] [ X138 X138 X124 + X129 + X138 + 1 X111 X124 + X129 + 1 X125 + X129 + X137 + 1 X138 + 1 X115 X116 1 X129 0] [ X126 + X129 + X142 X121 X124 + X126 + X129 X123 X124 X125 X126 X127 X128 X129 X130 X142] [ X138 + 1 X133 X138 X135 0 X137 X138 X139 X140 0 X142 1] [1 0 0 0 0 1 0 0 0 0 0 1] [0 1 0 0 0 0 0 1 1 0 0 0] [0 0 1 0 1 0 0 0 0 0 1 0] [0 0 0 1 0 0 1 0 0 1 0 0] [0 0 1 0 1 0 0 0 0 0 1 0] [1 0 0 0 0 1 0 0 0 0 0 1] [0 0 0 1 0 0 1 0 0 1 0 0] [0 1 0 0 0 0 0 1 1 0 0 0] [0 1 0 0 0 0 0 1 1 0 0 0] [0 0 0 1 0 0 1 0 0 1 0 0] [0 0 1 0 1 0 0 0 0 0 1 0] [1 0 0 0 0 1 0 0 0 0 0 1]

load('PuzzlePart2.sage')

81 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X80 = R[8,8] REI.gens() = 45 A 1st pass [ 1 X1 X2 X3 X4 X5 X6 X7 X8] [ X9 1 X11 X12 X13 X14 X15 X16 X17] [X18 X19 1 X21 X22 X23 X24 X25 X26] [X27 X28 X29 1 X31 X32 X33 X34 X35] [X36 X37 X38 X39 1 X41 X42 X43 X44] [X45 X46 X47 X48 X49 1 X51 X52 X53] [X54 X55 X56 X57 X58 X59 1 X61 X62] [X63 X64 X65 X66 X67 X68 X69 1 X71] [X72 X73 X74 X75 X76 X77 X78 X79 1]

Error in lines 1-1 Traceback (most recent call last): File "/projects/5511fe15-8085-4d1d-bdc7-c6bf6c99e693/.sagemathcloud/sage_server.py", line 865, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/5511fe15-8085-4d1d-bdc7-c6bf6c99e693/.sagemathcloud/sage_salvus.py", line 3026, in load exec 'salvus.namespace["%s"] = sage.structure.sage_object.load(*__args, **__kwds)'%t in salvus.namespace, {'__args':other_args, '__kwds':kwds} File "<string>", line 1, in <module> File "sage/structure/sage_object.pyx", line 916, in sage.structure.sage_object.load (build/cythonized/sage/structure/sage_object.c:10737) File "/usr/local/sage/sage-6.4/local/lib/python2.7/site-packages/sage/misc/preparser.py", line 1797, in load exec(preparse_file(open(fpath).read()) + "\n", globals) File "<string>", line 87, in <module> File "<string>", line 8, in newton_krylov File "/usr/local/sage/sage-6.4/local/lib/python2.7/site-packages/scipy/optimize/nonlin.py", line 323, in nonlin_solve line_search) File "/usr/local/sage/sage-6.4/local/lib/python2.7/site-packages/scipy/optimize/nonlin.py", line 401, in _nonlin_line_search amin=smin) File "/usr/local/sage/sage-6.4/local/lib/python2.7/site-packages/scipy/optimize/linesearch.py", line 611, in scalar_search_armijo phi_a0 = phi(alpha0) File "/usr/local/sage/sage-6.4/local/lib/python2.7/site-packages/scipy/optimize/nonlin.py", line 384, in phi v = func(xt) File "/usr/local/sage/sage-6.4/local/lib/python2.7/site-packages/scipy/optimize/nonlin.py", line 276, in <lambda> func = lambda z: _as_inexact(F(_array_like(z, x0))).flatten() File "<string>", line 82, in F File "sage/symbolic/expression.pyx", line 4382, in sage.symbolic.expression.Expression.__call__ (build/cythonized/sage/symbolic/expression.cpp:24669) File "/usr/local/sage/sage-6.4/local/lib/python2.7/site-packages/sage/symbolic/callable.py", line 474, in _call_element_ return SR(_the_element.substitute(**d)) File "sage/symbolic/expression.pyx", line 4233, in sage.symbolic.expression.Expression.substitute (build/cythonized/sage/symbolic/expression.cpp:23788) File "sage/symbolic/pynac.pyx", line 1568, in sage.symbolic.pynac.py_log (build/cythonized/sage/symbolic/pynac.cpp:14936) File "sage/rings/real_double.pyx", line 2029, in sage.rings.real_double.RealDoubleElement.log (build/cythonized/sage/rings/real_double.c:17088) File "sage/rings/real_double.pyx", line 1955, in sage.rings.real_double.RealDoubleElement._log_base (build/cythonized/sage/rings/real_double.c:16828) File "sage/ext/c_lib.pyx", line 73, in sage.ext.c_lib.sig_raise_exception (build/cythonized/sage/ext/c_lib.c:769) KeyboardInterrupt

R;constraints;clues;len(clues)

[ X0 X1 X2 X3 X4 X5 X6 X7 X8] [ X9 X10 X11 X12 X13 X14 X15 X16 X17] [X18 X19 X20 X21 X22 X23 X24 X25 X26] [X27 X28 X29 X30 X31 X32 X33 X34 X35] [X36 X37 X38 X39 X40 X41 X42 X43 X44] [X45 X46 X47 X48 X49 X50 X51 X52 X53] [X54 X55 X56 X57 X58 X59 X60 X61 X62] [X63 X64 X65 X66 X67 X68 X69 X70 X71] [X72 X73 X74 X75 X76 X77 X78 X79 X80] [X35*X48*X53 + X35*X48 + 1, X23*X38*X41 + X23*X38 + 1, X14*X17*X53 + X14*X53 + 1, X13*X64*X67 + X13*X64 + 1, X2*X72*X74 + X2*X72 + 1, X8*X45*X53 + X8*X45 + 1, X22*X47*X49 + X22*X47 + 1, X32*X33*X59 + X33*X59 + 1, X28*X33*X55 + X33*X55 + 1, X47*X51*X56 + X51*X56 + 1, X38*X47*X49 + X38*X49 + 1, X15*X46*X51 + X15*X46 + 1, X53*X77 + X77 + 1, X48*X75*X77 + X48*X77 + 1, X3*X8*X35 + X3*X35 + 1, X5*X27*X32 + X5*X27 + 1, X33*X35*X78 + X35*X78 + 1, X1*X8*X73 + X8*X73 + 1, X14*X46*X50 + X14*X46 + 1, X4*X5*X49 + X5*X49 + 1, X35*X75 + X75 + 1, X30*X34*X66 + X34*X66 + 1, X5*X8*X53 + X5*X53 + 1, X31*X39 + X39 + 1, X2*X5*X23 + X2*X23 + 1, X44*X49*X53 + X44*X49 + 1, X2*X8*X26 + X2*X26 + 1, X28*X32*X46 + X32*X46 + 1, X31*X35*X76 + X35*X76 + 1, X49*X52*X67 + X52*X67 + 1, X23*X65*X68 + X23*X65 + 1, X2*X7*X25 + X2*X25 + 1, X80, X42*X49*X51 + X42*X49 + 1, X11*X15*X56 + X15*X56 + 1, X2*X36*X38 + X2*X36 + 1, X2*X4*X22 + X2*X22 + 1, X23*X24*X59 + X24*X59 + 1, X24*X74*X78 + X24*X74 + 1, X59*X77*X78 + X59*X78 + 1, X4*X8*X76 + X8*X76 + 1, X26*X47*X53 + X26*X47 + 1, X15*X16*X69 + X16*X69 + 1, X45*X72*X77 + X45*X77 + 1, X39*X42*X57 + X42*X57 + 1, X37*X64*X67 + X37*X67 + 1, X27*X33*X54 + X33*X54 + 1, X23*X47 + X23 + 1, X7*X36*X43 + X7*X36 + 1, X34*X75*X79 + X34*X75 + 1, X39*X48*X49 + X39*X49 + 1, X30*X33*X57 + X33*X57 + 1, X27*X63*X66 + X27*X66 + 1, X43*X67 + X67 + 1, X45*X51*X54 + X51*X54 + 1, X57*X75*X78 + X57*X78 + 1, X46*X55*X59 + X46*X59 + 1, X25*X26*X79 + X26*X79 + 1, X71*X79 + X71 + 1, X25*X47*X52 + X25*X47 + 1, X5*X7*X68 + X7*X68 + 1, X9*X13*X36 + X13*X36 + 1, X39*X57*X58 + X39*X58 + 1, X22*X38 + X38 + 1, X21*X26*X35 + X21*X35 + 1, X22*X74*X76 + X22*X74 + 1, X22*X29*X31 + X22*X29 + 1, X53*X77 + X53 + 1, X41*X49*X50 + X41*X49 + 1, X34*X57*X61 + X34*X57 + 1, X30*X32*X48 + X32*X48 + 1, X1*X45*X46 + X1*X45 + 1, X4*X9*X13 + X4*X9 + 1, X37*X42*X55 + X42*X55 + 1, X27*X54*X57 + X27*X57 + 1, X34*X35*X71 + X34*X71 + 1, X7*X18*X25 + X7*X18 + 1, X13*X17*X44 + X13*X44 + 1, X39*X43*X66 + X43*X66 + 1, X32*X34*X52 + X32*X52 + 1, X12*X28*X30 + X12*X28 + 1, X8*X72 + X72 + 1, X22*X38*X40 + X22*X38 + 1, X18*X36*X38 + X18*X38 + 1, X11*X55*X56 + X11*X55 + 1, X5*X6*X59 + X6*X59 + 1, X12*X73*X75 + X12*X73 + 1, X51*X59*X60 + X51*X59 + 1, X1*X5*X14 + X1*X14 + 1, X43*X67 + X43 + 1, X1*X9 + X9 + 1, X31*X34*X67 + X34*X67 + 1, X5*X9*X14 + X5*X9 + 1, X2*X27*X29 + X2*X27 + 1, X29*X32*X47 + X32*X47 + 1, X24*X56*X60 + X24*X56 + 1, X3*X4*X39 + X4*X39 + 1, X1*X63*X64 + X1*X63 + 1, X10, X23*X26*X77 + X26*X77 + 1, X69*X78*X79 + X69*X79 + 1, X11*X13*X38 + X13*X38 + 1, X15*X17*X78 + X17*X78 + 1, X8*X63*X71 + X8*X63 + 1, X52*X68 + X68 + 1, X15*X55 + X15 + 1, X18*X54*X56 + X18*X56 + 1, X34*X39*X43 + X34*X39 + 1, X35*X66*X71 + X35*X66 + 1, X14*X46 + X14 + 1, X9*X14*X45 + X14*X45 + 1, X12*X16*X34 + X12*X34 + 1, X8*X18*X26 + X8*X18 + 1, X24*X47*X51 + X24*X47 + 1, X41*X44*X53 + X41*X53 + 1, X48*X66*X68 + X48*X68 + 1, X23*X24*X51 + X23*X51 + 1, X26*X74 + X74 + 1, X32*X48 + X48 + 1, X6*X27*X33 + X6*X27 + 1, X44*X76 + X44 + 1, X29*X34*X65 + X34*X65 + 1, X6*X7*X69 + X7*X69 + 1, X39*X44*X75 + X44*X75 + 1, X3*X5*X48 + X5*X48 + 1, X38*X56*X58 + X38*X58 + 1, X41*X49 + X49 + 1, X47*X74*X77 + X47*X77 + 1, X14*X15*X59 + X15*X59 + 1, X40, X4*X36 + X36 + 1, X6*X8*X62 + X6*X62 + 1, X3*X45*X48 + X3*X45 + 1, X16*X64 + X16 + 1, X18*X45*X47 + X18*X47 + 1, X27*X45*X48 + X27*X48 + 1, X25*X65*X70 + X25*X65 + 1, X6*X54*X60 + X6*X54 + 1, X34*X66 + X34 + 1, X17*X73*X80 + X17*X73 + 1, X26*X74*X80 + X26*X74 + 1, X3*X6*X57 + X6*X57 + 1, X8*X72*X80 + X8*X72 + 1, X9*X15*X54 + X15*X54 + 1, X5*X72*X77 + X5*X72 + 1, X0*X5*X45 + X5*X45 + 1, X40*X42*X58 + X42*X58 + 1, X17*X19*X26 + X17*X19 + 1, X61*X78*X79 + X61*X78 + 1, X3*X54*X57 + X3*X54 + 1, X36*X45*X49 + X36*X49 + 1, X13*X28*X31 + X13*X28 + 1, X19*X46*X47 + X19*X47 + 1, X12*X14*X32 + X12*X32 + 1, X2*X5*X47 + X5*X47 + 1, X21*X74*X75 + X21*X74 + 1, X4*X36*X40 + X4*X36 + 1, X18*X21*X27 + X21*X27 + 1, X64*X71*X73 + X71*X73 + 1, X31*X48*X49 + X31*X48 + 1, X59*X68*X69 + X59*X69 + 1, X4*X18*X22 + X4*X18 + 1, X7*X63*X70 + X7*X63 + 1, X12*X16*X66 + X16*X66 + 1, X45*X63*X68 + X45*X68 + 1, X57*X61*X66 + X61*X66 + 1, X24*X56 + X56 + 1, X21*X25*X34 + X21*X34 + 1, X8*X36*X44 + X8*X36 + 1, X3*X7*X66 + X7*X66 + 1, X2*X4*X38 + X4*X38 + 1, X40*X41*X49 + X41*X49 + 1, X32*X66*X68 + X32*X66 + 1, X29*X74*X75 + X29*X75 + 1, X23*X29*X32 + X23*X29 + 1, X1*X36*X37 + X1*X36 + 1, X29*X65*X66 + X29*X66 + 1, X3*X27 + X3 + 1, X50*X52*X68 + X52*X68 + 1, X33*X39*X42 + X33*X39 + 1, X62*X78 + X62 + 1, X22*X24*X42 + X22*X42 + 1, X36*X43*X63 + X43*X63 + 1, X12*X37*X39 + X12*X37 + 1, X42*X67*X69 + X42*X67 + 1, X47*X56*X59 + X47*X59 + 1, X27*X36*X39 + X27*X39 + 1, X71*X79 + X79 + 1, X60*X61*X69 + X61*X69 + 1, X41*X58*X59 + X41*X58 + 1, X46*X53*X73 + X53*X73 + 1, X19*X24*X55 + X24*X55 + 1, X53*X77*X80 + X53*X77 + 1, X48*X53*X75 + X53*X75 + 1, X29*X31*X38 + X31*X38 + 1, X3*X9*X12 + X3*X9 + 1, X13*X19*X22 + X13*X19 + 1, X61*X69 + X69 + 1, X2*X45*X47 + X2*X45 + 1, X51*X53*X62 + X51*X62 + 1, X45*X53*X72 + X53*X72 + 1, X22*X23*X41 + X22*X41 + 1, X19*X25*X64 + X25*X64 + 1, X10*X17*X73 + X17*X73 + 1, X24*X38*X42 + X24*X38 + 1, X26*X38*X44 + X26*X38 + 1, X5*X18*X23 + X5*X18 + 1, X2*X54*X56 + X2*X54 + 1, X3*X18*X21 + X3*X18 + 1, X1*X2*X19 + X2*X19 + 1, X27*X72*X75 + X27*X75 + 1, X20*X26*X74 + X26*X74 + 1, X14*X46 + X46 + 1, X20, X28*X64*X66 + X28*X66 + 1, X4*X5*X41 + X4*X41 + 1, X47*X52*X65 + X52*X65 + 1, X23*X25*X52 + X23*X52 + 1, X4*X7*X43 + X4*X43 + 1, X21*X24*X57 + X24*X57 + 1, X63*X72*X79 + X63*X79 + 1, X1*X3*X28 + X3*X28 + 1, X15*X19*X24 + X15*X19 + 1, X6*X7*X61 + X6*X61 + 1, X1*X27*X28 + X1*X27 + 1, X26*X74 + X26 + 1, X31*X75*X76 + X31*X75 + 1, X32*X48 + X32 + 1, X50*X51*X59 + X51*X59 + 1, X31*X39 + X31 + 1, X34*X48*X52 + X34*X48 + 1, X7*X72*X79 + X7*X72 + 1, X35*X75*X80 + X35*X75 + 1, X56*X62*X74 + X62*X74 + 1, X4*X27*X31 + X4*X27 + 1, X23*X56*X59 + X23*X56 + 1, X42*X58*X60 + X42*X58 + 1, X52*X68*X70 + X52*X68 + 1, X37*X44*X73 + X44*X73 + 1, X12*X55*X57 + X12*X55 + 1, X11*X17*X26 + X11*X26 + 1, X32*X48*X50 + X32*X48 + 1, X20*X23*X47 + X23*X47 + 1, X25*X26*X71 + X25*X71 + 1, X11*X37*X38 + X11*X37 + 1, X11*X73*X74 + X11*X73 + 1, X0*X1*X9 + X1*X9 + 1, X35*X57*X62 + X35*X57 + 1, X10*X11*X19 + X11*X19 + 1, X25*X56*X61 + X25*X56 + 1, X5*X7*X52 + X5*X52 + 1, X21*X47*X48 + X21*X47 + 1, X66*X75*X79 + X66*X79 + 1, X7*X63 + X63 + 1, X50, X19*X37*X38 + X19*X38 + 1, X44*X76 + X76 + 1, X36*X42*X54 + X42*X54 + 1, X6*X63*X69 + X6*X63 + 1, X14*X15*X51 + X14*X51 + 1, X65*X74*X79 + X65*X79 + 1, X11*X19 + X19 + 1, X58*X76*X78 + X58*X78 + 1, X32*X33*X51 + X32*X51 + 1, X41*X44*X77 + X44*X77 + 1, X52*X53*X71 + X52*X71 + 1, X42*X58 + 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X7*X27, X14*X17*X77 + X17*X77, X7*X8*X79 + X8*X79, X36*X41*X45 + X41*X45, X14*X73*X77 + X14*X73, X23*X47*X50 + X23*X47, X19*X26*X73 + X26*X73, X9*X11*X18 + X11*X18, X52*X77*X79 + X52*X77, X41*X42*X59 + X42*X59, X14*X28*X32 + X14*X28, X30*X31*X39 + X31*X39, X43*X58*X61 + X43*X58, X61*X62*X71 + X61*X71, X31*X33*X42 + X31*X42, X1*X4*X13 + X1*X13, X42*X58 + X42, X31*X34*X43 + X31*X43, X39*X75*X76 + X39*X76, X7*X9*X16 + X7*X9, X48*X51*X57 + X51*X57, X36*X63*X67 + X36*X67, X0*X8*X72 + X8*X72, X13*X16*X67 + X16*X67, X0*X4*X36 + X4*X36, X41*X49 + X41, X28*X37*X39 + X28*X39, X44*X58*X62 + X44*X58, X41*X76*X77 + X41*X76, X46*X64*X68 + X46*X68, X21*X38*X39 + X21*X38, X13*X37 + X13, X66*X71*X75 + X71*X75, X36*X54*X58 + X36*X58, X23*X47 + X47, X25*X74*X79 + X25*X74, X42*X43*X69 + X43*X69, X40*X44*X76 + X44*X76, X33*X48*X51 + X33*X48, X25*X65 + X25, X3*X8*X75 + X8*X75, X24*X25*X69 + X25*X69, X58*X61*X67 + X61*X67, X13*X46*X49 + X13*X46, X18*X24*X54 + X24*X54, X70 + 1, X49*X67*X68 + X49*X68, X18*X27*X29 + X18*X29, X4*X6*X58 + X6*X58, X36*X72*X76 + X36*X76, X30*X35*X75 + X35*X75, X1*X4*X37 + X4*X37, X6*X18*X24 + X6*X18, X27*X34*X63 + X34*X63, X1*X18*X19 + X1*X18, X1*X6*X15 + X1*X15, X11*X19 + X11, X62*X78 + X78, X0*X3*X27 + X3*X27, X25*X29*X34 + X25*X29, X33*X35*X62 + X33*X62, X3*X36*X39 + X3*X36, X12*X13*X31 + X12*X31, X1*X5*X46 + X5*X46, X54*X62*X72 + X62*X72, X4*X7*X67 + X7*X67, X13*X16*X43 + X13*X43, X31*X57*X58 + X31*X57, X10*X12*X28 + X12*X28, X13*X37*X40 + X13*X37, X41*X67*X68 + X41*X67, X16*X17*X71 + X16*X71, X21*X22*X39 + X22*X39, X56*X74*X78 + X56*X78, X1*X2*X11 + X1*X11, X27*X32*X45 + X32*X45, X52*X68 + X52, X20*X22*X38 + X22*X38, X35*X75 + X35, X16*X64 + X64, X5*X54*X59 + X5*X54, X50*X53*X77 + X53*X77, X33*X57 + X33, X11*X64*X65 + X11*X64, X22*X25*X67 + X25*X67, X38*X42*X56 + X42*X56, X15*X55*X60 + X15*X55, X62*X69*X71 + X62*X69, X3*X27 + X27, X20*X25*X65 + X25*X65, X58*X62*X76 + X62*X76, X9*X17*X72 + X17*X72, X28*X31*X37 + X31*X37, X5*X8*X77 + X8*X77, X15*X64*X69 + X15*X64, X11*X16*X65 + X16*X65, X2*X3*X29 + X3*X29, X24*X29*X33 + X24*X29, X14*X37*X41 + X14*X37, X49*X53*X76 + X53*X76, X24*X26*X62 + X24*X62, X2*X18*X20 + X2*X18, X31*X39*X40 + X31*X39, X12*X17*X35 + X12*X35, X19*X64*X65 + X19*X65, X67*X76*X79 + X67*X79, X8*X72 + X8, X19*X22*X37 + X22*X37, X5*X45 + X45, X33*X34*X69 + X34*X69, X25*X65 + X65, X45*X52*X63 + X52*X63, X28*X46*X48 + X28*X48, X6*X45*X51 + X6*X45, X11*X15*X24 + X11*X24, X8*X54*X62 + X8*X54, X2*X3*X21 + X2*X21, X28*X55*X57 + X28*X57, X11*X13*X22 + X11*X22, X24*X56 + X24, X32*X57*X59 + X32*X57, X31*X35*X44 + X31*X44, X31*X33*X58 + X33*X58, X15*X55 + X55, X56*X65*X69 + X56*X69, X55*X73*X78 + X55*X78, X43*X44*X71 + X43*X71, X42*X43*X61 + X42*X61, X32*X35*X77 + X35*X77, X38*X43*X65 + X43*X65, X64*X73*X79 + X64*X79, X38*X44*X74 + X44*X74, X11*X12*X29 + X12*X29, X2*X18 + X2, X61*X69 + X61, X14*X64*X68 + X14*X64, X51*X77*X78 + X51*X77, X17*X64*X71 + X17*X64, X29*X38*X39 + X29*X39, X10*X16*X64 + X16*X64, X16*X46*X52 + X16*X46, X13*X14*X49 + X14*X49, X22*X38 + X22, X33*X57 + X57, X3*X72*X75 + X3*X72, X24*X25*X61 + X24*X61, X63*X71*X72 + X71*X72, X18*X25*X63 + X25*X63, X2*X63*X65 + X2*X63, X21*X22*X31 + X21*X31, X21*X29 + X29, X6*X8*X78 + X8*X78, X40*X43*X67 + X43*X67, X52*X53*X79 + X53*X79, X21*X23*X32 + X21*X32, X39*X41*X48 + X41*X48, X7*X8*X71 + X7*X71, X54*X61*X63 + X61*X63, X15*X17*X62 + X15*X62, X25*X38*X43 + X25*X38, X11*X12*X21 + X11*X21, X44*X67*X71 + X44*X67, X53*X59*X62 + X53*X59, X12*X64*X66 + X12*X64, X1*X9*X10 + X1*X9, X6*X72*X78 + X6*X72, X37*X43*X64 + X43*X64, X11*X14*X47 + X14*X47, X19*X23*X46 + X23*X46, X19*X73*X74 + X19*X74, X4*X63*X67 + X4*X63, X33*X57*X60 + X33*X57, X4*X6*X42 + X4*X42, X11*X19*X20 + X11*X19, X23*X26*X53 + X23*X53, X14*X19*X23 + X14*X19, X9*X36*X37 + X9*X37, X24*X26*X78 + X26*X78, X11*X28*X29 + X11*X28, X16*X37*X43 + X16*X37, X34*X66*X70 + X34*X66, X38*X41*X47 + X41*X47, X0 + 1, X57*X62*X75 + X62*X75, X21*X25*X66 + X25*X66, X15*X16*X61 + X15*X61, X15*X37*X42 + X15*X37, X9*X18*X19 + X9*X19, X19*X28*X29 + X19*X29] 657

IF(AND(R[3,8],R[5,3]),R[5,8])
IF(R[0,0],R[0,0])

X35*X48*X53 + X35*X48 + 1 1

load('PuzzlePart3.sage')

objects = 12

gradH;gradg;gradL;gradL2;F(init)

