HW - Prolog

Define each of the following functions without using high-order functions.
- power(X,Y): X^Y.
- gcd(X,Y): The greatest common divisor of X and Y.
- sum(L,Sum): Sum is the sum of the elements in L.
- perms(L): The list of permutations of L.
Write a program to generate all perfect numbers between 1 and 100. A perfect number is a positive integer that is equal to the sum of its proper divisors. For example, 6(=1+2+3) is a perfect number.
Assume a set is represented as a list without duplicates. Define the following functions on sets:
1. contains(X,S): X is a member of S.
2. proper_subset(S1,S2): S1 is a proper subset of S2.
3. disjoint(S1,S2): S1 and S2 are disjoint.
4. union(S1,S2): the union of S1 and S2.
5. intersection(S1,S2): the intersection of S1 and S2.
6. difference(S1,S2): the difference of S1 and S2.
Consider the following type definition for binary trees of integers:
```
type Tree =
   | Void
   | Leaf of int
   | Node of int*Tree*Tree
```
Define the following predicates:
1. contains(T,X): T contains a node with the value X.
2. count(T): the number of nodes in T.
3. leaves(T): A set of all leaves in T.
4. inorder(T): the in-order traversal of the nodes in T.
Define each of the following functions using high-order functions.
- sum(L,Sum): Sum is the sum of the elements in L.
- perms(L): The list of permutations of L.
Find a DFA (Deterministic Finite Automaton) and write a program in F# to simulate it.