#############################################################
# Computing True and Expected Valuations of Cross functions #
#############################################################

# To determine the metric trees associated to all extremal curves we must settle the differences between expected and true valuations. In order to do so, we
# 1) compute the valuations of all Cross functions from all four expressions of Crosses as differences of Yoshidas, whenever possible.
# 2) list the Crosses with undertermined valuations for each cone.
# 3) Create dictionaries of valuations for Crosses for each cone.
# 4) compute true valuation of Crosses.
# 5) compute the expected valuation of Crosses.
# 6) verify the differences on examples

# For 1), we pick all 4 ways of writing a given Cross function as differences of Yoshidas, and determine if on a given cone, one of these ways gives no ties on the center of mass, hence determining the true valuation of the Cross function on the whole cone (it will agree with the valuation of a Yoshida function).
# If all ways yield tie, we record 'None' as the true valuation of the given Cross function.

# For 5): When computing expected valuations, we consider only the Cross with 'None' as true valuations and determine the largest among the expected valuations for each way. This would be our expected valuation.

# For 6): we compute parameters in ds to find differences between true and expected valuations.

###########################################################################
# Setting up the ambient rings with the W(E6) action.
###########################################################################

# A list of variables d_1, ..., d_6 for the coefficients
ds = [var("d%s"%i) for i in range(1,7)]

# A list of variables E_1, ..., E_6
Es = [var("E%s"%i) for i in range(1,7)]

# A list of variables F_12, ..., F_56
Fs = [var("F%s%s"%(i,j)) for i in range(1,7) for j in range(1,7) if i < j]

# A list of variables G_1, ..., G_6
Gs = [var("G%s"%i) for i in range(1,7)]

# A list of variables X_12, ..., X_65
Xs = [var("X%s%s"%(i,j)) for i in range(1,7) for j in range(1,7) if i != j]

# A list of variables Y_123456, ..., Y_162534
Ys = [var("Y%s%s%s%s%s%s"%(i,j,k,l,m,n)) for i in range(1,7) for j in range(1,7) for k in range(1,7) for l in range(1,7) for m in range(1,7) for n in range(1,7) if i<j and k<l and m<n and i<k and k<m and len(uniq([i,j,k,l,m,n])) == 6]

# A list of variables Yoshida_0, ..., Yoshida_39
Yoshidas = [var("Yoshida%s"%i) for i in range(0,40)]



# A polynomial ring on all these generators
Qd = PolynomialRing(QQ, ds)
basering = FractionField(Qd)
S = PolynomialRing(basering, Xs+Ys)
Sxy = PolynomialRing(QQ, Xs+Ys)
T = PolynomialRing(QQ, ds+Es+Fs+Gs)

# Yoshida expressions in ds
YM = load("../../Yoshida/Output/Yoshida_matrix.sobj")
#Yoshidas = [VectorToYoshida(YM[i],positive_roots) for i in range(0,40)]

roots=load("../../Yoshida/Scripts/Input/rootsE6.sobj")
y_to_d = {Yoshidas[i]: prod([roots[j]^YM[i][j] for j in range(0,36)]) for i in range(0,40)}

#Yoshidas = [VectorToYoshida(YM[i],positive_roots) for i in range(0,40)]
YRing = PolynomialRing(QQ, Yoshidas)
YField = FractionField(YRing)

#load("Yoshida.sage")

# The functions for computing valuations and substitutions.


# We need to work with the fraction field of Qt.
Qt = PolynomialRing(QQ,[var("t")])
QtField = FractionField(Qt)


# We use the Cross functions computed in "Yoshida.sage"

cross_variables = [var("Cross%s"%i) for i in range(0,135)]

cross_to_yoshidas = load("../../Yoshida/Scripts/Input/cross_to_yoshidas.sobj")


AllConeNames = ['aa2a3a4', 'aa2a3b', 'aa2a3', 'aa2a4', 'aa3a4','a2a3a4', 'aa2b', 'aa3b', 'a2a3b', 'aa2','aa3','aa4','a2a3', 'a2a4', 'a3a4', 'ab', 'a2b', 'a3b', 'a', 'a2', 'a3', 'a4','b']


# All the necessary functions are coded in the following script.
load("../../Yoshida/Scripts/allTropicalNodesPerNarukiConesMacros.sage")


# The entries of all135Nodes[keys] are factorized already.
all135Nodes = load("../../Yoshida/Scripts/Input/all135ClassicalNodes.sobj")

dictNodes = all135Nodes


# The following function takes a list of lists, and returns a list of all the elements (without repetitions)

def combineLists(listOfList):
    newList = []
    for x in listOfList:
    	for y in x:
    	    newList.append(y)
    return list(set(newList))


#########################################################
# Finding valuations of Cross functions on Naruki cones #
#########################################################

# We record the dictionary of true valuations for each Cross function on each cone in the Naruki fan. The necessary functions are coded in '../../Yoshida/Scripts/allTropicalNodesPerNarukiConesMacros.sage'

# val_cross_to_yoshidas = dict()
# for coneName in AllConeNames:
#     print str(coneName)
#     val_cross_to_yoshidas[coneName] = dict()
#     for key in cross_variables:
#     	cone = load('../../ComputeLines/Scripts/Input/' + str(coneName)+'Cell.sobj')
#     	# cone
#     	for l in range(0,4):
#     	    print str(l)
#        	    val_cross_to_yoshidas[coneName][SR(key)] = [yoshida_to_trop_yoshida_with_Nones(q, cone) for q in cross_to_yoshidas[SR(key)]]
# 	val_cross_to_yoshidas[coneName][-SR(key)] = val_cross_to_yoshidas[coneName][SR(key)]
#     	print 'Done with ' + str(key)	 
#     print 'Done with ' + str(coneName)
#     print ''

# # We save the dictionary as a sage object file:
# save(val_cross_to_yoshidas,'Input/val_cross.sobj')

# We load the dictionary:
val_cross_to_yoshidas = load('../../Yoshida/Scripts/Input/val_cross.sobj')



###################################
# Analysis of Issues on each Cone #
###################################

# We record the Cross functions with undefined valuations

# undefinedCrosses = dict()

#################
# MAXIMAL CONES #
#################

##############
# AAAA Types #
##############

# cone  = 'aa2a3a4'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True

# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# print badCrosses
# [Cross36, Cross37, Cross38]
# undefinedCrosses[cone] = badCrosses


# ##############
# # AAAB Types #
# ##############

# cone  = 'aa2a3b'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# False
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# print badCrosses
# []
# undefinedCrosses[cone] = badCrosses


# ###############################
# # TWO CONES
# ###############################

# #############
# # AAA Types #
# #############

# cone  = 'aa2a3'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone = 'aa2a4'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone = 'aa3a4'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone = 'a2a3a4'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# #############
# # AAB Types #
# #############

# cone  = 'aa2b'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# False
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses

# cone = 'aa3b'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone = 'a2a3b'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses



# ###############################
# # EDGES
# ###############################

# ############
# # AA Types #
# ############

# cone  = 'aa2'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone  = 'aa3'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone  = 'aa4'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses

# cone  = 'a2a3'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone  = 'a2a4'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone  = 'a3a4'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses



# ############
# # AB Types #
# ############

# cone  = 'ab'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses

# cone = 'a2b'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone = 'a3b'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# ###############################
# #  VERTICES
# ###############################

# ###########
# # A Types #
# ###########

# cone  = 'a'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses

# cone = 'a2'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses

# cone = 'a3'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# cone = 'a4'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses



# ###########
# # B Types #
# ###########

# cone  = 'b'
# 4*[None] in val_cross_to_yoshidas[cone].values()
# True
# badCrosses = [k for k in cross_variables if val_cross_to_yoshidas[cone][k]==4*[None]]
# undefinedCrosses[cone] = badCrosses


# We record the Crosses with undefined valuations for each cone:
# print undefinedCrosses
undefinedCrosses = {'aa2a3': [Cross36, Cross37, Cross38], 'ab': [Cross3, Cross4, Cross5, Cross51, Cross52, Cross53, Cross66, Cross67, Cross68, Cross72, Cross73, Cross74, Cross75, Cross76, Cross77, Cross123, Cross124, Cross125], 'aa2a4': [Cross15, Cross16, Cross17, Cross27, Cross28, Cross29, Cross36, Cross37, Cross38], 'aa3a4': [Cross36, Cross37, Cross38, Cross51, Cross52, Cross53, Cross123, Cross124, Cross125], 'a3b': [Cross39, Cross40, Cross41, Cross51, Cross52, Cross53, Cross78, Cross79, Cross80, Cross105, Cross106, Cross107, Cross120, Cross121, Cross122, Cross123, Cross124, Cross125], 'aa4': [Cross15, Cross16, Cross17, Cross27, Cross28, Cross29, Cross36, Cross37, Cross38, Cross51, Cross52, Cross53, Cross96, Cross97, Cross98, Cross99, Cross100, Cross101, Cross123, Cross124, Cross125], 'aa2a3a4': [Cross36, Cross37, Cross38], 'aa3': [Cross36, Cross37, Cross38, Cross51, Cross52, Cross53, Cross123, Cross124, Cross125], 'aa2': [Cross15, Cross16, Cross17, Cross27, Cross28, Cross29, Cross36, Cross37, Cross38], 'aa3b': [Cross51, Cross52, Cross53, Cross123, Cross124, Cross125], 'a2a3': [Cross36, Cross37, Cross38, Cross78, Cross79, Cross80, Cross105, Cross106, Cross107], 'a3a4': [Cross36, Cross37, Cross38, Cross39, Cross40, Cross41, Cross51, Cross52, Cross53, Cross78, Cross79, Cross80, Cross105, Cross106, Cross107, Cross120, Cross121, Cross122, Cross123, Cross124, Cross125], 'a2a4': [Cross15, Cross16, Cross17, Cross27, Cross28, Cross29, Cross36, Cross37, Cross38, Cross78, Cross79, Cross80, Cross105, Cross106, Cross107], 'a3': [Cross36, Cross37, Cross38, Cross39, Cross40, Cross41, Cross51, Cross52, Cross53, Cross78, Cross79, Cross80, Cross105, Cross106, Cross107, Cross120, Cross121, Cross122, Cross123, Cross124, Cross125], 'a2': [Cross15, Cross16, Cross17, Cross27, Cross28, Cross29, Cross36, Cross37, Cross38, Cross78, Cross79, Cross80, Cross105, Cross106, Cross107], 'a4': [Cross15, Cross16, Cross17, Cross27, Cross28, Cross29, Cross36, Cross37, Cross38, Cross39, Cross40, Cross41, Cross45, Cross46, Cross47, Cross51, Cross52, Cross53, Cross78, Cross79, Cross80, Cross96, Cross97, Cross98, Cross99, Cross100, Cross101, Cross105, Cross106, Cross107, Cross108, Cross109, Cross110, Cross120, Cross121, Cross122, Cross123, Cross124, Cross125], 'a': [Cross3, Cross4, Cross5, Cross15, Cross16, Cross17, Cross27, Cross28, Cross29, Cross36, Cross37, Cross38, Cross51, Cross52, Cross53, Cross57, Cross58, Cross59, Cross66, Cross67, Cross68, Cross72, Cross73, Cross74, Cross75, Cross76, Cross77, Cross81, Cross82, Cross83, Cross90, Cross91, Cross92, Cross96, Cross97, Cross98, Cross99, Cross100, Cross101, Cross102, Cross103, Cross104, Cross123, Cross124, Cross125], 'b': [Cross0, Cross1, Cross2, Cross3, Cross4, Cross5, Cross9, Cross10, Cross11, Cross12, Cross13, Cross14, Cross39, Cross40, Cross41, Cross51, Cross52, Cross53, Cross54, Cross55, Cross56, Cross60, Cross61, Cross62, Cross66, Cross67, Cross68, Cross69, Cross70, Cross71, Cross72, Cross73, Cross74, Cross75, Cross76, Cross77, Cross78, Cross79, Cross80, Cross84, Cross85, Cross86, Cross105, Cross106, Cross107, Cross114, Cross115, Cross116, Cross120, Cross121, Cross122, Cross123, Cross124, Cross125], 'aa2a3b': [], 'a2b': [Cross78, Cross79, Cross80, Cross105, Cross106, Cross107], 'a2a3b': [Cross78, Cross79, Cross80, Cross105, Cross106, Cross107], 'a2a3a4': [Cross36, Cross37, Cross38, Cross78, Cross79, Cross80, Cross105, Cross106, Cross107], 'aa2b': []}



