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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:
 Minimizing a <span style="font-variant
Step1:  The function cart_prod(A, B) computes the Cartesian product $A \times B$ of the sets $A$ and $B$ where $A \times B$ is defined as follows
Step2:  The function separate takes four arguments
Step3:  Given a state p and a Partition of the set of all states, the function find_equivalence_class(p, Partition) returns the equivalence class of p, i.e. it returns the set from Partition that contains x.
Step4:  The function reachable(q0, Σ, 𝛿) takes three arguments
Step5:  The function all_separable(Q, A, Σ, 𝛿) takes four arguments
Step6:  The function minimize(A) takes a deterministic 
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Python Code:
def arb(M):
    for x in M:
        return x
    assert False, 'Error: arb called with empty set!'
Explanation: Minimizing a <span style="font-variant:small-caps;">Fsm</span>
The function arb(M) takes a non-empty set M as its argument and returns an arbitrary element from this set.
The set M is not changed.
End of explanation
def cart_prod(A, B):
    return { (x, y) for x in A for y in B }
Explanation: The function cart_prod(A, B) computes the Cartesian product $A \times B$ of the sets $A$ and $B$ where $A \times B$ is defined as follows:
$$ A \times B := { (x, y) \mid x \in A \wedge y \in B }. $$
End of explanation
def separate(Pairs, States, Σ, 𝛿):
    Result = { (q1, q2) for q1 in States
                        for q2 in States
                        for c  in Σ 
                        if (𝛿[q1, c], 𝛿[q2, c]) in Pairs
             }
    return Result
Explanation: The function separate takes four arguments:
- Pairs  a set Pairs of pairs of states from some given <span style="font-variant:small-caps;">Fsm</span> $F$.
If $(p_1, p_2) \in \texttt{Pairs}$, then $p_1$ and $p_2$ are known to be separable.
- States is the set of all states of the <span style="font-variant:small-caps;">Fsm</span> $F$,
- Σ is the alphabet of the <span style="font-variant:small-caps;">Fsm</span> $F$.
- 𝛿 is the transition function of the <span style="font-variant:small-caps;">Fsm</span>
The function separate(Pairs, States, Σ, 𝛿) computes the set of pairs of states $(q_1, q_2)$ that are separable because there is some character $c \in \Sigma$ such that 
$$\delta(q_1,c) = p_1, \quad \textrm{but} \quad \delta(q_2,c) = p_2. $$
End of explanation
def find_equivalence_class(p, Partition):
    return arb({ C for C in Partition if p in C })
Explanation: Given a state p and a Partition of the set of all states, the function find_equivalence_class(p, Partition) returns the equivalence class of p, i.e. it returns the set from Partition that contains x.
End of explanation
def reachable(q0, Σ, 𝛿):
    Result = { q0 }
    while True:
        NewStates = { 𝛿[p, c] for p in Result for c in Σ }
        if NewStates <= Result:
            return Result
        Result |= NewStates
Explanation: The function reachable(q0, Σ, 𝛿) takes three arguments:
* q0 is the start state of an Fsm,
* Σ  is the alphabet.
* 𝛿  is the transition function.  The transition function is assumed to be complete. 𝛿 is represented as a dictionary.   
It returns the set of all states that can be reached from the start state q0 by reading strings of characters from Σ.
End of explanation
def all_separable(Q, A, Σ, 𝛿):
        Separable = cart_prod(Q - A, A) | cart_prod(A, Q - A)
        while True:
            NewPairs = separate(Separable, Q, Σ, 𝛿)
            if NewPairs <= Separable:
                return Separable
            Separable |= NewPairs
Explanation: The function all_separable(Q, A, Σ, 𝛿) takes four arguments:
* Q is the set of states of the Fsm.
* A is the set of all accepting states,
* Σ  is the alphabet.
* 𝛿 is the transition function.  
𝛿 is represented as a dictionary.   
The function computes the set of all Pairs (p, q) such that p and q are separable, i.e. all pairs such that
$$ \exists s \in \Sigma^: \bigl(\delta^(p, s) \in A \wedge \delta^(q,s) \not\in A\bigr) \vee 
                           \bigl(\delta^(p, s) \not\in A \wedge \delta^*(q,s) \in A\bigr). 
$$
End of explanation
def minimize(F):
    Q, Σ, 𝛿, q0, A = F
    Q            = reachable(q0, Σ, 𝛿)
    Separable    = all_separable(Q, A, Σ, 𝛿)
    Equivalent   = cart_prod(Q, Q) - Separable
    EquivClasses = { frozenset({ p for p in Q if (p, q) in Equivalent })
                     for q in Q
                   }
    newQ0        = arb({ M for M in EquivClasses if q0 in M })
    newAccept    = { M for M in EquivClasses if arb(M) in A }   
    newDelta     = {}
    for q in Q:
        for c in Σ:
            p = 𝛿.get((q, c))
            if p != None:
                classOfP = find_equivalence_class(p, EquivClasses)
                classOfQ = find_equivalence_class(q, EquivClasses)
                newDelta[(classOfQ, c)] = classOfP
            else:
                newDelta[(classOfQ, c)] = frozenset()
    return EquivClasses, Σ, newDelta, newQ0, newAccept
Explanation: The function minimize(A) takes a deterministic 
<span style="font-variant:small-caps;">Fsm</span> F as its input.
Here F is a 5-tuple of the form
$$ F = (Q, \Sigma, \delta, q_0, A) $$
The algorithm performs the following steps:
1. All unreachable states are eliminated.
2. All accepting states are separated form all non-accepting states.
3. States are separated as long as possible.
   Two states $p_1$ and $p_2$ are separable if there is a character 
   $c \in \Sigma$ such that 
   $$\delta(p_1,c) = q_1, \quad \delta(p_2,c) = q_2, \quad \textrm{and} \quad
     \mbox{$q_1$ and $q_2$ are separable.}
   $$
4. States that are not separable are equivalent and are therefore identified and grouped
   in equivalence classes.  The states in an equivalence class are then identified.
End of explanation
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