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(* Title: HOL/MicroJava/DFA/Typing_Framework.thy
Author: Tobias Nipkow
Copyright 2000 TUM
*)
section \<open>Typing and Dataflow Analysis Framework\<close>
theory Typing_Framework
imports Listn
begin
text \<open>
The relationship between dataflow analysis and a welltyped-instruction predicate.
\<close>
type_synonym 's step_type = "nat \ 's \ (nat \ 's) list"
definition stable :: "'s ord \ 's step_type \ 's list \ nat \ bool" where
"stable r step ss p == \(q,s')\set(step p (ss!p)). s' <=_r ss!q"
definition stables :: "'s ord \ 's step_type \ 's list \ bool" where
"stables r step ss == \p
definition wt_step ::
"'s ord \ 's \ 's step_type \ 's list \ bool" where
"wt_step r T step ts ==
\<forall>p<size(ts). ts!p ~= T & stable r step ts p"
definition is_bcv :: "'s ord \ 's \ 's step_type
\<Rightarrow> nat \<Rightarrow> 's set \<Rightarrow> ('s list \<Rightarrow> 's list) \<Rightarrow> bool" where
"is_bcv r T step n A bcv == \ss \ list n A.
(\<forall>p<n. (bcv ss)!p ~= T) =
(\<exists>ts \<in> list n A. ss <=[r] ts & wt_step r T step ts)"
end
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