--- a/CpsG.thy Fri Dec 18 19:13:19 2015 +0800
+++ b/CpsG.thy Fri Dec 18 22:47:32 2015 +0800
@@ -6,16 +6,38 @@
imports PrioG Max RTree
begin
+text {* @{text "the_preced"} is also the same as @{text "preced"}, the only
+ difference is the order of arguemts. *}
+definition "the_preced s th = preced th s"
+
+text {* @{term "the_thread"} extracts thread out of RAG node. *}
+fun the_thread :: "node \<Rightarrow> thread" where
+ "the_thread (Th th) = th"
+
+text {* The following @{text "wRAG"} is the waiting sub-graph of @{text "RAG"}. *}
definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"
+text {* The following @{text "hRAG"} is the holding sub-graph of @{text "RAG"}. *}
definition "hRAG (s::state) = {(Cs cs, Th th) | th cs. holding s th cs}"
-definition "tRAG s = wRAG s O hRAG s"
-
+text {* The following lemma splits @{term "RAG"} graph into the above two sub-graphs. *}
lemma RAG_split: "RAG s = (wRAG s \<union> hRAG s)"
by (unfold s_RAG_abv wRAG_def hRAG_def s_waiting_abv
s_holding_abv cs_RAG_def, auto)
+text {*
+ The following @{text "tRAG"} is the thread-graph derived from @{term "RAG"}.
+ It characterizes the dependency between threads when calculating current
+ precedences. It is defined as the composition of the above two sub-graphs,
+ names @{term "wRAG"} and @{term "hRAG"}.
+ *}
+definition "tRAG s = wRAG s O hRAG s"
+
+(* ccc *)
+
+definition "cp_gen s x =
+ Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"
+
lemma tRAG_alt_def:
"tRAG s = {(Th th1, Th th2) | th1 th2.
\<exists> cs. (Th th1, Cs cs) \<in> RAG s \<and> (Cs cs, Th th2) \<in> RAG s}"
@@ -126,47 +148,6 @@
} ultimately show ?thesis by auto
qed
-lemma readys_root:
- assumes "vt s"
- and "th \<in> readys s"
- shows "root (RAG s) (Th th)"
-proof -
- { fix x
- assume "x \<in> ancestors (RAG s) (Th th)"
- hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
- from tranclD[OF this]
- obtain z where "(Th th, z) \<in> RAG s" by auto
- with assms(2) have False
- apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
- by (fold wq_def, blast)
- } thus ?thesis by (unfold root_def, auto)
-qed
-
-lemma readys_in_no_subtree:
- assumes "vt s"
- and "th \<in> readys s"
- and "th' \<noteq> th"
- shows "Th th \<notin> subtree (RAG s) (Th th')"
-proof
- assume "Th th \<in> subtree (RAG s) (Th th')"
- thus False
- proof(cases rule:subtreeE)
- case 1
- with assms show ?thesis by auto
- next
- case 2
- with readys_root[OF assms(1,2)]
- show ?thesis by (auto simp:root_def)
- qed
-qed
-
-lemma image_id:
- assumes "\<And> x. x \<in> A \<Longrightarrow> f x = x"
- shows "f ` A = A"
- using assms by (auto simp:image_def)
-
-definition "the_preced s th = preced th s"
-
lemma cp_alt_def:
"cp s th =
Max ((the_preced s) ` {th'. Th th' \<in> (subtree (RAG s) (Th th))})"
@@ -182,12 +163,6 @@
thus ?thesis by (unfold cp_eq_cpreced cpreced_def, fold the_preced_def, simp)
qed
-fun the_thread :: "node \<Rightarrow> thread" where
- "the_thread (Th th) = th"
-
-definition "cp_gen s x =
- Max ((the_preced s \<circ> the_thread) ` subtree (tRAG s) x)"
-
lemma cp_gen_alt_def:
"cp_gen s = (Max \<circ> (\<lambda>x. (the_preced s \<circ> the_thread) ` subtree (tRAG s) x))"
by (auto simp:cp_gen_def)
@@ -346,8 +321,6 @@
by (unfold eq_a, simp, unfold cp_gen_def_cond[OF refl[of "Th th"]], simp)
