Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 4. VERIFYING CIRCUIT LAYOUTS 75
constdefs
ser :: " [(’ a⇒ ’b⇒ bool,’a⇒ ’b⇒ int)block,(’b⇒ ’c⇒ bool,’b⇒ ’c⇒ int)block]
⇒ (’a⇒ ’c⇒ bool, ’a⇒ ’c⇒ int)block" (infixl ";;" 48)
"ser ≡ (λ B1 B2. (| Def = λx y. ∃ s. (Def B1) x s ∧ (Def B2) s y,
Height = λx y. let s = (THE s. (Def B1) x s ∧ (Def B2) s y) in
max (Height B1 x s) (Height B2 s y),
Width = λx y. let s = (THE s. (Def B1) x s ∧ (Def B2) s y) in
(Width B1 x s) + (Width B2 s y)|))"
Figure 4.5: Part of the QuartzLayout theory of series composition
operation is identified by triple semi-colons in HOL. This eases the job of parsing Quart-
zLayout definitions and allows it to clearly distinguished from series compositions. A block
instantiation in QuartzLayout is therefore represented by the string a ;;; bid $ p1 . . . $
pn ;;; b. This is sufficiently similar to the Quartz syntax to be readable and yet sufficiently
simplified to not require the development of AST-transforming ML functions for Isabelle.
The contents of a block record are accessed using the functions Def, Height and Width, so
Height (a ;;; bid $ p1 . . . $ pn ;;; b) will return an integer value corresponding to
the height of that block instantiation. This is identical to the height() function of Quartz
expressions and is used to model this operation.
Block compositions are described in the SeriesComposition and ParallelComposition theories.
Parallel composition is defined as an operation on nested pairs, mirroring the definition of
Cartesian products used to model Quartz tuples and is syntactically represented using double
square brackets.
Series composition is defined as a left-associative binary operator (identified by double semi-
colons), as illustrated in Figure 4.5. The semantic definition of a series composition of
two blocks is defined with an explicit internal signal connecting them, while the definite
description operator is used to parameterise the width and height functions of the two blocks
by the correct value to determine the width and height of the composition.
The definitions of series and parallel composition implement the SIσ β function defined on
Page 43. Series compositions are laid out horizontally with the widths summed and the
maximum height selected, while parallel compositions are laid out vertically with heights
summed and the maximum width selected.