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Solving for Set Variables in Higher-Order Theorem Proving

Summary: Solving for Set Variables in Higher-Order
Theorem Proving
Chad E. Brown 
Department of Mathematical Sciences, Carnegie Mellon University,
Pittsburgh, PA 15213, USA. cebrown@andrew.cmu.edu
Abstract. In higher-order 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 Knaster-Tarski Fixed Point Theorem [ 15 ] , con-
straints whose solutions require recursive de nitions can be solved as
xed points of monotone set functions. In this paper, we consider an ap-
proach to higher-order 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 quanti cation 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


Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University


Collections: Mathematics