--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Test.thy Thu Sep 21 14:15:55 2017 +0100
@@ -0,0 +1,789 @@
+theory Test
+imports Precedence_ord Graphs
+begin
+
+type_synonym thread = nat -- {* Type for thread identifiers. *}
+type_synonym priority = nat -- {* Type for priorities. *}
+type_synonym cs = nat -- {* Type for critical sections (or resources). *}
+
+-- {* Schedulling Events *}
+
+datatype event =
+ Create thread priority
+| Exit thread
+| P thread cs
+| V thread cs
+| Set thread priority
+
+type_synonym state = "event list"
+
+fun threads :: "state \<Rightarrow> thread set"
+ where
+ "threads [] = {}"
+| "threads (Create th prio#s) = {th} \<union> threads s"
+| "threads (Exit th # s) = (threads s) - {th}"
+| "threads (_#s) = threads s"
+
+fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
+ where
+ "priority th [] = 0"
+| "priority th (Create th' prio#s) = (if th' = th then prio else priority th s)"
+| "priority th (Set th' prio#s) = (if th' = th then prio else priority th s)"
+| "priority th (_#s) = priority th s"
+
+fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
+ where
+ "last_set th [] = 0"
+| "last_set th ((Create th' prio)#s) = (if (th = th') then length s else last_set th s)"
+| "last_set th ((Set th' prio)#s) = (if (th = th') then length s else last_set th s)"
+| "last_set th (_#s) = last_set th s"
+
+
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+ where "preced th s \<equiv> Prc (priority th s) (last_set th s)"
+
+abbreviation
+ "preceds s ths \<equiv> {preced th s | th. th \<in> ths}"
+
+definition
+ "holds wq th cs \<equiv> th \<in> set (wq cs) \<and> th = hd (wq cs)"
+
+definition
+ "waits wq th cs \<equiv> th \<in> set (wq cs) \<and> th \<noteq> hd (wq cs)"
+
+--{* Nodes in Resource Graph *}
+datatype node =
+ Th "thread"
+| Cs "cs"
+
+definition
+ "RAG wq \<equiv> {(Th th, Cs cs) | th cs. waits wq th cs} \<union> {(Cs cs, Th th) | cs th. holds wq th cs}"
+
+definition
+ "dependants wq th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
+
+record schedule_state =
+ wq_fun :: "cs \<Rightarrow> thread list"
+ cprec_fun :: "thread \<Rightarrow> precedence"
+
+definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
+ where
+ "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
+
+abbreviation
+ "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
+
+abbreviation
+ "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
+
+abbreviation
+ "release qs \<equiv> case qs of
+ [] => []
+ | (_ # qs) => SOME q. distinct q \<and> set q = set qs"
+
+lemma [simp]:
+ "(SOME q. distinct q \<and> q = []) = []"
+by auto
+
+lemma [simp]:
+ "(x \<in> set (SOME q. distinct q \<and> set q = set p)) = (x \<in> set p)"
+apply(rule iffI)
+apply (metis (mono_tags, lifting) List.finite_set finite_distinct_list some_eq_ex)+
+done
+
+abbreviation
+ "next_to_run ths \<equiv> hd (SOME q::thread list. distinct q \<and> set q = set ths)"
+
+
+fun schs :: "state \<Rightarrow> schedule_state"
+ where
+ "schs [] = (| wq_fun = \<lambda> cs. [], cprec_fun = (\<lambda>_. Prc 0 0) |)"
+| "schs (Create th prio # s) =
+ (let wq = wq_fun (schs s) in
+ (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"
+| "schs (Exit th # s) =
+ (let wq = wq_fun (schs s) in
+ (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"
+| "schs (Set th prio # s) =
+ (let wq = wq_fun (schs s) in
+ (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"
+| "schs (P th cs # s) =
+ (let wq = wq_fun (schs s) in
+ let new_wq = wq(cs := (wq cs @ [th])) in
+ (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"
+| "schs (V th cs # s) =
+ (let wq = wq_fun (schs s) in
+ let new_wq = wq(cs := release (wq cs)) in
+ (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"
+
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
+ where "wq s = wq_fun (schs s)"
+
+definition cpreced2 :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+ where "cpreced2 s \<equiv> cprec_fun (schs s)"
+
+abbreviation
+ "cpreceds2 s ths \<equiv> {cpreced2 s th | th. th \<in> ths}"
+
+definition
+ "holds2 s \<equiv> holds (wq_fun (schs s))"
+
+definition
+ "waits2 s \<equiv> waits (wq_fun (schs s))"
+
+definition
+ "RAG2 s \<equiv> RAG (wq_fun (schs s))"
+
+definition
+ "dependants2 s \<equiv> dependants (wq_fun (schs s))"
+
+(* ready -> is a thread that is not waiting for any resource *)
+definition readys :: "state \<Rightarrow> thread set"
+ where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waits2 s th cs)}"
+
+definition runing :: "state \<Rightarrow> thread set"
+ where "runing s \<equiv> {th . th \<in> readys s \<and> cpreced2 s th = Max (cpreceds2 s (readys s))}"
+
+(* all resources a thread hols in a state *)
+definition holding :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+ where "holding s th \<equiv> {cs . holds2 s th cs}"
+
+
+lemma exists_distinct:
+ obtains ys where "distinct ys" "set ys = set xs"
+by (metis List.finite_set finite_distinct_list)
+
+lemma next_to_run_set [simp]:
+ "wts \<noteq> [] \<Longrightarrow> next_to_run wts \<in> set wts"
+apply(rule exists_distinct[of wts])
+by (metis (mono_tags, lifting) hd_in_set set_empty some_eq_ex)
+
+lemma holding_RAG:
+ "holding s th = {cs . (Cs cs, Th th) \<in> RAG2 s}"
+unfolding holding_def RAG2_def RAG_def
+unfolding holds2_def holds_def waits_def
+by auto
+
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+ where
+ step_Create: "\<lbrakk>th \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create th prio)"
+| step_Exit: "\<lbrakk>th \<in> runing s; holding s th = {}\<rbrakk> \<Longrightarrow> step s (Exit th)"
+| step_P: "\<lbrakk>th \<in> runing s; (Cs cs, Th th) \<notin> (RAG2 s)^+\<rbrakk> \<Longrightarrow> step s (P th cs)"
+| step_V: "\<lbrakk>th \<in> runing s; holds2 s th cs\<rbrakk> \<Longrightarrow> step s (V th cs)"
+| step_Set: "\<lbrakk>th \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set th prio)"
+
+(* valid states *)
+inductive vt :: "state \<Rightarrow> bool"
+ where
+ vt_nil[intro]: "vt []"
+| vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
+
+lemma runing_ready:
+ shows "runing s \<subseteq> readys s"
+ unfolding runing_def readys_def
+ by auto
+
+lemma readys_threads:
+ shows "readys s \<subseteq> threads s"
+ unfolding readys_def
+ by auto
+
+lemma wq_threads:
+ assumes vt: "vt s"
+ and h: "th \<in> set (wq s cs)"
+ shows "th \<in> threads s"
+using assms
+apply(induct)
+apply(simp add: wq_def)
+apply(erule step.cases)
+apply(auto simp add: wq_def Let_def holding_def holds2_def holds_def waits2_def runing_def readys_def)
+apply(simp add: waits_def)
+apply(auto simp add: waits_def split: if_splits)[1]
+apply(auto split: if_splits)
+apply(simp only: waits_def)
+by (metis insert_iff set_simps(2))
+
+
+
+lemma Domain_RAG_threads:
+ assumes vt: "vt s"
+ and in_dom: "(Th th) \<in> Domain (RAG2 s)"
+ shows "th \<in> threads s"
+proof -
+ from in_dom obtain n where "(Th th, n) \<in> RAG2 s" by auto
+ then obtain cs where "n = Cs cs" "(Th th, Cs cs) \<in> RAG2 s"
+ unfolding RAG2_def RAG_def by auto
+ then have "th \<in> set (wq s cs)"
+ unfolding wq_def RAG_def RAG2_def waits_def by auto
+ with wq_threads [OF vt] show ?thesis .
+qed
+
+lemma dependants_threads:
+ assumes vt: "vt s"
+ shows "dependants2 s th \<subseteq> threads s"
+proof
+ fix th1
+ assume "th1 \<in> dependants2 s th"
+ then have h: "(Th th1, Th th) \<in> (RAG2 s)\<^sup>+"
+ unfolding dependants2_def dependants_def RAG2_def by simp
+ then have "Th th1 \<in> Domain ((RAG2 s)\<^sup>+)" unfolding Domain_def by auto
+ then have "Th th1 \<in> Domain (RAG2 s)" using trancl_domain by simp
+ then show "th1 \<in> threads s" using vt by (rule_tac Domain_RAG_threads)
+qed
+
+lemma finite_threads:
+ assumes vt: "vt s"
+ shows "finite (threads s)"
+using vt by (induct) (auto elim: step.cases)
+
+
+section {* Distinctness of @{const wq} *}
+
+lemma wq_distinct_step:
+ assumes "step s e" "distinct (wq s cs)"
+ shows "distinct (wq (e # s) cs)"
+using assms
+unfolding wq_def
+apply(erule_tac step.cases)
+apply(auto simp add: RAG2_def RAG_def Let_def)[1]
+apply(auto simp add: wq_def Let_def RAG2_def RAG_def holds_def runing_def waits2_def waits_def readys_def)
+apply(auto split: list.split)
+apply(rule someI2)
+apply(auto)
+done
+
+lemma wq_distinct:
+ assumes "vt s"
+ shows "distinct (wq s cs)"
+using assms
+apply(induct)
+apply(simp add: wq_def)
+apply(simp add: wq_distinct_step)
+done
+
+
+section {* Single_Valuedness of @{const waits2}, @{const holds2}, @{const RAG2} *}
+
+lemma waits2_unique:
+ assumes "vt s"
+ and "waits2 s th cs1"
+ and "waits2 s th cs2"
+ shows "cs1 = cs2"
+using assms
+apply(induct)
+apply(simp add: waits2_def waits_def)
+apply(erule step.cases)
+apply(auto simp add: Let_def waits2_def waits_def holds_def RAG2_def RAG_def
+ readys_def runing_def split: if_splits)
+apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
+apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
+apply (metis distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
+by (metis (full_types, hide_lams) distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
+
+lemma single_valued_waits2:
+ assumes "vt s"
+ shows "single_valuedP (waits2 s)"
+using assms
+unfolding single_valued_def
+by (simp add: Product_Type.Collect_case_prodD waits2_unique)
+
+lemma single_valued_holds2:
+ assumes "vt s"
+ shows "single_valuedP (\<lambda>cs th. holds2 s th cs)"
+unfolding single_valued_def holds2_def holds_def by simp
+
+lemma single_valued_RAG2:
+ assumes "vt s"
+ shows "single_valued (RAG2 s)"
+using single_valued_waits2[OF assms] single_valued_holds2[OF assms]
+unfolding RAG2_def RAG_def
+apply(rule_tac single_valued_union)
+unfolding holds2_def[symmetric] waits2_def[symmetric]
+apply(rule single_valued_Collect)
+apply(simp)
+apply(simp add: inj_on_def)
+apply(rule single_valued_Collect)
+apply(simp)
+apply(simp add: inj_on_def)
+apply(auto)
+done
+
+
+section {* Properties of @{const RAG2} under events *}
+
+lemma RAG_Set [simp]:
+ shows "RAG2 (Set th prio # s) = RAG2 s"
+unfolding RAG2_def
+by (simp add: Let_def)
+
+lemma RAG_Create [simp]:
+ "RAG2 (Create th prio # s) = RAG2 s"
+unfolding RAG2_def
+by (simp add: Let_def)
+
+lemma RAG_Exit [simp]:
+ shows "RAG2 (Exit th # s) = RAG2 s"
+unfolding RAG2_def
+by (simp add: Let_def)
+
+lemma RAG_P1:
+ assumes "wq s cs = []"
+ shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Cs cs, Th th)}"
+using assms
+unfolding RAG2_def RAG_def wq_def Let_def waits_def holds_def
+by (auto simp add: Let_def)
+
+lemma RAG_P2:
+ assumes "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" "wq s cs \<noteq> []"
+ shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Th th, Cs cs)}"
+using assms
+unfolding RAG2_def RAG_def wq_def Let_def waits_def holds_def
+by (auto simp add: Let_def)
+
+
+lemma RAG_V1:
+assumes vt: "wq s cs = [th]"
+shows "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}"
+using assms
+unfolding RAG2_def RAG_def waits_def holds_def wq_def
+by (auto simp add: Let_def)
+
+lemma RAG_V2:
+assumes vt:"vt s" "wq s cs = th # wts \<and> wts \<noteq> []"
+shows "RAG2 (V th cs # s) \<subseteq>
+ RAG2 s - {(Cs cs, Th th), (Th (next_to_run wts), Cs cs)} \<union> {(Cs cs, Th (next_to_run wts))}"
+unfolding RAG2_def RAG_def waits_def holds_def
+using assms wq_distinct[OF vt(1), of"cs"]
+by (auto simp add: Let_def wq_def)
+
+
+
+section {* Acyclicity of @{const RAG2} *}
+
+lemma acyclic_RAG_step:
+ assumes vt: "vt s"
+ and stp: "step s e"
+ and ac: "acyclic (RAG2 s)"
+ shows "acyclic (RAG2 (e # s))"
+using stp vt ac
+proof (induct)
+ case (step_P th s cs)
+ have ac: "acyclic (RAG2 s)" by fact
+ have ds: "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" by fact
+ { assume wq_empty: "wq s cs = []" -- "case waiting queue is empty"
+ then have "(Th th, Cs cs) \<notin> (RAG2 s)\<^sup>+"
+ proof (rule_tac notI)
+ assume "(Th th, Cs cs) \<in> (RAG2 s)\<^sup>+"
+ then obtain x where "(x, Cs cs) \<in> RAG2 s" using tranclD2 by metis
+ with wq_empty show False by (auto simp: RAG2_def RAG_def wq_def waits_def)
+ qed
+ with ac have "acyclic (RAG2 s \<union> {(Cs cs, Th th)})" by simp
+ then have "acyclic (RAG2 (P th cs # s))" using RAG_P1[OF wq_empty]
+ by (rule acyclic_subset)
+ }
+ moreover
+ { assume wq_not_empty: "wq s cs \<noteq> []" -- "case waiting queue is not empty"
+ from ac ds
+ have "acyclic (RAG2 s \<union> {(Th th, Cs cs)})" by simp
+ then have "acyclic (RAG2 (P th cs # s))" using RAG_P2[OF ds wq_not_empty]
+ by (rule acyclic_subset)
+ }
+ ultimately show "acyclic (RAG2 (P th cs # s))" by metis
+next
+ case (step_V th s cs) -- "case for release of a lock"
+ have vt: "vt s" by fact
+ have ac: "acyclic (RAG2 s)" by fact
+ have hd: "holds2 s th cs" by fact
+ from vt have wq_distinct:"distinct (wq s cs)" by (rule wq_distinct)
+ from hd have "th \<in> set (wq s cs)" "th = hd (wq s cs)" unfolding holds2_def holds_def wq_def by auto
+ then obtain wts where eq_wq: "wq s cs = th # wts" by (cases "wq s cs") (auto)
+ -- "case no thread present in the waiting queue to take over"
+ { assume "wts = []"
+ with eq_wq have "wq s cs = [th]" by simp
+ then have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}" by (rule RAG_V1)
+ moreover have "acyclic (RAG2 s - {(Cs cs, Th th)})" using ac by (auto intro: acyclic_subset)
+ ultimately
+ have "acyclic (RAG2 (V th cs # s))" by (auto intro: acyclic_subset)
+ }
+ moreover
+ -- "at least one thread present to take over"
+ { def nth \<equiv> "next_to_run wts"
+ assume wq_not_empty: "wts \<noteq> []"
+ have "waits2 s nth cs"
+ using eq_wq wq_not_empty wq_distinct
