CS451/551      Artificial Intelligence       Spring 2002


Assignment #4                due: Friday,  4/19/2002


The Assignment:
--------------
Represent the following as first order clauses, and prove them
using OTTER.

1.  There is somebody who loves everybody and is loved by everybody.
Love is transitive (i.e., if a person loves somebody else, and that
person loves a third person, then the first person loves the third
person)  Prove that everybody loves everybody.

2.  Everybody has a parent.  Everybody is male or female.  A male 
parent of anybody is the person's father.  A female parent of anybody
is the person's mother.  Prove that everybody has a mother or a father.

3.  Note: This problem deals with integers > 1.  It is actually a proof that
there are infinitely many primes, because if there are only finitely many
primes, then there is a number that is the product of all the primes.
When I say that a number divides another number, I mean that it is a 
smaller number that divides into it evenly.  For example, 6 divides 24.  
Also, the successor of a number is the next number.  For example, 15
is the successor of 14.

Facts
-----
If a number divides another number, then it is smaller than that number.
(Example: 6 divides 24, so 6 is smaller than 24)
A number is smaller than its successor.
(Example: 14 is smaller than 15)
No number is less than itself.
(Example: 5 is not less than 5)
Less than is transitive.
(Example: 13 < 14 and 14 < 15.  Therefore 13 < 15)
If a number divides another number, then it does not divide the successor
of the number.
(Example: 2 divides 14, therefore 2 does not divide 15)
If a number is not prime, then some prime number divides it.
(Example: 15 is not prime, so 3 divides 15)
Every number has a successor.
(This is necessary if successor is viewed as a predicate,
but not necessary if successor is viewed as a function.)

To prove
--------
There is no number that is a product of all the primes.