--- a/CpsG.thy Tue May 20 12:49:21 2014 +0100
+++ b/CpsG.thy Thu May 22 17:40:39 2014 +0100
@@ -1,6 +1,6 @@
theory CpsG
-imports PrioG
+imports PrioG Max
begin
lemma not_thread_holdents:
@@ -23,7 +23,7 @@
and not_in': "thread \<notin> threads s"
have "holdents (e # s) th = holdents s th"
apply (unfold eq_e holdents_test)
- by (simp add:depend_create_unchanged)
+ by (simp add:RAG_create_unchanged)
moreover have "th \<notin> threads s"
proof -
from not_in eq_e show ?thesis by simp
@@ -38,13 +38,13 @@
case True
with nh eq_e
show ?thesis
- by (auto simp:holdents_test depend_exit_unchanged)
+ by (auto simp:holdents_test RAG_exit_unchanged)
next
case False
with not_in and eq_e
have "th \<notin> threads s" by simp
from ih[OF this] False eq_e show ?thesis
- by (auto simp:holdents_test depend_exit_unchanged)
+ by (auto simp:holdents_test RAG_exit_unchanged)
qed
next
case (thread_P thread cs)
@@ -60,7 +60,7 @@
qed
hence "holdents (e # s) th = holdents s th "
apply (unfold cntCS_def holdents_test eq_e)
- by (unfold step_depend_p[OF vtp], auto)
+ by (unfold step_RAG_p[OF vtp], auto)
moreover have "holdents s th = {}"
proof(rule ih)
from not_in eq_e show "th \<notin> threads s" by simp
@@ -102,7 +102,7 @@
qed
moreover note neq_th eq_wq
ultimately have "holdents (e # s) th = holdents s th"
- by (unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
+ by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
moreover have "holdents s th = {}"
proof(rule ih)
from not_in eq_e show "th \<notin> threads s" by simp
@@ -117,12 +117,12 @@
from ih [OF this] and eq_e
show ?thesis
apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:depend_set_unchanged)
+ by (simp add:RAG_set_unchanged)
qed
next
case vt_nil
show ?case
- by (auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
+ by (auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
qed
qed
@@ -161,97 +161,33 @@
shows "th1 = th2"
using assms by (unfold next_th_def, auto)
-lemma wf_depend:
+lemma wf_RAG:
assumes vt: "vt s"
- shows "wf (depend s)"
+ shows "wf (RAG s)"
proof(rule finite_acyclic_wf)
- from finite_depend[OF vt] show "finite (depend s)" .
+ from finite_RAG[OF vt] show "finite (RAG s)" .
next
- from acyclic_depend[OF vt] show "acyclic (depend s)" .
+ from acyclic_RAG[OF vt] show "acyclic (RAG s)" .
qed
-lemma Max_Union:
- assumes fc: "finite C"
- and ne: "C \<noteq> {}"
- and fa: "\<And> A. A \<in> C \<Longrightarrow> finite A \<and> A \<noteq> {}"
- shows "Max (\<Union> C) = Max (Max ` C)"
-proof -
- from fc ne fa show ?thesis
- proof(induct)
- case (insert x F)
- assume ih: "\<lbrakk>F \<noteq> {}; \<And>A. A \<in> F \<Longrightarrow> finite A \<and> A \<noteq> {}\<rbrakk> \<Longrightarrow> Max (\<Union>F) = Max (Max ` F)"
- and h: "\<And> A. A \<in> insert x F \<Longrightarrow> finite A \<and> A \<noteq> {}"
- show ?case (is "?L = ?R")
- proof(cases "F = {}")
- case False
- from Union_insert have "?L = Max (x \<union> (\<Union> F))" by simp
- also have "\<dots> = max (Max x) (Max(\<Union> F))"
- proof(rule Max_Un)
- from h[of x] show "finite x" by auto
- next
- from h[of x] show "x \<noteq> {}" by auto
- next
- show "finite (\<Union>F)"
- proof(rule finite_Union)
- show "finite F" by fact
- next
- from h show "\<And>M. M \<in> F \<Longrightarrow> finite M" by auto
- qed
- next
- from False and h show "\<Union>F \<noteq> {}" by auto
- qed
- also have "\<dots> = ?R"
- proof -
- have "?R = Max (Max ` ({x} \<union> F))" by simp
- also have "\<dots> = Max ({Max x} \<union> (Max ` F))" by simp
- also have "\<dots> = max (Max x) (Max (\<Union>F))"
- proof -
- have "Max ({Max x} \<union> Max ` F) = max (Max {Max x}) (Max (Max ` F))"
- proof(rule Max_Un)
- show "finite {Max x}" by simp
- next
- show "{Max x} \<noteq> {}" by simp
- next
- from insert show "finite (Max ` F)" by auto
- next
- from False show "Max ` F \<noteq> {}" by auto
- qed
- moreover have "Max {Max x} = Max x" by simp
- moreover have "Max (\<Union>F) = Max (Max ` F)"
- proof(rule ih)
- show "F \<noteq> {}" by fact
- next
- from h show "\<And>A. A \<in> F \<Longrightarrow> finite A \<and> A \<noteq> {}"
- by auto
- qed
- ultimately show ?thesis by auto
- qed
- finally show ?thesis by simp
- qed
- finally show ?thesis by simp
- next
- case True
- thus ?thesis by auto
- qed
- next
- case empty
- assume "{} \<noteq> {}" show ?case by auto
- qed
-qed
+
definition child :: "state \<Rightarrow> (node \<times> node) set"
where "child s \<equiv>
- {(Th th', Th th) | th th'. \<exists>cs. (Th th', Cs cs) \<in> depend s \<and> (Cs cs, Th th) \<in> depend s}"
+ {(Th th', Th th) | th th'. \<exists>cs. (Th th', Cs cs) \<in> RAG s \<and> (Cs cs, Th th) \<in> RAG s}"
definition children :: "state \<Rightarrow> thread \<Rightarrow> thread set"
where "children s th \<equiv> {th'. (Th th', Th th) \<in> child s}"
lemma children_def2:
- "children s th \<equiv> {th'. \<exists> cs. (Th th', Cs cs) \<in> depend s \<and> (Cs cs, Th th) \<in> depend s}"
+ "children s th \<equiv> {th'. \<exists> cs. (Th th', Cs cs) \<in> RAG s \<and> (Cs cs, Th th) \<in> RAG s}"
unfolding child_def children_def by simp
-lemma children_dependants: "children s th \<subseteq> dependants (wq s) th"
- by (unfold children_def child_def cs_dependants_def, auto simp:eq_depend)
+lemma children_dependants:
+ "children s th \<subseteq> dependants (wq s) th"
+ unfolding children_def2
+ unfolding cs_dependants_def
+ by (auto simp add: eq_RAG)
lemma child_unique:
assumes vt: "vt s"
@@ -261,92 +197,92 @@
using ch1 ch2
proof(unfold child_def, clarsimp)
fix cs csa
- assume h1: "(Th th, Cs cs) \<in> depend s"
- and h2: "(Cs cs, Th th1) \<in> depend s"
- and h3: "(Th th, Cs csa) \<in> depend s"
- and h4: "(Cs csa, Th th2) \<in> depend s"
- from unique_depend[OF vt h1 h3] have "cs = csa" by simp
- with h4 have "(Cs cs, Th th2) \<in> depend s" by simp
- from unique_depend[OF vt h2 this]
+ assume h1: "(Th th, Cs cs) \<in> RAG s"
+ and h2: "(Cs cs, Th th1) \<in> RAG s"
+ and h3: "(Th th, Cs csa) \<in> RAG s"
+ and h4: "(Cs csa, Th th2) \<in> RAG s"
+ from unique_RAG[OF vt h1 h3] have "cs = csa" by simp
+ with h4 have "(Cs cs, Th th2) \<in> RAG s" by simp
+ from unique_RAG[OF vt h2 this]
show "th1 = th2" by simp
qed
-lemma depend_children:
- assumes h: "(Th th1, Th th2) \<in> (depend s)^+"
- shows "th1 \<in> children s th2 \<or> (\<exists> th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)^+)"
+lemma RAG_children:
+ assumes h: "(Th th1, Th th2) \<in> (RAG s)^+"
+ shows "th1 \<in> children s th2 \<or> (\<exists> th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (RAG s)^+)"
proof -
from h show ?thesis
proof(induct rule: tranclE)
fix c th2
- assume h1: "(Th th1, c) \<in> (depend s)\<^sup>+"
- and h2: "(c, Th th2) \<in> depend s"
+ assume h1: "(Th th1, c) \<in> (RAG s)\<^sup>+"
+ and h2: "(c, Th th2) \<in> RAG s"
from h2 obtain cs where eq_c: "c = Cs cs"
- by (case_tac c, auto simp:s_depend_def)
- show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)\<^sup>+)"
+ by (case_tac c, auto simp:s_RAG_def)
+ show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (RAG s)\<^sup>+)"
proof(rule tranclE[OF h1])
fix ca
- assume h3: "(Th th1, ca) \<in> (depend s)\<^sup>+"
- and h4: "(ca, c) \<in> depend s"
- show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (depend s)\<^sup>+)"
+ assume h3: "(Th th1, ca) \<in> (RAG s)\<^sup>+"
+ and h4: "(ca, c) \<in> RAG s"
+ show "th1 \<in> children s th2 \<or> (\<exists>th3. th3 \<in> children s th2 \<and> (Th th1, Th th3) \<in> (RAG s)\<^sup>+)"
proof -
from eq_c and h4 obtain th3 where eq_ca: "ca = Th th3"
- by (case_tac ca, auto simp:s_depend_def)
+ by (case_tac ca, auto simp:s_RAG_def)
from eq_ca h4 h2 eq_c
have "th3 \<in> children s th2" by (auto simp:children_def child_def)
- moreover from h3 eq_ca have "(Th th1, Th th3) \<in> (depend s)\<^sup>+" by simp
+ moreover from h3 eq_ca have "(Th th1, Th th3) \<in> (RAG s)\<^sup>+" by simp
ultimately show ?thesis by auto
qed
next
- assume "(Th th1, c) \<in> depend s"
+ assume "(Th th1, c) \<in> RAG s"
with h2 eq_c
have "th1 \<in> children s th2"
by (auto simp:children_def child_def)
thus ?thesis by auto
qed
next
- assume "(Th th1, Th th2) \<in> depend s"
+ assume "(Th th1, Th th2) \<in> RAG s"
thus ?thesis
- by (auto simp:s_depend_def)
+ by (auto simp:s_RAG_def)
qed
qed
-lemma sub_child: "child s \<subseteq> (depend s)^+"
+lemma sub_child: "child s \<subseteq> (RAG s)^+"
by (unfold child_def, auto)
lemma wf_child:
assumes vt: "vt s"
shows "wf (child s)"
apply(rule wf_subset)
-apply(rule wf_trancl[OF wf_depend[OF vt]])
+apply(rule wf_trancl[OF wf_RAG[OF vt]])
apply(rule sub_child)
done
-lemma depend_child_pre:
+lemma RAG_child_pre:
assumes vt: "vt s"
shows
- "(Th th, n) \<in> (depend s)^+ \<longrightarrow> (\<forall> th'. n = (Th th') \<longrightarrow> (Th th, Th th') \<in> (child s)^+)" (is "?P n")
+ "(Th th, n) \<in> (RAG s)^+ \<longrightarrow> (\<forall> th'. n = (Th th') \<longrightarrow> (Th th, Th th') \<in> (child s)^+)" (is "?P n")
proof -
- from wf_trancl[OF wf_depend[OF vt]]
- have wf: "wf ((depend s)^+)" .
+ from wf_trancl[OF wf_RAG[OF vt]]
+ have wf: "wf ((RAG s)^+)" .
show ?thesis
proof(rule wf_induct[OF wf, of ?P], clarsimp)
fix th'
- assume ih[rule_format]: "\<forall>y. (y, Th th') \<in> (depend s)\<^sup>+ \<longrightarrow>
- (Th th, y) \<in> (depend s)\<^sup>+ \<longrightarrow> (\<forall>th'. y = Th th' \<longrightarrow> (Th th, Th th') \<in> (child s)\<^sup>+)"
- and h: "(Th th, Th th') \<in> (depend s)\<^sup>+"
+ assume ih[rule_format]: "\<forall>y. (y, Th th') \<in> (RAG s)\<^sup>+ \<longrightarrow>
+ (Th th, y) \<in> (RAG s)\<^sup>+ \<longrightarrow> (\<forall>th'. y = Th th' \<longrightarrow> (Th th, Th th') \<in> (child s)\<^sup>+)"
+ and h: "(Th th, Th th') \<in> (RAG s)\<^sup>+"
show "(Th th, Th th') \<in> (child s)\<^sup>+"
proof -
- from depend_children[OF h]
- have "th \<in> children s th' \<or> (\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (depend s)\<^sup>+)" .
+ from RAG_children[OF h]
+ have "th \<in> children s th' \<or> (\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (RAG s)\<^sup>+)" .
thus ?thesis
proof
assume "th \<in> children s th'"
thus "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def)
next
- assume "\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (depend s)\<^sup>+"
+ assume "\<exists>th3. th3 \<in> children s th' \<and> (Th th, Th th3) \<in> (RAG s)\<^sup>+"
then obtain th3 where th3_in: "th3 \<in> children s th'"
- and th_dp: "(Th th, Th th3) \<in> (depend s)\<^sup>+" by auto
- from th3_in have "(Th th3, Th th') \<in> (depend s)^+" by (auto simp:children_def child_def)
+ and th_dp: "(Th th, Th th3) \<in> (RAG s)\<^sup>+" by auto
+ from th3_in have "(Th th3, Th th') \<in> (RAG s)^+" by (auto simp:children_def child_def)
from ih[OF this th_dp, of th3] have "(Th th, Th th3) \<in> (child s)\<^sup>+" by simp
with th3_in show "(Th th, Th th') \<in> (child s)\<^sup>+" by (auto simp:children_def)
qed
@@ -354,12 +290,12 @@
qed
qed
-lemma depend_child: "\<lbrakk>vt s; (Th th, Th th') \<in> (depend s)^+\<rbrakk> \<Longrightarrow> (Th th, Th th') \<in> (child s)^+"
- by (insert depend_child_pre, auto)
+lemma RAG_child: "\<lbrakk>vt s; (Th th, Th th') \<in> (RAG s)^+\<rbrakk> \<Longrightarrow> (Th th, Th th') \<in> (child s)^+"
+ by (insert RAG_child_pre, auto)
-lemma child_depend_p:
+lemma child_RAG_p:
assumes "(n1, n2) \<in> (child s)^+"
- shows "(n1, n2) \<in> (depend s)^+"
+ shows "(n1, n2) \<in> (RAG s)^+"
proof -
from assms show ?thesis
proof(induct)
@@ -368,26 +304,26 @@
next
case (step y z)
assume "(y, z) \<in> child s"
- with sub_child have "(y, z) \<in> (depend s)^+" by auto
- moreover have "(n1, y) \<in> (depend s)^+" by fact
+ with sub_child have "(y, z) \<in> (RAG s)^+" by auto
+ moreover have "(n1, y) \<in> (RAG s)^+" by fact
ultimately show ?case by auto
qed
qed
-lemma child_depend_eq:
+lemma child_RAG_eq:
assumes vt: "vt s"
- shows "(Th th1, Th th2) \<in> (child s)^+ \<longleftrightarrow> (Th th1, Th th2) \<in> (depend s)^+"
- by (auto intro: depend_child[OF vt] child_depend_p)
+ shows "(Th th1, Th th2) \<in> (child s)^+ \<longleftrightarrow> (Th th1, Th th2) \<in> (RAG s)^+"
+ by (auto intro: RAG_child[OF vt] child_RAG_p)
lemma children_no_dep:
fixes s th th1 th2 th3
assumes vt: "vt s"
and ch1: "(Th th1, Th th) \<in> child s"
and ch2: "(Th th2, Th th) \<in> child s"
- and ch3: "(Th th1, Th th2) \<in> (depend s)^+"
+ and ch3: "(Th th1, Th th2) \<in> (RAG s)^+"
shows "False"
proof -
- from depend_child[OF vt ch3]
+ from RAG_child[OF vt ch3]
have "(Th th1, Th th2) \<in> (child s)\<^sup>+" .