+ unfolding nth_def waits2_def waits_def wq_def[symmetric] by auto
+ then have cs_in_RAG: "(Th nth, Cs cs) \<in> RAG2 s"
+ unfolding RAG2_def RAG_def waits2_def by auto
+ have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)}"
+ unfolding nth_def using vt wq_not_empty eq_wq by (rule_tac RAG_V2) (auto)
+ moreover
+ have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)})"
+ proof -
+ have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})" using ac by (auto intro: acyclic_subset)
+ moreover
+ have "(Th nth, Cs cs) \<notin> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+"
+ proof (rule notI)
+ assume "(Th nth, Cs cs) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+"
+ then obtain z where a: "(Th nth, z) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})"
+ by (metis converse_tranclE)
+ then have "(Th nth, z) \<in> RAG2 s" by simp
+ then have "z = Cs cs" using cs_in_RAG single_valued_RAG2[OF vt]
+ by (simp add: single_valued_def)
+ then show "False" using a by simp
+ qed
+ ultimately
+ show "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth) })" by simp
+ qed
+ ultimately have "acyclic (RAG2 (V th cs # s))"
+ by (rule_tac acyclic_subset)
+ }
+ ultimately show "acyclic (RAG2 (V th cs # s))" by metis
+qed (simp_all)
+
+
+lemma finite_RAG:
+ assumes "vt s"
+ shows "finite (RAG2 s)"
+using assms
+apply(induct)
+apply(simp add: RAG2_def RAG_def waits_def holds_def)
+apply(erule step.cases)
+apply(auto)
+apply(case_tac "wq sa cs = []")
+apply(rule finite_subset)
+apply(rule RAG_P1)
+apply(simp)
+apply(simp)
+apply(rule finite_subset)
+apply(rule RAG_P2)
+apply(simp)
+apply(simp)
+apply(simp)
+apply(subgoal_tac "\<exists>wts. wq sa cs = th # wts")
+apply(erule exE)
+apply(case_tac "wts = []")
+apply(rule finite_subset)
+apply(rule RAG_V1)
+apply(simp)
+apply(simp)
+apply(rule finite_subset)
+apply(rule RAG_V2)
+apply(simp)
+apply(simp)
+apply(simp)
+apply(subgoal_tac "th \<in> set (wq sa cs) \<and> th = hd (wq sa cs)")
+apply(case_tac "wq sa cs")
+apply(auto)[2]
+apply(auto simp add: holds2_def holds_def wq_def)
+done
+
+
+
+lemma dchain_unique:
+ assumes vt: "vt s"
+ and th1_d: "(n, Th th1) \<in> (RAG2 s)^+"
+ and th1_r: "th1 \<in> readys s"
+ and th2_d: "(n, Th th2) \<in> (RAG2 s)^+"
+ and th2_r: "th2 \<in> readys s"
+ shows "th1 = th2"
+proof(rule ccontr)
+ assume neq: "th1 \<noteq> th2"
+ with single_valued_confluent2 [OF single_valued_RAG2 [OF vt]] th1_d th2_d
+ have "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG2 s)\<^sup>+" by auto
+ moreover
+ { assume "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+"
+ then obtain n where dd: "(Th th1, n) \<in> RAG2 s" by (metis converse_tranclE)
+ then obtain cs where eq_n: "n = Cs cs"
+ unfolding RAG2_def RAG_def by (case_tac n) (auto)
+ from dd eq_n have "th1 \<notin> readys s"
+ unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
+ with th1_r have "False" by auto
+ }
+ moreover
+ { assume "(Th th2, Th th1) \<in> (RAG2 s)\<^sup>+"
+ then obtain n where dd: "(Th th2, n) \<in> RAG2 s" by (metis converse_tranclE)
+ then obtain cs where eq_n: "n = Cs cs"
+ unfolding RAG2_def RAG_def by (case_tac n) (auto)
+ from dd eq_n have "th2 \<notin> readys s"
+ unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
+ with th2_r have "False" by auto
+ }
+ ultimately show "False" by metis
+qed
+
+lemma cpreced2_cpreced: "cpreced2 s th = cpreced (wq s) s th"
+unfolding cpreced2_def wq_def
+apply(induct s rule: schs.induct)
+apply(simp add: Let_def cpreced_def dependants_def RAG_def waits_def holds_def preced_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+apply(subst (2) schs.simps)
+apply(simp add: Let_def)
+done
+
+lemma cpreced_Exit:
+ shows "cpreced2 (Exit th # s) th' = cpreced2 s th'"
+by (simp add: cpreced2_cpreced cpreced_def preced_def wq_def Let_def)
+
+lemma readys_Exit:
+ shows "readys (Exit th # s) = readys s - {th}"
+by (auto simp add: readys_def waits2_def Let_def)
+
+lemma readys_Create:
+ shows "readys (Create th prio # s) \<subseteq> {th} \<union> readys s"
+apply (auto simp add: readys_def waits2_def Let_def waits_def)
+done
+
+lemma readys_Set:
+ shows "readys (Set th prio # s) = readys s"
+by (auto simp add: readys_def waits2_def Let_def)
+
+
+lemma readys_P:
+ shows "readys (P th cs # s) \<subseteq> readys s"
+apply(auto simp add: readys_def waits2_def Let_def)
+apply(simp add: waits_def)
+apply(case_tac "csa = cs")
+apply(simp)
+apply(drule_tac x="cs" in spec)
+apply(simp)
+apply (metis hd_append2 in_set_insert insert_Nil list.sel(1))
+apply(drule_tac x="csa" in spec)
+apply(simp)
+done
+
+lemma readys_V:
+ shows "readys (V th cs # s) \<subseteq> readys s \<union> set (wq s cs)"
+apply(auto simp add: readys_def waits2_def waits_def Let_def wq_def)
+done
+
+
+fun the_th :: "node \<Rightarrow> thread"
+ where "the_th (Th th) = th"
+
+lemma image_Collect2:
+ "f ` A = {f x | x. x \<in> A}"
+apply(auto)
+done
+
+lemma Collect_disj_eq2:
+ "{f x | x. x = y \<or> x \<in> A} = {f y} \<union> {f x | x. x \<in> A}"
+by (auto)
+
+lemma last_set_lt:
+ "th \<in> threads s \<Longrightarrow> last_set th s < length s"
+ apply(induct rule: threads.induct)
+ apply(auto)
+ done
+
+lemma last_set_eq_iff:
+ assumes "th1 \<in> threads s" "th2 \<in> threads s"
+ shows "last_set th1 s = last_set th2 s \<longleftrightarrow> th1 = th2"
+ using assms
+ apply(induct s rule: threads.induct)
+ apply(auto split:if_splits dest:last_set_lt)
+ done
+
+lemma preced_eq_iff:
+ assumes th_in1: "th1 \<in> threads s"
+ and th_in2: "th2 \<in> threads s"
+ shows "preced th1 s = preced th2 s \<longleftrightarrow> th1 = th2"
+using assms
+by (auto simp add: preced_def last_set_eq_iff)
+
+lemma dm_RAG_threads:
+ assumes vt: "vt s"
+ and in_dom: "(Th th) \<in> Domain (RAG2 s)"
+ shows "th \<in> threads s"
+proof -
+ from in_dom obtain n where a: "(Th th, n) \<in> RAG2 s" by auto
+ then obtain cs where "n = Cs cs"
+ unfolding RAG2_def RAG_def
+ by auto
+ then have "(Th th, Cs cs) \<in> RAG2 s" using a by simp
+ hence "th \<in> set (wq s cs)"
+ unfolding RAG2_def wq_def RAG_def waits_def
+ by (auto)
+ then show ?thesis
+ apply(rule_tac wq_threads)
+ apply(rule assms)
+ apply(simp)
+ done
+qed
+
+lemma cpreced_eq_iff:
+ assumes "th1 \<in> readys s" "th2 \<in> readys s" "vt s"
+ shows "cpreced2 s th1 = cpreced2 s th2 \<longleftrightarrow> th1 = th2"
+proof
+ def S1\<equiv>"({th1} \<union> dependants (wq s) th1)"
+ def S2\<equiv>"({th2} \<union> dependants (wq s) th2)"
+ have fin: "finite ((the_th o fst) ` ((RAG (wq s))\<^sup>+))"
+ apply(rule)
+ apply(simp add: finite_trancl)
+ apply(simp add: wq_def)
+ apply(rule finite_RAG[simplified RAG2_def])
+ apply(rule assms)
+ done
+
+ from fin have h: "finite (preceds s S1)" "finite (preceds s S2)"
+ apply(simp_all add: S2_def S1_def Collect_disj_eq2 image_Collect[symmetric])
+ apply(rule)
+ apply(simp add: dependants_def)
+ apply(rule rev_finite_subset)
+ apply(assumption)
+ apply(auto simp add: image_def)[1]
+ apply(metis fst_conv the_th.simps)
+ apply(rule)
+ apply(simp add: dependants_def)
+ apply(rule rev_finite_subset)
+ apply(assumption)
+ apply(auto simp add: image_def)[1]
+ apply(metis fst_conv the_th.simps)
+ done
+ moreover have "S1 \<noteq> {}" "S2 \<noteq> {}" by (simp_all add: S1_def S2_def)
+ then have "(preceds s S1) \<noteq> {}" "(preceds s S2) \<noteq> {}" by simp_all
+ ultimately have m: "Max (preceds s S1) \<in> (preceds s S1)" "Max (preceds s S2) \<in> (preceds s S2)"
+ apply(rule_tac [!] Max_in)
+ apply(simp_all)
+ done
+
+ assume q: "cpreced2 s th1 = cpreced2 s th2"
+ then have eq_max: "Max (preceds s S1) = Max (preceds s S2)"
+ unfolding cpreced2_cpreced cpreced_def
+ apply(simp only: S1_def S2_def)
+ apply(simp add: Collect_disj_eq2)
+ done
+
+ obtain th0 where th0_in: "th0 \<in> S1" "th0 \<in> S2" and
+ eq_f_th1: "preced th0 s = Max (preceds s S1)"
+ "preced th0 s = Max (preceds s S2)"
+ using m
+ apply(clarify)
+ apply(simp add: eq_max)
+ apply(subst (asm) (2) preced_eq_iff)
+ apply(insert assms(2))[1]
+ apply(simp add: S2_def)
+ apply(auto)[1]
+ apply (metis contra_subsetD readys_threads)
+ apply(simp add: dependants_def)
+ apply(subgoal_tac "Th tha \<in> Domain ((RAG2 s)^+)")
+ apply(simp add: trancl_domain)
+ apply (metis Domain_RAG_threads assms(3))
+ apply(simp only: RAG2_def wq_def)
+ apply (metis Domain_iff)
+ apply(insert assms(1))[1]
+ apply(simp add: S1_def)
+ apply(auto)[1]
+ apply (metis contra_subsetD readys_threads)
+ apply(simp add: dependants_def)
+ apply(subgoal_tac "Th th \<in> Domain ((RAG2 s)^+)")
+ apply(simp add: trancl_domain)
+ apply (metis Domain_RAG_threads assms(3))
+ apply(simp only: RAG2_def wq_def)
+ apply (metis Domain_iff)
+ apply(simp)
+ done
+ then show "th1 = th2"
+ apply -
+ apply(insert th0_in assms(1, 2))[1]
+ apply(simp add: S1_def S2_def)
+ apply(auto)
+ --"first case"
+ prefer 2
+ apply(subgoal_tac "Th th2 \<in> Domain (RAG2 s)")
+ apply(subgoal_tac "\<exists>cs. (Th th2, Cs cs) \<in> RAG2 s")
+ apply(erule exE)
+ apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
+ apply(auto simp add: RAG2_def RAG_def)[1]
+ apply(subgoal_tac "Th th2 \<in> Domain ((RAG2 s)^+)")
+ apply (metis trancl_domain)
+ apply(subgoal_tac "(Th th2, Th th1) \<in> (RAG2 s)^+")
+ apply (metis Domain_iff)
+ apply(simp add: dependants_def RAG2_def wq_def)
+ --"second case"
+ apply(subgoal_tac "Th th1 \<in> Domain (RAG2 s)")
+ apply(subgoal_tac "\<exists>cs. (Th th1, Cs cs) \<in> RAG2 s")
+ apply(erule exE)
+ apply(insert assms(1))[1]
+ apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
+ apply(auto simp add: RAG2_def RAG_def)[1]
+ apply(subgoal_tac "Th th1 \<in> Domain ((RAG2 s)^+)")
+ apply (metis trancl_domain)
+ apply(subgoal_tac "(Th th1, Th th2) \<in> (RAG2 s)^+")
+ apply (metis Domain_iff)
+ apply(simp add: dependants_def RAG2_def wq_def)
+ --"third case"
+ apply(rule dchain_unique)
+ apply(rule assms(3))
+ apply(simp add: dependants_def RAG2_def wq_def)
+ apply(simp)
+ apply(simp add: dependants_def RAG2_def wq_def)
+ apply(simp)
+ done
+next
+ assume "th1 = th2"
+ then show "cpreced2 s th1 = cpreced2 s th2" by simp
+qed
+
+lemma at_most_one_running:
+ assumes "vt s"
+ shows "card (runing s) \<le> 1"
+proof (rule ccontr)
+ assume "\<not> card (runing s) \<le> 1"
+ then have "2 \<le> card (runing s)" by auto
+ moreover
+ have "finite (runing s)"
+ by (metis `\<not> card (runing s) \<le> 1` card_infinite le0)
+ ultimately obtain th1 th2 where a:
+ "th1 \<noteq> th2" "th1 \<in> runing s" "th2 \<in> runing s"
+ "cpreced2 s th1 = cpreced2 s th2"
+ apply(auto simp add: numerals card_le_Suc_iff runing_def)
+ apply(blast)
+ done
+ then have "th1 = th2"
+ apply(subst (asm) cpreced_eq_iff)
+ apply(auto intro: assms a)
+ apply (metis contra_subsetD runing_ready)+
+ done
+ then show "False" using a(1) by auto
+qed
+
+
+
+ (*
+ obtain th0 where th0_in: "th0 \<in> S1 \<and> th0 \<in> S2"
+ and eq_f_th0: "preced th0 s = Max ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
+ proof -
+ from fin have h1: "finite ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
+ apply(simp only: S1_def S2_def)
+ apply(rule)
+ apply(rule)
+ apply(rule)
+ apply(simp add: dependants_def)
+ apply(rule rev_finite_subset)
+ apply(assumption)
+ apply(auto simp add: image_def)
+ apply (metis fst_conv the_th.simps)
+ done
+ moreover
+ have "S1 \<inter> S2 \<noteq> {}" apply (simp add: S1_def S2_def)
+ apply(auto)
+
+ done
+ then have h2: "((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<noteq> {}" by simp
+ ultimately have "Max ((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<in> ((\<lambda>th. preced th s) ` (S1 \<union> S2))"
+ apply(rule Max_in)
+ done
+ then show ?thesis using that[intro] apply(auto)
+
+ apply(erule_tac preced_unique)
+ done
+ qed
+ *)
+
+thm waits_def waits2_def
+
+end
--- a/ExtGG.ty Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,922 +0,0 @@
-theory ExtGG
-imports PrioG CpsG
-begin
-
-text {*
- The following two auxiliary lemmas are used to reason about @{term Max}.
-*}
-lemma image_Max_eqI:
- assumes "finite B"
- and "b \<in> B"
- and "\<forall> x \<in> B. f x \<le> f b"
- shows "Max (f ` B) = f b"
- using assms
- using Max_eqI by blast
-
-lemma image_Max_subset:
- assumes "finite A"
- and "B \<subseteq> A"
- and "a \<in> B"
- and "Max (f ` A) = f a"
- shows "Max (f ` B) = f a"
-proof(rule image_Max_eqI)
- show "finite B"
- using assms(1) assms(2) finite_subset by auto
-next
- show "a \<in> B" using assms by simp
-next
- show "\<forall>x\<in>B. f x \<le> f a"
- by (metis Max_ge assms(1) assms(2) assms(4)
- finite_imageI image_eqI subsetCE)
-qed
-
-text {*
- The following locale @{text "highest_gen"} sets the basic context for our
- investigation: supposing thread @{text th} holds the highest @{term cp}-value
- in state @{text s}, which means the task for @{text th} is the
- most urgent. We want to show that
- @{text th} is treated correctly by PIP, which means
- @{text th} will not be blocked unreasonably by other less urgent
- threads.
-*}
-locale highest_gen =
- fixes s th prio tm
- assumes vt_s: "vt s"
- and threads_s: "th \<in> threads s"
- and highest: "preced th s = Max ((cp s)`threads s)"
- -- {* The internal structure of @{term th}'s precedence is exposed:*}
- and preced_th: "preced th s = Prc prio tm"
-
--- {* @{term s} is a valid trace, so it will inherit all results derived for
- a valid trace: *}
-sublocale highest_gen < vat_s: valid_trace "s"
- by (unfold_locales, insert vt_s, simp)
-
-context highest_gen
-begin
-
-text {*
- @{term tm} is the time when the precedence of @{term th} is set, so
- @{term tm} must be a valid moment index into @{term s}.