qed
-
-
locale valid_trace =
fixes s
assumes vt : "vt s"
@@ -355,6 +328,38 @@
context valid_trace
begin
+lemma readys_root:
+ assumes "th \<in> readys s"
+ shows "root (RAG s) (Th th)"
+proof -
+ { fix x
+ assume "x \<in> ancestors (RAG s) (Th th)"
+ hence h: "(Th th, x) \<in> (RAG s)^+" by (auto simp:ancestors_def)
+ from tranclD[OF this]
+ obtain z where "(Th th, z) \<in> RAG s" by auto
+ with assms(1) have False
+ apply (case_tac z, auto simp:readys_def s_RAG_def s_waiting_def cs_waiting_def)
+ by (fold wq_def, blast)
+ } thus ?thesis by (unfold root_def, auto)
+qed
+
+lemma readys_in_no_subtree:
+ assumes "th \<in> readys s"
+ and "th' \<noteq> th"
+ shows "Th th \<notin> subtree (RAG s) (Th th')"
+proof
+ assume "Th th \<in> subtree (RAG s) (Th th')"
+ thus False
+ proof(cases rule:subtreeE)
+ case 1
+ with assms show ?thesis by auto
+ next
+ case 2
+ with readys_root[OF assms(1)]
+ show ?thesis by (auto simp:root_def)
+ qed
+qed
+
lemma not_in_thread_isolated:
assumes "th \<notin> threads s"
shows "(Th th) \<notin> Field (RAG s)"
@@ -371,11 +376,6 @@
from acyclic_RAG[OF vt] show "acyclic (RAG s)" .
qed
-end
-
-context valid_trace
-begin
-
lemma sgv_wRAG: "single_valued (wRAG s)"
using waiting_unique[OF vt]
by (unfold single_valued_def wRAG_def, auto)
@@ -403,8 +403,19 @@
lemma rtree_RAG: "rtree (RAG s)"
using sgv_RAG acyclic_RAG[OF vt]
by (unfold rtree_def rtree_axioms_def sgv_def, auto)
+end
-end
+
+sublocale valid_trace < rtree_RAG: rtree "RAG s"
+proof
+ show "single_valued (RAG s)"
+ apply (intro_locales)
+ by (unfold single_valued_def,
+ auto intro:unique_RAG[OF vt])
+
+ show "acyclic (RAG s)"
+ by (rule acyclic_RAG[OF vt])
+qed
sublocale valid_trace < rtree_s: rtree "tRAG s"
proof(unfold_locales)
@@ -486,7 +497,6 @@
finally show ?thesis by simp
qed
-
context valid_trace
begin
@@ -568,521 +578,7 @@
end
-
-lemma eq_dependants: "dependants (wq s) = dependants s"
- by (simp add: s_dependants_abv wq_def)
-
-
-(* obvious lemma *)
-lemma not_thread_holdents:
- fixes th s
- assumes vt: "vt s"
- and not_in: "th \<notin> threads s"
- shows "holdents s th = {}"
-proof -
- from vt not_in show ?thesis
- proof(induct arbitrary:th)
- case (vt_cons s e th)
- assume vt: "vt s"
- and ih: "\<And>th. th \<notin> threads s \<Longrightarrow> holdents s th = {}"
- and stp: "step s e"
- and not_in: "th \<notin> threads (e # s)"
- from stp show ?case
- proof(cases)
- case (thread_create thread prio)
- assume eq_e: "e = Create thread prio"
- and not_in': "thread \<notin> threads s"
- have "holdents (e # s) th = holdents s th"
- apply (unfold eq_e holdents_test)
- by (simp add:RAG_create_unchanged)
- moreover have "th \<notin> threads s"
- proof -
- from not_in eq_e show ?thesis by simp
- qed
- moreover note ih ultimately show ?thesis by auto
- next
- case (thread_exit thread)
- assume eq_e: "e = Exit thread"
- and nh: "holdents s thread = {}"
- show ?thesis
- proof(cases "th = thread")
- case True
- with nh eq_e
- show ?thesis
- 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 RAG_exit_unchanged)
- qed
- next
- case (thread_P thread cs)
- assume eq_e: "e = P thread cs"
- and is_runing: "thread \<in> runing s"
- from assms thread_exit ih stp not_in vt eq_e have vtp: "vt (P thread cs#s)" by auto