thus ?thesis
proof(rule converse_tranclE)
@@ -412,15 +348,15 @@
qed
qed
-lemma unique_depend_p:
+lemma unique_RAG_p:
assumes vt: "vt s"
- and dp1: "(n, n1) \<in> (depend s)^+"
- and dp2: "(n, n2) \<in> (depend s)^+"
+ and dp1: "(n, n1) \<in> (RAG s)^+"
+ and dp2: "(n, n2) \<in> (RAG s)^+"
and neq: "n1 \<noteq> n2"
- shows "(n1, n2) \<in> (depend s)^+ \<or> (n2, n1) \<in> (depend s)^+"
+ shows "(n1, n2) \<in> (RAG s)^+ \<or> (n2, n1) \<in> (RAG s)^+"
proof(rule unique_chain [OF _ dp1 dp2 neq])
- from unique_depend[OF vt]
- show "\<And>a b c. \<lbrakk>(a, b) \<in> depend s; (a, c) \<in> depend s\<rbrakk> \<Longrightarrow> b = c" by auto
+ from unique_RAG[OF vt]
+ show "\<And>a b c. \<lbrakk>(a, b) \<in> RAG s; (a, c) \<in> RAG s\<rbrakk> \<Longrightarrow> b = c" by auto
qed
lemma dependants_child_unique:
@@ -433,47 +369,41 @@
shows "th1 = th2"
proof -
{ assume neq: "th1 \<noteq> th2"
- from dp1 have dp1: "(Th th3, Th th1) \<in> (depend s)^+"
- by (simp add:s_dependants_def eq_depend)
- from dp2 have dp2: "(Th th3, Th th2) \<in> (depend s)^+"
- by (simp add:s_dependants_def eq_depend)
- from unique_depend_p[OF vt dp1 dp2] and neq
- have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
+ from dp1 have dp1: "(Th th3, Th th1) \<in> (RAG s)^+"
+ by (simp add:s_dependants_def eq_RAG)
+ from dp2 have dp2: "(Th th3, Th th2) \<in> (RAG s)^+"
+ by (simp add:s_dependants_def eq_RAG)
+ from unique_RAG_p[OF vt dp1 dp2] and neq
+ have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
hence False
proof
- assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+ "
+ assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ "
from children_no_dep[OF vt ch1 ch2 this] show ?thesis .
next
- assume " (Th th2, Th th1) \<in> (depend s)\<^sup>+"
+ assume " (Th th2, Th th1) \<in> (RAG s)\<^sup>+"
from children_no_dep[OF vt ch2 ch1 this] show ?thesis .
qed
} thus ?thesis by auto
qed
-lemma depend_plus_elim:
+lemma RAG_plus_elim:
assumes "vt s"
fixes x
- assumes "(Th x, Th th) \<in> (depend (wq s))\<^sup>+"
- shows "\<exists>th'\<in>children s th. x = th' \<or> (Th x, Th th') \<in> (depend (wq s))\<^sup>+"
- using assms(2)[unfolded eq_depend, folded child_depend_eq[OF `vt s`]]
+ assumes "(Th x, Th th) \<in> (RAG (wq s))\<^sup>+"
+ shows "\<exists>th'\<in>children s th. x = th' \<or> (Th x, Th th') \<in> (RAG (wq s))\<^sup>+"
+ using assms(2)[unfolded eq_RAG, folded child_RAG_eq[OF `vt s`]]
apply (unfold children_def)
- by (metis assms(2) children_def depend_children eq_depend)
-
-lemma dependants_expand_pre:
- assumes "vt s"
- shows "dependants (wq s) th = (\<Union> th' \<in> children s th. {th'} \<union> dependants (wq s) th')"
- apply (unfold cs_dependants_def)
- apply (auto elim!:depend_plus_elim[OF assms])
- apply (metis children_def eq_depend mem_Collect_eq set_mp sub_child)
- apply (unfold children_def, auto)
- apply (unfold eq_depend, fold child_depend_eq[OF `vt s`])
- by (metis trancl.simps)
+ by (metis assms(2) children_def RAG_children eq_RAG)
lemma dependants_expand:
assumes "vt s"
- shows "dependants (wq s) th = (\<Union> ((\<lambda> th. {th} \<union> dependants (wq s) th) ` (children s th)))"
- apply (subst dependants_expand_pre[OF assms])
- by simp
+ shows "dependants (wq s) th = (children s th) \<union> (\<Union>((dependants (wq s)) ` children s th))"
+apply(simp add: image_def)
+unfolding cs_dependants_def
+apply(auto)
+apply (metis assms RAG_plus_elim mem_Collect_eq)
+apply (metis child_RAG_p children_def eq_RAG mem_Collect_eq r_into_trancl')
+by (metis assms child_RAG_eq children_def eq_RAG mem_Collect_eq trancl.trancl_into_trancl)
lemma finite_children:
assumes "vt s"
@@ -487,23 +417,11 @@
using dependants_threads[OF assms] finite_threads[OF assms]
by (metis rev_finite_subset)
-lemma Max_insert:
- assumes "finite B"
- and "B \<noteq> {}"
- shows "Max ({x} \<union> B) = max x (Max B)"
- by (metis Max_insert assms insert_is_Un)
-
-lemma dependands_expand2:
- assumes "vt s"
- shows "dependants (wq s) th = (children s th) \<union> (\<Union>((dependants (wq s)) ` children s th))"
- by (subst dependants_expand[OF assms]) (auto)
+abbreviation
+ "preceds s ths \<equiv> {preced th s| th. th \<in> ths}"
abbreviation
- "preceds s Ths \<equiv> {preced th s| th. th \<in> Ths}"
-
-lemma image_compr:
- "f ` A = {f x | x. x \<in> A}"
-by auto
+ "cpreceds s ths \<equiv> (cp s) ` ths"
lemma Un_compr:
"{f th | th. R th \<or> Q th} = ({f th | th. R th} \<union> {f th' | th'. Q th'})"
@@ -531,11 +449,11 @@
show ?thesis (is "?LHS = ?RHS")
proof -
have eq_cp: "cp s = (\<lambda>th. Max (preceds s ({th} \<union> dependants (wq s) th)))"
- by (simp add: fun_eq_iff cp_eq_cpreced cpreced_def Un_compr image_compr[symmetric])
+ by (simp add: fun_eq_iff cp_eq_cpreced cpreced_def Un_compr image_Collect[symmetric])
have not_emptyness_facts[simp]:
"dependants (wq s) th \<noteq> {}" "children s th \<noteq> {}"
- using False dependands_expand2[OF assms] by(auto simp only: Un_empty)
+ using False dependants_expand[OF assms] by(auto simp only: Un_empty)
have finiteness_facts[simp]:
"\<And>th. finite (dependants (wq s) th)" "\<And>th. finite (children s th)"
@@ -552,7 +470,7 @@
(* expanding dependants *)
also have "\<dots> = max (Max {preced th s})
(Max (preceds s (children s th \<union> \<Union>(dependants (wq s) ` children s th))))"
- by (subst dependands_expand2[OF `vt s`]) (simp)
+ by (subst dependants_expand[OF `vt s`]) (simp)
(* moving out big Union *)
also have "\<dots> = max (Max {preced th s})
@@ -602,8 +520,8 @@
by (unfold s_def, auto simp:preced_def)
qed
-lemma eq_dep: "depend s = depend s'"
- by (unfold s_def depend_set_unchanged, auto)
+lemma eq_dep: "RAG s = RAG s'"
+ by (unfold s_def RAG_set_unchanged, auto)
lemma eq_cp_pre:
fixes th'
@@ -613,7 +531,7 @@
apply (unfold cp_eq_cpreced cpreced_def)
proof -
have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th"
- by (unfold cs_dependants_def, auto simp:eq_dep eq_depend)
+ by (unfold cs_dependants_def, auto simp:eq_dep eq_RAG)
moreover {
fix th1
assume "th1 \<in> {th'} \<union> dependants (wq s') th'"
@@ -626,7 +544,7 @@
next
assume "th1 \<in> dependants (wq s') th'"
with nd and eq_dp have "th1 \<noteq> th"
- by (auto simp:eq_depend cs_dependants_def s_dependants_def eq_dep)
+ by (auto simp:eq_RAG cs_dependants_def s_dependants_def eq_dep)
from eq_preced[OF this] show "preced th1 s = preced th1 s'" by simp
qed
} ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) =
@@ -646,12 +564,12 @@
hence "th \<in> runing s'" by (cases, simp)
hence rd_th: "th \<in> readys s'"
by (simp add:readys_def runing_def)
- from h have "(Th th, Th th') \<in> (depend s')\<^sup>+"
- by (unfold s_dependants_def, unfold eq_depend, unfold eq_dep, auto)
+ from h have "(Th th, Th th') \<in> (RAG s')\<^sup>+"
+ by (unfold s_dependants_def, unfold eq_RAG, unfold eq_dep, auto)
from tranclD[OF this]
- obtain z where "(Th th, z) \<in> depend s'" by auto
+ obtain z where "(Th th, z) \<in> RAG s'" by auto
with rd_th show "False"
- apply (case_tac z, auto simp:readys_def s_waiting_def s_depend_def s_waiting_def cs_waiting_def)
+ apply (case_tac z, auto simp:readys_def s_waiting_def s_RAG_def s_waiting_def cs_waiting_def)
by (fold wq_def, blast)
qed
@@ -673,8 +591,8 @@
shows "cp s th'' = cp s' th''"
proof -
from dp2
- have "(Th th', Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependants_def)
- from depend_child[OF vt_s this[unfolded eq_depend]]
+ have "(Th th', Th th'') \<in> (RAG (wq s))\<^sup>+" by (simp add:s_dependants_def)
+ from RAG_child[OF vt_s this[unfolded eq_RAG]]
have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" .
moreover { fix n th''
have "\<lbrakk>(Th th', n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
@@ -686,10 +604,10 @@
and ch': "(Th th', y) \<in> (child s)\<^sup>+"
from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
with ih have eq_cpy:"cp s thy = cp s' thy" by blast
- from dp1 have "(Th th, Th th') \<in> (depend s)^+" by (auto simp:s_dependants_def eq_depend)
- moreover from child_depend_p[OF ch'] and eq_y
- have "(Th th', Th thy) \<in> (depend s)^+" by simp
- ultimately have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by auto
+ from dp1 have "(Th th, Th th') \<in> (RAG s)^+" by (auto simp:s_dependants_def eq_RAG)
+ moreover from child_RAG_p[OF ch'] and eq_y
+ have "(Th th', Th thy) \<in> (RAG s)^+" by simp
+ ultimately have dp_thy: "(Th th, Th thy) \<in> (RAG s)^+" by auto
show "cp s th'' = cp s' th''"
apply (subst cp_rec[OF vt_s])
proof -
@@ -699,9 +617,9 @@
proof
assume "th'' = th"
with dp_thy y_ch[unfolded eq_y]
- have "(Th th, Th th) \<in> (depend s)^+"
+ have "(Th th, Th th) \<in> (RAG s)^+"
by (auto simp:child_def)
- with wf_trancl[OF wf_depend[OF vt_s]]
+ with wf_trancl[OF wf_RAG[OF vt_s]]
show False by auto
qed
qed
@@ -717,14 +635,14 @@
have neq_th1: "th1 \<noteq> th"
proof
assume eq_th1: "th1 = th"
- with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+ with dp_thy have "(Th th1, Th thy) \<in> (RAG s)^+" by simp
from children_no_dep[OF vt_s _ _ this] and
th1_in y_ch eq_y show False by (auto simp:children_def)
qed
have "th \<notin> dependants s th1"
proof
assume h:"th \<in> dependants s th1"
- from eq_y dp_thy have "th \<in> dependants s thy" by (auto simp:s_dependants_def eq_depend)
+ from eq_y dp_thy have "th \<in> dependants s thy" by (auto simp:s_dependants_def eq_RAG)
from dependants_child_unique[OF vt_s _ _ h this]
th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
with False show False by auto
@@ -736,7 +654,7 @@
ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
{preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
moreover have "children s th'' = children s' th''"
- by (unfold children_def child_def s_def depend_set_unchanged, simp)
+ by (unfold children_def child_def s_def RAG_set_unchanged, simp)
ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
qed
@@ -752,9 +670,9 @@
proof
assume "th'' = th"
with dp1 dp'
- have "(Th th, Th th) \<in> (depend s)^+"
- by (auto simp:child_def s_dependants_def eq_depend)
- with wf_trancl[OF wf_depend[OF vt_s]]
+ have "(Th th, Th th) \<in> (RAG s)^+"
+ by (auto simp:child_def s_dependants_def eq_RAG)
+ with wf_trancl[OF wf_RAG[OF vt_s]]
show False by auto
qed
qed
@@ -770,8 +688,8 @@
have neq_th1: "th1 \<noteq> th"
proof
assume eq_th1: "th1 = th"
- with dp1 have "(Th th1, Th th') \<in> (depend s)^+"
- by (auto simp:s_dependants_def eq_depend)
+ with dp1 have "(Th th1, Th th') \<in> (RAG s)^+"
+ by (auto simp:s_dependants_def eq_RAG)
from children_no_dep[OF vt_s _ _ this]
th1_in dp'
show False by (auto simp:children_def)
@@ -792,7 +710,7 @@
ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
{preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
moreover have "children s th'' = children s' th''"
- by (unfold children_def child_def s_def depend_set_unchanged, simp)
+ by (unfold children_def child_def s_def RAG_set_unchanged, simp)
ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
qed
@@ -808,8 +726,8 @@
shows "cp s th'' = cp s' th''"
proof -
from dp
- have "(Th th, Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependants_def)
- from depend_child[OF vt_s this[unfolded eq_depend]]
+ have "(Th th, Th th'') \<in> (RAG (wq s))\<^sup>+" by (simp add:s_dependants_def)
+ from RAG_child[OF vt_s this[unfolded eq_RAG]]
have ch_th': "(Th th, Th th'') \<in> (child s)\<^sup>+" .
