1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
|
#----------------------------------------------------------------
# Problem: In a ring, if x*x*x = x for all x
# in the ring, then
# x*y = y*x and 6*x = 0 for all x,y in the ring.
#
# Funktionen: f : Multiplikation *
# J : Addition +
# g : Inverses
# e : Neutrales Element
# a,b : Konstanten
j (e,X) = X. # e ist a left identity for sum
j (X,e) = X. # e ist a right identity for sum
j (g (X),X) = e. # there exists a left inverse for sum
j (X,g (X)) = e. # there exists a right inverse for sum
j (j (X,Y),Z) = j (X,j (Y,Z)). # associativity of addition
j (X,Y) = j(Y,X). # commutativity of addition
f (f (X,Y),Z) = f (X,f (Y,Z)). # associativity of multiplication
f (X,j (Y,Z)) = j (f (X,Y),f (X,Z)). # distributivity axioms
f (j (X,Y),Z) = j (f (X,Z),f (Y,Z)). #
f (f(X,X),X) = X. # special hypothese: x*x*x = x
?- f (a,b) = f (b,a). # theorem
|