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- hence "holdents (e # s) th = holdents s th "
- apply (unfold cntCS_def holdents_test eq_e)
- 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
- qed
- ultimately show ?thesis by simp
- next
- case (thread_V thread cs)
- assume eq_e: "e = V thread cs"
- and is_runing: "thread \<in> runing s"
- and hold: "holding s thread cs"
- have neq_th: "th \<noteq> thread"
- proof -
- from not_in eq_e have "th \<notin> threads s" by simp
- moreover from is_runing have "thread \<in> threads s"
- by (simp add:runing_def readys_def)
- ultimately show ?thesis by auto
- qed
- from assms thread_V eq_e ih stp not_in vt have vtv: "vt (V thread cs#s)" by auto
- from hold obtain rest
- where eq_wq: "wq s cs = thread # rest"
- by (case_tac "wq s cs", auto simp: wq_def s_holding_def)
- from not_in eq_e eq_wq
- have "\<not> next_th s thread cs th"
- apply (auto simp:next_th_def)
- proof -
- assume ne: "rest \<noteq> []"
- and ni: "hd (SOME q. distinct q \<and> set q = set rest) \<notin> threads s" (is "?t \<notin> threads s")
- have "?t \<in> set rest"
- proof(rule someI2)
- from wq_distinct[OF step_back_vt[OF vtv], of cs] and eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x assume "distinct x \<and> set x = set rest" with ne
- show "hd x \<in> set rest" by (cases x, auto)
- qed
- with eq_wq have "?t \<in> set (wq s cs)" by simp
- from wq_threads[OF step_back_vt[OF vtv], OF this] and ni
- show False by auto
- 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_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
- qed
- ultimately show ?thesis by simp
- next
- case (thread_set thread prio)
- print_facts
- assume eq_e: "e = Set thread prio"
- and is_runing: "thread \<in> runing s"
- from not_in and eq_e have "th \<notin> threads s" by auto
- from ih [OF this] and eq_e
- show ?thesis
- apply (unfold eq_e cntCS_def holdents_test)
- by (simp add:RAG_set_unchanged)
- qed
- next
- case vt_nil
- show ?case
- by (auto simp:count_def holdents_test s_RAG_def wq_def cs_holding_def)
- qed
-qed
-
-(* obvious lemma *)
-lemma next_th_neq:
- assumes vt: "vt s"
- and nt: "next_th s th cs th'"
- shows "th' \<noteq> th"
-proof -
- from nt show ?thesis
- apply (auto simp:next_th_def)
- proof -
- fix rest
- assume eq_wq: "wq s cs = hd (SOME q. distinct q \<and> set q = set rest) # rest"
- and ne: "rest \<noteq> []"
- have "hd (SOME q. distinct q \<and> set q = set rest) \<in> set rest"
- proof(rule someI2)
- from wq_distinct[OF vt, of cs] eq_wq
- show "distinct rest \<and> set rest = set rest" by auto
- next
- fix x
- assume "distinct x \<and> set x = set rest"
- hence eq_set: "set x = set rest" by auto
- with ne have "x \<noteq> []" by auto
- hence "hd x \<in> set x" by auto
- with eq_set show "hd x \<in> set rest" by auto
- qed
- with wq_distinct[OF vt, of cs] eq_wq show False by auto
- qed
-qed
-
-(* obvious lemma *)
-lemma next_th_unique:
- assumes nt1: "next_th s th cs th1"
- and nt2: "next_th s th cs th2"
- shows "th1 = th2"
-using assms by (unfold next_th_def, auto)
-
-lemma wf_RAG:
- assumes vt: "vt s"
- shows "wf (RAG s)"
-proof(rule finite_acyclic_wf)
- from finite_RAG[OF vt] show "finite (RAG s)" .
-next
- from acyclic_RAG[OF vt] show "acyclic (RAG s)" .