(-log(x0) + log(-x0 + 1), -log(x1) + log(-x1 + 1), -log(x2) + log(-x2 + 1), -log(x3) + log(-x3 + 1), -log(x4) + log(-x4 + 1), -log(x5) + log(-x5 + 1), -log(x6) + log(-x6 + 1), -log(x7) + log(-x7 + 1), -log(x8) + log(-x8 + 1), -log(x9) + log(-x9 + 1), -log(x10) + log(-x10 + 1), -log(x11) + log(-x11 + 1), -log(x12) + log(-x12 + 1), -log(x13) + log(-x13 + 1), -log(x14) + log(-x14 + 1), -log(x15) + log(-x15 + 1), -log(x16) + log(-x16 + 1), -log(x17) + log(-x17 + 1), -log(x18) + log(-x18 + 1), -log(x19) + log(-x19 + 1), -log(x20) + log(-x20 + 1), -log(x21) + log(-x21 + 1), -log(x22) + log(-x22 + 1), -log(x23) + log(-x23 + 1), -log(x24) + log(-x24 + 1), -log(x25) + log(-x25 + 1), -log(x26) + log(-x26 + 1), -log(x27) + log(-x27 + 1), -log(x28) + log(-x28 + 1), -log(x29) + log(-x29 + 1), -log(x30) + log(-x30 + 1), -log(x31) + log(-x31 + 1), -log(x32) + log(-x32 + 1), -log(x33) + log(-x33 + 1), -log(x34) + log(-x34 + 1), -log(x35) + log(-x35 + 1), -log(x36) + log(-x36 + 1), -log(x37) + log(-x37 + 1), -log(x38) + log(-x38 + 1), -log(x39) + log(-x39 + 1), -log(x40) + log(-x40 + 1), -log(x41) + log(-x41 + 1), -log(x42) + log(-x42 + 1), -log(x43) + log(-x43 + 1), -log(x44) + log(-x44 + 1), -log(x45) + log(-x45 + 1), -log(x46) + log(-x46 + 1), -log(x47) + log(-x47 + 1), -log(x48) + log(-x48 + 1), -log(x49) + log(-x49 + 1), -log(x50) + log(-x50 + 1), -log(x51) + log(-x51 + 1), -log(x52) + log(-x52 + 1), -log(x53) + log(-x53 + 1), -log(x54) + log(-x54 + 1), -log(x55) + log(-x55 + 1), -log(x56) + log(-x56 + 1), -log(x57) + log(-x57 + 1), -log(x58) + log(-x58 + 1), -log(x59) + log(-x59 + 1), -log(x60) + log(-x60 + 1), -log(x61) + log(-x61 + 1), -log(x62) + log(-x62 + 1), -log(x63) + log(-x63 + 1), -log(x64) + log(-x64 + 1), -log(x65) + log(-x65 + 1), -log(x66) + log(-x66 + 1), -log(x67) + log(-x67 + 1), -log(x68) + log(-x68 + 1), -log(x69) + log(-x69 + 1), -log(x70) + log(-x70 + 1), -log(x71) + log(-x71 + 1), -log(x72) + log(-x72 + 1), -log(x73) + log(-x73 + 1), -log(x74) + log(-x74 + 1), -log(x75) + log(-x75 + 1), -log(x76) + log(-x76 + 1), -log(x77) + log(-x77 + 1), -log(x78) + log(-x78 + 1), -log(x79) + log(-x79 + 1), -log(x80) + log(-x80 + 1), -log(x81) + log(-x81 + 1), -log(x82) + log(-x82 + 1), -log(x83) + log(-x83 + 1), -log(x84) + log(-x84 + 1), -log(x85) + log(-x85 + 1), -log(x86) + log(-x86 + 1), -log(x87) + log(-x87 + 1), -log(x88) + log(-x88 + 1), -log(x89) + log(-x89 + 1), -log(x90) + log(-x90 + 1), -log(x91) + log(-x91 + 1), -log(x92) + log(-x92 + 1), -log(x93) + log(-x93 + 1), -log(x94) + log(-x94 + 1), -log(x95) + log(-x95 + 1), -log(x96) + log(-x96 + 1), -log(x97) + log(-x97 + 1), -log(x98) + log(-x98 + 1), -log(x99) + log(-x99 + 1), -log(x100) + log(-x100 + 1), -log(x101) + log(-x101 + 1), -log(x102) + log(-x102 + 1), -log(x103) + log(-x103 + 1), -log(x104) + log(-x104 + 1), -log(x105) + log(-x105 + 1), -log(x106) + log(-x106 + 1), -log(x107) + log(-x107 + 1), -log(x108) + log(-x108 + 1), -log(x109) + log(-x109 + 1), -log(x110) + log(-x110 + 1), -log(x111) + log(-x111 + 1), -log(x112) + log(-x112 + 1), -log(x113) + log(-x113 + 1), -log(x114) + log(-x114 + 1), -log(x115) + log(-x115 + 1), -log(x116) + log(-x116 + 1), -log(x117) + log(-x117 + 1), -log(x118) + log(-x118 + 1), -log(x119) + log(-x119 + 1), -log(x120) + log(-x120 + 1), -log(x121) + log(-x121 + 1), -log(x122) + log(-x122 + 1), -log(x123) + log(-x123 + 1), -log(x124) + log(-x124 + 1), -log(x125) + log(-x125 + 1), -log(x126) + log(-x126 + 1), -log(x127) + log(-x127 + 1), -log(x128) + log(-x128 + 1), -log(x129) + log(-x129 + 1), -log(x130) + log(-x130 + 1), -log(x131) + log(-x131 + 1), -log(x132) + log(-x132 + 1), -log(x133) + log(-x133 + 1), -log(x134) + log(-x134 + 1), -log(x135) + log(-x135 + 1), -log(x136) + log(-x136 + 1), -log(x137) + log(-x137 + 1), -log(x138) + log(-x138 + 1), -log(x139) + log(-x139 + 1), -log(x140) + log(-x140 + 1), -log(x141) + log(-x141 + 1), -log(x142) + log(-x142 + 1), -log(x143) + log(-x143 + 1), 0, 0, 0, 0, 0) (0, 0, 0, 0, (-2.0*x11*x28*x35 + x35)*lam4, 0, 0, 0, 0, 0, 0, (-2.0*x28*x35*x4 + x28)*lam4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-2.0*x11*x35*x4 + x11)*lam4, 0, 0, 0, 0, 0, 0, (-2.0*x11*x28*x4 + x4)*lam4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (x57*x58*x69*x70*x71 - x57*x58*x69*x71 - 2.0*x57*x58*x70*x71 - 2.0*x57*x69*x70*x71 - x58*x69*x70*x71 + x57*x69*x70 + 3.0*x57*x58*x71 + x57*x69*x71 + x58*x69*x71 + 4.0*x57*x70*x71 + x58*x70*x71 + 3.0*x69*x70*x71 - 2.0*x57*x70 - 2.0*x69*x70 - 2.0*x57*x71 - 2.0*x58*x71 - 2.0*x69*x71 - 2.0*x70*x71 + x69 + x70 + x71)*lam3, (3.0*x58*x59*x80*x81*x82 - 2.0*x58*x80*x81 - 2.0*x59*x80*x82 + x80)*lam0 + (x56*x58*x69*x70*x71 - x56*x58*x69*x71 - 2.0*x56*x58*x70*x71 - 2.0*x56*x69*x70*x71 + x56*x69*x70 + 3.0*x56*x58*x71 + x56*x69*x71 + 4.0*x56*x70*x71 + x58*x70*x71 - 2.0*x56*x70 - 2.0*x56*x71 - 2.0*x58*x71 - 2.0*x70*x71 + x70 + x71)*lam3, (3.0*x57*x59*x80*x81*x82 - 2.0*x57*x80*x81 - 2.0*x59*x81*x82 + x81)*lam0 + (x56*x57*x69*x70*x71 - x56*x57*x69*x71 - 2.0*x56*x57*x70*x71 - x56*x69*x70*x71 + 3.0*x56*x57*x71 + x56*x69*x71 + x56*x70*x71 + x57*x70*x71 - 2.0*x56*x71 - 2.0*x57*x71 + x71)*lam3, (3.0*x57*x58*x80*x81*x82 - 2.0*x57*x80*x82 - 2.0*x58*x81*x82 + x82)*lam0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (x56*x57*x58*x70*x71 - x56*x57*x58*x71 - 2.0*x56*x57*x70*x71 - x56*x58*x70*x71 + x56*x57*x70 + x56*x57*x71 + x56*x58*x71 + 3.0*x56*x70*x71 - 2.0*x56*x70 - 2.0*x56*x71 + x56)*lam3, (x56*x57*x58*x69*x71 - 2.0*x56*x57*x58*x71 - 2.0*x56*x57*x69*x71 - x56*x58*x69*x71 + x56*x57*x69 + 4.0*x56*x57*x71 + x56*x58*x71 + x57*x58*x71 + 3.0*x56*x69*x71 - 2.0*x56*x57 - 2.0*x56*x69 - 2.0*x56*x71 - 2.0*x57*x71 + x56 + x57)*lam3, (x56*x57*x58*x69*x70 - x56*x57*x58*x69 - 2.0*x56*x57*x58*x70 - 2.0*x56*x57*x69*x70 - x56*x58*x69*x70 + 3.0*x56*x57*x58 + x56*x57*x69 + x56*x58*x69 + 4.0*x56*x57*x70 + x56*x58*x70 + x57*x58*x70 + 3.0*x56*x69*x70 - 2.0*x56*x57 - 2.0*x56*x58 - 2.0*x57*x58 - 2.0*x56*x69 - 2.0*x56*x70 - 2.0*x57*x70 + x56 + x57 + x58)*lam3, 0, 0, 0, 0, 0, 0, 0, 0, (3.0*x57*x58*x59*x81*x82 - 2.0*x57*x58*x81 - 2.0*x57*x59*x82 + x57)*lam0, (3.0*x57*x58*x59*x80*x82 - 2.0*x57*x58*x80 - 2.0*x58*x59*x82 + x58)*lam0, (3.0*x57*x58*x59*x80*x81 - 2.0*x57*x59*x80 - 2.0*x58*x59*x81 + x59)*lam0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-2.0*x114*x132*x138 + x138)*lam1, lam2*(-2.0*x114 + 1), 0, 0, 0, 0, (-2.0*x108*x132*x138 + x132)*lam1 + lam2*(-2.0*x109 + 1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-2.0*x108*x114*x138 + x114)*lam1, 0, 0, 0, 0, 0, (-2.0*x108*x114*x132 + x108)*lam1, 0, 0, 0, 0, 0, 3.0*x57*x58*x59*x80*x81*x82 - 2.0*x57*x58*x80*x81 - 2.0*x57*x59*x80*x82 - 2.0*x58*x59*x81*x82 + x57*x80 + x58*x81 + x59*x82 - 0.7806915470435619, -2.0*x108*x114*x132*x138 + x114*x132 + x108*x138 - 0.8805876422424685, -2.0*x109*x114 + x109 + x114 - 0.48762300276547044, x56*x57*x58*x69*x70*x71 - x56*x57*x58*x69*x71 - 2.0*x56*x57*x58*x70*x71 - 2.0*x56*x57*x69*x70*x71 - x56*x58*x69*x70*x71 + x56*x57*x69*x70 + 3.0*x56*x57*x58*x71 + x56*x57*x69*x71 + x56*x58*x69*x71 + 4.0*x56*x57*x70*x71 + x56*x58*x70*x71 + x57*x58*x70*x71 + 3.0*x56*x69*x70*x71 - 2.0*x56*x57*x70 - 2.0*x56*x69*x70 - 2.0*x56*x57*x71 - 2.0*x56*x58*x71 - 2.0*x57*x58*x71 - 2.0*x56*x69*x71 - 2.0*x56*x70*x71 - 2.0*x57*x70*x71 + x56*x69 + x56*x70 + x57*x70 + x56*x71 + x57*x71 + x58*x71 - 0.3998090675557092, -2.0*x11*x28*x35*x4 + x11*x28 + x35*x4 - 0.29511818155036684) (-log(x0) + log(-x0 + 1), -log(x1) + log(-x1 + 1), -log(x2) + log(-x2 + 1), -log(x3) + log(-x3 + 1), (-2.0*x11*x28*x35 + x35)*lam4 - log(x4) + log(-x4 + 1), -log(x5) + log(-x5 + 1), -log(x6) + log(-x6 + 1), -log(x7) + log(-x7 + 1), -log(x8) + log(-x8 + 1), -log(x9) + log(-x9 + 1), -log(x10) + log(-x10 + 1), (-2.0*x28*x35*x4 + x28)*lam4 - log(x11) + log(-x11 + 1), -log(x12) + log(-x12 + 1), -log(x13) + log(-x13 + 1), -log(x14) + log(-x14 + 1), -log(x15) + log(-x15 + 1), -log(x16) + log(-x16 + 1), -log(x17) + log(-x17 + 1), -log(x18) + log(-x18 + 1), -log(x19) + log(-x19 + 1), -log(x20) + log(-x20 + 1), -log(x21) + log(-x21 + 1), -log(x22) + log(-x22 + 1), -log(x23) + log(-x23 + 1), -log(x24) + log(-x24 + 1), -log(x25) + log(-x25 + 1), -log(x26) + log(-x26 + 1), -log(x27) + log(-x27 + 1), (-2.0*x11*x35*x4 + x11)*lam4 - log(x28) + log(-x28 + 1), -log(x29) + log(-x29 + 1), -log(x30) + log(-x30 + 1), -log(x31) + log(-x31 + 1), -log(x32) + log(-x32 + 1), -log(x33) + log(-x33 + 1), -log(x34) + log(-x34 + 1), (-2.0*x11*x28*x4 + x4)*lam4 - log(x35) + log(-x35 + 1), -log(x36) + log(-x36 + 1), -log(x37) + log(-x37 + 1), -log(x38) + log(-x38 + 1), -log(x39) + log(-x39 + 1), -log(x40) + log(-x40 + 1), -log(x41) + log(-x41 + 1), -log(x42) + log(-x42 + 1), -log(x43) + log(-x43 + 1), -log(x44) + log(-x44 + 1), -log(x45) + log(-x45 + 1), -log(x46) + log(-x46 + 1), -log(x47) + log(-x47 + 1), -log(x48) + log(-x48 + 1), -log(x49) + log(-x49 + 1), -log(x50) + log(-x50 + 1), -log(x51) + log(-x51 + 1), -log(x52) + log(-x52 + 1), -log(x53) + log(-x53 + 1), -log(x54) + log(-x54 + 1), -log(x55) + log(-x55 + 1), (x57*x58*x69*x70*x71 - x57*x58*x69*x71 - 2.0*x57*x58*x70*x71 - 2.0*x57*x69*x70*x71 - x58*x69*x70*x71 + x57*x69*x70 + 3.0*x57*x58*x71 + x57*x69*x71 + x58*x69*x71 + 4.0*x57*x70*x71 + x58*x70*x71 + 3.0*x69*x70*x71 - 2.0*x57*x70 - 2.0*x69*x70 - 2.0*x57*x71 - 2.0*x58*x71 - 2.0*x69*x71 - 2.0*x70*x71 + x69 + x70 + x71)*lam3 - log(x56) + log(-x56 + 1), (3.0*x58*x59*x80*x81*x82 - 2.0*x58*x80*x81 - 2.0*x59*x80*x82 + x80)*lam0 + (x56*x58*x69*x70*x71 - x56*x58*x69*x71 - 2.0*x56*x58*x70*x71 - 2.0*x56*x69*x70*x71 + x56*x69*x70 + 3.0*x56*x58*x71 + x56*x69*x71 + 4.0*x56*x70*x71 + x58*x70*x71 - 2.0*x56*x70 - 2.0*x56*x71 - 2.0*x58*x71 - 2.0*x70*x71 + x70 + x71)*lam3 - log(x57) + log(-x57 + 1), (3.0*x57*x59*x80*x81*x82 - 2.0*x57*x80*x81 - 2.0*x59*x81*x82 + x81)*lam0 + (x56*x57*x69*x70*x71 - x56*x57*x69*x71 - 2.0*x56*x57*x70*x71 - x56*x69*x70*x71 + 3.0*x56*x57*x71 + x56*x69*x71 + x56*x70*x71 + x57*x70*x71 - 2.0*x56*x71 - 2.0*x57*x71 + x71)*lam3 - log(x58) + log(-x58 + 1), (3.0*x57*x58*x80*x81*x82 - 2.0*x57*x80*x82 - 2.0*x58*x81*x82 + x82)*lam0 - log(x59) + log(-x59 + 1), -log(x60) + log(-x60 + 1), -log(x61) + log(-x61 + 1), -log(x62) + log(-x62 + 1), -log(x63) + log(-x63 + 1), -log(x64) + log(-x64 + 1), -log(x65) + log(-x65 + 1), -log(x66) + log(-x66 + 1), -log(x67) + log(-x67 + 1), -log(x68) + log(-x68 + 1), (x56*x57*x58*x70*x71 - x56*x57*x58*x71 - 2.0*x56*x57*x70*x71 - x56*x58*x70*x71 + x56*x57*x70 + x56*x57*x71 + x56*x58*x71 + 3.0*x56*x70*x71 - 2.0*x56*x70 - 2.0*x56*x71 + x56)*lam3 - log(x69) + log(-x69 + 1), (x56*x57*x58*x69*x71 - 2.0*x56*x57*x58*x71 - 2.0*x56*x57*x69*x71 - x56*x58*x69*x71 + x56*x57*x69 + 4.0*x56*x57*x71 + x56*x58*x71 + x57*x58*x71 + 3.0*x56*x69*x71 - 2.0*x56*x57 - 2.0*x56*x69 - 2.0*x56*x71 - 2.0*x57*x71 + x56 + x57)*lam3 - log(x70) + log(-x70 + 1), (x56*x57*x58*x69*x70 - x56*x57*x58*x69 - 2.0*x56*x57*x58*x70 - 2.0*x56*x57*x69*x70 - x56*x58*x69*x70 + 3.0*x56*x57*x58 + x56*x57*x69 + x56*x58*x69 + 4.0*x56*x57*x70 + x56*x58*x70 + x57*x58*x70 + 3.0*x56*x69*x70 - 2.0*x56*x57 - 2.0*x56*x58 - 2.0*x57*x58 - 2.0*x56*x69 - 2.0*x56*x70 - 2.0*x57*x70 + x56 + x57 + x58)*lam3 - log(x71) + log(-x71 + 1), -log(x72) + log(-x72 + 1), -log(x73) + log(-x73 + 1), -log(x74) + log(-x74 + 1), -log(x75) + log(-x75 + 1), -log(x76) + log(-x76 + 1), -log(x77) + log(-x77 + 1), -log(x78) + log(-x78 + 1), -log(x79) + log(-x79 + 1), (3.0*x57*x58*x59*x81*x82 - 2.0*x57*x58*x81 - 2.0*x57*x59*x82 + x57)*lam0 - log(x80) + log(-x80 + 1), (3.0*x57*x58*x59*x80*x82 - 2.0*x57*x58*x80 - 2.0*x58*x59*x82 + x58)*lam0 - log(x81) + log(-x81 + 1), (3.0*x57*x58*x59*x80*x81 - 2.0*x57*x59*x80 - 2.0*x58*x59*x81 + x59)*lam0 - log(x82) + log(-x82 + 1), -log(x83) + log(-x83 + 1), -log(x84) + log(-x84 + 1), -log(x85) + log(-x85 + 1), -log(x86) + log(-x86 + 1), -log(x87) + log(-x87 + 1), -log(x88) + log(-x88 + 1), -log(x89) + log(-x89 + 1), -log(x90) + log(-x90 + 1), -log(x91) + log(-x91 + 1), -log(x92) + log(-x92 + 1), -log(x93) + log(-x93 + 1), -log(x94) + log(-x94 + 1), -log(x95) + log(-x95 + 1), -log(x96) + log(-x96 + 1), -log(x97) + log(-x97 + 1), -log(x98) + log(-x98 + 1), -log(x99) + log(-x99 + 1), -log(x100) + log(-x100 + 1), -log(x101) + log(-x101 + 1), -log(x102) + log(-x102 + 1), -log(x103) + log(-x103 + 1), -log(x104) + log(-x104 + 1), -log(x105) + log(-x105 + 1), -log(x106) + log(-x106 + 1), -log(x107) + log(-x107 + 1), (-2.0*x114*x132*x138 + x138)*lam1 - log(x108) + log(-x108 + 1), lam2*(-2.0*x114 + 1) - log(x109) + log(-x109 + 1), -log(x110) + log(-x110 + 1), -log(x111) + log(-x111 + 1), -log(x112) + log(-x112 + 1), -log(x113) + log(-x113 + 1), (-2.0*x108*x132*x138 + x132)*lam1 + lam2*(-2.0*x109 + 1) - log(x114) + log(-x114 + 1), -log(x115) + log(-x115 + 1), -log(x116) + log(-x116 + 1), -log(x117) + log(-x117 + 1), -log(x118) + log(-x118 + 1), -log(x119) + log(-x119 + 1), -log(x120) + log(-x120 + 1), -log(x121) + log(-x121 + 1), -log(x122) + log(-x122 + 1), -log(x123) + log(-x123 + 1), -log(x124) + log(-x124 + 1), -log(x125) + log(-x125 + 1), -log(x126) + log(-x126 + 1), -log(x127) + log(-x127 + 1), -log(x128) + log(-x128 + 1), -log(x129) + log(-x129 + 1), -log(x130) + log(-x130 + 1), -log(x131) + log(-x131 + 1), (-2.0*x108*x114*x138 + x114)*lam1 - log(x132) + log(-x132 + 1), -log(x133) + log(-x133 + 1), -log(x134) + log(-x134 + 1), -log(x135) + log(-x135 + 1), -log(x136) + log(-x136 + 1), -log(x137) + log(-x137 + 1), (-2.0*x108*x114*x132 + x108)*lam1 - log(x138) + log(-x138 + 1), -log(x139) + log(-x139 + 1), -log(x140) + log(-x140 + 1), -log(x141) + log(-x141 + 1), -log(x142) + log(-x142 + 1), -log(x143) + log(-x143 + 1), 3.0*x57*x58*x59*x80*x81*x82 - 2.0*x57*x58*x80*x81 - 2.0*x57*x59*x80*x82 - 2.0*x58*x59*x81*x82 + x57*x80 + x58*x81 + x59*x82 - 0.7806915470435619, -2.0*x108*x114*x132*x138 + x114*x132 + x108*x138 - 0.8805876422424685, -2.0*x109*x114 + x109 + x114 - 0.48762300276547044, x56*x57*x58*x69*x70*x71 - x56*x57*x58*x69*x71 - 2.0*x56*x57*x58*x70*x71 - 2.0*x56*x57*x69*x70*x71 - x56*x58*x69*x70*x71 + x56*x57*x69*x70 + 3.0*x56*x57*x58*x71 + x56*x57*x69*x71 + x56*x58*x69*x71 + 4.0*x56*x57*x70*x71 + x56*x58*x70*x71 + x57*x58*x70*x71 + 3.0*x56*x69*x70*x71 - 2.0*x56*x57*x70 - 2.0*x56*x69*x70 - 2.0*x56*x57*x71 - 2.0*x56*x58*x71 - 2.0*x57*x58*x71 - 2.0*x56*x69*x71 - 2.0*x56*x70*x71 - 2.0*x57*x70*x71 + x56*x69 + x56*x70 + x57*x70 + x56*x71 + x57*x71 + x58*x71 - 0.3998090675557092, -2.0*x11*x28*x35*x4 + x11*x28 + x35*x4 - 0.29511818155036684) [(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x0) + log(-x0 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x1) + log(-x1 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x2) + log(-x2 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x3) + log(-x3 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (-2.0*x11*x28*x35 + x35)*lam4 - log(x4) + log(-x4 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x5) + log(-x5 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x6) + log(-x6 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x7) + log(-x7 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x8) + log(-x8 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x9) + log(-x9 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x10) + log(-x10 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (-2.0*x28*x35*x4 + x28)*lam4 - log(x11) + log(-x11 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x12) + log(-x12 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x13) + log(-x13 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x14) + log(-x14 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x15) + log(-x15 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x16) + log(-x16 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x17) + log(-x17 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x18) + log(-x18 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x19) + log(-x19 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x20) + log(-x20 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x21) + log(-x21 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x22) + log(-x22 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x23) + log(-x23 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x24) + log(-x24 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x25) + log(-x25 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x26) + log(-x26 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x27) + log(-x27 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (-2.0*x11*x35*x4 + x11)*lam4 - log(x28) + log(-x28 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x29) + log(-x29 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x30) + log(-x30 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x31) + log(-x31 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x32) + log(-x32 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x33) + log(-x33 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x34) + log(-x34 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (-2.0*x11*x28*x4 + x4)*lam4 - log(x35) + log(-x35 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x36) + log(-x36 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x37) + log(-x37 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x38) + log(-x38 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x39) + log(-x39 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x40) + log(-x40 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x41) + log(-x41 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x42) + log(-x42 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x43) + log(-x43 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x44) + log(-x44 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x45) + log(-x45 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x46) + log(-x46 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x47) + log(-x47 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x48) + log(-x48 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x49) + log(-x49 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x50) + log(-x50 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x51) + log(-x51 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x52) + log(-x52 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x53) + log(-x53 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x54) + log(-x54 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x55) + log(-x55 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (x57*x58*x69*x70*x71 - x57*x58*x69*x71 - 2.0*x57*x58*x70*x71 - 2.0*x57*x69*x70*x71 - x58*x69*x70*x71 + x57*x69*x70 + 3.0*x57*x58*x71 + x57*x69*x71 + x58*x69*x71 + 4.0*x57*x70*x71 + x58*x70*x71 + 3.0*x69*x70*x71 - 2.0*x57*x70 - 2.0*x69*x70 - 2.0*x57*x71 - 2.0*x58*x71 - 2.0*x69*x71 - 2.0*x70*x71 + x69 + x70 + x71)*lam3 - log(x56) + log(-x56 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (3.0*x58*x59*x80*x81*x82 - 2.0*x58*x80*x81 - 2.0*x59*x80*x82 + x80)*lam0 + (x56*x58*x69*x70*x71 - x56*x58*x69*x71 - 2.0*x56*x58*x70*x71 - 2.0*x56*x69*x70*x71 + x56*x69*x70 + 3.0*x56*x58*x71 + x56*x69*x71 + 4.0*x56*x70*x71 + x58*x70*x71 - 2.0*x56*x70 - 2.0*x56*x71 - 2.0*x58*x71 - 2.0*x70*x71 + x70 + x71)*lam3 - log(x57) + log(-x57 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (3.0*x57*x59*x80*x81*x82 - 2.0*x57*x80*x81 - 2.0*x59*x81*x82 + x81)*lam0 + (x56*x57*x69*x70*x71 - x56*x57*x69*x71 - 2.0*x56*x57*x70*x71 - x56*x69*x70*x71 + 3.0*x56*x57*x71 + x56*x69*x71 + x56*x70*x71 + x57*x70*x71 - 2.0*x56*x71 - 2.0*x57*x71 + x71)*lam3 - log(x58) + log(-x58 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (3.0*x57*x58*x80*x81*x82 - 2.0*x57*x80*x82 - 2.0*x58*x81*x82 + x82)*lam0 - log(x59) + log(-x59 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x60) + log(-x60 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x61) + log(-x61 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x62) + log(-x62 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x63) + log(-x63 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x64) + log(-x64 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x65) + log(-x65 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x66) + log(-x66 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x67) + log(-x67 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x68) + log(-x68 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (x56*x57*x58*x70*x71 - x56*x57*x58*x71 - 2.0*x56*x57*x70*x71 - x56*x58*x70*x71 + x56*x57*x70 + x56*x57*x71 + x56*x58*x71 + 3.0*x56*x70*x71 - 2.0*x56*x70 - 2.0*x56*x71 + x56)*lam3 - log(x69) + log(-x69 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (x56*x57*x58*x69*x71 - 2.0*x56*x57*x58*x71 - 2.0*x56*x57*x69*x71 - x56*x58*x69*x71 + x56*x57*x69 + 4.0*x56*x57*x71 + x56*x58*x71 + x57*x58*x71 + 3.0*x56*x69*x71 - 2.0*x56*x57 - 2.0*x56*x69 - 2.0*x56*x71 - 2.0*x57*x71 + x56 + x57)*lam3 - log(x70) + log(-x70 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (x56*x57*x58*x69*x70 - x56*x57*x58*x69 - 2.0*x56*x57*x58*x70 - 2.0*x56*x57*x69*x70 - x56*x58*x69*x70 + 3.0*x56*x57*x58 + x56*x57*x69 + x56*x58*x69 + 4.0*x56*x57*x70 + x56*x58*x70 + x57*x58*x70 + 3.0*x56*x69*x70 - 2.0*x56*x57 - 2.0*x56*x58 - 2.0*x57*x58 - 2.0*x56*x69 - 2.0*x56*x70 - 2.0*x57*x70 + x56 + x57 + x58)*lam3 - log(x71) + log(-x71 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x72) + log(-x72 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x73) + log(-x73 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x74) + log(-x74 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x75) + log(-x75 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x76) + log(-x76 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x77) + log(-x77 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x78) + log(-x78 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x79) + log(-x79 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (3.0*x57*x58*x59*x81*x82 - 2.0*x57*x58*x81 - 2.0*x57*x59*x82 + x57)*lam0 - log(x80) + log(-x80 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (3.0*x57*x58*x59*x80*x82 - 2.0*x57*x58*x80 - 2.0*x58*x59*x82 + x58)*lam0 - log(x81) + log(-x81 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (3.0*x57*x58*x59*x80*x81 - 2.0*x57*x59*x80 - 2.0*x58*x59*x81 + x59)*lam0 - log(x82) + log(-x82 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x83) + log(-x83 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x84) + log(-x84 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x85) + log(-x85 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x86) + log(-x86 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x87) + log(-x87 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x88) + log(-x88 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x89) + log(-x89 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x90) + log(-x90 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x91) + log(-x91 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x92) + log(-x92 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x93) + log(-x93 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x94) + log(-x94 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x95) + log(-x95 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x96) + log(-x96 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x97) + log(-x97 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x98) + log(-x98 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x99) + log(-x99 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x100) + log(-x100 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x101) + log(-x101 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x102) + log(-x102 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x103) + log(-x103 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x104) + log(-x104 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x105) + log(-x105 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x106) + log(-x106 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x107) + log(-x107 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (-2.0*x114*x132*x138 + x138)*lam1 - log(x108) + log(-x108 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> lam2*(-2.0*x114 + 1) - log(x109) + log(-x109 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x110) + log(-x110 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x111) + log(-x111 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x112) + log(-x112 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x113) + log(-x113 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> (-2.0*x108*x132*x138 + x132)*lam1 + lam2*(-2.0*x109 + 1) - log(x114) + log(-x114 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x115) + log(-x115 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x116) + log(-x116 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x117) + log(-x117 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125, x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136, x137, x138, x139, x140, x141, x142, x143, lam0, lam1, lam2, lam3, lam4) |--> -log(x118) + log(-x118 + 1), (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54, x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65, x66, x67, x68, x69, x70, x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81, x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92, x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103, x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114, x115, x116, x117, x118,[...] [0.2543944676226072, -0.12584648087421624, 1.0622071698683853, 0.03205205473397277, -1.1808791636898353, 1.1765534176610453, 0.26340778183551916, -1.1922136787886524, -0.6603201761459235, -1.2809877322844454, 0.5305235637193543, 1.059448597122751, 0.9981259768256585, -0.6600800212290965, -0.862002357351683, -0.5127401731701646, -0.7443331978386591, 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0.3656652282594651, -1.175418152029056, 0.4904527700527382, -0.2051986896109298, 1.3245063032642095, 0.38646291014568646, 0.45195287310308957, -0.19975634367361084, 0.759454363699378, 0.7566436446894069, 0.9057238752146559, 0.9196254106137176, 0.38150465485204466, 1.3200883864211672, 1.126800031298564, 1.1253916572539588, 0.08888898613793694, -0.12848267868437227, -1.2997944202017533, -0.32081540484739335, -1.3467422045095685, -1.2273647817594275, -1.3238970461189141, 0.7732926923848134, 0.5984035944286944, 0.031660628555176484, 0.017398547365053085, 0.4737553415132414, -0.49104352437011256, 0.8198287361334007, -0.6515667510821533, 0.39838683421276744, 0.6669208448557125, 1.0677540285052178, 0.3909123211140437, 0.47319465052223514, -0.7368948136993231, -0.6999828822791174, 0.2799279018275449, -0.38682396305761657, 0.12370084834681694, 0.2849970771818656, -0.8447965896772167, 1.2987207722203655, -0.4246735989732671, 0.07600314177278433, -0.792384127618955, -1.0571471025834254, -0.19500981515727123, 1.2552228675690664, 0.4541251483597762, 0.8953066176062452, -0.4716692682751306, -0.6057428319237339, 0.9905112461616884, 0.4927953015674285, 0.3258966540019791, 0.07960098101363611, -0.5340803498803501, -0.9537751838555395, 0.637148739157273, 0.06338505352017976, -1.1601450414695984, -1.313274658318529, 0.15317685083494925, -1.0191503815964413, 0.0014141544166700326, -0.27016319153475365, 0.8900661494015956, -0.2696159765146544, -0.8378401616212467, 0.8453081381735765, -0.25068013933146216, 0.674675985625135, 0.998447143310397, -0.19243562855533636, -0.5681139457047484, 0.9594451063150383, -1.2662056222384872, -0.38584623903849097, -0.688839543088308, 0.09400394124740696, -0.12485061631942229, 0.1492779999391509]

ssoln = scipy.optimize.newton_krylov(F, init, method='lgmres', f_tol=1e-14)

(F(init)[0].n()).parent()

Real Field with 53 bits of precision

a=var('a');b=var('b')
solve(x^2-(a-b-1)*x+1-a==0,x)
solve(x^2-(b-a-1)*x+1-b==0,x)

[x == 1/2*a - 1/2*b - 1/2*sqrt(a^2 - 2*(a - 1)*b + b^2 + 2*a - 3) - 1/2, x == 1/2*a - 1/2*b + 1/2*sqrt(a^2 - 2*(a - 1)*b + b^2 + 2*a - 3) - 1/2] [x == -1/2*a + 1/2*b - 1/2*sqrt(a^2 - 2*(a - 1)*b + b^2 + 2*a - 3) - 1/2, x == -1/2*a + 1/2*b + 1/2*sqrt(a^2 - 2*(a - 1)*b + b^2 + 2*a - 3) - 1/2]

a,b=.75,.75
solve(x^2-(a-b-1)*x+1-a==0,x)
solve(x^2-(b-a-1)*x+1-b==0,x)

[x == (-1/2)] [x == (-1/2)]

IFF(P,P)

1

Prob(IF(Q,P)*IF(P,Q))/Prob(IF(P,Q))
GIVEN(IF(Q,P),IF(P,Q))

(2*p*q - p - q + 1)/(p*q - p + 1) (2*p*q - p - q + 1)/(p*q - p + 1)

load('maxentrocond2.sage')
g
gradH
gradg
gradL
ssoln
htt

2.00000000000000*x0*x1 - x0 - x1 + 0.350000000000000 (-log(x0) + log(-x0 + 1), -log(x1) + log(-x1 + 1), 0) (lam0*(2.00000000000000*x1 - 1.00000000000000), lam0*(2.00000000000000*x0 - 1.00000000000000), 2.00000000000000*x0*x1 - x0 - x1 + 0.350000000000000) (lam0*(2.00000000000000*x1 - 1.00000000000000) - log(x0) + log(-x0 + 1), lam0*(2.00000000000000*x0 - 1.00000000000000) - log(x1) + log(-x1 + 1), 2.00000000000000*x0*x1 - x0 - x1 + 0.350000000000000) [0.22613872124741694, 0.22613872124741691, 2.2461079837862061] 1.54243851797692

H
g
gradH
gradg
gradL
critical
ssoln
htt

-(x0 - 1)*(x1 - 1)*log((x0 - 1)*(x1 - 1)) + x0*(x1 - 1)*log(-x0*(x1 - 1)) + (x0 - 1)*x1*log(-(x0 - 1)*x1) - x0*x1*log(x0*x1) 2.00000000000000*x0*x1 - x0 - x1 + 0.700000000000000 (-log(x0) + log(-x0 + 1), -log(x1) + log(-x1 + 1), 0) (lam0*(2.00000000000000*x1 - 1.00000000000000), lam0*(2.00000000000000*x0 - 1.00000000000000), 2.00000000000000*x0*x1 - x0 - x1 + 0.700000000000000) (lam0*(2.00000000000000*x1 - 1.00000000000000) - log(x0) + log(-x0 + 1), lam0*(2.00000000000000*x0 - 1.00000000000000) - log(x1) + log(-x1 + 1), 2.00000000000000*x0*x1 - x0 - x1 + 0.700000000000000) [{2.0*x0*x1 - x0 - x1 + 0.7: 0}, {lam0*(2.0*x1 - 1.0) - log(x0) + log(-x0 + 1): 0}, {lam0*(2.0*x0 - 1.0) - log(x1) + log(-x1 + 1): 0}] [(0.18377223398316217+9.0564754739500348e-18j), (0.81622776601683789-8.099504146052118e-18j), (-2.3574721596638955+4.2550371721195694e-17j)] (0.95413112102032305+2.5579502290613354e-17j)

X = EX7.gens() #generators X[n] are independent random variables that can take on values 0 or 1
NewGens = str(X).lower()
NewGens = NewGens[1:len(NewGens) - 1]
p,q,r,s,v,w = var(NewGens) #x = Pr(X)
NewGens
MPR=(Prob(IF(P,Q))).parent()
NI=ideal([Prob(IF(P,Q))-QQ(.4),Prob(IF(Q,P))-QQ(.6)])
NGB=NI.groebner_basis();NGB;MPR.term_order()

'p, q, r, s, v, w' [p - q - 1/5, q^2 - 4/5*q + 2/5] Lexicographic term order

x=var('x')
b=.65
soln=solve(2*x^2-2*x+1-b==0,x);soln
ccsoln=[CC(s.rhs()) for s in soln];ccsoln

[x == -1/20*sqrt(30) + 1/2, x == 1/20*sqrt(30) + 1/2] [0.226138721247417, 0.773861278752583]

a=var('a')
solve(2*x^2-2*x+1-a==0,x)

[x == -1/2*sqrt(2*a - 1) + 1/2, x == 1/2*sqrt(2*a - 1) + 1/2]

RR(1-sqrt(1-.75))

0.500000000000000

load("probpuzzle1.sage")

36 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X35 = R[5,5] REI.gens() = 111 A 1st pass [ 1 X1 X2 X3 X4 X5] [ X1 1 X8 X9 X10 X11] [ X2 X8 1 X15 X16 X17] [ X3 X9 X15 1 X22 X23] [ X4 X10 X16 X22 1 X29] [ X5 X11 X17 X23 X29 1]

constraints;len(constraints);min(2,3)

[X2*X4*X14 + X2*X14 + 1, X3*X4*X19 + X3*X19 + 1, X1*X3*X8 + X1*X3 + 1, X7*X8*X13 + X7*X8 + 1, X2*X12 + X2 + 1, X12, X1*X2*X7 + X1*X2 + 1, X2*X4*X14 + X2*X4 + 1, X7*X9*X14 + X7*X9 + 1, X3*X4*X19 + X3*X4 + 1, X1*X3*X8 + X1*X8 + 1, X8*X9*X19 + X9*X19 + 1, X13*X18 + X13 + 1, X18*X19 + X19 + 1, X18, X20 + X4 + 1, X23 + X19 + 1, X1*X3*X8 + X3*X8 + 1, X13*X14*X19 + X13*X14 + 1, X1*X2*X7 + X1*X7 + 1, X0*X3 + X3 + 1, X15 + X3 + 1, X8*X9*X19 + X8*X9 + 1, X0*X4 + X4 + 1, X10 + X2 + 1, X9*X24 + X9 + 1, X7*X9*X14 + X7*X14 + 1, X7*X12 + X7 + 1, X1*X6 + X1 + 1, X0*X1 + X1 + 1, X3*X18 + X3 + 1, X24, X13*X14*X19 + X13*X19 + 1, X2*X3*X13 + X2*X3 + 1, X1*X2*X7 + X2*X7 + 1, X5 + X1 + 1, X7*X9*X14 + X9*X14 + 1, X7*X8*X13 + X7*X13 + 1, X1*X4*X9 + X4*X9 + 1, X4*X24 + X4 + 1, X12*X14 + X14 + 1, X1*X4*X9 + X1*X9 + 1, X6, X6*X7 + X7 + 1, X6*X8 + X8 + 1, X3*X4*X19 + X4*X19 + 1, X0, X16 + X8 + 1, X6*X9 + X9 + 1, X8*X9*X19 + X8*X19 + 1, X0*X2 + X2 + 1, X7*X8*X13 + X8*X13 + 1, X2*X4*X14 + X4*X14 + 1, X11 + X7 + 1, X17 + X13 + 1, X22 + X14 + 1, X8*X18 + X8 + 1, X2*X3*X13 + X3*X13 + 1, X14*X24 + X14 + 1, X1*X4*X9 + X1*X4 + 1, X21 + X9 + 1, X19*X24 + X19 + 1, X12*X13 + X13 + 1, X2*X3*X13 + X2*X13 + 1, X13*X14*X19 + X14*X19 + 1] 65 2

len(set(constraints))

65

x=var('x')
a=.25;b=.8
RR(2*sqrt(a)-a)
soln=solve(x^2-(a+b)*x+a==0,x);soln
ccsoln=[CC(s.rhs()) for s in soln];ccsoln

0.750000000000000 [x == -1/40*sqrt(41) + 21/40, x == 1/40*sqrt(41) + 21/40] [0.364921894064179, 0.685078105935821]

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
Pr1 = (P + R + P*R)*(P + Q + P*Q)                  # premise 1 of exercise 7
Pr2 = 1 + Q*R + Q*R*(1 + V + V*W)                  # premise 2 of exercise 7
Pr3 = 1 + P + P*S + (1 +P + P*S)*(S + S*W)         # premise 3 of exercise 7
Pr4 = 1 + W                                        # not W -- premise 4 of exercise 7
Concl = 1 + V + V*S                                # conclusion to be proved of exercise 7
prems=[Pr1,Pr2,Pr3,Pr4]
a=.1;b=.9
premprobs=[Prob(A) for A in prems]
prb = [QQ(int(100*numpy.random.uniform(a,b)+.5)/100) for A in prems]
premsys=[Prob(A)-prb[prems.index(A)] for A in prems]

premprobs
Prob(Concl)

[-p*q*r + p + q*r, q*r*v*w - q*r*v + 1, p*s - p + s*w - s + 1, -w + 1] s*v - v + 1

EX7.<P,Q,R,S,V,W> = BooleanPolynomialRing(6,order='lex')
X = EX7.gens() #generators X[n] are independent random variables that can take on values 0 or 1
NewGens = str(X).lower()
NewGens = NewGens[1:len(NewGens) - 1]
x = var(NewGens) ;x#x = Pr(X)

(p, q, r, s, v, w)

load("ProbExample.sage")

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_salvus.py", line 3228, in load exec 'salvus.namespace["%s"] = sage.structure.sage_object.load(*__args, **__kwds)'%t in salvus.namespace, {'__args':other_args, '__kwds':kwds} File "<string>", line 1, in <module> File "sage/structure/sage_object.pyx", line 983, in sage.structure.sage_object.load (/projects/sage/sage-6.9/src/build/cythonized/sage/structure/sage_object.c:11288) sage.repl.load.load(filename, globals()) File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/repl/load.py", line 290, in load exec(preparse_file(open(fpath).read()) + "\n", globals) File "<string>", line 58, in <module> File "<string>", line 8, in newton_krylov File "/projects/sage/sage-6.9/local/lib/python2.7/site-packages/scipy/optimize/nonlin.py", line 351, in nonlin_solve raise NoConvergence(_array_like(x, x0)) NoConvergence: [ 0.096000852885279 0.430999922915692 0.422088641987998 0.628291100695054 0.658085484772839 0.585103708551573 -3.336141787216485 -8.626185229209298 -1.684882825170378 -2.447623571889068]

prems;constraints;gradL

[P*Q*R + P + Q*R, Q*R*V*W + Q*R*V + 1, P*S + P + S*W + S + 1, W + 1] [-p*q*r + q*r + p - 0.520000000000000, q*r*v*w - q*r*v + 0.400000000000000, p*s + s*w - p - s + 0.870000000000000, -w + 0.240000000000000] ((-q*r + 1)*lam0 + lam2*(s - 1.00000000000000) - log(p) + log(-p + 1), (-p*r + r)*lam0 + (r*v*w - r*v)*lam1 - log(q) + log(-q + 1), (-p*q + q)*lam0 + (q*v*w - q*v)*lam1 - log(r) + log(-r + 1), lam2*(p + w - 1.00000000000000) - log(s) + log(-s + 1), (q*r*w - q*r)*lam1 - log(v) + log(-v + 1), lam1*q*r*v + lam2*s - lam3 - log(w) + log(-w + 1), -p*q*r + q*r + p - 0.520000000000000, q*r*v*w - q*r*v + 0.400000000000000, p*s + s*w - p - s + 0.870000000000000, -w + 0.240000000000000)

aa=.1;bb=.9
a=.5;b=.8;w=.4;c=.2
p,q,r,lam=var('p,q,r,lam')
x=[p,q,r,lam]
s(p)=(c-1+p)/(w-1+p);v(p)=(1-b)*(1-p)/(1-w)/(a-p)
H(p,q,r,lam)=-(p*ln(p)+(1-p)*ln(1-p)+q*ln(q)+(1-q)*ln(1-q)+r*ln(r)+(1-r)*ln(1-r)+s*ln(s)+(1-s)*ln(1-s)+v*ln(v)+(1-v)*ln(1-v)+w*ln(w)+(1-w)*ln(1-w))
g(p,q,r,lam)=(1-p)*q*r-a+p
L(p,q,r,lam)=H+lam*g
gradL=L.gradient();gradL
gradL2 = []
for gr in gradL:
    gradL2.append((gr).function(*x))
def F(s):
    tmp=[]
    for gr in gradL2:
        tmp.append(((gr)(*s)).n())
    return tmp
numpy.set_printoptions(precision=15)
init = [5.,.5,.5,.5]
ssoln = scipy.optimize.broyden1(F, init, f_tol=1e-14)
#ssoln = scipy.optimize.newton_krylov(F, init, method='lgmres')#, f_tol=1e-14)
ssoln = list(ssoln);ssoln;s;v
htt = 0.0 #the entropy of the truth table equals the sum of the entropies of the independent generators of the free boolean algebra
for ii in range(len(x)):
    htt = htt - (ssoln[ii]*ln(ssoln[ii])+(1-ssoln[ii])*ln(1-ssoln[ii]))
#htt = RR(htt/ln(2))

(p, q, r, lam) |--> (-(q*r - 1.00000000000000)*lam + (1/(p - 0.600000000000000) - (p - 0.800000000000000)/(p - 0.600000000000000)^2)*log(-(p - 0.800000000000000)/(p - 0.600000000000000) + 1) - (-0.333333333333333/(p - 0.500000000000000) - (-0.333333333333333*p + 0.333333333333333)/(p - 0.500000000000000)^2)*log((-0.333333333333333*p + 0.333333333333333)/(p - 0.500000000000000) + 1) - log((p - 0.800000000000000)/(p - 0.600000000000000))/(p - 0.600000000000000) + (p - 0.800000000000000)*log((p - 0.800000000000000)/(p - 0.600000000000000))/(p - 0.600000000000000)^2 - 0.333333333333333*log(-(-0.333333333333333*p + 0.333333333333333)/(p - 0.500000000000000))/(p - 0.500000000000000) - (-0.333333333333333*p + 0.333333333333333)*log(-(-0.333333333333333*p + 0.333333333333333)/(p - 0.500000000000000))/(p - 0.500000000000000)^2 - log(p) + log(-p + 1), -lam*(p - 1)*r - log(q) + log(-q + 1), -lam*(p - 1)*q - log(r) + log(-r + 1), -(p - 1)*q*r + p - 0.500000000000000) [(0.76103218893948343+0.14535503827500615j), (0.52005501791809494-0.89315519079074335j), (0.52005501791809472-0.89315519079074335j), (7.3352044606688747+0.20701673159837541j)] p |--> (p - 0.800000000000000)/(p - 0.600000000000000) p |--> -(-0.333333333333333*p + 0.333333333333333)/(p - 0.500000000000000)

ssoln

[(0.49819424363272691+1.1268603558711039j), (-0.92006532596286694+0.28976557801488817j), (-0.92006532596286683+0.28976557801488828j), (2.5602438593183972+0.29908797985823443j)]

import numpy
from scipy.optimize import minimize_scalar
#a=.9;b=.8;w=.4;c=.2
b=.5
w=numpy.random.uniform(b,1)
c=numpy.random.uniform(0,w)
a=numpy.random.uniform(max(1-c,1-w),1)
p=var('p')
r=sqrt((a-p)/(1-p))
s=(c-1+p)/(w-1+p)
v=(1-b)*(1-p)/(1-w)/(a-p)
H(p)=-(p*ln(p)+(1-p)*ln(1-p)+2*r*ln(r)+2*(1-r)*ln(1-r)+s*ln(s)+(1-s)*ln(1-s)+v*ln(v)+(1-v)*ln(1-v)+w*ln(w)+(1-w)*ln(1-w))
def HH(x):
    return -H(x).n()
dH=H.derivative();H;dH
res = minimize_scalar(HH, bounds=(max(1-c,1-w), a), method='bounded');res.x;-HH(res.x)

p |--> -p*log(p) + (p - 1)*log(-p + 1) + 2*(sqrt((p - 0.9314978256797797)/(p - 1)) - 1)*log(-sqrt((p - 0.9314978256797797)/(p - 1)) + 1) + ((p - 0.6745609927397498)/(p - 0.26443075756713086) - 1)*log(-(p - 0.6745609927397498)/(p - 0.26443075756713086) + 1) - ((-1.89085416764752*p + 1.89085416764752)/(p - 0.9314978256797797) + 1)*log((-1.89085416764752*p + 1.89085416764752)/(p - 0.9314978256797797) + 1) - 2*sqrt((p - 0.9314978256797797)/(p - 1))*log(sqrt((p - 0.9314978256797797)/(p - 1))) - (p - 0.6745609927397498)*log((p - 0.6745609927397498)/(p - 0.26443075756713086))/(p - 0.26443075756713086) + (-1.89085416764752*p + 1.89085416764752)*log(-(-1.89085416764752*p + 1.89085416764752)/(p - 0.9314978256797797))/(p - 0.9314978256797797) + 0.577640518610534 p |--> (1/(p - 0.26443075756713086) - (p - 0.6745609927397498)/(p - 0.26443075756713086)^2)*log(-(p - 0.6745609927397498)/(p - 0.26443075756713086) + 1) - (-1.89085416764752/(p - 0.9314978256797797) - (-1.89085416764752*p + 1.89085416764752)/(p - 0.9314978256797797)^2)*log((-1.89085416764752*p + 1.89085416764752)/(p - 0.9314978256797797) + 1) + (p - 1)*sqrt((p - 0.9314978256797797)/(p - 1))*((p - 0.9314978256797797)/(p - 1)^2 - 1/(p - 1))/(p - 0.9314978256797797) + ((p - 0.9314978256797797)/(p - 1)^2 - 1/(p - 1))*log(sqrt((p - 0.9314978256797797)/(p - 1)))/sqrt((p - 0.9314978256797797)/(p - 1)) - ((p - 0.9314978256797797)/(p - 1)^2 - 1/(p - 1))*log(-sqrt((p - 0.9314978256797797)/(p - 1)) + 1)/sqrt((p - 0.9314978256797797)/(p - 1)) - ((p - 0.9314978256797797)/(p - 1)^2 - 1/(p - 1))/sqrt((p - 0.9314978256797797)/(p - 1)) - log((p - 0.6745609927397498)/(p - 0.26443075756713086))/(p - 0.26443075756713086) + (p - 0.6745609927397498)*log((p - 0.6745609927397498)/(p - 0.26443075756713086))/(p - 0.26443075756713086)^2 - 1.89085416764752*log(-(-1.89085416764752*p + 1.89085416764752)/(p - 0.9314978256797797))/(p - 0.9314978256797797) - (-1.89085416764752*p + 1.89085416764752)*log(-(-1.89085416764752*p + 1.89085416764752)/(p - 0.9314978256797797))/(p - 0.9314978256797797)^2 - log(p) + log(-p + 1) 0.8042263314662732 0.734822392913082 + 5.99598422491310*I