# print [len(undefinedCrosses[cone]) for cone in AllConeNames]
# [3, 0, 3, 9, 9, 9, 0, 6, 6, 9, 9, 21, 9, 15, 21, 18, 6, 18, 45, 15, 21, 39, 54]
# set([len(undefinedCrosses[cone]) for cone in AllConeNames])
# {0, 3, 6, 9, 15, 18, 21, 39, 45, 54}

# CONCLUSION: Only two cones have all valuations of Crosses completely determined. The rest has some undefined crosses. Their number varies between 3 (for (aa2a3a4) and 54 (for (b)).

# We check that the same quintic appears as a factor in the Cross functions with undefined valuation on the top cell 'a2a3a4'

Qd = PolynomialRing(QQ,ds)
quintics = load('../../Yoshida/Scripts/Input/quintics.sobj')

undefinedCrosses['aa2a3a4']
[Cross36, Cross37, Cross38]
quintic36 = [f[0] for f in factor(basering(cross_to_yoshidas[Cross36][0].substitute(y_to_d))) if Qd(f[0]).total_degree() == 5][0]
quintic37 = [f[0] for f in factor(basering(cross_to_yoshidas[Cross37][0].substitute(y_to_d))) if Qd(f[0]).total_degree() == 5][0]
quintic38 = [f[0] for f in factor(basering(cross_to_yoshidas[Cross38][0].substitute(y_to_d))) if Qd(f[0]).total_degree() == 5][0]


[key for key in quintics.keys() if Qd(quintics[key]) == Qd(quintic36)]
[X63]
[key for key in quintics.keys() if Qd(quintics[key]) == Qd(quintic37)]
[X63]
[key for key in quintics.keys() if Qd(quintics[key]) == Qd(quintic38)]
[X63]


##################################
# Dictionary of Cross Valuations #
##################################

# We build the dictionaries of Cross valuations for each cone in the Naruki fan

# dict_cross_vals = dict()

# for cone in AllConeNames:
#     print cone
#     dict_cross_vals[cone] = dict()
#     for cross in cross_variables:
#     	# print cross
#     	if cross not in undefinedCrosses[cone]:
# 	   # we pick the first no 'None' entry 
# 	   dict_cross_vals[cone][SR(cross)]=[k for k in val_cross_to_yoshidas[cone][cross] if k !=None][0]
# 	   # dict_cross_vals[cone][-1*SR(cross)] = dict_cross_vals[cone][SR(cross)]
# 	else:
# 		dict_cross_vals[cone][SR(cross)] = None
# 		# dict_cross_vals[cone][-1*SR(cross)] = None

# # We record the data:
# save(dict_cross_vals, '../../TropicalConvexHulls/Scripts/Input/dict_cross_valuations_per_cone.sobj')

# We save the data as a plain text file:
# f = file('../../Yoshida/Output/dict_cross_valuations_per_cone.txt','w')
# f.write(str(dict_cross_vals))
# f.close()


# # We create dictionaries of true Cross valuations for each cone.
# for cone in AllConeNames:
#     save(dict_cross_vals[cone], '../../Yoshida/Scripts/Input/valCrosses_'+str(cone)+'.sobj')

# We load the dictionary of valuations of Crosses on each cone in the Naruki fan
dict_cross_vals=load('../../TropicalConvexHulls/Scripts/Input/dict_cross_valuations_per_cone.sobj')


###########################################
# Expected valuations for Cross functions #
###########################################

# We compute the expected valuation of the Cross functions for those that are undecided. Out of the four possible valuations, we pick the largest one.

# dict_exp_cross_vals = dict()

# for cone in AllConeNames:
#     print cone
#     dict_exp_cross_vals[cone] = dict()
#     for cross in cross_variables:
#     	# print cross
# 	# if the valuations are known, the expected agrees with the known valuations.
# 	if dict_cross_vals[cone][SR(cross)] != None:	   
# 	   dict_exp_cross_vals[cone][SR(cross)] = dict_cross_vals[cone][SR(cross)]
# 	else:
# 		# pick the largest of the possible valuations
# 		coneP = load('../../ComputeLines/Scripts/Input/' + str(cone)+'Cell.sobj')
# 		allValuesCross = [yoshida_to_trop_yoshida(q,coneP) for q in cross_to_yoshidas[SR(cross)]]
# 		print allValuesCross
# 		interior_point =sum([vector(r) for r in coneP.rays()])
# 		values_baricenter = [Yoshidas.index(allValuesCross[l]) for l in range(0,4)]
# 		max_value = max(values_baricenter)
# 		max_index = [l for l in range(0,4) if values_baricenter[l]==max_value][0]
# 		dict_exp_cross_vals[cone][SR(cross)] = allValuesCross[max_index]
#     	print 'Done with ' + str(cross)	 


# # We record the data:
# save(dict_exp_cross_vals, '../../TropicalConvexHulls/Scripts/Input/dict_exp_cross_valuations_per_cone.sobj')

# # We save the data as a plain text file:
# f = file('../../Yoshida/Output/dict_exp_cross_valuations_per_cone.txt','w')
# f.write(str(dict_exp_cross_vals))
# f.close()

# We load the dictionary of expected valuation of Cross functions:
dict_exp_cross_vals=load('../../TropicalConvexHulls/Scripts/Input/dict_exp_cross_valuations_per_cone.sobj')




#################################################################
# Expected vs True Valuations for Cross functions from examples #
#################################################################

# We use the computed examples of parameters d in Q[t] giving Yoshidas whose valuation is the center of mass of each cone in the Naruki fan to check which Cross functions have the expected valuations. We will use them to label the leaves of the tropical lines associated to each extremal curve.

# The examples for the parameters ds are constructed in "ConstructingDsForTheBergmanFan.sage"

# The computation for each cone takes about 9 minutes on a 2.4 GHz Intel(R) Core 2 Duo  with 3MB cache and 2GB RAM.

########################
# Relevant Naruki rays #
########################

# We start by loading the vertices of the Bergman fan:
vBerg=load("../../BergmanFan/Output/CanonicalCoordsOfVerticesBE6.sobj")

# We compute the relevant 4 rays in the Bergman fan yielding the generators of the 5 relevant rays in the Naruki fan.
v0=vector(vBerg[0])
v1=vector(vBerg[1])
v4=vector(vBerg[4])
v158=vector(vBerg[158])
v36=vector(vBerg[36])

va2 = v0 +v1
va3 = v0 + v1 +v4
va4 = v0 + v1 +v158
va = v0
vb = v36

###############################################################
# Produce valid parameters ds for each cone in the Naruki fan #
###############################################################

# We load the functions and the example generators for every vector in 'aa2a3a4' and 'aa2a3b' generated in "ConstructingDsForTheBergmanFan.sage".

load("ConstructingDsForTheBergmanFan.sage")

# The following functions unifies the constructions from "ConstructingDsForTheBergmanFan.sage" to depend on the cone in the Naruki fan we are interested in.

def buildRoots(cone,a,b,c,d):
    if 'b' in cone:
	   return buildRootsFor_aa2a3b(a,b,c,d)
    else:
	   return buildRootsFor_aa2a3a4(a,b,c,d)

def checkLowerFaces(cone, varus,D, a,b,c,d):
    if 'b' in cone:
	   return checkLowerFacesaa2a3b(varus, D, a, b, c, d)
    else:
	   return checkLowerFacesaa2a3a4(varus, D, a, b, c, d)
  

  
# We start with the two maximal cones, and then go to lower faces. As our representative, we choose the center of mass of each cell, in agreement with your choice for "../../ComputeLines/Scripts/ComputationsOfLinesForAllCones.sage".


# NOTE: The values of Yoshidas (rational functions in Q[t]) have been precomputed for choices of (a,b,c,d) that are 0/1 vectors giving the baricenter of each cone. If we wish to change this choice, we simply comment the line
# valYoshidas = load("Input/valYoshidas_aa2a3a4.sobj")
# and uncomment the line
# valYoshidas =  substituteValdsInYField(vald,Yoshidas)


def computeValsYoshidasNarukiVector_aa2a3a4(a,b,c,d):
    w = YM*vector(a*va + b*va2 + c*va3 + d*va4)
    valuationsYoshidas = {SR(Yoshidas[i]): QQ(w[i]) for i in range(0,len(w))}
    # print valuationsYoshidas
    return (w,valuationsYoshidas)

def computeValsYoshidasNarukiVector_aa2a3b(a,b,c,d):
    w = YM*vector(a*va + b*va2 + c*va3 + d*vb)
    valuationsYoshidas = {SR(Yoshidas[i]): QQ(w[i]) for i in range(0,len(w))}
    # print valuationsYoshidas
    return (w,valuationsYoshidas)
    

# We merge these two into a single function, depending on the cone and the scalars.
def computeValsYoshidasNarukiVector(cone,a,b,c,d):
    if 'b' in cone:
	   (w,valuationsYoshidas) = computeValsYoshidasNarukiVector_aa2a3b(a,b,c,d)
    else:
	(w,valuationsYoshidas) = computeValsYoshidasNarukiVector_aa2a3a4(a,b,c,d)
    return (w,valuationsYoshidas)



######################################################
# Compute valuations for Rational functions in QQ[t] #
######################################################

# The following function calculations the valuation of polynomials in Qt.

def valnsInQt(x):
    if x !=0:
       xcoeff = Qt(expand(x)).coefficients(sparse=False)
       a = min([k for k in range(0,len(xcoeff)) if xcoeff[k]!=0])
    else:
	a = +Infinity
    return a	




#######################################
# Computing expected and None entries #
#######################################

# In order to determine the differences between true and expected valuations of Cross functions from concrete examples, we need only consider functions within each cone with true valuations with values 'None'. We use the dictionary of expected valuations and specialize to dictionary of valuations for a given example.


