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