moreover { fix n th''
have "\<lbrakk>(Th th, n) \<in> (child s)^+\<rbrakk> \<Longrightarrow>
@@ -821,8 +739,8 @@
and ch': "(Th th, y) \<in> (child s)\<^sup>+"
from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
with ih have eq_cpy:"cp s thy = cp s' thy" by blast
- from child_depend_p[OF ch'] and eq_y
- have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by simp
+ from child_RAG_p[OF ch'] and eq_y
+ have dp_thy: "(Th th, Th thy) \<in> (RAG s)^+" by simp
show "cp s th'' = cp s' th''"
apply (subst cp_rec[OF vt_s])
proof -
@@ -832,9 +750,9 @@
proof
assume "th'' = th"
with dp_thy y_ch[unfolded eq_y]
- have "(Th th, Th th) \<in> (depend s)^+"
+ have "(Th th, Th th) \<in> (RAG s)^+"
by (auto simp:child_def)
- with wf_trancl[OF wf_depend[OF vt_s]]
+ with wf_trancl[OF wf_RAG[OF vt_s]]
show False by auto
qed
qed
@@ -850,14 +768,14 @@
have neq_th1: "th1 \<noteq> th"
proof
assume eq_th1: "th1 = th"
- with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+ with dp_thy have "(Th th1, Th thy) \<in> (RAG s)^+" by simp
from children_no_dep[OF vt_s _ _ this] and
th1_in y_ch eq_y show False by (auto simp:children_def)
qed
have "th \<notin> dependants s th1"
proof
assume h:"th \<in> dependants s th1"
- from eq_y dp_thy have "th \<in> dependants s thy" by (auto simp:s_dependants_def eq_depend)
+ from eq_y dp_thy have "th \<in> dependants s thy" by (auto simp:s_dependants_def eq_RAG)
from dependants_child_unique[OF vt_s _ _ h this]
th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
with False show False by auto
@@ -869,7 +787,7 @@
ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
{preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
moreover have "children s th'' = children s' th''"
- by (unfold children_def child_def s_def depend_set_unchanged, simp)
+ by (unfold children_def child_def s_def RAG_set_unchanged, simp)
ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
qed
@@ -885,9 +803,9 @@
proof
assume "th'' = th"
with dp dp'
- have "(Th th, Th th) \<in> (depend s)^+"
- by (auto simp:child_def s_dependants_def eq_depend)
- with wf_trancl[OF wf_depend[OF vt_s]]
+ have "(Th th, Th th) \<in> (RAG s)^+"
+ by (auto simp:child_def s_dependants_def eq_RAG)
+ with wf_trancl[OF wf_RAG[OF vt_s]]
show False by auto
qed
qed
@@ -906,7 +824,7 @@
show "th \<notin> dependants s th1"
proof
assume "th \<in> dependants s th1"
- hence "(Th th, Th th1) \<in> (depend s)^+" by (auto simp:s_dependants_def eq_depend)
+ hence "(Th th, Th th1) \<in> (RAG s)^+" by (auto simp:s_dependants_def eq_RAG)
from children_no_dep[OF vt_s _ _ this]
and th1_in dp' show False
by (auto simp:children_def)
@@ -917,7 +835,7 @@
ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
{preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
moreover have "children s th'' = children s' th''"
- by (unfold children_def child_def s_def depend_set_unchanged, simp)
+ by (unfold children_def child_def s_def RAG_set_unchanged, simp)
ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
qed
@@ -998,11 +916,11 @@
context step_v_cps_nt
begin
-lemma depend_s:
- "depend s = (depend s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
+lemma RAG_s:
+ "RAG s = (RAG s' - {(Cs cs, Th th), (Th th', Cs cs)}) \<union>
{(Cs cs, Th th')}"
proof -
- from step_depend_v[OF vt_s[unfolded s_def], folded s_def]
+ from step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
and nt show ?thesis by (auto intro:next_th_unique)
qed
@@ -1014,163 +932,163 @@
proof(auto)
fix x
assume "x \<in> dependants (wq s) th''"
- hence dp: "(Th x, Th th'') \<in> (depend s)^+"
- by (auto simp:cs_dependants_def eq_depend)
+ hence dp: "(Th x, Th th'') \<in> (RAG s)^+"
+ by (auto simp:cs_dependants_def eq_RAG)
{ fix n
- have "(n, Th th'') \<in> (depend s)^+ \<Longrightarrow> (n, Th th'') \<in> (depend s')^+"
+ have "(n, Th th'') \<in> (RAG s)^+ \<Longrightarrow> (n, Th th'') \<in> (RAG s')^+"
proof(induct rule:converse_trancl_induct)
fix y
- assume "(y, Th th'') \<in> depend s"
- with depend_s neq1 neq2
- have "(y, Th th'') \<in> depend s'" by auto
- thus "(y, Th th'') \<in> (depend s')\<^sup>+" by auto
+ assume "(y, Th th'') \<in> RAG s"
+ with RAG_s neq1 neq2
+ have "(y, Th th'') \<in> RAG s'" by auto
+ thus "(y, Th th'') \<in> (RAG s')\<^sup>+" by auto
next
fix y z
- assume yz: "(y, z) \<in> depend s"
- and ztp: "(z, Th th'') \<in> (depend s)\<^sup>+"
- and ztp': "(z, Th th'') \<in> (depend s')\<^sup>+"
+ assume yz: "(y, z) \<in> RAG s"
+ and ztp: "(z, Th th'') \<in> (RAG s)\<^sup>+"
+ and ztp': "(z, Th th'') \<in> (RAG s')\<^sup>+"
have "y \<noteq> Cs cs \<and> y \<noteq> Th th'"
proof
show "y \<noteq> Cs cs"
proof
assume eq_y: "y = Cs cs"
- with yz have dp_yz: "(Cs cs, z) \<in> depend s" by simp
- from depend_s
- have cst': "(Cs cs, Th th') \<in> depend s" by simp
- from unique_depend[OF vt_s this dp_yz]
+ with yz have dp_yz: "(Cs cs, z) \<in> RAG s" by simp
+ from RAG_s
+ have cst': "(Cs cs, Th th') \<in> RAG s" by simp
+ from unique_RAG[OF vt_s this dp_yz]
have eq_z: "z = Th th'" by simp
- with ztp have "(Th th', Th th'') \<in> (depend s)^+" by simp
+ with ztp have "(Th th', Th th'') \<in> (RAG s)^+" by simp
from converse_tranclE[OF this]
- obtain cs' where dp'': "(Th th', Cs cs') \<in> depend s"
- by (auto simp:s_depend_def)
- with depend_s have dp': "(Th th', Cs cs') \<in> depend s'" by auto
- from dp'' eq_y yz eq_z have "(Cs cs, Cs cs') \<in> (depend s)^+" by auto
+ obtain cs' where dp'': "(Th th', Cs cs') \<in> RAG s"
+ by (auto simp:s_RAG_def)
+ with RAG_s have dp': "(Th th', Cs cs') \<in> RAG s'" by auto
+ from dp'' eq_y yz eq_z have "(Cs cs, Cs cs') \<in> (RAG s)^+" by auto
moreover have "cs' = cs"
proof -
from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
- have "(Th th', Cs cs) \<in> depend s'"
- by (auto simp:s_waiting_def wq_def s_depend_def cs_waiting_def)
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this dp']
+ have "(Th th', Cs cs) \<in> RAG s'"
+ by (auto simp:s_waiting_def wq_def s_RAG_def cs_waiting_def)
+ from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] this dp']
show ?thesis by simp
qed
- ultimately have "(Cs cs, Cs cs) \<in> (depend s)^+" by simp
- moreover note wf_trancl[OF wf_depend[OF vt_s]]
+ ultimately have "(Cs cs, Cs cs) \<in> (RAG s)^+" by simp
+ moreover note wf_trancl[OF wf_RAG[OF vt_s]]
ultimately show False by auto
qed
next
show "y \<noteq> Th th'"
proof
assume eq_y: "y = Th th'"
- with yz have dps: "(Th th', z) \<in> depend s" by simp
- with depend_s have dps': "(Th th', z) \<in> depend s'" by auto
+ with yz have dps: "(Th th', z) \<in> RAG s" by simp
+ with RAG_s have dps': "(Th th', z) \<in> RAG s'" by auto
have "z = Cs cs"
proof -
from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
- have "(Th th', Cs cs) \<in> depend s'"
- by (auto simp:s_waiting_def wq_def s_depend_def cs_waiting_def)
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] dps' this]
+ have "(Th th', Cs cs) \<in> RAG s'"
+ by (auto simp:s_waiting_def wq_def s_RAG_def cs_waiting_def)
+ from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] dps' this]
show ?thesis .
qed
- with dps depend_s show False by auto
+ with dps RAG_s show False by auto
qed
qed
- with depend_s yz have "(y, z) \<in> depend s'" by auto
+ with RAG_s yz have "(y, z) \<in> RAG s'" by auto
with ztp'
- show "(y, Th th'') \<in> (depend s')\<^sup>+" by auto
+ show "(y, Th th'') \<in> (RAG s')\<^sup>+" by auto
qed
}
from this[OF dp]
show "x \<in> dependants (wq s') th''"
- by (auto simp:cs_dependants_def eq_depend)
+ by (auto simp:cs_dependants_def eq_RAG)
next
fix x
assume "x \<in> dependants (wq s') th''"
- hence dp: "(Th x, Th th'') \<in> (depend s')^+"
- by (auto simp:cs_dependants_def eq_depend)
+ hence dp: "(Th x, Th th'') \<in> (RAG s')^+"
+ by (auto simp:cs_dependants_def eq_RAG)
{ fix n
- have "(n, Th th'') \<in> (depend s')^+ \<Longrightarrow> (n, Th th'') \<in> (depend s)^+"
+ have "(n, Th th'') \<in> (RAG s')^+ \<Longrightarrow> (n, Th th'') \<in> (RAG s)^+"
proof(induct rule:converse_trancl_induct)
fix y
- assume "(y, Th th'') \<in> depend s'"
- with depend_s neq1 neq2
- have "(y, Th th'') \<in> depend s" by auto
- thus "(y, Th th'') \<in> (depend s)\<^sup>+" by auto
+ assume "(y, Th th'') \<in> RAG s'"
+ with RAG_s neq1 neq2
+ have "(y, Th th'') \<in> RAG s" by auto
+ thus "(y, Th th'') \<in> (RAG s)\<^sup>+" by auto
next
fix y z
- assume yz: "(y, z) \<in> depend s'"
- and ztp: "(z, Th th'') \<in> (depend s')\<^sup>+"
- and ztp': "(z, Th th'') \<in> (depend s)\<^sup>+"
+ assume yz: "(y, z) \<in> RAG s'"
+ and ztp: "(z, Th th'') \<in> (RAG s')\<^sup>+"
+ and ztp': "(z, Th th'') \<in> (RAG s)\<^sup>+"
have "y \<noteq> Cs cs \<and> y \<noteq> Th th'"
proof
show "y \<noteq> Cs cs"
proof
assume eq_y: "y = Cs cs"
- with yz have dp_yz: "(Cs cs, z) \<in> depend s'" by simp
+ with yz have dp_yz: "(Cs cs, z) \<in> RAG s'" by simp
from this have eq_z: "z = Th th"
proof -
from step_back_step[OF vt_s[unfolded s_def]]
- have "(Cs cs, Th th) \<in> depend s'"
- by(cases, auto simp: wq_def s_depend_def cs_holding_def s_holding_def)
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this dp_yz]
+ have "(Cs cs, Th th) \<in> RAG s'"
+ by(cases, auto simp: wq_def s_RAG_def cs_holding_def s_holding_def)
+ from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] this dp_yz]
show ?thesis by simp
qed
from converse_tranclE[OF ztp]
- obtain u where "(z, u) \<in> depend s'" by auto
+ obtain u where "(z, u) \<in> RAG s'" by auto
moreover
from step_back_step[OF vt_s[unfolded s_def]]
have "th \<in> readys s'" by (cases, simp add:runing_def)
moreover note eq_z
ultimately show False
- by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def)
+ by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
qed
next
show "y \<noteq> Th th'"
proof
assume eq_y: "y = Th th'"
- with yz have dps: "(Th th', z) \<in> depend s'" by simp
+ with yz have dps: "(Th th', z) \<in> RAG s'" by simp
have "z = Cs cs"
proof -
from next_waiting[OF step_back_vt[OF vt_s[unfolded s_def]] nt]
- have "(Th th', Cs cs) \<in> depend s'"
- by (auto simp:s_waiting_def wq_def s_depend_def cs_waiting_def)
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] dps this]
+ have "(Th th', Cs cs) \<in> RAG s'"
+ by (auto simp:s_waiting_def wq_def s_RAG_def cs_waiting_def)
+ from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] dps this]
show ?thesis .
qed
- with ztp have cs_i: "(Cs cs, Th th'') \<in> (depend s')\<^sup>+" by simp
+ with ztp have cs_i: "(Cs cs, Th th'') \<in> (RAG s')\<^sup>+" by simp
from step_back_step[OF vt_s[unfolded s_def]]
- have cs_th: "(Cs cs, Th th) \<in> depend s'"
- by(cases, auto simp: s_depend_def wq_def cs_holding_def s_holding_def)
- have "(Cs cs, Th th'') \<notin> depend s'"
+ have cs_th: "(Cs cs, Th th) \<in> RAG s'"
+ by(cases, auto simp: s_RAG_def wq_def cs_holding_def s_holding_def)
+ have "(Cs cs, Th th'') \<notin> RAG s'"
proof
- assume "(Cs cs, Th th'') \<in> depend s'"
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] this cs_th]
+ assume "(Cs cs, Th th'') \<in> RAG s'"
+ from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] this cs_th]
and neq1 show "False" by simp
qed
with converse_tranclE[OF cs_i]
- obtain u where cu: "(Cs cs, u) \<in> depend s'"
- and u_t: "(u, Th th'') \<in> (depend s')\<^sup>+" by auto
+ obtain u where cu: "(Cs cs, u) \<in> RAG s'"
+ and u_t: "(u, Th th'') \<in> (RAG s')\<^sup>+" by auto
have "u = Th th"
proof -
- from unique_depend[OF step_back_vt[OF vt_s[unfolded s_def]] cu cs_th]
+ from unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]] cu cs_th]
show ?thesis .