# We load the dictionary of valuations of Crosses on each cone in the Naruki fan
dict_cross_vals=load('../../TropicalConvexHulls/Scripts/Input/dict_cross_valuations_per_cone.sobj')


# We load the dictionary of expected valuation of Cross functions:
dict_exp_cross_vals=load('../../TropicalConvexHulls/Scripts/Input/dict_exp_cross_valuations_per_cone.sobj')


# Input: cross is a Cross function, "cone" is the name of one of the Naruki cones, a,b,c,d are scalars
# Output: a tuple of expected and true valuation for the given cross function with respect to parameters associated to a, b, c, d. First entry is the expected valuation specialized and the second is the true valuation from choices of parameters d1,...,d6.

def compare_exp_and_true_cross_vals_from_examples(cross,cone, a,b,c,d, dict_expected, dict_true):
    if cross not in cross_variables:
       print 'Input is not a cross variable!'
       return 'None'
    if dict_true[cone][SR(cross)] != None:
        if dict_expected[cone][SR(cross)] != dict_true[cone][SR(cross)]:
	   print 'we have a problem!'
	   return False
	else:
		true_val = SR(dict_true[cone][SR(cross)])
		print true_val
		i = Yoshidas.index(true_val)
		# print i
		neweq = computeValsYoshidasNarukiVector(cone,a,b,c,d)[0][i]
		return tuple([True, neweq, neweq])
    else:
	exp_val = SR(dict_expected[cone][SR(cross)])
	print exp_val
	i = Yoshidas.index(exp_val)
	g = cross_to_yoshidas[SR(cross)][0]
	print g
	(varus,D) = buildRoots(cone,a,b,c,d)
	vald = computeVald(D,varus)
	dictYInD = substituteValdsInYField(vald,Yoshidas)
	crossInD = valnsInQt(SR(g.substitute(dictYInD)))
	# print vald
	print checkLowerFaces(cone, varus,D, a,b,c,d)	
	return (vald,computeValsYoshidasNarukiVector(cone,a,b,c,d)[0][i],crossInD) #valnsInQt(substituteValdsInYField(vald,Yoshidas)[exp_val]))


# Next, we generate the dictionary of expected and true valuations from examples for those Cross functions where valuations cannot be determined on the corresponding cone.

param = dict()
param['aa2a3a4'] = (1,1,1,1)
param['aa2a3b'] =  (1,1,1,1)
param['aa2a3'] = (1,1,1,0)
param['aa2a4'] = (1,1,0,1)
param['aa3a4'] = (1,0,1,1)
param['a2a3a4'] = (0,1,1,1)
param['aa2b'] = (1,1,0,1)
param['aa3b'] =(1,0,1,1)
param['a2a3b'] = (0,1,1,1)
param['aa2'] = (1,1,0,0)
param['aa3'] = (1,0,1,0)
param['aa4'] = (1,0,0,1)
param['a2a3'] = (0,1,1,0)
param['a2a4'] = (0,1,0,1)
param['a3a4'] = (0,0,1,1)
param['ab']= (1,0,0,1)
param['a2b'] =  (0,1,0,1)
param['a3b'] = (0,0,1,1)
param['a'] = (1,0,0,0)
param['a2'] = (0,1,0,0)
param['a3'] = (0,0,1,0)
param['a4'] = (0,0,0,1)
param['b'] = (0,0,0,1)

dict_expected = dict_exp_cross_vals
dict_true = dict_cross_vals

exp_true_vals_fromExamples = dict()

# for cone in AllConeNames:
#     print str(cone)
#     exp_true_vals_fromExamples[cone] =dict()
#     toDecide = undefinedCrosses[cone]
#     print toDecide
#     for cross in toDecide:
#     	print str(cross)
#     	(a,b,c,d) = param[cone]
#     	exp_true_vals_fromExamples[cone][SR(cross)] = compare_exp_and_true_cross_vals_from_examples(cross,cone, a,b,c,d, dict_expected, dict_true)
#     save(exp_true_vals_fromExamples, '../../Yoshida/Output/exp_true_vals_fromExamplesv2.sobj')
#     print str(exp_true_vals_fromExamples)
#     print ''
#     print ''


# Sample output:
# exp_true_vals_fromExamples['aa2a3a4']
# {Cross38: ({d4: -t^5 + 2*t^4 + t + 14,
#    d3: -t^5 + 2*t^4 - 6,
#    d6: -t^5 + 2*t^4 + 17*t + 3,
#    d5: 57*t^8 - t^5 + 2*t^4 - 57*t + 14,
#    d2: 2*t^5 + 2*t^4 + 3,
#    d1: -t^5 + 2*t^4 + 3},
#   1,
#   1),
#  Cross37: ({d4: -t^5 + 2*t^4 + t + 14,
#    d3: -t^5 + 2*t^4 - 6,
#    d6: -t^5 + 2*t^4 + 17*t + 3,
#    d5: 57*t^8 - t^5 + 2*t^4 - 57*t + 14,
#    d2: 2*t^5 + 2*t^4 + 3,
#    d1: -t^5 + 2*t^4 + 3},
#   1,
#   1),
#  Cross36: ({d4: -t^5 + 2*t^4 + t + 14,
#    d3: -t^5 + 2*t^4 - 6,
#    d6: -t^5 + 2*t^4 + 17*t + 3,
#    d5: 57*t^8 - t^5 + 2*t^4 - 57*t + 14,
#    d2: 2*t^5 + 2*t^4 + 3,
#    d1: -t^5 + 2*t^4 + 3},
#   10,
#   11)}

# We load the precomputed data:
exp_true_vals_fromExamples = load('../../Yoshida/Output/exp_true_vals_fromExamplesv2.sobj')