-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> 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> 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"
- unfolding children_def2
- unfolding cs_dependants_def
- by (auto simp add: eq_RAG)
-
-lemma child_unique:
- assumes vt: "vt s"
- and ch1: "(Th th, Th th1) \<in> child s"
- and ch2: "(Th th, Th th2) \<in> child s"
- shows "th1 = th2"
-using ch1 ch2
-proof(unfold child_def, clarsimp)
- fix cs csa
- 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 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> (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_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> (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_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> (RAG s)\<^sup>+" by simp
- ultimately show ?thesis by auto
- qed
- next
- 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> RAG s"
- thus ?thesis
- by (auto simp:s_RAG_def)
- qed
-qed
-
-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_RAG[OF vt]])
-apply(rule sub_child)
-done
-
-lemma RAG_child_pre:
- assumes vt: "vt s"
- shows
- "(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_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> (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 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> (RAG s)\<^sup>+"
- then obtain th3 where th3_in: "th3 \<in> children s th'"
- 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
- qed
- qed
-qed
-
-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_RAG_p:
- assumes "(n1, n2) \<in> (child s)^+"
- shows "(n1, n2) \<in> (RAG s)^+"
-proof -
- from assms show ?thesis
- proof(induct)
- case (base y)
- with sub_child show ?case by auto
- next
- case (step y z)
- assume "(y, z) \<in> child s"
- 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
-
-text {* (* ddd *)
-*}
-lemma child_RAG_eq:
- assumes vt: "vt s"
- 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)
-
-text {* (* ddd *)
-*}
-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> (RAG s)^+"
- shows "False"
-proof -
- from RAG_child[OF vt ch3]
- have "(Th th1, Th th2) \<in> (child s)\<^sup>+" .
- thus ?thesis
- proof(rule converse_tranclE)
- assume "(Th th1, Th th2) \<in> child s"
- from child_unique[OF vt ch1 this] have "th = th2" by simp
- with ch2 have "(Th th2, Th th2) \<in> child s" by simp
- with wf_child[OF vt] show ?thesis by auto
- next
- fix c
- assume h1: "(Th th1, c) \<in> child s"
- and h2: "(c, Th th2) \<in> (child s)\<^sup>+"
- from h1 obtain th3 where eq_c: "c = Th th3" by (unfold child_def, auto)
- with h1 have "(Th th1, Th th3) \<in> child s" by simp
- from child_unique[OF vt ch1 this] have eq_th3: "th3 = th" by simp
- with eq_c and h2 have "(Th th, Th th2) \<in> (child s)\<^sup>+" by simp
- with ch2 have "(Th th, Th th) \<in> (child s)\<^sup>+" by auto
- moreover have "wf ((child s)\<^sup>+)"
- proof(rule wf_trancl)
- from wf_child[OF vt] show "wf (child s)" .
- qed
- ultimately show False by auto
- qed
-qed
-
-text {* (* ddd *)
-*}
-lemma unique_RAG_p:
- assumes vt: "vt s"
- and dp1: "(n, n1) \<in> (RAG s)^+"
- and dp2: "(n, n2) \<in> (RAG s)^+"
- and neq: "n1 \<noteq> n2"
- shows "(n1, n2) \<in> (RAG s)^+ \<or> (n2, n1) \<in> (RAG s)^+"
-proof(rule unique_chain [OF _ dp1 dp2 neq])
- 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
-
-text {* (* ddd *)
-*}
-lemma dependants_child_unique:
- 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 dp1: "th3 \<in> dependants s th1"
- and dp2: "th3 \<in> dependants s th2"
-shows "th1 = th2"
-proof -
- { assume neq: "th1 \<noteq> th2"
- 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> (RAG s)\<^sup>+ "
- from children_no_dep[OF vt ch1 ch2 this] show ?thesis .
- next
- 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 RAG_plus_elim:
- assumes "vt s"
- fixes x
- 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 RAG_children eq_RAG)
-
-text {* (* ddd *)
-*}
-lemma dependants_expand:
- assumes "vt s"
- 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"
- shows "finite (children s th)"
- using children_dependants dependants_threads[OF assms] finite_threads[OF assms]
- by (metis rev_finite_subset)
-
-lemma finite_dependants:
- assumes "vt s"
- shows "finite (dependants (wq s) th')"
- using dependants_threads[OF assms] finite_threads[OF assms]
- by (metis rev_finite_subset)
-
-abbreviation
- "preceds s ths \<equiv> {preced th s| th. th \<in> ths}"
-
-abbreviation
- "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'})"
-by auto
-
-lemma in_disj:
- shows "x \<in> A \<or> (\<exists>y \<in> A. x \<in> Q y) \<longleftrightarrow> (\<exists>y \<in> A. x = y \<or> x \<in> Q y)"
-by metis
-
-lemma UN_exists:
- shows "(\<Union>x \<in> A. {f y | y. Q y x}) = ({f y | y. (\<exists>x \<in> A. Q y x)})"
-by auto
-
-text {* (* ddd *)
- This is the recursive equation used to compute the current precedence of
- a thread (the @{text "th"}) here.