qed
- with u_t have "(Th th, Th th'') \<in> (depend s')\<^sup>+" by simp
+ with u_t have "(Th th, Th th'') \<in> (RAG s')\<^sup>+" by simp
from converse_tranclE[OF this]
- obtain v where "(Th th, v) \<in> (depend s')" by auto
+ obtain v where "(Th th, v) \<in> (RAG s')" by auto
moreover from step_back_step[OF vt_s[unfolded s_def]]
have "th \<in> readys s'" by (cases, simp add:runing_def)
ultimately show False
- by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def)
+ by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
qed
qed
- with depend_s yz have "(y, z) \<in> depend s" by auto
+ with RAG_s yz have "(y, z) \<in> RAG s" by auto
with ztp'
- show "(y, Th th'') \<in> (depend s)\<^sup>+" by auto
+ show "(y, Th th'') \<in> (RAG s)\<^sup>+" by auto
qed
}
from this[OF dp]
show "x \<in> dependants (wq s) th''"
- by (auto simp:cs_dependants_def eq_depend)
+ by (auto simp:cs_dependants_def eq_RAG)
qed
lemma cp_kept:
@@ -1204,11 +1122,11 @@
context step_v_cps_nnt
begin
-lemma nw_cs: "(Th th1, Cs cs) \<notin> depend s'"
+lemma nw_cs: "(Th th1, Cs cs) \<notin> RAG s'"
proof
- assume "(Th th1, Cs cs) \<in> depend s'"
+ assume "(Th th1, Cs cs) \<in> RAG s'"
thus "False"
- apply (auto simp:s_depend_def cs_waiting_def)
+ apply (auto simp:s_RAG_def cs_waiting_def)
proof -
assume h1: "th1 \<in> set (wq s' cs)"
and h2: "th1 \<noteq> hd (wq s' cs)"
@@ -1229,9 +1147,9 @@
qed
qed
-lemma depend_s: "depend s = depend s' - {(Cs cs, Th th)}"
+lemma RAG_s: "RAG s = RAG s' - {(Cs cs, Th th)}"
proof -
- from nnt and step_depend_v[OF vt_s[unfolded s_def], folded s_def]
+ from nnt and step_RAG_v[OF vt_s[unfolded s_def], folded s_def]
show ?thesis by auto
qed
@@ -1244,18 +1162,18 @@
proof(induct rule: converse_trancl_induct)
case (base y)
from base obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s'"
- and h2: "(Cs cs1, Th th2) \<in> depend s'"
+ where h1: "(Th th1, Cs cs1) \<in> RAG s'"
+ and h2: "(Cs cs1, Th th2) \<in> RAG s'"
and eq_y: "y = Th th1" and eq_n2: "n2 = Th th2" by (auto simp:child_def)
have "cs1 \<noteq> cs"
proof
assume eq_cs: "cs1 = cs"
- with h1 have "(Th th1, Cs cs1) \<in> depend s'" by simp
+ with h1 have "(Th th1, Cs cs1) \<in> RAG s'" by simp
with nw_cs eq_cs show False by auto
qed
- with h1 h2 depend_s have
- h1': "(Th th1, Cs cs1) \<in> depend s" and
- h2': "(Cs cs1, Th th2) \<in> depend s" by auto
+ with h1 h2 RAG_s have
+ h1': "(Th th1, Cs cs1) \<in> RAG s" and
+ h2': "(Cs cs1, Th th2) \<in> RAG s" by auto
hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
with eq_y eq_n2 have "(y, n2) \<in> child s" by simp
thus ?case by auto
@@ -1263,18 +1181,18 @@
case (step y z)
have "(y, z) \<in> child s'" by fact
then obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s'"
- and h2: "(Cs cs1, Th th2) \<in> depend s'"
+ where h1: "(Th th1, Cs cs1) \<in> RAG s'"
+ and h2: "(Cs cs1, Th th2) \<in> RAG s'"
and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def)
have "cs1 \<noteq> cs"
proof
assume eq_cs: "cs1 = cs"
- with h1 have "(Th th1, Cs cs1) \<in> depend s'" by simp
+ with h1 have "(Th th1, Cs cs1) \<in> RAG s'" by simp
with nw_cs eq_cs show False by auto
qed
- with h1 h2 depend_s have
- h1': "(Th th1, Cs cs1) \<in> depend s" and
- h2': "(Cs cs1, Th th2) \<in> depend s" by auto
+ with h1 h2 RAG_s have
+ h1': "(Th th1, Cs cs1) \<in> RAG s" and
+ h2': "(Cs cs1, Th th2) \<in> RAG s" by auto
hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
with eq_y eq_z have "(y, z) \<in> child s" by simp
moreover have "(z, n2) \<in> (child s)^+" by fact
@@ -1290,14 +1208,14 @@
from assms show ?thesis
proof(induct)
case (base y)
- from base and depend_s
+ from base and RAG_s
have "(n1, y) \<in> child s'"
by (auto simp:child_def)
thus ?case by auto
next
case (step y z)
have "(y, z) \<in> child s" by fact
- with depend_s have "(y, z) \<in> child s'"
+ with RAG_s have "(y, z) \<in> child s'"
by (auto simp:child_def)
moreover have "(n1, y) \<in> (child s')\<^sup>+" by fact
ultimately show ?case by auto
@@ -1313,13 +1231,13 @@
apply (unfold cp_eq_cpreced cpreced_def)
proof -
have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th"
- apply (unfold cs_dependants_def, unfold eq_depend)
+ apply (unfold cs_dependants_def, unfold eq_RAG)
proof -
from eq_child
have "\<And>th. {th'. (Th th', Th th) \<in> (child s)\<^sup>+} = {th'. (Th th', Th th) \<in> (child s')\<^sup>+}"
by simp
- with child_depend_eq[OF vt_s] child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
- show "\<And>th. {th'. (Th th', Th th) \<in> (depend s)\<^sup>+} = {th'. (Th th', Th th) \<in> (depend s')\<^sup>+}"
+ with child_RAG_eq[OF vt_s] child_RAG_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ show "\<And>th. {th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} = {th'. (Th th', Th th) \<in> (RAG s')\<^sup>+}"
by simp
qed
moreover {
@@ -1357,9 +1275,9 @@
context step_P_cps_e
begin
-lemma depend_s: "depend s = depend s' \<union> {(Cs cs, Th th)}"
+lemma RAG_s: "RAG s = RAG s' \<union> {(Cs cs, Th th)}"
proof -
- from ee and step_depend_p[OF vt_s[unfolded s_def], folded s_def]
+ from ee and step_RAG_p[OF vt_s[unfolded s_def], folded s_def]
show ?thesis by auto
qed
@@ -1372,19 +1290,19 @@
proof(induct rule: converse_trancl_induct)
case (base y)
from base obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s'"
- and h2: "(Cs cs1, Th th2) \<in> depend s'"
+ where h1: "(Th th1, Cs cs1) \<in> RAG s'"
+ and h2: "(Cs cs1, Th th2) \<in> RAG s'"
and eq_y: "y = Th th1" and eq_n2: "n2 = Th th2" by (auto simp:child_def)
have "cs1 \<noteq> cs"
proof
assume eq_cs: "cs1 = cs"
- with h1 have "(Th th1, Cs cs) \<in> depend s'" by simp
+ with h1 have "(Th th1, Cs cs) \<in> RAG s'" by simp
with ee show False
- by (auto simp:s_depend_def cs_waiting_def)
+ by (auto simp:s_RAG_def cs_waiting_def)
qed
- with h1 h2 depend_s have
- h1': "(Th th1, Cs cs1) \<in> depend s" and
- h2': "(Cs cs1, Th th2) \<in> depend s" by auto
+ with h1 h2 RAG_s have
+ h1': "(Th th1, Cs cs1) \<in> RAG s" and
+ h2': "(Cs cs1, Th th2) \<in> RAG s" by auto
hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
with eq_y eq_n2 have "(y, n2) \<in> child s" by simp
thus ?case by auto
@@ -1392,19 +1310,19 @@
case (step y z)
have "(y, z) \<in> child s'" by fact
then obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s'"
- and h2: "(Cs cs1, Th th2) \<in> depend s'"
+ where h1: "(Th th1, Cs cs1) \<in> RAG s'"
+ and h2: "(Cs cs1, Th th2) \<in> RAG s'"
and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def)
have "cs1 \<noteq> cs"
proof
assume eq_cs: "cs1 = cs"
- with h1 have "(Th th1, Cs cs) \<in> depend s'" by simp
+ with h1 have "(Th th1, Cs cs) \<in> RAG s'" by simp
with ee show False
- by (auto simp:s_depend_def cs_waiting_def)
+ by (auto simp:s_RAG_def cs_waiting_def)
qed
- with h1 h2 depend_s have
- h1': "(Th th1, Cs cs1) \<in> depend s" and
- h2': "(Cs cs1, Th th2) \<in> depend s" by auto
+ with h1 h2 RAG_s have
+ h1': "(Th th1, Cs cs1) \<in> RAG s" and
+ h2': "(Cs cs1, Th th2) \<in> RAG s" by auto
hence "(Th th1, Th th2) \<in> child s" by (auto simp:child_def)
with eq_y eq_z have "(y, z) \<in> child s" by simp
moreover have "(z, n2) \<in> (child s)^+" by fact
@@ -1420,28 +1338,28 @@
from assms show ?thesis
proof(induct)
case (base y)
- from base and depend_s
+ from base and RAG_s
have "(n1, y) \<in> child s'"
apply (auto simp:child_def)
proof -
fix th'
- assume "(Th th', Cs cs) \<in> depend s'"
+ assume "(Th th', Cs cs) \<in> RAG s'"
with ee have "False"
- by (auto simp:s_depend_def cs_waiting_def)
- thus "\<exists>cs. (Th th', Cs cs) \<in> depend s' \<and> (Cs cs, Th th) \<in> depend s'" by auto
+ by (auto simp:s_RAG_def cs_waiting_def)
+ thus "\<exists>cs. (Th th', Cs cs) \<in> RAG s' \<and> (Cs cs, Th th) \<in> RAG s'" by auto
qed
thus ?case by auto
next
case (step y z)
have "(y, z) \<in> child s" by fact
- with depend_s have "(y, z) \<in> child s'"
+ with RAG_s have "(y, z) \<in> child s'"
apply (auto simp:child_def)
proof -
fix th'
- assume "(Th th', Cs cs) \<in> depend s'"
+ assume "(Th th', Cs cs) \<in> RAG s'"
with ee have "False"
- by (auto simp:s_depend_def cs_waiting_def)
- thus "\<exists>cs. (Th th', Cs cs) \<in> depend s' \<and> (Cs cs, Th th) \<in> depend s'" by auto
+ by (auto simp:s_RAG_def cs_waiting_def)
+ thus "\<exists>cs. (Th th', Cs cs) \<in> RAG s' \<and> (Cs cs, Th th) \<in> RAG s'" by auto
qed
moreover have "(n1, y) \<in> (child s')\<^sup>+" by fact
ultimately show ?case by auto
@@ -1457,13 +1375,13 @@
apply (unfold cp_eq_cpreced cpreced_def)
proof -
have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th"
- apply (unfold cs_dependants_def, unfold eq_depend)
+ apply (unfold cs_dependants_def, unfold eq_RAG)
proof -
from eq_child
have "\<And>th. {th'. (Th th', Th th) \<in> (child s)\<^sup>+} = {th'. (Th th', Th th) \<in> (child s')\<^sup>+}"
by auto
- with child_depend_eq[OF vt_s] child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
- show "\<And>th. {th'. (Th th', Th th) \<in> (depend s)\<^sup>+} = {th'. (Th th', Th th) \<in> (depend s')\<^sup>+}"
+ with child_RAG_eq[OF vt_s] child_RAG_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ show "\<And>th. {th'. (Th th', Th th) \<in> (RAG s)\<^sup>+} = {th'. (Th th', Th th) \<in> (RAG s')\<^sup>+}"
by simp
qed
moreover {
@@ -1490,9 +1408,9 @@
context step_P_cps_ne
begin
-lemma depend_s: "depend s = depend s' \<union> {(Th th, Cs cs)}"
+lemma RAG_s: "RAG s = RAG s' \<union> {(Th th, Cs cs)}"
proof -
- from step_depend_p[OF vt_s[unfolded s_def]] and ne
+ from step_RAG_p[OF vt_s[unfolded s_def]] and ne
show ?thesis by (simp add:s_def)
qed
@@ -1502,8 +1420,8 @@
proof(induct rule:converse_trancl_induct)
case (base y)
from base obtain th1 cs1
- where h1: "(Th th1, Cs cs1) \<in> depend s"
- and h2: "(Cs cs1, Th th') \<in> depend s"
+ where h1: "(Th th1, Cs cs1) \<in> RAG s"
+ and h2: "(Cs cs1, Th th') \<in> RAG s"
and eq_y: "y = Th th1" by (auto simp:child_def)
have "th1 \<noteq> th"
proof
@@ -1511,16 +1429,16 @@
with base eq_y have "(Th th, Th th') \<in> child s" by simp
with nd show False by auto
qed
- with h1 h2 depend_s
- have h1': "(Th th1, Cs cs1) \<in> depend s'" and
- h2': "(Cs cs1, Th th') \<in> depend s'" by auto
+ with h1 h2 RAG_s
+ have h1': "(Th th1, Cs cs1) \<in> RAG s'" and
+ h2': "(Cs cs1, Th th') \<in> RAG s'" by auto
with eq_y show ?case by (auto simp:child_def)
next
case (step y z)
have yz: "(y, z) \<in> child s" by fact
then obtain th1 cs1 th2
- where h1: "(Th th1, Cs cs1) \<in> depend s"
- and h2: "(Cs cs1, Th th2) \<in> depend s"
+ where h1: "(Th th1, Cs cs1) \<in> RAG s"
+ and h2: "(Cs cs1, Th th2) \<in> RAG s"
and eq_y: "y = Th th1" and eq_z: "z = Th th2" by (auto simp:child_def)
have "th1 \<noteq> th"
proof
@@ -1530,8 +1448,8 @@
ultimately have "(Th th, Th th') \<in> (child s)^+" by auto
with nd show False by auto
qed
- with h1 h2 depend_s have h1': "(Th th1, Cs cs1) \<in> depend s'"
- and h2': "(Cs cs1, Th th2) \<in> depend s'" by auto
+ with h1 h2 RAG_s have h1': "(Th th1, Cs cs1) \<in> RAG s'"
+ and h2': "(Cs cs1, Th th2) \<in> RAG s'" by auto
with eq_y eq_z have "(y, z) \<in> child s'" by (auto simp:child_def)
moreover have "(z, Th th') \<in> (child s')^+" by fact
ultimately show ?case by auto
@@ -1541,11 +1459,11 @@
shows "(n1, Th th') \<in> (child s')^+ \<Longrightarrow> (n1, Th th') \<in> (child s)^+"
proof(induct rule:converse_trancl_induct)
case (base y)
- with depend_s show ?case by (auto simp:child_def)
+ with RAG_s show ?case by (auto simp:child_def)
next
case (step y z)
have "(y, z) \<in> child s'" by fact
- with depend_s have "(y, z) \<in> child s" by (auto simp:child_def)
+ with RAG_s have "(y, z) \<in> child s" by (auto simp:child_def)
moreover have "(z, Th th') \<in> (child s)^+" by fact
ultimately show ?case by auto
qed
@@ -1564,34 +1482,34 @@
have nd': "(Th th, Th th') \<notin> (child s)^+"
proof
assume "(Th th, Th th') \<in> (child s)\<^sup>+"
- with child_depend_eq[OF vt_s]
- have "(Th th, Th th') \<in> (depend s)\<^sup>+" by simp
+ with child_RAG_eq[OF vt_s]
+ have "(Th th, Th th') \<in> (RAG s)\<^sup>+" by simp
with nd show False
- by (simp add:s_dependants_def eq_depend)
+ by (simp add:s_dependants_def eq_RAG)
qed
have eq_dp: "dependants (wq s) th' = dependants (wq s') th'"
proof(auto)
fix x assume " x \<in> dependants (wq s) th'"
thus "x \<in> dependants (wq s') th'"
- apply (auto simp:cs_dependants_def eq_depend)
+ apply (auto simp:cs_dependants_def eq_RAG)
proof -
- assume "(Th x, Th th') \<in> (depend s)\<^sup>+"
- with child_depend_eq[OF vt_s] have "(Th x, Th th') \<in> (child s)\<^sup>+" by simp
+ assume "(Th x, Th th') \<in> (RAG s)\<^sup>+"
+ with child_RAG_eq[OF vt_s] have "(Th x, Th th') \<in> (child s)\<^sup>+" by simp
with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s')^+" by simp
- with child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
- show "(Th x, Th th') \<in> (depend s')\<^sup>+" by simp
+ with child_RAG_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ show "(Th x, Th th') \<in> (RAG s')\<^sup>+" by simp
qed
next
fix x assume "x \<in> dependants (wq s') th'"
thus "x \<in> dependants (wq s) th'"
- apply (auto simp:cs_dependants_def eq_depend)
+ apply (auto simp:cs_dependants_def eq_RAG)
proof -
- assume "(Th x, Th th') \<in> (depend s')\<^sup>+"
- with child_depend_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ assume "(Th x, Th th') \<in> (RAG s')\<^sup>+"
+ with child_RAG_eq[OF step_back_vt[OF vt_s[unfolded s_def]]]
have "(Th x, Th th') \<in> (child s')\<^sup>+" by simp
with eq_child[OF nd'] have "(Th x, Th th') \<in> (child s)^+" by simp
- with child_depend_eq[OF vt_s]
- show "(Th x, Th th') \<in> (depend s)\<^sup>+" by simp
+ with child_RAG_eq[OF vt_s]
+ show "(Th x, Th th') \<in> (RAG s)\<^sup>+" by simp
qed
qed
moreover {
@@ -1611,8 +1529,8 @@
shows "cp s th'' = cp s' th''"
proof -
from dp2
- have "(Th th', Th th'') \<in> (depend (wq s))\<^sup>+" by (simp add:s_dependants_def)
- from depend_child[OF vt_s this[unfolded eq_depend]]
+ have "(Th th', Th th'') \<in> (RAG (wq s))\<^sup>+" by (simp add:s_dependants_def)
+ from RAG_child[OF vt_s this[unfolded eq_RAG]]
have ch_th': "(Th th', Th th'') \<in> (child s)\<^sup>+" .
moreover {
fix n th''
@@ -1625,10 +1543,10 @@
and ch': "(Th th', y) \<in> (child s)\<^sup>+"
from y_ch obtain thy where eq_y: "y = Th thy" by (auto simp:child_def)
with ih have eq_cpy:"cp s thy = cp s' thy" by blast
- from dp1 have "(Th th, Th th') \<in> (depend s)^+" by (auto simp:s_dependants_def eq_depend)
- moreover from child_depend_p[OF ch'] and eq_y
- have "(Th th', Th thy) \<in> (depend s)^+" by simp
- ultimately have dp_thy: "(Th th, Th thy) \<in> (depend s)^+" by auto
+ from dp1 have "(Th th, Th th') \<in> (RAG s)^+" by (auto simp:s_dependants_def eq_RAG)
+ moreover from child_RAG_p[OF ch'] and eq_y
+ have "(Th th', Th thy) \<in> (RAG s)^+" by simp
+ ultimately have dp_thy: "(Th th, Th thy) \<in> (RAG s)^+" by auto
show "cp s th'' = cp s' th''"
apply (subst cp_rec[OF vt_s])
proof -
@@ -1646,14 +1564,14 @@
have neq_th1: "th1 \<noteq> th"
proof
assume eq_th1: "th1 = th"
- with dp_thy have "(Th th1, Th thy) \<in> (depend s)^+" by simp
+ with dp_thy have "(Th th1, Th thy) \<in> (RAG s)^+" by simp
from children_no_dep[OF vt_s _ _ this] and
th1_in y_ch eq_y show False by (auto simp:children_def)
qed
have "th \<notin> dependants s th1"
proof
assume h:"th \<in> dependants s th1"
- from eq_y dp_thy have "th \<in> dependants s thy" by (auto simp:s_dependants_def eq_depend)
+ from eq_y dp_thy have "th \<in> dependants s thy" by (auto simp:s_dependants_def eq_RAG)
from dependants_child_unique[OF vt_s _ _ h this]
th1_in y_ch eq_y have "th1 = thy" by (auto simp:children_def child_def)
with False show False by auto
@@ -1665,48 +1583,48 @@
ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
{preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
moreover have "children s th'' = children s' th''"
- apply (unfold children_def child_def s_def depend_set_unchanged, simp)
- apply (fold s_def, auto simp:depend_s)
+ apply (unfold children_def child_def s_def RAG_set_unchanged, simp)
+ apply (fold s_def, auto simp:RAG_s)
proof -
- assume "(Cs cs, Th th'') \<in> depend s'"
- with depend_s have cs_th': "(Cs cs, Th th'') \<in> depend s" by auto
- from dp1 have "(Th th, Th th') \<in> (depend s)^+"
- by (auto simp:s_dependants_def eq_depend)
+ assume "(Cs cs, Th th'') \<in> RAG s'"
+ with RAG_s have cs_th': "(Cs cs, Th th'') \<in> RAG s" by auto
+ from dp1 have "(Th th, Th th') \<in> (RAG s)^+"
+ by (auto simp:s_dependants_def eq_RAG)
from converse_tranclE[OF this]
- obtain cs1 where h1: "(Th th, Cs cs1) \<in> depend s"
- and h2: "(Cs cs1 , Th th') \<in> (depend s)\<^sup>+"
- by (auto simp:s_depend_def)
+ obtain cs1 where h1: "(Th th, Cs cs1) \<in> RAG s"
+ and h2: "(Cs cs1 , Th th') \<in> (RAG s)\<^sup>+"
+ by (auto simp:s_RAG_def)
have eq_cs: "cs1 = cs"
proof -
- from depend_s have "(Th th, Cs cs) \<in> depend s" by simp
- from unique_depend[OF vt_s this h1]
+ from RAG_s have "(Th th, Cs cs) \<in> RAG s" by simp
+ from unique_RAG[OF vt_s this h1]
show ?thesis by simp
qed
have False
proof(rule converse_tranclE[OF h2])
- assume "(Cs cs1, Th th') \<in> depend s"
- with eq_cs have "(Cs cs, Th th') \<in> depend s" by simp
- from unique_depend[OF vt_s this cs_th']
+ assume "(Cs cs1, Th th') \<in> RAG s"
+ with eq_cs have "(Cs cs, Th th') \<in> RAG s" by simp
+ from unique_RAG[OF vt_s this cs_th']
have "th' = th''" by simp
with ch' y_ch have "(Th th'', Th th'') \<in> (child s)^+" by auto
with wf_trancl[OF wf_child[OF vt_s]]
show False by auto
next
fix y
- assume "(Cs cs1, y) \<in> depend s"
- and ytd: " (y, Th th') \<in> (depend s)\<^sup>+"
- with eq_cs have csy: "(Cs cs, y) \<in> depend s" by simp
- from unique_depend[OF vt_s this cs_th']
+ assume "(Cs cs1, y) \<in> RAG s"
+ and ytd: " (y, Th th') \<in> (RAG s)\<^sup>+"
+ with eq_cs have csy: "(Cs cs, y) \<in> RAG s" by simp
+ from unique_RAG[OF vt_s this cs_th']
have "y = Th th''" .