# # We record the output:
# {'aa2a3': {Cross38: ({d4: -t^3 + 2*t^2 + t + 14, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 20, d5: 57*t^6 - t^3 + 2*t^2 - 57*t + 14, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 0, 0), Cross37: ({d4: -t^3 + 2*t^2 + t + 14, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 20, d5: 57*t^6 - t^3 + 2*t^2 - 57*t + 14, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 0, 0), Cross36: ({d4: -t^3 + 2*t^2 + t + 14, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 20, d5: 57*t^6 - t^3 + 2*t^2 - 57*t + 14, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 6, 6)}, 'ab': {Cross72: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 0, 1), Cross5: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 0, 1), Cross76: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 2, 2), Cross75: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 0, 1), Cross68: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 1, 1), Cross125: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 2, 2), Cross4: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 1, 1), Cross3: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 2, 2), Cross67: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 1, 1), Cross53: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 0, 1), Cross124: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 1, 1), Cross123: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 1, 1), Cross74: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 2, 2), Cross73: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 1, 1), Cross77: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 0, 1), Cross52: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 2, 2), Cross51: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 1, 1), Cross66: ({d4: -t^2 + 3*t + 14, d3: -t^2 - 6*t + 2, d6: -t^2 + 3*t + 19, d5: 57*t^4 - t^2 + 3*t - 44, d2: 2*t^2 + 3*t + 2, d1: -t^2 + 3*t + 2}, 2, 2)}, 'aa2a4': {Cross37: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 1, 1), Cross15: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 1, 2), Cross29: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 7, 7), Cross36: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 7, 8), Cross17: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 3, 7), Cross38: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 1, 1), Cross28: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 2, 2), Cross27: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 1, 2), Cross16: ({d4: -t^4 + 2*t^3 + 15, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 43, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 1, 2)}, 'aa3a4': {Cross37: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 1, 1), Cross125: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 7, 7), Cross36: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 8, 9), Cross38: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 1, 1), Cross53: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 1, 3), Cross124: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 3, 3), Cross123: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 3, 3), Cross52: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 7, 7), Cross51: ({d4: -t^4 + 2*t^3 + t + 14, d3: -t^4 + 2*t^3 - 6, d6: -t^4 + 2*t^3 + 17*t + 3, d5: 57*t^6 - t^4 + 2*t^3 - 57*t + 14, d2: 2*t^4 + 2*t^3 + 3, d1: -t^4 + 2*t^3 + 3}, 3, 3)}, 'a3b': {Cross40: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 0, 2), Cross107: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 0, 2), Cross122: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 0, 2), Cross121: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 3, 3), Cross79: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross78: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 3, 3), Cross125: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 3, 3), Cross41: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross120: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross39: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 3), Cross53: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 0, 2), Cross124: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross123: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross106: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross105: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 3, 3), Cross80: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 0, 2), Cross52: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 3, 3), Cross51: ({d4: -t^2 + 6*t + 11, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 - 52*t + 11, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2)}, 'aa4': {Cross37: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 1, 1), Cross100: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 5, 5), Cross96: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 1, 2), Cross15: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 1, 2), Cross125: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 5, 5), Cross36: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 5, 6), Cross99: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 2, 2), Cross29: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 5, 5), Cross17: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 2, 5), Cross38: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 1, 1), Cross53: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 1, 2), Cross28: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 2, 2), Cross27: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 1, 2), Cross16: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 1, 2), Cross124: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 2, 2), Cross101: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 2, 2), Cross123: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 2, 2), Cross52: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 5, 5), Cross51: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 2, 2), Cross98: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 2, 2), Cross97: ({d4: -t^3 + 2*t^2 + 15, d3: -t^3 + 2*t^2 - 6, d6: -t^3 + 2*t^2 + 17*t + 3, d5: 57*t^4 - t^3 + 2*t^2 - 43, d2: 2*t^3 + 2*t^2 + 3, d1: -t^3 + 2*t^2 + 3}, 5, 5)}, 'aa2a3a4': {Cross38: ({d4: -t^5 + 2*t^4 + t + 14, d3: -t^5 + 2*t^4 - 6, d6: -t^5 + 2*t^4 + 17*t + 3, d5: 57*t^8 - t^5 + 2*t^4 - 57*t + 14, d2: 2*t^5 + 2*t^4 + 3, d1: -t^5 + 2*t^4 + 3}, 1, 1), Cross37: ({d4: -t^5 + 2*t^4 + t + 14, d3: -t^5 + 2*t^4 - 6, d6: -t^5 + 2*t^4 + 17*t + 3, d5: 57*t^8 - t^5 + 2*t^4 - 57*t + 14, d2: 2*t^5 + 2*t^4 + 3, d1: -t^5 + 2*t^4 + 3}, 1, 1), Cross36: ({d4: -t^5 + 2*t^4 + t + 14, d3: -t^5 + 2*t^4 - 6, d6: -t^5 + 2*t^4 + 17*t + 3, d5: 57*t^8 - t^5 + 2*t^4 - 57*t + 14, d2: 2*t^5 + 2*t^4 + 3, d1: -t^5 + 2*t^4 + 3}, 10, 11)}, 'aa3': {Cross37: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 0), Cross125: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 3, 3), Cross36: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 4, 4), Cross38: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 0), Cross53: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 1), Cross124: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 1, 1), Cross123: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 1, 1), Cross52: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 3, 3), Cross51: ({d4: -t^2 + 3*t + 14, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 - 55*t + 14, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 1, 1)}, 'aa2': {Cross37: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 0), Cross15: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 0), Cross29: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 3, 3), Cross36: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 3, 3), Cross17: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 1, 3), Cross38: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 0), Cross28: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 0), Cross27: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 0), Cross16: ({d4: -t^2 + 2*t + 15, d3: -t^2 + 2*t - 6, d6: -t^2 + 2*t + 20, d5: 57*t^4 - t^2 + 2*t - 43, d2: 2*t^2 + 2*t + 3, d1: -t^2 + 2*t + 3}, 0, 0)}, 'aa3b': {Cross125: ({d4: -t^3 + 6*t + 11, d3: -t^3 - 4*t, d6: -t^3 + 5*t + 17, d5: 57*t^6 - t^3 - 52*t + 11, d2: 2*t^3 + 5*t, d1: -t^3 + 5*t}, 4, 4), Cross53: ({d4: -t^3 + 6*t + 11, d3: -t^3 - 4*t, d6: -t^3 + 5*t + 17, d5: 57*t^6 - t^3 - 52*t + 11, d2: 2*t^3 + 5*t, d1: -t^3 + 5*t}, 0, 2), Cross124: ({d4: -t^3 + 6*t + 11, d3: -t^3 - 4*t, d6: -t^3 + 5*t + 17, d5: 57*t^6 - t^3 - 52*t + 11, d2: 2*t^3 + 5*t, d1: -t^3 + 5*t}, 2, 2), Cross123: ({d4: -t^3 + 6*t + 11, d3: -t^3 - 4*t, d6: -t^3 + 5*t + 17, d5: 57*t^6 - t^3 - 52*t + 11, d2: 2*t^3 + 5*t, d1: -t^3 + 5*t}, 2, 2), Cross52: ({d4: -t^3 + 6*t + 11, d3: -t^3 - 4*t, d6: -t^3 + 5*t + 17, d5: 57*t^6 - t^3 - 52*t + 11, d2: 2*t^3 + 5*t, d1: -t^3 + 5*t}, 4, 4), Cross51: ({d4: -t^3 + 6*t + 11, d3: -t^3 - 4*t, d6: -t^3 + 5*t + 17, d5: 57*t^6 - t^3 - 52*t + 11, d2: 2*t^3 + 5*t, d1: -t^3 + 5*t}, 2, 2)}, 'a2a3': {Cross37: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 0, 0), Cross107: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 0, 2), Cross79: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross78: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 3, 3), Cross36: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 5, 5), Cross38: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 0, 0), Cross106: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross105: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 3, 3), Cross80: ({d4: t^2 + t + 14, d3: t^2 - 6, d6: t^2 + 20, d5: 57*t^4 + t^2 - 57*t + 14, d2: 4*t^2 + 3, d1: t^2 + 3}, 0, 2)}, 'a3a4': {Cross40: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 3), Cross37: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 1), Cross107: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 3), Cross122: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 3), Cross121: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 6, 6), Cross79: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross78: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 6, 6), Cross125: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 6, 6), Cross36: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 7, 8), Cross120: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross39: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 6), Cross38: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 1), Cross53: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 3), Cross124: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross123: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross106: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross41: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross80: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 3), Cross52: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 6, 6), Cross51: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross105: ({d4: t^3 + t + 14, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 57*t + 14, d2: 4*t^3 + 3, d1: t^3 + 3}, 6, 6)}, 'a2a4': {Cross37: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 1), Cross107: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 3), Cross15: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 2), Cross78: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 5, 5), Cross29: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 6, 6), Cross36: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 6, 7), Cross17: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 6), Cross80: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 3), Cross38: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 1), Cross28: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 2, 2), Cross27: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 2), Cross106: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross79: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 3, 3), Cross16: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 1, 2), Cross105: ({d4: t^3 + 15, d3: t^3 - 6, d6: t^3 + 17*t + 3, d5: 57*t^4 + t^3 - 43, d2: 4*t^3 + 3, d1: t^3 + 3}, 5, 5)}, 'a3': {Cross40: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 0, 1), Cross37: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 0, 0), Cross107: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 0, 1), Cross122: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 0, 1), Cross121: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 2, 2), Cross79: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross78: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 2, 2), Cross125: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 2, 2), Cross36: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 3, 3), Cross41: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross120: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross39: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 1, 2), Cross38: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 0, 0), Cross53: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 0, 1), Cross124: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross123: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross106: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross105: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 2, 2), Cross80: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 0, 1), Cross52: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 2, 2), Cross51: ({d4: 2*t + 14, d3: t - 6, d6: t + 20, d5: 57*t^2 - 56*t + 14, d2: 4*t + 3, d1: t + 3}, 1, 1)}, 'a2': {Cross37: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 0, 0), Cross107: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 0, 1), Cross15: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 0, 0), Cross78: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross29: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 2, 2), Cross36: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 2, 2), Cross17: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 1, 2), Cross16: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 0, 0), Cross38: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 0, 0), Cross79: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross28: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 0, 0), Cross27: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 0, 0), Cross106: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross105: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 1, 1), Cross80: ({d4: t + 15, d3: t - 6, d6: t + 20, d5: 57*t^2 + t - 43, d2: 4*t + 3, d1: t + 3}, 0, 1)}, 'a4': {Cross108: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 4), Cross47: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross46: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross36: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 5), Cross80: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross109: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross98: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross37: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 1), Cross15: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross120: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross39: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 4), Cross28: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross106: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross105: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4), Cross122: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross121: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4), Cross110: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross29: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4), Cross100: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4), Cross99: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross38: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 1), Cross124: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross27: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross16: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross45: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4), Cross52: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4), Cross51: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross40: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross101: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross107: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross96: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross79: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross78: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4), Cross125: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4), Cross17: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 4), Cross53: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 1, 2), Cross123: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross41: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 2, 2), Cross97: ({d4: t^2 + 15, d3: t^2 - 6, d6: t^2 + 17*t + 3, d5: 58*t^2 - 43, d2: 4*t^2 + 3, d1: t^2 + 3}, 4, 4)}, 'a': {Cross75: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross58: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross57: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross36: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross82: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross102: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross91: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross98: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross104: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross37: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross76: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross15: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross4: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross3: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross81: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross28: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross59: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross77: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross83: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross66: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross72: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross5: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross29: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross100: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross99: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross38: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross124: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross27: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross74: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross73: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross16: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross52: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross51: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross101: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross90: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross96: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross125: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross68: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross67: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross17: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 1), Cross103: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross53: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross92: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1), Cross123: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 0, 0), Cross97: ({d4: -t + 17, d3: -t - 4, d6: -t + 22, d5: 57*t^2 - t - 41, d2: 2*t + 5, d1: -t + 5}, 1, 1)}, 'b': {Cross86: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross69: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross75: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross71: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross60: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross10: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross9: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross80: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross116: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross115: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross54: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross76: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross14: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross61: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross4: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross3: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross120: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross39: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross70: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross106: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross105: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross77: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross84: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross66: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross72: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross55: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross5: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross11: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross122: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross121: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross0: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross85: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross124: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross74: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross73: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross62: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross52: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross51: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross1: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross40: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross12: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross107: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross79: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross78: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross125: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross68: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross67: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross114: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross56: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross53: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 0, 1), Cross123: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross41: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross13: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1), Cross2: ({d4: 2*t + 14, d3: -7*t + 2, d6: 2*t + 19, d5: 57*t^2 + 2*t - 44, d2: 5*t + 2, d1: 2*t + 2}, 1, 1)}, 'aa2a3b': {}, 'a2b': {Cross107: ({d4: -t^2 + 5*t + 12, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 + 5*t - 46, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 0, 2), Cross79: ({d4: -t^2 + 5*t + 12, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 + 5*t - 46, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross78: ({d4: -t^2 + 5*t + 12, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 + 5*t - 46, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross106: ({d4: -t^2 + 5*t + 12, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 + 5*t - 46, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross105: ({d4: -t^2 + 5*t + 12, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 + 5*t - 46, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 2, 2), Cross80: ({d4: -t^2 + 5*t + 12, d3: -t^2 - 4*t, d6: -t^2 + 5*t + 17, d5: 57*t^4 - t^2 + 5*t - 46, d2: 2*t^2 + 5*t, d1: -t^2 + 5*t}, 0, 2)}, 'a2a3b': {Cross107: ({d4: -t^3 + 2*t^2 + 4*t + 11, d3: -t^3 + 2*t^2 - 6*t, d6: -t^3 + 2*t^2 + 3*t + 17, d5: 57*t^6 - t^3 + 2*t^2 - 54*t + 11, d2: 2*t^3 + 2*t^2 + 3*t, d1: -t^3 + 2*t^2 + 3*t}, 0, 3), Cross79: ({d4: -t^3 + 2*t^2 + 4*t + 11, d3: -t^3 + 2*t^2 - 6*t, d6: -t^3 + 2*t^2 + 3*t + 17, d5: 57*t^6 - t^3 + 2*t^2 - 54*t + 11, d2: 2*t^3 + 2*t^2 + 3*t, d1: -t^3 + 2*t^2 + 3*t}, 3, 3), Cross78: ({d4: -t^3 + 2*t^2 + 4*t + 11, d3: -t^3 + 2*t^2 - 6*t, d6: -t^3 + 2*t^2 + 3*t + 17, d5: 57*t^6 - t^3 + 2*t^2 - 54*t + 11, d2: 2*t^3 + 2*t^2 + 3*t, d1: -t^3 + 2*t^2 + 3*t}, 4, 4), Cross106: ({d4: -t^3 + 2*t^2 + 4*t + 11, d3: -t^3 + 2*t^2 - 6*t, d6: -t^3 + 2*t^2 + 3*t + 17, d5: 57*t^6 - t^3 + 2*t^2 - 54*t + 11, d2: 2*t^3 + 2*t^2 + 3*t, d1: -t^3 + 2*t^2 + 3*t}, 3, 3), Cross105: ({d4: -t^3 + 2*t^2 + 4*t + 11, d3: -t^3 + 2*t^2 - 6*t, d6: -t^3 + 2*t^2 + 3*t + 17, d5: 57*t^6 - t^3 + 2*t^2 - 54*t + 11, d2: 2*t^3 + 2*t^2 + 3*t, d1: -t^3 + 2*t^2 + 3*t}, 4, 4), Cross80: ({d4: -t^3 + 2*t^2 + 4*t + 11, d3: -t^3 + 2*t^2 - 6*t, d6: -t^3 + 2*t^2 + 3*t + 17, d5: 57*t^6 - t^3 + 2*t^2 - 54*t + 11, d2: 2*t^3 + 2*t^2 + 3*t, d1: -t^3 + 2*t^2 + 3*t}, 0, 3)}, 'a2a3a4': {Cross37: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 1, 1), Cross107: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 1, 4), Cross79: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 4, 4), Cross78: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 7, 7), Cross36: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 9, 10), Cross38: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 1, 1), Cross106: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 4, 4), Cross105: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 7, 7), Cross80: ({d4: t^4 + t + 14, d3: t^4 - 6, d6: t^4 + 17*t + 3, d5: 57*t^6 + t^4 - 57*t + 14, d2: 4*t^4 + 3, d1: t^4 + 3}, 1, 4)}, 'aa2b': {}}



