--- a/src/HOL/MicroJava/BV/Typing_Framework_err.thy Fri Dec 04 11:44:57 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,255 +0,0 @@
-(* Title: HOL/MicroJava/BV/Typing_Framework_err.thy
- ID: $Id$
- 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
-