Verification of Parameterised FPGA Circuit Descriptions with Layout ...

APPENDIX B. THEORETICAL BASIS FOR LAYOUT REASONING 195
end
B.7 Functions
header {* Definition of Quartz expression functions: maxf, sum, etc *}
theory Functions = IntAlgebra:
section {* Useful logic *}
theorem conj_imp: "(A --> B & C) = ((A --> B) & (A --> C))"
by (auto)
section {* maxf function to find the maximum point of a function within in a range *}
consts
maxf :: "(int*int*(int=>int))=>int"
recdef maxf "measure (%(b, t, f). nat(t+1-b))"
"maxf (bot, top, fun) = (if (top < bot) then 0
else (
case (top = bot) of True =>
fun top
| False => (
let one = fun top in
let two = maxf (bot, top - 1, fun) in
max one two)
))"
theorem maxf_expand_if: "maxf(b,t,f) = (if (t < b) then 0 else (if t = b then f t
else max (f t) (maxf(b,t - 1,f))))"
by (simp add: Let_def)
subsection {* Correctness proof *}
constdefs
is_maxf :: "[int,int, int=>int,int]=>bool"
"is_maxf == % bot top fun max.
(EX y. bot C"
by (auto)
lemma logic_impand: "A & B = (A --> B) | ~A"
by (blast)
theorem maxf_nobigger: "!! b t f. [| b