Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 2. BACKGROUND AND RELATED WORK 30
The question can of course be raised as to whether the meta-logic representation is correct.
Paulson [60] defines the properties that a meta-logic formalisation is faithful if it admits no
incorrect object-level inferences and adequate if it admits all correct object-level inferences.
2.5.2 Theories
The basic building block of Isabelle mathematics is theories, which organise syntax, declara-
tions, axioms and proofs. Theories are built starting from the Pure theory which represents
the meta-logic, by extending and combining existing theories.
Isabelle theories support multiple inheritance and theory dependencies form a directed acyclic
graph (DAG). Theories can declare additional syntax for constants (operators) within a
logic using a priority grammar where each nonterminal is annotated with an integer priority
which controls how it is parsed. Mixfix annotations allow the formulation of sophisticated
grammar productions to produce readable notation. Special support exists for variable-
binding constructs such as quantifiers which can be declared as binders.
Figure 2.5 shows the definition of a minimal logic of implication in Isabelle. Line 1 begins
the MinLogic theory by stating that it inherits directly from the Pure meta-logic. Lines 2-5
declare a type o of object logic formulae and line 7 declares a coercion from formulae to
propositions. This allows object level operators to be defined over the type o rather than
the general meta-logic prop type, which is important to prevent object logic operations being
used on meta-logic propositions themselves.
The consts section defines the constants and operators in the logic, annotating them with
their types (the short double arrow ⇒ indicates a function type, and [a, b, c] ⇒ d abbreviates
a ⇒ b ⇒ c ⇒ d). Lines 7 and 8 demonstrate the use of mixfix syntax annotations to describe
how the constructs should be parsed, for example infixr indicates that the implication symbol
should be regarded as a right associative infix operator.
The axioms section declares the three inference rules of the logic as meta-logic axioms.
Isabelle provides a wide-range of existing theories, grouped into complete object logics. The
most commonly used Isabelle logics are first-order logic, ZF set theory (which is built as
extension of first-order logic) and an implementation of higher-order logic in the style of the