Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 2. BACKGROUND AND RELATED WORK 25
themes and concepts, than it is to switch between different proof methodologies or theorem
proving tools. It is questionable whether there is any benefit in attempting to enforce a
standard logic, though it seems likely that any moves in this direction in the future will be
driven by EDA tool builders.
Most mature theorem proving tools support both top-down/backward proof and bottom-
up/forward proof. Backward proof involves the statement of a theorem goal and the applica-
tion of rules to split the goal to be proved into subgoals. Each subgoal can then be handled
in the same manner, splitting the goals repeatedly until trivial subgoals are reached that
can be proved directly from logical axioms. Forward proof proceeds by starting from basic
axioms and combining them using rules of inference until eventually the goal to be proved is
deduced. Both styles have advantages and drawbacks, for example every step in a forward
proof is correct and proved, however it may not bring the user any closer to proving the main
proof goal. In a backward proof however, a proof is guaranteed to terminate at the correct
conclusion but it may not actually be a proof of anything at all, unless it can eventually be
reduced to something axiomatic.
There are two main styles of interacting with a theorem prover: declarative or imperative
(tactic-style). The imperative style effectively involves the creation of a proof-generating
program as a combination of prover tactics in a typically prover-specific format. This style,
typified by the descendants of the LCF theorem prover [21], is useful for finding proofs and
for programming verification algorithms but produces output that is generally unreadable.
As soon as the proof itself and not just the existence of a proof becomes important, the
declarative style becomes beneficial. This style, pioneered by the Mizar proof checker [71],
involves the statement of a series of lemmas or subgoals leading to a conclusion. Declarative
systems are good for mechanised checking of proofs and can produce proof scripts that are
easily readable by humans but are unwieldy and impractical for finding the original proofs.
2.4.5 Theorem Proving Tools
There are a range of theorem proving systems in widespread use. One is HOL, named after
Higher Order Logic, the underlying formalism. HOL [22] is intended to be a general platform
for the modelling of systems in higher-order logic with reasoning based on natural deduction