-*}
-lemma lt_tm: "tm < length s"
- by (insert preced_tm_lt[OF threads_s preced_th], simp)
-
-text {*
- Since @{term th} holds the highest precedence and @{text "cp"}
- is the highest precedence of all threads in the sub-tree of
- @{text "th"} and @{text th} is among these threads,
- its @{term cp} must equal to its precedence:
-*}
-lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
-proof -
- have "?L \<le> ?R"
- by (unfold highest, rule Max_ge,
- auto simp:threads_s finite_threads)
- moreover have "?R \<le> ?L"
- by (unfold vat_s.cp_rec, rule Max_ge,
- auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
- ultimately show ?thesis by auto
-qed
-
-(* ccc *)
-lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold max_cp_eq, unfold eq_cp_s_th, insert highest, simp)
-
-lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
- by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
-
-lemma highest': "cp s th = Max (cp s ` threads s)"
-proof -
- from highest_cp_preced max_cp_eq[symmetric]
- show ?thesis by simp
-qed
-
-end
-
-locale extend_highest_gen = highest_gen +
- fixes t
- assumes vt_t: "vt (t@s)"
- and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
- and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
- and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
-
-sublocale extend_highest_gen < vat_t: valid_trace "t@s"
- by (unfold_locales, insert vt_t, simp)
-
-lemma step_back_vt_app:
- assumes vt_ts: "vt (t@s)"
- shows "vt s"
-proof -
- from vt_ts show ?thesis
- proof(induct t)
- case Nil
- from Nil show ?case by auto
- next
- case (Cons e t)
- assume ih: " vt (t @ s) \<Longrightarrow> vt s"
- and vt_et: "vt ((e # t) @ s)"
- show ?case
- proof(rule ih)
- show "vt (t @ s)"
- proof(rule step_back_vt)
- from vt_et show "vt (e # t @ s)" by simp
- qed
- qed
- qed
-qed
-
-
-locale red_extend_highest_gen = extend_highest_gen +
- fixes i::nat
-
-sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
- apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
- apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
- by (unfold highest_gen_def, auto dest:step_back_vt_app)
-
-
-context extend_highest_gen
-begin
-
- lemma ind [consumes 0, case_names Nil Cons, induct type]:
- assumes
- h0: "R []"
- and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e;
- extend_highest_gen s th prio tm t;
- extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
- shows "R t"
-proof -
- from vt_t extend_highest_gen_axioms show ?thesis
- proof(induct t)
- from h0 show "R []" .
- next
- case (Cons e t')
- assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
- and vt_e: "vt ((e # t') @ s)"
- and et: "extend_highest_gen s th prio tm (e # t')"
- from vt_e and step_back_step have stp: "step (t'@s) e" by auto
- from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
- show ?case
- proof(rule h2 [OF vt_ts stp _ _ _ ])
- show "R t'"
- proof(rule ih)
- from et show ext': "extend_highest_gen s th prio tm t'"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
- next
- from vt_ts show "vt (t' @ s)" .
- qed
- next
- from et show "extend_highest_gen s th prio tm (e # t')" .
- next
- from et show ext': "extend_highest_gen s th prio tm t'"
- by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
- qed
- qed
-qed
-
-
-lemma th_kept: "th \<in> threads (t @ s) \<and>
- preced th (t@s) = preced th s" (is "?Q t")
-proof -
- show ?thesis
- proof(induct rule:ind)
- case Nil
- from threads_s
- show ?case
- by auto
- next
- case (Cons e t)
- interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
- interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
- show ?case
- proof(cases e)
- case (Create thread prio)
- show ?thesis
- proof -
- from Cons and Create have "step (t@s) (Create thread prio)" by auto
- hence "th \<noteq> thread"
- proof(cases)
- case thread_create
- with Cons show ?thesis by auto
- qed
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold Create, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:Create)
- qed
- next
- case (Exit thread)
- from h_e.exit_diff and Exit
- have neq_th: "thread \<noteq> th" by auto
- with Cons
- show ?thesis
- by (unfold Exit, auto simp:preced_def)
- next
- case (P thread cs)
- with Cons
- show ?thesis
- by (auto simp:P preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis
- by (auto simp:V preced_def)
- next
- case (Set thread prio')
- show ?thesis
- proof -
- from h_e.set_diff_low and Set
- have "th \<noteq> thread" by auto
- hence "preced th ((e # t) @ s) = preced th (t @ s)"
- by (unfold Set, auto simp:preced_def)
- moreover note Cons
- ultimately show ?thesis
- by (auto simp:Set)
- qed
- qed
- qed
-qed
-
-text {*
- According to @{thm th_kept}, thread @{text "th"} has its living status
- and precedence kept along the way of @{text "t"}. The following lemma
- shows that this preserved precedence of @{text "th"} remains as the highest
- along the way of @{text "t"}.
-
- The proof goes by induction over @{text "t"} using the specialized
- induction rule @{thm ind}, followed by case analysis of each possible
- operations of PIP. All cases follow the same pattern rendered by the
- generalized introduction rule @{thm "image_Max_eqI"}.
-
- The very essence is to show that precedences, no matter whether they are newly introduced
- or modified, are always lower than the one held by @{term "th"},
- which by @{thm th_kept} is preserved along the way.
-*}
-lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
-proof(induct rule:ind)
- case Nil
- from highest_preced_thread
- show ?case
- by (unfold the_preced_def, simp)
-next
- case (Cons e t)
- interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
- interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
- show ?case
- proof(cases e)
- case (Create thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- -- {* The following is the common pattern of each branch of the case analysis. *}
- -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
- have "Max (?f ` ?A) = ?f th"
- proof(rule image_Max_eqI)
- show "finite ?A" using h_e.finite_threads by auto
- next
- show "th \<in> ?A" using h_e.th_kept by auto
- next
- show "\<forall>x\<in>?A. ?f x \<le> ?f th"
- proof
- fix x
- assume "x \<in> ?A"
- hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
- thus "?f x \<le> ?f th"
- proof
- assume "x = thread"
- thus ?thesis
- apply (simp add:Create the_preced_def preced_def, fold preced_def)
- using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force
- next
- assume h: "x \<in> threads (t @ s)"
- from Cons(2)[unfolded Create]
- have "x \<noteq> thread" using h by (cases, auto)
- hence "?f x = the_preced (t@s) x"
- by (simp add:Create the_preced_def preced_def)
- hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
- by (simp add: h_t.finite_threads h)
- also have "... = ?f th"
- by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
- finally show ?thesis .
- qed
- qed
- qed
- -- {* The minor part is to show that the precedence of @{text "th"}
- equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
- also have "... = ?t" using h_e.th_kept the_preced_def by auto
- -- {* Then it follows trivially that the precedence preserved
- for @{term "th"} remains the maximum of all living threads along the way. *}
- finally show ?thesis .
- qed
- next
- case (Exit thread)
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = ?f th"
- proof(rule image_Max_eqI)
- show "finite ?A" using h_e.finite_threads by auto
- next
- show "th \<in> ?A" using h_e.th_kept by auto
- next
- show "\<forall>x\<in>?A. ?f x \<le> ?f th"
- proof
- fix x
- assume "x \<in> ?A"
- hence "x \<in> threads (t@s)" by (simp add: Exit)
- hence "?f x \<le> Max (?f ` threads (t@s))"
- by (simp add: h_t.finite_threads)
- also have "... \<le> ?f th"
- apply (simp add:Exit the_preced_def preced_def, fold preced_def)
- using Cons.hyps(5) h_t.th_kept the_preced_def by auto
- finally show "?f x \<le> ?f th" .
- qed
- qed
- also have "... = ?t" using h_e.th_kept the_preced_def by auto
- finally show ?thesis .
- qed
- next
- case (P thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def the_preced_def)
- next
- case (V thread cs)
- with Cons
- show ?thesis by (auto simp:preced_def the_preced_def)
- next
- case (Set thread prio')
- show ?thesis (is "Max (?f ` ?A) = ?t")
- proof -
- have "Max (?f ` ?A) = ?f th"
- proof(rule image_Max_eqI)
- show "finite ?A" using h_e.finite_threads by auto
- next
- show "th \<in> ?A" using h_e.th_kept by auto
- next
- show "\<forall>x\<in>?A. ?f x \<le> ?f th"
- proof
- fix x
- assume h: "x \<in> ?A"
- show "?f x \<le> ?f th"
- proof(cases "x = thread")
- case True
- moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
- proof -
- have "the_preced (t @ s) th = Prc prio tm"
- using h_t.th_kept preced_th by (simp add:the_preced_def)
- moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
- ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
- qed
- ultimately show ?thesis
- by (unfold Set, simp add:the_preced_def preced_def)
- next
- case False
- then have "?f x = the_preced (t@s) x"
- by (simp add:the_preced_def preced_def Set)
- also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
- using Set h h_t.finite_threads by auto
- also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
- finally show ?thesis .
- qed
- qed
- qed
- also have "... = ?t" using h_e.th_kept the_preced_def by auto
- finally show ?thesis .
- qed
- qed
-qed
-
-lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
- by (insert th_kept max_kept, auto)
-
-text {*
- The reason behind the following lemma is that:
- Since @{term "cp"} is defined as the maximum precedence
- of those threads contained in the sub-tree of node @{term "Th th"}
- in @{term "RAG (t@s)"}, and all these threads are living threads, and
- @{term "th"} is also among them, the maximum precedence of
- them all must be the one for @{text "th"}.
-*}
-lemma th_cp_max_preced:
- "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R")
-proof -
- let ?f = "the_preced (t@s)"
- have "?L = ?f th"
- proof(unfold cp_alt_def, rule image_Max_eqI)
- show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
- proof -
- have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} =
- the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
- (\<exists> th'. n = Th th')}"
- by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
- moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree)
- ultimately show ?thesis by simp
- qed
- next
- show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
- by (auto simp:subtree_def)
- next
- show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
- the_preced (t @ s) x \<le> the_preced (t @ s) th"
- proof
- fix th'
- assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
- hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
- moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
- by (meson subtree_Field)
- ultimately have "Th th' \<in> ..." by auto
- hence "th' \<in> threads (t@s)"
- proof
- assume "Th th' \<in> {Th th}"
- thus ?thesis using th_kept by auto
- next
- assume "Th th' \<in> Field (RAG (t @ s))"
- thus ?thesis using vat_t.not_in_thread_isolated by blast
- qed
- thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
- by (metis Max_ge finite_imageI finite_threads image_eqI
- max_kept th_kept the_preced_def)
- qed
- qed
- also have "... = ?R" by (simp add: max_preced the_preced_def)
- finally show ?thesis .
-qed
-
-lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
- using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
-
-lemma th_cp_preced: "cp (t@s) th = preced th s"
- by (fold max_kept, unfold th_cp_max_preced, simp)
-
-lemma preced_less:
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- shows "preced th' s < preced th s"
- using assms
-by (metis Max.coboundedI finite_imageI highest not_le order.trans
- preced_linorder rev_image_eqI threads_s vat_s.finite_threads
- vat_s.le_cp)
-
-text {*
- Counting of the number of @{term "P"} and @{term "V"} operations
- is the cornerstone of a large number of the following proofs.
- The reason is that this counting is quite easy to calculate and
- convenient to use in the reasoning.
-
- The following lemma shows that the counting controls whether
- a thread is running or not.
-*}
-
-lemma pv_blocked_pre:
- assumes th'_in: "th' \<in> threads (t@s)"
- and neq_th': "th' \<noteq> th"
- and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
- shows "th' \<notin> runing (t@s)"
-proof
- assume otherwise: "th' \<in> runing (t@s)"
- show False
- proof -
- have "th' = th"
- proof(rule preced_unique)
- show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
- proof -
- have "?L = cp (t@s) th'"
- by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
- also have "... = cp (t @ s) th" using otherwise
- by (metis (mono_tags, lifting) mem_Collect_eq
- runing_def th_cp_max vat_t.max_cp_readys_threads)
- also have "... = ?R" by (metis th_cp_preced th_kept)
- finally show ?thesis .
- qed
- qed (auto simp: th'_in th_kept)
- moreover have "th' \<noteq> th" using neq_th' .
- ultimately show ?thesis by simp
- qed
-qed
-
-lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
-
-lemma runing_precond_pre:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and eq_pv: "cntP s th' = cntV s th'"
- and neq_th': "th' \<noteq> th"
- shows "th' \<in> threads (t@s) \<and>
- cntP (t@s) th' = cntV (t@s) th'"
-proof(induct rule:ind)
- case (Cons e t)
- interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
- interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
- show ?case
- proof(cases e)
- case (P thread cs)
- show ?thesis
- proof -
- have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- proof -
- have "thread \<noteq> th'"
- proof -
- have "step (t@s) (P thread cs)" using Cons P by auto
- thus ?thesis
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" using Cons(5)
- by (metis neq_th' vat_t.pv_blocked_pre)
- ultimately show ?thesis by auto
- qed
- qed with Cons show ?thesis
- by (unfold P, simp add:cntP_def cntV_def count_def)
- qed
- moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (V thread cs)
- show ?thesis
- proof -
- have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- proof -
- have "thread \<noteq> th'"
- proof -
- have "step (t@s) (V thread cs)" using Cons V by auto
- thus ?thesis
- proof(cases)
- assume "thread \<in> runing (t@s)"
- moreover have "th' \<notin> runing (t@s)" using Cons(5)
- by (metis neq_th' vat_t.pv_blocked_pre)
- ultimately show ?thesis by auto
- qed
- qed with Cons show ?thesis
- by (unfold V, simp add:cntP_def cntV_def count_def)
- qed
- moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (Create thread prio')
- show ?thesis
- proof -
- have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
- proof -
- have "thread \<noteq> th'"
- proof -
- have "step (t@s) (Create thread prio')" using Cons Create by auto
- thus ?thesis using Cons(5) by (cases, auto)
- qed with Cons show ?thesis
- by (unfold Create, simp add:cntP_def cntV_def count_def)
- qed
- moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (Exit thread)
- show ?thesis
- proof -
- have neq_thread: "thread \<noteq> th'"
- proof -
- have "step (t@s) (Exit thread)" using Cons Exit by auto
- thus ?thesis apply (cases) using Cons(5)
- by (metis neq_th' vat_t.pv_blocked_pre)
- qed
- hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons
- by (unfold Exit, simp add:cntP_def cntV_def count_def)
- moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread
- by (unfold Exit, simp)
- ultimately show ?thesis by auto
- qed
- next
- case (Set thread prio')
- with Cons
- show ?thesis
- by (auto simp:cntP_def cntV_def count_def)
- qed
-next
- case Nil
- with assms
- show ?case by auto
-qed
-
-text {* Changing counting balance to detachedness *}
-lemmas runing_precond_pre_dtc = runing_precond_pre
- [folded vat_t.detached_eq vat_s.detached_eq]
-
-lemma runing_precond:
- fixes th'
- assumes th'_in: "th' \<in> threads s"
- and neq_th': "th' \<noteq> th"
- and is_runing: "th' \<in> runing (t@s)"
- shows "cntP s th' > cntV s th'"
- using assms
-proof -
- have "cntP s th' \<noteq> cntV s th'"
- by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)
- moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
- ultimately show ?thesis by auto
-qed
-
-lemma moment_blocked_pre:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
- th' \<in> threads ((moment (i+j) t)@s)"
-proof -
- interpret h_i: red_extend_highest_gen _ _ _ _ _ i
- by (unfold_locales)
- interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"
- by (unfold_locales)
- interpret h: extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"
- proof(unfold_locales)
- show "vt (moment i t @ s)" by (metis h_i.vt_t)
- next
- show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)
- next
- show "preced th (moment i t @ s) =
- Max (cp (moment i t @ s) ` threads (moment i t @ s))"
- by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)
- next
- show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th)
- next
- show "vt (moment j (restm i t) @ moment i t @ s)"
- using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)
- next
- fix th' prio'
- assume "Create th' prio' \<in> set (moment j (restm i t))"
- thus "prio' \<le> prio" using assms
- by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)
- next
- fix th' prio'
- assume "Set th' prio' \<in> set (moment j (restm i t))"
- thus "th' \<noteq> th \<and> prio' \<le> prio"
- by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)
- next
- fix th'
- assume "Exit th' \<in> set (moment j (restm i t))"
- thus "th' \<noteq> th"
- by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)
- qed
- show ?thesis
- by (metis add.commute append_assoc eq_pv h.runing_precond_pre
- moment_plus_split neq_th' th'_in)
-qed
-
-lemma moment_blocked_eqpv:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
- and le_ij: "i \<le> j"
- shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
- have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
- and h2: "th' \<in> threads ((moment j t)@s)" by auto
- moreover have "th' \<notin> runing ((moment j t)@s)"
- proof -
- interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
- show ?thesis
- using h.pv_blocked_pre h1 h2 neq_th' by auto
- qed
- ultimately show ?thesis by auto
-qed
-
-(* The foregoing two lemmas are preparation for this one, but
- in long run can be combined. Maybe I am wrong.
-*)
-lemma moment_blocked:
- assumes neq_th': "th' \<noteq> th"
- and th'_in: "th' \<in> threads ((moment i t)@s)"
- and dtc: "detached (moment i t @ s) th'"
- and le_ij: "i \<le> j"
- shows "detached (moment j t @ s) th' \<and>
- th' \<in> threads ((moment j t)@s) \<and>
- th' \<notin> runing ((moment j t)@s)"
-proof -
- interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
- interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
- have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"
- by (metis dtc h_i.detached_elim)
- from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]
- show ?thesis by (metis h_j.detached_intro)
-qed
-
-lemma runing_preced_inversion:
- assumes runing': "th' \<in> runing (t@s)"
- shows "cp (t@s) th' = preced th s" (is "?L = ?R")
-proof -
- have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
- by (unfold runing_def, auto)
- also have "\<dots> = ?R"
- by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads)
- finally show ?thesis .
-qed
-
-text {*
- The situation when @{term "th"} is blocked is analyzed by the following lemmas.