(CC(res.x)).real();(CC(res.x)).imag()
-HH((CC(res.x)).real())
r(p=(CC(res.x)).real())
s(p=(CC(res.x)).real());c-1+(CC(res.x)).real();w-1+(CC(res.x)).real()
v(p=(CC(res.x)).real())

0.804226331466273 0.000000000000000 0.734822392913082 + 5.99598422491310*I 0.806284724551966 0.240211933917692 0.129665338726523 0.539795573899142 2.90858105619189

import numpy
from scipy.optimize import minimize_scalar
upperb=0;lowerb=1
while lowerb>upperb:
    w=numpy.random.uniform(0,1)
    b=numpy.random.uniform(w,1)
    alph=(1-b)/(1-w);alph
    a=numpy.random.uniform(alph,1)
    bet=(1-b-a*(1-w))/(w-b);bet
    c=numpy.random.uniform(0,1)
    upperb=a;lowerb=0
    upperb=min(bet,upperb)
    if c<w: lowerb=max(1-c,lowerb)
    if c>w: upperb=min(1-c,upperb)
#p,q,r=var('p,q,r')
lowerb;upperb
p=var('p')
f(p)=(1-p)/(a-p)
r(p)=sqrt((a-p)/(1-p))
s(p)=(c-1+p)/(w-1+p)
v(p)=(1-b)*(1-p)/(1-w)/(a-p)
#H(p,q,r)=-(p*ln(p)+(1-p)*ln(1-p)+r*ln(r)+(1-r)*ln(1-r)+q*ln(q)+(1-q)*ln(1-q)+s*ln(s)+(1-s)*ln(1-s)+v*ln(v)+(1-v)*ln(1-v)+w*ln(w)+(1-w)*ln(1-w))
#g(p,q,r)=q*r*(1-p)+p-a
H(p)=-(p*ln(p)+(1-p)*ln(1-p)+2*r*ln(r)+2*(1-r)*ln(1-r)+s*ln(s)+(1-s)*ln(1-s)+v*ln(v)+(1-v)*ln(1-v)+w*ln(w)+(1-w)*ln(1-w))
def HH(x):
    return -H(x).n()
dH=H.derivative();H;dH
res = minimize_scalar(HH, bounds=(lowerb,upperb), method='bounded');-HH(real(res.x))
real(res.x);r(real(res.x));s(real(res.x));v(real(res.x));w

0.297557045724947 0.8896413509489796 0 0.6816348100249157 p |--> -p*log(p) + (p - 1)*log(-p + 1) + 2*(sqrt((p - 0.9224793445307975)/(p - 1)) - 1)*log(-sqrt((p - 0.9224793445307975)/(p - 1)) + 1) + ((p - 0.6816348100249157)/(p - 0.7496035254895789) - 1)*log(-(p - 0.6816348100249157)/(p - 0.7496035254895789) + 1) - ((-0.297557045724947*p + 0.297557045724947)/(p - 0.9224793445307975) + 1)*log((-0.297557045724947*p + 0.297557045724947)/(p - 0.9224793445307975) + 1) - 2*sqrt((p - 0.9224793445307975)/(p - 1))*log(sqrt((p - 0.9224793445307975)/(p - 1))) - (p - 0.6816348100249157)*log((p - 0.6816348100249157)/(p - 0.7496035254895789))/(p - 0.7496035254895789) + (-0.297557045724947*p + 0.297557045724947)*log(-(-0.297557045724947*p + 0.297557045724947)/(p - 0.9224793445307975))/(p - 0.9224793445307975) + 0.5627702973569254 p |--> -((p - 0.6816348100249157)/(p - 0.7496035254895789)^2 - 1/(p - 0.7496035254895789))*log(-(p - 0.6816348100249157)/(p - 0.7496035254895789) + 1) - (-0.297557045724947/(p - 0.9224793445307975) - (-0.297557045724947*p + 0.297557045724947)/(p - 0.9224793445307975)^2)*log((-0.297557045724947*p + 0.297557045724947)/(p - 0.9224793445307975) + 1) + (p - 1)*sqrt((p - 0.9224793445307975)/(p - 1))*((p - 0.9224793445307975)/(p - 1)^2 - 1/(p - 1))/(p - 0.9224793445307975) + ((p - 0.9224793445307975)/(p - 1)^2 - 1/(p - 1))*log(sqrt((p - 0.9224793445307975)/(p - 1)))/sqrt((p - 0.9224793445307975)/(p - 1)) - ((p - 0.9224793445307975)/(p - 1)^2 - 1/(p - 1))*log(-sqrt((p - 0.9224793445307975)/(p - 1)) + 1)/sqrt((p - 0.9224793445307975)/(p - 1)) - ((p - 0.9224793445307975)/(p - 1)^2 - 1/(p - 1))/sqrt((p - 0.9224793445307975)/(p - 1)) + (p - 0.6816348100249157)*log((p - 0.6816348100249157)/(p - 0.7496035254895789))/(p - 0.7496035254895789)^2 - log((p - 0.6816348100249157)/(p - 0.7496035254895789))/(p - 0.7496035254895789) - 0.297557045724947*log(-(-0.297557045724947*p + 0.297557045724947)/(p - 0.9224793445307975))/(p - 0.9224793445307975) - (-0.297557045724947*p + 0.297557045724947)*log(-(-0.297557045724947*p + 0.297557045724947)/(p - 0.9224793445307975))/(p - 0.9224793445307975)^2 - log(p) + log(-p + 1) 3.26529298115520 0.6263361845936647 0.8902466151001972 0.4486072712312715 0.37544781054617105

a;alph;bet2
lowerb;upperb;1-c
f(real(res.x))
r(p)=sqrt((a-p)/(1-p))
s(p)=(c-1+p)/(w-1+p)
v(p)=(1-b)*(1-p)/((1-w)*(a-p))
r(res.x);s(res.x);v(real(res.x))
alph*f(res.x)
real(res.x)
(a-alph)/(1-alph)

0.9319053692494879 0.385676818484241 0.889155036307603 0.7035370192168051 0.889155036307603 0.7035370192168051 1.8701950469107271 0.7312342900976023 0.18455912519549703 0.7212908756375155 0.7212908756375151 0.853653222225376 0.889155036307603

load('ProbExample3.sage')

status: 0 success: True njev: 24 nfev: 145 fun: -3.030889167879031 x: array([ 0.5027099273301 , 0.757271524632081, 0.75727167030936 ]) message: 'Optimization terminated successfully.' jac: array([ 1.288713842630386, 1.137775987386703, 1.13777768611908 , 0. ]) nit: 24 [0.50270992733009956, 0.75727152463208136, 0.75727167030936016, 0.8205003356389732, 0.8279384203401001, 0.08872309129847011] 4.372648771983155

a;b;c;w;alph;bet
-HH(sssoln[0:3])/RR(ln(2))
H(*sssoln[0:3])/RR(ln(2))

0.3105926673392683 0.8448414562493256 0.9901755105095976 0.24950518269147626 0.2067416591990797 0.13091700748502 3.813560014475027 3.813560014475027

load('maxentrocond2.sage')

[0.0, 0.0, 0.0, 0.0, 0.0, 0.0] [0.6810716549906166, 1.0493314940315797, 1.0493314940322065, 5.906386491005833e-14, 0.6658757796947674, 0.13547901061696477] [30.332398899622188, 1.523877932381924, 1.5238779323879013, 29.476361110236198, 1.2200606401016993, 0.5320507309549475] [4.080256120216759, 1.2697477849076275, 1.2697477848766276, 1.9171223706459337, 32.00941275096475, 0.6149581601139393] [30.512180261128126, 1.291455851274243, 1.2914558495380333, 1.5459339845597795, 32.265137319672625, 0.6149581601139393] [31.83839541812106, 1.2914558457877203, 1.2914558550247293, 1.5459339845584634, 31.274589284876154, 0.6149581601139393] status: 8 success: False njev: 6 nfev: 7 fun: (-2.1553717419345126-1.5088982354920233e-13j) x: array([ -1.738756708058453e-14, 7.843935042435221e-01, 7.843935058056881e-01, 1.756743031575382e-01, 1.000000000000031e+00, 6.490709941051962e-01]) message: 'Positive directional derivative for linesearch' jac: array([ 31.83839541812106 , 1.29145584578772 , 1.291455855024729, 1.545933984558463, 31.274589284876154, 0.614958160113939, 0. ]) nit: 10 [0.6132740863129872, 0.935945231252979, 0.7794192258133512, 0.3509290058948038] (0.18252913743534313, 0.5269239169286903) [-1.7387567080584532e-14, 0.78439350424352206, 0.78439350580568812, 0.17567430315753815, 1.0000000000000306, 0.6490709941051962] (3.10954412336+2.17688000155e-13j)

load('lsqcond.sage')

status: 0 success: True njev: 9 nfev: 10 fun: 1.243757753482089e-07 x: array([ 2.895763140240498e-04, 9.403408601794696e-01, 9.403408601794745e-01, 2.934528224617628e-02, 9.522729556165573e-01, 1.639422264780529e-01]) message: 'Optimization terminated successfully.' jac: array([-0.000615658658397, -0.000475308599404, -0.000475308599404, -0.000440445537553, 0.000363730947264, -0.001267837307625, 0. ]) nit: 9 (0.029282285743199472, 0.8807101722746475) [0.8842032510963583, 0.9755191080108951, 0.29608859277626376, 0.8360307731369744] [0.88427445440691255, 0.97518457004653158, 0.29600697664106301, 0.8360577735219471] [0.00028957631402404976, 0.94034086017946961, 0.94034086017947449, 0.029345282246176281, 0.95227295561655734, 0.16394222647805293] 1.76741953191

[str(c) for c in constraint]

str2='])(*s))},'
str1='{\'type\': \'eq\', \'fun\': lambda s: ((constraint['  
mystr='['
for ii in range(4):
    mystr=mystr+str1+str(ii)+str2
mystr=mystr+']'
mystr

['-x0*x1*x2 + x1*x2 + x0 - 0.961667339522', 'x0*x3 + x3*x5 - x0 - x3 + 0.490998333178', 'x1*x2*x4*x5 - x1*x2*x4 + 0.533328983816', '-x5 + 0.577912416122'] "[{'type': 'eq', 'fun': lambda s: ((constraint[0])(*s))},{'type': 'eq', 'fun': lambda s: ((constraint[1])(*s))},{'type': 'eq', 'fun': lambda s: ((constraint[2])(*s))},{'type': 'eq', 'fun': lambda s: ((constraint[3])(*s))},]"

HH(numpy.array([0.5, 0.5, 0.5, 0.5, 0.5, 0.5]))

-0.69314718055994529

cons

({'fun': <function <lambda> at 0x7fd6f8d837d0>, 'type': 'eq'}, {'fun': <function <lambda> at 0x7fd6f8d93aa0>, 'type': 'eq'}, {'fun': <function <lambda> at 0x7fd6f8f1fc08>, 'type': 'eq'}, {'fun': <function <lambda> at 0x7fd6f8db05f0>, 'type': 'eq'})

bnds = [(0.0, 1.0) for ii in range(nn)];tuple(bnds)

((0.000000000000000, 1.00000000000000), (0.000000000000000, 1.00000000000000), (0.000000000000000, 1.00000000000000), (0.000000000000000, 1.00000000000000), (0.000000000000000, 1.00000000000000), (0.000000000000000, 1.00000000000000))

print(probs);print(alph,bet)

[0.8394785302888622, 0.9845006314798125, 0.8893632971327695, 0.9802641807336469] (0.785342037794793, 0.25219885597495895)

({'type': 'eq', 'fun': lambda s:  ((constraint[0])(*s))}).parent()

Error in lines 1-1 Traceback (most recent call last): File "/projects/5511fe15-8085-4d1d-bdc7-c6bf6c99e693/.sagemathcloud/sage_server.py", line 865, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> AttributeError: 'dict' object has no attribute 'parent'

mytuple=([],[]);mytuple[0]={'type': 'eq', 'fun': lambda s:  ((constraint[0])(*s))};mytuple

Error in lines 1-1 Traceback (most recent call last): File "/projects/5511fe15-8085-4d1d-bdc7-c6bf6c99e693/.sagemathcloud/sage_server.py", line 865, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> TypeError: 'tuple' object does not support item assignment

load('PuzzlePart3c.sage')

objects = 12 types = 3 rel_classes = 4 144 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X143 = R[11,11] status: 0 success: True njev: 44 nfev: 87 fun: 1.5316938987775146 x: array([ 1.0001e+00, -1.9717e-06, -9.6458e-06, -2.2948e-06, -6.8672e-05, 1.6781e-02, 4.1152e-06, -2.2948e-06, -7.0747e-07, -1.4493e-05, 1.2391e-05, 1.5000e+00, 2.2125e-05, 1.2146e+00, -6.8904e-06, -2.3095e-04, 3.0879e-05, -9.7554e-04, 1.2105e-03, -2.3095e-04, -5.1737e-04, 3.3887e-02, 4.9388e-04, -7.9805e-07, 2.9779e-05, 3.3121e-05, 9.9994e-01, -2.9396e-05, 3.8770e-01, 4.1545e-03, 1.1854e-06, -2.9396e-05, 9.2111e-02, 7.3707e-05, 9.8850e-02, -1.3582e-04, 2.0530e-05, 2.9119e-04, 5.1019e-05, 1.2217e+00, 3.2517e-05, -9.5508e-04, 2.9872e-06, -2.4156e-04, -4.4501e-04, 2.0009e-06, 4.7806e-04, -1.2011e-06, 9.8009e-05, -9.1592e-05, -5.0000e-01, -9.3620e-05, 1.0000e+00, 3.4662e-02, 1.0359e-03, 3.2517e-05, 2.8622e-01, 3.3274e-02, 3.9077e-01, -5.7102e-06, -1.7392e-02, 1.0752e-03, -4.1831e-03, 1.0458e-03, -3.7381e-02, 1.0009e+00, 1.2772e-03, -9.5508e-04, 1.2830e-03, 3.5556e-02, 2.8714e-01, 1.1816e-01, 6.1287e-05, -1.1669e-03, 3.0850e-05, 2.2795e-05, -9.5020e-04, -1.2904e-03, 9.9997e-01, 2.9872e-06, -1.1079e-03, 1.5000e+00, -6.1033e-05, 1.1172e-05, 2.0530e-05, 2.9119e-04, 5.1019e-05, 2.9382e-04, -9.3620e-05, 1.0458e-03, 2.2795e-05, 1.2217e+00, -4.4501e-04, 2.0009e-06, 4.7806e-04, -1.2011e-06, 1.7785e-05, 5.9262e-04, -1.1271e-01, 5.0535e-04, -5.0000e-01, -1.2042e-03, 1.1221e-03, 5.0535e-04, 1.0004e+00, 2.8233e-05, 5.7010e-02, -2.5549e-06, -5.2444e-05, -1.0521e-01, -4.0963e-05, 2.4711e-05, -3.5610e-02, -3.8306e-02, 8.4054e-01, 2.4711e-05, 5.3888e-06, 9.9999e-01, 1.1447e-04, -1.9285e-05, 1.2744e-05, -5.7451e-04, -1.2296e-01, -5.6028e-04, -5.0000e-01, -5.0000e-01, 1.0928e-04, -5.6028e-04, -6.4258e-02, -7.5014e-05, 1.0001e+00, 4.6635e-04, 8.1602e-01, 1.9889e-05, 9.1768e-05, 1.8160e-05, 1.0880e-04, -1.5475e-01, -3.3414e-05, 1.8160e-05, 1.5590e-05, 6.1566e-05, -4.4166e-04, 1.0000e+00]) message: 'Optimization terminated successfully.' jac: array([ -5.9360e-04, 7.2248e-05, -6.2825e-05, 8.5252e-05, -7.4684e-04, 1.4554e-06, 1.6541e-04, 8.5252e-05, 5.9863e-05, 8.2293e-06, -1.0529e-05, -1.1651e-01, 4.5839e-05, 4.9395e-04, -1.8607e-04, -2.7614e-04, 3.3558e-04, -1.1430e-04, 4.9304e-05, -2.7614e-04, -1.9408e-04, -3.3071e-04, 1.9762e-04, 6.9935e-05, 1.0032e-05, -1.8514e-04, -2.8633e-05, -1.3237e-04, 1.2480e-04, 3.7091e-05, -4.6183e-05, -1.3237e-04, -2.0696e-04, -3.3667e-05, -8.3794e-04, -9.3593e-05, 5.1314e-05, -1.9652e-04, -1.2661e-04, 8.6773e-07, 3.3429e-04, -2.8402e-05, -2.0729e-05, -1.5120e-04, -7.1591e-05, -2.8476e-06, 2.1600e-04, 8.5584e-05, -8.4107e-05, 2.1347e-04, 1.2371e-01, 2.0750e-04, -2.5502e-05, 3.5990e-04, -3.5273e-04, 3.3429e-04, -3.4487e-04, 1.4043e-05, 1.0907e-05, -5.4126e-04, -4.1986e-05, 1.8433e-04, -1.7020e-06, 2.5931e-04, -9.1360e-05, 2.2519e-04, 8.8273e-05, -2.8402e-05, 7.4858e-05, 5.8668e-06, 2.9497e-04, -3.1172e-04, 1.5352e-04, 3.2408e-05, 2.1201e-07, 9.9455e-06, -6.8797e-05, 7.2831e-05, 1.8292e-04, -2.0729e-05, 2.1398e-04, -6.5910e-02, -3.5267e-04, -1.2236e-04, 5.1314e-05, -1.9652e-04, -1.2661e-04, -7.4435e-05, 2.0750e-04, 2.5931e-04, 9.9455e-06, 8.6773e-07, -7.1591e-05, -2.8476e-06, 2.1600e-04, 8.5584e-05, 3.8065e-05, -1.5459e-05, 2.5340e-04, 8.0236e-05, 6.1819e-02, 3.3463e-04, 1.1019e-04, 8.0236e-05, 1.8161e-04, 4.5366e-05, -2.0841e-05, 9.4263e-05, -1.7917e-04, -3.9731e-04, 4.0247e-06, 1.5052e-05, -9.3661e-07, 4.7392e-05, 3.4683e-04, 1.5052e-05, 8.7620e-05, -2.4459e-05, -1.2706e-04, -3.3353e-04, 7.7125e-06, -1.7982e-04, 6.9213e-05, -1.5716e-04, 1.2302e-01, 6.3319e-02, -1.7706e-05, -1.5716e-04, 2.3332e-04, -2.6579e-05, 9.0811e-05, 2.0269e-05, -1.1019e-04, 4.0043e-05, 1.2269e-04, 4.6123e-05, -6.2189e-05, -1.9897e-04, -1.9517e-04, 4.6123e-05, 4.6058e-05, 7.5155e-05, 1.8115e-05, -2.3513e-04, 0.0000e+00]) nit: 44 [ 1.0 -2.0e-6 -9.5e-6 -2.3e-6 -0.000069 0.017 4.1e-6 -2.3e-6 -7.2e-7 -0.000014 0.000012 1.5] [ 0.000022 1.2 -6.9e-6 -0.00023 0.000031 -0.00098 0.0012 -0.00023 -0.00052 0.033 0.00049 -8.0e-7] [ 0.000030 0.000032 1.0 -0.000030 0.39 0.0042 1.2e-6 -0.000030 0.094 0.000072 0.098 -0.00014] [ 0.000021 0.00029 0.000051 1.2 0.000032 -0.00095 3.0e-6 -0.00024 -0.00044 2.0e-6 0.00047 -1.2e-6] [ 0.000099 -0.000092 -0.50 -0.000095 1.0 0.035 0.0010 0.000032 0.28 0.033 0.39 -5.7e-6] [ -0.018 0.0011 -0.0042 0.0010 -0.037 1.0 0.0013 -0.00095 0.0013 0.035 0.28 0.12] [ 0.000061 -0.0012 0.000031 0.000023 -0.00095 -0.0013 1.0 3.0e-6 -0.0011 1.5 -0.000061 0.000011] [ 0.000021 0.00029 0.000051 0.00029 -0.000095 0.0010 0.000023 1.2 -0.00044 2.0e-6 0.00047 -1.2e-6] [ 0.000018 0.00058 -0.11 0.00052 -0.50 -0.0012 0.0011 0.00052 1.0 0.000029 0.057 -2.5e-6] [-0.000051 -0.11 -0.000040 0.000025 -0.035 -0.039 0.84 0.000025 5.5e-6 1.0 0.00011 -0.000019] [ 0.000013 -0.00058 -0.12 -0.00055 -0.50 -0.50 0.00011 -0.00055 -0.062 -0.000076 1.0 0.00047] [ 0.81 0.000020 0.000092 0.000018 0.00011 -0.16 -0.000034 0.000018 0.000015 0.000061 -0.00044 1.0]

def HH(s):
    hh=0
    for ii in range(len(x)):
        hh=hh+(s[ii]*ln(s[ii])+(1-s[ii])*ln(1-s[ii]))
    return hh
print -HH(sssoln)
A=matrix(RR, objects, objects, sssoln);A

(-2.77391461191+11.8927219896j) [ 1.00000505353045 0.0000294135582362149 0.0000241545836365197 0.0000245873648328289 0.0000316206048297949 0.0168192521136186 0.0000119265447893070 0.0000245812207092187 8.72812517039096e-6 -0.0000237672129465480 -6.56734280879684e-8 1.50000000000000] [-0.0000402395747841601 0.999962216901920 0.000313089918314325 -0.0194268379672321 0.000168904589561207 -0.000558509068753074 0.00130889605188983 -0.0194264199223155 0.0000712784182045985 0.0344918273515964 -0.000535573832509702 0.0000323899746612603] [-0.0000165823955525974 -0.000308548378172608 0.999850303196343 0.000330234938993502 0.387756548090587 0.00398867428607788 0.0000165558199145618 0.000330303177814151 0.0967807295559178 0.0000867864539594795 0.100386199945243 0.0000517917365388226] [-0.0000375183526004767 0.0187116810333552 -0.000321241831785089 0.999925704032538 0.000172150508908699 -0.000598405562579725 -0.0000235499658504104 -0.0276078596451742 0.000114789515418590 -0.0000279512603507040 -0.000545040927109923 0.0000255540468999237] [-0.0000484811620554004 -0.000281138465006914 -0.500000000000000 -0.000262715204891630 1.00003757696617 0.0341080679244395 0.00104588746658500 0.000172025347353270 0.286277002996331 0.0333450751960850 0.390724093845325 -0.0000363900065136603] [ -0.0173809934364802 0.000575026076632618 -0.00405480885737827 0.000611730561381510 -0.0363926926429300 1.00005970031797 0.00126657749040093 -0.000598365239911712 0.00261461094827320 0.0356927575926667 0.287161911641583 0.118160705599398] [-0.0000598755600456176 -0.00129722223504022 -0.0000333944070553237 -0.0000427977028527267 -0.00103182417988011 -0.00133384071626486 0.999968383993821 -0.0000235262091785784 -0.00103518668196755 1.50000000000000 0.0000262732984794275 -0.0000240017599925289] [-0.0000375248654537204 0.0187115093028608 -0.000321362552475939 0.0261195664020825 -0.000262821081020772 0.000611658835179675 -0.0000428128710476350 0.999925714803257 0.000115018151642744 -0.0000279020766793224 -0.000544995475436077 0.0000255516416352003] [-0.0000445720464987873 -0.0000731155022260420 -0.120344009894748 -0.000134339856041769 -0.500000000000000 -0.00267834155531896 0.00104024484646687 -0.000134545311472071 1.00005618915125 0.0000369860261644148 0.0581549591757848 0.0000247195940996991] [ 0.0000710958630431933 -0.105860588605843 -0.000103028368203521 -0.0000512574461475913 -0.0356773843243856 -0.0384397370194799 0.840553528779256 -0.0000512424937901841 -0.0000613610881914837 0.999976544875759 0.0000628417450175984 0.0000265717211743610] [-0.0000621502293826759 0.000623323445449488 -0.125549841679886 0.000605709737020917 -0.500000000000000 -0.500000000000000 0.000124611794046541 0.000605734798482475 -0.0658511655422306 -0.000143181994983768 1.00003434670099 0.000482834202673383] [ 0.816021778761429 -0.0000453811564374448 -0.0000467536032285355 -0.0000435816765517250 0.0000242284647840227 -0.154591986220805 0.0000631047966890037 -0.0000435786369207985 -0.0000515067799758254 -0.000109404376970598 -0.000513275881541789 0.999986096956555]

random.random()

Error in lines 1-1 Traceback (most recent call last): File "/projects/5511fe15-8085-4d1d-bdc7-c6bf6c99e693/.sagemathcloud/sage_server.py", line 865, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> AttributeError: 'function' object has no attribute 'random'

load('experiment.sage')

status: 0 success: True njev: 10 nfev: 15 fun: 7.0281352245113237e-06 x: array([ 0.21830376984273 , 0.603544858332129, 0.603544858332123, 0.318900143071508, 0.675670242292849, 0.200156723226423]) message: 'Optimization terminated successfully.' jac: array([-0.000508377913666, -0.001257347101805, -0.001257347101803, 0.000906919021823, -0.000328329045201, -0.001392108189744, 0. ]) nit: 10 [0.22888799935245807, 0.7139182393034512, 0.5025327555814896, 0.29613455548558176, 0.7073328092453715, 0.21184593565047494] [0.21830376984272995, 0.60354485833212901, 0.60354485833212268, 0.31890014307150832, 0.67567024229284867, 0.20015672322642333] [0.50304943838387151, 0.59624319819528859, 0.80313940209582801, 0.79984327677357669] 5.22918720915

results[0:10,:]

[1 0 0 1] [1 0 1 1] [1 0 0 1] [1 0 1 1] [1 0 1 1] [0 0 1 1] [1 0 1 1] [1 0 1 1] [1 0 1 1] [1 0 0 1]

obsrvprob;obsrvprobRR;beq;probbeq-obsrvprob;AgivenB;AgivenNotB

[877/1000 17/200 77/125 98/125] [ 114/125 3/25 1/10 49/1000] [ 1 759/1000 739/1000 319/500] [ 124/125 97/100 1 899/1000] [ 0.877000000000000 0.0850000000000000 0.616000000000000 0.784000000000000] [ 0.912000000000000 0.120000000000000 0.100000000000000 0.0490000000000000] [ 1.00000000000000 0.759000000000000 0.739000000000000 0.638000000000000] [ 0.992000000000000 0.970000000000000 1.00000000000000 0.899000000000000] [ X0*X1*X2 + X1*X2 + X0 X0*X1*X2*X3*X5 + X1*X2*X3*X5 + X0*X1*X2*X3 + X1*X2*X3 + X0*X3*X5 + X0*X1*X2 + X1*X2 X1*X2*X4*X5 + X1*X2*X4 + X0*X1*X2 + X1*X2 + X0 X0*X1*X2*X5 + X1*X2*X5 + X0*X1*X2 + X1*X2 + X0*X5 + X0] [ X0*X1*X2*X3*X5 + X1*X2*X3*X5 + X0*X1*X2*X3 + X1*X2*X3 + X0*X3*X5 + X3*X5 + X0*X3 + X3 + 1 X3*X5 + X0*X3 + X3 + X0 + 1 X0*X1*X2*X3*X4*X5 + X1*X2*X3*X4*X5 + X0*X1*X2*X4*X5 + X0*X1*X2*X3*X4 + X1*X2*X4*X5 + X1*X2*X3*X4 + X0*X1*X2*X4 + X1*X2*X4 + X3*X5 + X0*X3 + X3 + X0 + 1 X0*X3*X5 + X3*X5 + X0*X5 + X0*X3 + X5 + X3 + X0 + 1] [ 1 X0*X1*X2*X3*X4*X5 + X1*X2*X3*X4*X5 + X0*X1*X2*X4*X5 + X0*X1*X2*X3*X4 + X1*X2*X3*X4 + X0*X1*X2*X4 + 1 X1*X2*X4*X5 + X1*X2*X4 + 1 X1*X2*X4*X5 + X1*X2*X4 + X5 + 1] [ X0*X1*X2*X5 + X1*X2*X5 + X0*X5 + X5 + 1 X0*X3*X5 + X0*X5 + 1 1 X5 + 1] [ -x0*x1*x2 + x1*x2 + x0 - 877/1000 -x0*x1*x2*x3*x5 + x1*x2*x3*x5 + x0*x1*x2*x3 + x0*x3*x5 - x1*x2*x3 - x0*x1*x2 + x1*x2 - 17/200 x1*x2*x4*x5 - x1*x2*x4 - x0*x1*x2 + x1*x2 + x0 - 77/125 x0*x1*x2*x5 - x1*x2*x5 - x0*x1*x2 - x0*x5 + x1*x2 + x0 - 98/125] [ x0*x1*x2*x3*x5 - x1*x2*x3*x5 - x0*x1*x2*x3 - x0*x3*x5 + x1*x2*x3 + x3*x5 + x0*x3 - x3 + 11/125 x3*x5 + x0*x3 - x3 - x0 + 22/25 x0*x1*x2*x3*x4*x5 - x1*x2*x3*x4*x5 - x0*x1*x2*x4*x5 - x0*x1*x2*x3*x4 + x1*x2*x4*x5 + x1*x2*x3*x4 + x0*x1*x2*x4 - x1*x2*x4 + x3*x5 + x0*x3 - x3 - x0 + 9/10 -x0*x3*x5 + x3*x5 + x0*x5 + x0*x3 - x5 - x3 - x0 + 951/1000] [ 0 -x0*x1*x2*x3*x4*x5 + x1*x2*x3*x4*x5 + x0*x1*x2*x4*x5 + x0*x1*x2*x3*x4 - x1*x2*x3*x4 - x0*x1*x2*x4 + 241/1000 x1*x2*x4*x5 - x1*x2*x4 + 261/1000 x1*x2*x4*x5 - x1*x2*x4 - x5 + 181/500] [ -x0*x1*x2*x5 + x1*x2*x5 + x0*x5 - x5 + 1/125 x0*x3*x5 - x0*x5 + 3/100 0 -x5 + 101/1000] [ 1 17/24 616/739 784/899] [ 85/877 1 100/739 49/899] [616/877 5/6 1 22/31] [784/877 49/120 638/739 1] [ 0 9/10 1 93/101] [ 35/123 0 20/261 71/101] [ 1 639/880 0 1] [115/123 85/88 1 0]

Prob(X[0]).parent()

Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5 over Rational Field

U=CoordinatePatch(x);F=DifferentialForms(U);form1=DifferentialForm(F,0,Prob(prems[0]));Prob(prems[0]);form1.diff()

-x0*x1*x2 + x1*x2 + x0 (-x1*x2 + 1)*dx0 + (-x0*x2 + x2)*dx1 + (-x0*x1 + x1)*dx2

len(constraint);constraint

14 [x3*x5 + x0*x3 - x3 - x0 + 37/40, -x5 + 31/500, -x0*x1*x2*x5 + x1*x2*x5 + x0*x5 - x5 + 11/1000, -x0*x1*x2 + x1*x2 + x0 - 87/100, x0*x3*x5 - x0*x5 + 1/125, x1*x2*x4*x5 - x1*x2*x4 - x5 + 91/500, x0*x1*x2*x5 - x1*x2*x5 - x0*x1*x2 - x0*x5 + x1*x2 + x0 - 819/1000, x1*x2*x4*x5 - x1*x2*x4 - x0*x1*x2 + x1*x2 + x0 - 3/4, -x0*x3*x5 + x3*x5 + x0*x5 + x0*x3 - x5 - x3 - x0 + 979/1000, x0*x1*x2*x3*x5 - x1*x2*x3*x5 - x0*x1*x2*x3 - x0*x3*x5 + x1*x2*x3 + x3*x5 + x0*x3 - x3 + 1/10, -x0*x1*x2*x3*x4*x5 + x1*x2*x3*x4*x5 + x0*x1*x2*x4*x5 + x0*x1*x2*x3*x4 - x1*x2*x3*x4 - x0*x1*x2*x4 + 119/1000, -x0*x1*x2*x3*x5 + x1*x2*x3*x5 + x0*x1*x2*x3 + x0*x3*x5 - x1*x2*x3 - x0*x1*x2 + x1*x2 - 9/200, x0*x1*x2*x3*x4*x5 - x1*x2*x3*x4*x5 - x0*x1*x2*x4*x5 - x0*x1*x2*x3*x4 + x1*x2*x4*x5 + x1*x2*x3*x4 + x0*x1*x2*x4 - x1*x2*x4 + x3*x5 + x0*x3 - x3 - x0 + 463/500, x1*x2*x4*x5 - x1*x2*x4 + 3/25]

M=matrix(QQ,3,3)
M[0,0]=1;M[0,1]=1;M[1,0]=1;M[1,2]=1;M[2,0]=1;M[2,1]=1;M[2,2]=1
M

[1 1 0] [1 0 1] [1 1 1]

M=matrix(QQ,3,3)
M[0,0]=1;M[0,1]=1;M[1,0]=1;M[1,2]=1;M[2,0]=1;M[2,1]=1;M[2,2]=1
M
Minv=M.inverse()
Minv
var('a','b','x0','x1','x2','x3')
V=vector([a,b,1-x3])
X=Minv*V
X
x0=X[0];x1=X[1];x2=X[2]
1-(x0+x1+x2)
x0+x1+x2+x3

[ 1 1 -1] [ 0 -1 1] [-1 0 1] (a, b, x0, x1, x2, x3) (a + b + x3 - 1, -b - x3 + 1, -a - x3 + 1) 1

1-(x0+x1+x2)
1-x3
x0+x1+x2+x3

x3 -x3 + 1 1

M2=matrix(QQ,7,7)
M2[0,:]=[[1,1,1,1,0,0,0]]
M2[1,:]=[[1,1,0,0,1,1,0]]
M2[2,:]=[[1,0,1,0,1,0,1]]
M2[3,:]=[[1,1,0,0,0,0,0]]
M2[4,:]=[[1,0,0,0,1,0,0]]
M2[5,:]=[[1,0,1,0,0,0,0]]
M2[6,:]=[[1,1,1,1,1,1,1]]
M2
var('A','B','C','AB','BC','CA','X000','X001','X010','X011','X100','X101','X110','X111')
V2=vector([A,B,C,AB,BC,CA,1-X111])
M2inv=M2.inverse()
M2inv
X2=M2inv*V2
X2
(X000, X001, X010, X011, X100, X101, X110)=X2
X000+X001+X010+X011+X100+X101+X110+X111
#are M,M2 totally unimodular etc?

