Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 2. BACKGROUND AND RELATED WORK 28
A shallow embedding is typically much easier to construct than a deep embedding since the
meaning of the language can be encoded directly. With a deep embedding the meaning of
the language must be defined as a function over the abstract syntax in a way that can be
cumbersome however it does have the advantage that the theorem prover is able to reason
over syntactic structures and state theorems about all programs.
2.5 Isabelle: A Generic Theorem Prover
Although the work presented in this thesis is not dependent on a particular theorem prover,
we make extensive use of the Isabelle proof tool and a basic understanding of its capabilities
and how they can be used is useful for understanding the work presented in Chapter 4 in
particular.
2.5.1 Meta-logic
Isabelle’s distinctive feature is its representation of logics within a small fragment of a higher-
order logic, called the meta-logic [60].
Isabelle’s meta-logic is typed with basic and function types σ → τ. The basic types depend
on the logic being represented but always include the type prop for propositions. The terms of
the meta-logic are essentially those of the typed λ-calculus - constants, variables, abstractions
and function applications.
There are essentially three operations within the meta-logic: universal quantification, meta-
implication and meta-equality. Isabelle uses the symbols , =⇒ and ≡ for these operations to
avoid confusion with the equivalent operations in object logics. Implication expresses logical
entailment, quantification expresses generality in rules and axiom schemes and meta-equality
is intended for expressing definitions.
Isabelle’s meta-logic is simply a logic like any other complete with basic inference rules. For
example, Figure 2.5.1 shows the meta-logic rules for universal quantification and implication
introduction and elimination.