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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:
 <p>
<img src="http
Step1:  
Step2:  Non-commutative symbols
Step3:  Exponential gf recap
Step4:  Linear types
Step5:  occupancies
Step6:  
Step7:  
Step8:  
Step9:  Differences
Step10:  
Step11:  
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Python Code:
from itertools import repeat
from sympy import *
#from type_system import *
%run ../../src/commons.py
%run ./type-system.py
Explanation: <p>
<img src="http://www.cerm.unifi.it/chianti/images/logo%20unifi_positivo.jpg" 
        alt="UniFI logo" style="float: left; width: 20%; height: 20%;">
<div align="right">
Massimo Nocentini<br>
</div>
</p>
<br>
<div align="center">
<b>Abstract</b><br>
In this document we collect a naive <i>type system</i> based on sets.
</div>
End of explanation
init_printing()
x,y,m,n,t,z = symbols('x y m n t z', commutative=True)
alpha, beta, gamma, eta = symbols(r'\alpha \beta \gamma \eta', commutative=True)
f,g = Function('f'), Function('g')
Explanation: 
End of explanation
((1/(1-w[0]*z))*(1/(1-w[1]*z))).diff(z).series(z, n=6)
define(f(z), z/((1-z)**2),ctor=FEq).series(z,n=10)
define(f(z), 1/(1-alpha*z), ctor=FEq).series(z,n=10)
define(f(z), 1/(1-(u[0]+u[1])*z), ctor=FEq).series(z,n=4)
define(f(z), 1/(1-(o[0]+o[1])*z), ctor=FEq).series(z,n=4)
Explanation: Non-commutative symbols
End of explanation
define(f(z), z*(1/(1-z))*(1/(1-z)), ctor=FEq).series(z,n=10)
define(f(z), z**3,ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), exp(z),ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), z*exp(z), ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), z**2*exp(z)/factorial(2,evaluate=False), 
       ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), z**3*exp(z)/factorial(3, evaluate=False), 
       ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), (exp(z)+exp(-z))/2, ctor=FEq).series(z, n=20, kernel='exponential')
define(f(z), exp(m*z), ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), (exp(z)-1)/z, ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), 1/(1-z), ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), (1/(1-z))*(1/(1-z)), ctor=FEq).series(z, n=10, kernel='exponential')
define(f(z), exp(z)**2, ctor=FEq).series(z, n=10, kernel='exponential')
Explanation: Exponential gf recap
End of explanation
tyvar(x).gf()
(tyvar(u[0]) * tyvar(u[1]) * tyvar(u[2])).gf()
(tyvar(o[0]) * tyvar(o[1]) * tyvar(o[2])).gf()
(tyvar(u[0]) | tyvar(u[1]) | tyvar(u[2])).gf()
(tyvar(o[0]) | tyvar(o[1]) | tyvar(o[2])).gf()
truth.gf() + falsehood.gf()
boolean.gf()
maybe(tyvar(alpha)[z]).gf()
Explanation: Linear types
End of explanation
nel = 4
syms=[u[i] for i in range(nel)]
occ_prb, = cp(maybe(tyvar(u[i]*z)) for i in range(nel)).gf() # here we can use the `[z]` notation too.
occ_prb
occupancy(occ_prb, syms, objects='unlike', boxes='unlike').series(z)