[1 1 1 1 0 0 0] [1 1 0 0 1 1 0] [1 0 1 0 1 0 1] [1 1 0 0 0 0 0] [1 0 0 0 1 0 0] [1 0 1 0 0 0 0] [1 1 1 1 1 1 1] (A, B, C, AB, BC, CA, X000, X001, X010, X011, X100, X101, X110, X111) [-1 -1 -1 1 1 1 1] [ 1 1 1 0 -1 -1 -1] [ 1 1 1 -1 -1 0 -1] [ 0 -1 -1 0 1 0 1] [ 1 1 1 -1 0 -1 -1] [-1 0 -1 0 0 1 1] [-1 -1 0 1 0 0 1] (-A + AB - B + BC - C + CA - X111 + 1, A + B - BC + C - CA + X111 - 1, A - AB + B - BC + C + X111 - 1, -B + BC - C - X111 + 1, A - AB + B + C - CA + X111 - 1, -A - C + CA - X111 + 1, -A + AB - B - X111 + 1) 1

MA = lambda n: [MA(i) for i in range(0,n)]
MA3 = MA(3)
MA3

[[], [[]], [[], [[]]]]

[0 for i in range(2)]

[0, 0]

def MA(s):
    #s=s-1
    if s == 0:
        X = 0
    else:
        X = [MA(s-1) for ii in range(2)]
    return X

MA(3)

[[[0, 0], [0, 0]], [[0, 0], [0, 0]]]

MA3=MA(3)
MA3[1][0][1]

0

X=['X'+Integer(i).binary() for i in range(2^3)];X

['X0', 'X1', 'X10', 'X11', 'X100', 'X101', 'X110', 'X111']

nn=3;bnn=2**nn
X=['X'+Integer(ii+bnn).binary()[1:nn+1] for ii in range(bnn)];X
Y=['Y'+Integer(ii+bnn).binary()[1:nn+1] for ii in range(bnn)];Y
MPR=PolynomialRing(QQ,X+Y);MPR;MPR.gens()
XX=[MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'X')]];XX #these are the vertices of the hypercube
YY=[MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'Y')]];YY
sum([XX[k] for k in [i for i, j in enumerate(XX) if (str(j)[1]=='1')]])
IY=[[i for i,j in enumerate(str(YYY)) if j=='1'] for YYY in YY];IY
AND = lambda A, B: A*B
FA = lambda mylist: 1 if not mylist else Integer(reduce(AND, mylist))
MM = matrix([[Integer(FA([str(XXX)[i] == '1' for i in iy])) for XXX in XX] for iy in IY]);MM
MMinv = MM.inverse() # is MM totally unimodular? Integer matrix with integer inverse det = +/- 1
#vector(YY)=MM*vector(XX)
YY[0]-1
mylist=list(MM*vector(XX)-vector(YY));mylist.append((YY[0]-1));mylist

['X000', 'X001', 'X010', 'X011', 'X100', 'X101', 'X110', 'X111'] ['Y000', 'Y001', 'Y010', 'Y011', 'Y100', 'Y101', 'Y110', 'Y111'] Multivariate Polynomial Ring in X000, X001, X010, X011, X100, X101, X110, X111, Y000, Y001, Y010, Y011, Y100, Y101, Y110, Y111 over Rational Field (X000, X001, X010, X011, X100, X101, X110, X111, Y000, Y001, Y010, Y011, Y100, Y101, Y110, Y111) [X000, X001, X010, X011, X100, X101, X110, X111] [Y000, Y001, Y010, Y011, Y100, Y101, Y110, Y111] X100 + X101 + X110 + X111 [[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]] [1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1] Y000 - 1 [X000 + X001 + X010 + X011 + X100 + X101 + X110 + X111 - Y000, X001 + X011 + X101 + X111 - Y001, X010 + X011 + X110 + X111 - Y010, X011 + X111 - Y011, X100 + X101 + X110 + X111 - Y100, X101 + X111 - Y101, X110 + X111 - Y110, X111 - Y111, Y000 - 1]

sum(MPR.gens()[0:8])
mylist2=[QQ(normalvariate(0, 1)^2) for i in range(bnn)];mylistsum=sum(mylist2);mylist2=[ml2/mylistsum for ml2 in mylist2];mylist2
mylist2=list(vector(XX)-vector(mylist2));mylist2
sum(mylist2)

X000 + X001 + X010 + X011 + X100 + X101 + X110 + X111 [381787114317400647773823943513260629879301845499372396616490400/8725665956006962478592565209181526160335864496531258106880724971, 232163584823496079382773089712111108470447233228300567029189500/8725665956006962478592565209181526160335864496531258106880724971, 543916889439807816287150797336804717505999537515275045912066575/8725665956006962478592565209181526160335864496531258106880724971, 2506952347161838936679487280543154047018627230644834360232713616/8725665956006962478592565209181526160335864496531258106880724971, 3527286953466432831650864582900938289439518478561429445010825200/8725665956006962478592565209181526160335864496531258106880724971, 137484553196327889352991572165687286152339866711511087062800/8725665956006962478592565209181526160335864496531258106880724971, 820903274039015583390412988259578687813263056954309461246431680/8725665956006962478592565209181526160335864496531258106880724971, 712518308205774255538699535343512992922554774261025319745945200/8725665956006962478592565209181526160335864496531258106880724971] [X000 - 381787114317400647773823943513260629879301845499372396616490400/8725665956006962478592565209181526160335864496531258106880724971, X001 - 232163584823496079382773089712111108470447233228300567029189500/8725665956006962478592565209181526160335864496531258106880724971, X010 - 543916889439807816287150797336804717505999537515275045912066575/8725665956006962478592565209181526160335864496531258106880724971, X011 - 2506952347161838936679487280543154047018627230644834360232713616/8725665956006962478592565209181526160335864496531258106880724971, X100 - 3527286953466432831650864582900938289439518478561429445010825200/8725665956006962478592565209181526160335864496531258106880724971, X101 - 137484553196327889352991572165687286152339866711511087062800/8725665956006962478592565209181526160335864496531258106880724971, X110 - 820903274039015583390412988259578687813263056954309461246431680/8725665956006962478592565209181526160335864496531258106880724971, X111 - 712518308205774255538699535343512992922554774261025319745945200/8725665956006962478592565209181526160335864496531258106880724971] X000 + X001 + X010 + X011 + X100 + X101 + X110 + X111 - 1

#probability calculus for non independent generators (nn=3 in this case) modeled on unit nn-cube
#vertices are atoms 1-topes, the cube itself is 1 the n-tope, set (n-1)-topes is generator or its negation
#could just assign probabilities to the vertices non-negative and sum to one for empirical distribution
nn=3;bnn=2**nn
x=['x'+Integer(ii+bnn).binary()[1:nn+1] for ii in range(bnn)];x
y=['y'+Integer(ii+bnn).binary()[1:nn+1] for ii in range(bnn)];y
MPR=PolynomialRing(QQ,x+y);MPR;MPR.gens()
xx=[MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'x')]];xx # these correspond to the vertices of the hypercube ie 0-polytopes --- 0-cube
yy=[MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'y')]];yy # these correspond to certain k-polytopes YYY[0,0,0] = 1 is nn-polytope --- nn-cube
# the null-polytope is the 0 elt in the boolean poly ring  --- 0-cube are vertices, edges are 1-cubes, faces are 2-cubes
sum([xx[k] for k in [i for i, j in enumerate(xx) if (str(j)[1]=='1')]])
iy=[[i for i,j in enumerate(str(yyy)) if j=='1'] for yyy in yy];iy
AND = lambda A, B: A*B
FA = lambda mylist: 1 if not mylist else Integer(reduce(AND, mylist))
MM = matrix([[Integer(FA([str(xxx)[i] == '1' for i in iyy])) for xxx in xx] for iyy in iy]);MM
#need to add a row of 1's, roughly yy = MM*xx y000=1 then 1-x000 MM[row 1's]*<x001,x010,...,x111> gotta check this
#Prob(A*~B) = Prob(A*(1+B)) = Prob(A+A*B)=Prob(A)-Prob(A*B)
MMinv = MM.inverse() # well almost, is MM totally unimodular? Integer matrix with integer inverse det = +/- 1
#yy[0]-1
#MM*vector(xx)-vector(yy)
mylist=list(MM*vector(xx)-vector(yy));mylist.append((yy[0]-1));mylist
#euler characteristic? |0-polytopes|-|1-polytopes|+|2-polytopes|-|3-polytopes|+...=what
mylist2=[QQ(normalvariate(0, 1)^2) for i in range(bnn)]
mylistsum=sum(mylist2)
mylist2=[ml2/mylistsum for ml2 in mylist2]
mylist3=[RR(w) for w in mylist2]
mylist2=list(vector(xx)-vector(mylist2));mylist2 #just assign rational probabilities to the X variables
sum(mylist2)
# Y000 = Prob(1),Y100=Prob(A),Y010=Prob(B),Y001=Prob(C),Y110=Prob(A*B),Y011=Prob(B*C),Y101=Prob(C*A),Y111=Prob(A*B*C)
# X000 = Prob((1+A)*(1+B)*(1+C)),X100=Prob(A*(1+B)*(1+C)),X010=Prob((1+A)*B*(1+C)),X001=Prob((1+A)*(1+B)*C),X110=Prob(A*B*(1+C)),X011=Prob((1+A)*B*C),X101=Prob(A*(1+B)*C),X111=Prob(A*B*C)
MMinv
vyy=vector(yy);vyy[0]=1
mylist4 = list(MMinv*vyy-vector(xx));mylist4
mylist3

['x000', 'x001', 'x010', 'x011', 'x100', 'x101', 'x110', 'x111'] ['y000', 'y001', 'y010', 'y011', 'y100', 'y101', 'y110', 'y111'] Multivariate Polynomial Ring in x000, x001, x010, x011, x100, x101, x110, x111, y000, y001, y010, y011, y100, y101, y110, y111 over Rational Field (x000, x001, x010, x011, x100, x101, x110, x111, y000, y001, y010, y011, y100, y101, y110, y111) [x000, x001, x010, x011, x100, x101, x110, x111] [y000, y001, y010, y011, y100, y101, y110, y111] x100 + x101 + x110 + x111 [[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]] [1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1] [x000 + x001 + x010 + x011 + x100 + x101 + x110 + x111 - y000, x001 + x011 + x101 + x111 - y001, x010 + x011 + x110 + x111 - y010, x011 + x111 - y011, x100 + x101 + x110 + x111 - y100, x101 + x111 - y101, x110 + x111 - y110, x111 - y111, y000 - 1] [x000 - 2240669592224312079831168545282847992929641276763129480885787775/519640263466496508854748438273454045229306193415468193480738399453, x001 - 13191366174056316904896187780052819787754029939948873337229413305/519640263466496508854748438273454045229306193415468193480738399453, x010 - 86677927903400614252813281026458229997864266859572424586267063850/519640263466496508854748438273454045229306193415468193480738399453, x011 - 1750725016739661743831280463032223604433036987183776817085520625/519640263466496508854748438273454045229306193415468193480738399453, x100 - 165221027734534803411153364206961724578407508733854877958389988555/519640263466496508854748438273454045229306193415468193480738399453, x101 - 49294735052476700508452567862861521215181176755020336859005748510/519640263466496508854748438273454045229306193415468193480738399453, x110 - 641027644290642124172951503672072844493761451659579229236359938/519640263466496508854748438273454045229306193415468193480738399453, x111 - 200622784348773457829597636885132605208242771411465195212638516895/519640263466496508854748438273454045229306193415468193480738399453] x000 + x001 + x010 + x011 + x100 + x101 + x110 + x111 - 1 [ 1 -1 -1 1 -1 1 1 -1] [ 0 1 0 -1 0 -1 0 1] [ 0 0 1 -1 0 0 -1 1] [ 0 0 0 1 0 0 0 -1] [ 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 1] [-x000 - y001 - y010 + y011 - y100 + y101 + y110 - y111 + 1, -x001 + y001 - y011 - y101 + y111, -x010 + y010 - y011 - y110 + y111, -x011 + y011 - y111, -x100 + y100 - y101 - y110 + y111, -x101 + y101 - y111, -x110 + y110 - y111, -x111 + y111] [0.00431196300547788, 0.0253855736390735, 0.166803717874316, 0.00336910963184541, 0.317952705651315, 0.0948631938634504, 0.00123359887475689, 0.386080137459765]

# Y000 = Prob(1),Y100=Prob(A),Y010=Prob(B),Y001=Prob(C),Y110=Prob(A*B),Y011=Prob(B*C),Y101=Prob(C*A),Y111=Prob(A*B*C)
# X000 = Prob((1+A)*(1+B)*(1+C)),X100=Prob(A*(1+B)*(1+C)),X010=Prob((1+A)*B*(1+C)),X001=Prob((1+A)*(1+B)*C),X110=Prob(A*B*(1+C)),X011=Prob((1+A)*B*C),X101=Prob(A*(1+B)*C),X111=Prob(A*B*C)
MMinv
vYY=vector(YY);vYY[0]=1
mylist3 = list(MMinv*vYY-vector(XX));mylist3

[ 1 -1 -1 1 -1 1 1 -1] [ 0 1 0 -1 0 -1 0 1] [ 0 0 1 -1 0 0 -1 1] [ 0 0 0 1 0 0 0 -1] [ 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 1] [-X000 - Y001 - Y010 + Y011 - Y100 + Y101 + Y110 - Y111 + 1, -X001 + Y001 - Y011 - Y101 + Y111, -X010 + Y010 - Y011 - Y110 + Y111, -X011 + Y011 - Y111, -X100 + Y100 - Y101 - Y110 + Y111, -X101 + Y101 - Y111, -X110 + Y110 - Y111, -X111 + Y111]

#probability calculus for non independent generators (nn=3 in this case) modeled on unit nn-cube
#vertices are atoms 1-topes, the cube itself is 1 the n-tope, set (n-1)-topes is generator or its negation
#could just assign probabilities to the vertices non-negative and sum to one for empirical distribution
nn=3;bnn=2**nn
print('# of generators '+str(nn)+', # of atoms '+str(bnn))
x=['x'+Integer(ii+bnn).binary()[1:nn+1] for ii in range(bnn)];print('x probability variables');print(x);print(' ')
y=['y'+Integer(ii+bnn).binary()[1:nn+1] for ii in range(bnn)];print('y probability variables');print(y);print(' ')
MPR=PolynomialRing(QQ,x+y);print(MPR);print(' ')
xx=[MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'x')]];print('xx');print(xx);print(' ') # these correspond to the vertices of the hypercube ie 0-polytopes --- 0-cube
yy=[MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'y')]];print('yy');print(yy);print(' ') # these correspond to certain k-polytopes YYY[0,0,0] = 1 is nn-polytope --- nn-cube
# the null-polytope is the 0 elt in the boolean poly ring  --- 0-cube are vertices, edges are 1-cubes, faces are 2-cubes
iy=[[i for i,j in enumerate(str(yyy)) if j=='1'] for yyy in yy];print('iy');print(iy);print(' ')
AND = lambda A, B: A*B
FA = lambda mylist: 1 if not mylist else Integer(reduce(AND, mylist))
MM = matrix([[Integer(FA([str(xxx)[i] == '1' for i in iyy])) for xxx in xx] for iyy in iy]);print('MM');print('is MM totally unimodular? going out on limb and saying yes');print(MM);print(' ')
#need to add a row of 1's, roughly yy = MM*xx y000=1 then 1-x000 MM[row 1's]*<x001,x010,...,x111> gotta check this
#Prob(A*~B) = Prob(A*(1+B)) = Prob(A+A*B)=Prob(A)-Prob(A*B)
MMinv = MM.inverse();print('MMinv');print(MMinv);print(' ') # well almost, is MM totally unimodular? Integer matrix with integer inverse det = +/- 1
#yy[0]-1
#MM*vector(xx)-vector(yy)
print('example variable description for 3-cube with generators A,B,C')
print('y000 = Prob(1), y100=Prob(A), y010=Prob(B), y001=Prob(C), y110=Prob(A*B), y011=Prob(B*C), y101=Prob(C*A), y111=Prob(A*B*C)')
print('x000 = Prob((1+A)*(1+B)*(1+C)), x100=Prob(A*(1+B)*(1+C)), x010=Prob((1+A)*B*(1+C)), x001=Prob((1+A)*(1+B)*C), x110=Prob(A*B*(1+C)), x011=Prob((1+A)*B*C), x101=Prob(A*(1+B)*C), x111=Prob(A*B*C)')
print(' ')
vyy=vector(yy);vyy[0]=1
mylist4 = list(MMinv*vyy-vector(xx));print('x variables in terms of y variables');print(mylist4);print(' ')
mylist=list(MM*vector(xx)-vector(yy));mylist.append((yy[0]-1));print('y variables in terms of x variables');print(mylist);print(' ')
#euler characteristic? |0-polytopes|-|1-polytopes|+|2-polytopes|-|3-polytopes|+...=what
mylist2=[QQ(normalvariate(0, 1)^2) for i in range(bnn)]
mylistsum=sum(mylist2)
mylist2=[ml2/mylistsum for ml2 in mylist2]
mylist3=[RR(w) for w in mylist2]
print('random pick atom probabilities');print(mylist3);print(' ')
mylist5 = list(MM*vector(mylist3));print('y variable probabilities');print(mylist5);print(' ')
mylist2=list(vector(xx)-vector(mylist2)) #just assign rational probabilities to the X variables
print('sum(mylist2)');print(sum(mylist2));print(' ')

# of generators 3, # of atoms 8 x probability variables ['x000', 'x001', 'x010', 'x011', 'x100', 'x101', 'x110', 'x111'] y probability variables ['y000', 'y001', 'y010', 'y011', 'y100', 'y101', 'y110', 'y111'] Multivariate Polynomial Ring in x000, x001, x010, x011, x100, x101, x110, x111, y000, y001, y010, y011, y100, y101, y110, y111 over Rational Field xx [x000, x001, x010, x011, x100, x101, x110, x111] yy [y000, y001, y010, y011, y100, y101, y110, y111] iy [[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]] MM is MM totally unimodular? going out on limb and saying yes [1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1] MMinv [ 1 -1 -1 1 -1 1 1 -1] [ 0 1 0 -1 0 -1 0 1] [ 0 0 1 -1 0 0 -1 1] [ 0 0 0 1 0 0 0 -1] [ 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 1] example variable description for 3-cube with generators A,B,C y000 = Prob(1), y100=Prob(A), y010=Prob(B), y001=Prob(C), y110=Prob(A*B), y011=Prob(B*C), y101=Prob(C*A), y111=Prob(A*B*C) x000 = Prob((1+A)*(1+B)*(1+C)), x100=Prob(A*(1+B)*(1+C)), x010=Prob((1+A)*B*(1+C)), x001=Prob((1+A)*(1+B)*C), x110=Prob(A*B*(1+C)), x011=Prob((1+A)*B*C), x101=Prob(A*(1+B)*C), x111=Prob(A*B*C) x variables in terms of y variables [-x000 - y001 - y010 + y011 - y100 + y101 + y110 - y111 + 1, -x001 + y001 - y011 - y101 + y111, -x010 + y010 - y011 - y110 + y111, -x011 + y011 - y111, -x100 + y100 - y101 - y110 + y111, -x101 + y101 - y111, -x110 + y110 - y111, -x111 + y111] y variables in terms of x variables [x000 + x001 + x010 + x011 + x100 + x101 + x110 + x111 - y000, x001 + x011 + x101 + x111 - y001, x010 + x011 + x110 + x111 - y010, x011 + x111 - y011, x100 + x101 + x110 + x111 - y100, x101 + x111 - y101, x110 + x111 - y110, x111 - y111, y000 - 1] random pick atom probabilities [0.124607277095934, 0.00108636084690530, 0.0585603874682545, 0.562481250045919, 0.158444040380286, 0.0937703383308307, 0.00102792480386519, 0.0000224210280059958] y variable probabilities [1.00000000000000, 0.657360370251660, 0.622091983346044, 0.562503671073925, 0.253264724542988, 0.0937927593588367, 0.00105034583187119, 0.0000224210280059958] sum(mylist2) x000 + x001 + x010 + x011 + x100 + x101 + x110 + x111 - 1

#probability calculus for non independent generators (nn=3 in this case) modeled on unit nn-cube
#vertices are atoms 1-topes, the cube itself is 1 the n-tope, set (n-1)-topes is generator or its negation
#could just assign probabilities to the vertices non-negative and sum to one for empirical distribution
nn=4;bnn=2**nn
print('# of generators '+str(nn)+', # of atoms '+str(bnn))
x=['x'+Integer(ii+bnn).binary()[1:nn+1] for ii in range(bnn)];print('x probability variables');print(x);print(' ')
y=['y'+Integer(ii+bnn).binary()[1:nn+1] for ii in range(bnn)];print('y probability variables');print(y);print(' ')
MPR=PolynomialRing(QQ,x+y,order='deglex');print(MPR);print(' ')
MPR.inject_variables()
xx=[MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'x')]];print('xx');print(xx);print(' ') # these correspond to the vertices of the hypercube ie 0-polytopes --- 0-cube
yy=[MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'y')]];print('yy');print(yy);print(' ') # these correspond to certain k-polytopes YYY[0,0,0] = 1 is nn-polytope --- nn-cube
# the null-polytope is the 0 elt in the boolean poly ring  --- 0-cube are vertices, edges are 1-cubes, faces are 2-cubes
iy=[[i for i,j in enumerate(str(yyy)) if j=='1'] for yyy in yy];print('iy');print(iy);print(' ')
leniy = [len(c) for c in iy]
#AND = lambda A, B: A*B
#FA = lambda mylist: 1 if not mylist else Integer(reduce(AND, mylist))
#MM = matrix([[Integer(FA([str(xxx)[i] == '1' for i in iyy])) for xxx in xx] for iyy in iy]);print('MM');print('is MM totally unimodular? going out on limb and saying yes');print(MM);print(' ')
#need to add a row of 1's, roughly yy = MM*xx y000=1 then 1-x000 MM[row 1's]*<x001,x010,...,x111> gotta check this
#Prob(A*~B) = Prob(A*(1+B)) = Prob(A+A*B)=Prob(A)-Prob(A*B)
#MMinv = MM.inverse();print('MMinv');print(MMinv);print(' ') # well almost, is MM totally unimodular? Integer matrix with integer inverse det = +/- 1
# recursive way of defining MM and MMinv
MM=matrix([1])
for ii in range(1,nn+1):
    MM=block_matrix(2,2,[MM,MM,0,MM],subdivide=False)
print('MM');print('is MM totally unimodular? going out on limb and saying yes');print(MM);print(' ')
MMinv=matrix([1])
for ii in range(1,nn+1):
    MMinv=block_matrix(2,2,[MMinv,-MMinv,0,MMinv],subdivide=False)
print('MMinv');print(MMinv);print(' ')
#print(' ');print('both upper triangular, look at the 1×1 blocks, the 2×2 blocks, the 4×4 blocks, 8×8 blocks, 16×16 blocks etc. these can be written from the pattern recursively for 0,1,2,3,4,...generators')
print('don''t need the lines for FA and MM=matrix() nor do we need to actually invert MM to obtain MMinv')
#yy[0]-1
#MM*vector(xx)-vector(yy)
print('example variable description for 3-cube with generators A,B,C')
print('y000 = Prob(1), y100=Prob(A), y010=Prob(B), y001=Prob(C), y110=Prob(A*B), y011=Prob(B*C), y101=Prob(C*A), y111=Prob(A*B*C)')
print('x000 = Prob((1+A)*(1+B)*(1+C)), x100=Prob(A*(1+B)*(1+C)), x010=Prob((1+A)*B*(1+C)), x001=Prob((1+A)*(1+B)*C), x110=Prob(A*B*(1+C)), x011=Prob((1+A)*B*C), x101=Prob(A*(1+B)*C), x111=Prob(A*B*C)')
print(' ')
vyy=vector(yy);vyy[0]=1
mylist4 = list(MMinv*vyy-vector(xx));print('x variables in terms of y variables');print(mylist4);print(' ')
mylist=list(MM*vector(xx)-vector(yy));mylist.append((yy[0]-1));print('y variables in terms of x variables');print(mylist);print(' ')
#euler characteristic? |0-polytopes|-|1-polytopes|+|2-polytopes|-|3-polytopes|+...=what
mylist2=[QQ(normalvariate(0, 1)^2) for i in range(bnn)]
mylistsum=sum(mylist2)
mylist2=[ml2/mylistsum for ml2 in mylist2]
mylist3=[RR(w) for w in mylist2]
print('random pick atom probabilities');print(mylist3);print(' ')
mylist5 = list(MM*vector(mylist3));print('y variable probabilities');print(mylist5);print(' ')
mylist2=list(vector(xx)-vector(mylist2)) #just assign rational probabilities to the X variables
print('sum(mylist2)');print(sum(mylist2));print(' ')

print('define polynomials for independent generators, nn=3 then y100,y010,y001 are the generators')
yylist0=[]
for ii in range(len(yy)):
    if leniy[ii]>1:
        stryy = str(yy[ii])
        yylist=[]
        for jj in range(1,len(stryy)):
            if stryy[jj]=='1':
                yylist.append(yy[2^(nn-jj)])
        yylist0.append(reduce(lambda a,b:a*b ,yylist)-yy[ii])
print(yylist0);print(' ')
myideal=ideal(yylist0)
QR = MPR.quotient_ring(myideal)

#vctr=vector([1,0,1,1,0,0,0,1,0,1,1,1,0,1,1,0])
#vctr*vector(yy)
#testpoly2=vctr*MM*vector(xx);testpoly2
#sum([QQ(mod(testpoly2.coefficient(a),2))*a for a in xx])
#vector([QQ(mod(testpoly2.coefficient(a),2)) for a in xx])*MMinv*vector(yy) #just converted boolean polynomial in y vars into probabilty polynomial in y vars i think

#vctr=vector([0,0,1,1,0,0,0,1,0,1,1,1,0,1,1,0]) # negation since y[0]=1 ie y000=1
#vctr*vector(yy)
#testpoly2=vctr*MM*vector(xx);testpoly2
#sum([QQ(mod(testpoly2.coefficient(a),2))*a for a in xx])
#vector([QQ(mod(testpoly2.coefficient(a),2)) for a in xx])*MMinv*vector(yy)

#MPR.gens()
#MPR.inject_variables()
#y0101*y1100
testpoly2=y0101*y1100+y1111*y1000*y0110+2*y1110;testpoly2 # supposed to be arbitrary (poly1 in y)*(poly2 in y)+poly3 in y etc though in QQ use coefficients in ZZ and regard as boolean polynomial
testpoly2.coefficients()
testpoly2.monomials()
testpoly2=vector([QQ(mod(a,2)) for a in testpoly2.coefficients()])*vector(testpoly2.monomials()) #reduce modulo 2 (boolean polynomial remember) then coerce new coefficients back to QQ
s2=[[b.strip() for b in c] for c in [a.split('*') for a in str(testpoly2).split('+')]] #parsing and removing leading and trailing spaces
ll2=[]
for a in s2:
    ll=[]
    for b in a:
        ll.append('1'+b[1:])
    k=int(ll[0],2)
    for lll in ll:
        k = k | int(lll,2) # oring the bits in y101 with y110 gives y111 (A*C)*(A*B)=A*B*C for instance ie y101*y110=y111
    ll2.append('y'+Integer(k).binary()[1:])
#creating the equivalent reduced boolean polynomial
#y vars as string being converted back to the generators in the polynomiaring and we previously injected the variable so we can write expressions in the generators directly
testpoly2b = sum([MPR(y) for y in ll2]);testpoly2b 
#now the procedure to obtain the probability expression
vctr = vector([testpoly2b.coefficient(g) for g in yy])
testpoly3=vctr*MM*vector(xx);testpoly3
sum([QQ(mod(testpoly3.coefficient(a),2))*a for a in xx])
testpoly4=vector([QQ(mod(testpoly3.coefficient(a),2)) for a in xx])*MMinv*vector(yy);testpoly4

# of generators 4, # of atoms 16 x probability variables ['x0000', 'x0001', 'x0010', 'x0011', 'x0100', 'x0101', 'x0110', 'x0111', 'x1000', 'x1001', 'x1010', 'x1011', 'x1100', 'x1101', 'x1110', 'x1111'] y probability variables ['y0000', 'y0001', 'y0010', 'y0011', 'y0100', 'y0101', 'y0110', 'y0111', 'y1000', 'y1001', 'y1010', 'y1011', 'y1100', 'y1101', 'y1110', 'y1111'] Multivariate Polynomial Ring in x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111, y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111 over Rational Field Defining x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111, y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111 xx [x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111] yy [y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111] iy [[], [4], [3], [3, 4], [2], [2, 4], [2, 3], [2, 3, 4], [1], [1, 4], [1, 3], [1, 3, 4], [1, 2], [1, 2, 4], [1, 2, 3], [1, 2, 3, 4]] MM is MM totally unimodular? going out on limb and saying yes [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] MMinv [ 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1] [ 0 1 0 -1 0 -1 0 1 0 -1 0 1 0 1 0 -1] [ 0 0 1 -1 0 0 -1 1 0 0 -1 1 0 0 1 -1] [ 0 0 0 1 0 0 0 -1 0 0 0 -1 0 0 0 1] [ 0 0 0 0 1 -1 -1 1 0 0 0 0 -1 1 1 -1] [ 0 0 0 0 0 1 0 -1 0 0 0 0 0 -1 0 1] [ 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1] [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1] [ 0 0 0 0 0 0 0 0 1 -1 -1 1 -1 1 1 -1] [ 0 0 0 0 0 0 0 0 0 1 0 -1 0 -1 0 1] [ 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] dont need the lines for FA and MM=matrix() nor do we need to actually invert MM to obtain MMinv example variable description for 3-cube with generators A,B,C y000 = Prob(1), y100=Prob(A), y010=Prob(B), y001=Prob(C), y110=Prob(A*B), y011=Prob(B*C), y101=Prob(C*A), y111=Prob(A*B*C) x000 = Prob((1+A)*(1+B)*(1+C)), x100=Prob(A*(1+B)*(1+C)), x010=Prob((1+A)*B*(1+C)), x001=Prob((1+A)*(1+B)*C), x110=Prob(A*B*(1+C)), x011=Prob((1+A)*B*C), x101=Prob(A*(1+B)*C), x111=Prob(A*B*C) x variables in terms of y variables [-x0000 - y0001 - y0010 + y0011 - y0100 + y0101 + y0110 - y0111 - y1000 + y1001 + y1010 - y1011 + y1100 - y1101 - y1110 + y1111 + 1, -x0001 + y0001 - y0011 - y0101 + y0111 - y1001 + y1011 + y1101 - y1111, -x0010 + y0010 - y0011 - y0110 + y0111 - y1010 + y1011 + y1110 - y1111, -x0011 + y0011 - y0111 - y1011 + y1111, -x0100 + y0100 - y0101 - y0110 + y0111 - y1100 + y1101 + y1110 - y1111, -x0101 + y0101 - y0111 - y1101 + y1111, -x0110 + y0110 - y0111 - y1110 + y1111, -x0111 + y0111 - y1111, -x1000 + y1000 - y1001 - y1010 + y1011 - y1100 + y1101 + y1110 - y1111, -x1001 + y1001 - y1011 - y1101 + y1111, -x1010 + y1010 - y1011 - y1110 + y1111, -x1011 + y1011 - y1111, -x1100 + y1100 - y1101 - y1110 + y1111, -x1101 + y1101 - y1111, -x1110 + y1110 - y1111, -x1111 + y1111] y variables in terms of x variables [x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x0110 + x0111 + x1000 + x1001 + x1010 + x1011 + x1100 + x1101 + x1110 + x1111 - y0000, x0001 + x0011 + x0101 + x0111 + x1001 + x1011 + x1101 + x1111 - y0001, x0010 + x0011 + x0110 + x0111 + x1010 + x1011 + x1110 + x1111 - y0010, x0011 + x0111 + x1011 + x1111 - y0011, x0100 + x0101 + x0110 + x0111 + x1100 + x1101 + x1110 + x1111 - y0100, x0101 + x0111 + x1101 + x1111 - y0101, x0110 + x0111 + x1110 + x1111 - y0110, x0111 + x1111 - y0111, x1000 + x1001 + x1010 + x1011 + x1100 + x1101 + x1110 + x1111 - y1000, x1001 + x1011 + x1101 + x1111 - y1001, x1010 + x1011 + x1110 + x1111 - y1010, x1011 + x1111 - y1011, x1100 + x1101 + x1110 + x1111 - y1100, x1101 + x1111 - y1101, x1110 + x1111 - y1110, x1111 - y1111, y0000 - 1] random pick atom probabilities [0.0324384829167317, 0.0506461393820951, 0.0496552932559498, 0.240164456345467, 0.000349008335707834, 0.0576304920916400, 0.194392661898743, 0.0709946320380557, 0.0102441968725576, 0.00289876955555036, 0.00636596544588485, 0.0917769549427657, 0.00985887794297960, 0.0815733995873333, 0.0423212137143600, 0.0586894556741792] y variable probabilities [1.00000000000000, 0.654374299617086, 0.754360633315405, 0.461625499000467, 0.515809741282998, 0.268887979391208, 0.366397963325338, 0.129684087712235, 0.303728833735610, 0.234938579759828, 0.199153589777190, 0.150466410616945, 0.192442946918852, 0.140262855261512, 0.101010669388539, 0.0586894556741792] sum(mylist2) x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x0110 + x0111 + x1000 + x1001 + x1010 + x1011 + x1100 + x1101 + x1110 + x1111 - 1 define polynomials for independent generators, nn=3 then y100,y010,y001 are the generators [y0001*y0010 - y0011, y0001*y0100 - y0101, y0010*y0100 - y0110, y0001*y0010*y0100 - y0111, y0001*y1000 - y1001, y0010*y1000 - y1010, y0001*y0010*y1000 - y1011, y0100*y1000 - y1100, y0001*y0100*y1000 - y1101, y0010*y0100*y1000 - y1110, y0001*y0010*y0100*y1000 - y1111] y0110*y1000*y1111 + y0101*y1100 + 2*y1110 [1, 1, 2] [y0110*y1000*y1111, y0101*y1100, y1110] y1101 + y1111 x1101 + 2*x1111 x1101 y1101 - y1111

leniy

[0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4]

yylist0=[]
for ii in range(len(yy)):
    if leniy[ii]>1:
        stryy = str(yy[ii])
        yylist=[]
        for jj in range(1,len(stryy)):
            if stryy[jj]=='1':
                yylist.append(yy[2^(nn-jj)])
        yylist0.append(reduce(lambda a,b:a*b ,yylist)-yy[ii])                

yylist0
len(yylist0)

[y0001*y0010 - y0011, y0001*y0100 - y0101, y0010*y0100 - y0110, y0001*y0010*y0100 - y0111, y0001*y1000 - y1001, y0010*y1000 - y1010, y0001*y0010*y1000 - y1011, y0100*y1000 - y1100, y0001*y0100*y1000 - y1101, y0010*y0100*y1000 - y1110, y0001*y0010*y0100*y1000 - y1111] 11

vctr=vector([1,0,1,1,0,0,0,1,0,1,1,1,0,1,1,0])
vctr*vector(yy)
testpoly2=vctr*MM*vector(xx);testpoly2
sum([QQ(mod(testpoly2.coefficient(a),2))*a for a in xx])
vector([QQ(mod(testpoly2.coefficient(a),2)) for a in xx])*MMinv*vector(yy)

y0000 + y0010 + y0011 + y0111 + y1001 + y1010 + y1011 + y1101 + y1110 x0000 + x0001 + 2*x0010 + 3*x0011 + x0100 + x0101 + 2*x0110 + 4*x0111 + x1000 + 2*x1001 + 3*x1010 + 6*x1011 + x1100 + 3*x1101 + 4*x1110 + 9*x1111 x0000 + x0001 + x0011 + x0100 + x0101 + x1000 + x1010 + x1100 + x1101 + x1111 y0000 - y0010 + y0011 - y0111 - y1001 + y1010 - y1011 + y1101 - y1110 + 2*y1111

vctr=vector([0,0,1,1,0,0,0,1,0,1,1,1,0,1,1,0])
vctr*vector(yy)
testpoly2=vctr*MM*vector(xx);testpoly2
sum([QQ(mod(testpoly2.coefficient(a),2))*a for a in xx])
vector([QQ(mod(testpoly2.coefficient(a),2)) for a in xx])*MMinv*vector(yy)

y0010 + y0011 + y0111 + y1001 + y1010 + y1011 + y1101 + y1110 x0010 + 2*x0011 + x0110 + 3*x0111 + x1001 + 2*x1010 + 5*x1011 + 2*x1101 + 3*x1110 + 8*x1111 x0010 + x0110 + x0111 + x1001 + x1011 + x1110 y0010 - y0011 + y0111 + y1001 - y1010 + y1011 - y1101 + y1110 - 2*y1111

testpoly2=y0101*y1100+y1111*y1000*y0110+2*y1110;testpoly2
testpoly2.coefficients()
testpoly2.monomials()
testpoly2=vector([QQ(mod(a,2)) for a in testpoly2.coefficients()])*vector(testpoly2.monomials());testpoly2
s2=[[b.strip() for b in c] for c in [a.split('*') for a in str(testpoly2).split('+')]]
ll2=[]
for a in s2:
    ll=[]
    for b in a:
        ll.append('1'+b[1:])
    k=int(ll[0],2)
    for lll in ll:
        k = k | int(lll,2)
    ll2.append('y'+Integer(k).binary()[1:])
testpoly2b = sum([MPR(y) for y in ll2]);testpoly2b
vctr = vector([testpoly2b.coefficient(g) for g in yy])
testpoly3=vctr*MM*vector(xx);testpoly3
sum([QQ(mod(testpoly3.coefficient(a),2))*a for a in xx])
vector([QQ(mod(testpoly3.coefficient(a),2)) for a in xx])*MMinv*vector(yy)

y0110*y1000*y1111 + y0101*y1100 + 2*y1110 [1, 1, 2] [y0110*y1000*y1111, y0101*y1100, y1110] y0110*y1000*y1111 + y0101*y1100 y1101 + y1111 x1101 + 2*x1111 x1101 y1101 - y1111

MM
MM=matrix([1]);MM
for ii in range(1,nn+1):
    MM=block_matrix(2,2,[MM,MM,0,MM],subdivide=False);MM
MMinv
MMinv=matrix([1]);MMinv
for ii in range(1,nn+1):
    MMinv=block_matrix(2,2,[MMinv,-MMinv,0,MMinv],subdivide=False);MMinv

[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [1] [1 1] [0 1] [1 1 1 1] [0 1 0 1] [0 0 1 1] [0 0 0 1] [1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1] [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [ 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1] [ 0 1 0 -1 0 -1 0 1 0 -1 0 1 0 1 0 -1] [ 0 0 1 -1 0 0 -1 1 0 0 -1 1 0 0 1 -1] [ 0 0 0 1 0 0 0 -1 0 0 0 -1 0 0 0 1] [ 0 0 0 0 1 -1 -1 1 0 0 0 0 -1 1 1 -1] [ 0 0 0 0 0 1 0 -1 0 0 0 0 0 -1 0 1] [ 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1] [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1] [ 0 0 0 0 0 0 0 0 1 -1 -1 1 -1 1 1 -1] [ 0 0 0 0 0 0 0 0 0 1 0 -1 0 -1 0 1] [ 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [1] [ 1 -1] [ 0 1] [ 1 -1 -1 1] [ 0 1 0 -1] [ 0 0 1 -1] [ 0 0 0 1] [ 1 -1 -1 1 -1 1 1 -1] [ 0 1 0 -1 0 -1 0 1] [ 0 0 1 -1 0 0 -1 1] [ 0 0 0 1 0 0 0 -1] [ 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 1] [ 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1] [ 0 1 0 -1 0 -1 0 1 0 -1 0 1 0 1 0 -1] [ 0 0 1 -1 0 0 -1 1 0 0 -1 1 0 0 1 -1] [ 0 0 0 1 0 0 0 -1 0 0 0 -1 0 0 0 1] [ 0 0 0 0 1 -1 -1 1 0 0 0 0 -1 1 1 -1] [ 0 0 0 0 0 1 0 -1 0 0 0 0 0 -1 0 1] [ 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1] [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1] [ 0 0 0 0 0 0 0 0 1 -1 -1 1 -1 1 1 -1] [ 0 0 0 0 0 0 0 0 0 1 0 -1 0 -1 0 1] [ 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]

MM*MMinv

[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]

yylist0
myideal=ideal(yylist0+[y0000-1])
QR = MPR.quotient_ring(myideal)
QR.inject_variables()
testpoly4=vector([QQ(mod(testpoly3.coefficient(a),2)) for a in xx])*MMinv*vector(yy);testpoly4
#QRsub=QR.subring(accepting_variables=(y0000bar, y0001bar,y0010bar,y0100bar,y1000bar))
QR(y0001*y0100*y1000)
QR(y0000)