-*}
-lemma cp_rec:
- fixes s th
- assumes vt: "vt s"
- shows "cp s th = Max ({preced th s} \<union> (cp s ` children s th))"
-proof(cases "children s th = {}")
- case True
- show ?thesis
- unfolding cp_eq_cpreced cpreced_def
- by (subst dependants_expand[OF `vt s`]) (simp add: True)
-next
- case False
- 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_Collect[symmetric])
-
- have not_emptyness_facts[simp]:
- "dependants (wq s) th \<noteq> {}" "children s th \<noteq> {}"
- 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)"
- by (simp_all add: finite_dependants[OF `vt s`] finite_children[OF `vt s`])
-
- (* expanding definition *)
- have "?LHS = Max ({preced th s} \<union> preceds s (dependants (wq s) th))"
- unfolding eq_cp by (simp add: Un_compr)
-
- (* moving Max in *)
- also have "\<dots> = max (Max {preced th s}) (Max (preceds s (dependants (wq s) th)))"
- by (simp add: Max_Un)
-
- (* 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 dependants_expand[OF `vt s`]) (simp)
-
- (* moving out big Union *)
- also have "\<dots> = max (Max {preced th s})
- (Max (preceds s (\<Union> ({children s th} \<union> (dependants (wq s) ` children s th)))))"
- by simp
-
- (* moving in small union *)
- also have "\<dots> = max (Max {preced th s})
- (Max (preceds s (\<Union> ((\<lambda>th. {th} \<union> (dependants (wq s) th)) ` children s th))))"
- by (simp add: in_disj)
-
- (* moving in preceds *)
- also have "\<dots> = max (Max {preced th s})
- (Max (\<Union> ((\<lambda>th. preceds s ({th} \<union> (dependants (wq s) th))) ` children s th)))"
- by (simp add: UN_exists)
-
- (* moving in Max *)
- also have "\<dots> = max (Max {preced th s})
- (Max ((\<lambda>th. Max (preceds s ({th} \<union> (dependants (wq s) th)))) ` children s th))"
- by (subst Max_Union) (auto simp add: image_image)
-
- (* folding cp + moving out Max *)
- also have "\<dots> = ?RHS"
- unfolding eq_cp by (simp add: Max_insert)
-
- finally show "?LHS = ?RHS" .
- qed
-qed
-
+(* keep *)
lemma next_th_holding:
assumes vt: "vt s"
and nxt: "next_th s th cs th'"
@@ -1172,6 +668,16 @@
the legitimacy of @{text "s"} can be derived. *}
assumes vt_s: "vt s"
+sublocale step_set_cps < vat_s : valid_trace "s"
+proof
+ from vt_s show "vt s" .
+qed
+
+sublocale step_set_cps < vat_s' : valid_trace "s'"
+proof
+ from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
+qed
+
context step_set_cps
begin
@@ -1181,7 +687,6 @@
*}
lemma eq_preced:
- fixes th'
assumes "th' \<noteq> th"
shows "preced th' s = preced th' s'"
proof -
@@ -1255,7 +760,8 @@
hence "th \<in> runing s'" by (cases, simp)
thus ?thesis by (simp add:readys_def runing_def)
qed
- from readys_in_no_subtree[OF step_back_vt[OF vt_s[unfolded s_def]] this assms(1)]
+ find_theorems readys subtree
+ from vat_s'.readys_in_no_subtree[OF this assms(1)]
show ?thesis by blast
qed
@@ -1284,40 +790,26 @@
-- {* @{text "s"} is assumed to be valid, which implies the validity of @{text "s'"} *}
assumes vt_s: "vt s"
-context step_v_cps
-begin
-
-lemma rtree_RAGs: "rtree (RAG s)"
+sublocale step_v_cps < vat_s : valid_trace "s"
proof
- show "single_valued (RAG s)"
- apply (intro_locales)
- by (unfold single_valued_def, auto intro: unique_RAG[OF vt_s])
-
- show "acyclic (RAG s)"
- by (rule acyclic_RAG[OF vt_s])
+ from vt_s show "vt s" .