- with ytd have "(Th th'', Th th') \<in> (depend s)^+" by simp
- from depend_child[OF vt_s this]
+ with ytd have "(Th th'', Th th') \<in> (RAG s)^+" by simp
+ from RAG_child[OF vt_s this]
have "(Th th'', Th th') \<in> (child s)\<^sup>+" .
moreover from ch' y_ch have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto
ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto
with wf_trancl[OF wf_child[OF vt_s]]
show False by auto
qed
- thus "\<exists>cs. (Th th, Cs cs) \<in> depend s' \<and> (Cs cs, Th th'') \<in> depend s'" by auto
+ thus "\<exists>cs. (Th th, Cs cs) \<in> RAG s' \<and> (Cs cs, Th th'') \<in> RAG s'" by auto
qed
ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
@@ -1731,8 +1649,8 @@
have neq_th1: "th1 \<noteq> th"
proof
assume eq_th1: "th1 = th"
- with dp1 have "(Th th1, Th th') \<in> (depend s)^+"
- by (auto simp:s_dependants_def eq_depend)
+ with dp1 have "(Th th1, Th th') \<in> (RAG s)^+"
+ by (auto simp:s_dependants_def eq_RAG)
from children_no_dep[OF vt_s _ _ this]
th1_in dp'
show False by (auto simp:children_def)
@@ -1753,48 +1671,48 @@
ultimately have "{preced th'' s} \<union> (cp s ` children s th'') =
{preced th'' s'} \<union> (cp s' ` children s th'')" by (auto simp:image_def)
moreover have "children s th'' = children s' th''"
- apply (unfold children_def child_def s_def depend_set_unchanged, simp)
- apply (fold s_def, auto simp:depend_s)
+ apply (unfold children_def child_def s_def RAG_set_unchanged, simp)
+ apply (fold s_def, auto simp:RAG_s)
proof -
- assume "(Cs cs, Th th'') \<in> depend s'"
- with depend_s have cs_th': "(Cs cs, Th th'') \<in> depend s" by auto
- from dp1 have "(Th th, Th th') \<in> (depend s)^+"
- by (auto simp:s_dependants_def eq_depend)
+ assume "(Cs cs, Th th'') \<in> RAG s'"
+ with RAG_s have cs_th': "(Cs cs, Th th'') \<in> RAG s" by auto
+ from dp1 have "(Th th, Th th') \<in> (RAG s)^+"
+ by (auto simp:s_dependants_def eq_RAG)
from converse_tranclE[OF this]
- obtain cs1 where h1: "(Th th, Cs cs1) \<in> depend s"
- and h2: "(Cs cs1 , Th th') \<in> (depend s)\<^sup>+"
- by (auto simp:s_depend_def)
+ obtain cs1 where h1: "(Th th, Cs cs1) \<in> RAG s"
+ and h2: "(Cs cs1 , Th th') \<in> (RAG s)\<^sup>+"
+ by (auto simp:s_RAG_def)
have eq_cs: "cs1 = cs"
proof -
- from depend_s have "(Th th, Cs cs) \<in> depend s" by simp
- from unique_depend[OF vt_s this h1]
+ from RAG_s have "(Th th, Cs cs) \<in> RAG s" by simp
+ from unique_RAG[OF vt_s this h1]
show ?thesis by simp
qed
have False
proof(rule converse_tranclE[OF h2])
- assume "(Cs cs1, Th th') \<in> depend s"
- with eq_cs have "(Cs cs, Th th') \<in> depend s" by simp
- from unique_depend[OF vt_s this cs_th']
+ assume "(Cs cs1, Th th') \<in> RAG s"
+ with eq_cs have "(Cs cs, Th th') \<in> RAG s" by simp
+ from unique_RAG[OF vt_s this cs_th']
have "th' = th''" by simp
with dp' have "(Th th'', Th th'') \<in> (child s)^+" by auto
with wf_trancl[OF wf_child[OF vt_s]]
show False by auto
next
fix y
- assume "(Cs cs1, y) \<in> depend s"
- and ytd: " (y, Th th') \<in> (depend s)\<^sup>+"
- with eq_cs have csy: "(Cs cs, y) \<in> depend s" by simp
- from unique_depend[OF vt_s this cs_th']
+ assume "(Cs cs1, y) \<in> RAG s"
+ and ytd: " (y, Th th') \<in> (RAG s)\<^sup>+"
+ with eq_cs have csy: "(Cs cs, y) \<in> RAG s" by simp
+ from unique_RAG[OF vt_s this cs_th']
have "y = Th th''" .
- with ytd have "(Th th'', Th th') \<in> (depend s)^+" by simp
- from depend_child[OF vt_s this]
+ with ytd have "(Th th'', Th th') \<in> (RAG s)^+" by simp
+ from RAG_child[OF vt_s this]
have "(Th th'', Th th') \<in> (child s)\<^sup>+" .
moreover from dp' have ch'': "(Th th', Th th'') \<in> (child s)^+" by auto
ultimately have "(Th th'', Th th'') \<in> (child s)^+" by auto
with wf_trancl[OF wf_child[OF vt_s]]
show False by auto
qed
- thus "\<exists>cs. (Th th, Cs cs) \<in> depend s' \<and> (Cs cs, Th th'') \<in> depend s'" by auto
+ thus "\<exists>cs. (Th th, Cs cs) \<in> RAG s' \<and> (Cs cs, Th th'') \<in> RAG s'" by auto
qed
ultimately show "Max ({preced th'' s} \<union> cp s ` children s th'') = cp s' th''"
by (simp add:cp_rec[OF step_back_vt[OF vt_s[unfolded s_def]]])
@@ -1814,8 +1732,8 @@
context step_create_cps
begin
-lemma eq_dep: "depend s = depend s'"
- by (unfold s_def depend_create_unchanged, auto)
+lemma eq_dep: "RAG s = RAG s'"
+ by (unfold s_def RAG_create_unchanged, auto)
lemma eq_cp:
fixes th'
@@ -1826,11 +1744,11 @@
have nd: "th \<notin> dependants s th'"
proof
assume "th \<in> dependants s th'"
- hence "(Th th, Th th') \<in> (depend s)^+" by (simp add:s_dependants_def eq_depend)
- with eq_dep have "(Th th, Th th') \<in> (depend s')^+" by simp
+ hence "(Th th, Th th') \<in> (RAG s)^+" by (simp add:s_dependants_def eq_RAG)
+ with eq_dep have "(Th th, Th th') \<in> (RAG s')^+" by simp
from converse_tranclE[OF this]
- obtain y where "(Th th, y) \<in> depend s'" by auto
- with dm_depend_threads[OF step_back_vt[OF vt_s[unfolded s_def]]]
+ obtain y where "(Th th, y) \<in> RAG s'" by auto
+ with dm_RAG_threads[OF step_back_vt[OF vt_s[unfolded s_def]]]
have in_th: "th \<in> threads s'" by auto
from step_back_step[OF vt_s[unfolded s_def]]
show False
@@ -1840,7 +1758,7 @@
qed
qed
have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th"
- by (unfold cs_dependants_def, auto simp:eq_dep eq_depend)
+ by (unfold cs_dependants_def, auto simp:eq_dep eq_RAG)
moreover {
fix th1
assume "th1 \<in> {th'} \<union> dependants (wq s') th'"
@@ -1853,7 +1771,7 @@
next
assume "th1 \<in> dependants (wq s') th'"
with nd and eq_dp have "th1 \<noteq> th"
- by (auto simp:eq_depend cs_dependants_def s_dependants_def eq_dep)
+ by (auto simp:eq_RAG cs_dependants_def s_dependants_def eq_dep)
thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
qed
} ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) =
@@ -1874,16 +1792,16 @@
have "dependants s' th = {}"
proof -
{ assume "dependants s' th \<noteq> {}"
- then obtain th' where dp: "(Th th', Th th) \<in> (depend s')^+"
- by (auto simp:s_dependants_def eq_depend)
+ then obtain th' where dp: "(Th th', Th th) \<in> (RAG s')^+"
+ by (auto simp:s_dependants_def eq_RAG)
from tranclE[OF this] obtain cs' where
- "(Cs cs', Th th) \<in> depend s'" by (auto simp:s_depend_def)
+ "(Cs cs', Th th) \<in> RAG s'" by (auto simp:s_RAG_def)
with hdn
have False by (auto simp:holdents_test)
} thus ?thesis by auto
qed
thus ?thesis
- by (unfold s_def s_dependants_def eq_depend depend_create_unchanged, simp)
+ by (unfold s_def s_dependants_def eq_RAG RAG_create_unchanged, simp)
qed
qed
@@ -1902,8 +1820,8 @@
context step_exit_cps
begin
-lemma eq_dep: "depend s = depend s'"
- by (unfold s_def depend_exit_unchanged, auto)
+lemma eq_dep: "RAG s = RAG s'"
+ by (unfold s_def RAG_exit_unchanged, auto)
lemma eq_cp:
fixes th'
@@ -1914,22 +1832,22 @@
have nd: "th \<notin> dependants s th'"
proof
assume "th \<in> dependants s th'"
- hence "(Th th, Th th') \<in> (depend s)^+" by (simp add:s_dependants_def eq_depend)
- with eq_dep have "(Th th, Th th') \<in> (depend s')^+" by simp
+ hence "(Th th, Th th') \<in> (RAG s)^+" by (simp add:s_dependants_def eq_RAG)
+ with eq_dep have "(Th th, Th th') \<in> (RAG s')^+" by simp
from converse_tranclE[OF this]
- obtain cs' where bk: "(Th th, Cs cs') \<in> depend s'"
- by (auto simp:s_depend_def)
+ obtain cs' where bk: "(Th th, Cs cs') \<in> RAG s'"
+ by (auto simp:s_RAG_def)
from step_back_step[OF vt_s[unfolded s_def]]
show False
proof(cases)
assume "th \<in> runing s'"
with bk show ?thesis
- apply (unfold runing_def readys_def s_waiting_def s_depend_def)
+ apply (unfold runing_def readys_def s_waiting_def s_RAG_def)
by (auto simp:cs_waiting_def wq_def)
qed
qed
have eq_dp: "\<And> th. dependants (wq s) th = dependants (wq s') th"
- by (unfold cs_dependants_def, auto simp:eq_dep eq_depend)
+ by (unfold cs_dependants_def, auto simp:eq_dep eq_RAG)
moreover {
fix th1
assume "th1 \<in> {th'} \<union> dependants (wq s') th'"
@@ -1942,7 +1860,7 @@
next
assume "th1 \<in> dependants (wq s') th'"
with nd and eq_dp have "th1 \<noteq> th"
- by (auto simp:eq_depend cs_dependants_def s_dependants_def eq_dep)
+ by (auto simp:eq_RAG cs_dependants_def s_dependants_def eq_dep)
thus "preced th1 s = preced th1 s'" by (auto simp:s_def preced_def)
qed
} ultimately have "((\<lambda>th. preced th s) ` ({th'} \<union> dependants (wq s) th')) =
--- a/PrioG.thy Tue May 20 12:49:21 2014 +0100
+++ b/PrioG.thy Thu May 22 17:40:39 2014 +0100
@@ -24,7 +24,7 @@
thus "distinct (wq (e # s) cs)"
proof(induct rule:step.induct, auto simp: wq_def Let_def split:list.splits)
fix thread s
- assume h1: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+ assume h1: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
and h2: "thread \<in> set (wq_fun (schs s) cs)"
and h3: "thread \<in> runing s"
show "False"
@@ -34,8 +34,8 @@
by (simp add:runing_def readys_def s_waiting_def wq_def)
from this [OF h2] have "thread = hd (wq_fun (schs s) cs)" .