occupancy(occ_prb, syms, objects='unlike', boxes='like').series(z)
occupancy(occ_prb, syms, objects='like', boxes='unlike').series(z)
occupancy(occ_prb, syms, objects='like', boxes='like').series(z)
Explanation: occupancies
End of explanation
u_hat = symbols(r'␣_0:10')
nel = 3
occ_prb, = cp(tyvar(z*(sum(u[j] for j in range(nel) if j != i))) | tyvar(u_hat[i]) 
              for i in range(nel)).gf()
occ_prb
syms=[u[i] for i in range(nel)]+[u_hat[i] for i in range(nel)]
occupancy(occ_prb, syms, objects='unlike', boxes='unlike').series(z)
occupancy(occ_prb, syms, objects='unlike', boxes='like').series(z)
occupancy(occ_prb, syms, objects='like', boxes='unlike').series(z)
occupancy(occ_prb, syms, objects='like', boxes='like').series(z)
Explanation: 
End of explanation
occupancy_problem, = cp(maybe(du(tyvar((u[i]*z)**(j+1)) for j in range(i+1))) 
                        for i in range(3)).gf()
occupancy_problem
occupancy(occupancy_problem, syms=[u[i] for i in range(3)], objects='unlike', boxes='unlike').series(z)
occupancy(occupancy_problem, syms=[u[i] for i in range(3)], objects='unlike', boxes='like').series(z)
occupancy(occupancy_problem, syms=[u[i] for i in range(3)], objects='like', boxes='unlike').series(z)
occupancy(occupancy_problem, syms=[u[i] for i in range(3)], objects='like', boxes='like').series(z)
((1+t)*(1+t+t**2)*(1+t+t**2+t**3)).series(t,n=10) # just for checking
Explanation: 
End of explanation
def sums_of_powers(boxes, base):
    p = IndexedBase('\space')
    return cp(cp() | tyvar(p[j]*z**(base**i)) 
              for i in range(0,boxes) 
              for j in [Pow(base,i,evaluate=False)] # implicit let
             ).gf()
occupancy, = sums_of_powers(boxes=4, base=2)
occupancy.series(z, n=32)
occupancy, = sums_of_powers(boxes=4, base=3)
occupancy.series(z, n=100)
occupancy, = sums_of_powers(boxes=4, base=5)
occupancy.series(z, n=200)
occupancy, = sums_of_powers(boxes=4, base=7)
occupancy.series(z, n=500)
assert 393 == 7**0 + 7**2 + 7**3 # _.rhs.rhs.coeff(z, 393)
Explanation: 
End of explanation
difference = (cp() | tyvar(-gamma*z))
ones = nats * difference
ones_gf, = ones.gf()
ones_gf
ones_gf(z,1,1,1).series(z, n=10) # check!
one_gf, = (ones * difference).gf()
one_gf.series(z, n=10).rhs.rhs.subs({w[0]:1, w[1]:1, gamma:1})
Explanation: Differences
End of explanation
l = IndexedBase(r'\circ')
def linear_comb_of_powers(boxes, base):
    return cp(lst(tyvar(Mul(l[j], z**(base**i), evaluate=False)))
              for i in range(boxes) 
              for j in [Pow(base,i,evaluate=False)]).gf()
occupancy, = linear_comb_of_powers(boxes=4, base=Integer(2))
occupancy.series(z, n=8)
occupancy, = linear_comb_of_powers(boxes=4, base=3)
occupancy.series(z, n=9)
occupancy, = linear_comb_of_powers(boxes=4, base=5)
occupancy.series(z, n=10)
def uniform_rv(n):
    return tyvar(S(1)/nel) * lst(tyvar(x))
occupancy, = uniform_rv(n=10).gf()
occupancy.series(x,n=10)
class lst_structure_w(rec):
         