[y0001*y0010 - y0011, y0001*y0100 - y0101, y0010*y0100 - y0110, y0001*y0010*y0100 - y0111, y0001*y1000 - y1001, y0010*y1000 - y1010, y0001*y0010*y1000 - y1011, y0100*y1000 - y1100, y0001*y0100*y1000 - y1101, y0010*y0100*y1000 - y1110, y0001*y0010*y0100*y1000 - y1111] Defining x0000bar, x0001bar, x0010bar, x0011bar, x0100bar, x0101bar, x0110bar, x0111bar, x1000bar, x1001bar, x1010bar, x1011bar, x1100bar, x1101bar, x1110bar, x1111bar, y0000bar, y0001bar, y0010bar, y0011bar, y0100bar, y0101bar, y0110bar, y0111bar, y1000bar, y1001bar, y1010bar, y1011bar, y1100bar, y1101bar, y1110bar, y1111bar y1101 - y1111 y1101bar 1

yylist0

[y0001*y0010 - y0011, y0001*y0100 - y0101, y0010*y0100 - y0110, y0001*y0010*y0100 - y0111, y0001*y1000 - y1001, y0010*y1000 - y1010, y0001*y0010*y1000 - y1011, y0100*y1000 - y1100, y0001*y0100*y1000 - y1101, y0010*y0100*y1000 - y1110, y0001*y0010*y0100*y1000 - y1111]

tp4m=(testpoly4+y0000-y0001).monomials();tp4c=(testpoly4+y0000-y0001).coefficients();tp4m;tp4c
yylist5=[]
for a in tp4m:
    stryy = str(a)
    yylist4=[]
    for jj in range(1,len(stryy)):
        if stryy[jj]=='1':
            yylist4.append(yy[2^(nn-jj)])
    if len(yylist4)<>0:
        yylist5.append(reduce(lambda a,b:a*b ,yylist4))
    else:
        yylist5.append(yy[0])
probpolyind=vector(tp4c)*vector(yylist5);probpolyind

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'testpoly4' is not defined

len([])

0

print('probability calculus for non independent generators (4 in this case) modeled on unit n-cube the vertices are atoms of statement calculus')
print('example variable description for 3-cube with generators A,B,C')
print('y000 = Prob(1), y100=Prob(A), y010=Prob(B), y001=Prob(C), y110=Prob(A*B), y011=Prob(B*C), y101=Prob(C*A), y111=Prob(A*B*C)')
print('x000 = Prob((1+A)*(1+B)*(1+C)), x100=Prob(A*(1+B)*(1+C)), x010=Prob((1+A)*B*(1+C)), x001=Prob((1+A)*(1+B)*C), x110=Prob(A*B*(1+C)), x011=Prob((1+A)*B*C), x101=Prob(A*(1+B)*C), x111=Prob(A*B*C)')
print(' ')
num_gens = 4 # 4-cube
atoms = 2**num_gens
print('# of generators ' + str(num_gens) + ', # of atoms ' + str(atoms)); print(' ')
x = ['x' + Integer(ii+atoms).binary()[1:num_gens + 1] for ii in range(atoms)];print('x probability variables string');print(x);print(' ')
y = ['y' + Integer(ii+atoms).binary()[1:num_gens + 1] for ii in range(atoms)];print('y probability variables string');print(y);print(' ')
MPR = PolynomialRing(QQ,x + y);print(MPR);print(' ')
MPR.inject_variables(); print(' ')
xvars = [MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'x')]];print('xvars');print(xvars);print(' ') # atoms correspond to the vertices of the unit n-cube
yvars = [MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'y')]];print('yvars');print(yvars);print(' ') # these correspond to expressions formed by generators and *, y101 = a*c, y111=a*b*c,  y000 = 1 = x000+x001+x010+... also y101 = Prob(A*C) etc
#AND = lambda A, B: A*B
#FA = lambda mylist: 1 if not mylist else Integer(reduce(AND, mylist))
#MM = matrix([[Integer(FA([str(xvarsx)[i] == '1' for i in iyvars])) for xvarsx in xvars] for iyvars in iy]);print('MM');print('is MM totally unimodular? going out on limb and saying yes');print(MM);print(' ')
# just a note: Prob(A*~B) = Prob(A*(1+B)) = Prob(A+A*B)=Prob(A)-Prob(A*B)
#MMinv = MM.inverse();print('MMinv');print(MMinv);print(' ') # well almost, is MM totally unimodular? Integer matrix with integer inverse det = +/- 1
# recursive way of defining MM and MMinv
MM = matrix([1])
for ii in range(1,num_gens + 1):
    MM = block_matrix(2,2,[MM,MM,0,MM],subdivide=False)
print('Sierpinski triangle Matrix, M');print(MM);print(' ')
print('y = M*x, is M totally unimodular? going out on limb and saying yes');
MMinv = matrix([1])
for ii in range(1,num_gens+1):
    MMinv = block_matrix(2,2,[MMinv,-MMinv,0,MMinv],subdivide=False)
print('M^-1');print(MMinv);print(' ')
x_of_y = list(MMinv*vector(yvars) - vector(xvars));print('x variables in terms of y variables');print(x_of_y);print(' ')
y_of_x = list(MM*vector(xvars) - vector(yvars));print('y variables in terms of x variables');print(y_of_x);print(' ')
prob_atoms = [QQ(normalvariate(0, 1)^2) for i in range(atoms)]
normalize = sum(prob_atoms)
prob_atoms = [pa/normalize for pa in prob_atoms]
RR_prob_atoms = [RR(pa) for pa in prob_atoms]
print('random pick atom i.e. x variable probabilities'); print(RR_prob_atoms); print(' ')
y_var_prob = list(MM*vector(RR_prob_atoms)); print('y variable probabilities'); print(y_var_prob);print(' ')
atom_prob_polys = list(vector(xvars) - vector(prob_atoms)) #just assign rational probabilities to the X variables
print('sum probabilities check');print(sum(prob_atoms));print(' ')

iy = [[i for i,j in enumerate(str(a)) if j == '1'] for a in yvars]; print('iy'); print(iy); print(' ')
leniy = [len(c) for c in iy]
ind_poly_list = []
for ii in range(len(yvars)):
    if leniy[ii] > 1:
        str_yvars = str(yvars[ii])
        gens_list = []
        for jj in range(1,len(str_yvars)):
            if str_yvars[jj] == '1':
                gens_list.append(yvars[2^(num_gens - jj)])
        ind_poly_list.append(reduce(lambda a,b: a*b, gens_list) - yvars[ii])

print('define polynomials for independent generators, generators=3 then y100,y010,y001 are the generators')
print(ind_poly_list);print(' ')
#myideal=ideal(ind_poly_list)
#QR = MPR.quotient_ring(myideal)
#QR.inject_variables()

#bool_poly = y0101*y1100+y1111*y1000*y0110+2*y1110; print(bool_poly) # make sure coefficients are in ZZ
bool_poly = MPR.remove_var(*tuple(xvars)).change_ring(ZZ).random_element(); print('random integer polynomial in y variables'); print(bool_poly); print(' ')
bool_poly = vector([QQ(mod(a,2)) for a in bool_poly.coefficients()])*vector(bool_poly.monomials()); print('treated as boolean polynomial') ;print(bool_poly); print(' ') #reduce modulo 2 (boolean polynomial remember) then coerce new coefficients back to QQ
parse_bool_poly = [[b.strip() for b in c] for c in [a.split('*') for a in str(bool_poly).split('+')]]; print('parsing and removing leading and trailing spaces'); print(parse_bool_poly); print(' ') 
parse_bool_poly = [[a.split('^')[0] for a in c] for c in parse_bool_poly]; print('removing powers y^m --> y'); print(parse_bool_poly); print(' ')
listA = []
for pbp in parse_bool_poly:
    listB = []
    for b in pbp:
        listB.append('1' + b[1:])
    k = int(listB[0],2)
    for b in listB:
        k = k | int(b,2) # or-ing the bits in y101 with y110 gives y111 (A*C)*(A*B)=A*B*C for instance ie y101*y110=y111
    listA.append('y' + Integer(k).binary()[1:])
#creating the equivalent reduced boolean polynomial
#y vars as string being converted back to the generators in the polynomial ring and we previously injected the variable so we can write expressions in the generators directly
print('or-ing the bits in y101 with y110 gives y111 (A*C)*(A*B)=A*B*C for instance ie y101*y110=y111')
reduced_bool_poly = sum([MPR(y) for y in listA]); print('reduced to 1st order - linear - boolean polynomial in y variables'); print(reduced_bool_poly); print(' ')
#now the procedure to obtain the probability expression
vctr = vector([reduced_bool_poly.coefficient(g) for g in yvars])
atom_bool_poly = vctr*MM*vector(xvars); print('expressing in the x variables - still boolean - using vector(y var coefficients)*M*vector(xvars)'); print(atom_bool_poly); print(' ')
atom_bool_poly = sum([QQ(mod(atom_bool_poly.coefficient(a),2))*a for a in xvars]); print('reducing the x var polynomial'); print(atom_bool_poly); print(' ')
prob_poly = vector([atom_bool_poly.coefficient(a) for a in xvars])*MMinv*vector(yvars); print('converting to probability using vector(x var coefficients)*(M^-1)*vector(yvars)');print(prob_poly); print(' ')
#for independent generators prob_poly needs to be linear in the y variables use this with the output from above (testpoly4)
prob_poly_monomials = prob_poly.monomials(); print('prob_poly_monomials'); print(prob_poly_monomials); print(' ')
prob_poly_coefficients = prob_poly.coefficients(); print('prob_poly_coefficients'); print(prob_poly_coefficients); print(' ')
monomial_gens_list = []
for a in prob_poly_monomials:
    str_a = str(a)
    gens_list = []
    for jj in range(1,len(str_a)):
        if str_a[jj] == '1':
            gens_list.append(yvars[2^(num_gens - jj)])
    if len(gens_list) <> 0:
        monomial_gens_list.append(reduce(lambda a,b:a*b ,gens_list))
    else:
        monomial_gens_list.append(yvars[0]) #yvars[0]=y000 then gens_list = [] so we put yvars[0] back
prob_poly_ind = vector(prob_poly_coefficients)*vector(monomial_gens_list); print('if the generators are independent the probability poly becomes'); print(prob_poly_ind)

probability calculus for non independent generators (4 in this case) modeled on unit n-cube the vertices are atoms of statement calculus example variable description for 3-cube with generators A,B,C y000 = Prob(1), y100=Prob(A), y010=Prob(B), y001=Prob(C), y110=Prob(A*B), y011=Prob(B*C), y101=Prob(C*A), y111=Prob(A*B*C) x000 = Prob((1+A)*(1+B)*(1+C)), x100=Prob(A*(1+B)*(1+C)), x010=Prob((1+A)*B*(1+C)), x001=Prob((1+A)*(1+B)*C), x110=Prob(A*B*(1+C)), x011=Prob((1+A)*B*C), x101=Prob(A*(1+B)*C), x111=Prob(A*B*C) # of generators 4, # of atoms 16 x probability variables string ['x0000', 'x0001', 'x0010', 'x0011', 'x0100', 'x0101', 'x0110', 'x0111', 'x1000', 'x1001', 'x1010', 'x1011', 'x1100', 'x1101', 'x1110', 'x1111'] y probability variables string ['y0000', 'y0001', 'y0010', 'y0011', 'y0100', 'y0101', 'y0110', 'y0111', 'y1000', 'y1001', 'y1010', 'y1011', 'y1100', 'y1101', 'y1110', 'y1111'] Multivariate Polynomial Ring in x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111, y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111 over Rational Field Defining x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111, y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111 xvars [x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111] yvars [y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111] Sierpinski triangle Matrix, M [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] y = M*x, is M totally unimodular? going out on limb and saying yes M^-1 [ 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1] [ 0 1 0 -1 0 -1 0 1 0 -1 0 1 0 1 0 -1] [ 0 0 1 -1 0 0 -1 1 0 0 -1 1 0 0 1 -1] [ 0 0 0 1 0 0 0 -1 0 0 0 -1 0 0 0 1] [ 0 0 0 0 1 -1 -1 1 0 0 0 0 -1 1 1 -1] [ 0 0 0 0 0 1 0 -1 0 0 0 0 0 -1 0 1] [ 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1] [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1] [ 0 0 0 0 0 0 0 0 1 -1 -1 1 -1 1 1 -1] [ 0 0 0 0 0 0 0 0 0 1 0 -1 0 -1 0 1] [ 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] x variables in terms of y variables [-x0000 + y0000 - y0001 - y0010 + y0011 - y0100 + y0101 + y0110 - y0111 - y1000 + y1001 + y1010 - y1011 + y1100 - y1101 - y1110 + y1111, -x0001 + y0001 - y0011 - y0101 + y0111 - y1001 + y1011 + y1101 - y1111, -x0010 + y0010 - y0011 - y0110 + y0111 - y1010 + y1011 + y1110 - y1111, -x0011 + y0011 - y0111 - y1011 + y1111, -x0100 + y0100 - y0101 - y0110 + y0111 - y1100 + y1101 + y1110 - y1111, -x0101 + y0101 - y0111 - y1101 + y1111, -x0110 + y0110 - y0111 - y1110 + y1111, -x0111 + y0111 - y1111, -x1000 + y1000 - y1001 - y1010 + y1011 - y1100 + y1101 + y1110 - y1111, -x1001 + y1001 - y1011 - y1101 + y1111, -x1010 + y1010 - y1011 - y1110 + y1111, -x1011 + y1011 - y1111, -x1100 + y1100 - y1101 - y1110 + y1111, -x1101 + y1101 - y1111, -x1110 + y1110 - y1111, -x1111 + y1111] y variables in terms of x variables [x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x0110 + x0111 + x1000 + x1001 + x1010 + x1011 + x1100 + x1101 + x1110 + x1111 - y0000, x0001 + x0011 + x0101 + x0111 + x1001 + x1011 + x1101 + x1111 - y0001, x0010 + x0011 + x0110 + x0111 + x1010 + x1011 + x1110 + x1111 - y0010, x0011 + x0111 + x1011 + x1111 - y0011, x0100 + x0101 + x0110 + x0111 + x1100 + x1101 + x1110 + x1111 - y0100, x0101 + x0111 + x1101 + x1111 - y0101, x0110 + x0111 + x1110 + x1111 - y0110, x0111 + x1111 - y0111, x1000 + x1001 + x1010 + x1011 + x1100 + x1101 + x1110 + x1111 - y1000, x1001 + x1011 + x1101 + x1111 - y1001, x1010 + x1011 + x1110 + x1111 - y1010, x1011 + x1111 - y1011, x1100 + x1101 + x1110 + x1111 - y1100, x1101 + x1111 - y1101, x1110 + x1111 - y1110, x1111 - y1111] random pick atom i.e. x variable probabilities [0.0499286050247534, 0.00124006780192525, 0.0770562797441154, 0.244429385233403, 0.0428526043638640, 0.0478534087266897, 0.00257721034718066, 0.0478929010196389, 0.232913880506096, 0.000215368431558970, 0.0753243541255760, 0.000332654327962137, 0.0142291218211613, 0.0379301929733677, 0.123307182154035, 0.00191678339867260] y variable probabilities [1.00000000000000, 0.381810761913218, 0.572836750350584, 0.294571723979677, 0.318559404804610, 0.135593286118369, 0.175694076919527, 0.0498096844183115, 0.486169537738429, 0.0403949991315614, 0.200880974006245, 0.00224943772663473, 0.177383280347236, 0.0398469763720403, 0.125223965552707, 0.00191678339867260] sum probabilities check 1 iy [[], [4], [3], [3, 4], [2], [2, 4], [2, 3], [2, 3, 4], [1], [1, 4], [1, 3], [1, 3, 4], [1, 2], [1, 2, 4], [1, 2, 3], [1, 2, 3, 4]] define polynomials for independent generators, generators=3 then y100,y010,y001 are the generators [y0001*y0010 - y0011, y0001*y0100 - y0101, y0010*y0100 - y0110, y0001*y0010*y0100 - y0111, y0001*y1000 - y1001, y0010*y1000 - y1010, y0001*y0010*y1000 - y1011, y0100*y1000 - y1100, y0001*y0100*y1000 - y1101, y0010*y0100*y1000 - y1110, y0001*y0010*y0100*y1000 - y1111] random integer polynomial in y variables 3*y0000*y0001 - y0111*y1011 + y1100^2 - y0000 treated as boolean polynomial y0000*y0001 + y0111*y1011 + y1100^2 + y0000 parsing and removing leading and trailing spaces [['y0000', 'y0001'], ['y0111', 'y1011'], ['y1100^2'], ['y0000']] removing powers y^m --> y [['y0000', 'y0001'], ['y0111', 'y1011'], ['y1100'], ['y0000']] or-ing the bits in y101 with y110 gives y111 (A*C)*(A*B)=A*B*C for instance ie y101*y110=y111 reduced to 1st order - linear - boolean polynomial in y variables y0000 + y0001 + y1100 + y1111 expressing in the x variables - still boolean - using vector(y var coefficients)*M*vector(xvars) x0000 + 2*x0001 + x0010 + 2*x0011 + x0100 + 2*x0101 + x0110 + 2*x0111 + x1000 + 2*x1001 + x1010 + 2*x1011 + 2*x1100 + 3*x1101 + 2*x1110 + 4*x1111 reducing the x var polynomial x0000 + x0010 + x0100 + x0110 + x1000 + x1010 + x1101 converting to probability using vector(x var coefficients)*(M^-1)*vector(yvars) y0000 - y0001 - y1100 + 2*y1101 - y1111 prob_poly_monomials [y0000, y0001, y1100, y1101, y1111] prob_poly_coefficients [1, -1, -1, 2, -1] if the generators are independent the probability poly becomes -y0001*y0010*y0100*y1000 + 2*y0001*y0100*y1000 - y0100*y1000 + y0000 - y0001

MPR.remove_var(*tuple(xvars))
MPR
MPR.remove_var(*tuple(xvars)).change_ring(ZZ).random_element()

Multivariate Polynomial Ring in y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111 over Rational Field Multivariate Polynomial Ring in x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111, y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111 over Rational Field -y0001*y0010 - y1000*y1100 - y0001*y1110 + y0010

bool_poly = MPR.remove_var(*tuple(xvars)).change_ring(ZZ).random_element()+y0100^3; print(bool_poly); print(' ')
bool_poly = vector([QQ(mod(a,2)) for a in bool_poly.coefficients()])*vector(bool_poly.monomials()); print(bool_poly); print(' ') #reduce modulo 2 (boolean polynomial remember) then coerce new coefficients back to QQ
parse_bool_poly = [[b.strip() for b in c] for c in [a.split('*') for a in str(bool_poly).split('+')]]; print(parse_bool_poly); print(' ')

y0100^3 - 5*y0111^2 - y0100*y1101 + 16*y0000*y1110 + y1000*y1110 + 2*y1011 y0100^3 + y0111^2 + y0100*y1101 + y1000*y1110 [['y0100^3'], ['y0111^2'], ['y0100', 'y1101'], ['y1000', 'y1110']]

('y0100').find('^')
('y0100^3').find('^')
[[a.split('^')[0] for a in c] for c in parse_bool_poly]

-1 5 [['y0100'], ['y0111'], ['y0100', 'y1101'], ['y1000', 'y1110']]

prob_poly
[prob_poly.coefficient(a) for a in yvars]
[1 for i in range(atoms)]

y0000 - y0001 - y1100 + 2*y1101 - y1111 [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0, -1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

prob_poly
[prob_poly.coefficient(a) for a in yvars]
[1 for i in range(atoms)]
block_matrix(2,1,[matrix([prob_poly.coefficient(a) for a in yvars]),matrix([1 for i in range(atoms)])],subdivide=False)
zero_matrix(2,2)
2*identity_matrix(atoms,atoms)

y0000 - y0001 - y1100 + 2*y1101 - y1111 [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0, -1] [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] [ 1 -1 0 0 0 0 0 0 0 0 0 0 -1 2 0 -1] [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [0 0] [0 0] [2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2]

prob_poly_x = vector([prob_poly.coefficient(a) for a in yvars])*MM*vector(xvars);prob_poly_x
prob_poly_x_coefficients = matrix([prob_poly_x.coefficient(a) for a in xvars]);prob_poly_x_coefficients
lgr=block_matrix(2,1,[prob_poly_x_coefficients,matrix([1 for i in range(atoms)])],subdivide=False)
lagrange_matrix=block_matrix(2,2,[2*identity_matrix(atoms,atoms),lgr.transpose(),lgr,zero_matrix(2,2)],subdivide=False);lagrange_matrix
lagrange_vector = vector([2/atoms for i in range(atoms)]+[.6,1]);lagrange_vector
lagrange_inverse = lagrange_matrix.inverse();lagrange_inverse
(lagrange_inverse*lagrange_vector)[0:atoms]

x0000 + x0010 + x0100 + x0110 + x1000 + x1010 + x1101 [1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0] [2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1] [1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 0] [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0] (0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.125000000000000, 0.600000000000000, 1.00000000000000) [ 3/7 0 -1/14 0 -1/14 0 -1/14 0 -1/14 0 -1/14 0 0 -1/14 0 0 1/7 0] [ 0 4/9 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 -1/18 -1/18 0 -1/18 -1/18 -1/9 1/9] [ -1/14 0 3/7 0 -1/14 0 -1/14 0 -1/14 0 -1/14 0 0 -1/14 0 0 1/7 0] [ 0 -1/18 0 4/9 0 -1/18 0 -1/18 0 -1/18 0 -1/18 -1/18 0 -1/18 -1/18 -1/9 1/9] [ -1/14 0 -1/14 0 3/7 0 -1/14 0 -1/14 0 -1/14 0 0 -1/14 0 0 1/7 0] [ 0 -1/18 0 -1/18 0 4/9 0 -1/18 0 -1/18 0 -1/18 -1/18 0 -1/18 -1/18 -1/9 1/9] [ -1/14 0 -1/14 0 -1/14 0 3/7 0 -1/14 0 -1/14 0 0 -1/14 0 0 1/7 0] [ 0 -1/18 0 -1/18 0 -1/18 0 4/9 0 -1/18 0 -1/18 -1/18 0 -1/18 -1/18 -1/9 1/9] [ -1/14 0 -1/14 0 -1/14 0 -1/14 0 3/7 0 -1/14 0 0 -1/14 0 0 1/7 0] [ 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 4/9 0 -1/18 -1/18 0 -1/18 -1/18 -1/9 1/9] [ -1/14 0 -1/14 0 -1/14 0 -1/14 0 -1/14 0 3/7 0 0 -1/14 0 0 1/7 0] [ 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 4/9 -1/18 0 -1/18 -1/18 -1/9 1/9] [ 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 -1/18 4/9 0 -1/18 -1/18 -1/9 1/9] [ -1/14 0 -1/14 0 -1/14 0 -1/14 0 -1/14 0 -1/14 0 0 3/7 0 0 1/7 0] [ 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 -1/18 -1/18 0 4/9 -1/18 -1/9 1/9] [ 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 -1/18 0 -1/18 -1/18 0 -1/18 4/9 -1/9 1/9] [ 1/7 -1/9 1/7 -1/9 1/7 -1/9 1/7 -1/9 1/7 -1/9 1/7 -1/9 -1/9 1/7 -1/9 -1/9 -32/63 2/9] [ 0 1/9 0 1/9 0 1/9 0 1/9 0 1/9 0 1/9 1/9 0 1/9 1/9 2/9 -2/9] (0.0857142857142857, 0.0444444444444444, 0.0857142857142857, 0.0444444444444444, 0.0857142857142857, 0.0444444444444444, 0.0857142857142857, 0.0444444444444444, 0.0857142857142857, 0.0444444444444444, 0.0857142857142857, 0.0444444444444444, 0.0444444444444444, 0.0857142857142857, 0.0444444444444444, 0.0444444444444444)

print('probability calculus for non independent generators (4 in this case) modeled on unit n-cube the vertices are atoms of statement calculus')
print('example variable description for 3-cube with generators A,B,C')
print('y000 = Prob(1), y100=Prob(A), y010=Prob(B), y001=Prob(C), y110=Prob(A*B), y011=Prob(B*C), y101=Prob(C*A), y111=Prob(A*B*C)')
print('x000 = Prob((1+A)*(1+B)*(1+C)), x100=Prob(A*(1+B)*(1+C)), x010=Prob((1+A)*B*(1+C)), x001=Prob((1+A)*(1+B)*C), x110=Prob(A*B*(1+C)), x011=Prob((1+A)*B*C), x101=Prob(A*(1+B)*C), x111=Prob(A*B*C)')
print(' ')
num_gens = 4 # 4-cube
num_atoms = 2**num_gens
print('# of generators ' + str(num_gens) + ', # of atoms ' + str(num_atoms)); print(' ')
x = ['x' + Integer(ii+num_atoms).binary()[1:num_gens + 1] for ii in range(num_atoms)];print('x probability variables');print(x);print(' ')
y = ['y' + Integer(ii+num_atoms).binary()[1:num_gens + 1] for ii in range(num_atoms)];print('y probability variables');print(y);print(' ')
MPR = PolynomialRing(QQ,x + y);print(MPR);print(' ')
MPR.inject_variables(); print(' ')
xvars = [MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'x')]];print('xvars');print(xvars);print(' ') # atoms correspond to the vertices of the unit n-cube
yvars = [MPR.gens()[k] for k in [i for i, j in enumerate(MPR.gens()) if (str(j)[0] == 'y')]];print('yvars');print(yvars);print(' ') # these correspond to expressions formed by generators and *, y101 = a*c, y111=a*b*c,  y000 = 1 = x000+x001+x010+... also y101 = Prob(A*C) etc
#AND = lambda A, B: A*B
#FA = lambda mylist: 1 if not mylist else Integer(reduce(AND, mylist))
#MM = matrix([[Integer(FA([str(xvarsx)[i] == '1' for i in iyvars])) for xvarsx in xvars] for iyvars in iy]);print('MM');print('is MM totally unimodular? going out on limb and saying yes');print(MM);print(' ')
# just a note: Prob(A*~B) = Prob(A*(1+B)) = Prob(A+A*B)=Prob(A)-Prob(A*B)
#MMinv = MM.inverse();print('MMinv');print(MMinv);print(' ') # well almost, is MM totally unimodular? Integer matrix with integer inverse det = +/- 1
# recursive way of defining MM and MMinv
MM = matrix([1])
for ii in range(1,num_gens + 1):
    MM = block_matrix(2,2,[MM,MM,0,MM],subdivide=False)
print('Sierpinski triangle Matrix, M');print(MM);print(' ')
print('y = M*x, is M totally unimodular? going out on limb and saying yes');
MMinv = matrix([1])
for ii in range(1,num_gens+1):
    MMinv = block_matrix(2,2,[MMinv,-MMinv,0,MMinv],subdivide=False)
print('M^-1');print(MMinv);print(' ')
x_of_y = list(MMinv*vector(yvars) - vector(xvars));print('x variables in terms of y variables');print(x_of_y);print(' ')
y_of_x = list(MM*vector(xvars) - vector(yvars));print('y variables in terms of x variables');print(y_of_x);print(' ')
prob_atoms = [QQ(normalvariate(0, 1)^2) for i in range(num_atoms)]
normalize = sum(prob_atoms)
prob_atoms = [pa/normalize for pa in prob_atoms]
RR_prob_atoms = [RR(pa) for pa in prob_atoms]
print('random pick atom i.e. x variable probabilities'); print(RR_prob_atoms); print(' ')
y_var_prob = list(MM*vector(RR_prob_atoms)); print('y variable probabilities'); print(y_var_prob);print(' ')
atom_prob_polys = list(vector(xvars) - vector(prob_atoms)) #just assign rational probabilities to the X variables
print('sum probabilities check');print(sum(prob_atoms));print(' ')

iy = [[i for i,j in enumerate(str(a)) if j == '1'] for a in yvars]; print('iy'); print(iy); print(' ')
leniy = [len(c) for c in iy]
ind_poly_list = []
for ii in range(len(yvars)):
    if leniy[ii] > 1:
        str_yvars = str(yvars[ii])
        gens_list = []
        for jj in range(1,len(str_yvars)):
            if str_yvars[jj] == '1':
                gens_list.append(yvars[2^(num_gens - jj)])
        ind_poly_list.append(reduce(lambda a,b: a*b, gens_list) - yvars[ii])

print('define polynomials for independent generators, generators=3 then y100,y010,y001 are the generators')
print(ind_poly_list);print(' ')
#myideal=ideal(ind_poly_list)
#QR = MPR.quotient_ring(myideal)
#QR.inject_variables()
        
#bool_poly = y0101*y1100+y1111*y1000*y0110+2*y1110; print(bool_poly) # make sure coefficients are in ZZ
bool_poly = MPR.remove_var(*tuple(xvars)).change_ring(ZZ).random_element(); print('random integer polynomial in y variables'); print(bool_poly); print(' ')
bool_poly = vector([QQ(mod(a,2)) for a in bool_poly.coefficients()])*vector(bool_poly.monomials()); print('treated as boolean polynomial') ;print(bool_poly); print(' ') #reduce modulo 2 (boolean polynomial remember) then coerce new coefficients back to QQ
parse_bool_poly = [[b.strip() for b in c] for c in [a.split('*') for a in str(bool_poly).split('+')]]; print('parsing and removing leading and trailing spaces'); print(parse_bool_poly); print(' ') 
parse_bool_poly = [[a.split('^')[0] for a in c] for c in parse_bool_poly]; print('removing powers y^m --> y'); print(parse_bool_poly); print(' ')
listA = []
for pbp in parse_bool_poly:
    listB = []
    for b in pbp:
        listB.append('1' + b[1:])
    k = int(listB[0],2)
    for b in listB:
        k = k | int(b,2) # or-ing the bits in y101 with y110 gives y111 (A*C)*(A*B)=A*B*C for instance ie y101*y110=y111
    listA.append('y' + Integer(k).binary()[1:])
#creating the equivalent reduced boolean polynomial
#y vars as string being converted back to the generators in the polynomial ring and we previously injected the variable so we can write expressions in the generators directly
print('or-ing the bits in y101 with y110 gives y111 (A*C)*(A*B)=A*B*C for instance ie y101*y110=y111')
reduced_bool_poly = sum([MPR(y) for y in listA]); print('reduced to 1st order - linear - boolean polynomial in yvariables'); print(reduced_bool_poly); print(' ')
#now the procedure to obtain the probability expression
vctr = vector([reduced_bool_poly.coefficient(g) for g in yvars])
atom_bool_poly = vctr*MM*vector(xvars); print('expressing in the x variables - still boolean - using vector(y var coefficients)*M*vector(xvars)'); print(atom_bool_poly); print(' ')
atom_bool_poly = sum([QQ(mod(atom_bool_poly.coefficient(a),2))*a for a in xvars]); print('reducing the x var polynomial'); print(atom_bool_poly); print(' ')
prob_poly = vector([atom_bool_poly.coefficient(a) for a in xvars])*MMinv*vector(yvars); print('converting to probability using xvar_coeff_vector*(M^-1)*vector(yvars)');print(prob_poly); print(' ')
#for independent generators prob_poly needs to be linear in the y variables use this with the output from above (testpoly4)
prob_poly_monomials = prob_poly.monomials(); print('prob_poly_monomials'); print(prob_poly_monomials); print(' ')
prob_poly_coefficients = prob_poly.coefficients(); print('prob_poly_coefficients'); print(prob_poly_coefficients); print(' ')
monomial_gens_list = []
for a in prob_poly_monomials:
    str_a = str(a)
    gens_list = []
    for jj in range(1,len(str_a)):
        if str_a[jj] == '1':
            gens_list.append(yvars[2^(num_gens - jj)])
    if len(gens_list) <> 0:
        monomial_gens_list.append(reduce(lambda a,b:a*b ,gens_list))
    else:
        monomial_gens_list.append(yvars[0]) #yvars[0]=y000 then gens_list = [] so we put yvars[0] back
prob_poly_ind = vector(prob_poly_coefficients)*vector(monomial_gens_list); print('if the generators are independent the probability poly becomes'); print(prob_poly_ind); print(' ')

#lagrange multiplier problem
print('approximation to maximum entropy using least squares subject to constraints:')
print('minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value'); print(' ')
#atom_bool_poly2 = vector([prob_poly.coefficient(a) for a in yvars])*MM*vector(xvars) # should equal atom_bool_poly
#atom_bool_poly_coefficients = matrix([atom_bool_poly2.coefficient(a) for a in xvars])
atom_bool_poly_coefficients = matrix([atom_bool_poly.coefficient(a) for a in xvars]); print('coefficient vector of boolean polynomial in terms of atoms - xvars'); print(atom_bool_poly_coefficients); print(' ')
observed_prob_of_bool_poly = QQ(round(.05 + 0.9*random(), 3)); print('suppose observed probability of boolean polynomial = ' + str(observed_prob_of_bool_poly)); print(' ')
lgr = block_matrix(2,1,[atom_bool_poly_coefficients, matrix([1 for i in range(num_atoms)])], subdivide = False)
lagrange_matrix = block_matrix(QQ, 2, 2, [2*identity_matrix(num_atoms,num_atoms), lgr.transpose(), lgr, zero_matrix(2,2)],subdivide = False); print('lagrange multiplier matrix'); print(lagrange_matrix); print(' ')
lagrange_inverse = lagrange_matrix.inverse();print('inverse of lagrange matrix'); print(lagrange_inverse); print(' ')
lagrange_vector = vector([QQ(2/num_atoms) for i in range(num_atoms)] + [observed_prob_of_bool_poly, 1])
print('left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu])'); print(lagrange_vector); print(' ')
atom_probabilities = (lagrange_inverse*lagrange_vector)[0:num_atoms]; print('atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial'); print(atom_probabilities); print(' ')

