Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 4. VERIFYING CIRCUIT LAYOUTS 73
verification. The modified Quartz compiler ensures that this occurs (see Section 4.5).
Otherwise the HOL type system is generally suitable for modelling the Quartz type sys-
tem. Quartz boolean types can be represented as HOL booleans and Quartz integers can
be modelled as HOL integers. Isabelle/HOL also defines a theory of natural numbers with
an extensive range of useful theorems and we will make some use of this when modelling
recursive blocks, however in general we use the integer type which is a more accurate model
and supports true arithmetic.
Quartz tuples are represented using the HOL Cartesian product type, which defines tuples as
nested pairs. This means that the types of (a, b, c) and (a, (b, c)) are considered equivalent,
however this is not a major problem since this is more permissive than the Quartz type
system and designs must already type-check using the Quartz compiler’s type processor.
The Types theory defines the types of wires and of vectors. Wires are named types only, with
no properties. This is a useful model for our purposes since we are actually not interested
in wiring in any way and the values on wires in a generated circuit are not relevant to its
compilation.
Vectors are defined as functions from integers to some other type. In this model a two
dimensional Quartz vector wire a[m][n] is given type int → int → wire. There are significant
limitations to this model [52], essentially arising from the fact that it is not possible to
identify that a vector has a fixed number of elements and if each element of two vectors are
equivalent then the vectors themselves are equivalent. However, this theoretical model is
quite sufficient for layout proofs although it would likely be limiting when reasoning about
functional properties.
4.4.2 Blocks and Block Instantiation
There are three characteristics of blocks that are of interest when reasoning about Quartz
layouts: their semantic interpretation, width expression and height expression. It is necessary
to model the meaning of blocks as well as purely their height and width expressions since
the two are inter-related, with size expressions possibly involving internal signals defined in
a block.