    def definition(self, alpha):
        me = self.me()
        return alpha | lst(me)
    
    def label(self):
        return r'\mathcal{L}_{w}' # `_s` stands for "structure"
lst_structure_w(tyvar(alpha)).gf()
[gf.series(alpha) for gf in _]
class lst_structure(rec):
         
    def definition(self, alpha):
        me = self.me()
        return alpha | (lst(me) * me * me)
    
    def label(self):
        return r'\mathcal{L}_{s}' # `_s` stands for "structure"
lst_structure(tyvar(alpha)).gf()
_[0].series(alpha, n=10)
class structure(rec):
         
    def definition(self, alpha):
        me = self.me()
        return alpha | (bin_tree(me) * me * me)
    
    def label(self):
        return r'\mathcal{S}'
structure(tyvar(alpha)).gf()
gf = _[0]
gf.simplify()
nel = 7
s = gf.simplify().series(alpha, n=nel).rhs.rhs
[s.coeff(alpha, n=i).subs({pow(-1,S(1)/3):-1}).radsimp().powsimp() for i in range(nel)]
class structure(rec):
         
    def definition(self, alpha):
        me = self.me()
        return alpha | (nnbin_tree(me) * me)
    
    def label(self):
        return r'\mathcal{S}'
structure(tyvar(alpha)).gf()
gf = _[0]
gf.simplify()
nel = 20
s = gf.simplify().series(alpha, n=nel).rhs.rhs
[s.coeff(alpha, n=i).subs({pow(-1,S(1)/3):-1}).radsimp().powsimp() for i in range(nel)]
class nn_structure(rec):
         
    def definition(self, alpha):
        me = self.me()
        return alpha * bin_tree(nnbin_tree(me))
    
    def label(self):
        return r'\mathcal{L}_{s}^{+}' # `_s` stands for "structure"
nn_structure(tyvar(alpha)).gf()
_[0].series(alpha, n=10)
class nnlst_structure(rec):
         
    def definition(self, alpha):
        me = self.me()
        return alpha * lst(nnlst(me))
    
    def label(self):
        return r'\mathcal{L}_{s}^{+}' # `_s` stands for "structure"
nnlst_structure(tyvar(alpha)).gf()
_[0].series(alpha, n=10)
class tree(rec):
         
    def definition(self, alpha):
        return alpha * lst(self.me())
    
    def label(self):
        return r'\mathcal{T}'
tree(tyvar(alpha)).gf()
_[0].series(alpha, n=10)
class combination(rec):
         
    def definition(self, alpha):
        me = self.me()
        return alpha | (me * me)
    
    def label(self):
        return r'\mathcal{C}'
combination(tyvar(alpha)).gf()
_[0].series(alpha, n=10)
class ab_tree(rec):
         
    def definition(self, alpha, beta):
        me = self.me()
        return beta | (alpha * me * me)
    
    def label(self):
        return r'\mathcal{T}_{a,b}'
ab_tree_gfs = ab_tree(tyvar(alpha), tyvar(beta)).gf()
ab_tree_gfs
ab_tree_gf = ab_tree_gfs[0]
fab_eq = FEq(ab_tree_gf.lhs, ab_tree_gf.rhs.series(beta, n=20).removeO(), evaluate=False)
fab_eq
fab_eq(x,x)
(_*alpha).expand()
#with lift_to_Lambda(fab_eq) as F:
B = fab_eq(x,1)
A = fab_eq(1,x)
A,B,
(A+B).expand()
((1+x)*A).expand()
class dyck(rec):
         
    def definition(self, alpha, beta):
        me = self.me()
        return cp() | (alpha * me * beta * me)
    
    def label(self):
        return r'\mathcal{D}'
dyck_gfs = dyck(tyvar(alpha*x), tyvar(beta*x)).gf()
dyck_gfs
dyck_gf = dyck_gfs[0]
dyck_gf.series(x,n=10)
class motzkin(rec):
         
    def definition(self, alpha, beta, gamma):
        me = self.me()
        return cp() | (alpha * me * beta * me) | (gamma * me)
    
    def label(self):
        return r'\mathcal{M}'
motzkin_gfs = motzkin(tyvar(alpha*x), tyvar(beta*x), tyvar(gamma*x),).gf()
motzkin_gfs
motzkin_gf = motzkin_gfs[0]
motzkin_gf.series(x,n=10)
motzkin_gf(x,1,1,1).series(x,n=10)
class motzkin_p(rec):
         
    def definition(self, alpha, beta, gamma, eta):
        me = self.me()
        return cp() | (alpha * me * beta * me) | (gamma * me) | (eta * me)
    
    def label(self):
        return r'\mathcal{M}^{+}'
motzkinp_gfs = motzkin_p(tyvar(alpha*x), tyvar(beta*x), tyvar(gamma*x), tyvar(eta*x),).gf()
motzkinp_gfs
motzkinp_gf = motzkinp_gfs[0]
motzkinp_gf.series(x,n=6)
motzkinp_gf(x,1,1,1,1).series(x,n=10)
class fibo(rec):
         
    def definition(self, alpha, beta):
        me = self.me()
        return cp() | alpha | ((beta | (alpha * beta)) * me)
    