###################################
# Analysis of expected valuations #
###################################

# Next, we analyze the data. We look for Cross functions that achieve unexpected valuation:
unexpectedValuations = dict()
for coneName in AllConeNames:
    unexpectedValuations[coneName] = [cross for cross in exp_true_vals_fromExamples[coneName].keys() if exp_true_vals_fromExamples[coneName][cross][1] != exp_true_vals_fromExamples[coneName][cross][2]]

# # We record the data:
# for coneName in AllConeNames:
#     print coneName
#     print unexpectedValuations[coneName]

# aa2a3a4
# [Cross36]
# aa2a3b
# []
# aa2a3
# []
# aa2a4
# [Cross15, Cross36, Cross17, Cross27, Cross16]
# aa3a4
# [Cross36, Cross53]
# a2a3a4
# [Cross107, Cross36, Cross80]
# aa2b
# []
# aa3b
# [Cross53]
# a2a3b
# [Cross107, Cross80]
# aa2
# [Cross17]
# aa3
# [Cross53]
# aa4
# [Cross96, Cross15, Cross36, Cross17, Cross53, Cross27, Cross16]
# a2a3
# [Cross107, Cross80]
# a2a4
# [Cross107, Cross15, Cross36, Cross17, Cross80, Cross27, Cross16]
# a3a4
# [Cross40, Cross107, Cross122, Cross36, Cross39, Cross53, Cross80]
# ab
# [Cross72, Cross5, Cross75, Cross53, Cross77]
# a2b
# [Cross107, Cross80]
# a3b
# [Cross40, Cross107, Cross122, Cross39, Cross53, Cross80]
# a
# [Cross17]
# a2
# [Cross107, Cross17, Cross80]
# a3
# [Cross40, Cross107, Cross122, Cross39, Cross53, Cross80]
# a4
# [Cross108, Cross46, Cross36, Cross80, Cross109, Cross15, Cross39, Cross122, Cross27, Cross16, Cross40, Cross107, Cross96, Cross17, Cross53]
# b
# [Cross75, Cross10, Cross80, Cross77, Cross72, Cross5, Cross122, Cross40, Cross107, Cross53]

# # We collect all the unexpected valuations:
BadChoices = list(set(combineLists(unexpectedValuations.values())))
# len(BadChoices)
# 20
# badIndices = sorted([k for k in range(0,len(cross_variables)) if cross_variables[k] in BadChoices])
# print badIndices
# [5, 10, 15, 16, 17, 27, 36, 39, 40, 46, 53, 72, 75, 77, 80, 96, 107, 108, 109, 122]
# print [cross_variables[k] for k in badIndices]
# [Cross5, Cross10, Cross15, Cross16, Cross17, Cross27, Cross36, Cross39, Cross40, Cross46, Cross53, Cross72, Cross75, Cross77, Cross80, Cross96, Cross107, Cross108, Cross109, Cross122]

# # We record all the coneNames where the BadChoices of Cross functions have unexpected valuations:
# conesForBadCrosses = dict()
# for cross in  [cross_variables[k] for k in badIndices]:
#     print cross
#     conesForBadCrosses[cross] = [coneName for coneName in AllConeNames if cross in unexpectedValuations[coneName]]
#     print conesForBadCrosses[cross]
# Cross5
# ['ab', 'b']
# Cross10
# ['b']
# Cross15
# ['aa2a4', 'aa4', 'a2a4', 'a4']
# Cross16
# ['aa2a4', 'aa4', 'a2a4', 'a4']
# Cross17
# ['aa2a4', 'aa2', 'aa4', 'a2a4', 'a', 'a2', 'a4']
# Cross27
# ['aa2a4', 'aa4', 'a2a4', 'a4']
# Cross36
# ['aa2a3a4', 'aa2a4', 'aa3a4', 'a2a3a4', 'aa4', 'a2a4', 'a3a4', 'a4']
# Cross39
# ['a3a4', 'a3b', 'a3', 'a4']
# Cross40
# ['a3a4', 'a3b', 'a3', 'a4', 'b']
# Cross46
# ['a4']
# Cross53
# ['aa3a4', 'aa3b', 'aa3', 'aa4', 'a3a4', 'ab', 'a3b', 'a3', 'a4', 'b']
# Cross72
# ['ab', 'b']
# Cross75
# ['ab', 'b']
# Cross77
# ['ab', 'b']
# Cross80
# ['a2a3a4', 'a2a3b', 'a2a3', 'a2a4', 'a3a4', 'a2b', 'a3b', 'a2', 'a3', 'a4', 'b']
# Cross96
# ['aa4', 'a4']
# Cross107
# ['a2a3a4', 'a2a3b', 'a2a3', 'a2a4', 'a3a4', 'a2b', 'a3b', 'a2', 'a3', 'a4', 'b']
# Cross108
# ['a4']
# Cross109
# ['a4']
# Cross122
# ['a3a4', 'a3b', 'a3', 'a4', 'b']


# We analyze expected vs true valuation for the unique triple where all three crosses have unexpected valuation for our choice of parameteres d1,...,d6:

# print {cone: exp_true_vals_fromExamples[cone][Cross15][1:3] for cone in conesForBadCrosses[Cross15]}
# {'a2a4': (1, 2), 'a4': (1, 2), 'aa2a4': (1, 2), 'aa4': (1, 2)}

# print {cone: exp_true_vals_fromExamples[cone][Cross16][1:3] for cone in conesForBadCrosses[Cross16]}
# {'a2a4': (1, 2), 'a4': (1, 2), 'aa2a4': (1, 2), 'aa4': (1, 2)}

# print {cone: exp_true_vals_fromExamples[cone][Cross17][1:3] for cone in conesForBadCrosses[Cross17]}
# {'a': (0, 1), 'aa2a4': (3, 7), 'a2a4': (3, 6), 'aa4': (2, 5), 'a2': (1, 2), 'a4': (2, 4), 'aa2': (1, 3)}


########################################################################
# Computing 45 classes of Cross functions with expected valuation a.e. #
########################################################################

# By construction, we know that the cross functions associated to the same Eckardt quintic have related valuations. More precisely, we show that ratios of pairs of triples are monomials in cube-roots of Yoshida functions.Therefore, the valuation of these ratios is uniquely determined by the valuations of the Yoshida functions.

# We construct a dictionary that leaves the Cross[3*k] as the "independent variables" and rewrites the valuations of the remaining Cross functions in terms of these Cross functions and linear expressions in Yoshidas with coefficients in 1/3*Z.

#########################################
# Loading Auxilliary functions and data #
#########################################

os.chdir('../../Yoshida/Scripts')
load('Yoshida.sage')

os.chdir('../../ComputeLines/Scripts')

DField = FractionField(PolynomialRing(QQ, ds))
YtoD = YField.hom([DField(y_to_d[y]) for y in y_to_d.keys()])

# The following is a basis of 16 vectors giving independent columns in the Yoshida matrix:
uyvs_basis = load("Input/expVectorsZBasisIndex3.sobj")

os.chdir('../../Yoshida/Scripts')
# os.chdir('../../TropicalConvexHulls/Scripts')


###########################################################################
# Functions for writing ratios of Cross functions as monomial in Yoshidas #
###########################################################################

# The next function expresses a product of positive roots as a Laurent monomial in the 16 exponent vectors of a Yoshida Q-basis, whenever possible. The result is correct up to sign, and we check the sign on a different function

def constructMonomialFromProduct(expressionInRoots, roots, basisOfYoshidas):
     v_exp = exponent_vector(expressionInRoots, roots)
     # print v_exp
     soln = matrix(basisOfYoshidas).transpose() \ vector(v_exp)
     return soln

# The next function finds the correct sign in the monomial expression of Yoshidas. We multiply the input vertex by 3 to avoid cube roots of Yoshidas.

labels = [5]+[17..31]

def findConstant(ratioInY):
    factorInYs = 1
    tripleV = 3*vector(ratioInY)	
    for k in range(0,16):
    	factorInYs = factorInYs*(Yoshidas[labels[k]].factor())^tripleV[k]
    return factorInYs


# The following function takes a two crosses and exponents with 'None' and returns a factorization of their weighted ratio into product of Yoshidas whenever possible.
# cr0 = (cross variable, integer)
# cr1 = (cross variable, integer)
# y_to_d = dictionary that converts Yoshidas into polynomials in ds
# Qd = polynomial ring in ds
# roots = list of 36 positive roots of E6
# basisOfYoshidas = 16 yoshida functions generating the column span of the Yoshida matrix (we will always use uyvs_basis)
# OUTPUT =  product of Yoshidas if the ratio of the cross variables (with exponents) can be written that way, or False if not.