with h2
- have "(Cs cs, Th thread) \<in> (depend s)"
- by (simp add:s_depend_def s_holding_def wq_def cs_holding_def)
+ have "(Cs cs, Th thread) \<in> (RAG s)"
+ by (simp add:s_RAG_def s_holding_def wq_def cs_holding_def)
with h1 show False by auto
qed
next
@@ -112,7 +112,7 @@
qed
lemma p_pre: "\<lbrakk>vt ((P thread cs)#s)\<rbrakk> \<Longrightarrow>
- thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (depend s)^+"
+ thread \<in> runing s \<and> (Cs cs, Th thread) \<notin> (RAG s)^+"
apply (ind_cases "vt ((P thread cs)#s)")
apply (ind_cases "step s (P thread cs)")
by auto
@@ -501,16 +501,16 @@
qed
qed
-lemma depend_set_unchanged: "(depend (Set th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
+lemma RAG_set_unchanged: "(RAG (Set th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
by (simp add:Let_def)
-lemma depend_create_unchanged: "(depend (Create th prio # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
+lemma RAG_create_unchanged: "(RAG (Create th prio # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
by (simp add:Let_def)
-lemma depend_exit_unchanged: "(depend (Exit th # s)) = depend s"
-apply (unfold s_depend_def s_waiting_def wq_def)
+lemma RAG_exit_unchanged: "(RAG (Exit th # s)) = RAG s"
+apply (unfold s_RAG_def s_waiting_def wq_def)
by (simp add:Let_def)
@@ -773,16 +773,16 @@
qed
qed
-lemma step_depend_v:
+lemma step_RAG_v:
fixes th::thread
assumes vt:
"vt (V th cs#s)"
shows "
- depend (V th cs # s) =
- depend s - {(Cs cs, Th th)} -
+ RAG (V th cs # s) =
+ RAG s - {(Cs cs, Th th)} -
{(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
{(Cs cs, Th th') |th'. next_th s th cs th'}"
- apply (insert vt, unfold s_depend_def)
+ apply (insert vt, unfold s_RAG_def)
apply (auto split:if_splits list.splits simp:Let_def)
apply (auto elim: step_v_waiting_mono step_v_hold_inv
step_v_release step_v_wait_inv
@@ -790,17 +790,17 @@
apply (erule_tac step_v_not_wait, auto)
done
-lemma step_depend_p:
+lemma step_RAG_p:
"vt (P th cs#s) \<Longrightarrow>
- depend (P th cs # s) = (if (wq s cs = []) then depend s \<union> {(Cs cs, Th th)}
- else depend s \<union> {(Th th, Cs cs)})"
- apply(simp only: s_depend_def wq_def)
+ RAG (P th cs # s) = (if (wq s cs = []) then RAG s \<union> {(Cs cs, Th th)}
+ else RAG s \<union> {(Th th, Cs cs)})"
+ apply(simp only: s_RAG_def wq_def)
apply (auto split:list.splits prod.splits simp:Let_def wq_def cs_waiting_def cs_holding_def)
apply(case_tac "csa = cs", auto)
apply(fold wq_def)
apply(drule_tac step_back_step)
apply(ind_cases " step s (P (hd (wq s cs)) cs)")
- apply(auto simp:s_depend_def wq_def cs_holding_def)
+ apply(auto simp:s_RAG_def wq_def cs_holding_def)
done
lemma simple_A:
@@ -815,35 +815,35 @@
thus ?thesis by simp
qed
-lemma depend_target_th: "(Th th, x) \<in> depend (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
- by (unfold s_depend_def, auto)
+lemma RAG_target_th: "(Th th, x) \<in> RAG (s::state) \<Longrightarrow> \<exists> cs. x = Cs cs"
+ by (unfold s_RAG_def, auto)
-lemma acyclic_depend:
+lemma acyclic_RAG:
fixes s
assumes vt: "vt s"
- shows "acyclic (depend s)"
+ shows "acyclic (RAG s)"
proof -
from vt show ?thesis
proof(induct)
case (vt_cons s e)
- assume ih: "acyclic (depend s)"
+ assume ih: "acyclic (RAG s)"
and stp: "step s e"
and vt: "vt s"
show ?case
proof(cases e)
case (Create th prio)
with ih
- show ?thesis by (simp add:depend_create_unchanged)
+ show ?thesis by (simp add:RAG_create_unchanged)
next
case (Exit th)
- with ih show ?thesis by (simp add:depend_exit_unchanged)
+ with ih show ?thesis by (simp add:RAG_exit_unchanged)
next
case (V th cs)
from V vt stp have vtt: "vt (V th cs#s)" by auto
- from step_depend_v [OF this]
+ from step_RAG_v [OF this]
have eq_de:
- "depend (e # s) =
- depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+ "RAG (e # s) =
+ RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
{(Cs cs, Th th') |th'. next_th s th cs th'}"
(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
from ih have ac: "acyclic (?A - ?B - ?C)" by (auto elim:acyclic_subset)
@@ -871,11 +871,11 @@
from tranclD [OF this]
obtain x where th'_e: "(Th ?th', x) \<in> ?E" by blast
hence th_d: "(Th ?th', x) \<in> ?A" by simp
- from depend_target_th [OF this]
+ from RAG_target_th [OF this]
obtain cs' where eq_x: "x = Cs cs'" by auto
with th_d have "(Th ?th', Cs cs') \<in> ?A" by simp
hence wt_th': "waiting s ?th' cs'"
- unfolding s_depend_def s_waiting_def cs_waiting_def wq_def by simp
+ unfolding s_RAG_def s_waiting_def cs_waiting_def wq_def by simp
hence "cs' = cs"
proof(rule waiting_unique [OF vt])
from eq_wq wq_distinct[OF vt, of cs]
@@ -945,35 +945,35 @@
next
case (P th cs)
from P vt stp have vtt: "vt (P th cs#s)" by auto
- from step_depend_p [OF this] P
- have "depend (e # s) =
- (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
- depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+ from step_RAG_p [OF this] P
+ have "RAG (e # s) =
+ (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
+ RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
by simp
moreover have "acyclic ?R"
proof(cases "wq s cs = []")
case True
- hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
- have "(Th th, Cs cs) \<notin> (depend s)\<^sup>*"
+ hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
+ have "(Th th, Cs cs) \<notin> (RAG s)\<^sup>*"
proof
- assume "(Th th, Cs cs) \<in> (depend s)\<^sup>*"
- hence "(Th th, Cs cs) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+ assume "(Th th, Cs cs) \<in> (RAG s)\<^sup>*"
+ hence "(Th th, Cs cs) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
from tranclD2 [OF this]
- obtain x where "(x, Cs cs) \<in> depend s" by auto
- with True show False by (auto simp:s_depend_def cs_waiting_def)
+ obtain x where "(x, Cs cs) \<in> RAG s" by auto
+ with True show False by (auto simp:s_RAG_def cs_waiting_def)
qed
with acyclic_insert ih eq_r show ?thesis by auto
next
case False
- hence eq_r: "?R = depend s \<union> {(Th th, Cs cs)}" by simp
- have "(Cs cs, Th th) \<notin> (depend s)\<^sup>*"
+ hence eq_r: "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
+ have "(Cs cs, Th th) \<notin> (RAG s)\<^sup>*"
proof
- assume "(Cs cs, Th th) \<in> (depend s)\<^sup>*"
- hence "(Cs cs, Th th) \<in> (depend s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
+ assume "(Cs cs, Th th) \<in> (RAG s)\<^sup>*"
+ hence "(Cs cs, Th th) \<in> (RAG s)\<^sup>+" by (simp add: rtrancl_eq_or_trancl)
moreover from step_back_step [OF vtt] have "step s (P th cs)" .
ultimately show False
proof -
- show " \<lbrakk>(Cs cs, Th th) \<in> (depend s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
+ show " \<lbrakk>(Cs cs, Th th) \<in> (RAG s)\<^sup>+; step s (P th cs)\<rbrakk> \<Longrightarrow> False"
by (ind_cases "step s (P th cs)", simp)
qed
qed
@@ -983,42 +983,42 @@
next
case (Set thread prio)
with ih
- thm depend_set_unchanged
- show ?thesis by (simp add:depend_set_unchanged)
+ thm RAG_set_unchanged
+ show ?thesis by (simp add:RAG_set_unchanged)
qed
next
case vt_nil
- show "acyclic (depend ([]::state))"
- by (auto simp: s_depend_def cs_waiting_def
+ show "acyclic (RAG ([]::state))"
+ by (auto simp: s_RAG_def cs_waiting_def
cs_holding_def wq_def acyclic_def)
qed
qed
-lemma finite_depend:
+lemma finite_RAG:
fixes s
assumes vt: "vt s"
- shows "finite (depend s)"
+ shows "finite (RAG s)"
proof -
from vt show ?thesis
proof(induct)
case (vt_cons s e)
- assume ih: "finite (depend s)"
+ assume ih: "finite (RAG s)"
and stp: "step s e"
and vt: "vt s"
show ?case
proof(cases e)
case (Create th prio)
with ih
- show ?thesis by (simp add:depend_create_unchanged)
+ show ?thesis by (simp add:RAG_create_unchanged)
next
case (Exit th)
- with ih show ?thesis by (simp add:depend_exit_unchanged)
+ with ih show ?thesis by (simp add:RAG_exit_unchanged)
next
case (V th cs)
from V vt stp have vtt: "vt (V th cs#s)" by auto
- from step_depend_v [OF this]
- have eq_de: "depend (e # s) =
- depend s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
+ from step_RAG_v [OF this]
+ have eq_de: "RAG (e # s) =
+ RAG s - {(Cs cs, Th th)} - {(Th th', Cs cs) |th'. next_th s th cs th'} \<union>
{(Cs cs, Th th') |th'. next_th s th cs th'}
"
(is "?L = (?A - ?B - ?C) \<union> ?D") by (simp add:V)
@@ -1041,31 +1041,31 @@
next
case (P th cs)
from P vt stp have vtt: "vt (P th cs#s)" by auto
- from step_depend_p [OF this] P
- have "depend (e # s) =
- (if wq s cs = [] then depend s \<union> {(Cs cs, Th th)} else
- depend s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
+ from step_RAG_p [OF this] P
+ have "RAG (e # s) =
+ (if wq s cs = [] then RAG s \<union> {(Cs cs, Th th)} else
+ RAG s \<union> {(Th th, Cs cs)})" (is "?L = ?R")
by simp
moreover have "finite ?R"
proof(cases "wq s cs = []")
case True
- hence eq_r: "?R = depend s \<union> {(Cs cs, Th th)}" by simp
+ hence eq_r: "?R = RAG s \<union> {(Cs cs, Th th)}" by simp
with True and ih show ?thesis by auto
next
case False
- hence "?R = depend s \<union> {(Th th, Cs cs)}" by simp
+ hence "?R = RAG s \<union> {(Th th, Cs cs)}" by simp
with False and ih show ?thesis by auto
qed
ultimately show ?thesis by auto
next
case (Set thread prio)
with ih
- show ?thesis by (simp add:depend_set_unchanged)
+ show ?thesis by (simp add:RAG_set_unchanged)
qed
next
case vt_nil
- show "finite (depend ([]::state))"
- by (auto simp: s_depend_def cs_waiting_def
+ show "finite (RAG ([]::state))"
+ by (auto simp: s_RAG_def cs_waiting_def
cs_holding_def wq_def acyclic_def)
qed
qed
@@ -1075,20 +1075,20 @@
lemma wf_dep_converse:
fixes s
assumes vt: "vt s"
- shows "wf ((depend s)^-1)"
+ shows "wf ((RAG s)^-1)"
proof(rule finite_acyclic_wf_converse)
- from finite_depend [OF vt]
- show "finite (depend s)" .
+ from finite_RAG [OF vt]
+ show "finite (RAG s)" .
next
- from acyclic_depend[OF vt]
- show "acyclic (depend s)" .
+ from acyclic_RAG[OF vt]
+ show "acyclic (RAG s)" .
qed
lemma hd_np_in: "x \<in> set l \<Longrightarrow> hd l \<in> set l"
by (induct l, auto)
-lemma th_chasing: "(Th th, Cs cs) \<in> depend (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> depend s"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+lemma th_chasing: "(Th th, Cs cs) \<in> RAG (s::state) \<Longrightarrow> \<exists> th'. (Cs cs, Th th') \<in> RAG s"
+ by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
lemma wq_threads:
fixes s cs
@@ -1114,7 +1114,7 @@
apply (auto simp:wq_def Let_def)
apply (ind_cases "step s (Exit th')")
apply (auto simp:runing_def readys_def s_holding_def s_waiting_def holdents_def
- s_depend_def s_holding_def cs_holding_def)
+ s_RAG_def s_holding_def cs_holding_def)
done
next
case (V th' cs')
@@ -1192,8 +1192,8 @@
qed
qed
-lemma range_in: "\<lbrakk>vt s; (Th th) \<in> Range (depend (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
- apply(unfold s_depend_def cs_waiting_def cs_holding_def)
+lemma range_in: "\<lbrakk>vt s; (Th th) \<in> Range (RAG (s::state))\<rbrakk> \<Longrightarrow> th \<in> threads s"
+ apply(unfold s_RAG_def cs_waiting_def cs_holding_def)
by (auto intro:wq_threads)
lemma readys_v_eq:
@@ -1231,33 +1231,33 @@
lemma chain_building:
assumes vt: "vt s"
- shows "node \<in> Domain (depend s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (depend s)^+)"
+ shows "node \<in> Domain (RAG s) \<longrightarrow> (\<exists> th'. th' \<in> readys s \<and> (node, Th th') \<in> (RAG s)^+)"
proof -
from wf_dep_converse [OF vt]
- have h: "wf ((depend s)\<inverse>)" .
+ have h: "wf ((RAG s)\<inverse>)" .
show ?thesis
proof(induct rule:wf_induct [OF h])
fix x
assume ih [rule_format]:
- "\<forall>y. (y, x) \<in> (depend s)\<inverse> \<longrightarrow>
- y \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (depend s)\<^sup>+)"
- show "x \<in> Domain (depend s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+)"
+ "\<forall>y. (y, x) \<in> (RAG s)\<inverse> \<longrightarrow>
+ y \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (y, Th th') \<in> (RAG s)\<^sup>+)"
+ show "x \<in> Domain (RAG s) \<longrightarrow> (\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+)"
proof
- assume x_d: "x \<in> Domain (depend s)"
- show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (depend s)\<^sup>+"
+ assume x_d: "x \<in> Domain (RAG s)"
+ show "\<exists>th'. th' \<in> readys s \<and> (x, Th th') \<in> (RAG s)\<^sup>+"
proof(cases x)
case (Th th)
- from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> depend s" by (auto simp:s_depend_def)
- with Th have x_in_r: "(Cs cs, x) \<in> (depend s)^-1" by simp
- from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> depend s" by blast
- hence "Cs cs \<in> Domain (depend s)" by auto
+ from x_d Th obtain cs where x_in: "(Th th, Cs cs) \<in> RAG s" by (auto simp:s_RAG_def)
+ with Th have x_in_r: "(Cs cs, x) \<in> (RAG s)^-1" by simp
+ from th_chasing [OF x_in] obtain th' where "(Cs cs, Th th') \<in> RAG s" by blast
+ hence "Cs cs \<in> Domain (RAG s)" by auto
from ih [OF x_in_r this] obtain th'
- where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (depend s)\<^sup>+" by auto
- have "(x, Th th') \<in> (depend s)\<^sup>+" using Th x_in cs_in by auto
+ where th'_ready: " th' \<in> readys s" and cs_in: "(Cs cs, Th th') \<in> (RAG s)\<^sup>+" by auto
+ have "(x, Th th') \<in> (RAG s)\<^sup>+" using Th x_in cs_in by auto
with th'_ready show ?thesis by auto
next
case (Cs cs)
- from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (depend s)^-1" by (auto simp:s_depend_def)
+ from x_d Cs obtain th' where th'_d: "(Th th', x) \<in> (RAG s)^-1" by (auto simp:s_RAG_def)
show ?thesis
proof(cases "th' \<in> readys s")
case True
@@ -1265,14 +1265,14 @@
next
case False
from th'_d and range_in [OF vt] have "th' \<in> threads s" by auto
- with False have "Th th' \<in> Domain (depend s)"
- by (auto simp:readys_def wq_def s_waiting_def s_depend_def cs_waiting_def Domain_def)
+ with False have "Th th' \<in> Domain (RAG s)"
+ by (auto simp:readys_def wq_def s_waiting_def s_RAG_def cs_waiting_def Domain_def)
from ih [OF th'_d this]
obtain th'' where
th''_r: "th'' \<in> readys s" and
- th''_in: "(Th th', Th th'') \<in> (depend s)\<^sup>+" by auto
+ th''_in: "(Th th', Th th'') \<in> (RAG s)\<^sup>+" by auto
from th'_d and th''_in
- have "(x, Th th'') \<in> (depend s)\<^sup>+" by auto
+ have "(x, Th th'') \<in> (RAG s)\<^sup>+" by auto
with th''_r show ?thesis by auto
qed
qed
@@ -1284,14 +1284,14 @@
fixes s th
assumes vt: "vt s"
and th_in: "th \<in> threads s"
- shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (depend s)^+)"
+ shows "th \<in> readys s \<or> (\<exists> th'. th' \<in> readys s \<and> (Th th, Th th') \<in> (RAG s)^+)"
proof(cases "th \<in> readys s")
case True
thus ?thesis by auto
next
case False
- from False and th_in have "Th th \<in> Domain (depend s)"
- by (auto simp:readys_def s_waiting_def s_depend_def wq_def cs_waiting_def Domain_def)
+ from False and th_in have "Th th \<in> Domain (RAG s)"
+ by (auto simp:readys_def s_waiting_def s_RAG_def wq_def cs_waiting_def Domain_def)
from chain_building [rule_format, OF vt this]
show ?thesis by auto
qed
@@ -1305,8 +1305,8 @@
lemma holding_unique: "\<lbrakk>holding (s::state) th1 cs; holding s th2 cs\<rbrakk> \<Longrightarrow> th1 = th2"
by (unfold s_holding_def cs_holding_def, auto)
-lemma unique_depend: "\<lbrakk>vt s; (n, n1) \<in> depend s; (n, n2) \<in> depend s\<rbrakk> \<Longrightarrow> n1 = n2"
- apply(unfold s_depend_def, auto, fold waiting_eq holding_eq)
+lemma unique_RAG: "\<lbrakk>vt s; (n, n1) \<in> RAG s; (n, n2) \<in> RAG s\<rbrakk> \<Longrightarrow> n1 = n2"
+ apply(unfold s_RAG_def, auto, fold waiting_eq holding_eq)
by(auto elim:waiting_unique holding_unique)
lemma trancl_split: "(a, b) \<in> r^+ \<Longrightarrow> \<exists> c. (a, c) \<in> r"
@@ -1314,34 +1314,34 @@
lemma dchain_unique:
assumes vt: "vt s"
- and th1_d: "(n, Th th1) \<in> (depend s)^+"
+ and th1_d: "(n, Th th1) \<in> (RAG s)^+"
and th1_r: "th1 \<in> readys s"
- and th2_d: "(n, Th th2) \<in> (depend s)^+"
+ and th2_d: "(n, Th th2) \<in> (RAG s)^+"
and th2_r: "th2 \<in> readys s"
shows "th1 = th2"
proof -
{ assume neq: "th1 \<noteq> th2"
hence "Th th1 \<noteq> Th th2" by simp
- from unique_chain [OF _ th1_d th2_d this] and unique_depend [OF vt]
- have "(Th th1, Th th2) \<in> (depend s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (depend s)\<^sup>+" by auto
+ from unique_chain [OF _ th1_d th2_d this] and unique_RAG [OF vt]
+ have "(Th th1, Th th2) \<in> (RAG s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG s)\<^sup>+" by auto
hence "False"
proof
- assume "(Th th1, Th th2) \<in> (depend s)\<^sup>+"
+ assume "(Th th1, Th th2) \<in> (RAG s)\<^sup>+"
from trancl_split [OF this]
- obtain n where dd: "(Th th1, n) \<in> depend s" by auto
+ obtain n where dd: "(Th th1, n) \<in> RAG s" by auto
then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+ by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
from dd eq_n have "th1 \<notin> readys s"
- by (auto simp:readys_def s_depend_def wq_def s_waiting_def cs_waiting_def)
+ by (auto simp:readys_def s_RAG_def wq_def s_waiting_def cs_waiting_def)
with th1_r show ?thesis by auto
next
- assume "(Th th2, Th th1) \<in> (depend s)\<^sup>+"
+ assume "(Th th2, Th th1) \<in> (RAG s)\<^sup>+"
from trancl_split [OF this]
- obtain n where dd: "(Th th2, n) \<in> depend s" by auto
+ obtain n where dd: "(Th th2, n) \<in> RAG s" by auto
then obtain cs where eq_n: "n = Cs cs"
- by (auto simp:s_depend_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
+ by (auto simp:s_RAG_def s_holding_def cs_holding_def cs_waiting_def wq_def dest:hd_np_in)
from dd eq_n have "th2 \<notin> readys s"
- by (auto simp:readys_def wq_def s_depend_def s_waiting_def cs_waiting_def)
+ by (auto simp:readys_def wq_def s_RAG_def s_waiting_def cs_waiting_def)
with th2_r show ?thesis by auto
qed
} thus ?thesis by auto
@@ -1355,7 +1355,7 @@
shows "holdents (P th cs#s) th = holdents s th \<union> {cs}"
proof -
from assms show ?thesis
- unfolding holdents_test step_depend_p[OF vt] by (auto)
+ unfolding holdents_test step_RAG_p[OF vt] by (auto)
qed
lemma step_holdents_p_eq:
@@ -1365,7 +1365,7 @@
shows "holdents (P th cs#s) th = holdents s th"
proof -
from assms show ?thesis
- unfolding holdents_test step_depend_p[OF vt] by auto
+ unfolding holdents_test step_RAG_p[OF vt] by auto
qed
@@ -1375,16 +1375,16 @@
shows "finite (holdents s th)"
proof -
let ?F = "\<lambda> (x, y). the_cs x"
- from finite_depend [OF vt]
- have "finite (depend s)" .
