--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MicroJava/DFA/Typing_Framework_err.thy Tue Nov 24 14:37:23 2009 +0100
@@ -0,0 +1,253 @@
+(* Title: HOL/MicroJava/BV/Typing_Framework_err.thy
+ Author: Gerwin Klein
+ Copyright 2000 TUM
+*)
+
+header {* \isaheader{Lifting the Typing Framework to err, app, and eff} *}
+
+theory Typing_Framework_err
+imports Typing_Framework SemilatAlg
+begin
+
+constdefs
+
+wt_err_step :: "'s ord \<Rightarrow> 's err step_type \<Rightarrow> 's err list \<Rightarrow> bool"
+"wt_err_step r step ts \<equiv> wt_step (Err.le r) Err step ts"
+
+wt_app_eff :: "'s ord \<Rightarrow> (nat \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> 's step_type \<Rightarrow> 's list \<Rightarrow> bool"
+"wt_app_eff r app step ts \<equiv>
+ \<forall>p < size ts. app p (ts!p) \<and> (\<forall>(q,t) \<in> set (step p (ts!p)). t <=_r ts!q)"
+
+map_snd :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'c) list"
+"map_snd f \<equiv> map (\<lambda>(x,y). (x, f y))"
+
+error :: "nat \<Rightarrow> (nat \<times> 'a err) list"
+"error n \<equiv> map (\<lambda>x. (x,Err)) [0..<n]"
+
+err_step :: "nat \<Rightarrow> (nat \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> 's step_type \<Rightarrow> 's err step_type"
+"err_step n app step p t \<equiv>
+ case t of
+ Err \<Rightarrow> error n
+ | OK t' \<Rightarrow> if app p t' then map_snd OK (step p t') else error n"
+
+app_mono :: "'s ord \<Rightarrow> (nat \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 's set \<Rightarrow> bool"
+"app_mono r app n A \<equiv>
+ \<forall>s p t. s \<in> A \<and> p < n \<and> s <=_r t \<longrightarrow> app p t \<longrightarrow> app p s"
+
+
+lemmas err_step_defs = err_step_def map_snd_def error_def
+
+
+lemma bounded_err_stepD:
+ "bounded (err_step n app step) n \<Longrightarrow>
+ p < n \<Longrightarrow> app p a \<Longrightarrow> (q,b) \<in> set (step p a) \<Longrightarrow>
+ q < n"
+ apply (simp add: bounded_def err_step_def)
+ apply (erule allE, erule impE, assumption)
+ apply (erule_tac x = "OK a" in allE, drule bspec)
+ apply (simp add: map_snd_def)
+ apply fast
+ apply simp
+ done
+
+
+lemma in_map_sndD: "(a,b) \<in> set (map_snd f xs) \<Longrightarrow> \<exists>b'. (a,b') \<in> set xs"
+ apply (induct xs)
+ apply (auto simp add: map_snd_def)
+ done
+
+
+lemma bounded_err_stepI:
+ "\<forall>p. p < n \<longrightarrow> (\<forall>s. ap p s \<longrightarrow> (\<forall>(q,s') \<in> set (step p s). q < n))
+ \<Longrightarrow> bounded (err_step n ap step) n"
+apply (clarsimp simp: bounded_def err_step_def split: err.splits)
+apply (simp add: error_def image_def)
+apply (blast dest: in_map_sndD)
+done
+
+
+lemma bounded_lift:
+ "bounded step n \<Longrightarrow> bounded (err_step n app step) n"
+ apply (unfold bounded_def err_step_def error_def)
+ apply clarify
+ apply (erule allE, erule impE, assumption)
+ apply (case_tac s)
+ apply (auto simp add: map_snd_def split: split_if_asm)
+ done
+
+
+lemma le_list_map_OK [simp]:
+ "\<And>b. map OK a <=[Err.le r] map OK b = (a <=[r] b)"
+ apply (induct a)
+ apply simp
+ apply simp
+ apply (case_tac b)
+ apply simp
+ apply simp
+ done
+
+
+lemma map_snd_lessI:
+ "x <=|r| y \<Longrightarrow> map_snd OK x <=|Err.le r| map_snd OK y"
+ apply (induct x)
+ apply (unfold lesubstep_type_def map_snd_def)
+ apply auto
+ done
+
+
+lemma mono_lift:
+ "order r \<Longrightarrow> app_mono r app n A \<Longrightarrow> bounded (err_step n app step) n \<Longrightarrow>
+ \<forall>s p t. s \<in> A \<and> p < n \<and> s <=_r t \<longrightarrow> app p t \<longrightarrow> step p s <=|r| step p t \<Longrightarrow>
+ mono (Err.le r) (err_step n app step) n (err A)"
+apply (unfold app_mono_def mono_def err_step_def)
+apply clarify
+apply (case_tac s)
+ apply simp
+apply simp
+apply (case_tac t)
+ apply simp
+ apply clarify
+ apply (simp add: lesubstep_type_def error_def)
+ apply clarify
+ apply (drule in_map_sndD)
+ apply clarify
+ apply (drule bounded_err_stepD, assumption+)
+ apply (rule exI [of _ Err])
+ apply simp
+apply simp
+apply (erule allE, erule allE, erule allE, erule impE)
+ apply (rule conjI, assumption)
+ apply (rule conjI, assumption)
+ apply assumption
+apply (rule conjI)
+apply clarify
+apply (erule allE, erule allE, erule allE, erule impE)
+ apply (rule conjI, assumption)
+ apply (rule conjI, assumption)
+ apply assumption
+apply (erule impE, assumption)
+apply (rule map_snd_lessI, assumption)
+apply clarify
+apply (simp add: lesubstep_type_def error_def)
+apply clarify
+apply (drule in_map_sndD)
+apply clarify
+apply (drule bounded_err_stepD, assumption+)
+apply (rule exI [of _ Err])
+apply simp
+done
+
+lemma in_errorD:
+ "(x,y) \<in> set (error n) \<Longrightarrow> y = Err"
+ by (auto simp add: error_def)
+
+lemma pres_type_lift:
+ "\<forall>s\<in>A. \<forall>p. p < n \<longrightarrow> app p s \<longrightarrow> (\<forall>(q, s')\<in>set (step p s). s' \<in> A)
+ \<Longrightarrow> pres_type (err_step n app step) n (err A)"
+apply (unfold pres_type_def err_step_def)
+apply clarify
+apply (case_tac b)
+ apply simp
+apply (case_tac s)
+ apply simp
+ apply (drule in_errorD)
+ apply simp
+apply (simp add: map_snd_def split: split_if_asm)
+ apply fast
+apply (drule in_errorD)
+apply simp
+done
+
+
+
+text {*
+ There used to be a condition here that each instruction must have a
+ successor. This is not needed any more, because the definition of
+ @{term error} trivially ensures that there is a successor for
+ the critical case where @{term app} does not hold.