probability calculus for non independent generators (4 in this case) modeled on unit n-cube the vertices are atoms of statement calculus example variable description for 3-cube with generators A,B,C y000 = Prob(1), y100=Prob(A), y010=Prob(B), y001=Prob(C), y110=Prob(A*B), y011=Prob(B*C), y101=Prob(C*A), y111=Prob(A*B*C) x000 = Prob((1+A)*(1+B)*(1+C)), x100=Prob(A*(1+B)*(1+C)), x010=Prob((1+A)*B*(1+C)), x001=Prob((1+A)*(1+B)*C), x110=Prob(A*B*(1+C)), x011=Prob((1+A)*B*C), x101=Prob(A*(1+B)*C), x111=Prob(A*B*C) # of generators 4, # of atoms 16 x probability variables ['x0000', 'x0001', 'x0010', 'x0011', 'x0100', 'x0101', 'x0110', 'x0111', 'x1000', 'x1001', 'x1010', 'x1011', 'x1100', 'x1101', 'x1110', 'x1111'] y probability variables ['y0000', 'y0001', 'y0010', 'y0011', 'y0100', 'y0101', 'y0110', 'y0111', 'y1000', 'y1001', 'y1010', 'y1011', 'y1100', 'y1101', 'y1110', 'y1111'] Multivariate Polynomial Ring in x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111, y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111 over Rational Field Defining x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111, y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111 xvars [x0000, x0001, x0010, x0011, x0100, x0101, x0110, x0111, x1000, x1001, x1010, x1011, x1100, x1101, x1110, x1111] yvars [y0000, y0001, y0010, y0011, y0100, y0101, y0110, y0111, y1000, y1001, y1010, y1011, y1100, y1101, y1110, y1111] Sierpinski triangle Matrix, M [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1] [0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1] [0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1] [0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1] [0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] y = M*x, is M totally unimodular? going out on limb and saying yes M^-1 [ 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1] [ 0 1 0 -1 0 -1 0 1 0 -1 0 1 0 1 0 -1] [ 0 0 1 -1 0 0 -1 1 0 0 -1 1 0 0 1 -1] [ 0 0 0 1 0 0 0 -1 0 0 0 -1 0 0 0 1] [ 0 0 0 0 1 -1 -1 1 0 0 0 0 -1 1 1 -1] [ 0 0 0 0 0 1 0 -1 0 0 0 0 0 -1 0 1] [ 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1] [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1] [ 0 0 0 0 0 0 0 0 1 -1 -1 1 -1 1 1 -1] [ 0 0 0 0 0 0 0 0 0 1 0 -1 0 -1 0 1] [ 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1] [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] x variables in terms of y variables [-x0000 + y0000 - y0001 - y0010 + y0011 - y0100 + y0101 + y0110 - y0111 - y1000 + y1001 + y1010 - y1011 + y1100 - y1101 - y1110 + y1111, -x0001 + y0001 - y0011 - y0101 + y0111 - y1001 + y1011 + y1101 - y1111, -x0010 + y0010 - y0011 - y0110 + y0111 - y1010 + y1011 + y1110 - y1111, -x0011 + y0011 - y0111 - y1011 + y1111, -x0100 + y0100 - y0101 - y0110 + y0111 - y1100 + y1101 + y1110 - y1111, -x0101 + y0101 - y0111 - y1101 + y1111, -x0110 + y0110 - y0111 - y1110 + y1111, -x0111 + y0111 - y1111, -x1000 + y1000 - y1001 - y1010 + y1011 - y1100 + y1101 + y1110 - y1111, -x1001 + y1001 - y1011 - y1101 + y1111, -x1010 + y1010 - y1011 - y1110 + y1111, -x1011 + y1011 - y1111, -x1100 + y1100 - y1101 - y1110 + y1111, -x1101 + y1101 - y1111, -x1110 + y1110 - y1111, -x1111 + y1111] y variables in terms of x variables [x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x0110 + x0111 + x1000 + x1001 + x1010 + x1011 + x1100 + x1101 + x1110 + x1111 - y0000, x0001 + x0011 + x0101 + x0111 + x1001 + x1011 + x1101 + x1111 - y0001, x0010 + x0011 + x0110 + x0111 + x1010 + x1011 + x1110 + x1111 - y0010, x0011 + x0111 + x1011 + x1111 - y0011, x0100 + x0101 + x0110 + x0111 + x1100 + x1101 + x1110 + x1111 - y0100, x0101 + x0111 + x1101 + x1111 - y0101, x0110 + x0111 + x1110 + x1111 - y0110, x0111 + x1111 - y0111, x1000 + x1001 + x1010 + x1011 + x1100 + x1101 + x1110 + x1111 - y1000, x1001 + x1011 + x1101 + x1111 - y1001, x1010 + x1011 + x1110 + x1111 - y1010, x1011 + x1111 - y1011, x1100 + x1101 + x1110 + x1111 - y1100, x1101 + x1111 - y1101, x1110 + x1111 - y1110, x1111 - y1111] random pick atom i.e. x variable probabilities [0.400447430848116, 0.0471982467364013, 0.00731937150571459, 0.00369538183082823, 0.144034680703587, 0.0736927279463103, 0.0776219939341513, 0.0216040394769143, 0.135864634148865, 0.0701051147160910, 0.00996209002643161, 0.00349084697594845, 0.000409105229442846, 0.00229786940149651, 0.000752356293173467, 0.00150411022652813] y variable probabilities [1.00000000000000, 0.223588337310518, 0.125950190269690, 0.0302943785102191, 0.321916883211604, 0.0990987470512493, 0.101482499930767, 0.0231081497034425, 0.224386127017977, 0.0773979413200641, 0.0157094035220817, 0.00499495720247658, 0.00496344115064095, 0.00380197962802464, 0.00225646651970160, 0.00150411022652813] sum probabilities check 1 iy [[], [4], [3], [3, 4], [2], [2, 4], [2, 3], [2, 3, 4], [1], [1, 4], [1, 3], [1, 3, 4], [1, 2], [1, 2, 4], [1, 2, 3], [1, 2, 3, 4]] define polynomials for independent generators, generators=3 then y100,y010,y001 are the generators [y0001*y0010 - y0011, y0001*y0100 - y0101, y0010*y0100 - y0110, y0001*y0010*y0100 - y0111, y0001*y1000 - y1001, y0010*y1000 - y1010, y0001*y0010*y1000 - y1011, y0100*y1000 - y1100, y0001*y0100*y1000 - y1101, y0010*y0100*y1000 - y1110, y0001*y0010*y0100*y1000 - y1111] random integer polynomial in y variables -10*y0001*y0010 + 9*y1011*y1101 + 4*y0010*y1111 - 3*y1100 treated as boolean polynomial y1011*y1101 + y1100 parsing and removing leading and trailing spaces [['y1011', 'y1101'], ['y1100']] removing powers y^m --> y [['y1011', 'y1101'], ['y1100']] or-ing the bits in y101 with y110 gives y111 (A*C)*(A*B)=A*B*C for instance ie y101*y110=y111 reduced to 1st order - linear - boolean polynomial in yvariables y1100 + y1111 expressing in the x variables - still boolean - using vector(y var coefficients)*M*vector(xvars) x1100 + x1101 + x1110 + 2*x1111 reducing the x var polynomial x1100 + x1101 + x1110 converting to probability using xvar_coeff_vector*(M^-1)*vector(yvars) y1100 - y1111 prob_poly_monomials [y1100, y1111] prob_poly_coefficients [1, -1] if the generators are independent the probability poly becomes -y0001*y0010*y0100*y1000 + y0100*y1000 approximation to maximum entropy using least squares subject to constraints: minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value coefficient vector of boolean polynomial in terms of atoms - xvars [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0] suppose observed probability of boolean polynomial = 341/1000 lagrange multiplier matrix [2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0] [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0] inverse of lagrange matrix [ 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 0 0 0 -1/26 -1/13 1/13] [ 0 0 0 0 0 0 0 0 0 0 0 0 1/3 -1/6 -1/6 0 1/3 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 -1/6 1/3 -1/6 0 1/3 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 -1/6 -1/6 1/3 0 1/3 0] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 6/13 -1/13 1/13] [ -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 1/3 1/3 1/3 -1/13 -32/39 2/13] [ 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 0 0 0 1/13 2/13 -2/13] left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu]) (1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 341/1000, 1) atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial (659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 341/3000, 341/3000, 341/3000, 659/13000) y1011*y1101 + y1100 parsing and removing leading and trailing spaces [['y1011', 'y1101'], ['y1100']] removing powers y^m --> y [['y1011', 'y1101'], ['y1100']] or-ing the bits in y101 with y110 gives y111 (A*C)*(A*B)=A*B*C for instance ie y101*y110=y111 reduced to 1st order - linear - boolean polynomial in yvariables y1100 + y1111 expressing in the x variables - still boolean - using vector(y var coefficients)*M*vector(xvars) x1100 + x1101 + x1110 + 2*x1111 reducing the x var polynomial x1100 + x1101 + x1110 converting to probability using xvar_coeff_vector*(M^-1)*vector(yvars) y1100 - y1111 prob_poly_monomials [y1100, y1111] prob_poly_coefficients [1, -1] if the generators are independent the probability poly becomes -y0001*y0010*y0100*y1000 + y0100*y1000 approximation to maximum entropy using least squares subject to constraints: minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value coefficient vector of boolean polynomial in terms of atoms - xvars [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0] suppose observed probability of boolean polynomial = 341/1000 lagrange multiplier matrix [2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0] [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0] inverse of lagrange matrix [ 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 -1/26 0 0 0 -1/26 -1/13 1/13] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 6/13 0 0 0 -1/26 -1/13 1/13] [ 0 0 0 0 0 0 0 0 0 0 0 0 1/3 -1/6 -1/6 0 1/3 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 -1/6 1/3 -1/6 0 1/3 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 -1/6 -1/6 1/3 0 1/3 0] [ -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 -1/26 0 0 0 6/13 -1/13 1/13] [ -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 1/3 1/3 1/3 -1/13 -32/39 2/13] [ 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 1/13 0 0 0 1/13 2/13 -2/13] left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu]) (1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 341/1000, 1) atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial (659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 659/13000, 341/3000, 341/3000, 341/3000, 659/13000)

#atom_bool_poly=x1010 + x1011 + x1101 + x1110 + x1111
#lagrange multiplier problem
print('approximation to maximum entropy using least squares subject to constraints:')
print('minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value'); print(' ')
#atom_bool_poly2 = vector([prob_poly.coefficient(a) for a in yvars])*MM*vector(xvars) # should equal atom_bool_poly
#atom_bool_poly_coefficients = matrix([atom_bool_poly2.coefficient(a) for a in xvars])
atom_bool_poly_coefficients = matrix([atom_bool_poly.coefficient(a) for a in xvars]); print('coefficient vector of boolean polynomial in terms of atoms - xvars'); print(atom_bool_poly_coefficients); print(' ')
observed_prob_of_bool_poly = 0 #QQ(round(.05 + 0.9*random(), 3)); print('suppose observed probability of boolean polynomial = ' + str(observed_prob_of_bool_poly)); print(' ')
lgr = block_matrix(2,1,[atom_bool_poly_coefficients, matrix([1 for i in range(num_atoms)])], subdivide = False)
lagrange_matrix = block_matrix(QQ, 2, 2, [2*identity_matrix(num_atoms,num_atoms), lgr.transpose(), lgr, zero_matrix(2,2)],subdivide = False); print('lagrange multiplier matrix'); print(lagrange_matrix); print(' ')
lagrange_inverse = lagrange_matrix.inverse();print('inverse of lagrange matrix'); print(lagrange_inverse); print(' ')
lagrange_vector = vector([QQ(2/num_atoms) for i in range(num_atoms)] + [observed_prob_of_bool_poly, 1])
print('left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu])'); print(lagrange_vector); print(' ')
atom_probabilities = (lagrange_inverse*lagrange_vector)[0:num_atoms]; print('atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial'); print(atom_probabilities); print(' ')

approximation to maximum entropy using least squares subject to constraints: minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value coefficient vector of boolean polynomial in terms of atoms - xvars [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1] lagrange multiplier matrix [2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1] [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0] [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0] inverse of lagrange matrix [11/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 11/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 11/24 -1/24 -1/24 -1/24 0 0 -1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 -1/24 11/24 -1/24 -1/24 0 0 -1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 -1/24 -1/24 11/24 -1/24 0 0 -1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 -1/24 -1/24 -1/24 11/24 0 0 -1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [ 0 0 0 0 0 0 3/8 -1/8 0 0 0 0 0 0 -1/8 -1/8 1/4 0] [ 0 0 0 0 0 0 -1/8 3/8 0 0 0 0 0 0 -1/8 -1/8 1/4 0] [-1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 11/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/24 11/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/24 -1/24 11/24 -1/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/24 -1/24 -1/24 11/24 -1/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/24 -1/24 -1/24 -1/24 11/24 -1/24 0 0 -1/12 1/12] [-1/24 -1/24 -1/24 -1/24 -1/24 -1/24 0 0 -1/24 -1/24 -1/24 -1/24 -1/24 11/24 0 0 -1/12 1/12] [ 0 0 0 0 0 0 -1/8 -1/8 0 0 0 0 0 0 3/8 -1/8 1/4 0] [ 0 0 0 0 0 0 -1/8 -1/8 0 0 0 0 0 0 -1/8 3/8 1/4 0] [-1/12 -1/12 -1/12 -1/12 -1/12 -1/12 1/4 1/4 -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 1/4 1/4 -2/3 1/6] [ 1/12 1/12 1/12 1/12 1/12 1/12 0 0 1/12 1/12 1/12 1/12 1/12 1/12 0 0 1/6 -1/6] left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu]) (1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 0, 1) atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial (1/12, 1/12, 1/12, 1/12, 1/12, 1/12, 0, 0, 1/12, 1/12, 1/12, 1/12, 1/12, 1/12, 0, 0)

atom_bool_poly2=x1010 + x1011 + x1101 + x1110 + x1111
atom_bool_poly=x0110 + x0111 + x1110 + x1111 
nabp=sum(xvars)-atom_bool_poly
nabp2=sum(xvars)-atom_bool_poly2
#lagrange multiplier problem
print('approximation to maximum entropy using least squares subject to constraints:')
print('minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value'); print(' ')
#atom_bool_poly2 = vector([prob_poly.coefficient(a) for a in yvars])*MM*vector(xvars) # should equal atom_bool_poly
#atom_bool_poly_coefficients = matrix([atom_bool_poly2.coefficient(a) for a in xvars])
atom_bool_poly_coefficients = matrix([atom_bool_poly.coefficient(a) for a in xvars]); print('coefficient vector of boolean polynomial in terms of atoms - xvars'); print(atom_bool_poly_coefficients); print(' ')
nabpc=matrix([nabp.coefficient(a) for a in xvars])
observed_prob_of_bool_poly = 1 #QQ(round(.05 + 0.9*random(), 3)); print('suppose observed probability of boolean polynomial = ' + str(observed_prob_of_bool_poly)); print(' ')
atom_bool_poly_coefficients2 = matrix([atom_bool_poly2.coefficient(a) for a in xvars]); print('coefficient vector of boolean polynomial in terms of atoms - xvars'); print(atom_bool_poly_coefficients2); print(' ')
nabpc2=matrix([nabp2.coefficient(a) for a in xvars])
observed_prob_of_bool_poly2 = 1
lgr = block_matrix(3,1,[atom_bool_poly_coefficients, atom_bool_poly_coefficients2, matrix([1 for i in range(num_atoms)])], subdivide = False)
lagrange_matrix = block_matrix(QQ, 2, 2, [2*identity_matrix(num_atoms,num_atoms), lgr.transpose(), lgr, zero_matrix(3,3)],subdivide = False); print('lagrange multiplier matrix'); print(lagrange_matrix); print(' ')
lagrange_inverse = lagrange_matrix.inverse();print('inverse of lagrange matrix'); print(lagrange_inverse); print(' ')
lagrange_vector = vector([QQ(2/num_atoms) for i in range(num_atoms)] + [observed_prob_of_bool_poly,observed_prob_of_bool_poly2,1])
print('left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu])'); print(lagrange_vector); print(' ')
atom_probabilities = (lagrange_inverse*lagrange_vector)[0:num_atoms]; print('atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial'); print(atom_probabilities); print(' ')

approximation to maximum entropy using least squares subject to constraints: minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value coefficient vector of boolean polynomial in terms of atoms - xvars [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1] coefficient vector of boolean polynomial in terms of atoms - xvars [0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1] lagrange multiplier matrix [2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 1] [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0] [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0] inverse of lagrange matrix [ 35/78 -2/39 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 35/78 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 35/78 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 -2/39 35/78 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 -2/39 -2/39 35/78 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 -2/39 -2/39 -2/39 35/78 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -1/52 -1/52 -1/52 -1/52 -1/52 -1/52 35/104 -17/104 -1/52 -1/52 3/52 3/52 -1/52 3/52 -9/104 -9/104 15/52 -2/13 1/26] [ -1/52 -1/52 -1/52 -1/52 -1/52 -1/52 -17/104 35/104 -1/52 -1/52 3/52 3/52 -1/52 3/52 -9/104 -9/104 15/52 -2/13 1/26] [ -2/39 -2/39 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 35/78 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 35/78 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -1/78 -1/78 -1/78 -1/78 -1/78 -1/78 3/52 3/52 -1/78 -1/78 29/78 -5/39 -1/78 -5/39 -3/52 -3/52 -11/78 3/13 1/39] [ -1/78 -1/78 -1/78 -1/78 -1/78 -1/78 3/52 3/52 -1/78 -1/78 -5/39 29/78 -1/78 -5/39 -3/52 -3/52 -11/78 3/13 1/39] [ -2/39 -2/39 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 35/78 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -1/78 -1/78 -1/78 -1/78 -1/78 -1/78 3/52 3/52 -1/78 -1/78 -5/39 -5/39 -1/78 29/78 -3/52 -3/52 -11/78 3/13 1/39] [ 1/52 1/52 1/52 1/52 1/52 1/52 -9/104 -9/104 1/52 1/52 -3/52 -3/52 1/52 -3/52 35/104 -17/104 11/52 2/13 -1/26] [ 1/52 1/52 1/52 1/52 1/52 1/52 -9/104 -9/104 1/52 1/52 -3/52 -3/52 1/52 -3/52 -17/104 35/104 11/52 2/13 -1/26] [ -5/78 -5/78 -5/78 -5/78 -5/78 -5/78 15/52 15/52 -5/78 -5/78 -11/78 -11/78 -5/78 -11/78 11/52 11/52 -55/78 2/13 5/39] [ -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -2/13 -2/13 -1/13 -1/13 3/13 3/13 -1/13 3/13 2/13 2/13 2/13 -8/13 2/13] [ 4/39 4/39 4/39 4/39 4/39 4/39 1/26 1/26 4/39 4/39 1/39 1/39 4/39 1/39 -1/26 -1/26 5/39 2/13 -8/39] left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu]) (1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1, 1, 1) atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial (-1/26, -1/26, -1/26, -1/26, -1/26, -1/26, 9/52, 9/52, -1/26, -1/26, 3/26, 3/26, -1/26, 3/26, 17/52, 17/52) minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value coefficient vector of boolean polynomial in terms of atoms - xvars [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1] coefficient vector of boolean polynomial in terms of atoms - xvars [0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1] lagrange multiplier matrix [2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 1] [0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0] [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0] inverse of lagrange matrix [ 35/78 -2/39 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 35/78 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 35/78 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 -2/39 35/78 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 -2/39 -2/39 35/78 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 -2/39 -2/39 -2/39 35/78 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -1/52 -1/52 -1/52 -1/52 -1/52 -1/52 35/104 -17/104 -1/52 -1/52 3/52 3/52 -1/52 3/52 -9/104 -9/104 15/52 -2/13 1/26] [ -1/52 -1/52 -1/52 -1/52 -1/52 -1/52 -17/104 35/104 -1/52 -1/52 3/52 3/52 -1/52 3/52 -9/104 -9/104 15/52 -2/13 1/26] [ -2/39 -2/39 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 35/78 -2/39 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -2/39 -2/39 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 35/78 -1/78 -1/78 -2/39 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -1/78 -1/78 -1/78 -1/78 -1/78 -1/78 3/52 3/52 -1/78 -1/78 29/78 -5/39 -1/78 -5/39 -3/52 -3/52 -11/78 3/13 1/39] [ -1/78 -1/78 -1/78 -1/78 -1/78 -1/78 3/52 3/52 -1/78 -1/78 -5/39 29/78 -1/78 -5/39 -3/52 -3/52 -11/78 3/13 1/39] [ -2/39 -2/39 -2/39 -2/39 -2/39 -2/39 -1/52 -1/52 -2/39 -2/39 -1/78 -1/78 35/78 -1/78 1/52 1/52 -5/78 -1/13 4/39] [ -1/78 -1/78 -1/78 -1/78 -1/78 -1/78 3/52 3/52 -1/78 -1/78 -5/39 -5/39 -1/78 29/78 -3/52 -3/52 -11/78 3/13 1/39] [ 1/52 1/52 1/52 1/52 1/52 1/52 -9/104 -9/104 1/52 1/52 -3/52 -3/52 1/52 -3/52 35/104 -17/104 11/52 2/13 -1/26] [ 1/52 1/52 1/52 1/52 1/52 1/52 -9/104 -9/104 1/52 1/52 -3/52 -3/52 1/52 -3/52 -17/104 35/104 11/52 2/13 -1/26] [ -5/78 -5/78 -5/78 -5/78 -5/78 -5/78 15/52 15/52 -5/78 -5/78 -11/78 -11/78 -5/78 -11/78 11/52 11/52 -55/78 2/13 5/39] [ -1/13 -1/13 -1/13 -1/13 -1/13 -1/13 -2/13 -2/13 -1/13 -1/13 3/13 3/13 -1/13 3/13 2/13 2/13 2/13 -8/13 2/13] [ 4/39 4/39 4/39 4/39 4/39 4/39 1/26 1/26 4/39 4/39 1/39 1/39 4/39 1/39 -1/26 -1/26 5/39 2/13 -8/39] left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu]) (1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1, 1, 1) atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial (-1/26, -1/26, -1/26, -1/26, -1/26, -1/26, 9/52, 9/52, -1/26, -1/26, 3/26, 3/26, -1/26, 3/26, 17/52, 17/52)

atom_bool_poly;atom_bool_poly2;print(' ')
atom_bool_poly;sum(xvars)-atom_bool_poly2;print(' ')
sum(xvars)-atom_bool_poly;atom_bool_poly2;print(' ')
sum(xvars)-atom_bool_poly;sum(xvars)-atom_bool_poly2;print(' ')

x0110 + x0111 + x1110 + x1111 x1010 + x1011 + x1101 + x1110 + x1111 x0110 + x0111 + x1110 + x1111 x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x0110 + x0111 + x1000 + x1001 + x1100 x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x1000 + x1001 + x1010 + x1011 + x1100 + x1101 x1010 + x1011 + x1101 + x1110 + x1111 x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x1000 + x1001 + x1010 + x1011 + x1100 + x1101 x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x0110 + x0111 + x1000 + x1001 + x1100

x1=sum(xvars)
atom_bool_poly2=x1-x1010 + x1011 + x1101 + x1110 + x1111
atom_bool_poly=x1-x0110 + x0111 + x1110 + x1111
bool_poly = atom_bool_poly*atom_bool_poly2;bool_poly
bool_poly = sum([QQ(mod(bool_poly.coefficient(a),2))*a for a in bool_poly.monomials()]);bool_poly
parse_bool_poly = [[b.strip() for b in c] for c in [a.split('*') for a in str(bool_poly).split('+')]]; print('parsing and removing leading and trailing spaces'); print(parse_bool_poly); print(' ') 
parse_bool_poly = [[a.split('^')[0] for a in c] for c in parse_bool_poly]; print('removing powers x^m --> x'); print(parse_bool_poly); print(' ')
[len(a) for a in parse_bool_poly]
listA = []
for a in parse_bool_poly:
    if len(a) < 2:
        listA.append(a[0])
listA
atom_bool_poly = sum([MPR(x) for x in listA]);atom_bool_poly
atom_bool_poly = sum([QQ(mod(atom_bool_poly.coefficient(a),2))*a for a in xvars]); print('reducing the x var polynomial'); print(atom_bool_poly); print(' ')
print('approximation to maximum entropy using least squares subject to constraints:')
print('minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value'); print(' ')
atom_bool_poly_coefficients = matrix([atom_bool_poly.coefficient(a) for a in xvars]); print('coefficient vector of boolean polynomial in terms of atoms - xvars'); print(atom_bool_poly_coefficients); print(' ')
observed_prob_of_bool_poly = 1 #QQ(round(.05 + 0.9*random(), 3)); print('suppose observed probability of boolean polynomial = ' + str(observed_prob_of_bool_poly)); print(' ')
lgr = block_matrix(2,1,[atom_bool_poly_coefficients, matrix([1 for i in range(num_atoms)])], subdivide = False)
lagrange_matrix = block_matrix(QQ, 2, 2, [2*identity_matrix(num_atoms,num_atoms), lgr.transpose(), lgr, zero_matrix(2,2)],subdivide = False); print('lagrange multiplier matrix'); print(lagrange_matrix); print(' ')
lagrange_inverse = lagrange_matrix.inverse();print('inverse of lagrange matrix'); print(lagrange_inverse); print(' ')
lagrange_vector = vector([QQ(2/num_atoms) for i in range(num_atoms)] + [observed_prob_of_bool_poly, 1])
print('left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu])'); print(lagrange_vector); print(' ')
atom_probabilities = (lagrange_inverse*lagrange_vector)[0:num_atoms]; print('atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial'); print(atom_probabilities); print(' ')

x0000^2 + 2*x0000*x0001 + x0001^2 + 2*x0000*x0010 + 2*x0001*x0010 + x0010^2 + 2*x0000*x0011 + 2*x0001*x0011 + 2*x0010*x0011 + x0011^2 + 2*x0000*x0100 + 2*x0001*x0100 + 2*x0010*x0100 + 2*x0011*x0100 + x0100^2 + 2*x0000*x0101 + 2*x0001*x0101 + 2*x0010*x0101 + 2*x0011*x0101 + 2*x0100*x0101 + x0101^2 + x0000*x0110 + x0001*x0110 + x0010*x0110 + x0011*x0110 + x0100*x0110 + x0101*x0110 + 3*x0000*x0111 + 3*x0001*x0111 + 3*x0010*x0111 + 3*x0011*x0111 + 3*x0100*x0111 + 3*x0101*x0111 + 2*x0110*x0111 + 2*x0111^2 + 2*x0000*x1000 + 2*x0001*x1000 + 2*x0010*x1000 + 2*x0011*x1000 + 2*x0100*x1000 + 2*x0101*x1000 + x0110*x1000 + 3*x0111*x1000 + x1000^2 + 2*x0000*x1001 + 2*x0001*x1001 + 2*x0010*x1001 + 2*x0011*x1001 + 2*x0100*x1001 + 2*x0101*x1001 + x0110*x1001 + 3*x0111*x1001 + 2*x1000*x1001 + x1001^2 + x0000*x1010 + x0001*x1010 + x0010*x1010 + x0011*x1010 + x0100*x1010 + x0101*x1010 + x0110*x1010 + x0111*x1010 + x1000*x1010 + x1001*x1010 + 3*x0000*x1011 + 3*x0001*x1011 + 3*x0010*x1011 + 3*x0011*x1011 + 3*x0100*x1011 + 3*x0101*x1011 + x0110*x1011 + 5*x0111*x1011 + 3*x1000*x1011 + 3*x1001*x1011 + 2*x1010*x1011 + 2*x1011^2 + 2*x0000*x1100 + 2*x0001*x1100 + 2*x0010*x1100 + 2*x0011*x1100 + 2*x0100*x1100 + 2*x0101*x1100 + x0110*x1100 + 3*x0111*x1100 + 2*x1000*x1100 + 2*x1001*x1100 + x1010*x1100 + 3*x1011*x1100 + x1100^2 + 3*x0000*x1101 + 3*x0001*x1101 + 3*x0010*x1101 + 3*x0011*x1101 + 3*x0100*x1101 + 3*x0101*x1101 + x0110*x1101 + 5*x0111*x1101 + 3*x1000*x1101 + 3*x1001*x1101 + 2*x1010*x1101 + 4*x1011*x1101 + 3*x1100*x1101 + 2*x1101^2 + 4*x0000*x1110 + 4*x0001*x1110 + 4*x0010*x1110 + 4*x0011*x1110 + 4*x0100*x1110 + 4*x0101*x1110 + 2*x0110*x1110 + 6*x0111*x1110 + 4*x1000*x1110 + 4*x1001*x1110 + 2*x1010*x1110 + 6*x1011*x1110 + 4*x1100*x1110 + 6*x1101*x1110 + 4*x1110^2 + 4*x0000*x1111 + 4*x0001*x1111 + 4*x0010*x1111 + 4*x0011*x1111 + 4*x0100*x1111 + 4*x0101*x1111 + 2*x0110*x1111 + 6*x0111*x1111 + 4*x1000*x1111 + 4*x1001*x1111 + 2*x1010*x1111 + 6*x1011*x1111 + 4*x1100*x1111 + 6*x1101*x1111 + 8*x1110*x1111 + 4*x1111^2 x0000^2 + x0001^2 + x0010^2 + x0011^2 + x0100^2 + x0101^2 + x0000*x0110 + x0001*x0110 + x0010*x0110 + x0011*x0110 + x0100*x0110 + x0101*x0110 + x0000*x0111 + x0001*x0111 + x0010*x0111 + x0011*x0111 + x0100*x0111 + x0101*x0111 + x0110*x1000 + x0111*x1000 + x1000^2 + x0110*x1001 + x0111*x1001 + x1001^2 + x0000*x1010 + x0001*x1010 + x0010*x1010 + x0011*x1010 + x0100*x1010 + x0101*x1010 + x0110*x1010 + x0111*x1010 + x1000*x1010 + x1001*x1010 + x0000*x1011 + x0001*x1011 + x0010*x1011 + x0011*x1011 + x0100*x1011 + x0101*x1011 + x0110*x1011 + x0111*x1011 + x1000*x1011 + x1001*x1011 + x0110*x1100 + x0111*x1100 + x1010*x1100 + x1011*x1100 + x1100^2 + x0000*x1101 + x0001*x1101 + x0010*x1101 + x0011*x1101 + x0100*x1101 + x0101*x1101 + x0110*x1101 + x0111*x1101 + x1000*x1101 + x1001*x1101 + x1100*x1101 parsing and removing leading and trailing spaces [['x0000^2'], ['x0001^2'], ['x0010^2'], ['x0011^2'], ['x0100^2'], ['x0101^2'], ['x0000', 'x0110'], ['x0001', 'x0110'], ['x0010', 'x0110'], ['x0011', 'x0110'], ['x0100', 'x0110'], ['x0101', 'x0110'], ['x0000', 'x0111'], ['x0001', 'x0111'], ['x0010', 'x0111'], ['x0011', 'x0111'], ['x0100', 'x0111'], ['x0101', 'x0111'], ['x0110', 'x1000'], ['x0111', 'x1000'], ['x1000^2'], ['x0110', 'x1001'], ['x0111', 'x1001'], ['x1001^2'], ['x0000', 'x1010'], ['x0001', 'x1010'], ['x0010', 'x1010'], ['x0011', 'x1010'], ['x0100', 'x1010'], ['x0101', 'x1010'], ['x0110', 'x1010'], ['x0111', 'x1010'], ['x1000', 'x1010'], ['x1001', 'x1010'], ['x0000', 'x1011'], ['x0001', 'x1011'], ['x0010', 'x1011'], ['x0011', 'x1011'], ['x0100', 'x1011'], ['x0101', 'x1011'], ['x0110', 'x1011'], ['x0111', 'x1011'], ['x1000', 'x1011'], ['x1001', 'x1011'], ['x0110', 'x1100'], ['x0111', 'x1100'], ['x1010', 'x1100'], ['x1011', 'x1100'], ['x1100^2'], ['x0000', 'x1101'], ['x0001', 'x1101'], ['x0010', 'x1101'], ['x0011', 'x1101'], ['x0100', 'x1101'], ['x0101', 'x1101'], ['x0110', 'x1101'], ['x0111', 'x1101'], ['x1000', 'x1101'], ['x1001', 'x1101'], ['x1100', 'x1101']] removing powers x^m --> x [['x0000'], ['x0001'], ['x0010'], ['x0011'], ['x0100'], ['x0101'], ['x0000', 'x0110'], ['x0001', 'x0110'], ['x0010', 'x0110'], ['x0011', 'x0110'], ['x0100', 'x0110'], ['x0101', 'x0110'], ['x0000', 'x0111'], ['x0001', 'x0111'], ['x0010', 'x0111'], ['x0011', 'x0111'], ['x0100', 'x0111'], ['x0101', 'x0111'], ['x0110', 'x1000'], ['x0111', 'x1000'], ['x1000'], ['x0110', 'x1001'], ['x0111', 'x1001'], ['x1001'], ['x0000', 'x1010'], ['x0001', 'x1010'], ['x0010', 'x1010'], ['x0011', 'x1010'], ['x0100', 'x1010'], ['x0101', 'x1010'], ['x0110', 'x1010'], ['x0111', 'x1010'], ['x1000', 'x1010'], ['x1001', 'x1010'], ['x0000', 'x1011'], ['x0001', 'x1011'], ['x0010', 'x1011'], ['x0011', 'x1011'], ['x0100', 'x1011'], ['x0101', 'x1011'], ['x0110', 'x1011'], ['x0111', 'x1011'], ['x1000', 'x1011'], ['x1001', 'x1011'], ['x0110', 'x1100'], ['x0111', 'x1100'], ['x1010', 'x1100'], ['x1011', 'x1100'], ['x1100'], ['x0000', 'x1101'], ['x0001', 'x1101'], ['x0010', 'x1101'], ['x0011', 'x1101'], ['x0100', 'x1101'], ['x0101', 'x1101'], ['x0110', 'x1101'], ['x0111', 'x1101'], ['x1000', 'x1101'], ['x1001', 'x1101'], ['x1100', 'x1101']] [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] ['x0000', 'x0001', 'x0010', 'x0011', 'x0100', 'x0101', 'x1000', 'x1001', 'x1100'] x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x1000 + x1001 + x1100 reducing the x var polynomial x0000 + x0001 + x0010 + x0011 + x0100 + x0101 + x1000 + x1001 + x1100 approximation to maximum entropy using least squares subject to constraints: minimize sum of squares of atom probabilities to an equiprobable distribution subject to constraints sum(prob(atoms)) = 1.0 and prob(boolean_polynomial) = observed_value coefficient vector of boolean polynomial in terms of atoms - xvars [1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0] lagrange multiplier matrix [2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1] [1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0] [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0] inverse of lagrange matrix [ 4/9 -1/18 -1/18 -1/18 -1/18 -1/18 0 0 -1/18 -1/18 0 0 -1/18 0 0 0 1/9 0] [ -1/18 4/9 -1/18 -1/18 -1/18 -1/18 0 0 -1/18 -1/18 0 0 -1/18 0 0 0 1/9 0] [ -1/18 -1/18 4/9 -1/18 -1/18 -1/18 0 0 -1/18 -1/18 0 0 -1/18 0 0 0 1/9 0] [ -1/18 -1/18 -1/18 4/9 -1/18 -1/18 0 0 -1/18 -1/18 0 0 -1/18 0 0 0 1/9 0] [ -1/18 -1/18 -1/18 -1/18 4/9 -1/18 0 0 -1/18 -1/18 0 0 -1/18 0 0 0 1/9 0] [ -1/18 -1/18 -1/18 -1/18 -1/18 4/9 0 0 -1/18 -1/18 0 0 -1/18 0 0 0 1/9 0] [ 0 0 0 0 0 0 3/7 -1/14 0 0 -1/14 -1/14 0 -1/14 -1/14 -1/14 -1/7 1/7] [ 0 0 0 0 0 0 -1/14 3/7 0 0 -1/14 -1/14 0 -1/14 -1/14 -1/14 -1/7 1/7] [ -1/18 -1/18 -1/18 -1/18 -1/18 -1/18 0 0 4/9 -1/18 0 0 -1/18 0 0 0 1/9 0] [ -1/18 -1/18 -1/18 -1/18 -1/18 -1/18 0 0 -1/18 4/9 0 0 -1/18 0 0 0 1/9 0] [ 0 0 0 0 0 0 -1/14 -1/14 0 0 3/7 -1/14 0 -1/14 -1/14 -1/14 -1/7 1/7] [ 0 0 0 0 0 0 -1/14 -1/14 0 0 -1/14 3/7 0 -1/14 -1/14 -1/14 -1/7 1/7] [ -1/18 -1/18 -1/18 -1/18 -1/18 -1/18 0 0 -1/18 -1/18 0 0 4/9 0 0 0 1/9 0] [ 0 0 0 0 0 0 -1/14 -1/14 0 0 -1/14 -1/14 0 3/7 -1/14 -1/14 -1/7 1/7] [ 0 0 0 0 0 0 -1/14 -1/14 0 0 -1/14 -1/14 0 -1/14 3/7 -1/14 -1/7 1/7] [ 0 0 0 0 0 0 -1/14 -1/14 0 0 -1/14 -1/14 0 -1/14 -1/14 3/7 -1/7 1/7] [ 1/9 1/9 1/9 1/9 1/9 1/9 -1/7 -1/7 1/9 1/9 -1/7 -1/7 1/9 -1/7 -1/7 -1/7 -32/63 2/7] [ 0 0 0 0 0 0 1/7 1/7 0 0 1/7 1/7 0 1/7 1/7 1/7 2/7 -2/7] left hand side of vector([2*equiprobable, observed_prob, 1]) = lagrange_matrix*vector([prob(atoms), lambda, mu]) (1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1/8, 1, 1) atom probabilities that minimize sum of squares subject to observed probability of boolean polynomial (1/9, 1/9, 1/9, 1/9, 1/9, 1/9, 0, 0, 1/9, 1/9, 0, 0, 1/9, 0, 0, 0)

tuple([(0,1) for x in range(5)])

((0, 1), (0, 1), (0, 1), (0, 1), (0, 1))

from scipy.optimize import minimize
pnn=1/num_atoms
x1 = sum([x for x in xvars])
xc = matrix([1 for x in xvars])
atom_bool_poly=x0110 + x0111 + x1110 + x1111
atom_bool_poly2=x1010 + x1011 + x1101 + x1110 + x1111
atom_bool_poly_coefficients = matrix([atom_bool_poly.coefficient(a) for a in xvars])
atom_bool_poly_coefficients2 = matrix([atom_bool_poly2.coefficient(a) for a in xvars])
bnds = tuple([(0,1) for x in xvars])
MPR3=MPR.remove_var(*tuple(yvars))
cons = ({'type':'eq','fun': lambda x: MPR3(atom_bool_poly)(*x)-0},{'type':'eq','fun': lambda x: MPR3(atom_bool_poly2)(*x)-1},{'type':'eq','fun': lambda x: MPR3(x1)(*x)-1})
#cons = ({'type':'eq','fun': lambda x: sum([atom_bool_poly_coefficients[0,i]*x[i] for i in range(num_atoms)])-1},{'type':'eq','fun': lambda x: sum([atom_bool_poly_coefficients2[0,i]*x[i] for i in range(num_atoms)])-1},{'type':'eq','fun': lambda x: sum([x[i] for i in range(num_atoms)])-1})
fun = lambda x: sum([(y-pnn)**2 for y in x])
jacob = lambda x: 2*x
res = minimize(fun, vector([.5 for x in xvars]), method='SLSQP', jac = jacob, bounds=bnds, constraints=cons)
tuple(res.x.round(4))

(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.33329999999999999, 0.33329999999999999, 0.0, 0.33329999999999999, 0.0, 0.0)

MPR3=MPR.remove_var(*tuple(yvars))
(MPR3(atom_bool_poly))(*[.5 for x in xvars])

2.00000)

BPR.<A,B,C,D>=BooleanPolynomialRing(4)

#BPR.<A,B,C,D>=BooleanPolynomialRing(4)
f=A*B*C*D + A*B*C
f.degree(D)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()])
convert(convert(A*B*C))

1 A*B*C

load('ProbRecurse.py')
load('Prob.py')
import numpy
import scipy.optimize
from operator import add
from operator import mul