-*}
-
-text {*
- The following lemmas shows the running thread @{text "th'"}, if it is different from
- @{term th}, must be live at the very beginning. By the term {\em the very beginning},
- we mean the moment where the formal investigation starts, i.e. the moment (or state)
- @{term s}.
-*}
-
-lemma runing_inversion_0:
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- shows "th' \<in> threads s"
-proof -
- -- {* The proof is by contradiction: *}
- { assume otherwise: "\<not> ?thesis"
- have "th' \<notin> runing (t @ s)"
- proof -
- -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}
- have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)
- -- {* However, @{text "th'"} does not exist at very beginning. *}
- have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise
- by (metis append.simps(1) moment_zero)
- -- {* Therefore, there must be a moment during @{text "t"}, when
- @{text "th'"} came into being. *}
- -- {* Let us suppose the moment being @{text "i"}: *}
- from p_split_gen[OF th'_in th'_notin]
- obtain i where lt_its: "i < length t"
- and le_i: "0 \<le> i"
- and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
- and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by (auto)
- interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
- interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)
- from lt_its have "Suc i \<le> length t" by auto
- -- {* Let us also suppose the event which makes this change is @{text e}: *}
- from moment_head[OF this] obtain e where
- eq_me: "moment (Suc i) t = e # moment i t" by blast
- hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t)
- hence "PIP (moment i t @ s) e" by (cases, simp)
- -- {* It can be derived that this event @{text "e"}, which
- gives birth to @{term "th'"} must be a @{term "Create"}: *}
- from create_pre[OF this, of th']
- obtain prio where eq_e: "e = Create th' prio"
- by (metis append_Cons eq_me lessI post pre)
- have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto
- have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
- proof -
- have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
- by (metis h_i.cnp_cnv_eq pre)
- thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)
- qed
- show ?thesis
- using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge
- by auto
- qed
- with `th' \<in> runing (t@s)`
- have False by simp
- } thus ?thesis by auto
-qed
-
-text {*
- The second lemma says, if the running thread @{text th'} is different from
- @{term th}, then this @{text th'} must in the possession of some resources
- at the very beginning.
-
- To ease the reasoning of resource possession of one particular thread,
- we used two auxiliary functions @{term cntV} and @{term cntP},
- which are the counters of @{term P}-operations and
- @{term V}-operations respectively.
- If the number of @{term V}-operation is less than the number of
- @{term "P"}-operations, the thread must have some unreleased resource.
-*}
-
-lemma runing_inversion_1: (* ddd *)
- assumes neq_th': "th' \<noteq> th"
- and runing': "th' \<in> runing (t@s)"
- -- {* thread @{term "th'"} is a live on in state @{term "s"} and
- it has some unreleased resource. *}
- shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
-proof -
- -- {* The proof is a simple composition of @{thm runing_inversion_0} and
- @{thm runing_precond}: *}
- -- {* By applying @{thm runing_inversion_0} to assumptions,
- it can be shown that @{term th'} is live in state @{term s}: *}
- have "th' \<in> threads s" using runing_inversion_0[OF assms(1,2)] .
- -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}
- with runing_precond [OF this neq_th' runing'] show ?thesis by simp
-qed
-
-text {*
- The following lemma is just a rephrasing of @{thm runing_inversion_1}:
-*}
-lemma runing_inversion_2:
- assumes runing': "th' \<in> runing (t@s)"
- shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
-proof -
- from runing_inversion_1[OF _ runing']
- show ?thesis by auto
-qed
-
-lemma runing_inversion_3:
- assumes runing': "th' \<in> runing (t@s)"
- and neq_th: "th' \<noteq> th"
- shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
- by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
-
-lemma runing_inversion_4:
- assumes runing': "th' \<in> runing (t@s)"
- and neq_th: "th' \<noteq> th"
- shows "th' \<in> threads s"
- and "\<not>detached s th'"
- and "cp (t@s) th' = preced th s"
- apply (metis neq_th runing' runing_inversion_2)
- apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
- by (metis neq_th runing' runing_inversion_3)
-
-
-text {*
- Suppose @{term th} is not running, it is first shown that
- there is a path in RAG leading from node @{term th} to another thread @{text "th'"}
- in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
-
- Now, since @{term readys}-set is non-empty, there must be
- one in it which holds the highest @{term cp}-value, which, by definition,
- is the @{term runing}-thread. However, we are going to show more: this running thread
- is exactly @{term "th'"}.
- *}
-lemma th_blockedE: (* ddd *)
- assumes "th \<notin> runing (t@s)"
- obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
- "th' \<in> runing (t@s)"
-proof -
- -- {* According to @{thm vat_t.th_chain_to_ready}, either
- @{term "th"} is in @{term "readys"} or there is path leading from it to
- one thread in @{term "readys"}. *}
- have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)"
- using th_kept vat_t.th_chain_to_ready by auto
- -- {* However, @{term th} can not be in @{term readys}, because otherwise, since
- @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
- moreover have "th \<notin> readys (t@s)"
- using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto
- -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in
- term @{term readys}: *}
- ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
- and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
- -- {* We are going to show that this @{term th'} is running. *}
- have "th' \<in> runing (t@s)"
- proof -
- -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
- have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
- proof -
- have "?L = Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
- by (unfold cp_alt_def1, simp)
- also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
- proof(rule image_Max_subset)
- show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
- next
- show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
- by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread)
- next
- show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
- by (unfold tRAG_subtree_eq, auto simp:subtree_def)
- next
- show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
- (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
- proof -
- have "?L = the_preced (t @ s) ` threads (t @ s)"
- by (unfold image_comp, rule image_cong, auto)
- thus ?thesis using max_preced the_preced_def by auto
- qed
- qed
- also have "... = ?R"
- using th_cp_max th_cp_preced th_kept
- the_preced_def vat_t.max_cp_readys_threads by auto
- finally show ?thesis .
- qed
- -- {* Now, since @{term th'} holds the highest @{term cp}
- and we have already show it is in @{term readys},
- it is @{term runing} by definition. *}
- with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def)
- qed
- -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
- moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
- using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
- ultimately show ?thesis using that by metis
-qed
-
-text {*
- Now it is easy to see there is always a thread to run by case analysis
- on whether thread @{term th} is running: if the answer is Yes, the
- the running thread is obviously @{term th} itself; otherwise, the running
- thread is the @{text th'} given by lemma @{thm th_blockedE}.
-*}
-lemma live: "runing (t@s) \<noteq> {}"
-proof(cases "th \<in> runing (t@s)")
- case True thus ?thesis by auto
-next
- case False
- thus ?thesis using th_blockedE by auto
-qed
-
-end
-end
-
-
-
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Paper/ExtGG.ty Thu Sep 21 14:15:55 2017 +0100
@@ -0,0 +1,922 @@
+theory ExtGG
+imports PrioG CpsG
+begin
+
+text {*
+ The following two auxiliary lemmas are used to reason about @{term Max}.
+*}
+lemma image_Max_eqI:
+ assumes "finite B"
+ and "b \<in> B"
+ and "\<forall> x \<in> B. f x \<le> f b"
+ shows "Max (f ` B) = f b"
+ using assms
+ using Max_eqI by blast
+
+lemma image_Max_subset:
+ assumes "finite A"
+ and "B \<subseteq> A"
+ and "a \<in> B"
+ and "Max (f ` A) = f a"
+ shows "Max (f ` B) = f a"
+proof(rule image_Max_eqI)
+ show "finite B"
+ using assms(1) assms(2) finite_subset by auto
+next
+ show "a \<in> B" using assms by simp
+next
+ show "\<forall>x\<in>B. f x \<le> f a"
+ by (metis Max_ge assms(1) assms(2) assms(4)
+ finite_imageI image_eqI subsetCE)
+qed
+
+text {*
+ The following locale @{text "highest_gen"} sets the basic context for our
+ investigation: supposing thread @{text th} holds the highest @{term cp}-value
+ in state @{text s}, which means the task for @{text th} is the
+ most urgent. We want to show that
+ @{text th} is treated correctly by PIP, which means
+ @{text th} will not be blocked unreasonably by other less urgent
+ threads.
+*}
+locale highest_gen =
+ fixes s th prio tm
+ assumes vt_s: "vt s"
+ and threads_s: "th \<in> threads s"
+ and highest: "preced th s = Max ((cp s)`threads s)"
+ -- {* The internal structure of @{term th}'s precedence is exposed:*}
+ and preced_th: "preced th s = Prc prio tm"
+
+-- {* @{term s} is a valid trace, so it will inherit all results derived for
+ a valid trace: *}
+sublocale highest_gen < vat_s: valid_trace "s"
+ by (unfold_locales, insert vt_s, simp)
+
+context highest_gen
+begin
+
+text {*
+ @{term tm} is the time when the precedence of @{term th} is set, so
+ @{term tm} must be a valid moment index into @{term s}.
+*}
+lemma lt_tm: "tm < length s"
+ by (insert preced_tm_lt[OF threads_s preced_th], simp)
+
+text {*
+ Since @{term th} holds the highest precedence and @{text "cp"}
+ is the highest precedence of all threads in the sub-tree of
+ @{text "th"} and @{text th} is among these threads,
+ its @{term cp} must equal to its precedence:
+*}
+lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R")
+proof -
+ have "?L \<le> ?R"
+ by (unfold highest, rule Max_ge,
+ auto simp:threads_s finite_threads)
+ moreover have "?R \<le> ?L"
+ by (unfold vat_s.cp_rec, rule Max_ge,
+ auto simp:the_preced_def vat_s.fsbttRAGs.finite_children)
+ ultimately show ?thesis by auto
+qed
+
+(* ccc *)
+lemma highest_cp_preced: "cp s th = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold max_cp_eq, unfold eq_cp_s_th, insert highest, simp)
+
+lemma highest_preced_thread: "preced th s = Max ((\<lambda> th'. preced th' s) ` threads s)"
+ by (fold eq_cp_s_th, unfold highest_cp_preced, simp)
+
+lemma highest': "cp s th = Max (cp s ` threads s)"
+proof -
+ from highest_cp_preced max_cp_eq[symmetric]
+ show ?thesis by simp
+qed
+
+end
+
+locale extend_highest_gen = highest_gen +
+ fixes t
+ assumes vt_t: "vt (t@s)"
+ and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio"
+ and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio"
+ and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th"
+
+sublocale extend_highest_gen < vat_t: valid_trace "t@s"
+ by (unfold_locales, insert vt_t, simp)
+
+lemma step_back_vt_app:
+ assumes vt_ts: "vt (t@s)"
+ shows "vt s"
+proof -
+ from vt_ts show ?thesis
+ proof(induct t)
+ case Nil
+ from Nil show ?case by auto
+ next
+ case (Cons e t)
+ assume ih: " vt (t @ s) \<Longrightarrow> vt s"
+ and vt_et: "vt ((e # t) @ s)"
+ show ?case
+ proof(rule ih)
+ show "vt (t @ s)"
+ proof(rule step_back_vt)
+ from vt_et show "vt (e # t @ s)" by simp
+ qed
+ qed
+ qed
+qed
+
+
+locale red_extend_highest_gen = extend_highest_gen +
+ fixes i::nat
+
+sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)"
+ apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric])
+ apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp)
+ by (unfold highest_gen_def, auto dest:step_back_vt_app)
+
+
+context extend_highest_gen
+begin
+
+ lemma ind [consumes 0, case_names Nil Cons, induct type]:
+ assumes
+ h0: "R []"
+ and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e;
+ extend_highest_gen s th prio tm t;
+ extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)"
+ shows "R t"
+proof -
+ from vt_t extend_highest_gen_axioms show ?thesis
+ proof(induct t)
+ from h0 show "R []" .
+ next
+ case (Cons e t')
+ assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'"
+ and vt_e: "vt ((e # t') @ s)"
+ and et: "extend_highest_gen s th prio tm (e # t')"
+ from vt_e and step_back_step have stp: "step (t'@s) e" by auto
+ from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto
+ show ?case
+ proof(rule h2 [OF vt_ts stp _ _ _ ])
+ show "R t'"
+ proof(rule ih)
+ from et show ext': "extend_highest_gen s th prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ next
+ from vt_ts show "vt (t' @ s)" .
+ qed
+ next
+ from et show "extend_highest_gen s th prio tm (e # t')" .
+ next
+ from et show ext': "extend_highest_gen s th prio tm t'"
+ by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt)
+ qed
+ qed
+qed
+
+
+lemma th_kept: "th \<in> threads (t @ s) \<and>
+ preced th (t@s) = preced th s" (is "?Q t")
+proof -
+ show ?thesis
+ proof(induct rule:ind)
+ case Nil
+ from threads_s
+ show ?case
+ by auto
+ next
+ case (Cons e t)
+ interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+ interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+ show ?case
+ proof(cases e)
+ case (Create thread prio)
+ show ?thesis
+ proof -
+ from Cons and Create have "step (t@s) (Create thread prio)" by auto
+ hence "th \<noteq> thread"
+ proof(cases)
+ case thread_create
+ with Cons show ?thesis by auto
+ qed
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold Create, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:Create)
+ qed
+ next
+ case (Exit thread)
+ from h_e.exit_diff and Exit
+ have neq_th: "thread \<noteq> th" by auto
+ with Cons
+ show ?thesis
+ by (unfold Exit, auto simp:preced_def)
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis
+ by (auto simp:P preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis
+ by (auto simp:V preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis
+ proof -
+ from h_e.set_diff_low and Set
+ have "th \<noteq> thread" by auto
+ hence "preced th ((e # t) @ s) = preced th (t @ s)"
+ by (unfold Set, auto simp:preced_def)
+ moreover note Cons
+ ultimately show ?thesis
+ by (auto simp:Set)
+ qed
+ qed
+ qed
+qed
+
+text {*
+ According to @{thm th_kept}, thread @{text "th"} has its living status
+ and precedence kept along the way of @{text "t"}. The following lemma
+ shows that this preserved precedence of @{text "th"} remains as the highest
+ along the way of @{text "t"}.
+
+ The proof goes by induction over @{text "t"} using the specialized
+ induction rule @{thm ind}, followed by case analysis of each possible
+ operations of PIP. All cases follow the same pattern rendered by the
+ generalized introduction rule @{thm "image_Max_eqI"}.
+
+ The very essence is to show that precedences, no matter whether they are newly introduced
+ or modified, are always lower than the one held by @{term "th"},
+ which by @{thm th_kept} is preserved along the way.
+*}
+lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s"
+proof(induct rule:ind)
+ case Nil
+ from highest_preced_thread
+ show ?case
+ by (unfold the_preced_def, simp)
+next
+ case (Cons e t)
+ interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto
+ interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto
+ show ?case
+ proof(cases e)
+ case (Create thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ -- {* The following is the common pattern of each branch of the case analysis. *}
+ -- {* The major part is to show that @{text "th"} holds the highest precedence: *}
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
+ next
+ show "th \<in> ?A" using h_e.th_kept by auto
+ next
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume "x \<in> ?A"
+ hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create)
+ thus "?f x \<le> ?f th"
+ proof
+ assume "x = thread"
+ thus ?thesis
+ apply (simp add:Create the_preced_def preced_def, fold preced_def)
+ using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 preced_th by force
+ next
+ assume h: "x \<in> threads (t @ s)"
+ from Cons(2)[unfolded Create]
+ have "x \<noteq> thread" using h by (cases, auto)
+ hence "?f x = the_preced (t@s) x"
+ by (simp add:Create the_preced_def preced_def)
+ hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))"
+ by (simp add: h_t.finite_threads h)
+ also have "... = ?f th"
+ by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
+ finally show ?thesis .
+ qed
+ qed
+ qed
+ -- {* The minor part is to show that the precedence of @{text "th"}
+ equals to preserved one, given by the foregoing lemma @{thm th_kept} *}
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ -- {* Then it follows trivially that the precedence preserved
+ for @{term "th"} remains the maximum of all living threads along the way. *}
+ finally show ?thesis .
+ qed
+ next
+ case (Exit thread)
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
+ next
+ show "th \<in> ?A" using h_e.th_kept by auto
+ next
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume "x \<in> ?A"
+ hence "x \<in> threads (t@s)" by (simp add: Exit)
+ hence "?f x \<le> Max (?f ` threads (t@s))"
+ by (simp add: h_t.finite_threads)
+ also have "... \<le> ?f th"
+ apply (simp add:Exit the_preced_def preced_def, fold preced_def)
+ using Cons.hyps(5) h_t.th_kept the_preced_def by auto
+ finally show "?f x \<le> ?f th" .
+ qed
+ qed
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ finally show ?thesis .