    def label(self):
        return r'\mathcal{F}'
fibo_gf, = fibo(tyvar(alpha*x), tyvar(beta*x),).gf()
fibo_gf
fibo_gf.series(x,n=10)
fibo_gf(1,x,1).series(x,n=10)
lst_of_truth_gf, = lst(tyvar(x)).gf()
lst_of_truth_gf.series(x, n=10, is_exp=True)
lst_of_boolean_gf.series(x,n=10,is_exp=True)
_.rhs.rhs.subs({w[0]:1,w[1]:1})
sum((_.rhs.rhs.coeff(x,i)/factorial(i))*x**i for i in range(1,10))
class powerset(ty):
        
    def gf_rhs(self, ty):
        return [exp(self.mulfactor() * gf.rhs) for gf in ty.gf()]
    
    def mulfactor(self):
        return 1
    
    def label(self):
        return r'\mathcal{P}'
powerset_of_tyvar_gf, = (2**(nnlst(tyvar(alpha)))).gf()
powerset_of_tyvar_gf
powerset_of_tyvar_gf.series(alpha, n=10, is_exp=True)
powerset_of_tyvar_gf, = (2**(nnlst(boolean))).gf()
powerset_of_tyvar_gf
powerset_of_tyvar_gf.series(x, n=5, is_exp=True)
_.rhs.rhs.subs({w[0]:1,w[1]:1})
powerset_of_tyvar_gf, _ = (2**(bin_tree(tyvar(alpha)))).gf()
powerset_of_tyvar_gf
powerset_of_tyvar_gf.series(alpha, n=10, is_exp=True)
l, = (2**(2**(nnlst(tyvar(alpha))))).gf()
define(l.lhs, l.rhs.ratsimp(), ctor=FEq).series(alpha,n=8,is_exp=True)
class cycle(ty):
         
    def gf_rhs(self, ty):
        return [log(gf.rhs) for gf in ty.gf()]
    
    def label(self):
        return r'\mathcal{C}'
cycle_of_tyvar_gf, = (~(lst(tyvar(alpha)))).gf()
cycle_of_tyvar_gf
cycle_of_tyvar_gf.series(alpha, n=10, is_exp=True)
cycle_of_tyvar_gf, = (~(lst(boolean))).gf()
cycle_of_tyvar_gf
cycle_of_tyvar_gf.series(x, n=8, is_exp=True)
_.rhs.rhs.subs({w[0]:1,w[1]:1})
Pstar_gf, = (2**(~(lst(tyvar(alpha))))).gf()
Pstar_gf.series(alpha, n=10, is_exp=True)
class ipowerset(powerset):
    
    def mulfactor(self):
        return -1
derangements_gf, = ((-2)**tyvar(alpha)).gf()
derangements_gf.series(alpha, n=10, is_exp=True)
derangements_gf, = ((-2)**nnlst(tyvar(alpha))).gf()
derangements_gf.series(alpha, n=10, is_exp=True)
[1,2][1:]
def foldr(f, l, i):
    if not l:
        return i
    else:
        car, *cdr = l
        return f(car, foldr(f, cdr, i))
    
class arrow(ty):
    
    def label(self):
        return r'\rightarrow'
        
    def gf_rhs(self, alpha, beta):
        v = Dummy()
        return [foldr(lambda gf, acc: Lambda([x], acc(gf.rhs)), 
                      gfs[:-1], 
                      Lambda([x], gfs[-1].rhs))(x)
                for gfs in self.gfs_space()]
        return [foldr(lambda gf, acc: acc**gf.rhs, gfs[:-1], gfs[-1].rhs)
                for gfs in self.gfs_space()]
arr, = arrow(boolean, boolean).gf()
arr
arr.series(x,n=5,is_exp=False)
_.rhs.rhs.removeO().subs({w[0]:1,w[1]:1})
arr, = arrow(lst(boolean), lst(boolean)).gf()
arr
arr.series(x,n=5,is_exp=False)
_.rhs.rhs.removeO().subs({w[0]:1,w[1]:1})
Explanation: 
End of explanation
lamda_gf = lamda(tyvar(x)).gf_rhs(tyvar(x))
lamda_gf
lamda_gf.rhs.series(x,n=10)
Explanation: 
End of explanation
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