def checkRatioCrosses(cr0,cr1,y_to_d,Qd,roots,basisOfYoshidas):
    test0 =  prod([cross_to_yoshidas[cr0[k][0]][0]^cr0[k][1] for k in range(0,le n(cr0))])
    test1 =  prod([cross_to_yoshidas[cr1[k][0]][0]^cr1[k][1] for k in range(0,len(cr1))])
    test = (test0/test1).substitute(y_to_d)
    testF = test.factor_list()
    # test =  (cross_to_yoshidas[cr0[0]][0]^cr0[1]/cross_to_yoshidas[cr1[0]][0]^cr1[1]).substitute(y_to_d)
    # testF = test.factor_list()
    if all([Qd(x[0]).total_degree() !=5 for x in testF]) == True:
       print 'we are good!'
       newTest =  constructMonomialFromProduct(DField(test), roots, basisOfYoshidas)
       return findConstant(newTest)
    else:
	print 'we are in trouble!'
	return False


#########################
# Create new dictionary #
#########################

# By design, our indexing of Cross functions is such that cross_variables[3*k], cross_variables[3*k+1] and cross_variables[3*k+2] are all associated to the same quintic. We would like to pick cross_variables[3*k] as our independent choice for all k=0,...,44, but due to the information of bad choices of Cross functions to modify this assignement of 45 relevant Cross functions. Our modified choices are:
# replace Cross27 by Cross28 (k=9)
# replace Cross36 by Cross37 (k=12)
# replace Cross39 by Cross41 (k=13)
# replace Cross72 by Cross73 (k=24)
# replace Cross75 by Cross76 (k=25)
# replace Cross96 by Cross97 (k=32)
# replace Cross108 by Cross110 (k=36)

def createPairsForRatioTest(cross_variables):
    allPairs = dict()
    for k in range(0,len(cross_variables)/3):
    	if k not in [9,12,13,24,25,32,36]:
    	   allPairs[k] = [([(cross_variables[3*k+1],1)],[(cross_variables[3*k],1)]), ([(cross_variables[3*k+2],1)],[(cross_variables[3*k],1)])]
	if k in [9, 12, 24, 25, 32]:
	   allPairs[k] = [([(cross_variables[3*k],1)],[(cross_variables[3*k+1],1)]), ([(cross_variables[3*k+2],1)],[(cross_variables[3*k+1],1)])]
	if k in [13, 36]:
	   allPairs[k] = [([(cross_variables[3*k],1)],[(cross_variables[3*k+2],1)]), ([(cross_variables[3*k+1],1)],[(cross_variables[3*k+2],1)])]
    return allPairs



# # We create the pairs to compute their ratios later:
# pairs_crosses = createPairsForRatioTest(cross_variables)

dictionary_cross_ratios = dict()
for k in pairs_crosses.keys():
    print k
    for x in pairs_crosses[k]:
    	print x[0][0]
        dictionary_cross_ratios[x[0][0][0]] = (x[1][0][0],checkRatioCrosses(x[0],x[1],y_to_d,Qd,roots,uyvs_basis)) 