+ qed
+ next
+ case (P thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def the_preced_def)
+ next
+ case (V thread cs)
+ with Cons
+ show ?thesis by (auto simp:preced_def the_preced_def)
+ next
+ case (Set thread prio')
+ show ?thesis (is "Max (?f ` ?A) = ?t")
+ proof -
+ have "Max (?f ` ?A) = ?f th"
+ proof(rule image_Max_eqI)
+ show "finite ?A" using h_e.finite_threads by auto
+ next
+ show "th \<in> ?A" using h_e.th_kept by auto
+ next
+ show "\<forall>x\<in>?A. ?f x \<le> ?f th"
+ proof
+ fix x
+ assume h: "x \<in> ?A"
+ show "?f x \<le> ?f th"
+ proof(cases "x = thread")
+ case True
+ moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th"
+ proof -
+ have "the_preced (t @ s) th = Prc prio tm"
+ using h_t.th_kept preced_th by (simp add:the_preced_def)
+ moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto
+ ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def)
+ qed
+ ultimately show ?thesis
+ by (unfold Set, simp add:the_preced_def preced_def)
+ next
+ case False
+ then have "?f x = the_preced (t@s) x"
+ by (simp add:the_preced_def preced_def Set)
+ also have "... \<le> Max (the_preced (t@s) ` threads (t@s))"
+ using Set h h_t.finite_threads by auto
+ also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def)
+ finally show ?thesis .
+ qed
+ qed
+ qed
+ also have "... = ?t" using h_e.th_kept the_preced_def by auto
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))"
+ by (insert th_kept max_kept, auto)
+
+text {*
+ The reason behind the following lemma is that:
+ Since @{term "cp"} is defined as the maximum precedence
+ of those threads contained in the sub-tree of node @{term "Th th"}
+ in @{term "RAG (t@s)"}, and all these threads are living threads, and
+ @{term "th"} is also among them, the maximum precedence of
+ them all must be the one for @{text "th"}.
+*}
+lemma th_cp_max_preced:
+ "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R")
+proof -
+ let ?f = "the_preced (t@s)"
+ have "?L = ?f th"
+ proof(unfold cp_alt_def, rule image_Max_eqI)
+ show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ proof -
+ have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} =
+ the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and>
+ (\<exists> th'. n = Th th')}"
+ by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps)
+ moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree)
+ ultimately show ?thesis by simp
+ qed
+ next
+ show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ by (auto simp:subtree_def)
+ next
+ show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}.
+ the_preced (t @ s) x \<le> the_preced (t @ s) th"
+ proof
+ fix th'
+ assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}"
+ hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto
+ moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}"
+ by (meson subtree_Field)
+ ultimately have "Th th' \<in> ..." by auto
+ hence "th' \<in> threads (t@s)"
+ proof
+ assume "Th th' \<in> {Th th}"
+ thus ?thesis using th_kept by auto
+ next
+ assume "Th th' \<in> Field (RAG (t @ s))"
+ thus ?thesis using vat_t.not_in_thread_isolated by blast
+ qed
+ thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th"
+ by (metis Max_ge finite_imageI finite_threads image_eqI
+ max_kept th_kept the_preced_def)
+ qed
+ qed
+ also have "... = ?R" by (simp add: max_preced the_preced_def)
+ finally show ?thesis .
+qed
+
+lemma th_cp_max: "cp (t@s) th = Max (cp (t@s) ` threads (t@s))"
+ using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger
+
+lemma th_cp_preced: "cp (t@s) th = preced th s"
+ by (fold max_kept, unfold th_cp_max_preced, simp)
+
+lemma preced_less:
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ shows "preced th' s < preced th s"
+ using assms
+by (metis Max.coboundedI finite_imageI highest not_le order.trans
+ preced_linorder rev_image_eqI threads_s vat_s.finite_threads
+ vat_s.le_cp)
+
+text {*
+ Counting of the number of @{term "P"} and @{term "V"} operations
+ is the cornerstone of a large number of the following proofs.
+ The reason is that this counting is quite easy to calculate and
+ convenient to use in the reasoning.
+
+ The following lemma shows that the counting controls whether
+ a thread is running or not.
+*}
+
+lemma pv_blocked_pre:
+ assumes th'_in: "th' \<in> threads (t@s)"
+ and neq_th': "th' \<noteq> th"
+ and eq_pv: "cntP (t@s) th' = cntV (t@s) th'"
+ shows "th' \<notin> runing (t@s)"
+proof
+ assume otherwise: "th' \<in> runing (t@s)"
+ show False
+ proof -
+ have "th' = th"
+ proof(rule preced_unique)
+ show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R")
+ proof -
+ have "?L = cp (t@s) th'"
+ by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp)
+ also have "... = cp (t @ s) th" using otherwise
+ by (metis (mono_tags, lifting) mem_Collect_eq
+ runing_def th_cp_max vat_t.max_cp_readys_threads)
+ also have "... = ?R" by (metis th_cp_preced th_kept)
+ finally show ?thesis .
+ qed
+ qed (auto simp: th'_in th_kept)
+ moreover have "th' \<noteq> th" using neq_th' .
+ ultimately show ?thesis by simp
+ qed
+qed
+
+lemmas pv_blocked = pv_blocked_pre[folded detached_eq]
+
+lemma runing_precond_pre:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and eq_pv: "cntP s th' = cntV s th'"
+ and neq_th': "th' \<noteq> th"
+ shows "th' \<in> threads (t@s) \<and>
+ cntP (t@s) th' = cntV (t@s) th'"
+proof(induct rule:ind)
+ case (Cons e t)
+ interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp
+ interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp
+ show ?case
+ proof(cases e)
+ case (P thread cs)
+ show ?thesis
+ proof -
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (P thread cs)" using Cons P by auto
+ thus ?thesis
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ ultimately show ?thesis by auto
+ qed
+ qed with Cons show ?thesis
+ by (unfold P, simp add:cntP_def cntV_def count_def)
+ qed
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold P, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (V thread cs)
+ show ?thesis
+ proof -
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (V thread cs)" using Cons V by auto
+ thus ?thesis
+ proof(cases)
+ assume "thread \<in> runing (t@s)"
+ moreover have "th' \<notin> runing (t@s)" using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ ultimately show ?thesis by auto
+ qed
+ qed with Cons show ?thesis
+ by (unfold V, simp add:cntP_def cntV_def count_def)
+ qed
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold V, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (Create thread prio')
+ show ?thesis
+ proof -
+ have "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'"
+ proof -
+ have "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (Create thread prio')" using Cons Create by auto
+ thus ?thesis using Cons(5) by (cases, auto)
+ qed with Cons show ?thesis
+ by (unfold Create, simp add:cntP_def cntV_def count_def)
+ qed
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons by (unfold Create, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (Exit thread)
+ show ?thesis
+ proof -
+ have neq_thread: "thread \<noteq> th'"
+ proof -
+ have "step (t@s) (Exit thread)" using Cons Exit by auto
+ thus ?thesis apply (cases) using Cons(5)
+ by (metis neq_th' vat_t.pv_blocked_pre)
+ qed
+ hence "cntP ((e # t) @ s) th' = cntV ((e # t) @ s) th'" using Cons
+ by (unfold Exit, simp add:cntP_def cntV_def count_def)
+ moreover have "th' \<in> threads ((e # t) @ s)" using Cons neq_thread
+ by (unfold Exit, simp)
+ ultimately show ?thesis by auto
+ qed
+ next
+ case (Set thread prio')
+ with Cons
+ show ?thesis
+ by (auto simp:cntP_def cntV_def count_def)
+ qed
+next
+ case Nil
+ with assms
+ show ?case by auto
+qed
+
+text {* Changing counting balance to detachedness *}
+lemmas runing_precond_pre_dtc = runing_precond_pre
+ [folded vat_t.detached_eq vat_s.detached_eq]
+
+lemma runing_precond:
+ fixes th'
+ assumes th'_in: "th' \<in> threads s"
+ and neq_th': "th' \<noteq> th"
+ and is_runing: "th' \<in> runing (t@s)"
+ shows "cntP s th' > cntV s th'"
+ using assms
+proof -
+ have "cntP s th' \<noteq> cntV s th'"
+ by (metis is_runing neq_th' pv_blocked_pre runing_precond_pre th'_in)
+ moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma moment_blocked_pre:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ shows "cntP ((moment (i+j) t)@s) th' = cntV ((moment (i+j) t)@s) th' \<and>
+ th' \<in> threads ((moment (i+j) t)@s)"
+proof -
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i
+ by (unfold_locales)
+ interpret h_j: red_extend_highest_gen _ _ _ _ _ "i+j"
+ by (unfold_locales)
+ interpret h: extend_highest_gen "((moment i t)@s)" th prio tm "moment j (restm i t)"
+ proof(unfold_locales)
+ show "vt (moment i t @ s)" by (metis h_i.vt_t)
+ next
+ show "th \<in> threads (moment i t @ s)" by (metis h_i.th_kept)
+ next
+ show "preced th (moment i t @ s) =
+ Max (cp (moment i t @ s) ` threads (moment i t @ s))"
+ by (metis h_i.th_cp_max h_i.th_cp_preced h_i.th_kept)
+ next
+ show "preced th (moment i t @ s) = Prc prio tm" by (metis h_i.th_kept preced_th)
+ next
+ show "vt (moment j (restm i t) @ moment i t @ s)"
+ using moment_plus_split by (metis add.commute append_assoc h_j.vt_t)
+ next
+ fix th' prio'
+ assume "Create th' prio' \<in> set (moment j (restm i t))"
+ thus "prio' \<le> prio" using assms
+ by (metis Un_iff add.commute h_j.create_low moment_plus_split set_append)
+ next
+ fix th' prio'
+ assume "Set th' prio' \<in> set (moment j (restm i t))"
+ thus "th' \<noteq> th \<and> prio' \<le> prio"
+ by (metis Un_iff add.commute h_j.set_diff_low moment_plus_split set_append)
+ next
+ fix th'
+ assume "Exit th' \<in> set (moment j (restm i t))"
+ thus "th' \<noteq> th"
+ by (metis Un_iff add.commute h_j.exit_diff moment_plus_split set_append)
+ qed
+ show ?thesis
+ by (metis add.commute append_assoc eq_pv h.runing_precond_pre
+ moment_plus_split neq_th' th'_in)
+qed
+
+lemma moment_blocked_eqpv:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and eq_pv: "cntP ((moment i t)@s) th' = cntV ((moment i t)@s) th'"
+ and le_ij: "i \<le> j"
+ shows "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ from moment_blocked_pre [OF neq_th' th'_in eq_pv, of "j-i"] and le_ij
+ have h1: "cntP ((moment j t)@s) th' = cntV ((moment j t)@s) th'"
+ and h2: "th' \<in> threads ((moment j t)@s)" by auto
+ moreover have "th' \<notin> runing ((moment j t)@s)"
+ proof -
+ interpret h: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+ show ?thesis
+ using h.pv_blocked_pre h1 h2 neq_th' by auto
+ qed
+ ultimately show ?thesis by auto
+qed
+
+(* The foregoing two lemmas are preparation for this one, but
+ in long run can be combined. Maybe I am wrong.
+*)
+lemma moment_blocked:
+ assumes neq_th': "th' \<noteq> th"
+ and th'_in: "th' \<in> threads ((moment i t)@s)"
+ and dtc: "detached (moment i t @ s) th'"
+ and le_ij: "i \<le> j"
+ shows "detached (moment j t @ s) th' \<and>
+ th' \<in> threads ((moment j t)@s) \<and>
+ th' \<notin> runing ((moment j t)@s)"
+proof -
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+ interpret h_j: red_extend_highest_gen _ _ _ _ _ j by (unfold_locales)
+ have cnt_i: "cntP (moment i t @ s) th' = cntV (moment i t @ s) th'"
+ by (metis dtc h_i.detached_elim)
+ from moment_blocked_eqpv[OF neq_th' th'_in cnt_i le_ij]
+ show ?thesis by (metis h_j.detached_intro)
+qed
+
+lemma runing_preced_inversion:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "cp (t@s) th' = preced th s" (is "?L = ?R")
+proof -
+ have "?L = Max (cp (t @ s) ` readys (t @ s))" using assms
+ by (unfold runing_def, auto)
+ also have "\<dots> = ?R"
+ by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads)
+ finally show ?thesis .
+qed
+
+text {*
+ The situation when @{term "th"} is blocked is analyzed by the following lemmas.
+*}
+
+text {*
+ The following lemmas shows the running thread @{text "th'"}, if it is different from
+ @{term th}, must be live at the very beginning. By the term {\em the very beginning},
+ we mean the moment where the formal investigation starts, i.e. the moment (or state)
+ @{term s}.
+*}
+
+lemma runing_inversion_0:
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ shows "th' \<in> threads s"
+proof -
+ -- {* The proof is by contradiction: *}
+ { assume otherwise: "\<not> ?thesis"
+ have "th' \<notin> runing (t @ s)"
+ proof -
+ -- {* Since @{term "th'"} is running at time @{term "t@s"}, so it exists that time. *}
+ have th'_in: "th' \<in> threads (t@s)" using runing' by (simp add:runing_def readys_def)
+ -- {* However, @{text "th'"} does not exist at very beginning. *}
+ have th'_notin: "th' \<notin> threads (moment 0 t @ s)" using otherwise
+ by (metis append.simps(1) moment_zero)
+ -- {* Therefore, there must be a moment during @{text "t"}, when
+ @{text "th'"} came into being. *}
+ -- {* Let us suppose the moment being @{text "i"}: *}
+ from p_split_gen[OF th'_in th'_notin]
+ obtain i where lt_its: "i < length t"
+ and le_i: "0 \<le> i"
+ and pre: " th' \<notin> threads (moment i t @ s)" (is "th' \<notin> threads ?pre")
+ and post: "(\<forall>i'>i. th' \<in> threads (moment i' t @ s))" by (auto)
+ interpret h_i: red_extend_highest_gen _ _ _ _ _ i by (unfold_locales)
+ interpret h_i': red_extend_highest_gen _ _ _ _ _ "(Suc i)" by (unfold_locales)
+ from lt_its have "Suc i \<le> length t" by auto
+ -- {* Let us also suppose the event which makes this change is @{text e}: *}
+ from moment_head[OF this] obtain e where
+ eq_me: "moment (Suc i) t = e # moment i t" by blast
+ hence "vt (e # (moment i t @ s))" by (metis append_Cons h_i'.vt_t)
+ hence "PIP (moment i t @ s) e" by (cases, simp)
+ -- {* It can be derived that this event @{text "e"}, which
+ gives birth to @{term "th'"} must be a @{term "Create"}: *}
+ from create_pre[OF this, of th']
+ obtain prio where eq_e: "e = Create th' prio"
+ by (metis append_Cons eq_me lessI post pre)
+ have h1: "th' \<in> threads (moment (Suc i) t @ s)" using post by auto
+ have h2: "cntP (moment (Suc i) t @ s) th' = cntV (moment (Suc i) t@ s) th'"
+ proof -
+ have "cntP (moment i t@s) th' = cntV (moment i t@s) th'"
+ by (metis h_i.cnp_cnv_eq pre)
+ thus ?thesis by (simp add:eq_me eq_e cntP_def cntV_def count_def)
+ qed
+ show ?thesis
+ using moment_blocked_eqpv [OF neq_th' h1 h2, of "length t"] lt_its moment_ge
+ by auto
+ qed
+ with `th' \<in> runing (t@s)`
+ have False by simp
+ } thus ?thesis by auto
+qed
+
+text {*
+ The second lemma says, if the running thread @{text th'} is different from
+ @{term th}, then this @{text th'} must in the possession of some resources
+ at the very beginning.
+
+ To ease the reasoning of resource possession of one particular thread,
+ we used two auxiliary functions @{term cntV} and @{term cntP},
+ which are the counters of @{term P}-operations and
+ @{term V}-operations respectively.
+ If the number of @{term V}-operation is less than the number of
+ @{term "P"}-operations, the thread must have some unreleased resource.
+*}
+
+lemma runing_inversion_1: (* ddd *)
+ assumes neq_th': "th' \<noteq> th"
+ and runing': "th' \<in> runing (t@s)"
+ -- {* thread @{term "th'"} is a live on in state @{term "s"} and
+ it has some unreleased resource. *}
+ shows "th' \<in> threads s \<and> cntV s th' < cntP s th'"
+proof -
+ -- {* The proof is a simple composition of @{thm runing_inversion_0} and
+ @{thm runing_precond}: *}
+ -- {* By applying @{thm runing_inversion_0} to assumptions,
+ it can be shown that @{term th'} is live in state @{term s}: *}
+ have "th' \<in> threads s" using runing_inversion_0[OF assms(1,2)] .
