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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Using hypothesis to find interesting examples
Hypothesis is a powerful and unique library for testing code. It also includes a find function for finding examples that satisfy an arbitrary predicate. Here, we will explore some of the neat things that can be found using this function.
Step1:  To illustrate what the distribution source looks like, here we instantiate it with a size of 3 and an alphabet of 2
Step2:  Negativity of co-information
Step3:  The Gács-Körner common information is bound from above by the dual total correlation
As we will see, hypothesis can not find an example of $K > B$, because one does not exist.
Step4:  BROJA is not Proj
We know that the BROJA and Proj PID measures are not the same, but the BROJA paper did not provide any simple examples of this. Here, we find one.
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Python Code:
from hypothesis import find
import dit
from dit.abc import *
from dit.pid import *
from dit.utils.testing import distribution_structures
dit.ditParams['repr.print'] = dit.ditParams['print.exact'] = True
Explanation: Using hypothesis to find interesting examples
Hypothesis is a powerful and unique library for testing code. It also includes a find function for finding examples that satisfy an arbitrary predicate. Here, we will explore some of the neat things that can be found using this function.
End of explanation
a = distribution_structures(size=3, alphabet=2)
a.example()
Explanation: To illustrate what the distribution source looks like, here we instantiate it with a size of 3 and an alphabet of 2:
End of explanation
def pred(value):
    return lambda d: dit.multivariate.coinformation(d) < value
ce = find(distribution_structures(3, 2), pred(-1e-5))
print(ce)
print("The coinformation is: {}".format(dit.multivariate.coinformation(ce)))
ce = find(distribution_structures(3, 2), pred(-0.5))
print(ce)
print("The coinformation is: {}".format(dit.multivariate.coinformation(ce)))
Explanation: Negativity of co-information
End of explanation
def b_lt_k(d):
    k = dit.multivariate.gk_common_information(d)
    b = dit.multivariate.dual_total_correlation(d)
    return  k > b
find(distribution_structures(size=3, alphabet=3, uniform=True), b_lt_k)
Explanation: The Gács-Körner common information is bound from above by the dual total correlation
As we will see, hypothesis can not find an example of $K > B$, because one does not exist.
End of explanation
ce = find(distribution_structures(3, 2, True), lambda d: PID_BROJA(d) != PID_Proj(d))
ce
print(PID_BROJA(ce))
print(PID_Proj(ce))
Explanation: BROJA is not Proj
We know that the BROJA and Proj PID measures are not the same, but the BROJA paper did not provide any simple examples of this. Here, we find one.
End of explanation
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