# # We save the data:
# save(dictionary_cross_ratios,'../../TropicalConvexHulls/Scripts/Input/dict_cross_ratiosv2.sobj')

# # We record the data:
# dictionary_cross_ratios
# {Cross108: (Cross110,
#   Yoshida17^3*Yoshida18^2*Yoshida21*Yoshida26*Yoshida27*Yoshida30^2/(Yoshida20^2*Yoshida22*Yoshida28^3*Yoshida29^3*Yoshida5)),
#  Cross35: (Cross33,
#   Yoshida18*Yoshida20^2*Yoshida22*Yoshida24^3*Yoshida28^3/(Yoshida21*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2*Yoshida5^2)),
#  Cross82: (Cross81,
#   Yoshida21^2*Yoshida22*Yoshida25^3*Yoshida27^2*Yoshida29^3*Yoshida30/(Yoshida18^2*Yoshida20*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2)),
#  Cross113: (Cross111,
#   Yoshida22*Yoshida26^2*Yoshida31^3*Yoshida5^4/(Yoshida17^3*Yoshida18^2*Yoshida20*Yoshida21*Yoshida27*Yoshida30^2)),
#  Cross98: (Cross97,
#   Yoshida19^3*Yoshida21^2*Yoshida22*Yoshida29^3*Yoshida5/(Yoshida17^3*Yoshida18^2*Yoshida20*Yoshida26*Yoshida27*Yoshida30^2)),
#  Cross25: (Cross24,
#   Yoshida18*Yoshida22*Yoshida26^2*Yoshida5/(Yoshida20*Yoshida21*Yoshida27*Yoshida30^2)),
#  Cross11: (Cross9,
#   Yoshida18*Yoshida26^2*Yoshida30*Yoshida31^3*Yoshida5/(Yoshida20*Yoshida21*Yoshida22^2*Yoshida27*Yoshida29^3)),
#  Cross122: (Cross120,
#   Yoshida21^4*Yoshida23^3*Yoshida25^3*Yoshida27*Yoshida29^3*Yoshida30^2/(Yoshida18*Yoshida20^2*Yoshida22*Yoshida24^3*Yoshida26^2*Yoshida28^3*Yoshida31^3*Yoshida5)),
#  Cross52: (Cross51,
#   Yoshida17^3*Yoshida18^2*Yoshida20*Yoshida26*Yoshida27*Yoshida30^2/(Yoshida19^3*Yoshida21^2*Yoshida22*Yoshida31^3*Yoshida5)),
#  Cross34: (Cross33,
#   Yoshida20*Yoshida21*Yoshida26*Yoshida27/(Yoshida18*Yoshida22*Yoshida30*Yoshida5)),
#  Cross96: (Cross97,
#   Yoshida21*Yoshida26*Yoshida27*Yoshida5^2/(Yoshida18*Yoshida20^2*Yoshida22*Yoshida30)),
#  Cross125: (Cross123,
#   Yoshida18*Yoshida20^2*Yoshida22*Yoshida27^2*Yoshida30/(Yoshida21*Yoshida26*Yoshida31^3*Yoshida5^2)),
#  Cross20: (Cross18,
#   Yoshida17^3*Yoshida18*Yoshida21^2*Yoshida23^3*Yoshida25^3*Yoshida27^2*Yoshida30/(Yoshida20*Yoshida22^2*Yoshida24^3*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2)),
#  Cross8: (Cross6,
#   Yoshida18^2*Yoshida20*Yoshida22^2*Yoshida27/(Yoshida21^2*Yoshida26^2*Yoshida30*Yoshida5)),
#  Cross119: (Cross117,
#   Yoshida17^3*Yoshida18*Yoshida22*Yoshida27^2*Yoshida30/(Yoshida20*Yoshida21*Yoshida26*Yoshida29^3*Yoshida5^2)),
#  Cross64: (Cross63,
#   Yoshida18^2*Yoshida20^4*Yoshida22^2*Yoshida24^3*Yoshida28^6/(Yoshida17^3*Yoshida21^2*Yoshida25^3*Yoshida26^2*Yoshida27^2*Yoshida30^4*Yoshida5)),
#  Cross10: (Cross9,
#   Yoshida26^4*Yoshida27*Yoshida31^3*Yoshida5^2/(Yoshida18*Yoshida20^2*Yoshida21^2*Yoshida22*Yoshida29^3*Yoshida30)),
#  Cross109: (Cross110,
#   Yoshida17^3*Yoshida18*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5/(Yoshida20*Yoshida21*Yoshida22^2*Yoshida28^3*Yoshida29^3)),
#  Cross32: (Cross30,
#   Yoshida17^3*Yoshida18*Yoshida21^2*Yoshida27^2*Yoshida30^4/(Yoshida20*Yoshida22^2*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2)),
#  Cross4: (Cross3,
#   Yoshida19^3*Yoshida21*Yoshida22^2*Yoshida27/(Yoshida18*Yoshida20^2*Yoshida26^2*Yoshida30*Yoshida5)),
#  Cross95: (Cross93,
#   Yoshida19^3*Yoshida21*Yoshida22^2*Yoshida26*Yoshida31^3*Yoshida5^2/(Yoshida17^3*Yoshida18^4*Yoshida20^2*Yoshida27^2*Yoshida30)),
#  Cross77: (Cross76,
#   Yoshida21*Yoshida25^3*Yoshida26*Yoshida27*Yoshida29^3*Yoshida5^2/(Yoshida18*Yoshida20^2*Yoshida22*Yoshida24^3*Yoshida28^3*Yoshida30)),
#  Cross22: (Cross21,
#   Yoshida20*Yoshida21*Yoshida22^2*Yoshida24^3*Yoshida28^3/(Yoshida17^3*Yoshida18*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5)),
#  Cross121: (Cross120,
#   Yoshida18*Yoshida21^2*Yoshida22*Yoshida27^2*Yoshida30/(Yoshida20*Yoshida26*Yoshida31^3*Yoshida5^2)),
#  Cross7: (Cross6,
#   Yoshida18*Yoshida19^3*Yoshida20^2*Yoshida22^4*Yoshida24^3*Yoshida28^3/(Yoshida17^3*Yoshida21*Yoshida25^3*Yoshida26^4*Yoshida27*Yoshida30^2*Yoshida5^2)),
#  Cross16: (Cross15,
#   Yoshida20*Yoshida21*Yoshida22^2*Yoshida27*Yoshida29^3/(Yoshida17^3*Yoshida18*Yoshida26^2*Yoshida30*Yoshida5)),
#  Cross107: (Cross105,
#   Yoshida20*Yoshida21*Yoshida26*Yoshida5^2/(Yoshida18*Yoshida22*Yoshida27^2*Yoshida30)),
#  Cross53: (Cross51,
#   Yoshida17^3*Yoshida18*Yoshida25^3*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5/(Yoshida19^3*Yoshida20*Yoshida21*Yoshida22^2*Yoshida28^3*Yoshida31^3)),
#  Cross92: (Cross90,
#   Yoshida20*Yoshida22^2*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2/(Yoshida17^3*Yoshida18*Yoshida21^2*Yoshida25^3*Yoshida27^2*Yoshida30)),
#  Cross31: (Cross30,
#   Yoshida17^6*Yoshida18^2*Yoshida21^4*Yoshida23^3*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^5/(Yoshida19^3*Yoshida20^2*Yoshida22^7*Yoshida24^3*Yoshida28^6*Yoshida31^3*Yoshida5)),
#  Cross19: (Cross18,
#   Yoshida18^2*Yoshida20*Yoshida21*Yoshida22^2*Yoshida27/(Yoshida26^2*Yoshida30*Yoshida31^3*Yoshida5)),
#  Cross75: (Cross76,
#   Yoshida17^3*Yoshida21^2*Yoshida25^3*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5/(Yoshida18^2*Yoshida20^4*Yoshida22^2*Yoshida24^3*Yoshida28^3)),
#  Cross47: (Cross45,
#   Yoshida20*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2/(Yoshida18*Yoshida21^2*Yoshida22*Yoshida25^3*Yoshida27^2*Yoshida30)),
#  Cross91: (Cross90,
#   Yoshida18*Yoshida20^2*Yoshida22*Yoshida28^3*Yoshida31^3*Yoshida5/(Yoshida17^3*Yoshida21*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2)),
#  Cross116: (Cross114,
#   Yoshida18*Yoshida20^2*Yoshida22*Yoshida28^3*Yoshida5/(Yoshida21*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2)),
#  Cross65: (Cross63,
#   Yoshida18*Yoshida20^2*Yoshida26^2*Yoshida5/(Yoshida21*Yoshida22^2*Yoshida27*Yoshida30^2)),
#  Cross104: (Cross102,
#   Yoshida21^2*Yoshida22*Yoshida25^3*Yoshida27^2*Yoshida30/(Yoshida18^2*Yoshida20*Yoshida26*Yoshida31^3*Yoshida5^2)),
#  Cross43: (Cross42,
#   Yoshida21^2*Yoshida22*Yoshida25^3*Yoshida27^2*Yoshida29^3*Yoshida30*Yoshida5/(Yoshida18^2*Yoshida20*Yoshida24^3*Yoshida26*Yoshida28^3*Yoshida31^3)),
#  Cross50: (Cross48,
#   Yoshida20*Yoshida21*Yoshida22^2*Yoshida27/(Yoshida18*Yoshida26^2*Yoshida30*Yoshida5)),
#  Cross59: (Cross57,
#   Yoshida18^2*Yoshida20^4*Yoshida22^2*Yoshida28^3/(Yoshida21^2*Yoshida25^3*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5)),
#  Cross106: (Cross105,
#   Yoshida19^3*Yoshida20^2*Yoshida22^4*Yoshida24^3*Yoshida28^6*Yoshida31^3*Yoshida5/(Yoshida17^6*Yoshida18^2*Yoshida21*Yoshida25^3*Yoshida26*Yoshida27^4*Yoshida30^5)),
#  Cross94: (Cross93,
#   Yoshida21^2*Yoshida25^3*Yoshida26^2*Yoshida30*Yoshida5/(Yoshida18^2*Yoshida20*Yoshida22^2*Yoshida27*Yoshida28^3)),
#  Cross72: (Cross73,
#   Yoshida18*Yoshida20^2*Yoshida22*Yoshida28^3*Yoshida5/(Yoshida17^3*Yoshida21*Yoshida26*Yoshida27*Yoshida30^2)),
#  Cross5: (Cross3,
#   Yoshida18*Yoshida22*Yoshida27^2*Yoshida5/(Yoshida20*Yoshida21*Yoshida26*Yoshida30^2)),
#  Cross44: (Cross42,
#   Yoshida17^3*Yoshida21*Yoshida25^3*Yoshida26*Yoshida27^4*Yoshida30^2/(Yoshida18*Yoshida20^2*Yoshida22*Yoshida24^3*Yoshida28^3*Yoshida31^3*Yoshida5)),
#  Cross128: (Cross126,
#   Yoshida19^3*Yoshida20*Yoshida21*Yoshida22^5*Yoshida24^3*Yoshida28^3/(Yoshida17^3*Yoshida18*Yoshida23^3*Yoshida25^3*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5)),
#  Cross100: (Cross99,
#   Yoshida17^3*Yoshida18*Yoshida21^2*Yoshida23^3*Yoshida25^3*Yoshida27^2*Yoshida30^4/(Yoshida19^3*Yoshida20*Yoshida22^2*Yoshida24^3*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2)),
#  Cross49: (Cross48,
#   Yoshida17^3*Yoshida18*Yoshida21^2*Yoshida23^3*Yoshida25^3*Yoshida27^2*Yoshida30/(Yoshida19^3*Yoshida20*Yoshida22^2*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2)),
#  Cross88: (Cross87,
#   Yoshida21*Yoshida22^2*Yoshida26*Yoshida27/(Yoshida18*Yoshida20^2*Yoshida30*Yoshida5)),
#  Cross27: (Cross28,
#   Yoshida17^3*Yoshida18*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5/(Yoshida19^3*Yoshida20*Yoshida21*Yoshida22^2*Yoshida28^3)),
#  Cross74: (Cross73,
#   Yoshida17^3*Yoshida18^2*Yoshida20*Yoshida21*Yoshida23^3*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2/(Yoshida19^3*Yoshida22^4*Yoshida24^3*Yoshida28^3*Yoshida31^3*Yoshida5)),
#  Cross62: (Cross60,
#   Yoshida18*Yoshida21^2*Yoshida22*Yoshida24^3*Yoshida30/(Yoshida20*Yoshida26*Yoshida27*Yoshida31^3*Yoshida5^2)),
#  Cross1: (Cross0,
#   Yoshida17^6*Yoshida18^2*Yoshida21^4*Yoshida23^3*Yoshida25^6*Yoshida27^4*Yoshida30^5/(Yoshida19^3*Yoshida20^2*Yoshida22^4*Yoshida24^3*Yoshida26^2*Yoshida28^6*Yoshida31^6*Yoshida5^4)),
#  Cross40: (Cross41,
#   Yoshida17^6*Yoshida18*Yoshida21^2*Yoshida23^3*Yoshida25^3*Yoshida26^2*Yoshida27^2*Yoshida30^4*Yoshida5/(Yoshida19^3*Yoshida20^4*Yoshida22^5*Yoshida24^3*Yoshida28^6*Yoshida31^3)),
#  Cross118: (Cross117,
#   Yoshida18^2*Yoshida20*Yoshida22^2*Yoshida24^3*Yoshida27*Yoshida28^3/(Yoshida21^2*Yoshida25^3*Yoshida26^2*Yoshida29^3*Yoshida30*Yoshida5)),
#  Cross68: (Cross66,
#   Yoshida19^3*Yoshida22*Yoshida26^2*Yoshida31^3*Yoshida5^4/(Yoshida17^3*Yoshida18^2*Yoshida20*Yoshida21*Yoshida23^3*Yoshida27*Yoshida30^2)),
#  Cross17: (Cross15,
#   Yoshida18*Yoshida20^2*Yoshida22^4*Yoshida24^3*Yoshida28^3/(Yoshida17^3*Yoshida21*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2*Yoshida5^2)),
#  Cross56: (Cross54,
#   Yoshida22*Yoshida27^2*Yoshida30*Yoshida5/(Yoshida18^2*Yoshida20*Yoshida21*Yoshida26)),
#  Cross103: (Cross102,
#   Yoshida20*Yoshida21*Yoshida22^2*Yoshida27*Yoshida30^2/(Yoshida18*Yoshida26^2*Yoshida31^3*Yoshida5)),
#  Cross134: (Cross132,
#   Yoshida18^2*Yoshida20*Yoshida21*Yoshida27*Yoshida30^2/(Yoshida22*Yoshida26^2*Yoshida31^3*Yoshida5)),
#  Cross112: (Cross111,
#   Yoshida20*Yoshida21*Yoshida26*Yoshida31^3*Yoshida5^2/(Yoshida17^3*Yoshida18*Yoshida22*Yoshida27^2*Yoshida30)),
#  Cross2: (Cross0,
#   Yoshida18*Yoshida20^2*Yoshida21^2*Yoshida22*Yoshida27^2*Yoshida30/(Yoshida26^4*Yoshida31^3*Yoshida5^2)),
#  Cross86: (Cross84,
#   Yoshida19^3*Yoshida20^2*Yoshida22^4*Yoshida24^3*Yoshida28^3*Yoshida5/(Yoshida17^3*Yoshida18^2*Yoshida21*Yoshida23^3*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2)),
#  Cross58: (Cross57,
#   Yoshida18*Yoshida19^3*Yoshida20^2*Yoshida22*Yoshida28^3*Yoshida31^3*Yoshida5/(Yoshida17^3*Yoshida21*Yoshida25^3*Yoshida26*Yoshida27^4*Yoshida30^2)),
#  Cross46: (Cross45,
#   Yoshida19^3*Yoshida20^2*Yoshida22*Yoshida28^3*Yoshida31^3*Yoshida5/(Yoshida17^3*Yoshida18^2*Yoshida21*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2)),
#  Cross36: (Cross37,
#   Yoshida17^6*Yoshida18^4*Yoshida26^2*Yoshida27^2*Yoshida30^4/(Yoshida19^3*Yoshida20*Yoshida21*Yoshida22^2*Yoshida28^3*Yoshida29^3*Yoshida31^3*Yoshida5^2)),
#  Cross71: (Cross69,
#   Yoshida19^3*Yoshida21^2*Yoshida22*Yoshida30*Yoshida5/(Yoshida17^3*Yoshida18^2*Yoshida20*Yoshida26*Yoshida27)),
#  Cross80: (Cross78,
#   Yoshida20^2*Yoshida21^2*Yoshida22*Yoshida29^3*Yoshida5/(Yoshida17^3*Yoshida18^2*Yoshida26*Yoshida27*Yoshida30^2)),
#  Cross127: (Cross126,
#   Yoshida19^3*Yoshida20^2*Yoshida22^4*Yoshida24^3*Yoshida28^6*Yoshida31^3*Yoshida5/(Yoshida17^6*Yoshida18^2*Yoshida21*Yoshida23^3*Yoshida25^3*Yoshida26*Yoshida27^4*Yoshida30^2)),
#  Cross115: (Cross114,
#   Yoshida19^3*Yoshida20*Yoshida22^2*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2/(Yoshida17^3*Yoshida18*Yoshida21^2*Yoshida25^3*Yoshida27^2*Yoshida30^4)),
#  Cross26: (Cross24,
#   Yoshida17^6*Yoshida18^2*Yoshida21*Yoshida23^3*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2/(Yoshida19^3*Yoshida20^2*Yoshida22^4*Yoshida24^3*Yoshida28^3*Yoshida31^3*Yoshida5)),
#  Cross14: (Cross12,
#   Yoshida21^2*Yoshida22*Yoshida29^3*Yoshida30*Yoshida5/(Yoshida17^3*Yoshida18^2*Yoshida20*Yoshida26*Yoshida27)),
#  Cross61: (Cross60,
#   Yoshida21*Yoshida26*Yoshida30^2*Yoshida5^2/(Yoshida18*Yoshida20^2*Yoshida22*Yoshida27^2)),
#  Cross131: (Cross129,
#   Yoshida18^2*Yoshida20*Yoshida21*Yoshida23^3/(Yoshida22*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5)),
#  Cross39: (Cross41,
#   Yoshida17^3*Yoshida18^2*Yoshida21*Yoshida26*Yoshida27*Yoshida30^2/(Yoshida20^2*Yoshida22*Yoshida28^3*Yoshida31^3*Yoshida5)),
#  Cross70: (Cross69,
#   Yoshida21^4*Yoshida25^3*Yoshida26*Yoshida27*Yoshida30^2/(Yoshida18*Yoshida20^2*Yoshida22*Yoshida28^3*Yoshida31^3*Yoshida5)),
#  Cross83: (Cross81,
#   Yoshida20*Yoshida22^2*Yoshida26*Yoshida27/(Yoshida18*Yoshida21^2*Yoshida30*Yoshida5)),
#  Cross55: (Cross54,
#   Yoshida20*Yoshida22^2*Yoshida27*Yoshida28^3/(Yoshida18*Yoshida21^2*Yoshida26^2*Yoshida30*Yoshida5)),
#  Cross133: (Cross132,
#   Yoshida17^3*Yoshida18*Yoshida21^5*Yoshida23^3*Yoshida25^6*Yoshida27^2*Yoshida29^3*Yoshida30^4/(Yoshida19^3*Yoshida20*Yoshida22^5*Yoshida24^3*Yoshida26*Yoshida28^6*Yoshida31^6*Yoshida5^2)),
#  Cross29: (Cross28,
#   Yoshida17^3*Yoshida18^2*Yoshida20*Yoshida21*Yoshida26*Yoshida27*Yoshida30^2/(Yoshida19^3*Yoshida22*Yoshida28^3*Yoshida31^3*Yoshida5)),
#  Cross38: (Cross37,
#   Yoshida17^6*Yoshida18^2*Yoshida26^4*Yoshida27*Yoshida30^2*Yoshida5^2/(Yoshida19^3*Yoshida20^2*Yoshida21^2*Yoshida22^4*Yoshida28^3*Yoshida29^3)),
#  Cross85: (Cross84,
#   Yoshida19^3*Yoshida20*Yoshida22^5*Yoshida24^3*Yoshida26*Yoshida28^6*Yoshida31^3*Yoshida5^2/(Yoshida17^6*Yoshida18^4*Yoshida21^2*Yoshida23^3*Yoshida25^3*Yoshida27^2*Yoshida30^4)),
#  Cross124: (Cross123,
#   Yoshida20*Yoshida21*Yoshida22^2*Yoshida25^3*Yoshida27*Yoshida29^3/(Yoshida17^3*Yoshida18*Yoshida26^2*Yoshida30*Yoshida31^3*Yoshida5)),
#  Cross23: (Cross21,
#   Yoshida21^2*Yoshida22*Yoshida30*Yoshida5/(Yoshida18^2*Yoshida20*Yoshida26*Yoshida27)),
#  Cross101: (Cross99,
#   Yoshida20*Yoshida21*Yoshida22^2*Yoshida27*Yoshida28^3/(Yoshida17^3*Yoshida18*Yoshida26^2*Yoshida30*Yoshida5)),
#  Cross89: (Cross87,
#   Yoshida17^3*Yoshida18*Yoshida21^2*Yoshida23^3*Yoshida25^3*Yoshida27^2*Yoshida29^3*Yoshida30/(Yoshida19^3*Yoshida20*Yoshida22^2*Yoshida24^3*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2)),
#  Cross79: (Cross78,
#   Yoshida20*Yoshida22^2*Yoshida24^3*Yoshida26*Yoshida28^3*Yoshida31^3*Yoshida5^2/(Yoshida17^3*Yoshida18*Yoshida21^2*Yoshida25^3*Yoshida27^2*Yoshida30^4)),
#  Cross67: (Cross66,
#   Yoshida19^3*Yoshida21*Yoshida22^2*Yoshida26*Yoshida27*Yoshida31^3*Yoshida5^2/(Yoshida17^3*Yoshida18^4*Yoshida20^2*Yoshida23^3*Yoshida30)),
#  Cross13: (Cross12,
#   Yoshida18^2*Yoshida20*Yoshida21*Yoshida22^2*Yoshida24^3*Yoshida28^3/(Yoshida17^3*Yoshida25^3*Yoshida26^2*Yoshida27^2*Yoshida30*Yoshida5)),
#  Cross130: (Cross129,
#   Yoshida18*Yoshida20^2*Yoshida22*Yoshida5/(Yoshida21*Yoshida26*Yoshida27*Yoshida30^2))}