+ -- {* Then the thesis is derived easily by applying @{thm runing_precond}: *}
+ with runing_precond [OF this neq_th' runing'] show ?thesis by simp
+qed
+
+text {*
+ The following lemma is just a rephrasing of @{thm runing_inversion_1}:
+*}
+lemma runing_inversion_2:
+ assumes runing': "th' \<in> runing (t@s)"
+ shows "th' = th \<or> (th' \<noteq> th \<and> th' \<in> threads s \<and> cntV s th' < cntP s th')"
+proof -
+ from runing_inversion_1[OF _ runing']
+ show ?thesis by auto
+qed
+
+lemma runing_inversion_3:
+ assumes runing': "th' \<in> runing (t@s)"
+ and neq_th: "th' \<noteq> th"
+ shows "th' \<in> threads s \<and> (cntV s th' < cntP s th' \<and> cp (t@s) th' = preced th s)"
+ by (metis neq_th runing' runing_inversion_2 runing_preced_inversion)
+
+lemma runing_inversion_4:
+ assumes runing': "th' \<in> runing (t@s)"
+ and neq_th: "th' \<noteq> th"
+ shows "th' \<in> threads s"
+ and "\<not>detached s th'"
+ and "cp (t@s) th' = preced th s"
+ apply (metis neq_th runing' runing_inversion_2)
+ apply (metis neq_th pv_blocked runing' runing_inversion_2 runing_precond_pre_dtc)
+ by (metis neq_th runing' runing_inversion_3)
+
+
+text {*
+ Suppose @{term th} is not running, it is first shown that
+ there is a path in RAG leading from node @{term th} to another thread @{text "th'"}
+ in the @{term readys}-set (So @{text "th'"} is an ancestor of @{term th}}).
+
+ Now, since @{term readys}-set is non-empty, there must be
+ one in it which holds the highest @{term cp}-value, which, by definition,
+ is the @{term runing}-thread. However, we are going to show more: this running thread
+ is exactly @{term "th'"}.
+ *}
+lemma th_blockedE: (* ddd *)
+ assumes "th \<notin> runing (t@s)"
+ obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ "th' \<in> runing (t@s)"
+proof -
+ -- {* According to @{thm vat_t.th_chain_to_ready}, either
+ @{term "th"} is in @{term "readys"} or there is path leading from it to
+ one thread in @{term "readys"}. *}
+ have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)"
+ using th_kept vat_t.th_chain_to_ready by auto
+ -- {* However, @{term th} can not be in @{term readys}, because otherwise, since
+ @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *}
+ moreover have "th \<notin> readys (t@s)"
+ using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto
+ -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in
+ term @{term readys}: *}
+ ultimately obtain th' where th'_in: "th' \<in> readys (t@s)"
+ and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto
+ -- {* We are going to show that this @{term th'} is running. *}
+ have "th' \<in> runing (t@s)"
+ proof -
+ -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *}
+ have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R")
+ proof -
+ have "?L = Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))"
+ by (unfold cp_alt_def1, simp)
+ also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)"
+ proof(rule image_Max_subset)
+ show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads)
+ next
+ show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)"
+ by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread)
+ next
+ show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp
+ by (unfold tRAG_subtree_eq, auto simp:subtree_def)
+ next
+ show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) =
+ (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _")
+ proof -
+ have "?L = the_preced (t @ s) ` threads (t @ s)"
+ by (unfold image_comp, rule image_cong, auto)
+ thus ?thesis using max_preced the_preced_def by auto
+ qed
+ qed
+ also have "... = ?R"
+ using th_cp_max th_cp_preced th_kept
+ the_preced_def vat_t.max_cp_readys_threads by auto
+ finally show ?thesis .
+ qed
+ -- {* Now, since @{term th'} holds the highest @{term cp}
+ and we have already show it is in @{term readys},
+ it is @{term runing} by definition. *}
+ with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def)
+ qed
+ -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *}
+ moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)"
+ using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def)
+ ultimately show ?thesis using that by metis
+qed
+
+text {*
+ Now it is easy to see there is always a thread to run by case analysis
+ on whether thread @{term th} is running: if the answer is Yes, the
+ the running thread is obviously @{term th} itself; otherwise, the running
+ thread is the @{text th'} given by lemma @{thm th_blockedE}.
+*}
+lemma live: "runing (t@s) \<noteq> {}"
+proof(cases "th \<in> runing (t@s)")
+ case True thus ?thesis by auto
+next
+ case False
+ thus ?thesis using th_blockedE by auto
+qed
+
+end
+end
+
+
+
--- a/Test.thy Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,789 +0,0 @@
-theory Test
-imports Precedence_ord Graphs
-begin
-
-type_synonym thread = nat -- {* Type for thread identifiers. *}
-type_synonym priority = nat -- {* Type for priorities. *}
-type_synonym cs = nat -- {* Type for critical sections (or resources). *}
-
--- {* Schedulling Events *}
-
-datatype event =
- Create thread priority
-| Exit thread
-| P thread cs
-| V thread cs
-| Set thread priority
-
-type_synonym state = "event list"
-
-fun threads :: "state \<Rightarrow> thread set"
- where
- "threads [] = {}"
-| "threads (Create th prio#s) = {th} \<union> threads s"
-| "threads (Exit th # s) = (threads s) - {th}"
-| "threads (_#s) = threads s"
-
-fun priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
- where
- "priority th [] = 0"
-| "priority th (Create th' prio#s) = (if th' = th then prio else priority th s)"
-| "priority th (Set th' prio#s) = (if th' = th then prio else priority th s)"
-| "priority th (_#s) = priority th s"
-
-fun last_set :: "thread \<Rightarrow> state \<Rightarrow> nat"
- where
- "last_set th [] = 0"
-| "last_set th ((Create th' prio)#s) = (if (th = th') then length s else last_set th s)"
-| "last_set th ((Set th' prio)#s) = (if (th = th') then length s else last_set th s)"
-| "last_set th (_#s) = last_set th s"
-
-
-definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
- where "preced th s \<equiv> Prc (priority th s) (last_set th s)"
-
-abbreviation
- "preceds s ths \<equiv> {preced th s | th. th \<in> ths}"
-
-definition
- "holds wq th cs \<equiv> th \<in> set (wq cs) \<and> th = hd (wq cs)"
-
-definition
- "waits wq th cs \<equiv> th \<in> set (wq cs) \<and> th \<noteq> hd (wq cs)"
-
---{* Nodes in Resource Graph *}
-datatype node =
- Th "thread"
-| Cs "cs"
-
-definition
- "RAG wq \<equiv> {(Th th, Cs cs) | th cs. waits wq th cs} \<union> {(Cs cs, Th th) | cs th. holds wq th cs}"
-
-definition
- "dependants wq th \<equiv> {th' . (Th th', Th th) \<in> (RAG wq)^+}"
-
-record schedule_state =
- wq_fun :: "cs \<Rightarrow> thread list"
- cprec_fun :: "thread \<Rightarrow> precedence"
-
-definition cpreced :: "(cs \<Rightarrow> thread list) \<Rightarrow> state \<Rightarrow> thread \<Rightarrow> precedence"
- where
- "cpreced wq s th \<equiv> Max ({preced th s} \<union> {preced th' s | th'. th' \<in> dependants wq th})"
-
-abbreviation
- "all_unlocked \<equiv> \<lambda>_::cs. ([]::thread list)"
-
-abbreviation
- "initial_cprec \<equiv> \<lambda>_::thread. Prc 0 0"
-
-abbreviation
- "release qs \<equiv> case qs of
- [] => []
- | (_ # qs) => SOME q. distinct q \<and> set q = set qs"
-
-lemma [simp]:
- "(SOME q. distinct q \<and> q = []) = []"
-by auto
-
-lemma [simp]:
- "(x \<in> set (SOME q. distinct q \<and> set q = set p)) = (x \<in> set p)"
-apply(rule iffI)
-apply (metis (mono_tags, lifting) List.finite_set finite_distinct_list some_eq_ex)+
-done
-
-abbreviation
- "next_to_run ths \<equiv> hd (SOME q::thread list. distinct q \<and> set q = set ths)"
-
-
-fun schs :: "state \<Rightarrow> schedule_state"
- where
- "schs [] = (| wq_fun = \<lambda> cs. [], cprec_fun = (\<lambda>_. Prc 0 0) |)"
-| "schs (Create th prio # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))"
-| "schs (Exit th # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))"
-| "schs (Set th prio # s) =
- (let wq = wq_fun (schs s) in
- (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))"
-| "schs (P th cs # s) =
- (let wq = wq_fun (schs s) in
- let new_wq = wq(cs := (wq cs @ [th])) in
- (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))"
-| "schs (V th cs # s) =
- (let wq = wq_fun (schs s) in
- let new_wq = wq(cs := release (wq cs)) in
- (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))"
-
-definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
- where "wq s = wq_fun (schs s)"
-
-definition cpreced2 :: "state \<Rightarrow> thread \<Rightarrow> precedence"
- where "cpreced2 s \<equiv> cprec_fun (schs s)"
-
-abbreviation
- "cpreceds2 s ths \<equiv> {cpreced2 s th | th. th \<in> ths}"
-
-definition
- "holds2 s \<equiv> holds (wq_fun (schs s))"
-
-definition
- "waits2 s \<equiv> waits (wq_fun (schs s))"
-
-definition
- "RAG2 s \<equiv> RAG (wq_fun (schs s))"
-
-definition
- "dependants2 s \<equiv> dependants (wq_fun (schs s))"
-
-(* ready -> is a thread that is not waiting for any resource *)
-definition readys :: "state \<Rightarrow> thread set"
- where "readys s \<equiv> {th . th \<in> threads s \<and> (\<forall> cs. \<not> waits2 s th cs)}"
-
-definition runing :: "state \<Rightarrow> thread set"
- where "runing s \<equiv> {th . th \<in> readys s \<and> cpreced2 s th = Max (cpreceds2 s (readys s))}"
-
-(* all resources a thread hols in a state *)
-definition holding :: "state \<Rightarrow> thread \<Rightarrow> cs set"
- where "holding s th \<equiv> {cs . holds2 s th cs}"
-
-
-lemma exists_distinct:
- obtains ys where "distinct ys" "set ys = set xs"
-by (metis List.finite_set finite_distinct_list)
-
-lemma next_to_run_set [simp]:
- "wts \<noteq> [] \<Longrightarrow> next_to_run wts \<in> set wts"
-apply(rule exists_distinct[of wts])
-by (metis (mono_tags, lifting) hd_in_set set_empty some_eq_ex)
-
-lemma holding_RAG:
- "holding s th = {cs . (Cs cs, Th th) \<in> RAG2 s}"
-unfolding holding_def RAG2_def RAG_def
-unfolding holds2_def holds_def waits_def
-by auto
-
-inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
- where
- step_Create: "\<lbrakk>th \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create th prio)"
-| step_Exit: "\<lbrakk>th \<in> runing s; holding s th = {}\<rbrakk> \<Longrightarrow> step s (Exit th)"
-| step_P: "\<lbrakk>th \<in> runing s; (Cs cs, Th th) \<notin> (RAG2 s)^+\<rbrakk> \<Longrightarrow> step s (P th cs)"
-| step_V: "\<lbrakk>th \<in> runing s; holds2 s th cs\<rbrakk> \<Longrightarrow> step s (V th cs)"
-| step_Set: "\<lbrakk>th \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set th prio)"
-
-(* valid states *)
-inductive vt :: "state \<Rightarrow> bool"
- where
- vt_nil[intro]: "vt []"
-| vt_cons[intro]: "\<lbrakk>vt s; step s e\<rbrakk> \<Longrightarrow> vt (e#s)"
-
-lemma runing_ready:
- shows "runing s \<subseteq> readys s"
- unfolding runing_def readys_def
- by auto
-
-lemma readys_threads:
- shows "readys s \<subseteq> threads s"
- unfolding readys_def
- by auto
-
-lemma wq_threads:
- assumes vt: "vt s"
- and h: "th \<in> set (wq s cs)"
- shows "th \<in> threads s"
-using assms
-apply(induct)
-apply(simp add: wq_def)
-apply(erule step.cases)
-apply(auto simp add: wq_def Let_def holding_def holds2_def holds_def waits2_def runing_def readys_def)
-apply(simp add: waits_def)
-apply(auto simp add: waits_def split: if_splits)[1]
-apply(auto split: if_splits)
-apply(simp only: waits_def)
-by (metis insert_iff set_simps(2))
-
-
-
-lemma Domain_RAG_threads:
- assumes vt: "vt s"
- and in_dom: "(Th th) \<in> Domain (RAG2 s)"
- shows "th \<in> threads s"
-proof -
- from in_dom obtain n where "(Th th, n) \<in> RAG2 s" by auto
- then obtain cs where "n = Cs cs" "(Th th, Cs cs) \<in> RAG2 s"
- unfolding RAG2_def RAG_def by auto
- then have "th \<in> set (wq s cs)"
- unfolding wq_def RAG_def RAG2_def waits_def by auto
- with wq_threads [OF vt] show ?thesis .
