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properties, there is e.g. a theory of bit vector machine arithmetic. It defines a lot
of the integer operations found in computer architectures, including addition,
multiplication, two’s complement and rotation. Moreover, the HOL system allows
the user to access external programs within the HOL system by an oracle
mechanism. In this way, SAT (e.g. SATO, GRASP and ZCHAFF) and BDD
engines can be used in proofs.
The HOL system is well documented[4, 6, 3]. After having learned some HOL
basics and got used to the system (which takes some weeks), the documentation
provides a lot of useful information for further work. A lot of material in this
article has been borrowed from there.
2.2 Foundations
The following paragraphs present the logic supported by the HOL system. Introducing
only the some basic concepts, they are a summary of the HOL description
which presents the logic extensively.
Types The HOL logic is based on Church’s simple theory of types[2]. It contains
syntactic categories of types and terms whose elements denote respectively certain
sets and elements of sets. These are taken from a fixed set of sets U, the
universe, that has the following properties.
– Inhab: Each element of U is a non-empty set
– Sub: IfX ∈U and ¬Y ⊆ X, thenY ∈ U
– Prod: IfX ∈Uand Y ∈U,thenX × Y ∈U. X × Y is the cartesian product,
a set that consists of ordered pairs (x, y) withx ∈ X and y ∈ Y
– Pow: IfX ∈U,thenthepowersetP(X) ={Y : Y ⊆ X} is also in U
– Infty: U contains a distinguished infinite set I.
– Choice Thereisadistinguishedelementch ∈ Π X∈U X. The elements of the
product Π X∈U X are (dependently) typed functions: thus, for all X ∈ U, X
is non-empty by Inhab and ch(X) ∈ X witnesses this.
From these properties, it follows:
– Fun: The universe contains the set of functions X → Y ∈U (if X ∈U and
Y ∈U.
– Bool: There is a distinguished two-element set B = {0, 1}.
The types of the HOL logic represent sets in the universe U. Therearefourkinds
of types:
– Atomic types: These denote fixed sets in the universe. Each theory determines
a particular collection of atomic types for example, the standard
atomic types B and I denote, respectively the distinguished two-element
set 2 and the distinguished infinite set I.