 
Summary: Solving for Set Variables in HigherOrder
Theorem Proving
Chad E. Brown
Department of Mathematical Sciences, Carnegie Mellon University,
Pittsburgh, PA 15213, USA. cebrown@andrew.cmu.edu
Abstract. In higherorder logic, we must consider literals with
exi
ble (set variable) heads. Set variables may be instantiated with logical
formulas of arbitrary complexity. An alternative to guessing the logical
structures of instantiations for set variables is to solve for sets satisfying
constraints. Using the KnasterTarski Fixed Point Theorem [ 15 ] , con
straints whose solutions require recursive denitions can be solved as
xed points of monotone set functions. In this paper, we consider an ap
proach to higherorder theorem proving which intertwines conventional
theorem proving in the form of mating search with generating and solving
set constraints.
1 Introduction
Church's simple type theory [ 10 ] allows quantication over set variables. This
expressive power allows one to express mathematical statements and represent
mathematical objects in a natural way. For example, one can express a statement
asserting the existence of a least set containing a base element 0 and closed under