- hence "finite (?F `(depend s))" by simp
- moreover have "{cs . (Cs cs, Th th) \<in> depend s} \<subseteq> \<dots>"
+ from finite_RAG [OF vt]
+ have "finite (RAG s)" .
+ hence "finite (?F `(RAG s))" by simp
+ moreover have "{cs . (Cs cs, Th th) \<in> RAG s} \<subseteq> \<dots>"
proof -
{ have h: "\<And> a A f. a \<in> A \<Longrightarrow> f a \<in> f ` A" by auto
- fix x assume "(Cs x, Th th) \<in> depend s"
- hence "?F (Cs x, Th th) \<in> ?F `(depend s)" by (rule h)
+ fix x assume "(Cs x, Th th) \<in> RAG s"
+ hence "?F (Cs x, Th th) \<in> ?F `(RAG s)" by (rule h)
moreover have "?F (Cs x, Th th) = x" by simp
- ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` depend s" by simp
+ ultimately have "x \<in> (\<lambda>(x, y). the_cs x) ` RAG s" by simp
} thus ?thesis by auto
qed
ultimately show ?thesis by (unfold holdents_test, auto intro:finite_subset)
@@ -1397,12 +1397,12 @@
proof -
from step_back_step[OF vtv]
have cs_in: "cs \<in> holdents s thread"
- apply (cases, unfold holdents_test s_depend_def, simp)
+ apply (cases, unfold holdents_test s_RAG_def, simp)
by (unfold cs_holding_def s_holding_def wq_def, auto)
moreover have cs_not_in:
"(holdents (V thread cs#s) thread) = holdents s thread - {cs}"
apply (insert wq_distinct[OF step_back_vt[OF vtv], of cs])
- apply (unfold holdents_test, unfold step_depend_v[OF vtv],
+ apply (unfold holdents_test, unfold step_RAG_v[OF vtv],
auto simp:next_th_def)
proof -
fix rest
@@ -1425,7 +1425,7 @@
show "x \<noteq> []" by auto
qed
ultimately
- show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"
+ show "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
by auto
next
fix rest
@@ -1494,7 +1494,7 @@
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_test
- by (simp add:depend_create_unchanged eq_e)
+ by (simp add:RAG_create_unchanged eq_e)
{ assume "th \<noteq> thread"
with eq_readys eq_e
have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
@@ -1519,7 +1519,7 @@
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_test
- by (simp add:depend_exit_unchanged eq_e)
+ by (simp add:RAG_exit_unchanged eq_e)
{ assume "th \<noteq> thread"
with eq_e
have "(th \<in> readys (e # s) \<or> th \<notin> threads (e # s)) =
@@ -1544,7 +1544,7 @@
case (thread_P thread cs)
assume eq_e: "e = P thread cs"
and is_runing: "thread \<in> runing s"
- and no_dep: "(Cs cs, Th thread) \<notin> (depend s)\<^sup>+"
+ and no_dep: "(Cs cs, Th thread) \<notin> (RAG s)\<^sup>+"
from thread_P vt stp ih have vtp: "vt (P thread cs#s)" by auto
show ?thesis
proof -
@@ -1561,7 +1561,7 @@
by (case_tac "(wq_fun (schs s) cs)", auto)
moreover from neq_th eq_e have "cntCS (e # s) th = cntCS s th"
apply (simp add:cntCS_def holdents_test)
- by (unfold step_depend_p [OF vtp], auto)
+ by (unfold step_RAG_p [OF vtp], auto)
moreover from eq_e neq_th have "cntP (e # s) th = cntP s th"
by (simp add:cntP_def count_def)
moreover from eq_e neq_th have "cntV (e#s) th = cntV s th"
@@ -1582,26 +1582,26 @@
case True
with is_runing
have "th \<in> readys (e#s)"
- apply (unfold eq_e wq_def, unfold readys_def s_depend_def)
+ apply (unfold eq_e wq_def, unfold readys_def s_RAG_def)
apply (simp add: wq_def[symmetric] runing_def eq_th s_waiting_def)
by (auto simp:readys_def wq_def Let_def s_waiting_def wq_def)
moreover have "cntCS (e # s) th = 1 + cntCS s th"
proof -
- have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> depend s} =
- Suc (card {cs. (Cs cs, Th thread) \<in> depend s})" (is "card ?L = Suc (card ?R)")
+ have "card {csa. csa = cs \<or> (Cs csa, Th thread) \<in> RAG s} =
+ Suc (card {cs. (Cs cs, Th thread) \<in> RAG s})" (is "card ?L = Suc (card ?R)")
proof -
have "?L = insert cs ?R" by auto
moreover have "card \<dots> = Suc (card (?R - {cs}))"
proof(rule card_insert)
from finite_holding [OF vt, of thread]
- show " finite {cs. (Cs cs, Th thread) \<in> depend s}"
+ show " finite {cs. (Cs cs, Th thread) \<in> RAG s}"
by (unfold holdents_test, simp)
qed
moreover have "?R - {cs} = ?R"
proof -
have "cs \<notin> ?R"
proof
- assume "cs \<in> {cs. (Cs cs, Th thread) \<in> depend s}"
+ assume "cs \<in> {cs. (Cs cs, Th thread) \<in> RAG s}"
with no_dep show False by auto
qed
thus ?thesis by auto
@@ -1611,7 +1611,7 @@
thus ?thesis
apply (unfold eq_e eq_th cntCS_def)
apply (simp add: holdents_test)
- by (unfold step_depend_p [OF vtp], auto simp:True)
+ by (unfold step_RAG_p [OF vtp], auto simp:True)
qed
moreover from is_runing have "th \<in> readys s"
by (simp add:runing_def eq_th)
@@ -1638,7 +1638,7 @@
moreover from is_runing have "th \<in> threads (e#s)"
by (unfold eq_e, auto simp:runing_def readys_def eq_th)
moreover have "cntCS (e # s) th = cntCS s th"
- apply (unfold cntCS_def holdents_test eq_e step_depend_p[OF vtp])
+ apply (unfold cntCS_def holdents_test eq_e step_RAG_p[OF vtp])
by (auto simp:False)
moreover note eq_cnp eq_cnv ih[of th]
moreover from is_runing have "th \<in> readys s"
@@ -1735,13 +1735,13 @@
apply (insert step_back_vt[OF vtv])
by (unfold eq_e, rule readys_v_eq [OF _ neq_th eq_wq False], auto)
moreover have "cntCS (e#s) th = cntCS s th"
- apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
+ apply (insert neq_th, unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
proof -
- have "{csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
- {cs. (Cs cs, Th th) \<in> depend s}"
+ have "{csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+ {cs. (Cs cs, Th th) \<in> RAG s}"
proof -
from False eq_wq
- have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> depend s"
+ have " next_th s thread cs th \<Longrightarrow> (Cs cs, Th th) \<in> RAG s"
apply (unfold next_th_def, auto)
proof -
assume ne: "rest \<noteq> []"
@@ -1759,13 +1759,13 @@
with ne show "x \<noteq> []" by auto
qed
ultimately show
- "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> depend s"
+ "(Cs cs, Th (hd (SOME q. distinct q \<and> set q = set rest))) \<in> RAG s"
by auto
qed
thus ?thesis by auto
qed
- thus "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs \<and> next_th s thread cs th} =
- card {cs. (Cs cs, Th th) \<in> depend s}" by simp
+ thus "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs \<and> next_th s thread cs th} =
+ card {cs. (Cs cs, Th th) \<in> RAG s}" by simp
qed
moreover note ih eq_cnp eq_cnv eq_threads
ultimately show ?thesis by auto
@@ -1796,8 +1796,8 @@
proof -
from eq_wq and th_in and neq_hd
have "(holdents (e # s) th) = (holdents s th)"
- apply (unfold eq_e step_depend_v[OF vtv],
- auto simp:next_th_def eq_set s_depend_def holdents_test wq_def
+ apply (unfold eq_e step_RAG_v[OF vtv],
+ auto simp:next_th_def eq_set s_RAG_def holdents_test wq_def
Let_def cs_holding_def)
by (insert wq_distinct[OF step_back_vt[OF vtv], of cs], auto simp:wq_def)
thus ?thesis by (simp add:cntCS_def)
@@ -1862,28 +1862,28 @@
ultimately show ?thesis using ih by auto
qed
moreover from True neq_th have "cntCS (e # s) th = 1 + cntCS s th"
- apply (unfold cntCS_def holdents_test eq_e step_depend_v[OF vtv], auto)
+ apply (unfold cntCS_def holdents_test eq_e step_RAG_v[OF vtv], auto)
proof -
- show "card {csa. (Cs csa, Th th) \<in> depend s \<or> csa = cs} =
- Suc (card {cs. (Cs cs, Th th) \<in> depend s})"
+ show "card {csa. (Cs csa, Th th) \<in> RAG s \<or> csa = cs} =
+ Suc (card {cs. (Cs cs, Th th) \<in> RAG s})"
(is "card ?A = Suc (card ?B)")
proof -
have "?A = insert cs ?B" by auto
hence "card ?A = card (insert cs ?B)" by simp
also have "\<dots> = Suc (card ?B)"
proof(rule card_insert_disjoint)
- have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` depend s)"
+ have "?B \<subseteq> ((\<lambda> (x, y). the_cs x) ` RAG s)"
apply (auto simp:image_def)
by (rule_tac x = "(Cs x, Th th)" in bexI, auto)
- with finite_depend[OF step_back_vt[OF vtv]]
- show "finite {cs. (Cs cs, Th th) \<in> depend s}" by (auto intro:finite_subset)
+ with finite_RAG[OF step_back_vt[OF vtv]]
+ show "finite {cs. (Cs cs, Th th) \<in> RAG s}" by (auto intro:finite_subset)
next
- show "cs \<notin> {cs. (Cs cs, Th th) \<in> depend s}"
+ show "cs \<notin> {cs. (Cs cs, Th th) \<in> RAG s}"
proof
- assume "cs \<in> {cs. (Cs cs, Th th) \<in> depend s}"
- hence "(Cs cs, Th th) \<in> depend s" by simp
+ assume "cs \<in> {cs. (Cs cs, Th th) \<in> RAG s}"
+ hence "(Cs cs, Th th) \<in> RAG s" by simp
with True neq_th eq_wq show False
- by (auto simp:next_th_def s_depend_def cs_holding_def)
+ by (auto simp:next_th_def s_RAG_def cs_holding_def)
qed
qed
finally show ?thesis .
@@ -1905,7 +1905,7 @@
from eq_e have eq_cnv: "cntV (e#s) th = cntV s th" by (simp add:cntV_def count_def)
have eq_cncs: "cntCS (e#s) th = cntCS s th"
unfolding cntCS_def holdents_test
- by (simp add:depend_set_unchanged eq_e)
+ by (simp add:RAG_set_unchanged eq_e)
from eq_e have eq_readys: "readys (e#s) = readys s"
by (simp add:readys_def cs_waiting_def s_waiting_def wq_def,
auto simp:Let_def)
@@ -1931,7 +1931,7 @@
case vt_nil
show ?case
by (unfold cntP_def cntV_def cntCS_def,
- auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
+ auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
qed
qed
@@ -1955,7 +1955,7 @@
and not_in': "thread \<notin> threads s"
have "cntCS (e # s) th = cntCS s th"
apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:depend_create_unchanged)
+ by (simp add:RAG_create_unchanged)
moreover have "th \<notin> threads s"
proof -
from not_in eq_e show ?thesis by simp
@@ -1967,7 +1967,7 @@
and nh: "holdents s thread = {}"
have eq_cns: "cntCS (e # s) th = cntCS s th"
apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:depend_exit_unchanged)
+ by (simp add:RAG_exit_unchanged)
show ?thesis
proof(cases "th = thread")
case True
@@ -1993,7 +1993,7 @@
qed
hence "cntCS (e # s) th = cntCS s th "
apply (unfold cntCS_def holdents_test eq_e)
- by (unfold step_depend_p[OF vtp], auto)
+ by (unfold step_RAG_p[OF vtp], auto)
moreover have "cntCS s th = 0"
proof(rule ih)
from not_in eq_e show "th \<notin> threads s" by simp
@@ -2035,7 +2035,7 @@
qed
moreover note neq_th eq_wq
ultimately have "cntCS (e # s) th = cntCS s th"
- by (unfold eq_e cntCS_def holdents_test step_depend_v[OF vtv], auto)
+ by (unfold eq_e cntCS_def holdents_test step_RAG_v[OF vtv], auto)
moreover have "cntCS s th = 0"
proof(rule ih)
from not_in eq_e show "th \<notin> threads s" by simp
@@ -2050,30 +2050,30 @@
from ih [OF this] and eq_e
show ?thesis
apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:depend_set_unchanged)
+ by (simp add:RAG_set_unchanged)
qed
next
case vt_nil
show ?case
by (unfold cntCS_def,
- auto simp:count_def holdents_test s_depend_def wq_def cs_holding_def)
+ auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
qed
qed
lemma eq_waiting: "waiting (wq (s::state)) th cs = waiting s th cs"
by (auto simp:s_waiting_def cs_waiting_def wq_def)
-lemma dm_depend_threads:
+lemma dm_RAG_threads:
fixes th s
assumes vt: "vt s"
- and in_dom: "(Th th) \<in> Domain (depend s)"
+ and in_dom: "(Th th) \<in> Domain (RAG s)"
shows "th \<in> threads s"
proof -
- from in_dom obtain n where "(Th th, n) \<in> depend s" by auto
- moreover from depend_target_th[OF this] obtain cs where "n = Cs cs" by auto
- ultimately have "(Th th, Cs cs) \<in> depend s" by simp
+ from in_dom obtain n where "(Th th, n) \<in> RAG s" by auto
+ moreover from RAG_target_th[OF this] obtain cs where "n = Cs cs" by auto
+ ultimately have "(Th th, Cs cs) \<in> RAG s" by simp
hence "th \<in> set (wq s cs)"
- by (unfold s_depend_def, auto simp:cs_waiting_def)
+ by (unfold s_RAG_def, auto simp:cs_waiting_def)
from wq_threads [OF vt this] show ?thesis .
qed
@@ -2112,18 +2112,18 @@
proof -
have "finite (dependants (wq s) th1)"
proof-
- have "finite {th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+}"
+ have "finite {th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th1) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+ have "{th'. (Th th', Th th1) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th1)" in bexI, auto)
moreover have "finite \<dots>"
proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
+ from finite_RAG[OF vt] have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
@@ -2151,18 +2151,18 @@
proof -
have "finite (dependants (wq s) th2)"
proof-
- have "finite {th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+}"
+ have "finite {th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th2) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+ have "{th'. (Th th', Th th2) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th2)" in bexI, auto)
moreover have "finite \<dots>"
proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
+ from finite_RAG[OF vt] have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
@@ -2190,11 +2190,11 @@
thus "th1' \<in> threads s"
proof
assume "th1' \<in> dependants (wq s) th1"
- hence "(Th th1') \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependants_def cs_depend_def s_depend_def)
+ hence "(Th th1') \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
by (auto simp:Domain_def)
- hence "(Th th1') \<in> Domain (depend s)" by (simp add:trancl_domain)
- from dm_depend_threads[OF vt this] show ?thesis .