-qed
-
-lemma dependants_threads:
- assumes vt: "vt s"
- shows "dependants2 s th \<subseteq> threads s"
-proof
- fix th1
- assume "th1 \<in> dependants2 s th"
- then have h: "(Th th1, Th th) \<in> (RAG2 s)\<^sup>+"
- unfolding dependants2_def dependants_def RAG2_def by simp
- then have "Th th1 \<in> Domain ((RAG2 s)\<^sup>+)" unfolding Domain_def by auto
- then have "Th th1 \<in> Domain (RAG2 s)" using trancl_domain by simp
- then show "th1 \<in> threads s" using vt by (rule_tac Domain_RAG_threads)
-qed
-
-lemma finite_threads:
- assumes vt: "vt s"
- shows "finite (threads s)"
-using vt by (induct) (auto elim: step.cases)
-
-
-section {* Distinctness of @{const wq} *}
-
-lemma wq_distinct_step:
- assumes "step s e" "distinct (wq s cs)"
- shows "distinct (wq (e # s) cs)"
-using assms
-unfolding wq_def
-apply(erule_tac step.cases)
-apply(auto simp add: RAG2_def RAG_def Let_def)[1]
-apply(auto simp add: wq_def Let_def RAG2_def RAG_def holds_def runing_def waits2_def waits_def readys_def)
-apply(auto split: list.split)
-apply(rule someI2)
-apply(auto)
-done
-
-lemma wq_distinct:
- assumes "vt s"
- shows "distinct (wq s cs)"
-using assms
-apply(induct)
-apply(simp add: wq_def)
-apply(simp add: wq_distinct_step)
-done
-
-
-section {* Single_Valuedness of @{const waits2}, @{const holds2}, @{const RAG2} *}
-
-lemma waits2_unique:
- assumes "vt s"
- and "waits2 s th cs1"
- and "waits2 s th cs2"
- shows "cs1 = cs2"
-using assms
-apply(induct)
-apply(simp add: waits2_def waits_def)
-apply(erule step.cases)
-apply(auto simp add: Let_def waits2_def waits_def holds_def RAG2_def RAG_def
- readys_def runing_def split: if_splits)
-apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
-apply (metis Nil_is_append_conv hd_append2 list.distinct(1) split_list)
-apply (metis distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
-by (metis (full_types, hide_lams) distinct.simps(2) distinct_length_2_or_more list.sel(1) wq_def wq_distinct)
-
-lemma single_valued_waits2:
- assumes "vt s"
- shows "single_valuedP (waits2 s)"
-using assms
-unfolding single_valued_def
-by (simp add: Product_Type.Collect_case_prodD waits2_unique)
-
-lemma single_valued_holds2:
- assumes "vt s"
- shows "single_valuedP (\<lambda>cs th. holds2 s th cs)"
-unfolding single_valued_def holds2_def holds_def by simp
-
-lemma single_valued_RAG2:
- assumes "vt s"
- shows "single_valued (RAG2 s)"
-using single_valued_waits2[OF assms] single_valued_holds2[OF assms]
-unfolding RAG2_def RAG_def
-apply(rule_tac single_valued_union)
-unfolding holds2_def[symmetric] waits2_def[symmetric]
-apply(rule single_valued_Collect)
-apply(simp)
-apply(simp add: inj_on_def)
-apply(rule single_valued_Collect)
-apply(simp)
-apply(simp add: inj_on_def)
-apply(auto)
-done
-
-
-section {* Properties of @{const RAG2} under events *}
-
-lemma RAG_Set [simp]:
- shows "RAG2 (Set th prio # s) = RAG2 s"
-unfolding RAG2_def
-by (simp add: Let_def)
-
-lemma RAG_Create [simp]:
- "RAG2 (Create th prio # s) = RAG2 s"
-unfolding RAG2_def
-by (simp add: Let_def)
-
-lemma RAG_Exit [simp]:
- shows "RAG2 (Exit th # s) = RAG2 s"
-unfolding RAG2_def
-by (simp add: Let_def)
-
-lemma RAG_P1:
- assumes "wq s cs = []"
- shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Cs cs, Th th)}"
-using assms
-unfolding RAG2_def RAG_def wq_def Let_def waits_def holds_def
-by (auto simp add: Let_def)
-
-lemma RAG_P2:
- assumes "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" "wq s cs \<noteq> []"
- shows "RAG2 (P th cs # s) \<subseteq> RAG2 s \<union> {(Th th, Cs cs)}"
-using assms
-unfolding RAG2_def RAG_def wq_def Let_def waits_def holds_def
-by (auto simp add: Let_def)
-
-
-lemma RAG_V1:
-assumes vt: "wq s cs = [th]"
-shows "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}"
-using assms
-unfolding RAG2_def RAG_def waits_def holds_def wq_def
-by (auto simp add: Let_def)
-
-lemma RAG_V2:
-assumes vt:"vt s" "wq s cs = th # wts \<and> wts \<noteq> []"
-shows "RAG2 (V th cs # s) \<subseteq>
- RAG2 s - {(Cs cs, Th th), (Th (next_to_run wts), Cs cs)} \<union> {(Cs cs, Th (next_to_run wts))}"
-unfolding RAG2_def RAG_def waits_def holds_def
-using assms wq_distinct[OF vt(1), of"cs"]
-by (auto simp add: Let_def wq_def)
-
-
-
-section {* Acyclicity of @{const RAG2} *}
-
-lemma acyclic_RAG_step:
- assumes vt: "vt s"
- and stp: "step s e"
- and ac: "acyclic (RAG2 s)"
- shows "acyclic (RAG2 (e # s))"
-using stp vt ac
-proof (induct)
- case (step_P th s cs)
- have ac: "acyclic (RAG2 s)" by fact
- have ds: "(Cs cs, Th th) \<notin> (RAG2 s)\<^sup>+" by fact
- { assume wq_empty: "wq s cs = []" -- "case waiting queue is empty"
- then have "(Th th, Cs cs) \<notin> (RAG2 s)\<^sup>+"
- proof (rule_tac notI)
- assume "(Th th, Cs cs) \<in> (RAG2 s)\<^sup>+"
- then obtain x where "(x, Cs cs) \<in> RAG2 s" using tranclD2 by metis
- with wq_empty show False by (auto simp: RAG2_def RAG_def wq_def waits_def)
- qed
- with ac have "acyclic (RAG2 s \<union> {(Cs cs, Th th)})" by simp
- then have "acyclic (RAG2 (P th cs # s))" using RAG_P1[OF wq_empty]
- by (rule acyclic_subset)
- }
- moreover
- { assume wq_not_empty: "wq s cs \<noteq> []" -- "case waiting queue is not empty"
- from ac ds
- have "acyclic (RAG2 s \<union> {(Th th, Cs cs)})" by simp
- then have "acyclic (RAG2 (P th cs # s))" using RAG_P2[OF ds wq_not_empty]
- by (rule acyclic_subset)
- }
- ultimately show "acyclic (RAG2 (P th cs # s))" by metis
-next
- case (step_V th s cs) -- "case for release of a lock"
- have vt: "vt s" by fact
- have ac: "acyclic (RAG2 s)" by fact
- have hd: "holds2 s th cs" by fact
- from vt have wq_distinct:"distinct (wq s cs)" by (rule wq_distinct)
- from hd have "th \<in> set (wq s cs)" "th = hd (wq s cs)" unfolding holds2_def holds_def wq_def by auto
- then obtain wts where eq_wq: "wq s cs = th # wts" by (cases "wq s cs") (auto)
- -- "case no thread present in the waiting queue to take over"
- { assume "wts = []"
- with eq_wq have "wq s cs = [th]" by simp
- then have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th)}" by (rule RAG_V1)
- moreover have "acyclic (RAG2 s - {(Cs cs, Th th)})" using ac by (auto intro: acyclic_subset)
- ultimately
- have "acyclic (RAG2 (V th cs # s))" by (auto intro: acyclic_subset)
- }
- moreover
- -- "at least one thread present to take over"
- { def nth \<equiv> "next_to_run wts"
- assume wq_not_empty: "wts \<noteq> []"
- have "waits2 s nth cs"
- using eq_wq wq_not_empty wq_distinct
- unfolding nth_def waits2_def waits_def wq_def[symmetric] by auto
- then have cs_in_RAG: "(Th nth, Cs cs) \<in> RAG2 s"
- unfolding RAG2_def RAG_def waits2_def by auto
- have "RAG2 (V th cs # s) \<subseteq> RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)}"
- unfolding nth_def using vt wq_not_empty eq_wq by (rule_tac RAG_V2) (auto)
- moreover
- have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth)})"
- proof -
- have "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})" using ac by (auto intro: acyclic_subset)
- moreover
- have "(Th nth, Cs cs) \<notin> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+"
- proof (rule notI)
- assume "(Th nth, Cs cs) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})\<^sup>+"
- then obtain z where a: "(Th nth, z) \<in> (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)})"
- by (metis converse_tranclE)
- then have "(Th nth, z) \<in> RAG2 s" by simp
- then have "z = Cs cs" using cs_in_RAG single_valued_RAG2[OF vt]
- by (simp add: single_valued_def)
- then show "False" using a by simp
- qed
- ultimately
- show "acyclic (RAG2 s - {(Cs cs, Th th), (Th nth, Cs cs)} \<union> {(Cs cs, Th nth) })" by simp
- qed
- ultimately have "acyclic (RAG2 (V th cs # s))"
- by (rule_tac acyclic_subset)
- }
- ultimately show "acyclic (RAG2 (V th cs # s))" by metis
-qed (simp_all)
-
-
-lemma finite_RAG:
- assumes "vt s"
- shows "finite (RAG2 s)"
-using assms
-apply(induct)
-apply(simp add: RAG2_def RAG_def waits_def holds_def)
-apply(erule step.cases)
-apply(auto)
-apply(case_tac "wq sa cs = []")
-apply(rule finite_subset)
-apply(rule RAG_P1)
-apply(simp)
-apply(simp)
-apply(rule finite_subset)
-apply(rule RAG_P2)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(subgoal_tac "\<exists>wts. wq sa cs = th # wts")
-apply(erule exE)
-apply(case_tac "wts = []")
-apply(rule finite_subset)
-apply(rule RAG_V1)
-apply(simp)
-apply(simp)
-apply(rule finite_subset)
-apply(rule RAG_V2)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(subgoal_tac "th \<in> set (wq sa cs) \<and> th = hd (wq sa cs)")
-apply(case_tac "wq sa cs")
-apply(auto)[2]
-apply(auto simp add: holds2_def holds_def wq_def)
-done
-
-
-
-lemma dchain_unique:
- assumes vt: "vt s"
- and th1_d: "(n, Th th1) \<in> (RAG2 s)^+"
- and th1_r: "th1 \<in> readys s"
- and th2_d: "(n, Th th2) \<in> (RAG2 s)^+"
- and th2_r: "th2 \<in> readys s"
- shows "th1 = th2"
-proof(rule ccontr)
- assume neq: "th1 \<noteq> th2"
- with single_valued_confluent2 [OF single_valued_RAG2 [OF vt]] th1_d th2_d
- have "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+ \<or> (Th th2, Th th1) \<in> (RAG2 s)\<^sup>+" by auto
- moreover
- { assume "(Th th1, Th th2) \<in> (RAG2 s)\<^sup>+"
- then obtain n where dd: "(Th th1, n) \<in> RAG2 s" by (metis converse_tranclE)
- then obtain cs where eq_n: "n = Cs cs"
- unfolding RAG2_def RAG_def by (case_tac n) (auto)
- from dd eq_n have "th1 \<notin> readys s"
- unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
- with th1_r have "False" by auto
- }
- moreover
- { assume "(Th th2, Th th1) \<in> (RAG2 s)\<^sup>+"
- then obtain n where dd: "(Th th2, n) \<in> RAG2 s" by (metis converse_tranclE)
- then obtain cs where eq_n: "n = Cs cs"
- unfolding RAG2_def RAG_def by (case_tac n) (auto)
- from dd eq_n have "th2 \<notin> readys s"
- unfolding RAG2_def RAG_def waits2_def readys_def by (auto)
- with th2_r have "False" by auto
- }
- ultimately show "False" by metis
-qed
-
-lemma cpreced2_cpreced: "cpreced2 s th = cpreced (wq s) s th"
-unfolding cpreced2_def wq_def
-apply(induct s rule: schs.induct)
-apply(simp add: Let_def cpreced_def dependants_def RAG_def waits_def holds_def preced_def)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-apply(subst (2) schs.simps)
-apply(simp add: Let_def)
-done
-
-lemma cpreced_Exit:
- shows "cpreced2 (Exit th # s) th' = cpreced2 s th'"
-by (simp add: cpreced2_cpreced cpreced_def preced_def wq_def Let_def)
-
-lemma readys_Exit:
- shows "readys (Exit th # s) = readys s - {th}"
-by (auto simp add: readys_def waits2_def Let_def)
-
-lemma readys_Create:
- shows "readys (Create th prio # s) \<subseteq> {th} \<union> readys s"
-apply (auto simp add: readys_def waits2_def Let_def waits_def)
-done
-
-lemma readys_Set:
- shows "readys (Set th prio # s) = readys s"
-by (auto simp add: readys_def waits2_def Let_def)
-
-
-lemma readys_P:
- shows "readys (P th cs # s) \<subseteq> readys s"
-apply(auto simp add: readys_def waits2_def Let_def)
-apply(simp add: waits_def)
-apply(case_tac "csa = cs")
-apply(simp)
-apply(drule_tac x="cs" in spec)
-apply(simp)
-apply (metis hd_append2 in_set_insert insert_Nil list.sel(1))
-apply(drule_tac x="csa" in spec)
-apply(simp)
-done
-
-lemma readys_V:
- shows "readys (V th cs # s) \<subseteq> readys s \<union> set (wq s cs)"
-apply(auto simp add: readys_def waits2_def waits_def Let_def wq_def)
-done
-
-
-fun the_th :: "node \<Rightarrow> thread"
- where "the_th (Th th) = th"
-
-lemma image_Collect2:
- "f ` A = {f x | x. x \<in> A}"
-apply(auto)
-done
-
-lemma Collect_disj_eq2:
- "{f x | x. x = y \<or> x \<in> A} = {f y} \<union> {f x | x. x \<in> A}"
-by (auto)
-
-lemma last_set_lt:
- "th \<in> threads s \<Longrightarrow> last_set th s < length s"
- apply(induct rule: threads.induct)
- apply(auto)
- done
-
-lemma last_set_eq_iff:
- assumes "th1 \<in> threads s" "th2 \<in> threads s"
- shows "last_set th1 s = last_set th2 s \<longleftrightarrow> th1 = th2"
- using assms
- apply(induct s rule: threads.induct)
- apply(auto split:if_splits dest:last_set_lt)
- done
-
-lemma preced_eq_iff:
- assumes th_in1: "th1 \<in> threads s"
- and th_in2: "th2 \<in> threads s"
- shows "preced th1 s = preced th2 s \<longleftrightarrow> th1 = th2"
-using assms
-by (auto simp add: preced_def last_set_eq_iff)
-
-lemma dm_RAG_threads:
- assumes vt: "vt s"
- and in_dom: "(Th th) \<in> Domain (RAG2 s)"
- shows "th \<in> threads s"
-proof -
- from in_dom obtain n where a: "(Th th, n) \<in> RAG2 s" by auto
- then obtain cs where "n = Cs cs"
- unfolding RAG2_def RAG_def
- by auto
- then have "(Th th, Cs cs) \<in> RAG2 s" using a by simp
- hence "th \<in> set (wq s cs)"
- unfolding RAG2_def wq_def RAG_def waits_def
- by (auto)
- then show ?thesis
- apply(rule_tac wq_threads)
- apply(rule assms)
- apply(simp)
- done
-qed
-
-lemma cpreced_eq_iff:
- assumes "th1 \<in> readys s" "th2 \<in> readys s" "vt s"
- shows "cpreced2 s th1 = cpreced2 s th2 \<longleftrightarrow> th1 = th2"
-proof
- def S1\<equiv>"({th1} \<union> dependants (wq s) th1)"
- def S2\<equiv>"({th2} \<union> dependants (wq s) th2)"
- have fin: "finite ((the_th o fst) ` ((RAG (wq s))\<^sup>+))"
- apply(rule)
- apply(simp add: finite_trancl)
- apply(simp add: wq_def)
- apply(rule finite_RAG[simplified RAG2_def])
- apply(rule assms)
- done
-
- from fin have h: "finite (preceds s S1)" "finite (preceds s S2)"
- apply(simp_all add: S2_def S1_def Collect_disj_eq2 image_Collect[symmetric])
- apply(rule)
- apply(simp add: dependants_def)
- apply(rule rev_finite_subset)
- apply(assumption)
- apply(auto simp add: image_def)[1]
- apply(metis fst_conv the_th.simps)
- apply(rule)
- apply(simp add: dependants_def)
- apply(rule rev_finite_subset)
- apply(assumption)
- apply(auto simp add: image_def)[1]
- apply(metis fst_conv the_th.simps)
- done
- moreover have "S1 \<noteq> {}" "S2 \<noteq> {}" by (simp_all add: S1_def S2_def)
- then have "(preceds s S1) \<noteq> {}" "(preceds s S2) \<noteq> {}" by simp_all
- ultimately have m: "Max (preceds s S1) \<in> (preceds s S1)" "Max (preceds s S2) \<in> (preceds s S2)"
- apply(rule_tac [!] Max_in)
- apply(simp_all)
- done
-
- assume q: "cpreced2 s th1 = cpreced2 s th2"
- then have eq_max: "Max (preceds s S1) = Max (preceds s S2)"
- unfolding cpreced2_cpreced cpreced_def
- apply(simp only: S1_def S2_def)
- apply(simp add: Collect_disj_eq2)
- done
-
- obtain th0 where th0_in: "th0 \<in> S1" "th0 \<in> S2" and
- eq_f_th1: "preced th0 s = Max (preceds s S1)"
- "preced th0 s = Max (preceds s S2)"
- using m
- apply(clarify)
- apply(simp add: eq_max)
- apply(subst (asm) (2) preced_eq_iff)
- apply(insert assms(2))[1]
- apply(simp add: S2_def)
- apply(auto)[1]
- apply (metis contra_subsetD readys_threads)
- apply(simp add: dependants_def)
- apply(subgoal_tac "Th tha \<in> Domain ((RAG2 s)^+)")
- apply(simp add: trancl_domain)
- apply (metis Domain_RAG_threads assms(3))
- apply(simp only: RAG2_def wq_def)
- apply (metis Domain_iff)
- apply(insert assms(1))[1]
- apply(simp add: S1_def)
- apply(auto)[1]
- apply (metis contra_subsetD readys_threads)
- apply(simp add: dependants_def)
- apply(subgoal_tac "Th th \<in> Domain ((RAG2 s)^+)")
- apply(simp add: trancl_domain)
- apply (metis Domain_RAG_threads assms(3))
- apply(simp only: RAG2_def wq_def)
- apply (metis Domain_iff)
- apply(simp)
- done
- then show "th1 = th2"
- apply -
- apply(insert th0_in assms(1, 2))[1]
- apply(simp add: S1_def S2_def)
- apply(auto)
- --"first case"
- prefer 2
- apply(subgoal_tac "Th th2 \<in> Domain (RAG2 s)")
- apply(subgoal_tac "\<exists>cs. (Th th2, Cs cs) \<in> RAG2 s")
- apply(erule exE)
- apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
- apply(auto simp add: RAG2_def RAG_def)[1]
- apply(subgoal_tac "Th th2 \<in> Domain ((RAG2 s)^+)")
- apply (metis trancl_domain)
- apply(subgoal_tac "(Th th2, Th th1) \<in> (RAG2 s)^+")
- apply (metis Domain_iff)
- apply(simp add: dependants_def RAG2_def wq_def)
- --"second case"
- apply(subgoal_tac "Th th1 \<in> Domain (RAG2 s)")
- apply(subgoal_tac "\<exists>cs. (Th th1, Cs cs) \<in> RAG2 s")
- apply(erule exE)
- apply(insert assms(1))[1]
- apply(simp add: runing_def RAG2_def RAG_def readys_def waits2_def)[1]
- apply(auto simp add: RAG2_def RAG_def)[1]
- apply(subgoal_tac "Th th1 \<in> Domain ((RAG2 s)^+)")
- apply (metis trancl_domain)
- apply(subgoal_tac "(Th th1, Th th2) \<in> (RAG2 s)^+")
- apply (metis Domain_iff)
- apply(simp add: dependants_def RAG2_def wq_def)
- --"third case"
- apply(rule dchain_unique)
- apply(rule assms(3))
- apply(simp add: dependants_def RAG2_def wq_def)
- apply(simp)
- apply(simp add: dependants_def RAG2_def wq_def)
- apply(simp)
- done
-next
- assume "th1 = th2"
- then show "cpreced2 s th1 = cpreced2 s th2" by simp
-qed
-
-lemma at_most_one_running:
- assumes "vt s"
- shows "card (runing s) \<le> 1"
-proof (rule ccontr)
- assume "\<not> card (runing s) \<le> 1"
- then have "2 \<le> card (runing s)" by auto
- moreover
- have "finite (runing s)"
- by (metis `\<not> card (runing s) \<le> 1` card_infinite le0)
- ultimately obtain th1 th2 where a:
- "th1 \<noteq> th2" "th1 \<in> runing s" "th2 \<in> runing s"
- "cpreced2 s th1 = cpreced2 s th2"
- apply(auto simp add: numerals card_le_Suc_iff runing_def)
- apply(blast)
- done
- then have "th1 = th2"
- apply(subst (asm) cpreced_eq_iff)
- apply(auto intro: assms a)
- apply (metis contra_subsetD runing_ready)+
- done
- then show "False" using a(1) by auto
-qed
-
-
-
- (*
- obtain th0 where th0_in: "th0 \<in> S1 \<and> th0 \<in> S2"
- and eq_f_th0: "preced th0 s = Max ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
- proof -
- from fin have h1: "finite ((\<lambda>th. preced th s) ` (S1 \<inter> S2))"
- apply(simp only: S1_def S2_def)
- apply(rule)
- apply(rule)
- apply(rule)
- apply(simp add: dependants_def)
- apply(rule rev_finite_subset)
- apply(assumption)
- apply(auto simp add: image_def)
- apply (metis fst_conv the_th.simps)
- done
- moreover
- have "S1 \<inter> S2 \<noteq> {}" apply (simp add: S1_def S2_def)
- apply(auto)
-
- done
- then have h2: "((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<noteq> {}" by simp
- ultimately have "Max ((\<lambda>th. preced th s) ` (S1 \<union> S2)) \<in> ((\<lambda>th. preced th s) ` (S1 \<union> S2))"
- apply(rule Max_in)
- done
- then show ?thesis using that[intro] apply(auto)
-
- apply(erule_tac preced_unique)
- done
- qed
- *)
-
-thm waits_def waits2_def
-
-end
--- a/draf.txt Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,11 +0,0 @@
-There are low priority threads,
-which do not hold any resources,
-such thread will not block th.