qed
-lemma rtree_RAGs': "rtree (RAG s')"
+sublocale step_v_cps < vat_s' : valid_trace "s'"
proof
- show "single_valued (RAG s')"
- apply (intro_locales)
- by (unfold single_valued_def,
- auto intro:unique_RAG[OF step_back_vt[OF vt_s[unfolded s_def]]])
-
- show "acyclic (RAG s')"
- by (rule acyclic_RAG[OF step_back_vt[OF vt_s[unfolded s_def]]])
+ from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
qed
-lemmas vt_s' = step_back_vt[OF vt_s[unfolded s_def]]
+context step_v_cps
+begin
lemma ready_th_s': "th \<in> readys s'"
using step_back_step[OF vt_s[unfolded s_def]]
by (cases, simp add:runing_def)
-
lemma ancestors_th: "ancestors (RAG s') (Th th) = {}"
proof -
- from readys_root[OF vt_s' ready_th_s']
+ from vat_s'.readys_root[OF ready_th_s']
show ?thesis
by (unfold root_def, simp)
qed
@@ -1340,9 +832,8 @@
lemma ancestors_cs:
"ancestors (RAG s') (Cs cs) = {Th th}"
proof -
- find_theorems ancestors
have "ancestors (RAG s') (Cs cs) = ancestors (RAG s') (Th th) \<union> {Th th}"
- proof(rule RTree.rtree.ancestors_accum[OF rtree_RAGs'])
+ proof(rule vat_s'.rtree_RAG.ancestors_accum)
from vt_s[unfolded s_def]
have " PIP s' (V th cs)" by (cases, simp)
thus "(Cs cs, Th th) \<in> RAG s'"
@@ -1360,7 +851,6 @@
end
-
text {*
The following @{text "step_v_cps_nt"} is the sub-locale for @{text "V"}-operation,
which represents the case when there is another thread @{text "th'"}
@@ -1415,13 +905,13 @@
*)
lemma sub_RAGs': "{(Cs cs, Th th), (Th th', Cs cs)} \<subseteq> RAG s'"
- using next_th_RAG[OF vt_s' nt] .
+ using next_th_RAG[OF vat_s'.vt nt] .
lemma ancestors_th':
"ancestors (RAG s') (Th th') = {Th th, Cs cs}"
proof -
have "ancestors (RAG s') (Th th') = ancestors (RAG s') (Cs cs) \<union> {Cs cs}"
- proof(rule RTree.rtree.ancestors_accum[OF rtree_RAGs'])
+ proof(rule vat_s'.rtree_RAG.ancestors_accum)
from sub_RAGs' show "(Th th', Cs cs) \<in> RAG s'" by auto
qed
thus ?thesis using ancestors_th ancestors_cs by auto
@@ -1552,7 +1042,8 @@
lemma subtree_th:
"subtree (RAG s) (Th th) = subtree (RAG s') (Th th) - {Cs cs}"
-proof(unfold RAG_s, fold subtree_cs, rule RTree.rtree.subtree_del_inside[OF rtree_RAGs'])
+find_theorems "subtree" "_ - _" RAG
+proof(unfold RAG_s, fold subtree_cs, rule vat_s'.rtree_RAG.subtree_del_inside)
from edge_of_th
show "(Cs cs, Th th) \<in> edges_in (RAG s') (Th th)"
by (unfold edges_in_def, auto simp:subtree_def)
@@ -1567,18 +1058,8 @@
shows "cp s th' = cp s' th'"
using cp_kept_1 cp_kept_2
by (cases "th' = th", auto)
-
end
-find_theorems "_`_" "\<Union> _"
-
-find_theorems "Max" "\<Union> _"
-
-find_theorems wf RAG
-
-thm wf_def
-
-thm image_Union
locale step_P_cps =
fixes s' th cs s
@@ -1595,7 +1076,6 @@
from step_back_vt[OF vt_s[unfolded s_def]] show "vt s'" .
qed
-
context step_P_cps
begin
@@ -1608,7 +1088,7 @@
qed
lemma root_th: "root (RAG s') (Th th)"
- using readys_root[OF vat_s'.vt readys_th] .
+ using readys_root[OF readys_th] .
lemma in_no_others_subtree:
assumes "th' \<noteq> th"
@@ -2039,7 +1519,8 @@
qed auto
have neq_th_a: "th_a \<noteq> th"
proof -
- from readys_in_no_subtree[OF vat_s'.vt th_ready assms]
+ find_theorems readys subtree s'
+ from vat_s'.readys_in_no_subtree[OF th_ready assms]
have "(Th th) \<notin> subtree (RAG s') (Th th')" .
with tRAG_subtree_RAG[of s' "Th th'"]
have "(Th th) \<notin> subtree (tRAG s') (Th th')" by auto