Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 2. BACKGROUND AND RELATED WORK 29
[φ]
ψ
(⇒-I)
φ ⇒ ψ
φ ⇒ ψ φ
(⇒-E)
ψ
(Î-I)
φ
x.φ
x.φ
(Î-E)
φ[b/x]
Figure 2.4: Isabelle meta-logic inference rules
The power of Isabelle lies in the ability to use the meta-logic to represent the inference
rules of other logics. Object logics are formalised by extending Isabelle’s meta-logic with
types, constants and axioms. The natural deduction rules of object logics are represented by
meta-level axioms. For example, the rules for introduction and elimination of the logical and
operation in first order logic can be expressed as:
P Q
(∧-I)
P ∧ Q
P ∧ Q
(∧-E1)
P
P ∧ Q
(∧-E2)
Q
Declared as axioms in the Isabelle meta-logic these inference rules can be described by:
P ; Q =⇒ P ∧ Q
P ∧ Q =⇒ P
P ∧ Q =⇒ Q
Where the nested implication φ1 =⇒ (· · · φn =⇒ ψ) can be abbreviated as φ1 ; . . . ; φn =⇒ ψ
which allows the easy expression of a rule with n premises. The syntactic resemblance
between the meta-level axioms and the original inference rules is a happy coincidence arising
from the similarities of the logic being represented to the meta-logic. In general, Isabelle
possesses sophisticated mechanisms for supporting object logics with syntax independent
from the meta-logic through syntax declarations and transformation rules which can be used
to rewrite parsed abstract syntax trees.