Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 4. VERIFYING CIRCUIT LAYOUTS 74
record (’a,’b)block =
Def :: "’a"
Height :: "’b"
Width :: "’b"
constdefs
ap :: "[(’ a⇒ ’b,’a⇒ ’c)block,’a]⇒ (’b,’c)block"
( infixl "$" 49)
"ap ≡ λB x. (| Def = Def B x, Height = Height B x, Width = Width B x |)"
constdefs
inst :: "[’a, (’a⇒ ’b⇒ bool,’a⇒ ’b⇒ int)block,’b]⇒ (bool, int)block"
(" ;;; ;;; " [45, 46, 47] 45)
"inst ≡ (λ x B y. (| Def = Def B x y, Height = Height B x y, Width = Width B x y |))"
Figure 4.4: Part of the QuartzLayout theory of blocks
Because Quartz blocks are relational, rather than functional, we must model block semantics
as logical predicates on their domain and range signals. A predicate block returns the boolean
true for all valid combinations of inputs and outputs and false for invalid combinations. This
means that a predicate can not calculate a block’s output values from its inputs in the same
way as a function, however it can confirm that values are correct. Within block semantic
definitions, internal signals can be represented using existential quantifiers ∃ (written “EX”
in Isabelle/HOL).
We model block size functions, rather than size expressions. A size function includes the
block’s semantic definition within it to define the values of any internal variables, bound
using the definite description operator.
A block is modelled as a record consisting of a logical predicate, height function and width
function, as shown in Figure 4.4. A block with type δ1 δ2 . . .δn ∼ ρ will be modelled as a
record of type (δ1 → δ2 → · · · → δn → ρ → bool, δ1 → δ2 → · · · → δn → ρ → int)block.
While the general form of the type is the same for both the size functions (which return
integers) and the semantic definition predicate (which returns a boolean) they have to be
defined separately because of the way types are described.
Two operations are defined on block records to model the behaviour described by statements
of the type a ; bid p1 . . .pn ; b. Block application describes the application of an inner
(curried) parameter to a block. So as not to confuse this with HOL function application
in QuartzLayout this requires the use of a dollar sign operator. The block instantiation