+ hence "(Th th1') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ from dm_RAG_threads[OF vt this] show ?thesis .
next
assume "th1' = th1"
with runing_1 show ?thesis
@@ -2205,11 +2205,11 @@
thus "th2' \<in> threads s"
proof
assume "th2' \<in> dependants (wq s) th2"
- hence "(Th th2') \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependants_def cs_depend_def s_depend_def)
+ hence "(Th th2') \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
by (auto simp:Domain_def)
- hence "(Th th2') \<in> Domain (depend s)" by (simp add:trancl_domain)
- from dm_depend_threads[OF vt this] show ?thesis .
+ hence "(Th th2') \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ from dm_RAG_threads[OF vt this] show ?thesis .
next
assume "th2' = th2"
with runing_2 show ?thesis
@@ -2227,18 +2227,18 @@
next
assume "th2' \<in> dependants (wq s) th2"
with eq_th12 eq_th' have "th1 \<in> dependants (wq s) th2" by simp
- hence "(Th th1, Th th2) \<in> (depend s)^+"
- by (unfold cs_dependants_def s_depend_def cs_depend_def, simp)
- hence "Th th1 \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependants_def cs_depend_def s_depend_def)
+ hence "(Th th1, Th th2) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ hence "Th th1 \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
by (auto simp:Domain_def)
- hence "Th th1 \<in> Domain (depend s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th1, n) \<in> depend s" by (auto simp:Domain_def)
- from depend_target_th [OF this]
+ hence "Th th1 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ then obtain n where d: "(Th th1, n) \<in> RAG s" by (auto simp:Domain_def)
+ from RAG_target_th [OF this]
obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th1, Cs cs') \<in> depend s" by simp
+ with d have "(Th th1, Cs cs') \<in> RAG s" by simp
with runing_1 have "False"
- apply (unfold runing_def readys_def s_depend_def)
+ apply (unfold runing_def readys_def s_RAG_def)
by (auto simp:eq_waiting)
thus ?thesis by simp
qed
@@ -2249,27 +2249,27 @@
proof
assume "th2' = th2"
with th1'_in eq_th12 have "th2 \<in> dependants (wq s) th1" by simp
- hence "(Th th2, Th th1) \<in> (depend s)^+"
- by (unfold cs_dependants_def s_depend_def cs_depend_def, simp)
- hence "Th th2 \<in> Domain ((depend s)^+)"
- apply (unfold cs_dependants_def cs_depend_def s_depend_def)
+ hence "(Th th2, Th th1) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ hence "Th th2 \<in> Domain ((RAG s)^+)"
+ apply (unfold cs_dependants_def cs_RAG_def s_RAG_def)
by (auto simp:Domain_def)
- hence "Th th2 \<in> Domain (depend s)" by (simp add:trancl_domain)
- then obtain n where d: "(Th th2, n) \<in> depend s" by (auto simp:Domain_def)
- from depend_target_th [OF this]
+ hence "Th th2 \<in> Domain (RAG s)" by (simp add:trancl_domain)
+ then obtain n where d: "(Th th2, n) \<in> RAG s" by (auto simp:Domain_def)
+ from RAG_target_th [OF this]
obtain cs' where "n = Cs cs'" by auto
- with d have "(Th th2, Cs cs') \<in> depend s" by simp
+ with d have "(Th th2, Cs cs') \<in> RAG s" by simp
with runing_2 have "False"
- apply (unfold runing_def readys_def s_depend_def)
+ apply (unfold runing_def readys_def s_RAG_def)
by (auto simp:eq_waiting)
thus ?thesis by simp
next
assume "th2' \<in> dependants (wq s) th2"
with eq_th12 have "th1' \<in> dependants (wq s) th2" by simp
- hence h1: "(Th th1', Th th2) \<in> (depend s)^+"
- by (unfold cs_dependants_def s_depend_def cs_depend_def, simp)
- from th1'_in have h2: "(Th th1', Th th1) \<in> (depend s)^+"
- by (unfold cs_dependants_def s_depend_def cs_depend_def, simp)
+ hence h1: "(Th th1', Th th2) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
+ from th1'_in have h2: "(Th th1', Th th1) \<in> (RAG s)^+"
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, simp)
show ?thesis
proof(rule dchain_unique[OF vt h1 _ h2, symmetric])
from runing_1 show "th1 \<in> readys s" by (simp add:runing_def)
@@ -2411,9 +2411,9 @@
qed
qed
-lemma eq_depend:
- "depend (wq s) = depend s"
-by (unfold cs_depend_def s_depend_def, auto)
+lemma eq_RAG:
+ "RAG (wq s) = RAG s"
+by (unfold cs_RAG_def s_RAG_def, auto)
lemma count_eq_dependants:
assumes vt: "vt s"
@@ -2423,28 +2423,28 @@
from cnp_cnv_cncs[OF vt] and eq_pv
have "cntCS s th = 0"
by (auto split:if_splits)
- moreover have "finite {cs. (Cs cs, Th th) \<in> depend s}"
+ moreover have "finite {cs. (Cs cs, Th th) \<in> RAG s}"
proof -
from finite_holding[OF vt, of th] show ?thesis
by (simp add:holdents_test)
qed
- ultimately have h: "{cs. (Cs cs, Th th) \<in> depend s} = {}"
+ ultimately have h: "{cs. (Cs cs, Th th) \<in> RAG s} = {}"
by (unfold cntCS_def holdents_test cs_dependants_def, auto)
show ?thesis
proof(unfold cs_dependants_def)
- { assume "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}"
- then obtain th' where "(Th th', Th th) \<in> (depend (wq s))\<^sup>+" by auto
+ { assume "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}"
+ then obtain th' where "(Th th', Th th) \<in> (RAG (wq s))\<^sup>+" by auto
hence "False"
proof(cases)
- assume "(Th th', Th th) \<in> depend (wq s)"
- thus "False" by (auto simp:cs_depend_def)
+ assume "(Th th', Th th) \<in> RAG (wq s)"
+ thus "False" by (auto simp:cs_RAG_def)
next
fix c
- assume "(c, Th th) \<in> depend (wq s)"
- with h and eq_depend show "False"
- by (cases c, auto simp:cs_depend_def)
+ assume "(c, Th th) \<in> RAG (wq s)"
+ with h and eq_RAG show "False"
+ by (cases c, auto simp:cs_RAG_def)
qed
- } thus "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} = {}" by auto
+ } thus "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} = {}" by auto
qed
qed
@@ -2454,20 +2454,20 @@
shows "dependants (wq s) th \<subseteq> threads s"
proof
{ fix th th'
- assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (depend (wq s))\<^sup>+}"
- have "Th th \<in> Domain (depend s)"
+ assume h: "th \<in> {th'a. (Th th'a, Th th') \<in> (RAG (wq s))\<^sup>+}"
+ have "Th th \<in> Domain (RAG s)"
proof -
- from h obtain th' where "(Th th, Th th') \<in> (depend (wq s))\<^sup>+" by auto
- hence "(Th th) \<in> Domain ( (depend (wq s))\<^sup>+)" by (auto simp:Domain_def)
- with trancl_domain have "(Th th) \<in> Domain (depend (wq s))" by simp
- thus ?thesis using eq_depend by simp
+ from h obtain th' where "(Th th, Th th') \<in> (RAG (wq s))\<^sup>+" by auto
+ hence "(Th th) \<in> Domain ( (RAG (wq s))\<^sup>+)" by (auto simp:Domain_def)
+ with trancl_domain have "(Th th) \<in> Domain (RAG (wq s))" by simp
+ thus ?thesis using eq_RAG by simp
qed
- from dm_depend_threads[OF vt this]
+ from dm_RAG_threads[OF vt this]
have "th \<in> threads s" .
} note hh = this
fix th1
assume "th1 \<in> dependants (wq s) th"
- hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (depend (wq s))\<^sup>+}"
+ hence "th1 \<in> {th'a. (Th th'a, Th th) \<in> (RAG (wq s))\<^sup>+}"
by (unfold cs_dependants_def, simp)
from hh [OF this] show "th1 \<in> threads s" .
qed
@@ -2496,21 +2496,21 @@
and th_in: "th \<in> threads s"
shows "cp s th \<le> Max ((\<lambda> th. (preced th s)) ` threads s)"
proof(unfold cp_eq_cpreced cpreced_def cs_dependants_def)
- show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}))
+ show "Max ((\<lambda>th. preced th s) ` ({th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}))
\<le> Max ((\<lambda>th. preced th s) ` threads s)"
(is "Max (?f ` ?A) \<le> Max (?f ` ?B)")
proof(rule Max_f_mono)
- show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<noteq> {}" by simp
+ show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<noteq> {}" by simp
next
from finite_threads [OF vt]
show "finite (threads s)" .
next
from th_in
- show "{th} \<union> {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> threads s"
+ show "{th} \<union> {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> threads s"
apply (auto simp:Domain_def)
- apply (rule_tac dm_depend_threads[OF vt])
- apply (unfold trancl_domain [of "depend s", symmetric])
- by (unfold cs_depend_def s_depend_def, auto simp:Domain_def)
+ apply (rule_tac dm_RAG_threads[OF vt])
+ apply (unfold trancl_domain [of "RAG s", symmetric])
+ by (unfold cs_RAG_def s_RAG_def, auto simp:Domain_def)
qed
qed
@@ -2528,18 +2528,18 @@
proof -
have "finite ?B"
proof-
- have "finite {th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+}"
+ have "finite {th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+}"
proof -
let ?F = "\<lambda> (x, y). the_th x"
- have "{th'. (Th th', Th th) \<in> (depend (wq s))\<^sup>+} \<subseteq> ?F ` ((depend (wq s))\<^sup>+)"
+ have "{th'. (Th th', Th th) \<in> (RAG (wq s))\<^sup>+} \<subseteq> ?F ` ((RAG (wq s))\<^sup>+)"
apply (auto simp:image_def)
by (rule_tac x = "(Th x, Th th)" in bexI, auto)
moreover have "finite \<dots>"
proof -
- from finite_depend[OF vt] have "finite (depend s)" .
- hence "finite ((depend (wq s))\<^sup>+)"
+ from finite_RAG[OF vt] have "finite (RAG s)" .
+ hence "finite ((RAG (wq s))\<^sup>+)"
apply (unfold finite_trancl)
- by (auto simp: s_depend_def cs_depend_def wq_def)
+ by (auto simp: s_RAG_def cs_RAG_def wq_def)
thus ?thesis by auto
qed
ultimately show ?thesis by (auto intro:finite_subset)
@@ -2621,12 +2621,12 @@
then obtain tm where tm_max: "?f tm = ?p" and tm_in: "tm \<in> threads s"
by (auto simp:Image_def)
from th_chain_to_ready [OF vt tm_in]
- have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+)" .
+ have "tm \<in> readys s \<or> (\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+)" .
thus ?thesis
proof
- assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (depend s)\<^sup>+ "
+ assume "\<exists>th'. th' \<in> readys s \<and> (Th tm, Th th') \<in> (RAG s)\<^sup>+ "
then obtain th' where th'_in: "th' \<in> readys s"
- and tm_chain:"(Th tm, Th th') \<in> (depend s)\<^sup>+" by auto
+ and tm_chain:"(Th tm, Th th') \<in> (RAG s)\<^sup>+" by auto
have "cp s th' = ?f tm"
proof(subst cp_eq_cpreced, subst cpreced_def, rule Max_eqI)
from dependants_threads[OF vt] finite_threads[OF vt]
@@ -2650,7 +2650,7 @@
proof -
from tm_chain
have "tm \<in> dependants (wq s) th'"
- by (unfold cs_dependants_def s_depend_def cs_depend_def, auto)
+ by (unfold cs_dependants_def s_RAG_def cs_RAG_def, auto)
thus ?thesis by auto
qed
qed
@@ -2780,9 +2780,9 @@
lemma detached_test:
- shows "detached s th = (Th th \<notin> Field (depend s))"
+ shows "detached s th = (Th th \<notin> Field (RAG s))"
apply(simp add: detached_def Field_def)
-apply(simp add: s_depend_def)
+apply(simp add: s_RAG_def)
apply(simp add: s_holding_abv s_waiting_abv)
apply(simp add: Domain_iff Range_iff)
apply(simp add: wq_def)
@@ -2805,12 +2805,12 @@
thus ?thesis
proof
assume "th \<notin> threads s"
- with range_in[OF vt] dm_depend_threads[OF vt]
+ with range_in[OF vt] dm_RAG_threads[OF vt]
show ?thesis
- by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
+ by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def Domain_iff Range_iff)
next
assume "th \<in> readys s"
- moreover have "Th th \<notin> Range (depend s)"
+ moreover have "Th th \<notin> Range (RAG s)"
proof -
from card_0_eq [OF finite_holding [OF vt]] and cncs_zero
have "holdents s th = {}"
@@ -2818,11 +2818,11 @@
thus ?thesis
apply(auto simp:holdents_test)
apply(case_tac a)
- apply(auto simp:holdents_test s_depend_def)
+ apply(auto simp:holdents_test s_RAG_def)
done
qed
ultimately show ?thesis
- by (auto simp add: detached_def s_depend_def s_waiting_abv s_holding_abv wq_def readys_def)
+ by (auto simp add: detached_def s_RAG_def s_waiting_abv s_holding_abv wq_def readys_def)
qed
qed
@@ -2838,7 +2838,7 @@
have cncs_z: "cntCS s th = 0"
proof -
from dtc have "holdents s th = {}"
- unfolding detached_def holdents_test s_depend_def
+ unfolding detached_def holdents_test s_RAG_def
by (simp add: s_waiting_abv wq_def s_holding_abv Domain_iff Range_iff)
thus ?thesis by (auto simp:cntCS_def)
qed
@@ -2848,7 +2848,7 @@
with dtc
have "th \<in> readys s"
by (unfold readys_def detached_def Field_def Domain_def Range_def,
- auto simp:eq_waiting s_depend_def)
+ auto simp:eq_waiting s_RAG_def)
with cncs_z and eq_pv show ?thesis by simp
next
case False
@@ -2862,4 +2862,4 @@
shows "(detached s th) = (cntP s th = cntV s th)"
by (insert vt, auto intro:detached_intro detached_elim)
-end
\ No newline at end of file
+end