+*}
+lemma wt_err_imp_wt_app_eff:
+ assumes wt: "wt_err_step r (err_step (size ts) app step) ts"
+ assumes b: "bounded (err_step (size ts) app step) (size ts)"
+ shows "wt_app_eff r app step (map ok_val ts)"
+proof (unfold wt_app_eff_def, intro strip, rule conjI)
+ fix p assume "p < size (map ok_val ts)"
+ hence lp: "p < size ts" by simp
+ hence ts: "0 < size ts" by (cases p) auto
+ hence err: "(0,Err) \<in> set (error (size ts))" by (simp add: error_def)
+
+ from wt lp
+ have [intro?]: "\<And>p. p < size ts \<Longrightarrow> ts ! p \<noteq> Err"
+ by (unfold wt_err_step_def wt_step_def) simp
+
+ show app: "app p (map ok_val ts ! p)"
+ proof (rule ccontr)
+ from wt lp obtain s where
+ OKp: "ts ! p = OK s" and
+ less: "\<forall>(q,t) \<in> set (err_step (size ts) app step p (ts!p)). t <=_(Err.le r) ts!q"
+ by (unfold wt_err_step_def wt_step_def stable_def)
+ (auto iff: not_Err_eq)
+ assume "\<not> app p (map ok_val ts ! p)"
+ with OKp lp have "\<not> app p s" by simp
+ with OKp have "err_step (size ts) app step p (ts!p) = error (size ts)"
+ by (simp add: err_step_def)
+ with err ts obtain q where
+ "(q,Err) \<in> set (err_step (size ts) app step p (ts!p))" and
+ q: "q < size ts" by auto
+ with less have "ts!q = Err" by auto
+ moreover from q have "ts!q \<noteq> Err" ..
+ ultimately show False by blast
+ qed
+
+ show "\<forall>(q,t)\<in>set(step p (map ok_val ts ! p)). t <=_r map ok_val ts ! q"
+ proof clarify
+ fix q t assume q: "(q,t) \<in> set (step p (map ok_val ts ! p))"
+
+ from wt lp q
+ obtain s where
+ OKp: "ts ! p = OK s" and
+ less: "\<forall>(q,t) \<in> set (err_step (size ts) app step p (ts!p)). t <=_(Err.le r) ts!q"
+ by (unfold wt_err_step_def wt_step_def stable_def)
+ (auto iff: not_Err_eq)
+
+ from b lp app q have lq: "q < size ts" by (rule bounded_err_stepD)
+ hence "ts!q \<noteq> Err" ..
+ then obtain s' where OKq: "ts ! q = OK s'" by (auto iff: not_Err_eq)
+
+ from lp lq OKp OKq app less q
+ show "t <=_r map ok_val ts ! q"
+ by (auto simp add: err_step_def map_snd_def)
+ qed
+qed
+
+
+lemma wt_app_eff_imp_wt_err:
+ assumes app_eff: "wt_app_eff r app step ts"
+ assumes bounded: "bounded (err_step (size ts) app step) (size ts)"
+ shows "wt_err_step r (err_step (size ts) app step) (map OK ts)"
+proof (unfold wt_err_step_def wt_step_def, intro strip, rule conjI)
+ fix p assume "p < size (map OK ts)"
+ hence p: "p < size ts" by simp
+ thus "map OK ts ! p \<noteq> Err" by simp
+ { fix q t
+ assume q: "(q,t) \<in> set (err_step (size ts) app step p (map OK ts ! p))"
+ with p app_eff obtain
+ "app p (ts ! p)" "\<forall>(q,t) \<in> set (step p (ts!p)). t <=_r ts!q"
+ by (unfold wt_app_eff_def) blast
+ moreover
+ from q p bounded have "q < size ts"
+ by - (rule boundedD)
+ hence "map OK ts ! q = OK (ts!q)" by simp
+ moreover
+ have "p < size ts" by (rule p)
+ moreover note q
+ ultimately
+ have "t <=_(Err.le r) map OK ts ! q"
+ by (auto simp add: err_step_def map_snd_def)
+ }
+ thus "stable (Err.le r) (err_step (size ts) app step) (map OK ts) p"
+ by (unfold stable_def) blast
+qed
+
+end
+