# We record the 45 relevant crosses:
relevantCrosses = [pairs_crosses[k][1][1][0][0] for  k in range(0,len(cross_variables)/3)]

# print relevantCrosses
# [Cross0, Cross3, Cross6, Cross9, Cross12, Cross15, Cross18, Cross21, Cross24, Cross28, Cross30, Cross33, Cross37, Cross41, Cross42, Cross45, Cross48, Cross51, Cross54, Cross57, Cross60, Cross63, Cross66, Cross69, Cross73, Cross76, Cross78, Cross81, Cross84, Cross87, Cross90, Cross93, Cross97, Cross99, Cross102, Cross105, Cross110, Cross111, Cross114, Cross117, Cross120, Cross123, Cross126, Cross129, Cross132]


####################################
# Factorization of Cross functions #
####################################

# We start by loading the cross to ds expressions:

cross_to_ds = load("../../Yoshida/Scripts/Input/cross_to_ds.sobj")

# We find the quintic factors for each cross function

anticanonicalTrianglesPerCross = dict()
for cross in cross_variables:
    print cross
    factorCr = Qd(cross_to_ds[cross]).factor()
    quinticFactor = [x[0] for x in factorCr if Qd(x[0]).total_degree() ==5][0]
    posAntic = [k for k in quintics.keys() if bool(SR(quintics[k]) == SR(quinticFactor)) == True]
    if posAntic !=[]:
       print 'pos found!'
       anticanonicalTrianglesPerCross[cross] = posAntic[0]
    else:
	negAntic = [k for k in quintics.keys() if bool(SR(-1*quintics[k]) == SR(quinticFactor)) == True]
	if negAntic !=[]:
	   print 'neg found!'
	   anticanonicalTrianglesPerCross[cross] = negAntic[0]
        else:
		print 'we are in trouble!'


# print  anticanonicalTrianglesPerCross
# {Cross30: Y152346, Cross108: X62, Cross96: Y123645, Cross86: X32, Cross4: X45, Cross35: X16, Cross74: X46, Cross121: Y142536, Cross130: Y142356, Cross48: Y132456, Cross20: X41, Cross76: X54, Cross64: X26, Cross111: X15, Cross3: X45, Cross42: X14, Cross89: X52, Cross120: Y142536, Cross133: Y132546, Cross105: X23, Cross44: X14, Cross32: Y152346, Cross79: X13, Cross88: X52, Cross123: X64, Cross95: X25, Cross73: X46, Cross18: X41, Cross129: Y142356, Cross0: X31, Cross47: X61, Cross117: Y132645, Cross91: X35, Cross63: X26, Cross41: Y152436, Cross119: Y132645, Cross107: X23, Cross85: X32, Cross132: Y132546, Cross31: Y152346, Cross87: X52, Cross126: X42, Cross75: X54, Cross131: Y142356, Cross17: X53, Cross116: X21, Cross61: Y162435, Cross43: X14, Cross102: X34, Cross84: X32, Cross29: X43, Cross128: X42, Cross14: Y142635, Cross70: Y162534, Cross19: X41, Cross58: Y123456, Cross114: X21, Cross13: Y142635, Cross60: Y162435, Cross38: X63, Cross26: X51, Cross104: X34, Cross82: Y123546, Cross54: Y152634, Cross28: X43, Cross98: Y123645, Cross16: X53, Cross127: X42, Cross6: Y162345, Cross72: X46, Cross50: Y132456, Cross22: X24, Cross10: X12, Cross57: Y123456, Cross66: X56, Cross113: X15, Cross101: X36, Cross40: Y152436, Cross12: Y142635, Cross25: X51, Cross56: Y152634, Cross103: X34, Cross34: X16, Cross81: Y123546, Cross69: Y162534, Cross8: Y162345, Cross125: X64, Cross97: Y123645, Cross15: X53, Cross24: X51, Cross71: Y162534, Cross110: X62, Cross59: Y123456, Cross2: X31, Cross49: Y132456, Cross37: X63, Cross115: X21, Cross9: X12, Cross93: X25, Cross65: X26, Cross53: X65, Cross100: X36, Cross39: Y152436, Cross78: X13, Cross109: X62, Cross27: X43, Cross134: Y132546, Cross5: X45, Cross83: Y123546, Cross122: Y142536, Cross55: Y152634, Cross94: X25, Cross33: X16, Cross21: X24, Cross68: X56, Cross99: X36, Cross7: Y162345, Cross46: X61, Cross77: X54, Cross124: X64, Cross112: X15, Cross51: X65, Cross90: X35, Cross23: X24, Cross62: Y162435, Cross11: X12, Cross1: X31, Cross118: Y132645, Cross36: X63, Cross67: X56, Cross106: X23, Cross45: X61, Cross92: X35, Cross80: X13, Cross52: X65}


# We provide the labeling anticanonical coordinates for each Cross function:
indicesCrossAntic = {k: anticanonicalTrianglesPerCross[cross_variables[3*k]] for k in range(0,45)}

# We record the output:
# print indicesCrossAntic
# {0: X31, 1: X45, 2: Y162345, 3: X12, 4: Y142635, 5: X53, 6: X41, 7: X24, 8: X51, 9: X43, 10: Y152346, 11: X16, 12: X63, 13: Y152436, 14: X14, 15: X61, 16: Y132456, 17: X65, 18: Y152634, 19: Y123456, 20: Y162435, 21: X26, 22: X56, 23: Y162534, 24: X46, 25: X54, 26: X13, 27: Y123546, 28: X32, 29: X52, 30: X35, 31: X25, 32: Y123645, 33: X36, 34: X34, 35: X23, 36: X62, 37: X15, 38: X21, 39: Y132645, 40: Y142536, 41: X64, 42: X42, 43: Y142356, 44: Y132546}

# [cross for cross in cross_variables if anticanonicalTrianglesPerCross[cross] == S(Y152346)]
# [Cross30, Cross31, Cross32]