Theory Typing_Framework

(*  Title:      HOL/MicroJava/BV/Typing_Framework.thy
    Author:     Tobias Nipkow
    Copyright   2000 TUM
*)

header {* \isaheader{Typing and Dataflow Analysis Framework} *}

theory Typing_Framework imports Semilattices begin

text {* 
  The relationship between dataflow analysis and a welltyped-instruction predicate. 
*}
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 τs p <-> (∀(q,τ) ∈ set (step p (τs!p)). τ \<sqsubseteq>⇩r τs!q)"

definition stables :: "'s ord => 's step_type => 's list => bool"
where
  "stables r step τs <-> (∀p < size τs. stable r step τs p)"

definition wt_step :: "'s ord => 's => 's step_type => 's list => bool"
where
  "wt_step r T step τs <-> (∀p<size τs. τs!p ≠ T ∧ stable r step τs p)"

definition is_bcv :: "'s ord => 's => 's step_type => nat => 's set => ('s list => 's list) => bool"
where
  "is_bcv r T step n A bcv <-> (∀τs⇣0 ∈ list n A.
  (∀p<n. (bcv τs⇣0)!p ≠ T) = (∃τs ∈ list n A. τs⇣0 [\<sqsubseteq>⇩r] τs ∧ wt_step r T step τs))"

end