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C*1*k) in Boolean Ring i,j,k=0,1
rel_classes = 4
types = 3
objects = rel_classes*types; print('objects = ' + str(objects))
BPR = BooleanPolynomialRing(objects^2,'X',order='degneglex')
print('types = ' + str(types))
print('rel_classes = ' + str(rel_classes))
print(str(objects^2) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
X = BPR.gens()
NewGens = str(X).lower()
NewGens = NewGens[1:len(NewGens) - 1]
x = var(NewGens)
R = matrix(BPR, objects, objects, BPR.gens())
prems = []
#for i in range(0, objects):    #reflexive
#    poly = R[i,i]
#    prems.append(poly)
#for i in range(0,objects):    #symmetric
#    for j in range(i, objects):
#        poly = IFF(R[i,j],R[j,i])
#        prems.append(poly)  
#for j in range(0, objects):    #transitive
#    for i in range(0, objects):
#        for k in range(0, objects):
#            poly = IF(AND(R[min(i,j),max(i,j)],R[min(j,k),max(j,k)]),R[min(i,k),max(i,k)])
#            prems.append(poly)
prems.append(UNIQUE([R[9,1],R[9,6]]))    # clue 1
prems.append(UNIQUE([AND(R[4,11],R[6,10]), AND(R[4,10],R[6,9]), AND(R[4,9],R[6,8])]))    # clue 2
prems.append(UNIQUE([AND(R[11,6],R[9,0]), AND(R[11,0],R[9,6])]))    # clue 3
prems.append(UNIQUE([AND(R[0,11],R[2,4]), AND(R[0,4],R[2,11])]))    # clue 4
#prems.append(UNIQUE([AND(R[5,11],R[4,10]), AND(R[5,11],R[4,9]), AND(R[5,11],R[4,8]), AND(R[5,10],R[4,9]), AND(R[5,10],R[4,8]), AND(R[5,9],R[4,8])]))    # clue 5
if BPR(0) in prems: 
    print "explicit contradiction in premises"
prems = list(set(prems)) # remove dup
if BPR(1) in prems: prems.remove(BPR(1))
#probs = [a for ii in range(len(prems))]
prems = [FORALL(prems)]
probprems=[Prob(p) for p in prems]
#constraint=[(probprems[ii]-probs[ii]) for ii in range(len(prems))]
constraint=[(p-1) for p in probprems]

numpy.set_printoptions(precision=4)
y0 = [.5 for xx in x]
bnds = tuple([(0, 1) for xx in x])
#cons = list([{'type':'eq','fun': lambda x: c(*x)} for c in constraint])
str1='{\'type\': \'eq\', \'fun\': lambda s: ((constraint['
str2='])(*s))},'
mystr='['
for ii in range(len(constraint)):
    mystr=mystr+str1+str(ii)+str2
mystr=mystr+']'
cons = eval(mystr)
#FJAC=lambda s: numpy.array([ln(s[ii]+ln(1-s[ii])+2) for ii in range(len(s))])
def dHH(s):
    tmp=[]
    for ii in range(len(s)):
        t=ln(s[ii])-ln(1-s[ii])
        tmp.append(t)
    #print tmp
    return tmp
def HH(s):
    hh=0
    for ii in range(len(x)):
        hh=hh+(s[ii]*ln(s[ii])+(1-s[ii])*ln(1-s[ii]))
    return hh
ssoln = scipy.optimize.minimize(HH, y0, jac=dHH, method='SLSQP', bounds=bnds, constraints=cons)
sssoln=[s for s in ssoln.x]
A=matrix(RR, objects, objects, sssoln)
RR5 = RealField(prec=5)
MatPrint = MatrixSpace(RR5,objects,objects)
print(MatPrint(A))

objects = 12 types = 3 rel_classes = 4 144 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X143 = R[11,11] [ 0.47 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 1.0 3.7e-7 0.27] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 1.0 2.1e-7 0.010 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50] [ 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50]

load('ProbRecurse.py')
load('Prob.py')
import numpy
import scipy.optimize

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
CONVERT = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()])
BPR = BooleanPolynomialRing(6,'X', order='degneglex')
X = BPR.gens() #generators X[n] are independent random variables that can take on values 0 or 1
NewGens = str(X).lower()
NewGens = NewGens[1:len(NewGens) - 1]
x = var(NewGens) #x = Pr(X)
C1=AND(OR(X[0],X[1]),OR(X[0],X[2]))
C2=IF(AND(X[1],X[2]),IF(X[4],X[5]))
C3=NOT(IF(IF(X[0],X[3]),NOT(IF(X[3],X[5]))))
C4=NOT(X[5])
prems=[C1,C2,C3,C4]
if BPR(0) in prems: 
    print "explicit contradiction in premises"
prems = list(set(prems)) # remove dup
if BPR(1) in prems: prems.remove(BPR(1)) # remove tautologies
prb = 1.0
constraint=[Prob(FORALL(prems)) - prb]
HH = lambda s: sum([s[ii]*ln(s[ii])+(1-s[ii])*ln(1-s[ii]) for ii in range(len(s))])
dHH = lambda s: [ln(s[ii])-ln(1-s[ii]) for  ii in range(len(s))]
str1='{\'type\': \'eq\', \'fun\': lambda s: ((constraint['
str2='])(*s))},'
mystr='['
for ii in range(len(constraint)):
    mystr=mystr+str1+str(ii)+str2
mystr=mystr+']'
cons = eval(mystr)
numpy.set_printoptions(precision=15)
y0 = [.5 for xx in x]
bnds = tuple([(0.0, 1.0) for xx in x])

ssoln = scipy.optimize.minimize(HH, y0, jac=dHH, method='SLSQP', bounds=bnds, constraints=cons)
print('ssoln.x');print(ssoln.x);print(' ')
print('entropy ssoln.x');print(-HH(ssoln.x)/RR(ln(2)));print(' ')

pvar = lambda s: sum([(2*s[i]-1)^2 for i in range(len(s))])
dpvar = lambda s: [4*(2*s[i]-1) for i in range(len(s))]
pvarsoln = scipy.optimize.minimize(pvar, y0, jac=dpvar, method='SLSQP', bounds=bnds, constraints=cons)
print('pvarsoln.x');print(pvarsoln.x);print(' ')
print('entropy pvarsoln.x');print( -HH(pvarsoln.x)/RR(ln(2)));print(' ')

constraint=[(Prob(prm)-1.0) for prm in prems]
str1='{\'type\': \'eq\', \'fun\': lambda s: ((constraint['
str2='])(*s))},'
mystr='['
for ii in range(len(constraint)):
    mystr=mystr+str1+str(ii)+str2
mystr=mystr+']'
cons = eval(mystr)
ssoln = scipy.optimize.minimize(HH, y0, jac=dHH, method='SLSQP', bounds=bnds, constraints=cons)
print('ssoln.x');print(ssoln.x);print(' ')
print('entropy ssoln.x');print(-HH(ssoln.x)/RR(ln(2)));print(' ')

ssoln.x [ 5.693223670277803e-13 9.999999999996521e-01 9.999999999993416e-01 7.759903830617532e-13 1.550981565401344e-13 1.461053500406706e-13] entropy ssoln.x 1.12073139297e-10 pvarsoln.x [ 5.693223670277803e-13 9.999999999996521e-01 9.999999999993416e-01 7.759903830617532e-13 1.550981565401344e-13 1.461053500406706e-13] entropy pvarsoln.x 1.12073139297e-10 ssoln.x [ 1.749321734770491e-01 8.946392213506320e-01 9.478151143471312e-01 3.031949691312263e-13 6.505213051068898e-20 3.087204888824203e-01] entropy ssoln.x 2.34192380411

Prob(FORALL(prems)) - prb
mystr
C1
CONVERT(C1)
CONVERT(CONVERT(C1))
CONVERT(1+C1)

x0*x1*x2*x3*x4*x5 - x0*x1*x2*x3*x4 - x0*x1*x2*x3*x5 - x0*x1*x2*x4*x5 - x1*x2*x3*x4*x5 + x0*x1*x2*x3 + x0*x1*x2*x4 + x1*x2*x3*x4 + x0*x1*x2*x5 + x1*x2*x3*x5 + x1*x2*x4*x5 - x0*x1*x2 - x1*x2*x3 - x1*x2*x4 - x1*x2*x5 + x1*x2 - 1.00000000000000 "[{'type': 'eq', 'fun': lambda s: ((constraint[0])(*s))},{'type': 'eq', 'fun': lambda s: ((constraint[1])(*s))},{'type': 'eq', 'fun': lambda s: ((constraint[2])(*s))},{'type': 'eq', 'fun': lambda s: ((constraint[3])(*s))},]" X0*X1*X2 + X1*X2 + X0 X0*X1*X2*X3*X4*X5 + X1*X2*X3*X4*X5 + X0*X2*X3*X4*X5 + X0*X1*X3*X4*X5 + X0*X1*X2*X4*X5 + X0*X1*X2*X3*X5 + X0*X1*X2*X3*X4 + X1*X2*X4*X5 + X1*X2*X3*X5 + X1*X2*X3*X4 + X0*X3*X4*X5 + X0*X2*X4*X5 + X0*X2*X3*X5 + X0*X2*X3*X4 + X0*X1*X4*X5 + X0*X1*X3*X5 + X0*X1*X3*X4 + X0*X1*X2*X5 + X0*X1*X2*X4 + X0*X1*X2*X3 + X1*X2*X5 + X1*X2*X4 + X1*X2*X3 + X0*X4*X5 + X0*X3*X5 + X0*X3*X4 + X0*X2*X5 + X0*X2*X4 + X0*X2*X3 + X0*X1*X5 + X0*X1*X4 + X0*X1*X3 + X0*X1*X2 + X1*X2 + X0*X5 + X0*X4 + X0*X3 + X0*X2 + X0*X1 + X0 X0*X1*X2 + X1*X2 + X0 X2*X3*X4*X5 + X1*X3*X4*X5 + X3*X4*X5 + X2*X4*X5 + X2*X3*X5 + X2*X3*X4 + X1*X4*X5 + X1*X3*X5 + X1*X3*X4 + X4*X5 + X3*X5 + X3*X4 + X2*X5 + X2*X4 + X2*X3 + X1*X5 + X1*X4 + X1*X3 + X5 + X4 + X3 + X2 + X1 + 1

import scipy.linalg
A=identity_matrix(RR,5,5)
for i in range(0,5):
    for j in range(i+1,5):
        A[i,j]=uniform(0,1)
        A[j,i]=A[i,j]
A
B=scipy.linalg.eigh(A);Lambda=diagonal_matrix(B[0]);Q=matrix(B[1]);Lambda;Q
Q*Lambda*Q.transpose()
Q.determinant()
#suppose A represents the probabilities independent generators that represent possible relatios between n objects ( yes or no) that maximize entropy (or minimze variance from 50/50) 
#with underlying assumption of equivalence relation (guarantees symmetric matrix w main diagonal probs = 1.0) subject to a single prob polynomial constraint for the generators
#determinine eigenvalues and orthogonal matrix to find A, each 0-1 n by n matrix (is adjacency matrix, is an atom, defines a relation among the n objects tho not nec equiv)
#2^(n^2) possible monomials in such a polynomial kind of huge for a large problem AND all the clues together hmmm, each connected component a clique, complete graph for equiv relation
#why can't we use a system of poly's instead of AND all together (n^2) is manageable but 2^(n^2) is not is not

[ 1.00000000000000 0.496265160862274 0.418898591140118 0.123147810090251 0.369712209165948] [ 0.496265160862274 1.00000000000000 0.361575207423723 0.558515005617884 0.150674048629061] [ 0.418898591140118 0.361575207423723 1.00000000000000 0.444900960380892 0.0307824883493855] [ 0.123147810090251 0.558515005617884 0.444900960380892 1.00000000000000 0.0480831030531087] [ 0.369712209165948 0.150674048629061 0.0307824883493855 0.0480831030531087 1.00000000000000] [0.2410877688814176 0.0 0.0 0.0 0.0] [ 0.0 0.5900591076900569 0.0 0.0 0.0] [ 0.0 0.0 0.7316858723181444 0.0 0.0] [ 0.0 0.0 0.0 1.1566013794531638 0.0] [ 0.0 0.0 0.0 0.0 2.2805658716572137] [ -0.5414540245968408 -0.3258044510560972 0.435947295076377 0.4310503161449515 -0.474156703829912] [ 0.5398293486331674 -0.5826259195446164 -0.25550143654850926 -0.10013468392689064 -0.5420547695810735] [ 0.33895385568154923 0.5511529504875798 0.5515107152733957 -0.23413084800274628 -0.4715499823344173] [ -0.5192634104708683 0.24210623475582088 -0.5056865942119444 -0.4577252633604659 -0.45444332248855746] [ 0.17574870658684355 0.43819344336493726 -0.4298673879351518 0.7347373675819859 -0.22907188346354163] [ 1.0 0.49626516086227335 0.41889859114011796 0.12314781009025028 0.3697122091659482] [ 0.49626516086227346 0.9999999999999989 0.3615752074237223 0.5585150056178826 0.15067404862906142] [ 0.41889859114011796 0.3615752074237222 0.9999999999999993 0.44490096038089133 0.030782488349385156] [ 0.12314781009025039 0.5585150056178826 0.44490096038089133 1.0 0.04808310305310792] [ 0.3697122091659482 0.15067404862906142 0.03078248834938513 0.04808310305310795 0.9999999999999991] 1.0000000000000007

load('ProbRecurse.py')
load('Prob.py')
import numpy
import scipy.optimize
from operator import add
from operator import mul

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C*1*k) in Boolean Ring i,j,k=0,1

objects = 8 #rel_classes*types; print('objects = ' + str(objects))
BPR = BooleanPolynomialRing(objects^2,'X',order='degneglex')
BPR.inject_variables()
print(str(objects^2) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
X = BPR.gens()
NewGens = str(X).lower()
NewGens = NewGens[1:len(NewGens) - 1]
x = var(NewGens)
R = matrix(BPR, objects, objects, BPR.gens())
prems = [];reflex=[];symmtrc=[];trans=[];direct=[]
for i in range(0, objects):    #reflexive
    reflex.append(R[i,i])

for i in range(0,objects):    
    for j in range(0, objects):
        #symmtrc.append(IF(R[i,j],R[j,i]))
        direct.append(IF(R[i,j],NOT(R[j,i])))

for i in range(0,objects):    #symmetric
    for j in range(i+1, objects):
        symmtrc.append(IFF(R[i,j],R[j,i]))
symmtrc=list(set(symmtrc))
if BPR(1) in symmtrc: symmtrc.remove(BPR(1)) # get rid on tautologies

trans2=[]
for j in range(0, objects):    #transitive
    for i in range(0, objects):
        for k in range(0, objects):
            poly = IF(AND(R[min(i,j),max(i,j)],R[min(j,k),max(j,k)]),R[min(i,k),max(i,k)])
            trans2.append(poly)
#len(trans2)
trans2 = list(set(trans2))#;len(trans2)

for i in range(0, objects):    #transitive
    for j in range(0, objects):
        for k in range(0, objects):
            trans.append(IF(AND(R[i,j],R[j,k]),R[i,k]))
#print(len(trans))
trans = list(set(trans))
M=matrix([[125,25,5,1],[64,16,4,1],[27,9,3,1],[8,4,2,1]])#;M

A=vector([81,37,13,3])#;A
#M.inverse()
B=M.inverse()*A # B=(1, -2, 1, 1)
var('t')
mypoly=B*vector([t^3,t^2,t,1])#;mypoly # gives number of unique transitive polynomials

B=M.inverse()*vector([35,18,8,3])#;B
mypoly2=vector([t^3,t^2,t,1])*B;mypoly2
#mypoly2.subs(t=6);mypoly2.subs(t=7) #seems to give the number of polynomials in groebner basis for equiv relation with gen order degneglexx

prems=reflex+symmtrc+trans2
print(len(reflex));print(len(symmtrc));print(len(trans2));print(len(prems))
if BPR(0) in prems: 
    print "explicit contradiction in premises"
prems = list(set(prems)) # remove dup
premstrue = [p+1 for p in prems] #make true
#if BPR(1) in prems: prems.remove(BPR(1))
REI = ideal(premstrue); print('REI.gens() = ' + str(len(REI.gens())))
GB = REI.groebner_basis();GB
#compute groebner basis directly assuming equivalence relation
#correspond to trans2 objects=5: X1*(X2+X(2+1*5));X1*(X3+X(3+1*5));X1*(X4+X(4+1*5));X2*(X3+X(3+2*5));X2*(X4+X(4+2*5));X3*(X4+X(4+3*5));X7*(X8+X(8+1*5));X7*(X9+X(9+1*5));X8*(X9+X(9+2*5));X13*(X14+X(14+1*5)) then
#X2*(X1+X(2+1*5));X3*(X1+X(3+1*5));X4*(X1+X(4+1*5));X3*(X2+X(3+2*5));X4*(X2+X(4+2*5));X4*(X3+X(4+3*5));X8*(X7+X(8+1*5));X9*(X7+X(9+1*5));X9*(X8+X(9+2*5));X14*(X13+X(14+1*5)) that 20 polynomials
GB2=[]
for i in range(objects):
    for j in range(i+1,objects):
        for k in range(j+1,objects):
            GB2.append(R[i,j]*(R[i,k]+R[j,k]))
            GB2.append(R[i,k]*(R[i,j]+R[j,k]))
GB2=GB2+[p+1 for p in reflex]+[p+1 for p in symmtrc]
set(GB)==set(GB2)
QR = BPR.quotient_ring(REI)
#print(premstrue);print(len(premstrue));print(' ')
print(matrix(objects,objects,QR.gens()));print(' ')
if False:
    prb = 1.0
    constraint=[Prob(p) for p in GB]
    HH = lambda s: sum([s[ii]*ln(s[ii])+(1-s[ii])*ln(1-s[ii]) for ii in range(len(s))])
    dHH = lambda s: [ln(s[ii])-ln(1-s[ii]) for  ii in range(len(s))]
    str1='{\'type\': \'eq\', \'fun\': lambda s: ((constraint['
    str2='])(*s))},'
    mystr='['
    for ii in range(len(constraint)):
        mystr=mystr+str1+str(ii)+str2
    mystr=mystr+']'
    cons = eval(mystr)
    numpy.set_printoptions(precision=15)
    y0 = [.5 for xx in x]
    bnds = tuple([(0.0, 1.0) for xx in x])

    ssoln = scipy.optimize.minimize(HH, y0, jac=dHH, method='SLSQP', bounds=bnds, constraints=cons)
    A=matrix(RR, objects, objects, list(ssoln.x))
    RR5 = RealField(prec=5)
    MatPrint = MatrixSpace(RR5,objects,objects)
    print(MatPrint(A))
    #print('ssoln.x');print(ssoln.x);print(' ')
    print('entropy ssoln.x');print(-HH(ssoln.x)/RR(ln(2)));print(' ')

    pvar = lambda s: sum([(2*s[i]-1)^2 for i in range(len(s))])
    dpvar = lambda s: [4*(2*s[i]-1) for i in range(len(s))]
    pvarsoln = scipy.optimize.minimize(pvar, y0, jac=dpvar, method='SLSQP', bounds=bnds, constraints=cons)
    A=matrix(RR, objects, objects, list(pvarsoln.x))
    print(MatPrint(A))
    #print('pvarsoln.x');print(pvarsoln.x);print(' ')
    print('entropy pvarsoln.x');print( -HH(pvarsoln.x)/RR(ln(2)));print(' ')

Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47, X48, X49, X50, X51, X52, X53, X54, X55, X56, X57, X58, X59, X60, X61, X62, X63 64 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X63 = R[7,7] t 1/3*t^3 - 1/2*t^2 + 7/6*t 8 28 225 261 REI.gens() = 261 Polynomial Sequence with 148 Polynomials in 64 Variables True [ 1 X1 X2 X3 X4 X5 X6 X7] [ X1 1 X10 X11 X12 X13 X14 X15] [ X2 X10 1 X19 X20 X21 X22 X23] [ X3 X11 X19 1 X28 X29 X30 X31] [ X4 X12 X20 X28 1 X37 X38 X39] [ X5 X13 X21 X29 X37 1 X46 X47] [ X6 X14 X22 X30 X38 X46 1 X55] [ X7 X15 X23 X31 X39 X47 X55 1]

trans2=[]
for j in range(0, objects):    #transitive
    for i in range(0, objects):
        for k in range(0, objects):
            poly = IF(AND(R[min(i,j),max(i,j)],R[min(j,k),max(j,k)]),R[min(i,k),max(i,k)])
            trans2.append(poly)
len(trans2)
trans2 = list(set(trans2));len(trans2);trans2

64 25 [X0*X3 + X3 + 1, X6*X7*X11 + X7*X11 + 1, X1*X3*X7 + X3*X7 + 1, X1*X2*X6 + X2*X6 + 1, X5*X6 + X6 + 1, X10*X11 + X11 + 1, X2*X3*X11 + X2*X3 + 1, X5*X7 + X7 + 1, X1*X2*X6 + X1*X6 + 1, X1*X5 + X1 + 1, X11*X15 + X11 + 1, X7*X15 + X7 + 1, 1, X2*X3*X11 + X2*X11 + 1, X1*X3*X7 + X1*X3 + 1, X2*X3*X11 + X3*X11 + 1, X1*X3*X7 + X1*X7 + 1, X2*X10 + X2 + 1, X0*X2 + X2 + 1, X6*X10 + X6 + 1, X0*X1 + X1 + 1, X6*X7*X11 + X6*X11 + 1, X3*X15 + X3 + 1, X6*X7*X11 + X6*X7 + 1, X1*X2*X6 + X1*X2 + 1]

M=matrix([[125,25,5,1],[64,16,4,1],[27,9,3,1],[8,4,2,1]]);M
A=vector([81,37,13,3]);A
M.inverse()
B=M.inverse()*A;B
var('t')
mypoly=B*vector([t^3,t^2,t,1]);mypoly
mypoly(6)
B2=M.inverse()*vector([51,25,10,3]);B2
((1/2)*(t^3+2*t^2+t+2))(6)

[125 25 5 1] [ 64 16 4 1] [ 27 9 3 1] [ 8 4 2 1] (81, 37, 13, 3) [ 1/6 -1/2 1/2 -1/6] [ -3/2 5 -11/2 2] [ 13/3 -31/2 19 -47/6] [ -4 15 -20 10] (1, -2, 1, 1) t t^3 - 2*t^2 + t + 1 151 (1/2, -1/2, 0, 1) 148

((1/2)*(t^3-t^2+2))(6)

91

symmtrc2=[]
for i in range(0,objects):    #symmetric
    for j in range(i+1, objects):
        symmtrc2.append(IFF(R[i,j],R[j,i]))
        
symmtrc2=list(set(symmtrc2));symmtrc2;len(symmtrc2)

[X18 + X3 + 1, X24 + X4 + 1, X31 + X11 + 1, X27 + X22 + 1, X30 + X5 + 1, X33 + X23 + 1, X13 + X8 + 1, X26 + X16 + 1, X19 + X9 + 1, X12 + X2 + 1, X34 + X29 + 1, X25 + X10 + 1, X32 + X17 + 1, X6 + X1 + 1, X20 + X15 + 1] 15

Prob(symmtrc[1]*symmtrc[2]);Prob(symmtrc[1])*Prob(symmtrc[2])
#constraint
GB
GB[0];GB[1];GB[2];GB[34]
Prob(GB[2])

4*x4*x19*x20*x23 - 2*x19*x20*x23 - 2*x4*x20*x23 - 2*x4*x19*x23 - 2*x4*x19*x20 + x20*x23 + 2*x19*x23 + x4*x23 + x19*x20 + 2*x4*x20 + x4*x19 - x23 - x20 - x19 - x4 + 1 4*x4*x19*x20*x23 - 2*x19*x20*x23 - 2*x4*x20*x23 - 2*x4*x19*x23 - 2*x4*x19*x20 + x20*x23 + 2*x19*x23 + x4*x23 + x19*x20 + 2*x4*x20 + x4*x19 - x23 - x20 - x19 - x4 + 1 Polynomial Sequence with 35 Polynomials in 25 Variables X14*X19 + X13*X14 X13*X19 + X13*X14 X9*X19 + X8*X9 X0 + 1 -2*x8*x9*x19 + x9*x19 + x8*x9

len(GB)

3

B=M.inverse()*vector([35,18,8,3]);B
mypoly2=vector([t^3,t^2,t,1])*B;mypoly2
mypoly2(6);mypoly2(8)
mypoly2.subs(t=6)

(1/3, -1/2, 7/6, 0) 1/3*t^3 - 1/2*t^2 + 7/6*t 61 148

/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py:905: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...) See http://trac.sagemath.org/5930 for details. exec compile(block+'\n', '', 'single') in namespace, locals

61

#Prob(GB[2])*Prob(GB[3]);Prob(GB[2]*GB[3])
#Prob(GB[21])*Prob(GB[0]);Prob(GB[21]*GB[0])
for g in GB:
    g;Prob(g)

X7*X11 + X6*X7 -2*x6*x7*x11 + x7*x11 + x6*x7 X6*X11 + X6*X7 -2*x6*x7*x11 + x6*x11 + x6*x7 X3*X11 + X2*X3 -2*x2*x3*x11 + x3*x11 + x2*x3 X3*X7 + X1*X3 -2*x1*x3*x7 + x3*x7 + x1*x3 X2*X11 + X2*X3 -2*x2*x3*x11 + x2*x11 + x2*x3 X2*X6 + X1*X2 -2*x1*x2*x6 + x2*x6 + x1*x2 X1*X7 + X1*X3 -2*x1*x3*x7 + x1*x7 + x1*x3 X1*X6 + X1*X2 -2*x1*x2*x6 + x1*x6 + x1*x2 X15 + 1 -x15 + 1 X14 + X11 -2*x11*x14 + x14 + x11 X13 + X7 -2*x7*x13 + x13 + x7 X12 + X3 -2*x3*x12 + x12 + x3 X10 + 1 -x10 + 1 X9 + X6 -2*x6*x9 + x9 + x6 X8 + X2 -2*x2*x8 + x8 + x2 X5 + 1 -x5 + 1 X4 + X1 -2*x1*x4 + x4 + x1 X0 + 1 -x0 + 1

len(set(GB2));len(set(GB))

10 35

load('ProbRecurse.py')
load('Prob.py')
import numpy
import scipy.optimize
from operator import add
from operator import mul

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C*1*k) in Boolean Ring i,j,k=0,1

objects = 6 #rel_classes*types; print('objects = ' + str(objects))
BPR = BooleanPolynomialRing(objects^2,'X',order='degneglex')
BPR.inject_variables()
print(str(objects^2) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
X = BPR.gens()
NewGens = str(X).lower()
NewGens = NewGens[1:len(NewGens) - 1]
PPR = PolynomialRing(QQ,objects^2,NewGens,order='degneglex')
PPR.inject_variables()
#x = var(NewGens)
R = matrix(BPR, objects, objects, BPR.gens())
prems = [];reflex=[];symmtrc=[];trans2=[];direct=[]
for i in range(0, objects):    #reflexive
    reflex.append(R[i,i])

for i in range(0,objects):    
    for j in range(0, objects):
        #symmtrc.append(IF(R[i,j],R[j,i]))
        direct.append(IF(R[i,j],NOT(R[j,i])))

for i in range(0,objects):    #symmetric
    for j in range(i+1, objects):
        symmtrc.append(IFF(R[i,j],R[j,i]))
symmtrc=list(set(symmtrc))

for j in range(0, objects):    #transitive
    for i in range(0, objects):
        for k in range(0, objects):
            poly = IF(AND(R[min(i,j),max(i,j)],R[min(j,k),max(j,k)]),R[min(i,k),max(i,k)])
            trans2.append(poly)
trans2 = list(set(trans2))

M=matrix([[125,25,5,1],[64,16,4,1],[27,9,3,1],[8,4,2,1]])#;M
var('t')
B=M.inverse()*vector([35,18,8,3])#;B
mypoly2=vector([t^3,t^2,t,1])*B;mypoly2
#mypoly2.subs(t=6);mypoly2.subs(t=7) #seems to give the number of polynomials in groebner basis for equiv relation with gen order degneglexx

prems=reflex+symmtrc+trans2
if BPR(0) in prems: 
    print "explicit contradiction in premises"
prems = list(set(prems)) # remove dup
premstrue = [p+1 for p in prems] #make true
REI = ideal(premstrue);# print('REI.gens() = ' + str(len(REI.gens())))
GB = REI.groebner_basis();GB
#compute groebner basis directly assuming equivalence relation
#correspond to trans2 objects=5: X1*(X2+X(2+1*5));X1*(X3+X(3+1*5));X1*(X4+X(4+1*5));X2*(X3+X(3+2*5));X2*(X4+X(4+2*5));X3*(X4+X(4+3*5));X7*(X8+X(8+1*5));X7*(X9+X(9+1*5));X8*(X9+X(9+2*5));X13*(X14+X(14+1*5)) then
#X2*(X1+X(2+1*5));X3*(X1+X(3+1*5));X4*(X1+X(4+1*5));X3*(X2+X(3+2*5));X4*(X2+X(4+2*5));X4*(X3+X(4+3*5));X8*(X7+X(8+1*5));X9*(X7+X(9+1*5));X9*(X8+X(9+2*5));X14*(X13+X(14+1*5)) that 20 polynomials
GB2=[];PGB=[]
for i in range(objects):
    for j in range(i+1,objects):
        for k in range(j+1,objects):
            GB2.append(R[i,j]*(R[i,k]+R[j,k]));PGB.append(Prob(R[i,k])-Prob(R[j,k])) #unless R[i,j]=0 ****************************
            GB2.append(R[i,k]*(R[i,j]+R[j,k]));PGB.append(Prob(R[i,j])-Prob(R[j,k]))
GB2=GB2+[p+1 for p in reflex]+[p+1 for p in symmtrc]
set(GB)==set(GB2)
REI2=ideal(GB2)
set(GB2)==set(GB) #check
QR = BPR.quotient_ring(REI2)
print(len(GB2));print(str(mypoly2.subs(t=objects))+' polynomials in Groebner Basis for equivalence relation in '+str(objects^2)+' variables')
print(matrix(objects,objects,QR.gens()));print(' ')
symmtrc2=[Prob(R[i,j])-Prob(R[j,i]) for i in range(objects) for j in range(i+1,objects)]
constraint=[Prob(p)-1.0 for p in reflex]+symmtrc2+PGB
mymat=matrix([[PPR(c).coefficient(g) for g in PPR.gens()] for c in constraint]) # mymat is 24 by 25 for objects=5 19 rows corresponding to not reflexive, 20 columns 
myvect = matrix([PPR(c).constant_coefficient() for c in constraint ])
pv=mymat.pivot_rows()
mymat = mymat[pv,:];myvect=myvect[:,pv]
lincons=matrix(mymat*vector(PPR.gens()))+myvect
lincons=[lincons[0,i].change_ring(RR) for i in range(len(lincons.columns()))] #fing awful to convert to list
lincons.append(Prob(IF(R[3,4],R[4,2]))-.9)
if True:
    HH = lambda s: sum([s[ii]*ln(s[ii])+(1-s[ii])*ln(1-s[ii]) for ii in range(len(s))])
    dHH = lambda s: [ln(s[ii])-ln(1-s[ii]) for  ii in range(len(s))]
    str1='{\'type\': \'eq\', \'fun\': lambda s: (lincons['
    str2='])(*s)},'
    mystr='['
    for ii in range(len(lincons)):
        mystr=mystr+str1+str(ii)+str2
    mystr=mystr+']'
    cons = eval(mystr)
    numpy.set_printoptions(precision=15)
    y0 = [.5 for xx in PPR.gens()]
    bnds = tuple([(0.0, 1.0) for xx in PPR.gens()])
    ssoln = scipy.optimize.minimize(HH, y0, jac=dHH, method='SLSQP', bounds=bnds, constraints=cons)
    A=matrix(RR, objects, objects, list(ssoln.x))
    RR5 = RealField(prec=5)
    MatPrint = MatrixSpace(RR5,objects,objects)
    print(MatPrint(A))
    #print('ssoln.x');print(ssoln.x);print(' ')
    print('entropy ssoln.x');print(-HH(ssoln.x)/RR(ln(2)));print(' ')

    pvar = lambda s: sum([(2*s[i]-1)^2 for i in range(len(s))])
    dpvar = lambda s: [4*(2*s[i]-1) for i in range(len(s))]
    pvarsoln = scipy.optimize.minimize(pvar, y0, jac=dpvar, method='SLSQP', bounds=bnds, constraints=cons)
    A=matrix(RR, objects, objects, list(pvarsoln.x))
    print(MatPrint(A))
    #print('pvarsoln.x');print(pvarsoln.x);print(' ')
    print('entropy pvarsoln.x');print( -HH(pvarsoln.x)/RR(ln(2)));print(' ')
#let's say 3 generators a,b,c and we minimize (2a-1)^2+(2b-1)^2+(2c-1)^2 where we know a*b*c=p0, a*b*(1-c)=p1,a*(1-b)*c=p2,(1-a)*b*c=p3 yields 3 linear constraints and do lagrange multipliers lambda,mu,nu, 0<=p0+p1+p2+p3<=1
#oops but that completely determines the probs of the generators so no point minimizing
#the probabilty polynomial for these boolean polynomials correspond on the vertices of the hypercube defined by the atoms. Setting an atom equal to 1 defines the truth values of the generators and the prob poly = boolean poly 
#with these as arguments
#say X1*X7+X1*X2=0 then x1*x7-x1*x2=0 not Prob(X1*X7+X1*X2) = -2*x1*x2*x7 + x1*x7 + x1*x2 = 0 or whatever
#then lagrange mult pvar for linear system

Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35 36 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X35 = R[5,5] Defining x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35 t 1/3*t^3 - 1/2*t^2 + 7/6*t Polynomial Sequence with 61 Polynomials in 36 Variables True True 61 61 polynomials in Groebner Basis for equivalence relation in 36 variables [ 1 X1 X2 X3 X4 X5] [ X1 1 X8 X9 X10 X11] [ X2 X8 1 X15 X16 X17] [ X3 X9 X15 1 X22 X23] [ X4 X10 X16 X22 1 X29] [ X5 X11 X17 X23 X29 1] [ 1.0 0.88 0.88 0.88 0.88 0.88] [0.88 1.0 0.88 0.88 0.88 0.88] [0.88 0.88 1.0 0.88 0.88 0.88] [0.88 0.88 0.88 1.0 0.88 0.88] [0.88 0.88 0.88 0.88 1.0 0.88] [0.88 0.88 0.88 0.88 0.88 1.0] entropy ssoln.x 15.2446441202 [0.50 0.50 0.50 0.50 0.50 0.50] [0.50 0.50 0.50 0.50 0.50 0.50] [0.50 0.50 0.50 0.50 0.50 0.50] [0.50 0.50 0.50 0.50 0.50 0.50] [0.50 0.50 0.50 0.50 0.50 0.50] [0.50 0.50 0.50 0.50 0.50 0.50] entropy pvarsoln.x 36.0

for g in GB2:
    g;Prob(g)

X1*X8 + X1*X2 -2*x1*x2*x8 + x1*x8 + x1*x2 X2*X8 + X1*X2 -2*x1*x2*x8 + x2*x8 + x1*x2 X1*X9 + X1*X3 -2*x1*x3*x9 + x1*x9 + x1*x3 X3*X9 + X1*X3 -2*x1*x3*x9 + x3*x9 + x1*x3 X1*X10 + X1*X4 -2*x1*x4*x10 + x1*x10 + x1*x4 X4*X10 + X1*X4 -2*x1*x4*x10 + x4*x10 + x1*x4 X1*X11 + X1*X5 -2*x1*x5*x11 + x1*x11 + x1*x5 X5*X11 + X1*X5 -2*x1*x5*x11 + x5*x11 + x1*x5 X2*X15 + X2*X3 -2*x2*x3*x15 + x2*x15 + x2*x3 X3*X15 + X2*X3 -2*x2*x3*x15 + x3*x15 + x2*x3 X2*X16 + X2*X4 -2*x2*x4*x16 + x2*x16 + x2*x4 X4*X16 + X2*X4 -2*x2*x4*x16 + x4*x16 + x2*x4 X2*X17 + X2*X5 -2*x2*x5*x17 + x2*x17 + x2*x5 X5*X17 + X2*X5 -2*x2*x5*x17 + x5*x17 + x2*x5 X3*X22 + X3*X4 -2*x3*x4*x22 + x3*x22 + x3*x4 X4*X22 + X3*X4 -2*x3*x4*x22 + x4*x22 + x3*x4 X3*X23 + X3*X5 -2*x3*x5*x23 + x3*x23 + x3*x5 X5*X23 + X3*X5 -2*x3*x5*x23 + x5*x23 + x3*x5 X4*X29 + X4*X5 -2*x4*x5*x29 + x4*x29 + x4*x5 X5*X29 + X4*X5 -2*x4*x5*x29 + x5*x29 + x4*x5 X8*X15 + X8*X9 -2*x8*x9*x15 + x8*x15 + x8*x9 X9*X15 + X8*X9 -2*x8*x9*x15 + x9*x15 + x8*x9 X8*X16 + X8*X10 -2*x8*x10*x16 + x8*x16 + x8*x10 X10*X16 + X8*X10 -2*x8*x10*x16 + x10*x16 + x8*x10 X8*X17 + X8*X11 -2*x8*x11*x17 + x8*x17 + x8*x11 X11*X17 + X8*X11 -2*x8*x11*x17 + x11*x17 + x8*x11 X9*X22 + X9*X10 -2*x9*x10*x22 + x9*x22 + x9*x10 X10*X22 + X9*X10 -2*x9*x10*x22 + x10*x22 + x9*x10 X9*X23 + X9*X11 -2*x9*x11*x23 + x9*x23 + x9*x11 X11*X23 + X9*X11 -2*x9*x11*x23 + x11*x23 + x9*x11 X10*X29 + X10*X11 -2*x10*x11*x29 + x10*x29 + x10*x11 X11*X29 + X10*X11 -2*x10*x11*x29 + x11*x29 + x10*x11 X15*X22 + X15*X16 -2*x15*x16*x22 + x15*x22 + x15*x16 X16*X22 + X15*X16 -2*x15*x16*x22 + x16*x22 + x15*x16 X15*X23 + X15*X17 -2*x15*x17*x23 + x15*x23 + x15*x17 X17*X23 + X15*X17 -2*x15*x17*x23 + x17*x23 + x15*x17 X16*X29 + X16*X17 -2*x16*x17*x29 + x16*x29 + x16*x17 X17*X29 + X16*X17 -2*x16*x17*x29 + x17*x29 + x16*x17 X22*X29 + X22*X23 -2*x22*x23*x29 + x22*x29 + x22*x23 X23*X29 + X22*X23 -2*x22*x23*x29 + x23*x29 + x22*x23 X0 + 1 -x0 + 1 X7 + 1 -x7 + 1 X14 + 1 -x14 + 1 X21 + 1 -x21 + 1 X28 + 1 -x28 + 1 X35 + 1 -x35 + 1 X18 + X3 -2*x3*x18 + x18 + x3 X24 + X4 -2*x4*x24 + x24 + x4 X31 + X11 -2*x11*x31 + x31 + x11 X27 + X22 -2*x22*x27 + x27 + x22 X30 + X5 -2*x5*x30 + x30 + x5 X33 + X23 -2*x23*x33 + x33 + x23 X13 + X8 -2*x8*x13 + x13 + x8 X26 + X16 -2*x16*x26 + x26 + x16 X19 + X9 -2*x9*x19 + x19 + x9 X12 + X2 -2*x2*x12 + x12 + x2 X34 + X29 -2*x29*x34 + x34 + x29 X25 + X10 -2*x10*x25 + x25 + x10 X32 + X17 -2*x17*x32 + x32 + x17 X6 + X1 -2*x1*x6 + x6 + x1 X20 + X15 -2*x15*x20 + x20 + x15

z=BPR.random_element();z;QR(z)
Prob(z)
QR(GB2[2])
b=[int(getrandbits(1)) for i in range(objects^2)];matrix(objects,objects,b);(GB2[1])(*b);(Prob(GB2[1]))(*b)
QR(X0*X6)

