#Experiments in lambda calculus


#SET UP

ident = lambda x:x

#Booleans

true =  lambda x:lambda y:x
false = lambda x:lambda y:y
And =   lambda x:lambda y:x(y)(x)

#Church Numerals

zero =  lambda x: lambda y:y

one  =  lambda x: lambda y:x(y)
two =   lambda x: lambda y:x(x(y)) #Note not our usual order!
three = lambda x: lambda y:x(x(x(y)))

succ = lambda x: lambda y: lambda z:y(x(y)(z))

#Pairing

pair = lambda x: lambda y: lambda z: z(x)(y)
first = lambda x:x(true)
second = lambda x: x(false)





#Arithmetic
add = lambda m: lambda n:lambda u: lambda v:m(u)(n(u)(v))

#TESTS

#Testing helpers. In Python we cannot reduce lambda expressions to
#some normal form, so to test that we are getting the right thing,
#we will apply them to some arguments so as to get a result that is
#a Python boolean, or a Python int, or a Python tuple.

#Note that these next two functions are not pure lambda calculus! 
inc = lambda x:x+1
tuplize = lambda x: lambda y: (x,y)
#To test something that should give a boolean result, apply it to
#(True)(False).  For instance, here is how we test that Not(true) is false
#and Not(false) is true.

print( Not(true)(True)(False))
print( Not(false)(True)(False))

#To test something that should give a numeral as a result, apply it to
#(inc)(0)  For example, here is how we test that three is the right thing--
#It should print 3. (It does)

print(three(inc)(0))

#To test something that should return a pair, apply it to tuplize.__

print(pair(3)(4)(tuplize))



#test addition 
print (add(three)(two)(inc)(0))
print (add(three)(zero)(inc)(0))
print (multiply(three)(two)(inc)(0))
print (multiply(three)(succ(three))(inc)(0))
print (multiply(three)(zero)(inc)(0))

#test factorial

print (factorial(zero)(inc)(0))
print (factorial(one)(inc)(0))
print (factorial(two)(inc)(0))
print (factorial(three)(inc)(0))
print (factorial(succ(three))(inc)(0))