-Theorem 3 does not exclude such threads.
-
-There are resources, which are not held by any low prioirty threads,
-such resources can not cause blockage of th neither. And similiary,
-theorem 6 does not exlude them.
-
-Our one bound excudle them by using a different formaulation.
-
--- a/red_1.thy Thu Sep 07 16:04:03 2017 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,359 +0,0 @@
-section {*
- This file contains lemmas used to guide the recalculation of current precedence
- after every system call (or system operation)
-*}
-theory CpsG
-imports PrioG Max RTree
-begin
-
-
-definition "wRAG (s::state) = {(Th th, Cs cs) | th cs. waiting s th cs}"
-
-definition "hRAG (s::state) = {(Cs cs, Th th) | th cs. holding s th cs}"
-
-definition "tRAG s = wRAG s O hRAG s"
-
-definition "pairself f = (\<lambda>(a, b). (f a, f b))"
-
-definition "rel_map f r = (pairself f ` r)"
-
-fun the_thread :: "node \<Rightarrow> thread" where
- "the_thread (Th th) = th"
-
-definition "tG s = rel_map the_thread (tRAG s)"
-
-locale pip =
- fixes s
- assumes vt: "vt s"
-
-
-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)
-
-lemma relpow_mult:
- "((r::'a rel) ^^ m) ^^ n = r ^^ (m*n)"
-proof(induct n arbitrary:m)
- case (Suc k m)
- thus ?case (is "?L = ?R")
- proof -
- have h: "(m * k + m) = (m + m * k)" by auto
- show ?thesis
- apply (simp add:Suc relpow_add[symmetric])
- by (unfold h, simp)
- qed
-qed simp
-
-lemma compose_relpow_2:
- assumes "r1 \<subseteq> r"
- and "r2 \<subseteq> r"
- shows "r1 O r2 \<subseteq> r ^^ (2::nat)"
-proof -
- { fix a b
- assume "(a, b) \<in> r1 O r2"
- then obtain e where "(a, e) \<in> r1" "(e, b) \<in> r2"
- by auto
- with assms have "(a, e) \<in> r" "(e, b) \<in> r" by auto
- hence "(a, b) \<in> r ^^ (Suc (Suc 0))" by auto
- } thus ?thesis by (auto simp:numeral_2_eq_2)
-qed
-
-
-lemma acyclic_compose:
- assumes "acyclic r"
- and "r1 \<subseteq> r"
- and "r2 \<subseteq> r"
- shows "acyclic (r1 O r2)"
-proof -
- { fix a
- assume "(a, a) \<in> (r1 O r2)^+"
- from trancl_mono[OF this compose_relpow_2[OF assms(2, 3)]]
- have "(a, a) \<in> (r ^^ 2) ^+" .
- from trancl_power[THEN iffD1, OF this]
- obtain n where h: "(a, a) \<in> (r ^^ 2) ^^ n" "n > 0" by blast
- from this(1)[unfolded relpow_mult] have h2: "(a, a) \<in> r ^^ (2 * n)" .
- have "(a, a) \<in> r^+"
- proof(cases rule:trancl_power[THEN iffD2])
- from h(2) h2 show "\<exists>n>0. (a, a) \<in> r ^^ n"
- by (rule_tac x = "2*n" in exI, auto)
- qed
- with assms have "False" by (auto simp:acyclic_def)
- } thus ?thesis by (auto simp:acyclic_def)
-qed
-
-lemma range_tRAG: "Range (tRAG s) \<subseteq> {Th th | th. True}"
-proof -
- have "Range (wRAG s O hRAG s) \<subseteq> {Th th |th. True}" (is "?L \<subseteq> ?R")
- proof -
- have "?L \<subseteq> Range (hRAG s)" by auto
- also have "... \<subseteq> ?R"
- by (unfold hRAG_def, auto)
- finally show ?thesis by auto
- qed
- thus ?thesis by (simp add:tRAG_def)
-qed
-
-lemma domain_tRAG: "Domain (tRAG s) \<subseteq> {Th th | th. True}"
-proof -
- have "Domain (wRAG s O hRAG s) \<subseteq> {Th th |th. True}" (is "?L \<subseteq> ?R")
- proof -
- have "?L \<subseteq> Domain (wRAG s)" by auto
- also have "... \<subseteq> ?R"
- by (unfold wRAG_def, auto)
- finally show ?thesis by auto
- qed
- thus ?thesis by (simp add:tRAG_def)
-qed
-
-lemma rel_mapE:
- assumes "(a, b) \<in> rel_map f r"
- obtains c d
- where "(c, d) \<in> r" "(a, b) = (f c, f d)"
- using assms
- by (unfold rel_map_def pairself_def, auto)
-
-lemma rel_mapI:
- assumes "(a, b) \<in> r"
- and "c = f a"
- and "d = f b"
- shows "(c, d) \<in> rel_map f r"
- using assms
- by (unfold rel_map_def pairself_def, auto)
-
-lemma map_appendE:
- assumes "map f zs = xs @ ys"
- obtains xs' ys'
- where "zs = xs' @ ys'" "xs = map f xs'" "ys = map f ys'"
-proof -
- have "\<exists> xs' ys'. zs = xs' @ ys' \<and> xs = map f xs' \<and> ys = map f ys'"
- using assms
- proof(induct xs arbitrary:zs ys)
- case (Nil zs ys)
- thus ?case by auto
- next
- case (Cons x xs zs ys)
- note h = this
- show ?case
- proof(cases zs)
- case (Cons e es)
- with h have eq_x: "map f es = xs @ ys" "x = f e" by auto
- from h(1)[OF this(1)]
- obtain xs' ys' where "es = xs' @ ys'" "xs = map f xs'" "ys = map f ys'"
- by blast
- with Cons eq_x
- have "zs = (e#xs') @ ys' \<and> x # xs = map f (e#xs') \<and> ys = map f ys'" by auto
- thus ?thesis by metis
- qed (insert h, auto)
- qed
- thus ?thesis by (auto intro!:that)
-qed
-
-lemma rel_map_mono:
- assumes "r1 \<subseteq> r2"
- shows "rel_map f r1 \<subseteq> rel_map f r2"
- using assms
- by (auto simp:rel_map_def pairself_def)
-
-lemma rel_map_compose [simp]:
- shows "rel_map f1 (rel_map f2 r) = rel_map (f1 o f2) r"
- by (auto simp:rel_map_def pairself_def)
-
-lemma edges_on_map: "edges_on (map f xs) = rel_map f (edges_on xs)"
-proof -
- { fix a b
- assume "(a, b) \<in> edges_on (map f xs)"
- then obtain l1 l2 where eq_map: "map f xs = l1 @ [a, b] @ l2"
- by (unfold edges_on_def, auto)
- hence "(a, b) \<in> rel_map f (edges_on xs)"
- by (auto elim!:map_appendE intro!:rel_mapI simp:edges_on_def)
- } moreover {
- fix a b
- assume "(a, b) \<in> rel_map f (edges_on xs)"
- then obtain c d where
- h: "(c, d) \<in> edges_on xs" "(a, b) = (f c, f d)"
- by (elim rel_mapE, auto)
- then obtain l1 l2 where
- eq_xs: "xs = l1 @ [c, d] @ l2"
- by (auto simp:edges_on_def)
- hence eq_map: "map f xs = map f l1 @ [f c, f d] @ map f l2" by auto
- have "(a, b) \<in> edges_on (map f xs)"
- proof -
- from h(2) have "[f c, f d] = [a, b]" by simp
- from eq_map[unfolded this] show ?thesis by (auto simp:edges_on_def)
- qed
- } ultimately show ?thesis by auto
-qed
-
-lemma plus_rpath:
- assumes "(a, b) \<in> r^+"
- obtains xs where "rpath r a xs b" "xs \<noteq> []"
-proof -
- from assms obtain m where h: "(a, m) \<in> r" "(m, b) \<in> r^*"
- by (auto dest!:tranclD)
- from star_rpath[OF this(2)] obtain xs where "rpath r m xs b" by auto
- from rstepI[OF h(1) this] have "rpath r a (m # xs) b" .
- from that[OF this] show ?thesis by auto
-qed
-
-lemma edges_on_unfold:
- "edges_on (a # b # xs) = {(a, b)} \<union> edges_on (b # xs)" (is "?L = ?R")
-proof -
- { fix c d
- assume "(c, d) \<in> ?L"
- then obtain l1 l2 where h: "(a # b # xs) = l1 @ [c, d] @ l2"
- by (auto simp:edges_on_def)
- have "(c, d) \<in> ?R"
- proof(cases "l1")
- case Nil
- with h have "(c, d) = (a, b)" by auto
- thus ?thesis by auto
- next
- case (Cons e es)
- from h[unfolded this] have "b#xs = es@[c, d]@l2" by auto
- thus ?thesis by (auto simp:edges_on_def)
- qed
- } moreover
- { fix c d
- assume "(c, d) \<in> ?R"
- moreover have "(a, b) \<in> ?L"
- proof -
- have "(a # b # xs) = []@[a,b]@xs" by simp
- hence "\<exists> l1 l2. (a # b # xs) = l1@[a,b]@l2" by auto
- thus ?thesis by (unfold edges_on_def, simp)
- qed
- moreover {
- assume "(c, d) \<in> edges_on (b#xs)"
- then obtain l1 l2 where "b#xs = l1@[c, d]@l2" by (unfold edges_on_def, auto)
- hence "a#b#xs = (a#l1)@[c,d]@l2" by simp
- hence "\<exists> l1 l2. (a # b # xs) = l1@[c,d]@l2" by metis
- hence "(c,d) \<in> ?L" by (unfold edges_on_def, simp)
- }
- ultimately have "(c, d) \<in> ?L" by auto
- } ultimately show ?thesis by auto
-qed
-
-lemma edges_on_rpathI:
- assumes "edges_on (a#xs@[b]) \<subseteq> r"
- shows "rpath r a (xs@[b]) b"
- using assms
-proof(induct xs arbitrary: a b)
- case Nil
- moreover have "(a, b) \<in> edges_on (a # [] @ [b])"
- by (unfold edges_on_def, auto)
- ultimately have "(a, b) \<in> r" by auto
- thus ?case by auto
-next
- case (Cons x xs a b)
- from this(2) have "edges_on (x # xs @ [b]) \<subseteq> r" by (simp add:edges_on_unfold)
- from Cons(1)[OF this] have " rpath r x (xs @ [b]) b" .
- moreover from Cons(2) have "(a, x) \<in> r" by (auto simp:edges_on_unfold)
- ultimately show ?case by (auto intro!:rstepI)
-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)
-
-lemma rel_map_inv_id:
- assumes "inj_on f ((Domain r) \<union> (Range r))"
- shows "(rel_map (inv_into ((Domain r) \<union> (Range r)) f \<circ> f) r) = r"
-proof -
- let ?f = "(inv_into (Domain r \<union> Range r) f \<circ> f)"
- {
- fix a b
- assume h0: "(a, b) \<in> r"
- have "pairself ?f (a, b) = (a, b)"
- proof -
- from assms h0 have "?f a = a" by (auto intro:inv_into_f_f)
- moreover have "?f b = b"
- by (insert h0, simp, intro inv_into_f_f[OF assms], auto intro!:RangeI)
- ultimately show ?thesis by (auto simp:pairself_def)
- qed
- } thus ?thesis by (unfold rel_map_def, intro image_id, case_tac x, auto)
-qed
-
-lemma rel_map_acyclic:
- assumes "acyclic r"
- and "inj_on f ((Domain r) \<union> (Range r))"
- shows "acyclic (rel_map f r)"
-proof -
- let ?D = "Domain r \<union> Range r"
- { fix a
- assume "(a, a) \<in> (rel_map f r)^+"
- from plus_rpath[OF this]
- obtain xs where rp: "rpath (rel_map f r) a xs a" "xs \<noteq> []" by auto
- from rpath_nnl_lastE[OF this] obtain xs' where eq_xs: "xs = xs'@[a]" by auto
- from rpath_edges_on[OF rp(1)]
- have h: "edges_on (a # xs) \<subseteq> rel_map f r" .
- from edges_on_map[of "inv_into ?D f" "a#xs"]
- have "edges_on (map (inv_into ?D f) (a # xs)) = rel_map (inv_into ?D f) (edges_on (a # xs))" .
- with rel_map_mono[OF h, of "inv_into ?D f"]
- have "edges_on (map (inv_into ?D f) (a # xs)) \<subseteq> rel_map ((inv_into ?D f) o f) r" by simp
- from this[unfolded eq_xs]
- have subr: "edges_on (map (inv_into ?D f) (a # xs' @ [a])) \<subseteq> rel_map (inv_into ?D f \<circ> f) r" .
- have "(map (inv_into ?D f) (a # xs' @ [a])) = (inv_into ?D f a) # map (inv_into ?D f) xs' @ [inv_into ?D f a]"
- by simp
- from edges_on_rpathI[OF subr[unfolded this]]
- have "rpath (rel_map (inv_into ?D f \<circ> f) r)
- (inv_into ?D f a) (map (inv_into ?D f) xs' @ [inv_into ?D f a]) (inv_into ?D f a)" .
- hence "(inv_into ?D f a, inv_into ?D f a) \<in> (rel_map (inv_into ?D f \<circ> f) r)^+"
- by (rule rpath_plus, simp)
- moreover have "(rel_map (inv_into ?D f \<circ> f) r) = r" by (rule rel_map_inv_id[OF assms(2)])
- moreover note assms(1)
- ultimately have False by (unfold acyclic_def, auto)
- } thus ?thesis by (auto simp:acyclic_def)
-qed
-
-context pip
-begin
-
-interpretation rtree_RAG: rtree "RAG s"
-proof
- show "single_valued (RAG s)"
- by (unfold single_valued_def, auto intro: unique_RAG[OF vt])
-
- show "acyclic (RAG s)"
- by (rule acyclic_RAG[OF vt])
-qed
-
-lemma sgv_wRAG:
- shows "single_valued (wRAG s)"
- using waiting_unique[OF vt]
- by (unfold single_valued_def wRAG_def, auto)
-
-lemma sgv_hRAG:
- shows "single_valued (hRAG s)"
- using held_unique
- by (unfold single_valued_def hRAG_def, auto)
-
-lemma sgv_tRAG: shows "single_valued (tRAG s)"
- by (unfold tRAG_def, rule Relation.single_valued_relcomp,
- insert sgv_hRAG sgv_wRAG, auto)
-
-lemma acyclic_hRAG:
- shows "acyclic (hRAG s)"
- by (rule acyclic_subset[OF acyclic_RAG[OF vt]], insert RAG_split, auto)
-
-lemma acyclic_wRAG:
- shows "acyclic (wRAG s)"
- by (rule acyclic_subset[OF acyclic_RAG[OF vt]], insert RAG_split, auto)
-
-lemma acyclic_tRAG:
- shows "acyclic (tRAG s)"
- by (unfold tRAG_def, rule acyclic_compose[OF acyclic_RAG[OF vt]],
- unfold RAG_split, auto)
-
-lemma acyclic_tG:
- shows "acyclic (tG s)"
-proof(unfold tG_def, rule rel_map_acyclic[OF acyclic_tRAG])
- show "inj_on the_thread (Domain (tRAG s) \<union> Range (tRAG s))"
- proof(rule subset_inj_on)
- show " inj_on the_thread {Th th |th. True}" by (unfold inj_on_def, auto)
- next
- from domain_tRAG range_tRAG
- show " Domain (tRAG s) \<union> Range (tRAG s) \<subseteq> {Th th |th. True}" by auto
- qed
-qed
-
-end