X15*X24 + X3*X20 + X0*X24 + X0*X4 + X20 X3*X4 + X3 + 1 4*x0*x3*x4*x15*x20*x24 - 4*x0*x4*x15*x20*x24 - 4*x0*x3*x15*x20*x24 - 4*x0*x3*x4*x20*x24 + 2*x3*x15*x20*x24 + 4*x0*x15*x20*x24 + 4*x0*x4*x20*x24 + 2*x0*x3*x20*x24 + 2*x0*x4*x15*x24 + 2*x0*x3*x4*x20 - 2*x15*x20*x24 - 2*x0*x20*x24 - 2*x0*x15*x24 - 2*x0*x4*x24 - 2*x0*x4*x20 + x15*x24 + x0*x24 - x3*x20 + x0*x4 + x20 0 [1 0 0 0 1] [1 0 0 1 1] [1 1 1 0 1] [1 1 1 0 1] [1 0 0 0 1] 0 0 1

Prob(UNIQUE([X0,X1,X2,X3]))

-4*x0*x1*x2*x3 + 3*x1*x2*x3 + 3*x0*x2*x3 + 3*x0*x1*x3 + 3*x0*x1*x2 - 2*x2*x3 - 2*x1*x3 - 2*x0*x3 - 2*x1*x2 - 2*x0*x2 - 2*x0*x1 + x3 + x2 + x1 + x0

(reflex+symmtrc)

[X0, X6, X12, X18, X24, X16 + X8 + 1, X20 + X4 + 1, X23 + X19 + 1, X22 + X14 + 1, X15 + X3 + 1, X11 + X7 + 1, X5 + X1 + 1, X17 + X13 + 1, X10 + X2 + 1, X21 + X9 + 1]

mymat=matrix([[PPR(c).coefficient(g) for g in PPR.gens()] for c in constraint]); mymat = mymat[mymat.pivot_rows(),:]

24 x 25 dense matrix over Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24 over Rational Field

mymat;myvect;pv

35 x 36 dense matrix over Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35 over Rational Field [-1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 37, 39)

myvect
lincons=matrix(mymat*vector(PPR.gens()))+myvect;lincons=[lincons[0,i] for i in range(len(lincons.columns()))];lincons

[-1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [x0 - 1, x7 - 1, x14 - 1, x21 - 1, x28 - 1, x35 - 1, -x6 + x1, -x12 + x2, -x18 + x3, -x24 + x4, -x30 + x5, -x13 + x8, -x19 + x9, -x25 + x10, -x31 + x11, -x20 + x15, -x26 + x16, -x32 + x17, -x27 + x22, -x33 + x23, -x34 + x29, -x8 + x2, -x8 + x1, -x9 + x3, -x9 + x1, -x10 + x4, -x10 + x1, -x11 + x5, -x11 + x1, -x15 + x3, -x16 + x4, -x17 + x5, -x22 + x4, -x23 + x5, -x29 + x5]

constraint[1](*(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36));lincons[1](*(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36))

7.00000000000000 7.00000000000000

ssoln.x
Prob(IF(R[3,4],R[4,2]))
(Prob(IF(R[3,4],R[4,2])))(*ssoln.x)

array([ 0.999969148174041, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.999969148174041, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.999969148174041, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.999969148174041, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.999969148174041, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.887284277041075, 0.999969148174041]) x22*x26 - x22 + 1 0.89998911124322811

IF(X1,X7);Prob(IF(X1,X7))
QR(trans2[2]);QR(IF(X1,X8))
XOR(R[4,2],R[4,1])+1

X1*X7 + X1 + 1 x1*x7 - x1 + 1 1 X1*X2 + X1 + 1 X26 + X25 + 1

constraint2=constraint+[x17-.75]
mymat=matrix([[PPR(c).coefficient(g) for g in PPR.gens()] for c in constraint2]) # mymat is 24 by 25 for objects=5 19 rows corresponding to not reflexive, 20 columns 
myvect = matrix([PPR(c).constant_coefficient() for c in constraint2 ])
pv=mymat.pivot_rows()
mymat = mymat[pv,:];myvect=myvect[:,pv]
lincons=matrix(mymat*vector(PPR.gens()))+myvect
lincons=[lincons[0,i].change_ring(RR) for i in range(len(lincons.columns()))] #fing awful to convert to list
mymat;myvect;mymat.rank();mymat.inverse()*([-a for a in myvect][0])

36 x 36 dense matrix over Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35 over Rational Field [ -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3/4] 36 (1, 3/4, 3/4, 3/4, 3/4, 3/4, 3/4, 1, 3/4, 3/4, 3/4, 3/4, 3/4, 3/4, 1, 3/4, 3/4, 3/4, 3/4, 3/4, 3/4, 1, 3/4, 3/4, 3/4, 3/4, 3/4, 3/4, 1, 3/4, 3/4, 3/4, 3/4, 3/4, 3/4, 1)

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C+1+k) in Boolean Ring i,j,k=0,1

rel_classes = 4
types = 3
objects = rel_classes*types; print('objects = ' + str(objects))
BPR = BooleanPolynomialRing(objects^2,'X',order='degneglex')
BPR.inject_variables()
print(str(objects^2) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
R = matrix(BPR, objects, objects, BPR.gens())
reflex=[];symmtrc=[];trans=[];exactlyone=[];clues = [];

for i in range(0, objects):    #reflexive
    reflex.append(R[i,i])
reflex=[r+1 for r in reflex]   #assert truth

for i in range(0,objects):    #symmetric
    for j in range(i+1, objects):
        symmtrc.append(IFF(R[i,j],R[j,i]))
symmtrc=[s+1 for s in symmtrc]   #assert truth

for i in range(objects):     #transitive truth assertion built into these which assume symmtrc and reflex in place
    for j in range(i+1,objects):
        for k in range(j+1,objects):
            trans.append(R[i,j]*(R[i,k]+R[j,k]))
            trans.append(R[i,k]*(R[i,j]+R[j,k]))

for i in range(0,types):
    for j in range(i,types):
        for k in range(0,rel_classes):
            poly = UNIQUE([R[i*rel_classes+k,j*rel_classes+b] for b in range(0,rel_classes)])
            exactlyone.append(poly)
            poly = UNIQUE([R[j*rel_classes+b,i*rel_classes+k] for b in range(0,rel_classes)])
            exactlyone.append(poly)
exactlyone=[e+1 for e in exactlyone] #assert truth
            
clues.append(UNIQUE([R[9,1],R[9,6]]))    # clue 1
clues.append(UNIQUE([AND(R[4,11],R[6,10]), AND(R[4,10],R[6,9]), AND(R[4,9],R[6,8])]))    # clue 2
clues.append(UNIQUE([AND(R[11,6],R[9,0]), AND(R[11,0],R[9,6])]))    # clue 3
clues.append(UNIQUE([AND(R[0,11],R[2,4]), AND(R[0,4],R[2,11])]))    # clue 4
clues.append(UNIQUE([AND(R[5,11],R[4,10]), AND(R[5,11],R[4,9]), AND(R[5,11],R[4,8]), AND(R[5,10],R[4,9]), AND(R[5,10],R[4,8]), AND(R[5,9],R[4,8])]))    # clue 5
clues = [c+1 for c in clues] #assert truth

prems=reflex+symmtrc+trans+exactlyone+clues
REI=ideal(prems)
QR = BPR.quotient_ring(REI)
solnmat=matrix(objects,objects,QR.gens())
sd=[j*rel_classes for j in range(1,types)]
solnmat.subdivide(sd,sd)
print(solnmat);print(' ')

objects = 12 Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47, X48, X49, X50, X51, X52, X53, X54, X55, X56, X57, X58, X59, X60, X61, X62, X63, X64, X65, X66, X67, X68, X69, X70, X71, X72, X73, X74, X75, X76, X77, X78, X79, X80, X81, X82, X83, X84, X85, X86, X87, X88, X89, X90, X91, X92, X93, X94, X95, X96, X97, X98, X99, X100, X101, X102, X103, X104, X105, X106, X107, X108, X109, X110, X111, X112, X113, X114, X115, X116, X117, X118, X119, X120, X121, X122, X123, X124, X125, X126, X127, X128, X129, X130, X131, X132, X133, X134, X135, X136, X137, X138, X139, X140, X141, X142, X143 144 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X143 = R[11,11] [1 0 0 0|0 1 0 0|0 0 0 1] [0 1 0 0|0 0 0 1|1 0 0 0] [0 0 1 0|1 0 0 0|0 0 1 0] [0 0 0 1|0 0 1 0|0 1 0 0] [-------+-------+-------] [0 0 1 0|1 0 0 0|0 0 1 0] [1 0 0 0|0 1 0 0|0 0 0 1] [0 0 0 1|0 0 1 0|0 1 0 0] [0 1 0 0|0 0 0 1|1 0 0 0] [-------+-------+-------] [0 1 0 0|0 0 0 1|1 0 0 0] [0 0 0 1|0 0 1 0|0 1 0 0] [0 0 1 0|1 0 0 0|0 0 1 0] [1 0 0 0|0 1 0 0|0 0 0 1]

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C+1+k) in Boolean Ring i,j,k=0,1

rel_classes = 6
types = 4
objects = rel_classes*types; print('objects = ' + str(objects))
BPR = BooleanPolynomialRing(objects^2,'X',order='degneglex')
BPR.inject_variables()
print(str(objects^2) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
R = matrix(BPR, objects, objects, BPR.gens())
reflex=[];symmtrc=[];trans=[];exactlyone=[];clues = []

for i in range(0, objects):    #reflexive
    reflex.append(R[i,i])
reflex = [r+1 for r in reflex]
    
for i in range(0,objects):    #symmetric
    for j in range(i+1, objects):
        symmtrc.append(IFF(R[i,j],R[j,i]))
symmtrc = [s+1 for s in symmtrc]
        
for i in range(objects): #transitive
    for j in range(i+1,objects):
        for k in range(j+1,objects):
            trans.append(R[i,j]*(R[i,k]+R[j,k])) #1+ built in
            trans.append(R[i,k]*(R[i,j]+R[j,k]))

for i in range(0,types):
    for j in range(i,types):
        for k in range(0,rel_classes):
            poly = UNIQUE([R[i*rel_classes+k,j*rel_classes+b] for b in range(0,rel_classes)])
            exactlyone.append(poly)
            poly = UNIQUE([R[j*rel_classes+b,i*rel_classes+k] for b in range(0,rel_classes)])
            exactlyone.append(poly)
exactlyone = [e+1 for e in exactlyone]
            
clues.append(UNIQUE([AND(R[11,22],R[16,1]), AND(R[11,1],R[16,22])]))    # clue 1
clues.append(UNIQUE([AND(R[1,18],R[15,19]), AND(R[1,18],R[15,20]), AND(R[1,18],R[15,21]), AND(R[1,18],R[15,22]), AND(R[1,18],R[15,23]),    # clue 2
                   AND(R[1,19],R[15,20]), AND(R[1,19],R[15,21]), AND(R[1,19],R[15,22]), AND(R[1,19],R[15,23]),
                   AND(R[1,20],R[15,21]), AND(R[1,20],R[15,22]), AND(R[1,20],R[15,23]),
                   AND(R[1,21],R[15,22]), AND(R[1,21],R[15,23]),
                   AND(R[1,22],R[15,23])]))
clues.append(UNIQUE([AND(R[13,23],R[9,21]), AND(R[13,22],R[9,20]), AND(R[13,21],R[9,19]), AND(R[13,20],R[9,18])]))    # clue 3
clues.append(R[5,11])    # clue 4
clues.append(NOT(R[4,23]))    # clue 5
clues.append(NOT(R[1,10]))    # clue 6.1
clues.append(NOT(R[1,14]))    # clue 6.2
clues.append(NOT(R[1,17]))    # clue 6.3
clues.append(NOT(R[1,9]))    # clue 6.4
clues.append(NOT(R[1,8]))    # clue 6.5
clues.append(NOT(R[10,14]))    # clue 6.6
clues.append(NOT(R[10,17]))    # clue 6.7
clues.append(NOT(R[14,9]))    # clue 6.8
clues.append(NOT(R[14,8]))    # clue 6.9
clues.append(NOT(R[17,9]))    # clue 6.10
clues.append(NOT(R[17,8]))    # clue 6.11
clues.append(R[21,6])    # clue 7
clues.append(UNIQUE([R[0,8], R[0,7]]))    # clue 8
clues.append(NOT(R[13,10]))    # clue 9
clues.append(NOT(R[3,14]))    # clue 10
clues.append(UNIQUE([AND(R[16,23],R[3,21]), AND(R[16,22],R[3,20]), AND(R[16,21],R[3,19]), AND(R[16,20],R[3,18])]))    # clue 11
clues = [c+1 for c in clues]

prems=reflex+symmtrc+trans+exactlyone+clues
REI=ideal(prems)
QR = BPR.quotient_ring(REI)
solnmat=matrix(objects,objects,QR.gens())
sd=[j*rel_classes for j in range(1,types)]
solnmat.subdivide(sd,sd)
print(solnmat);print(' ')

objects = 24 Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47, X48, X49, X50, X51, X52, X53, X54, X55, X56, X57, X58, X59, X60, X61, X62, X63, X64, X65, X66, X67, X68, X69, X70, X71, X72, X73, X74, X75, X76, X77, X78, X79, X80, X81, X82, X83, X84, X85, X86, X87, X88, X89, X90, X91, X92, X93, X94, X95, X96, X97, X98, X99, X100, X101, X102, X103, X104, X105, X106, X107, X108, X109, X110, X111, X112, X113, X114, X115, X116, X117, X118, X119, X120, X121, X122, X123, X124, X125, X126, X127, X128, X129, X130, X131, X132, X133, X134, X135, X136, X137, X138, X139, X140, X141, X142, X143, X144, X145, X146, X147, X148, X149, X150, X151, X152, X153, X154, X155, X156, X157, X158, X159, X160, X161, X162, X163, X164, X165, X166, X167, X168, X169, X170, X171, X172, X173, X174, X175, X176, X177, X178, X179, X180, X181, X182, X183, X184, X185, X186, X187, X188, X189, X190, X191, X192, X193, X194, X195, X196, X197, X198, X199, X200, X201, X202, X203, X204, X205, X206, X207, X208, X209, X210, X211, X212, X213, X214, X215, X216, X217, X218, X219, X220, X221, X222, X223, X224, X225, X226, X227, X228, X229, X230, X231, X232, X233, X234, X235, X236, X237, X238, X239, X240, X241, X242, X243, X244, X245, X246, X247, X248, X249, X250, X251, X252, X253, X254, X255, X256, X257, X258, X259, X260, X261, X262, X263, X264, X265, X266, X267, X268, X269, X270, X271, X272, X273, X274, X275, X276, X277, X278, X279, X280, X281, X282, X283, X284, X285, X286, X287, X288, X289, X290, X291, X292, X293, X294, X295, X296, X297, X298, X299, X300, X301, X302, X303, X304, X305, X306, X307, X308, X309, X310, X311, X312, X313, X314, X315, X316, X317, X318, X319, X320, X321, X322, X323, X324, X325, X326, X327, X328, X329, X330, X331, X332, X333, X334, X335, X336, X337, X338, X339, X340, X341, X342, X343, X344, X345, X346, X347, X348, X349, X350, X351, X352, X353, X354, X355, X356, X357, X358, X359, X360, X361, X362, X363, X364, X365, X366, X367, X368, X369, X370, X371, X372, X373, X374, X375, X376, X377, X378, X379, X380, X381, X382, X383, X384, X385, X386, X387, X388, X389, X390, X391, X392, X393, X394, X395, X396, X397, X398, X399, X400, X401, X402, X403, X404, X405, X406, X407, X408, X409, X410, X411, X412, X413, X414, X415, X416, X417, X418, X419, X420, X421, X422, X423, X424, X425, X426, X427, X428, X429, X430, X431, X432, X433, X434, X435, X436, X437, X438, X439, X440, X441, X442, X443, X444, X445, X446, X447, X448, X449, X450, X451, X452, X453, X454, X455, X456, X457, X458, X459, X460, X461, X462, X463, X464, X465, X466, X467, X468, X469, X470, X471, X472, X473, X474, X475, X476, X477, X478, X479, X480, X481, X482, X483, X484, X485, X486, X487, X488, X489, X490, X491, X492, X493, X494, X495, X496, X497, X498, X499, X500, X501, X502, X503, X504, X505, X506, X507, X508, X509, X510, X511, X512, X513, X514, X515, X516, X517, X518, X519, X520, X521, X522, X523, X524, X525, X526, X527, X528, X529, X530, X531, X532, X533, X534, X535, X536, X537, X538, X539, X540, X541, X542, X543, X544, X545, X546, X547, X548, X549, X550, X551, X552, X553, X554, X555, X556, X557, X558, X559, X560, X561, X562, X563, X564, X565, X566, X567, X568, X569, X570, X571, X572, X573, X574, X575 576 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X575 = R[23,23]

solnmat1=solnmat

exactlyone

[X57*X58*X59 + X56*X58*X59 + X56*X57*X59 + X56*X57*X58 + X59 + X58 + X57 + X56, X112*X124*X136 + X100*X124*X136 + X100*X112*X136 + X100*X112*X124 + X136 + X124 + X112 + X100, X69*X70*X71 + X68*X70*X71 + X68*X69*X71 + X68*X69*X70 + X71 + X70 + X69 + X68, X113*X125*X137 + X101*X125*X137 + X101*X113*X137 + X101*X113*X125 + X137 + X125 + X113 + X101, X81*X82*X83 + X80*X82*X83 + X80*X81*X83 + X80*X81*X82 + X83 + X82 + X81 + X80, X114*X126*X138 + X102*X126*X138 + X102*X114*X138 + X102*X114*X126 + X138 + X126 + X114 + X102, X93*X94*X95 + X92*X94*X95 + X92*X93*X95 + X92*X93*X94 + X95 + X94 + X93 + X92, X115*X127*X139 + X103*X127*X139 + X103*X115*X139 + X103*X115*X127 + X139 + X127 + X115 + X103]

sd=[j*rel_classes for j in range(1,types)]
solnmat.subdivide(sd,sd);solnmat

[1 0 0 0|0 1 0 0|0 0 0 1] [0 1 0 0|0 0 0 1|1 0 0 0] [0 0 1 0|1 0 0 0|0 0 1 0] [0 0 0 1|0 0 1 0|0 1 0 0] [-------+-------+-------] [0 0 1 0|1 0 0 0|0 0 1 0] [1 0 0 0|0 1 0 0|0 0 0 1] [0 0 0 1|0 0 1 0|0 1 0 0] [0 1 0 0|0 0 0 1|1 0 0 0] [-------+-------+-------] [0 1 0 0|0 0 0 1|1 0 0 0] [0 0 0 1|0 0 1 0|0 1 0 0] [0 0 1 0|1 0 0 0|0 0 1 0] [1 0 0 0|0 1 0 0|0 0 0 1]

solnmat.submatrix(0,18,6,6)

[0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 0 1] [0 1 0 0 0 0] [1 0 0 0 0 0] [0 0 0 0 1 0]

solnmat2

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'solnmat2' is not defined

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C+1+k) in Boolean Ring i,j,k=0,1

rel_classes = 4
types = 3
objects = rel_classes*types; print('objects = ' + str(objects))
BPR = BooleanPolynomialRing(objects^2,'X',order='degneglex')
BPR.inject_variables()
print(str(objects^2) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
R = matrix(BPR, objects, objects, BPR.gens())
reflex=[];symmtrc=[];trans=[];exactlyone=[];clues = [];

for i in range(0, objects):    #reflexive
    reflex.append(R[i,i])
#reflex=[r+1 for r in reflex]   #assert truth

for i in range(0,objects):    #symmetric
    for j in range(i+1, objects):
        symmtrc.append(IFF(R[i,j],R[j,i]))
#symmtrc=[s+1 for s in symmtrc]   #assert truth

for i in range(objects):     #transitive truth assertion built into these which assume symmtrc and reflex in place
    for j in range(i+1,objects):
        for k in range(j+1,objects):
            trans.append(R[i,j]*(R[i,k]+R[j,k]))
            trans.append(R[i,k]*(R[i,j]+R[j,k]))

for i in range(0,types):
    for j in range(i,types):
        for k in range(0,rel_classes):
            poly = UNIQUE([R[i*rel_classes+k,j*rel_classes+b] for b in range(0,rel_classes)])
            exactlyone.append(poly)
            poly = UNIQUE([R[j*rel_classes+b,i*rel_classes+k] for b in range(0,rel_classes)])
            exactlyone.append(poly)
exactlyone=[e+1 for e in exactlyone] #assert truth
            
clues.append(UNIQUE([R[9,1],R[9,6]]))    # clue 1
clues.append(UNIQUE([AND(R[4,11],R[6,10]), AND(R[4,10],R[6,9]), AND(R[4,9],R[6,8])]))    # clue 2
clues.append(UNIQUE([AND(R[11,6],R[9,0]), AND(R[11,0],R[9,6])]))    # clue 3
clues.append(UNIQUE([AND(R[0,11],R[2,4]), AND(R[0,4],R[2,11])]))    # clue 4
clues.append(UNIQUE([AND(R[5,11],R[4,10]), AND(R[5,11],R[4,9]), AND(R[5,11],R[4,8]), AND(R[5,10],R[4,9]), AND(R[5,10],R[4,8]), AND(R[5,9],R[4,8])]))    # clue 5

            #reflex+symmtrc+trans

objects = 12 Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47, X48, X49, X50, X51, X52, X53, X54, X55, X56, X57, X58, X59, X60, X61, X62, X63, X64, X65, X66, X67, X68, X69, X70, X71, X72, X73, X74, X75, X76, X77, X78, X79, X80, X81, X82, X83, X84, X85, X86, X87, X88, X89, X90, X91, X92, X93, X94, X95, X96, X97, X98, X99, X100, X101, X102, X103, X104, X105, X106, X107, X108, X109, X110, X111, X112, X113, X114, X115, X116, X117, X118, X119, X120, X121, X122, X123, X124, X125, X126, X127, X128, X129, X130, X131, X132, X133, X134, X135, X136, X137, X138, X139, X140, X141, X142, X143 144 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X143 = R[11,11]

clues
mypoly=FORALL(clues);#mypoly
#clues[0]*clues[1]
#convert(clues[0])

[X114 + X109, X57*X58*X59*X80*X81*X82 + X59*X82 + X58*X81 + X57*X80, X114*X132 + X108*X138, X11*X28 + X4*X35, X56*X57*X58*X69*X70*X71 + X56*X58*X69*X70*X71 + X56*X57*X58*X69*X71 + X57*X58*X70*X71 + X56*X69*X70*X71 + X56*X58*X70*X71 + X56*X58*X69*X71 + X56*X57*X69*X71 + X56*X57*X69*X70 + X56*X57*X58*X71 + X58*X71 + X57*X71 + X57*X70 + X56*X71 + X56*X70 + X56*X69]

mystring = "R0_1"
[int(x) for x in mystring.split("R")[1].split("_")]
flatten = lambda l: [item for sublist in l for item in sublist]
n=5;myvars=flatten([["R"+str(i).rjust(len(str(n)),'0')+"_"+str(j).rjust(len(str(n)),'0') for j in range(i+1,n) ] for i in range(0,n-1)]);myvars;[[int(x) for x in mystring.split("R")[1].split("_")] for mystring in myvars]
BPR = BooleanPolynomialRing(n*(n-1)/2,myvars,order='degneglex')
BPR.inject_variables()
trans=[]
for i in range(0,n):     #transitive truth assertion built into these which assume symmtrc and reflex in place
    for j in range(i+1,n):
        for k in range(j+1,n):
            trans.append(BPR("R"+str(i).rjust(len(str(n)),'0')+"_"+str(j).rjust(len(str(n)),'0'))*(BPR("R"+str(i).rjust(len(str(n)),'0')+"_"+str(k).rjust(len(str(n)),'0'))+BPR("R"+str(j).rjust(len(str(n)),'0')+"_"+str(k).rjust(len(str(n)),'0'))))
            trans.append(BPR("R"+str(i).rjust(len(str(n)),'0')+"_"+str(k).rjust(len(str(n)),'0'))*(BPR("R"+str(i).rjust(len(str(n)),'0')+"_"+str(j).rjust(len(str(n)),'0'))+BPR("R"+str(j).rjust(len(str(n)),'0')+"_"+str(k).rjust(len(str(n)),'0'))))
trans

[0, 1] ['R0_1', 'R0_2', 'R0_3', 'R0_4', 'R1_2', 'R1_3', 'R1_4', 'R2_3', 'R2_4', 'R3_4'] [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]] Defining R0_1, R0_2, R0_3, R0_4, R1_2, R1_3, R1_4, R2_3, R2_4, R3_4 [R0_1*R1_2 + R0_1*R0_2, R0_2*R1_2 + R0_1*R0_2, R0_1*R1_3 + R0_1*R0_3, R0_3*R1_3 + R0_1*R0_3, R0_1*R1_4 + R0_1*R0_4, R0_4*R1_4 + R0_1*R0_4, R0_2*R2_3 + R0_2*R0_3, R0_3*R2_3 + R0_2*R0_3, R0_2*R2_4 + R0_2*R0_4, R0_4*R2_4 + R0_2*R0_4, R0_3*R3_4 + R0_3*R0_4, R0_4*R3_4 + R0_3*R0_4, R1_2*R2_3 + R1_2*R1_3, R1_3*R2_3 + R1_2*R1_3, R1_2*R2_4 + R1_2*R1_4, R1_4*R2_4 + R1_2*R1_4, R1_3*R3_4 + R1_3*R1_4, R1_4*R3_4 + R1_3*R1_4, R2_3*R3_4 + R2_3*R2_4, R2_4*R3_4 + R2_3*R2_4]

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C+1+k) in Boolean Ring i,j,k=0,1

rel_classes = 4
types = 3
objects = rel_classes*types; print('objects = ' + str(objects))
BPR = BooleanPolynomialRing(types*rel_classes*rel_classes,'X',order='degneglex')
BPR.inject_variables()
#print(str(objects^2) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
R = matrix(BPR, objects, objects)
trans=[];exactlyone=[];clues = [];

for i in range(0, objects):    #reflexive
    R[i,i] = BPR(1)

nn=-1
for i in range(0,types):    #symmetric
    for j in range(i+1, types):
        nn=nn+1
        mm=0
        for k in range(0,rel_classes):
            for l in range(0,rel_classes):
                R[i*rel_classes+k,rel_classes*j+l] = BPR.gens()[nn*16+mm]
                R[rel_classes*j+l,i*rel_classes+k] = BPR.gens()[nn*16+mm]
                mm=mm+1

for i in range(objects):     #transitive truth assertion built into these which assume symmtrc and reflex in place
    for j in range(objects):
        for k in range(objects):
            trans.append(IF(R[i,j]*R[j,k],R[i,k]))

trans = list(set(trans))
trans.remove(1)

for i in range(0,types):
    for j in range(i,types):
        for k in range(0,rel_classes):
            poly = UNIQUE([R[i*rel_classes+k,j*rel_classes+b] for b in range(0,rel_classes)])
            exactlyone.append(poly)
            poly = UNIQUE([R[j*rel_classes+b,i*rel_classes+k] for b in range(0,rel_classes)])
            exactlyone.append(poly)
            
exactlyone = list(set(exactlyone))
exactlyone.remove(1)
            
clues.append(UNIQUE([R[9,1],R[9,6]]))    # clue 1
clues.append(UNIQUE([AND(R[4,11],R[6,10]), AND(R[4,10],R[6,9]), AND(R[4,9],R[6,8])]))    # clue 2
clues.append(UNIQUE([AND(R[11,6],R[9,0]), AND(R[11,0],R[9,6])]))    # clue 3
clues.append(UNIQUE([AND(R[0,11],R[2,4]), AND(R[0,4],R[2,11])]))    # clue 4
clues.append(UNIQUE([AND(R[5,11],R[4,10]), AND(R[5,11],R[4,9]), AND(R[5,11],R[4,8]), AND(R[5,10],R[4,9]), AND(R[5,10],R[4,8]), AND(R[5,9],R[4,8])]))    # clue 5


prems = trans + exactlyone + clues
prems = [p+1 for p in prems]
REI=ideal(prems)
QR = BPR.quotient_ring(REI)
solnmat=matrix([[QR(R[i,j]) for i in range(objects)] for j in range(objects)])
sd=[j*rel_classes for j in range(1,types)]
solnmat.subdivide(sd,sd)
print(solnmat);print(' ')
#exactlyone

objects = 12 Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47 [1 0 0 0|0 1 0 0|0 0 0 1] [0 1 0 0|0 0 0 1|1 0 0 0] [0 0 1 0|1 0 0 0|0 0 1 0] [0 0 0 1|0 0 1 0|0 1 0 0] [-------+-------+-------] [0 0 1 0|1 0 0 0|0 0 1 0] [1 0 0 0|0 1 0 0|0 0 0 1] [0 0 0 1|0 0 1 0|0 1 0 0] [0 1 0 0|0 0 0 1|1 0 0 0] [-------+-------+-------] [0 1 0 0|0 0 0 1|1 0 0 0] [0 0 0 1|0 0 1 0|0 1 0 0] [0 0 1 0|1 0 0 0|0 0 1 0] [1 0 0 0|0 1 0 0|0 0 0 1]

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C+1+k) in Boolean Ring i,j,k=0,1

rel_classes = 6
types = 4
objects = rel_classes*types; print('objects = ' + str(objects))
BPR = BooleanPolynomialRing(types*rel_classes*rel_classes,'X',order='degneglex')
BPR.inject_variables()
#print(str(objects^2) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
R = matrix(BPR, objects, objects)
trans=[];exactlyone=[];clues = [];

for i in range(0, objects):    #reflexive
    R[i,i] = BPR(1)

nn=-1
for i in range(0,types):    #symmetric
    for j in range(i+1, types):
        nn=nn+1
        mm=0
        for k in range(0,rel_classes):
            for l in range(0,rel_classes):
                R[i*rel_classes+k,rel_classes*j+l] = BPR.gens()[nn*16+mm]
                R[rel_classes*j+l,i*rel_classes+k] = BPR.gens()[nn*16+mm]
                mm=mm+1

for i in range(objects):     #transitive truth assertion built into these which assume symmtrc and reflex in place
    for j in range(objects):
        for k in range(objects):
            trans.append(IF(R[i,j]*R[j,k],R[i,k]))

trans = list(set(trans))
trans.remove(1)

for i in range(0,types):
    for j in range(i,types):
        for k in range(0,rel_classes):
            poly = UNIQUE([R[i*rel_classes+k,j*rel_classes+b] for b in range(0,rel_classes)])
            exactlyone.append(poly)
            poly = UNIQUE([R[j*rel_classes+b,i*rel_classes+k] for b in range(0,rel_classes)])
            exactlyone.append(poly)

exactlyone = list(set(exactlyone))
exactlyone.remove(1)
            
clues.append(UNIQUE([AND(R[11,22],R[16,1]), AND(R[11,1],R[16,22])]))    # clue 1
clues.append(UNIQUE([AND(R[1,18],R[15,19]), AND(R[1,18],R[15,20]), AND(R[1,18],R[15,21]), AND(R[1,18],R[15,22]), AND(R[1,18],R[15,23]),    # clue 2
                   AND(R[1,19],R[15,20]), AND(R[1,19],R[15,21]), AND(R[1,19],R[15,22]), AND(R[1,19],R[15,23]),
                   AND(R[1,20],R[15,21]), AND(R[1,20],R[15,22]), AND(R[1,20],R[15,23]),
                   AND(R[1,21],R[15,22]), AND(R[1,21],R[15,23]),
                   AND(R[1,22],R[15,23])]))
clues.append(UNIQUE([AND(R[13,23],R[9,21]), AND(R[13,22],R[9,20]), AND(R[13,21],R[9,19]), AND(R[13,20],R[9,18])]))    # clue 3
clues.append(R[5,11])    # clue 4
clues.append(NOT(R[4,23]))    # clue 5
clues.append(NOT(R[1,10]))    # clue 6.1
clues.append(NOT(R[1,14]))    # clue 6.2
clues.append(NOT(R[1,17]))    # clue 6.3
clues.append(NOT(R[1,9]))    # clue 6.4
clues.append(NOT(R[1,8]))    # clue 6.5
clues.append(NOT(R[10,14]))    # clue 6.6
clues.append(NOT(R[10,17]))    # clue 6.7
clues.append(NOT(R[14,9]))    # clue 6.8
clues.append(NOT(R[14,8]))    # clue 6.9
clues.append(NOT(R[17,9]))    # clue 6.10
clues.append(NOT(R[17,8]))    # clue 6.11
clues.append(R[21,6])    # clue 7
clues.append(UNIQUE([R[0,8], R[0,7]]))    # clue 8
clues.append(NOT(R[13,10]))    # clue 9
clues.append(NOT(R[3,14]))    # clue 10
clues.append(UNIQUE([AND(R[16,23],R[3,21]), AND(R[16,22],R[3,20]), AND(R[16,21],R[3,19]), AND(R[16,20],R[3,18])]))    # clue 11


prems=trans+exactlyone+clues
prems = [p+1 for p in prems]
REI=ideal(prems)
QR = BPR.quotient_ring(REI)
solnmat=matrix([[QR(R[i,j]) for i in range(objects)] for j in range(objects)])
#sd=[j*rel_classes for j in range(1,types)]
#solnmat.subdivide(sd,sd)
#print(solnmat);print(' ')

objects = 24 Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47, X48, X49, X50, X51, X52, X53, X54, X55, X56, X57, X58, X59, X60, X61, X62, X63, X64, X65, X66, X67, X68, X69, X70, X71, X72, X73, X74, X75, X76, X77, X78, X79, X80, X81, X82, X83, X84, X85, X86, X87, X88, X89, X90, X91, X92, X93, X94, X95, X96, X97, X98, X99, X100, X101, X102, X103, X104, X105, X106, X107, X108, X109, X110, X111, X112, X113, X114, X115, X116, X117, X118, X119, X120, X121, X122, X123, X124, X125, X126, X127, X128, X129, X130, X131, X132, X133, X134, X135, X136, X137, X138, X139, X140, X141, X142, X143

load('ProbRecurse.py')
load('Prob.py')
import numpy
import scipy.optimize

KDELTA = lambda A, B: A.parent(A == B)
NOT = lambda A: A.parent(1) + A
XOR = lambda A, B: A + B
OR = lambda A, B: A + B + A*B
IF = lambda A, B: A.parent(1) + A + A*B
IFF = lambda A, B: A.parent(1) + A + B
AND = lambda A, B: A*B
NAND = lambda A, B: A.parent(1) + A*B
NOR = lambda A, B: A.parent(1) + A + B + A*B
FORALL = lambda mylist: reduce(AND, mylist)
EXISTS = lambda mylist: reduce(OR, mylist)
UNIQUE = lambda mylist: reduce(XOR,[FORALL([A + NOT(KDELTA(A,B)) for A in mylist]) for B in mylist])
GIVEN = lambda A, B: Prob(AND(A,B))/Prob(B)
convert = lambda f: reduce(lambda a,b:a+b,[reduce(lambda a,b:a*b,[(m.degree(g)+1+g) for g in BPR.gens()]) for m in f.monomials()]) #self inverse convert poly in gens to poly in atoms (missing var x implies, interpret as ~x)
# convert monomial terms A^i*B^j*C^k --> (A+1+i)*(B+1+j)*(C+1+k) in Boolean Ring i,j,k=0,1

objects = 5; print('objects = ' + str(objects))
UD=objects*(objects-1)/2
BPR = BooleanPolynomialRing(UD,'X',order='degneglex')
BPR.inject_variables()
print(str(UD) + ' Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X' + str(objects^2-1) + ' = R[' + str(objects-1) + ',' + str(objects-1) + ']')
R = matrix(BPR, objects, objects)
trans=[];clues = [];

for i in range(0, objects):    #reflexive
    R[i,i] = BPR(1)

mm=-1
for i in range(0,objects):    #symmetric
    for j in range(i+1, objects):
        mm=mm+1
        R[i,j] = BPR.gens()[mm]
        R[j,i] = BPR.gens()[mm]

for i in range(objects):     #transitive
    for j in range(objects):
        for k in range(objects):
            trans.append(IF(R[i,j]*R[j,k],R[i,k]))

trans = list(set(trans))
trans.remove(1)

prems=trans
prems = [p+1 for p in prems]
REI=ideal(prems)
QR = BPR.quotient_ring(REI)

NewGens = str(BPR.gens()).lower()
NewGens = NewGens[1:len(NewGens) - 1]
PPR = PolynomialRing(QQ,NewGens,order='degneglex')
PPR.inject_variables()

objects = 5 Defining X0, X1, X2, X3, X4, X5, X6, X7, X8, X9 10 Free Boolean Algebra generators: X0 = R[0,0], X1 = R[0,1], X2 = R[0,2],..., X24 = R[4,4] Defining x0, x1, x2, x3, x4, x5, x6, x7, x8, x9

#MyProbPoly=Prob(FORALL(trans))
lincons = [Prob(trans[0])-.9]
str1='{\'type\': \'eq\', \'fun\': lambda s: (lincons['
str2='])(*s)},'
mystr='['
for ii in range(len(lincons)):
    mystr=mystr+str1+str(ii)+str2
mystr=mystr+']'
cons = eval(mystr)
bnds = tuple([(0.0, 1.0) for xx in range(10)])
y0 = [.5 for xx in range(10)]
#HH = lambda s: sum([s[ii]*ln(s[ii])+(1-s[ii])*ln(1-s[ii]) for ii in range(len(s))])
#dHH = lambda s: [ln(s[ii])-ln(1-s[ii]) for  ii in range(len(s))]
pvar = lambda s: sum([(2*s[i]-1)^2 for i in range(len(s))])
dpvar = lambda s: [4*(2*s[i]-1) for i in range(len(s))]
pvarsoln = scipy.optimize.minimize(pvar, y0, jac=dpvar, method='SLSQP', bounds=bnds, constraints=cons)

trans[0];Prob(trans[0]);cons
pvarsoln.x

X1*X3*X8 + X1*X3 + 1 x1*x3*x8 - x1*x3 + 1 [{'fun': <function <lambda> at 0x7f5d9fc818c0>, 'type': 'eq'}] array([ 0.5 , 0.46415888, 0.5 , 0.46415888, 0.5 , 0.5 , 0.5 , 0.5 , 0.53584111, 0.5 ])